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image | page_id
string | expressions
sequence |
|---|---|---|
0001015_page02 | {
"latex": [
"$S(\\phi )$",
"$\\phi $",
"$\\exp (-Ht)$",
"$\\psi (x)$",
"$d\\mu $",
"$x$",
"$x(t)$",
"$V$",
"$H= L +V(x)$",
"$L$",
"$x(t)$",
"\\begin {equation} H = \\half p^2 + V(x) \\end {equation}",
"\\begin {equation}\\label {EVeq} \\exp (-Ht) \\psi (x) = \\int d\\mu \\exp \\left ( -\\int _0^t V((x(s)) ds \\right ) \\psi (x(t)) \\end {equation}"
],
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"$ x ( t ) $",
"$ V $",
"$ H = L + V ( x ) $",
"$ L $",
"$ x ( t ) $",
"\\begin{equation*} H = \\frac { 1 } { 2 } p ^ { 2 } + V ( x ) \\end{equation*}",
"\\begin{equation*} \\operatorname { e x p } ( - H t ) \\psi ( x ) = \\int d \\mu \\operatorname { e x p } ( - \\int _ { 0 } ^ { t } V ( ( x ( s ) ) d s ) \\psi ( x ( t ) ) \\end{equation*}"
],
"latex_expand": [
"$ \\mitS ( \\mitphi ) $",
"$ \\mitphi $",
"$ \\mathrm { e x p } ( - \\mitH \\mitt ) $",
"$ \\mitpsi ( \\mitx ) $",
"$ \\mitd \\mitmu $",
"$ \\mitx $",
"$ \\mitx ( \\mitt ) $",
"$ \\mitV $",
"$ \\mitH = \\mitL + \\mitV ( \\mitx ) $",
"$ \\mitL $",
"$ \\mitx ( \\mitt ) $",
"\\begin{equation*} \\mitH = \\frac { 1 } { 2 } \\mitp ^ { 2 } + \\mitV ( \\mitx ) \\end{equation*}",
"\\begin{equation*} \\operatorname { e x p } ( - \\mitH \\mitt ) \\mitpsi ( \\mitx ) = \\int \\mitd \\mitmu \\operatorname { e x p } \\left( - \\int _ { 0 } ^ { \\mitt } \\mitV ( ( \\mitx ( \\mits ) ) \\mitd \\mits \\right) \\mitpsi ( \\mitx ( \\mitt ) ) \\end{equation*}"
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|
0001015_page03 | {
"latex": [
"$x: I \\to M$",
"$I$",
"$[0,t]$",
"$M$",
"$n$",
"$g$",
"$\\omega =dh$",
"$M$",
"$\\omega = \\Omu (x) d\\Xmu $",
"$\\omega = \\Omu (x) d\\Xmu $",
"$\\dot {x}^{\\mu }(t')= \\frac {d{x}^{\\mu }}{d t'}$",
"$S[x(.)] = i(h(x(t))-h(x(0)))$",
"$x(t)$",
"$\\omega =0$",
"$h$",
"$M$",
"$\\omega =dh$",
"$\\Pmu $",
"$\\Xmu $",
"$n$",
"$H(p,x)= \\Pmu \\Xmu - \\Lag (x,\\dot {x})$",
"$H(p,x)= \\Pmu \\Xmu - \\Lag (x,\\dot {x})$",
"$\\omega $",
"$\\Pb {\\Tmu }{T_{\\nu }}=0$",
"$\\Pb {\\Tmu }{H_c}=0$",
"$\\Tmu $",
"$\\psi (x)$",
"$\\Pmu =-i \\Dmu $",
"$\\Pmu $",
"$-i\\DDmu $",
"$\\XXmu \\psi =0$",
"$\\XXmu =g^{\\mu \\nu } (p_{\\nu } + i\\omega _{\\nu })$",
"\\begin {equation}\\label {ACeq} S[x(.)] = \\Intot i\\Omu (x(t'))\\dot {x}^{\\mu }(t') \\, dt' \\end {equation}",
"\\begin {equation}\\label {MOMeq} \\Pmu = \\frac {\\delta \\Lag }{\\delta \\dot {x}^\\mu } = i\\Omu , \\end {equation}",
"\\begin {equation} \\Tmu \\equiv \\Pmu - i\\Omu . \\end {equation}",
"\\begin {equation}\\label {GTeq} \\delta _{\\epsilon }\\psi (x) =-i \\epsilon (\\Dmu \\psi (x) + \\Omu (x) \\psi (x)) \\end {equation}"
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"$ M $",
"$ \\omega = \\omega _ { \\mu } ( x ) d x ^ { \\mu } $",
"$ \\omega = \\omega _ { \\mu } ( x ) d x ^ { \\mu } $",
"$ \\dot { x } ^ { \\mu } ( t ^ { \\prime } ) = \\frac { d x ^ { \\mu } } { d t ^ { \\prime } } $",
"$ S [ x ( . ) ] = i ( h ( x ( t ) ) - h ( x ( 0 ) ) ) $",
"$ x ( t ) $",
"$ \\omega = 0 $",
"$ h $",
"$ M $",
"$ \\omega = d h $",
"$ p _ { \\mu } $",
"$ x ^ { \\mu } $",
"$ n $",
"$ H ( p , x ) = p _ { \\mu } x ^ { \\mu } - L ( x , \\dot { x } ) $",
"$ H ( p , x ) = p _ { \\mu } x ^ { \\mu } - L ( x , \\dot { x } ) $",
"$ \\omega $",
"$ \\{ T _ { \\mu } , T _ { \\nu } \\} = 0 $",
"$ \\{ T _ { \\mu } , H _ { c } \\} = 0 $",
"$ T _ { \\mu } $",
"$ \\psi ( x ) $",
"$ p _ { \\mu } = - i \\partial _ { \\mu } $",
"$ p _ { \\mu } $",
"$ - i \\nabla _ { \\mu } $",
"$ X ^ { \\mu } \\psi = 0 $",
"$ X ^ { \\mu } = g ^ { \\mu \\nu } ( p _ { \\nu } + i \\omega _ { \\nu } ) $",
"\\begin{equation*} S [ x ( . ) ] = \\int _ { 0 } ^ { t } i \\omega _ { \\mu } ( x ( t ^ { \\prime } ) ) \\dot { x } ^ { \\mu } ( t ^ { \\prime } ) \\, d t ^ { \\prime } \\end{equation*}",
"\\begin{equation*} p _ { \\mu } = \\frac { \\delta L } { \\delta \\dot { x } ^ { \\mu } } = i \\omega _ { \\mu } , \\end{equation*}",
"\\begin{equation*} T _ { \\mu } \\equiv p _ { \\mu } - i \\omega _ { \\mu } . \\end{equation*}",
"\\begin{equation*} \\delta _ { \\epsilon } \\psi ( x ) = - i \\epsilon ( \\partial _ { \\mu } \\psi ( x ) + \\omega _ { \\mu } ( x ) \\psi ( x ) ) \\end{equation*}"
],
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"$ \\mitI $",
"$ [ 0 , \\mitt ] $",
"$ \\mitM $",
"$ \\mitn $",
"$ \\mitg $",
"$ \\mitomega = \\mitd \\Planckconst $",
"$ \\mitM $",
"$ \\mitomega = \\mitomega _ { \\mitmu } ( \\mitx ) \\mitd \\mitx ^ { \\mitmu } $",
"$ \\mitomega = \\mitomega _ { \\mitmu } ( \\mitx ) \\mitd \\mitx ^ { \\mitmu } $",
"$ \\dot { \\mitx } ^ { \\mitmu } ( \\mitt ^ { \\prime } ) = \\frac { \\mitd \\mitx ^ { \\mitmu } } { \\mitd \\mitt ^ { \\prime } } $",
"$ \\mitS [ \\mitx ( . ) ] = \\miti ( \\Planckconst ( \\mitx ( \\mitt ) ) - \\Planckconst ( \\mitx ( 0 ) ) ) $",
"$ \\mitx ( \\mitt ) $",
"$ \\mitomega = 0 $",
"$ \\Planckconst $",
"$ \\mitM $",
"$ \\mitomega = \\mitd \\Planckconst $",
"$ \\mitp _ { \\mitmu } $",
"$ \\mitx ^ { \\mitmu } $",
"$ \\mitn $",
"$ \\mitH ( \\mitp , \\mitx ) = \\mitp _ { \\mitmu } \\mitx ^ { \\mitmu } - \\mitL ( \\mitx , \\dot { \\mitx } ) $",
"$ \\mitH ( \\mitp , \\mitx ) = \\mitp _ { \\mitmu } \\mitx ^ { \\mitmu } - \\mitL ( \\mitx , \\dot { \\mitx } ) $",
"$ \\mitomega $",
"$ \\left\\{ \\mitT _ { \\mitmu } , \\mitT _ { \\mitnu } \\right \\} = 0 $",
"$ \\left\\{ \\mitT _ { \\mitmu } , \\mitH _ { \\mitc } \\right \\} = 0 $",
"$ \\mitT _ { \\mitmu } $",
"$ \\mitpsi ( \\mitx ) $",
"$ \\mitp _ { \\mitmu } = - \\miti \\mitpartial _ { \\mitmu } $",
"$ \\mitp _ { \\mitmu } $",
"$ - \\miti \\nabla _ { \\mitmu } $",
"$ \\mitX ^ { \\mitmu } \\mitpsi = 0 $",
"$ \\mitX ^ { \\mitmu } = \\mitg ^ { \\mitmu \\mitnu } ( \\mitp _ { \\mitnu } + \\miti \\mitomega _ { \\mitnu } ) $",
"\\begin{equation*} \\mitS [ \\mitx ( . ) ] = \\int _ { 0 } ^ { \\mitt } \\miti \\mitomega _ { \\mitmu } ( \\mitx ( \\mitt ^ { \\prime } ) ) \\dot { \\mitx } ^ { \\mitmu } ( \\mitt ^ { \\prime } ) \\, \\mitd \\mitt ^ { \\prime } \\end{equation*}",
"\\begin{equation*} \\mitp _ { \\mitmu } = \\frac { \\mitdelta \\mitL } { \\mitdelta \\dot { \\mitx } ^ { \\mitmu } } = \\miti \\mitomega _ { \\mitmu } , \\end{equation*}",
"\\begin{equation*} \\mitT _ { \\mitmu } \\equiv \\mitp _ { \\mitmu } - \\miti \\mitomega _ { \\mitmu } . \\end{equation*}",
"\\begin{equation*} \\mitdelta _ { \\mitepsilon } \\mitpsi ( \\mitx ) = - \\miti \\mitepsilon ( \\mitpartial _ { \\mitmu } \\mitpsi ( \\mitx ) + \\mitomega _ { \\mitmu } ( \\mitx ) \\mitpsi ( \\mitx ) ) \\end{equation*}"
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|
0001015_page04 | {
"latex": [
"$\\Etamu $",
"$\\Pimu $",
"$(2n,2n)$",
"$\\Etamu ,\\Pimu $",
"$\\nabla $",
"$\\psi (x,\\eta )$",
"$\\Pmu =-i \\DDmu $",
"$\\Pimu = -i\\frac {\\partial }{\\partial \\Etamu }$",
"$\\psi (x,\\eta )$",
"$(n,n)$",
"$SM$",
"$\\Xmu ,\\Etamu $",
"$Q$",
"$Q=\\Etamu \\Tmu =-i\\Etamu (\\Dmu + \\omega )$",
"$Q=\\Etamu \\Tmu =-i\\Etamu (\\Dmu + \\omega )$",
"$\\chi $",
"$\\chi = \\Pimu \\XXmu = -ig^{\\mu \\nu } \\Pimu (\\nabla _{\\nu }-\\omega _{\\nu })$",
"$M$",
"$\\psi (x,\\eta )$",
"$Q=-i\\Emh d \\Eph $",
"$\\chi = \\Eph \\delta \\Emh $",
"$d$",
"$\\delta = *d*$",
"$h$",
"$\\chi =\\Pimu \\XXmu $",
"$h$",
"$Q$",
"$h$",
"$H_g$",
"\\begin {equation}\\label {SPBeq} d\\Pmu \\wedge d \\Xmu + \\nabla \\Pimu \\wedge \\nabla \\Etamu + \\frac 12 dx^{\\mu } \\wedge dx^{\\nu } \\Curv {\\mu }{\\nu }{\\kappa }{\\lambda }\\eta ^{\\kappa }\\pi _{\\lambda }, \\end {equation}",
"\\begin {eqnarray}\\DDmu \\psi (x,\\eta ) = \\Dmu \\psi (x,\\eta ) + \\Gam {\\mu }{\\nu }{\\lambda } \\eta ^{\\nu } \\frac {\\partial }{\\partial \\eta ^{\\lambda }}\\psi (x,\\eta ). \\end {eqnarray}",
"\\begin {eqnarray}H_g &=& i( Q \\chi + \\chi Q) \\End &=& d \\delta + \\delta d + g^{\\mu \\nu }\\Omu \\omega _{\\nu } -i (\\Pimu \\eta ^{\\nu } - \\eta ^{\\nu }\\Pimu ) \\frac {\\partial ^2 h }{\\partial \\Xmu \\partial x_{\\nu }}. \\end {eqnarray}"
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"$ ( 2 n , 2 n ) $",
"$ \\eta ^ { \\mu } , \\pi _ { \\mu } $",
"$ \\nabla $",
"$ \\psi ( x , \\eta ) $",
"$ p _ { \\mu } = - i \\nabla _ { \\mu } $",
"$ \\pi _ { \\mu } = - i \\frac { \\partial } { \\partial \\eta ^ { \\mu } } $",
"$ \\psi ( x , \\eta ) $",
"$ ( n , n ) $",
"$ S M $",
"$ x ^ { \\mu } , \\eta ^ { \\mu } $",
"$ Q $",
"$ Q = \\eta ^ { \\mu } T _ { \\mu } = - i \\eta ^ { \\mu } ( \\partial _ { \\mu } + \\omega ) $",
"$ Q = \\eta ^ { \\mu } T _ { \\mu } = - i \\eta ^ { \\mu } ( \\partial _ { \\mu } + \\omega ) $",
"$ \\chi $",
"$ \\chi = \\pi _ { \\mu } X ^ { \\mu } = - i g ^ { \\mu \\nu } \\pi _ { \\mu } ( \\nabla _ { \\nu } - \\omega _ { \\nu } ) $",
"$ M $",
"$ \\psi ( x , \\eta ) $",
"$ Q = - i e ^ { - h } d e ^ { h } $",
"$ \\chi = e ^ { h } \\delta e ^ { - h } $",
"$ d $",
"$ \\delta = \\ast d \\ast $",
"$ h $",
"$ \\chi = \\pi _ { \\mu } X ^ { \\mu } $",
"$ h $",
"$ Q $",
"$ h $",
"$ H _ { g } $",
"\\begin{equation*} d p _ { \\mu } \\wedge d x ^ { \\mu } + \\nabla \\pi _ { \\mu } \\wedge \\nabla \\eta ^ { \\mu } + \\frac { 1 } { 2 } d x ^ { \\mu } \\wedge d x ^ { \\nu } R _ { \\mu \\nu \\kappa } { } ^ { \\lambda } \\eta ^ { \\kappa } \\pi _ { \\lambda } , \\end{equation*}",
"\\begin{equation*} \\nabla _ { \\mu } \\psi ( x , \\eta ) = \\partial _ { \\mu } \\psi ( x , \\eta ) + \\Gamma _ { \\mu \\nu } ^ { \\lambda } \\eta ^ { \\nu } \\frac { \\partial } { \\partial \\eta ^ { \\lambda } } \\psi ( x , \\eta ) . \\end{equation*}",
"\\begin{align*} H _ { g } & = & i ( Q \\chi + \\chi Q ) \\\\ & = & d \\delta + \\delta d + g ^ { \\mu \\nu } \\omega _ { \\mu } \\omega _ { \\nu } - i ( \\pi _ { \\mu } \\eta ^ { \\nu } - \\eta ^ { \\nu } \\pi _ { \\mu } ) \\frac { \\partial ^ { 2 } h } { \\partial x ^ { \\mu } \\partial x _ { \\nu } } . \\end{align*}"
],
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"$ \\miteta ^ { \\mitmu } $",
"$ \\mitpi _ { \\mitmu } $",
"$ ( 2 \\mitn , 2 \\mitn ) $",
"$ \\miteta ^ { \\mitmu } , \\mitpi _ { \\mitmu } $",
"$ \\nabla $",
"$ \\mitpsi ( \\mitx , \\miteta ) $",
"$ \\mitp _ { \\mitmu } = - \\miti \\nabla _ { \\mitmu } $",
"$ \\mitpi _ { \\mitmu } = - \\miti \\frac { \\mitpartial } { \\mitpartial \\miteta ^ { \\mitmu } } $",
"$ \\mitpsi ( \\mitx , \\miteta ) $",
"$ ( \\mitn , \\mitn ) $",
"$ \\mitS \\mitM $",
"$ \\mitx ^ { \\mitmu } , \\miteta ^ { \\mitmu } $",
"$ \\mitQ $",
"$ \\mitQ = \\miteta ^ { \\mitmu } \\mitT _ { \\mitmu } = - \\miti \\miteta ^ { \\mitmu } ( \\mitpartial _ { \\mitmu } + \\mitomega ) $",
"$ \\mitQ = \\miteta ^ { \\mitmu } \\mitT _ { \\mitmu } = - \\miti \\miteta ^ { \\mitmu } ( \\mitpartial _ { \\mitmu } + \\mitomega ) $",
"$ \\mitchi $",
"$ \\mitchi = \\mitpi _ { \\mitmu } \\mitX ^ { \\mitmu } = - \\miti \\mitg ^ { \\mitmu \\mitnu } \\mitpi _ { \\mitmu } ( \\nabla _ { \\mitnu } - \\mitomega _ { \\mitnu } ) $",
"$ \\mitM $",
"$ \\mitpsi ( \\mitx , \\miteta ) $",
"$ \\mitQ = - \\miti \\mite ^ { - \\Planckconst } \\mitd \\mite ^ { \\Planckconst } $",
"$ \\mitchi = \\mite ^ { \\Planckconst } \\mitdelta \\mite ^ { - \\Planckconst } $",
"$ \\mitd $",
"$ \\mitdelta = \\ast \\mitd \\ast $",
"$ \\Planckconst $",
"$ \\mitchi = \\mitpi _ { \\mitmu } \\mitX ^ { \\mitmu } $",
"$ \\Planckconst $",
"$ \\mitQ $",
"$ \\Planckconst $",
"$ \\mitH _ { \\mitg } $",
"\\begin{equation*} \\mitd \\mitp _ { \\mitmu } \\wedge \\mitd \\mitx ^ { \\mitmu } + \\nabla \\mitpi _ { \\mitmu } \\wedge \\nabla \\miteta ^ { \\mitmu } + \\frac { 1 } { 2 } \\mitd \\mitx ^ { \\mitmu } \\wedge \\mitd \\mitx ^ { \\mitnu } \\mitR _ { \\mitmu \\mitnu \\mitkappa } { } ^ { \\mitlambda } \\miteta ^ { \\mitkappa } \\mitpi _ { \\mitlambda } , \\end{equation*}",
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0001015_page05 | {
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0001015_page06 | {
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0001073_page01 | {
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0001073_page02 | {
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0001073_page03 | {
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"$\\star $",
"$O(\\theta ^2)$",
"$F$",
"$i$",
"$j$",
"$O(\\theta ^2)$",
"$\\theta $",
"$F$",
"$i$",
"$F$",
"$A$",
"$A$",
"$A\\simeq A^0+A^1+A^2$",
"$\\theta $",
"$F_{ij}\\simeq F_{ij}^0+F_{ij}^1+F_{ij}^2$",
"$f^{1, 2}=dA^{1, 2}$",
"$g$",
"$\\ast 1$",
"$F$",
"$\\theta $",
"$f(x)\\star g(x)=\\exp (\\frac {i}{2}\\theta _{ij}\\partial _i\\partial _j') f(x)g(x')|_{x=x'}$",
"$[x_i, x_j]=i\\theta _{ij}$",
"$A_0=0$",
"\\begin {equation} F=dA-\\frac {i}{2}[A, A]_{\\star }. \\end {equation}",
"\\begin {equation} \\label {eq:F_ij} F_{ij}\\simeq \\partial _i A_j-\\partial _j A_i+\\theta _{mn}\\partial _m A_i \\partial _n A_j, \\end {equation}",
"\\begin {eqnarray} F_{ij}^0 & = & \\partial _i A_j^0-\\partial _j A_i^0 \\\\ F_{ij}^1 & = & \\partial _i A_j^1-\\partial _j A_i^1+\\theta _{mn}\\partial _m A_i^0 \\partial _n A_j^0 \\\\ F_{ij}^2 & = & \\partial _i A_j^2-\\partial _j A_i^2+\\theta _{mn}\\partial _m A_i^0 \\partial _n A_j^1+\\theta _{mn}\\partial _m A_i^1 \\partial _n A_j^0. \\end {eqnarray}",
"\\begin {equation} \\label {eq:DF} DF=4\\pi g\\delta ^3(\\vec r\\,)\\ast \\!1 \\end {equation}",
"\\begin {equation} DF=dF-i[A, F]_{\\star }. \\end {equation}",
"\\begin {eqnarray} dF^0 & = & 4\\pi g\\delta ^3(\\vec r\\,)\\ast \\!1 \\\\ dF^1 & = & -\\theta _{mn}\\partial _m A^0\\wedge \\partial _n F^0 \\\\ dF^2 & = & -\\theta _{mn}\\partial _m A^1\\wedge \\partial _n F^0 -\\theta _{mn}\\partial _m A^0\\wedge \\partial _n F^1. \\end {eqnarray}"
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"$ A $",
"$ A \\sime A ^ { 0 } + A ^ { 1 } + A ^ { 2 } $",
"$ \\theta $",
"$ F _ { i j } \\sime F _ { i j } ^ { 0 } + F _ { i j } ^ { 1 } + F _ { i j } ^ { 2 } $",
"$ f ^ { 1 , 2 } = d A ^ { 1 , 2 } $",
"$ g $",
"$ \\ast 1 $",
"$ F $",
"$ \\theta $",
"$ f ( x ) \\star g ( x ) = e x p ( \\frac { i } { 2 } \\theta _ { i j } \\partial _ { i } \\partial _ { j } ^ { \\prime } ) f ( x ) g ( x ^ { \\prime } ) \\vert _ { x = x ^ { \\prime } } $",
"$ [ x _ { i } , x _ { j } ] = i \\theta _ { i j } $",
"$ A _ { 0 } = 0 $",
"\\begin{equation*} F = d A - \\frac { i } { 2 } [ A , A ] _ { \\star } . \\end{equation*}",
"\\begin{equation*} F _ { i j } \\sime \\partial _ { i } A _ { j } - \\partial _ { j } A _ { i } + \\theta _ { m n } \\partial _ { m } A _ { i } \\partial _ { n } A _ { j } , \\end{equation*}",
"\\begin{align*} F _ { i j } ^ { 0 } & = & \\partial _ { i } A _ { j } ^ { 0 } - \\partial _ { j } A _ { i } ^ { 0 } \\\\ F _ { i j } ^ { 1 } & = & \\partial _ { i } A _ { j } ^ { 1 } - \\partial _ { j } A _ { i } ^ { 1 } + \\theta _ { m n } \\partial _ { m } A _ { i } ^ { 0 } \\partial _ { n } A _ { j } ^ { 0 } \\\\ F _ { i j } ^ { 2 } & = & \\partial _ { i } A _ { j } ^ { 2 } - \\partial _ { j } A _ { i } ^ { 2 } + \\theta _ { m n } \\partial _ { m } A _ { i } ^ { 0 } \\partial _ { n } A _ { j } ^ { 1 } + \\theta _ { m n } \\partial _ { m } A _ { i } ^ { 1 } \\partial _ { n } A _ { j } ^ { 0 } . \\end{align*}",
"\\begin{equation*} D F = 4 \\pi g \\delta ^ { 3 } ( \\vec { r } \\, ) \\ast \\! 1 \\end{equation*}",
"\\begin{equation*} D F = d F - i [ A , F ] _ { \\star } . \\end{equation*}",
"\\begin{align*} d F ^ { 0 } & = & 4 \\pi g \\delta ^ { 3 } ( \\vec { r } \\, ) \\ast \\! 1 \\\\ d F ^ { 1 } & = & - \\theta _ { m n } \\partial _ { m } A ^ { 0 } \\wedge \\partial _ { n } F ^ { 0 } \\\\ d F ^ { 2 } & = & - \\theta _ { m n } \\partial _ { m } A ^ { 1 } \\wedge \\partial _ { n } F ^ { 0 } - \\theta _ { m n } \\partial _ { m } A ^ { 0 } \\wedge \\partial _ { n } F ^ { 1 } . \\end{align*}"
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"$ \\miti $",
"$ \\mitj $",
"$ \\mitO ( \\mittheta ^ { 2 } ) $",
"$ \\mittheta $",
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"$ \\miti $",
"$ \\mitF $",
"$ \\mitA $",
"$ \\mitA $",
"$ \\mitA \\sime \\mitA ^ { 0 } + \\mitA ^ { 1 } + \\mitA ^ { 2 } $",
"$ \\mittheta $",
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"$ \\ast 1 $",
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"$ \\mittheta $",
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"$ [ \\mitx _ { \\miti } , \\mitx _ { \\mitj } ] = \\miti \\mittheta _ { \\miti \\mitj } $",
"$ \\mitA _ { 0 } = 0 $",
"\\begin{equation*} \\mitF = \\mitd \\mitA - \\frac { \\miti } { 2 } [ \\mitA , \\mitA ] _ { \\star } . \\end{equation*}",
"\\begin{equation*} \\mitF _ { \\miti \\mitj } \\sime \\mitpartial _ { \\miti } \\mitA _ { \\mitj } - \\mitpartial _ { \\mitj } \\mitA _ { \\miti } + \\mittheta _ { \\mitm \\mitn } \\mitpartial _ { \\mitm } \\mitA _ { \\miti } \\mitpartial _ { \\mitn } \\mitA _ { \\mitj } , \\end{equation*}",
"\\begin{align*} \\mitF _ { \\miti \\mitj } ^ { 0 } & = & \\mitpartial _ { \\miti } \\mitA _ { \\mitj } ^ { 0 } - \\mitpartial _ { \\mitj } \\mitA _ { \\miti } ^ { 0 } \\\\ \\mitF _ { \\miti \\mitj } ^ { 1 } & = & \\mitpartial _ { \\miti } \\mitA _ { \\mitj } ^ { 1 } - \\mitpartial _ { \\mitj } \\mitA _ { \\miti } ^ { 1 } + \\mittheta _ { \\mitm \\mitn } \\mitpartial _ { \\mitm } \\mitA _ { \\miti } ^ { 0 } \\mitpartial _ { \\mitn } \\mitA _ { \\mitj } ^ { 0 } \\\\ \\mitF _ { \\miti \\mitj } ^ { 2 } & = & \\mitpartial _ { \\miti } \\mitA _ { \\mitj } ^ { 2 } - \\mitpartial _ { \\mitj } \\mitA _ { \\miti } ^ { 2 } + \\mittheta _ { \\mitm \\mitn } \\mitpartial _ { \\mitm } \\mitA _ { \\miti } ^ { 0 } \\mitpartial _ { \\mitn } \\mitA _ { \\mitj } ^ { 1 } + \\mittheta _ { \\mitm \\mitn } \\mitpartial _ { \\mitm } \\mitA _ { \\miti } ^ { 1 } \\mitpartial _ { \\mitn } \\mitA _ { \\mitj } ^ { 0 } . \\end{align*}",
"\\begin{equation*} \\mitD \\mitF = 4 \\mitpi \\mitg \\mitdelta ^ { 3 } ( \\vec { \\mitr } \\, ) \\ast \\! 1 \\end{equation*}",
"\\begin{equation*} \\mitD \\mitF = \\mitd \\mitF - \\miti [ \\mitA , \\mitF ] _ { \\star } . \\end{equation*}",
"\\begin{align*} \\mitd \\mitF ^ { 0 } & = & 4 \\mitpi \\mitg \\mitdelta ^ { 3 } ( \\vec { \\mitr } \\, ) \\ast \\! 1 \\\\ \\mitd \\mitF ^ { 1 } & = & - \\mittheta _ { \\mitm \\mitn } \\mitpartial _ { \\mitm } \\mitA ^ { 0 } \\wedge \\mitpartial _ { \\mitn } \\mitF ^ { 0 } \\\\ \\mitd \\mitF ^ { 2 } & = & - \\mittheta _ { \\mitm \\mitn } \\mitpartial _ { \\mitm } \\mitA ^ { 1 } \\wedge \\mitpartial _ { \\mitn } \\mitF ^ { 0 } - \\mittheta _ { \\mitm \\mitn } \\mitpartial _ { \\mitm } \\mitA ^ { 0 } \\wedge \\mitpartial _ { \\mitn } \\mitF ^ { 1 } . \\end{align*}"
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|
0001073_page04 | {
"latex": [
"$\\nabla \\cdot \\vec B^0=4\\pi g\\delta ^3(\\vec r\\,)$",
"$B^0=\\ast F^0$",
"$\\vec B^0=g\\vec r/r^3$",
"$A^0$",
"$A^{1, 2}$",
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"$d f^1=0$",
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"$A^1$",
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"$F^1$",
"$-2\\epsilon _{ijk}(\\theta _{mn} \\theta _{pq}\\partial _m A^0_k\\partial _q A^0_j \\partial _n \\partial _p A^0_i)$",
"$j$",
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"$f^{1, 2}$",
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"$S^2$",
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"$A$",
"$A$",
"$g\\simeq g^0+g^1+g^2$",
"$|A^0|\\sim 1/r$",
"$|F^0|\\sim 1/r^2$",
"$|F^1|\\sim 1/r^4$",
"$|F^2|=0$",
"\\begin {eqnarray} \\epsilon _{ijk}\\partial _i f^1_{jk} & = & -\\epsilon _{ijk}\\partial _i (\\theta _{mn}\\partial _m A^0_j\\partial _n A^0_k) -\\epsilon _{ijk}\\theta _{nm}\\partial _n A^0_k\\partial _m F^0_{ij} \\\\ &= & -\\epsilon _{ijk}\\theta _{mn}\\Big (\\partial _m \\partial _i A^0_j \\partial _n A^0_k +\\partial _mA^0_j\\partial _n\\partial _i A^0_k -\\partial _n A^0_k\\partial _m (\\partial _i A_j^0- \\partial _j A_i^0)\\Big ). \\end {eqnarray}",
"\\begin {eqnarray} \\epsilon _{ijk}\\partial _i f^2_{jk} & = & -\\epsilon _{ijk}\\partial _i \\Big (\\theta _{mn}(\\partial _m A^0_j\\partial _n A^1_k- \\partial _m A^0_k\\partial _n A^1_j)\\Big ) \\\\ && -\\epsilon _{ijk}\\theta _{nm}\\partial _n A^1_k\\partial _m F^0_{ij} -\\epsilon _{ijk}\\theta _{mn}\\partial _m A^0_k\\partial _n F^1_{ij}. \\end {eqnarray}",
"\\begin {equation} \\epsilon _{ijk}\\partial _i f^2_{jk}=-\\epsilon _{ijk}\\theta _{mn}\\theta _{pq} \\partial _m A^0_k\\partial _n (\\partial _p A^0_i\\partial _q A^0_j). \\end {equation}",
"\\begin {equation} m=\\int |F|^2 \\sim \\int _0^{\\infty } r^2dr\\left | \\frac {1}{r^2}+\\frac {1}{r^4} \\right |^2=\\infty . \\end {equation}"
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"$ \\vec { B } ^ { 0 } = g \\vec { r } \\slash r ^ { 3 } $",
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"$ S ^ { 2 } $",
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"$ \\vert A ^ { 0 } \\vert \\sim 1 \\slash r $",
"$ \\vert F ^ { 0 } \\vert \\sim 1 \\slash r ^ { 2 } $",
"$ \\vert F ^ { 1 } \\vert \\sim 1 \\slash r ^ { 4 } $",
"$ \\vert F ^ { 2 } \\vert = 0 $",
"\\begin{align*} \\epsilon _ { i j k } \\partial _ { i } f _ { j k } ^ { 1 } & = & - \\epsilon _ { i j k } \\partial _ { i } ( \\theta _ { m n } \\partial _ { m } A _ { j } ^ { 0 } \\partial _ { n } A _ { k } ^ { 0 } ) - \\epsilon _ { i j k } \\theta _ { n m } \\partial _ { n } A _ { k } ^ { 0 } \\partial _ { m } F _ { i j } ^ { 0 } \\\\ & = & - \\epsilon _ { i j k } \\theta _ { m n } ( \\partial _ { m } \\partial _ { i } A _ { j } ^ { 0 } \\partial _ { n } A _ { k } ^ { 0 } + \\partial _ { m } A _ { j } ^ { 0 } \\partial _ { n } \\partial _ { i } A _ { k } ^ { 0 } - \\partial _ { n } A _ { k } ^ { 0 } \\partial _ { m } ( \\partial _ { i } A _ { j } ^ { 0 } - \\partial _ { j } A _ { i } ^ { 0 } ) ) . \\end{align*}",
"\\begin{align*} \\epsilon _ { i j k } \\partial _ { i } f _ { j k } ^ { 2 } & = & - \\epsilon _ { i j k } \\partial _ { i } ( \\theta _ { m n } ( \\partial _ { m } A _ { j } ^ { 0 } \\partial _ { n } A _ { k } ^ { 1 } - \\partial _ { m } A _ { k } ^ { 0 } \\partial _ { n } A _ { j } ^ { 1 } ) ) \\\\ & & - \\epsilon _ { i j k } \\theta _ { n m } \\partial _ { n } A _ { k } ^ { 1 } \\partial _ { m } F _ { i j } ^ { 0 } - \\epsilon _ { i j k } \\theta _ { m n } \\partial _ { m } A _ { k } ^ { 0 } \\partial _ { n } F _ { i j } ^ { 1 } . \\end{align*}",
"\\begin{equation*} \\epsilon _ { i j k } \\partial _ { i } f _ { j k } ^ { 2 } = - \\epsilon _ { i j k } \\theta _ { m n } \\theta _ { p q } \\partial _ { m } A _ { k } ^ { 0 } \\partial _ { n } ( \\partial _ { p } A _ { i } ^ { 0 } \\partial _ { q } A _ { j } ^ { 0 } ) . \\end{equation*}",
"\\begin{equation*} m = \\int \\vert F \\vert ^ { 2 } \\sim \\int _ { 0 } ^ { \\infty } r ^ { 2 } d r { \\vert \\frac { 1 } { r ^ { 2 } } + \\frac { 1 } { r ^ { 4 } } \\vert } ^ { 2 } = \\infty . \\end{equation*}"
],
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"$ \\nabla \\cdot \\vec { \\mitB } ^ { 0 } = 4 \\mitpi \\mitg \\mitdelta ^ { 3 } ( \\vec { \\mitr } \\, ) $",
"$ \\mitB ^ { 0 } = \\ast \\mitF ^ { 0 } $",
"$ \\vec { \\mitB } ^ { 0 } = \\mitg \\vec { \\mitr } \\slash \\mitr ^ { 3 } $",
"$ \\mitA ^ { 0 } $",
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"$ \\mitA ^ { 0 } $",
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"$ \\mitk \\rightarrow \\mitj $",
"$ \\mitm \\leftrightarrow \\mitn $",
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"$ - 2 \\mitepsilon _ { \\miti \\mitj \\mitk } ( \\mittheta _ { \\mitm \\mitn } \\mittheta _ { \\mitp \\mitq } \\mitpartial _ { \\mitm } \\mitA _ { \\mitk } ^ { 0 } \\mitpartial _ { \\mitq } \\mitA _ { \\mitj } ^ { 0 } \\mitpartial _ { \\mitn } \\mitpartial _ { \\mitp } \\mitA _ { \\miti } ^ { 0 } ) $",
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"$ \\mitf ^ { 1 , 2 } $",
"$ \\mitf ^ { 1 , 2 } = \\mitd \\mitA ^ { 1 , 2 } $",
"$ \\mitF ^ { 0 } = \\mitd \\mitA ^ { 0 } $",
"$ \\mitS ^ { 2 } $",
"$ \\mitA ^ { 0 } $",
"$ \\mitA $",
"$ \\mitA $",
"$ \\mitg \\sime \\mitg ^ { 0 } + \\mitg ^ { 1 } + \\mitg ^ { 2 } $",
"$ \\vert \\mitA ^ { 0 } \\vert \\sim 1 \\slash \\mitr $",
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"\\begin{align*} \\mitepsilon _ { \\miti \\mitj \\mitk } \\mitpartial _ { \\miti } \\mitf _ { \\mitj \\mitk } ^ { 2 } & = & - \\mitepsilon _ { \\miti \\mitj \\mitk } \\mitpartial _ { \\miti } \\Big ( \\mittheta _ { \\mitm \\mitn } ( \\mitpartial _ { \\mitm } \\mitA _ { \\mitj } ^ { 0 } \\mitpartial _ { \\mitn } \\mitA _ { \\mitk } ^ { 1 } - \\mitpartial _ { \\mitm } \\mitA _ { \\mitk } ^ { 0 } \\mitpartial _ { \\mitn } \\mitA _ { \\mitj } ^ { 1 } ) \\Big ) \\\\ & & - \\mitepsilon _ { \\miti \\mitj \\mitk } \\mittheta _ { \\mitn \\mitm } \\mitpartial _ { \\mitn } \\mitA _ { \\mitk } ^ { 1 } \\mitpartial _ { \\mitm } \\mitF _ { \\miti \\mitj } ^ { 0 } - \\mitepsilon _ { \\miti \\mitj \\mitk } \\mittheta _ { \\mitm \\mitn } \\mitpartial _ { \\mitm } \\mitA _ { \\mitk } ^ { 0 } \\mitpartial _ { \\mitn } \\mitF _ { \\miti \\mitj } ^ { 1 } . \\end{align*}",
"\\begin{equation*} \\mitepsilon _ { \\miti \\mitj \\mitk } \\mitpartial _ { \\miti } \\mitf _ { \\mitj \\mitk } ^ { 2 } = - \\mitepsilon _ { \\miti \\mitj \\mitk } \\mittheta _ { \\mitm \\mitn } \\mittheta _ { \\mitp \\mitq } \\mitpartial _ { \\mitm } \\mitA _ { \\mitk } ^ { 0 } \\mitpartial _ { \\mitn } ( \\mitpartial _ { \\mitp } \\mitA _ { \\miti } ^ { 0 } \\mitpartial _ { \\mitq } \\mitA _ { \\mitj } ^ { 0 } ) . \\end{equation*}",
"\\begin{equation*} \\mitm = \\int \\vert \\mitF \\vert ^ { 2 } \\sim \\int _ { 0 } ^ { \\infty } \\mitr ^ { 2 } \\mitd \\mitr { \\left\\vert \\frac { 1 } { \\mitr ^ { 2 } } + \\frac { 1 } { \\mitr ^ { 4 } } \\right\\vert } ^ { 2 } = \\infty . \\end{equation*}"
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0001073_page05 | {
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0001073_page06 | {
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0001073_page07 | {
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0001101_page01 | {
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0001101_page03 | {
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0001101_page04 | {
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0001101_page05 | {
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0001101_page06 | {
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0001101_page07 | {
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0001101_page08 | {
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0001101_page09 | {
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"\\begin {equation} \\tau _B(\\cS ) = \\cS , \\qquad \\tau _B(\\cE ) = \\cE . \\label {eq:hello} \\end {equation}",
"\\begin {equation} \\begin {aligned} \\text {solution 1:} & \\quad \\eta = 14\\mathcal {S} + 22\\mathcal {E}, \\quad \\l = \\sfrac {3}{2}, \\\\ {}& \\sum _i\\k _i = \\eta \\cdot c_1 = 44, \\quad \\sum _i \\k _i^2 \\leq 60 , \\\\ \\text {solution 2:} & \\quad \\eta = 24\\mathcal {S} + 30\\mathcal {E}, \\quad \\l = -\\sfrac {1}{2}, \\\\ {}& \\sum _i\\k _i = \\eta \\cdot c_1 = 60, \\quad \\sum _i \\k _i^2 \\leq 76 . \\end {aligned} \\label {solF2} \\end {equation}",
"\\begin {equation} \\begin {aligned} \\text {solution 1:} \\quad & [W] = \\s _{*}\\left (10\\cS +26\\cE \\right ) + \\left (112-k\\right )\\left (F-N\\right ) + \\left (60-k\\right ) N, \\\\ \\text {solution 2:} \\quad & [W] = \\s _{*}\\left (18\\cE \\right ) + \\left (132-k\\right )\\left (F-N\\right ) + \\left (76-k\\right ) N, \\end {aligned} \\label {eq:branes} \\end {equation}",
"\\begin {equation} k = \\sum _i \\k _i^2 \\end {equation}",
"\\begin {equation} \\begin {aligned} \\text {solution 1:} \\quad & W_{B} = 10\\cS + 26\\cE , \\\\ \\text {solution 2:} \\quad & W_{B} = 18\\cE , \\end {aligned} \\end {equation}"
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"\\begin{equation*} \\begin{array}{ll} s o l u t i o n ~ 1 : & \\quad \\eta = 1 4 S + 2 2 E , \\quad \\lambda = \\frac { 3 } { 2 } , \\\\ & \\sum _ { i } \\kappa _ { i } = \\eta \\cdot c _ { 1 } = 4 4 , \\quad \\sum _ { i } \\kappa _ { i } ^ { 2 } \\leq 6 0 , \\\\ s o l u t i o n ~ 2 : & \\quad \\eta = 2 4 S + 3 0 E , \\quad \\lambda = - \\frac { 1 } { 2 } , \\\\ & \\sum _ { i } \\kappa _ { i } = \\eta \\cdot c _ { 1 } = 6 0 , \\quad \\sum _ { i } \\kappa _ { i } ^ { 2 } \\leq 7 6 . \\end{array} \\end{equation*}",
"\\begin{align*} \\begin{array}{ll} s o l u t i o n ~ 1 : \\quad & [ W ] = \\sigma _ { \\ast } ( 1 0 S + 2 6 E ) + ( 1 1 2 - k ) ( F - N ) + ( 6 0 - k ) N , \\\\ s o l u t i o n ~ 2 : \\quad & [ W ] = \\sigma _ { \\ast } ( 1 8 E ) + ( 1 3 2 - k ) ( F - N ) + ( 7 6 - k ) N , \\end{array} \\end{align*}",
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"\\begin{align*} \\begin{array}{ll} s o l u t i o n ~ 1 : \\quad & W _ { B } = 1 0 S + 2 6 E , \\\\ s o l u t i o n ~ 2 : \\quad & W _ { B } = 1 8 E , \\end{array} \\end{align*}"
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"\\begin{align*} \\begin{array}{ll} \\mathrm { s o l u t i o n ~ 1 } : \\quad & [ \\mitW ] = \\mitsigma _ { \\ast } \\left( 1 0 \\mscrS + 2 6 \\mscrE \\right) + \\left( 1 1 2 - \\mitk \\right) \\left( \\mitF - \\mitN \\right) + \\left( 6 0 - \\mitk \\right) \\mitN , \\\\ \\mathrm { s o l u t i o n ~ 2 } : \\quad & [ \\mitW ] = \\mitsigma _ { \\ast } \\left( 1 8 \\mscrE \\right) + \\left( 1 3 2 - \\mitk \\right) \\left( \\mitF - \\mitN \\right) + \\left( 7 6 - \\mitk \\right) \\mitN , \\end{array} \\end{align*}",
"\\begin{equation*} \\mitk = \\sum _ { \\miti } \\mitkappa _ { \\miti } ^ { 2 } \\end{equation*}",
"\\begin{align*} \\begin{array}{ll} \\mathrm { s o l u t i o n ~ 1 } : \\quad & \\mitW _ { \\mitB } = 1 0 \\mscrS + 2 6 \\mscrE , \\\\ \\mathrm { s o l u t i o n ~ 2 } : \\quad & \\mitW _ { \\mitB } = 1 8 \\mscrE , \\end{array} \\end{align*}"
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|
0001101_page10 | {
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0001113_page02 | {
"latex": [
"$F_{MN}=\\partial _MA_N-\\partial _NA_M$",
"$D_M \\phi =\\partial _M +ieA_M$",
"$A(r)$",
"$\\varphi (r)$",
"$O(\\epsilon )$",
"$(\\epsilon =1/\\sqrt {a})$",
"$(X^5,X^6)$",
"$(X^0-X^3)$",
"$a$",
"$R<<a$",
"$A_M^0$",
"$\\phi ^0$",
"$X^M=Y^M(\\xi ^\\mu )$",
"$(\\mu =0-3)$",
"$x^M$",
"$X^M$",
"$n_m^M$",
"$\\Psi _i$",
"$\\Psi _f$",
"\\begin {equation} {\\cal L}=-{1\\over 4}F_{MN}F^{MN}+D_M\\phi ^\\dagger D^M\\phi +a|\\phi |^2-b|\\phi |^4 +c \\label {1} \\end {equation}",
"\\begin {equation} A_M=\\epsilon _{0123MN}A(r)X^N/r,\\ \\phi =\\varphi (r)e^{in\\theta },\\ (r^2=(x^5)^2+(x^6)^2) \\label {2} \\end {equation}",
"\\begin {eqnarray}\\displaystyle &-\\frac {1}{r}\\frac {d}{dr}\\left (r{d \\over dr}\\varphi \\right ) +\\left [\\left ({n \\over r}+eA\\right )^2-a+2b\\varphi ^2\\right ]\\varphi =0\\cr &-{d \\over dr}\\left ({1\\over r}{d \\over dr}rA\\right ) +\\varphi ^2\\left (e^2A^2+{en \\over r}\\right )=0 \\end {eqnarray}",
"\\begin {equation} X^M=Y^M(x^\\mu )+n_m^M x^m,\\ \\ (M=0-3,5,6,\\ \\mu =0-3,\\ m=5,6) \\label {4} \\end {equation}",
"\\begin {equation} A_M^0=\\epsilon _{0123MN}A(r)x^N/r,\\ \\phi ^0=\\varphi (r)e^{in\\theta }.\\ (r^2=x^m x^m) \\label {5} \\end {equation}",
"\\begin {equation} S_{fi}=\\int \\prod _{X^M}dA_Md\\phi d\\phi ^\\dagger \\exp \\left [i\\int {\\cal L}d^6X \\right ]\\Psi _f^*\\Psi _i\\prod _{X^M}\\delta (\\partial _MA^M) \\label {6} \\end {equation}"
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"$ A ( r ) $",
"$ \\varphi ( r ) $",
"$ O ( \\epsilon ) $",
"$ ( \\epsilon = 1 \\slash \\sqrt { a } ) $",
"$ ( X ^ { 5 } , X ^ { 6 } ) $",
"$ ( X ^ { 0 } - X ^ { 3 } ) $",
"$ a $",
"$ R < < a $",
"$ A _ { M } ^ { 0 } $",
"$ \\phi ^ { 0 } $",
"$ X ^ { M } = Y ^ { M } ( \\xi ^ { \\mu } ) $",
"$ ( \\mu = 0 - 3 ) $",
"$ x ^ { M } $",
"$ X ^ { M } $",
"$ n _ { m } ^ { M } $",
"$ \\Psi _ { i } $",
"$ \\Psi _ { f } $",
"\\begin{equation*} L = - \\frac { 1 } { 4 } F _ { M N } F ^ { M N } + D _ { M } \\phi ^ { \\dagger } D ^ { M } \\phi + a \\vert \\phi \\vert ^ { 2 } - b \\vert \\phi \\vert ^ { 4 } + c \\end{equation*}",
"\\begin{equation*} A _ { M } = \\epsilon _ { 0 1 2 3 M N } A ( r ) X ^ { N } \\slash r , ~ \\phi = \\varphi ( r ) e ^ { i n \\theta } , ~ ( r ^ { 2 } = ( x ^ { 5 } ) ^ { 2 } + ( x ^ { 6 } ) ^ { 2 } ) \\end{equation*}",
"\\begin{align*} & - \\frac { 1 } { r } \\frac { d } { d r } ( r \\frac { d } { d r } \\varphi ) + [ { ( \\frac { n } { r } + e A ) } ^ { 2 } - a + 2 b \\varphi ^ { 2 } ] \\varphi = 0 \\\\ & - \\frac { d } { d r } ( \\frac { 1 } { r } \\frac { d } { d r } r A ) + \\varphi ^ { 2 } ( e ^ { 2 } A ^ { 2 } + \\frac { e n } { r } ) = 0 \\end{align*}",
"\\begin{equation*} X ^ { M } = Y ^ { M } ( x ^ { \\mu } ) + n _ { m } ^ { M } x ^ { m } , ~ ~ ( M = 0 - 3 , 5 , 6 , ~ \\mu = 0 - 3 , ~ m = 5 , 6 ) \\end{equation*}",
"\\begin{equation*} A _ { M } ^ { 0 } = \\epsilon _ { 0 1 2 3 M N } A ( r ) x ^ { N } \\slash r , ~ \\phi ^ { 0 } = \\varphi ( r ) e ^ { i n \\theta } . ~ ( r ^ { 2 } = x ^ { m } x ^ { m } ) \\end{equation*}",
"\\begin{equation*} S _ { f i } = \\int \\prod _ { X ^ { M } } d A _ { M } d \\phi d \\phi ^ { \\dagger } \\operatorname { e x p } [ i \\int L d ^ { 6 } X ] \\Psi _ { f } ^ { \\ast } \\Psi _ { i } \\prod _ { X ^ { M } } \\delta ( \\partial _ { M } A ^ { M } ) \\end{equation*}"
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"$ \\mitD _ { \\mitM } \\mitphi = \\mitpartial _ { \\mitM } + \\miti \\mite \\mitA _ { \\mitM } $",
"$ \\mitA ( \\mitr ) $",
"$ \\mitvarphi ( \\mitr ) $",
"$ \\mitO ( \\mitepsilon ) $",
"$ ( \\mitepsilon = 1 \\slash \\sqrt { \\mita } ) $",
"$ ( \\mitX ^ { 5 } , \\mitX ^ { 6 } ) $",
"$ ( \\mitX ^ { 0 } - \\mitX ^ { 3 } ) $",
"$ \\mita $",
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"$ \\mitphi ^ { 0 } $",
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"$ ( \\mitmu = 0 - 3 ) $",
"$ \\mitx ^ { \\mitM } $",
"$ \\mitX ^ { \\mitM } $",
"$ \\mitn _ { \\mitm } ^ { \\mitM } $",
"$ \\mupPsi _ { \\miti } $",
"$ \\mupPsi _ { \\mitf } $",
"\\begin{equation*} \\mitL = - \\frac { 1 } { 4 } \\mitF _ { \\mitM \\mitN } \\mitF ^ { \\mitM \\mitN } + \\mitD _ { \\mitM } \\mitphi ^ { \\dagger } \\mitD ^ { \\mitM } \\mitphi + \\mita \\vert \\mitphi \\vert ^ { 2 } - \\mitb \\vert \\mitphi \\vert ^ { 4 } + \\mitc \\end{equation*}",
"\\begin{equation*} \\mitA _ { \\mitM } = \\mitepsilon _ { 0 1 2 3 \\mitM \\mitN } \\mitA ( \\mitr ) \\mitX ^ { \\mitN } \\slash \\mitr , ~ \\mitphi = \\mitvarphi ( \\mitr ) \\mite ^ { \\miti \\mitn \\mittheta } , ~ ( \\mitr ^ { 2 } = ( \\mitx ^ { 5 } ) ^ { 2 } + ( \\mitx ^ { 6 } ) ^ { 2 } ) \\end{equation*}",
"\\begin{align*} & - \\frac { 1 } { \\mitr } \\frac { \\mitd } { \\mitd \\mitr } \\left( \\mitr \\frac { \\mitd } { \\mitd \\mitr } \\mitvarphi \\right) + \\left[ { \\left( \\frac { \\mitn } { \\mitr } + \\mite \\mitA \\right) } ^ { 2 } - \\mita + 2 \\mitb \\mitvarphi ^ { 2 } \\right] \\mitvarphi = 0 \\\\ & - \\frac { \\mitd } { \\mitd \\mitr } \\left( \\frac { 1 } { \\mitr } \\frac { \\mitd } { \\mitd \\mitr } \\mitr \\mitA \\right) + \\mitvarphi ^ { 2 } \\left( \\mite ^ { 2 } \\mitA ^ { 2 } + \\frac { \\mite \\mitn } { \\mitr } \\right) = 0 \\end{align*}",
"\\begin{equation*} \\mitX ^ { \\mitM } = \\mitY ^ { \\mitM } ( \\mitx ^ { \\mitmu } ) + \\mitn _ { \\mitm } ^ { \\mitM } \\mitx ^ { \\mitm } , ~ ~ ( \\mitM = 0 - 3 , 5 , 6 , ~ \\mitmu = 0 - 3 , ~ \\mitm = 5 , 6 ) \\end{equation*}",
"\\begin{equation*} \\mitA _ { \\mitM } ^ { 0 } = \\mitepsilon _ { 0 1 2 3 \\mitM \\mitN } \\mitA ( \\mitr ) \\mitx ^ { \\mitN } \\slash \\mitr , ~ \\mitphi ^ { 0 } = \\mitvarphi ( \\mitr ) \\mite ^ { \\miti \\mitn \\mittheta } . ~ ( \\mitr ^ { 2 } = \\mitx ^ { \\mitm } \\mitx ^ { \\mitm } ) \\end{equation*}",
"\\begin{equation*} \\mitS _ { \\mitf \\miti } = \\int \\prod _ { \\mitX ^ { \\mitM } } \\mitd \\mitA _ { \\mitM } \\mitd \\mitphi \\mitd \\mitphi ^ { \\dagger } \\operatorname { e x p } \\left[ \\miti \\int \\mitL \\mitd ^ { 6 } \\mitX \\right] \\mupPsi _ { \\mitf } ^ { \\ast } \\mupPsi _ { \\miti } \\prod _ { \\mitX ^ { \\mitM } } \\mitdelta ( \\mitpartial _ { \\mitM } \\mitA ^ { \\mitM } ) \\end{equation*}"
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|
0001113_page03 | {
"latex": [
"$C^M(\\xi ^\\mu )$",
"$|\\widetilde \\phi |^2$",
"$(\\widetilde \\phi =\\phi -\\sqrt {a/2b})$",
"$N(\\xi ^\\mu )$",
"$x^\\mu =\\xi ^\\mu $",
"$\\prod _{X_{/\\!/}}$",
"$\\xi ^\\mu $",
"$x^M$",
"$A_M$",
"$\\phi $",
"$B_{\\bar N}= A_{\\bar N}-A^0_{\\bar N}$",
"$\\sigma =\\phi -\\phi ^0$",
"$\\widetilde C^\\mu =0$",
"$V^{\\bar N M}$",
"$g^{LM}$",
"$\\nabla _M$",
"$D_M^0=\\nabla _M+ieA_M^0$",
"$J_0=\\int |\\widetilde \\phi ^0|^2dx^5dx^6$",
"${\\cal L}_2$",
"$|\\phi ^0|^2$",
"$g_{m\\mu }=O(R/a)<<1$",
"$g_{mn}=-\\delta _{mn}+O(R/a)$",
"$B_{\\bar M}$",
"$B_{\\bar \\mu }$",
"$B_{\\bar m}$",
"$S^{\\rm eff}$",
"$\\delta $",
"$\\delta =\\int dk e^{ikx}$",
"\\begin {equation} 1=\\int \\prod _{X_{/\\!/}}dY^M(\\xi ^\\mu )\\delta \\left (Y^M(\\xi ^\\mu )-C^M(\\xi ^\\mu )\\right ) \\label {7} \\end {equation}",
"\\begin {equation} C^M(\\xi ^\\mu )=\\int _{N(\\xi ^\\mu )} X^M |\\widetilde \\phi |^2 d^2X_{\\perp }\\Bigg / \\int _{N(\\xi ^\\mu )} |\\widetilde \\phi |^2 d^2X_{\\perp } \\label {8} \\end {equation}",
"\\begin {equation} S_{fi}=\\int \\prod _{X_{/\\!/}}dY^M\\prod _{X^M}dB_{\\bar N} d\\sigma d\\sigma ^\\dagger \\delta (\\sqrt {-g} \\nabla _{\\bar N}B^{\\bar N} \\prod _{X_{/\\!/}}\\delta (\\widetilde C^M) \\exp \\left [i\\int \\left ({\\cal L}_0+{\\cal L}_1 \\right )\\sqrt {-g}d^6x\\right ]\\Psi _f^*\\Psi _i \\label {9} \\end {equation}",
"\\begin {eqnarray} {\\cal L}_0 &=& {\\cal L}(\\phi =\\phi _0, A_M=A_M^0) \\\\ {\\cal L}_2 &=& -\\frac {1}{2}g^{LM}\\nabla _L B_{\\bar N} \\nabla _M B^{\\bar N} +B_{\\bar N} B^{\\bar N} e^2 |\\phi ^0|^2 +g^{LM}(D_L^0\\sigma )^\\dagger (D_M^0\\sigma )\\cr &&-4ieV^{\\bar N M} B_{\\bar N} {\\rm Im} \\left ( \\sigma ^\\dagger D_M^0 \\phi ^0 \\right ) +a|\\sigma |^2 -b\\left [ 4|\\phi ^0\\sigma |^2 +2{\\rm Re}(\\sigma ^\\dagger \\phi ^0)^2\\right ], \\\\ \\widetilde C^m &=& \\int x^m |\\widetilde \\phi |^2 dx^5 dx^6 \\Bigg / \\int |\\widetilde \\phi |^2 dx^5 dx^6 \\\\ &=& \\frac {1}{J_0}\\int x^m \\left [|\\sigma |^2 + {\\rm Re}(\\widetilde \\phi ^0\\sigma ^\\dagger ) \\left \\{1-\\frac {2}{J_0}\\int {\\rm Re}(\\widetilde \\phi ^0\\sigma ^\\dagger )dx^5 dx^6\\right \\}\\right ]dx^5 dx^6, \\end {eqnarray}",
"\\begin {equation} S^{\\rm eff} = -i\\ln \\int \\prod _{X^M}dB_{\\bar N} d\\sigma d\\sigma ^\\dagger \\delta \\left (\\sqrt {-g}\\nabla _{\\bar N}B^{\\bar N}\\right )\\prod _{X_{/\\!/}}\\delta (\\widetilde C^M) \\exp \\left [i\\int \\sqrt {-g}{\\cal L}_2 d^6 x \\right ]. \\label {14} \\end {equation}",
"\\begin {equation} S^{\\rm eff} = -i\\ln \\int \\prod _{\\xi ^\\mu }dw_m\\prod _{x^M}dB_{\\bar M} d\\sigma d\\sigma ^\\dagger dv \\exp \\left [i\\int (\\Xi \\Phi +\\Phi ^\\dagger \\Delta \\Phi )d^6x\\right ] \\label {15} \\end {equation}"
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"$ \\vert \\widetilde { \\phi } \\vert ^ { 2 } $",
"$ ( \\widetilde { \\phi } = \\phi - \\sqrt { a \\slash 2 b } ) $",
"$ N ( \\xi ^ { \\mu } ) $",
"$ x ^ { \\mu } = \\xi ^ { \\mu } $",
"$ \\prod _ { X _ { \\slash \\! \\slash } } $",
"$ \\xi ^ { \\mu } $",
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"$ A _ { M } $",
"$ \\phi $",
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"$ \\sigma = \\phi - \\phi ^ { 0 } $",
"$ \\widetilde { C } ^ { \\mu } = 0 $",
"$ V ^ { \\bar { N } M } $",
"$ g ^ { L M } $",
"$ \\nabla _ { M } $",
"$ D _ { M } ^ { 0 } = \\nabla _ { M } + i e A _ { M } ^ { 0 } $",
"$ J _ { 0 } = \\int \\vert \\widetilde { \\phi } ^ { 0 } \\vert ^ { 2 } d x ^ { 5 } d x ^ { 6 } $",
"$ L _ { 2 } $",
"$ \\vert \\phi ^ { 0 } \\vert ^ { 2 } $",
"$ g _ { m \\mu } = O ( R \\slash a ) < < 1 $",
"$ g _ { m n } = - \\delta _ { m n } + O ( R \\slash a ) $",
"$ B _ { \\bar { M } } $",
"$ B _ { \\bar { \\mu } } $",
"$ B _ { \\bar { m } } $",
"$ S ^ { e f f } $",
"$ \\delta $",
"$ \\delta = \\int d k e ^ { i k x } $",
"\\begin{equation*} 1 = \\int \\prod _ { X _ { \\slash \\! \\slash } } d Y ^ { M } ( \\xi ^ { \\mu } ) \\delta ( Y ^ { M } ( \\xi ^ { \\mu } ) - C ^ { M } ( \\xi ^ { \\mu } ) ) \\end{equation*}",
"\\begin{equation*} C ^ { M } ( \\xi ^ { \\mu } ) = \\int _ { N ( \\xi ^ { \\mu } ) } X ^ { M } \\vert \\widetilde { \\phi } \\vert ^ { 2 } d ^ { 2 } X _ { \\perp } \\slash \\int _ { N ( \\xi ^ { \\mu } ) } \\vert \\widetilde { \\phi } \\vert ^ { 2 } d ^ { 2 } X _ { \\perp } \\end{equation*}",
"\\begin{equation*} S _ { f i } = \\int \\prod _ { X _ { \\slash \\! \\slash } } d Y ^ { M } \\prod _ { X ^ { M } } d B _ { \\bar { N } } d \\sigma d \\sigma ^ { \\dagger } \\delta ( \\sqrt { - g } \\nabla _ { \\bar { N } } B ^ { \\bar { N } } \\prod _ { X _ { \\slash \\! \\slash } } \\delta ( \\widetilde { C } ^ { M } ) \\operatorname { e x p } [ i \\int ( L _ { 0 } + L _ { 1 } ) \\sqrt { - g } d ^ { 6 } x ] \\Psi _ { f } ^ { \\ast } \\Psi _ { i } \\end{equation*}",
"\\begin{align*} L _ { 0 } & = & L ( \\phi = \\phi _ { 0 } , A _ { M } = A _ { M } ^ { 0 } ) \\\\ L _ { 2 } & = & - \\frac { 1 } { 2 } g ^ { L M } \\nabla _ { L } B _ { \\bar { N } } \\nabla _ { M } B ^ { \\bar { N } } + B _ { \\bar { N } } B ^ { \\bar { N } } e ^ { 2 } \\vert \\phi ^ { 0 } \\vert ^ { 2 } + g ^ { L M } ( D _ { L } ^ { 0 } \\sigma ) ^ { \\dagger } ( D _ { M } ^ { 0 } \\sigma ) \\\\ & & - 4 i e V ^ { \\bar { N } M } B _ { \\bar { N } } I m ( \\sigma ^ { \\dagger } D _ { M } ^ { 0 } \\phi ^ { 0 } ) + a \\vert \\sigma \\vert ^ { 2 } - b [ 4 \\vert \\phi ^ { 0 } \\sigma \\vert ^ { 2 } + 2 R e ( \\sigma ^ { \\dagger } \\phi ^ { 0 } ) ^ { 2 } ] , \\\\ \\widetilde { C } ^ { m } & = & \\int x ^ { m } \\vert \\widetilde { \\phi } \\vert ^ { 2 } d x ^ { 5 } d x ^ { 6 } \\slash \\int \\vert \\widetilde { \\phi } \\vert ^ { 2 } d x ^ { 5 } d x ^ { 6 } \\\\ & = & \\frac { 1 } { J _ { 0 } } \\int x ^ { m } [ \\vert \\sigma \\vert ^ { 2 } + R e ( \\widetilde { \\phi } ^ { 0 } \\sigma ^ { \\dagger } ) \\{ 1 - \\frac { 2 } { J _ { 0 } } \\int R e ( \\widetilde { \\phi } ^ { 0 } \\sigma ^ { \\dagger } ) d x ^ { 5 } d x ^ { 6 } \\} ] d x ^ { 5 } d x ^ { 6 } , \\end{align*}",
"\\begin{equation*} S ^ { e f f } = - i \\operatorname { l n } \\int \\prod _ { X ^ { M } } d B _ { \\bar { N } } d \\sigma d \\sigma ^ { \\dagger } \\delta ( \\sqrt { - g } \\nabla _ { \\bar { N } } B ^ { \\bar { N } } ) \\prod _ { X _ { \\slash \\! \\slash } } \\delta ( \\widetilde { C } ^ { M } ) \\operatorname { e x p } [ i \\int \\sqrt { - g } L _ { 2 } d ^ { 6 } x ] . \\end{equation*}",
"\\begin{equation*} S ^ { e f f } = - i \\operatorname { l n } \\int \\prod _ { \\xi ^ { \\mu } } d w _ { m } \\prod _ { x ^ { M } } d B _ { \\bar { M } } d \\sigma d \\sigma ^ { \\dagger } d v \\operatorname { e x p } [ i \\int ( \\Xi \\Phi + \\Phi ^ { \\dagger } \\Delta \\Phi ) d ^ { 6 } x ] \\end{equation*}"
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"$ \\mitC ^ { \\mitM } ( \\mitxi ^ { \\mitmu } ) $",
"$ \\vert \\widetilde { \\mitphi } \\vert ^ { 2 } $",
"$ ( \\widetilde { \\mitphi } = \\mitphi - \\sqrt { \\mita \\slash 2 \\mitb } ) $",
"$ \\mitN ( \\mitxi ^ { \\mitmu } ) $",
"$ \\mitx ^ { \\mitmu } = \\mitxi ^ { \\mitmu } $",
"$ \\prod _ { \\mitX _ { \\slash \\! \\slash } } $",
"$ \\mitxi ^ { \\mitmu } $",
"$ \\mitx ^ { \\mitM } $",
"$ \\mitA _ { \\mitM } $",
"$ \\mitphi $",
"$ \\mitB _ { \\bar { \\mitN } } = \\mitA _ { \\bar { \\mitN } } - \\mitA _ { \\bar { \\mitN } } ^ { 0 } $",
"$ \\mitsigma = \\mitphi - \\mitphi ^ { 0 } $",
"$ \\widetilde { \\mitC } ^ { \\mitmu } = 0 $",
"$ \\mitV ^ { \\bar { \\mitN } \\mitM } $",
"$ \\mitg ^ { \\mitL \\mitM } $",
"$ \\nabla _ { \\mitM } $",
"$ \\mitD _ { \\mitM } ^ { 0 } = \\nabla _ { \\mitM } + \\miti \\mite \\mitA _ { \\mitM } ^ { 0 } $",
"$ \\mitJ _ { 0 } = \\int \\nolimits \\vert \\widetilde { \\mitphi } ^ { 0 } \\vert ^ { 2 } \\mitd \\mitx ^ { 5 } \\mitd \\mitx ^ { 6 } $",
"$ \\mitL _ { 2 } $",
"$ \\vert \\mitphi ^ { 0 } \\vert ^ { 2 } $",
"$ \\mitg _ { \\mitm \\mitmu } = \\mitO ( \\mitR \\slash \\mita ) < < 1 $",
"$ \\mitg _ { \\mitm \\mitn } = - \\mitdelta _ { \\mitm \\mitn } + \\mitO ( \\mitR \\slash \\mita ) $",
"$ \\mitB _ { \\bar { \\mitM } } $",
"$ \\mitB _ { \\bar { \\mitmu } } $",
"$ \\mitB _ { \\bar { \\mitm } } $",
"$ \\mitS ^ { \\mathrm { e f f } } $",
"$ \\mitdelta $",
"$ \\mitdelta = \\int \\nolimits \\mitd \\mitk \\mite ^ { \\miti \\mitk \\mitx } $",
"\\begin{equation*} 1 = \\int \\prod _ { \\mitX _ { \\slash \\! \\slash } } \\mitd \\mitY ^ { \\mitM } ( \\mitxi ^ { \\mitmu } ) \\mitdelta \\left( \\mitY ^ { \\mitM } ( \\mitxi ^ { \\mitmu } ) - \\mitC ^ { \\mitM } ( \\mitxi ^ { \\mitmu } ) \\right) \\end{equation*}",
"\\begin{equation*} \\mitC ^ { \\mitM } ( \\mitxi ^ { \\mitmu } ) = \\int _ { \\mitN ( \\mitxi ^ { \\mitmu } ) } \\mitX ^ { \\mitM } \\vert \\widetilde { \\mitphi } \\vert ^ { 2 } \\mitd ^ { 2 } \\mitX _ { \\perp } \\Biggl / \\int _ { \\mitN ( \\mitxi ^ { \\mitmu } ) } \\vert \\widetilde { \\mitphi } \\vert ^ { 2 } \\mitd ^ { 2 } \\mitX _ { \\perp } \\end{equation*}",
"\\begin{equation*} \\mitS _ { \\mitf \\miti } = \\int \\prod _ { \\mitX _ { \\slash \\! \\slash } } \\mitd \\mitY ^ { \\mitM } \\prod _ { \\mitX ^ { \\mitM } } \\mitd \\mitB _ { \\bar { \\mitN } } \\mitd \\mitsigma \\mitd \\mitsigma ^ { \\dagger } \\mitdelta ( \\sqrt { - \\mitg } \\nabla _ { \\bar { \\mitN } } \\mitB ^ { \\bar { \\mitN } } \\prod _ { \\mitX _ { \\slash \\! \\slash } } \\mitdelta ( \\widetilde { \\mitC } ^ { \\mitM } ) \\operatorname { e x p } \\left[ \\miti \\int \\left( \\mitL _ { 0 } + \\mitL _ { 1 } \\right) \\sqrt { - \\mitg } \\mitd ^ { 6 } \\mitx \\right] \\mupPsi _ { \\mitf } ^ { \\ast } \\mupPsi _ { \\miti } \\end{equation*}",
"\\begin{align*} \\mitL _ { 0 } & = & \\mitL ( \\mitphi = \\mitphi _ { 0 } , \\mitA _ { \\mitM } = \\mitA _ { \\mitM } ^ { 0 } ) \\\\ \\mitL _ { 2 } & = & - \\frac { 1 } { 2 } \\mitg ^ { \\mitL \\mitM } \\nabla _ { \\mitL } \\mitB _ { \\bar { \\mitN } } \\nabla _ { \\mitM } \\mitB ^ { \\bar { \\mitN } } + \\mitB _ { \\bar { \\mitN } } \\mitB ^ { \\bar { \\mitN } } \\mite ^ { 2 } \\vert \\mitphi ^ { 0 } \\vert ^ { 2 } + \\mitg ^ { \\mitL \\mitM } ( \\mitD _ { \\mitL } ^ { 0 } \\mitsigma ) ^ { \\dagger } ( \\mitD _ { \\mitM } ^ { 0 } \\mitsigma ) \\\\ & & - 4 \\miti \\mite \\mitV ^ { \\bar { \\mitN } \\mitM } \\mitB _ { \\bar { \\mitN } } \\mathrm { I m } \\left( \\mitsigma ^ { \\dagger } \\mitD _ { \\mitM } ^ { 0 } \\mitphi ^ { 0 } \\right) + \\mita \\vert \\mitsigma \\vert ^ { 2 } - \\mitb \\left[ 4 \\vert \\mitphi ^ { 0 } \\mitsigma \\vert ^ { 2 } + 2 \\mathrm { R e } ( \\mitsigma ^ { \\dagger } \\mitphi ^ { 0 } ) ^ { 2 } \\right] , \\\\ \\widetilde { \\mitC } ^ { \\mitm } & = & \\int \\mitx ^ { \\mitm } \\vert \\widetilde { \\mitphi } \\vert ^ { 2 } \\mitd \\mitx ^ { 5 } \\mitd \\mitx ^ { 6 } \\Biggl / \\int \\vert \\widetilde { \\mitphi } \\vert ^ { 2 } \\mitd \\mitx ^ { 5 } \\mitd \\mitx ^ { 6 } \\\\ & = & \\frac { 1 } { \\mitJ _ { 0 } } \\int \\mitx ^ { \\mitm } \\left[ \\vert \\mitsigma \\vert ^ { 2 } + \\mathrm { R e } ( \\widetilde { \\mitphi } ^ { 0 } \\mitsigma ^ { \\dagger } ) \\left\\{ 1 - \\frac { 2 } { \\mitJ _ { 0 } } \\int \\mathrm { R e } ( \\widetilde { \\mitphi } ^ { 0 } \\mitsigma ^ { \\dagger } ) \\mitd \\mitx ^ { 5 } \\mitd \\mitx ^ { 6 } \\right\\} \\right] \\mitd \\mitx ^ { 5 } \\mitd \\mitx ^ { 6 } , \\end{align*}",
"\\begin{equation*} \\mitS ^ { \\mathrm { e f f } } = - \\miti \\operatorname { l n } \\int \\prod _ { \\mitX ^ { \\mitM } } \\mitd \\mitB _ { \\bar { \\mitN } } \\mitd \\mitsigma \\mitd \\mitsigma ^ { \\dagger } \\mitdelta \\left( \\sqrt { - \\mitg } \\nabla _ { \\bar { \\mitN } } \\mitB ^ { \\bar { \\mitN } } \\right) \\prod _ { \\mitX _ { \\slash \\! \\slash } } \\mitdelta ( \\widetilde { \\mitC } ^ { \\mitM } ) \\operatorname { e x p } \\left[ \\miti \\int \\sqrt { - \\mitg } \\mitL _ { 2 } \\mitd ^ { 6 } \\mitx \\right] . \\end{equation*}",
"\\begin{equation*} \\mitS ^ { \\mathrm { e f f } } = - \\miti \\operatorname { l n } \\int \\prod _ { \\mitxi ^ { \\mitmu } } \\mitd \\mitw _ { \\mitm } \\prod _ { \\mitx ^ { \\mitM } } \\mitd \\mitB _ { \\bar { \\mitM } } \\mitd \\mitsigma \\mitd \\mitsigma ^ { \\dagger } \\mitd \\mitv \\operatorname { e x p } \\left[ \\miti \\int ( \\mupXi \\mupPhi + \\mupPhi ^ { \\dagger } \\mupDelta \\mupPhi ) \\mitd ^ { 6 } \\mitx \\right] \\end{equation*}"
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0001113_page04 | {
"latex": [
"$\\delta ^m$",
"$B_{\\bar N}$",
"$\\sigma $",
"$\\sigma ^\\dagger $",
"$v$",
"$\\Xi _0=\\Xi |_{v=0}$",
"$S^{\\rm eff}$",
"$h^{MN}=g^{MN}-\\eta ^{MN}$",
"$\\eta ^{MN}={\\rm diag}(1,-1,-1,-1,-1,-1)$",
"$w$",
"$\\Delta |_{h^{MN}=0,w=0}\\equiv \\Delta _0$",
"$\\Delta _0$",
"$\\Delta _0^{\\rm sp}$",
"$\\Delta _0^{\\rm ex}$",
"$x^\\mu $",
"$x^m$",
"$\\Delta _0$",
"$\\Delta _0^{\\rm V}$",
"$\\Delta _0^{\\rm S}$",
"$B^\\mu $",
"$(S^{(1)},S^{(2)},S^{(3)},S^{(4)})=(B^5,B^6,\\sigma ,\\sigma ^\\dagger )$",
"$\\square \\,= \\eta ^{\\mu \\nu }\\partial _\\mu \\partial _\\nu $",
"$V_k$",
"$S_k^{(0)}$",
"${m_k}^2$",
"${m'_k}^2$",
"$\\Delta _{\\rm int}$",
"$\\Delta '_{\\rm int}$",
"$h^{\\mu \\nu }$",
"$w$",
"\\begin {eqnarray} &&\\hskip -5mm \\Phi ^\\dagger =(B^{\\bar M},\\sigma ,\\sigma ^\\dagger ), \\\\ &&\\hskip -5mm \\Xi =\\sqrt {-g}(\\nabla _{\\bar M}v, \\ w_m x^m \\widetilde \\phi ^{0\\dagger }/J_0, \\ w_m x^m \\widetilde \\phi ^0/J_0), \\\\ &&\\hskip -5mm \\Delta =\\sqrt {-g} \\\\ &&\\hskip-5mm \\times \\begin{pmatrix} \\hskip-1mm \\eta _{\\bar M \\bar N}\\left (\\frac {1}{2}\\nabla _L\\nabla ^L+e^2|\\phi ^0|^2\\right ) & ieD_{\\bar M}^0\\phi ^{0\\dagger } & -ieD_{\\bar M}^0\\phi ^{0} \\cr -ieD_{\\bar N}^0\\phi ^{0} & \\hskip -3mm\\frac {1}{2}D_L^0 D^{0L} +\\frac {a}{2}-2b|\\phi ^0|^2+\\delta _{11}^mw_m & -b(\\phi ^0)^2+\\delta _{12}^m w_m \\cr ieD_{\\bar N}^0\\phi ^{0\\dagger } & -b(\\phi ^0\\dagger )^2+\\delta _{21}^m w_m & \\hskip -3mm\\frac {1}{2}D_L^0 D^{0L} +\\frac {a}{2}-2b|\\phi ^0|^2+\\delta _{22}^mw_m \\end{pmatrix} \\end {eqnarray}",
"\\begin {equation} \\delta ^m (x,x') = \\frac {1}{2J_0}x^m\\delta (x-x') +\\frac {1}{2{J_0}^2}(x^m+x'^m) \\begin{pmatrix} \\widetilde \\phi ^0(x)\\cr \\widetilde \\phi ^0(x)^\\dagger \\end{pmatrix} \\begin{pmatrix} \\widetilde \\phi ^0(x')^\\dagger &\\widetilde \\phi ^0(x') \\end{pmatrix} \\label {19} \\end {equation}",
"\\begin {equation} S^{\\rm eff} = \\frac {1}{2}i{\\rm Tr}\\ln \\Delta +\\frac {1}{2}i{\\rm Tr}\\ln \\left [\\partial _M\\sqrt {-g}(\\Delta ^{-1})^{MN}\\sqrt {-g}\\partial _N\\right ] -\\frac {1}{4}\\int \\Xi _0^\\dagger \\Delta ^{-1}\\Xi _0d^6x \\label {20} \\end {equation}",
"\\begin {eqnarray} &&\\hskip -13mm \\Delta _0^{\\rm V,sp}=\\frac {1}{2}\\ \\square \\,, \\ \\ \\^^M\\Delta _0^{\\rm S,sp}=\\frac {1}{2}\\ \\square \\,, \\ \\ \\^^M\\Delta _0^{\\rm V,ex}=-\\frac {1}{2}\\partial _l \\partial _l+e^2|\\phi ^0|^2, \\cr &&\\hskip -13mm \\Delta _0^{\\rm S,ex} =\\begin{pmatrix} \\left (-\\frac {1}{2}\\partial _l\\partial _l+e^2|\\phi ^0|^2\\right )\\eta _{mn} & ieD_{n}^0\\phi ^{0\\dagger } & -ieD_{n}^0\\phi ^{0} \\cr -ieD_{m}^0\\phi ^{0} & -\\frac {1}{2}D_l^0 D_l^0 +\\frac {a}{2}-b|\\phi ^0|^2 & -b(\\phi ^0)^2 \\cr ieD_{m}^0\\phi ^{0\\dagger } & -b(\\phi ^{0\\dagger })^2 & -\\frac {1}{2}D_l^0 D_l^0 +\\frac {a}{2}-b|\\phi ^0|^2 \\end{pmatrix} \\end {eqnarray}",
"\\begin {eqnarray} \\left [({\\Delta _0^{\\rm V}})^{-1}\\right ]^{\\mu \\nu } &=&\\eta ^{\\mu \\nu }\\sum _{k(}\\,\\square \\,+{m_k}^2)^{-1}V_k(x^m)V_k(x'^m), \\cr \\left [({\\Delta _0^{\\rm V}})^{-1}\\right ]^{\\mu \\nu } &=&\\sum _{k(}\\,\\square \\,+{m'_k}^2)^{-1}S_k^{(a)}(x^m)S_k^{(b)}(x'^m), \\end {eqnarray}",
"\\begin {equation} \\Delta _0^{\\rm V,ex} V_k = {m_k}^2 V_k, \\ \\ \\ \\^^M\\Delta _0^{{\\rm S,ex}(a)(b)} S_k^{(b)} = {m'_k}^2 S_k^{(a)}. \\label {23} \\end {equation}",
"\\begin {eqnarray} &&\\hskip -10mm \\Delta =\\Delta _0(1+\\Delta _0^{-1}\\Delta _{\\rm int}), \\\\ &&\\hskip -10mm \\partial _M\\sqrt {-g}(\\Delta ^{-1})^{MN}\\sqrt {-g}\\partial _N =1+{\\Delta '_0}^{-1} +\\partial _m(\\Delta _0^{-1})^{mn}\\partial _n+\\Delta '_{\\rm int}, \\end {eqnarray}",
"\\begin {equation} {\\Delta '_0}^{-1}=\\sum _k {m_k}^2(\\,\\square \\,+{m_k}^2)^{-1}V_k(x^m)V_k(x'^m). \\label {26} \\end {equation}"
],
"latex_norm": [
"$ \\delta ^ { m } $",
"$ B _ { \\bar { N } } $",
"$ \\sigma $",
"$ \\sigma ^ { \\dagger } $",
"$ v $",
"$ \\Xi _ { 0 } = \\Xi \\vert _ { v = 0 } $",
"$ S ^ { e f f } $",
"$ h ^ { M N } = g ^ { M N } - \\eta ^ { M N } $",
"$ \\eta ^ { M N } = d i a g ( 1 , - 1 , - 1 , - 1 , - 1 , - 1 ) $",
"$ w $",
"$ \\Delta \\vert _ { h ^ { M N } = 0 , w = 0 } \\equiv \\Delta _ { 0 } $",
"$ \\Delta _ { 0 } $",
"$ \\Delta _ { 0 } ^ { s p } $",
"$ \\Delta _ { 0 } ^ { e x } $",
"$ x ^ { \\mu } $",
"$ x ^ { m } $",
"$ \\Delta _ { 0 } $",
"$ \\Delta _ { 0 } ^ { V } $",
"$ \\Delta _ { 0 } ^ { S } $",
"$ B ^ { \\mu } $",
"$ ( S ^ { ( 1 ) } , S ^ { ( 2 ) } , S ^ { ( 3 ) } , S ^ { ( 4 ) } ) = ( B ^ { 5 } , B ^ { 6 } , \\sigma , \\sigma ^ { \\dagger } ) $",
"$ \\square \\, = \\eta ^ { \\mu \\nu } \\partial _ { \\mu } \\partial _ { \\nu } $",
"$ V _ { k } $",
"$ S _ { k } ^ { ( 0 ) } $",
"$ { m _ { k } } ^ { 2 } $",
"$ { m _ { k } ^ { \\prime } } ^ { 2 } $",
"$ \\Delta _ { i n t } $",
"$ \\Delta _ { i n t } ^ { \\prime } $",
"$ h ^ { \\mu \\nu } $",
"$ w $",
"\\begin{align*} & & \\hspace{-14.23pt} \\Phi ^ { \\dagger } = ( B ^ { \\bar { M } } , \\sigma , \\sigma ^ { \\dagger } ) , \\\\ & & \\hspace{-14.23pt} \\Xi = \\sqrt { - g } ( \\nabla _ { \\bar { M } } v , ~ w _ { m } x ^ { m } \\widetilde { \\phi } ^ { 0 \\dagger } \\slash J _ { 0 } , ~ w _ { m } x ^ { m } \\widetilde { \\phi } ^ { 0 } \\slash J _ { 0 } ) , \\\\ & & \\hspace{-14.23pt} \\Delta = \\sqrt { - g } \\\\ \\hspace{-14.23pt} \\times ( \\begin{array}{ccc} \\hspace{-2.85pt} \\eta _ { \\bar { M } \\bar { N } } ( \\frac { 1 } { 2 } \\nabla _ { L } \\nabla ^ { L } + e ^ { 2 } \\vert \\phi ^ { 0 } \\vert ^ { 2 } ) & i e D _ { \\bar { M } } ^ { 0 } \\phi ^ { 0 \\dagger } & - i e D _ { \\bar { M } } ^ { 0 } \\phi ^ { 0 } \\\\ - i e D _ { \\bar { N } } ^ { 0 } \\phi ^ { 0 } & \\hspace{-8.54pt} \\frac { 1 } { 2 } D _ { L } ^ { 0 } D ^ { 0 L } + \\frac { a } { 2 } - 2 b \\vert \\phi ^ { 0 } \\vert ^ { 2 } + \\delta _ { 1 1 } ^ { m } w _ { m } & - b ( \\phi ^ { 0 } ) ^ { 2 } + \\delta _ { 1 2 } ^ { m } w _ { m } \\\\ i e D _ { \\bar { N } } ^ { 0 } \\phi ^ { 0 \\dagger } & - b ( \\phi ^ { 0 } \\dagger ) ^ { 2 } + \\delta _ { 2 1 } ^ { m } w _ { m } & \\hspace{-8.54pt} \\frac { 1 } { 2 } D _ { L } ^ { 0 } D ^ { 0 L } + \\frac { a } { 2 } - 2 b \\vert \\phi ^ { 0 } \\vert ^ { 2 } + \\delta _ { 2 2 } ^ { m } w _ { m } \\end{array} ) \\end{align*}",
"\\begin{align*} \\delta ^ { m } ( x , x ^ { \\prime } ) = \\frac { 1 } { 2 J _ { 0 } } x ^ { m } \\delta ( x - x ^ { \\prime } ) + \\frac { 1 } { 2 { J _ { 0 } } ^ { 2 } } ( x ^ { m } + x ^ { \\prime m } ) ( \\begin{array}{c} \\widetilde { \\phi } ^ { 0 } ( x ) \\\\ \\widetilde { \\phi } ^ { 0 } ( x ) ^ { \\dagger } \\end{array} ) ( \\begin{array}{cc} \\widetilde { \\phi } ^ { 0 } ( x ^ { \\prime } ) ^ { \\dagger } & \\widetilde { \\phi } ^ { 0 } ( x ^ { \\prime } ) \\end{array} ) \\end{align*}",
"\\begin{equation*} S ^ { e f f } = \\frac { 1 } { 2 } i T r \\operatorname { l n } \\Delta + \\frac { 1 } { 2 } i T r \\operatorname { l n } [ \\partial _ { M } \\sqrt { - g } ( \\Delta ^ { - 1 } ) ^ { M N } \\sqrt { - g } \\partial _ { N } ] - \\frac { 1 } { 4 } \\int \\Xi _ { 0 } ^ { \\dagger } \\Delta ^ { - 1 } \\Xi _ { 0 } d ^ { 6 } x \\end{equation*}",
"\\begin{align*} & \\hspace{-36.99pt} \\Delta _ { 0 } ^ { V , s p } = \\frac { 1 } { 2 } ~ \\square \\, , ~ ~ ~ \\Delta _ { 0 } ^ { S , s p } = \\frac { 1 } { 2 } ~ \\square \\, , ~ ~ ~ \\Delta _ { 0 } ^ { V , e x } = - \\frac { 1 } { 2 } \\partial _ { l } \\partial _ { l } + e ^ { 2 } \\vert \\phi ^ { 0 } \\vert ^ { 2 } , \\\\ \\hspace{-36.99pt} \\Delta _ { 0 } ^ { S , e x } = ( \\begin{array}{ccc} & i e D _ { n } ^ { 0 } \\phi ^ { 0 \\dagger } & - i e D _ { n } ^ { 0 } \\phi ^ { 0 } \\\\ - i e D _ { m } ^ { 0 } \\phi ^ { 0 } & - \\frac { 1 } { 2 } D _ { l } ^ { 0 } D _ { l } ^ { 0 } + \\frac { a } { 2 } - b \\vert \\phi ^ { 0 } \\vert ^ { 2 } & - b ( \\phi ^ { 0 } ) ^ { 2 } \\\\ i e D _ { m } ^ { 0 } \\phi ^ { 0 \\dagger } & - b ( \\phi ^ { 0 \\dagger } ) ^ { 2 } & - \\frac { 1 } { 2 } D _ { l } ^ { 0 } D _ { l } ^ { 0 } + \\frac { a } { 2 } - b \\vert \\phi ^ { 0 } \\vert ^ { 2 } \\end{array} ) \\end{align*}",
"\\begin{align*} { [ ( \\Delta _ { 0 } ^ { V } ) ^ { - 1 } ] } ^ { \\mu \\nu } & = & \\eta ^ { \\mu \\nu } \\sum _ { k ( } \\, \\square \\, + { m _ { k } } ^ { 2 } ) ^ { - 1 } V _ { k } ( x ^ { m } ) V _ { k } ( x ^ { \\prime m } ) , \\\\ { [ ( \\Delta _ { 0 } ^ { V } ) ^ { - 1 } ] } ^ { \\mu \\nu } & = & \\sum _ { k ( } \\, \\square \\, + { m _ { k } ^ { \\prime } } ^ { 2 } ) ^ { - 1 } S _ { k } ^ { ( a ) } ( x ^ { m } ) S _ { k } ^ { ( b ) } ( x ^ { \\prime m } ) , \\end{align*}",
"\\begin{equation*} \\Delta _ { 0 } ^ { V , e x } V _ { k } = { m _ { k } } ^ { 2 } V _ { k } , ~ ~ ~ ~ \\Delta _ { 0 } ^ { S , e x ( a ) ( b ) } S _ { k } ^ { ( b ) } = { m _ { k } ^ { \\prime } } ^ { 2 } S _ { k } ^ { ( a ) } . \\end{equation*}",
"\\begin{align*} & & \\hspace{-28.45pt} \\Delta = \\Delta _ { 0 } ( 1 + \\Delta _ { 0 } ^ { - 1 } \\Delta _ { i n t } ) , \\\\ & & \\hspace{-28.45pt} \\partial _ { M } \\sqrt { - g } ( \\Delta ^ { - 1 } ) ^ { M N } \\sqrt { - g } \\partial _ { N } = 1 + { \\Delta _ { 0 } ^ { \\prime } } ^ { - 1 } + \\partial _ { m } ( \\Delta _ { 0 } ^ { - 1 } ) ^ { m n } \\partial _ { n } + \\Delta _ { i n t } ^ { \\prime } , \\end{align*}",
"\\begin{equation*} { \\Delta _ { 0 } ^ { \\prime } } ^ { - 1 } = \\sum _ { k } { m _ { k } } ^ { 2 } ( \\, \\square \\, + { m _ { k } } ^ { 2 } ) ^ { - 1 } V _ { k } ( x ^ { m } ) V _ { k } ( x ^ { \\prime m } ) . \\end{equation*}"
],
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"$ \\mitdelta ^ { \\mitm } $",
"$ \\mitB _ { \\bar { \\mitN } } $",
"$ \\mitsigma $",
"$ \\mitsigma ^ { \\dagger } $",
"$ \\mitv $",
"$ \\mupXi _ { 0 } = \\mupXi \\vert _ { \\mitv = 0 } $",
"$ \\mitS ^ { \\mathrm { e f f } } $",
"$ \\Planckconst ^ { \\mitM \\mitN } = \\mitg ^ { \\mitM \\mitN } - \\miteta ^ { \\mitM \\mitN } $",
"$ \\miteta ^ { \\mitM \\mitN } = \\mathrm { d i a g } ( 1 , - 1 , - 1 , - 1 , - 1 , - 1 ) $",
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"$ \\mupDelta _ { 0 } ^ { \\mathrm { V } } $",
"$ \\mupDelta _ { 0 } ^ { \\mathrm { S } } $",
"$ \\mitB ^ { \\mitmu } $",
"$ ( \\mitS ^ { ( 1 ) } , \\mitS ^ { ( 2 ) } , \\mitS ^ { ( 3 ) } , \\mitS ^ { ( 4 ) } ) = ( \\mitB ^ { 5 } , \\mitB ^ { 6 } , \\mitsigma , \\mitsigma ^ { \\dagger } ) $",
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"$ \\mitS _ { \\mitk } ^ { ( 0 ) } $",
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"$ \\mupDelta _ { \\mathrm { i n t } } ^ { \\prime } $",
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"\\begin{align*} & & \\displaystyle \\hspace{-14.23pt} \\mupPhi ^ { \\dagger } = ( \\mitB ^ { \\bar { \\mitM } } , \\mitsigma , \\mitsigma ^ { \\dagger } ) , \\\\ & & \\displaystyle \\hspace{-14.23pt} \\mupXi = \\sqrt { - \\mitg } ( \\nabla _ { \\bar { \\mitM } } \\mitv , ~ \\mitw _ { \\mitm } \\mitx ^ { \\mitm } \\widetilde { \\mitphi } ^ { 0 \\dagger } \\slash \\mitJ _ { 0 } , ~ \\mitw _ { \\mitm } \\mitx ^ { \\mitm } \\widetilde { \\mitphi } ^ { 0 } \\slash \\mitJ _ { 0 } ) , \\\\ & & \\displaystyle \\hspace{-14.23pt} \\mupDelta = \\sqrt { - \\mitg } \\\\ \\displaystyle \\hspace{-14.23pt} \\times \\left( \\begin{array}{ccc} \\hspace{-2.85pt} \\miteta _ { \\bar { \\mitM } \\bar { \\mitN } } \\left( \\frac { 1 } { 2 } \\nabla _ { \\mitL } \\nabla ^ { \\mitL } + \\mite ^ { 2 } \\vert \\mitphi ^ { 0 } \\vert ^ { 2 } \\right) & \\miti \\mite \\mitD _ { \\bar { \\mitM } } ^ { 0 } \\mitphi ^ { 0 \\dagger } & - \\miti \\mite \\mitD _ { \\bar { \\mitM } } ^ { 0 } \\mitphi ^ { 0 } \\\\ - \\miti \\mite \\mitD _ { \\bar { \\mitN } } ^ { 0 } \\mitphi ^ { 0 } & \\hspace{-8.54pt} \\frac { 1 } { 2 } \\mitD _ { \\mitL } ^ { 0 } \\mitD ^ { 0 \\mitL } + \\frac { \\mita } { 2 } - 2 \\mitb \\vert \\mitphi ^ { 0 } \\vert ^ { 2 } + \\mitdelta _ { 1 1 } ^ { \\mitm } \\mitw _ { \\mitm } & - \\mitb ( \\mitphi ^ { 0 } ) ^ { 2 } + \\mitdelta _ { 1 2 } ^ { \\mitm } \\mitw _ { \\mitm } \\\\ \\miti \\mite \\mitD _ { \\bar { \\mitN } } ^ { 0 } \\mitphi ^ { 0 \\dagger } & - \\mitb ( \\mitphi ^ { 0 } \\dagger ) ^ { 2 } + \\mitdelta _ { 2 1 } ^ { \\mitm } \\mitw _ { \\mitm } & \\hspace{-8.54pt} \\frac { 1 } { 2 } \\mitD _ { \\mitL } ^ { 0 } \\mitD ^ { 0 \\mitL } + \\frac { \\mita } { 2 } - 2 \\mitb \\vert \\mitphi ^ { 0 } \\vert ^ { 2 } + \\mitdelta _ { 2 2 } ^ { \\mitm } \\mitw _ { \\mitm } \\end{array} \\right) \\end{align*}",
"\\begin{align*} \\displaystyle \\mitdelta ^ { \\mitm } ( \\mitx , \\mitx ^ { \\prime } ) = \\frac { 1 } { 2 \\mitJ _ { 0 } } \\mitx ^ { \\mitm } \\mitdelta ( \\mitx - \\mitx ^ { \\prime } ) + \\frac { 1 } { 2 { \\mitJ _ { 0 } } ^ { 2 } } ( \\mitx ^ { \\mitm } + \\mitx ^ { \\prime \\mitm } ) \\left( \\begin{array}{c} \\widetilde { \\mitphi } ^ { 0 } ( \\mitx ) \\\\ \\widetilde { \\mitphi } ^ { 0 } ( \\mitx ) ^ { \\dagger } \\end{array} \\right) \\left( \\begin{array}{cc} \\widetilde { \\mitphi } ^ { 0 } ( \\mitx ^ { \\prime } ) ^ { \\dagger } & \\widetilde { \\mitphi } ^ { 0 } ( \\mitx ^ { \\prime } ) \\end{array} \\right) \\end{align*}",
"\\begin{equation*} \\mitS ^ { \\mathrm { e f f } } = \\frac { 1 } { 2 } \\miti \\mathrm { T r } \\operatorname { l n } \\mupDelta + \\frac { 1 } { 2 } \\miti \\mathrm { T r } \\operatorname { l n } \\left[ \\mitpartial _ { \\mitM } \\sqrt { - \\mitg } ( \\mupDelta ^ { - 1 } ) ^ { \\mitM \\mitN } \\sqrt { - \\mitg } \\mitpartial _ { \\mitN } \\right] - \\frac { 1 } { 4 } \\int \\mupXi _ { 0 } ^ { \\dagger } \\mupDelta ^ { - 1 } \\mupXi _ { 0 } \\mitd ^ { 6 } \\mitx \\end{equation*}",
"\\begin{align*} & \\displaystyle \\hspace{-36.99pt} \\mupDelta _ { 0 } ^ { \\mathrm { V } , \\mathrm { s p } } = \\frac { 1 } { 2 } ~ \\square \\, , ~ ~ ~ \\mupDelta _ { 0 } ^ { \\mathrm { S } , \\mathrm { s p } } = \\frac { 1 } { 2 } ~ \\square \\, , ~ ~ ~ \\mupDelta _ { 0 } ^ { \\mathrm { V } , \\mathrm { e x } } = - \\frac { 1 } { 2 } \\mitpartial _ { \\mitl } \\mitpartial _ { \\mitl } + \\mite ^ { 2 } \\vert \\mitphi ^ { 0 } \\vert ^ { 2 } , \\\\ \\displaystyle \\hspace{-36.99pt} \\mupDelta _ { 0 } ^ { \\mathrm { S } , \\mathrm { e x } } = \\left( \\begin{array}{ccc} & \\miti \\mite \\mitD _ { \\mitn } ^ { 0 } \\mitphi ^ { 0 \\dagger } & - \\miti \\mite \\mitD _ { \\mitn } ^ { 0 } \\mitphi ^ { 0 } \\\\ - \\miti \\mite \\mitD _ { \\mitm } ^ { 0 } \\mitphi ^ { 0 } & - \\frac { 1 } { 2 } \\mitD _ { \\mitl } ^ { 0 } \\mitD _ { \\mitl } ^ { 0 } + \\frac { \\mita } { 2 } - \\mitb \\vert \\mitphi ^ { 0 } \\vert ^ { 2 } & - \\mitb ( \\mitphi ^ { 0 } ) ^ { 2 } \\\\ \\miti \\mite \\mitD _ { \\mitm } ^ { 0 } \\mitphi ^ { 0 \\dagger } & - \\mitb ( \\mitphi ^ { 0 \\dagger } ) ^ { 2 } & - \\frac { 1 } { 2 } \\mitD _ { \\mitl } ^ { 0 } \\mitD _ { \\mitl } ^ { 0 } + \\frac { \\mita } { 2 } - \\mitb \\vert \\mitphi ^ { 0 } \\vert ^ { 2 } \\end{array} \\right) \\end{align*}",
"\\begin{align*} \\displaystyle { \\left[ ( \\mupDelta _ { 0 } ^ { \\mathrm { V } } ) ^ { - 1 } \\right] } ^ { \\mitmu \\mitnu } & = & \\displaystyle \\miteta ^ { \\mitmu \\mitnu } \\sum _ { \\mitk ( } \\, \\square \\, + { \\mitm _ { \\mitk } } ^ { 2 } ) ^ { - 1 } \\mitV _ { \\mitk } ( \\mitx ^ { \\mitm } ) \\mitV _ { \\mitk } ( \\mitx ^ { \\prime \\mitm } ) , \\\\ \\displaystyle { \\left[ ( \\mupDelta _ { 0 } ^ { \\mathrm { V } } ) ^ { - 1 } \\right] } ^ { \\mitmu \\mitnu } & = & \\displaystyle \\sum _ { \\mitk ( } \\, \\square \\, + { \\mitm _ { \\mitk } ^ { \\prime } } ^ { 2 } ) ^ { - 1 } \\mitS _ { \\mitk } ^ { ( \\mita ) } ( \\mitx ^ { \\mitm } ) \\mitS _ { \\mitk } ^ { ( \\mitb ) } ( \\mitx ^ { \\prime \\mitm } ) , \\end{align*}",
"\\begin{equation*} \\mupDelta _ { 0 } ^ { \\mathrm { V } , \\mathrm { e x } } \\mitV _ { \\mitk } = { \\mitm _ { \\mitk } } ^ { 2 } \\mitV _ { \\mitk } , ~ ~ ~ ~ \\mupDelta _ { 0 } ^ { \\mathrm { S } , \\mathrm { e x } ( \\mita ) ( \\mitb ) } \\mitS _ { \\mitk } ^ { ( \\mitb ) } = { \\mitm _ { \\mitk } ^ { \\prime } } ^ { 2 } \\mitS _ { \\mitk } ^ { ( \\mita ) } . \\end{equation*}",
"\\begin{align*} & & \\hspace{-28.45pt} \\mupDelta = \\mupDelta _ { 0 } ( 1 + \\mupDelta _ { 0 } ^ { - 1 } \\mupDelta _ { \\mathrm { i n t } } ) , \\\\ & & \\hspace{-28.45pt} \\mitpartial _ { \\mitM } \\sqrt { - \\mitg } ( \\mupDelta ^ { - 1 } ) ^ { \\mitM \\mitN } \\sqrt { - \\mitg } \\mitpartial _ { \\mitN } = 1 + { \\mupDelta _ { 0 } ^ { \\prime } } ^ { - 1 } + \\mitpartial _ { \\mitm } ( \\mupDelta _ { 0 } ^ { - 1 } ) ^ { \\mitm \\mitn } \\mitpartial _ { \\mitn } + \\mupDelta _ { \\mathrm { i n t } } ^ { \\prime } , \\end{align*}",
"\\begin{equation*} \\displaystyle { \\mupDelta _ { 0 } ^ { \\prime } } ^ { - 1 } = \\sum _ { \\mitk } { \\mitm _ { \\mitk } } ^ { 2 } ( \\, \\square \\, + { \\mitm _ { \\mitk } } ^ { 2 } ) ^ { - 1 } \\mitV _ { \\mitk } ( \\mitx ^ { \\mitm } ) \\mitV _ { \\mitk } ( \\mitx ^ { \\prime \\mitm } ) . \\end{equation*}"
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0001113_page05 | {
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"$h^{\\mu \\nu }$",
"$w$",
"$\\Lambda $",
"$\\sqrt {a}$",
"$N_0$",
"$N_1$",
"$\\alpha _0$",
"$\\alpha _1$",
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"\\begin {equation} S^{\\rm eff} = \\int \\sqrt {-g}\\left [ (N_0\\alpha _0+N_1\\alpha _1+\\alpha _c)\\Lambda ^4+ (N_0\\beta _0+N_1\\beta _1+\\beta _c)\\Lambda ^2R\\right ] d^4x \\label {27} \\end {equation}",
"\\begin {equation} S = \\int \\sqrt {-g}\\left (\\lambda + \\frac {1}{16\\pi G}R\\right )d^4x \\label {28} \\end {equation}",
"\\begin {equation} \\lambda = \\int {\\cal L}_0 dx^5 dx^6 + (N_0\\alpha _0+N_1\\alpha _1+\\alpha _c)\\Lambda ^4, \\ \\ \\ \\^^M\\frac {1}{16\\pi G}=(N_0\\beta _0+N_1\\beta _1+\\beta _c)\\Lambda ^2. \\label {29} \\end {equation}"
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"$ N _ { 0 } $",
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"\\begin{equation*} S ^ { e f f } = \\int \\sqrt { - g } [ ( N _ { 0 } \\alpha _ { 0 } + N _ { 1 } \\alpha _ { 1 } + \\alpha _ { c } ) \\Lambda ^ { 4 } + ( N _ { 0 } \\beta _ { 0 } + N _ { 1 } \\beta _ { 1 } + \\beta _ { c } ) \\Lambda ^ { 2 } R ] d ^ { 4 } x \\end{equation*}",
"\\begin{equation*} S = \\int \\sqrt { - g } ( \\lambda + \\frac { 1 } { 1 6 \\pi G } R ) d ^ { 4 } x \\end{equation*}",
"\\begin{equation*} \\lambda = \\int L _ { 0 } d x ^ { 5 } d x ^ { 6 } + ( N _ { 0 } \\alpha _ { 0 } + N _ { 1 } \\alpha _ { 1 } + \\alpha _ { c } ) \\Lambda ^ { 4 } , ~ ~ ~ ~ \\frac { 1 } { 1 6 \\pi G } = ( N _ { 0 } \\beta _ { 0 } + N _ { 1 } \\beta _ { 1 } + \\beta _ { c } ) \\Lambda ^ { 2 } . \\end{equation*}"
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"\\begin{equation*} \\mitS = \\int \\sqrt { - \\mitg } \\left( \\mitlambda + \\frac { 1 } { 1 6 \\mitpi \\mitG } \\mitR \\right) \\mitd ^ { 4 } \\mitx \\end{equation*}",
"\\begin{equation*} \\mitlambda = \\int \\mitL _ { 0 } \\mitd \\mitx ^ { 5 } \\mitd \\mitx ^ { 6 } + ( \\mitN _ { 0 } \\mitalpha _ { 0 } + \\mitN _ { 1 } \\mitalpha _ { 1 } + \\mitalpha _ { \\mitc } ) \\mupLambda ^ { 4 } , ~ ~ ~ ~ \\frac { 1 } { 1 6 \\mitpi \\mitG } = ( \\mitN _ { 0 } \\mitbeta _ { 0 } + \\mitN _ { 1 } \\mitbeta _ { 1 } + \\mitbeta _ { \\mitc } ) \\mupLambda ^ { 2 } . \\end{equation*}"
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0001125_page02 | {
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"\\( G_{\\mu \\nu }=\\delta _{\\mu \\nu } \\)",
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"\\( N^{a} \\)",
"\\( \\partial {\\cal M} \\)",
"\\( N_{a}dz^{a},\\, d\\tau \\)",
"\\( i \\)",
"\\( \\sqrt {-h}\\rightarrow i\\sqrt {h} \\)",
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"\\( i \\)",
"\\begin {equation} S=\\frac {1}{2\\pi \\alpha ' }\\left [ \\frac {1}{2}\\int _{\\cal M}d^{2}z\\sqrt {h} h^{ab}\\partial _{a}X_{\\mu }\\partial _{b}X^{\\mu } +\\int _{\\partial {\\cal M}}d\\tau A_{\\mu }\\partial _{\\tau } X^{\\mu }\\right ] \\plabel {act} \\end {equation}"
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0001125_page05 | {
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"\\( b_{0} \\)",
"\\( \\sigma _{1} \\)",
"\\( b_{0}+\\sigma _{1}\\Gamma ^{2} \\)",
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"\\( i \\)",
"\\begin {eqnarray} && \\gamma =\\frac {1}{4}\\left [ \\frac {2}{\\sqrt {1+\\Gamma ^{2}}}-1\\right ]\\,, \\\\ && b_{1}=\\frac {1}{\\sqrt {-\\Gamma ^{2}}}{\\textrm {Artanh}} (\\sqrt {-\\Gamma ^{2}})-\\frac {1}{2}\\,,\\\\ && b_{2}=\\frac {2}{1+\\Gamma ^{2}}\\,,\\\\ && b_{0}+\\sigma _{1}\\Gamma ^{2}=\\frac {1}{3}\\, \\,, \\end {eqnarray}",
"\\begin {equation} \\zeta _{D}(s)={\\textrm {Tr}}(D^{-s})\\, \\, .\\plabel {defzeta} \\end {equation}",
"\\begin {equation} W=-\\frac {1}{2s}\\zeta _{D}(0)-\\frac {1}{2}\\zeta ' _{D}(0)\\,,\\label {W2} \\end {equation}",
"\\begin {equation} W_{{\\mbox {\\scriptsize {div}}}}=-\\frac {1}{2s}\\frac {1}{4\\pi } \\int _{\\partial {\\cal M}}d\\tau \\, \\left [ -\\dot {\\bar {X}}^{\\rho }(\\partial _{\\nu }F_{\\mu \\rho } +\\partial _{\\mu }F_{\\nu \\rho })(1+F^{2})^{-1}_{\\nu \\mu } +\\frac {1}{3}k\\delta _{\\nu }^{\\nu }\\right ] .\\label {Wdiv} \\end {equation}",
"\\begin {equation} \\beta _{\\mu }^{A}\\propto (\\partial _{\\rho }F_{\\nu \\mu })(1+F^{2})^{-1}_{\\nu \\rho }\\,.\\label {beta} \\end {equation}"
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"$ s \\rightarrow 0 $",
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"\\begin{equation*} W = - \\frac { 1 } { 2 s } \\zeta _ { D } ( 0 ) - \\frac { 1 } { 2 } \\zeta _ { D } ^ { \\prime } ( 0 ) \\, , \\end{equation*}",
"\\begin{equation*} W _ { d i v } = - \\frac { 1 } { 2 s } \\frac { 1 } { 4 \\pi } \\int _ { \\partial M } d \\tau \\, [ - \\dot { \\bar { X } } ^ { \\rho } ( \\partial _ { \\nu } F _ { \\mu \\rho } + \\partial _ { \\mu } F _ { \\nu \\rho } ) ( 1 + F ^ { 2 } ) _ { \\nu \\mu } ^ { - 1 } + \\frac { 1 } { 3 } k \\delta _ { \\nu } ^ { \\nu } ] . \\end{equation*}",
"\\begin{equation*} \\beta _ { \\mu } ^ { A } \\propto ( \\partial _ { \\rho } F _ { \\nu \\mu } ) ( 1 + F ^ { 2 } ) _ { \\nu \\rho } ^ { - 1 } \\, . \\end{equation*}"
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"\\begin{equation*} \\mitW = - \\frac { 1 } { 2 \\mits } \\mitzeta _ { \\mitD } ( 0 ) - \\frac { 1 } { 2 } \\mitzeta _ { \\mitD } ^ { \\prime } ( 0 ) \\, , \\end{equation*}",
"\\begin{equation*} \\mitW _ { \\mathrm { d i v } } = - \\frac { 1 } { 2 \\mits } \\frac { 1 } { 4 \\mitpi } \\int _ { \\mitpartial \\mitM } \\mitd \\mittau \\, \\left[ - \\dot { \\bar { \\mitX } } ^ { \\mitrho } ( \\mitpartial _ { \\mitnu } \\mitF _ { \\mitmu \\mitrho } + \\mitpartial _ { \\mitmu } \\mitF _ { \\mitnu \\mitrho } ) ( 1 + \\mitF ^ { 2 } ) _ { \\mitnu \\mitmu } ^ { - 1 } + \\frac { 1 } { 3 } \\mitk \\mitdelta _ { \\mitnu } ^ { \\mitnu } \\right] . \\end{equation*}",
"\\begin{equation*} \\mitbeta _ { \\mitmu } ^ { \\mitA } \\propto ( \\mitpartial _ { \\mitrho } \\mitF _ { \\mitnu \\mitmu } ) ( 1 + \\mitF ^ { 2 } ) _ { \\mitnu \\mitrho } ^ { - 1 } \\, . \\end{equation*}"
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0001125_page06 | {
"latex": [
"\\( A_{\\mu } \\)",
"\\( F_{\\mu \\nu } \\)",
"\\( \\delta _{\\nu }^{\\nu } \\)",
"\\( F \\)",
"\\( \\mathcal {M} \\)",
"\\( {\\cal M} \\)",
"\\( 2\\pi \\chi (\\mathcal {M})=\\int _{\\partial {\\cal M}}d\\tau k \\)",
"\\( \\delta h_{ab}=(\\delta k)h_{ab} \\)",
"\\( \\delta k \\)",
"\\( W_{{\\mbox {\\scriptsize {ren}}}} \\)",
"\\( N_{a}dz^{a},d\\tau \\)",
"\\( \\zeta \\)",
"\\( \\delta \\zeta _{D_{k}}(s)=s\\mbox {Tr}(D^{-s}\\delta k) \\)",
"\\( \\zeta (0|\\delta k,D)=a_{1}(\\delta k,D,{\\mathcal {B}}) \\)",
"\\begin {equation} \\delta W_{{\\mbox {\\scriptsize {ren}}}}= \\frac {1}{2}\\int _{\\cal M}d^{2}z\\sqrt {h}\\delta h^{ab}T_{ab} =-\\frac {1}{2}\\int _{\\cal M}d^{2}z\\sqrt {h}\\delta k(x)T_{a}^{a}(x)\\,,\\label {T} \\end {equation}",
"\\begin {eqnarray} && \\Delta \\rightarrow (1-k+\\dots )\\Delta ,\\\\ && {\\mathcal {B}}\\rightarrow (1-\\frac {k}{2}+\\dots ){\\mathcal {B}}\\,. \\end {eqnarray}",
"\\begin {equation} \\zeta (s|\\delta k,D)={\\textrm {Tr}}(\\delta kD^{-s})\\plabel {varW} \\end {equation}",
"\\begin {equation} \\plabel {TX}\\delta W_{{\\mbox {\\scriptsize {ren}}}}=-\\frac {1}{2}\\zeta (0|\\delta k,D)\\quad , \\end {equation}",
"\\begin {equation} \\zeta (0|\\delta k,D)=\\int d^{2}z\\sqrt {h}\\delta k(z)T_{a}^{a}(x)\\; .\\plabel {T2} \\end {equation}",
"\\begin {eqnarray} && \\int _{\\cal M}\\sqrt {h}d^{2}zf(z)T_{a}^{a}(z)= \\frac {1}{4\\pi }\\int _{\\partial {\\cal M}}d\\tau \\, \\left [ f(\\tau )\\left ( \\frac {1}{3}k\\delta _{\\nu }^{\\nu } -2\\dot {\\bar {X}}^{\\rho }(\\partial _{\\nu }F_{\\mu \\rho }) (1+F^{2})^{-1}_{\\nu \\mu }\\right ) \\right . \\\\ && \\qquad \\qquad \\left . +(\\nabla _{N}f)\\left ( (-F^{2})_{\\mu \\nu }^{-1/2}{\\textrm {Artanh}}(\\sqrt {-F^{2}})_{\\nu \\mu }- \\frac {1}{2}\\delta _{\\mu }^{\\mu }\\right ) \\right ]\\,. \\end {eqnarray}"
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"$ \\zeta ( 0 \\vert \\delta k , D ) = a _ { 1 } ( \\delta k , D , B ) $",
"\\begin{equation*} \\delta W _ { r e n } = \\frac { 1 } { 2 } \\int _ { M } d ^ { 2 } z \\sqrt { h } \\delta h ^ { a b } T _ { a b } = - \\frac { 1 } { 2 } \\int _ { M } d ^ { 2 } z \\sqrt { h } \\delta k ( x ) T _ { a } ^ { a } ( x ) \\, , \\end{equation*}",
"\\begin{align*} & & \\Delta \\rightarrow ( 1 - k + \\ldots \\, ) \\Delta , \\\\ & & B \\rightarrow ( 1 - \\frac { k } { 2 } + \\ldots \\, ) B \\, . \\end{align*}",
"\\begin{equation*} \\zeta ( s \\vert \\delta k , D ) = T r ( \\delta k D ^ { - s } ) \\end{equation*}",
"\\begin{equation*} \\delta W _ { r e n } = - \\frac { 1 } { 2 } \\zeta ( 0 \\vert \\delta k , D ) \\quad , \\end{equation*}",
"\\begin{equation*} \\zeta ( 0 \\vert \\delta k , D ) = \\int d ^ { 2 } z \\sqrt { h } \\delta k ( z ) T _ { a } ^ { a } ( x ) \\; . \\end{equation*}",
"\\begin{align*} & & \\int _ { M } \\sqrt { h } d ^ { 2 } z f ( z ) T _ { a } ^ { a } ( z ) = \\frac { 1 } { 4 \\pi } \\int _ { \\partial M } d \\tau \\, [ f ( \\tau ) ( \\frac { 1 } { 3 } k \\delta _ { \\nu } ^ { \\nu } - 2 \\dot { \\bar { X } } ^ { \\rho } ( \\partial _ { \\nu } F _ { \\mu \\rho } ) ( 1 + F ^ { 2 } ) _ { \\nu \\mu } ^ { - 1 } ) \\\\ & & \\qquad \\qquad + ( \\nabla _ { N } f ) ( ( - F ^ { 2 } ) _ { \\mu \\nu } ^ { - 1 \\slash 2 } A r t a n h ( \\sqrt { - F ^ { 2 } } ) _ { \\nu \\mu } - \\frac { 1 } { 2 } \\delta _ { \\mu } ^ { \\mu } ) ] \\, . \\end{align*}"
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"\\begin{align*} & & \\mupDelta \\rightarrow ( 1 - \\mitk + \\ldots \\, ) \\mupDelta , \\\\ & & \\mscrB \\rightarrow ( 1 - \\frac { \\mitk } { 2 } + \\ldots \\, ) \\mscrB \\, . \\end{align*}",
"\\begin{equation*} \\mitzeta ( \\mits \\vert \\mitdelta \\mitk , \\mitD ) = \\mathrm { T r } ( \\mitdelta \\mitk \\mitD ^ { - \\mits } ) \\end{equation*}",
"\\begin{equation*} \\mitdelta \\mitW _ { \\mathrm { r e n } } = - \\frac { 1 } { 2 } \\mitzeta ( 0 \\vert \\mitdelta \\mitk , \\mitD ) \\quad , \\end{equation*}",
"\\begin{equation*} \\mitzeta ( 0 \\vert \\mitdelta \\mitk , \\mitD ) = \\int \\mitd ^ { 2 } \\mitz \\sqrt { \\Planckconst } \\mitdelta \\mitk ( \\mitz ) \\mitT _ { \\mita } ^ { \\mita } ( \\mitx ) \\; . \\end{equation*}",
"\\begin{align*} & & \\int _ { \\mitM } \\sqrt { \\Planckconst } \\mitd ^ { 2 } \\mitz \\mitf ( \\mitz ) \\mitT _ { \\mita } ^ { \\mita } ( \\mitz ) = \\frac { 1 } { 4 \\mitpi } \\int _ { \\mitpartial \\mitM } \\mitd \\mittau \\, \\left[ \\mitf ( \\mittau ) \\left( \\frac { 1 } { 3 } \\mitk \\mitdelta _ { \\mitnu } ^ { \\mitnu } - 2 \\dot { \\bar { \\mitX } } ^ { \\mitrho } ( \\mitpartial _ { \\mitnu } \\mitF _ { \\mitmu \\mitrho } ) ( 1 + \\mitF ^ { 2 } ) _ { \\mitnu \\mitmu } ^ { - 1 } \\right) \\right. \\\\ & & \\qquad \\qquad \\left. + ( \\nabla _ { \\mitN } \\mitf ) \\left( ( - \\mitF ^ { 2 } ) _ { \\mitmu \\mitnu } ^ { - 1 \\slash 2 } \\mathrm { A r t a n h } ( \\sqrt { - \\mitF ^ { 2 } } ) _ { \\mitnu \\mitmu } - \\frac { 1 } { 2 } \\mitdelta _ { \\mitmu } ^ { \\mitmu } \\right) \\right] \\, . \\end{align*}"
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End of preview. Expand
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IBEM Full: Mathematical Expression Annotations on Scientific Pages
A structured version of the IBEM Dataset (Anitei et al., 2023), designed for training and evaluation of mathematical expression detection models. This variant retains full page images and provides bounding box annotations for all mathematical expressions, enabling end-to-end layout analysis and object detection tasks.
Dataset Summary
This dataset is derived from the original IBEM corpus by preserving full-page scientific document images and annotating all mathematical expressions using their corresponding bounding boxes and LaTeX transcriptions. It is intended for:
- Mathematical expression detection (localization and classification)
- Document understanding and layout parsing
- End-to-end systems for recognizing math expressions in context
Each record in the dataset includes:
A full-page image from a LaTeX source document
A unique page identifier
A list of expressions (
expressions), where each item contains:- The raw LaTeX string (
latex) - A normalized version of the LaTeX string (
latex_norm) - An expanded version with macros resolved (
latex_expand) - Normalized bounding box coordinates (
x_min,y_min,x_max,y_max) relative to image dimensions - The expression type (
expr_type: either"isolated"or"embedded")
- The raw LaTeX string (
Dataset Structure
Features
features = Features({
"image": Image(),
"page_id": Value("string"),
"expressions": Sequence({
"latex": Value("string"),
"latex_norm": Value("string"),
"latex_expand": Value("string"),
"x_min": Value("float32"),
"y_min": Value("float32"),
"x_max": Value("float32"),
"y_max": Value("float32"),
"expr_type": Value("string"),
}),
})
Splits
The dataset is split into train, val, and test subsets using the official partition files (Tr*.lst, Va*.lst, Ts*.lst) provided by the original authors.
Source
Citation:
Anitei, D., Sánchez, J. A., & Benedí, J. M. (2023). The IBEM Dataset: a large printed scientific image dataset for indexing and searching mathematical expressions (1.0) [Data set]. Zenodo. https://doi.org/10.5281/zenodo.7963703
Original Dataset Description
The original IBEM dataset consists of 600 LaTeX documents with a total of 8,272 pages, containing:
- 29,603 isolated (displayed) mathematical expressions
- 137,089 embedded (in-line) mathematical expressions
It was created by parsing LaTeX source files from the KDD Cup Collection and supports a wide range of tasks such as:
- Mathematical expression detection and extraction
- LaTeX OCR / recognition
- Search and indexing in scientific literature
Preprocessing Notes
- Bounding boxes are provided in relative coordinates (percentage of image width/height).
- No cropping was applied — images retain their full-page content.
- Multiple expressions per page are stored in a single list under
"expressions". - Expression-level
latex,latex_expand, andlatex_normwere directly taken from the original ground truth. - Expressions split across lines are preserved with their full content.
Licensing
This dataset is distributed under the same license as the original IBEM dataset: Creative Commons Attribution 4.0 International (CC BY 4.0)
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