image
image | page_id
string | expressions
dict |
|---|---|---|
0002194_page22
|
{
"latex": [
"$J_4$",
"$J_5$",
"$J_6$",
"$J_7$",
"$\\partial _j F_{ji}$",
"$F$",
"$B+F$",
"$\\partial _j F_{ji}$",
"$\\hat {T}_4$",
"$b_1$",
"$b_2$",
"$b_3$",
"$O(B^n)$",
"$J_i$",
"$F$",
"$B+F$",
"$J_i (B^n)$",
"$J_i (B)$",
"$J_i (B^2)$",
"$i=1, 2, 3$",
"$J_1 (B^2)$",
"$J_2 (B^2)$",
"$J_3 (B^2)$",
"$J_1 (B)$",
"$J_2 (B)$",
"$J_3 (B)$",
"$O(B, \\zeta ^3, k^5)$",
"\\begin {equation} {\\cal L} = \\sum _{i=1}^{7} b_i J_i, \\end {equation}",
"\\begin {eqnarray} && J_1 = \\partial _n F_{ij} \\partial _n F_{ji} F_{kl} F_{lk}, \\quad J_2 = \\partial _n F_{ij} \\partial _n F_{jk} F_{kl} F_{li}, \\\\ && J_3 = F_{ni} F_{im} \\partial _n F_{kl} \\partial _m F_{lk}, \\quad J_4 = \\partial _n F_{ni} \\partial _m F_{im} F_{kl} F_{lk}, \\\\ && J_5 = -\\partial _n F_{ni} \\partial _m F_{ij} F_{jk} F_{km}, \\quad J_6 = \\partial ^2 F_{ij} F_{ji} F_{kl} F_{lk}, \\\\ && J_7 = \\partial ^2 F_{ij} F_{jk} F_{kl} F_{li}, \\quad \\partial ^2 F_{ij} = \\partial _i \\partial _k F_{kj} - \\partial _j \\partial _k F_{ki}. \\end {eqnarray}",
"\\begin {eqnarray} && J_1 (B) = 2 \\partial _n F_{ij} \\partial _n F_{ji} B_{kl} F_{lk}, \\quad J_1 (B^2) = \\partial _n F_{ij} \\partial _n F_{ji} B_{kl} B_{lk}, \\\\ && J_2 (B) = 2 B_{ij} F_{jk} \\partial _n F_{kl} \\partial _n F_{li}, \\quad J_2 (B^2) = \\partial _n F_{ij} \\partial _n F_{jk} B_{kl} B_{li}, \\\\ && J_3 (B) = 2 B_{ni} F_{im} \\partial _n F_{kl} \\partial _m F_{lk}, \\quad J_3 (B^2) = B_{ni} B_{im} \\partial _n F_{kl} \\partial _m F_{lk}. \\end {eqnarray}",
"\\begin {eqnarray} J_3 (B) &=& 4 B_{ni} \\partial _i A_m \\partial _n \\partial _k A_l \\partial _m \\partial _l A_k + {\\rm ~terms~with~} \\partial ^2 A + {\\rm total~derivative} \\\\ &=& -4 B_{ni} \\partial _i \\partial _l A_m \\partial _n \\partial _k A_l \\partial _m A_k \\\\ && + {\\rm ~a~term~with~} \\partial _l A_l + {\\rm ~terms~with~} \\partial ^2 A + {\\rm total~derivative} \\\\ &=& -4 B_{nm} \\partial _i A_j \\partial _n \\partial _j A_k \\partial _m \\partial _k A_i \\\\ && + {\\rm ~a~term~with~} \\partial \\cdot A + {\\rm ~terms~with~} \\partial ^2 A + {\\rm total~derivative}. \\end {eqnarray}"
],
"latex_norm": [
"$ J _ { 4 } $",
"$ J _ { 5 } $",
"$ J _ { 6 } $",
"$ J _ { 7 } $",
"$ \\partial _ { j } F _ { j i } $",
"$ F $",
"$ B + F $",
"$ \\partial _ { j } F _ { j i } $",
"$ \\hat { T } _ { 4 } $",
"$ b _ { 1 } $",
"$ b _ { 2 } $",
"$ b _ { 3 } $",
"$ O ( B ^ { n } ) $",
"$ J _ { i } $",
"$ F $",
"$ B + F $",
"$ J _ { i } ( B ^ { n } ) $",
"$ J _ { i } ( B ) $",
"$ J _ { i } ( B ^ { 2 } ) $",
"$ i = 1 , 2 , 3 $",
"$ J _ { 1 } ( B ^ { 2 } ) $",
"$ J _ { 2 } ( B ^ { 2 } ) $",
"$ J _ { 3 } ( B ^ { 2 } ) $",
"$ J _ { 1 } ( B ) $",
"$ J _ { 2 } ( B ) $",
"$ J _ { 3 } ( B ) $",
"$ O ( B , \\zeta ^ { 3 } , k ^ { 5 } ) $",
"\\begin{equation*} L = \\sum _ { i = 1 } ^ { 7 } b _ { i } J _ { i } , \\end{equation*}",
"\\begin{align*} & & J _ { 1 } = \\partial _ { n } F _ { i j } \\partial _ { n } F _ { j i } F _ { k l } F _ { l k } , \\quad J _ { 2 } = \\partial _ { n } F _ { i j } \\partial _ { n } F _ { j k } F _ { k l } F _ { l i } , \\\\ & & J _ { 3 } = F _ { n i } F _ { i m } \\partial _ { n } F _ { k l } \\partial _ { m } F _ { l k } , \\quad J _ { 4 } = \\partial _ { n } F _ { n i } \\partial _ { m } F _ { i m } F _ { k l } F _ { l k } , \\\\ & & J _ { 5 } = - \\partial _ { n } F _ { n i } \\partial _ { m } F _ { i j } F _ { j k } F _ { k m } , \\quad J _ { 6 } = \\partial ^ { 2 } F _ { i j } F _ { j i } F _ { k l } F _ { l k } , \\\\ & & J _ { 7 } = \\partial ^ { 2 } F _ { i j } F _ { j k } F _ { k l } F _ { l i } , \\quad \\partial ^ { 2 } F _ { i j } = \\partial _ { i } \\partial _ { k } F _ { k j } - \\partial _ { j } \\partial _ { k } F _ { k i } . \\end{align*}",
"\\begin{align*} & & J _ { 1 } ( B ) = 2 \\partial _ { n } F _ { i j } \\partial _ { n } F _ { j i } B _ { k l } F _ { l k } , \\quad J _ { 1 } ( B ^ { 2 } ) = \\partial _ { n } F _ { i j } \\partial _ { n } F _ { j i } B _ { k l } B _ { l k } , \\\\ & & J _ { 2 } ( B ) = 2 B _ { i j } F _ { j k } \\partial _ { n } F _ { k l } \\partial _ { n } F _ { l i } , \\quad J _ { 2 } ( B ^ { 2 } ) = \\partial _ { n } F _ { i j } \\partial _ { n } F _ { j k } B _ { k l } B _ { l i } , \\\\ & & J _ { 3 } ( B ) = 2 B _ { n i } F _ { i m } \\partial _ { n } F _ { k l } \\partial _ { m } F _ { l k } , \\quad J _ { 3 } ( B ^ { 2 } ) = B _ { n i } B _ { i m } \\partial _ { n } F _ { k l } \\partial _ { m } F _ { l k } . \\end{align*}",
"\\begin{align*} J _ { 3 } ( B ) & = & 4 B _ { n i } \\partial _ { i } A _ { m } \\partial _ { n } \\partial _ { k } A _ { l } \\partial _ { m } \\partial _ { l } A _ { k } + ~ t e r m s ~ w i t h ~ \\partial ^ { 2 } A + t o t a l ~ d e r i v a t i v e \\\\ & = & - 4 B _ { n i } \\partial _ { i } \\partial _ { l } A _ { m } \\partial _ { n } \\partial _ { k } A _ { l } \\partial _ { m } A _ { k } \\\\ & & + ~ a ~ t e r m ~ w i t h ~ \\partial _ { l } A _ { l } + ~ t e r m s ~ w i t h ~ \\partial ^ { 2 } A + t o t a l ~ d e r i v a t i v e \\\\ & = & - 4 B _ { n m } \\partial _ { i } A _ { j } \\partial _ { n } \\partial _ { j } A _ { k } \\partial _ { m } \\partial _ { k } A _ { i } \\\\ & & + ~ a ~ t e r m ~ w i t h ~ \\partial \\cdot A + ~ t e r m s ~ w i t h ~ \\partial ^ { 2 } A + t o t a l ~ d e r i v a t i v e . \\end{align*}"
],
"latex_expand": [
"$ \\mitJ _ { 4 } $",
"$ \\mitJ _ { 5 } $",
"$ \\mitJ _ { 6 } $",
"$ \\mitJ _ { 7 } $",
"$ \\mitpartial _ { \\mitj } \\mitF _ { \\mitj \\miti } $",
"$ \\mitF $",
"$ \\mitB + \\mitF $",
"$ \\mitpartial _ { \\mitj } \\mitF _ { \\mitj \\miti } $",
"$ \\hat { \\mitT } _ { 4 } $",
"$ \\mitb _ { 1 } $",
"$ \\mitb _ { 2 } $",
"$ \\mitb _ { 3 } $",
"$ \\mitO ( \\mitB ^ { \\mitn } ) $",
"$ \\mitJ _ { \\miti } $",
"$ \\mitF $",
"$ \\mitB + \\mitF $",
"$ \\mitJ _ { \\miti } ( \\mitB ^ { \\mitn } ) $",
"$ \\mitJ _ { \\miti } ( \\mitB ) $",
"$ \\mitJ _ { \\miti } ( \\mitB ^ { 2 } ) $",
"$ \\miti = 1 , 2 , 3 $",
"$ \\mitJ _ { 1 } ( \\mitB ^ { 2 } ) $",
"$ \\mitJ _ { 2 } ( \\mitB ^ { 2 } ) $",
"$ \\mitJ _ { 3 } ( \\mitB ^ { 2 } ) $",
"$ \\mitJ _ { 1 } ( \\mitB ) $",
"$ \\mitJ _ { 2 } ( \\mitB ) $",
"$ \\mitJ _ { 3 } ( \\mitB ) $",
"$ \\mitO ( \\mitB , \\mitzeta ^ { 3 } , \\mitk ^ { 5 } ) $",
"\\begin{equation*} \\mitL = \\sum _ { \\miti = 1 } ^ { 7 } \\mitb _ { \\miti } \\mitJ _ { \\miti } , \\end{equation*}",
"\\begin{align*} & & \\displaystyle \\mitJ _ { 1 } = \\mitpartial _ { \\mitn } \\mitF _ { \\miti \\mitj } \\mitpartial _ { \\mitn } \\mitF _ { \\mitj \\miti } \\mitF _ { \\mitk \\mitl } \\mitF _ { \\mitl \\mitk } , \\quad \\mitJ _ { 2 } = \\mitpartial _ { \\mitn } \\mitF _ { \\miti \\mitj } \\mitpartial _ { \\mitn } \\mitF _ { \\mitj \\mitk } \\mitF _ { \\mitk \\mitl } \\mitF _ { \\mitl \\miti } , \\\\ & & \\displaystyle \\mitJ _ { 3 } = \\mitF _ { \\mitn \\miti } \\mitF _ { \\miti \\mitm } \\mitpartial _ { \\mitn } \\mitF _ { \\mitk \\mitl } \\mitpartial _ { \\mitm } \\mitF _ { \\mitl \\mitk } , \\quad \\mitJ _ { 4 } = \\mitpartial _ { \\mitn } \\mitF _ { \\mitn \\miti } \\mitpartial _ { \\mitm } \\mitF _ { \\miti \\mitm } \\mitF _ { \\mitk \\mitl } \\mitF _ { \\mitl \\mitk } , \\\\ & & \\displaystyle \\mitJ _ { 5 } = - \\mitpartial _ { \\mitn } \\mitF _ { \\mitn \\miti } \\mitpartial _ { \\mitm } \\mitF _ { \\miti \\mitj } \\mitF _ { \\mitj \\mitk } \\mitF _ { \\mitk \\mitm } , \\quad \\mitJ _ { 6 } = \\mitpartial ^ { 2 } \\mitF _ { \\miti \\mitj } \\mitF _ { \\mitj \\miti } \\mitF _ { \\mitk \\mitl } \\mitF _ { \\mitl \\mitk } , \\\\ & & \\displaystyle \\mitJ _ { 7 } = \\mitpartial ^ { 2 } \\mitF _ { \\miti \\mitj } \\mitF _ { \\mitj \\mitk } \\mitF _ { \\mitk \\mitl } \\mitF _ { \\mitl \\miti } , \\quad \\mitpartial ^ { 2 } \\mitF _ { \\miti \\mitj } = \\mitpartial _ { \\miti } \\mitpartial _ { \\mitk } \\mitF _ { \\mitk \\mitj } - \\mitpartial _ { \\mitj } \\mitpartial _ { \\mitk } \\mitF _ { \\mitk \\miti } . \\end{align*}",
"\\begin{align*} & & \\displaystyle \\mitJ _ { 1 } ( \\mitB ) = 2 \\mitpartial _ { \\mitn } \\mitF _ { \\miti \\mitj } \\mitpartial _ { \\mitn } \\mitF _ { \\mitj \\miti } \\mitB _ { \\mitk \\mitl } \\mitF _ { \\mitl \\mitk } , \\quad \\mitJ _ { 1 } ( \\mitB ^ { 2 } ) = \\mitpartial _ { \\mitn } \\mitF _ { \\miti \\mitj } \\mitpartial _ { \\mitn } \\mitF _ { \\mitj \\miti } \\mitB _ { \\mitk \\mitl } \\mitB _ { \\mitl \\mitk } , \\\\ & & \\displaystyle \\mitJ _ { 2 } ( \\mitB ) = 2 \\mitB _ { \\miti \\mitj } \\mitF _ { \\mitj \\mitk } \\mitpartial _ { \\mitn } \\mitF _ { \\mitk \\mitl } \\mitpartial _ { \\mitn } \\mitF _ { \\mitl \\miti } , \\quad \\mitJ _ { 2 } ( \\mitB ^ { 2 } ) = \\mitpartial _ { \\mitn } \\mitF _ { \\miti \\mitj } \\mitpartial _ { \\mitn } \\mitF _ { \\mitj \\mitk } \\mitB _ { \\mitk \\mitl } \\mitB _ { \\mitl \\miti } , \\\\ & & \\displaystyle \\mitJ _ { 3 } ( \\mitB ) = 2 \\mitB _ { \\mitn \\miti } \\mitF _ { \\miti \\mitm } \\mitpartial _ { \\mitn } \\mitF _ { \\mitk \\mitl } \\mitpartial _ { \\mitm } \\mitF _ { \\mitl \\mitk } , \\quad \\mitJ _ { 3 } ( \\mitB ^ { 2 } ) = \\mitB _ { \\mitn \\miti } \\mitB _ { \\miti \\mitm } \\mitpartial _ { \\mitn } \\mitF _ { \\mitk \\mitl } \\mitpartial _ { \\mitm } \\mitF _ { \\mitl \\mitk } . \\end{align*}",
"\\begin{align*} \\displaystyle \\mitJ _ { 3 } ( \\mitB ) & = & \\displaystyle 4 \\mitB _ { \\mitn \\miti } \\mitpartial _ { \\miti } \\mitA _ { \\mitm } \\mitpartial _ { \\mitn } \\mitpartial _ { \\mitk } \\mitA _ { \\mitl } \\mitpartial _ { \\mitm } \\mitpartial _ { \\mitl } \\mitA _ { \\mitk } + ~ \\mathrm { t e r m s } ~ \\mathrm { w i t h } ~ \\mitpartial ^ { 2 } \\mitA + \\mathrm { t o t a l ~ d e r i v a t i v e } \\\\ & = & \\displaystyle - 4 \\mitB _ { \\mitn \\miti } \\mitpartial _ { \\miti } \\mitpartial _ { \\mitl } \\mitA _ { \\mitm } \\mitpartial _ { \\mitn } \\mitpartial _ { \\mitk } \\mitA _ { \\mitl } \\mitpartial _ { \\mitm } \\mitA _ { \\mitk } \\\\ & & \\displaystyle + ~ \\mathrm { a } ~ \\mathrm { t e r m } ~ \\mathrm { w i t h } ~ \\mitpartial _ { \\mitl } \\mitA _ { \\mitl } + ~ \\mathrm { t e r m s } ~ \\mathrm { w i t h } ~ \\mitpartial ^ { 2 } \\mitA + \\mathrm { t o t a l ~ d e r i v a t i v e } \\\\ & = & \\displaystyle - 4 \\mitB _ { \\mitn \\mitm } \\mitpartial _ { \\miti } \\mitA _ { \\mitj } \\mitpartial _ { \\mitn } \\mitpartial _ { \\mitj } \\mitA _ { \\mitk } \\mitpartial _ { \\mitm } \\mitpartial _ { \\mitk } \\mitA _ { \\miti } \\\\ & & \\displaystyle + ~ \\mathrm { a } ~ \\mathrm { t e r m } ~ \\mathrm { w i t h } ~ \\mitpartial \\cdot \\mitA + ~ \\mathrm { t e r m s } ~ \\mathrm { w i t h } ~ \\mitpartial ^ { 2 } \\mitA + \\mathrm { t o t a l ~ d e r i v a t i v e } . \\end{align*}"
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|
|
0002194_page23
|
{
"latex": [
"$J_3 (B)$",
"$\\hat {T}_4$",
"$\\hat {T}_4$",
"$J_3 (B)$",
"$J_3$",
"$J_1$",
"$J_2$",
"$J_1 (B)$",
"$J_2 (B)$",
"$f(F)$",
"$\\alpha '$",
"$f(F)$",
"$J_1$",
"$J_2$",
"$J_1$",
"$J_2$",
"$J_1$",
"$J_2$",
"$J_3$",
"$\\hat {T}_4$",
"$\\hat {T}_4$",
"$J_3$",
"$O(\\partial ^2 F^4)$",
"$\\alpha '$",
"\\begin {eqnarray} J_3 (B) &=& 4 B_{ni} \\partial _i A_m \\partial _n \\partial _k A_l \\partial _m \\partial _l A_k + {\\rm ~terms~with~} \\partial ^2 A + {\\rm total~derivative} \\\\ &=& -4 B_{ni} \\partial _i \\partial _l A_m \\partial _n \\partial _k A_l \\partial _m A_k \\\\ && + {\\rm ~a~term~with~} \\partial _l A_l + {\\rm ~terms~with~} \\partial ^2 A + {\\rm total~derivative} \\\\ &=& -4 B_{nm} \\partial _i A_j \\partial _n \\partial _j A_k \\partial _m \\partial _k A_i \\\\ && + {\\rm ~a~term~with~} \\partial \\cdot A + {\\rm ~terms~with~} \\partial ^2 A + {\\rm total~derivative}. \\end {eqnarray}",
"\\begin {equation} \\hat {T}_4 \\sim -\\frac {1}{4} (2 \\pi \\alpha ')^2 J_3 (B). \\label {T_4-J_3} \\end {equation}",
"\\begin {equation} f(B+F) = f(F) + {\\rm total~derivative~using~the~equation~of~motion}, \\label {on-shell-initial} \\end {equation}"
],
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"$ J _ { 1 } $",
"$ J _ { 2 } $",
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"$ J _ { 2 } ( B ) $",
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"$ \\hat { T } _ { 4 } $",
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"$ O ( \\partial ^ { 2 } F ^ { 4 } ) $",
"$ \\alpha ^ { \\prime } $",
"\\begin{align*} J _ { 3 } ( B ) & = & 4 B _ { n i } \\partial _ { i } A _ { m } \\partial _ { n } \\partial _ { k } A _ { l } \\partial _ { m } \\partial _ { l } A _ { k } + ~ t e r m s ~ w i t h ~ \\partial ^ { 2 } A + t o t a l ~ d e r i v a t i v e \\\\ & = & - 4 B _ { n i } \\partial _ { i } \\partial _ { l } A _ { m } \\partial _ { n } \\partial _ { k } A _ { l } \\partial _ { m } A _ { k } \\\\ & & + ~ a ~ t e r m ~ w i t h ~ \\partial _ { l } A _ { l } + ~ t e r m s ~ w i t h ~ \\partial ^ { 2 } A + t o t a l ~ d e r i v a t i v e \\\\ & = & - 4 B _ { n m } \\partial _ { i } A _ { j } \\partial _ { n } \\partial _ { j } A _ { k } \\partial _ { m } \\partial _ { k } A _ { i } \\\\ & & + ~ a ~ t e r m ~ w i t h ~ \\partial \\cdot A + ~ t e r m s ~ w i t h ~ \\partial ^ { 2 } A + t o t a l ~ d e r i v a t i v e . \\end{align*}",
"\\begin{equation*} \\hat { T } _ { 4 } \\sim - \\frac { 1 } { 4 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } J _ { 3 } ( B ) . \\end{equation*}",
"\\begin{equation*} f ( B + F ) = f ( F ) + t o t a l ~ d e r i v a t i v e ~ u s i n g ~ t h e ~ e q u a t i o n ~ o f ~ m o t i o n , \\end{equation*}"
],
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"$ \\mitJ _ { 3 } ( \\mitB ) $",
"$ \\hat { \\mitT } _ { 4 } $",
"$ \\hat { \\mitT } _ { 4 } $",
"$ \\mitJ _ { 3 } ( \\mitB ) $",
"$ \\mitJ _ { 3 } $",
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"$ \\mitJ _ { 1 } ( \\mitB ) $",
"$ \\mitJ _ { 2 } ( \\mitB ) $",
"$ \\mitf ( \\mitF ) $",
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"$ \\mitJ _ { 2 } $",
"$ \\mitJ _ { 1 } $",
"$ \\mitJ _ { 2 } $",
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0002194_page24
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{
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"$J_3$",
"$\\hat {T}_4$",
"$F^3$",
"$O(D^2 F^4)$",
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"$J_2$",
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"$O(\\partial ^2 F^4)$",
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"$+$",
"\\begin {equation} -\\frac {1}{4} J_1 +2 J_2 + J_3. \\label {Okawa-J} \\end {equation}",
"\\begin {equation} -\\frac {1}{4} J_1 -2 J_2 + J_3. \\label {AT-J} \\end {equation}"
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"\\begin{equation*} - \\frac { 1 } { 4 } J _ { 1 } - 2 J _ { 2 } + J _ { 3 } . \\end{equation*}"
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"$ \\mitO ( \\mitpartial ^ { 2 } \\mitF ^ { 4 } ) $",
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"\\begin{equation*} - \\frac { 1 } { 4 } \\mitJ _ { 1 } + 2 \\mitJ _ { 2 } + \\mitJ _ { 3 } . \\end{equation*}",
"\\begin{equation*} - \\frac { 1 } { 4 } \\mitJ _ { 1 } - 2 \\mitJ _ { 2 } + \\mitJ _ { 3 } . \\end{equation*}"
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0002194_page25
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{
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"$J_1$",
"$J_2$",
"$J_1 (B)$",
"$J_1 (B^2)$",
"$J_2 (B)$",
"$J_2 (B^2)$",
"$J_4$",
"$J_5$",
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"$J_1$",
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"$O(\\alpha '^2)$",
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"$O(\\alpha ')$",
"$\\hat {{\\cal L}} (\\hat {F})$",
"$O(\\alpha '^2)$",
"$\\hat {{\\cal L}} (\\hat {F})$",
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"$J_2 (B^2)$",
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"$O(\\alpha '^3)$",
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"$J_2 (B^2)$",
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0002194_page26
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"$c_2 (2 \\pi \\alpha ')^3 ( J_2 (B) + J_2 (B^2) )$",
"$\\hat {F}^2$",
"$\\hat {D} \\hat {F}$",
"$\\hat {F}$",
"$O(\\alpha '^3)$",
"$G_{ij}$",
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"\\begin {eqnarray} \\hat {A}_i &=& A_i + \\frac {1}{2} (2 \\pi \\alpha ')^2 B_{kl} A_k (\\partial _l A_i + F_{li}) \\\\ && -\\frac {1}{4} c_2 (2 \\pi \\alpha ')^3 \\partial _j ( 2 F B F + B^2 F + F B^2 )_{ji} + O(\\alpha '^4) \\end {eqnarray}",
"\\begin {eqnarray} {\\cal L}(B+F) &=& \\frac {\\sqrt {\\det g}}{g_s} \\Biggl [ {\\rm Tr} (B+F)^2 + (2 \\pi \\alpha ')^2 \\left [ \\frac {1}{2} {\\rm Tr} (B+F)^4 -\\frac {1}{8} [ {\\rm Tr} (B+F)^2 ]^2 \\right ] \\\\ && \\qquad \\qquad + c_2 (2 \\pi \\alpha ')^3 \\partial _n (B+F)_{ij} \\partial _n (B+F)_{jk} (B+F)_{kl} (B+F)_{li} \\\\ && \\qquad \\qquad + ~O(\\alpha '^4) \\Biggr ], \\end {eqnarray}",
"\\begin {eqnarray} \\hat {{\\cal L}}(\\hat {F}) &=& \\frac {\\sqrt {\\det G}}{G_s} \\Biggl [ {\\rm Tr} ( G^{-1} \\hat {F} \\ast G^{-1} \\hat {F} ) \\\\ && \\qquad \\quad + (2 \\pi \\alpha ')^2 \\biggl [ \\frac {1}{2} {\\rm Tr} ( G^{-1} \\hat {F} )^4_{\\rm arbitrary} -\\frac {1}{8} ( {\\rm Tr} ( G^{-1} \\hat {F} )^2 )^2_{\\rm arbitrary} \\biggr ] \\\\ && \\qquad \\quad + c_2 (2 \\pi \\alpha ')^3 ( \\hat {D}_n \\hat {F}_{ij} \\ast \\hat {D}_n \\hat {F}_{jk} \\ast \\hat {F}_{kl} \\ast \\hat {F}_{li} )_{G, \\rm ~arbitrary} + ~O(\\alpha '^4) \\Biggr ], \\end {eqnarray}",
"\\begin {equation} B_{ij} = -\\frac {1}{(2 \\pi \\alpha ')^2} ( g \\theta g )_{ij} + O( \\theta ^2 ), \\end {equation}",
"\\begin {equation} \\hat {A}_i \\to \\tilde {A}_i \\to A_i, \\end {equation}",
"\\begin {eqnarray} \\hat {A}_i &=& \\tilde {A}_i + \\frac {1}{2} (2 \\pi \\alpha ')^2 B_{kl} \\tilde {A}_k (\\partial _l \\tilde {A}_i + \\tilde {F}_{li}) + O(\\alpha '^4), \\\\ \\tilde {A}_i &=& A_i -\\frac {1}{4} c_2 (2 \\pi \\alpha ')^3 \\partial _j ( 2 F B F + B^2 F + F B^2 )_{ji} + O(\\alpha '^4). \\end {eqnarray}"
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"$ \\hat { D } \\hat { F } $",
"$ \\hat { F } $",
"$ O ( \\alpha ^ { \\prime 3 } ) $",
"$ G _ { i j } $",
"$ \\alpha ^ { \\prime } $",
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"\\begin{align*} \\hat { A } _ { i } & = & A _ { i } + \\frac { 1 } { 2 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } B _ { k l } A _ { k } ( \\partial _ { l } A _ { i } + F _ { l i } ) \\\\ & & - \\frac { 1 } { 4 } c _ { 2 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 3 } \\partial _ { j } ( 2 F B F + B ^ { 2 } F + F B ^ { 2 } ) _ { j i } + O ( \\alpha ^ { \\prime 4 } ) \\end{align*}",
"\\begin{align*} L ( B + F ) & = & \\frac { \\sqrt { \\operatorname { d e t } g } } { g _ { s } } [ T r ( B + F ) ^ { 2 } + ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } [ \\frac { 1 } { 2 } T r ( B + F ) ^ { 4 } - \\frac { 1 } { 8 } [ T r ( B + F ) ^ { 2 } ] ^ { 2 } ] \\\\ & & \\qquad \\qquad + c _ { 2 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 3 } \\partial _ { n } ( B + F ) _ { i j } \\partial _ { n } ( B + F ) _ { j k } ( B + F ) _ { k l } ( B + F ) _ { l i } \\\\ & & \\qquad \\qquad + ~ O ( \\alpha ^ { \\prime 4 } ) ] , \\end{align*}",
"\\begin{align*} \\hat { L } ( \\hat { F } ) & = & \\frac { \\sqrt { \\operatorname { d e t } G } } { G _ { s } } [ T r ( G ^ { - 1 } \\hat { F } \\ast G ^ { - 1 } \\hat { F } ) \\\\ & & \\qquad \\quad + ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } [ \\frac { 1 } { 2 } T r ( G ^ { - 1 } \\hat { F } ) _ { a r b i t r a r y } ^ { 4 } - \\frac { 1 } { 8 } ( T r ( G ^ { - 1 } \\hat { F } ) ^ { 2 } ) _ { a r b i t r a r y } ^ { 2 } ] \\\\ & & \\qquad \\quad + c _ { 2 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 3 } ( \\hat { D } _ { n } \\hat { F } _ { i j } \\ast \\hat { D } _ { n } \\hat { F } _ { j k } \\ast \\hat { F } _ { k l } \\ast \\hat { F } _ { l i } ) _ { G , ~ a r b i t r a r y } + ~ O ( \\alpha ^ { \\prime 4 } ) ] , \\end{align*}",
"\\begin{equation*} B _ { i j } = - \\frac { 1 } { ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } } ( g \\theta g ) _ { i j } + O ( \\theta ^ { 2 } ) , \\end{equation*}",
"\\begin{equation*} \\hat { A } _ { i } \\rightarrow \\widetilde { A } _ { i } \\rightarrow A _ { i } , \\end{equation*}",
"\\begin{align*} \\hat { A } _ { i } & = & \\widetilde { A } _ { i } + \\frac { 1 } { 2 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } B _ { k l } \\widetilde { A } _ { k } ( \\partial _ { l } \\widetilde { A } _ { i } + \\widetilde { F } _ { l i } ) + O ( \\alpha ^ { \\prime 4 } ) , \\\\ \\widetilde { A } _ { i } & = & A _ { i } - \\frac { 1 } { 4 } c _ { 2 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 3 } \\partial _ { j } ( 2 F B F + B ^ { 2 } F + F B ^ { 2 } ) _ { j i } + O ( \\alpha ^ { \\prime 4 } ) . \\end{align*}"
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"$ \\hat { \\mitF } ^ { 2 } $",
"$ \\hat { \\mitD } \\hat { \\mitF } $",
"$ \\hat { \\mitF } $",
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"\\begin{align*} \\displaystyle \\hat { \\mitL } ( \\hat { \\mitF } ) & = & \\displaystyle \\frac { \\sqrt { \\operatorname { d e t } \\mitG } } { \\mitG _ { \\mits } } \\Bigg [ \\mathrm { T r } ( \\mitG ^ { - 1 } \\hat { \\mitF } \\ast \\mitG ^ { - 1 } \\hat { \\mitF } ) \\\\ & & \\displaystyle \\qquad \\quad + ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\bigg [ \\frac { 1 } { 2 } \\mathrm { T r } ( \\mitG ^ { - 1 } \\hat { \\mitF } ) _ { \\mathrm { a r b i t r a r y } } ^ { 4 } - \\frac { 1 } { 8 } ( \\mathrm { T r } ( \\mitG ^ { - 1 } \\hat { \\mitF } ) ^ { 2 } ) _ { \\mathrm { a r b i t r a r y } } ^ { 2 } \\bigg ] \\\\ & & \\displaystyle \\qquad \\quad + \\mitc _ { 2 } ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 3 } ( \\hat { \\mitD } _ { \\mitn } \\hat { \\mitF } _ { \\miti \\mitj } \\ast \\hat { \\mitD } _ { \\mitn } \\hat { \\mitF } _ { \\mitj \\mitk } \\ast \\hat { \\mitF } _ { \\mitk \\mitl } \\ast \\hat { \\mitF } _ { \\mitl \\miti } ) _ { \\mitG , ~ \\mathrm { a r b i t r a r y } } + ~ \\mitO ( \\mitalpha ^ { \\prime 4 } ) \\Bigg ] , \\end{align*}",
"\\begin{equation*} \\mitB _ { \\miti \\mitj } = - \\frac { 1 } { ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } } ( \\mitg \\mittheta \\mitg ) _ { \\miti \\mitj } + \\mitO ( \\mittheta ^ { 2 } ) , \\end{equation*}",
"\\begin{equation*} \\hat { \\mitA } _ { \\miti } \\rightarrow \\tilde { \\mitA } _ { \\miti } \\rightarrow \\mitA _ { \\miti } , \\end{equation*}",
"\\begin{align*} \\displaystyle \\hat { \\mitA } _ { \\miti } & = & \\displaystyle \\tilde { \\mitA } _ { \\miti } + \\frac { 1 } { 2 } ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mitB _ { \\mitk \\mitl } \\tilde { \\mitA } _ { \\mitk } ( \\mitpartial _ { \\mitl } \\tilde { \\mitA } _ { \\miti } + \\tilde { \\mitF } _ { \\mitl \\miti } ) + \\mitO ( \\mitalpha ^ { \\prime 4 } ) , \\\\ \\displaystyle \\tilde { \\mitA } _ { \\miti } & = & \\displaystyle \\mitA _ { \\miti } - \\frac { 1 } { 4 } \\mitc _ { 2 } ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 3 } \\mitpartial _ { \\mitj } ( 2 \\mitF \\mitB \\mitF + \\mitB ^ { 2 } \\mitF + \\mitF \\mitB ^ { 2 } ) _ { \\mitj \\miti } + \\mitO ( \\mitalpha ^ { \\prime 4 } ) . \\end{align*}"
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0002194_page27
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{
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"$\\hat {A}_i$",
"$\\tilde {A}_i$",
"$\\tilde {A}_i$",
"$A_i$",
"$\\tilde {A}_i$",
"$A_i$",
"$B$",
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"$\\hat {T}_4$",
"$\\hat {{\\cal L}} (\\hat {F})$",
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"$O(\\alpha '^2)$",
"$O(\\alpha ')$",
"$\\hat {T}_4$",
"$\\hat {{\\cal L}} (\\hat {F}) = {\\cal L} (B+F)$",
"$G_s$",
"$\\hat {J}_i$",
"$J_i$",
"\\begin {equation} \\hat {{\\cal L}}_1 (\\hat {F}) = \\frac {\\sqrt {\\det G}}{G_s} \\left [ \\hat {T}_3 + (2 \\pi \\alpha ')^2 \\left ( -\\frac {1}{4} \\hat {J}_1 + 2 \\hat {J}_2 + \\hat {J}_3 \\right ) + O(\\alpha '^4) \\right ], \\end {equation}",
"\\begin {equation} \\hat {{\\cal L}}_2 (\\hat {F}) = \\frac {\\sqrt {\\det G}}{G_s} \\left [ \\hat {T}_1 + (2 \\pi \\alpha ')^2 \\left ( \\hat {J}_5 -\\frac {1}{8} \\hat {J}_6 + \\frac {1}{2} \\hat {J}_7 \\right ) + O(\\alpha '^4) \\right ], \\end {equation}"
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"\\begin{equation*} \\hat { L } _ { 2 } ( \\hat { F } ) = \\frac { \\sqrt { \\operatorname { d e t } G } } { G _ { s } } [ \\hat { T } _ { 1 } + ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } ( \\hat { J } _ { 5 } - \\frac { 1 } { 8 } \\hat { J } _ { 6 } + \\frac { 1 } { 2 } \\hat { J } _ { 7 } ) + O ( \\alpha ^ { \\prime 4 } ) ] , \\end{equation*}"
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"\\begin{equation*} \\hat { \\mitL } _ { 2 } ( \\hat { \\mitF } ) = \\frac { \\sqrt { \\operatorname { d e t } \\mitG } } { \\mitG _ { \\mits } } \\left[ \\hat { \\mitT } _ { 1 } + ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\left( \\hat { \\mitJ } _ { 5 } - \\frac { 1 } { 8 } \\hat { \\mitJ } _ { 6 } + \\frac { 1 } { 2 } \\hat { \\mitJ } _ { 7 } \\right) + \\mitO ( \\mitalpha ^ { \\prime 4 } ) \\right] , \\end{equation*}"
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0002194_page28
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"latex": [
"${\\cal L} (B+F) \\to \\hat {{\\cal L}} (\\hat {F})$",
"$\\{ \\hat {T}_1, \\hat {T}_4 \\}$",
"$\\alpha '$",
"$\\hat {{\\cal L}} (\\hat {F}) = {\\cal L} (B+F)$",
"$G_s$",
"$\\hat {{\\cal L}}' (\\hat {F}) = {\\cal L}' (B+F)$",
"$G_s$",
"${\\cal L}' (F) - {\\cal L} (F)$",
"$O(\\partial ^2 F^4)$",
"$O(\\partial ^2 F^4)$",
"$O(\\partial ^2 F^4)$",
"${\\cal F} (F)$",
"$a=0$",
"$\\hat {{\\cal L}} (\\hat {F}) = {\\cal L} (B+F)$",
"$G_s$",
"\\begin {eqnarray} \\hat {{\\cal L}} (\\hat {F}) &=& \\frac {\\sqrt {\\det G}}{G_s} \\Biggl [ a \\hat {T}_1 + b \\hat {T}_4 + a (2 \\pi \\alpha ')^2 \\left ( \\hat {J}_5 -\\frac {1}{8} \\hat {J}_6 + \\frac {1}{2} \\hat {J}_7 \\right ) \\\\ && \\quad - \\frac {1}{4} b (2 \\pi \\alpha ')^2 \\left ( -\\frac {1}{4} \\hat {J}_1 + 2 \\hat {J}_2 + \\hat {J}_3 +2 \\hat {J}_5 -\\frac {1}{4} \\hat {J}_6 + \\hat {J}_7 \\right ) + O(\\alpha '^4) \\Biggr ]. \\end {eqnarray}",
"\\begin {eqnarray} \\hat {{\\cal L}}' (\\hat {F}) &=& \\frac {\\sqrt {\\det G}}{G_s} \\Biggl [ a \\hat {T}_1 + b \\hat {T}_4 \\\\ && + ~O(\\alpha '^2) {\\rm ~terms~different~from~those~of~} \\hat {{\\cal L}} (\\hat {F}) + O(\\alpha '^4) \\Biggr ], \\end {eqnarray}",
"\\begin {equation} {\\cal F} (F) \\equiv -\\frac {1}{4} J_1 + 2 J_2 + J_3 +2 J_5 -\\frac {1}{4} J_6 + J_7, \\label {necessary} \\end {equation}",
"\\begin {equation} -\\frac {1}{4} {\\cal F} (B+F) = \\frac {1}{2} B_{nm} F_{ij} \\partial _n F_{jk} \\partial _m F_{ki} -\\frac {1}{4} {\\cal F} (F) + {\\rm total~derivative}, \\end {equation}"
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"$ \\alpha ^ { \\prime } $",
"$ \\hat { L } ( \\hat { F } ) = L ( B + F ) $",
"$ G _ { s } $",
"$ \\hat { L } ^ { \\prime } ( \\hat { F } ) = L ^ { \\prime } ( B + F ) $",
"$ G _ { s } $",
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"$ O ( \\partial ^ { 2 } F ^ { 4 } ) $",
"$ O ( \\partial ^ { 2 } F ^ { 4 } ) $",
"$ F ( F ) $",
"$ a = 0 $",
"$ \\hat { L } ( \\hat { F } ) = L ( B + F ) $",
"$ G _ { s } $",
"\\begin{align*} \\hat { L } ( \\hat { F } ) & = & \\frac { \\sqrt { \\operatorname { d e t } G } } { G _ { s } } [ a \\hat { T } _ { 1 } + b \\hat { T } _ { 4 } + a ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } ( \\hat { J } _ { 5 } - \\frac { 1 } { 8 } \\hat { J } _ { 6 } + \\frac { 1 } { 2 } \\hat { J } _ { 7 } ) \\\\ & & \\quad - \\frac { 1 } { 4 } b ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } ( - \\frac { 1 } { 4 } \\hat { J } _ { 1 } + 2 \\hat { J } _ { 2 } + \\hat { J } _ { 3 } + 2 \\hat { J } _ { 5 } - \\frac { 1 } { 4 } \\hat { J } _ { 6 } + \\hat { J } _ { 7 } ) + O ( \\alpha ^ { \\prime 4 } ) ] . \\end{align*}",
"\\begin{align*} \\hat { L } ^ { \\prime } ( \\hat { F } ) & = & \\frac { \\sqrt { \\operatorname { d e t } G } } { G _ { s } } [ a \\hat { T } _ { 1 } + b \\hat { T } _ { 4 } \\\\ & & + ~ O ( \\alpha ^ { \\prime 2 } ) ~ t e r m s ~ d i f f e r e n t ~ f r o m ~ t h o s e ~ o f ~ \\hat { L } ( \\hat { F } ) + O ( \\alpha ^ { \\prime 4 } ) ] , \\end{align*}",
"\\begin{equation*} F ( F ) \\equiv - \\frac { 1 } { 4 } J _ { 1 } + 2 J _ { 2 } + J _ { 3 } + 2 J _ { 5 } - \\frac { 1 } { 4 } J _ { 6 } + J _ { 7 } , \\end{equation*}",
"\\begin{equation*} - \\frac { 1 } { 4 } F ( B + F ) = \\frac { 1 } { 2 } B _ { n m } F _ { i j } \\partial _ { n } F _ { j k } \\partial _ { m } F _ { k i } - \\frac { 1 } { 4 } F ( F ) + t o t a l ~ d e r i v a t i v e , \\end{equation*}"
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"$ \\mitalpha ^ { \\prime } $",
"$ \\hat { \\mitL } ( \\hat { \\mitF } ) = \\mitL ( \\mitB + \\mitF ) $",
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"$ \\hat { \\mitL } ^ { \\prime } ( \\hat { \\mitF } ) = \\mitL ^ { \\prime } ( \\mitB + \\mitF ) $",
"$ \\mitG _ { \\mits } $",
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"$ \\mitO ( \\mitpartial ^ { 2 } \\mitF ^ { 4 } ) $",
"$ \\mitO ( \\mitpartial ^ { 2 } \\mitF ^ { 4 } ) $",
"$ \\mitF ( \\mitF ) $",
"$ \\mita = 0 $",
"$ \\hat { \\mitL } ( \\hat { \\mitF } ) = \\mitL ( \\mitB + \\mitF ) $",
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"\\begin{align*} \\displaystyle \\hat { \\mitL } ^ { \\prime } ( \\hat { \\mitF } ) & = & \\displaystyle \\frac { \\sqrt { \\operatorname { d e t } \\mitG } } { \\mitG _ { \\mits } } \\Bigg [ \\mita \\hat { \\mitT } _ { 1 } + \\mitb \\hat { \\mitT } _ { 4 } \\\\ & & \\displaystyle + ~ \\mitO ( \\mitalpha ^ { \\prime 2 } ) ~ \\mathrm { t e r m s } ~ \\mathrm { d i f f e r e n t } ~ \\mathrm { f r o m } ~ \\mathrm { t h o s e } ~ \\mathrm { o f } ~ \\hat { \\mitL } ( \\hat { \\mitF } ) + \\mitO ( \\mitalpha ^ { \\prime 4 } ) \\Bigg ] , \\end{align*}",
"\\begin{equation*} \\mitF ( \\mitF ) \\equiv - \\frac { 1 } { 4 } \\mitJ _ { 1 } + 2 \\mitJ _ { 2 } + \\mitJ _ { 3 } + 2 \\mitJ _ { 5 } - \\frac { 1 } { 4 } \\mitJ _ { 6 } + \\mitJ _ { 7 } , \\end{equation*}",
"\\begin{equation*} - \\frac { 1 } { 4 } \\mitF ( \\mitB + \\mitF ) = \\frac { 1 } { 2 } \\mitB _ { \\mitn \\mitm } \\mitF _ { \\miti \\mitj } \\mitpartial _ { \\mitn } \\mitF _ { \\mitj \\mitk } \\mitpartial _ { \\mitm } \\mitF _ { \\mitk \\miti } - \\frac { 1 } { 4 } \\mitF ( \\mitF ) + \\mathrm { t o t a l ~ d e r i v a t i v e } , \\end{equation*}"
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0002194_page29
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{
"latex": [
"${\\cal F} (F)$",
"$O(\\partial ^2 F^4)$",
"$\\alpha '$",
"$\\hat {{\\cal L}} (\\hat {F}) = {\\cal L} (B+F)$",
"$G_s$",
"$J_3$",
"${\\cal L} (B+F)$",
"${\\hat {\\cal L}} (\\hat {F})$",
"$J_i$",
"$J_3$",
"$\\alpha '^3$",
"$J_2$",
"$J_1$",
"$J_2$",
"$J_3$",
"$b (2 \\pi \\alpha ')^3 \\hat {J_2}$",
"$J_4$",
"$J_5$",
"$J_6$",
"$J_7$",
"$O(\\alpha '^3)$",
"\\begin {eqnarray} \\hat {{\\cal L}}(\\hat {F}) &=& \\frac {\\sqrt {\\det G}}{G_s} \\Biggl [ {\\rm Tr} ( G^{-1} \\hat {F} \\ast G^{-1} \\hat {F} ) + b (2 \\pi \\alpha ') \\hat {T}_4 \\\\ && \\qquad \\qquad + (2 \\pi \\alpha ')^2 \\biggl [ \\frac {1}{2} {\\rm Tr} ( G^{-1} \\hat {F} )^4_{\\rm arbitrary} -\\frac {1}{8} ( {\\rm Tr} ( G^{-1} \\hat {F} )^2 )^2_{\\rm arbitrary} \\biggr ] \\\\ && \\qquad \\qquad - \\frac {1}{4} b (2 \\pi \\alpha ')^3 \\left ( -\\frac {1}{4} \\hat {J}_1 + 2 \\hat {J}_2 + \\hat {J}_3 +2 \\hat {J}_5 -\\frac {1}{4} \\hat {J}_6 + \\hat {J}_7 \\right ) \\\\ && \\qquad \\qquad + ~O(\\alpha '^4) \\Biggr ], \\end {eqnarray}",
"\\begin {eqnarray} \\hat {A}_i &=& A_i + \\frac {1}{2} (2 \\pi \\alpha ')^2 B_{kl} A_k (\\partial _l A_i + F_{li}) \\\\ && -\\frac {1}{4} b (2 \\pi \\alpha ')^3 \\partial _j ( 2 F B F + B^2 F + F B^2 )_{ji} + O(\\alpha '^4). \\end {eqnarray}"
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"\\begin{align*} \\displaystyle \\hat { \\mitA } _ { \\miti } & = & \\displaystyle \\mitA _ { \\miti } + \\frac { 1 } { 2 } ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mitB _ { \\mitk \\mitl } \\mitA _ { \\mitk } ( \\mitpartial _ { \\mitl } \\mitA _ { \\miti } + \\mitF _ { \\mitl \\miti } ) \\\\ & & \\displaystyle - \\frac { 1 } { 4 } \\mitb ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 3 } \\mitpartial _ { \\mitj } ( 2 \\mitF \\mitB \\mitF + \\mitB ^ { 2 } \\mitF + \\mitF \\mitB ^ { 2 } ) _ { \\mitj \\miti } + \\mitO ( \\mitalpha ^ { \\prime 4 } ) . \\end{align*}"
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0002194_page30
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0002194_page31
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0002194_page32
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0002194_page33
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"\\begin {eqnarray} {\\bf F}_{ij} &\\equiv & \\partial _i {\\bf A}_j - \\partial _j {\\bf A}_i, \\\\ {\\bf \\hat {F}}_{ij} &\\equiv & \\partial _i {\\bf \\hat {A}}_j - \\partial _j {\\bf \\hat {A}}_i -i \\sqrt {G_s} {\\bf \\hat {A}}_i \\ast {\\bf \\hat {A}}_j +i \\sqrt {G_s} {\\bf \\hat {A}}_j \\ast {\\bf \\hat {A}}_i, \\end {eqnarray}",
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"\\begin {eqnarray} && \\sqrt {\\det G} \\, {\\rm Tr } (G^{-1} {\\bf \\hat {F}} \\ast G^{-1} {\\bf \\hat {F}}) \\\\ &=& \\sqrt {\\det g} \\Biggl [ ( \\partial _i {\\bf \\hat {A}}_j - \\partial _j {\\bf \\hat {A}}_i ) ( \\partial _j {\\bf \\hat {A}}_i - \\partial _i {\\bf \\hat {A}}_j ) -4 (2 \\pi \\alpha ')^2 \\sqrt {G_s} B_{kl} \\partial _k {\\bf \\hat {A}}_i \\partial _l {\\bf \\hat {A}}_j \\partial _j {\\bf \\hat {A}}_i \\\\ && +2 (2 \\pi \\alpha ')^2 (B^2)_{ij} ( \\partial _j {\\bf \\hat {A}}_k - \\partial _k {\\bf \\hat {A}}_j ) ( \\partial _k {\\bf \\hat {A}}_i - \\partial _i {\\bf \\hat {A}}_k ) \\\\ && -\\frac {1}{2} (2 \\pi \\alpha ')^2 {\\rm Tr} B^2 ( \\partial _i {\\bf \\hat {A}}_j - \\partial _j {\\bf \\hat {A}}_i ) ( \\partial _j {\\bf \\hat {A}}_i - \\partial _i {\\bf \\hat {A}}_j ) + O(\\alpha '^4) \\Biggr ]. \\end {eqnarray}"
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"\\begin{align*} & & \\displaystyle \\sqrt { \\operatorname { d e t } \\mitG } \\, \\mathrm { T r } ( \\mitG ^ { - 1 } \\hat { \\mitF } \\ast \\mitG ^ { - 1 } \\hat { \\mitF } ) \\\\ & = & \\displaystyle \\sqrt { \\operatorname { d e t } \\mitg } \\Bigg [ ( \\mitpartial _ { \\miti } \\hat { \\mitA } _ { \\mitj } - \\mitpartial _ { \\mitj } \\hat { \\mitA } _ { \\miti } ) ( \\mitpartial _ { \\mitj } \\hat { \\mitA } _ { \\miti } - \\mitpartial _ { \\miti } \\hat { \\mitA } _ { \\mitj } ) - 4 ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\sqrt { \\mitG _ { \\mits } } \\mitB _ { \\mitk \\mitl } \\mitpartial _ { \\mitk } \\hat { \\mitA } _ { \\miti } \\mitpartial _ { \\mitl } \\hat { \\mitA } _ { \\mitj } \\mitpartial _ { \\mitj } \\hat { \\mitA } _ { \\miti } \\\\ & & \\displaystyle + 2 ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } ( \\mitB ^ { 2 } ) _ { \\miti \\mitj } ( \\mitpartial _ { \\mitj } \\hat { \\mitA } _ { \\mitk } - \\mitpartial _ { \\mitk } \\hat { \\mitA } _ { \\mitj } ) ( \\mitpartial _ { \\mitk } \\hat { \\mitA } _ { \\miti } - \\mitpartial _ { \\miti } \\hat { \\mitA } _ { \\mitk } ) \\\\ & & \\displaystyle - \\frac { 1 } { 2 } ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mathrm { T r } \\mitB ^ { 2 } ( \\mitpartial _ { \\miti } \\hat { \\mitA } _ { \\mitj } - \\mitpartial _ { \\mitj } \\hat { \\mitA } _ { \\miti } ) ( \\mitpartial _ { \\mitj } \\hat { \\mitA } _ { \\miti } - \\mitpartial _ { \\miti } \\hat { \\mitA } _ { \\mitj } ) + \\mitO ( \\mitalpha ^ { \\prime 4 } ) \\Bigg ] . \\end{align*}"
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0002194_page34
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"$\\hat {{\\cal L}} (\\hat {F})$",
"${\\cal L} (B+F)$",
"$O(B,\\zeta ^3,k^3)$",
"${\\cal L} (B+F)$",
"$t$",
"$O(B)$",
"$O(B^2)$",
"$O(\\alpha '^2)$",
"$B$",
"\\begin {eqnarray} && \\frac {\\sqrt {\\det g}}{g_s} {\\rm Tr} (B+F)^4 = \\sqrt {\\det g} \\, g_s {\\rm Tr} \\left ( \\frac {B}{\\sqrt {g_s}} + {\\bf F} \\right )^4 \\\\ &=& \\sqrt {\\det g} \\, \\left [ g_s {\\rm Tr} \\, {\\bf F}^4 + 4 \\sqrt {g_s} \\, {\\rm Tr} B {\\bf F}^3 + O(B^2) \\right ], \\\\ && \\frac {\\sqrt {\\det g}}{g_s} [{\\rm Tr} (B+F)^2]^2 = \\sqrt {\\det g} \\, g_s \\left [ {\\rm Tr} \\left ( \\frac {B}{\\sqrt {g_s}} + {\\bf F} \\right )^2 \\right ]^2 \\\\ &=& \\sqrt {\\det g} \\, \\left [ g_s ( {\\rm Tr} \\, {\\bf F}^2 )^2 + 4 \\sqrt {g_s} \\, {\\rm Tr} B {\\bf F} \\, {\\rm Tr} \\, {\\bf F}^2 + O(B^2) \\right ]. \\end {eqnarray}",
"\\begin {eqnarray} {\\cal L} (B+F) &=& \\sqrt {\\det g} \\left [ {\\rm Tr} \\left ( \\frac {B}{\\sqrt {g_s}} + {\\bf F} \\right )^2 + (2 \\pi \\alpha ')^2 \\sqrt {G_s \\, g_s} \\left [ \\frac {1}{2} {\\rm Tr} \\left ( \\frac {B}{\\sqrt {g_s}} + {\\bf F} \\right )^4 \\right . \\right . \\\\ && \\qquad \\qquad \\left . \\left . -\\frac {1}{8} \\left ( {\\rm Tr} \\left ( \\frac {B}{\\sqrt {g_s}} + {\\bf F} \\right )^2 \\right )^2 \\right ] + O(\\alpha '^4) + {\\rm derivative~corrections} \\right ] \\\\ &=& \\frac {\\sqrt {\\det g}}{g_s} \\Biggl [ {\\rm Tr} (B+F)^2 + \\sqrt {t} \\, (2 \\pi \\alpha ')^2 \\left [ \\frac {1}{2} {\\rm Tr} (B+F)^4 \\right . \\\\ && \\qquad \\qquad \\left . -\\frac {1}{8} [ {\\rm Tr} (B+F)^2 ]^2 \\right ] + ~O(\\alpha '^4) + {\\rm derivative~corrections} \\Biggr ], \\end {eqnarray}",
"\\begin {eqnarray} && \\sqrt {\\det g} \\, (2 \\pi \\alpha ')^2 \\sqrt {G_s \\, g_s} \\left [ \\frac {1}{2} {\\rm Tr} \\left ( \\frac {B}{\\sqrt {g_s}} + {\\bf F} \\right )^4 -\\frac {1}{8} \\left ( {\\rm Tr} \\left ( \\frac {B}{\\sqrt {g_s}} + {\\bf F} \\right )^2 \\right )^2 \\right ] \\\\ &=& \\sqrt {\\det g} \\, (2 \\pi \\alpha ')^2 \\Biggl [ \\sqrt {t} \\, g_s \\left [ \\frac {1}{2} {\\rm Tr} \\, {\\bf F}^4 -\\frac {1}{8} ( {\\rm Tr} \\, {\\bf F}^2 )^2 \\right ] \\\\ && + \\sqrt {t \\, g_s} \\left ( 2 {\\rm Tr} B {\\bf F}^3 -\\frac {1}{2} {\\rm Tr} B {\\bf F} \\, {\\rm Tr} \\, {\\bf F}^2 \\right ) \\\\ && + \\sqrt {t} \\left ( 2 {\\rm Tr} B^2 {\\bf F}^2 -\\frac {1}{4} {\\rm Tr} B^2 \\, {\\rm Tr} \\, {\\bf F}^2 \\right ) + {\\rm total~derivative} + {\\rm const.} \\Biggr ]. \\end {eqnarray}"
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"$ L ( B + F ) $",
"$ O ( B , \\zeta ^ { 3 } , k ^ { 3 } ) $",
"$ L ( B + F ) $",
"$ t $",
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"\\begin{align*} L ( B + F ) & = & \\sqrt { \\operatorname { d e t } g } [ T r { ( \\frac { B } { \\sqrt { g _ { s } } } + F ) } ^ { 2 } + ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } \\sqrt { G _ { s } \\, g _ { s } } [ \\frac { 1 } { 2 } T r { ( \\frac { B } { \\sqrt { g _ { s } } } + F ) } ^ { 4 } \\\\ & & \\qquad \\qquad - \\frac { 1 } { 8 } { ( T r { ( \\frac { B } { \\sqrt { g _ { s } } } + F ) } ^ { 2 } ) } ^ { 2 } ] + O ( \\alpha ^ { \\prime 4 } ) + d e r i v a t i v e ~ c o r r e c t i o n s ] \\\\ & = & \\frac { \\sqrt { \\operatorname { d e t } g } } { g _ { s } } [ T r ( B + F ) ^ { 2 } + \\sqrt { t } \\, ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } [ \\frac { 1 } { 2 } T r ( B + F ) ^ { 4 } \\\\ & & \\qquad \\qquad - \\frac { 1 } { 8 } [ T r ( B + F ) ^ { 2 } ] ^ { 2 } ] + ~ O ( \\alpha ^ { \\prime 4 } ) + d e r i v a t i v e ~ c o r r e c t i o n s ] , \\end{align*}",
"\\begin{align*} & & \\sqrt { \\operatorname { d e t } g } \\, ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } \\sqrt { G _ { s } \\, g _ { s } } [ \\frac { 1 } { 2 } T r { ( \\frac { B } { \\sqrt { g _ { s } } } + F ) } ^ { 4 } - \\frac { 1 } { 8 } { ( T r { ( \\frac { B } { \\sqrt { g _ { s } } } + F ) } ^ { 2 } ) } ^ { 2 } ] \\\\ & = & \\sqrt { \\operatorname { d e t } g } \\, ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } [ \\sqrt { t } \\, g _ { s } [ \\frac { 1 } { 2 } T r \\, F ^ { 4 } - \\frac { 1 } { 8 } ( T r \\, F ^ { 2 } ) ^ { 2 } ] \\\\ & & + \\sqrt { t \\, g _ { s } } ( 2 T r B F ^ { 3 } - \\frac { 1 } { 2 } T r B F \\, T r \\, F ^ { 2 } ) \\\\ & & + \\sqrt { t } ( 2 T r B ^ { 2 } F ^ { 2 } - \\frac { 1 } { 4 } T r B ^ { 2 } \\, T r \\, F ^ { 2 } ) + t o t a l ~ d e r i v a t i v e + c o n s t . ] . \\end{align*}"
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"$ \\mitO ( \\mitB , \\mitzeta ^ { 3 } , \\mitk ^ { 3 } ) $",
"$ \\mitL ( \\mitB + \\mitF ) $",
"$ \\mitt $",
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"$ \\mitO ( \\mitB ^ { 2 } ) $",
"$ \\mitO ( \\mitalpha ^ { \\prime 2 } ) $",
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"\\begin{align*} \\displaystyle \\mitL ( \\mitB + \\mitF ) & = & \\displaystyle \\sqrt { \\operatorname { d e t } \\mitg } \\left[ \\mathrm { T r } { \\left( \\frac { \\mitB } { \\sqrt { \\mitg _ { \\mits } } } + \\mitF \\right) } ^ { 2 } + ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\sqrt { \\mitG _ { \\mits } \\, \\mitg _ { \\mits } } \\left[ \\frac { 1 } { 2 } \\mathrm { T r } { \\left( \\frac { \\mitB } { \\sqrt { \\mitg _ { \\mits } } } + \\mitF \\right) } ^ { 4 } \\right. \\right. \\\\ & & \\displaystyle \\qquad \\qquad \\left. \\left. - \\frac { 1 } { 8 } { \\left( \\mathrm { T r } { \\left( \\frac { \\mitB } { \\sqrt { \\mitg _ { \\mits } } } + \\mitF \\right) } ^ { 2 } \\right) } ^ { 2 } \\right] + \\mitO ( \\mitalpha ^ { \\prime 4 } ) + \\mathrm { d e r i v a t i v e ~ c o r r e c t i o n s } \\right] \\\\ & = & \\displaystyle \\frac { \\sqrt { \\operatorname { d e t } \\mitg } } { \\mitg _ { \\mits } } \\Bigg [ \\mathrm { T r } ( \\mitB + \\mitF ) ^ { 2 } + \\sqrt { \\mitt } \\, ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\left[ \\frac { 1 } { 2 } \\mathrm { T r } ( \\mitB + \\mitF ) ^ { 4 } \\right. \\\\ & & \\displaystyle \\qquad \\qquad \\left. - \\frac { 1 } { 8 } [ \\mathrm { T r } ( \\mitB + \\mitF ) ^ { 2 } ] ^ { 2 } \\right] + ~ \\mitO ( \\mitalpha ^ { \\prime 4 } ) + \\mathrm { d e r i v a t i v e ~ c o r r e c t i o n s } \\Bigg ] , \\end{align*}",
"\\begin{align*} & & \\displaystyle \\sqrt { \\operatorname { d e t } \\mitg } \\, ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\sqrt { \\mitG _ { \\mits } \\, \\mitg _ { \\mits } } \\left[ \\frac { 1 } { 2 } \\mathrm { T r } { \\left( \\frac { \\mitB } { \\sqrt { \\mitg _ { \\mits } } } + \\mitF \\right) } ^ { 4 } - \\frac { 1 } { 8 } { \\left( \\mathrm { T r } { \\left( \\frac { \\mitB } { \\sqrt { \\mitg _ { \\mits } } } + \\mitF \\right) } ^ { 2 } \\right) } ^ { 2 } \\right] \\\\ & = & \\displaystyle \\sqrt { \\operatorname { d e t } \\mitg } \\, ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\Bigg [ \\sqrt { \\mitt } \\, \\mitg _ { \\mits } \\left[ \\frac { 1 } { 2 } \\mathrm { T r } \\, \\mitF ^ { 4 } - \\frac { 1 } { 8 } ( \\mathrm { T r } \\, \\mitF ^ { 2 } ) ^ { 2 } \\right] \\\\ & & \\displaystyle + \\sqrt { \\mitt \\, \\mitg _ { \\mits } } \\left( 2 \\mathrm { T r } \\mitB \\mitF ^ { 3 } - \\frac { 1 } { 2 } \\mathrm { T r } \\mitB \\mitF \\, \\mathrm { T r } \\, \\mitF ^ { 2 } \\right) \\\\ & & \\displaystyle + \\sqrt { \\mitt } \\left( 2 \\mathrm { T r } \\mitB ^ { 2 } \\mitF ^ { 2 } - \\frac { 1 } { 4 } \\mathrm { T r } \\mitB ^ { 2 } \\, \\mathrm { T r } \\, \\mitF ^ { 2 } \\right) + \\mathrm { t o t a l ~ d e r i v a t i v e } + \\mathrm { c o n s t } . \\Bigg ] . \\end{align*}"
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0002194_page35
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{
"latex": [
"$O(B^2)$",
"${\\rm Tr} B^2 {\\rm Tr} F^2$",
"$t$",
"${\\rm Tr} B^2 F^2$",
"${\\rm Tr} B^2 F^2$",
"$t$",
"$O(\\alpha '^2)$",
"$G_s$",
"$\\alpha '^2$",
"$O(\\alpha '^2)$",
"$G_s$",
"$\\hat {A}_i$",
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"$c$",
"$c$",
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"$O(\\alpha '^4)$",
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"\\begin {equation} t=1. \\end {equation}",
"\\begin {eqnarray} G_s &=& g_s \\left [ 1 +\\frac {c-1}{4} (2 \\pi \\alpha ')^2 {\\rm Tr} B^2 + O(\\alpha '^4) \\right ], \\\\ \\hat {A}_i &=& A_i + \\frac {1}{2} (2 \\pi \\alpha ')^2 B_{kl} A_k (\\partial _l A_i + F_{li}) + \\frac {c}{8} (2 \\pi \\alpha ')^2 {\\rm Tr} B^2 A_i + O(\\alpha '^4), \\end {eqnarray}",
"\\begin {eqnarray} (2 \\pi \\alpha ')^2 {\\rm Tr} ( G^{-1} \\hat {F} )^4_{\\rm arbitrary} &=& (2 \\pi \\alpha ')^2 {\\rm Tr} ( G^{-1} \\hat {F} G^{-1} \\hat {F} G^{-1} \\hat {F} G^{-1} \\hat {F} ) + O(\\alpha '^6), \\\\ (2 \\pi \\alpha ')^2 ( {\\rm Tr} ( G^{-1} \\hat {F} )^2 )^2_{\\rm arbitrary} &=& (2 \\pi \\alpha ')^2 [ {\\rm Tr} ( G^{-1} \\hat {F} G^{-1} \\hat {F} ) ]^2 + O(\\alpha '^6), \\end {eqnarray}",
"\\begin {eqnarray} && \\frac {\\sqrt {\\det G}}{G_s} \\biggl [ \\frac {1}{2} (2 \\pi \\alpha ')^2 {\\rm Tr} ( G^{-1} \\hat {F} )^4_{\\rm arbitrary} -\\frac {1}{8} (2 \\pi \\alpha ')^2 ( {\\rm Tr} ( G^{-1} \\hat {F} )^2 )^2_{\\rm arbitrary} \\biggr ] \\\\ &=& \\frac {\\sqrt {\\det g}}{g_s} \\Biggl [ \\frac {1}{2} (2 \\pi \\alpha ')^2 {\\rm Tr} F^4 -\\frac {1}{8} (2 \\pi \\alpha ')^2 ( {\\rm Tr} F^2 )^2 \\\\ && + (2 \\pi \\alpha ')^4 \\biggl [ 2 {\\rm Tr} B F^5 -\\frac {1}{4} {\\rm Tr} BF {\\rm Tr} F^4 -\\frac {1}{2} {\\rm Tr} BF^3 {\\rm Tr} F^2 +\\frac {1}{16} {\\rm Tr} BF ( {\\rm Tr} F^2 )^2 \\biggr ] \\\\ && + (2 \\pi \\alpha ')^4 \\biggl [ 2 {\\rm Tr} B^2 F^4 + \\frac {c-1}{8} {\\rm Tr} B^2 {\\rm Tr} F^4 -\\frac {1}{2} {\\rm Tr} B^2 F^2 {\\rm Tr} F^2 +\\frac {1-c}{32} {\\rm Tr} B^2 ( {\\rm Tr} F^2 )^2 \\biggr ] \\\\ && + {\\rm ~total~derivative} + O(\\alpha '^6) \\Biggr ]. \\end {eqnarray}"
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"$ \\alpha ^ { \\prime } $",
"\\begin{equation*} t = 1 . \\end{equation*}",
"\\begin{align*} G _ { s } & = & g _ { s } [ 1 + \\frac { c - 1 } { 4 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } T r B ^ { 2 } + O ( \\alpha ^ { \\prime 4 } ) ] , \\\\ \\hat { A } _ { i } & = & A _ { i } + \\frac { 1 } { 2 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } B _ { k l } A _ { k } ( \\partial _ { l } A _ { i } + F _ { l i } ) + \\frac { c } { 8 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } T r B ^ { 2 } A _ { i } + O ( \\alpha ^ { \\prime 4 } ) , \\end{align*}",
"\\begin{align*} ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } T r ( G ^ { - 1 } \\hat { F } ) _ { a r b i t r a r y } ^ { 4 } & = & ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } T r ( G ^ { - 1 } \\hat { F } G ^ { - 1 } \\hat { F } G ^ { - 1 } \\hat { F } G ^ { - 1 } \\hat { F } ) + O ( \\alpha ^ { \\prime 6 } ) , \\\\ ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } ( T r ( G ^ { - 1 } \\hat { F } ) ^ { 2 } ) _ { a r b i t r a r y } ^ { 2 } & = & ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } [ T r ( G ^ { - 1 } \\hat { F } G ^ { - 1 } \\hat { F } ) ] ^ { 2 } + O ( \\alpha ^ { \\prime 6 } ) , \\end{align*}",
"\\begin{align*} & & \\frac { \\sqrt { \\operatorname { d e t } G } } { G _ { s } } [ \\frac { 1 } { 2 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } T r ( G ^ { - 1 } \\hat { F } ) _ { a r b i t r a r y } ^ { 4 } - \\frac { 1 } { 8 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } ( T r ( G ^ { - 1 } \\hat { F } ) ^ { 2 } ) _ { a r b i t r a r y } ^ { 2 } ] \\\\ & = & \\frac { \\sqrt { \\operatorname { d e t } g } } { g _ { s } } [ \\frac { 1 } { 2 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } T r F ^ { 4 } - \\frac { 1 } { 8 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } ( T r F ^ { 2 } ) ^ { 2 } \\\\ & & + ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 4 } [ 2 T r B F ^ { 5 } - \\frac { 1 } { 4 } T r B F T r F ^ { 4 } - \\frac { 1 } { 2 } T r B F ^ { 3 } T r F ^ { 2 } + \\frac { 1 } { 1 6 } T r B F ( T r F ^ { 2 } ) ^ { 2 } ] \\\\ & & + ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 4 } [ 2 T r B ^ { 2 } F ^ { 4 } + \\frac { c - 1 } { 8 } T r B ^ { 2 } T r F ^ { 4 } - \\frac { 1 } { 2 } T r B ^ { 2 } F ^ { 2 } T r F ^ { 2 } + \\frac { 1 - c } { 3 2 } T r B ^ { 2 } ( T r F ^ { 2 } ) ^ { 2 } ] \\\\ & & + ~ t o t a l ~ d e r i v a t i v e + O ( \\alpha ^ { \\prime 6 } ) ] . \\end{align*}"
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"$ \\mathrm { T r } \\mitB ^ { 2 } \\mathrm { T r } \\mitF ^ { 2 } $",
"$ \\mitt $",
"$ \\mathrm { T r } \\mitB ^ { 2 } \\mitF ^ { 2 } $",
"$ \\mathrm { T r } \\mitB ^ { 2 } \\mitF ^ { 2 } $",
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"$ \\mitG _ { \\mits } $",
"$ \\mitalpha ^ { \\prime 2 } $",
"$ \\mitO ( \\mitalpha ^ { \\prime 2 } ) $",
"$ \\mitG _ { \\mits } $",
"$ \\hat { \\mitA } _ { \\miti } $",
"$ \\mitA _ { \\miti } $",
"$ \\mitc $",
"$ \\mitc $",
"$ \\mitalpha ^ { \\prime 4 } $",
"$ \\ast $",
"$ \\hat { \\mitF } ^ { 4 } $",
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"$ \\ast $",
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"$ \\mitalpha ^ { \\prime } $",
"\\begin{equation*} \\mitt = 1 . \\end{equation*}",
"\\begin{align*} \\displaystyle \\mitG _ { \\mits } & = & \\displaystyle \\mitg _ { \\mits } \\left[ 1 + \\frac { \\mitc - 1 } { 4 } ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mathrm { T r } \\mitB ^ { 2 } + \\mitO ( \\mitalpha ^ { \\prime 4 } ) \\right] , \\\\ \\displaystyle \\hat { \\mitA } _ { \\miti } & = & \\displaystyle \\mitA _ { \\miti } + \\frac { 1 } { 2 } ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mitB _ { \\mitk \\mitl } \\mitA _ { \\mitk } ( \\mitpartial _ { \\mitl } \\mitA _ { \\miti } + \\mitF _ { \\mitl \\miti } ) + \\frac { \\mitc } { 8 } ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mathrm { T r } \\mitB ^ { 2 } \\mitA _ { \\miti } + \\mitO ( \\mitalpha ^ { \\prime 4 } ) , \\end{align*}",
"\\begin{align*} \\displaystyle ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mathrm { T r } ( \\mitG ^ { - 1 } \\hat { \\mitF } ) _ { \\mathrm { a r b i t r a r y } } ^ { 4 } & = & \\displaystyle ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mathrm { T r } ( \\mitG ^ { - 1 } \\hat { \\mitF } \\mitG ^ { - 1 } \\hat { \\mitF } \\mitG ^ { - 1 } \\hat { \\mitF } \\mitG ^ { - 1 } \\hat { \\mitF } ) + \\mitO ( \\mitalpha ^ { \\prime 6 } ) , \\\\ \\displaystyle ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } ( \\mathrm { T r } ( \\mitG ^ { - 1 } \\hat { \\mitF } ) ^ { 2 } ) _ { \\mathrm { a r b i t r a r y } } ^ { 2 } & = & \\displaystyle ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } [ \\mathrm { T r } ( \\mitG ^ { - 1 } \\hat { \\mitF } \\mitG ^ { - 1 } \\hat { \\mitF } ) ] ^ { 2 } + \\mitO ( \\mitalpha ^ { \\prime 6 } ) , \\end{align*}",
"\\begin{align*} & & \\displaystyle \\frac { \\sqrt { \\operatorname { d e t } \\mitG } } { \\mitG _ { \\mits } } \\bigg [ \\frac { 1 } { 2 } ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mathrm { T r } ( \\mitG ^ { - 1 } \\hat { \\mitF } ) _ { \\mathrm { a r b i t r a r y } } ^ { 4 } - \\frac { 1 } { 8 } ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } ( \\mathrm { T r } ( \\mitG ^ { - 1 } \\hat { \\mitF } ) ^ { 2 } ) _ { \\mathrm { a r b i t r a r y } } ^ { 2 } \\bigg ] \\\\ & = & \\displaystyle \\frac { \\sqrt { \\operatorname { d e t } \\mitg } } { \\mitg _ { \\mits } } \\Bigg [ \\frac { 1 } { 2 } ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mathrm { T r } \\mitF ^ { 4 } - \\frac { 1 } { 8 } ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } ( \\mathrm { T r } \\mitF ^ { 2 } ) ^ { 2 } \\\\ & & \\displaystyle + ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 4 } \\bigg [ 2 \\mathrm { T r } \\mitB \\mitF ^ { 5 } - \\frac { 1 } { 4 } \\mathrm { T r } \\mitB \\mitF \\mathrm { T r } \\mitF ^ { 4 } - \\frac { 1 } { 2 } \\mathrm { T r } \\mitB \\mitF ^ { 3 } \\mathrm { T r } \\mitF ^ { 2 } + \\frac { 1 } { 1 6 } \\mathrm { T r } \\mitB \\mitF ( \\mathrm { T r } \\mitF ^ { 2 } ) ^ { 2 } \\bigg ] \\\\ & & \\displaystyle + ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 4 } \\bigg [ 2 \\mathrm { T r } \\mitB ^ { 2 } \\mitF ^ { 4 } + \\frac { \\mitc - 1 } { 8 } \\mathrm { T r } \\mitB ^ { 2 } \\mathrm { T r } \\mitF ^ { 4 } - \\frac { 1 } { 2 } \\mathrm { T r } \\mitB ^ { 2 } \\mitF ^ { 2 } \\mathrm { T r } \\mitF ^ { 2 } + \\frac { 1 - \\mitc } { 3 2 } \\mathrm { T r } \\mitB ^ { 2 } ( \\mathrm { T r } \\mitF ^ { 2 } ) ^ { 2 } \\bigg ] \\\\ & & \\displaystyle + ~ \\mathrm { t o t a l ~ d e r i v a t i v e } + \\mitO ( \\mitalpha ^ { \\prime 6 } ) \\Bigg ] . \\end{align*}"
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0002194_page36
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"$O(B F^5)$",
"$O(BF^5)$",
"$O(BF^5)$",
"${\\cal L}(B+F)$",
"$F^6$",
"$F^6$",
"$O(\\alpha '^2)$",
"$G_s$",
"$B$",
"$O(B^2)$",
"\\begin {eqnarray} && \\frac {\\sqrt {\\det G}}{G_s} \\biggl [ \\frac {1}{2} (2 \\pi \\alpha ')^2 {\\rm Tr} ( G^{-1} \\hat {F} )^4_{\\rm arbitrary} -\\frac {1}{8} (2 \\pi \\alpha ')^2 ( {\\rm Tr} ( G^{-1} \\hat {F} )^2 )^2_{\\rm arbitrary} \\biggr ] \\\\ &=& \\frac {\\sqrt {\\det g}}{g_s} \\Biggl [ \\frac {1}{2} (2 \\pi \\alpha ')^2 {\\rm Tr} F^4 -\\frac {1}{8} (2 \\pi \\alpha ')^2 ( {\\rm Tr} F^2 )^2 \\\\ && + (2 \\pi \\alpha ')^4 \\biggl [ 2 {\\rm Tr} B F^5 -\\frac {1}{4} {\\rm Tr} BF {\\rm Tr} F^4 -\\frac {1}{2} {\\rm Tr} BF^3 {\\rm Tr} F^2 +\\frac {1}{16} {\\rm Tr} BF ( {\\rm Tr} F^2 )^2 \\biggr ] \\\\ && + (2 \\pi \\alpha ')^4 \\biggl [ 2 {\\rm Tr} B^2 F^4 + \\frac {c-1}{8} {\\rm Tr} B^2 {\\rm Tr} F^4 -\\frac {1}{2} {\\rm Tr} B^2 F^2 {\\rm Tr} F^2 +\\frac {1-c}{32} {\\rm Tr} B^2 ( {\\rm Tr} F^2 )^2 \\biggr ] \\\\ && + {\\rm ~total~derivative} + O(\\alpha '^6) \\Biggr ]. \\end {eqnarray}",
"\\begin {eqnarray} {\\rm Tr} (B+F)^6 &=& {\\rm Tr} F^6 + 6 {\\rm Tr} B F^5 + O(B^2), \\\\ {\\rm Tr} (B+F)^2 {\\rm Tr} (B+F)^4 &=& {\\rm Tr} F^2 {\\rm Tr} F^4 \\\\ && + 2 {\\rm Tr} B F {\\rm Tr} F^4 + 4 {\\rm Tr} F^2 {\\rm Tr} B F^3 + O(B^2), \\\\ \\left [ {\\rm Tr} (B+F)^2 \\right ]^3 &=& ( {\\rm Tr} F^2 )^3 + 6 {\\rm Tr} B F ( {\\rm Tr} F^2 )^2 + O(B^2). \\end {eqnarray}",
"\\begin {equation} \\frac {(2 \\pi \\alpha ')^4 \\sqrt {\\det g}}{g_s} \\left [ \\frac {1}{3} {\\rm Tr} (B+F)^6 - \\frac {1}{8} {\\rm Tr} (B+F)^2 {\\rm Tr} (B+F)^4 + \\frac {1}{96} \\left [ {\\rm Tr} (B+F)^2 \\right ]^3 \\right ]. \\label {(B+F)^6} \\end {equation}",
"\\begin {eqnarray} && \\frac {(2 \\pi \\alpha ')^4 \\sqrt {\\det g}}{g_s} \\left [ 2 {\\rm Tr} B^2 F^4 - \\frac {1}{8} {\\rm Tr} B^2 {\\rm Tr} F^4 - \\frac {1}{2} {\\rm Tr} F^2 {\\rm Tr} B^2 F^2 + \\frac {1}{32} {\\rm Tr} B^2 ( {\\rm Tr} F^2 )^2 \\right . \\\\ && \\qquad \\qquad \\qquad \\qquad + 2 {\\rm Tr} B F B F^3 + {\\rm Tr} B F^2 B F^2 - {\\rm Tr} B F {\\rm Tr} B F^3 \\\\ && \\qquad \\qquad \\qquad \\qquad \\left . - \\frac {1}{4} {\\rm Tr} F^2 {\\rm Tr} B F B F + \\frac {1}{8} {\\rm Tr} F^2 ( {\\rm Tr} B F )^2 \\right ], \\end {eqnarray}"
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"$ O ( B F ^ { 5 } ) $",
"$ O ( B F ^ { 5 } ) $",
"$ L ( B + F ) $",
"$ F ^ { 6 } $",
"$ F ^ { 6 } $",
"$ O ( \\alpha ^ { \\prime 2 } ) $",
"$ G _ { s } $",
"$ B $",
"$ O ( B ^ { 2 } ) $",
"\\begin{align*} & & \\frac { \\sqrt { \\operatorname { d e t } G } } { G _ { s } } [ \\frac { 1 } { 2 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } T r ( G ^ { - 1 } \\hat { F } ) _ { a r b i t r a r y } ^ { 4 } - \\frac { 1 } { 8 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } ( T r ( G ^ { - 1 } \\hat { F } ) ^ { 2 } ) _ { a r b i t r a r y } ^ { 2 } ] \\\\ & = & \\frac { \\sqrt { \\operatorname { d e t } g } } { g _ { s } } [ \\frac { 1 } { 2 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } T r F ^ { 4 } - \\frac { 1 } { 8 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } ( T r F ^ { 2 } ) ^ { 2 } \\\\ & & + ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 4 } [ 2 T r B F ^ { 5 } - \\frac { 1 } { 4 } T r B F T r F ^ { 4 } - \\frac { 1 } { 2 } T r B F ^ { 3 } T r F ^ { 2 } + \\frac { 1 } { 1 6 } T r B F ( T r F ^ { 2 } ) ^ { 2 } ] \\\\ & & + ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 4 } [ 2 T r B ^ { 2 } F ^ { 4 } + \\frac { c - 1 } { 8 } T r B ^ { 2 } T r F ^ { 4 } - \\frac { 1 } { 2 } T r B ^ { 2 } F ^ { 2 } T r F ^ { 2 } + \\frac { 1 - c } { 3 2 } T r B ^ { 2 } ( T r F ^ { 2 } ) ^ { 2 } ] \\\\ & & + ~ t o t a l ~ d e r i v a t i v e + O ( \\alpha ^ { \\prime 6 } ) ] . \\end{align*}",
"\\begin{align*} T r ( B + F ) ^ { 6 } & = & T r F ^ { 6 } + 6 T r B F ^ { 5 } + O ( B ^ { 2 } ) , \\\\ T r ( B + F ) ^ { 2 } T r ( B + F ) ^ { 4 } & = & T r F ^ { 2 } T r F ^ { 4 } \\\\ & & + 2 T r B F T r F ^ { 4 } + 4 T r F ^ { 2 } T r B F ^ { 3 } + O ( B ^ { 2 } ) , \\\\ { [ T r ( B + F ) ^ { 2 } ] } ^ { 3 } & = & ( T r F ^ { 2 } ) ^ { 3 } + 6 T r B F ( T r F ^ { 2 } ) ^ { 2 } + O ( B ^ { 2 } ) . \\end{align*}",
"\\begin{equation*} \\frac { ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 4 } \\sqrt { \\operatorname { d e t } g } } { g _ { s } } [ \\frac { 1 } { 3 } T r ( B + F ) ^ { 6 } - \\frac { 1 } { 8 } T r ( B + F ) ^ { 2 } T r ( B + F ) ^ { 4 } + \\frac { 1 } { 9 6 } { [ T r ( B + F ) ^ { 2 } ] } ^ { 3 } ] . \\end{equation*}",
"\\begin{align*} & & \\frac { ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 4 } \\sqrt { \\operatorname { d e t } g } } { g _ { s } } [ 2 T r B ^ { 2 } F ^ { 4 } - \\frac { 1 } { 8 } T r B ^ { 2 } T r F ^ { 4 } - \\frac { 1 } { 2 } T r F ^ { 2 } T r B ^ { 2 } F ^ { 2 } + \\frac { 1 } { 3 2 } T r B ^ { 2 } ( T r F ^ { 2 } ) ^ { 2 } \\\\ & & \\qquad \\qquad \\qquad \\qquad + 2 T r B F B F ^ { 3 } + T r B F ^ { 2 } B F ^ { 2 } - T r B F T r B F ^ { 3 } \\\\ & & \\qquad \\qquad \\qquad \\qquad - \\frac { 1 } { 4 } T r F ^ { 2 } T r B F B F + \\frac { 1 } { 8 } T r F ^ { 2 } ( T r B F ) ^ { 2 } ] , \\end{align*}"
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"$ \\mitO ( \\mitB \\mitF ^ { 5 } ) $",
"$ \\mitO ( \\mitB \\mitF ^ { 5 } ) $",
"$ \\mitL ( \\mitB + \\mitF ) $",
"$ \\mitF ^ { 6 } $",
"$ \\mitF ^ { 6 } $",
"$ \\mitO ( \\mitalpha ^ { \\prime 2 } ) $",
"$ \\mitG _ { \\mits } $",
"$ \\mitB $",
"$ \\mitO ( \\mitB ^ { 2 } ) $",
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"\\begin{align*} \\displaystyle \\mathrm { T r } ( \\mitB + \\mitF ) ^ { 6 } & = & \\displaystyle \\mathrm { T r } \\mitF ^ { 6 } + 6 \\mathrm { T r } \\mitB \\mitF ^ { 5 } + \\mitO ( \\mitB ^ { 2 } ) , \\\\ \\displaystyle \\mathrm { T r } ( \\mitB + \\mitF ) ^ { 2 } \\mathrm { T r } ( \\mitB + \\mitF ) ^ { 4 } & = & \\displaystyle \\mathrm { T r } \\mitF ^ { 2 } \\mathrm { T r } \\mitF ^ { 4 } \\\\ & & \\displaystyle + 2 \\mathrm { T r } \\mitB \\mitF \\mathrm { T r } \\mitF ^ { 4 } + 4 \\mathrm { T r } \\mitF ^ { 2 } \\mathrm { T r } \\mitB \\mitF ^ { 3 } + \\mitO ( \\mitB ^ { 2 } ) , \\\\ \\displaystyle { \\left[ \\mathrm { T r } ( \\mitB + \\mitF ) ^ { 2 } \\right] } ^ { 3 } & = & \\displaystyle ( \\mathrm { T r } \\mitF ^ { 2 } ) ^ { 3 } + 6 \\mathrm { T r } \\mitB \\mitF ( \\mathrm { T r } \\mitF ^ { 2 } ) ^ { 2 } + \\mitO ( \\mitB ^ { 2 } ) . \\end{align*}",
"\\begin{equation*} \\frac { ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 4 } \\sqrt { \\operatorname { d e t } \\mitg } } { \\mitg _ { \\mits } } \\left[ \\frac { 1 } { 3 } \\mathrm { T r } ( \\mitB + \\mitF ) ^ { 6 } - \\frac { 1 } { 8 } \\mathrm { T r } ( \\mitB + \\mitF ) ^ { 2 } \\mathrm { T r } ( \\mitB + \\mitF ) ^ { 4 } + \\frac { 1 } { 9 6 } { \\left[ \\mathrm { T r } ( \\mitB + \\mitF ) ^ { 2 } \\right] } ^ { 3 } \\right] . \\end{equation*}",
"\\begin{align*} & & \\displaystyle \\frac { ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 4 } \\sqrt { \\operatorname { d e t } \\mitg } } { \\mitg _ { \\mits } } \\left[ 2 \\mathrm { T r } \\mitB ^ { 2 } \\mitF ^ { 4 } - \\frac { 1 } { 8 } \\mathrm { T r } \\mitB ^ { 2 } \\mathrm { T r } \\mitF ^ { 4 } - \\frac { 1 } { 2 } \\mathrm { T r } \\mitF ^ { 2 } \\mathrm { T r } \\mitB ^ { 2 } \\mitF ^ { 2 } + \\frac { 1 } { 3 2 } \\mathrm { T r } \\mitB ^ { 2 } ( \\mathrm { T r } \\mitF ^ { 2 } ) ^ { 2 } \\right. \\\\ & & \\displaystyle \\qquad \\qquad \\qquad \\qquad + 2 \\mathrm { T r } \\mitB \\mitF \\mitB \\mitF ^ { 3 } + \\mathrm { T r } \\mitB \\mitF ^ { 2 } \\mitB \\mitF ^ { 2 } - \\mathrm { T r } \\mitB \\mitF \\mathrm { T r } \\mitB \\mitF ^ { 3 } \\\\ & & \\displaystyle \\qquad \\qquad \\qquad \\qquad \\left. - \\frac { 1 } { 4 } \\mathrm { T r } \\mitF ^ { 2 } \\mathrm { T r } \\mitB \\mitF \\mitB \\mitF + \\frac { 1 } { 8 } \\mathrm { T r } \\mitF ^ { 2 } ( \\mathrm { T r } \\mitB \\mitF ) ^ { 2 } \\right] , \\end{align*}"
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0002194_page37
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{
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"$O(B^2)$",
"$\\alpha '^4$",
"$\\hat {F}^2$",
"${\\rm Tr} B^2 {\\rm Tr} F^4$",
"${\\rm Tr} B^2 ( {\\rm Tr} F^2 )^2$",
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"$\\alpha '^4$",
"$O(\\theta ^2)$",
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0003051_page01
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0003051_page02
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0003051_page03
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0003051_page04
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{
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"$AdS_4$",
"$OSp(8/4,\\mathbb {R})$",
"$OSp(8/4,\\mathbb {R})$",
"$OSp(8/4,\\mathbb {R})$",
"$N=8$",
"$Q^i_\\alpha $",
"$N=8$",
"$\\alpha =1,2$",
"$SO(8)$",
"$i=1,\\ldots ,8$",
"$S^i_\\alpha $",
"$P_\\mu $",
"$\\mu =0,1,2$",
"$K_\\mu $",
"$M_{\\alpha \\beta } = M_{\\beta \\alpha }$",
"$SO(2,1)\\sim SL(2,\\mathbb {R})$",
"$SO(2,1)\\sim SL(2,\\mathbb {R})$",
"$D$",
"$T^{ij}=-T^{ji}$",
"$SO(8)$",
"$Q^i_\\alpha $",
"$SO(8)$",
"$8_v$",
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"$8_c$",
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"\\begin {eqnarray} && \\{Q^i_\\alpha , Q^j_\\beta \\} = 2\\delta ^{ij} \\Gamma ^\\mu _{\\alpha \\beta } P_\\mu \\;, \\qquad \\{S^i_\\alpha , S^j_\\beta \\} = 2\\delta ^{ij} \\Gamma ^\\mu _{\\alpha \\beta } K_\\mu \\;, \\\\ && \\{Q^i_\\alpha , S^j_\\beta \\} = \\delta ^{ij} M_{\\alpha \\beta } + 2\\epsilon _{\\alpha \\beta } (T^{ij} + \\delta ^{ij} D) \\;, \\\\ && [D,Q^i_\\alpha ] = {i\\over 2}Q^i_\\alpha \\;, \\qquad [D,S^i_\\alpha ] = -{i\\over 2}S^i_\\alpha \\;,\\\\ && [M_{\\alpha \\beta }, Q^i_\\gamma ] = i(\\epsilon _{\\gamma \\alpha } Q^i_\\beta + \\epsilon _{\\gamma \\beta } Q^i_\\alpha )\\;,\\qquad [M_{\\alpha \\beta }, S^i_\\gamma ] = i(\\epsilon _{\\gamma \\alpha } S^i_\\beta + \\epsilon _{\\gamma \\beta } S^i_\\alpha )\\;,\\\\ && [T^{ij}, Q^k_\\alpha ] = i(\\delta ^{ki} Q^j_\\alpha - \\delta ^{kj} Q^i_\\alpha )\\;,\\qquad [T^{ij}, S^k_\\alpha ] = i(\\delta ^{ki} S^j_\\alpha - \\delta ^{kj} S^i_\\alpha )\\;,\\\\ && [T^{ij}, T^{kl}] = i(\\delta ^{ik} T^{jl} + \\delta ^{jl} T^{ik} - \\delta ^{jk} T^{il} - \\delta ^{il} T^{jk})\\;. \\end {eqnarray}"
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"$ O S p ( 8 \\slash 4 , R ) $",
"$ O S p ( 8 \\slash 4 , R ) $",
"$ N = 8 $",
"$ Q _ { \\alpha } ^ { i } $",
"$ N = 8 $",
"$ \\alpha = 1 , 2 $",
"$ S O ( 8 ) $",
"$ i = 1 , \\ldots , 8 $",
"$ S _ { \\alpha } ^ { i } $",
"$ P _ { \\mu } $",
"$ \\mu = 0 , 1 , 2 $",
"$ K _ { \\mu } $",
"$ M _ { \\alpha \\beta } = M _ { \\beta \\alpha } $",
"$ S O ( 2 , 1 ) \\sim S L ( 2 , R ) $",
"$ S O ( 2 , 1 ) \\sim S L ( 2 , R ) $",
"$ D $",
"$ T ^ { i j } = - T ^ { j i } $",
"$ S O ( 8 ) $",
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"$ S O ( 8 ) $",
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"$ S O ( 8 ) $",
"$ N $",
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"$ i $",
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"$ \\mitO \\mitS \\mitp ( 8 \\slash 4 , \\BbbR ) $",
"$ \\mitN = 8 $",
"$ \\mitQ _ { \\mitalpha } ^ { \\miti } $",
"$ \\mitN = 8 $",
"$ \\mitalpha = 1 , 2 $",
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"$ \\mitS _ { \\mitalpha } ^ { \\miti } $",
"$ \\mitP _ { \\mitmu } $",
"$ \\mitmu = 0 , 1 , 2 $",
"$ \\mitK _ { \\mitmu } $",
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"$ \\mitS \\mitO ( 2 , 1 ) \\sim \\mitS \\mitL ( 2 , \\BbbR ) $",
"$ \\mitD $",
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"$ \\mitQ _ { \\mitalpha } ^ { \\miti } $",
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0003051_page05
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{
"latex": [
"$SO(8)$",
"$[SO(2)]^4\\sim [U(1)]^4$",
"$SO(8)$",
"$Q^i_\\alpha $",
"$U(1)$",
"$H_1 = 2iT^{12}$",
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"$8_v$",
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"$Q^{\\underline {i}} \\ \\rightarrow \\^^MQ^{\\underline {a}\\underline {a}'}= Q^{\\underline {i}} (\\sigma ^{\\underline {i}})^{\\underline {a}\\underline {a}'}$",
"$SU(2)$",
"$t$",
"\\begin {equation}\\label {2.2} Q^{\\pm \\pm }_\\alpha = {1\\over 2}(Q^1_\\alpha \\pm Q^2_\\alpha ) \\end {equation}",
"\\begin {equation}\\label {2.3} [H_1, Q^{\\pm \\pm }_\\alpha ] = \\pm 2i Q^{\\pm \\pm }_\\alpha \\;. \\end {equation}",
"\\begin {equation}\\label {2.4} \\{Q^{++}_\\alpha , S^{--}_\\beta \\} = {1\\over 2} M_{\\alpha \\beta } + \\epsilon _{\\alpha \\beta } (D-{1\\over 2}H_1) \\; . \\end {equation}",
"\\begin {equation}\\label {2.5} Q^{(\\pm \\pm )}_\\alpha = {1\\over 2}(Q^3_\\alpha \\pm Q^4_\\alpha ) \\end {equation}",
"\\begin {equation}\\label {2.6} [H_2, Q^{(\\pm \\pm )}_\\alpha ] = \\pm 2i Q^{(\\pm \\pm )}_\\alpha \\end {equation}",
"\\begin {equation}\\label {2.7} \\{Q^{(++)}_\\alpha , S^{(--)}_\\beta \\} = {1\\over 2} M_{\\alpha \\beta } + \\epsilon _{\\alpha \\beta } (D - {1\\over 2}H_2) \\; . \\end {equation}",
"\\begin {equation}\\label {2.8} \\{Q^{\\underline {a}\\underline {a}'}_\\alpha , S^{\\underline {b}\\underline {b}'}_\\beta \\} = {1\\over 2} \\epsilon ^{\\underline {a}\\underline {b}} \\epsilon ^{\\underline {a}'\\underline {b}'} M_{\\alpha \\beta } -{1\\over 2} \\epsilon _{\\alpha \\beta } (t^{\\underline {a}\\underline {b}} \\epsilon ^{\\underline {a}'\\underline {b}'} + \\epsilon ^{\\underline {a}\\underline {b}} t^{\\underline {a}'\\underline {b}'} -2 \\epsilon ^{\\underline {a}\\underline {b}} \\epsilon ^{\\underline {a}'\\underline {b}'} D) \\end {equation}",
"\\begin {equation}\\label {2.9} [t^{\\underline {a}\\underline {b}} ,Q^{\\underline {c}\\underline {c}'}] = i (\\epsilon ^{\\underline {c}\\underline {a}}Q^{\\underline {b}\\underline {c}'} +\\epsilon ^{\\underline {c}\\underline {b}}Q^{\\underline {a}\\underline {c}'})\\;, \\qquad [t^{\\underline {a}'\\underline {b}'} , Q^{\\underline {c}\\underline {c}'}] = i(\\epsilon ^{\\underline {c}'\\underline {a}'}Q^{\\underline {c}\\underline {b}'} +\\epsilon ^{\\underline {c}'\\underline {b}'}Q^{\\underline {c}\\underline {a}'})\\;. \\end {equation}"
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"$ U ( 1 ) $",
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"$ H _ { 2 } = 2 i T ^ { 3 4 } $",
"$ \\underline { i } = 5 , 6 , 7 , 8 $",
"$ 8 _ { v } $",
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"\\begin{equation*} [ H _ { 1 } , Q _ { \\alpha } ^ { \\pm \\pm } ] = \\pm 2 i Q _ { \\alpha } ^ { \\pm \\pm } \\; . \\end{equation*}",
"\\begin{equation*} \\{ Q _ { \\alpha } ^ { + + } , S _ { \\beta } ^ { - - } \\} = \\frac { 1 } { 2 } M _ { \\alpha \\beta } + \\epsilon _ { \\alpha \\beta } ( D - \\frac { 1 } { 2 } H _ { 1 } ) \\; . \\end{equation*}",
"\\begin{equation*} Q _ { \\alpha } ^ { ( \\pm \\pm ) } = \\frac { 1 } { 2 } ( Q _ { \\alpha } ^ { 3 } \\pm Q _ { \\alpha } ^ { 4 } ) \\end{equation*}",
"\\begin{equation*} [ H _ { 2 } , Q _ { \\alpha } ^ { ( \\pm \\pm ) } ] = \\pm 2 i Q _ { \\alpha } ^ { ( \\pm \\pm ) } \\end{equation*}",
"\\begin{equation*} \\{ Q _ { \\alpha } ^ { ( + + ) } , S _ { \\beta } ^ { ( - - ) } \\} = \\frac { 1 } { 2 } M _ { \\alpha \\beta } + \\epsilon _ { \\alpha \\beta } ( D - \\frac { 1 } { 2 } H _ { 2 } ) \\; . \\end{equation*}",
"\\begin{equation*} \\{ Q _ { \\alpha } ^ { \\underline { a } \\underline { a } ^ { \\prime } } , S _ { \\beta } ^ { \\underline { b } \\underline { b } ^ { \\prime } } \\} = \\frac { 1 } { 2 } \\epsilon ^ { \\underline { a } \\underline { b } } \\epsilon ^ { \\underline { a } ^ { \\prime } \\underline { b } ^ { \\prime } } M _ { \\alpha \\beta } - \\frac { 1 } { 2 } \\epsilon _ { \\alpha \\beta } ( t ^ { \\underline { a } \\underline { b } } \\epsilon ^ { \\underline { a } ^ { \\prime } \\underline { b } ^ { \\prime } } + \\epsilon ^ { \\underline { a } \\underline { b } } t ^ { \\underline { a } ^ { \\prime } \\underline { b } ^ { \\prime } } - 2 \\epsilon ^ { \\underline { a } \\underline { b } } \\epsilon ^ { \\underline { a } ^ { \\prime } \\underline { b } ^ { \\prime } } D ) \\end{equation*}",
"\\begin{equation*} [ t ^ { \\underline { a } \\underline { b } } , Q ^ { \\underline { c } \\underline { c } ^ { \\prime } } ] = i ( \\epsilon ^ { \\underline { c } \\underline { a } } Q ^ { \\underline { b } \\underline { c } ^ { \\prime } } + \\epsilon ^ { \\underline { c } \\underline { b } } Q ^ { \\underline { a } \\underline { c } ^ { \\prime } } ) \\; , \\qquad [ t ^ { \\underline { a } ^ { \\prime } \\underline { b } ^ { \\prime } } , Q ^ { \\underline { c } \\underline { c } ^ { \\prime } } ] = i ( \\epsilon ^ { \\underline { c } ^ { \\prime } \\underline { a } ^ { \\prime } } Q ^ { \\underline { c } \\underline { b } ^ { \\prime } } + \\epsilon ^ { \\underline { c } ^ { \\prime } \\underline { b } ^ { \\prime } } Q ^ { \\underline { c } \\underline { a } ^ { \\prime } } ) \\; . \\end{equation*}"
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"$ [ \\mitS \\mitO ( 2 ) ] ^ { 4 } \\sim [ \\mitU ( 1 ) ] ^ { 4 } $",
"$ \\mitS \\mitO ( 8 ) $",
"$ \\mitQ _ { \\mitalpha } ^ { \\miti } $",
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"$ \\mitS \\mitO ( 4 ) \\sim \\mitS \\mitU ( 2 ) \\times \\mitS \\mitU ( 2 ) $",
"$ \\mitQ ^ { \\underline { \\miti } } ~ \\rightarrow ~ \\mitQ ^ { \\underline { \\mita } \\underline { \\mita } ^ { \\prime } } = \\mitQ ^ { \\underline { \\miti } } ( \\mitsigma ^ { \\underline { \\miti } } ) ^ { \\underline { \\mita } \\underline { \\mita } ^ { \\prime } } $",
"$ \\mitS \\mitU ( 2 ) $",
"$ \\mitt $",
"\\begin{equation*} \\mitQ _ { \\mitalpha } ^ { \\pm \\pm } = \\frac { 1 } { 2 } ( \\mitQ _ { \\mitalpha } ^ { 1 } \\pm \\mitQ _ { \\mitalpha } ^ { 2 } ) \\end{equation*}",
"\\begin{equation*} [ \\mitH _ { 1 } , \\mitQ _ { \\mitalpha } ^ { \\pm \\pm } ] = \\pm 2 \\miti \\mitQ _ { \\mitalpha } ^ { \\pm \\pm } \\; . \\end{equation*}",
"\\begin{equation*} \\{ \\mitQ _ { \\mitalpha } ^ { + + } , \\mitS _ { \\mitbeta } ^ { - - } \\} = \\frac { 1 } { 2 } \\mitM _ { \\mitalpha \\mitbeta } + \\mitepsilon _ { \\mitalpha \\mitbeta } ( \\mitD - \\frac { 1 } { 2 } \\mitH _ { 1 } ) \\; . \\end{equation*}",
"\\begin{equation*} \\mitQ _ { \\mitalpha } ^ { ( \\pm \\pm ) } = \\frac { 1 } { 2 } ( \\mitQ _ { \\mitalpha } ^ { 3 } \\pm \\mitQ _ { \\mitalpha } ^ { 4 } ) \\end{equation*}",
"\\begin{equation*} [ \\mitH _ { 2 } , \\mitQ _ { \\mitalpha } ^ { ( \\pm \\pm ) } ] = \\pm 2 \\miti \\mitQ _ { \\mitalpha } ^ { ( \\pm \\pm ) } \\end{equation*}",
"\\begin{equation*} \\{ \\mitQ _ { \\mitalpha } ^ { ( + + ) } , \\mitS _ { \\mitbeta } ^ { ( - - ) } \\} = \\frac { 1 } { 2 } \\mitM _ { \\mitalpha \\mitbeta } + \\mitepsilon _ { \\mitalpha \\mitbeta } ( \\mitD - \\frac { 1 } { 2 } \\mitH _ { 2 } ) \\; . \\end{equation*}",
"\\begin{equation*} \\{ \\mitQ _ { \\mitalpha } ^ { \\underline { \\mita } \\underline { \\mita } ^ { \\prime } } , \\mitS _ { \\mitbeta } ^ { \\underline { \\mitb } \\underline { \\mitb } ^ { \\prime } } \\} = \\frac { 1 } { 2 } \\mitepsilon ^ { \\underline { \\mita } \\underline { \\mitb } } \\mitepsilon ^ { \\underline { \\mita } ^ { \\prime } \\underline { \\mitb } ^ { \\prime } } \\mitM _ { \\mitalpha \\mitbeta } - \\frac { 1 } { 2 } \\mitepsilon _ { \\mitalpha \\mitbeta } ( \\mitt ^ { \\underline { \\mita } \\underline { \\mitb } } \\mitepsilon ^ { \\underline { \\mita } ^ { \\prime } \\underline { \\mitb } ^ { \\prime } } + \\mitepsilon ^ { \\underline { \\mita } \\underline { \\mitb } } \\mitt ^ { \\underline { \\mita } ^ { \\prime } \\underline { \\mitb } ^ { \\prime } } - 2 \\mitepsilon ^ { \\underline { \\mita } \\underline { \\mitb } } \\mitepsilon ^ { \\underline { \\mita } ^ { \\prime } \\underline { \\mitb } ^ { \\prime } } \\mitD ) \\end{equation*}",
"\\begin{equation*} [ \\mitt ^ { \\underline { \\mita } \\underline { \\mitb } } , \\mitQ ^ { \\underline { \\mitc } \\underline { \\mitc } ^ { \\prime } } ] = \\miti ( \\mitepsilon ^ { \\underline { \\mitc } \\underline { \\mita } } \\mitQ ^ { \\underline { \\mitb } \\underline { \\mitc } ^ { \\prime } } + \\mitepsilon ^ { \\underline { \\mitc } \\underline { \\mitb } } \\mitQ ^ { \\underline { \\mita } \\underline { \\mitc } ^ { \\prime } } ) \\; , \\qquad [ \\mitt ^ { \\underline { \\mita } ^ { \\prime } \\underline { \\mitb } ^ { \\prime } } , \\mitQ ^ { \\underline { \\mitc } \\underline { \\mitc } ^ { \\prime } } ] = \\miti ( \\mitepsilon ^ { \\underline { \\mitc } ^ { \\prime } \\underline { \\mita } ^ { \\prime } } \\mitQ ^ { \\underline { \\mitc } \\underline { \\mitb } ^ { \\prime } } + \\mitepsilon ^ { \\underline { \\mitc } ^ { \\prime } \\underline { \\mitb } ^ { \\prime } } \\mitQ ^ { \\underline { \\mitc } \\underline { \\mita } ^ { \\prime } } ) \\; . \\end{equation*}"
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0003051_page06
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{
"latex": [
"${\\underline {1}}\\equiv {[+]}\\;, \\^^M{\\underline {2}}\\equiv {[-]}$",
"${\\underline {1}'}\\equiv {\\{+\\}}\\;, \\ {\\underline {2}'}\\equiv {\\{-\\}}$",
"$SO(2)$",
"$SO(8)$",
"$28-4 = 24$",
"$28-4 = 24$",
"$SO(8)$",
"$T^{[++]}, \\ T^{\\{++\\}}, \\ T^{++(--)}, \\ T^{(++)[-]\\{-\\}}$",
"$SO(8)$",
"$[SO(2)]^4$",
"$8_v$",
"$Q^i$",
"$28$",
"$SO(8)$",
"$T^{ij}$",
"$8_s$",
"$\\phi ^a$",
"$a=1,\\ldots ,8$",
"$8_c$",
"$\\psi ^{\\dot a}$",
"$\\dot a=1,\\ldots ,8$",
"$SO(8)\\^^M\\rightarrow \\ SO(2)\\times SO(6)\\sim U(1)\\times SU(4) \\ \\rightarrow \\ [SO(2)]^2\\times SO(4)\\sim [U(1)]^2\\times SU(2)\\times SU(2) \\^^M\\rightarrow \\ [SO(2)]^4 \\sim [U(1)]^4$",
"$SO(8)\\^^M\\rightarrow \\ SO(2)\\times SO(6)\\sim U(1)\\times SU(4) \\ \\rightarrow \\ [SO(2)]^2\\times SO(4)\\sim [U(1)]^2\\times SU(2)\\times SU(2) \\^^M\\rightarrow \\ [SO(2)]^4 \\sim [U(1)]^4$",
"$SO(8)\\^^M\\rightarrow \\ SO(2)\\times SO(6)\\sim U(1)\\times SU(4) \\ \\rightarrow \\ [SO(2)]^2\\times SO(4)\\sim [U(1)]^2\\times SU(2)\\times SU(2) \\^^M\\rightarrow \\ [SO(2)]^4 \\sim [U(1)]^4$",
"$OSp(8/4,\\mathbb {R})$",
"$OSp(8/4,\\mathbb {R})$",
"$\\ell $",
"$J$",
"$d_1,d_2,d_3,d_4$",
"$SO(8)$",
"\\begin {equation}\\label {2.10} H_3 = t^{\\underline {1}\\underline {2}}\\;, \\qquad H_4 = t^{\\underline {1}'\\underline {2}'}\\;, \\end {equation}",
"\\begin {equation}\\label {2.11} [H_3, Q^{[\\pm ]\\{\\pm \\}}] = [H_4, Q^{[\\pm ]\\{\\pm \\}}] = \\pm i Q^{[\\pm ]\\{\\pm \\}}\\;. \\end {equation}",
"\\begin {equation}\\label {2.12} \\{Q^{[+]\\{+\\}}_\\alpha , S^{[-]\\{-\\}}_\\beta \\} = {1\\over 2} M_{\\alpha \\beta } + \\epsilon _{\\alpha \\beta } (D - {1\\over 2}H_3 - {1\\over 2}H_4) \\; , \\end {equation}",
"\\begin {equation}\\label {2.13} \\{Q^{[+]\\{-\\}}_\\alpha , S^{[-]\\{+\\}}_\\beta \\} = -{1\\over 2} M_{\\alpha \\beta } - \\epsilon _{\\alpha \\beta } (D - {1\\over 2}H_3 + {1\\over 2}H_4) \\; . \\end {equation}",
"\\begin {equation}\\label {2.14} \\{{\\cal T}\\}_+ = \\left \\{ \\begin {array}{l} T^{++(++)}\\;, \\ T^{++(--)}\\;, \\ T^{++[\\pm ]\\{\\pm \\}}\\;; \\\\ T^{(++)[\\pm ]\\{\\pm \\}}\\;; \\\\ T^{[++]} \\equiv T^{[+]\\{+\\}[+]\\{-\\}}\\;, \\ T^{\\{++\\}} \\equiv T^{[+]\\{+\\}[-]\\{+\\}} \\end {array} \\right . \\end {equation}",
"\\begin {eqnarray} \\phi ^a &\\rightarrow & \\phi ^{+(+)[\\pm ]}, \\ \\phi ^{-(-)[\\pm ]}, \\ \\phi ^{+(-)\\{\\pm \\}}, \\ \\phi ^{-(+)\\{\\pm \\}}\\\\ \\sigma ^{\\dot a} &\\rightarrow & \\sigma ^{+(+)\\{\\pm \\}}, \\ \\sigma ^{-(-)\\{\\pm \\}}, \\ \\sigma ^{+(-)[\\pm ]}, \\^^M\\sigma ^{-(+)[\\pm ]}\\end {eqnarray}",
"\\begin {equation}\\label {555} {\\cal D}(\\ell , J; d_1,d_2,d_3,d_4) \\end {equation}"
],
"latex_norm": [
"$ \\underline { 1 } \\equiv [ + ] \\; , ~ \\underline { 2 } \\equiv [ - ] $",
"$ \\underline { 1 } ^ { \\prime } \\equiv \\{ + \\} \\; , ~ \\underline { 2 } ^ { \\prime } \\equiv \\{ - \\} $",
"$ S O ( 2 ) $",
"$ S O ( 8 ) $",
"$ 2 8 - 4 = 2 4 $",
"$ 2 8 - 4 = 2 4 $",
"$ S O ( 8 ) $",
"$ T ^ { [ + + ] } , ~ T ^ { \\{ + + \\} } , ~ T ^ { + + ( - - ) } , ~ T ^ { ( + + ) [ - ] \\{ - \\} } $",
"$ S O ( 8 ) $",
"$ [ S O ( 2 ) ] ^ { 4 } $",
"$ 8 _ { v } $",
"$ Q ^ { i } $",
"$ 2 8 $",
"$ S O ( 8 ) $",
"$ T ^ { i j } $",
"$ 8 _ { s } $",
"$ \\phi ^ { a } $",
"$ a = 1 , \\ldots , 8 $",
"$ 8 _ { c } $",
"$ \\psi ^ { \\dot { a } } $",
"$ \\dot { a } = 1 , \\ldots , 8 $",
"$ S O ( 8 ) ~ \\rightarrow ~ S O ( 2 ) \\times S O ( 6 ) \\sim U ( 1 ) \\times S U ( 4 ) ~ \\rightarrow ~ [ S O ( 2 ) ] ^ { 2 } \\times S O ( 4 ) \\sim [ U ( 1 ) ] ^ { 2 } \\times S U ( 2 ) \\times S U ( 2 ) ~ \\rightarrow ~ [ S O ( 2 ) ] ^ { 4 } \\sim [ U ( 1 ) ] ^ { 4 } $",
"$ S O ( 8 ) ~ \\rightarrow ~ S O ( 2 ) \\times S O ( 6 ) \\sim U ( 1 ) \\times S U ( 4 ) ~ \\rightarrow ~ [ S O ( 2 ) ] ^ { 2 } \\times S O ( 4 ) \\sim [ U ( 1 ) ] ^ { 2 } \\times S U ( 2 ) \\times S U ( 2 ) ~ \\rightarrow ~ [ S O ( 2 ) ] ^ { 4 } \\sim [ U ( 1 ) ] ^ { 4 } $",
"$ S O ( 8 ) ~ \\rightarrow ~ S O ( 2 ) \\times S O ( 6 ) \\sim U ( 1 ) \\times S U ( 4 ) ~ \\rightarrow ~ [ S O ( 2 ) ] ^ { 2 } \\times S O ( 4 ) \\sim [ U ( 1 ) ] ^ { 2 } \\times S U ( 2 ) \\times S U ( 2 ) ~ \\rightarrow ~ [ S O ( 2 ) ] ^ { 4 } \\sim [ U ( 1 ) ] ^ { 4 } $",
"$ O S p ( 8 \\slash 4 , R ) $",
"$ O S p ( 8 \\slash 4 , R ) $",
"$ l $",
"$ J $",
"$ d _ { 1 } , d _ { 2 } , d _ { 3 } , d _ { 4 } $",
"$ S O ( 8 ) $",
"\\begin{equation*} H _ { 3 } = t ^ { \\underline { 1 } \\underline { 2 } } \\; , \\qquad H _ { 4 } = t ^ { \\underline { 1 } ^ { \\prime } \\underline { 2 } ^ { \\prime } } \\; , \\end{equation*}",
"\\begin{equation*} [ H _ { 3 } , Q ^ { [ \\pm ] \\{ \\pm \\} } ] = [ H _ { 4 } , Q ^ { [ \\pm ] \\{ \\pm \\} } ] = \\pm i Q ^ { [ \\pm ] \\{ \\pm \\} } \\; . \\end{equation*}",
"\\begin{equation*} \\{ Q _ { \\alpha } ^ { [ + ] \\{ + \\} } , S _ { \\beta } ^ { [ - ] \\{ - \\} } \\} = \\frac { 1 } { 2 } M _ { \\alpha \\beta } + \\epsilon _ { \\alpha \\beta } ( D - \\frac { 1 } { 2 } H _ { 3 } - \\frac { 1 } { 2 } H _ { 4 } ) \\; , \\end{equation*}",
"\\begin{equation*} \\{ Q _ { \\alpha } ^ { [ + ] \\{ - \\} } , S _ { \\beta } ^ { [ - ] \\{ + \\} } \\} = - \\frac { 1 } { 2 } M _ { \\alpha \\beta } - \\epsilon _ { \\alpha \\beta } ( D - \\frac { 1 } { 2 } H _ { 3 } + \\frac { 1 } { 2 } H _ { 4 } ) \\; . \\end{equation*}",
"\\begin{align*} \\{ T \\} _ { + } = \\{ \\begin{array}{l} T ^ { + + ( + + ) } \\; , ~ T ^ { + + ( - - ) } \\; , ~ T ^ { + + [ \\pm ] \\{ \\pm \\} } \\; ; \\\\ T ^ { ( + + ) [ \\pm ] \\{ \\pm \\} } \\; ; \\\\ T ^ { [ + + ] } \\equiv T ^ { [ + ] \\{ + \\} [ + ] \\{ - \\} } \\; , ~ T ^ { \\{ + + \\} } \\equiv T ^ { [ + ] \\{ + \\} [ - ] \\{ + \\} } \\end{array} \\end{align*}",
"\\begin{align*} \\phi ^ { a } & \\rightarrow & \\phi ^ { + ( + ) [ \\pm ] } , ~ \\phi ^ { - ( - ) [ \\pm ] } , ~ \\phi ^ { + ( - ) \\{ \\pm \\} } , ~ \\phi ^ { - ( + ) \\{ \\pm \\} } \\\\ \\sigma ^ { \\dot { a } } & \\rightarrow & \\sigma ^ { + ( + ) \\{ \\pm \\} } , ~ \\sigma ^ { - ( - ) \\{ \\pm \\} } , ~ \\sigma ^ { + ( - ) [ \\pm ] } , ~ \\sigma ^ { - ( + ) [ \\pm ] } \\end{align*}",
"\\begin{equation*} D ( l , J ; d _ { 1 } , d _ { 2 } , d _ { 3 } , d _ { 4 } ) \\end{equation*}"
],
"latex_expand": [
"$ \\underline { 1 } \\equiv [ + ] \\; , ~ \\underline { 2 } \\equiv [ - ] $",
"$ \\underline { 1 } ^ { \\prime } \\equiv \\{ + \\} \\; , ~ \\underline { 2 } ^ { \\prime } \\equiv \\{ - \\} $",
"$ \\mitS \\mitO ( 2 ) $",
"$ \\mitS \\mitO ( 8 ) $",
"$ 2 8 - 4 = 2 4 $",
"$ 2 8 - 4 = 2 4 $",
"$ \\mitS \\mitO ( 8 ) $",
"$ \\mitT ^ { [ + + ] } , ~ \\mitT ^ { \\{ + + \\} } , ~ \\mitT ^ { + + ( - - ) } , ~ \\mitT ^ { ( + + ) [ - ] \\{ - \\} } $",
"$ \\mitS \\mitO ( 8 ) $",
"$ [ \\mitS \\mitO ( 2 ) ] ^ { 4 } $",
"$ 8 _ { \\mitv } $",
"$ \\mitQ ^ { \\miti } $",
"$ 2 8 $",
"$ \\mitS \\mitO ( 8 ) $",
"$ \\mitT ^ { \\miti \\mitj } $",
"$ 8 _ { \\mits } $",
"$ \\mitphi ^ { \\mita } $",
"$ \\mita = 1 , \\ldots , 8 $",
"$ 8 _ { \\mitc } $",
"$ \\mitpsi ^ { \\dot { \\mita } } $",
"$ \\dot { \\mita } = 1 , \\ldots , 8 $",
"$ \\mitS \\mitO ( 8 ) ~ \\rightarrow ~ \\mitS \\mitO ( 2 ) \\times \\mitS \\mitO ( 6 ) \\sim \\mitU ( 1 ) \\times \\mitS \\mitU ( 4 ) ~ \\rightarrow ~ [ \\mitS \\mitO ( 2 ) ] ^ { 2 } \\times \\mitS \\mitO ( 4 ) \\sim [ \\mitU ( 1 ) ] ^ { 2 } \\times \\mitS \\mitU ( 2 ) \\times \\mitS \\mitU ( 2 ) ~ \\rightarrow ~ [ \\mitS \\mitO ( 2 ) ] ^ { 4 } \\sim [ \\mitU ( 1 ) ] ^ { 4 } $",
"$ \\mitS \\mitO ( 8 ) ~ \\rightarrow ~ \\mitS \\mitO ( 2 ) \\times \\mitS \\mitO ( 6 ) \\sim \\mitU ( 1 ) \\times \\mitS \\mitU ( 4 ) ~ \\rightarrow ~ [ \\mitS \\mitO ( 2 ) ] ^ { 2 } \\times \\mitS \\mitO ( 4 ) \\sim [ \\mitU ( 1 ) ] ^ { 2 } \\times \\mitS \\mitU ( 2 ) \\times \\mitS \\mitU ( 2 ) ~ \\rightarrow ~ [ \\mitS \\mitO ( 2 ) ] ^ { 4 } \\sim [ \\mitU ( 1 ) ] ^ { 4 } $",
"$ \\mitS \\mitO ( 8 ) ~ \\rightarrow ~ \\mitS \\mitO ( 2 ) \\times \\mitS \\mitO ( 6 ) \\sim \\mitU ( 1 ) \\times \\mitS \\mitU ( 4 ) ~ \\rightarrow ~ [ \\mitS \\mitO ( 2 ) ] ^ { 2 } \\times \\mitS \\mitO ( 4 ) \\sim [ \\mitU ( 1 ) ] ^ { 2 } \\times \\mitS \\mitU ( 2 ) \\times \\mitS \\mitU ( 2 ) ~ \\rightarrow ~ [ \\mitS \\mitO ( 2 ) ] ^ { 4 } \\sim [ \\mitU ( 1 ) ] ^ { 4 } $",
"$ \\mitO \\mitS \\mitp ( 8 \\slash 4 , \\BbbR ) $",
"$ \\mitO \\mitS \\mitp ( 8 \\slash 4 , \\BbbR ) $",
"$ \\ell $",
"$ \\mitJ $",
"$ \\mitd _ { 1 } , \\mitd _ { 2 } , \\mitd _ { 3 } , \\mitd _ { 4 } $",
"$ \\mitS \\mitO ( 8 ) $",
"\\begin{equation*} \\mitH _ { 3 } = \\mitt ^ { \\underline { 1 } \\underline { 2 } } \\; , \\qquad \\mitH _ { 4 } = \\mitt ^ { \\underline { 1 } ^ { \\prime } \\underline { 2 } ^ { \\prime } } \\; , \\end{equation*}",
"\\begin{equation*} [ \\mitH _ { 3 } , \\mitQ ^ { [ \\pm ] \\{ \\pm \\} } ] = [ \\mitH _ { 4 } , \\mitQ ^ { [ \\pm ] \\{ \\pm \\} } ] = \\pm \\miti \\mitQ ^ { [ \\pm ] \\{ \\pm \\} } \\; . \\end{equation*}",
"\\begin{equation*} \\{ \\mitQ _ { \\mitalpha } ^ { [ + ] \\{ + \\} } , \\mitS _ { \\mitbeta } ^ { [ - ] \\{ - \\} } \\} = \\frac { 1 } { 2 } \\mitM _ { \\mitalpha \\mitbeta } + \\mitepsilon _ { \\mitalpha \\mitbeta } ( \\mitD - \\frac { 1 } { 2 } \\mitH _ { 3 } - \\frac { 1 } { 2 } \\mitH _ { 4 } ) \\; , \\end{equation*}",
"\\begin{equation*} \\{ \\mitQ _ { \\mitalpha } ^ { [ + ] \\{ - \\} } , \\mitS _ { \\mitbeta } ^ { [ - ] \\{ + \\} } \\} = - \\frac { 1 } { 2 } \\mitM _ { \\mitalpha \\mitbeta } - \\mitepsilon _ { \\mitalpha \\mitbeta } ( \\mitD - \\frac { 1 } { 2 } \\mitH _ { 3 } + \\frac { 1 } { 2 } \\mitH _ { 4 } ) \\; . \\end{equation*}",
"\\begin{align*} \\{ \\mitT \\} _ { + } = \\left\\{ \\begin{array}{l} \\mitT ^ { + + ( + + ) } \\; , ~ \\mitT ^ { + + ( - - ) } \\; , ~ \\mitT ^ { + + [ \\pm ] \\{ \\pm \\} } \\; ; \\\\ \\mitT ^ { ( + + ) [ \\pm ] \\{ \\pm \\} } \\; ; \\\\ \\mitT ^ { [ + + ] } \\equiv \\mitT ^ { [ + ] \\{ + \\} [ + ] \\{ - \\} } \\; , ~ \\mitT ^ { \\{ + + \\} } \\equiv \\mitT ^ { [ + ] \\{ + \\} [ - ] \\{ + \\} } \\end{array} \\right. \\end{align*}",
"\\begin{align*} \\displaystyle \\mitphi ^ { \\mita } & \\displaystyle \\rightarrow & \\displaystyle \\mitphi ^ { + ( + ) [ \\pm ] } , ~ \\mitphi ^ { - ( - ) [ \\pm ] } , ~ \\mitphi ^ { + ( - ) \\{ \\pm \\} } , ~ \\mitphi ^ { - ( + ) \\{ \\pm \\} } \\\\ \\displaystyle \\mitsigma ^ { \\dot { \\mita } } & \\displaystyle \\rightarrow & \\displaystyle \\mitsigma ^ { + ( + ) \\{ \\pm \\} } , ~ \\mitsigma ^ { - ( - ) \\{ \\pm \\} } , ~ \\mitsigma ^ { + ( - ) [ \\pm ] } , ~ \\mitsigma ^ { - ( + ) [ \\pm ] } \\end{align*}",
"\\begin{equation*} \\mitD ( \\ell , \\mitJ ; \\mitd _ { 1 } , \\mitd _ { 2 } , \\mitd _ { 3 } , \\mitd _ { 4 } ) \\end{equation*}"
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0003051_page07
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"$q_1,q_2,q_3,q_4$",
"$H_1,\\ldots ,H_4$",
"$|\\ell , J, q_i\\rangle $",
"$[d_1,d_2,d_3,d_4]$",
"$U(1)$",
"$(q_1,q_2,q_3,q_4)$",
"$8_v: \\ [1,0,0,0] \\ \\leftrightarrow (2,0,0,0)$",
"$8_v: \\ [1,0,0,0] \\ \\leftrightarrow (2,0,0,0)$",
"$28: \\ [0,1,0,0] \\ \\leftrightarrow (2,2,0,0)$",
"$8_s: \\ [0,0,1,0] \\^^M\\leftrightarrow (1,1,1,0)$",
"$8_c: \\ [0,0,0,1] \\ \\leftrightarrow (1,1,0,1)$",
"$K_\\mu $",
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"$SO(8)$",
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"$\\ell $",
"$J$",
"$q_i$",
"$Q^i_\\alpha $",
"$Q$",
"$Q$",
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"$SO(8)$",
"$\\{{\\cal T}\\}_+$",
"$4= {1\\over 2}\\;8$",
"\\begin {equation}\\label {172} d_1 = {1\\over 2} (q_1-q_2)\\;, \\ d_2 = {1\\over 2} (q_2-q_3-q_4)\\;, \\^^Md_3 = q_3\\;, \\ d_4=q_4\\;. \\end {equation}",
"\\begin {equation}\\label {173} q_1-q_2 =2n \\geq 0 \\;, \\quad q_2 - q_3 - q_4 =2k \\geq 0 \\;, \\quad q_3 \\geq 0\\;, \\quad q_4 \\geq 0\\;. \\end {equation}",
"\\begin {equation}\\label {2.15} S^i_\\alpha |\\ell , J, q_i\\rangle = 0 \\end {equation}",
"\\begin {equation}\\label {2.16} \\{{\\cal T}\\}_+ |\\ell , J, q_i\\rangle = 0\\;. \\end {equation}",
"\\begin {eqnarray} \\mbox {type I 1/2 BPS:} &&Q^{++}|\\ell , J, q_i\\rangle = Q^{(++)}|\\ell , J, q_i\\rangle = Q^{[+]\\{+\\}}|\\ell , J, q_i\\rangle = \\\\ && Q^{[+]\\{-\\}}|\\ell , J, q_i\\rangle = 0 \\end {eqnarray}"
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"$ [ d _ { 1 } , d _ { 2 } , d _ { 3 } , d _ { 4 } ] $",
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"$ 8 _ { v } : ~ [ 1 , 0 , 0 , 0 ] ~ \\leftrightarrow ( 2 , 0 , 0 , 0 ) $",
"$ 8 _ { v } : ~ [ 1 , 0 , 0 , 0 ] ~ \\leftrightarrow ( 2 , 0 , 0 , 0 ) $",
"$ 2 8 : ~ [ 0 , 1 , 0 , 0 ] ~ \\leftrightarrow ( 2 , 2 , 0 , 0 ) $",
"$ 8 _ { s } : ~ [ 0 , 0 , 1 , 0 ] ~ \\leftrightarrow ( 1 , 1 , 1 , 0 ) $",
"$ 8 _ { c } : ~ [ 0 , 0 , 0 , 1 ] ~ \\leftrightarrow ( 1 , 1 , 0 , 1 ) $",
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"$ \\{ T \\} _ { + } $",
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"\\begin{align*} t y p e ~ I ~ 1 \\slash 2 ~ B P S : & & Q ^ { + + } \\vert l , J , q _ { i } \\rangle = Q ^ { ( + + ) } \\vert l , J , q _ { i } \\rangle = Q ^ { [ + ] \\{ + \\} } \\vert l , J , q _ { i } \\rangle = \\\\ & & Q ^ { [ + ] \\{ - \\} } \\vert l , J , q _ { i } \\rangle = 0 \\end{align*}"
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"$ 8 _ { \\mitv } : ~ [ 1 , 0 , 0 , 0 ] ~ \\leftrightarrow ( 2 , 0 , 0 , 0 ) $",
"$ 2 8 : ~ [ 0 , 1 , 0 , 0 ] ~ \\leftrightarrow ( 2 , 2 , 0 , 0 ) $",
"$ 8 _ { \\mits } : ~ [ 0 , 0 , 1 , 0 ] ~ \\leftrightarrow ( 1 , 1 , 1 , 0 ) $",
"$ 8 _ { \\mitc } : ~ [ 0 , 0 , 0 , 1 ] ~ \\leftrightarrow ( 1 , 1 , 0 , 1 ) $",
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"\\begin{equation*} \\mitq _ { 1 } - \\mitq _ { 2 } = 2 \\mitn \\geq 0 \\; , \\quad \\mitq _ { 2 } - \\mitq _ { 3 } - \\mitq _ { 4 } = 2 \\mitk \\geq 0 \\; , \\quad \\mitq _ { 3 } \\geq 0 \\; , \\quad \\mitq _ { 4 } \\geq 0 \\; . \\end{equation*}",
"\\begin{equation*} \\mitS _ { \\mitalpha } ^ { \\miti } \\vert \\ell , \\mitJ , \\mitq _ { \\miti } \\rangle = 0 \\end{equation*}",
"\\begin{equation*} \\{ \\mitT \\} _ { + } \\vert \\ell , \\mitJ , \\mitq _ { \\miti } \\rangle = 0 \\; . \\end{equation*}",
"\\begin{align*} \\displaystyle \\mathrm { t y p e } ~ \\mathrm { I } ~ 1 \\slash 2 ~ \\mathrm { B P S } : & & \\displaystyle \\mitQ ^ { + + } \\vert \\ell , \\mitJ , \\mitq _ { \\miti } \\rangle = \\mitQ ^ { ( + + ) } \\vert \\ell , \\mitJ , \\mitq _ { \\miti } \\rangle = \\mitQ ^ { [ + ] \\{ + \\} } \\vert \\ell , \\mitJ , \\mitq _ { \\miti } \\rangle = \\\\ & & \\displaystyle \\mitQ ^ { [ + ] \\{ - \\} } \\vert \\ell , \\mitJ , \\mitq _ { \\miti } \\rangle = 0 \\end{align*}"
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0003051_page08
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{
"latex": [
"$2\\ell \\equiv m$",
"$q_3=m$",
"$q_4=n$",
"$m,n$",
"\\begin {eqnarray} \\mbox {type II 1/2 BPS:} &&Q^{++}|\\ell , J, q_i\\rangle = Q^{(++)}|\\ell , J, q_i\\rangle = Q^{[+]\\{+\\}}|\\ell , J, q_i\\rangle = \\\\ && Q^{[-]\\{+\\}}|\\ell , J, q_i\\rangle = 0\\;. \\end {eqnarray}",
"\\begin {equation}\\label {2.18} \\mbox {type I 1/2 BPS:} \\qquad q_1=q_2=q_3 = 2\\ell \\;, \\quad q_4=0\\;, \\quad J=0\\;; \\end {equation}",
"\\begin {equation}\\label {2.18''} \\mbox {type II 1/2 BPS:} \\qquad q_1=q_2=q_4 = 2\\ell \\;, \\quad q_3=0\\;, \\quad J=0\\;, \\end {equation}",
"\\begin {equation}\\label {2.18'} \\mbox {type I 1/2 BPS:} \\qquad {\\cal D}(m/2, 0; 0,0,m,0)\\;; \\end {equation}",
"\\begin {equation}\\label {2.18'''} \\mbox {type II 1/2 BPS:} \\qquad {\\cal D}(m/2, 0; 0,0,0,m)\\;. \\end {equation}",
"\\begin {equation}\\label {2.19} \\mbox {3/8 BPS:} \\qquad Q^{++}|\\ell , J, q_i\\rangle = Q^{(++)}|\\ell , J, q_i\\rangle = Q^{[+]\\{+\\}}|\\ell , J, q_i\\rangle = 0\\;. \\end {equation}",
"\\begin {equation}\\label {2.20} q_1=q_2 = q_3+q_4= 2\\ell \\;, \\quad J=0\\;. \\end {equation}",
"\\begin {equation}\\label {2.21} \\mbox {3/8 BPS:} \\qquad {\\cal D}(1/2(m+n), 0; 0,0,m,n)\\;. \\end {equation}",
"\\begin {equation}\\label {2.22} \\mbox {1/4 BPS:} \\qquad Q^{++}|\\ell , J, q_i\\rangle = Q^{(++)}|\\ell , J, q_i\\rangle = 0\\;. \\end {equation}",
"\\begin {equation}\\label {2.23} q_1=q_2 = 2\\ell \\;, \\quad J=0\\;, \\end {equation}"
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"\\begin{equation*} t y p e ~ I ~ 1 \\slash 2 ~ B P S : \\qquad q _ { 1 } = q _ { 2 } = q _ { 3 } = 2 l \\; , \\quad q _ { 4 } = 0 \\; , \\quad J = 0 \\; ; \\end{equation*}",
"\\begin{equation*} t y p e ~ I I ~ 1 \\slash 2 ~ B P S : \\qquad q _ { 1 } = q _ { 2 } = q _ { 4 } = 2 l \\; , \\quad q _ { 3 } = 0 \\; , \\quad J = 0 \\; , \\end{equation*}",
"\\begin{equation*} t y p e ~ I ~ 1 \\slash 2 ~ B P S : \\qquad D ( m \\slash 2 , 0 ; 0 , 0 , m , 0 ) \\; ; \\end{equation*}",
"\\begin{equation*} t y p e ~ I I ~ 1 \\slash 2 ~ B P S : \\qquad D ( m \\slash 2 , 0 ; 0 , 0 , 0 , m ) \\; . \\end{equation*}",
"\\begin{equation*} 3 \\slash 8 ~ B P S : \\qquad Q ^ { + + } \\vert l , J , q _ { i } \\rangle = Q ^ { ( + + ) } \\vert l , J , q _ { i } \\rangle = Q ^ { [ + ] \\{ + \\} } \\vert l , J , q _ { i } \\rangle = 0 \\; . \\end{equation*}",
"\\begin{equation*} q _ { 1 } = q _ { 2 } = q _ { 3 } + q _ { 4 } = 2 l \\; , \\quad J = 0 \\; . \\end{equation*}",
"\\begin{equation*} 3 \\slash 8 ~ B P S : \\qquad D ( 1 \\slash 2 ( m + n ) , 0 ; 0 , 0 , m , n ) \\; . \\end{equation*}",
"\\begin{equation*} 1 \\slash 4 ~ B P S : \\qquad Q ^ { + + } \\vert l , J , q _ { i } \\rangle = Q ^ { ( + + ) } \\vert l , J , q _ { i } \\rangle = 0 \\; . \\end{equation*}",
"\\begin{equation*} q _ { 1 } = q _ { 2 } = 2 l \\; , \\quad J = 0 \\; , \\end{equation*}"
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"\\begin{equation*} \\mathrm { t y p e } ~ \\mathrm { I } ~ 1 \\slash 2 ~ \\mathrm { B P S } : \\qquad \\mitq _ { 1 } = \\mitq _ { 2 } = \\mitq _ { 3 } = 2 \\ell \\; , \\quad \\mitq _ { 4 } = 0 \\; , \\quad \\mitJ = 0 \\; ; \\end{equation*}",
"\\begin{equation*} \\mathrm { t y p e } ~ \\mathrm { I I } ~ 1 \\slash 2 ~ \\mathrm { B P S } : \\qquad \\mitq _ { 1 } = \\mitq _ { 2 } = \\mitq _ { 4 } = 2 \\ell \\; , \\quad \\mitq _ { 3 } = 0 \\; , \\quad \\mitJ = 0 \\; , \\end{equation*}",
"\\begin{equation*} \\mathrm { t y p e } ~ \\mathrm { I } ~ 1 \\slash 2 ~ \\mathrm { B P S } : \\qquad \\mitD ( \\mitm \\slash 2 , 0 ; 0 , 0 , \\mitm , 0 ) \\; ; \\end{equation*}",
"\\begin{equation*} \\mathrm { t y p e } ~ \\mathrm { I I } ~ 1 \\slash 2 ~ \\mathrm { B P S } : \\qquad \\mitD ( \\mitm \\slash 2 , 0 ; 0 , 0 , 0 , \\mitm ) \\; . \\end{equation*}",
"\\begin{equation*} 3 \\slash 8 ~ \\mathrm { B P S } : \\qquad \\mitQ ^ { + + } \\vert \\ell , \\mitJ , \\mitq _ { \\miti } \\rangle = \\mitQ ^ { ( + + ) } \\vert \\ell , \\mitJ , \\mitq _ { \\miti } \\rangle = \\mitQ ^ { [ + ] \\{ + \\} } \\vert \\ell , \\mitJ , \\mitq _ { \\miti } \\rangle = 0 \\; . \\end{equation*}",
"\\begin{equation*} \\mitq _ { 1 } = \\mitq _ { 2 } = \\mitq _ { 3 } + \\mitq _ { 4 } = 2 \\ell \\; , \\quad \\mitJ = 0 \\; . \\end{equation*}",
"\\begin{equation*} 3 \\slash 8 ~ \\mathrm { B P S } : \\qquad \\mitD ( 1 \\slash 2 ( \\mitm + \\mitn ) , 0 ; 0 , 0 , \\mitm , \\mitn ) \\; . \\end{equation*}",
"\\begin{equation*} 1 \\slash 4 ~ \\mathrm { B P S } : \\qquad \\mitQ ^ { + + } \\vert \\ell , \\mitJ , \\mitq _ { \\miti } \\rangle = \\mitQ ^ { ( + + ) } \\vert \\ell , \\mitJ , \\mitq _ { \\miti } \\rangle = 0 \\; . \\end{equation*}",
"\\begin{equation*} \\mitq _ { 1 } = \\mitq _ { 2 } = 2 \\ell \\; , \\quad \\mitJ = 0 \\; , \\end{equation*}"
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0003051_page09
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"$Q^i$",
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"$OSp(8/4,\\mathbb {R})$",
"$OSp(8/4,\\mathbb {R})$",
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"\\begin {equation}\\label {2.25} \\mbox {1/8 BPS:} \\qquad Q^{++}|\\ell , J, q_i\\rangle = 0\\;. \\end {equation}",
"\\begin {equation}\\label {2.26} q_1 = 2\\ell \\;, \\quad J=0\\;, \\end {equation}",
"\\begin {equation}\\label {2.27} \\mbox {1/8 BPS:} \\qquad {\\cal D}(1/2(m+n)+k+l, 0; l,k,m,n)\\;. \\end {equation}"
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"\\begin{equation*} 1 \\slash 8 ~ B P S : \\qquad Q ^ { + + } \\vert l , J , q _ { i } \\rangle = 0 \\; . \\end{equation*}",
"\\begin{equation*} q _ { 1 } = 2 l \\; , \\quad J = 0 \\; , \\end{equation*}",
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"$ \\mitO \\mitS \\mitp ( 8 \\slash 4 , \\BbbR ) $",
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0003051_page10
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{
"latex": [
"$8_s$",
"$SO(8)$",
"$\\Phi _a(x^\\mu , \\theta ^\\alpha _i)$",
"$8_s$",
"$a$",
"$\\Phi _a$",
"$N=8$",
"$SO(8)$",
"$\\psi _{\\alpha \\; ia} \\ \\rightarrow \\^^M8_v \\otimes 8_s = 8_c \\oplus 56_s$",
"$56_s$",
"$\\psi _{\\alpha \\; ia}$",
"$D^i_\\alpha $",
"$SO(8)$",
"$\\gamma ^i_{a\\dot b}$",
"$\\tilde \\gamma ^i_{\\dot a b} = (\\gamma ^{iT})_{\\dot a b}$",
"$\\partial _{\\alpha \\beta }= \\partial _{\\beta \\alpha } = (\\Gamma ^\\mu )_{\\alpha \\beta }\\partial _\\mu $",
"$\\gamma _{ij\\ldots }$",
"$SO(8)$",
"$8_s$",
"$8_c$",
"$SO(8)$",
"$OSp(N/4)$",
"\\begin {equation}\\label {2} \\Phi _a(x^\\mu , \\theta ^\\alpha _i) = \\phi _a(x) + \\theta ^\\alpha _i \\psi _{\\alpha \\; ia}(x) + \\ldots \\;, \\end {equation}",
"\\begin {equation}\\label {3} \\mbox {type I:}\\qquad D^i_\\alpha \\Phi _a = {1\\over 8}\\gamma ^i_{a\\dot b}\\tilde \\gamma ^j_{\\dot b c} D^j_\\alpha \\Phi _c\\;. \\end {equation}",
"\\begin {equation}\\label {4} \\{D^i_\\alpha ,D^j_\\beta \\}= 2i\\delta ^{ij}(\\Gamma ^\\mu )_{\\alpha \\beta }\\partial _\\mu \\;. \\end {equation}",
"\\begin {equation}\\label {5} \\gamma ^i_{a\\dot b}\\tilde \\gamma ^j_{\\dot b c} + \\gamma ^j_{a\\dot b}\\tilde \\gamma ^i_{\\dot b c} = 2\\delta ^{ij} \\delta _{ac}\\;, \\qquad \\tilde \\gamma ^i_{\\dot ab}\\gamma ^j_{b\\dot c} + \\tilde \\gamma ^j_{\\dot ab}\\gamma ^i_{b\\dot c} = 2\\delta ^{ij} \\delta _{\\dot a\\dot c}\\;. \\end {equation}",
"\\begin {eqnarray} \\Phi _a(x^\\mu , \\theta ^\\alpha _i)&=&\\phi _a(x) + \\theta ^\\alpha _i (\\gamma _i)_{a\\dot b}\\; \\psi _{\\alpha \\; \\dot b}(x) \\\\ && +\\theta ^{\\alpha }_i\\theta ^{\\beta }_j (\\gamma _{ij})_{ab} \\; i\\partial _{\\alpha \\beta } \\phi _b \\\\ && +\\theta ^{\\alpha }_i\\theta ^\\beta _i\\theta ^{\\gamma }_k (\\gamma _{ijk})_{a\\dot b} \\; i\\partial _{(\\alpha \\beta } \\psi _{\\gamma )\\dot b} \\\\ && +\\theta ^{\\alpha }_i\\theta ^\\beta _i\\theta ^\\gamma _k\\theta ^{\\delta }_l (\\gamma _{ijkl})_{ab}\\; \\partial _{(\\alpha \\beta } \\partial _{\\gamma \\delta )} \\phi _b \\end {eqnarray}",
"\\begin {equation}\\label {7} \\square \\phi _a = 0\\;, \\qquad \\partial ^{\\alpha \\beta }\\psi _{\\beta \\; \\dot a} = 0\\;. \\end {equation}"
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"$ a $",
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"$ 5 6 _ { s } $",
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"$ S O ( 8 ) $",
"$ \\gamma _ { a \\dot { b } } ^ { i } $",
"$ \\widetilde { \\gamma } _ { \\dot { a } b } ^ { i } = ( \\gamma ^ { i T } ) _ { \\dot { a } b } $",
"$ \\partial _ { \\alpha \\beta } = \\partial _ { \\beta \\alpha } = ( \\Gamma ^ { \\mu } ) _ { \\alpha \\beta } \\partial _ { \\mu } $",
"$ \\gamma _ { i j \\ldots } $",
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"$ 8 _ { c } $",
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"$ O S p ( N \\slash 4 ) $",
"\\begin{equation*} \\Phi _ { a } ( x ^ { \\mu } , \\theta _ { i } ^ { \\alpha } ) = \\phi _ { a } ( x ) + \\theta _ { i } ^ { \\alpha } \\psi _ { \\alpha \\; i a } ( x ) + \\ldots \\; , \\end{equation*}",
"\\begin{equation*} t y p e ~ I : \\qquad D _ { \\alpha } ^ { i } \\Phi _ { a } = \\frac { 1 } { 8 } \\gamma _ { a \\dot { b } } ^ { i } \\widetilde { \\gamma } _ { \\dot { b } c } ^ { j } D _ { \\alpha } ^ { j } \\Phi _ { c } \\; . \\end{equation*}",
"\\begin{equation*} \\{ D _ { \\alpha } ^ { i } , D _ { \\beta } ^ { j } \\} = 2 i \\delta ^ { i j } ( \\Gamma ^ { \\mu } ) _ { \\alpha \\beta } \\partial _ { \\mu } \\; . \\end{equation*}",
"\\begin{equation*} \\gamma _ { a \\dot { b } } ^ { i } \\widetilde { \\gamma } _ { \\dot { b } c } ^ { j } + \\gamma _ { a \\dot { b } } ^ { j } \\widetilde { \\gamma } _ { \\dot { b } c } ^ { i } = 2 \\delta ^ { i j } \\delta _ { a c } \\; , \\qquad \\widetilde { \\gamma } _ { \\dot { a } b } ^ { i } \\gamma _ { b \\dot { c } } ^ { j } + \\widetilde { \\gamma } _ { \\dot { a } b } ^ { j } \\gamma _ { b \\dot { c } } ^ { i } = 2 \\delta ^ { i j } \\delta _ { \\dot { a } \\dot { c } } \\; . \\end{equation*}",
"\\begin{align*} \\Phi _ { a } ( x ^ { \\mu } , \\theta _ { i } ^ { \\alpha } ) & = & \\phi _ { a } ( x ) + \\theta _ { i } ^ { \\alpha } ( \\gamma _ { i } ) _ { a \\dot { b } } \\; \\psi _ { \\alpha \\; \\dot { b } } ( x ) \\\\ & & + \\theta _ { i } ^ { \\alpha } \\theta _ { j } ^ { \\beta } ( \\gamma _ { i j } ) _ { a b } \\; i \\partial _ { \\alpha \\beta } \\phi _ { b } \\\\ & & + \\theta _ { i } ^ { \\alpha } \\theta _ { i } ^ { \\beta } \\theta _ { k } ^ { \\gamma } ( \\gamma _ { i j k } ) _ { a \\dot { b } } \\; i \\partial _ { ( \\alpha \\beta } \\psi _ { \\gamma ) \\dot { b } } \\\\ & & + \\theta _ { i } ^ { \\alpha } \\theta _ { i } ^ { \\beta } \\theta _ { k } ^ { \\gamma } \\theta _ { l } ^ { \\delta } ( \\gamma _ { i j k l } ) _ { a b } \\; \\partial _ { ( \\alpha \\beta } \\partial _ { \\gamma \\delta ) } \\phi _ { b } \\end{align*}",
"\\begin{equation*} \\square \\phi _ { a } = 0 \\; , \\qquad \\partial ^ { \\alpha \\beta } \\psi _ { \\beta \\; \\dot { a } } = 0 \\; . \\end{equation*}"
],
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"$ 8 _ { \\mits } $",
"$ \\mitS \\mitO ( 8 ) $",
"$ \\mupPhi _ { \\mita } ( \\mitx ^ { \\mitmu } , \\mittheta _ { \\miti } ^ { \\mitalpha } ) $",
"$ 8 _ { \\mits } $",
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"$ 5 6 _ { \\mits } $",
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"$ \\mitD _ { \\mitalpha } ^ { \\miti } $",
"$ \\mitS \\mitO ( 8 ) $",
"$ \\mitgamma _ { \\mita \\dot { \\mitb } } ^ { \\miti } $",
"$ \\tilde { \\mitgamma } _ { \\dot { \\mita } \\mitb } ^ { \\miti } = ( \\mitgamma ^ { \\miti \\mitT } ) _ { \\dot { \\mita } \\mitb } $",
"$ \\mitpartial _ { \\mitalpha \\mitbeta } = \\mitpartial _ { \\mitbeta \\mitalpha } = ( \\mupGamma ^ { \\mitmu } ) _ { \\mitalpha \\mitbeta } \\mitpartial _ { \\mitmu } $",
"$ \\mitgamma _ { \\miti \\mitj \\ldots } $",
"$ \\mitS \\mitO ( 8 ) $",
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"$ 8 _ { \\mitc } $",
"$ \\mitS \\mitO ( 8 ) $",
"$ \\mitO \\mitS \\mitp ( \\mitN \\slash 4 ) $",
"\\begin{equation*} \\mupPhi _ { \\mita } ( \\mitx ^ { \\mitmu } , \\mittheta _ { \\miti } ^ { \\mitalpha } ) = \\mitphi _ { \\mita } ( \\mitx ) + \\mittheta _ { \\miti } ^ { \\mitalpha } \\mitpsi _ { \\mitalpha \\; \\miti \\mita } ( \\mitx ) + \\ldots \\; , \\end{equation*}",
"\\begin{equation*} \\mathrm { t y p e } ~ \\mathrm { I } : \\qquad \\mitD _ { \\mitalpha } ^ { \\miti } \\mupPhi _ { \\mita } = \\frac { 1 } { 8 } \\mitgamma _ { \\mita \\dot { \\mitb } } ^ { \\miti } \\tilde { \\mitgamma } _ { \\dot { \\mitb } \\mitc } ^ { \\mitj } \\mitD _ { \\mitalpha } ^ { \\mitj } \\mupPhi _ { \\mitc } \\; . \\end{equation*}",
"\\begin{equation*} \\{ \\mitD _ { \\mitalpha } ^ { \\miti } , \\mitD _ { \\mitbeta } ^ { \\mitj } \\} = 2 \\miti \\mitdelta ^ { \\miti \\mitj } ( \\mupGamma ^ { \\mitmu } ) _ { \\mitalpha \\mitbeta } \\mitpartial _ { \\mitmu } \\; . \\end{equation*}",
"\\begin{equation*} \\mitgamma _ { \\mita \\dot { \\mitb } } ^ { \\miti } \\tilde { \\mitgamma } _ { \\dot { \\mitb } \\mitc } ^ { \\mitj } + \\mitgamma _ { \\mita \\dot { \\mitb } } ^ { \\mitj } \\tilde { \\mitgamma } _ { \\dot { \\mitb } \\mitc } ^ { \\miti } = 2 \\mitdelta ^ { \\miti \\mitj } \\mitdelta _ { \\mita \\mitc } \\; , \\qquad \\tilde { \\mitgamma } _ { \\dot { \\mita } \\mitb } ^ { \\miti } \\mitgamma _ { \\mitb \\dot { \\mitc } } ^ { \\mitj } + \\tilde { \\mitgamma } _ { \\dot { \\mita } \\mitb } ^ { \\mitj } \\mitgamma _ { \\mitb \\dot { \\mitc } } ^ { \\miti } = 2 \\mitdelta ^ { \\miti \\mitj } \\mitdelta _ { \\dot { \\mita } \\dot { \\mitc } } \\; . \\end{equation*}",
"\\begin{align*} \\displaystyle \\mupPhi _ { \\mita } ( \\mitx ^ { \\mitmu } , \\mittheta _ { \\miti } ^ { \\mitalpha } ) & = & \\displaystyle \\mitphi _ { \\mita } ( \\mitx ) + \\mittheta _ { \\miti } ^ { \\mitalpha } ( \\mitgamma _ { \\miti } ) _ { \\mita \\dot { \\mitb } } \\; \\mitpsi _ { \\mitalpha \\; \\dot { \\mitb } } ( \\mitx ) \\\\ & & \\displaystyle + \\mittheta _ { \\miti } ^ { \\mitalpha } \\mittheta _ { \\mitj } ^ { \\mitbeta } ( \\mitgamma _ { \\miti \\mitj } ) _ { \\mita \\mitb } \\; \\miti \\mitpartial _ { \\mitalpha \\mitbeta } \\mitphi _ { \\mitb } \\\\ & & \\displaystyle + \\mittheta _ { \\miti } ^ { \\mitalpha } \\mittheta _ { \\miti } ^ { \\mitbeta } \\mittheta _ { \\mitk } ^ { \\mitgamma } ( \\mitgamma _ { \\miti \\mitj \\mitk } ) _ { \\mita \\dot { \\mitb } } \\; \\miti \\mitpartial _ { ( \\mitalpha \\mitbeta } \\mitpsi _ { \\mitgamma ) \\dot { \\mitb } } \\\\ & & \\displaystyle + \\mittheta _ { \\miti } ^ { \\mitalpha } \\mittheta _ { \\miti } ^ { \\mitbeta } \\mittheta _ { \\mitk } ^ { \\mitgamma } \\mittheta _ { \\mitl } ^ { \\mitdelta } ( \\mitgamma _ { \\miti \\mitj \\mitk \\mitl } ) _ { \\mita \\mitb } \\; \\mitpartial _ { ( \\mitalpha \\mitbeta } \\mitpartial _ { \\mitgamma \\mitdelta ) } \\mitphi _ { \\mitb } \\end{align*}",
"\\begin{equation*} \\square \\mitphi _ { \\mita } = 0 \\; , \\qquad \\mitpartial ^ { \\mitalpha \\mitbeta } \\mitpsi _ { \\mitbeta \\; \\dot { \\mita } } = 0 \\; . \\end{equation*}"
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0003051_page11
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"$\\psi _{\\alpha \\; \\dot a}$",
"$1/2$",
"$1$",
"$\\Phi _a$",
"$1/2$",
"${\\cal D}(1/2, 0; 0,0,1,0)$",
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"$Spin(8)/U(4)$",
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"\\begin {equation}\\label {8} {SO(8)\\over [SO(2)]^4} \\ \\sim \\ {Spin(8)\\over [U(1)]^4}\\;. \\end {equation}",
"\\begin {equation}\\label {9} u_a^A\\;, \\ w^{\\dot A}_{\\dot a}\\;, \\ v^I_i \\end {equation}"
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"$ \\chi _ { \\alpha \\; a } ( x ) $",
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"\\begin{equation*} u _ { a } ^ { A } \\; , ~ w _ { \\dot { a } } ^ { \\dot { A } } \\; , ~ v _ { i } ^ { I } \\end{equation*}"
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"\\begin{equation*} \\mitu _ { \\mita } ^ { \\mitA } \\; , ~ \\mitw _ { \\dot { \\mita } } ^ { \\dot { \\mitA } } \\; , ~ \\mitv _ { \\miti } ^ { \\mitI } \\end{equation*}"
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0003051_page12
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{
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"$8_v$",
"$U(1)$",
"$[U(1)]^4$",
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"$U(1)$",
"$[U(1)]^4$",
"$SO(8)\\sim Spin(8)$",
"$SO(8)$",
"$SO(8)$",
"\\begin {equation}\\label {10} u_a^A u_a^B = \\delta ^{AB}\\;, \\quad w^{\\dot A}_{\\dot a} w^{\\dot B}_{\\dot a} = \\delta ^{\\dot A\\dot B}\\;, \\quad v^I_i v^J_i = \\delta ^{IJ} \\end {equation}",
"\\begin {equation}\\label {11} u_a^A (\\gamma ^I)_{A\\dot A} w^{\\dot A}_{\\dot a} = v^I_i (\\gamma ^i)_{a\\dot a}\\;. \\end {equation}",
"\\begin {eqnarray} \\phi ^{+(+)[+]}(u,w) &=&\\phi _a u^{+(+)[+]}_a \\\\ && + \\phi _{abc} u^{+(+)[+]}_a u^{+(+)[+]}_b u^{-(-)[-]}_c \\\\ && + \\phi _{a\\dot b\\dot c} u^{+(+)[+]}_a w^{+(+)\\{+\\}}_{\\dot b} w^{-(-)\\{-\\}}_{\\dot c} + \\ldots \\;. \\end {eqnarray}",
"\\begin {equation}\\label {14} D^{IJ} = u^A_a (\\gamma ^{IJ})^{AB}{\\partial \\over \\partial u^B_a} + w^{\\dot A}_{\\dot a} (\\gamma ^{IJ})^{\\dot A\\dot B}{\\partial \\over \\partial w^{\\dot B}_{\\dot a}} + v^{[I}_i {\\partial \\over \\partial v^{J]}_{i}}\\;. \\end {equation}"
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"$ [ U ( 1 ) ] ^ { 4 } $",
"$ 8 \\times 8 $",
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"$ S O ( 8 ) $",
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"$ 8 _ { \\mitc } $",
"$ 8 _ { \\mitv } $",
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"$ \\mitS \\mitO ( 8 ) $",
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"$ \\mitU ( 1 ) $",
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"$ \\mitS \\mitO ( 8 ) $",
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"\\begin{equation*} \\mitD ^ { \\mitI \\mitJ } = \\mitu _ { \\mita } ^ { \\mitA } ( \\mitgamma ^ { \\mitI \\mitJ } ) ^ { \\mitA \\mitB } \\frac { \\mitpartial } { \\mitpartial \\mitu _ { \\mita } ^ { \\mitB } } + \\mitw _ { \\dot { \\mita } } ^ { \\dot { \\mitA } } ( \\mitgamma ^ { \\mitI \\mitJ } ) ^ { \\dot { \\mitA } \\dot { \\mitB } } \\frac { \\mitpartial } { \\mitpartial \\mitw _ { \\dot { \\mita } } ^ { \\dot { \\mitB } } } + \\mitv _ { \\miti } ^ { [ \\mitI } \\frac { \\mitpartial } { \\mitpartial \\mitv _ { \\miti } ^ { \\mitJ ] } } \\; . \\end{equation*}"
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0003051_page13
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{
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"\\begin {equation}\\label {151} H_n f^{(q_1,q_2,q_3,q_4)} (u,w) = q_n f^{(q_1,q_2,q_3,q_4)} (u,w)\\;, \\quad n=1,2,3,4\\;. \\end {equation}",
"\\begin {equation}\\label {16} \\{{\\cal D}\\}_+ = \\left \\{ \\begin {array}{l} D^{++(++)}\\;, \\ D^{++(--)}\\;, \\ D^{++[\\pm ]\\{\\pm \\}}\\;; \\\\ D^{(++)[\\pm ]\\{\\pm \\}}\\;; \\\\ D^{[++]} \\equiv D^{[+]\\{+\\}[+]\\{-\\}}\\;, \\ D^{\\{++\\}} \\equiv D^{[+]\\{+\\}[-]\\{+\\}} \\end {array} \\right . \\end {equation}",
"\\begin {equation}\\label {17} \\{{\\cal D}\\}_+\\phi ^{+(+)[+]}(u,w) = 0 \\ \\Rightarrow \\^^M\\phi ^{+(+)[+]}(u,w) = \\phi _a u^{+(+)[+]}_a \\;, \\end {equation}",
"\\begin {equation}\\label {171} \\{{\\cal D}\\}_+ f^{(q_1,q_2,q_3,q_4)} (u,w) = 0\\;. \\end {equation}",
"\\begin {eqnarray} && f^{(q_1,q_2,q_3,q_4)} (u,w) = f^{(2d_1+2d_2+d_3+d_4,2d_2+d_3+d_4,d_3,d_4)} (u,w) = \\\\ &&f_{a\\ldots b\\ldots c\\ldots \\dot d\\ldots } (u^{+(+)[+]}_a)^{d_2+d_3} (u^{+(+)[-]}_b)^{d_2} (u^{+(-)\\{-\\}}_c)^{d_1} (w^{+(+)\\{+\\}}_{\\dot d})^{d_1+d_4} \\;. \\end {eqnarray}"
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"\\begin{equation*} H _ { n } f ^ { ( q _ { 1 } , q _ { 2 } , q _ { 3 } , q _ { 4 } ) } ( u , w ) = q _ { n } f ^ { ( q _ { 1 } , q _ { 2 } , q _ { 3 } , q _ { 4 } ) } ( u , w ) \\; , \\quad n = 1 , 2 , 3 , 4 \\; . \\end{equation*}",
"\\begin{align*} \\{ D \\} _ { + } = \\{ \\begin{array}{l} D ^ { + + ( + + ) } \\; , ~ D ^ { + + ( - - ) } \\; , ~ D ^ { + + [ \\pm ] \\{ \\pm \\} } \\; ; \\\\ D ^ { ( + + ) [ \\pm ] \\{ \\pm \\} } \\; ; \\\\ D ^ { [ + + ] } \\equiv D ^ { [ + ] \\{ + \\} [ + ] \\{ - \\} } \\; , ~ D ^ { \\{ + + \\} } \\equiv D ^ { [ + ] \\{ + \\} [ - ] \\{ + \\} } \\end{array} \\end{align*}",
"\\begin{equation*} \\{ D \\} _ { + } \\phi ^ { + ( + ) [ + ] } ( u , w ) = 0 ~ \\Rightarrow ~ \\phi ^ { + ( + ) [ + ] } ( u , w ) = \\phi _ { a } u _ { a } ^ { + ( + ) [ + ] } \\; , \\end{equation*}",
"\\begin{equation*} \\{ D \\} _ { + } f ^ { ( q _ { 1 } , q _ { 2 } , q _ { 3 } , q _ { 4 } ) } ( u , w ) = 0 \\; . \\end{equation*}",
"\\begin{align*} & & f ^ { ( q _ { 1 } , q _ { 2 } , q _ { 3 } , q _ { 4 } ) } ( u , w ) = f ^ { ( 2 d _ { 1 } + 2 d _ { 2 } + d _ { 3 } + d _ { 4 } , 2 d _ { 2 } + d _ { 3 } + d _ { 4 } , d _ { 3 } , d _ { 4 } ) } ( u , w ) = \\\\ & & f _ { a \\ldots b \\ldots c \\ldots \\dot { d } \\ldots } ( u _ { a } ^ { + ( + ) [ + ] } ) ^ { d _ { 2 } + d _ { 3 } } ( u _ { b } ^ { + ( + ) [ - ] } ) ^ { d _ { 2 } } ( u _ { c } ^ { + ( - ) \\{ - \\} } ) ^ { d _ { 1 } } ( w _ { \\dot { d } } ^ { + ( + ) \\{ + \\} } ) ^ { d _ { 1 } + d _ { 4 } } \\; . \\end{align*}"
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"$ \\mupPhi _ { \\mita } ~ \\rightarrow ~ \\mupPhi ^ { \\mitA } = \\mitu _ { \\mita } ^ { \\mitA } \\mupPhi _ { \\mita } $",
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"\\begin{align*} \\{ \\mitD \\} _ { + } = \\left\\{ \\begin{array}{l} \\mitD ^ { + + ( + + ) } \\; , ~ \\mitD ^ { + + ( - - ) } \\; , ~ \\mitD ^ { + + [ \\pm ] \\{ \\pm \\} } \\; ; \\\\ \\mitD ^ { ( + + ) [ \\pm ] \\{ \\pm \\} } \\; ; \\\\ \\mitD ^ { [ + + ] } \\equiv \\mitD ^ { [ + ] \\{ + \\} [ + ] \\{ - \\} } \\; , ~ \\mitD ^ { \\{ + + \\} } \\equiv \\mitD ^ { [ + ] \\{ + \\} [ - ] \\{ + \\} } \\end{array} \\right. \\end{align*}",
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"\\begin{equation*} \\{ \\mitD \\} _ { + } \\mitf ^ { ( \\mitq _ { 1 } , \\mitq _ { 2 } , \\mitq _ { 3 } , \\mitq _ { 4 } ) } ( \\mitu , \\mitw ) = 0 \\; . \\end{equation*}",
"\\begin{align*} & & \\displaystyle \\mitf ^ { ( \\mitq _ { 1 } , \\mitq _ { 2 } , \\mitq _ { 3 } , \\mitq _ { 4 } ) } ( \\mitu , \\mitw ) = \\mitf ^ { ( 2 \\mitd _ { 1 } + 2 \\mitd _ { 2 } + \\mitd _ { 3 } + \\mitd _ { 4 } , 2 \\mitd _ { 2 } + \\mitd _ { 3 } + \\mitd _ { 4 } , \\mitd _ { 3 } , \\mitd _ { 4 } ) } ( \\mitu , \\mitw ) = \\\\ & & \\displaystyle \\mitf _ { \\mita \\ldots \\mitb \\ldots \\mitc \\ldots \\dot { \\mitd } \\ldots } ( \\mitu _ { \\mita } ^ { + ( + ) [ + ] } ) ^ { \\mitd _ { 2 } + \\mitd _ { 3 } } ( \\mitu _ { \\mitb } ^ { + ( + ) [ - ] } ) ^ { \\mitd _ { 2 } } ( \\mitu _ { \\mitc } ^ { + ( - ) \\{ - \\} } ) ^ { \\mitd _ { 1 } } ( \\mitw _ { \\dot { \\mitd } } ^ { + ( + ) \\{ + \\} } ) ^ { \\mitd _ { 1 } + \\mitd _ { 4 } } \\; . \\end{align*}"
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0003051_page14
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"$\\Phi ^{+(+)[+]}$",
"$\\Phi ^{+(+)[+]}$",
"$\\Phi ^{+(+)[+]}$",
"$\\Phi ^{+(+)[+]}$",
"$\\{{\\cal D}\\}_+$",
"$D^{++(++)} = \\partial ^{++(++)}_{u,w} + i\\theta ^{++}\\Gamma ^\\mu \\theta ^{(++)}\\partial _\\mu $",
"$D^{++[+]\\{\\pm \\}} = \\partial ^{++[+]\\{\\pm \\}}_{u,w} + i\\theta ^{++}\\Gamma ^\\mu \\theta ^{[+]\\{\\pm \\}}\\partial _\\mu $",
"$D^{++[+]\\{\\pm \\}} = \\partial ^{++[+]\\{\\pm \\}}_{u,w} + i\\theta ^{++}\\Gamma ^\\mu \\theta ^{[+]\\{\\pm \\}}\\partial _\\mu $",
"$\\Phi ^{+(+)[+]}\\;$",
"$\\Phi ^{+(+)[+]}$",
"\\begin {equation}\\label {18} D^{++}\\Phi ^{+(+)[+]} = D^{(++)}\\Phi ^{+(+)[+]} = D^{[+]\\{\\pm \\}}\\Phi ^{+(+)[+]} = 0\\;. \\end {equation}",
"\\begin {equation}\\label {19} \\mbox {type I:}\\qquad \\Phi ^{+(+)[+]} = \\Phi ^{+(+)[+]} (x_A,\\theta ^{++}, \\theta ^{(++)}, \\theta ^{[+]\\{\\pm \\}}, u,w) \\end {equation}",
"\\begin {equation}\\label {191} x_{A\\alpha \\beta } = x_{\\alpha \\beta } + i\\theta ^{++}_{(\\alpha }\\theta ^{--}_{\\beta )} + i\\theta ^{(++)}_{(\\alpha }\\theta ^{(--)}_{\\beta )} + i\\theta ^{[+]\\{+\\}}_{(\\alpha }\\theta ^{[-]\\{-\\}}_{\\beta )} + i\\theta ^{[+]\\{-\\}}_{(\\alpha }\\theta ^{[-]\\{+\\}}_{\\beta )}\\;. \\end {equation}",
"\\begin {equation}\\label {20} \\{{\\cal D}\\}_+\\Phi ^{+(+)[+]} (x,\\theta ^{++}, \\theta ^{(++)}, \\theta ^{[+]\\{\\pm \\}}, u,w) = 0\\;. \\end {equation}",
"\\begin {eqnarray} \\Phi ^{+(+)[+]} &=& u^{+(+)[+]}_a \\phi _a(x) \\\\ &&+(\\theta ^{[+]\\{-\\}\\alpha }w^{+(+)\\{+\\}}_{\\dot a} - \\theta ^{[+]\\{+\\}\\alpha }w^{+(+)\\{-\\}}_{\\dot a} \\\\ &&\\phantom {+(} - \\theta ^{++\\alpha } w^{-(+)[+]}_{\\dot a} - \\theta ^{(++)\\alpha } w^{+(-)[+]}_{\\dot a})\\psi _{\\dot a\\;\\alpha }(x) \\\\ && + \\mbox { derivative terms}\\end {eqnarray}"
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"$ \\{ D \\} _ { + } $",
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"$ D ^ { + + [ + ] \\{ \\pm \\} } = \\partial _ { u , w } ^ { + + [ + ] \\{ \\pm \\} } + i \\theta ^ { + + } \\Gamma ^ { \\mu } \\theta ^ { [ + ] \\{ \\pm \\} } \\partial _ { \\mu } $",
"$ D ^ { + + [ + ] \\{ \\pm \\} } = \\partial _ { u , w } ^ { + + [ + ] \\{ \\pm \\} } + i \\theta ^ { + + } \\Gamma ^ { \\mu } \\theta ^ { [ + ] \\{ \\pm \\} } \\partial _ { \\mu } $",
"$ \\Phi ^ { + ( + ) [ + ] } \\; $",
"$ \\Phi ^ { + ( + ) [ + ] } $",
"\\begin{equation*} D ^ { + + } \\Phi ^ { + ( + ) [ + ] } = D ^ { ( + + ) } \\Phi ^ { + ( + ) [ + ] } = D ^ { [ + ] \\{ \\pm \\} } \\Phi ^ { + ( + ) [ + ] } = 0 \\; . \\end{equation*}",
"\\begin{equation*} t y p e ~ I : \\qquad \\Phi ^ { + ( + ) [ + ] } = \\Phi ^ { + ( + ) [ + ] } ( x _ { A } , \\theta ^ { + + } , \\theta ^ { ( + + ) } , \\theta ^ { [ + ] \\{ \\pm \\} } , u , w ) \\end{equation*}",
"\\begin{equation*} x _ { A \\alpha \\beta } = x _ { \\alpha \\beta } + i \\theta _ { ( \\alpha } ^ { + + } \\theta _ { \\beta ) } ^ { - - } + i \\theta _ { ( \\alpha } ^ { ( + + ) } \\theta _ { \\beta ) } ^ { ( - - ) } + i \\theta _ { ( \\alpha } ^ { [ + ] \\{ + \\} } \\theta _ { \\beta ) } ^ { [ - ] \\{ - \\} } + i \\theta _ { ( \\alpha } ^ { [ + ] \\{ - \\} } \\theta _ { \\beta ) } ^ { [ - ] \\{ + \\} } \\; . \\end{equation*}",
"\\begin{equation*} \\{ D \\} _ { + } \\Phi ^ { + ( + ) [ + ] } ( x , \\theta ^ { + + } , \\theta ^ { ( + + ) } , \\theta ^ { [ + ] \\{ \\pm \\} } , u , w ) = 0 \\; . \\end{equation*}",
"\\begin{align*} \\Phi ^ { + ( + ) [ + ] } & = & u _ { a } ^ { + ( + ) [ + ] } \\phi _ { a } ( x ) \\\\ & & + ( \\theta ^ { [ + ] \\{ - \\} \\alpha } w _ { \\dot { a } } ^ { + ( + ) \\{ + \\} } - \\theta ^ { [ + ] \\{ + \\} \\alpha } w _ { \\dot { a } } ^ { + ( + ) \\{ - \\} } \\\\ & & - \\theta ^ { + + \\alpha } w _ { \\dot { a } } ^ { - ( + ) [ + ] } - \\theta ^ { ( + + ) \\alpha } w _ { \\dot { a } } ^ { + ( - ) [ + ] } ) \\psi _ { \\dot { a } \\; \\alpha } ( x ) \\\\ & & + ~ d e r i v a t i v e ~ t e r m s \\end{align*}"
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"$ \\mupPhi ^ { + ( + ) [ + ] } $",
"$ \\mupPhi ^ { + ( + ) [ + ] } $",
"$ \\mupPhi ^ { + ( + ) [ + ] } $",
"$ \\mupPhi ^ { + ( + ) [ + ] } $",
"$ \\{ \\mitD \\} _ { + } $",
"$ \\mitD ^ { + + ( + + ) } = \\mitpartial _ { \\mitu , \\mitw } ^ { + + ( + + ) } + \\miti \\mittheta ^ { + + } \\mupGamma ^ { \\mitmu } \\mittheta ^ { ( + + ) } \\mitpartial _ { \\mitmu } $",
"$ \\mitD ^ { + + [ + ] \\{ \\pm \\} } = \\mitpartial _ { \\mitu , \\mitw } ^ { + + [ + ] \\{ \\pm \\} } + \\miti \\mittheta ^ { + + } \\mupGamma ^ { \\mitmu } \\mittheta ^ { [ + ] \\{ \\pm \\} } \\mitpartial _ { \\mitmu } $",
"$ \\mitD ^ { + + [ + ] \\{ \\pm \\} } = \\mitpartial _ { \\mitu , \\mitw } ^ { + + [ + ] \\{ \\pm \\} } + \\miti \\mittheta ^ { + + } \\mupGamma ^ { \\mitmu } \\mittheta ^ { [ + ] \\{ \\pm \\} } \\mitpartial _ { \\mitmu } $",
"$ \\mupPhi ^ { + ( + ) [ + ] } \\; $",
"$ \\mupPhi ^ { + ( + ) [ + ] } $",
"\\begin{equation*} \\mitD ^ { + + } \\mupPhi ^ { + ( + ) [ + ] } = \\mitD ^ { ( + + ) } \\mupPhi ^ { + ( + ) [ + ] } = \\mitD ^ { [ + ] \\{ \\pm \\} } \\mupPhi ^ { + ( + ) [ + ] } = 0 \\; . \\end{equation*}",
"\\begin{equation*} \\mathrm { t y p e } ~ \\mathrm { I } : \\qquad \\mupPhi ^ { + ( + ) [ + ] } = \\mupPhi ^ { + ( + ) [ + ] } ( \\mitx _ { \\mitA } , \\mittheta ^ { + + } , \\mittheta ^ { ( + + ) } , \\mittheta ^ { [ + ] \\{ \\pm \\} } , \\mitu , \\mitw ) \\end{equation*}",
"\\begin{equation*} \\mitx _ { \\mitA \\mitalpha \\mitbeta } = \\mitx _ { \\mitalpha \\mitbeta } + \\miti \\mittheta _ { ( \\mitalpha } ^ { + + } \\mittheta _ { \\mitbeta ) } ^ { - - } + \\miti \\mittheta _ { ( \\mitalpha } ^ { ( + + ) } \\mittheta _ { \\mitbeta ) } ^ { ( - - ) } + \\miti \\mittheta _ { ( \\mitalpha } ^ { [ + ] \\{ + \\} } \\mittheta _ { \\mitbeta ) } ^ { [ - ] \\{ - \\} } + \\miti \\mittheta _ { ( \\mitalpha } ^ { [ + ] \\{ - \\} } \\mittheta _ { \\mitbeta ) } ^ { [ - ] \\{ + \\} } \\; . \\end{equation*}",
"\\begin{equation*} \\{ \\mitD \\} _ { + } \\mupPhi ^ { + ( + ) [ + ] } ( \\mitx , \\mittheta ^ { + + } , \\mittheta ^ { ( + + ) } , \\mittheta ^ { [ + ] \\{ \\pm \\} } , \\mitu , \\mitw ) = 0 \\; . \\end{equation*}",
"\\begin{align*} \\displaystyle \\mupPhi ^ { + ( + ) [ + ] } & = & \\displaystyle \\mitu _ { \\mita } ^ { + ( + ) [ + ] } \\mitphi _ { \\mita } ( \\mitx ) \\\\ & & \\displaystyle + ( \\mittheta ^ { [ + ] \\{ - \\} \\mitalpha } \\mitw _ { \\dot { \\mita } } ^ { + ( + ) \\{ + \\} } - \\mittheta ^ { [ + ] \\{ + \\} \\mitalpha } \\mitw _ { \\dot { \\mita } } ^ { + ( + ) \\{ - \\} } \\\\ & & \\displaystyle - \\mittheta ^ { + + \\mitalpha } \\mitw _ { \\dot { \\mita } } ^ { - ( + ) [ + ] } - \\mittheta ^ { ( + + ) \\mitalpha } \\mitw _ { \\dot { \\mita } } ^ { + ( - ) [ + ] } ) \\mitpsi _ { \\dot { \\mita } \\; \\mitalpha } ( \\mitx ) \\\\ & & \\displaystyle + ~ \\mathrm { d e r i v a t i v e ~ t e r m s } \\end{align*}"
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0003051_page15
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{
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"$\\theta $",
"$8_s$",
"$SO(8)$",
"$\\theta $",
"$\\theta ^{[+]\\{-\\}}$",
"$\\{{\\cal D}\\}_+$",
"$SO(8)$",
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"$(1,1,0,1)\\ \\leftrightarrow \\^^M[0,0,0,1]$",
"$8_c$",
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"$8_c$",
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"$\\Phi ^{+(+)[+]}$",
"$A^{(1,1,-1,2)}=0$",
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"\\begin {equation}\\label {211} \\theta ^{[+]\\{-\\}}\\ \\stackrel {D^{\\{++\\}}}{\\rightarrow } \\^^M\\theta ^{[+]\\{+\\}}\\ \\stackrel {D^{(++)[-]\\{-\\}}}{\\rightarrow } \\^^M\\theta ^{(++)}\\ \\stackrel {D^{++(--)}}{\\rightarrow } \\ \\theta ^{++}\\;. \\end {equation}",
"\\begin {equation}\\label {212} \\theta ^{[+]\\{-\\}\\alpha }\\theta ^{[+]\\{-\\}}_\\alpha A^{(1,1,-1,2)}\\;, \\qquad \\theta ^{[+]\\{-\\}\\alpha } \\theta ^{[+]\\{+\\}\\beta } B_{(\\alpha \\beta )}^{(1,1,-1,0)} \\end {equation}",
"\\begin {equation}\\label {212'} \\mbox {type II:}\\qquad \\Sigma ^{+(+)\\{+\\}}(\\theta ^{++},\\theta ^{(++)},\\theta ^{[\\pm ]\\{+\\}}) \\end {equation}"
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"\\begin{equation*} \\theta ^ { [ + ] \\{ - \\} \\alpha } \\theta _ { \\alpha } ^ { [ + ] \\{ - \\} } A ^ { ( 1 , 1 , - 1 , 2 ) } \\; , \\qquad \\theta ^ { [ + ] \\{ - \\} \\alpha } \\theta ^ { [ + ] \\{ + \\} \\beta } B _ { ( \\alpha \\beta ) } ^ { ( 1 , 1 , - 1 , 0 ) } \\end{equation*}",
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"\\begin{equation*} \\mittheta ^ { [ + ] \\{ - \\} \\mitalpha } \\mittheta _ { \\mitalpha } ^ { [ + ] \\{ - \\} } \\mitA ^ { ( 1 , 1 , - 1 , 2 ) } \\; , \\qquad \\mittheta ^ { [ + ] \\{ - \\} \\mitalpha } \\mittheta ^ { [ + ] \\{ + \\} \\mitbeta } \\mitB _ { ( \\mitalpha \\mitbeta ) } ^ { ( 1 , 1 , - 1 , 0 ) } \\end{equation*}",
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0003051_page16
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0003051_page17
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"\\begin {eqnarray} (\\Phi ^{+(+)[+]})^p &=& \\phi ^{[0,0,p,0]} \\\\ &+& \\theta ^{[+]\\{-\\}\\alpha }\\psi ^{[0,0,p-1,1]}_\\alpha + \\ldots \\\\ &+& (\\theta ^{[+]\\{-\\}})^2 A^{[0,0,p-2,2]} + \\ldots \\\\ &+& \\theta ^{[+]\\{-\\}\\alpha } \\theta ^{[+]\\{+\\}\\beta } B_{(\\alpha \\beta )}^{[0,1,p-2,0]} + \\ldots \\\\ &+& (\\theta ^{[+]\\{-\\}})^2\\theta ^{[+]\\{+\\}\\alpha }\\chi ^{[0,1,p-3,1]}_\\alpha + \\ldots \\\\ &+& \\theta ^{[+]\\{-\\}\\alpha } \\theta ^{[+]\\{+\\}\\beta } \\theta ^{(++)\\gamma } \\rho _{(\\alpha \\beta \\gamma )}^{[1,0,p-2,0]} + \\ldots \\\\ &+& (\\theta ^{[+]\\{-\\}})^2(\\theta ^{[+]\\{+\\}})^2 C^{[0,2,p-4,0]} + \\ldots \\\\ &+& (\\theta ^{[+]\\{-\\}})^2\\theta ^{[+]\\{+\\}\\alpha } \\theta ^{(++)\\beta } D^{[1,0,p-3,1]}_{(\\alpha \\beta )} + \\ldots \\\\ &+& \\theta ^{[+]\\{-\\}\\alpha } \\theta ^{[+]\\{+\\}\\beta } \\theta ^{(++)\\gamma } \\theta ^{++\\delta } E^{[0,0,p-2,0]}_{(\\alpha \\beta \\gamma \\delta )} + \\ldots \\\\ &+& (\\theta ^{[+]\\{-\\}})^2 (\\theta ^{[+]\\{+\\}})^2 \\theta ^{(++)\\alpha } \\sigma ^{[1,1,p-4,0]}_{\\alpha } + \\ldots \\\\ &+& (\\theta ^{[+]\\{-\\}})^2 \\theta ^{[+]\\{+\\}\\alpha } \\theta ^{(++)\\beta } \\theta ^{++\\gamma } \\omega ^{[0,0,p-3,1]}_{(\\alpha \\beta \\gamma )} + \\ldots \\\\ &+& (\\theta ^{[+]\\{-\\}})^2 (\\theta ^{[+]\\{+\\}})^2 (\\theta ^{(++)})^2 F^{[2,0,p-4,0]} + \\ldots \\\\ &+& (\\theta ^{[+]\\{-\\}})^2 (\\theta ^{[+]\\{+\\}})^2 \\theta ^{(++)\\alpha } \\theta ^{++\\beta } G^{[0,1,p-4,0]}_{(\\alpha \\beta )} + \\ldots \\\\ &+& (\\theta ^{[+]\\{-\\}})^2 (\\theta ^{[+]\\{+\\}})^2 (\\theta ^{(++)})^2 \\theta ^{++\\alpha }\\tau _\\alpha ^{[1,0,p-4,0]} + \\ldots \\\\ &+& (\\theta ^{[+]\\{-\\}})^2 (\\theta ^{[+]\\{+\\}})^2 (\\theta ^{(++)})^2 (\\theta ^{++})^2 H^{[0,0,p-4,0]} + \\ldots \\\\ &+& \\mbox { derivative terms} \\end {eqnarray}",
"\\begin {eqnarray} &&\\Phi ^{+(+)[+]} (\\theta ^{++}, \\theta ^{(++)}, \\theta ^{[+]\\{+\\}}, \\theta ^{[+]\\{-\\}})\\;, \\\\ &&\\Phi ^{+(+)[-]} (\\theta ^{++}, \\theta ^{(++)}, \\theta ^{[-]\\{+\\}}, \\theta ^{[-]\\{-\\}})\\;, \\\\ &&\\Phi ^{+(-)\\{+\\}} (\\theta ^{++}, \\theta ^{(--)}, \\theta ^{[+]\\{+\\}}, \\theta ^{[-]\\{+\\}})\\;, \\\\ &&\\Phi ^{+(-)\\{-\\}} (\\theta ^{++}, \\theta ^{(--)}, \\theta ^{[+]\\{-\\}}, \\theta ^{[-]\\{-\\}})\\;. \\end {eqnarray}"
],
"latex_norm": [
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"$ p \\slash 2 $",
"$ \\theta $",
"$ - 1 \\slash 2 $",
"$ ( \\Phi ^ { + ( + ) [ + ] } ) ^ { p } $",
"$ p = 1 $",
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"$ p = 2 $",
"$ \\partial ^ { \\alpha \\beta } E _ { ( \\alpha \\beta \\gamma \\delta ) } ^ { [ 0 , 0 , 0 , 0 ] } = \\partial ^ { \\alpha \\beta } \\rho _ { ( \\alpha \\beta \\gamma ) } ^ { [ 1 , 0 , 0 , 0 ] } = \\partial ^ { \\alpha \\beta } B _ { ( \\alpha \\beta ) } ^ { [ 0 , 1 , 0 , 0 ] } = 0 $",
"$ S O ( 8 ) $",
"$ O S p ( 8 \\slash 4 , R ) $",
"$ \\Phi ^ { + ( + ) [ + ] } $",
"$ \\theta ^ { - - } $",
"$ 8 _ { v } $",
"$ \\theta $",
"$ S O ( 8 ) $",
"\\begin{align*} ( \\Phi ^ { + ( + ) [ + ] } ) ^ { p } & = & \\phi ^ { [ 0 , 0 , p , 0 ] } \\\\ & + & \\theta ^ { [ + ] \\{ - \\} \\alpha } \\psi _ { \\alpha } ^ { [ 0 , 0 , p - 1 , 1 ] } + \\ldots \\\\ & + & ( \\theta ^ { [ + ] \\{ - \\} } ) ^ { 2 } A ^ { [ 0 , 0 , p - 2 , 2 ] } + \\ldots \\\\ & + & \\theta ^ { [ + ] \\{ - \\} \\alpha } \\theta ^ { [ + ] \\{ + \\} \\beta } B _ { ( \\alpha \\beta ) } ^ { [ 0 , 1 , p - 2 , 0 ] } + \\ldots \\\\ & + & ( \\theta ^ { [ + ] \\{ - \\} } ) ^ { 2 } \\theta ^ { [ + ] \\{ + \\} \\alpha } \\chi _ { \\alpha } ^ { [ 0 , 1 , p - 3 , 1 ] } + \\ldots \\\\ & + & \\theta ^ { [ + ] \\{ - \\} \\alpha } \\theta ^ { [ + ] \\{ + \\} \\beta } \\theta ^ { ( + + ) \\gamma } \\rho _ { ( \\alpha \\beta \\gamma ) } ^ { [ 1 , 0 , p - 2 , 0 ] } + \\ldots \\\\ & + & ( \\theta ^ { [ + ] \\{ - \\} } ) ^ { 2 } ( \\theta ^ { [ + ] \\{ + \\} } ) ^ { 2 } C ^ { [ 0 , 2 , p - 4 , 0 ] } + \\ldots \\\\ & + & ( \\theta ^ { [ + ] \\{ - \\} } ) ^ { 2 } \\theta ^ { [ + ] \\{ + \\} \\alpha } \\theta ^ { ( + + ) \\beta } D _ { ( \\alpha \\beta ) } ^ { [ 1 , 0 , p - 3 , 1 ] } + \\ldots \\\\ & + & \\theta ^ { [ + ] \\{ - \\} \\alpha } \\theta ^ { [ + ] \\{ + \\} \\beta } \\theta ^ { ( + + ) \\gamma } \\theta ^ { + + \\delta } E _ { ( \\alpha \\beta \\gamma \\delta ) } ^ { [ 0 , 0 , p - 2 , 0 ] } + \\ldots \\\\ & + & ( \\theta ^ { [ + ] \\{ - \\} } ) ^ { 2 } ( \\theta ^ { [ + ] \\{ + \\} } ) ^ { 2 } \\theta ^ { ( + + ) \\alpha } \\sigma _ { \\alpha } ^ { [ 1 , 1 , p - 4 , 0 ] } + \\ldots \\\\ & + & ( \\theta ^ { [ + ] \\{ - \\} } ) ^ { 2 } \\theta ^ { [ + ] \\{ + \\} \\alpha } \\theta ^ { ( + + ) \\beta } \\theta ^ { + + \\gamma } \\omega _ { ( \\alpha \\beta \\gamma ) } ^ { [ 0 , 0 , p - 3 , 1 ] } + \\ldots \\\\ & + & ( \\theta ^ { [ + ] \\{ - \\} } ) ^ { 2 } ( \\theta ^ { [ + ] \\{ + \\} } ) ^ { 2 } ( \\theta ^ { ( + + ) } ) ^ { 2 } F ^ { [ 2 , 0 , p - 4 , 0 ] } + \\ldots \\\\ & + & ( \\theta ^ { [ + ] \\{ - \\} } ) ^ { 2 } ( \\theta ^ { [ + ] \\{ + \\} } ) ^ { 2 } \\theta ^ { ( + + ) \\alpha } \\theta ^ { + + \\beta } G _ { ( \\alpha \\beta ) } ^ { [ 0 , 1 , p - 4 , 0 ] } + \\ldots \\\\ & + & ( \\theta ^ { [ + ] \\{ - \\} } ) ^ { 2 } ( \\theta ^ { [ + ] \\{ + \\} } ) ^ { 2 } ( \\theta ^ { ( + + ) } ) ^ { 2 } \\theta ^ { + + \\alpha } \\tau _ { \\alpha } ^ { [ 1 , 0 , p - 4 , 0 ] } + \\ldots \\\\ & + & ( \\theta ^ { [ + ] \\{ - \\} } ) ^ { 2 } ( \\theta ^ { [ + ] \\{ + \\} } ) ^ { 2 } ( \\theta ^ { ( + + ) } ) ^ { 2 } ( \\theta ^ { + + } ) ^ { 2 } H ^ { [ 0 , 0 , p - 4 , 0 ] } + \\ldots \\\\ & + & ~ d e r i v a t i v e ~ t e r m s \\end{align*}",
"\\begin{align*} & & \\Phi ^ { + ( + ) [ + ] } ( \\theta ^ { + + } , \\theta ^ { ( + + ) } , \\theta ^ { [ + ] \\{ + \\} } , \\theta ^ { [ + ] \\{ - \\} } ) \\; , \\\\ & & \\Phi ^ { + ( + ) [ - ] } ( \\theta ^ { + + } , \\theta ^ { ( + + ) } , \\theta ^ { [ - ] \\{ + \\} } , \\theta ^ { [ - ] \\{ - \\} } ) \\; , \\\\ & & \\Phi ^ { + ( - ) \\{ + \\} } ( \\theta ^ { + + } , \\theta ^ { ( - - ) } , \\theta ^ { [ + ] \\{ + \\} } , \\theta ^ { [ - ] \\{ + \\} } ) \\; , \\\\ & & \\Phi ^ { + ( - ) \\{ - \\} } ( \\theta ^ { + + } , \\theta ^ { ( - - ) } , \\theta ^ { [ + ] \\{ - \\} } , \\theta ^ { [ - ] \\{ - \\} } ) \\; . \\end{align*}"
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"$ \\mittheta $",
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"\\begin{align*} & & \\displaystyle \\mupPhi ^ { + ( + ) [ + ] } ( \\mittheta ^ { + + } , \\mittheta ^ { ( + + ) } , \\mittheta ^ { [ + ] \\{ + \\} } , \\mittheta ^ { [ + ] \\{ - \\} } ) \\; , \\\\ & & \\displaystyle \\mupPhi ^ { + ( + ) [ - ] } ( \\mittheta ^ { + + } , \\mittheta ^ { ( + + ) } , \\mittheta ^ { [ - ] \\{ + \\} } , \\mittheta ^ { [ - ] \\{ - \\} } ) \\; , \\\\ & & \\displaystyle \\mupPhi ^ { + ( - ) \\{ + \\} } ( \\mittheta ^ { + + } , \\mittheta ^ { ( - - ) } , \\mittheta ^ { [ + ] \\{ + \\} } , \\mittheta ^ { [ - ] \\{ + \\} } ) \\; , \\\\ & & \\displaystyle \\mupPhi ^ { + ( - ) \\{ - \\} } ( \\mittheta ^ { + + } , \\mittheta ^ { ( - - ) } , \\mittheta ^ { [ + ] \\{ - \\} } , \\mittheta ^ { [ - ] \\{ - \\} } ) \\; . \\end{align*}"
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0003051_page18
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{
"latex": [
"$SO(8)$",
"$[r+2s,q,p,r]$",
"$\\ell = {1\\over 2}(p+2q+3r+4s)$",
"$1/2$",
"$J_{top}=2$",
"$q=r=s=0$",
"$J_{top}=3$",
"$r=s=0$",
"$J_{top}=7/2$",
"$r\\neq 0$",
"$s\\neq 0$",
"$\\ell [J_{top}] = {1\\over 2}(p+2q+3r+4s)+J_{top}$",
"$\\theta $",
"$-1/2$",
"$J=5/2$",
"$\\theta $",
"$OSp(8/4,\\mathbb {R})$",
"$1/8$",
"$1/4$",
"$1/2$",
"$\\Phi $",
"$\\Sigma $",
"$d_1-d_4 = 2s$",
"\\begin {eqnarray} &&(\\Phi ^{+(+)[+]})^{p+q+r+s}(\\Phi ^{+(+)[-]})^{q+r+s} (\\Phi ^{+(-)\\{+\\}})^{r+s}(\\Phi ^{+(-)\\{-\\}})^{s}\\\\ &&\\ =\\phi ^{[r+2s,q,p,r]} +\\ldots \\\\ &&\\ +\\theta ^{[+]\\{-\\}}_{\\alpha _1} \\theta ^{[+]\\{+\\}}_{\\alpha _2} \\theta ^{(++)}_{\\alpha _3} \\theta ^{++}_{\\alpha _4} A^{[r+2s,q,p-2,r](\\alpha _1\\ldots \\alpha _4)} +\\ldots \\\\ &&\\ +\\theta ^{[+]\\{-\\}}_{\\alpha _1} \\theta ^{[+]\\{+\\}}_{\\alpha _2} \\theta ^{(++)}_{\\alpha _3} \\theta ^{++}_{\\alpha _4} \\theta ^{[-]\\{+\\}}_{\\alpha _5} \\theta ^{[-]\\{-\\}}_{\\alpha _6} B^{[r+2s,q-1,p,r](\\alpha _1\\ldots \\alpha _6)} +\\ldots \\\\ &&\\ +\\theta ^{[+]\\{-\\}}_{\\alpha _1} \\theta ^{[+]\\{+\\}}_{\\alpha _2} \\theta ^{(++)}_{\\alpha _3} \\theta ^{++}_{\\alpha _4} \\theta ^{[-]\\{+\\}}_{\\alpha _5} \\theta ^{[-]\\{-\\}}_{\\alpha _6} \\theta ^{(--)}_{\\alpha _7} \\chi ^{[r+2s-1,q,p,r](\\alpha _1\\ldots \\alpha _7)} +\\ldots \\end {eqnarray}",
"\\begin {eqnarray} {1\\over 8} \\mbox { BPS:} && {\\cal D}(d_1+d_2 + {1\\over 2}(d_3+d_4), 0; d_1,d_2,d_3,d_4)\\;, \\quad d_1-d_4 = 2s \\geq 0\\;; \\\\ {1\\over 4} \\mbox { BPS:} && {\\cal D}(d_2 + {1\\over 2}d_3, 0; 0,d_2,d_3,0)\\;; \\\\ {1\\over 2} \\mbox { BPS:} && {\\cal D}({1\\over 2}d_3, 0; 0,0,d_3,0)\\;. \\end {eqnarray}",
"\\begin {equation}\\label {009} [\\Phi ^{+(+)[+]}(\\theta ^{++},\\theta ^{(++)},\\theta ^{[+]\\{\\pm \\}})]^{p+q} [\\Sigma ^{+(+)\\{+\\}}(\\theta ^{++},\\theta ^{(++)},\\theta ^{[\\pm ]\\{+\\}})]^{q}\\;, \\end {equation}",
"\\begin {equation}\\label {00777} [\\Phi ^{+(+)[+]}]^{m+k}[\\Phi ^{+(+)[-]}]^{k} [\\Sigma ^{+(+)\\{+\\}}]^{n} \\end {equation}"
],
"latex_norm": [
"$ S O ( 8 ) $",
"$ [ r + 2 s , q , p , r ] $",
"$ l = \\frac { 1 } { 2 } ( p + 2 q + 3 r + 4 s ) $",
"$ 1 \\slash 2 $",
"$ J _ { t o p } = 2 $",
"$ q = r = s = 0 $",
"$ J _ { t o p } = 3 $",
"$ r = s = 0 $",
"$ J _ { t o p } = 7 \\slash 2 $",
"$ r \\ne 0 $",
"$ s \\ne 0 $",
"$ l [ J _ { t o p } ] = \\frac { 1 } { 2 } ( p + 2 q + 3 r + 4 s ) + J _ { t o p } $",
"$ \\theta $",
"$ - 1 \\slash 2 $",
"$ J = 5 \\slash 2 $",
"$ \\theta $",
"$ O S p ( 8 \\slash 4 , R ) $",
"$ 1 \\slash 8 $",
"$ 1 \\slash 4 $",
"$ 1 \\slash 2 $",
"$ \\Phi $",
"$ \\Sigma $",
"$ d _ { 1 } - d _ { 4 } = 2 s $",
"\\begin{align*} & & ( \\Phi ^ { + ( + ) [ + ] } ) ^ { p + q + r + s } ( \\Phi ^ { + ( + ) [ - ] } ) ^ { q + r + s } ( \\Phi ^ { + ( - ) \\{ + \\} } ) ^ { r + s } ( \\Phi ^ { + ( - ) \\{ - \\} } ) ^ { s } \\\\ & & ~ = \\phi ^ { [ r + 2 s , q , p , r ] } + \\ldots \\\\ & & ~ + \\theta _ { \\alpha _ { 1 } } ^ { [ + ] \\{ - \\} } \\theta _ { \\alpha _ { 2 } } ^ { [ + ] \\{ + \\} } \\theta _ { \\alpha _ { 3 } } ^ { ( + + ) } \\theta _ { \\alpha _ { 4 } } ^ { + + } A ^ { [ r + 2 s , q , p - 2 , r ] ( \\alpha _ { 1 } \\ldots \\alpha _ { 4 } ) } + \\ldots \\\\ & & ~ + \\theta _ { \\alpha _ { 1 } } ^ { [ + ] \\{ - \\} } \\theta _ { \\alpha _ { 2 } } ^ { [ + ] \\{ + \\} } \\theta _ { \\alpha _ { 3 } } ^ { ( + + ) } \\theta _ { \\alpha _ { 4 } } ^ { + + } \\theta _ { \\alpha _ { 5 } } ^ { [ - ] \\{ + \\} } \\theta _ { \\alpha _ { 6 } } ^ { [ - ] \\{ - \\} } B ^ { [ r + 2 s , q - 1 , p , r ] ( \\alpha _ { 1 } \\ldots \\alpha _ { 6 } ) } + \\ldots \\\\ & & ~ + \\theta _ { \\alpha _ { 1 } } ^ { [ + ] \\{ - \\} } \\theta _ { \\alpha _ { 2 } } ^ { [ + ] \\{ + \\} } \\theta _ { \\alpha _ { 3 } } ^ { ( + + ) } \\theta _ { \\alpha _ { 4 } } ^ { + + } \\theta _ { \\alpha _ { 5 } } ^ { [ - ] \\{ + \\} } \\theta _ { \\alpha _ { 6 } } ^ { [ - ] \\{ - \\} } \\theta _ { \\alpha _ { 7 } } ^ { ( - - ) } \\chi ^ { [ r + 2 s - 1 , q , p , r ] ( \\alpha _ { 1 } \\ldots \\alpha _ { 7 } ) } + \\ldots \\end{align*}",
"\\begin{align*} \\frac { 1 } { 8 } ~ B P S : & & D ( d _ { 1 } + d _ { 2 } + \\frac { 1 } { 2 } ( d _ { 3 } + d _ { 4 } ) , 0 ; d _ { 1 } , d _ { 2 } , d _ { 3 } , d _ { 4 } ) \\; , \\quad d _ { 1 } - d _ { 4 } = 2 s \\geq 0 \\; ; \\\\ \\frac { 1 } { 4 } ~ B P S : & & D ( d _ { 2 } + \\frac { 1 } { 2 } d _ { 3 } , 0 ; 0 , d _ { 2 } , d _ { 3 } , 0 ) \\; ; \\\\ \\frac { 1 } { 2 } ~ B P S : & & D ( \\frac { 1 } { 2 } d _ { 3 } , 0 ; 0 , 0 , d _ { 3 } , 0 ) \\; . \\end{align*}",
"\\begin{equation*} [ \\Phi ^ { + ( + ) [ + ] } ( \\theta ^ { + + } , \\theta ^ { ( + + ) } , \\theta ^ { [ + ] \\{ \\pm \\} } ) ] ^ { p + q } [ \\Sigma ^ { + ( + ) \\{ + \\} } ( \\theta ^ { + + } , \\theta ^ { ( + + ) } , \\theta ^ { [ \\pm ] \\{ + \\} } ) ] ^ { q } \\; , \\end{equation*}",
"\\begin{equation*} [ \\Phi ^ { + ( + ) [ + ] } ] ^ { m + k } [ \\Phi ^ { + ( + ) [ - ] } ] ^ { k } [ \\Sigma ^ { + ( + ) \\{ + \\} } ] ^ { n } \\end{equation*}"
],
"latex_expand": [
"$ \\mitS \\mitO ( 8 ) $",
"$ [ \\mitr + 2 \\mits , \\mitq , \\mitp , \\mitr ] $",
"$ \\ell = \\frac { 1 } { 2 } ( \\mitp + 2 \\mitq + 3 \\mitr + 4 \\mits ) $",
"$ 1 \\slash 2 $",
"$ \\mitJ _ { \\mitt \\mito \\mitp } = 2 $",
"$ \\mitq = \\mitr = \\mits = 0 $",
"$ \\mitJ _ { \\mitt \\mito \\mitp } = 3 $",
"$ \\mitr = \\mits = 0 $",
"$ \\mitJ _ { \\mitt \\mito \\mitp } = 7 \\slash 2 $",
"$ \\mitr \\ne 0 $",
"$ \\mits \\ne 0 $",
"$ \\ell [ \\mitJ _ { \\mitt \\mito \\mitp } ] = \\frac { 1 } { 2 } ( \\mitp + 2 \\mitq + 3 \\mitr + 4 \\mits ) + \\mitJ _ { \\mitt \\mito \\mitp } $",
"$ \\mittheta $",
"$ - 1 \\slash 2 $",
"$ \\mitJ = 5 \\slash 2 $",
"$ \\mittheta $",
"$ \\mitO \\mitS \\mitp ( 8 \\slash 4 , \\BbbR ) $",
"$ 1 \\slash 8 $",
"$ 1 \\slash 4 $",
"$ 1 \\slash 2 $",
"$ \\mupPhi $",
"$ \\mupSigma $",
"$ \\mitd _ { 1 } - \\mitd _ { 4 } = 2 \\mits $",
"\\begin{align*} & & \\displaystyle ( \\mupPhi ^ { + ( + ) [ + ] } ) ^ { \\mitp + \\mitq + \\mitr + \\mits } ( \\mupPhi ^ { + ( + ) [ - ] } ) ^ { \\mitq + \\mitr + \\mits } ( \\mupPhi ^ { + ( - ) \\{ + \\} } ) ^ { \\mitr + \\mits } ( \\mupPhi ^ { + ( - ) \\{ - \\} } ) ^ { \\mits } \\\\ & & \\displaystyle ~ = \\mitphi ^ { [ \\mitr + 2 \\mits , \\mitq , \\mitp , \\mitr ] } + \\ldots \\\\ & & \\displaystyle ~ + \\mittheta _ { \\mitalpha _ { 1 } } ^ { [ + ] \\{ - \\} } \\mittheta _ { \\mitalpha _ { 2 } } ^ { [ + ] \\{ + \\} } \\mittheta _ { \\mitalpha _ { 3 } } ^ { ( + + ) } \\mittheta _ { \\mitalpha _ { 4 } } ^ { + + } \\mitA ^ { [ \\mitr + 2 \\mits , \\mitq , \\mitp - 2 , \\mitr ] ( \\mitalpha _ { 1 } \\ldots \\mitalpha _ { 4 } ) } + \\ldots \\\\ & & \\displaystyle ~ + \\mittheta _ { \\mitalpha _ { 1 } } ^ { [ + ] \\{ - \\} } \\mittheta _ { \\mitalpha _ { 2 } } ^ { [ + ] \\{ + \\} } \\mittheta _ { \\mitalpha _ { 3 } } ^ { ( + + ) } \\mittheta _ { \\mitalpha _ { 4 } } ^ { + + } \\mittheta _ { \\mitalpha _ { 5 } } ^ { [ - ] \\{ + \\} } \\mittheta _ { \\mitalpha _ { 6 } } ^ { [ - ] \\{ - \\} } \\mitB ^ { [ \\mitr + 2 \\mits , \\mitq - 1 , \\mitp , \\mitr ] ( \\mitalpha _ { 1 } \\ldots \\mitalpha _ { 6 } ) } + \\ldots \\\\ & & \\displaystyle ~ + \\mittheta _ { \\mitalpha _ { 1 } } ^ { [ + ] \\{ - \\} } \\mittheta _ { \\mitalpha _ { 2 } } ^ { [ + ] \\{ + \\} } \\mittheta _ { \\mitalpha _ { 3 } } ^ { ( + + ) } \\mittheta _ { \\mitalpha _ { 4 } } ^ { + + } \\mittheta _ { \\mitalpha _ { 5 } } ^ { [ - ] \\{ + \\} } \\mittheta _ { \\mitalpha _ { 6 } } ^ { [ - ] \\{ - \\} } \\mittheta _ { \\mitalpha _ { 7 } } ^ { ( - - ) } \\mitchi ^ { [ \\mitr + 2 \\mits - 1 , \\mitq , \\mitp , \\mitr ] ( \\mitalpha _ { 1 } \\ldots \\mitalpha _ { 7 } ) } + \\ldots \\end{align*}",
"\\begin{align*} \\displaystyle \\frac { 1 } { 8 } ~ \\mathrm { B P S } : & & \\displaystyle \\mitD ( \\mitd _ { 1 } + \\mitd _ { 2 } + \\frac { 1 } { 2 } ( \\mitd _ { 3 } + \\mitd _ { 4 } ) , 0 ; \\mitd _ { 1 } , \\mitd _ { 2 } , \\mitd _ { 3 } , \\mitd _ { 4 } ) \\; , \\quad \\mitd _ { 1 } - \\mitd _ { 4 } = 2 \\mits \\geq 0 \\; ; \\\\ \\displaystyle \\frac { 1 } { 4 } ~ \\mathrm { B P S } : & & \\displaystyle \\mitD ( \\mitd _ { 2 } + \\frac { 1 } { 2 } \\mitd _ { 3 } , 0 ; 0 , \\mitd _ { 2 } , \\mitd _ { 3 } , 0 ) \\; ; \\\\ \\displaystyle \\frac { 1 } { 2 } ~ \\mathrm { B P S } : & & \\displaystyle \\mitD ( \\frac { 1 } { 2 } \\mitd _ { 3 } , 0 ; 0 , 0 , \\mitd _ { 3 } , 0 ) \\; . \\end{align*}",
"\\begin{equation*} [ \\mupPhi ^ { + ( + ) [ + ] } ( \\mittheta ^ { + + } , \\mittheta ^ { ( + + ) } , \\mittheta ^ { [ + ] \\{ \\pm \\} } ) ] ^ { \\mitp + \\mitq } [ \\mupSigma ^ { + ( + ) \\{ + \\} } ( \\mittheta ^ { + + } , \\mittheta ^ { ( + + ) } , \\mittheta ^ { [ \\pm ] \\{ + \\} } ) ] ^ { \\mitq } \\; , \\end{equation*}",
"\\begin{equation*} [ \\mupPhi ^ { + ( + ) [ + ] } ] ^ { \\mitm + \\mitk } [ \\mupPhi ^ { + ( + ) [ - ] } ] ^ { \\mitk } [ \\mupSigma ^ { + ( + ) \\{ + \\} } ] ^ { \\mitn } \\end{equation*}"
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0003051_page19
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{
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"$OSp(8/4,\\mathbb {R})$",
"$N=1$",
"$AdS_4$",
"$AdS_4\\times S^7$",
"$M^4\\times T^7$",
"$AdS_4$",
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"$E_{7(7)}$",
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"$ \\mitM ^ { 4 } \\times \\mitT ^ { 7 } $",
"$ \\mitA \\mitd \\mitS _ { 4 } $",
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0003051_page20
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0003060_page01
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{
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"$d_{k}=2$",
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"$(d-d_{k})$",
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"\\begin {eqnarray} 0&=& R_{\\mu \\nu } \\,-\\, 2\\nabla _{\\mu }\\nabla _{\\nu }\\phi \\,+\\, \\textstyle {\\frac {1}{4}} {H_{\\mu }}^{\\kappa \\rho }H_{\\nu \\kappa \\rho } \\,-\\, \\textstyle {\\frac {1}{2}}e^{2\\phi } \\left [ F_{(2)\\mu \\kappa }{F_{(2)\\nu }}^{\\kappa } \\,-\\, \\textstyle {\\frac {1}{4}}g_{\\mu \\nu }F_{(2)}^{2} \\right ] \\; ,\\\\ && \\\\ 0 &=& R \\,+\\, 4\\left (\\partial \\phi \\right )^{2} \\,-\\, 4 \\nabla ^{2}\\phi \\,+\\, \\textstyle {\\frac {1}{2\\cdot 3!}}H^{2} \\; , \\\\ && \\\\ 0 &=& \\nabla _{\\mu }\\left ( e^{-2\\phi }H^{\\mu \\kappa \\rho }\\right ) \\;=\\; \\nabla _{\\mu }F_{(2)}^{\\mu \\nu } \\; , \\end {eqnarray}"
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"$ \\mitd $",
"$ \\mitd - \\mitd _ { \\mitk } $",
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"$ \\mitd _ { \\mitk } = 2 $",
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0003060_page02
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{
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"$dH=dF_{(2)}=0$",
"$\\overline {g}_{ij}$",
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"$h_{mn}$",
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"\\begin {equation} B \\;=\\; f(\\sigma )\\, \\mathcal {J} \\;=\\; \\textstyle {\\frac {1}{2}} f(\\sigma ) \\mathcal {J}_{mn} dy^{m}\\wedge dy^{n} \\; , \\end {equation}",
"\\begin {equation} {\\mathcal {J}^{m}}_{p}{\\mathcal {J}^{p}}_{n} \\;=\\; -{\\delta ^{m}}_{n} \\hspace {.5cm},\\hspace {.5cm} \\overline {\\nabla }_{m}\\, \\mathcal {J}_{np}\\;=\\; 0 \\; , \\label {eq:KDef} \\end {equation}",
"\\begin {equation} \\dot {f} \\;=\\; \\aleph \\, Ne^{2\\phi }\\eta ^{-D}R^{4-d_{k}} \\; , \\end {equation}",
"\\begin {equation} \\psi \\;=\\; \\phi \\,+\\, \\textstyle {\\frac {1}{2}}\\log \\left ( N\\right ) \\,-\\, \\textstyle {\\frac {D}{2}}\\log \\left ( \\eta \\right ) \\,-\\, \\textstyle {\\frac {d_{k}}{2}}\\log \\left ( R\\right ) \\; , \\end {equation}",
"\\begin {eqnarray} 0 &=& \\left (\\log R\\right )^{\\prime \\prime } \\,+\\, \\textstyle {\\frac {\\aleph ^{2}}{2}}R^{4} \\; ,\\\\ 0 &=& \\left (\\log \\eta \\right )^{\\prime \\prime } \\,-\\, \\lambda \\, \\eta ^{-2}M^{2} \\; , \\\\ 0 &=& \\left (\\log M\\right )^{\\prime \\prime } \\,-\\, D\\lambda \\, \\eta ^{-2}M^{2} \\; , \\\\ 0 &=& \\left [ \\left (\\log M\\right )^{\\prime }\\right ]^{2} \\,-\\, D \\left [ \\left (\\log \\eta \\right )^{\\prime }\\right ]^{2} \\,-\\, d_{k}\\left [ \\left (\\log R\\right )^{\\prime }\\right ]^{2} \\,-\\, \\textstyle {\\frac {d_{k}\\aleph ^{2}}{4}}R^{4} \\; , \\end {eqnarray}",
"\\begin {equation} R(t)\\;=\\; R_{0}\\, \\cosh ^{-1/2}\\left ( \\aleph R_{0}^{2}t\\right ) \\; , \\end {equation}"
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"\\begin{equation*} { J ^ { m } } _ { p } { J ^ { p } } _ { n } \\; = \\; - { \\delta ^ { m } } _ { n } \\hspace{14.23pt} , \\hspace{14.23pt} \\overline { \\nabla } _ { m } \\, J _ { n p } \\; = \\; 0 \\; , \\end{equation*}",
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"\\begin{equation*} \\psi \\; = \\; \\phi \\, + \\, \\frac { 1 } { 2 } \\operatorname { l o g } ( N ) \\, - \\, \\frac { D } { 2 } \\operatorname { l o g } ( \\eta ) \\, - \\, \\frac { d _ { k } } { 2 } \\operatorname { l o g } ( R ) \\; , \\end{equation*}",
"\\begin{align*} 0 & = & { ( \\operatorname { l o g } R ) } ^ { \\prime \\prime } \\, + \\, \\frac { \\aleph ^ { 2 } } { 2 } R ^ { 4 } \\; , \\\\ 0 & = & { ( \\operatorname { l o g } \\eta ) } ^ { \\prime \\prime } \\, - \\, \\lambda \\, \\eta ^ { - 2 } M ^ { 2 } \\; , \\\\ 0 & = & { ( \\operatorname { l o g } M ) } ^ { \\prime \\prime } \\, - \\, D \\lambda \\, \\eta ^ { - 2 } M ^ { 2 } \\; , \\\\ 0 & = & { [ { ( \\operatorname { l o g } M ) } ^ { \\prime } ] } ^ { 2 } \\, - \\, D { [ { ( \\operatorname { l o g } \\eta ) } ^ { \\prime } ] } ^ { 2 } \\, - \\, d _ { k } { [ { ( \\operatorname { l o g } R ) } ^ { \\prime } ] } ^ { 2 } \\, - \\, \\frac { d _ { k } \\aleph ^ { 2 } } { 4 } R ^ { 4 } \\; , \\end{align*}",
"\\begin{equation*} R ( t ) \\; = \\; R _ { 0 } \\, { \\operatorname { c o s h } } ^ { - 1 \\slash 2 } ( \\aleph R _ { 0 } ^ { 2 } t ) \\; , \\end{equation*}"
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"$ \\mitd \\mitH = \\mitd \\mitF _ { ( 2 ) } = 0 $",
"$ \\overline { \\mitg } _ { \\miti \\mitj } $",
"$ \\mitR ( \\overline { \\mitg } ) _ { \\miti \\mitj } \\equiv \\mitlambda \\overline { \\mitg } _ { \\miti \\mitj } $",
"$ \\Planckconst _ { \\mitm \\mitn } $",
"$ \\mitd _ { \\mitk } $",
"$ \\mitD \\equiv \\mitd - 1 - \\mitd _ { \\mitk } $",
"$ \\mitd $",
"$ \\mscrJ $",
"$ \\overline { \\nabla } $",
"$ \\mitsigma $",
"$ \\aleph $",
"$ \\mitM = \\mite ^ { - 2 \\mitpsi } \\mitN $",
"$ \\mite ^ { 2 \\mitpsi } \\mitd \\mitsigma = \\mitd \\mitt $",
"$ \\mitt $",
"$ \\mitlambda = 0 $",
"$ \\mitM = \\mite ^ { \\mitalpha \\mitt } $",
"$ \\miteta = \\mite ^ { \\mitbeta \\mitt } $",
"$ \\Planckconst _ { \\mitm \\mitn } $",
"\\begin{equation*} \\mitd \\mits ^ { 2 } \\; = \\; \\mitN ^ { 2 } ( \\mitsigma ) \\mitd \\mitsigma ^ { 2 } \\, - \\, \\miteta ^ { 2 } ( \\mitsigma ) \\overline { \\mitg } _ { \\miti \\mitj } \\mitd \\mitx ^ { \\miti } \\mitd \\mitx ^ { \\mitj } \\, - \\, \\mitR ^ { 2 } ( \\mitsigma ) \\Planckconst _ { \\mitm \\mitn } \\mitd \\mity ^ { \\mitm } \\mitd \\mity ^ { \\mitn } \\; , \\end{equation*}",
"\\begin{equation*} \\mitB \\; = \\; \\mitf ( \\mitsigma ) \\, \\mscrJ \\; = \\; \\textstyle \\frac { 1 } { 2 } \\mitf ( \\mitsigma ) \\mscrJ _ { \\mitm \\mitn } \\mitd \\mity ^ { \\mitm } \\wedge \\mitd \\mity ^ { \\mitn } \\; , \\end{equation*}",
"\\begin{equation*} { \\mscrJ ^ { \\mitm } } _ { \\mitp } { \\mscrJ ^ { \\mitp } } _ { \\mitn } \\; = \\; - { \\mitdelta ^ { \\mitm } } _ { \\mitn } \\hspace{14.23pt} , \\hspace{14.23pt} \\overline { \\nabla } _ { \\mitm } \\, \\mscrJ _ { \\mitn \\mitp } \\; = \\; 0 \\; , \\end{equation*}",
"\\begin{equation*} \\dot { \\mitf } \\; = \\; \\aleph \\, \\mitN \\mite ^ { 2 \\mitphi } \\miteta ^ { - \\mitD } \\mitR ^ { 4 - \\mitd _ { \\mitk } } \\; , \\end{equation*}",
"\\begin{equation*} \\mitpsi \\; = \\; \\mitphi \\, + \\, \\textstyle \\frac { 1 } { 2 } \\operatorname { l o g } \\left( \\mitN \\right) \\, - \\, \\textstyle \\frac { \\mitD } { 2 } \\operatorname { l o g } \\left( \\miteta \\right) \\, - \\, \\textstyle \\frac { \\mitd _ { \\mitk } } { 2 } \\operatorname { l o g } \\left( \\mitR \\right) \\; , \\end{equation*}",
"\\begin{align*} \\displaystyle 0 & = & \\displaystyle { \\left( \\operatorname { l o g } \\mitR \\right) } ^ { \\prime \\prime } \\, + \\, \\textstyle \\frac { \\aleph ^ { 2 } } { 2 } \\mitR ^ { 4 } \\; , \\\\ \\displaystyle 0 & = & \\displaystyle { \\left( \\operatorname { l o g } \\miteta \\right) } ^ { \\prime \\prime } \\, - \\, \\mitlambda \\, \\miteta ^ { - 2 } \\mitM ^ { 2 } \\; , \\\\ \\displaystyle 0 & = & \\displaystyle { \\left( \\operatorname { l o g } \\mitM \\right) } ^ { \\prime \\prime } \\, - \\, \\mitD \\mitlambda \\, \\miteta ^ { - 2 } \\mitM ^ { 2 } \\; , \\\\ \\displaystyle 0 & = & \\displaystyle { \\left[ { \\left( \\operatorname { l o g } \\mitM \\right) } ^ { \\prime } \\right] } ^ { 2 } \\, - \\, \\mitD { \\left[ { \\left( \\operatorname { l o g } \\miteta \\right) } ^ { \\prime } \\right] } ^ { 2 } \\, - \\, \\mitd _ { \\mitk } { \\left[ { \\left( \\operatorname { l o g } \\mitR \\right) } ^ { \\prime } \\right] } ^ { 2 } \\, - \\, \\textstyle \\frac { \\mitd _ { \\mitk } \\aleph ^ { 2 } } { 4 } \\mitR ^ { 4 } \\; , \\end{align*}",
"\\begin{equation*} \\mitR ( \\mitt ) \\; = \\; \\mitR _ { 0 } \\, { \\operatorname { c o s h } } ^ { - 1 \\slash 2 } \\left( \\aleph \\mitR _ { 0 } ^ { 2 } \\mitt \\right) \\; , \\end{equation*}"
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0003060_page03
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{
"latex": [
"$g_{s}^{2}= e^{2\\phi }=M^{-1}\\eta ^{D}R^{d_{k}}$",
"$g_{s}^{2}= e^{2\\phi }=M^{-1}\\eta ^{D}R^{d_{k}}$",
"$H=\\aleph R^{4}dt\\wedge \\mathcal {J}$",
"$\\beta =0$",
"$\\alpha =1$",
"$\\tau $",
"$R$",
"$f$",
"$SL(2,\\mathbb {R})/U(1)$",
"$d_{k}=6$",
"$SL(2,\\mathbb {R})$",
"$d_{k}=4$",
"$R=R_{0}e^{\\alpha t}$",
"$M$",
"$\\eta $",
"$M=\\eta ^{D+2}$",
"\\begin {equation} \\alpha ^{2}\\,-\\, D\\beta ^{2} \\;=\\; \\frac {d_{k}\\aleph ^{2}R_{0}^{4}}{4} \\; . \\end {equation}",
"\\begin {eqnarray} ds^{2} &=& d\\tau ^{2}\\,-\\, d\\vec {x}_{(D)} \\,-\\, 2R^{2}_{0} B(\\tau )^{-1} \\,h_{mn}dy^{m}dy^{n} \\; , \\\\ e^{2\\phi } &=& \\left ( \\sqrt {2}R_{0} \\right )^{d_{k}} \\tau ^{-1}B(\\tau )^{-d_{k}/2} \\; ,\\\\ H &=& 8R_{0}^{2} d_{k}^{-1/2} \\tau ^{-1}B(\\tau )^{-2} \\, d\\tau \\wedge \\mathcal {J} \\; , \\\\ B(\\tau ) &=& \\tau ^{2/\\sqrt {d_{k}}}\\,+\\, \\tau ^{-2/\\sqrt {d_{k}}} \\; . \\end {eqnarray}",
"\\begin {equation} F_{(2)} \\;=\\; \\aleph \\mathcal {J} \\;=\\; \\textstyle {\\frac {1}{2}} \\aleph \\mathcal {J}_{mn} dy^{m}\\wedge dy^{n} \\; . \\label {eq:F2Ansatz} \\end {equation}",
"\\begin {eqnarray} 0 &=& \\left (\\log R\\right )^{\\prime \\prime } \\,+\\, \\textstyle {\\frac {(d_{k}-4)\\aleph ^{2}}{8}} M\\eta ^{D}R^{d_{k}-4} \\; ,\\\\ 0 &=& \\left (\\log \\eta \\right )^{\\prime \\prime } \\,+\\, \\textstyle {\\frac {d_{k}\\aleph ^{2}}{8}} M\\eta ^{D}R^{d_{k}-4} \\,-\\, \\lambda \\, \\eta ^{-2}M^{2} \\; , \\\\ 0 &=& \\left (\\log M\\right )^{\\prime \\prime } \\,-\\, \\textstyle {\\frac {d_{k}\\aleph ^{2}}{8}} M\\eta ^{D}R^{d_{k}-4} \\,-\\, D\\lambda \\, \\eta ^{-2}M^{2} \\; , \\\\ 0 &=& \\left [ \\left (\\log M\\right )^{\\prime }\\right ]^{2} \\,-\\, D \\left [ \\left (\\log \\eta \\right )^{\\prime }\\right ]^{2} \\,-\\, d_{k}\\left [ \\left (\\log R\\right )^{\\prime }\\right ]^{2} \\,-\\, \\textstyle {\\frac {d_{k}\\aleph ^{2}}{4}}M\\eta ^{D}R^{d_{k}-4} \\,-\\, D\\lambda M^{2}\\eta ^{-2} \\; , \\end {eqnarray}",
"\\begin {equation} \\lambda \\;=\\; \\frac {\\aleph ^{2}}{4}\\left ( D+3 \\right ) \\; . \\end {equation}"
],
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"$ g _ { s } ^ { 2 } = e ^ { 2 \\phi } = M ^ { - 1 } \\eta ^ { D } R ^ { d _ { k } } $",
"$ g _ { s } ^ { 2 } = e ^ { 2 \\phi } = M ^ { - 1 } \\eta ^ { D } R ^ { d _ { k } } $",
"$ H = \\aleph R ^ { 4 } d t \\wedge J $",
"$ \\beta = 0 $",
"$ \\alpha = 1 $",
"$ \\tau $",
"$ R $",
"$ f $",
"$ S L ( 2 , R ) \\slash U ( 1 ) $",
"$ d _ { k } = 6 $",
"$ S L ( 2 , R ) $",
"$ d _ { k } = 4 $",
"$ R = R _ { 0 } e ^ { \\alpha t } $",
"$ M $",
"$ \\eta $",
"$ M = \\eta ^ { D + 2 } $",
"\\begin{equation*} \\alpha ^ { 2 } \\, - \\, D \\beta ^ { 2 } \\; = \\; \\frac { d _ { k } \\aleph ^ { 2 } R _ { 0 } ^ { 4 } } { 4 } \\; . \\end{equation*}",
"\\begin{align*} d s ^ { 2 } & = & d \\tau ^ { 2 } \\, - \\, d \\vec { x } _ { ( D ) } \\, - \\, 2 R _ { 0 } ^ { 2 } B ( \\tau ) ^ { - 1 } \\, h _ { m n } d y ^ { m } d y ^ { n } \\; , \\\\ e ^ { 2 \\phi } & = & { ( \\sqrt { 2 } R _ { 0 } ) } ^ { d _ { k } } \\tau ^ { - 1 } B ( \\tau ) ^ { - d _ { k } \\slash 2 } \\; , \\\\ H & = & 8 R _ { 0 } ^ { 2 } d _ { k } ^ { - 1 \\slash 2 } \\tau ^ { - 1 } B ( \\tau ) ^ { - 2 } \\, d \\tau \\wedge J \\; , \\\\ B ( \\tau ) & = & \\tau ^ { 2 \\slash \\sqrt { d _ { k } } } \\, + \\, \\tau ^ { - 2 \\slash \\sqrt { d _ { k } } } \\; . \\end{align*}",
"\\begin{equation*} F _ { ( 2 ) } \\; = \\; \\aleph J \\; = \\; \\frac { 1 } { 2 } \\aleph J _ { m n } d y ^ { m } \\wedge d y ^ { n } \\; . \\end{equation*}",
"\\begin{align*} 0 & = & { ( \\operatorname { l o g } R ) } ^ { \\prime \\prime } \\, + \\, \\frac { ( d _ { k } - 4 ) \\aleph ^ { 2 } } { 8 } M \\eta ^ { D } R ^ { d _ { k } - 4 } \\; , \\\\ 0 & = & { ( \\operatorname { l o g } \\eta ) } ^ { \\prime \\prime } \\, + \\, \\frac { d _ { k } \\aleph ^ { 2 } } { 8 } M \\eta ^ { D } R ^ { d _ { k } - 4 } \\, - \\, \\lambda \\, \\eta ^ { - 2 } M ^ { 2 } \\; , \\\\ 0 & = & { ( \\operatorname { l o g } M ) } ^ { \\prime \\prime } \\, - \\, \\frac { d _ { k } \\aleph ^ { 2 } } { 8 } M \\eta ^ { D } R ^ { d _ { k } - 4 } \\, - \\, D \\lambda \\, \\eta ^ { - 2 } M ^ { 2 } \\; , \\\\ 0 & = & { [ { ( \\operatorname { l o g } M ) } ^ { \\prime } ] } ^ { 2 } \\, - \\, D { [ { ( \\operatorname { l o g } \\eta ) } ^ { \\prime } ] } ^ { 2 } \\, - \\, d _ { k } { [ { ( \\operatorname { l o g } R ) } ^ { \\prime } ] } ^ { 2 } \\, - \\, \\frac { d _ { k } \\aleph ^ { 2 } } { 4 } M \\eta ^ { D } R ^ { d _ { k } - 4 } \\, - \\, D \\lambda M ^ { 2 } \\eta ^ { - 2 } \\; , \\end{align*}",
"\\begin{equation*} \\lambda \\; = \\; \\frac { \\aleph ^ { 2 } } { 4 } ( D + 3 ) \\; . \\end{equation*}"
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"$ \\mitg _ { \\mits } ^ { 2 } = \\mite ^ { 2 \\mitphi } = \\mitM ^ { - 1 } \\miteta ^ { \\mitD } \\mitR ^ { \\mitd _ { \\mitk } } $",
"$ \\mitg _ { \\mits } ^ { 2 } = \\mite ^ { 2 \\mitphi } = \\mitM ^ { - 1 } \\miteta ^ { \\mitD } \\mitR ^ { \\mitd _ { \\mitk } } $",
"$ \\mitH = \\aleph \\mitR ^ { 4 } \\mitd \\mitt \\wedge \\mscrJ $",
"$ \\mitbeta = 0 $",
"$ \\mitalpha = 1 $",
"$ \\mittau $",
"$ \\mitR $",
"$ \\mitf $",
"$ \\mitS \\mitL ( 2 , \\BbbR ) \\slash \\mitU ( 1 ) $",
"$ \\mitd _ { \\mitk } = 6 $",
"$ \\mitS \\mitL ( 2 , \\BbbR ) $",
"$ \\mitd _ { \\mitk } = 4 $",
"$ \\mitR = \\mitR _ { 0 } \\mite ^ { \\mitalpha \\mitt } $",
"$ \\mitM $",
"$ \\miteta $",
"$ \\mitM = \\miteta ^ { \\mitD + 2 } $",
"\\begin{equation*} \\mitalpha ^ { 2 } \\, - \\, \\mitD \\mitbeta ^ { 2 } \\; = \\; \\frac { \\mitd _ { \\mitk } \\aleph ^ { 2 } \\mitR _ { 0 } ^ { 4 } } { 4 } \\; . \\end{equation*}",
"\\begin{align*} \\displaystyle \\mitd \\mits ^ { 2 } & = & \\displaystyle \\mitd \\mittau ^ { 2 } \\, - \\, \\mitd \\vec { \\mitx } _ { ( \\mitD ) } \\, - \\, 2 \\mitR _ { 0 } ^ { 2 } \\mitB ( \\mittau ) ^ { - 1 } \\, \\Planckconst _ { \\mitm \\mitn } \\mitd \\mity ^ { \\mitm } \\mitd \\mity ^ { \\mitn } \\; , \\\\ \\displaystyle \\mite ^ { 2 \\mitphi } & = & \\displaystyle { \\left( \\sqrt { 2 } \\mitR _ { 0 } \\right) } ^ { \\mitd _ { \\mitk } } \\mittau ^ { - 1 } \\mitB ( \\mittau ) ^ { - \\mitd _ { \\mitk } \\slash 2 } \\; , \\\\ \\displaystyle \\mitH & = & \\displaystyle 8 \\mitR _ { 0 } ^ { 2 } \\mitd _ { \\mitk } ^ { - 1 \\slash 2 } \\mittau ^ { - 1 } \\mitB ( \\mittau ) ^ { - 2 } \\, \\mitd \\mittau \\wedge \\mscrJ \\; , \\\\ \\displaystyle \\mitB ( \\mittau ) & = & \\displaystyle \\mittau ^ { 2 \\slash \\sqrt { \\mitd _ { \\mitk } } } \\, + \\, \\mittau ^ { - 2 \\slash \\sqrt { \\mitd _ { \\mitk } } } \\; . \\end{align*}",
"\\begin{equation*} \\mitF _ { ( 2 ) } \\; = \\; \\aleph \\mscrJ \\; = \\; \\textstyle \\frac { 1 } { 2 } \\aleph \\mscrJ _ { \\mitm \\mitn } \\mitd \\mity ^ { \\mitm } \\wedge \\mitd \\mity ^ { \\mitn } \\; . \\end{equation*}",
"\\begin{align*} \\displaystyle 0 & = & \\displaystyle { \\left( \\operatorname { l o g } \\mitR \\right) } ^ { \\prime \\prime } \\, + \\, \\textstyle \\frac { ( \\mitd _ { \\mitk } - 4 ) \\aleph ^ { 2 } } { 8 } \\mitM \\miteta ^ { \\mitD } \\mitR ^ { \\mitd _ { \\mitk } - 4 } \\; , \\\\ \\displaystyle 0 & = & \\displaystyle { \\left( \\operatorname { l o g } \\miteta \\right) } ^ { \\prime \\prime } \\, + \\, \\textstyle \\frac { \\mitd _ { \\mitk } \\aleph ^ { 2 } } { 8 } \\mitM \\miteta ^ { \\mitD } \\mitR ^ { \\mitd _ { \\mitk } - 4 } \\, - \\, \\mitlambda \\, \\miteta ^ { - 2 } \\mitM ^ { 2 } \\; , \\\\ \\displaystyle 0 & = & \\displaystyle { \\left( \\operatorname { l o g } \\mitM \\right) } ^ { \\prime \\prime } \\, - \\, \\textstyle \\frac { \\mitd _ { \\mitk } \\aleph ^ { 2 } } { 8 } \\mitM \\miteta ^ { \\mitD } \\mitR ^ { \\mitd _ { \\mitk } - 4 } \\, - \\, \\mitD \\mitlambda \\, \\miteta ^ { - 2 } \\mitM ^ { 2 } \\; , \\\\ \\displaystyle 0 & = & \\displaystyle { \\left[ { \\left( \\operatorname { l o g } \\mitM \\right) } ^ { \\prime } \\right] } ^ { 2 } \\, - \\, \\mitD { \\left[ { \\left( \\operatorname { l o g } \\miteta \\right) } ^ { \\prime } \\right] } ^ { 2 } \\, - \\, \\mitd _ { \\mitk } { \\left[ { \\left( \\operatorname { l o g } \\mitR \\right) } ^ { \\prime } \\right] } ^ { 2 } \\, - \\, \\textstyle \\frac { \\mitd _ { \\mitk } \\aleph ^ { 2 } } { 4 } \\mitM \\miteta ^ { \\mitD } \\mitR ^ { \\mitd _ { \\mitk } - 4 } \\, - \\, \\mitD \\mitlambda \\mitM ^ { 2 } \\miteta ^ { - 2 } \\; , \\end{align*}",
"\\begin{equation*} \\mitlambda \\; = \\; \\frac { \\aleph ^ { 2 } } { 4 } \\left( \\mitD + 3 \\right) \\; . \\end{equation*}"
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0003060_page04
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{
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"$\\alpha \\neq 0$",
"$\\alpha =0$",
"$t$",
"$Mdt=d\\tau $",
"$\\tau \\rightarrow \\infty $",
"$d_{k}$",
"$R =\\eta ^{\\alpha }$",
"$M=\\eta ^{\\beta }$",
"$\\beta = D+2+\\alpha (d_{k}-4)$",
"$e^{\\phi }=\\eta ^{2\\alpha -1}$",
"$A$",
"$t$",
"$\\tau $",
"$\\eta = B\\tau $",
"\\begin {equation} \\left (\\log \\eta \\right )^{\\prime } \\;=\\; \\pm \\left [ \\frac {\\aleph ^{2}}{4} \\eta ^{2(D+1)}\\,+\\, \\frac {4\\alpha ^{2}}{D(D+3)+4} \\right ]^{1/2} \\; . \\end {equation}",
"\\begin {equation} \\eta \\;=\\; \\left ( A \\,\\mp \\, \\frac {(D+1)\\aleph ^{2}}{2}t \\right )^{-\\frac {1}{D+1}} \\; , \\end {equation}",
"\\begin {eqnarray} ds^{2} &=& d\\tau ^{2} \\,-\\, \\left ( \\frac {\\aleph ^{2}}{2}\\tau \\right )^{2}d\\Omega ^{2}_{\\lambda } \\,-\\, R_{0}^{2}\\,h_{mn}dy^{m}dy^{n} \\; ,\\\\ e^{\\phi } &=& 4R_{0}^{2}\\aleph ^{-2}\\, \\tau ^{-1} \\; . \\end {eqnarray}",
"\\begin {eqnarray} \\lambda &=& \\frac {\\aleph ^{2}}{16}\\left ( d_{k}(D+3)\\,+\\, (d_{k}-4)^{2} \\right ) \\; , \\\\ && \\\\ \\alpha &=& -2\\frac {d_{k}-4}{d_{k}(D+1)+(d_{k}-4)^{2}} \\; ,\\\\ && \\\\ \\beta &=& \\frac {d_{k}(D+1)(D+2)+D(d_{k}-4)^{2}}{d_{k}(D+1)+(d_{k}-4)^{2}} \\; , \\end {eqnarray}",
"\\begin {equation} \\left (\\log \\eta \\right )^{\\prime } \\;=\\; \\pm B\\, \\eta ^{\\beta -1} \\; , \\end {equation}",
"\\begin {equation} B^{2} \\;=\\; \\frac {\\aleph ^{2}}{16}\\, \\frac {\\left [ d_{k}(D+1) +(d_{k}-4)^{2} \\right ]^{2}}{d_{k}(D+1)+(D-1)(d_{k}-4)^{2}} \\; . \\end {equation}",
"\\begin {equation} \\eta \\;=\\; \\left [ A\\mp (\\beta -1)B\\, t\\right ]^{\\frac {1}{\\beta -1}} \\; , \\end {equation}",
"\\begin {eqnarray} ds^{2} &=& d\\tau ^{2} \\,-\\, B^{2}\\tau ^{2}d\\Omega ^{2}_{\\lambda } \\,-\\, \\left ( B\\tau \\right )^{2\\alpha }h_{mn}dy^{m}dy^{n} \\; ,\\\\ e^{\\phi } &=& \\left ( B\\tau \\right )^{2\\alpha -1} \\; . \\end {eqnarray}"
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"$ \\alpha = 0 $",
"$ t $",
"$ M d t = d \\tau $",
"$ \\tau \\rightarrow \\infty $",
"$ d _ { k } $",
"$ R = \\eta ^ { \\alpha } $",
"$ M = \\eta ^ { \\beta } $",
"$ \\beta = D + 2 + \\alpha ( d _ { k } - 4 ) $",
"$ e ^ { \\phi } = \\eta ^ { 2 \\alpha - 1 } $",
"$ A $",
"$ t $",
"$ \\tau $",
"$ \\eta = B \\tau $",
"\\begin{equation*} { ( \\operatorname { l o g } \\eta ) } ^ { \\prime } \\; = \\; \\pm { [ \\frac { \\aleph ^ { 2 } } { 4 } \\eta ^ { 2 ( D + 1 ) } \\, + \\, \\frac { 4 \\alpha ^ { 2 } } { D ( D + 3 ) + 4 } ] } ^ { 1 \\slash 2 } \\; . \\end{equation*}",
"\\begin{equation*} \\eta \\; = \\; { ( A \\, \\mp \\, \\frac { ( D + 1 ) \\aleph ^ { 2 } } { 2 } t ) } ^ { - \\frac { 1 } { D + 1 } } \\; , \\end{equation*}",
"\\begin{align*} d s ^ { 2 } & = & d \\tau ^ { 2 } \\, - \\, { ( \\frac { \\aleph ^ { 2 } } { 2 } \\tau ) } ^ { 2 } d \\Omega _ { \\lambda } ^ { 2 } \\, - \\, R _ { 0 } ^ { 2 } \\, h _ { m n } d y ^ { m } d y ^ { n } \\; , \\\\ e ^ { \\phi } & = & 4 R _ { 0 } ^ { 2 } \\aleph ^ { - 2 } \\, \\tau ^ { - 1 } \\; . \\end{align*}",
"\\begin{align*} \\lambda & = & \\frac { \\aleph ^ { 2 } } { 1 6 } ( d _ { k } ( D + 3 ) \\, + \\, ( d _ { k } - 4 ) ^ { 2 } ) \\; , \\\\ \\alpha & = & - 2 \\frac { d _ { k } - 4 } { d _ { k } ( D + 1 ) + ( d _ { k } - 4 ) ^ { 2 } } \\; , \\\\ \\beta & = & \\frac { d _ { k } ( D + 1 ) ( D + 2 ) + D ( d _ { k } - 4 ) ^ { 2 } } { d _ { k } ( D + 1 ) + ( d _ { k } - 4 ) ^ { 2 } } \\; , \\end{align*}",
"\\begin{equation*} { ( \\operatorname { l o g } \\eta ) } ^ { \\prime } \\; = \\; \\pm B \\, \\eta ^ { \\beta - 1 } \\; , \\end{equation*}",
"\\begin{equation*} B ^ { 2 } \\; = \\; \\frac { \\aleph ^ { 2 } } { 1 6 } \\, \\frac { { [ d _ { k } ( D + 1 ) + ( d _ { k } - 4 ) ^ { 2 } ] } ^ { 2 } } { d _ { k } ( D + 1 ) + ( D - 1 ) ( d _ { k } - 4 ) ^ { 2 } } \\; . \\end{equation*}",
"\\begin{equation*} \\eta \\; = \\; { [ A \\mp ( \\beta - 1 ) B \\, t ] } ^ { \\frac { 1 } { \\beta - 1 } } \\; , \\end{equation*}",
"\\begin{align*} d s ^ { 2 } & = & d \\tau ^ { 2 } \\, - \\, B ^ { 2 } \\tau ^ { 2 } d \\Omega _ { \\lambda } ^ { 2 } \\, - \\, { ( B \\tau ) } ^ { 2 \\alpha } h _ { m n } d y ^ { m } d y ^ { n } \\; , \\\\ e ^ { \\phi } & = & { ( B \\tau ) } ^ { 2 \\alpha - 1 } \\; . \\end{align*}"
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"$ \\mittau \\rightarrow \\infty $",
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"\\begin{equation*} { \\left( \\operatorname { l o g } \\miteta \\right) } ^ { \\prime } \\; = \\; \\pm { \\left[ \\frac { \\aleph ^ { 2 } } { 4 } \\miteta ^ { 2 ( \\mitD + 1 ) } \\, + \\, \\frac { 4 \\mitalpha ^ { 2 } } { \\mitD ( \\mitD + 3 ) + 4 } \\right] } ^ { 1 \\slash 2 } \\; . \\end{equation*}",
"\\begin{equation*} \\miteta \\; = \\; { \\left( \\mitA \\, \\mp \\, \\frac { ( \\mitD + 1 ) \\aleph ^ { 2 } } { 2 } \\mitt \\right) } ^ { - \\frac { 1 } { \\mitD + 1 } } \\; , \\end{equation*}",
"\\begin{align*} \\displaystyle \\mitd \\mits ^ { 2 } & = & \\displaystyle \\mitd \\mittau ^ { 2 } \\, - \\, { \\left( \\frac { \\aleph ^ { 2 } } { 2 } \\mittau \\right) } ^ { 2 } \\mitd \\mupOmega _ { \\mitlambda } ^ { 2 } \\, - \\, \\mitR _ { 0 } ^ { 2 } \\, \\Planckconst _ { \\mitm \\mitn } \\mitd \\mity ^ { \\mitm } \\mitd \\mity ^ { \\mitn } \\; , \\\\ \\displaystyle \\mite ^ { \\mitphi } & = & \\displaystyle 4 \\mitR _ { 0 } ^ { 2 } \\aleph ^ { - 2 } \\, \\mittau ^ { - 1 } \\; . \\end{align*}",
"\\begin{align*} \\displaystyle \\mitlambda & = & \\displaystyle \\frac { \\aleph ^ { 2 } } { 1 6 } \\left( \\mitd _ { \\mitk } ( \\mitD + 3 ) \\, + \\, ( \\mitd _ { \\mitk } - 4 ) ^ { 2 } \\right) \\; , \\\\ \\displaystyle \\mitalpha & = & \\displaystyle - 2 \\frac { \\mitd _ { \\mitk } - 4 } { \\mitd _ { \\mitk } ( \\mitD + 1 ) + ( \\mitd _ { \\mitk } - 4 ) ^ { 2 } } \\; , \\\\ \\displaystyle \\mitbeta & = & \\displaystyle \\frac { \\mitd _ { \\mitk } ( \\mitD + 1 ) ( \\mitD + 2 ) + \\mitD ( \\mitd _ { \\mitk } - 4 ) ^ { 2 } } { \\mitd _ { \\mitk } ( \\mitD + 1 ) + ( \\mitd _ { \\mitk } - 4 ) ^ { 2 } } \\; , \\end{align*}",
"\\begin{equation*} { \\left( \\operatorname { l o g } \\miteta \\right) } ^ { \\prime } \\; = \\; \\pm \\mitB \\, \\miteta ^ { \\mitbeta - 1 } \\; , \\end{equation*}",
"\\begin{equation*} \\mitB ^ { 2 } \\; = \\; \\frac { \\aleph ^ { 2 } } { 1 6 } \\, \\frac { { \\left[ \\mitd _ { \\mitk } ( \\mitD + 1 ) + ( \\mitd _ { \\mitk } - 4 ) ^ { 2 } \\right] } ^ { 2 } } { \\mitd _ { \\mitk } ( \\mitD + 1 ) + ( \\mitD - 1 ) ( \\mitd _ { \\mitk } - 4 ) ^ { 2 } } \\; . \\end{equation*}",
"\\begin{equation*} \\miteta \\; = \\; { \\left[ \\mitA \\mp ( \\mitbeta - 1 ) \\mitB \\, \\mitt \\right] } ^ { \\frac { 1 } { \\mitbeta - 1 } } \\; , \\end{equation*}",
"\\begin{align*} \\displaystyle \\mitd \\mits ^ { 2 } & = & \\displaystyle \\mitd \\mittau ^ { 2 } \\, - \\, \\mitB ^ { 2 } \\mittau ^ { 2 } \\mitd \\mupOmega _ { \\mitlambda } ^ { 2 } \\, - \\, { \\left( \\mitB \\mittau \\right) } ^ { 2 \\mitalpha } \\Planckconst _ { \\mitm \\mitn } \\mitd \\mity ^ { \\mitm } \\mitd \\mity ^ { \\mitn } \\; , \\\\ \\displaystyle \\mite ^ { \\mitphi } & = & \\displaystyle { \\left( \\mitB \\mittau \\right) } ^ { 2 \\mitalpha - 1 } \\; . \\end{align*}"
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0003060_page05
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0003079_page02
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0003079_page09
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0003087_page02
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0003087_page03
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"\\begin {equation}\\label {2bbadr} b(q,s,t)=D_\\soft (q,t)b(q,s)\\,, \\end {equation}",
"\\begin {equation}\\label {2dsoft} D_{\\soft }(q,t)=\\exp \\left \\{-e\\!\\!\\!\\intks \\frac 1{2\\omega _k} \\left ( \\frac {q\\cd a(k)}{q\\cd k}\\ee ^{-itk\\ecd q/E_q}- \\frac {q\\cd \\ad (k)}{q\\cd k}\\ee ^{itk\\ecd q/E_q} \\right )\\right \\}\\,, \\end {equation}",
"\\begin {equation}\\label {4de} u\\cd \\pa h^{-1}(x)=-ieh^{-1}(x)u\\cd A(x)\\,, \\end {equation}"
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"\\begin{equation*} \\mitu \\cdot \\mitpartial \\Planckconst ^ { - 1 } ( \\mitx ) = - \\miti \\mite \\Planckconst ^ { - 1 } ( \\mitx ) \\mitu \\cdot \\mitA ( \\mitx ) \\, , \\end{equation*}"
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0003087_page04
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"$u^\\mu =\\gamma (\\eta +v)^\\mu $",
"$\\eta $",
"$v=(0,\\vb )$",
"$\\gamma =(1-|\\vb |^2)^{-1/2}$",
"$K$",
"$\\chi $",
"$\\G ^\\mu =(\\eta +v)^\\mu (\\eta -v)\\cd \\pa -\\pa ^\\mu $",
"$D_\\soft (q,t)$",
"$ V^\\mu =(\\eta +v)^\\mu (\\eta -v)\\cd k-k^\\mu $",
"$\\G ^\\mu $",
"\\begin {equation}\\label {4sol} h^{-1}(x)\\psi (x)=\\ee ^{-ieK(x)}\\ee ^{-ie\\chi (x)}\\psi (x)\\,. \\end {equation}",
"\\begin {equation}\\label {4min} \\chi (x)=\\frac {\\G \\cd A}{\\G \\cd \\pa }\\,, \\end {equation}",
"\\begin {equation}\\label {2bgood} b(q,s,v,t):=\\intx \\frac 1{\\sqrt {2E_{\\smash {q}}}}u^{\\dag s}(q)\\ee ^{-ie\\chi (x)} \\psi (x)\\ee ^{iq\\ecd x}\\,. \\end {equation}",
"\\begin {equation}\\label {3bexp} b(q,s,t,v)\\to h^{-1}_\\soft (q,t,v)D_\\soft (q,t)b(q,s)\\,, \\end {equation}",
"\\begin {equation}\\label {3ddress} h^{-1}_{\\soft }(q,t,v)=\\exp \\left \\{e\\!\\!\\!\\intks \\frac 1{2\\omega _k} \\left ( \\frac {V\\cd a(k)}{V\\cd k}\\ee ^{-itk\\ecd q/E_q}- \\frac {V\\cd \\ad (k)}{V\\cd k}\\ee ^{itk\\ecd q/E_q} \\right )\\right \\}\\,, \\end {equation}",
"\\begin {eqnarray} h^{-1}_\\soft (q,t,v)D_{\\soft }(q,t)&=&\\exp \\Bigg (e\\!\\!\\intks \\frac 1{2\\omega _k} \\bigg [\\bigg ( \\frac {V\\cd a(k)}{V\\cd k}-\\frac {q\\cd a(k)}{q\\cd k}\\bigg )\\ee ^{-itk\\ecd q/E_q}\\\\ &&\\qquad \\qquad \\qquad - \\bigg (\\frac {V\\cd \\ad (k)}{V\\cd k}-\\frac {q\\cd \\ad (k)}{q\\cd k}\\bigg )\\ee ^{itk\\ecd q/E_q} \\bigg ]\\Bigg )\\,. \\end {eqnarray}"
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"$ K $",
"$ \\chi $",
"$ G ^ { \\mu } = ( \\eta + v ) ^ { \\mu } ( \\eta - v ) \\cdot \\partial - \\partial ^ { \\mu } $",
"$ D _ { s o f t } ( q , t ) $",
"$ V ^ { \\mu } = ( \\eta + v ) ^ { \\mu } ( \\eta - v ) \\cdot k - k ^ { \\mu } $",
"$ G ^ { \\mu } $",
"\\begin{equation*} h ^ { - 1 } ( x ) \\psi ( x ) = e ^ { - i e K ( x ) } e ^ { - i e \\chi ( x ) } \\psi ( x ) \\, . \\end{equation*}",
"\\begin{equation*} \\chi ( x ) = \\frac { G \\cdot A } { G \\cdot \\partial } \\, , \\end{equation*}",
"\\begin{equation*} b ( q , s , v , t ) : = \\intx \\frac { 1 } { \\sqrt { 2 E _ { q } } } u ^ { \\dagger s } ( q ) e ^ { - i e \\chi ( x ) } \\psi ( x ) e ^ { i q \\cdot x } \\, . \\end{equation*}",
"\\begin{equation*} b ( q , s , t , v ) \\rightarrow h _ { s o f t } ^ { - 1 } ( q , t , v ) D _ { s o f t } ( q , t ) b ( q , s ) \\, , \\end{equation*}",
"\\begin{equation*} h _ { s o f t } ^ { - 1 } ( q , t , v ) = \\operatorname { e x p } \\{ e \\! \\! \\! \\int _ { s o f t } \\! \\frac { d ^ { 3 } k } { ( 2 \\pi ) ^ { 3 } } \\; \\frac { 1 } { 2 \\omega _ { k } } ( \\frac { V \\cdot a ( k ) } { V \\cdot k } e ^ { - i t k \\cdot q \\slash E _ { q } } - \\frac { V \\cdot a ^ { \\dagger } ( k ) } { V \\cdot k } e ^ { i t k \\cdot q \\slash E _ { q } } ) \\} \\, , \\end{equation*}",
"\\begin{align*} h _ { s o f t } ^ { - 1 } ( q , t , v ) D _ { s o f t } ( q , t ) & = & \\operatorname { e x p } ( e \\! \\! \\int _ { s o f t } \\! \\frac { d ^ { 3 } k } { ( 2 \\pi ) ^ { 3 } } \\; \\frac { 1 } { 2 \\omega _ { k } } [ ( \\frac { V \\cdot a ( k ) } { V \\cdot k } - \\frac { q \\cdot a ( k ) } { q \\cdot k } ) e ^ { - i t k \\cdot q \\slash E _ { q } } \\\\ & & \\qquad \\qquad \\qquad - ( \\frac { V \\cdot a ^ { \\dagger } ( k ) } { V \\cdot k } - \\frac { q \\cdot a ^ { \\dagger } ( k ) } { q \\cdot k } ) e ^ { i t k \\cdot q \\slash E _ { q } } ] ) \\, . \\end{align*}"
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"$ \\mitv=(0,\\textstyle \\mitv ) $",
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"$ \\mitD _ { \\mathrm { s o f t } } ( \\mitq , \\mitt ) $",
"$ \\mitV ^ { \\mitmu } = ( \\miteta + \\mitv ) ^ { \\mitmu } ( \\miteta - \\mitv ) \\cdot \\mitk - \\mitk ^ { \\mitmu } $",
"$ \\mitG ^ { \\mitmu } $",
"\\begin{equation*} \\Planckconst ^ { - 1 } ( \\mitx ) \\mitpsi ( \\mitx ) = \\mathrm { e } ^ { - \\miti \\mite \\mitK ( \\mitx ) } \\mathrm { e } ^ { - \\miti \\mite \\mitchi ( \\mitx ) } \\mitpsi ( \\mitx ) \\, . \\end{equation*}",
"\\begin{equation*} \\mitchi ( \\mitx ) = \\frac { \\mitG \\cdot \\mitA } { \\mitG \\cdot \\mitpartial } \\, , \\end{equation*}",
"\\begin{equation*} \\mitb ( \\mitq , \\mits , \\mitv , \\mitt ) : = \\intx \\frac { 1 } { \\sqrt { 2 \\mitE _ { \\mitq } } } \\mitu ^ { \\dagger \\mits } ( \\mitq ) \\mathrm { e } ^ { - \\miti \\mite \\mitchi ( \\mitx ) } \\mitpsi ( \\mitx ) \\mathrm { e } ^ { \\miti \\mitq \\cdot \\mitx } \\, . \\end{equation*}",
"\\begin{equation*} \\mitb ( \\mitq , \\mits , \\mitt , \\mitv ) \\rightarrow \\Planckconst _ { \\mathrm { s o f t } } ^ { - 1 } ( \\mitq , \\mitt , \\mitv ) \\mitD _ { \\mathrm { s o f t } } ( \\mitq , \\mitt ) \\mitb ( \\mitq , \\mits ) \\, , \\end{equation*}",
"\\begin{equation*} \\Planckconst _ { \\mathrm { s o f t } } ^ { - 1 } ( \\mitq , \\mitt , \\mitv ) = \\operatorname { e x p } \\left\\{ \\mite \\! \\! \\! \\int \\limits _ { \\mathrm { s o f t } } \\! \\frac { \\mitd ^ { 3 } \\mitk } { ( 2 \\mitpi ) ^ { 3 } } \\; \\frac { 1 } { 2 \\mitomega _ { \\mitk } } \\left( \\frac { \\mitV \\cdot \\mita ( \\mitk ) } { \\mitV \\cdot \\mitk } \\mathrm { e } ^ { - \\miti \\mitt \\mitk \\cdot \\mitq \\slash \\mitE _ { \\mitq } } - \\frac { \\mitV \\cdot \\mita ^ { \\dagger } ( \\mitk ) } { \\mitV \\cdot \\mitk } \\mathrm { e } ^ { \\miti \\mitt \\mitk \\cdot \\mitq \\slash \\mitE _ { \\mitq } } \\right) \\right\\} \\, , \\end{equation*}",
"\\begin{align*} \\displaystyle \\Planckconst _ { \\mathrm { s o f t } } ^ { - 1 } ( \\mitq , \\mitt , \\mitv ) \\mitD _ { \\mathrm { s o f t } } ( \\mitq , \\mitt ) & = & \\displaystyle \\operatorname { e x p } \\Bigg ( \\mite \\! \\! \\int \\limits _ { \\mathrm { s o f t } } \\! \\frac { \\mitd ^ { 3 } \\mitk } { ( 2 \\mitpi ) ^ { 3 } } \\; \\frac { 1 } { 2 \\mitomega _ { \\mitk } } \\bigg [ \\bigg ( \\frac { \\mitV \\cdot \\mita ( \\mitk ) } { \\mitV \\cdot \\mitk } - \\frac { \\mitq \\cdot \\mita ( \\mitk ) } { \\mitq \\cdot \\mitk } \\bigg ) \\mathrm { e } ^ { - \\miti \\mitt \\mitk \\cdot \\mitq \\slash \\mitE _ { \\mitq } } \\\\ & & \\displaystyle \\qquad \\qquad \\qquad - \\bigg ( \\frac { \\mitV \\cdot \\mita ^ { \\dagger } ( \\mitk ) } { \\mitV \\cdot \\mitk } - \\frac { \\mitq \\cdot \\mita ^ { \\dagger } ( \\mitk ) } { \\mitq \\cdot \\mitk } \\bigg ) \\mathrm { e } ^ { \\miti \\mitt \\mitk \\cdot \\mitq \\slash \\mitE _ { \\mitq } } \\bigg ] \\Bigg ) \\, . \\end{align*}"
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0003087_page05
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{
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"$k$",
"$q^\\mu =m\\gamma (\\eta +v)^\\mu $",
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"\\begin {eqnarray} \\frac {V^\\mu }{V\\cd k}-\\frac {q^\\mu }{q\\cd k}&=& \\frac {(\\eta +v)^\\mu (\\eta -v)\\cd k-k^\\mu }{(\\eta +v)\\cd k(\\eta -v)\\cd k}-\\frac {q^\\mu }{q\\cd k}\\\\ &=&\\frac {(\\eta +v)^\\mu }{(\\eta +v)\\cd k}-\\frac {q^\\mu }{q\\cd k}-\\frac {k^\\mu }{V\\cd k}\\,. \\end {eqnarray}",
"\\begin {equation} -e\\intk \\frac 1{2\\omega _k} \\bigg ( \\frac {k\\cd a(k)}{V\\cd k}\\ee ^{-it\\omega _k} -\\frac {k\\cd a^\\dag (k)}{V\\cd k} \\ee ^{it\\omega _k}\\bigg )\\,, \\end {equation}",
"\\begin {eqnarray}\\Aas _\\mu (x)&=&\\exp \\!\\!\\left (i\\!\\!\\int _{-\\infty }^t\\!\\!\\!\\!\\!\\! d\\tau \\,\\Haas (\\tau )\\right )\\Afree _\\mu (x) \\exp \\!\\!\\left (-i\\!\\!\\int _{-\\infty }^t\\!\\!\\!\\!\\!\\!d\\tau \\, \\Haas (\\tau )\\right )\\\\ &=&\\Afree _\\mu (x)-e\\int _{-\\infty }^t\\!\\!d\\tau d^3y\\,D(\\tau -t, \\yb -\\xb )J^\\as _\\mu (\\tau ,\\yb )\\,, \\end {eqnarray}",
"\\begin {equation}\\label {2jas} J^\\mu _{\\as }(t,\\xb )=\\intp \\frac {p^\\mu }{E_p}\\rho (p)\\delta ^3 \\bigl (\\xb -t\\pb /E_p\\bigr )\\,. \\end {equation}"
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"\\begin{align*} \\frac { V ^ { \\mu } } { V \\cdot k } - \\frac { q ^ { \\mu } } { q \\cdot k } & = & \\frac { ( \\eta + v ) ^ { \\mu } ( \\eta - v ) \\cdot k - k ^ { \\mu } } { ( \\eta + v ) \\cdot k ( \\eta - v ) \\cdot k } - \\frac { q ^ { \\mu } } { q \\cdot k } \\\\ & = & \\frac { ( \\eta + v ) ^ { \\mu } } { ( \\eta + v ) \\cdot k } - \\frac { q ^ { \\mu } } { q \\cdot k } - \\frac { k ^ { \\mu } } { V \\cdot k } \\, . \\end{align*}",
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"\\begin{align*} A _ { \\mu } ^ { a s } ( x ) & = & \\operatorname { e x p } \\! \\! ( i \\! \\! \\int _ { - \\infty } ^ { t } \\! \\! \\! \\! \\! \\! d \\tau \\, H _ { i n t } ^ { a s } ( \\tau ) ) A _ { \\mu } ^ { f } ( x ) \\operatorname { e x p } \\! \\! ( - i \\! \\! \\int _ { - \\infty } ^ { t } \\! \\! \\! \\! \\! \\! d \\tau \\, H _ { i n t } ^ { a s } ( \\tau ) ) \\\\ & = & Af\\mu(x)-et-\\inftyd\\taud3yD(\\tau-t,y -x )Jas\\mu(\\tau,y ), \\end{align*}",
"\\begin{equation*} J _ { a s } ^ { \\mu } ( t , x ) = \\int \\! \\frac { d ^ { 3 } p } { ( 2 \\pi ) ^ { 3 } } \\; \\frac { p ^ { \\mu } } { E _ { p } } \\rho ( p ) \\delta ^ { 3 } ( x - t p \\slash E _ { p } ) \\, . \\end{equation*}"
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"\\begin{equation*} - \\mite \\int \\! \\frac { \\mitd ^ { 3 } \\mitk } { ( 2 \\mitpi ) ^ { 3 } } \\; \\frac { 1 } { 2 \\mitomega _ { \\mitk } } \\bigg ( \\frac { \\mitk \\cdot \\mita ( \\mitk ) } { \\mitV \\cdot \\mitk } \\mathrm { e } ^ { - \\miti \\mitt \\mitomega _ { \\mitk } } - \\frac { \\mitk \\cdot \\mita ^ { \\dagger } ( \\mitk ) } { \\mitV \\cdot \\mitk } \\mathrm { e } ^ { \\miti \\mitt \\mitomega _ { \\mitk } } \\bigg ) \\, , \\end{equation*}",
"\\begin{align*} \\displaystyle \\mitA _ { \\mitmu } ^ { \\mathrm { a s } } ( \\mitx ) & = & \\displaystyle \\operatorname { e x p } \\! \\! \\left( \\miti \\! \\! \\int _ { - \\infty } ^ { \\mitt } \\! \\! \\! \\! \\! \\! \\mitd \\mittau \\, \\mitH _ { \\mathrm { i n t } } ^ { \\mathrm { a s } } ( \\mittau ) \\right) \\mitA _ { \\mitmu } ^ { \\mathrm { f } } ( \\mitx ) \\operatorname { e x p } \\! \\! \\left( - \\miti \\! \\! \\int _ { - \\infty } ^ { \\mitt } \\! \\! \\! \\! \\! \\! \\mitd \\mittau \\, \\mitH _ { \\mathrm { i n t } } ^ { \\mathrm { a s } } ( \\mittau ) \\right) \\\\ & = & \\mitA ^ { f } _ { \\mitmu } ( \\mitx ) - \\mite \\mitt-\\infty\\mitd\\mittau\\mitd3\\mity\\mitD(\\mittau-\\mitt,\\displaystyle \\mity -\\displaystyle \\mitx )\\mitJas\\mitmu(\\mittau,\\displaystyle \\mity ), \\end{align*}",
"\\begin{equation*} \\mitJ _ { \\mathrm { a s } } ^ { \\mitmu } ( \\mitt , \\mitx ) = \\int \\! \\frac { \\mitd ^ { 3 } \\mitp } { ( 2 \\mitpi ) ^ { 3 } } \\; \\frac { \\mitp ^ { \\mitmu } } { \\mitE _ { \\mitp } } \\mitrho ( \\mitp ) \\mitdelta ^ { 3 } \\big ( \\mitx - \\mitt \\mitp \\slash \\mitE _ { \\mitp } \\big ) \\, . \\end{equation*}"
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0003087_page06
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0003126_page01
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0003126_page02
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0003126_page07
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{
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"$P_{j}^{-}$",
"$P_{j}^{+}$",
"$F_{pr} $",
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"\\begin {equation} P_{j}^{+}\\circ S^{p}T^{r}=\\alpha ^{j}S^{p}T^{r} \\end {equation}",
"\\begin {equation} \\{S^{p},T^{r}\\}=\\left ( \\sum _{k_{p}=0}^{r}...\\sum _{k_{2}=0}^{k_{3}}\\sum _{k_{1}=0}^{k_{2}}\\alpha ^{k_{1}}\\alpha ^{k_{2}}...\\alpha ^{k_{p}}\\right ) S^{p}T^{r}. \\label {gaa26} \\end {equation}",
"\\begin {equation} \\{S^{p},T^{r}\\}=\\left ( \\sum _{k_{r}=0}^{p}...\\sum _{k_{2}=0}^{k_{3}}\\sum _{k_{1}=0}^{k_{2}}\\alpha ^{k_{1}}\\alpha ^{k_{2}}...\\alpha ^{k_{r}}\\right ) S^{p}T^{r}. \\label {gab26} \\end {equation}",
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"\\begin {equation} Q_{pr}=S^{p}T^{r},\\qquad p,r=1,2,...,n \\label {ga27} \\end {equation}",
"\\begin {equation} \\label {gaa27} Q_{rs}Q_{pq}=\\alpha ^{s\\cdot p}Q_{kl};\\ k=\\text {mod}(r+p-1,n)+1,\\ l=\\text {mod }(s+q-1,n)+1, \\end {equation}",
"\\begin {equation} \\label {ga28} Q_{rs}Q_{pq}=\\alpha ^{s\\cdot p-r\\cdot q}Q_{pq}Q_{rs}, \\end {equation}",
"\\begin {equation} \\label {ga29} \\left ( Q_{rs}\\right ) ^n=(-1)^{(n-1)r\\cdot s}I, \\end {equation}",
"\\begin {equation} \\label {ga30} Q_{rs}^{\\dagger }Q_{rs}=Q_{rs}Q_{rs}^{\\dagger }=I, \\end {equation}",
"\\begin {equation} \\label {gaa30} Q_{rs}^{\\dagger }=\\alpha ^{r\\cdot s}Q_{kl};\\qquad k=n-r,\\qquad l=n-s, \\end {equation}",
"\\begin {equation} \\label {ga31} \\det Q_{rs}=(-1)^{(n-1)(r+s)} \\end {equation}",
"\\begin {equation} \\label {ga32} \\text {\\textrm {Tr }}Q_{rs}=0. \\end {equation}",
"\\begin {equation} A=\\sum _{k,l=1}^{n}a_{kl}Q_{kl},\\qquad a_{kl}=\\frac {1}{n}\\text {\\textrm {Tr}} (Q_{kl}^{\\dagger }A). \\label {ga33} \\end {equation}"
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"\\begin{equation*} \\{ \\mitS ^ { \\mitp } , \\mitT ^ { \\mitr } \\} = \\left( \\sum _ { \\mitk _ { \\mitr } = 0 } ^ { \\mitp } . . . \\sum _ { \\mitk _ { 2 } = 0 } ^ { \\mitk _ { 3 } } \\sum _ { \\mitk _ { 1 } = 0 } ^ { \\mitk _ { 2 } } \\mitalpha ^ { \\mitk _ { 1 } } \\mitalpha ^ { \\mitk _ { 2 } } . . . \\mitalpha ^ { \\mitk _ { \\mitr } } \\right) \\mitS ^ { \\mitp } \\mitT ^ { \\mitr } . \\end{equation*}",
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"\\begin{equation*} \\mitQ _ { \\mitr \\mits } ^ { \\dagger } \\mitQ _ { \\mitr \\mits } = \\mitQ _ { \\mitr \\mits } \\mitQ _ { \\mitr \\mits } ^ { \\dagger } = \\mitI , \\end{equation*}",
"\\begin{equation*} \\mitQ _ { \\mitr \\mits } ^ { \\dagger } = \\mitalpha ^ { \\mitr \\cdot \\mits } \\mitQ _ { \\mitk \\mitl } ; \\qquad \\mitk = \\mitn - \\mitr , \\qquad \\mitl = \\mitn - \\mits , \\end{equation*}",
"\\begin{equation*} \\operatorname { d e t } \\mitQ _ { \\mitr \\mits } = ( - 1 ) ^ { ( \\mitn - 1 ) ( \\mitr + \\mits ) } \\end{equation*}",
"\\begin{equation*} \\mathrm { T r } \\hspace{3.33pt} \\mitQ _ { \\mitr \\mits } = 0 . \\end{equation*}",
"\\begin{equation*} \\mitA = \\sum _ { \\mitk , \\mitl = 1 } ^ { \\mitn } \\mita _ { \\mitk \\mitl } \\mitQ _ { \\mitk \\mitl } , \\qquad \\mita _ { \\mitk \\mitl } = \\frac { 1 } { \\mitn } \\mathrm { T r } ( \\mitQ _ { \\mitk \\mitl } ^ { \\dagger } \\mitA ) . \\end{equation*}"
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0003126_page08
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{
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"$Q_{kl}$",
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"\\begin {equation} \\label {ga34} \\{Q_\\lambda ^p,Q_\\mu ^r\\}=\\{Q_\\mu ^p,Q_\\nu ^r\\}=\\{Q_\\nu ^p,Q_\\lambda ^r\\}=0;\\qquad 0<p,r,\\qquad p+r=n \\end {equation}",
"\\begin {equation} \\label {ga35} \\{Q_\\lambda ^p,Q_\\mu ^r,Q_\\nu ^s\\}=0;\\qquad 0<p,r,s,\\qquad p+r+s=n. \\end {equation}",
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"\\begin {equation} YX=\\alpha XY,\\qquad ZX=\\alpha XZ,\\qquad ZY=\\alpha YZ. \\label {gaa35} \\end {equation}",
"\\begin {equation} \\{X^{p},Y^{r},Z^{s}\\} \\label {ga36} \\end {equation}",
"\\begin {equation} =\\sum _{j_{p}=0}^{r+s}...\\sum _{j_{2}=0}^{j_{3}} \\sum _{j_{1}=0}^{j_{2}}P_{j_{1}}^{+}\\circ P_{j_{2}}^{+}...P_{j_{p}}^{+}\\circ X^{p}\\sum _{k_{p}=0}^{s}...\\sum _{k_{2}=0}^{k_{3}} \\sum _{k_{1}=0}^{k_{2}}P_{k_{1}}^{+}\\circ P_{k_{2}}^{+}...P_{k_{r}}^{+}\\circ Y^{r}Z^{s}, \\end {equation}",
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"\\begin{equation*} \\{ \\mitQ _ { \\mitlambda } ^ { \\mitp } , \\mitQ _ { \\mitmu } ^ { \\mitr } , \\mitQ _ { \\mitnu } ^ { \\mits } \\} = 0 ; \\qquad 0 < \\mitp , \\mitr , \\mits , \\qquad \\mitp + \\mitr + \\mits = \\mitn . \\end{equation*}",
"\\begin{equation*} \\mitX = \\mitQ _ { 1 \\mitn } = \\mitS , \\qquad \\mitY = \\mitQ _ { 1 1 } , \\qquad \\mitZ = \\mitQ _ { \\mitn 1 } = \\mitT , \\end{equation*}",
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0003126_page09
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"$Q_{\\lambda }$",
"$a,b,c$",
"$n>2$",
"$Q_{rs}$",
"$X,Y,Z$",
"$XY=YX,$",
"$Y\\sim X^{p},2\\leq p<n$",
"$Q_{rs}$",
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"$p+r+s\\leq n$",
"$X^{p}Y^{r}Z^{s}\\sim I$",
"$\\beta _{k}$",
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"$p<n$",
"$\\beta _{k}^{p}=1$",
"$n$",
"$X,Y,Z$",
"$X^{\\prime },Y^{\\prime },Z^{\\prime }$",
"$X,Y,Z$",
"$\\beta _{k}\\neq \\beta _{k}^{\\prime }.$",
"$N$",
"$2\\leq n\\leq 10$",
"$N=4$",
"$U_{l},l=1,2,3$",
"$n^{2}\\times n^{2}$",
"\\begin {equation} (aQ_{\\lambda }+bQ_{\\mu }+cQ_{\\nu })^{n}=(a^{n}+b^{n}+c^{n})I. \\label {ga37} \\end {equation}",
"\\begin {equation} XY=\\beta _{3}YX\\qquad YZ=\\beta _{1}ZY\\qquad ZX=\\beta _{2}XZ \\label {gaa37} \\end {equation}",
"\\begin {equation} \\begin {array}{cccccccccccccccccccc} n: & \\text {{\\footnotesize 2}} & \\text {{\\footnotesize 3}} & \\text { {\\footnotesize 4}} & \\text {{\\footnotesize 5}} & \\text {{\\footnotesize 6}} & \\text {{\\footnotesize 7}} & \\text {{\\footnotesize 8}} & \\text {{\\footnotesize 9} } & \\text {{\\footnotesize 10}} & \\text {{\\footnotesize 11}} & \\text { {\\footnotesize 12}} & \\text {{\\footnotesize 13}} & \\text {{\\footnotesize 14}} & \\text {{\\footnotesize 15}} & \\text {{\\footnotesize 16}} & \\text { {\\footnotesize 17}} & \\text {{\\footnotesize 18}} & \\text {{\\footnotesize 19}} & \\text {{\\footnotesize 20}} \\\\ \\#3: & \\text {{\\footnotesize 1}} & \\text {{\\footnotesize 1}} & \\text { {\\footnotesize 1}} & \\text {{\\footnotesize 4}} & \\text {{\\footnotesize 1}} & \\text {{\\footnotesize 9}} & \\text {{\\footnotesize 4}} & \\text {{\\footnotesize 9} } & \\text {{\\footnotesize 4}} & \\text {{\\footnotesize 25}} & \\text { {\\footnotesize 4}} & \\text {{\\footnotesize 36}} & \\text {{\\footnotesize 9}} & \\text {{\\footnotesize 16}} & \\text {{\\footnotesize 16}} & \\text {{\\footnotesize 64}} & \\text {{\\footnotesize 9}} & \\text {{\\footnotesize 81}} & \\text { {\\footnotesize 16}} \\end {array} \\end {equation}",
"\\begin {equation} \\left ( \\sum _{\\lambda =0}^{N-1}a_{\\lambda }Q_{\\lambda }\\right ) ^{n}=\\sum _{\\lambda =0}^{N-1}a_{\\lambda }^{n}. \\label {ga38} \\end {equation}",
"\\begin {equation} Q_{0}=I\\otimes T=\\left ( \\begin {array}{cccccc} I & \\qquad & \\qquad & \\qquad & \\qquad & \\\\ \\qquad & \\alpha I & & & & \\\\ & & \\alpha ^{2}I & & & \\\\ & & & \\cdot & & \\\\ & & & & \\cdot & \\\\ & & & & & \\alpha ^{n-1}I \\end {array} \\right ) , \\label {ga39} \\end {equation}",
"\\begin {equation} Q_{l}=U_{l}\\otimes S=\\left ( \\begin {array}{cccccc} 0 & \\qquad & \\qquad & \\qquad & \\qquad & U_{l} \\\\ U_{l} & & & & & \\qquad \\\\ \\qquad & U_{l} & & & & \\\\ & & \\cdot & & & \\\\ & & & \\cdot & & \\\\ & & & & U_{l} & 0 \\end {array} \\right ) \\label {gb39} \\end {equation}"
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"$ X Y = Y X , $",
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"$ p + r + s \\leq n $",
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"$ \\beta _ { k } $",
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"$ N $",
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"$ N = 4 $",
"$ U _ { l } , l = 1 , 2 , 3 $",
"$ n ^ { 2 } \\times n ^ { 2 } $",
"\\begin{equation*} ( a Q _ { \\lambda } + b Q _ { \\mu } + c Q _ { \\nu } ) ^ { n } = ( a ^ { n } + b ^ { n } + c ^ { n } ) I . \\end{equation*}",
"\\begin{equation*} X Y = \\beta _ { 3 } Y X \\qquad Y Z = \\beta _ { 1 } Z Y \\qquad Z X = \\beta _ { 2 } X Z \\end{equation*}",
"\\begin{align*} \\begin{array}{cccccccccccccccccccc} n : & 2 & 3 & ~ 4 & 5 & 6 & 7 & 8 & 9 ~ & 1 0 & 1 1 & ~ 1 2 & 1 3 & 1 4 & 1 5 & 1 6 & ~ 1 7 & 1 8 & 1 9 & 2 0 \\\\ \\# 3 : & 1 & 1 & ~ 1 & 4 & 1 & 9 & 4 & 9 ~ & 4 & 2 5 & ~ 4 & 3 6 & 9 & 1 6 & 1 6 & 6 4 & 9 & 8 1 & ~ 1 6 \\end{array} \\end{align*}",
"\\begin{equation*} { ( \\sum _ { \\lambda = 0 } ^ { N - 1 } a _ { \\lambda } Q _ { \\lambda } ) } ^ { n } = \\sum _ { \\lambda = 0 } ^ { N - 1 } a _ { \\lambda } ^ { n } . \\end{equation*}",
"\\begin{align*} Q _ { 0 } = I \\otimes T = ( \\begin{array}{cccccc} I & \\qquad & \\qquad & \\qquad & \\qquad & \\\\ \\qquad & \\alpha I & & & & \\\\ & & \\alpha ^ { 2 } I & & & \\\\ & & & \\cdot & & \\\\ & & & & \\cdot & \\\\ & & & & & \\alpha ^ { n - 1 } I \\end{array} ) , \\end{align*}",
"\\begin{align*} Q _ { l } = U _ { l } \\otimes S = ( \\begin{array}{cccccc} 0 & \\qquad & \\qquad & \\qquad & \\qquad & U _ { l } \\\\ U _ { l } & & & & & \\qquad \\\\ \\qquad & U _ { l } & & & & \\\\ & & \\cdot & & & \\\\ & & & \\cdot & & \\\\ & & & & U _ { l } & 0 \\end{array} ) \\end{align*}"
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"$ \\mitX , \\mitY , \\mitZ $",
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"$ \\mitN $",
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"$ \\mitN = 4 $",
"$ \\mitU _ { \\mitl } , \\mitl = 1 , 2 , 3 $",
"$ \\mitn ^ { 2 } \\times \\mitn ^ { 2 } $",
"\\begin{equation*} ( \\mita \\mitQ _ { \\mitlambda } + \\mitb \\mitQ _ { \\mitmu } + \\mitc \\mitQ _ { \\mitnu } ) ^ { \\mitn } = ( \\mita ^ { \\mitn } + \\mitb ^ { \\mitn } + \\mitc ^ { \\mitn } ) \\mitI . \\end{equation*}",
"\\begin{equation*} \\mitX \\mitY = \\mitbeta _ { 3 } \\mitY \\mitX \\qquad \\mitY \\mitZ = \\mitbeta _ { 1 } \\mitZ \\mitY \\qquad \\mitZ \\mitX = \\mitbeta _ { 2 } \\mitX \\mitZ \\end{equation*}",
"\\begin{align*} \\begin{array}{cccccccccccccccccccc} \\mitn : & 2 & 3 & ~ 4 & 5 & 6 & 7 & 8 & 9 ~ & 1 0 & 1 1 & ~ 1 2 & 1 3 & 1 4 & 1 5 & 1 6 & ~ 1 7 & 1 8 & 1 9 & 2 0 \\\\ \\# 3 : & 1 & 1 & ~ 1 & 4 & 1 & 9 & 4 & 9 ~ & 4 & 2 5 & ~ 4 & 3 6 & 9 & 1 6 & 1 6 & 6 4 & 9 & 8 1 & ~ 1 6 \\end{array} \\end{align*}",
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"\\begin{align*} \\mitQ _ { \\mitl } = \\mitU _ { \\mitl } \\otimes \\mitS = \\left( \\begin{array}{cccccc} 0 & \\qquad & \\qquad & \\qquad & \\qquad & \\mitU _ { \\mitl } \\\\ \\mitU _ { \\mitl } & & & & & \\qquad \\\\ \\qquad & \\mitU _ { \\mitl } & & & & \\\\ & & \\cdot & & & \\\\ & & & \\cdot & & \\\\ & & & & \\mitU _ { \\mitl } & 0 \\end{array} \\right) \\end{align*}"
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0003126_page10
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{
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"\\begin {equation} =\\sum _{j_{p_{N}}=0}^{n-p_{N}}...\\sum _{j_{1}=0}^{j_{2}} \\sum _{j_{0}=0}^{j_{1}}P_{j_{0}}^{-}\\circ P_{j_{1}}^{-}...P_{j_{p_{N}}}^{-}\\circ \\{(U_{0}\\otimes S)^{p_{0}},...(U_{N-1}\\otimes S)^{p_{N-1}}\\}(I\\otimes T)^{p_{N}} \\end {equation}",
"\\begin {equation} =\\sum _{j_{p_{N}}=0}^{n-p_{N}}...\\sum _{j_{1}=0}^{j_{2}}\\sum _{j_{0}=0}^{j_{1}} \\alpha ^{j_{0}}\\alpha ^{j_{1}}...\\alpha ^{j_{p_{N}}}\\{(U_{0}\\otimes S)^{p_{0}},...(U_{N-1}\\otimes S)^{p_{N}-1}\\}(I\\otimes T)^{p_{N}} \\end {equation}",
"\\begin {equation} =\\left ( \\sum _{j_{p_{N}}=0}^{n-p_{N}}...\\sum _{j_{1}=0}^{j_{2}}\\sum _{j_{0}=0}^{j_{1}} \\alpha ^{j_{0}}\\alpha ^{j_{1}}...\\alpha ^{j_{p_{N}}}\\right ) \\{U_{1}^{p_{1}},...U_{N-1}^{p_{N-1}}\\}\\otimes S^{n-p_{N}}T^{p_{N}}, \\end {equation}",
"\\begin {equation} G_{rs}=a_{rs}Q_{rs}+a_{rs}^{\\ast }Q_{rs}^{+}, \\label {ga40} \\end {equation}",
"\\begin {equation} a_{kl}=\\frac {1}{\\sqrt {2}}\\alpha ^{\\lbrack kl+n(k+l-1/4)]/2} \\label {ga41} \\end {equation}",
"\\begin {equation} \\left [ G_{kl},G_{rs}\\right ] =i\\sin \\left ( \\pi (ks-lr)/n\\right ) \\label {ga42} \\end {equation}",
"\\begin {equation} \\cdot \\left \\{ \\mathrm {sg}(k+r,l+s,n)\\left ( G_{k+r,l+s}-(-1)^{n+k+l+r+s}G_{-k-r,-l-s}\\right ) \\right . \\end {equation}",
"\\begin {equation} -\\left . \\mathrm {sg}(k-r,l-s,n)\\left ( G_{k-r,l-s}-(-1)^{n+k+l+r+s}G_{r-k,s-l}\\right ) \\right \\} , \\end {equation}",
"\\begin {equation} \\mathrm {sg}(p,q,n)=(-1)^{p\\cdot m_{q}+q\\cdot m_{p}-n},\\qquad m_{x}=\\frac {x- \\text {mod}(x-1,n)-1}{n}. \\end {equation}"
],
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"\\begin{equation*} \\sum _ { k = 0 } ^ { N } p _ { k } = n , \\end{equation*}",
"\\begin{equation*} \\{ Q _ { 0 } ^ { p _ { 0 } } , Q _ { 1 } ^ { p _ { 1 } } , . . . Q _ { N } ^ { p _ { N } } \\} = \\sum _ { j _ { p _ { N } } = 0 } ^ { n - p _ { N } } . . . \\sum _ { j _ { 1 } = 0 } ^ { j _ { 2 } } \\sum _ { j _ { 0 } = 0 } ^ { j _ { 1 } } P _ { j _ { 0 } } ^ { - } \\circ P _ { j _ { 1 } } ^ { - } . . . P _ { j _ { p _ { N } } } ^ { - } \\circ \\{ Q _ { 0 } ^ { p _ { 0 } } , . . . Q _ { N - 1 } ^ { p _ { N - 1 } } \\} Q _ { N } ^ { p _ { N } } \\end{equation*}",
"\\begin{equation*} = \\sum _ { j _ { p _ { N } } = 0 } ^ { n - p _ { N } } . . . \\sum _ { j _ { 1 } = 0 } ^ { j _ { 2 } } \\sum _ { j _ { 0 } = 0 } ^ { j _ { 1 } } P _ { j _ { 0 } } ^ { - } \\circ P _ { j _ { 1 } } ^ { - } . . . P _ { j _ { p _ { N } } } ^ { - } \\circ \\{ ( U _ { 0 } \\otimes S ) ^ { p _ { 0 } } , . . . ( U _ { N - 1 } \\otimes S ) ^ { p _ { N - 1 } } \\} ( I \\otimes T ) ^ { p _ { N } } \\end{equation*}",
"\\begin{equation*} = \\sum _ { j _ { p _ { N } } = 0 } ^ { n - p _ { N } } . . . \\sum _ { j _ { 1 } = 0 } ^ { j _ { 2 } } \\sum _ { j _ { 0 } = 0 } ^ { j _ { 1 } } \\alpha ^ { j _ { 0 } } \\alpha ^ { j _ { 1 } } . . . \\alpha ^ { j _ { p _ { N } } } \\{ ( U _ { 0 } \\otimes S ) ^ { p _ { 0 } } , . . . ( U _ { N - 1 } \\otimes S ) ^ { p _ { N } - 1 } \\} ( I \\otimes T ) ^ { p _ { N } } \\end{equation*}",
"\\begin{equation*} = ( \\sum _ { j _ { p _ { N } } = 0 } ^ { n - p _ { N } } . . . \\sum _ { j _ { 1 } = 0 } ^ { j _ { 2 } } \\sum _ { j _ { 0 } = 0 } ^ { j _ { 1 } } \\alpha ^ { j _ { 0 } } \\alpha ^ { j _ { 1 } } . . . \\alpha ^ { j _ { p _ { N } } } ) \\{ U _ { 1 } ^ { p _ { 1 } } , . . . U _ { N - 1 } ^ { p _ { N - 1 } } \\} \\otimes S ^ { n - p _ { N } } T ^ { p _ { N } } , \\end{equation*}",
"\\begin{equation*} G _ { r s } = a _ { r s } Q _ { r s } + a _ { r s } ^ { \\ast } Q _ { r s } ^ { + } , \\end{equation*}",
"\\begin{equation*} a _ { k l } = \\frac { 1 } { \\sqrt { 2 } } \\alpha ^ { [ k l + n ( k + l - 1 \\slash 4 ) ] \\slash 2 } \\end{equation*}",
"\\begin{equation*} [ G _ { k l } , G _ { r s } ] = i \\operatorname { s i n } ( \\pi ( k s - l r ) \\slash n ) \\end{equation*}",
"\\begin{equation*} \\cdot \\{ s g ( k + r , l + s , n ) ( G _ { k + r , l + s } - ( - 1 ) ^ { n + k + l + r + s } G _ { - k - r , - l - s } ) \\end{equation*}",
"\\begin{equation*} - s g ( k - r , l - s , n ) ( G _ { k - r , l - s } - ( - 1 ) ^ { n + k + l + r + s } G _ { r - k , s - l } ) \\} , \\end{equation*}",
"\\begin{equation*} s g ( p , q , n ) = ( - 1 ) ^ { p \\cdot m _ { q } + q \\cdot m _ { p } - n } , \\qquad m _ { x } = \\frac { x - m o d ( x - 1 , n ) - 1 } { n } . \\end{equation*}"
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"\\begin{equation*} = \\sum _ { \\mitj _ { \\mitp _ { \\mitN } } = 0 } ^ { \\mitn - \\mitp _ { \\mitN } } . . . \\sum _ { \\mitj _ { 1 } = 0 } ^ { \\mitj _ { 2 } } \\sum _ { \\mitj _ { 0 } = 0 } ^ { \\mitj _ { 1 } } \\mitP _ { \\mitj _ { 0 } } ^ { - } \\vysmwhtcircle \\mitP _ { \\mitj _ { 1 } } ^ { - } . . . \\mitP _ { \\mitj _ { \\mitp _ { \\mitN } } } ^ { - } \\vysmwhtcircle \\{ ( \\mitU _ { 0 } \\otimes \\mitS ) ^ { \\mitp _ { 0 } } , . . . ( \\mitU _ { \\mitN - 1 } \\otimes \\mitS ) ^ { \\mitp _ { \\mitN - 1 } } \\} ( \\mitI \\otimes \\mitT ) ^ { \\mitp _ { \\mitN } } \\end{equation*}",
"\\begin{equation*} = \\sum _ { \\mitj _ { \\mitp _ { \\mitN } } = 0 } ^ { \\mitn - \\mitp _ { \\mitN } } . . . \\sum _ { \\mitj _ { 1 } = 0 } ^ { \\mitj _ { 2 } } \\sum _ { \\mitj _ { 0 } = 0 } ^ { \\mitj _ { 1 } } \\mitalpha ^ { \\mitj _ { 0 } } \\mitalpha ^ { \\mitj _ { 1 } } . . . \\mitalpha ^ { \\mitj _ { \\mitp _ { \\mitN } } } \\{ ( \\mitU _ { 0 } \\otimes \\mitS ) ^ { \\mitp _ { 0 } } , . . . ( \\mitU _ { \\mitN - 1 } \\otimes \\mitS ) ^ { \\mitp _ { \\mitN } - 1 } \\} ( \\mitI \\otimes \\mitT ) ^ { \\mitp _ { \\mitN } } \\end{equation*}",
"\\begin{equation*} = \\left( \\sum _ { \\mitj _ { \\mitp _ { \\mitN } } = 0 } ^ { \\mitn - \\mitp _ { \\mitN } } . . . \\sum _ { \\mitj _ { 1 } = 0 } ^ { \\mitj _ { 2 } } \\sum _ { \\mitj _ { 0 } = 0 } ^ { \\mitj _ { 1 } } \\mitalpha ^ { \\mitj _ { 0 } } \\mitalpha ^ { \\mitj _ { 1 } } . . . \\mitalpha ^ { \\mitj _ { \\mitp _ { \\mitN } } } \\right) \\{ \\mitU _ { 1 } ^ { \\mitp _ { 1 } } , . . . \\mitU _ { \\mitN - 1 } ^ { \\mitp _ { \\mitN - 1 } } \\} \\otimes \\mitS ^ { \\mitn - \\mitp _ { \\mitN } } \\mitT ^ { \\mitp _ { \\mitN } } , \\end{equation*}",
"\\begin{equation*} \\mitG _ { \\mitr \\mits } = \\mita _ { \\mitr \\mits } \\mitQ _ { \\mitr \\mits } + \\mita _ { \\mitr \\mits } ^ { \\ast } \\mitQ _ { \\mitr \\mits } ^ { + } , \\end{equation*}",
"\\begin{equation*} \\mita _ { \\mitk \\mitl } = \\frac { 1 } { \\sqrt { 2 } } \\mitalpha ^ { [ \\mitk \\mitl + \\mitn ( \\mitk + \\mitl - 1 \\slash 4 ) ] \\slash 2 } \\end{equation*}",
"\\begin{equation*} \\left[ \\mitG _ { \\mitk \\mitl } , \\mitG _ { \\mitr \\mits } \\right] = \\miti \\operatorname { s i n } \\left( \\mitpi ( \\mitk \\mits - \\mitl \\mitr ) \\slash \\mitn \\right) \\end{equation*}",
"\\begin{equation*} \\cdot \\left\\{ \\mathrm { s g } ( \\mitk + \\mitr , \\mitl + \\mits , \\mitn ) \\left( \\mitG _ { \\mitk + \\mitr , \\mitl + \\mits } - ( - 1 ) ^ { \\mitn + \\mitk + \\mitl + \\mitr + \\mits } \\mitG _ { - \\mitk - \\mitr , - \\mitl - \\mits } \\right) \\right. \\end{equation*}",
"\\begin{equation*} - \\left. \\mathrm { s g } ( \\mitk - \\mitr , \\mitl - \\mits , \\mitn ) \\left( \\mitG _ { \\mitk - \\mitr , \\mitl - \\mits } - ( - 1 ) ^ { \\mitn + \\mitk + \\mitl + \\mitr + \\mits } \\mitG _ { \\mitr - \\mitk , \\mits - \\mitl } \\right) \\right\\} , \\end{equation*}",
"\\begin{equation*} \\mathrm { s g } ( \\mitp , \\mitq , \\mitn ) = ( - 1 ) ^ { \\mitp \\cdot \\mitm _ { \\mitq } + \\mitq \\cdot \\mitm _ { \\mitp } - \\mitn } , \\qquad \\mitm _ { \\mitx } = \\frac { \\mitx - \\mathrm { m o d } ( \\mitx - 1 , \\mitn ) - 1 } { \\mitn } . \\end{equation*}"
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0003126_page11
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{
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"$\\square ^{1/n}$",
"$\\mu ,\\ \\pi _{\\lambda }$",
"$n-1$",
"$\\Gamma $",
"$p-$",
"$n^{2}$",
"$\\Psi $",
"$n^{2}$",
"$n^{2}=4$",
"$n>2$",
"$U_{l}$",
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"$p$",
"$\\pi _{\\lambda }$",
"\\begin {equation} \\left ( Q_{0}\\right ) ^{n}=-\\left ( Q_{l}\\right ) ^{n}=I,\\qquad l=1,2,3, \\label {s1} \\end {equation}",
"\\begin {equation} \\left ( \\Gamma (p)-\\mu I\\right ) \\Psi (p)=0, \\label {s2} \\end {equation}",
"\\begin {equation} \\Gamma (p)=\\sum _{\\lambda =0}^{3}\\pi _{\\lambda }Q_{\\lambda }. \\label {s3} \\end {equation}",
"\\begin {equation} \\mu ^{n}=m^{2},\\qquad \\pi _{\\lambda }^{n}=p_{\\lambda }^{2}, \\label {sa3} \\end {equation}",
"\\begin {equation} \\Gamma (p)^{n}=p_{0}^{2}-p_{1}^{2}-p_{2}^{2}-p_{3}^{2}\\equiv p^{2} \\label {sb3} \\end {equation}",
"\\begin {equation} \\left ( p^{2}-m^{2}\\right ) \\Psi (p)=0. \\label {s4} \\end {equation}",
"\\begin {equation} \\Psi (p)=\\left ( \\begin {array}{c} \\mathbf {h} \\\\ \\frac {U(p)}{\\alpha \\pi _{0}-\\mu }\\mathbf {h} \\\\ \\frac {U^{2}(p)}{(\\alpha \\pi _{0}-\\mu )(\\alpha ^{2}\\pi _{0}-\\mu )}\\mathbf {h} \\\\ \\cdot \\\\ \\cdot \\\\ \\frac {U^{n-1}(p)}{(\\alpha \\pi _{0}-\\mu )...(\\alpha ^{n-1}\\pi _{0}-\\mu )} \\mathbf {h} \\end {array} \\right ) ,\\qquad \\mathbf {h}=\\left ( \\begin {array}{c} h_{1} \\\\ h_{2} \\\\ \\cdot \\\\ \\cdot \\\\ h_{n} \\end {array} \\right ) , \\label {sa4} \\end {equation}",
"\\begin {equation} U(p)=\\sum _{l=1}^{3}\\pi _{l}U_{l},\\qquad \\left ( U_{l}\\right ) ^{n}=-I, \\end {equation}",
"\\begin {equation} \\pi _{0}^{n}-\\pi _{1}^{n}-\\pi _{2}^{n}-\\pi _{3}^{n}=\\mu ^{n}=m^{2}. \\label {sb4} \\end {equation}"
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"\\begin{equation*} { ( Q _ { 0 } ) } ^ { n } = - { ( Q _ { l } ) } ^ { n } = I , \\qquad l = 1 , 2 , 3 , \\end{equation*}",
"\\begin{equation*} ( \\Gamma ( p ) - \\mu I ) \\Psi ( p ) = 0 , \\end{equation*}",
"\\begin{equation*} \\Gamma ( p ) = \\sum _ { \\lambda = 0 } ^ { 3 } \\pi _ { \\lambda } Q _ { \\lambda } . \\end{equation*}",
"\\begin{equation*} \\mu ^ { n } = m ^ { 2 } , \\qquad \\pi _ { \\lambda } ^ { n } = p _ { \\lambda } ^ { 2 } , \\end{equation*}",
"\\begin{equation*} \\Gamma ( p ) ^ { n } = p _ { 0 } ^ { 2 } - p _ { 1 } ^ { 2 } - p _ { 2 } ^ { 2 } - p _ { 3 } ^ { 2 } \\equiv p ^ { 2 } \\end{equation*}",
"\\begin{equation*} ( p ^ { 2 } - m ^ { 2 } ) \\Psi ( p ) = 0 . \\end{equation*}",
"\\begin{align*} \\Psi ( p ) = ( \\begin{array}{c} h \\\\ \\frac { U ( p ) } { \\alpha \\pi _ { 0 } - \\mu } h \\\\ \\frac { U ^ { 2 } ( p ) } { ( \\alpha \\pi _ { 0 } - \\mu ) ( \\alpha ^ { 2 } \\pi _ { 0 } - \\mu ) } h \\\\ \\cdot \\\\ \\cdot \\\\ \\frac { U ^ { n - 1 } ( p ) } { ( \\alpha \\pi _ { 0 } - \\mu ) . . . ( \\alpha ^ { n - 1 } \\pi _ { 0 } - \\mu ) } h \\end{array} ) , \\qquad h = ( \\begin{array}{c} h _ { 1 } \\\\ h _ { 2 } \\\\ \\cdot \\\\ \\cdot \\\\ h _ { n } \\end{array} ) , \\end{align*}",
"\\begin{equation*} U ( p ) = \\sum _ { l = 1 } ^ { 3 } \\pi _ { l } U _ { l } , \\qquad { ( U _ { l } ) } ^ { n } = - I , \\end{equation*}",
"\\begin{equation*} \\pi _ { 0 } ^ { n } - \\pi _ { 1 } ^ { n } - \\pi _ { 2 } ^ { n } - \\pi _ { 3 } ^ { n } = \\mu ^ { n } = m ^ { 2 } . \\end{equation*}"
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"\\begin{equation*} { \\left( \\mitQ _ { 0 } \\right) } ^ { \\mitn } = - { \\left( \\mitQ _ { \\mitl } \\right) } ^ { \\mitn } = \\mitI , \\qquad \\mitl = 1 , 2 , 3 , \\end{equation*}",
"\\begin{equation*} \\left( \\mupGamma ( \\mitp ) - \\mitmu \\mitI \\right) \\mupPsi ( \\mitp ) = 0 , \\end{equation*}",
"\\begin{equation*} \\mupGamma ( \\mitp ) = \\sum _ { \\mitlambda = 0 } ^ { 3 } \\mitpi _ { \\mitlambda } \\mitQ _ { \\mitlambda } . \\end{equation*}",
"\\begin{equation*} \\mitmu ^ { \\mitn } = \\mitm ^ { 2 } , \\qquad \\mitpi _ { \\mitlambda } ^ { \\mitn } = \\mitp _ { \\mitlambda } ^ { 2 } , \\end{equation*}",
"\\begin{equation*} \\mupGamma ( \\mitp ) ^ { \\mitn } = \\mitp _ { 0 } ^ { 2 } - \\mitp _ { 1 } ^ { 2 } - \\mitp _ { 2 } ^ { 2 } - \\mitp _ { 3 } ^ { 2 } \\equiv \\mitp ^ { 2 } \\end{equation*}",
"\\begin{equation*} \\left( \\mitp ^ { 2 } - \\mitm ^ { 2 } \\right) \\mupPsi ( \\mitp ) = 0 . \\end{equation*}",
"\\begin{align*} \\mupPsi ( \\mitp ) = \\left( \\begin{array}{c} \\mbfh \\\\ \\frac { \\mitU ( \\mitp ) } { \\mitalpha \\mitpi _ { 0 } - \\mitmu } \\mbfh \\\\ \\frac { \\mitU ^ { 2 } ( \\mitp ) } { ( \\mitalpha \\mitpi _ { 0 } - \\mitmu ) ( \\mitalpha ^ { 2 } \\mitpi _ { 0 } - \\mitmu ) } \\mbfh \\\\ \\cdot \\\\ \\cdot \\\\ \\frac { \\mitU ^ { \\mitn - 1 } ( \\mitp ) } { ( \\mitalpha \\mitpi _ { 0 } - \\mitmu ) . . . ( \\mitalpha ^ { \\mitn - 1 } \\mitpi _ { 0 } - \\mitmu ) } \\mbfh \\end{array} \\right) , \\qquad \\mbfh = \\left( \\begin{array}{c} \\Planckconst _ { 1 } \\\\ \\Planckconst _ { 2 } \\\\ \\cdot \\\\ \\cdot \\\\ \\Planckconst _ { \\mitn } \\end{array} \\right) , \\end{align*}",
"\\begin{equation*} \\mitU ( \\mitp ) = \\sum _ { \\mitl = 1 } ^ { 3 } \\mitpi _ { \\mitl } \\mitU _ { \\mitl } , \\qquad { \\left( \\mitU _ { \\mitl } \\right) } ^ { \\mitn } = - \\mitI , \\end{equation*}",
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0003126_page12
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"\\begin {equation} \\pi _{\\lambda }=(p_{\\lambda })^{2/n}\\rightarrow (i\\partial _{\\lambda })^{2/n}. \\label {s5} \\end {equation}",
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"\\begin {equation} \\Lambda (d\\omega )=I+id\\omega \\cdot L_{\\omega }, \\label {s8} \\end {equation}",
"\\begin {equation} p_{i}^{\\prime }=p_{i}+\\epsilon _{ijk}p_{j}d\\varphi _{k},\\qquad i=1,2,3 \\label {s9} \\end {equation}",
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"\\begin {equation} p_{1}^{\\prime }=p_{1}\\cos \\varphi _{3}+p_{2}\\sin \\varphi _{3},\\qquad p_{2}^{\\prime }=p_{2}\\cos \\varphi _{3}-p_{1}\\sin \\varphi _{3},\\qquad p_{3}^{\\prime }=p_{3}, \\label {sa10} \\end {equation}",
"\\begin {equation} p_{2}^{\\prime }=p_{2}\\cos \\varphi _{1}+p_{3}\\sin \\varphi _{1},\\qquad p_{3}^{\\prime }=p_{3}\\cos \\varphi _{1}-p_{2}\\sin \\varphi _{1},\\qquad p_{1}^{\\prime }=p_{1}, \\label {sb10} \\end {equation}",
"\\begin {equation} p_{3}^{\\prime }=p_{3}\\cos \\varphi _{2}+p_{1}\\sin \\varphi _{2},\\qquad p_{1}^{\\prime }=p_{1}\\cos \\varphi _{2}-p_{3}\\sin \\varphi _{2},\\qquad p_{2}^{\\prime }=p_{2} \\label {sc10} \\end {equation}",
"\\begin {equation} p_{0}^{\\prime }=p_{0}\\cosh \\psi _{i}+p_{i}\\sinh \\psi _{i},\\qquad i=1,2,3, \\label {sd10} \\end {equation}",
"\\begin {equation} \\cosh \\psi _{i}=\\frac {1}{\\sqrt {1-\\beta _{i}^{2}}},\\qquad \\sinh \\psi _{i}= \\frac {\\beta _{i}}{\\sqrt {1-\\beta _{i}^{2}}}. \\label {se10} \\end {equation}"
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"\\begin{equation*} \\Gamma ( p ) \\rightarrow \\Gamma ( p ^ { \\prime } ) = \\Lambda \\Gamma ( p ) \\Lambda ^ { - 1 } \\end{equation*}",
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"\\begin{equation*} p _ { 2 } ^ { \\prime } = p _ { 2 } \\operatorname { c o s } \\varphi _ { 1 } + p _ { 3 } \\operatorname { s i n } \\varphi _ { 1 } , \\qquad p _ { 3 } ^ { \\prime } = p _ { 3 } \\operatorname { c o s } \\varphi _ { 1 } - p _ { 2 } \\operatorname { s i n } \\varphi _ { 1 } , \\qquad p _ { 1 } ^ { \\prime } = p _ { 1 } , \\end{equation*}",
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"\\begin{equation*} \\mupGamma ( \\mitp ) \\rightarrow \\mupGamma ( \\mitp ^ { \\prime } ) = \\mupLambda \\mupGamma ( \\mitp ) \\mupLambda ^ { - 1 } \\end{equation*}",
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"\\begin{equation*} \\mitp _ { \\miti } ^ { \\prime } = \\mitp _ { \\miti } + \\mitepsilon _ { \\miti \\mitj \\mitk } \\mitp _ { \\mitj } \\mitd \\mitvarphi _ { \\mitk } , \\qquad \\miti = 1 , 2 , 3 \\end{equation*}",
"\\begin{equation*} \\mitp _ { \\miti } ^ { \\prime } = \\mitp _ { \\miti } + \\mitp _ { 0 } \\mitd \\mitpsi _ { \\miti } , \\qquad \\mitp _ { 0 } ^ { \\prime } = \\mitp _ { 0 } + \\mitp _ { \\miti } \\mitd \\mitpsi _ { \\miti } , \\qquad \\miti = 1 , 2 , 3 , \\end{equation*}",
"\\begin{equation*} \\mitp _ { 1 } ^ { \\prime } = \\mitp _ { 1 } \\operatorname { c o s } \\mitvarphi _ { 3 } + \\mitp _ { 2 } \\operatorname { s i n } \\mitvarphi _ { 3 } , \\qquad \\mitp _ { 2 } ^ { \\prime } = \\mitp _ { 2 } \\operatorname { c o s } \\mitvarphi _ { 3 } - \\mitp _ { 1 } \\operatorname { s i n } \\mitvarphi _ { 3 } , \\qquad \\mitp _ { 3 } ^ { \\prime } = \\mitp _ { 3 } , \\end{equation*}",
"\\begin{equation*} \\mitp _ { 2 } ^ { \\prime } = \\mitp _ { 2 } \\operatorname { c o s } \\mitvarphi _ { 1 } + \\mitp _ { 3 } \\operatorname { s i n } \\mitvarphi _ { 1 } , \\qquad \\mitp _ { 3 } ^ { \\prime } = \\mitp _ { 3 } \\operatorname { c o s } \\mitvarphi _ { 1 } - \\mitp _ { 2 } \\operatorname { s i n } \\mitvarphi _ { 1 } , \\qquad \\mitp _ { 1 } ^ { \\prime } = \\mitp _ { 1 } , \\end{equation*}",
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"\\begin{equation*} \\mitp _ { 0 } ^ { \\prime } = \\mitp _ { 0 } \\operatorname { c o s h } \\mitpsi _ { \\miti } + \\mitp _ { \\miti } \\operatorname { s i n h } \\mitpsi _ { \\miti } , \\qquad \\miti = 1 , 2 , 3 , \\end{equation*}",
"\\begin{equation*} \\operatorname { c o s h } \\mitpsi _ { \\miti } = \\frac { 1 } { \\sqrt { 1 - \\mitbeta _ { \\miti } ^ { 2 } } } , \\qquad \\operatorname { s i n h } \\mitpsi _ { \\miti } = \\frac { \\mitbeta _ { \\miti } } { \\sqrt { 1 - \\mitbeta _ { \\miti } ^ { 2 } } } . \\end{equation*}"
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0003126_page13
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{
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"\\begin {equation} \\frac {d}{d\\varphi _{i}}=-\\epsilon _{ijk}p_{j}\\frac {\\partial }{\\partial p_{k}} ,\\qquad \\frac {d}{d\\psi _{i}}=p_{0}\\frac {\\partial }{\\partial p_{i}}+p_{i} \\frac {\\partial }{\\partial p_{0}},\\qquad i=1,2,3. \\label {s12} \\end {equation}",
"\\begin {equation} p^{\\prime }=p+\\frac {dp}{d\\omega }d\\omega \\label {s13} \\end {equation}",
"\\begin {equation} \\Gamma (p^{\\prime })=\\Gamma (p)+\\frac {d\\Gamma (p)}{d\\omega }d\\omega =\\left ( I+id\\omega \\cdot L_{\\omega }\\right ) \\Gamma (p)\\left ( I-id\\omega \\cdot L_{\\omega }\\right ) , \\label {s14} \\end {equation}",
"\\begin {equation} \\Psi ^{\\prime }(p^{\\prime })=\\Psi ^{\\prime }(p)+\\frac {d\\Psi ^{\\prime }(p)}{ d\\omega }d\\omega =\\left ( I+id\\omega \\cdot L_{\\omega }\\right ) \\Psi (p). \\label {s15} \\end {equation}",
"\\begin {equation} \\mathbf {L_{\\omega }}=L_{\\omega }+i\\frac {d}{d\\omega }, \\label {s16} \\end {equation}",
"\\begin {equation} \\lbrack \\mathbf {L_{\\omega }},\\Gamma ]=0, \\label {s17} \\end {equation}",
"\\begin {equation} \\Psi ^{\\prime }(p)=\\left ( I+id\\omega \\cdot \\mathbf {L_{\\omega }}\\right ) \\Psi (p). \\label {s18} \\end {equation}",
"\\begin {equation} \\lbrack \\mathbf {L_{\\varphi _{j}}},\\mathbf {L_{\\varphi _{k}}}]=i\\epsilon _{jkl} \\mathbf {L_{\\varphi _{l}},} \\label {s19} \\end {equation}",
"\\begin {equation} \\lbrack \\mathbf {L_{\\psi _{j}}},\\mathbf {L_{\\psi _{k}}}]=-i\\epsilon _{jkl} \\mathbf {L_{\\varphi _{l}},} \\label {s20} \\end {equation}",
"\\begin {equation} \\lbrack \\mathbf {L_{\\varphi _{j}}},\\mathbf {L_{\\psi _{k}}}]=i\\epsilon _{jkl} \\mathbf {L_{\\psi _{l}},\\qquad }j,k,l=1,2,3. \\label {s21} \\end {equation}"
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"\\begin{equation*} \\Psi ^ { \\prime } ( p ^ { \\prime } ) = \\Psi ^ { \\prime } ( p ) + \\frac { d \\Psi ^ { \\prime } ( p ) } { d \\omega } d \\omega = ( I + i d \\omega \\cdot L _ { \\omega } ) \\Psi ( p ) . \\end{equation*}",
"\\begin{equation*} L _ { \\omega } = L _ { \\omega } + i \\frac { d } { d \\omega } , \\end{equation*}",
"\\begin{equation*} [ L _ { \\omega } , \\Gamma ] = 0 , \\end{equation*}",
"\\begin{equation*} \\Psi ^ { \\prime } ( p ) = ( I + i d \\omega \\cdot L _ { \\omega } ) \\Psi ( p ) . \\end{equation*}",
"\\begin{equation*} [ L _ { \\varphi _ { j } } , L _ { \\varphi _ { k } } ] = i \\epsilon _ { j k l } L _ { \\varphi _ { l } } , \\end{equation*}",
"\\begin{equation*} [ L _ { \\psi _ { j } } , L _ { \\psi _ { k } } ] = - i \\epsilon _ { j k l } L _ { \\varphi _ { l } } , \\end{equation*}",
"\\begin{equation*} [ L _ { \\varphi _ { j } } , L _ { \\psi _ { k } } ] = i \\epsilon _ { j k l } L _ { \\psi _ { l } } , \\qquad j , k , l = 1 , 2 , 3 . \\end{equation*}"
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"$ \\mitL _ { \\mitomega } $",
"$ \\mitp _ { \\miti } $",
"\\begin{equation*} \\mitf ( \\mitp ) \\rightarrow \\mitf ( \\mitp ^ { \\prime } ) = \\mitf ( \\mitp + \\mitdelta \\mitp ) = \\mitf ( \\mitp ) + \\frac { \\mitd \\mitf } { \\mitd \\mitomega } \\mitd \\mitomega , \\end{equation*}",
"\\begin{equation*} \\frac { \\mitd } { \\mitd \\mitvarphi _ { \\miti } } = - \\mitepsilon _ { \\miti \\mitj \\mitk } \\mitp _ { \\mitj } \\frac { \\mitpartial } { \\mitpartial \\mitp _ { \\mitk } } , \\qquad \\frac { \\mitd } { \\mitd \\mitpsi _ { \\miti } } = \\mitp _ { 0 } \\frac { \\mitpartial } { \\mitpartial \\mitp _ { \\miti } } + \\mitp _ { \\miti } \\frac { \\mitpartial } { \\mitpartial \\mitp _ { 0 } } , \\qquad \\miti = 1 , 2 , 3 . \\end{equation*}",
"\\begin{equation*} \\mitp ^ { \\prime } = \\mitp + \\frac { \\mitd \\mitp } { \\mitd \\mitomega } \\mitd \\mitomega \\end{equation*}",
"\\begin{equation*} \\mupGamma ( \\mitp ^ { \\prime } ) = \\mupGamma ( \\mitp ) + \\frac { \\mitd \\mupGamma ( \\mitp ) } { \\mitd \\mitomega } \\mitd \\mitomega = \\left( \\mitI + \\miti \\mitd \\mitomega \\cdot \\mitL _ { \\mitomega } \\right) \\mupGamma ( \\mitp ) \\left( \\mitI - \\miti \\mitd \\mitomega \\cdot \\mitL _ { \\mitomega } \\right) , \\end{equation*}",
"\\begin{equation*} \\mupPsi ^ { \\prime } ( \\mitp ^ { \\prime } ) = \\mupPsi ^ { \\prime } ( \\mitp ) + \\frac { \\mitd \\mupPsi ^ { \\prime } ( \\mitp ) } { \\mitd \\mitomega } \\mitd \\mitomega = \\left( \\mitI + \\miti \\mitd \\mitomega \\cdot \\mitL _ { \\mitomega } \\right) \\mupPsi ( \\mitp ) . \\end{equation*}",
"\\begin{equation*} \\mbfL _ { \\mbfitomega } = \\mitL _ { \\mitomega } + \\miti \\frac { \\mitd } { \\mitd \\mitomega } , \\end{equation*}",
"\\begin{equation*} [ \\mbfL _ { \\mbfitomega } , \\mupGamma ] = 0 , \\end{equation*}",
"\\begin{equation*} \\mupPsi ^ { \\prime } ( \\mitp ) = \\left( \\mitI + \\miti \\mitd \\mitomega \\cdot \\mbfL _ { \\mbfitomega } \\right) \\mupPsi ( \\mitp ) . \\end{equation*}",
"\\begin{equation*} [ \\mbfL _ { \\mbfitvarphi _ { \\mbfj } } , \\mbfL _ { \\mbfitvarphi _ { \\mbfk } } ] = \\miti \\mitepsilon _ { \\mitj \\mitk \\mitl } \\mbfL _ { \\mbfitvarphi _ { \\mbfl } } , \\end{equation*}",
"\\begin{equation*} [ \\mbfL _ { \\mbfitpsi _ { \\mbfj } } , \\mbfL _ { \\mbfitpsi _ { \\mbfk } } ] = - \\miti \\mitepsilon _ { \\mitj \\mitk \\mitl } \\mbfL _ { \\mbfitvarphi _ { \\mbfl } } , \\end{equation*}",
"\\begin{equation*} [ \\mbfL _ { \\mbfitvarphi _ { \\mbfj } } , \\mbfL _ { \\mbfitpsi _ { \\mbfk } } ] = \\miti \\mitepsilon _ { \\mitj \\mitk \\mitl } \\mbfL _ { \\mbfitpsi _ { \\mbfl } } , \\qquad \\mitj , \\mitk , \\mitl = 1 , 2 , 3 . \\end{equation*}"
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0003126_page16
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"$Y,Y^{-1}$",
"$(p^{2})^{1/n}$",
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"$p$",
"$p^{2}=m^{2}$",
"$2\\times 2$",
"$j,k,l=1,2,3.$",
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"\\begin {equation} =\\frac {1-p_{0}^{2}/p^{2}}{n\\left [ 1-\\left ( p_{0}^{2}/p^{2}\\right ) ^{1/n} \\right ] }, \\end {equation}",
"\\begin {equation} \\Psi _{0}(p)=\\left ( \\begin {array}{c} \\mathbf {0} \\\\ \\cdot \\\\ \\mathbf {0} \\\\ \\mathbf {g} \\\\ \\mathbf {0} \\\\ \\cdot \\\\ \\cdot \\\\ \\mathbf {0} \\end {array} \\right ) ;\\qquad \\mathbf {g}\\equiv \\left ( \\begin {array}{c} g_{1} \\\\ g_{2} \\\\ \\cdot \\\\ \\cdot \\\\ g_{n} \\end {array} \\right ) ,\\qquad \\mathbf {0}\\equiv \\left ( \\begin {array}{c} 0 \\\\ 0 \\\\ \\cdot \\\\ \\cdot \\\\ 0 \\end {array} \\right ) , \\label {s38} \\end {equation}",
"\\begin {equation} \\lbrack \\mathbf {L_{\\omega }},\\Gamma _{0}(p)]=0 \\label {s39} \\end {equation}",
"\\begin {equation} \\sigma _{1}=\\left ( \\begin {array}{cc} 0 & 1 \\\\ 1 & 0 \\end {array} \\right ) ,\\qquad \\sigma _{2}=\\left ( \\begin {array}{cc} 0 & -i \\\\ i & 0 \\end {array} \\right ) ,\\qquad \\sigma _{3}=\\left ( \\begin {array}{cc} 1 & 0 \\\\ 0 & -1 \\end {array} \\right ) , \\label {s40} \\end {equation}",
"\\begin {equation} \\gamma _{0}=\\left ( \\begin {array}{cc} \\mathbf {1} & \\mathbf {0} \\\\ \\mathbf {0} & \\mathbf {-1} \\end {array} \\right ) ,\\qquad \\gamma _{j}=\\left ( \\begin {array}{cc} \\mathbf {0} & \\sigma _{j} \\\\ -\\sigma _{j} & \\mathbf {0} \\end {array} \\right ) ;\\qquad j=1,2,3, \\label {s41} \\end {equation}",
"\\begin {equation} \\left ( \\Gamma (p)-m\\right ) \\Psi (p)=0,\\qquad \\Gamma (p)\\equiv \\sum _{\\lambda =0}^{3}p_{\\lambda }\\gamma _{\\lambda } \\label {s42} \\end {equation}",
"\\begin {equation} \\mathbf {L_{\\varphi _{j}}=}\\frac {i}{4}\\epsilon _{jkl}\\gamma _{k}\\gamma _{l}+i \\frac {d}{d\\varphi _{j}}=L_{\\varphi _{j}}+i\\frac {d}{d\\varphi _{j}};\\qquad L_{\\varphi _{j}}=\\frac {1}{2}\\left ( \\begin {array}{cc} \\sigma _{j} & \\mathbf {0} \\\\ \\mathbf {0} & \\sigma _{j} \\end {array} \\right ) , \\label {s43} \\end {equation}",
"\\begin {equation} \\mathbf {L_{\\psi _{j}}=}\\frac {i}{2}\\gamma _{0}\\gamma _{j}+i\\frac {d}{d\\psi _{j} }=L_{\\psi _{j}}+i\\frac {d}{d\\psi _{j}};\\qquad L_{\\psi _{j}}=\\frac {i}{2}\\left ( \\begin {array}{cc} \\mathbf {0} & \\sigma _{j} \\\\ \\sigma _{j} & \\mathbf {0} \\end {array} \\right ) , \\label {s44} \\end {equation}",
"\\begin {equation} \\mathbf {L_{\\omega }\\rightarrow M_{\\omega }}=Y(p)\\mathbf {L_{\\omega }} Y^{-1}(p). \\label {s45} \\end {equation}"
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"\\begin{equation*} = \\frac { 1 - p _ { 0 } ^ { 2 } \\slash p ^ { 2 } } { n [ 1 - { ( p _ { 0 } ^ { 2 } \\slash p ^ { 2 } ) } ^ { 1 \\slash n } ] } , \\end{equation*}",
"\\begin{align*} \\Psi _ { 0 } ( p ) = ( \\begin{array}{c} 0 \\\\ \\cdot \\\\ 0 \\\\ g \\\\ 0 \\\\ \\cdot \\\\ \\cdot \\\\ 0 \\end{array} ) ; \\qquad g \\equiv ( \\begin{array}{c} g _ { 1 } \\\\ g _ { 2 } \\\\ \\cdot \\\\ \\cdot \\\\ g _ { n } \\end{array} ) , \\qquad 0 \\equiv ( \\begin{array}{c} 0 \\\\ 0 \\\\ \\cdot \\\\ \\cdot \\\\ 0 \\end{array} ) , \\end{align*}",
"\\begin{equation*} [ L _ { \\omega } , \\Gamma _ { 0 } ( p ) ] = 0 \\end{equation*}",
"\\begin{align*} \\sigma _ { 1 } = ( \\begin{array}{cc} 0 & 1 \\\\ 1 & 0 \\end{array} ) , \\qquad \\sigma _ { 2 } = ( \\begin{array}{cc} 0 & - i \\\\ i & 0 \\end{array} ) , \\qquad \\sigma _ { 3 } = ( \\begin{array}{cc} 1 & 0 \\\\ 0 & - 1 \\end{array} ) , \\end{align*}",
"\\begin{align*} \\gamma _ { 0 } = ( \\begin{array}{cc} 1 & 0 \\\\ 0 & - 1 \\end{array} ) , \\qquad \\gamma _ { j } = ( \\begin{array}{cc} 0 & \\sigma _ { j } \\\\ - \\sigma _ { j } & 0 \\end{array} ) ; \\qquad j = 1 , 2 , 3 , \\end{align*}",
"\\begin{equation*} ( \\Gamma ( p ) - m ) \\Psi ( p ) = 0 , \\qquad \\Gamma ( p ) \\equiv \\sum _ { \\lambda = 0 } ^ { 3 } p _ { \\lambda } \\gamma _ { \\lambda } \\end{equation*}",
"\\begin{align*} L _ { \\varphi _ { j } } = \\frac { i } { 4 } \\epsilon _ { j k l } \\gamma _ { k } \\gamma _ { l } + i \\frac { d } { d \\varphi _ { j } } = L _ { \\varphi _ { j } } + i \\frac { d } { d \\varphi _ { j } } ; \\qquad L _ { \\varphi _ { j } } = \\frac { 1 } { 2 } ( \\begin{array}{cc} \\sigma _ { j } & 0 \\\\ 0 & \\sigma _ { j } \\end{array} ) , \\end{align*}",
"\\begin{align*} L _ { \\psi _ { j } } = \\frac { i } { 2 } \\gamma _ { 0 } \\gamma _ { j } + i \\frac { d } { d \\psi _ { j } } = L _ { \\psi _ { j } } + i \\frac { d } { d \\psi _ { j } } ; \\qquad L _ { \\psi _ { j } } = \\frac { i } { 2 } ( \\begin{array}{cc} 0 & \\sigma _ { j } \\\\ \\sigma _ { j } & 0 \\end{array} ) , \\end{align*}",
"\\begin{equation*} L _ { \\omega } \\rightarrow M _ { \\omega } = Y ( p ) L _ { \\omega } Y ^ { - 1 } ( p ) . \\end{equation*}"
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"$ \\mitp $",
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"$ 2 \\times 2 $",
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"\\begin{equation*} = \\frac { 1 - \\mitp _ { 0 } ^ { 2 } \\slash \\mitp ^ { 2 } } { \\mitn \\left[ 1 - { \\left( \\mitp _ { 0 } ^ { 2 } \\slash \\mitp ^ { 2 } \\right) } ^ { 1 \\slash \\mitn } \\right] } , \\end{equation*}",
"\\begin{align*} \\mupPsi _ { 0 } ( \\mitp ) = \\left( \\begin{array}{c} \\mbfzero \\\\ \\cdot \\\\ \\mbfzero \\\\ \\mbfg \\\\ \\mbfzero \\\\ \\cdot \\\\ \\cdot \\\\ \\mbfzero \\end{array} \\right) ; \\qquad \\mbfg \\equiv \\left( \\begin{array}{c} \\mitg _ { 1 } \\\\ \\mitg _ { 2 } \\\\ \\cdot \\\\ \\cdot \\\\ \\mitg _ { \\mitn } \\end{array} \\right) , \\qquad \\mbfzero \\equiv \\left( \\begin{array}{c} 0 \\\\ 0 \\\\ \\cdot \\\\ \\cdot \\\\ 0 \\end{array} \\right) , \\end{align*}",
"\\begin{equation*} [ \\mbfL _ { \\mbfitomega } , \\mupGamma _ { 0 } ( \\mitp ) ] = 0 \\end{equation*}",
"\\begin{align*} \\mitsigma _ { 1 } = \\left( \\begin{array}{cc} 0 & 1 \\\\ 1 & 0 \\end{array} \\right) , \\qquad \\mitsigma _ { 2 } = \\left( \\begin{array}{cc} 0 & - \\miti \\\\ \\miti & 0 \\end{array} \\right) , \\qquad \\mitsigma _ { 3 } = \\left( \\begin{array}{cc} 1 & 0 \\\\ 0 & - 1 \\end{array} \\right) , \\end{align*}",
"\\begin{align*} \\mitgamma _ { 0 } = \\left( \\begin{array}{cc} \\mbfone & \\mbfzero \\\\ \\mbfzero & - \\mbfone \\end{array} \\right) , \\qquad \\mitgamma _ { \\mitj } = \\left( \\begin{array}{cc} \\mbfzero & \\mitsigma _ { \\mitj } \\\\ - \\mitsigma _ { \\mitj } & \\mbfzero \\end{array} \\right) ; \\qquad \\mitj = 1 , 2 , 3 , \\end{align*}",
"\\begin{equation*} \\left( \\mupGamma ( \\mitp ) - \\mitm \\right) \\mupPsi ( \\mitp ) = 0 , \\qquad \\mupGamma ( \\mitp ) \\equiv \\sum _ { \\mitlambda = 0 } ^ { 3 } \\mitp _ { \\mitlambda } \\mitgamma _ { \\mitlambda } \\end{equation*}",
"\\begin{align*} \\mbfL _ { \\mbfitvarphi _ { \\mbfj } } = \\frac { \\miti } { 4 } \\mitepsilon _ { \\mitj \\mitk \\mitl } \\mitgamma _ { \\mitk } \\mitgamma _ { \\mitl } + \\miti \\frac { \\mitd } { \\mitd \\mitvarphi _ { \\mitj } } = \\mitL _ { \\mitvarphi _ { \\mitj } } + \\miti \\frac { \\mitd } { \\mitd \\mitvarphi _ { \\mitj } } ; \\qquad \\mitL _ { \\mitvarphi _ { \\mitj } } = \\frac { 1 } { 2 } \\left( \\begin{array}{cc} \\mitsigma _ { \\mitj } & \\mbfzero \\\\ \\mbfzero & \\mitsigma _ { \\mitj } \\end{array} \\right) , \\end{align*}",
"\\begin{align*} \\mbfL _ { \\mbfitpsi _ { \\mbfj } } = \\frac { \\miti } { 2 } \\mitgamma _ { 0 } \\mitgamma _ { \\mitj } + \\miti \\frac { \\mitd } { \\mitd \\mitpsi _ { \\mitj } } = \\mitL _ { \\mitpsi _ { \\mitj } } + \\miti \\frac { \\mitd } { \\mitd \\mitpsi _ { \\mitj } } ; \\qquad \\mitL _ { \\mitpsi _ { \\mitj } } = \\frac { \\miti } { 2 } \\left( \\begin{array}{cc} \\mbfzero & \\mitsigma _ { \\mitj } \\\\ \\mitsigma _ { \\mitj } & \\mbfzero \\end{array} \\right) , \\end{align*}",
"\\begin{equation*} \\mbfL _ { \\mbfitomega } \\rightarrow \\mbfM _ { \\mbfitomega } = \\mitY ( \\mitp ) \\mbfL _ { \\mbfitomega } \\mitY ^ { - 1 } ( \\mitp ) . \\end{equation*}"
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0003126_page17
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{
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"$\\mathbf {L_{\\varphi _{j}}\\ }$",
"$\\Gamma _{0},\\Gamma $",
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"\\begin {equation} \\mathbf {M_{\\psi _{j}}=}M_{\\psi _{j}}(p)+i\\frac {d}{d\\psi _{j}};\\qquad M_{\\psi _{j}}(p)=\\epsilon _{jkl}\\frac {p_{k}L_{\\varphi _{l}}}{p_{0}+\\sqrt {p^{2}}}. \\label {s47} \\end {equation}",
"\\begin {equation} \\mathbf {M_{\\psi _{j}}=}M_{\\psi _{j}}(p)+i\\frac {d}{d\\psi _{j}};\\qquad M_{\\psi _{j}}(p)=\\frac {\\kappa L_{\\varphi _{j}}+\\epsilon _{jkl}p_{k}L_{\\varphi _{l}}}{ p_{0}+\\sqrt {p^{2}-\\kappa ^{2}}}, \\label {s48} \\end {equation}",
"\\begin {equation} M_{\\psi _{j}}=iL_{\\varphi _{j}}. \\label {s49} \\end {equation}",
"\\begin {equation} \\mathbf {M}_{\\omega }(\\kappa ^{\\prime })=X^{-1}(p)\\mathbf {M}_{\\omega }(\\kappa )X(p). \\label {sa49} \\end {equation}"
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"$ [ L _ { \\psi _ { 1 } } , L _ { \\psi _ { 3 } } ] , $",
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"\\begin{equation*} M _ { \\psi _ { j } } = M _ { \\psi _ { j } } ( p ) + i \\frac { d } { d \\psi _ { j } } ; \\qquad M _ { \\psi _ { j } } ( p ) = \\epsilon _ { j k l } \\frac { p _ { k } L _ { \\varphi _ { l } } } { p _ { 0 } + \\sqrt { p ^ { 2 } } } . \\end{equation*}",
"\\begin{equation*} M _ { \\psi _ { j } } = M _ { \\psi _ { j } } ( p ) + i \\frac { d } { d \\psi _ { j } } ; \\qquad M _ { \\psi _ { j } } ( p ) = \\frac { \\kappa L _ { \\varphi _ { j } } + \\epsilon _ { j k l } p _ { k } L _ { \\varphi _ { l } } } { p _ { 0 } + \\sqrt { p ^ { 2 } - \\kappa ^ { 2 } } } , \\end{equation*}",
"\\begin{equation*} M _ { \\psi _ { j } } = i L _ { \\varphi _ { j } } . \\end{equation*}",
"\\begin{equation*} M _ { \\omega } ( \\kappa ^ { \\prime } ) = X ^ { - 1 } ( p ) M _ { \\omega } ( \\kappa ) X ( p ) . \\end{equation*}"
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"\\begin{equation*} \\mbfM _ { \\mbfitpsi _ { \\mbfj } } = \\mitM _ { \\mitpsi _ { \\mitj } } ( \\mitp ) + \\miti \\frac { \\mitd } { \\mitd \\mitpsi _ { \\mitj } } ; \\qquad \\mitM _ { \\mitpsi _ { \\mitj } } ( \\mitp ) = \\mitepsilon _ { \\mitj \\mitk \\mitl } \\frac { \\mitp _ { \\mitk } \\mitL _ { \\mitvarphi _ { \\mitl } } } { \\mitp _ { 0 } + \\sqrt { \\mitp ^ { 2 } } } . \\end{equation*}",
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0003126_page19
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{
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"$M_{\\psi _{j}}(p)$",
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"$p\\rightarrow p^{\\prime }$",
"$\\Lambda $",
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"\\begin {equation} M_{\\psi _{j}}(p)=\\sum _{k=1}^{3}c_{jk}(p)L_{\\varphi _{k}}+\\sum _{\\xi }c_{j\\xi }k_{\\xi } \\label {s53} \\end {equation}",
"\\begin {equation} \\Psi _{0}^{\\prime }(p^{\\prime })=\\Lambda (\\omega )\\Psi _{0}(p),\\qquad \\Psi _{0}^{\\prime }(p)=\\Lambda (\\xi )\\Psi _{0}(p), \\label {s54} \\end {equation}",
"\\begin {equation} \\Lambda (\\omega +d\\omega )=\\Lambda (\\omega )\\Lambda (d\\omega ),\\qquad \\Lambda (\\xi +d\\xi )=\\Lambda (\\xi )\\Lambda (d\\xi ), \\label {s55} \\end {equation}",
"\\begin {equation} \\frac {d\\Lambda (\\varphi _{j})}{d\\varphi _{j}}=i\\Lambda (\\varphi _{j})R_{\\varphi _{j}},\\qquad \\frac {d\\Lambda (\\xi )}{d\\xi }=i\\Lambda (\\xi )K_{\\xi }. \\label {s56} \\end {equation}",
"\\begin {equation} \\Lambda (\\varphi _{j})=\\exp (i\\varphi _{j}R_{\\varphi _{j}}),\\qquad \\Lambda (\\xi )=\\exp (i\\xi K_{\\xi }). \\label {s57} \\end {equation}",
"\\begin {equation} \\Lambda (\\varphi ,\\vec {u})=\\exp \\left [ i\\varphi \\left ( \\vec {u}\\cdot \\vec {R} _{\\varphi }\\right ) \\right ] ;\\qquad \\vec {R}_{\\varphi }=\\left ( R_{\\varphi _{1}},R_{\\varphi _{2}},R_{\\varphi _{3}}\\right ) . \\label {sa57} \\end {equation}",
"\\begin {equation} \\frac {d\\Lambda (\\psi _{j})}{d\\psi _{j}}=if_{j}(\\psi _{j})\\Lambda (\\psi _{j})N_{j}, \\label {s58} \\end {equation}",
"\\begin {equation} f_{j}(\\psi _{j})=\\frac {1}{p_{0}\\cosh \\psi _{j}+p_{j}\\sinh \\psi _{j}+\\sqrt { p^{2}-\\kappa ^{2}}}, \\label {s59} \\end {equation}",
"\\begin {equation} N_{j}=\\kappa R_{\\varphi _{j}}+\\epsilon _{jkl}p_{k}R_{\\varphi _{l}}. \\label {sa59} \\end {equation}",
"\\begin {equation} \\Lambda (\\psi _{j})=\\exp \\left ( iF(\\psi _{j})N_{j}\\right ) ;\\qquad F(\\psi _{j})=\\int _{0}^{\\psi _{j}}f_{j}(\\eta )d\\eta . \\label {s60} \\end {equation}"
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"\\begin{equation*} \\Psi _ { 0 } ^ { \\prime } ( p ^ { \\prime } ) = \\Lambda ( \\omega ) \\Psi _ { 0 } ( p ) , \\qquad \\Psi _ { 0 } ^ { \\prime } ( p ) = \\Lambda ( \\xi ) \\Psi _ { 0 } ( p ) , \\end{equation*}",
"\\begin{equation*} \\Lambda ( \\omega + d \\omega ) = \\Lambda ( \\omega ) \\Lambda ( d \\omega ) , \\qquad \\Lambda ( \\xi + d \\xi ) = \\Lambda ( \\xi ) \\Lambda ( d \\xi ) , \\end{equation*}",
"\\begin{equation*} \\frac { d \\Lambda ( \\varphi _ { j } ) } { d \\varphi _ { j } } = i \\Lambda ( \\varphi _ { j } ) R _ { \\varphi _ { j } } , \\qquad \\frac { d \\Lambda ( \\xi ) } { d \\xi } = i \\Lambda ( \\xi ) K _ { \\xi } . \\end{equation*}",
"\\begin{equation*} \\Lambda ( \\varphi _ { j } ) = \\operatorname { e x p } ( i \\varphi _ { j } R _ { \\varphi _ { j } } ) , \\qquad \\Lambda ( \\xi ) = \\operatorname { e x p } ( i \\xi K _ { \\xi } ) . \\end{equation*}",
"\\begin{equation*} \\Lambda ( \\varphi , \\vec { u } ) = \\operatorname { e x p } [ i \\varphi ( \\vec { u } \\cdot \\vec { R } _ { \\varphi } ) ] ; \\qquad \\vec { R } _ { \\varphi } = ( R _ { \\varphi _ { 1 } } , R _ { \\varphi _ { 2 } } , R _ { \\varphi _ { 3 } } ) . \\end{equation*}",
"\\begin{equation*} \\frac { d \\Lambda ( \\psi _ { j } ) } { d \\psi _ { j } } = i f _ { j } ( \\psi _ { j } ) \\Lambda ( \\psi _ { j } ) N _ { j } , \\end{equation*}",
"\\begin{equation*} f _ { j } ( \\psi _ { j } ) = \\frac { 1 } { p _ { 0 } \\operatorname { c o s h } \\psi _ { j } + p _ { j } \\operatorname { s i n h } \\psi _ { j } + \\sqrt { p ^ { 2 } - \\kappa ^ { 2 } } } , \\end{equation*}",
"\\begin{equation*} N _ { j } = \\kappa R _ { \\varphi _ { j } } + \\epsilon _ { j k l } p _ { k } R _ { \\varphi _ { l } } . \\end{equation*}",
"\\begin{equation*} \\Lambda ( \\psi _ { j } ) = \\operatorname { e x p } ( i F ( \\psi _ { j } ) N _ { j } ) ; \\qquad F ( \\psi _ { j } ) = \\int _ { 0 } ^ { \\psi _ { j } } f _ { j } ( \\eta ) d \\eta . \\end{equation*}"
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"\\begin{equation*} \\mupPsi _ { 0 } ^ { \\prime } ( \\mitp ^ { \\prime } ) = \\mupLambda ( \\mitomega ) \\mupPsi _ { 0 } ( \\mitp ) , \\qquad \\mupPsi _ { 0 } ^ { \\prime } ( \\mitp ) = \\mupLambda ( \\mitxi ) \\mupPsi _ { 0 } ( \\mitp ) , \\end{equation*}",
"\\begin{equation*} \\mupLambda ( \\mitomega + \\mitd \\mitomega ) = \\mupLambda ( \\mitomega ) \\mupLambda ( \\mitd \\mitomega ) , \\qquad \\mupLambda ( \\mitxi + \\mitd \\mitxi ) = \\mupLambda ( \\mitxi ) \\mupLambda ( \\mitd \\mitxi ) , \\end{equation*}",
"\\begin{equation*} \\frac { \\mitd \\mupLambda ( \\mitvarphi _ { \\mitj } ) } { \\mitd \\mitvarphi _ { \\mitj } } = \\miti \\mupLambda ( \\mitvarphi _ { \\mitj } ) \\mitR _ { \\mitvarphi _ { \\mitj } } , \\qquad \\frac { \\mitd \\mupLambda ( \\mitxi ) } { \\mitd \\mitxi } = \\miti \\mupLambda ( \\mitxi ) \\mitK _ { \\mitxi } . \\end{equation*}",
"\\begin{equation*} \\mupLambda ( \\mitvarphi _ { \\mitj } ) = \\operatorname { e x p } ( \\miti \\mitvarphi _ { \\mitj } \\mitR _ { \\mitvarphi _ { \\mitj } } ) , \\qquad \\mupLambda ( \\mitxi ) = \\operatorname { e x p } ( \\miti \\mitxi \\mitK _ { \\mitxi } ) . \\end{equation*}",
"\\begin{equation*} \\mupLambda ( \\mitvarphi , \\vec { \\mitu } ) = \\operatorname { e x p } \\left[ \\miti \\mitvarphi \\left( \\vec { \\mitu } \\cdot \\vec { \\mitR } _ { \\mitvarphi } \\right) \\right] ; \\qquad \\vec { \\mitR } _ { \\mitvarphi } = \\left( \\mitR _ { \\mitvarphi _ { 1 } } , \\mitR _ { \\mitvarphi _ { 2 } } , \\mitR _ { \\mitvarphi _ { 3 } } \\right) . \\end{equation*}",
"\\begin{equation*} \\frac { \\mitd \\mupLambda ( \\mitpsi _ { \\mitj } ) } { \\mitd \\mitpsi _ { \\mitj } } = \\miti \\mitf _ { \\mitj } ( \\mitpsi _ { \\mitj } ) \\mupLambda ( \\mitpsi _ { \\mitj } ) \\mitN _ { \\mitj } , \\end{equation*}",
"\\begin{equation*} \\mitf _ { \\mitj } ( \\mitpsi _ { \\mitj } ) = \\frac { 1 } { \\mitp _ { 0 } \\operatorname { c o s h } \\mitpsi _ { \\mitj } + \\mitp _ { \\mitj } \\operatorname { s i n h } \\mitpsi _ { \\mitj } + \\sqrt { \\mitp ^ { 2 } - \\mitkappa ^ { 2 } } } , \\end{equation*}",
"\\begin{equation*} \\mitN _ { \\mitj } = \\mitkappa \\mitR _ { \\mitvarphi _ { \\mitj } } + \\mitepsilon _ { \\mitj \\mitk \\mitl } \\mitp _ { \\mitk } \\mitR _ { \\mitvarphi _ { \\mitl } } . \\end{equation*}",
"\\begin{equation*} \\mupLambda ( \\mitpsi _ { \\mitj } ) = \\operatorname { e x p } \\left( \\miti \\mitF ( \\mitpsi _ { \\mitj } ) \\mitN _ { \\mitj } \\right) ; \\qquad \\mitF ( \\mitpsi _ { \\mitj } ) = \\int _ { 0 } ^ { \\mitpsi _ { \\mitj } } \\mitf _ { \\mitj } ( \\miteta ) \\mitd \\miteta . \\end{equation*}"
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0003126_page20
|
{
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"\\begin {equation} \\Lambda (\\psi ,\\vec {u})=\\exp \\left ( iF(\\psi )N\\right ) ,\\qquad \\tanh \\psi =\\beta , \\label {s61} \\end {equation}",
"\\begin {equation} F(\\psi )=\\int _{0}^{\\psi }\\frac {d\\eta }{p_{0}\\cosh \\eta +\\vec {p}\\vec {u}\\sinh \\eta +\\sqrt {p^{2}-\\kappa ^{2}}}, \\label {s62} \\end {equation}",
"\\begin {equation} N=\\kappa \\vec {u}\\vec {R}_{\\varphi }+\\left ( \\vec {u}\\times \\vec {p}\\right ) \\cdot \\vec {R}_{\\varphi }. \\label {sa62} \\end {equation}",
"\\begin {equation} \\frac {d\\Lambda (t)}{dt}=\\Omega (t)\\Lambda (t), \\label {s63} \\end {equation}",
"\\begin {equation} \\Lambda (t)=\\exp \\left ( \\int _{0}^{t}\\Omega (\\eta )d\\eta \\right ) \\label {s64} \\end {equation}",
"\\begin {equation} \\left [ \\Omega (t),\\int _{0}^{t}\\Omega (\\eta )d\\eta \\right ] =0. \\label {s65} \\end {equation}",
"\\begin {equation} \\frac {d\\Lambda (t)}{dt}=\\frac {d}{dt}\\sum _{j=0}^{\\infty }\\frac {\\left ( \\int _{0}^{t}\\Omega (\\eta )d\\eta \\right ) ^{j}}{j!}=\\Omega (t)\\sum _{j=0}^{\\infty }\\frac {\\left ( \\int _{0}^{t}\\Omega (\\eta )d\\eta \\right ) ^{j}}{j!} \\label {s66} \\end {equation}",
"\\begin {equation} =\\Omega (t)\\Lambda (t)=\\Lambda (t)\\Omega (t). \\end {equation}",
"\\begin {equation} \\mathbf {R}_{\\omega }(\\Gamma )=Y^{-1}(p)\\mathbf {R}_{\\omega }(\\Gamma _{0})Y(p)=R_{\\omega }(\\Gamma )+i\\frac {d}{d\\omega }, \\label {s67} \\end {equation}",
"\\begin {equation} R_{\\omega }(\\Gamma )=Y^{-1}(p)R_{\\omega }(\\Gamma _{0})Y(p)+iY^{-1}(p)\\frac { dY(p)}{d\\omega }, \\end {equation}",
"\\begin {equation} K_{\\xi }(\\Gamma )=Y^{-1}(p)K_{\\xi }(\\Gamma _{0})Y(p). \\label {s68} \\end {equation}"
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"\\begin{equation*} \\Lambda ( t ) = \\operatorname { e x p } ( \\int _ { 0 } ^ { t } \\Omega ( \\eta ) d \\eta ) \\end{equation*}",
"\\begin{equation*} [ \\Omega ( t ) , \\int _ { 0 } ^ { t } \\Omega ( \\eta ) d \\eta ] = 0 . \\end{equation*}",
"\\begin{equation*} \\frac { d \\Lambda ( t ) } { d t } = \\frac { d } { d t } \\sum _ { j = 0 } ^ { \\infty } \\frac { { ( \\int _ { 0 } ^ { t } \\Omega ( \\eta ) d \\eta ) } ^ { j } } { j ! } = \\Omega ( t ) \\sum _ { j = 0 } ^ { \\infty } \\frac { { ( \\int _ { 0 } ^ { t } \\Omega ( \\eta ) d \\eta ) } ^ { j } } { j ! } \\end{equation*}",
"\\begin{equation*} = \\Omega ( t ) \\Lambda ( t ) = \\Lambda ( t ) \\Omega ( t ) . \\end{equation*}",
"\\begin{equation*} R _ { \\omega } ( \\Gamma ) = Y ^ { - 1 } ( p ) R _ { \\omega } ( \\Gamma _ { 0 } ) Y ( p ) = R _ { \\omega } ( \\Gamma ) + i \\frac { d } { d \\omega } , \\end{equation*}",
"\\begin{equation*} R _ { \\omega } ( \\Gamma ) = Y ^ { - 1 } ( p ) R _ { \\omega } ( \\Gamma _ { 0 } ) Y ( p ) + i Y ^ { - 1 } ( p ) \\frac { d Y ( p ) } { d \\omega } , \\end{equation*}",
"\\begin{equation*} K _ { \\xi } ( \\Gamma ) = Y ^ { - 1 } ( p ) K _ { \\xi } ( \\Gamma _ { 0 } ) Y ( p ) . \\end{equation*}"
],
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"$ \\vec { \\mitu } $",
"$ \\mitbeta $",
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"$ \\mitN $",
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"$ \\mitY ( \\mitp ) $",
"$ \\mitK _ { \\mitxi } ( \\mupGamma _ { 0 } ) $",
"\\begin{equation*} \\mupLambda ( \\mitpsi , \\vec { \\mitu } ) = \\operatorname { e x p } \\left( \\miti \\mitF ( \\mitpsi ) \\mitN \\right) , \\qquad \\operatorname { t a n h } \\mitpsi = \\mitbeta , \\end{equation*}",
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"\\begin{equation*} \\mupLambda ( \\mitt ) = \\operatorname { e x p } \\left( \\int _ { 0 } ^ { \\mitt } \\mupOmega ( \\miteta ) \\mitd \\miteta \\right) \\end{equation*}",
"\\begin{equation*} \\left[ \\mupOmega ( \\mitt ) , \\int _ { 0 } ^ { \\mitt } \\mupOmega ( \\miteta ) \\mitd \\miteta \\right] = 0 . \\end{equation*}",
"\\begin{equation*} \\frac { \\mitd \\mupLambda ( \\mitt ) } { \\mitd \\mitt } = \\frac { \\mitd } { \\mitd \\mitt } \\sum _ { \\mitj = 0 } ^ { \\infty } \\frac { { \\left( \\int _ { 0 } ^ { \\mitt } \\mupOmega ( \\miteta ) \\mitd \\miteta \\right) } ^ { \\mitj } } { \\mitj ! } = \\mupOmega ( \\mitt ) \\sum _ { \\mitj = 0 } ^ { \\infty } \\frac { { \\left( \\int _ { 0 } ^ { \\mitt } \\mupOmega ( \\miteta ) \\mitd \\miteta \\right) } ^ { \\mitj } } { \\mitj ! } \\end{equation*}",
"\\begin{equation*} = \\mupOmega ( \\mitt ) \\mupLambda ( \\mitt ) = \\mupLambda ( \\mitt ) \\mupOmega ( \\mitt ) . \\end{equation*}",
"\\begin{equation*} \\mbfR _ { \\mitomega } ( \\mupGamma ) = \\mitY ^ { - 1 } ( \\mitp ) \\mbfR _ { \\mitomega } ( \\mupGamma _ { 0 } ) \\mitY ( \\mitp ) = \\mitR _ { \\mitomega } ( \\mupGamma ) + \\miti \\frac { \\mitd } { \\mitd \\mitomega } , \\end{equation*}",
"\\begin{equation*} \\mitR _ { \\mitomega } ( \\mupGamma ) = \\mitY ^ { - 1 } ( \\mitp ) \\mitR _ { \\mitomega } ( \\mupGamma _ { 0 } ) \\mitY ( \\mitp ) + \\miti \\mitY ^ { - 1 } ( \\mitp ) \\frac { \\mitd \\mitY ( \\mitp ) } { \\mitd \\mitomega } , \\end{equation*}",
"\\begin{equation*} \\mitK _ { \\mitxi } ( \\mupGamma ) = \\mitY ^ { - 1 } ( \\mitp ) \\mitK _ { \\mitxi } ( \\mupGamma _ { 0 } ) \\mitY ( \\mitp ) . \\end{equation*}"
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0003126_page22
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{
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"$X_{3}$",
"$X_{3}=Z(\\Gamma _{0},X_{1},\\Gamma )^{-1}Z(\\Gamma _{0},X_{2},\\Gamma )$",
"$\\mathbf {R}_{\\omega }(\\Gamma ,X_{1}),\\mathbf {R}_{\\omega }(\\Gamma ,X_{2})$",
"$W$",
"$\\mathbf {R}_{\\omega }$",
"$K_{\\xi }.$",
"$W(p)$",
"\\begin {equation} \\lbrack Z(\\Gamma _{0},X_{1},\\Gamma )^{-1}Z(\\Gamma _{0},X_{2},\\Gamma ),\\Gamma ]=0. \\label {s80} \\end {equation}",
"\\begin {equation} Z(\\Gamma ,X_{3},\\Gamma )=Z(\\Gamma _{0},X_{1},\\Gamma )^{-1}Z(\\Gamma _{0},X_{2},\\Gamma ), \\label {s81} \\end {equation}",
"\\begin {equation} \\mathbf {R}_{\\omega }(\\Gamma ,X_{1})=Z(\\Gamma ,X_{3},\\Gamma )\\mathbf {R} _{\\omega }(\\Gamma ,X_{2})Z(\\Gamma ,X_{3},\\Gamma )^{-1}, \\label {s82} \\end {equation}",
"\\begin {equation} \\left ( \\Phi (p),\\Psi (q)\\right ) =\\left \\{ \\begin {array}{c} 0 \\\\ \\Phi ^{\\dagger }(p)W(p)\\Psi (q) \\end {array} \\right . \\quad \\mathrm {for}\\quad \\begin {array}{c} p\\neq q \\\\ p=q \\end {array} , \\label {s94} \\end {equation}",
"\\begin {equation} W^{\\dagger }(p)=W(p), \\label {s95} \\end {equation}",
"\\begin {equation} R_{\\omega }^{\\dagger }(p)W(p)-W(p)R_{\\omega }(p)+i\\frac {dW}{d\\omega }=0, \\label {s96} \\end {equation}",
"\\begin {equation} K_{\\xi }^{\\dagger }(p)W(p)-W(p)K_{\\xi }(p)=0. \\label {s97} \\end {equation}",
"\\begin {equation} \\Phi ^{\\prime \\dagger }(p^{\\prime })W(p^{\\prime })\\Psi ^{\\prime }(p^{\\prime }) \\label {s98} \\end {equation}",
"\\begin {equation} =\\Phi ^{\\dagger }(p)\\left ( I-id\\omega R_{\\omega }^{\\dagger }(p)\\right ) \\left ( W(p)+d\\omega \\frac {dW}{d\\omega }\\right ) \\left ( I+id\\omega R_{\\omega }(p)\\right ) \\Psi (p) \\end {equation}",
"\\begin {equation} \\Phi ^{\\prime \\dagger }(p^{\\prime })W(p^{\\prime })\\Psi ^{\\prime }(p^{\\prime })=\\Phi ^{\\dagger }(p)W(p)\\Psi (p). \\label {s99} \\end {equation}",
"\\begin {equation} R_{\\varphi _{j}}^{\\dagger }(\\Gamma _{0})=R_{\\varphi _{j}}(\\Gamma _{0}). \\label {s102} \\end {equation}"
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"$ R _ { \\omega } ( \\Gamma , X _ { 1 } ) , R _ { \\omega } ( \\Gamma , X _ { 2 } ) $",
"$ W $",
"$ R _ { \\omega } $",
"$ K _ { \\xi } . $",
"$ W ( p ) $",
"\\begin{equation*} [ Z ( \\Gamma _ { 0 } , X _ { 1 } , \\Gamma ) ^ { - 1 } Z ( \\Gamma _ { 0 } , X _ { 2 } , \\Gamma ) , \\Gamma ] = 0 . \\end{equation*}",
"\\begin{equation*} Z ( \\Gamma , X _ { 3 } , \\Gamma ) = Z ( \\Gamma _ { 0 } , X _ { 1 } , \\Gamma ) ^ { - 1 } Z ( \\Gamma _ { 0 } , X _ { 2 } , \\Gamma ) , \\end{equation*}",
"\\begin{equation*} R _ { \\omega } ( \\Gamma , X _ { 1 } ) = Z ( \\Gamma , X _ { 3 } , \\Gamma ) R _ { \\omega } ( \\Gamma , X _ { 2 } ) Z ( \\Gamma , X _ { 3 } , \\Gamma ) ^ { - 1 } , \\end{equation*}",
"\\begin{align*} ( \\Phi ( p ) , \\Psi ( q ) ) = \\{ \\begin{array}{c} 0 \\\\ \\Phi ^ { \\dagger } ( p ) W ( p ) \\Psi ( q ) \\end{array} \\quad f o r \\quad \\begin{array}{c} p \\ne q \\\\ p = q \\end{array} , \\end{align*}",
"\\begin{equation*} W ^ { \\dagger } ( p ) = W ( p ) , \\end{equation*}",
"\\begin{equation*} R _ { \\omega } ^ { \\dagger } ( p ) W ( p ) - W ( p ) R _ { \\omega } ( p ) + i \\frac { d W } { d \\omega } = 0 , \\end{equation*}",
"\\begin{equation*} K _ { \\xi } ^ { \\dagger } ( p ) W ( p ) - W ( p ) K _ { \\xi } ( p ) = 0 . \\end{equation*}",
"\\begin{equation*} \\Phi ^ { \\prime \\dagger } ( p ^ { \\prime } ) W ( p ^ { \\prime } ) \\Psi ^ { \\prime } ( p ^ { \\prime } ) \\end{equation*}",
"\\begin{equation*} = \\Phi ^ { \\dagger } ( p ) ( I - i d \\omega R _ { \\omega } ^ { \\dagger } ( p ) ) ( W ( p ) + d \\omega \\frac { d W } { d \\omega } ) ( I + i d \\omega R _ { \\omega } ( p ) ) \\Psi ( p ) \\end{equation*}",
"\\begin{equation*} \\Phi ^ { \\prime \\dagger } ( p ^ { \\prime } ) W ( p ^ { \\prime } ) \\Psi ^ { \\prime } ( p ^ { \\prime } ) = \\Phi ^ { \\dagger } ( p ) W ( p ) \\Psi ( p ) . \\end{equation*}",
"\\begin{equation*} R _ { \\varphi _ { j } } ^ { \\dagger } ( \\Gamma _ { 0 } ) = R _ { \\varphi _ { j } } ( \\Gamma _ { 0 } ) . \\end{equation*}"
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"$ \\mitX _ { 3 } = \\mitZ ( \\mupGamma _ { 0 } , \\mitX _ { 1 } , \\mupGamma ) ^ { - 1 } \\mitZ ( \\mupGamma _ { 0 } , \\mitX _ { 2 } , \\mupGamma ) $",
"$ \\mbfR _ { \\mitomega } ( \\mupGamma , \\mitX _ { 1 } ) , \\mbfR _ { \\mitomega } ( \\mupGamma , \\mitX _ { 2 } ) $",
"$ \\mitW $",
"$ \\mbfR _ { \\mitomega } $",
"$ \\mitK _ { \\mitxi } . $",
"$ \\mitW ( \\mitp ) $",
"\\begin{equation*} [ \\mitZ ( \\mupGamma _ { 0 } , \\mitX _ { 1 } , \\mupGamma ) ^ { - 1 } \\mitZ ( \\mupGamma _ { 0 } , \\mitX _ { 2 } , \\mupGamma ) , \\mupGamma ] = 0 . \\end{equation*}",
"\\begin{equation*} \\mitZ ( \\mupGamma , \\mitX _ { 3 } , \\mupGamma ) = \\mitZ ( \\mupGamma _ { 0 } , \\mitX _ { 1 } , \\mupGamma ) ^ { - 1 } \\mitZ ( \\mupGamma _ { 0 } , \\mitX _ { 2 } , \\mupGamma ) , \\end{equation*}",
"\\begin{equation*} \\mbfR _ { \\mitomega } ( \\mupGamma , \\mitX _ { 1 } ) = \\mitZ ( \\mupGamma , \\mitX _ { 3 } , \\mupGamma ) \\mbfR _ { \\mitomega } ( \\mupGamma , \\mitX _ { 2 } ) \\mitZ ( \\mupGamma , \\mitX _ { 3 } , \\mupGamma ) ^ { - 1 } , \\end{equation*}",
"\\begin{align*} \\left( \\mupPhi ( \\mitp ) , \\mupPsi ( \\mitq ) \\right) = \\left\\{ \\begin{array}{c} 0 \\\\ \\mupPhi ^ { \\dagger } ( \\mitp ) \\mitW ( \\mitp ) \\mupPsi ( \\mitq ) \\end{array} \\right. \\quad \\mathrm { f o r } \\quad \\begin{array}{c} \\mitp \\ne \\mitq \\\\ \\mitp = \\mitq \\end{array} , \\end{align*}",
"\\begin{equation*} \\mitW ^ { \\dagger } ( \\mitp ) = \\mitW ( \\mitp ) , \\end{equation*}",
"\\begin{equation*} \\mitR _ { \\mitomega } ^ { \\dagger } ( \\mitp ) \\mitW ( \\mitp ) - \\mitW ( \\mitp ) \\mitR _ { \\mitomega } ( \\mitp ) + \\miti \\frac { \\mitd \\mitW } { \\mitd \\mitomega } = 0 , \\end{equation*}",
"\\begin{equation*} \\mitK _ { \\mitxi } ^ { \\dagger } ( \\mitp ) \\mitW ( \\mitp ) - \\mitW ( \\mitp ) \\mitK _ { \\mitxi } ( \\mitp ) = 0 . \\end{equation*}",
"\\begin{equation*} \\mupPhi ^ { \\prime \\dagger } ( \\mitp ^ { \\prime } ) \\mitW ( \\mitp ^ { \\prime } ) \\mupPsi ^ { \\prime } ( \\mitp ^ { \\prime } ) \\end{equation*}",
"\\begin{equation*} = \\mupPhi ^ { \\dagger } ( \\mitp ) \\left( \\mitI - \\miti \\mitd \\mitomega \\mitR _ { \\mitomega } ^ { \\dagger } ( \\mitp ) \\right) \\left( \\mitW ( \\mitp ) + \\mitd \\mitomega \\frac { \\mitd \\mitW } { \\mitd \\mitomega } \\right) \\left( \\mitI + \\miti \\mitd \\mitomega \\mitR _ { \\mitomega } ( \\mitp ) \\right) \\mupPsi ( \\mitp ) \\end{equation*}",
"\\begin{equation*} \\mupPhi ^ { \\prime \\dagger } ( \\mitp ^ { \\prime } ) \\mitW ( \\mitp ^ { \\prime } ) \\mupPsi ^ { \\prime } ( \\mitp ^ { \\prime } ) = \\mupPhi ^ { \\dagger } ( \\mitp ) \\mitW ( \\mitp ) \\mupPsi ( \\mitp ) . \\end{equation*}",
"\\begin{equation*} \\mitR _ { \\mitvarphi _ { \\mitj } } ^ { \\dagger } ( \\mupGamma _ { 0 } ) = \\mitR _ { \\mitvarphi _ { \\mitj } } ( \\mupGamma _ { 0 } ) . \\end{equation*}"
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0003126_page24
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{
"latex": [
"$K_{\\xi },W,Z,Z^{-1}$",
"$\\Lambda $",
"$x-$",
"$p_{\\alpha }\\rightarrow i\\partial _{\\alpha }$",
"\\begin {equation} =\\frac {1}{(2\\pi )^{4}}\\int D(p)\\exp (-ipx)\\tilde {\\Psi }(y)\\exp (ipy)d^{4}yd^{4}p=\\frac {1}{(2\\pi )^{4}}\\int \\tilde {D}(x-y)\\tilde {\\Psi } (y)d^{4}y. \\end {equation}",
"\\begin {equation} \\Gamma _{0}(p)\\rightarrow \\tilde {\\Gamma }_{0}(z)=Q_{0}\\frac {1}{(2\\pi )^{4}} \\int (p^{2})^{1/n}\\exp (-ipz)d^{4}p;\\quad pz\\equiv p_{0}z_{0}-\\vec {p}\\vec {z}, \\label {s83} \\end {equation}",
"\\begin {equation} \\Gamma (p)\\rightarrow \\tilde {\\Gamma }(z)=\\sum _{\\lambda =0}^{3}Q_{\\lambda } \\frac {1}{(2\\pi )^{4}}\\int p_{\\lambda }^{2/n}\\exp (-ipz)d^{4}p, \\label {s84} \\end {equation}",
"\\begin {equation} \\mathbf {R}_{\\varphi _{j}}(\\Gamma _{0})\\rightarrow \\mathbf {\\tilde {R}} _{\\varphi _{j}}(\\Gamma _{0})=R_{\\varphi _{j}}(\\Gamma _{0})+i\\frac {d}{d\\tilde { \\varphi }_{j}};\\qquad \\frac {d}{d\\tilde {\\varphi }_{j}}=-\\epsilon _{jkl}x_{k} \\frac {\\partial }{\\partial x_{l}}, \\label {s85} \\end {equation}",
"\\begin {equation} \\mathbf {R}_{\\psi _{j}}(\\Gamma _{0})\\rightarrow \\mathbf {\\tilde {R}}_{\\psi _{j}}(z) \\label {s86} \\end {equation}",
"\\begin {equation} =\\frac {1}{(2\\pi )^{4}}\\int \\frac {\\kappa R_{\\varphi _{j}}(\\Gamma _{0})+\\epsilon _{jkl}p_{k}R_{\\varphi _{l}}(\\Gamma _{0})}{p_{0}+\\sqrt { p^{2}-\\kappa ^{2}}}\\exp (-ipz)d^{4}p+i\\frac {d}{d\\tilde {\\psi }_{j}}; \\end {equation}",
"\\begin {equation} \\frac {d}{d\\tilde {\\psi }_{j}}=-x_{0}\\frac {\\partial }{\\partial x_{j}}-x_{j} \\frac {\\partial }{\\partial x_{0}} \\end {equation}",
"\\begin {equation} \\mathbf {R}_{\\omega }(\\Gamma )\\rightarrow \\mathbf {\\tilde {R}}_{\\omega }(z) \\label {s87} \\end {equation}",
"\\begin {equation} =\\frac {1}{(2\\pi )^{4}}\\int Z(\\Gamma _{0},X,\\Gamma )^{-1}\\mathbf {R}_{\\omega }(\\Gamma _{0})Z(\\Gamma _{0},X,\\Gamma )\\exp (-ipz)d^{4}p. \\end {equation}",
"\\begin {equation} \\left ( \\Gamma _{0}(p)-\\mu \\right ) G_{0}(p)=I,\\qquad \\left ( \\Gamma (p)-\\mu \\right ) G(p)=I \\label {s88} \\end {equation}",
"\\begin {equation} G_{0}(p)=\\frac {(\\Gamma _{0}-\\alpha \\mu )(\\Gamma _{0}-\\alpha ^{2}\\mu )...(\\Gamma _{0}-\\alpha ^{n-1}\\mu )}{p^{2}-m^{2}}, \\label {s89} \\end {equation}",
"\\begin {equation} G(p)=\\frac {(\\Gamma -\\alpha \\mu )(\\Gamma -\\alpha ^{2}\\mu )...(\\Gamma -\\alpha ^{n-1}\\mu )}{p^{2}-m^{2}} \\label {sa89} \\end {equation}",
"\\begin {equation} G(p)=Z(\\Gamma _{0},X,\\Gamma )^{-1}G_{0}(p)Z(\\Gamma _{0},X,\\Gamma ). \\label {s90} \\end {equation}",
"\\begin {equation} \\tilde {G}_{0}(x)=\\frac {1}{(2\\pi )^{4}}\\int G_{0}(p)\\exp (-ipx)d^{4}p, \\label {sa90} \\end {equation}",
"\\begin {equation} \\tilde {G}(x)=\\frac {1}{(2\\pi )^{4}}\\int G(p)\\exp (-ipx)d^{4}p \\label {s91} \\end {equation}"
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"\\begin{equation*} \\Gamma _ { 0 } ( p ) \\rightarrow \\widetilde { \\Gamma } _ { 0 } ( z ) = Q _ { 0 } \\frac { 1 } { ( 2 \\pi ) ^ { 4 } } \\int ( p ^ { 2 } ) ^ { 1 \\slash n } \\operatorname { e x p } ( - i p z ) d ^ { 4 } p ; \\quad p z \\equiv p _ { 0 } z _ { 0 } - \\vec { p } \\vec { z } , \\end{equation*}",
"\\begin{equation*} \\Gamma ( p ) \\rightarrow \\widetilde { \\Gamma } ( z ) = \\sum _ { \\lambda = 0 } ^ { 3 } Q _ { \\lambda } \\frac { 1 } { ( 2 \\pi ) ^ { 4 } } \\int p _ { \\lambda } ^ { 2 \\slash n } \\operatorname { e x p } ( - i p z ) d ^ { 4 } p , \\end{equation*}",
"\\begin{equation*} R _ { \\varphi _ { j } } ( \\Gamma _ { 0 } ) \\rightarrow \\widetilde { R } _ { \\varphi _ { j } } ( \\Gamma _ { 0 } ) = R _ { \\varphi _ { j } } ( \\Gamma _ { 0 } ) + i \\frac { d } { d \\widetilde { \\varphi } _ { j } } ; \\qquad \\frac { d } { d \\widetilde { \\varphi } _ { j } } = - \\epsilon _ { j k l } x _ { k } \\frac { \\partial } { \\partial x _ { l } } , \\end{equation*}",
"\\begin{equation*} R _ { \\psi _ { j } } ( \\Gamma _ { 0 } ) \\rightarrow \\widetilde { R } _ { \\psi _ { j } } ( z ) \\end{equation*}",
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"\\begin{equation*} \\frac { d } { d \\widetilde { \\psi } _ { j } } = - x _ { 0 } \\frac { \\partial } { \\partial x _ { j } } - x _ { j } \\frac { \\partial } { \\partial x _ { 0 } } \\end{equation*}",
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"\\begin{equation*} = \\frac { 1 } { ( 2 \\pi ) ^ { 4 } } \\int Z ( \\Gamma _ { 0 } , X , \\Gamma ) ^ { - 1 } R _ { \\omega } ( \\Gamma _ { 0 } ) Z ( \\Gamma _ { 0 } , X , \\Gamma ) \\operatorname { e x p } ( - i p z ) d ^ { 4 } p . \\end{equation*}",
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0003126_page25
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0003194_page03
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"$ 2 . 2 $",
"\\begin{equation*} [ \\hat { Q } _ { i } , \\hat { H } ] = 0 , \\quad \\{ \\hat { Q } _ { i } , \\hat { Q } _ { j } \\} = \\delta _ { i j } \\hat { H } , \\qquad i = 1 , \\cdots N . \\end{equation*}"
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"$ \\mitS ^ { 1 } $",
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"$ \\hat { \\mitQ } _ { \\miti } $",
"$ \\hat { \\mitH } $",
"$ \\mitN = 2 $",
"$ \\hat { \\mitQ } _ { \\miti } \\vert \\mupPsi \\rangle = 0 $",
"$ \\hat { \\mitH } \\vert \\mupPsi \\rangle = 0 $",
"$ \\hat { \\mitQ } _ { \\miti } ( \\miti = 1 , 2 ) $",
"$ \\mitS ^ { 1 } $",
"$ 2 . 2 $",
"\\begin{equation*} [ \\hat { \\mitQ } _ { \\miti } , \\hat { \\mitH } ] = 0 , \\quad \\{ \\hat { \\mitQ } _ { \\miti } , \\hat { \\mitQ } _ { \\mitj } \\} = \\mitdelta _ { \\miti \\mitj } \\hat { \\mitH } , \\qquad \\miti = 1 , \\cdots \\mitN . \\end{equation*}"
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0003194_page04
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"$S^1$",
"$S^1$",
"$S^1$",
"${\\bf 2.2}$",
"$S^1$",
"$S^1$",
"$S^1$",
"$\\hg $",
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"$\\alpha $",
"$\\hw (\\hwd )$",
"$\\hg $",
"$\\hg $",
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"${\\cal H}_{\\alpha }$",
"${\\bf 1}_{\\alpha }$",
"${\\cal H}_{\\alpha }$",
"$\\hw \\ket {n+\\alpha }=\\ket {n+1+\\alpha }$",
"${\\cal H}_{\\alpha }$",
"${\\cal H}_{\\alpha }$",
"${\\cal H}_{\\beta }$",
"\\begin {equation} [\\hat G, \\hat W]=\\hbar ~{\\hat W}. \\label {comm} \\end {equation}",
"\\begin {equation} {\\hg } \\ket {\\alpha }=\\hbar ~\\alpha \\ket {\\alpha }\\qquad {\\rm with} \\qquad \\norm {\\alpha }{\\alpha }=1, \\end {equation}",
"\\begin {equation} {\\hg }{\\hw }\\ket {\\alpha }=\\hbar ~(\\alpha +1)\\hw \\ket {\\alpha },\\qquad {\\hg }{\\hwd }\\ket {\\alpha }=\\hbar ~(\\alpha -1)\\hwd \\ket {\\alpha }. \\end {equation}",
"\\begin {equation} \\ket {n+\\alpha }\\equiv {\\hw }^n \\ket {\\alpha }, \\qquad n={\\rm integer}, \\end {equation}",
"\\begin {equation} \\hg \\ket {n+\\alpha }=\\hbar ~(n+\\alpha )\\ket {n+\\alpha }. \\label {irred} \\end {equation}",
"\\begin {equation} \\norm {m+\\alpha }{n+\\alpha }=\\delta _{mn}, \\quad \\sum _{n=-\\infty }^{+\\infty }\\ket {n+\\alpha }\\bra {n+\\alpha }={\\bf 1}_{\\alpha }, \\label {irred2} \\end {equation}"
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"$ S ^ { 1 } $",
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"$ S ^ { 1 } $",
"$ S ^ { 1 } $",
"$ S ^ { 1 } $",
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"$ \\hat { W } $",
"$ \\hat { G } , \\hat { W } $",
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"$ \\alpha $",
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"$ \\hat { G } $",
"$ \\hat { G } $",
"$ \\alpha $",
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"$ \\hat { G } , \\hat { W } $",
"$ \\vert n + \\alpha \\rangle ( n = 0 , \\pm 1 , \\pm 2 , \\cdots ) $",
"$ H _ { \\alpha } $",
"$ 1 _ { \\alpha } $",
"$ H _ { \\alpha } $",
"$ \\hat { W } \\vert n + \\alpha \\rangle = \\vert n + 1 + \\alpha \\rangle $",
"$ H _ { \\alpha } $",
"$ H _ { \\alpha } $",
"$ H _ { \\beta } $",
"\\begin{equation*} [ \\hat { G } , \\hat { W } ] = \\hbar ~ \\hat { W } . \\end{equation*}",
"\\begin{equation*} \\hat { G } \\vert \\alpha \\rangle = \\hbar ~ \\alpha \\vert \\alpha \\rangle \\qquad w i t h \\qquad \\langle \\alpha \\vert \\alpha \\rangle = 1 , \\end{equation*}",
"\\begin{equation*} \\hat { G } \\hat { W } \\vert \\alpha \\rangle = \\hbar ~ ( \\alpha + 1 ) \\hat { W } \\vert \\alpha \\rangle , \\qquad \\hat { G } \\hat { W } ^ { \\dagger } \\vert \\alpha \\rangle = \\hbar ~ ( \\alpha - 1 ) \\hat { W } ^ { \\dagger } \\vert \\alpha \\rangle . \\end{equation*}",
"\\begin{equation*} \\vert n + \\alpha \\rangle \\equiv \\hat { W } ^ { n } \\vert \\alpha \\rangle , \\qquad n = i n t e g e r , \\end{equation*}",
"\\begin{equation*} \\hat { G } \\vert n + \\alpha \\rangle = \\hbar ~ ( n + \\alpha ) \\vert n + \\alpha \\rangle . \\end{equation*}",
"\\begin{equation*} \\langle m + \\alpha \\vert n + \\alpha \\rangle = \\delta _ { m n } , \\quad \\sum _ { n = - \\infty } ^ { + \\infty } \\vert n + \\alpha \\rangle \\langle n + \\alpha \\vert = 1 _ { \\alpha } , \\end{equation*}"
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"$ \\mitS ^ { 1 } $",
"$ \\mitS ^ { 1 } $",
"$ 2 . 2 $",
"$ \\mitS ^ { 1 } $",
"$ \\mitS ^ { 1 } $",
"$ \\mitS ^ { 1 } $",
"$ \\hat { \\mitG } $",
"$ \\hat { \\mitW } $",
"$ \\hat { \\mitG } , \\hat { \\mitW } $",
"$ \\hat { \\mitW } ^ { \\dagger } $",
"$ \\mitalpha $",
"$ \\hat { \\mitW } ( \\hat { \\mitW } ^ { \\dagger } ) $",
"$ \\hat { \\mitG } $",
"$ \\hat { \\mitG } $",
"$ \\mitalpha $",
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"$ \\hat { \\mitG } , \\hat { \\mitW } $",
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"$ \\mitH _ { \\mitalpha } $",
"$ 1 _ { \\mitalpha } $",
"$ \\mitH _ { \\mitalpha } $",
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"$ \\mitH _ { \\mitalpha } $",
"$ \\mitH _ { \\mitalpha } $",
"$ \\mitH _ { \\mitbeta } $",
"\\begin{equation*} [ \\hat { \\mitG } , \\hat { \\mitW } ] = \\hslash ~ \\hat { \\mitW } . \\end{equation*}",
"\\begin{equation*} \\hat { \\mitG } \\vert \\mitalpha \\rangle = \\hslash ~ \\mitalpha \\vert \\mitalpha \\rangle \\qquad \\mathrm { w i t h } \\qquad \\langle \\mitalpha \\vert \\mitalpha \\rangle = 1 , \\end{equation*}",
"\\begin{equation*} \\hat { \\mitG } \\hat { \\mitW } \\vert \\mitalpha \\rangle = \\hslash ~ ( \\mitalpha + 1 ) \\hat { \\mitW } \\vert \\mitalpha \\rangle , \\qquad \\hat { \\mitG } \\hat { \\mitW } ^ { \\dagger } \\vert \\mitalpha \\rangle = \\hslash ~ ( \\mitalpha - 1 ) \\hat { \\mitW } ^ { \\dagger } \\vert \\mitalpha \\rangle . \\end{equation*}",
"\\begin{equation*} \\vert \\mitn + \\mitalpha \\rangle \\equiv \\hat { \\mitW } ^ { \\mitn } \\vert \\mitalpha \\rangle , \\qquad \\mitn = \\mathrm { i n t e g e r } , \\end{equation*}",
"\\begin{equation*} \\hat { \\mitG } \\vert \\mitn + \\mitalpha \\rangle = \\hslash ~ ( \\mitn + \\mitalpha ) \\vert \\mitn + \\mitalpha \\rangle . \\end{equation*}",
"\\begin{equation*} \\langle \\mitm + \\mitalpha \\vert \\mitn + \\mitalpha \\rangle = \\mitdelta _ { \\mitm \\mitn } , \\quad \\sum _ { \\mitn = - \\infty } ^ { + \\infty } \\vert \\mitn + \\mitalpha \\rangle \\langle \\mitn + \\mitalpha \\vert = 1 _ { \\mitalpha } , \\end{equation*}"
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|
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0003194_page05
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{
"latex": [
"$\\alpha - \\beta ={\\rm integer}$",
"$\\cal H$",
"$\\alpha $",
"$\\cal H$",
"${\\cal H}_{\\alpha }$",
"${\\cal H}_{\\alpha } (0 \\leq \\alpha < 1)$",
"$\\alpha $",
"$\\hg $",
"$\\hw $",
"$S^1$",
"${\\cal H}_{\\alpha }$",
"$\\hw $",
"$\\theta $",
"$\\kappa (\\theta )$",
"$\\abs {\\kappa (\\theta )}=1$",
"$\\kappa (\\theta +2\\pi )=\\kappa (\\theta )$",
"${\\bf 1}_{\\alpha }$",
"${\\cal H}_{\\alpha }$",
"$\\hw $",
"$\\hg $",
"$\\hw $",
"$S^1$",
"$\\ket {\\psi }$",
"$\\psi (\\theta )$",
"$S^1$",
"$\\ket {\\psi }$",
"$\\hw $",
"$\\hg $",
"\\begin {equation} \\hw \\ket {\\theta }= \\e ^{i\\theta }\\ket {\\theta }. \\label {eigen} \\end {equation}",
"\\begin {equation} \\ket {\\theta } = \\kappa (\\theta )\\sum _{n=-\\infty }^{+\\infty } \\e ^{-i n \\theta }\\ket {n+\\alpha }, \\end {equation}",
"\\begin {eqnarray} \\ket {\\theta +2\\pi n}&=&\\ket {\\theta }, \\qquad n={\\rm integer}, \\\\ \\norm {\\theta }{\\theta ^{\\prime }}&=&2\\pi \\sum _{n=-\\infty }^{{}n=+\\infty } \\delta (\\theta -\\theta ^{\\prime }+2\\pi n), \\\\ \\int _0^{2\\pi }{{d\\theta }\\over {2\\pi }}\\ket {\\theta }\\bra {\\theta }&=& \\sum _{n=-\\infty }^{{}+\\infty }\\ket {n+\\alpha }\\bra {n+\\alpha }={\\bf 1}_{\\alpha }, \\\\ {\\rm exp}(-i\\lambda {\\hg \\over \\hbar })\\ket {\\theta }&=& \\e ^{-i\\lambda \\alpha }\\kappa (\\theta )\\kappa ^*(\\theta +\\lambda ) \\ket {\\theta +\\lambda }, \\end {eqnarray}",
"\\begin {equation} \\psi (\\theta )\\equiv \\norm {\\theta }{\\psi }. \\end {equation}",
"\\begin {equation} \\bra {\\theta }{\\rm exp}(i\\lambda {\\hg \\over \\hbar })\\ket {\\psi } =\\e ^{i\\lambda \\alpha }\\kappa ^*(\\theta )\\kappa (\\theta +\\lambda ) \\norm {\\theta +\\lambda }{\\psi }, \\end {equation}",
"\\begin {equation} \\bra {\\theta }{\\hg }\\ket {\\psi }= \\Bigl [-i\\hbar ~{\\del \\over {\\del \\theta }}-i\\hbar ~\\kappa ^*(\\theta ) {{\\del \\kappa (\\theta )}\\over {\\del \\theta }}+\\hbar ~\\alpha \\Bigr ]\\psi (\\theta ). \\label {repreg} \\end {equation}"
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"$ \\hat { W } $",
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"$ \\hat { G } $",
"\\begin{equation*} \\hat { W } \\vert \\theta \\rangle = e ^ { i \\theta } \\vert \\theta \\rangle . \\end{equation*}",
"\\begin{equation*} \\vert \\theta \\rangle = \\kappa ( \\theta ) \\sum _ { n = - \\infty } ^ { + \\infty } e ^ { - i n \\theta } \\vert n + \\alpha \\rangle , \\end{equation*}",
"\\begin{align*} \\vert \\theta + 2 \\pi n \\rangle & = & \\vert \\theta \\rangle , \\qquad n = i n t e g e r , \\\\ \\langle \\theta \\vert \\theta ^ { \\prime } \\rangle & = & 2 \\pi \\sum _ { n = - \\infty } ^ { n = + \\infty } \\delta ( \\theta - \\theta ^ { \\prime } + 2 \\pi n ) , \\\\ \\int _ { 0 } ^ { 2 \\pi } \\frac { d \\theta } { 2 \\pi } \\vert \\theta \\rangle \\langle \\theta \\vert & = & \\sum _ { n = - \\infty } ^ { + \\infty } \\vert n + \\alpha \\rangle \\langle n + \\alpha \\vert = 1 _ { \\alpha } , \\\\ e x p ( - i \\lambda \\frac { \\hat { G } } { \\hbar } ) \\vert \\theta \\rangle & = & e ^ { - i \\lambda \\alpha } \\kappa ( \\theta ) \\kappa ^ { \\ast } ( \\theta + \\lambda ) \\vert \\theta + \\lambda \\rangle , \\end{align*}",
"\\begin{equation*} \\psi ( \\theta ) \\equiv \\langle \\theta \\vert \\psi \\rangle . \\end{equation*}",
"\\begin{equation*} \\langle \\theta \\vert e x p ( i \\lambda \\frac { \\hat { G } } { \\hbar } ) \\vert \\psi \\rangle = e ^ { i \\lambda \\alpha } \\kappa ^ { \\ast } ( \\theta ) \\kappa ( \\theta + \\lambda ) \\langle \\theta + \\lambda \\vert \\psi \\rangle , \\end{equation*}",
"\\begin{equation*} \\langle \\theta \\vert \\hat { G } \\vert \\psi \\rangle = [ - i \\hbar ~ \\frac { \\partial } { \\partial \\theta } - i \\hbar ~ \\kappa ^ { \\ast } ( \\theta ) \\frac { \\partial \\kappa ( \\theta ) } { \\partial \\theta } + \\hbar ~ \\alpha ] \\psi ( \\theta ) . \\end{equation*}"
],
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"$ \\mitH $",
"$ \\mitalpha $",
"$ \\mitH $",
"$ \\mitH _ { \\mitalpha } $",
"$ \\mitH _ { \\mitalpha } ( 0 \\leq \\mitalpha < 1 ) $",
"$ \\mitalpha $",
"$ \\hat { \\mitG } $",
"$ \\hat { \\mitW } $",
"$ \\mitS ^ { 1 } $",
"$ \\mitH _ { \\mitalpha } $",
"$ \\hat { \\mitW } $",
"$ \\mittheta $",
"$ \\mitkappa ( \\mittheta ) $",
"$ \\left\\vert \\mitkappa ( \\mittheta ) \\right\\vert = 1 $",
"$ \\mitkappa ( \\mittheta + 2 \\mitpi ) = \\mitkappa ( \\mittheta ) $",
"$ 1 _ { \\mitalpha } $",
"$ \\mitH _ { \\mitalpha } $",
"$ \\hat { \\mitW } $",
"$ \\hat { \\mitG } $",
"$ \\hat { \\mitW } $",
"$ \\mitS ^ { 1 } $",
"$ \\vert \\mitpsi \\rangle $",
"$ \\mitpsi ( \\mittheta ) $",
"$ \\mitS ^ { 1 } $",
"$ \\vert \\mitpsi \\rangle $",
"$ \\hat { \\mitW } $",
"$ \\hat { \\mitG } $",
"\\begin{equation*} \\hat { \\mitW } \\vert \\mittheta \\rangle = \\mathrm { e } ^ { \\miti \\mittheta } \\vert \\mittheta \\rangle . \\end{equation*}",
"\\begin{equation*} \\vert \\mittheta \\rangle = \\mitkappa ( \\mittheta ) \\sum _ { \\mitn = - \\infty } ^ { + \\infty } \\mathrm { e } ^ { - \\miti \\mitn \\mittheta } \\vert \\mitn + \\mitalpha \\rangle , \\end{equation*}",
"\\begin{align*} \\displaystyle \\vert \\mittheta + 2 \\mitpi \\mitn \\rangle & = & \\displaystyle \\vert \\mittheta \\rangle , \\qquad \\mitn = \\mathrm { i n t e g e r } , \\\\ \\displaystyle \\langle \\mittheta \\vert \\mittheta ^ { \\prime } \\rangle & = & \\displaystyle 2 \\mitpi \\sum _ { \\mitn = - \\infty } ^ { \\mitn = + \\infty } \\mitdelta ( \\mittheta - \\mittheta ^ { \\prime } + 2 \\mitpi \\mitn ) , \\\\ \\displaystyle \\int _ { 0 } ^ { 2 \\mitpi } \\frac { \\mitd \\mittheta } { 2 \\mitpi } \\vert \\mittheta \\rangle \\langle \\mittheta \\vert & = & \\displaystyle \\sum _ { \\mitn = - \\infty } ^ { + \\infty } \\vert \\mitn + \\mitalpha \\rangle \\langle \\mitn + \\mitalpha \\vert = 1 _ { \\mitalpha } , \\\\ \\displaystyle \\mathrm { e x p } ( - \\miti \\mitlambda \\frac { \\hat { \\mitG } } { \\hslash } ) \\vert \\mittheta \\rangle & = & \\displaystyle \\mathrm { e } ^ { - \\miti \\mitlambda \\mitalpha } \\mitkappa ( \\mittheta ) \\mitkappa ^ { \\ast } ( \\mittheta + \\mitlambda ) \\vert \\mittheta + \\mitlambda \\rangle , \\end{align*}",
"\\begin{equation*} \\mitpsi ( \\mittheta ) \\equiv \\langle \\mittheta \\vert \\mitpsi \\rangle . \\end{equation*}",
"\\begin{equation*} \\langle \\mittheta \\vert \\mathrm { e x p } ( \\miti \\mitlambda \\frac { \\hat { \\mitG } } { \\hslash } ) \\vert \\mitpsi \\rangle = \\mathrm { e } ^ { \\miti \\mitlambda \\mitalpha } \\mitkappa ^ { \\ast } ( \\mittheta ) \\mitkappa ( \\mittheta + \\mitlambda ) \\langle \\mittheta + \\mitlambda \\vert \\mitpsi \\rangle , \\end{equation*}",
"\\begin{equation*} \\langle \\mittheta \\vert \\hat { \\mitG } \\vert \\mitpsi \\rangle = \\Big [ - \\miti \\hslash ~ \\frac { \\mitpartial } { \\mitpartial \\mittheta } - \\miti \\hslash ~ \\mitkappa ^ { \\ast } ( \\mittheta ) \\frac { \\mitpartial \\mitkappa ( \\mittheta ) } { \\mitpartial \\mittheta } + \\hslash ~ \\mitalpha \\Big ] \\mitpsi ( \\mittheta ) . \\end{equation*}"
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0003194_page06
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"$S^1$",
"$S^1$",
"$\\psi (\\theta +2\\pi n) =\\psi (\\theta )$",
"$S^1$",
"$\\alpha $",
"$\\kappa (\\theta )$",
"$\\kappa (\\theta )=\\omega (\\theta )\\kappa ^{\\prime }(\\theta )$",
"$\\omega (\\theta )$",
"$\\abs {\\omega (\\theta )}=1$",
"$\\omega (\\theta +2\\pi )=\\omega (\\theta )$",
"$\\ket {\\theta }=\\omega (\\theta )\\ket {\\theta }^{\\prime }$",
"$\\psi ^{\\prime }(\\theta )$",
"$\\hw $",
"$\\hg $",
"$\\alpha $",
"$A(\\theta )$",
"$A(\\theta +2\\pi )=A(\\theta )$",
"$A^{\\prime }(\\theta )=\\alpha $",
"$A^{\\prime }(\\theta )=\\alpha $",
"$A^{\\prime }(\\theta )=\\beta $",
"$\\beta -\\alpha $",
"$A_{\\alpha }\\equiv \\alpha (0\\leq \\alpha < 1)$",
"$\\kappa (\\theta )=1$",
"$S^1$",
"$0\\leq \\alpha <1$",
"$ \\psi (\\theta )\\rightarrow \\psi ^{\\prime }(\\theta )=\\e ^{in\\theta }\\psi (\\theta ), $",
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"$A(\\theta )-n\\hbar $",
"$n=$",
"$n$",
"$\\alpha =0$",
"$S^1$",
"${\\bf R}^2$",
"\\begin {equation} \\bra {\\theta }{\\hw }\\ket {\\psi }=\\e ^{i\\theta }\\psi (\\theta ). \\label {reprew} \\end {equation}",
"\\begin {equation} \\norm {\\chi }{\\psi }=\\int _0^{2\\pi }{{d\\theta }\\over {2\\pi }} \\chi ^*(\\theta )\\psi (\\theta ). \\end {equation}",
"\\begin {equation} \\psi ^{\\prime }(\\theta )=\\omega (\\theta )\\psi (\\theta ). \\label {gtrf1} \\end {equation}",
"\\begin {equation} '\\bra {\\theta }{\\hg }\\ket {\\psi }= \\Bigl [-i\\hbar ~{\\del \\over {\\del \\theta }}+A^{\\prime }(\\theta )\\Bigr ] \\psi (\\theta ), \\label {newrepreg} \\end {equation}",
"\\begin {equation} A^{\\prime }(\\theta ) \\equiv A(\\theta )+i\\hbar ~\\omega ^*(\\theta ) {{\\del \\omega (\\theta )}\\over {\\del \\theta }}, \\quad A(\\theta )\\equiv -i\\hbar ~\\kappa ^*({\\theta }) {{\\del \\kappa (\\theta )}\\over {\\del \\theta }}+\\hbar ~\\alpha . \\label {gtrf2} \\end {equation}"
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"$ S ^ { 1 } $",
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"$ S ^ { 1 } $",
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"$ \\kappa ( \\theta ) $",
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"$ \\omega ( \\theta ) $",
"$ \\vert \\omega ( \\theta ) \\vert = 1 $",
"$ \\omega ( \\theta + 2 \\pi ) = \\omega ( \\theta ) $",
"$ \\vert \\theta \\rangle = \\omega ( \\theta ) \\vert \\theta \\rangle ^ { \\prime } $",
"$ \\psi ^ { \\prime } ( \\theta ) $",
"$ \\hat { W } $",
"$ \\hat { G } $",
"$ \\alpha $",
"$ A ( \\theta ) $",
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"$ A ^ { \\prime } ( \\theta ) = \\alpha $",
"$ A ^ { \\prime } ( \\theta ) = \\alpha $",
"$ A ^ { \\prime } ( \\theta ) = \\beta $",
"$ \\beta - \\alpha $",
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"$ \\kappa ( \\theta ) = 1 $",
"$ S ^ { 1 } $",
"$ 0 \\leq \\alpha < 1 $",
"$ \\psi ( \\theta ) \\rightarrow \\psi ^ { \\prime } ( \\theta ) = { e } ^ { i n \\theta } \\psi ( \\theta ) , $",
"$ A ( \\theta ) $",
"$ A ( \\theta ) - n \\hbar $",
"$ n = $",
"$ n $",
"$ \\alpha = 0 $",
"$ S ^ { 1 } $",
"$ R ^ { 2 } $",
"\\begin{equation*} \\langle \\theta \\vert \\hat { W } \\vert \\psi \\rangle = e ^ { i \\theta } \\psi ( \\theta ) . \\end{equation*}",
"\\begin{equation*} \\langle \\chi \\vert \\psi \\rangle = \\int _ { 0 } ^ { 2 \\pi } \\frac { d \\theta } { 2 \\pi } \\chi ^ { \\ast } ( \\theta ) \\psi ( \\theta ) . \\end{equation*}",
"\\begin{equation*} \\psi ^ { \\prime } ( \\theta ) = \\omega ( \\theta ) \\psi ( \\theta ) . \\end{equation*}",
"\\begin{equation*} { } ^ { \\prime } \\langle \\theta \\vert \\hat { G } \\vert \\psi \\rangle = [ - i \\hbar ~ \\frac { \\partial } { \\partial \\theta } + A ^ { \\prime } ( \\theta ) ] \\psi ( \\theta ) , \\end{equation*}",
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"$ \\mitR ^ { 2 } $",
"\\begin{equation*} \\langle \\mittheta \\vert \\hat { \\mitW } \\vert \\mitpsi \\rangle = \\mathrm { e } ^ { \\miti \\mittheta } \\mitpsi ( \\mittheta ) . \\end{equation*}",
"\\begin{equation*} \\langle \\mitchi \\vert \\mitpsi \\rangle = \\int _ { 0 } ^ { 2 \\mitpi } \\frac { \\mitd \\mittheta } { 2 \\mitpi } \\mitchi ^ { \\ast } ( \\mittheta ) \\mitpsi ( \\mittheta ) . \\end{equation*}",
"\\begin{equation*} \\mitpsi ^ { \\prime } ( \\mittheta ) = \\mitomega ( \\mittheta ) \\mitpsi ( \\mittheta ) . \\end{equation*}",
"\\begin{equation*} { } ^ { \\prime } \\langle \\mittheta \\vert \\hat { \\mitG } \\vert \\mitpsi \\rangle = \\Big [ - \\miti \\hslash ~ \\frac { \\mitpartial } { \\mitpartial \\mittheta } + \\mitA ^ { \\prime } ( \\mittheta ) \\Big ] \\mitpsi ( \\mittheta ) , \\end{equation*}",
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0003194_page07
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{
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"$\\psi ^{\\prime }(\\theta )$",
"$0\\leq A(\\theta ) < \\hbar $",
"$0\\leq \\alpha <1$",
"$\\e ^{in\\theta }$",
"$n=$",
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"\\begin {equation} \\{\\hxi ,\\hxib \\}=1,\\qquad \\hxi ^2=\\hxib ^2=0. \\label {anticomm} \\end {equation}",
"\\begin {eqnarray} \\hat H &=& \\{{\\hat Q},{\\hat {\\bar Q}}\\} \\\\ &=& {1\\over {2m R^2}}\\hg ^2+V^2(\\hw ,\\hwd )\\\\ &-&{i\\over {{\\sqrt {2m}R}}} \\Bigl (\\hg V(\\hw ,\\hwd )-V(\\hw ,\\hwd )\\hg \\Bigr )\\Bigl [\\hxi , \\hxib \\Bigr ], \\end {eqnarray}"
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"\\begin{align*} \\hat { Q } & \\equiv & \\frac { 1 } { \\sqrt { 2 } } ( \\hat { Q } _ { 1 } + i \\hat { Q } _ { 2 } ) \\\\ & = & ( \\frac { 1 } { \\sqrt { 2 m } R } \\hat { G } + i V ( \\hat { W } , \\hat { W } ^ { \\dagger } ) ) \\hat { \\xi } \\equiv \\hat { q } ~ \\hat { \\xi } , \\\\ \\hat { \\bar { Q } } & \\equiv & \\frac { 1 } { \\sqrt { 2 } } ( \\hat { Q } _ { 1 } - i \\hat { Q } _ { 2 } ) \\\\ & = & ( \\frac { 1 } { \\sqrt { 2 m } R } \\hat { G } - i V ( \\hat { W } , \\hat { W } ^ { \\dagger } ) ) \\hat { \\bar { \\xi } } \\equiv \\hat { q } ^ { \\dagger } ~ \\hat { \\bar { \\xi } } . \\end{align*}",
"\\begin{equation*} \\{ \\hat { \\xi } , \\hat { \\bar { \\xi } } \\} = 1 , \\qquad \\hat { \\xi } ^ { 2 } = \\hat { \\bar { \\xi } } ^ { 2 } = 0 . \\end{equation*}",
"\\begin{align*} \\hat { H } & = & \\{ \\hat { Q } , \\hat { \\bar { Q } } \\} \\\\ & = & \\frac { 1 } { 2 m R ^ { 2 } } \\hat { G } ^ { 2 } + V ^ { 2 } ( \\hat { W } , \\hat { W } ^ { \\dagger } ) \\\\ & - & \\frac { i } { \\sqrt { 2 m } R } ( \\hat { G } V ( \\hat { W } , \\hat { W } ^ { \\dagger } ) - V ( \\hat { W } , \\hat { W } ^ { \\dagger } ) \\hat { G } ) [ \\hat { \\xi } , \\hat { \\bar { \\xi } } ] , \\end{align*}"
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"$ 0 \\leq \\mitA ( \\mittheta ) < \\hslash $",
"$ 0 \\leq \\mitalpha < 1 $",
"$ { \\mathrm { e } } ^ { \\miti \\mitn \\mittheta } $",
"$ \\mitn = $",
"$ \\mitS ^ { 1 } $",
"$ \\mitalpha $",
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"$ \\mitS ^ { 1 } $",
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"$ \\hat { \\mitW } $",
"$ \\mitS ^ { 1 } $",
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"$ \\hat { \\mitQ } _ { \\miti } ( \\miti = 1 , 2 ) $",
"$ \\mitV ( \\hat { \\mitW } , \\hat { \\mitW } ^ { \\dagger } ) $",
"$ \\hat { \\mitW } $",
"$ \\hat { \\mitW } ^ { \\dagger } $",
"$ \\mitm $",
"$ \\mitR $",
"$ \\mitS ^ { 1 } $",
"$ \\hat { \\mitxi } , \\hat { \\bar { \\mitxi } } $",
"\\begin{align*} \\displaystyle \\hat { \\mitQ } & \\displaystyle \\equiv & \\displaystyle \\frac { 1 } { \\sqrt { 2 } } \\Big ( \\hat { \\mitQ } _ { 1 } + \\miti \\hat { \\mitQ } _ { 2 } \\Big ) \\\\ & = & \\displaystyle \\Big ( \\frac { 1 } { \\sqrt { 2 \\mitm } \\mitR } \\hat { \\mitG } + \\miti \\mitV ( \\hat { \\mitW } , \\hat { \\mitW } ^ { \\dagger } ) \\Big ) \\hat { \\mitxi } \\equiv \\hat { \\mitq } ~ \\hat { \\mitxi } , \\\\ \\displaystyle \\hat { \\bar { \\mitQ } } & \\displaystyle \\equiv & \\displaystyle \\frac { 1 } { \\sqrt { 2 } } \\Big ( \\hat { \\mitQ } _ { 1 } - \\miti \\hat { \\mitQ } _ { 2 } \\Big ) \\\\ & = & \\displaystyle \\Big ( \\frac { 1 } { \\sqrt { 2 \\mitm } \\mitR } \\hat { \\mitG } - \\miti \\mitV ( \\hat { \\mitW } , \\hat { \\mitW } ^ { \\dagger } ) \\Big ) \\hat { \\bar { \\mitxi } } \\equiv \\hat { \\mitq } ^ { \\dagger } ~ \\hat { \\bar { \\mitxi } } . \\end{align*}",
"\\begin{equation*} \\{ \\hat { \\mitxi } , \\hat { \\bar { \\mitxi } } \\} = 1 , \\qquad \\hat { \\mitxi } ^ { 2 } = \\hat { \\bar { \\mitxi } } ^ { 2 } = 0 . \\end{equation*}",
"\\begin{align*} \\displaystyle \\hat { \\mitH } & = & \\displaystyle \\{ \\hat { \\mitQ } , \\hat { \\bar { \\mitQ } } \\} \\\\ & = & \\displaystyle \\frac { 1 } { 2 \\mitm \\mitR ^ { 2 } } \\hat { \\mitG } ^ { 2 } + \\mitV ^ { 2 } ( \\hat { \\mitW } , \\hat { \\mitW } ^ { \\dagger } ) \\\\ & \\displaystyle - & \\displaystyle \\frac { \\miti } { \\sqrt { 2 \\mitm } \\mitR } \\Big ( \\hat { \\mitG } \\mitV ( \\hat { \\mitW } , \\hat { \\mitW } ^ { \\dagger } ) - \\mitV ( \\hat { \\mitW } , \\hat { \\mitW } ^ { \\dagger } ) \\hat { \\mitG } \\Big ) \\Big [ \\hat { \\mitxi } , \\hat { \\bar { \\mitxi } } \\Big ] , \\end{align*}"
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0003194_page08
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"$P_{\\theta }$",
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"$2\\theta $",
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"$S^1$",
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"$\\hg =-i\\hbar ~\\del /\\del \\theta +\\hbar ~\\alpha $",
"$\\alpha $",
"\\begin {equation} \\{P_{\\theta }, \\e ^{i\\theta }\\}_P=-i \\e ^{i\\theta }. \\end {equation}",
"\\begin {equation} \\hat H\\rightarrow H_{cl}={P_{\\theta }^2\\over {2mR^2}}+ V^2(\\e ^{i\\theta },\\e ^{-i\\theta }). \\end {equation}",
"\\begin {equation} V(\\e ^{i\\theta },\\e ^{-i\\theta })=\\sqrt {{{mg_NR}\\over 2}}\\sin \\theta , \\label {poten1} \\end {equation}",
"\\begin {equation} V(\\hw ,\\hwd )=\\sqrt {{mg_NR}\\over {2}}\\Bigl ({{\\hw -\\hwd }\\over {2i}}\\Bigr ). \\end {equation}",
"\\begin {equation} \\hat H= {1\\over {2mR^2}}\\hg ^2+{{mg_NR}\\over 2}\\sin ^2\\theta -{\\hbar \\over 2}\\sqrt {{g_N\\over R}}\\cos \\theta \\Bigl [\\hxi , \\hxib \\Bigr ] \\label {hamilton2} \\end {equation}",
"\\begin {equation} \\hq \\ket {\\Psi }=0\\qquad {\\rm and}\\qquad \\hqb \\ket {\\Psi }=0. \\label {ground1} \\end {equation}",
"\\begin {equation} \\hxi =\\left (\\begin {array}{cc} 0&0\\\\ 1&0\\end {array}\\right ),\\qquad \\hxib =\\left (\\begin {array}{cc} 0&1\\\\ 0&0\\end {array}\\right ) \\label {supercharge2} \\end {equation}"
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"$ i \\hbar $",
"$ { e } ^ { i \\theta } $",
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"$ \\hat { W } $",
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"$ \\hat { \\xi } , \\hat { \\bar { \\xi } } $",
"$ V ( { e } ^ { i \\theta } , e ^ { - i \\theta } ) $",
"$ 2 \\theta $",
"$ g _ { N } $",
"$ S ^ { 1 } $",
"$ \\hat { W } $",
"$ \\hat { G } = - i \\hbar ~ \\partial \\slash \\partial \\theta + \\hbar ~ \\alpha $",
"$ \\alpha $",
"\\begin{equation*} \\{ P _ { \\theta } , e ^ { i \\theta } \\} _ { P } = - i e ^ { i \\theta } . \\end{equation*}",
"\\begin{equation*} \\hat { H } \\rightarrow H _ { c l } = \\frac { P _ { \\theta } ^ { 2 } } { 2 m R ^ { 2 } } + V ^ { 2 } ( e ^ { i \\theta } , e ^ { - i \\theta } ) . \\end{equation*}",
"\\begin{equation*} V ( e ^ { i \\theta } , e ^ { - i \\theta } ) = \\sqrt { \\frac { m g _ { N } R } { 2 } } \\operatorname { s i n } \\theta , \\end{equation*}",
"\\begin{equation*} V ( \\hat { W } , \\hat { W } ^ { \\dagger } ) = \\sqrt { \\frac { m g _ { N } R } { 2 } } ( \\frac { \\hat { W } - \\hat { W } ^ { \\dagger } } { 2 i } ) . \\end{equation*}",
"\\begin{equation*} \\hat { H } = \\frac { 1 } { 2 m R ^ { 2 } } \\hat { G } ^ { 2 } + \\frac { m g _ { N } R } { 2 } { \\operatorname { s i n } } ^ { 2 } \\theta - \\frac { \\hbar } { 2 } \\sqrt { \\frac { g _ { N } } { R } } \\operatorname { c o s } \\theta [ \\hat { \\xi } , \\hat { \\bar { \\xi } } ] \\end{equation*}",
"\\begin{equation*} \\hat { Q } \\vert \\Psi \\rangle = 0 \\qquad a n d \\qquad \\hat { \\bar { Q } } \\vert \\Psi \\rangle = 0 . \\end{equation*}",
"\\begin{align*} \\hat { \\xi } = ( \\begin{array}{cc} 0 & 0 \\\\ 1 & 0 \\end{array} ) , \\qquad \\hat { \\bar { \\xi } } = ( \\begin{array}{cc} 0 & 1 \\\\ 0 & 0 \\end{array} ) \\end{align*}"
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"$ \\mittheta $",
"$ \\mitP _ { \\mittheta } $",
"$ \\miti \\hslash $",
"$ { \\mathrm { e } } ^ { \\miti \\mittheta } $",
"$ \\mitP _ { \\mittheta } $",
"$ \\hat { \\mitW } $",
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"$ \\hat { \\mitW } $",
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"$ \\mitP _ { \\mittheta } $",
"$ \\hat { \\mitxi } , \\hat { \\bar { \\mitxi } } $",
"$ \\mitV ( { \\mathrm { e } } ^ { \\miti \\mittheta } , \\mathrm { e } ^ { - \\miti \\mittheta } ) $",
"$ 2 \\mittheta $",
"$ \\mitg _ { \\mitN } $",
"$ \\mitS ^ { 1 } $",
"$ \\hat { \\mitW } $",
"$ \\hat { \\mitG } = - \\miti \\hslash ~ \\mitpartial \\slash \\mitpartial \\mittheta + \\hslash ~ \\mitalpha $",
"$ \\mitalpha $",
"\\begin{equation*} \\{ \\mitP _ { \\mittheta } , \\mathrm { e } ^ { \\miti \\mittheta } \\} _ { \\mitP } = - \\miti \\mathrm { e } ^ { \\miti \\mittheta } . \\end{equation*}",
"\\begin{equation*} \\hat { \\mitH } \\rightarrow \\mitH _ { \\mitc \\mitl } = \\frac { \\mitP _ { \\mittheta } ^ { 2 } } { 2 \\mitm \\mitR ^ { 2 } } + \\mitV ^ { 2 } ( \\mathrm { e } ^ { \\miti \\mittheta } , \\mathrm { e } ^ { - \\miti \\mittheta } ) . \\end{equation*}",
"\\begin{equation*} \\mitV ( \\mathrm { e } ^ { \\miti \\mittheta } , \\mathrm { e } ^ { - \\miti \\mittheta } ) = \\sqrt { \\frac { \\mitm \\mitg _ { \\mitN } \\mitR } { 2 } } \\operatorname { s i n } \\mittheta , \\end{equation*}",
"\\begin{equation*} \\mitV ( \\hat { \\mitW } , \\hat { \\mitW } ^ { \\dagger } ) = \\sqrt { \\frac { \\mitm \\mitg _ { \\mitN } \\mitR } { 2 } } \\Big ( \\frac { \\hat { \\mitW } - \\hat { \\mitW } ^ { \\dagger } } { 2 \\miti } \\Big ) . \\end{equation*}",
"\\begin{equation*} \\hat { \\mitH } = \\frac { 1 } { 2 \\mitm \\mitR ^ { 2 } } \\hat { \\mitG } ^ { 2 } + \\frac { \\mitm \\mitg _ { \\mitN } \\mitR } { 2 } { \\operatorname { s i n } } ^ { 2 } \\mittheta - \\frac { \\hslash } { 2 } \\sqrt { \\frac { \\mitg _ { \\mitN } } { \\mitR } } \\operatorname { c o s } \\mittheta \\Big [ \\hat { \\mitxi } , \\hat { \\bar { \\mitxi } } \\Big ] \\end{equation*}",
"\\begin{equation*} \\hat { \\mitQ } \\vert \\mupPsi \\rangle = 0 \\qquad \\mathrm { a n d } \\qquad \\hat { \\bar { \\mitQ } } \\vert \\mupPsi \\rangle = 0 . \\end{equation*}",
"\\begin{align*} \\hat { \\mitxi } = \\left( \\begin{array}{cc} 0 & 0 \\\\ 1 & 0 \\end{array} \\right) , \\qquad \\hat { \\bar { \\mitxi } } = \\left( \\begin{array}{cc} 0 & 1 \\\\ 0 & 0 \\end{array} \\right) \\end{align*}"
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0003194_page09
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{
"latex": [
"$2\\times 2$",
"${\\hat S}^{F}\\equiv \\sigma ^3/2$",
"${\\hat S}^F$",
"$\\ket {+}$",
"$\\ket {-}$",
"$+1/2$",
"$-1/2$",
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"\\begin {eqnarray} \\hat H&=& \\Bigl ({1\\over {2mR^2}}\\hg ^2+{{mg_NR}\\over 2}\\sin ^2\\theta \\Bigr ) {\\bf 1}_{2\\times 2}+{\\hbar \\over 2}\\sqrt {{g_N\\over R}}\\sigma ^3\\cos \\theta \\\\ &=&\\left (\\begin {array}{cc} {\\hat q}^{\\dagger }{\\hat q}&0\\\\ 0&{\\hat q}{\\hat q}^{\\dagger }\\end {array}\\right )\\equiv \\left (\\begin {array}{cc} {\\hat H}_{+}&0\\\\ 0&{\\hat H}_{-}\\end {array}\\right ). \\end {eqnarray}",
"\\begin {equation} \\ket {\\Psi }={\\ket {+}\\choose \\ket {-}}. \\label {spinor} \\end {equation}",
"\\begin {equation} \\Psi (\\theta )={\\psi _{+\\half }(\\theta )\\choose \\psi _{-\\half }(\\theta )}. \\end {equation}",
"\\begin {eqnarray} {\\hat q}~\\psi _{+\\half }(\\theta )= \\Bigl ({1\\over {\\sqrt {2m}R}}(-i\\hbar ~{\\del \\over {\\del \\theta }}+\\hbar ~\\alpha ) + i\\sqrt {{mg_NR\\over 2}}\\sin \\theta \\Bigr )~\\psi _{+\\half }(\\theta )=0, \\\\ {\\hat q}^{\\dagger }\\psi _{-\\half }(\\theta )= \\Bigl ({1\\over {\\sqrt {2m}R}}(-i\\hbar ~{\\del \\over {\\del \\theta }}+\\hbar ~\\alpha ) - i\\sqrt {{mg_NR\\over 2}}\\sin \\theta \\Bigr )~\\psi _{-\\half }(\\theta )=0. \\end {eqnarray}",
"\\begin {equation} \\psi _{+\\half }(\\theta ) ={1\\over \\sqrt {I_0(2z)}}{\\rm exp}(-i\\alpha \\theta -{z\\over {\\hbar }}\\cos \\theta ), \\quad \\psi _{-\\half }(\\theta ) ={1\\over \\sqrt {I_0(2z)}}{\\rm exp}(-i\\alpha \\theta +{z\\over {\\hbar }}\\cos \\theta ), \\label {ground3} \\end {equation}",
"$$ {{mR^2}\\over \\hbar }\\sqrt {{g_N\\over R}} \\equiv {mR^2\\over \\hbar }\\omega \\equiv {z\\over {\\hbar }}~~. $$"
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"\\begin{align*} \\hat { H } & = & ( \\frac { 1 } { 2 m R ^ { 2 } } \\hat { G } ^ { 2 } + \\frac { m g _ { N } R } { 2 } { \\operatorname { s i n } } ^ { 2 } \\theta ) 1 _ { 2 \\times 2 } + \\frac { \\hbar } { 2 } \\sqrt { \\frac { g _ { N } } { R } } \\sigma ^ { 3 } \\operatorname { c o s } \\theta \\\\ & = & \\\\ & \\\\ & \\\\ & \\end{align*}",
"\\begin{equation*} \\vert \\Psi \\rangle = { \\vert + \\rangle \\atopwithdelims ( ) \\vert - \\rangle } . \\end{equation*}",
"\\begin{equation*} \\Psi ( \\theta ) = { \\psi _ { + \\frac { 1 } { 2 } } ( \\theta ) \\atopwithdelims ( ) \\psi _ { - \\frac { 1 } { 2 } } ( \\theta ) } . \\end{equation*}",
"\\begin{align*} \\hat { q } ~ \\psi _ { + \\frac { 1 } { 2 } } ( \\theta ) = ( \\frac { 1 } { \\sqrt { 2 m } R } ( - i \\hbar ~ \\frac { \\partial } { \\partial \\theta } + \\hbar ~ \\alpha ) + i \\sqrt { \\frac { m g _ { N } R } { 2 } } \\operatorname { s i n } \\theta ) ~ \\psi _ { + \\frac { 1 } { 2 } } ( \\theta ) = 0 , \\\\ \\hat { q } ^ { \\dagger } \\psi _ { - \\frac { 1 } { 2 } } ( \\theta ) = ( \\frac { 1 } { \\sqrt { 2 m } R } ( - i \\hbar ~ \\frac { \\partial } { \\partial \\theta } + \\hbar ~ \\alpha ) - i \\sqrt { \\frac { m g _ { N } R } { 2 } } \\operatorname { s i n } \\theta ) ~ \\psi _ { - \\frac { 1 } { 2 } } ( \\theta ) = 0 . \\end{align*}",
"\\begin{equation*} \\psi _ { + \\frac { 1 } { 2 } } ( \\theta ) = \\frac { 1 } { \\sqrt { I _ { 0 } ( 2 z ) } } e x p ( - i \\alpha \\theta - \\frac { z } { \\hbar } \\operatorname { c o s } \\theta ) , \\quad \\psi _ { - \\frac { 1 } { 2 } } ( \\theta ) = \\frac { 1 } { \\sqrt { I _ { 0 } ( 2 z ) } } e x p ( - i \\alpha \\theta + \\frac { z } { \\hbar } \\operatorname { c o s } \\theta ) , \\end{equation*}",
"\\begin{equation*} \\frac { m R ^ { 2 } } { \\hbar } \\sqrt { \\frac { g _ { N } } { R } } \\equiv \\frac { m R ^ { 2 } } { \\hbar } \\omega \\equiv \\frac { z } { \\hbar } ~ ~ . \\end{equation*}"
],
"latex_expand": [
"$ 2 \\times 2 $",
"$ \\hat { \\mitS } ^ { \\mitF } \\equiv \\mitsigma ^ { 3 } \\slash 2 $",
"$ \\hat { \\mitS } ^ { \\mitF } $",
"$ \\vert + \\rangle $",
"$ \\vert - \\rangle $",
"$ + 1 \\slash 2 $",
"$ - 1 \\slash 2 $",
"$ \\hat { \\mitW } $",
"$ \\hat { \\mitH } $",
"$ \\pm 1 \\slash 2 $",
"$ \\mitI _ { 0 } ( 2 \\mitz ) $",
"$ \\hat { \\mitf } = \\frac { 1 } { 2 } + \\frac { 1 } { 2 } [ \\hat { \\mitxi } , \\hat { \\bar { \\mitxi } } ] $",
"$ 0 , 1 $",
"$ 0 , 1 $",
"\\begin{align*} \\Big [ \\hat { \\mitxi } , \\hat { \\bar { \\mitxi } } \\Big ] = - \\left( \\begin{array}{cc} 1 & 0 \\\\ 0 & - 1 \\end{array} \\right) \\equiv - \\mitsigma ^ { 3 } . \\end{align*}",
"\\begin{align*} \\displaystyle \\hat { \\mitH } & = & \\displaystyle \\Big ( \\frac { 1 } { 2 \\mitm \\mitR ^ { 2 } } \\hat { \\mitG } ^ { 2 } + \\frac { \\mitm \\mitg _ { \\mitN } \\mitR } { 2 } { \\operatorname { s i n } } ^ { 2 } \\mittheta \\Big ) 1 _ { 2 \\times 2 } + \\frac { \\hslash } { 2 } \\sqrt { \\frac { \\mitg _ { \\mitN } } { \\mitR } } \\mitsigma ^ { 3 } \\operatorname { c o s } \\mittheta \\\\ & = & \\\\ & \\\\ & \\\\ & \\end{align*}",
"\\begin{equation*} \\vert \\mupPsi \\rangle = { \\vert + \\rangle \\atopwithdelims ( ) \\vert - \\rangle } . \\end{equation*}",
"\\begin{equation*} \\mupPsi ( \\mittheta ) = { \\mitpsi _ { + \\frac { 1 } { 2 } } ( \\mittheta ) \\atopwithdelims ( ) \\mitpsi _ { - \\frac { 1 } { 2 } } ( \\mittheta ) } . \\end{equation*}",
"\\begin{align*} \\displaystyle \\hat { \\mitq } ~ \\mitpsi _ { + \\frac { 1 } { 2 } } ( \\mittheta ) = \\Big ( \\frac { 1 } { \\sqrt { 2 \\mitm } \\mitR } ( - \\miti \\hslash ~ \\frac { \\mitpartial } { \\mitpartial \\mittheta } + \\hslash ~ \\mitalpha ) + \\miti \\sqrt { \\frac { \\mitm \\mitg _ { \\mitN } \\mitR } { 2 } } \\operatorname { s i n } \\mittheta \\Big ) ~ \\mitpsi _ { + \\frac { 1 } { 2 } } ( \\mittheta ) = 0 , \\\\ \\displaystyle \\hat { \\mitq } ^ { \\dagger } \\mitpsi _ { - \\frac { 1 } { 2 } } ( \\mittheta ) = \\Big ( \\frac { 1 } { \\sqrt { 2 \\mitm } \\mitR } ( - \\miti \\hslash ~ \\frac { \\mitpartial } { \\mitpartial \\mittheta } + \\hslash ~ \\mitalpha ) - \\miti \\sqrt { \\frac { \\mitm \\mitg _ { \\mitN } \\mitR } { 2 } } \\operatorname { s i n } \\mittheta \\Big ) ~ \\mitpsi _ { - \\frac { 1 } { 2 } } ( \\mittheta ) = 0 . \\end{align*}",
"\\begin{equation*} \\mitpsi _ { + \\frac { 1 } { 2 } } ( \\mittheta ) = \\frac { 1 } { \\sqrt { \\mitI _ { 0 } ( 2 \\mitz ) } } \\mathrm { e x p } ( - \\miti \\mitalpha \\mittheta - \\frac { \\mitz } { \\hslash } \\operatorname { c o s } \\mittheta ) , \\quad \\mitpsi _ { - \\frac { 1 } { 2 } } ( \\mittheta ) = \\frac { 1 } { \\sqrt { \\mitI _ { 0 } ( 2 \\mitz ) } } \\mathrm { e x p } ( - \\miti \\mitalpha \\mittheta + \\frac { \\mitz } { \\hslash } \\operatorname { c o s } \\mittheta ) , \\end{equation*}",
"\\begin{equation*} \\frac { \\mitm \\mitR ^ { 2 } } { \\hslash } \\sqrt { \\frac { \\mitg _ { \\mitN } } { \\mitR } } \\equiv \\frac { \\mitm \\mitR ^ { 2 } } { \\hslash } \\mitomega \\equiv \\frac { \\mitz } { \\hslash } ~ ~ . \\end{equation*}"
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0003194_page10
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{
"latex": [
"$\\Psi (\\theta +2\\pi )=\\Psi (\\theta )$",
"$\\psi _{\\pm \\half }(\\theta +2\\pi ) =\\psi _{\\pm \\half }(\\theta )$",
"$\\alpha $",
"$\\alpha ={\\rm integer}$",
"$0\\leq \\alpha <1$",
"$0\\leq \\alpha <1$",
"$0< \\alpha < 1$",
"$\\alpha $",
"${\\rm Tr}(-1)^{{\\hat f}}=n_B^{E=0}-n_F^{E=0}$",
"$n_B^{E=0}=n_F^{E=0}=1$",
"$\\alpha ={\\rm integer}$",
"$n_B^{E=0}=n_F^{E=0}=0$",
"$\\alpha =$",
"$\\alpha $",
"$S^1$",
"$S^1$",
"$\\alpha $",
"$\\alpha $",
"$\\alpha $",
"$0\\leq \\alpha < 1$",
"$\\e ^{in\\theta }(n={\\rm integer})$",
"$\\alpha ={\\rm integer}$",
"$0< \\alpha < 1$",
"$2\\pi $",
"\\begin {equation} \\psi _{\\pm \\half }(\\theta +2\\pi )=\\e ^{-i2\\pi \\alpha }\\psi _{\\pm \\half }(\\theta ). \\label {boundary} \\end {equation}",
"\\begin {equation} \\psi _{\\pm \\half }(\\theta )\\longrightarrow \\psi _{\\pm \\half }^{\\prime }(\\theta )=\\e ^{-i\\alpha \\theta }\\psi _{\\pm \\half }(\\theta ). \\label {gaugetrf} \\end {equation}"
],
"latex_norm": [
"$ \\Psi ( \\theta + 2 \\pi ) = \\Psi ( \\theta ) $",
"$ \\psi _ { \\pm \\frac { 1 } { 2 } } ( \\theta + 2 \\pi ) = \\psi _ { \\pm \\frac { 1 } { 2 } } ( \\theta ) $",
"$ \\alpha $",
"$ \\alpha = i n t e g e r $",
"$ 0 \\leq \\alpha < 1 $",
"$ 0 \\leq \\alpha < 1 $",
"$ 0 < \\alpha < 1 $",
"$ \\alpha $",
"$ T r ( - 1 ) ^ { \\hat { f } } = n _ { B } ^ { E = 0 } - n _ { F } ^ { E = 0 } $",
"$ n _ { B } ^ { E = 0 } = n _ { F } ^ { E = 0 } = 1 $",
"$ \\alpha = i n t e g e r $",
"$ n _ { B } ^ { E = 0 } = n _ { F } ^ { E = 0 } = 0 $",
"$ \\alpha = $",
"$ \\alpha $",
"$ S ^ { 1 } $",
"$ S ^ { 1 } $",
"$ \\alpha $",
"$ \\alpha $",
"$ \\alpha $",
"$ 0 \\leq \\alpha < 1 $",
"$ { e } ^ { i n \\theta } ( n = i n t e g e r ) $",
"$ \\alpha = i n t e g e r $",
"$ 0 < \\alpha < 1 $",
"$ 2 \\pi $",
"\\begin{equation*} \\psi _ { \\pm \\frac { 1 } { 2 } } ( \\theta + 2 \\pi ) = e ^ { - i 2 \\pi \\alpha } \\psi _ { \\pm \\frac { 1 } { 2 } } ( \\theta ) . \\end{equation*}",
"\\begin{equation*} \\psi _ { \\pm \\frac { 1 } { 2 } } ( \\theta ) \\longrightarrow \\psi _ { \\pm \\frac { 1 } { 2 } } ^ { \\prime } ( \\theta ) = e ^ { - i \\alpha \\theta } \\psi _ { \\pm \\frac { 1 } { 2 } } ( \\theta ) . \\end{equation*}"
],
"latex_expand": [
"$ \\mupPsi ( \\mittheta + 2 \\mitpi ) = \\mupPsi ( \\mittheta ) $",
"$ \\mitpsi _ { \\pm \\frac { 1 } { 2 } } ( \\mittheta + 2 \\mitpi ) = \\mitpsi _ { \\pm \\frac { 1 } { 2 } } ( \\mittheta ) $",
"$ \\mitalpha $",
"$ \\mitalpha = \\mathrm { i n t e g e r } $",
"$ 0 \\leq \\mitalpha < 1 $",
"$ 0 \\leq \\mitalpha < 1 $",
"$ 0 < \\mitalpha < 1 $",
"$ \\mitalpha $",
"$ \\mathrm { T r } ( - 1 ) ^ { \\hat { \\mitf } } = \\mitn _ { \\mitB } ^ { \\mitE = 0 } - \\mitn _ { \\mitF } ^ { \\mitE = 0 } $",
"$ \\mitn _ { \\mitB } ^ { \\mitE = 0 } = \\mitn _ { \\mitF } ^ { \\mitE = 0 } = 1 $",
"$ \\mitalpha = \\mathrm { i n t e g e r } $",
"$ \\mitn _ { \\mitB } ^ { \\mitE = 0 } = \\mitn _ { \\mitF } ^ { \\mitE = 0 } = 0 $",
"$ \\mitalpha = $",
"$ \\mitalpha $",
"$ \\mitS ^ { 1 } $",
"$ \\mitS ^ { 1 } $",
"$ \\mitalpha $",
"$ \\mitalpha $",
"$ \\mitalpha $",
"$ 0 \\leq \\mitalpha < 1 $",
"$ { \\mathrm { e } } ^ { \\miti \\mitn \\mittheta } ( \\mitn = \\mathrm { i n t e g e r } ) $",
"$ \\mitalpha = \\mathrm { i n t e g e r } $",
"$ 0 < \\mitalpha < 1 $",
"$ 2 \\mitpi $",
"\\begin{equation*} \\mitpsi _ { \\pm \\frac { 1 } { 2 } } ( \\mittheta + 2 \\mitpi ) = \\mathrm { e } ^ { - \\miti 2 \\mitpi \\mitalpha } \\mitpsi _ { \\pm \\frac { 1 } { 2 } } ( \\mittheta ) . \\end{equation*}",
"\\begin{equation*} \\mitpsi _ { \\pm \\frac { 1 } { 2 } } ( \\mittheta ) \\longrightarrow \\mitpsi _ { \\pm \\frac { 1 } { 2 } } ^ { \\prime } ( \\mittheta ) = \\mathrm { e } ^ { - \\miti \\mitalpha \\mittheta } \\mitpsi _ { \\pm \\frac { 1 } { 2 } } ( \\mittheta ) . \\end{equation*}"
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0003194_page11
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{
"latex": [
"${{mg_N R}\\over 2}\\sin ^2\\theta $",
"$(\\pm 1/2)$",
"$ \\cos \\theta _{cl}(\\tau )=\\pm \\tanh (\\omega (\\tau -\\tau _0)) $",
"$-2z/\\hbar $",
"${\\rm exp}(-2z/\\hbar )\\times \\cos {2\\pi \\alpha }$",
"$z$",
"$E_0\\sim {1\\over {2mR^2}}(\\alpha ^2+O(z^2))$",
"$E_0\\sim {1\\over {2mR^2}}(\\alpha ^2+O(z^2))$",
"$\\hbar =1$",
"$\\alpha $",
"$R\\rightarrow \\infty $",
"$R$",
"$S^1$",
"$R$",
"$S^1$",
"$R\\rightarrow \\infty $",
"$x\\equiv R\\theta $",
"$\\hg =-i\\hbar ~\\del /\\del \\theta +\\hbar ~\\alpha $",
"$\\hw $",
"$R\\rightarrow \\infty $",
"$S^1$",
"$\\omega =\\sqrt {g_N/R}$",
"$\\cos {2\\pi \\alpha }$",
"$K(\\theta _f,t;\\theta _i,0) =\\bra {\\theta _f}{\\exp (-i{\\hat H}t/\\hbar )\\ket {\\theta _i}}= \\sum _{n=-\\infty }^{+\\infty }\\int _{n-winding} {\\cal D}\\theta \\exp (iS_{eff}/\\hbar )$",
"$S_{eff}=\\int dt {{mR^2}\\over 2}({{d\\theta }\\over {dt}})^2 -{{mg_NR}\\over 2}\\sin ^2\\theta +\\half \\sqrt {g_N\\over R}\\cos \\theta [\\xi , \\xi ^*]+i\\xi ^*{{d\\xi }\\over {dt}} -\\alpha {d\\theta \\over {dt}}$",
"$\\alpha {\\dot \\theta }$",
"\\begin {equation} \\hat H= \\Bigl ({-\\hbar ^2\\over {2m}}\\Bigl ({\\del \\over {\\del x}} +i{\\alpha \\over R}\\Bigr )^2 +{{mg_NR}\\over 2}\\sin ^2({x\\over R})\\Bigr ){\\bf 1}_{2\\times 2} +{\\hbar \\over 2}\\sqrt {{g_N\\over R}}\\sigma ^3\\cos ({x\\over R}). \\label {hamilton4} \\end {equation}",
"\\begin {equation} {\\hat H}={-\\hbar ^2\\over {2m}}{\\del ^2\\over {\\del x^2}}{\\bf 1}_{2\\times 2}. \\end {equation}",
"\\begin {equation} {m\\over \\hbar }\\sqrt {g_N\\over R}\\equiv {{m\\omega }\\over \\hbar }=\\Bigl ({\\rm strength~of~oscillator} \\Bigr )^2 ={\\rm const}. \\label {relation} \\end {equation}",
"\\begin {equation} \\hat H = \\Bigl ({-{\\hbar }^2\\over {2m}}{\\del ^2\\over {\\del x^2}} +{{m\\omega ^2}\\over 2} x^2\\Bigr ){\\bf 1}_{2\\times 2} +{{\\hbar ~\\omega }\\over 2}\\sigma ^3 + O({1\\over R^2}). \\label {harmo} \\end {equation}"
],
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"$ \\frac { m g _ { N } R } { 2 } { s i n } ^ { 2 } \\theta $",
"$ ( \\pm 1 \\slash 2 ) $",
"$ c o s \\theta _ { c l } ( \\tau ) = \\pm \\operatorname { t a n h } ( \\omega ( \\tau - \\tau _ { 0 } ) ) $",
"$ - 2 z \\slash \\hbar $",
"$ e x p ( - 2 z \\slash \\hbar ) \\times \\operatorname { c o s } 2 \\pi \\alpha $",
"$ z $",
"$ E _ { 0 } \\sim \\frac { 1 } { 2 m R ^ { 2 } } ( \\alpha ^ { 2 } + O ( z ^ { 2 } ) ) $",
"$ E _ { 0 } \\sim \\frac { 1 } { 2 m R ^ { 2 } } ( \\alpha ^ { 2 } + O ( z ^ { 2 } ) ) $",
"$ \\hbar = 1 $",
"$ \\alpha $",
"$ R \\rightarrow \\infty $",
"$ R $",
"$ S ^ { 1 } $",
"$ R $",
"$ S ^ { 1 } $",
"$ R \\rightarrow \\infty $",
"$ x \\equiv R \\theta $",
"$ \\hat { G } = - i \\hbar ~ \\partial \\slash \\partial \\theta + \\hbar ~ \\alpha $",
"$ \\hat { W } $",
"$ R \\rightarrow \\infty $",
"$ S ^ { 1 } $",
"$ \\omega = \\sqrt { g _ { N } \\slash R } $",
"$ c o s 2 \\pi \\alpha $",
"$ K ( \\theta _ { f } , t ; \\theta _ { i } , 0 ) = \\langle \\theta _ { f } \\vert e x p ( - i \\hat { H } t \\slash \\hbar ) \\vert \\theta _ { i } \\rangle = \\sum _ { n = - \\infty } ^ { + \\infty } \\int _ { n - w i n d i n g } D \\theta \\operatorname { e x p } ( i S _ { e f f } \\slash \\hbar ) $",
"$ S _ { e f f } = \\int d t \\frac { m R ^ { 2 } } { 2 } ( \\frac { d \\theta } { d t } ) ^ { 2 } - \\frac { m g _ { N } R } { 2 } { s i n } ^ { 2 } \\theta + \\frac { 1 } { 2 } \\sqrt { \\frac { g _ { N } } { R } } \\operatorname { c o s } \\theta [ \\xi , \\xi ^ { \\ast } ] + i \\xi ^ { \\ast } \\frac { d \\xi } { d t } - \\alpha \\frac { d \\theta } { d t } $",
"$ \\alpha \\dot { \\theta } $",
"\\begin{equation*} \\hat { H } = ( \\frac { - \\hbar ^ { 2 } } { 2 m } ( \\frac { \\partial } { \\partial x } + i \\frac { \\alpha } { R } ) ^ { 2 } + \\frac { m g _ { N } R } { 2 } { \\operatorname { s i n } } ^ { 2 } ( \\frac { x } { R } ) ) 1 _ { 2 \\times 2 } + \\frac { \\hbar } { 2 } \\sqrt { \\frac { g _ { N } } { R } } \\sigma ^ { 3 } \\operatorname { c o s } ( \\frac { x } { R } ) . \\end{equation*}",
"\\begin{equation*} \\hat { H } = \\frac { - \\hbar ^ { 2 } } { 2 m } \\frac { \\partial ^ { 2 } } { \\partial x ^ { 2 } } 1 _ { 2 \\times 2 } . \\end{equation*}",
"\\begin{equation*} \\frac { m } { \\hbar } \\sqrt { \\frac { g _ { N } } { R } } \\equiv \\frac { m \\omega } { \\hbar } = ( s t r e n g t h ~ o f ~ o s c i l l a t o r ) ^ { 2 } = c o n s t . \\end{equation*}",
"\\begin{equation*} \\hat { H } = ( \\frac { - \\hbar ^ { 2 } } { 2 m } \\frac { \\partial ^ { 2 } } { \\partial x ^ { 2 } } + \\frac { m \\omega ^ { 2 } } { 2 } x ^ { 2 } ) 1 _ { 2 \\times 2 } + \\frac { \\hbar ~ \\omega } { 2 } \\sigma ^ { 3 } + O ( \\frac { 1 } { R ^ { 2 } } ) . \\end{equation*}"
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"$ ( \\pm 1 \\slash 2 ) $",
"$ \\mathrm { c o s } \\mittheta _ { \\mitc \\mitl } ( \\mittau ) = \\pm \\operatorname { t a n h } ( \\mitomega ( \\mittau - \\mittau _ { 0 } ) ) $",
"$ - 2 \\mitz \\slash \\hslash $",
"$ \\mathrm { e x p } ( - 2 \\mitz \\slash \\hslash ) \\times \\operatorname { c o s } 2 \\mitpi \\mitalpha $",
"$ \\mitz $",
"$ \\mitE _ { 0 } \\sim \\frac { 1 } { 2 \\mitm \\mitR ^ { 2 } } ( \\mitalpha ^ { 2 } + \\mitO ( \\mitz ^ { 2 } ) ) $",
"$ \\mitE _ { 0 } \\sim \\frac { 1 } { 2 \\mitm \\mitR ^ { 2 } } ( \\mitalpha ^ { 2 } + \\mitO ( \\mitz ^ { 2 } ) ) $",
"$ \\hslash = 1 $",
"$ \\mitalpha $",
"$ \\mitR \\rightarrow \\infty $",
"$ \\mitR $",
"$ \\mitS ^ { 1 } $",
"$ \\mitR $",
"$ \\mitS ^ { 1 } $",
"$ \\mitR \\rightarrow \\infty $",
"$ \\mitx \\equiv \\mitR \\mittheta $",
"$ \\hat { \\mitG } = - \\miti \\hslash ~ \\mitpartial \\slash \\mitpartial \\mittheta + \\hslash ~ \\mitalpha $",
"$ \\hat { \\mitW } $",
"$ \\mitR \\rightarrow \\infty $",
"$ \\mitS ^ { 1 } $",
"$ \\mitomega = \\sqrt { \\mitg _ { \\mitN } \\slash \\mitR } $",
"$ \\mathrm { c o s } 2 \\mitpi \\mitalpha $",
"$ \\mitK ( \\mittheta _ { \\mitf } , \\mitt ; \\mittheta _ { \\miti } , 0 ) = \\langle \\mittheta _ { \\mitf } \\vert \\mathrm { e x p } ( - \\miti \\hat { \\mitH } \\mitt \\slash \\hslash ) \\vert \\mittheta _ { \\miti } \\rangle = \\sum _ { \\mitn = - \\infty } ^ { + \\infty } \\int \\nolimits _ { \\mitn - \\mitw \\miti \\mitn \\mitd \\miti \\mitn \\mitg } \\mitD \\mittheta \\operatorname { e x p } ( \\miti \\mitS _ { \\mite \\mitf \\mitf } \\slash \\hslash ) $",
"$ \\mitS _ { \\mite \\mitf \\mitf } = \\int \\nolimits \\mitd \\mitt \\frac { \\mitm \\mitR ^ { 2 } } { 2 } ( \\frac { \\mitd \\mittheta } { \\mitd \\mitt } ) ^ { 2 } - \\frac { \\mitm \\mitg _ { \\mitN } \\mitR } { 2 } { \\mathrm { s i n } } ^ { 2 } \\mittheta + \\frac { 1 } { 2 } \\sqrt { \\frac { \\mitg _ { \\mitN } } { \\mitR } } \\operatorname { c o s } \\mittheta [ \\mitxi , \\mitxi ^ { \\ast } ] + \\miti \\mitxi ^ { \\ast } \\frac { \\mitd \\mitxi } { \\mitd \\mitt } - \\mitalpha \\frac { \\mitd \\mittheta } { \\mitd \\mitt } $",
"$ \\mitalpha \\dot { \\mittheta } $",
"\\begin{equation*} \\hat { \\mitH } = \\Big ( \\frac { - \\hslash ^ { 2 } } { 2 \\mitm } \\Big ( \\frac { \\mitpartial } { \\mitpartial \\mitx } + \\miti \\frac { \\mitalpha } { \\mitR } \\Big ) ^ { 2 } + \\frac { \\mitm \\mitg _ { \\mitN } \\mitR } { 2 } { \\operatorname { s i n } } ^ { 2 } ( \\frac { \\mitx } { \\mitR } ) \\Big ) 1 _ { 2 \\times 2 } + \\frac { \\hslash } { 2 } \\sqrt { \\frac { \\mitg _ { \\mitN } } { \\mitR } } \\mitsigma ^ { 3 } \\operatorname { c o s } ( \\frac { \\mitx } { \\mitR } ) . \\end{equation*}",
"\\begin{equation*} \\hat { \\mitH } = \\frac { - \\hslash ^ { 2 } } { 2 \\mitm } \\frac { \\mitpartial ^ { 2 } } { \\mitpartial \\mitx ^ { 2 } } 1 _ { 2 \\times 2 } . \\end{equation*}",
"\\begin{equation*} \\frac { \\mitm } { \\hslash } \\sqrt { \\frac { \\mitg _ { \\mitN } } { \\mitR } } \\equiv \\frac { \\mitm \\mitomega } { \\hslash } = \\Big ( \\mathrm { s t r e n g t h } ~ \\mathrm { o f } ~ \\mathrm { o s c i l l a t o r } \\Big ) ^ { 2 } = \\mathrm { c o n s t } . \\end{equation*}",
"\\begin{equation*} \\hat { \\mitH } = \\Big ( \\frac { - \\hslash ^ { 2 } } { 2 \\mitm } \\frac { \\mitpartial ^ { 2 } } { \\mitpartial \\mitx ^ { 2 } } + \\frac { \\mitm \\mitomega ^ { 2 } } { 2 } \\mitx ^ { 2 } \\Big ) 1 _ { 2 \\times 2 } + \\frac { \\hslash ~ \\mitomega } { 2 } \\mitsigma ^ { 3 } + \\mitO ( \\frac { 1 } { \\mitR ^ { 2 } } ) . \\end{equation*}"
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0003194_page12
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{
"latex": [
"$\\hq , \\hqb $",
"$W({\\hat x})\\equiv m{\\omega } {\\hat x}$",
"${\\hat p}\\equiv -i{\\hbar }~\\del /\\del x$",
"$[{\\hat p},{\\hat x}]=-i\\hbar $",
"$({\\hat x})$",
"$(\\hxi , \\hxib )$",
"$R\\rightarrow \\infty $",
"$\\hq _{susy}\\ket {\\Psi }=0$",
"$\\hqb _{susy}\\ket {\\Psi }=0$",
"${\\psi }_{-\\half }^{h.o}(x)$",
"${\\psi }_{+\\half }^{h.o}(x)$",
"$R\\rightarrow \\infty $",
"$I_0(z)\\sim \\e ^z/\\sqrt {2\\pi z}$",
"$z$",
"${\\tilde \\psi }_{\\pm \\half }(x)dx\\equiv \\psi _{\\pm \\half }(\\theta ){{d\\theta }\\over {\\sqrt {2\\pi R}}}$",
"${\\rm Tr}(-1)^{{\\hat f}}=1$",
"$N$",
"${\\hat V}_a\\equiv {\\hat V}_a(\\theta _1, \\cdots , \\theta _N)$",
"$\\hg _a=-i\\hbar ~\\del /\\del \\theta _a+\\hbar ~\\alpha _a$",
"$\\hw $",
"\\begin {eqnarray} \\hq &=&{1\\over {\\sqrt {2m}}}\\Bigl ({\\hat p} + i W({\\hat x})\\Bigr )\\hxi +O({1\\over R^2})\\equiv \\hq _{susy}+O({1\\over R^2}),\\\\ \\hqb &=&{1\\over {\\sqrt {2m}}}\\Bigl ({\\hat p} - i W({\\hat x})\\Bigr )\\hxib +O({1\\over R^2})\\equiv \\hqb _{susy}+O({1\\over R^2}), \\end {eqnarray}",
"\\begin {equation} {\\psi }_{+\\half }^{h.o}(x)\\sim {\\rm exp}(+{{m\\omega }\\over {2\\hbar }} x^2), \\qquad {\\psi }_{-\\half }^{h.o}(x)= \\Bigl ({{m\\omega }\\over {\\pi \\hbar }}\\Bigr )^{1/4} {\\rm exp}(-{{m\\omega }\\over {2\\hbar }} x^2). \\label {harmosol} \\end {equation}",
"\\begin {equation} {\\tilde \\psi }_{+\\half }\\sim {\\rm exp}(+{{m\\omega }\\over {2\\hbar }} x^2),\\qquad {\\tilde \\psi }_{-\\half }(x)= \\Bigl ({{m\\omega }\\over {\\pi \\hbar }}\\Bigr )^{1/4} {\\rm exp}(-{{m\\omega }\\over {2\\hbar }} x^2), \\end {equation}",
"\\begin {eqnarray} \\hq &=&\\sum _{a=1}^{{}N} \\Bigl ({1\\over {\\sqrt {2m}R_a}}\\hg _a +i {\\hat V}_a\\Bigr )\\hxi _a \\equiv \\sum _{a=1}^{{}N}{\\hat q}_a~\\hxi _a, \\\\ \\hqb &=&\\sum _{a=1}^{{}N} \\Bigl ({1\\over {\\sqrt {2m}R_a}}\\hg _a-i{\\hat V}_a\\Bigr )\\hxib \\equiv \\sum _{a=1}^{{}N}{\\hat q}_a^{\\dagger }~\\hxib _a, \\end {eqnarray}",
"\\begin {equation} [\\hg _a, \\hw _b]=\\hbar ~\\delta _{ab}\\hw _b, \\quad \\{\\hxi _a, \\hxib _b\\}=\\delta _{ab}, \\quad \\{\\hxi _a, \\hxi _b\\}=0, \\quad \\{\\hxib _a, \\hxib _b\\}=0. \\quad \\end {equation}"
],
"latex_norm": [
"$ \\hat { Q } , \\hat { \\bar { Q } } $",
"$ W ( \\hat { x } ) \\equiv m \\omega \\hat { x } $",
"$ \\hat { p } \\equiv - i \\hbar ~ \\partial \\slash \\partial x $",
"$ [ \\hat { p } , \\hat { x } ] = - i \\hbar $",
"$ ( \\hat { x } ) $",
"$ ( \\hat { \\xi } , \\hat { \\bar { \\xi } } ) $",
"$ R \\rightarrow \\infty $",
"$ \\hat { Q } _ { s u s y } \\vert \\Psi \\rangle = 0 $",
"$ \\hat { \\bar { Q } } _ { s u s y } \\vert \\Psi \\rangle = 0 $",
"$ \\psi _ { - \\frac { 1 } { 2 } } ^ { h . o } ( x ) $",
"$ \\psi _ { + \\frac { 1 } { 2 } } ^ { h . o } ( x ) $",
"$ R \\rightarrow \\infty $",
"$ I _ { 0 } ( z ) \\sim { e } ^ { z } \\slash \\sqrt { 2 \\pi z } $",
"$ z $",
"$ \\widetilde { \\psi } _ { \\pm \\frac { 1 } { 2 } } ( x ) d x \\equiv \\psi _ { \\pm \\frac { 1 } { 2 } } ( \\theta ) \\frac { d \\theta } { \\sqrt { 2 \\pi R } } $",
"$ T r ( - 1 ) ^ { \\hat { f } } = 1 $",
"$ N $",
"$ \\hat { V } _ { a } \\equiv \\hat { V } _ { a } ( \\theta _ { 1 } , \\cdots , \\theta _ { N } ) $",
"$ \\hat { G } _ { a } = - i \\hbar ~ \\partial \\slash \\partial \\theta _ { a } + \\hbar ~ \\alpha _ { a } $",
"$ \\hat { W } $",
"\\begin{align*} \\hat { Q } & = & \\frac { 1 } { \\sqrt { 2 m } } ( \\hat { p } + i W ( \\hat { x } ) ) \\hat { \\xi } + O ( \\frac { 1 } { R ^ { 2 } } ) \\equiv \\hat { Q } _ { s u s y } + O ( \\frac { 1 } { R ^ { 2 } } ) , \\\\ \\hat { \\bar { Q } } & = & \\frac { 1 } { \\sqrt { 2 m } } ( \\hat { p } - i W ( \\hat { x } ) ) \\hat { \\bar { \\xi } } + O ( \\frac { 1 } { R ^ { 2 } } ) \\equiv \\hat { \\bar { Q } } _ { s u s y } + O ( \\frac { 1 } { R ^ { 2 } } ) , \\end{align*}",
"\\begin{equation*} \\psi _ { + \\frac { 1 } { 2 } } ^ { h . o } ( x ) \\sim e x p ( + \\frac { m \\omega } { 2 \\hbar } x ^ { 2 } ) , \\qquad \\psi _ { - \\frac { 1 } { 2 } } ^ { h . o } ( x ) = ( \\frac { m \\omega } { \\pi \\hbar } ) ^ { 1 \\slash 4 } e x p ( - \\frac { m \\omega } { 2 \\hbar } x ^ { 2 } ) . \\end{equation*}",
"\\begin{equation*} \\widetilde { \\psi } _ { + \\frac { 1 } { 2 } } \\sim e x p ( + \\frac { m \\omega } { 2 \\hbar } x ^ { 2 } ) , \\qquad \\widetilde { \\psi } _ { - \\frac { 1 } { 2 } } ( x ) = ( \\frac { m \\omega } { \\pi \\hbar } ) ^ { 1 \\slash 4 } e x p ( - \\frac { m \\omega } { 2 \\hbar } x ^ { 2 } ) , \\end{equation*}",
"\\begin{align*} \\hat { Q } & = & \\sum _ { a = 1 } ^ { N } ( \\frac { 1 } { \\sqrt { 2 m } R _ { a } } \\hat { G } _ { a } + i \\hat { V } _ { a } ) \\hat { \\xi } _ { a } \\equiv \\sum _ { a = 1 } ^ { N } \\hat { q } _ { a } ~ \\hat { \\xi } _ { a } , \\\\ \\hat { \\bar { Q } } & = & \\sum _ { a = 1 } ^ { N } ( \\frac { 1 } { \\sqrt { 2 m } R _ { a } } \\hat { G } _ { a } - i \\hat { V } _ { a } ) \\hat { \\bar { \\xi } } \\equiv \\sum _ { a = 1 } ^ { N } \\hat { q } _ { a } ^ { \\dagger } ~ \\hat { \\bar { \\xi } } _ { a } , \\end{align*}",
"\\begin{equation*} [ \\hat { G } _ { a } , \\hat { W } _ { b } ] = \\hbar ~ \\delta _ { a b } \\hat { W } _ { b } , \\quad \\{ \\hat { \\xi } _ { a } , \\hat { \\bar { \\xi } } _ { b } \\} = \\delta _ { a b } , \\quad \\{ \\hat { \\xi } _ { a } , \\hat { \\xi } _ { b } \\} = 0 , \\quad \\{ \\hat { \\bar { \\xi } } _ { a } , \\hat { \\bar { \\xi } } _ { b } \\} = 0 . \\quad \\end{equation*}"
],
"latex_expand": [
"$ \\hat { \\mitQ } , \\hat { \\bar { \\mitQ } } $",
"$ \\mitW ( \\hat { \\mitx } ) \\equiv \\mitm \\mitomega \\hat { \\mitx } $",
"$ \\hat { \\mitp } \\equiv - \\miti \\hslash ~ \\mitpartial \\slash \\mitpartial \\mitx $",
"$ [ \\hat { \\mitp } , \\hat { \\mitx } ] = - \\miti \\hslash $",
"$ ( \\hat { \\mitx } ) $",
"$ ( \\hat { \\mitxi } , \\hat { \\bar { \\mitxi } } ) $",
"$ \\mitR \\rightarrow \\infty $",
"$ \\hat { \\mitQ } _ { \\mits \\mitu \\mits \\mity } \\vert \\mupPsi \\rangle = 0 $",
"$ \\hat { \\bar { \\mitQ } } _ { \\mits \\mitu \\mits \\mity } \\vert \\mupPsi \\rangle = 0 $",
"$ \\mitpsi _ { - \\frac { 1 } { 2 } } ^ { \\Planckconst . \\mito } ( \\mitx ) $",
"$ \\mitpsi _ { + \\frac { 1 } { 2 } } ^ { \\Planckconst . \\mito } ( \\mitx ) $",
"$ \\mitR \\rightarrow \\infty $",
"$ \\mitI _ { 0 } ( \\mitz ) \\sim { \\mathrm { e } } ^ { \\mitz } \\slash \\sqrt { 2 \\mitpi \\mitz } $",
"$ \\mitz $",
"$ \\tilde { \\mitpsi } _ { \\pm \\frac { 1 } { 2 } } ( \\mitx ) \\mitd \\mitx \\equiv \\mitpsi _ { \\pm \\frac { 1 } { 2 } } ( \\mittheta ) \\frac { \\mitd \\mittheta } { \\sqrt { 2 \\mitpi \\mitR } } $",
"$ \\mathrm { T r } ( - 1 ) ^ { \\hat { \\mitf } } = 1 $",
"$ \\mitN $",
"$ \\hat { \\mitV } _ { \\mita } \\equiv \\hat { \\mitV } _ { \\mita } ( \\mittheta _ { 1 } , \\cdots , \\mittheta _ { \\mitN } ) $",
"$ \\hat { \\mitG } _ { \\mita } = - \\miti \\hslash ~ \\mitpartial \\slash \\mitpartial \\mittheta _ { \\mita } + \\hslash ~ \\mitalpha _ { \\mita } $",
"$ \\hat { \\mitW } $",
"\\begin{align*} \\displaystyle \\hat { \\mitQ } & = & \\displaystyle \\frac { 1 } { \\sqrt { 2 \\mitm } } \\Big ( \\hat { \\mitp } + \\miti \\mitW ( \\hat { \\mitx } ) \\Big ) \\hat { \\mitxi } + \\mitO ( \\frac { 1 } { \\mitR ^ { 2 } } ) \\equiv \\hat { \\mitQ } _ { \\mits \\mitu \\mits \\mity } + \\mitO ( \\frac { 1 } { \\mitR ^ { 2 } } ) , \\\\ \\displaystyle \\hat { \\bar { \\mitQ } } & = & \\displaystyle \\frac { 1 } { \\sqrt { 2 \\mitm } } \\Big ( \\hat { \\mitp } - \\miti \\mitW ( \\hat { \\mitx } ) \\Big ) \\hat { \\bar { \\mitxi } } + \\mitO ( \\frac { 1 } { \\mitR ^ { 2 } } ) \\equiv \\hat { \\bar { \\mitQ } } _ { \\mits \\mitu \\mits \\mity } + \\mitO ( \\frac { 1 } { \\mitR ^ { 2 } } ) , \\end{align*}",
"\\begin{equation*} \\mitpsi _ { + \\frac { 1 } { 2 } } ^ { \\Planckconst . \\mito } ( \\mitx ) \\sim \\mathrm { e x p } ( + \\frac { \\mitm \\mitomega } { 2 \\hslash } \\mitx ^ { 2 } ) , \\qquad \\mitpsi _ { - \\frac { 1 } { 2 } } ^ { \\Planckconst . \\mito } ( \\mitx ) = \\Big ( \\frac { \\mitm \\mitomega } { \\mitpi \\hslash } \\Big ) ^ { 1 \\slash 4 } \\mathrm { e x p } ( - \\frac { \\mitm \\mitomega } { 2 \\hslash } \\mitx ^ { 2 } ) . \\end{equation*}",
"\\begin{equation*} \\tilde { \\mitpsi } _ { + \\frac { 1 } { 2 } } \\sim \\mathrm { e x p } ( + \\frac { \\mitm \\mitomega } { 2 \\hslash } \\mitx ^ { 2 } ) , \\qquad \\tilde { \\mitpsi } _ { - \\frac { 1 } { 2 } } ( \\mitx ) = \\Big ( \\frac { \\mitm \\mitomega } { \\mitpi \\hslash } \\Big ) ^ { 1 \\slash 4 } \\mathrm { e x p } ( - \\frac { \\mitm \\mitomega } { 2 \\hslash } \\mitx ^ { 2 } ) , \\end{equation*}",
"\\begin{align*} \\displaystyle \\hat { \\mitQ } & = & \\displaystyle \\sum _ { \\mita = 1 } ^ { \\mitN } \\Big ( \\frac { 1 } { \\sqrt { 2 \\mitm } \\mitR _ { \\mita } } \\hat { \\mitG } _ { \\mita } + \\miti \\hat { \\mitV } _ { \\mita } \\Big ) \\hat { \\mitxi } _ { \\mita } \\equiv \\sum _ { \\mita = 1 } ^ { \\mitN } \\hat { \\mitq } _ { \\mita } ~ \\hat { \\mitxi } _ { \\mita } , \\\\ \\displaystyle \\hat { \\bar { \\mitQ } } & = & \\displaystyle \\sum _ { \\mita = 1 } ^ { \\mitN } \\Big ( \\frac { 1 } { \\sqrt { 2 \\mitm } \\mitR _ { \\mita } } \\hat { \\mitG } _ { \\mita } - \\miti \\hat { \\mitV } _ { \\mita } \\Big ) \\hat { \\bar { \\mitxi } } \\equiv \\sum _ { \\mita = 1 } ^ { \\mitN } \\hat { \\mitq } _ { \\mita } ^ { \\dagger } ~ \\hat { \\bar { \\mitxi } } _ { \\mita } , \\end{align*}",
"\\begin{equation*} [ \\hat { \\mitG } _ { \\mita } , \\hat { \\mitW } _ { \\mitb } ] = \\hslash ~ \\mitdelta _ { \\mita \\mitb } \\hat { \\mitW } _ { \\mitb } , \\quad \\{ \\hat { \\mitxi } _ { \\mita } , \\hat { \\bar { \\mitxi } } _ { \\mitb } \\} = \\mitdelta _ { \\mita \\mitb } , \\quad \\{ \\hat { \\mitxi } _ { \\mita } , \\hat { \\mitxi } _ { \\mitb } \\} = 0 , \\quad \\{ \\hat { \\bar { \\mitxi } } _ { \\mita } , \\hat { \\bar { \\mitxi } } _ { \\mitb } \\} = 0 . \\quad \\end{equation*}"
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0003194_page13
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"latex": [
"$T^N=S^1\\otimes \\cdots \\otimes S^1$",
"$\\alpha _a (a=1, \\cdots , N)$",
"$S^1$",
"$\\hq \\ket {-}=\\hqb \\ket {+}=0$",
"$\\Psi ^{\\pm }(\\cdots , \\theta _a+2\\pi , \\cdots ) =\\Psi ^{\\pm }(\\cdots , \\theta _a, \\cdots )~(a=1,\\cdots , N)$",
"$\\Psi ^{\\pm }(\\cdots , \\theta _a+2\\pi , \\cdots ) =\\Psi ^{\\pm }(\\cdots , \\theta _a, \\cdots )~(a=1,\\cdots , N)$",
"$\\alpha _a (a=1, \\cdots , N)$",
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"$S^1$",
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"\\begin {equation} {\\hat H}=\\sum _{a=1}^N{1\\over {2 m R_a^2}}\\hg _a\\hg _a + {\\hat V}_a{\\hat V}_a -\\sum _{a, b=1}^N{i\\over {{\\sqrt {2m}R_a}}} [\\hg _a, {\\hat V}_b][\\hxi _a, \\hxib _b]. \\label {torus} \\end {equation}",
"\\begin {equation} \\ket {-}\\equiv \\ket {0},\\quad \\ket {+}\\equiv \\prod _{a=1}^N\\hxib _a~\\ket {0}\\quad {\\rm with}\\quad \\hxi _a \\ket {0}=0~~(a=1, \\cdots , N). \\end {equation}",
"\\begin {eqnarray} \\Psi ^{+}(\\theta _1,\\cdots , \\theta _N)&=& {\\rm exp}\\sum _{a=1}^{N}\\Bigl (-i\\alpha _a\\theta _a +{{\\sqrt {2m}R_a}\\over \\hbar }\\int ^{\\theta _a}d{\\bar \\theta }_a~ V_a({\\bar \\theta }_1,\\cdots , {\\bar \\theta }_a, \\cdots , {\\bar \\theta }_N)\\Bigr ) \\ket {+}, \\\\ \\Psi ^{-}(\\theta _1, \\cdots , \\theta _N)&=& {\\rm exp}\\sum _{a=1}^{N}\\Bigl (-i\\alpha _a\\theta _a -{{\\sqrt {2m}R_a}\\over \\hbar }\\int ^{\\theta _a}d{\\bar \\theta }_a~ V_a({\\bar \\theta }_1,\\cdots , {\\bar \\theta }_a, \\cdots , {\\bar \\theta }_N)\\Bigr ) \\ket {-}. \\end {eqnarray}"
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"$ \\Psi ^ { \\pm } ( \\cdots , \\theta _ { a } + 2 \\pi , \\cdots ) = \\Psi ^ { \\pm } ( \\cdots , \\theta _ { a } , \\cdots ) ~ ( a = 1 , \\cdots , N ) $",
"$ \\Psi ^ { \\pm } ( \\cdots , \\theta _ { a } + 2 \\pi , \\cdots ) = \\Psi ^ { \\pm } ( \\cdots , \\theta _ { a } , \\cdots ) ~ ( a = 1 , \\cdots , N ) $",
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"$ S ^ { 1 } $",
"$ S ^ { 1 } $",
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"\\begin{equation*} \\hat { H } = \\sum _ { a = 1 } ^ { N } \\frac { 1 } { 2 m R _ { a } ^ { 2 } } \\hat { G } _ { a } \\hat { G } _ { a } + \\hat { V } _ { a } \\hat { V } _ { a } - \\sum _ { a , b = 1 } ^ { N } \\frac { i } { \\sqrt { 2 m } R _ { a } } [ \\hat { G } _ { a } , \\hat { V } _ { b } ] [ \\hat { \\xi } _ { a } , \\hat { \\bar { \\xi } } _ { b } ] . \\end{equation*}",
"\\begin{equation*} \\vert - \\rangle \\equiv \\vert 0 \\rangle , \\quad \\vert + \\rangle \\equiv \\prod _ { a = 1 } ^ { N } \\hat { \\bar { \\xi } } _ { a } ~ \\vert 0 \\rangle \\quad w i t h \\quad \\hat { \\xi } _ { a } \\vert 0 \\rangle = 0 ~ ~ ( a = 1 , \\cdots , N ) . \\end{equation*}",
"\\begin{align*} \\Psi ^ { + } ( \\theta _ { 1 } , \\cdots , \\theta _ { N } ) & = & e x p \\sum _ { a = 1 } ^ { N } ( - i \\alpha _ { a } \\theta _ { a } + \\frac { \\sqrt { 2 m } R _ { a } } { \\hbar } \\int ^ { \\theta _ { a } } d \\bar { \\theta } _ { a } ~ V _ { a } ( \\bar { \\theta } _ { 1 } , \\cdots , \\bar { \\theta } _ { a } , \\cdots , \\bar { \\theta } _ { N } ) ) \\vert + \\rangle , \\\\ \\Psi ^ { - } ( \\theta _ { 1 } , \\cdots , \\theta _ { N } ) & = & e x p \\sum _ { a = 1 } ^ { N } ( - i \\alpha _ { a } \\theta _ { a } - \\frac { \\sqrt { 2 m } R _ { a } } { \\hbar } \\int ^ { \\theta _ { a } } d \\bar { \\theta } _ { a } ~ V _ { a } ( \\bar { \\theta } _ { 1 } , \\cdots , \\bar { \\theta } _ { a } , \\cdots , \\bar { \\theta } _ { N } ) ) \\vert - \\rangle . \\end{align*}"
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"\\begin{equation*} \\vert - \\rangle \\equiv \\vert 0 \\rangle , \\quad \\vert + \\rangle \\equiv \\prod _ { \\mita = 1 } ^ { \\mitN } \\hat { \\bar { \\mitxi } } _ { \\mita } ~ \\vert 0 \\rangle \\quad \\mathrm { w i t h } \\quad \\hat { \\mitxi } _ { \\mita } \\vert 0 \\rangle = 0 ~ ~ ( \\mita = 1 , \\cdots , \\mitN ) . \\end{equation*}",
"\\begin{align*} \\displaystyle \\mupPsi ^ { + } ( \\mittheta _ { 1 } , \\cdots , \\mittheta _ { \\mitN } ) & = & \\displaystyle \\mathrm { e x p } \\sum _ { \\mita = 1 } ^ { \\mitN } \\Big ( - \\miti \\mitalpha _ { \\mita } \\mittheta _ { \\mita } + \\frac { \\sqrt { 2 \\mitm } \\mitR _ { \\mita } } { \\hslash } \\int ^ { \\mittheta _ { \\mita } } \\mitd \\bar { \\mittheta } _ { \\mita } ~ \\mitV _ { \\mita } ( \\bar { \\mittheta } _ { 1 } , \\cdots , \\bar { \\mittheta } _ { \\mita } , \\cdots , \\bar { \\mittheta } _ { \\mitN } ) \\Big ) \\vert + \\rangle , \\\\ \\displaystyle \\mupPsi ^ { - } ( \\mittheta _ { 1 } , \\cdots , \\mittheta _ { \\mitN } ) & = & \\displaystyle \\mathrm { e x p } \\sum _ { \\mita = 1 } ^ { \\mitN } \\Big ( - \\miti \\mitalpha _ { \\mita } \\mittheta _ { \\mita } - \\frac { \\sqrt { 2 \\mitm } \\mitR _ { \\mita } } { \\hslash } \\int ^ { \\mittheta _ { \\mita } } \\mitd \\bar { \\mittheta } _ { \\mita } ~ \\mitV _ { \\mita } ( \\bar { \\mittheta } _ { 1 } , \\cdots , \\bar { \\mittheta } _ { \\mita } , \\cdots , \\bar { \\mittheta } _ { \\mitN } ) \\Big ) \\vert - \\rangle . \\end{align*}"
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0003194_page14
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{
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"$S^1$",
"$V(\\hw ,\\hwd )$",
"$\\hw $",
"$\\hwd $",
"${\\rm exp}(\\int ^{\\theta }d{\\bar \\theta } V(\\e ^{i{\\bar \\theta }},\\e ^{-i{\\bar \\theta }}))$",
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"$\\e ^{im\\theta }$",
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"$\\alpha $",
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"$(\\hxi , \\hxib )$",
"\\begin {equation} \\psi _{\\pm \\half }({\\theta })={\\rm exp}\\Bigl (-i\\alpha \\theta \\mp {{\\sqrt {2m}R}\\over \\hbar }\\int ^{\\theta }d{\\bar \\theta }~V(\\e ^{i{\\bar \\theta }}, \\e ^{-i{\\bar \\theta }}) \\Bigr ), \\label {closed} \\end {equation}"
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"$ ( \\hat { \\mitxi } , \\hat { \\bar { \\mitxi } } ) $",
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0003194_page15
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0003194_page16
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0003204_page01
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0003204_page02
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"\\begin{equation*} S = \\frac { 1 } { 4 \\pi \\alpha ^ { \\prime } } \\int _ { \\Sigma } d ^ { 2 } \\sigma ( g _ { \\mu \\nu } \\partial _ { a } X ^ { \\mu } \\partial ^ { a } X ^ { \\nu } - 2 \\pi \\alpha ^ { \\prime } B _ { \\mu \\nu } \\epsilon ^ { a b } \\partial _ { a } X ^ { \\mu } \\partial _ { b } X ^ { \\nu } ) + \\oint _ { \\partial \\Sigma } d \\tau A _ { i } ( X ) \\partial _ { \\tau } X ^ { i } , \\end{equation*}"
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0003204_page03
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"$A_i, i = 0,1,\\cdots ,p$",
"$p$",
"$g_{\\mu \\nu }, B_{\\mu \\nu }, \\mu = 0,1,\\cdots ,9$",
"$g_{\\mu \\nu }, B_{\\mu \\nu }, \\mu = 0,1,\\cdots ,9$",
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"$M_{ij} = g_{ij} - (2\\pi \\al )^2(\\F g^{-1}\\F )_{ij}$",
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"$M^{-1ik}\\F _{kl}g^{-1lj}$",
"$X^k(\\sigma ,\\tau )$",
"$\\tau = -i\\tau '$",
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"$M^{-1} - (2\\pi \\al )^2g^{-1}\\F M^{-1}\\F g^{-1} = g^{-1}, M^{-1}\\F g^{-1} = g^{-1}\\F M^{-1}$",
"\\begin {equation} S = \\frac {1}{4\\pi \\al }\\int _{\\Sigma }d^2\\sigma (g_{\\mu \\nu }\\partial _aX^{\\mu } \\partial ^aX^{\\nu } - 2\\pi \\al \\F _{ij}\\e ^{ab}\\partial _aX^{i}\\partial _b X^{j}) \\end {equation}",
"\\begin {equation} X^k = x_0^k + ( p_0^k\\tau + 2\\pi \\al p_0^j\\F _j^k\\sigma ) + \\sum _{n\\neq 0}\\frac {e^{-in\\tau }}{n}(ia_n^k \\cos n\\sigma + 2\\pi \\al a_n^j\\F _j^k \\sin n\\sigma ). \\label {mod}\\end {equation}",
"\\[ [a_m^i, a_n^j] = 2\\al mM^{-1ij}\\delta _{m+n}, \\hspace {1cm} [x_0^i, p_0^j] =i2\\al M^{-1ij}, \\]",
"\\begin {equation} [x_0^i, x_0^j] = i2\\pi \\al (M^{-1}\\F )^{ij}, \\hspace {1cm} [p_0^i, p_0^j] = 0, \\label {com}\\end {equation}",
"\\begin {equation} X^k(z) = x_0^k - \\frac {i}{2}(p_0^k\\ln z\\bar {z} + 2\\pi \\al p_0^j\\F _j^k \\ln \\frac {z}{\\bar {z}}) + i\\sum _{n\\neq 0} (a_n^k(z^{-n} + \\bar {z}^{-n}) + 2\\pi \\al a_n^j\\F _j^k(z^{-n} - \\bar {z}^{-n})). \\end {equation}",
"\\begin {eqnarray} <0|X^i(z)X^j(z')|0> = \\al ( -M^{-1ij}\\ln z\\bar {z} +2\\pi \\al (\\F M^{-1})^{ij} \\ln \\frac {z}{\\bar {z}} \\hspace {3cm} {}\\\\ -\\frac {1}{2}M^{-1ij}(\\ln |1-\\frac {z'}{z}|^2 + \\ln |1-\\frac {\\bar {z}'}{z}|^2) + \\frac {(2\\pi \\al )^2}{2}(\\F M^{-1}\\F )^{ij}(\\ln |1-\\frac {z'}{z}|^2 - \\ln |1-\\frac {\\bar {z}'}{z}|^2) \\\\ - \\frac {2\\pi \\al }{2}(M^{-1}\\F )^{ij}\\ln \\frac {(z-z')(\\bar {z}-z')} {(z-\\bar {z}')(\\bar {z}-\\bar {z}')} + \\frac {2\\pi \\al }{2}(\\F M^{-1})^{ij}(\\ln \\frac {(z-z')(z-\\bar {z}')}{(\\bar {z}-z')(\\bar {z}-\\bar {z}')} - 2\\ln \\frac {z}{\\bar {z}})), \\end {eqnarray}",
"\\begin {equation} -\\al (g^{-1ij}(\\ln |z-z'|-\\ln |z-\\bar {z}'|) + M^{-1ij}\\ln |z-z'|^2 -2\\pi \\al (M^{-1}\\F g^{-1})^{ij}\\ln \\frac {z-\\bar {z}'}{\\bar {z}-z'}). \\label {pro}\\end {equation}"
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"$ X ^ { i } $",
"$ \\partial _ { \\sigma } X ^ { i } - 2 \\pi \\alpha ^ { \\prime } \\partial _ { \\tau } X ^ { j } F _ { j } ^ { i } = 0 $",
"$ \\sigma = 0 , \\pi $",
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"$ M _ { i j } = g _ { i j } - ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } ( F g ^ { - 1 } F ) _ { i j } $",
"$ ( M ^ { - 1 } F ) ^ { i j } $",
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"$ X ^ { k } ( \\sigma , \\tau ) $",
"$ \\tau = - i \\tau ^ { \\prime } $",
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"$ M ^ { - 1 } - ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } g ^ { - 1 } F M ^ { - 1 } F g ^ { - 1 } = g ^ { - 1 } , M ^ { - 1 } F g ^ { - 1 } = g ^ { - 1 } F M ^ { - 1 } $",
"\\begin{equation*} S = \\frac { 1 } { 4 \\pi \\alpha ^ { \\prime } } \\int _ { \\Sigma } d ^ { 2 } \\sigma ( g _ { \\mu \\nu } \\partial _ { a } X ^ { \\mu } \\partial ^ { a } X ^ { \\nu } - 2 \\pi \\alpha ^ { \\prime } F _ { i j } \\epsilon ^ { a b } \\partial _ { a } X ^ { i } \\partial _ { b } X ^ { j } ) \\end{equation*}",
"\\begin{equation*} X ^ { k } = x _ { 0 } ^ { k } + ( p _ { 0 } ^ { k } \\tau + 2 \\pi \\alpha ^ { \\prime } p _ { 0 } ^ { j } F _ { j } ^ { k } \\sigma ) + \\sum _ { n \\ne 0 } \\frac { e ^ { - i n \\tau } } { n } ( i a _ { n } ^ { k } \\operatorname { c o s } n \\sigma + 2 \\pi \\alpha ^ { \\prime } a _ { n } ^ { j } F _ { j } ^ { k } \\operatorname { s i n } n \\sigma ) . \\end{equation*}",
"\\begin{equation*} [ a _ { m } ^ { i } , a _ { n } ^ { j } ] = 2 \\alpha ^ { \\prime } m M ^ { - 1 i j } \\delta _ { m + n } , \\hspace{28.45pt} [ x _ { 0 } ^ { i } , p _ { 0 } ^ { j } ] = i 2 \\alpha ^ { \\prime } M ^ { - 1 i j } , \\end{equation*}",
"\\begin{equation*} [ x _ { 0 } ^ { i } , x _ { 0 } ^ { j } ] = i 2 \\pi \\alpha ^ { \\prime } ( M ^ { - 1 } F ) ^ { i j } , \\hspace{28.45pt} [ p _ { 0 } ^ { i } , p _ { 0 } ^ { j } ] = 0 , \\end{equation*}",
"\\begin{equation*} X ^ { k } ( z ) = x _ { 0 } ^ { k } - \\frac { i } { 2 } ( p _ { 0 } ^ { k } \\operatorname { l n } z \\bar { z } + 2 \\pi \\alpha ^ { \\prime } p _ { 0 } ^ { j } F _ { j } ^ { k } \\operatorname { l n } \\frac { z } { \\bar { z } } ) + i \\sum _ { n \\ne 0 } ( a _ { n } ^ { k } ( z ^ { - n } + \\bar { z } ^ { - n } ) + 2 \\pi \\alpha ^ { \\prime } a _ { n } ^ { j } F _ { j } ^ { k } ( z ^ { - n } - \\bar { z } ^ { - n } ) ) . \\end{equation*}",
"\\begin{align*} < 0 \\vert X ^ { i } ( z ) X ^ { j } ( z ^ { \\prime } ) \\vert 0 > = \\alpha ^ { \\prime } ( - M ^ { - 1 i j } \\operatorname { l n } z \\bar { z } + 2 \\pi \\alpha ^ { \\prime } ( F M ^ { - 1 } ) ^ { i j } \\operatorname { l n } \\frac { z } { \\bar { z } } \\hspace{85.36pt} \\\\ - \\frac { 1 } { 2 } M ^ { - 1 i j } ( \\operatorname { l n } \\vert 1 - \\frac { z ^ { \\prime } } { z } \\vert ^ { 2 } + \\operatorname { l n } \\vert 1 - \\frac { \\bar { z } ^ { \\prime } } { z } \\vert ^ { 2 } ) + \\frac { ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } } { 2 } ( F M ^ { - 1 } F ) ^ { i j } ( \\operatorname { l n } \\vert 1 - \\frac { z ^ { \\prime } } { z } \\vert ^ { 2 } - \\operatorname { l n } \\vert 1 - \\frac { \\bar { z } ^ { \\prime } } { z } \\vert ^ { 2 } ) \\\\ - \\frac { 2 \\pi \\alpha ^ { \\prime } } { 2 } ( M ^ { - 1 } F ) ^ { i j } \\operatorname { l n } \\frac { ( z - z ^ { \\prime } ) ( \\bar { z } - z ^ { \\prime } ) } { ( z - \\bar { z } ^ { \\prime } ) ( \\bar { z } - \\bar { z } ^ { \\prime } ) } + \\frac { 2 \\pi \\alpha ^ { \\prime } } { 2 } ( F M ^ { - 1 } ) ^ { i j } ( \\operatorname { l n } \\frac { ( z - z ^ { \\prime } ) ( z - \\bar { z } ^ { \\prime } ) } { ( \\bar { z } - z ^ { \\prime } ) ( \\bar { z } - \\bar { z } ^ { \\prime } ) } - 2 \\operatorname { l n } \\frac { z } { \\bar { z } } ) ) , \\end{align*}",
"\\begin{equation*} - \\alpha ^ { \\prime } ( g ^ { - 1 i j } ( \\operatorname { l n } \\vert z - z ^ { \\prime } \\vert - \\operatorname { l n } \\vert z - \\bar { z } ^ { \\prime } \\vert ) + M ^ { - 1 i j } \\operatorname { l n } \\vert z - z ^ { \\prime } \\vert ^ { 2 } - 2 \\pi \\alpha ^ { \\prime } ( M ^ { - 1 } F g ^ { - 1 } ) ^ { i j } \\operatorname { l n } \\frac { z - \\bar { z } ^ { \\prime } } { \\bar { z } - z ^ { \\prime } } ) . \\end{equation*}"
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"$ \\mitA _ { \\miti } , \\miti = 0 , 1 , \\cdots , \\mitp $",
"$ \\mitp $",
"$ \\mitg _ { \\mitmu \\mitnu } , \\mitB _ { \\mitmu \\mitnu } , \\mitmu = 0 , 1 , \\cdots , 9 $",
"$ \\mitg _ { \\mitmu \\mitnu } , \\mitB _ { \\mitmu \\mitnu } , \\mitmu = 0 , 1 , \\cdots , 9 $",
"$ \\mitB _ { \\mitmu \\mitnu } $",
"$ \\mitp $",
"$ \\mscrF _ { \\miti \\mitj } = \\mitB _ { \\miti \\mitj } + \\mitF _ { \\miti \\mitj } $",
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"$ \\mitX ^ { \\mitmu } , \\mitmu = \\mitp + 1 , \\cdots , 9 $",
"$ \\mitX ^ { \\miti } $",
"$ \\mitX ^ { \\miti } $",
"$ \\mitpartial _ { \\mitsigma } \\mitX ^ { \\miti } - 2 \\mitpi \\mitalpha ^ { \\prime } \\mitpartial _ { \\mittau } \\mitX ^ { \\mitj } \\mscrF _ { \\mitj } ^ { \\miti } = 0 $",
"$ \\mitsigma = 0 , \\mitpi $",
"$ \\mitX ^ { \\mitk } $",
"$ \\mitM _ { \\miti \\mitj } = \\mitg _ { \\miti \\mitj } - ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } ( \\mscrF \\mitg ^ { - 1 } \\mscrF ) _ { \\miti \\mitj } $",
"$ ( \\mitM ^ { - 1 } \\mscrF ) ^ { \\miti \\mitj } $",
"$ \\mitM ^ { - 1 \\miti \\mitk } \\mscrF _ { \\mitk \\mitl } \\mitg ^ { - 1 \\mitl \\mitj } $",
"$ \\mitX ^ { \\mitk } ( \\mitsigma , \\mittau ) $",
"$ \\mittau = - \\miti \\mittau ^ { \\prime } $",
"$ \\mitz = \\mite ^ { \\mittau ^ { \\prime } + \\miti \\mitsigma } $",
"$ \\mitM ^ { - 1 } - ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mitg ^ { - 1 } \\mscrF \\mitM ^ { - 1 } \\mscrF \\mitg ^ { - 1 } = \\mitg ^ { - 1 } , \\mitM ^ { - 1 } \\mscrF \\mitg ^ { - 1 } = \\mitg ^ { - 1 } \\mscrF \\mitM ^ { - 1 } $",
"\\begin{equation*} \\mitS = \\frac { 1 } { 4 \\mitpi \\mitalpha ^ { \\prime } } \\int _ { \\mupSigma } \\mitd ^ { 2 } \\mitsigma ( \\mitg _ { \\mitmu \\mitnu } \\mitpartial _ { \\mita } \\mitX ^ { \\mitmu } \\mitpartial ^ { \\mita } \\mitX ^ { \\mitnu } - 2 \\mitpi \\mitalpha ^ { \\prime } \\mscrF _ { \\miti \\mitj } \\mitepsilon ^ { \\mita \\mitb } \\mitpartial _ { \\mita } \\mitX ^ { \\miti } \\mitpartial _ { \\mitb } \\mitX ^ { \\mitj } ) \\end{equation*}",
"\\begin{equation*} \\mitX ^ { \\mitk } = \\mitx _ { 0 } ^ { \\mitk } + ( \\mitp _ { 0 } ^ { \\mitk } \\mittau + 2 \\mitpi \\mitalpha ^ { \\prime } \\mitp _ { 0 } ^ { \\mitj } \\mscrF _ { \\mitj } ^ { \\mitk } \\mitsigma ) + \\sum _ { \\mitn \\ne 0 } \\frac { \\mite ^ { - \\miti \\mitn \\mittau } } { \\mitn } ( \\miti \\mita _ { \\mitn } ^ { \\mitk } \\operatorname { c o s } \\mitn \\mitsigma + 2 \\mitpi \\mitalpha ^ { \\prime } \\mita _ { \\mitn } ^ { \\mitj } \\mscrF _ { \\mitj } ^ { \\mitk } \\operatorname { s i n } \\mitn \\mitsigma ) . \\end{equation*}",
"\\begin{equation*} [ \\mita _ { \\mitm } ^ { \\miti } , \\mita _ { \\mitn } ^ { \\mitj } ] = 2 \\mitalpha ^ { \\prime } \\mitm \\mitM ^ { - 1 \\miti \\mitj } \\mitdelta _ { \\mitm + \\mitn } , \\hspace{28.45pt} [ \\mitx _ { 0 } ^ { \\miti } , \\mitp _ { 0 } ^ { \\mitj } ] = \\miti 2 \\mitalpha ^ { \\prime } \\mitM ^ { - 1 \\miti \\mitj } , \\end{equation*}",
"\\begin{equation*} [ \\mitx _ { 0 } ^ { \\miti } , \\mitx _ { 0 } ^ { \\mitj } ] = \\miti 2 \\mitpi \\mitalpha ^ { \\prime } ( \\mitM ^ { - 1 } \\mscrF ) ^ { \\miti \\mitj } , \\hspace{28.45pt} [ \\mitp _ { 0 } ^ { \\miti } , \\mitp _ { 0 } ^ { \\mitj } ] = 0 , \\end{equation*}",
"\\begin{equation*} \\mitX ^ { \\mitk } ( \\mitz ) = \\mitx _ { 0 } ^ { \\mitk } - \\frac { \\miti } { 2 } ( \\mitp _ { 0 } ^ { \\mitk } \\operatorname { l n } \\mitz \\bar { \\mitz } + 2 \\mitpi \\mitalpha ^ { \\prime } \\mitp _ { 0 } ^ { \\mitj } \\mscrF _ { \\mitj } ^ { \\mitk } \\operatorname { l n } \\frac { \\mitz } { \\bar { \\mitz } } ) + \\miti \\sum _ { \\mitn \\ne 0 } ( \\mita _ { \\mitn } ^ { \\mitk } ( \\mitz ^ { - \\mitn } + \\bar { \\mitz } ^ { - \\mitn } ) + 2 \\mitpi \\mitalpha ^ { \\prime } \\mita _ { \\mitn } ^ { \\mitj } \\mscrF _ { \\mitj } ^ { \\mitk } ( \\mitz ^ { - \\mitn } - \\bar { \\mitz } ^ { - \\mitn } ) ) . \\end{equation*}",
"\\begin{align*} \\displaystyle < 0 \\vert \\mitX ^ { \\miti } ( \\mitz ) \\mitX ^ { \\mitj } ( \\mitz ^ { \\prime } ) \\vert 0 > = \\mitalpha ^ { \\prime } ( - \\mitM ^ { - 1 \\miti \\mitj } \\operatorname { l n } \\mitz \\bar { \\mitz } + 2 \\mitpi \\mitalpha ^ { \\prime } ( \\mscrF \\mitM ^ { - 1 } ) ^ { \\miti \\mitj } \\operatorname { l n } \\frac { \\mitz } { \\bar { \\mitz } } \\hspace{85.36pt} \\\\ \\displaystyle - \\frac { 1 } { 2 } \\mitM ^ { - 1 \\miti \\mitj } ( \\operatorname { l n } \\vert 1 - \\frac { \\mitz ^ { \\prime } } { \\mitz } \\vert ^ { 2 } + \\operatorname { l n } \\vert 1 - \\frac { \\bar { \\mitz } ^ { \\prime } } { \\mitz } \\vert ^ { 2 } ) + \\frac { ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } } { 2 } ( \\mscrF \\mitM ^ { - 1 } \\mscrF ) ^ { \\miti \\mitj } ( \\operatorname { l n } \\vert 1 - \\frac { \\mitz ^ { \\prime } } { \\mitz } \\vert ^ { 2 } - \\operatorname { l n } \\vert 1 - \\frac { \\bar { \\mitz } ^ { \\prime } } { \\mitz } \\vert ^ { 2 } ) \\\\ \\displaystyle - \\frac { 2 \\mitpi \\mitalpha ^ { \\prime } } { 2 } ( \\mitM ^ { - 1 } \\mscrF ) ^ { \\miti \\mitj } \\operatorname { l n } \\frac { ( \\mitz - \\mitz ^ { \\prime } ) ( \\bar { \\mitz } - \\mitz ^ { \\prime } ) } { ( \\mitz - \\bar { \\mitz } ^ { \\prime } ) ( \\bar { \\mitz } - \\bar { \\mitz } ^ { \\prime } ) } + \\frac { 2 \\mitpi \\mitalpha ^ { \\prime } } { 2 } ( \\mscrF \\mitM ^ { - 1 } ) ^ { \\miti \\mitj } ( \\operatorname { l n } \\frac { ( \\mitz - \\mitz ^ { \\prime } ) ( \\mitz - \\bar { \\mitz } ^ { \\prime } ) } { ( \\bar { \\mitz } - \\mitz ^ { \\prime } ) ( \\bar { \\mitz } - \\bar { \\mitz } ^ { \\prime } ) } - 2 \\operatorname { l n } \\frac { \\mitz } { \\bar { \\mitz } } ) ) , \\end{align*}",
"\\begin{equation*} - \\mitalpha ^ { \\prime } ( \\mitg ^ { - 1 \\miti \\mitj } ( \\operatorname { l n } \\vert \\mitz - \\mitz ^ { \\prime } \\vert - \\operatorname { l n } \\vert \\mitz - \\bar { \\mitz } ^ { \\prime } \\vert ) + \\mitM ^ { - 1 \\miti \\mitj } \\operatorname { l n } \\vert \\mitz - \\mitz ^ { \\prime } \\vert ^ { 2 } - 2 \\mitpi \\mitalpha ^ { \\prime } ( \\mitM ^ { - 1 } \\mscrF \\mitg ^ { - 1 } ) ^ { \\miti \\mitj } \\operatorname { l n } \\frac { \\mitz - \\bar { \\mitz } ^ { \\prime } } { \\bar { \\mitz } - \\mitz ^ { \\prime } } ) . \\end{equation*}"
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0003204_page04
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"latex": [
"$M^{-1} = (g+2\\pi \\al \\F )^{-1}g(g-2\\pi \\al \\F )^{-1}$",
"$M^{-1}\\F g^{-1} = (g+2\\pi \\al \\F )^{-1}\\F (g-2\\pi \\al \\F )^{-1}$",
"$M^{-1}\\F g^{-1} = (g+2\\pi \\al \\F )^{-1}\\F (g-2\\pi \\al \\F )^{-1}$",
"$\\te $",
"$G, \\Phi $",
"$G$",
"$\\Phi $",
"$g, B$",
"$\\te $",
"$(p+1)\\times (p+1)$",
"$B_{0i}=0$",
"$g_{0i}=0$",
"$i=1,\\cdots ,p$",
"$p$",
"$p$",
"$T^p$",
"$x^i \\sim x^i + 2\\pi r$",
"$i = 1,\\cdots ,p$",
"$g_{ij}$",
"$SO(p,p,Z)$",
"$E = r^2(g + 2\\pi \\al B)/\\al $",
"$E' = (aE + b)(cE + d)^{-1}$",
"$c^ta + a^tc = 0, d^tb + b^td = 0, c^tb + a^td = 1$",
"$a, b, c$",
"$d$",
"$p\\times p$",
"$\\te $",
"$\\te ^{0i}$",
"$p\\times p$",
"$G_{00}=g_{00}$",
"$\\Phi _{0i}=0$",
"$\\Phi $",
"$p\\times p$",
"$p\\times p$",
"$\\Te = \\te /2\\pi r^2$",
"$G$",
"$\\Phi $",
"$g$",
"$B$",
"$G$",
"$\\te $",
"$\\Phi $",
"$SO(p)$",
"$\\Te $",
"$\\Te \\rightarrow \\Te ' = (c + d\\Te )(a + b\\Te )^{-1}$",
"$G$",
"$\\Phi $",
"$E' + E'^t = (E^tc^t + d^t)^{-1} (E + E^t)(cE+ d)^{-1}$",
"${\\Te '}^t = -\\Te '$",
"$\\Phi '$",
"\\begin {equation} \\frac {1}{G + 2\\pi \\al \\Phi } = - \\frac {\\te }{2\\pi \\al } + \\frac {1}{g + 2\\pi \\al B}, \\label {swf}\\end {equation}",
"\\begin {eqnarray} G &=& \\frac {\\al }{2r^2} \\frac {1}{1+E^t\\Te }( E + E^t )\\frac {1}{1-\\Te E}, \\\\ \\Phi &=& \\frac {1}{4\\pi r^2} \\frac {1}{1+E^t\\Te }( 2E^t\\Te E + E - E^t ) \\frac {1}{1-\\Te E}. \\end {eqnarray}",
"\\begin {equation} 1- \\Te 'E' = \\frac {1}{a^t - \\Te b^t}(1-\\Te E)\\frac {1}{cE + d}. \\label {the}\\end {equation}",
"\\begin {equation} G' = ( a + b\\Te ) G ( a + b \\Te )^t. \\label {tg}\\end {equation}",
"\\begin {equation} \\Phi ' = \\frac {1}{4\\pi r^2}(a+b\\Te )\\frac {1}{1+E^t\\Te }(2X+Y) \\frac {1}{1-\\Te E}(a+b\\Te )^t, \\end {equation}"
],
"latex_norm": [
"$ M ^ { - 1 } = ( g + 2 \\pi \\alpha ^ { \\prime } F ) ^ { - 1 } g ( g - 2 \\pi \\alpha ^ { \\prime } F ) ^ { - 1 } $",
"$ M ^ { - 1 } F g ^ { - 1 } = ( g + 2 \\pi \\alpha ^ { \\prime } F ) ^ { - 1 } F ( g - 2 \\pi \\alpha ^ { \\prime } F ) ^ { - 1 } $",
"$ M ^ { - 1 } F g ^ { - 1 } = ( g + 2 \\pi \\alpha ^ { \\prime } F ) ^ { - 1 } F ( g - 2 \\pi \\alpha ^ { \\prime } F ) ^ { - 1 } $",
"$ \\theta $",
"$ G , \\Phi $",
"$ G $",
"$ \\Phi $",
"$ g , B $",
"$ \\theta $",
"$ ( p + 1 ) \\times ( p + 1 ) $",
"$ B _ { 0 i } = 0 $",
"$ g _ { 0 i } = 0 $",
"$ i = 1 , \\cdots , p $",
"$ p $",
"$ p $",
"$ T ^ { p } $",
"$ x ^ { i } \\sim x ^ { i } + 2 \\pi r $",
"$ i = 1 , \\cdots , p $",
"$ g _ { i j } $",
"$ S O ( p , p , Z ) $",
"$ E = r ^ { 2 } ( g + 2 \\pi \\alpha ^ { \\prime } B ) \\slash \\alpha ^ { \\prime } $",
"$ E ^ { \\prime } = ( a E + b ) ( c E + d ) ^ { - 1 } $",
"$ c ^ { t } a + a ^ { t } c = 0 , d ^ { t } b + b ^ { t } d = 0 , c ^ { t } b + a ^ { t } d = 1 $",
"$ a , b , c $",
"$ d $",
"$ p \\times p $",
"$ \\theta $",
"$ \\theta ^ { 0 i } $",
"$ p \\times p $",
"$ G _ { 0 0 } = g _ { 0 0 } $",
"$ \\Phi _ { 0 i } = 0 $",
"$ \\Phi $",
"$ p \\times p $",
"$ p \\times p $",
"$ \\Theta = \\theta \\slash 2 \\pi r ^ { 2 } $",
"$ G $",
"$ \\Phi $",
"$ g $",
"$ B $",
"$ G $",
"$ \\theta $",
"$ \\Phi $",
"$ S O ( p ) $",
"$ \\Theta $",
"$ \\Theta \\rightarrow \\Theta ^ { \\prime } = ( c + d \\Theta ) ( a + b \\Theta ) ^ { - 1 } $",
"$ G $",
"$ \\Phi $",
"$ E ^ { \\prime } + E ^ { \\prime t } = ( E ^ { t } c ^ { t } + d ^ { t } ) ^ { - 1 } ( E + E ^ { t } ) ( c E + d ) ^ { - 1 } $",
"$ { \\Theta ^ { \\prime } } ^ { t } = - \\Theta ^ { \\prime } $",
"$ \\Phi ^ { \\prime } $",
"\\begin{equation*} \\frac { 1 } { G + 2 \\pi \\alpha ^ { \\prime } \\Phi } = - \\frac { \\theta } { 2 \\pi \\alpha ^ { \\prime } } + \\frac { 1 } { g + 2 \\pi \\alpha ^ { \\prime } B } , \\end{equation*}",
"\\begin{align*} G & = & \\frac { \\alpha ^ { \\prime } } { 2 r ^ { 2 } } \\frac { 1 } { 1 + E ^ { t } \\Theta } ( E + E ^ { t } ) \\frac { 1 } { 1 - \\Theta E } , \\\\ \\Phi & = & \\frac { 1 } { 4 \\pi r ^ { 2 } } \\frac { 1 } { 1 + E ^ { t } \\Theta } ( 2 E ^ { t } \\Theta E + E - E ^ { t } ) \\frac { 1 } { 1 - \\Theta E } . \\end{align*}",
"\\begin{equation*} 1 - \\Theta ^ { \\prime } E ^ { \\prime } = \\frac { 1 } { a ^ { t } - \\Theta b ^ { t } } ( 1 - \\Theta E ) \\frac { 1 } { c E + d } . \\end{equation*}",
"\\begin{equation*} G ^ { \\prime } = ( a + b \\Theta ) G ( a + b \\Theta ) ^ { t } . \\end{equation*}",
"\\begin{equation*} \\Phi ^ { \\prime } = \\frac { 1 } { 4 \\pi r ^ { 2 } } ( a + b \\Theta ) \\frac { 1 } { 1 + E ^ { t } \\Theta } ( 2 X + Y ) \\frac { 1 } { 1 - \\Theta E } ( a + b \\Theta ) ^ { t } , \\end{equation*}"
],
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"$ \\mitM ^ { - 1 } = ( \\mitg + 2 \\mitpi \\mitalpha ^ { \\prime } \\mscrF ) ^ { - 1 } \\mitg ( \\mitg - 2 \\mitpi \\mitalpha ^ { \\prime } \\mscrF ) ^ { - 1 } $",
"$ \\mitM ^ { - 1 } \\mscrF \\mitg ^ { - 1 } = ( \\mitg + 2 \\mitpi \\mitalpha ^ { \\prime } \\mscrF ) ^ { - 1 } \\mscrF ( \\mitg - 2 \\mitpi \\mitalpha ^ { \\prime } \\mscrF ) ^ { - 1 } $",
"$ \\mitM ^ { - 1 } \\mscrF \\mitg ^ { - 1 } = ( \\mitg + 2 \\mitpi \\mitalpha ^ { \\prime } \\mscrF ) ^ { - 1 } \\mscrF ( \\mitg - 2 \\mitpi \\mitalpha ^ { \\prime } \\mscrF ) ^ { - 1 } $",
"$ \\mittheta $",
"$ \\mitG , \\mupPhi $",
"$ \\mitG $",
"$ \\mupPhi $",
"$ \\mitg , \\mitB $",
"$ \\mittheta $",
"$ ( \\mitp + 1 ) \\times ( \\mitp + 1 ) $",
"$ \\mitB _ { 0 \\miti } = 0 $",
"$ \\mitg _ { 0 \\miti } = 0 $",
"$ \\miti = 1 , \\cdots , \\mitp $",
"$ \\mitp $",
"$ \\mitp $",
"$ \\mitT ^ { \\mitp } $",
"$ \\mitx ^ { \\miti } \\sim \\mitx ^ { \\miti } + 2 \\mitpi \\mitr $",
"$ \\miti = 1 , \\cdots , \\mitp $",
"$ \\mitg _ { \\miti \\mitj } $",
"$ \\mitS \\mitO ( \\mitp , \\mitp , \\mitZ ) $",
"$ \\mitE = \\mitr ^ { 2 } ( \\mitg + 2 \\mitpi \\mitalpha ^ { \\prime } \\mitB ) \\slash \\mitalpha ^ { \\prime } $",
"$ \\mitE ^ { \\prime } = ( \\mita \\mitE + \\mitb ) ( \\mitc \\mitE + \\mitd ) ^ { - 1 } $",
"$ \\mitc ^ { \\mitt } \\mita + \\mita ^ { \\mitt } \\mitc = 0 , \\mitd ^ { \\mitt } \\mitb + \\mitb ^ { \\mitt } \\mitd = 0 , \\mitc ^ { \\mitt } \\mitb + \\mita ^ { \\mitt } \\mitd = 1 $",
"$ \\mita , \\mitb , \\mitc $",
"$ \\mitd $",
"$ \\mitp \\times \\mitp $",
"$ \\mittheta $",
"$ \\mittheta ^ { 0 \\miti } $",
"$ \\mitp \\times \\mitp $",
"$ \\mitG _ { 0 0 } = \\mitg _ { 0 0 } $",
"$ \\mupPhi _ { 0 \\miti } = 0 $",
"$ \\mupPhi $",
"$ \\mitp \\times \\mitp $",
"$ \\mitp \\times \\mitp $",
"$ \\mupTheta = \\mittheta \\slash 2 \\mitpi \\mitr ^ { 2 } $",
"$ \\mitG $",
"$ \\mupPhi $",
"$ \\mitg $",
"$ \\mitB $",
"$ \\mitG $",
"$ \\mittheta $",
"$ \\mupPhi $",
"$ \\mitS \\mitO ( \\mitp ) $",
"$ \\mupTheta $",
"$ \\mupTheta \\rightarrow \\mupTheta ^ { \\prime } = ( \\mitc + \\mitd \\mupTheta ) ( \\mita + \\mitb \\mupTheta ) ^ { - 1 } $",
"$ \\mitG $",
"$ \\mupPhi $",
"$ \\mitE ^ { \\prime } + \\mitE ^ { \\prime \\mitt } = ( \\mitE ^ { \\mitt } \\mitc ^ { \\mitt } + \\mitd ^ { \\mitt } ) ^ { - 1 } ( \\mitE + \\mitE ^ { \\mitt } ) ( \\mitc \\mitE + \\mitd ) ^ { - 1 } $",
"$ { \\mupTheta ^ { \\prime } } ^ { \\mitt } = - \\mupTheta ^ { \\prime } $",
"$ \\mupPhi ^ { \\prime } $",
"\\begin{equation*} \\frac { 1 } { \\mitG + 2 \\mitpi \\mitalpha ^ { \\prime } \\mupPhi } = - \\frac { \\mittheta } { 2 \\mitpi \\mitalpha ^ { \\prime } } + \\frac { 1 } { \\mitg + 2 \\mitpi \\mitalpha ^ { \\prime } \\mitB } , \\end{equation*}",
"\\begin{align*} \\displaystyle \\mitG & = & \\displaystyle \\frac { \\mitalpha ^ { \\prime } } { 2 \\mitr ^ { 2 } } \\frac { 1 } { 1 + \\mitE ^ { \\mitt } \\mupTheta } ( \\mitE + \\mitE ^ { \\mitt } ) \\frac { 1 } { 1 - \\mupTheta \\mitE } , \\\\ \\displaystyle \\mupPhi & = & \\displaystyle \\frac { 1 } { 4 \\mitpi \\mitr ^ { 2 } } \\frac { 1 } { 1 + \\mitE ^ { \\mitt } \\mupTheta } ( 2 \\mitE ^ { \\mitt } \\mupTheta \\mitE + \\mitE - \\mitE ^ { \\mitt } ) \\frac { 1 } { 1 - \\mupTheta \\mitE } . \\end{align*}",
"\\begin{equation*} 1 - \\mupTheta ^ { \\prime } \\mitE ^ { \\prime } = \\frac { 1 } { \\mita ^ { \\mitt } - \\mupTheta \\mitb ^ { \\mitt } } ( 1 - \\mupTheta \\mitE ) \\frac { 1 } { \\mitc \\mitE + \\mitd } . \\end{equation*}",
"\\begin{equation*} \\mitG ^ { \\prime } = ( \\mita + \\mitb \\mupTheta ) \\mitG ( \\mita + \\mitb \\mupTheta ) ^ { \\mitt } . \\end{equation*}",
"\\begin{equation*} \\mupPhi ^ { \\prime } = \\frac { 1 } { 4 \\mitpi \\mitr ^ { 2 } } ( \\mita + \\mitb \\mupTheta ) \\frac { 1 } { 1 + \\mitE ^ { \\mitt } \\mupTheta } ( 2 \\mitX + \\mitY ) \\frac { 1 } { 1 - \\mupTheta \\mitE } ( \\mita + \\mitb \\mupTheta ) ^ { \\mitt } , \\end{equation*}"
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0003204_page05
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"latex": [
"$X = (E^ta^t+b^t)(c+d\\Te )(a+b\\Te )^{-1}(aE+b)$",
"$Y$",
"$Y = E-E^t + 2Y_0$",
"$Y_0 = -(E^ta^t + b^t)cE + (E^tc^t+d^t)b$",
"$X=(E^ta^t + b^t)\\Te '(aE + b)$",
"${\\Te '}^t = - \\Te '$",
"$X=( E^t + (1+E^t\\Te )b^t(a^t-\\Te b^t)^{-1})(\\Te +X_0)(E+ (a+b\\Te )^{-1} b(1-\\Te E))$",
"$X=( E^t + (1+E^t\\Te )b^t(a^t-\\Te b^t)^{-1})(\\Te +X_0)(E+ (a+b\\Te )^{-1} b(1-\\Te E))$",
"$X_0 = -(c^t-\\Te d^t)b\\Te + (a^t - \\Te b^t)c$",
"$2X + Y = 2E^t\\Te E + E - E^t + 2Z,$",
"$2X + Y = 2E^t\\Te E + E - E^t + 2Z,$",
"$Z_0 = (1+ E^t\\Te )b^t(a^t- \\Te b^t)^{-1} (\\Te +X_0)(a+b\\Te )^{-1}b(1-\\Te E)$",
"$X_0$",
"$E^t(a^tc+(a^t-\\Te b^t)d\\Te (a+b\\Te )^{-1}b\\Te )E$",
"$Y_0$",
"$b^t(c+d\\Te )(a+b\\Te )^{-1} = (a+b\\Te )^{-1} - d^t$",
"$[(1+E^t\\Te ) (b^t(a^t-\\Te b^t)^{-1} - d^tb) + (a+b\\Te )^{-1}b]\\Te E + b^tcE$",
"$Y_0$",
"$E^t[(\\Te + (a^t-\\Te b^t)c) (a+b\\Te )^{-1}b - (c^t-\\Te d^t)b\\Te (a+b\\Te )^{-1}b(1-\\Te E)]$",
"$E^t[(a^tc + \\Te d^ta)(a+b\\Te )^{-1}b -(c^t-\\Te d^t)(1-a(a+b\\Te )^{-1})b(1-\\Te E)]$",
"$E^t(a^tc+c^ta)(a+b\\Te )^{-1}b$",
"$Z_0=(E^t\\Te +1)b^t (a^t-\\Te b^t)^{-1}(\\Te +X_0)(-E + (a+b\\Te )^{-1}(aE+b))$",
"$Z_0=(E^t\\Te +1)b^t (a^t-\\Te b^t)^{-1}(\\Te +X_0)(-E + (a+b\\Te )^{-1}(aE+b))$",
"$-E^t(c^t-\\Te d^t)b \\Te E$",
"$Z_0$",
"$X_0$",
"$b$",
"\\begin {equation} X = \\frac {1}{2}(E^ta^t+ b^t)((c+d\\Te )\\frac {1}{a+b\\Te }- \\frac {1}{a^t-\\Te b^t}(c^t-\\Te d^t))(aE + b) \\end {equation}",
"\\begin {eqnarray} Z &=& E^tX_0E + E^t(\\Te + X_0)\\frac {1}{a + b\\Te } b(1-\\Te E) \\\\ &+& (1 + E^t\\Te )b^t\\frac {1}{a^t -\\Te b^t}(\\Te + X_0)E + Y_0 + Z_0 \\end {eqnarray}",
"\\begin {equation} c^t - \\Te d^t = - (a^t - \\Te b^t)(c + d\\Te )\\frac {1}{a + b\\Te } \\label {cd}\\end {equation}",
"\\begin {eqnarray} Z &=& E^t(c^tb + (a^t-\\Te b^t)(c + d\\Te )\\frac {1}{a+b\\Te }b + \\Te \\frac {1}{a+b\\Te }b -2\\Te d^tb)\\Te E \\\\ &+& E^t\\Te d^tb + (\\frac {1}{a+b\\Te } - d^tb)\\Te E + d^tb + Z_0 + (1+ E^t\\Te )b^t\\frac {1}{a^t -\\Te b^t}\\Te E, \\end {eqnarray}",
"\\begin {eqnarray} Z_0 &=& (E^t\\Te +1)b^t[ \\frac {1}{a^t-\\Te b^t}\\Te \\frac {1}{a+b\\Te }(aE+b) \\\\ &+& ((c+d\\Te )\\frac {1}{a+b\\Te }b\\Te + c)\\frac {1}{a+b\\Te } b(1-\\Te E)]. \\end {eqnarray}",
"\\begin {equation} \\frac {1}{a^t - \\Te b^t}\\Te + c = (c + d\\Te )\\frac {1}{a + b\\Te }a, \\label {ab}\\end {equation}"
],
"latex_norm": [
"$ X = ( E ^ { t } a ^ { t } + b ^ { t } ) ( c + d \\Theta ) ( a + b \\Theta ) ^ { - 1 } ( a E + b ) $",
"$ Y $",
"$ Y = E - E ^ { t } + 2 Y _ { 0 } $",
"$ Y _ { 0 } = - ( E ^ { t } a ^ { t } + b ^ { t } ) c E + ( E ^ { t } c ^ { t } + d ^ { t } ) b $",
"$ X = ( E ^ { t } a ^ { t } + b ^ { t } ) \\Theta ^ { \\prime } ( a E + b ) $",
"$ { \\Theta ^ { \\prime } } ^ { t } = - \\Theta ^ { \\prime } $",
"$ X = ( E ^ { t } + ( 1 + E ^ { t } \\Theta ) b ^ { t } ( a ^ { t } - \\Theta b ^ { t } ) ^ { - 1 } ) ( \\Theta + X _ { 0 } ) ( E + ( a + b \\Theta ) ^ { - 1 } b ( 1 - \\Theta E ) ) $",
"$ X = ( E ^ { t } + ( 1 + E ^ { t } \\Theta ) b ^ { t } ( a ^ { t } - \\Theta b ^ { t } ) ^ { - 1 } ) ( \\Theta + X _ { 0 } ) ( E + ( a + b \\Theta ) ^ { - 1 } b ( 1 - \\Theta E ) ) $",
"$ X _ { 0 } = - ( c ^ { t } - \\Theta d ^ { t } ) b \\Theta + ( a ^ { t } - \\Theta b ^ { t } ) c $",
"$ 2 X + Y = 2 E ^ { t } \\Theta E + E - E ^ { t } + 2 Z , $",
"$ 2 X + Y = 2 E ^ { t } \\Theta E + E - E ^ { t } + 2 Z , $",
"$ Z _ { 0 } = ( 1 + E ^ { t } \\Theta ) b ^ { t } ( a ^ { t } - \\Theta b ^ { t } ) ^ { - 1 } ( \\Theta + X _ { 0 } ) ( a + b \\Theta ) ^ { - 1 } b ( 1 - \\Theta E ) $",
"$ X _ { 0 } $",
"$ E ^ { t } ( a ^ { t } c + ( a ^ { t } - \\Theta b ^ { t } ) d \\Theta ( a + b \\Theta ) ^ { - 1 } b \\Theta ) E $",
"$ Y _ { 0 } $",
"$ b ^ { t } ( c + d \\Theta ) ( a + b \\Theta ) ^ { - 1 } = ( a + b \\Theta ) ^ { - 1 } - d ^ { t } $",
"$ [ ( 1 + E ^ { t } \\Theta ) ( b ^ { t } ( a ^ { t } - \\Theta b ^ { t } ) ^ { - 1 } - d ^ { t } b ) + ( a + b \\Theta ) ^ { - 1 } b ] \\Theta E + b ^ { t } c E $",
"$ Y _ { 0 } $",
"$ E ^ { t } [ ( \\Theta + ( a ^ { t } - \\Theta b ^ { t } ) c ) ( a + b \\Theta ) ^ { - 1 } b - ( c ^ { t } - \\Theta d ^ { t } ) b \\Theta ( a + b \\Theta ) ^ { - 1 } b ( 1 - \\Theta E ) ] $",
"$ E ^ { t } [ ( a ^ { t } c + \\Theta d ^ { t } a ) ( a + b \\Theta ) ^ { - 1 } b - ( c ^ { t } - \\Theta d ^ { t } ) ( 1 - a ( a + b \\Theta ) ^ { - 1 } ) b ( 1 - \\Theta E ) ] $",
"$ E ^ { t } ( a ^ { t } c + c ^ { t } a ) ( a + b \\Theta ) ^ { - 1 } b $",
"$ Z _ { 0 } = ( E ^ { t } \\Theta + 1 ) b ^ { t } ( a ^ { t } - \\Theta b ^ { t } ) ^ { - 1 } ( \\Theta + X _ { 0 } ) ( - E + ( a + b \\Theta ) ^ { - 1 } ( a E + b ) ) $",
"$ Z _ { 0 } = ( E ^ { t } \\Theta + 1 ) b ^ { t } ( a ^ { t } - \\Theta b ^ { t } ) ^ { - 1 } ( \\Theta + X _ { 0 } ) ( - E + ( a + b \\Theta ) ^ { - 1 } ( a E + b ) ) $",
"$ - E ^ { t } ( c ^ { t } - \\Theta d ^ { t } ) b \\Theta E $",
"$ Z _ { 0 } $",
"$ X _ { 0 } $",
"$ b $",
"\\begin{equation*} X = \\frac { 1 } { 2 } ( E ^ { t } a ^ { t } + b ^ { t } ) ( ( c + d \\Theta ) \\frac { 1 } { a + b \\Theta } - \\frac { 1 } { a ^ { t } - \\Theta b ^ { t } } ( c ^ { t } - \\Theta d ^ { t } ) ) ( a E + b ) \\end{equation*}",
"\\begin{align*} Z & = & E ^ { t } X _ { 0 } E + E ^ { t } ( \\Theta + X _ { 0 } ) \\frac { 1 } { a + b \\Theta } b ( 1 - \\Theta E ) \\\\ & + & ( 1 + E ^ { t } \\Theta ) b ^ { t } \\frac { 1 } { a ^ { t } - \\Theta b ^ { t } } ( \\Theta + X _ { 0 } ) E + Y _ { 0 } + Z _ { 0 } \\end{align*}",
"\\begin{equation*} c ^ { t } - \\Theta d ^ { t } = - ( a ^ { t } - \\Theta b ^ { t } ) ( c + d \\Theta ) \\frac { 1 } { a + b \\Theta } \\end{equation*}",
"\\begin{align*} Z & = & E ^ { t } ( c ^ { t } b + ( a ^ { t } - \\Theta b ^ { t } ) ( c + d \\Theta ) \\frac { 1 } { a + b \\Theta } b + \\Theta \\frac { 1 } { a + b \\Theta } b - 2 \\Theta d ^ { t } b ) \\Theta E \\\\ & + & E ^ { t } \\Theta d ^ { t } b + ( \\frac { 1 } { a + b \\Theta } - d ^ { t } b ) \\Theta E + d ^ { t } b + Z _ { 0 } + ( 1 + E ^ { t } \\Theta ) b ^ { t } \\frac { 1 } { a ^ { t } - \\Theta b ^ { t } } \\Theta E , \\end{align*}",
"\\begin{align*} Z _ { 0 } & = & ( E ^ { t } \\Theta + 1 ) b ^ { t } [ \\frac { 1 } { a ^ { t } - \\Theta b ^ { t } } \\Theta \\frac { 1 } { a + b \\Theta } ( a E + b ) \\\\ & + & ( ( c + d \\Theta ) \\frac { 1 } { a + b \\Theta } b \\Theta + c ) \\frac { 1 } { a + b \\Theta } b ( 1 - \\Theta E ) ] . \\end{align*}",
"\\begin{equation*} \\frac { 1 } { a ^ { t } - \\Theta b ^ { t } } \\Theta + c = ( c + d \\Theta ) \\frac { 1 } { a + b \\Theta } a , \\end{equation*}"
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"$ \\mitX = ( \\mitE ^ { \\mitt } \\mita ^ { \\mitt } + \\mitb ^ { \\mitt } ) ( \\mitc + \\mitd \\mupTheta ) ( \\mita + \\mitb \\mupTheta ) ^ { - 1 } ( \\mita \\mitE + \\mitb ) $",
"$ \\mitY $",
"$ \\mitY = \\mitE - \\mitE ^ { \\mitt } + 2 \\mitY _ { 0 } $",
"$ \\mitY _ { 0 } = - ( \\mitE ^ { \\mitt } \\mita ^ { \\mitt } + \\mitb ^ { \\mitt } ) \\mitc \\mitE + ( \\mitE ^ { \\mitt } \\mitc ^ { \\mitt } + \\mitd ^ { \\mitt } ) \\mitb $",
"$ \\mitX = ( \\mitE ^ { \\mitt } \\mita ^ { \\mitt } + \\mitb ^ { \\mitt } ) \\mupTheta ^ { \\prime } ( \\mita \\mitE + \\mitb ) $",
"$ { \\mupTheta ^ { \\prime } } ^ { \\mitt } = - \\mupTheta ^ { \\prime } $",
"$ \\mitX = ( \\mitE ^ { \\mitt } + ( 1 + \\mitE ^ { \\mitt } \\mupTheta ) \\mitb ^ { \\mitt } ( \\mita ^ { \\mitt } - \\mupTheta \\mitb ^ { \\mitt } ) ^ { - 1 } ) ( \\mupTheta + \\mitX _ { 0 } ) ( \\mitE + ( \\mita + \\mitb \\mupTheta ) ^ { - 1 } \\mitb ( 1 - \\mupTheta \\mitE ) ) $",
"$ \\mitX = ( \\mitE ^ { \\mitt } + ( 1 + \\mitE ^ { \\mitt } \\mupTheta ) \\mitb ^ { \\mitt } ( \\mita ^ { \\mitt } - \\mupTheta \\mitb ^ { \\mitt } ) ^ { - 1 } ) ( \\mupTheta + \\mitX _ { 0 } ) ( \\mitE + ( \\mita + \\mitb \\mupTheta ) ^ { - 1 } \\mitb ( 1 - \\mupTheta \\mitE ) ) $",
"$ \\mitX _ { 0 } = - ( \\mitc ^ { \\mitt } - \\mupTheta \\mitd ^ { \\mitt } ) \\mitb \\mupTheta + ( \\mita ^ { \\mitt } - \\mupTheta \\mitb ^ { \\mitt } ) \\mitc $",
"$ 2 \\mitX + \\mitY = 2 \\mitE ^ { \\mitt } \\mupTheta \\mitE + \\mitE - \\mitE ^ { \\mitt } + 2 \\mitZ , $",
"$ 2 \\mitX + \\mitY = 2 \\mitE ^ { \\mitt } \\mupTheta \\mitE + \\mitE - \\mitE ^ { \\mitt } + 2 \\mitZ , $",
"$ \\mitZ _ { 0 } = ( 1 + \\mitE ^ { \\mitt } \\mupTheta ) \\mitb ^ { \\mitt } ( \\mita ^ { \\mitt } - \\mupTheta \\mitb ^ { \\mitt } ) ^ { - 1 } ( \\mupTheta + \\mitX _ { 0 } ) ( \\mita + \\mitb \\mupTheta ) ^ { - 1 } \\mitb ( 1 - \\mupTheta \\mitE ) $",
"$ \\mitX _ { 0 } $",
"$ \\mitE ^ { \\mitt } ( \\mita ^ { \\mitt } \\mitc + ( \\mita ^ { \\mitt } - \\mupTheta \\mitb ^ { \\mitt } ) \\mitd \\mupTheta ( \\mita + \\mitb \\mupTheta ) ^ { - 1 } \\mitb \\mupTheta ) \\mitE $",
"$ \\mitY _ { 0 } $",
"$ \\mitb ^ { \\mitt } ( \\mitc + \\mitd \\mupTheta ) ( \\mita + \\mitb \\mupTheta ) ^ { - 1 } = ( \\mita + \\mitb \\mupTheta ) ^ { - 1 } - \\mitd ^ { \\mitt } $",
"$ [ ( 1 + \\mitE ^ { \\mitt } \\mupTheta ) ( \\mitb ^ { \\mitt } ( \\mita ^ { \\mitt } - \\mupTheta \\mitb ^ { \\mitt } ) ^ { - 1 } - \\mitd ^ { \\mitt } \\mitb ) + ( \\mita + \\mitb \\mupTheta ) ^ { - 1 } \\mitb ] \\mupTheta \\mitE + \\mitb ^ { \\mitt } \\mitc \\mitE $",
"$ \\mitY _ { 0 } $",
"$ \\mitE ^ { \\mitt } [ ( \\mupTheta + ( \\mita ^ { \\mitt } - \\mupTheta \\mitb ^ { \\mitt } ) \\mitc ) ( \\mita + \\mitb \\mupTheta ) ^ { - 1 } \\mitb - ( \\mitc ^ { \\mitt } - \\mupTheta \\mitd ^ { \\mitt } ) \\mitb \\mupTheta ( \\mita + \\mitb \\mupTheta ) ^ { - 1 } \\mitb ( 1 - \\mupTheta \\mitE ) ] $",
"$ \\mitE ^ { \\mitt } [ ( \\mita ^ { \\mitt } \\mitc + \\mupTheta \\mitd ^ { \\mitt } \\mita ) ( \\mita + \\mitb \\mupTheta ) ^ { - 1 } \\mitb - ( \\mitc ^ { \\mitt } - \\mupTheta \\mitd ^ { \\mitt } ) ( 1 - \\mita ( \\mita + \\mitb \\mupTheta ) ^ { - 1 } ) \\mitb ( 1 - \\mupTheta \\mitE ) ] $",
"$ \\mitE ^ { \\mitt } ( \\mita ^ { \\mitt } \\mitc + \\mitc ^ { \\mitt } \\mita ) ( \\mita + \\mitb \\mupTheta ) ^ { - 1 } \\mitb $",
"$ \\mitZ _ { 0 } = ( \\mitE ^ { \\mitt } \\mupTheta + 1 ) \\mitb ^ { \\mitt } ( \\mita ^ { \\mitt } - \\mupTheta \\mitb ^ { \\mitt } ) ^ { - 1 } ( \\mupTheta + \\mitX _ { 0 } ) ( - \\mitE + ( \\mita + \\mitb \\mupTheta ) ^ { - 1 } ( \\mita \\mitE + \\mitb ) ) $",
"$ \\mitZ _ { 0 } = ( \\mitE ^ { \\mitt } \\mupTheta + 1 ) \\mitb ^ { \\mitt } ( \\mita ^ { \\mitt } - \\mupTheta \\mitb ^ { \\mitt } ) ^ { - 1 } ( \\mupTheta + \\mitX _ { 0 } ) ( - \\mitE + ( \\mita + \\mitb \\mupTheta ) ^ { - 1 } ( \\mita \\mitE + \\mitb ) ) $",
"$ - \\mitE ^ { \\mitt } ( \\mitc ^ { \\mitt } - \\mupTheta \\mitd ^ { \\mitt } ) \\mitb \\mupTheta \\mitE $",
"$ \\mitZ _ { 0 } $",
"$ \\mitX _ { 0 } $",
"$ \\mitb $",
"\\begin{equation*} \\mitX = \\frac { 1 } { 2 } ( \\mitE ^ { \\mitt } \\mita ^ { \\mitt } + \\mitb ^ { \\mitt } ) ( ( \\mitc + \\mitd \\mupTheta ) \\frac { 1 } { \\mita + \\mitb \\mupTheta } - \\frac { 1 } { \\mita ^ { \\mitt } - \\mupTheta \\mitb ^ { \\mitt } } ( \\mitc ^ { \\mitt } - \\mupTheta \\mitd ^ { \\mitt } ) ) ( \\mita \\mitE + \\mitb ) \\end{equation*}",
"\\begin{align*} \\displaystyle \\mitZ & = & \\displaystyle \\mitE ^ { \\mitt } \\mitX _ { 0 } \\mitE + \\mitE ^ { \\mitt } ( \\mupTheta + \\mitX _ { 0 } ) \\frac { 1 } { \\mita + \\mitb \\mupTheta } \\mitb ( 1 - \\mupTheta \\mitE ) \\\\ & \\displaystyle + & \\displaystyle ( 1 + \\mitE ^ { \\mitt } \\mupTheta ) \\mitb ^ { \\mitt } \\frac { 1 } { \\mita ^ { \\mitt } - \\mupTheta \\mitb ^ { \\mitt } } ( \\mupTheta + \\mitX _ { 0 } ) \\mitE + \\mitY _ { 0 } + \\mitZ _ { 0 } \\end{align*}",
"\\begin{equation*} \\mitc ^ { \\mitt } - \\mupTheta \\mitd ^ { \\mitt } = - ( \\mita ^ { \\mitt } - \\mupTheta \\mitb ^ { \\mitt } ) ( \\mitc + \\mitd \\mupTheta ) \\frac { 1 } { \\mita + \\mitb \\mupTheta } \\end{equation*}",
"\\begin{align*} \\displaystyle \\mitZ & = & \\displaystyle \\mitE ^ { \\mitt } ( \\mitc ^ { \\mitt } \\mitb + ( \\mita ^ { \\mitt } - \\mupTheta \\mitb ^ { \\mitt } ) ( \\mitc + \\mitd \\mupTheta ) \\frac { 1 } { \\mita + \\mitb \\mupTheta } \\mitb + \\mupTheta \\frac { 1 } { \\mita + \\mitb \\mupTheta } \\mitb - 2 \\mupTheta \\mitd ^ { \\mitt } \\mitb ) \\mupTheta \\mitE \\\\ & \\displaystyle + & \\displaystyle \\mitE ^ { \\mitt } \\mupTheta \\mitd ^ { \\mitt } \\mitb + ( \\frac { 1 } { \\mita + \\mitb \\mupTheta } - \\mitd ^ { \\mitt } \\mitb ) \\mupTheta \\mitE + \\mitd ^ { \\mitt } \\mitb + \\mitZ _ { 0 } + ( 1 + \\mitE ^ { \\mitt } \\mupTheta ) \\mitb ^ { \\mitt } \\frac { 1 } { \\mita ^ { \\mitt } - \\mupTheta \\mitb ^ { \\mitt } } \\mupTheta \\mitE , \\end{align*}",
"\\begin{align*} \\displaystyle \\mitZ _ { 0 } & = & \\displaystyle ( \\mitE ^ { \\mitt } \\mupTheta + 1 ) \\mitb ^ { \\mitt } [ \\frac { 1 } { \\mita ^ { \\mitt } - \\mupTheta \\mitb ^ { \\mitt } } \\mupTheta \\frac { 1 } { \\mita + \\mitb \\mupTheta } ( \\mita \\mitE + \\mitb ) \\\\ & \\displaystyle + & \\displaystyle ( ( \\mitc + \\mitd \\mupTheta ) \\frac { 1 } { \\mita + \\mitb \\mupTheta } \\mitb \\mupTheta + \\mitc ) \\frac { 1 } { \\mita + \\mitb \\mupTheta } \\mitb ( 1 - \\mupTheta \\mitE ) ] . \\end{align*}",
"\\begin{equation*} \\frac { 1 } { \\mita ^ { \\mitt } - \\mupTheta \\mitb ^ { \\mitt } } \\mupTheta + \\mitc = ( \\mitc + \\mitd \\mupTheta ) \\frac { 1 } { \\mita + \\mitb \\mupTheta } \\mita , \\end{equation*}"
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0003204_page06
|
{
"latex": [
"$(E^t\\Te +1)b^t(c +d\\Te ) (a+b\\Te )^{-1}b$",
"$(E^t\\Te + 1)(a +b\\Te )^{-1}b -E^t\\Te d^tb -d^tb$",
"$Z$",
"$E$",
"$(E^t\\Te +1)(d^tb - 2(a+b\\Te )^{-1}b) \\Te E$",
"$SO(p,p,Z)$",
"$g_s$",
"$g'_s = g_s/\\det (cE+d)^{1/2}$",
"$g'_s = g_s/\\det (cE+d)^{1/2}$",
"$G_s = g_s(\\det (G+2\\pi \\al \\Phi )/\\det (g+2\\pi \\al B))^{1/2}$",
"$G_s = g_s(\\det (G+2\\pi \\al \\Phi )/\\det (g+2\\pi \\al B))^{1/2}$",
"$(p+1)\\times (p+1)$",
"$G_s = g_s(\\det (1-\\Te E))^{-1/2}$",
"$G_{00}=g_{00}, \\Phi _{0i} =B_{0i}=0$",
"$SO(p,p,Z)$",
"$G_s$",
"$g_{YM} =((2\\pi )^{p-2}G_s/(\\al )^{(3-p)/2})^{1/2}$",
"$g_{YM} =((2\\pi )^{p-2}G_s/(\\al )^{(3-p)/2})^{1/2}$",
"$\\Phi $",
"$\\Phi = 0$",
"$g$",
"$B$",
"$E^{-1} \\approx \\Te $",
"$\\Phi $",
"$p$",
"$\\Phi $",
"$p$",
"$G=g, \\Phi =B, G_s=g_s$",
"$\\Phi =0$",
"$Tr_{\\te }$",
"$S_{WZ}$",
"$S_{WZ}=\\int Tr_{\\te }(\\sum C^{(n)})e^{2\\pi \\al F}$",
"$n$",
"$C^{(n)}$",
"$F$",
"$B$",
"$\\te $",
"$\\sqrt {\\det (G+2\\pi \\al (F+\\Phi ))_{ij}} (-(G_{00}- (2\\pi \\al )^2 F_{0i}(G+ 2\\pi \\al (F+\\Phi ))^{-1ij}F_{j0}))^{1/2}$",
"\\begin {equation} Z = (E^t\\Te + 1)\\frac {1}{a + b\\Te } b(1 - \\Te E), \\end {equation}",
"\\begin {equation} \\Phi ' = (a + b\\Te )\\Phi (a + b\\Te )^t + \\frac {1}{2\\pi r^2}b(a + b\\Te )^t. \\label {phi}\\end {equation}",
"\\begin {equation} G_s' = \\sqrt {\\det (a + b\\Te )}G_s, \\label {gs}\\end {equation}",
"\\begin {equation} g'_{YM} = g_{YM}(\\det (a + b\\Te ))^{\\frac {1}{4}}. \\label {gym}\\end {equation}",
"\\begin {equation} S = - \\frac {1}{G_s(2\\pi )^p\\al ^{\\frac {p+1}{2}}}\\int d^{p+1}\\sigma Tr_{\\te }\\sqrt {-\\det (G + 2\\pi \\al (F + \\Phi ))} + S_{WZ}. \\end {equation}"
],
"latex_norm": [
"$ ( E ^ { t } \\Theta + 1 ) b ^ { t } ( c + d \\Theta ) ( a + b \\Theta ) ^ { - 1 } b $",
"$ ( E ^ { t } \\Theta + 1 ) ( a + b \\Theta ) ^ { - 1 } b - E ^ { t } \\Theta d ^ { t } b - d ^ { t } b $",
"$ Z $",
"$ E $",
"$ ( E ^ { t } \\Theta + 1 ) ( d ^ { t } b - 2 ( a + b \\Theta ) ^ { - 1 } b ) \\Theta E $",
"$ S O ( p , p , Z ) $",
"$ g _ { s } $",
"$ g _ { s } ^ { \\prime } = g _ { s } \\slash d e t ( c E + d ) ^ { 1 \\slash 2 } $",
"$ g _ { s } ^ { \\prime } = g _ { s } \\slash d e t ( c E + d ) ^ { 1 \\slash 2 } $",
"$ G _ { s } = g _ { s } ( d e t ( G + 2 \\pi \\alpha ^ { \\prime } \\Phi ) \\slash \\operatorname { d e t } ( g + 2 \\pi \\alpha ^ { \\prime } B ) ) ^ { 1 \\slash 2 } $",
"$ G _ { s } = g _ { s } ( d e t ( G + 2 \\pi \\alpha ^ { \\prime } \\Phi ) \\slash \\operatorname { d e t } ( g + 2 \\pi \\alpha ^ { \\prime } B ) ) ^ { 1 \\slash 2 } $",
"$ ( p + 1 ) \\times ( p + 1 ) $",
"$ G _ { s } = g _ { s } ( d e t ( 1 - \\Theta E ) ) ^ { - 1 \\slash 2 } $",
"$ G _ { 0 0 } = g _ { 0 0 } , \\Phi _ { 0 i } = B _ { 0 i } = 0 $",
"$ S O ( p , p , Z ) $",
"$ G _ { s } $",
"$ g _ { Y M } = ( ( 2 \\pi ) ^ { p - 2 } G _ { s } \\slash ( \\alpha ^ { \\prime } ) ^ { ( 3 - p ) \\slash 2 } ) ^ { 1 \\slash 2 } $",
"$ g _ { Y M } = ( ( 2 \\pi ) ^ { p - 2 } G _ { s } \\slash ( \\alpha ^ { \\prime } ) ^ { ( 3 - p ) \\slash 2 } ) ^ { 1 \\slash 2 } $",
"$ \\Phi $",
"$ \\Phi = 0 $",
"$ g $",
"$ B $",
"$ E ^ { - 1 } \\approx \\Theta $",
"$ \\Phi $",
"$ p $",
"$ \\Phi $",
"$ p $",
"$ G = g , \\Phi = B , G _ { s } = g _ { s } $",
"$ \\Phi = 0 $",
"$ T r _ { \\theta } $",
"$ S _ { W Z } $",
"$ S _ { W Z } = \\int T r _ { \\theta } ( \\sum C ^ { ( n ) } ) e ^ { 2 \\pi \\alpha ^ { \\prime } F } $",
"$ n $",
"$ C ^ { ( n ) } $",
"$ F $",
"$ B $",
"$ \\theta $",
"$ \\sqrt { d e t ( G + 2 \\pi \\alpha ^ { \\prime } ( F + \\Phi ) ) _ { i j } } ( - ( G _ { 0 0 } - ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } F _ { 0 i } ( G + 2 \\pi \\alpha ^ { \\prime } ( F + \\Phi ) ) ^ { - 1 i j } F _ { j 0 } ) ) ^ { 1 \\slash 2 } $",
"\\begin{equation*} Z = ( E ^ { t } \\Theta + 1 ) \\frac { 1 } { a + b \\Theta } b ( 1 - \\Theta E ) , \\end{equation*}",
"\\begin{equation*} \\Phi ^ { \\prime } = ( a + b \\Theta ) \\Phi ( a + b \\Theta ) ^ { t } + \\frac { 1 } { 2 \\pi r ^ { 2 } } b ( a + b \\Theta ) ^ { t } . \\end{equation*}",
"\\begin{equation*} G _ { s } ^ { \\prime } = \\sqrt { \\operatorname { d e t } ( a + b \\Theta ) } G _ { s } , \\end{equation*}",
"\\begin{equation*} g _ { Y M } ^ { \\prime } = g _ { Y M } ( \\operatorname { d e t } ( a + b \\Theta ) ) ^ { \\frac { 1 } { 4 } } . \\end{equation*}",
"\\begin{equation*} S = - \\frac { 1 } { G _ { s } ( 2 \\pi ) ^ { p } \\alpha ^ { \\prime \\frac { p + 1 } { 2 } } } \\int d ^ { p + 1 } \\sigma T r _ { \\theta } \\sqrt { - \\operatorname { d e t } ( G + 2 \\pi \\alpha ^ { \\prime } ( F + \\Phi ) ) } + S _ { W Z } . \\end{equation*}"
],
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"$ ( \\mitE ^ { \\mitt } \\mupTheta + 1 ) \\mitb ^ { \\mitt } ( \\mitc + \\mitd \\mupTheta ) ( \\mita + \\mitb \\mupTheta ) ^ { - 1 } \\mitb $",
"$ ( \\mitE ^ { \\mitt } \\mupTheta + 1 ) ( \\mita + \\mitb \\mupTheta ) ^ { - 1 } \\mitb - \\mitE ^ { \\mitt } \\mupTheta \\mitd ^ { \\mitt } \\mitb - \\mitd ^ { \\mitt } \\mitb $",
"$ \\mitZ $",
"$ \\mitE $",
"$ ( \\mitE ^ { \\mitt } \\mupTheta + 1 ) ( \\mitd ^ { \\mitt } \\mitb - 2 ( \\mita + \\mitb \\mupTheta ) ^ { - 1 } \\mitb ) \\mupTheta \\mitE $",
"$ \\mitS \\mitO ( \\mitp , \\mitp , \\mitZ ) $",
"$ \\mitg _ { \\mits } $",
"$ \\mitg _ { \\mits } ^ { \\prime } = \\mitg _ { \\mits } \\slash \\mathrm { d e t } ( \\mitc \\mitE + \\mitd ) ^ { 1 \\slash 2 } $",
"$ \\mitg _ { \\mits } ^ { \\prime } = \\mitg _ { \\mits } \\slash \\mathrm { d e t } ( \\mitc \\mitE + \\mitd ) ^ { 1 \\slash 2 } $",
"$ \\mitG _ { \\mits } = \\mitg _ { \\mits } ( \\mathrm { d e t } ( \\mitG + 2 \\mitpi \\mitalpha ^ { \\prime } \\mupPhi ) \\slash \\operatorname { d e t } ( \\mitg + 2 \\mitpi \\mitalpha ^ { \\prime } \\mitB ) ) ^ { 1 \\slash 2 } $",
"$ \\mitG _ { \\mits } = \\mitg _ { \\mits } ( \\mathrm { d e t } ( \\mitG + 2 \\mitpi \\mitalpha ^ { \\prime } \\mupPhi ) \\slash \\operatorname { d e t } ( \\mitg + 2 \\mitpi \\mitalpha ^ { \\prime } \\mitB ) ) ^ { 1 \\slash 2 } $",
"$ ( \\mitp + 1 ) \\times ( \\mitp + 1 ) $",
"$ \\mitG _ { \\mits } = \\mitg _ { \\mits } ( \\mathrm { d e t } ( 1 - \\mupTheta \\mitE ) ) ^ { - 1 \\slash 2 } $",
"$ \\mitG _ { 0 0 } = \\mitg _ { 0 0 } , \\mupPhi _ { 0 \\miti } = \\mitB _ { 0 \\miti } = 0 $",
"$ \\mitS \\mitO ( \\mitp , \\mitp , \\mitZ ) $",
"$ \\mitG _ { \\mits } $",
"$ \\mitg _ { \\mitY \\mitM } = ( ( 2 \\mitpi ) ^ { \\mitp - 2 } \\mitG _ { \\mits } \\slash ( \\mitalpha ^ { \\prime } ) ^ { ( 3 - \\mitp ) \\slash 2 } ) ^ { 1 \\slash 2 } $",
"$ \\mitg _ { \\mitY \\mitM } = ( ( 2 \\mitpi ) ^ { \\mitp - 2 } \\mitG _ { \\mits } \\slash ( \\mitalpha ^ { \\prime } ) ^ { ( 3 - \\mitp ) \\slash 2 } ) ^ { 1 \\slash 2 } $",
"$ \\mupPhi $",
"$ \\mupPhi = 0 $",
"$ \\mitg $",
"$ \\mitB $",
"$ \\mitE ^ { - 1 } \\approx \\mupTheta $",
"$ \\mupPhi $",
"$ \\mitp $",
"$ \\mupPhi $",
"$ \\mitp $",
"$ \\mitG = \\mitg , \\mupPhi = \\mitB , \\mitG _ { \\mits } = \\mitg _ { \\mits } $",
"$ \\mupPhi = 0 $",
"$ \\mitT \\mitr _ { \\mittheta } $",
"$ \\mitS _ { \\mitW \\mitZ } $",
"$ \\mitS _ { \\mitW \\mitZ } = \\int \\nolimits \\mitT \\mitr _ { \\mittheta } ( \\sum \\mitC ^ { ( \\mitn ) } ) \\mite ^ { 2 \\mitpi \\mitalpha ^ { \\prime } \\mitF } $",
"$ \\mitn $",
"$ \\mitC ^ { ( \\mitn ) } $",
"$ \\mitF $",
"$ \\mitB $",
"$ \\mittheta $",
"$ \\sqrt { \\mathrm { d e t } ( \\mitG + 2 \\mitpi \\mitalpha ^ { \\prime } ( \\mitF + \\mupPhi ) ) _ { \\miti \\mitj } } ( - ( \\mitG _ { 0 0 } - ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mitF _ { 0 \\miti } ( \\mitG + 2 \\mitpi \\mitalpha ^ { \\prime } ( \\mitF + \\mupPhi ) ) ^ { - 1 \\miti \\mitj } \\mitF _ { \\mitj 0 } ) ) ^ { 1 \\slash 2 } $",
"\\begin{equation*} \\mitZ = ( \\mitE ^ { \\mitt } \\mupTheta + 1 ) \\frac { 1 } { \\mita + \\mitb \\mupTheta } \\mitb ( 1 - \\mupTheta \\mitE ) , \\end{equation*}",
"\\begin{equation*} \\mupPhi ^ { \\prime } = ( \\mita + \\mitb \\mupTheta ) \\mupPhi ( \\mita + \\mitb \\mupTheta ) ^ { \\mitt } + \\frac { 1 } { 2 \\mitpi \\mitr ^ { 2 } } \\mitb ( \\mita + \\mitb \\mupTheta ) ^ { \\mitt } . \\end{equation*}",
"\\begin{equation*} \\mitG _ { \\mits } ^ { \\prime } = \\sqrt { \\operatorname { d e t } ( \\mita + \\mitb \\mupTheta ) } \\mitG _ { \\mits } , \\end{equation*}",
"\\begin{equation*} \\mitg _ { \\mitY \\mitM } ^ { \\prime } = \\mitg _ { \\mitY \\mitM } ( \\operatorname { d e t } ( \\mita + \\mitb \\mupTheta ) ) ^ { \\frac { 1 } { 4 } } . \\end{equation*}",
"\\begin{equation*} \\mitS = - \\frac { 1 } { \\mitG _ { \\mits } ( 2 \\mitpi ) ^ { \\mitp } \\mitalpha ^ { \\prime \\frac { \\mitp + 1 } { 2 } } } \\int \\mitd ^ { \\mitp + 1 } \\mitsigma \\mitT \\mitr _ { \\mittheta } \\sqrt { - \\operatorname { d e t } ( \\mitG + 2 \\mitpi \\mitalpha ^ { \\prime } ( \\mitF + \\mupPhi ) ) } + \\mitS _ { \\mitW \\mitZ } . \\end{equation*}"
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|
0003204_page07
|
{
"latex": [
"$\\sqrt {\\det (G+2\\pi \\al (F+\\Phi ))_{ij}} (-(G_{00}- (2\\pi \\al )^2 F_{0i}(G+ 2\\pi \\al (F+\\Phi ))^{-1ij}F_{j0}))^{1/2}$",
"$i=1,\\cdots ,p$",
"$\\Phi _{0i}=G_{0i}=0$",
"$p$",
"$F_{ij}$",
"$F_{ij} \\rightarrow ((a+b\\Te )F(a+b\\Te )^t -\\frac {1}{2\\pi r^2}b (a+b\\Te )^t)_{ij}$",
"$F_{ij} \\rightarrow ((a+b\\Te )F(a+b\\Te )^t -\\frac {1}{2\\pi r^2}b (a+b\\Te )^t)_{ij}$",
"$F_{0i}$",
"$F_{0i} \\rightarrow (F(a+b\\Te )^t)_{0i}$",
"$G_{00}=g_{00}$",
"$p$",
"$Tr_{\\te }$",
"$Tr_{\\te } \\rightarrow (\\det (a+b\\Te ))^{-1/2}Tr_{\\te }$",
"$D5$",
"$C, C_{lm}$",
"$C_{jklm}$",
"$A = a + b\\Te $",
"$A_i^a$",
"$A_i^{\\:\\:a}$",
"$S_{WZ}$",
"$C_{0i}, C_{0ijk}$",
"$C_{0ijklm}$",
"${}_3C_1, {}_5C_3$",
"$3\\cdot {}_5C_1$",
"\\begin {eqnarray} S_{WZ} &=& \\int d^6\\sigma Tr_{\\te }\\e ^{ilklm}(\\frac {2\\pi \\al }{24}F_{0i} C_{jklm} + \\frac {(2\\pi \\al )^2}{4}F_{0i}F_{jk}C_{lm} + \\frac {(2\\pi \\al )^3}{8}F_{0i}F_{jk}F_{lm}C \\\\ &+& \\frac {1}{5!}C_{0ijklm}+ \\frac {2\\pi \\al }{12}C_{0ijk}F_{lm} + \\frac {(2\\pi \\al )^2}{8}C_{0i}F_{jk}F_{lm}). \\end {eqnarray}",
"\\begin {eqnarray} C &\\rightarrow & \\frac {1}{\\sqrt {\\det A}}C, \\hspace {1cm} C_{ij} \\; \\rightarrow \\; \\frac {1} {\\sqrt {\\det A}}((ACA^t)_{ij} + \\frac {\\al }{r^2}(bA^t)_{[ij]}C), \\\\ C_{ijkl} &\\rightarrow & \\frac {1}{\\sqrt {\\det A}}(A_{[i}^a A_j^b A_k^c A_{l]}^d C_{abcd} + \\frac {6\\al }{r^2}A_{[i}^a A_j^b (bA^t)_{kl]}C_{ab} + 3(\\frac {\\al }{r^2})^2 C(bA^t)_{[ij}(bA^t)_{kl]} ), \\end {eqnarray}",
"\\begin {eqnarray} C_{0i} &\\rightarrow & \\frac {1}{\\sqrt {\\det A}}A_i^a C_{0a}, \\hspace {1cm} C_{0ijk} \\;\\rightarrow \\;\\frac {1}{\\sqrt {\\det A}}(A_{[i}^aA_j^b A_{k]}^c C_{0abc} + \\frac {3\\al }{r^2} A_{[i}^a(bA^t)_{jk]}C_{0a} ), \\\\ C_{0ijklm} &\\rightarrow & \\frac {1}{\\sqrt {\\det A}}(A_{[i}^a \\cdots A_{m]}^e C_{0a\\cdots e} + \\frac {10\\al }{r^2}A_{[i}^a A_j^b A_k^c (bA^t)_{lm]} C_{0abc} \\\\ &+& 15(\\frac {\\al }{r^2})^2 A_{[i}^a(bA^t)_{jk}(bA^t)_{lm]}C_{0a} ), \\end {eqnarray}",
"\\begin {eqnarray} \\la ^i = \\frac {1}{4!}\\e ^{ijklm}C_{jklm}, \\; \\la ^{ijk} = \\frac {1}{2!} \\e ^{ijklm}C_{lm}, \\; \\la ^{ijklm}=\\e ^{ijklm}C, \\\\ \\la =\\frac {1}{5!}\\e ^{ijklm}C_{0ijklm},\\; \\la ^{ij} = \\frac {1}{3!} \\e ^{ijklm}C_{0klm}, \\; \\la ^{ijkl}=\\e ^{ijklm}C_{0m} \\end {eqnarray}"
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"$ \\sqrt { d e t ( G + 2 \\pi \\alpha ^ { \\prime } ( F + \\Phi ) ) _ { i j } } ( - ( G _ { 0 0 } - ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } F _ { 0 i } ( G + 2 \\pi \\alpha ^ { \\prime } ( F + \\Phi ) ) ^ { - 1 i j } F _ { j 0 } ) ) ^ { 1 \\slash 2 } $",
"$ i = 1 , \\cdots , p $",
"$ \\Phi _ { 0 i } = G _ { 0 i } = 0 $",
"$ p $",
"$ F _ { i j } $",
"$ F _ { i j } \\rightarrow ( ( a + b \\Theta ) F ( a + b \\Theta ) ^ { t } - \\frac { 1 } { 2 \\pi r ^ { 2 } } b ( a + b \\Theta ) ^ { t } ) _ { i j } $",
"$ F _ { i j } \\rightarrow ( ( a + b \\Theta ) F ( a + b \\Theta ) ^ { t } - \\frac { 1 } { 2 \\pi r ^ { 2 } } b ( a + b \\Theta ) ^ { t } ) _ { i j } $",
"$ F _ { 0 i } $",
"$ F _ { 0 i } \\rightarrow ( F ( a + b \\Theta ) ^ { t } ) _ { 0 i } $",
"$ G _ { 0 0 } = g _ { 0 0 } $",
"$ p $",
"$ T r _ { \\theta } $",
"$ T r _ { \\theta } \\rightarrow ( d e t ( a + b \\Theta ) ) ^ { - 1 \\slash 2 } T r _ { \\theta } $",
"$ D 5 $",
"$ C , C _ { l m } $",
"$ C _ { j k l m } $",
"$ A = a + b \\Theta $",
"$ A _ { i } ^ { a } $",
"$ A _ { i } ^ { \\> \\> a } $",
"$ S _ { W Z } $",
"$ C _ { 0 i } , C _ { 0 i j k } $",
"$ C _ { 0 i j k l m } $",
"$ { } _ { 3 } C _ { 1 } , { } _ { 5 } C _ { 3 } $",
"$ 3 \\cdot { } _ { 5 } C _ { 1 } $",
"\\begin{align*} S _ { W Z } & = & \\int d ^ { 6 } \\sigma T r _ { \\theta } \\epsilon ^ { i l k l m } ( \\frac { 2 \\pi \\alpha ^ { \\prime } } { 2 4 } F _ { 0 i } C _ { j k l m } + \\frac { ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } } { 4 } F _ { 0 i } F _ { j k } C _ { l m } + \\frac { ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 3 } } { 8 } F _ { 0 i } F _ { j k } F _ { l m } C \\\\ & + & \\frac { 1 } { 5 ! } C _ { 0 i j k l m } + \\frac { 2 \\pi \\alpha ^ { \\prime } } { 1 2 } C _ { 0 i j k } F _ { l m } + \\frac { ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } } { 8 } C _ { 0 i } F _ { j k } F _ { l m } ) . \\end{align*}",
"\\begin{align*} C & \\rightarrow & \\frac { 1 } { \\sqrt { \\operatorname { d e t } A } } C , \\hspace{28.45pt} C _ { i j } \\; \\rightarrow \\; \\frac { 1 } { \\sqrt { \\operatorname { d e t } A } } ( ( A C A ^ { t } ) _ { i j } + \\frac { \\alpha ^ { \\prime } } { r ^ { 2 } } ( b A ^ { t } ) _ { [ i j ] } C ) , \\\\ C _ { i j k l } & \\rightarrow & \\frac { 1 } { \\sqrt { \\operatorname { d e t } A } } ( A _ { [ i } ^ { a } A _ { j } ^ { b } A _ { k } ^ { c } A _ { l ] } ^ { d } C _ { a b c d } + \\frac { 6 \\alpha ^ { \\prime } } { r ^ { 2 } } A _ { [ i } ^ { a } A _ { j } ^ { b } ( b A ^ { t } ) _ { k l ] } C _ { a b } + 3 ( \\frac { \\alpha ^ { \\prime } } { r ^ { 2 } } ) ^ { 2 } C ( b A ^ { t } ) _ { [ i j } ( b A ^ { t } ) _ { k l ] } ) , \\end{align*}",
"\\begin{align*} C _ { 0 i } & \\rightarrow & \\frac { 1 } { \\sqrt { \\operatorname { d e t } A } } A _ { i } ^ { a } C _ { 0 a } , \\hspace{28.45pt} C _ { 0 i j k } \\; \\rightarrow \\; \\frac { 1 } { \\sqrt { \\operatorname { d e t } A } } ( A _ { [ i } ^ { a } A _ { j } ^ { b } A _ { k ] } ^ { c } C _ { 0 a b c } + \\frac { 3 \\alpha ^ { \\prime } } { r ^ { 2 } } A _ { [ i } ^ { a } ( b A ^ { t } ) _ { j k ] } C _ { 0 a } ) , \\\\ C _ { 0 i j k l m } & \\rightarrow & \\frac { 1 } { \\sqrt { \\operatorname { d e t } A } } ( A _ { [ i } ^ { a } \\cdots A _ { m ] } ^ { e } C _ { 0 a \\cdots e } + \\frac { 1 0 \\alpha ^ { \\prime } } { r ^ { 2 } } A _ { [ i } ^ { a } A _ { j } ^ { b } A _ { k } ^ { c } ( b A ^ { t } ) _ { l m ] } C _ { 0 a b c } \\\\ & + & 1 5 ( \\frac { \\alpha ^ { \\prime } } { r ^ { 2 } } ) ^ { 2 } A _ { [ i } ^ { a } ( b A ^ { t } ) _ { j k } ( b A ^ { t } ) _ { l m ] } C _ { 0 a } ) , \\end{align*}",
"\\begin{align*} \\lambda ^ { i } = \\frac { 1 } { 4 ! } \\epsilon ^ { i j k l m } C _ { j k l m } , \\; \\lambda ^ { i j k } = \\frac { 1 } { 2 ! } \\epsilon ^ { i j k l m } C _ { l m } , \\; \\lambda ^ { i j k l m } = \\epsilon ^ { i j k l m } C , \\\\ \\lambda = \\frac { 1 } { 5 ! } \\epsilon ^ { i j k l m } C _ { 0 i j k l m } , \\; \\lambda ^ { i j } = \\frac { 1 } { 3 ! } \\epsilon ^ { i j k l m } C _ { 0 k l m } , \\; \\lambda ^ { i j k l } = \\epsilon ^ { i j k l m } C _ { 0 m } \\end{align*}"
],
"latex_expand": [
"$ \\sqrt { \\mathrm { d e t } ( \\mitG + 2 \\mitpi \\mitalpha ^ { \\prime } ( \\mitF + \\mupPhi ) ) _ { \\miti \\mitj } } ( - ( \\mitG _ { 0 0 } - ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mitF _ { 0 \\miti } ( \\mitG + 2 \\mitpi \\mitalpha ^ { \\prime } ( \\mitF + \\mupPhi ) ) ^ { - 1 \\miti \\mitj } \\mitF _ { \\mitj 0 } ) ) ^ { 1 \\slash 2 } $",
"$ \\miti = 1 , \\cdots , \\mitp $",
"$ \\mupPhi _ { 0 \\miti } = \\mitG _ { 0 \\miti } = 0 $",
"$ \\mitp $",
"$ \\mitF _ { \\miti \\mitj } $",
"$ \\mitF _ { \\miti \\mitj } \\rightarrow ( ( \\mita + \\mitb \\mupTheta ) \\mitF ( \\mita + \\mitb \\mupTheta ) ^ { \\mitt } - \\frac { 1 } { 2 \\mitpi \\mitr ^ { 2 } } \\mitb ( \\mita + \\mitb \\mupTheta ) ^ { \\mitt } ) _ { \\miti \\mitj } $",
"$ \\mitF _ { \\miti \\mitj } \\rightarrow ( ( \\mita + \\mitb \\mupTheta ) \\mitF ( \\mita + \\mitb \\mupTheta ) ^ { \\mitt } - \\frac { 1 } { 2 \\mitpi \\mitr ^ { 2 } } \\mitb ( \\mita + \\mitb \\mupTheta ) ^ { \\mitt } ) _ { \\miti \\mitj } $",
"$ \\mitF _ { 0 \\miti } $",
"$ \\mitF _ { 0 \\miti } \\rightarrow ( \\mitF ( \\mita + \\mitb \\mupTheta ) ^ { \\mitt } ) _ { 0 \\miti } $",
"$ \\mitG _ { 0 0 } = \\mitg _ { 0 0 } $",
"$ \\mitp $",
"$ \\mitT \\mitr _ { \\mittheta } $",
"$ \\mitT \\mitr _ { \\mittheta } \\rightarrow ( \\mathrm { d e t } ( \\mita + \\mitb \\mupTheta ) ) ^ { - 1 \\slash 2 } \\mitT \\mitr _ { \\mittheta } $",
"$ \\mitD 5 $",
"$ \\mitC , \\mitC _ { \\mitl \\mitm } $",
"$ \\mitC _ { \\mitj \\mitk \\mitl \\mitm } $",
"$ \\mitA = \\mita + \\mitb \\mupTheta $",
"$ \\mitA _ { \\miti } ^ { \\mita } $",
"$ \\mitA _ { \\miti } ^ { \\> \\> \\mita } $",
"$ \\mitS _ { \\mitW \\mitZ } $",
"$ \\mitC _ { 0 \\miti } , \\mitC _ { 0 \\miti \\mitj \\mitk } $",
"$ \\mitC _ { 0 \\miti \\mitj \\mitk \\mitl \\mitm } $",
"$ { } _ { 3 } \\mitC _ { 1 } , { } _ { 5 } \\mitC _ { 3 } $",
"$ 3 \\cdot { } _ { 5 } \\mitC _ { 1 } $",
"\\begin{align*} \\displaystyle \\mitS _ { \\mitW \\mitZ } & = & \\displaystyle \\int \\mitd ^ { 6 } \\mitsigma \\mitT \\mitr _ { \\mittheta } \\mitepsilon ^ { \\miti \\mitl \\mitk \\mitl \\mitm } ( \\frac { 2 \\mitpi \\mitalpha ^ { \\prime } } { 2 4 } \\mitF _ { 0 \\miti } \\mitC _ { \\mitj \\mitk \\mitl \\mitm } + \\frac { ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } } { 4 } \\mitF _ { 0 \\miti } \\mitF _ { \\mitj \\mitk } \\mitC _ { \\mitl \\mitm } + \\frac { ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 3 } } { 8 } \\mitF _ { 0 \\miti } \\mitF _ { \\mitj \\mitk } \\mitF _ { \\mitl \\mitm } \\mitC \\\\ & \\displaystyle + & \\displaystyle \\frac { 1 } { 5 ! } \\mitC _ { 0 \\miti \\mitj \\mitk \\mitl \\mitm } + \\frac { 2 \\mitpi \\mitalpha ^ { \\prime } } { 1 2 } \\mitC _ { 0 \\miti \\mitj \\mitk } \\mitF _ { \\mitl \\mitm } + \\frac { ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } } { 8 } \\mitC _ { 0 \\miti } \\mitF _ { \\mitj \\mitk } \\mitF _ { \\mitl \\mitm } ) . \\end{align*}",
"\\begin{align*} \\displaystyle \\mitC & \\displaystyle \\rightarrow & \\displaystyle \\frac { 1 } { \\sqrt { \\operatorname { d e t } \\mitA } } \\mitC , \\hspace{28.45pt} \\mitC _ { \\miti \\mitj } \\; \\rightarrow \\; \\frac { 1 } { \\sqrt { \\operatorname { d e t } \\mitA } } ( ( \\mitA \\mitC \\mitA ^ { \\mitt } ) _ { \\miti \\mitj } + \\frac { \\mitalpha ^ { \\prime } } { \\mitr ^ { 2 } } ( \\mitb \\mitA ^ { \\mitt } ) _ { [ \\miti \\mitj ] } \\mitC ) , \\\\ \\displaystyle \\mitC _ { \\miti \\mitj \\mitk \\mitl } & \\displaystyle \\rightarrow & \\displaystyle \\frac { 1 } { \\sqrt { \\operatorname { d e t } \\mitA } } ( \\mitA _ { [ \\miti } ^ { \\mita } \\mitA _ { \\mitj } ^ { \\mitb } \\mitA _ { \\mitk } ^ { \\mitc } \\mitA _ { \\mitl ] } ^ { \\mitd } \\mitC _ { \\mita \\mitb \\mitc \\mitd } + \\frac { 6 \\mitalpha ^ { \\prime } } { \\mitr ^ { 2 } } \\mitA _ { [ \\miti } ^ { \\mita } \\mitA _ { \\mitj } ^ { \\mitb } ( \\mitb \\mitA ^ { \\mitt } ) _ { \\mitk \\mitl ] } \\mitC _ { \\mita \\mitb } + 3 ( \\frac { \\mitalpha ^ { \\prime } } { \\mitr ^ { 2 } } ) ^ { 2 } \\mitC ( \\mitb \\mitA ^ { \\mitt } ) _ { [ \\miti \\mitj } ( \\mitb \\mitA ^ { \\mitt } ) _ { \\mitk \\mitl ] } ) , \\end{align*}",
"\\begin{align*} \\displaystyle \\mitC _ { 0 \\miti } & \\displaystyle \\rightarrow & \\displaystyle \\frac { 1 } { \\sqrt { \\operatorname { d e t } \\mitA } } \\mitA _ { \\miti } ^ { \\mita } \\mitC _ { 0 \\mita } , \\hspace{28.45pt} \\mitC _ { 0 \\miti \\mitj \\mitk } \\; \\rightarrow \\; \\frac { 1 } { \\sqrt { \\operatorname { d e t } \\mitA } } ( \\mitA _ { [ \\miti } ^ { \\mita } \\mitA _ { \\mitj } ^ { \\mitb } \\mitA _ { \\mitk ] } ^ { \\mitc } \\mitC _ { 0 \\mita \\mitb \\mitc } + \\frac { 3 \\mitalpha ^ { \\prime } } { \\mitr ^ { 2 } } \\mitA _ { [ \\miti } ^ { \\mita } ( \\mitb \\mitA ^ { \\mitt } ) _ { \\mitj \\mitk ] } \\mitC _ { 0 \\mita } ) , \\\\ \\displaystyle \\mitC _ { 0 \\miti \\mitj \\mitk \\mitl \\mitm } & \\displaystyle \\rightarrow & \\displaystyle \\frac { 1 } { \\sqrt { \\operatorname { d e t } \\mitA } } ( \\mitA _ { [ \\miti } ^ { \\mita } \\cdots \\mitA _ { \\mitm ] } ^ { \\mite } \\mitC _ { 0 \\mita \\cdots \\mite } + \\frac { 1 0 \\mitalpha ^ { \\prime } } { \\mitr ^ { 2 } } \\mitA _ { [ \\miti } ^ { \\mita } \\mitA _ { \\mitj } ^ { \\mitb } \\mitA _ { \\mitk } ^ { \\mitc } ( \\mitb \\mitA ^ { \\mitt } ) _ { \\mitl \\mitm ] } \\mitC _ { 0 \\mita \\mitb \\mitc } \\\\ & \\displaystyle + & \\displaystyle 1 5 ( \\frac { \\mitalpha ^ { \\prime } } { \\mitr ^ { 2 } } ) ^ { 2 } \\mitA _ { [ \\miti } ^ { \\mita } ( \\mitb \\mitA ^ { \\mitt } ) _ { \\mitj \\mitk } ( \\mitb \\mitA ^ { \\mitt } ) _ { \\mitl \\mitm ] } \\mitC _ { 0 \\mita } ) , \\end{align*}",
"\\begin{align*} \\displaystyle \\mitlambda ^ { \\miti } = \\frac { 1 } { 4 ! } \\mitepsilon ^ { \\miti \\mitj \\mitk \\mitl \\mitm } \\mitC _ { \\mitj \\mitk \\mitl \\mitm } , \\; \\mitlambda ^ { \\miti \\mitj \\mitk } = \\frac { 1 } { 2 ! } \\mitepsilon ^ { \\miti \\mitj \\mitk \\mitl \\mitm } \\mitC _ { \\mitl \\mitm } , \\; \\mitlambda ^ { \\miti \\mitj \\mitk \\mitl \\mitm } = \\mitepsilon ^ { \\miti \\mitj \\mitk \\mitl \\mitm } \\mitC , \\\\ \\displaystyle \\mitlambda = \\frac { 1 } { 5 ! } \\mitepsilon ^ { \\miti \\mitj \\mitk \\mitl \\mitm } \\mitC _ { 0 \\miti \\mitj \\mitk \\mitl \\mitm } , \\; \\mitlambda ^ { \\miti \\mitj } = \\frac { 1 } { 3 ! } \\mitepsilon ^ { \\miti \\mitj \\mitk \\mitl \\mitm } \\mitC _ { 0 \\mitk \\mitl \\mitm } , \\; \\mitlambda ^ { \\miti \\mitj \\mitk \\mitl } = \\mitepsilon ^ { \\miti \\mitj \\mitk \\mitl \\mitm } \\mitC _ { 0 \\mitm } \\end{align*}"
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0003204_page08
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{
"latex": [
"$B =(A^{-1})^t$",
"$B_a^i$",
"$B_{\\:\\:a}^i$",
"$B$",
"$\\Phi $",
"\\begin {eqnarray} S_{WZ} &=& \\int d^6\\sigma Tr_{\\te }(2\\pi \\al F_{0i}\\la ^i + \\frac {(2\\pi \\al )^2}{2}F_{0i}F_{jk}\\la ^{ijk} + \\frac {(2\\pi \\al )^3}{8} F_{0i}F_{jk}F_{lm}\\la ^{ijklm} \\\\ &+& \\la + \\frac {2\\pi \\al }{2}F_{ij}\\la ^{ij} + \\frac {(2\\pi \\al )^2}{8} F_{ij}F_{kl}\\la ^{ijkl} ). \\end {eqnarray}",
"\\begin {eqnarray} \\la ^i &\\rightarrow & \\sqrt {\\det A}(B_a^{i}\\la ^a + \\frac {\\al }{2r^2} B_a^iB_b^jB_c^k\\la ^{abc}(bA^t)_{jk}) + \\frac {1}{8} (\\frac {\\al }{r^2})^2 \\frac {1}{\\sqrt {\\det A}}\\la ^{ijklm}(bA^t)_{jk}(bA^t)_{lm}, \\\\ \\la ^{ijk} &\\rightarrow & \\sqrt {\\det A}B_a^i B_b^j B_c^k \\la ^{abc} + \\frac {\\al }{2r^2} \\frac {1}{\\sqrt {\\det A}}\\la ^{ijklm}(bA^t)_{lm}, \\; \\la ^{ijklm} \\rightarrow \\frac {1}{\\sqrt {\\det A}}\\la ^{ijklm}, \\end {eqnarray}",
"\\begin {eqnarray} \\la &\\rightarrow & \\sqrt {\\det A}(\\la + \\frac {\\al }{2r^2}B_a^i B_b^j (bA^t)_{ij}\\la ^{ab} + \\frac {1}{8} (\\frac {\\al }{r^2})^2B_a^i B_b^j B_c^k B_d^l (bA^t)_{ij}(bA^t)_{kl}\\la ^{abcd}), \\\\ \\la ^{ij} &\\rightarrow & \\sqrt {\\det A}(B_a^i B_b^j \\la ^{ab} + \\frac {\\al }{2r^2}B_a^i B_b^j B_c^k B_d^l(bA^t)_{kl}\\la ^{abcd}),\\; \\la ^{ijkl} \\rightarrow \\sqrt {\\det A}B_a^i B_b^j B_c^k B_d^l\\la ^{abcd}, \\end {eqnarray}"
],
"latex_norm": [
"$ B = ( A ^ { - 1 } ) ^ { t } $",
"$ B _ { a } ^ { i } $",
"$ B _ { \\> \\> a } ^ { i } $",
"$ B $",
"$ \\Phi $",
"\\begin{align*} S _ { W Z } & = & \\int d ^ { 6 } \\sigma T r _ { \\theta } ( 2 \\pi \\alpha ^ { \\prime } F _ { 0 i } \\lambda ^ { i } + \\frac { ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } } { 2 } F _ { 0 i } F _ { j k } \\lambda ^ { i j k } + \\frac { ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 3 } } { 8 } F _ { 0 i } F _ { j k } F _ { l m } \\lambda ^ { i j k l m } \\\\ & + & \\lambda + \\frac { 2 \\pi \\alpha ^ { \\prime } } { 2 } F _ { i j } \\lambda ^ { i j } + \\frac { ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } } { 8 } F _ { i j } F _ { k l } \\lambda ^ { i j k l } ) . \\end{align*}",
"\\begin{align*} \\lambda ^ { i } & \\rightarrow & \\sqrt { \\operatorname { d e t } A } ( B _ { a } ^ { i } \\lambda ^ { a } + \\frac { \\alpha ^ { \\prime } } { 2 r ^ { 2 } } B _ { a } ^ { i } B _ { b } ^ { j } B _ { c } ^ { k } \\lambda ^ { a b c } ( b A ^ { t } ) _ { j k } ) + \\frac { 1 } { 8 } ( \\frac { \\alpha ^ { \\prime } } { r ^ { 2 } } ) ^ { 2 } \\frac { 1 } { \\sqrt { \\operatorname { d e t } A } } \\lambda ^ { i j k l m } ( b A ^ { t } ) _ { j k } ( b A ^ { t } ) _ { l m } , \\\\ \\lambda ^ { i j k } & \\rightarrow & \\sqrt { \\operatorname { d e t } A } B _ { a } ^ { i } B _ { b } ^ { j } B _ { c } ^ { k } \\lambda ^ { a b c } + \\frac { \\alpha ^ { \\prime } } { 2 r ^ { 2 } } \\frac { 1 } { \\sqrt { \\operatorname { d e t } A } } \\lambda ^ { i j k l m } ( b A ^ { t } ) _ { l m } , \\; \\lambda ^ { i j k l m } \\rightarrow \\frac { 1 } { \\sqrt { \\operatorname { d e t } A } } \\lambda ^ { i j k l m } , \\end{align*}",
"\\begin{align*} \\lambda & \\rightarrow & \\sqrt { \\operatorname { d e t } A } ( \\lambda + \\frac { \\alpha ^ { \\prime } } { 2 r ^ { 2 } } B _ { a } ^ { i } B _ { b } ^ { j } ( b A ^ { t } ) _ { i j } \\lambda ^ { a b } + \\frac { 1 } { 8 } ( \\frac { \\alpha ^ { \\prime } } { r ^ { 2 } } ) ^ { 2 } B _ { a } ^ { i } B _ { b } ^ { j } B _ { c } ^ { k } B _ { d } ^ { l } ( b A ^ { t } ) _ { i j } ( b A ^ { t } ) _ { k l } \\lambda ^ { a b c d } ) , \\\\ \\lambda ^ { i j } & \\rightarrow & \\sqrt { \\operatorname { d e t } A } ( B _ { a } ^ { i } B _ { b } ^ { j } \\lambda ^ { a b } + \\frac { \\alpha ^ { \\prime } } { 2 r ^ { 2 } } B _ { a } ^ { i } B _ { b } ^ { j } B _ { c } ^ { k } B _ { d } ^ { l } ( b A ^ { t } ) _ { k l } \\lambda ^ { a b c d } ) , \\; \\lambda ^ { i j k l } \\rightarrow \\sqrt { \\operatorname { d e t } A } B _ { a } ^ { i } B _ { b } ^ { j } B _ { c } ^ { k } B _ { d } ^ { l } \\lambda ^ { a b c d } , \\end{align*}"
],
"latex_expand": [
"$ \\mitB = ( \\mitA ^ { - 1 } ) ^ { \\mitt } $",
"$ \\mitB _ { \\mita } ^ { \\miti } $",
"$ \\mitB _ { \\> \\> \\mita } ^ { \\miti } $",
"$ \\mitB $",
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"\\begin{align*} \\displaystyle \\mitS _ { \\mitW \\mitZ } & = & \\displaystyle \\int \\mitd ^ { 6 } \\mitsigma \\mitT \\mitr _ { \\mittheta } ( 2 \\mitpi \\mitalpha ^ { \\prime } \\mitF _ { 0 \\miti } \\mitlambda ^ { \\miti } + \\frac { ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } } { 2 } \\mitF _ { 0 \\miti } \\mitF _ { \\mitj \\mitk } \\mitlambda ^ { \\miti \\mitj \\mitk } + \\frac { ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 3 } } { 8 } \\mitF _ { 0 \\miti } \\mitF _ { \\mitj \\mitk } \\mitF _ { \\mitl \\mitm } \\mitlambda ^ { \\miti \\mitj \\mitk \\mitl \\mitm } \\\\ & \\displaystyle + & \\displaystyle \\mitlambda + \\frac { 2 \\mitpi \\mitalpha ^ { \\prime } } { 2 } \\mitF _ { \\miti \\mitj } \\mitlambda ^ { \\miti \\mitj } + \\frac { ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } } { 8 } \\mitF _ { \\miti \\mitj } \\mitF _ { \\mitk \\mitl } \\mitlambda ^ { \\miti \\mitj \\mitk \\mitl } ) . \\end{align*}",
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"\\begin{align*} \\displaystyle \\mitlambda & \\displaystyle \\rightarrow & \\displaystyle \\sqrt { \\operatorname { d e t } \\mitA } ( \\mitlambda + \\frac { \\mitalpha ^ { \\prime } } { 2 \\mitr ^ { 2 } } \\mitB _ { \\mita } ^ { \\miti } \\mitB _ { \\mitb } ^ { \\mitj } ( \\mitb \\mitA ^ { \\mitt } ) _ { \\miti \\mitj } \\mitlambda ^ { \\mita \\mitb } + \\frac { 1 } { 8 } ( \\frac { \\mitalpha ^ { \\prime } } { \\mitr ^ { 2 } } ) ^ { 2 } \\mitB _ { \\mita } ^ { \\miti } \\mitB _ { \\mitb } ^ { \\mitj } \\mitB _ { \\mitc } ^ { \\mitk } \\mitB _ { \\mitd } ^ { \\mitl } ( \\mitb \\mitA ^ { \\mitt } ) _ { \\miti \\mitj } ( \\mitb \\mitA ^ { \\mitt } ) _ { \\mitk \\mitl } \\mitlambda ^ { \\mita \\mitb \\mitc \\mitd } ) , \\\\ \\displaystyle \\mitlambda ^ { \\miti \\mitj } & \\displaystyle \\rightarrow & \\displaystyle \\sqrt { \\operatorname { d e t } \\mitA } ( \\mitB _ { \\mita } ^ { \\miti } \\mitB _ { \\mitb } ^ { \\mitj } \\mitlambda ^ { \\mita \\mitb } + \\frac { \\mitalpha ^ { \\prime } } { 2 \\mitr ^ { 2 } } \\mitB _ { \\mita } ^ { \\miti } \\mitB _ { \\mitb } ^ { \\mitj } \\mitB _ { \\mitc } ^ { \\mitk } \\mitB _ { \\mitd } ^ { \\mitl } ( \\mitb \\mitA ^ { \\mitt } ) _ { \\mitk \\mitl } \\mitlambda ^ { \\mita \\mitb \\mitc \\mitd } ) , \\; \\mitlambda ^ { \\miti \\mitj \\mitk \\mitl } \\rightarrow \\sqrt { \\operatorname { d e t } \\mitA } \\mitB _ { \\mita } ^ { \\miti } \\mitB _ { \\mitb } ^ { \\mitj } \\mitB _ { \\mitc } ^ { \\mitk } \\mitB _ { \\mitd } ^ { \\mitl } \\mitlambda ^ { \\mita \\mitb \\mitc \\mitd } , \\end{align*}"
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0003204_page09
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0003221_page01
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0003221_page02
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0003221_page03
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0003221_page04
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"\\begin{align*} \\displaystyle \\mitg _ { \\miti \\mitj } & = & \\displaystyle \\miteta _ { \\mitmu \\mitnu } \\mupPi _ { \\miti } ^ { \\mitmu } \\mupPi _ { \\mitj } ^ { \\mitnu } \\, ; \\qquad \\mupPi _ { \\miti } ^ { \\mitmu } = \\mitpartial _ { \\miti } \\mitX ^ { \\mitmu } - \\bar { \\mittheta } \\mupGamma ^ { \\mitmu } \\mitpartial _ { \\miti } \\mittheta \\, , \\\\ \\displaystyle \\mitF _ { \\miti \\mitj } & = & \\displaystyle \\mitpartial _ { \\miti } \\mitV _ { \\mitj } - \\mitpartial _ { \\mitj } \\mitV _ { \\miti } - \\left\\{ \\bar { \\mittheta } \\mupGamma _ { 1 1 } \\mupGamma _ { \\mitmu } \\mitpartial _ { \\miti } \\mittheta \\left( \\mitpartial _ { \\mitj } \\mitX ^ { \\mitmu } - { \\textstyle \\frac { 1 } { 2 } } \\bar { \\mittheta } \\mupGamma ^ { \\mitmu } \\mitpartial _ { \\mitj } \\mittheta \\right) - ( \\miti \\leftrightarrow \\mitj ) \\right\\} \\, , \\\\ \\displaystyle \\mitC _ { \\miti _ { 1 } \\ldots \\miti _ { \\mitp } } & = & \\displaystyle \\mitpartial _ { [ \\miti _ { 1 } } \\mitX ^ { \\mitmu _ { 1 } } \\cdots \\mitpartial _ { \\miti _ { \\mitp - 1 } } \\mitX ^ { \\mitmu _ { \\mitp - 1 } } \\, \\bar { \\mittheta } \\mitP _ { ( \\mitp ) } \\mupGamma _ { \\mitmu _ { 1 } \\ldots \\mitmu _ { \\mitp - 1 } } \\mitpartial _ { \\miti _ { \\mitp } ] } \\mittheta + \\ldots \\, . \\end{align*}",
"\\begin{equation*} \\mitL _ { \\mathrm { B I } } = - \\sqrt { \\vert \\mathrm { d e t } \\, \\mitG _ { 1 0 \\, \\miti \\mitj } ^ { ( \\mitp ) } \\vert } \\, , \\end{equation*}",
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"\\begin{equation*} \\mupGamma ^ { \\mitmu } \\rightarrow \\left( \\begin{array}{cc} 0 & \\mitgamma ^ { \\mitmu } \\\\ \\mitgamma ^ { \\mitmu } & 0 \\end{array} \\right) \\, , \\quad ( \\mitmu = 0 , \\ldots , 8 ) , \\quad \\mupGamma ^ { 9 } \\rightarrow \\left( \\begin{array}{cc} 0 & \\Bbbone \\\\ - \\Bbbone & 0 \\end{array} \\right) \\, , \\quad \\mupGamma ^ { 1 1 } = \\left( \\begin{array}{cc} \\Bbbone & 0 \\\\ 0 & - \\Bbbone \\end{array} \\right) \\, . \\end{equation*}"
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0003221_page05
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"$X^9$",
"$S$",
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"\\begin {eqnarray} {G}^{(p)}_{10\\,ij} &\\to & G^{(p)}_{9\\,ij} - \\partial _iS\\partial _jS + 2\\bar \\theta _2\\partial _i\\theta _2\\partial _jS - 2\\bar \\theta _1\\partial _j\\theta _1\\partial _iS + 2\\bar \\theta _2\\partial _i\\theta _2\\bar \\theta _1\\partial _j\\theta _1\\,, \\end {eqnarray}",
"\\begin {eqnarray} G^{(p)}_{9\\,ij} &=& g_{ij} + F_{ij} -2\\bar \\theta _2\\gamma _\\mu \\partial _i\\theta _2\\partial _jX^\\mu -2\\bar \\theta _1\\gamma _\\mu \\partial _j\\theta _1\\partial _iX^\\mu \\\\&& + \\bar \\theta _2\\gamma _\\mu \\partial _i\\theta _2 \\bar \\theta _2\\gamma ^\\mu \\partial _j\\theta _2 + \\bar \\theta _1\\gamma _\\mu \\partial _i\\theta _1 \\bar \\theta _1\\gamma ^\\mu \\partial _j\\theta _1 + 2 \\bar \\theta _2\\gamma _\\mu \\partial _i\\theta _2 \\bar \\theta _1\\gamma ^\\mu \\partial _j\\theta _1 \\,. \\end {eqnarray}",
"\\begin {equation} \\theta _{1R} \\to \\left (\\theta _1 \\atop 0\\right )\\,,\\quad \\theta _{2R} \\to \\left (\\theta _2 \\atop 0\\right )\\,, \\end {equation}",
"\\begin {equation} G^{(p+1)}_{10} \\to \\left (\\begin {array}{cc} G^{(p)}_{9\\,ij} - 2\\bar \\theta _2\\partial _i\\theta _2 \\bar \\theta _1\\partial _j\\theta _1 & \\partial _iS -2\\bar \\theta _2\\partial _i\\theta _2\\\\ -\\partial _jS -2\\bar \\theta _1\\partial _j\\theta _1& -1 \\end {array}\\right )\\,. \\end {equation}",
"\\begin {equation} \\label {matrix-id} {\\rm det} \\left (\\begin {array}{cc} A & B \\\\ C & D \\end {array}\\right ) = {\\rm det} \\left (\\begin {array}{cc} A - BD^{-1}C & B \\\\ 0 & D \\end {array}\\right ) \\left (\\begin {array}{cc} \\unitmatrixDT & 0 \\\\ D^{-1} C & \\unitmatrixDT \\end {array}\\right ) = {\\rm det}\\, ( A - BD^{-1}C) {\\rm det}\\, D\\,. \\end {equation}",
"\\begin {equation} \\label {LnonBPS} {\\cal L}_{\\rm BI}^{(p)} = -\\sqrt { |{\\rm det}\\, ({G}^{(p)}_{10\\,ij} + \\partial _iT\\partial _jT)|}\\,g(T)\\,. \\end {equation}",
"\\begin {equation} G^{(p)}_{9\\,ij} \\to G^{(p)}_{9\\,ij} + \\partial _iT\\partial _jT\\,. \\end {equation}"
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"\\begin{align*} G _ { 1 0 \\, i j } ^ { ( p ) } & \\rightarrow & G _ { 9 \\, i j } ^ { ( p ) } - \\partial _ { i } S \\partial _ { j } S + 2 \\bar { \\theta } _ { 2 } \\partial _ { i } \\theta _ { 2 } \\partial _ { j } S - 2 \\bar { \\theta } _ { 1 } \\partial _ { j } \\theta _ { 1 } \\partial _ { i } S + 2 \\bar { \\theta } _ { 2 } \\partial _ { i } \\theta _ { 2 } \\bar { \\theta } _ { 1 } \\partial _ { j } \\theta _ { 1 } \\, , \\end{align*}",
"\\begin{align*} G _ { 9 \\, i j } ^ { ( p ) } & = & g _ { i j } + F _ { i j } - 2 \\bar { \\theta } _ { 2 } \\gamma _ { \\mu } \\partial _ { i } \\theta _ { 2 } \\partial _ { j } X ^ { \\mu } - 2 \\bar { \\theta } _ { 1 } \\gamma _ { \\mu } \\partial _ { j } \\theta _ { 1 } \\partial _ { i } X ^ { \\mu } \\\\ & & + \\bar { \\theta } _ { 2 } \\gamma _ { \\mu } \\partial _ { i } \\theta _ { 2 } \\bar { \\theta } _ { 2 } \\gamma ^ { \\mu } \\partial _ { j } \\theta _ { 2 } + \\bar { \\theta } _ { 1 } \\gamma _ { \\mu } \\partial _ { i } \\theta _ { 1 } \\bar { \\theta } _ { 1 } \\gamma ^ { \\mu } \\partial _ { j } \\theta _ { 1 } + 2 \\bar { \\theta } _ { 2 } \\gamma _ { \\mu } \\partial _ { i } \\theta _ { 2 } \\bar { \\theta } _ { 1 } \\gamma ^ { \\mu } \\partial _ { j } \\theta _ { 1 } \\, . \\end{align*}",
"\\begin{equation*} \\theta _ { 1 R } \\rightarrow ( { \\theta _ { 1 } \\atop 0 } ) \\, , \\quad \\theta _ { 2 R } \\rightarrow ( { \\theta _ { 2 } \\atop 0 } ) \\, , \\end{equation*}",
"\\begin{align*} G _ { 1 0 } ^ { ( p + 1 ) } \\rightarrow ( \\begin{array}{cc} G _ { 9 \\, i j } ^ { ( p ) } - 2 \\bar { \\theta } _ { 2 } \\partial _ { i } \\theta _ { 2 } \\bar { \\theta } _ { 1 } \\partial _ { j } \\theta _ { 1 } & \\partial _ { i } S - 2 \\bar { \\theta } _ { 2 } \\partial _ { i } \\theta _ { 2 } \\\\ - \\partial _ { j } S - 2 \\bar { \\theta } _ { 1 } \\partial _ { j } \\theta _ { 1 } & - 1 \\end{array} ) \\, . \\end{align*}",
"\\begin{align*} d e t ( \\begin{array}{cc} A & B \\\\ C & D \\end{array} ) = d e t ( \\begin{array}{cc} A - B D ^ { - 1 } C & B \\\\ 0 & D \\end{array} ) ( \\begin{array}{cc} 1 & 0 \\\\ D ^ { - 1 } C & 1 \\end{array} ) = d e t \\, ( A - B D ^ { - 1 } C ) d e t \\, D \\, . \\end{align*}",
"\\begin{equation*} L _ { B I } ^ { ( p ) } = - \\sqrt { \\vert d e t \\, ( G _ { 1 0 \\, i j } ^ { ( p ) } + \\partial _ { i } T \\partial _ { j } T ) \\vert } \\, g ( T ) \\, . \\end{equation*}",
"\\begin{equation*} G _ { 9 \\, i j } ^ { ( p ) } \\rightarrow G _ { 9 \\, i j } ^ { ( p ) } + \\partial _ { i } T \\partial _ { j } T \\, . \\end{equation*}"
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"$ \\mitS $",
"$ \\mitX ^ { 9 } $",
"$ \\mitsigma $",
"$ \\mitS $",
"$ \\mitT $",
"$ \\mitT $",
"\\begin{align*} \\displaystyle \\mitG _ { 1 0 \\, \\miti \\mitj } ^ { ( \\mitp ) } & \\displaystyle \\rightarrow & \\displaystyle \\mitG _ { 9 \\, \\miti \\mitj } ^ { ( \\mitp ) } - \\mitpartial _ { \\miti } \\mitS \\mitpartial _ { \\mitj } \\mitS + 2 \\bar { \\mittheta } _ { 2 } \\mitpartial _ { \\miti } \\mittheta _ { 2 } \\mitpartial _ { \\mitj } \\mitS - 2 \\bar { \\mittheta } _ { 1 } \\mitpartial _ { \\mitj } \\mittheta _ { 1 } \\mitpartial _ { \\miti } \\mitS + 2 \\bar { \\mittheta } _ { 2 } \\mitpartial _ { \\miti } \\mittheta _ { 2 } \\bar { \\mittheta } _ { 1 } \\mitpartial _ { \\mitj } \\mittheta _ { 1 } \\, , \\end{align*}",
"\\begin{align*} \\displaystyle \\mitG _ { 9 \\, \\miti \\mitj } ^ { ( \\mitp ) } & = & \\displaystyle \\mitg _ { \\miti \\mitj } + \\mitF _ { \\miti \\mitj } - 2 \\bar { \\mittheta } _ { 2 } \\mitgamma _ { \\mitmu } \\mitpartial _ { \\miti } \\mittheta _ { 2 } \\mitpartial _ { \\mitj } \\mitX ^ { \\mitmu } - 2 \\bar { \\mittheta } _ { 1 } \\mitgamma _ { \\mitmu } \\mitpartial _ { \\mitj } \\mittheta _ { 1 } \\mitpartial _ { \\miti } \\mitX ^ { \\mitmu } \\\\ & & \\displaystyle + \\bar { \\mittheta } _ { 2 } \\mitgamma _ { \\mitmu } \\mitpartial _ { \\miti } \\mittheta _ { 2 } \\bar { \\mittheta } _ { 2 } \\mitgamma ^ { \\mitmu } \\mitpartial _ { \\mitj } \\mittheta _ { 2 } + \\bar { \\mittheta } _ { 1 } \\mitgamma _ { \\mitmu } \\mitpartial _ { \\miti } \\mittheta _ { 1 } \\bar { \\mittheta } _ { 1 } \\mitgamma ^ { \\mitmu } \\mitpartial _ { \\mitj } \\mittheta _ { 1 } + 2 \\bar { \\mittheta } _ { 2 } \\mitgamma _ { \\mitmu } \\mitpartial _ { \\miti } \\mittheta _ { 2 } \\bar { \\mittheta } _ { 1 } \\mitgamma ^ { \\mitmu } \\mitpartial _ { \\mitj } \\mittheta _ { 1 } \\, . \\end{align*}",
"\\begin{equation*} \\mittheta _ { 1 \\mitR } \\rightarrow \\left( { \\mittheta _ { 1 } \\atop 0 } \\right) \\, , \\quad \\mittheta _ { 2 \\mitR } \\rightarrow \\left( { \\mittheta _ { 2 } \\atop 0 } \\right) \\, , \\end{equation*}",
"\\begin{align*} \\mitG _ { 1 0 } ^ { ( \\mitp + 1 ) } \\rightarrow \\left( \\begin{array}{cc} \\mitG _ { 9 \\, \\miti \\mitj } ^ { ( \\mitp ) } - 2 \\bar { \\mittheta } _ { 2 } \\mitpartial _ { \\miti } \\mittheta _ { 2 } \\bar { \\mittheta } _ { 1 } \\mitpartial _ { \\mitj } \\mittheta _ { 1 } & \\mitpartial _ { \\miti } \\mitS - 2 \\bar { \\mittheta } _ { 2 } \\mitpartial _ { \\miti } \\mittheta _ { 2 } \\\\ - \\mitpartial _ { \\mitj } \\mitS - 2 \\bar { \\mittheta } _ { 1 } \\mitpartial _ { \\mitj } \\mittheta _ { 1 } & - 1 \\end{array} \\right) \\, . \\end{align*}",
"\\begin{align*} \\mathrm { d e t } \\left( \\begin{array}{cc} \\mitA & \\mitB \\\\ \\mitC & \\mitD \\end{array} \\right) = \\mathrm { d e t } \\left( \\begin{array}{cc} \\mitA - \\mitB \\mitD ^ { - 1 } \\mitC & \\mitB \\\\ 0 & \\mitD \\end{array} \\right) \\left( \\begin{array}{cc} \\Bbbone & 0 \\\\ \\mitD ^ { - 1 } \\mitC & \\Bbbone \\end{array} \\right) = \\mathrm { d e t } \\, ( \\mitA - \\mitB \\mitD ^ { - 1 } \\mitC ) \\mathrm { d e t } \\, \\mitD \\, . \\end{align*}",
"\\begin{equation*} \\mitL _ { \\mathrm { B I } } ^ { ( \\mitp ) } = - \\sqrt { \\vert \\mathrm { d e t } \\, ( \\mitG _ { 1 0 \\, \\miti \\mitj } ^ { ( \\mitp ) } + \\mitpartial _ { \\miti } \\mitT \\mitpartial _ { \\mitj } \\mitT ) \\vert } \\, \\mitg ( \\mitT ) \\, . \\end{equation*}",
"\\begin{equation*} \\mitG _ { 9 \\, \\miti \\mitj } ^ { ( \\mitp ) } \\rightarrow \\mitG _ { 9 \\, \\miti \\mitj } ^ { ( \\mitp ) } + \\mitpartial _ { \\miti } \\mitT \\mitpartial _ { \\mitj } \\mitT \\, . \\end{equation*}"
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0003221_page06
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{
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"\\begin {eqnarray} {\\cal L}^{(p)}_{\\rm BI} &=& g(T)\\,\\sqrt { |{\\rm det}\\, G_{ij} |}\\times \\bigg \\{1+ {1\\over 2}G^{ji}(-2\\bar \\theta _L\\Gamma _\\mu \\partial _i\\theta _L\\partial _jX^\\mu -2\\bar \\theta _R\\Gamma _\\mu \\partial _j\\theta _R\\partial _iX^\\mu +\\partial _i T\\partial _jT) \\\\ &&-{1\\over 2} G^{ki}(-2\\bar \\theta _L\\Gamma _\\mu \\partial _i\\theta _L\\partial _lX^\\mu -2\\bar \\theta _R\\Gamma _\\mu \\partial _l\\theta _R\\partial _iX^\\mu ) G^{lm}\\partial _mT\\partial _kT\\\\ &&+{1\\over 4} G^{ji}(-2\\bar \\theta _L\\Gamma _\\mu \\partial _i\\theta _L\\partial _jX^\\mu -2\\bar \\theta _R\\Gamma _\\mu \\partial _j\\theta _R\\partial _iX^\\mu ) G^{kl}\\partial _lT\\partial _kT + \\ldots \\bigg \\}\\,, \\end {eqnarray}",
"\\begin {equation} \\partial _i\\bar \\theta \\partial _j\\theta G^{ij}\\,, \\end {equation}",
"\\begin {equation} \\label {LWZIIA} {\\cal L}^{(p)}_{\\rm WZ} = \\epsilon ^{i_1\\ldots i_{p+1}} \\sum _{k=0}^{(p-1)/2} \\,\\,a_{p,k} C_{i_1\\ldots i_{p-2k}}(F^k)_{i_{p-2k+1}\\ldots i_p} \\partial _{i_{p+1}}T\\,, \\end {equation}"
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"\\begin{equation*} \\partial _ { i } \\bar { \\theta } \\partial _ { j } \\theta G ^ { i j } \\, , \\end{equation*}",
"\\begin{equation*} L _ { W Z } ^ { ( p ) } = \\epsilon ^ { i _ { 1 } \\ldots i _ { p + 1 } } \\sum _ { k = 0 } ^ { ( p - 1 ) \\slash 2 } \\, \\, a _ { p , k } C _ { i _ { 1 } \\ldots i _ { p - 2 k } } ( F ^ { k } ) _ { i _ { p - 2 k + 1 } \\ldots i _ { p } } \\partial _ { i _ { p + 1 } } T \\, , \\end{equation*}"
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"\\begin{equation*} \\mitpartial _ { \\miti } \\bar { \\mittheta } \\mitpartial _ { \\mitj } \\mittheta \\mitG ^ { \\miti \\mitj } \\, , \\end{equation*}",
"\\begin{equation*} \\mitL _ { \\mathrm { W Z } } ^ { ( \\mitp ) } = \\mitepsilon ^ { \\miti _ { 1 } \\ldots \\miti _ { \\mitp + 1 } } \\sum _ { \\mitk = 0 } ^ { ( \\mitp - 1 ) \\slash 2 } \\, \\, \\mita _ { \\mitp , \\mitk } \\mitC _ { \\miti _ { 1 } \\ldots \\miti _ { \\mitp - 2 \\mitk } } ( \\mitF ^ { \\mitk } ) _ { \\miti _ { \\mitp - 2 \\mitk + 1 } \\ldots \\miti _ { \\mitp } } \\mitpartial _ { \\miti _ { \\mitp + 1 } } \\mitT \\, , \\end{equation*}"
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0003221_page07
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"\\begin {equation} C_{i_1\\ldots i_{p-2k}}= 2\\partial _{[i_1}X^{\\mu _1}\\cdots \\partial _{i_{p-2k-1}}X^{\\mu _{p-2k-1}}\\, \\bar \\theta _L\\Gamma _{\\mu _1\\ldots \\mu _{p-2k-1}} \\partial _{i_{p-2k}]}\\theta _R\\,. \\end {equation}",
"\\begin {eqnarray} {\\cal L}^{(p)}_{\\rm WZ}&\\to & 2\\epsilon ^{i_1\\ldots i_{p+1}} \\sum _{k=0}^{{}(p-1)/2} \\,a_{p,k}\\,\\bigg \\{ \\tilde C_{i_1\\ldots i_{p-2k}} (F^k)_{i_{p-2k+1}\\ldots i_p} \\partial _{i_{p+1}}T \\\\ &&\\ + (p-2k-1) \\tilde C_{i_1\\ldots i_{p-2k-1}}\\partial _{i_{p-2k}}S (F^k)_{i_{p-2k+1}\\ldots i_p} \\partial _{i_{p+1}}T\\bigg \\} \\,. \\end {eqnarray}",
"\\begin {equation} \\tilde C_{i_1\\ldots i_m}= \\partial _{[i_1}X^{\\mu _1}\\cdots \\partial _{i_{m-1}}X^{\\mu _{m-1}}\\, \\bar \\theta _2\\gamma _{\\mu _1\\ldots \\mu _{m-1}} \\partial _{i_{m}]}\\theta _1\\,, \\end {equation}",
"\\begin {eqnarray} {\\cal L}^{(p+1)}_{\\rm WZ}&\\to & - 2\\epsilon ^{i_1\\ldots i_{p+1}} \\,s_{p-2k+1} b_{p+1,k}\\bigg \\{ \\sum _{k=0}^{{}(p-1)/2} \\, (p-2k)\\tilde C_{i_1\\ldots i_{p-2k}} (F^k)_{i_{p-2k+1}\\ldots i_p} \\partial _{i_{p+1}}T \\\\ &&\\ + \\sum _{k=1}^{{}(p+1)/2}\\,2k \\tilde C_{i_1\\ldots i_{p-2k+1}} (F^k)_{i_{p-2k+2}\\ldots i_{p-1}}\\partial _{i_p}S \\partial _{i_{p+1}}T\\bigg \\} \\,. \\end {eqnarray}",
"\\begin {eqnarray} a_{p,k} &=& {(p-1)!\\over 2^k\\, k!\\, (p-2k-1)!}\\, a_{p,0}\\,, \\\\ b_{p+1,k}&=& {(-1)^k p!\\over 2^k\\, k! \\,(p-2k)!}\\, b_{p+1,0}\\,, \\\\ a_{p,0} &=& -s_{p+1}\\, p\\, b_{p+1,0}\\,. \\end {eqnarray}"
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"\\begin{equation*} \\widetilde { C } _ { i _ { 1 } \\ldots i _ { m } } = \\partial _ { [ i _ { 1 } } X ^ { \\mu _ { 1 } } \\cdots \\partial _ { i _ { m - 1 } } X ^ { \\mu _ { m - 1 } } \\, \\bar { \\theta } _ { 2 } \\gamma _ { \\mu _ { 1 } \\ldots \\mu _ { m - 1 } } \\partial _ { i _ { m } ] } \\theta _ { 1 } \\, , \\end{equation*}",
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"\\begin{align*} a _ { p , k } & = & \\frac { ( p - 1 ) ! } { 2 ^ { k } \\, k ! \\, ( p - 2 k - 1 ) ! } \\, a _ { p , 0 } \\, , \\\\ b _ { p + 1 , k } & = & \\frac { ( - 1 ) ^ { k } p ! } { 2 ^ { k } \\, k ! \\, ( p - 2 k ) ! } \\, b _ { p + 1 , 0 } \\, , \\\\ a _ { p , 0 } & = & - s _ { p + 1 } \\, p \\, b _ { p + 1 , 0 } \\, . \\end{align*}"
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0003221_page08
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0003221_page09
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"\\begin{equation*} \\mitL _ { \\mitm } = \\mitm ^ { 2 } \\mitB ^ { \\mitmu \\mitnu } \\mitB _ { \\mitmu \\mitnu } \\, . \\end{equation*}"
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|
0003232_page01
|
{
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|
0003232_page02
|
{
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