image
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)$",
"\\begin {equation} H = \\half p^2 + V(x) \\end {equation}",
"\\begin {equation}\\label {EVeq} \\exp (-Ht) \\psi (x) = \\int d\\mu \\exp \\left ( -\\int _0^t V((x(s)) ds \\right ) \\psi (x(t)) \\end {equation}"
],
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"$ \\phi $",
"$ e x p ( - H t ) $",
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"$ d \\mu $",
"$ x $",
"$ 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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0001129_page06 | {
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"${\\cal A}_{\\theta {\\cal L}}({\\cal O}_1)\\subset {\\cal A}_{\\theta {\\cal L}}({\\cal O}_2)$",
"$\\theta \\in \\Theta ({\\cal O}_2)$",
"$\\Theta ({\\cal O}_1)$",
"$\\Theta ({\\cal O}_2)$",
"$i_{12}\\circ i_{23}=i_{13}$",
"${\\cal O}_3\\subset {\\cal O}_2 \\subset {\\cal O}_1$",
"$V\\in {\\cal U}(\\theta ,\\theta ^\\prime )$",
"$A,B\\in {\\cal A}_{\\cal L}({\\cal O})$",
"$n$",
"$n$",
"$(A_\\theta -B_\\theta )={\\cal O}(g^{n+1})$",
"$(A_\\theta -B_\\theta )={\\cal O}(g^{n+1})$",
"$A_{\\theta ^\\prime }-B_{\\theta ^\\prime }= V^{-1}(A_\\theta -B_\\theta )V={\\cal O}(g^{n+1})$",
"$S$",
"$S_{\\theta {\\cal L}}(h)$",
"$U$",
"${\\cal P}_+^\\uparrow $",
"$\\forall L\\in {\\cal P}_+^\\uparrow $",
"${\\cal L}$",
"${\\cal V}$",
"$D$",
"$A\\in {\\cal A}_{\\cal L}({\\cal O}),\\>\\theta \\in \\Theta (L{\\cal O})$",
"$\\alpha _L(A)$",
"$\\alpha _L$",
"${\\cal A}_{\\cal L}({\\cal O})$",
"$T_{\\cal L}(f^{\\otimes n})$",
"$f\\in {\\cal D} ({\\cal O},{\\cal V}),\\>n\\in \\NN _0$",
"\\begin {equation} {\\cal A}_{\\cal L}\\=d \\cup _{\\cal O}{\\cal A}_{\\cal L}({\\cal O}).\\label {global} \\end {equation}",
"\\begin {equation} U(L)S_{\\theta {\\cal L}}(h)U(L)^{-1}=S_{\\theta _L{\\cal L}}(h_L),\\quad \\theta _L(x):=\\theta (L^{-1}x),\\>h_L(x):=D(L)h(L^{-1}x),\\label {3.9} \\end {equation}",
"\\begin {equation} (\\alpha _L(A))_\\theta \\=d U(L)A_{\\theta _{L^{-1}}}U(L)^{-1}.\\label {3.10} \\end {equation}",
"\\begin {equation} \\alpha _L{\\cal A}_{\\cal L}({\\cal O})={\\cal A}_{\\cal L}(L{\\cal O}),\\quad \\quad \\alpha _{L_1L_2}=\\alpha _{L_1}\\alpha _{L_2}. \\label {3.10a}\\end {equation}",
"\\begin {equation} S_{\\cal L}(\\lambda f)=\\sum _{n=0}^\\infty \\frac {i^n\\lambda ^n}{n!}T_{\\cal L}(f^{\\otimes n}) \\label {E:timeordered products} \\end {equation}",
"\\begin {equation} T_{\\cal L}(hA)=:A_{\\cal L}(h)\\ ,\\ A\\in {\\cal V},\\ h\\in {\\cal D}(\\RR ^4)\\ , \\label {E:interacting fields} \\end {equation}"
],
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"$ \\Theta ( O _ { 1 } ) $",
"$ \\Theta ( O _ { 2 } ) $",
"$ i _ { 1 2 } \\circ i _ { 2 3 } = i _ { 1 3 } $",
"$ O _ { 3 } \\subset O _ { 2 } \\subset O _ { 1 } $",
"$ V \\in U ( \\theta , \\theta ^ { \\prime } ) $",
"$ A , B \\in A _ { L } ( O ) $",
"$ n $",
"$ n $",
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"$ ( A _ { \\theta } - B _ { \\theta } ) = O ( g ^ { n + 1 } ) $",
"$ A _ { \\theta ^ { \\prime } } - B _ { \\theta ^ { \\prime } } = V ^ { - 1 } ( A _ { \\theta } - B _ { \\theta } ) V = O ( g ^ { n + 1 } ) $",
"$ S $",
"$ S _ { \\theta L } ( h ) $",
"$ U $",
"$ P _ { + } ^ { \\uparrow } $",
"$ \\forall L \\in P _ { + } ^ { \\uparrow } $",
"$ L $",
"$ V $",
"$ D $",
"$ A \\in A _ { L } ( O ) , \\> \\theta \\in \\Theta ( L O ) $",
"$ \\alpha _ { L } ( A ) $",
"$ \\alpha _ { L } $",
"$ A _ { L } ( O ) $",
"$ T _ { L } ( f ^ { \\otimes n } ) $",
"$ f \\in D ( O , V ) , \\> n \\in N _ { 0 } $",
"\\begin{equation*} A _ { L } \\, \\overset { d e f } { = } \\, \\cup _ { O } A _ { L } ( O ) . \\end{equation*}",
"\\begin{equation*} U ( L ) S _ { \\theta L } ( h ) U ( L ) ^ { - 1 } = S _ { \\theta _ { L } L } ( h _ { L } ) , \\quad \\theta _ { L } ( x ) : = \\theta ( L ^ { - 1 } x ) , \\> h _ { L } ( x ) : = D ( L ) h ( L ^ { - 1 } x ) , \\end{equation*}",
"\\begin{equation*} ( \\alpha _ { L } ( A ) ) _ { \\theta } \\, \\overset { d e f } { = } \\, U ( L ) A _ { \\theta _ { L ^ { - 1 } } } U ( L ) ^ { - 1 } . \\end{equation*}",
"\\begin{equation*} \\alpha _ { L } A _ { L } ( O ) = A _ { L } ( L O ) , \\quad \\quad \\alpha _ { L _ { 1 } L _ { 2 } } = \\alpha _ { L _ { 1 } } \\alpha _ { L _ { 2 } } . \\end{equation*}",
"\\begin{equation*} S _ { L } ( \\lambda f ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { i ^ { n } \\lambda ^ { n } } { n ! } T _ { L } ( f ^ { \\otimes n } ) \\end{equation*}",
"\\begin{equation*} T _ { L } ( h A ) = : A _ { L } ( h ) ~ , ~ A \\in V , ~ h \\in D ( R ^ { 4 } ) ~ , \\end{equation*}"
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"$ \\mupTheta ( \\mitO _ { 1 } ) $",
"$ \\mupTheta ( \\mitO _ { 2 } ) $",
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"$ \\mitO _ { 3 } \\subset \\mitO _ { 2 } \\subset \\mitO _ { 1 } $",
"$ \\mitV \\in \\mitU ( \\mittheta , \\mittheta ^ { \\prime } ) $",
"$ \\mitA , \\mitB \\in \\mitA _ { \\mitL } ( \\mitO ) $",
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"$ ( \\mitA _ { \\mittheta } - \\mitB _ { \\mittheta } ) = \\mitO ( \\mitg ^ { \\mitn + 1 } ) $",
"$ \\mitA _ { \\mittheta ^ { \\prime } } - \\mitB _ { \\mittheta ^ { \\prime } } = \\mitV ^ { - 1 } ( \\mitA _ { \\mittheta } - \\mitB _ { \\mittheta } ) \\mitV = \\mitO ( \\mitg ^ { \\mitn + 1 } ) $",
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"$ \\mitS _ { \\mittheta \\mitL } ( \\Planckconst ) $",
"$ \\mitU $",
"$ \\mitP _ { + } ^ { \\uparrow } $",
"$ \\forall \\mitL \\in \\mitP _ { + } ^ { \\uparrow } $",
"$ \\mitL $",
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"$ \\mitD $",
"$ \\mitA \\in \\mitA _ { \\mitL } ( \\mitO ) , \\> \\mittheta \\in \\mupTheta ( \\mitL \\mitO ) $",
"$ \\mitalpha _ { \\mitL } ( \\mitA ) $",
"$ \\mitalpha _ { \\mitL } $",
"$ \\mitA _ { \\mitL } ( \\mitO ) $",
"$ \\mitT _ { \\mitL } ( \\mitf ^ { \\otimes \\mitn } ) $",
"$ \\mitf \\in \\mitD ( \\mitO , \\mitV ) , \\> \\mitn \\in \\BbbN _ { 0 } $",
"\\begin{equation*} \\mitA _ { \\mitL } \\, \\overset { \\mathrm { d e f } } { = } \\, \\cup _ { \\mitO } \\mitA _ { \\mitL } ( \\mitO ) . \\end{equation*}",
"\\begin{equation*} \\mitU ( \\mitL ) \\mitS _ { \\mittheta \\mitL } ( \\Planckconst ) \\mitU ( \\mitL ) ^ { - 1 } = \\mitS _ { \\mittheta _ { \\mitL } \\mitL } ( \\Planckconst _ { \\mitL } ) , \\quad \\mittheta _ { \\mitL } ( \\mitx ) : = \\mittheta ( \\mitL ^ { - 1 } \\mitx ) , \\> \\Planckconst _ { \\mitL } ( \\mitx ) : = \\mitD ( \\mitL ) \\Planckconst ( \\mitL ^ { - 1 } \\mitx ) , \\end{equation*}",
"\\begin{equation*} ( \\mitalpha _ { \\mitL } ( \\mitA ) ) _ { \\mittheta } \\, \\overset { \\mathrm { d e f } } { = } \\, \\mitU ( \\mitL ) \\mitA _ { \\mittheta _ { \\mitL ^ { - 1 } } } \\mitU ( \\mitL ) ^ { - 1 } . \\end{equation*}",
"\\begin{equation*} \\mitalpha _ { \\mitL } \\mitA _ { \\mitL } ( \\mitO ) = \\mitA _ { \\mitL } ( \\mitL \\mitO ) , \\quad \\quad \\mitalpha _ { \\mitL _ { 1 } \\mitL _ { 2 } } = \\mitalpha _ { \\mitL _ { 1 } } \\mitalpha _ { \\mitL _ { 2 } } . \\end{equation*}",
"\\begin{equation*} \\mitS _ { \\mitL } ( \\mitlambda \\mitf ) = \\sum _ { \\mitn = 0 } ^ { \\infty } \\frac { \\miti ^ { \\mitn } \\mitlambda ^ { \\mitn } } { \\mitn ! } \\mitT _ { \\mitL } ( \\mitf ^ { \\otimes \\mitn } ) \\end{equation*}",
"\\begin{equation*} \\mitT _ { \\mitL } ( \\Planckconst \\mitA ) = : \\mitA _ { \\mitL } ( \\Planckconst ) ~ , ~ \\mitA \\in \\mitV , ~ \\Planckconst \\in \\mitD ( \\BbbR ^ { 4 } ) ~ , \\end{equation*}"
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|
0001129_page08 | {
"latex": [
"$(x_i-y_j)^2<0\\quad \\forall (i,j)$",
"${\\cal J}$",
"$I$",
"$\\{1,...,n\\}$",
"${\\cal C}_I\\=d \\{(x_1,...,x_n)\\in \\RR ^{4n}\\>|\\>x_i\\not \\in J^-(x_j),\\,i\\in I,\\,j\\in I^c\\}$",
"$I\\in {\\cal J}$",
"${\\cal D}({\\cal C}_I)$",
"$I\\in {\\cal C}_I$",
"$I_1,I_2\\in {\\cal J},\\>{\\cal C}_{I_1}\\cap C_{I_2}\\not =\\emptyset $",
"$\\{f_I\\}_{I\\in {\\cal J}}\\cup \\{f_0\\}$",
"$\\RR ^{4n}$",
"$\\{{\\cal C}_I\\}_ {I\\in {\\cal J}}\\cup {\\cal U}_n$",
"${\\rm supp}\\>f_I\\subset {\\cal C}_I, {\\rm supp}\\>f_0\\subset {\\cal U}_n$",
"$\\{f_I\\}_{I\\in {\\cal J}}\\cup \\{f_0\\}$",
"$T_{n}$",
"$n-1$",
"$n$",
"$T$",
"$S$",
"$S(g)$",
"$g\\in {\\cal D}(\\RR ^4,{\\cal V})$",
"$S(g)$",
"$g\\in {\\cal D}(\\RR ^4,{\\cal V})$",
"$\\hbar $",
"$\\hbar $",
"$T_n(x_1,...,x_n)$",
"${\\bf N1}-{\\bf N4}$",
"$\\Delta _n \\equiv \\{(x_1,...,x_n)\\> |\\> x_1=...=x_n\\}$",
"${\\cal U}_n$",
"$\\Delta _n$",
"$T$",
"$T$",
"\\begin {equation} \\bigcup _{I\\in {\\cal J}}{\\cal C}_I\\>\\cup \\> {\\cal U}_n=\\RR ^{4n}.\\label {2.24} \\end {equation}",
"\\begin {equation} T^I(x_I)=T(\\prod _{i\\in I}A_i(x_i)),\\quad \\quad x_I=(x_i,i\\in I).\\label {2.25} \\end {equation}",
"\\begin {equation} T_I(x)\\=d T^I(x_I)T^{I^c}(x_{I^c})\\label {2.26} \\end {equation}",
"\\begin {equation} T_{I_1}\\vert _{{\\cal C}_{I_1}\\cap {\\cal C}_{I_2}}= T_{I_2}\\vert _{{\\cal C}_{I_1}\\cap {\\cal C}_{I_2}}.\\label {2.27} \\end {equation}",
"\\begin {equation} T_n(h)\\=d T_n\\vert _{{\\cal U}_n}(f_0h)+\\sum _{I\\in {\\cal J}}T_I(f_Ih),\\quad \\quad h\\in {\\cal D}(\\RR ^{4n},{\\cal V}^{\\otimes n}).\\label {2.28} \\end {equation}"
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"$ \\hbar $",
"$ \\hbar $",
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"$ N 1 - N 4 $",
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"\\begin{equation*} \\cup _ { I \\in J } C _ { I } \\> \\cup \\> U _ { n } = R ^ { 4 n } . \\end{equation*}",
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|
0001129_page09 | {
"latex": [
"$\\hbar $",
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"$\\hbar $",
"$\\varphi _{\\rm class} (x)$",
"$(\\w +m^2)\\varphi _{\\rm class}=0$",
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"$N<\\infty $",
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"$:\\varphi (x_1)...\\varphi (x_n):$",
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"\\begin {equation} a\\>\\times _\\hbar \\> b\\quad \\buildrel \\hbar \\rightarrow 0\\over \\longrightarrow \\quad ab,\\quad \\quad \\quad \\frac {1}{\\hbar }(a\\>\\times _\\hbar \\> b\\>-\\> b\\>\\times _\\hbar \\> a) \\quad \\buildrel \\hbar \\rightarrow 0\\over \\longrightarrow \\quad \\{a,b\\}.\\label {C1} \\end {equation}",
"\\begin {equation} \\{\\varphi _{\\rm class} (x),\\varphi _{\\rm class} (y)\\}=\\Delta (x-y)\\label {C2} \\end {equation}",
"\\begin {equation} \\phi (t)=t_0+\\sum _{n=1}^N\\int \\varphi _{\\rm class}(x_1)...\\varphi _{\\rm class} (x_n)t_n(x_1,...,x_n)dx_1...dx_n,\\quad t\\equiv (t_0,t_1,...),\\label {C3} \\end {equation}",
"\\begin {eqnarray} :\\prod _{i\\in I}\\varphi (x_i):\\times _\\hbar \\>:\\prod _{j\\in J}\\varphi (x_j): &=& \\\\ \\sum _{K\\subset I}\\sum _{\\alpha :K\\rightarrow J\\>{\\rm injective}}\\prod _{j\\in K} i\\hbar \\Delta _+(x_j-x_{\\alpha (j)})&&:\\prod _{l\\in (I\\setminus K)\\cup (J\\setminus \\alpha (K))}\\varphi (x_l):\\end {eqnarray}",
"\\begin {equation} {\\cal W}_{n}\\=d\\{t\\in {\\cal D}^{\\prime }(\\RR ^{4n})_{\\rm symm}\\>,\\>\\supp t \\mbox { compact },\\> {\\rm WF}(t)\\>\\cap \\>(\\RR ^{4n}\\times \\overline {V_+^n\\cup V_-^n})=\\emptyset \\} \\end {equation}"
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"\\begin{equation*} \\{ \\varphi _ { c l a s s } ( x ) , \\varphi _ { c l a s s } ( y ) \\} = \\Delta ( x - y ) \\end{equation*}",
"\\begin{equation*} \\phi ( t ) = t _ { 0 } + \\sum _ { n = 1 } ^ { N } \\int \\varphi _ { c l a s s } ( x _ { 1 } ) . . . \\varphi _ { c l a s s } ( x _ { n } ) t _ { n } ( x _ { 1 } , . . . , x _ { n } ) d x _ { 1 } . . . d x _ { n } , \\quad t \\equiv ( t _ { 0 } , t _ { 1 } , . . . ) , \\end{equation*}",
"\\begin{align*} : \\prod _ { i \\in I } \\varphi ( x _ { i } ) : \\times _ { \\hbar } \\> : \\prod _ { j \\in J } \\varphi ( x _ { j } ) : & = \\\\ \\sum _ { K \\subset I } \\sum _ { \\alpha : K \\rightarrow J \\> i n j e c t i v e } \\prod _ { j \\in K } i \\hbar \\Delta _ { + } ( x _ { j } - x _ { \\alpha ( j ) } ) & & : \\prod _ { l \\in ( I \\setminus K ) \\cup ( J \\setminus \\alpha ( K ) ) } \\varphi ( x _ { l } ) : \\end{align*}",
"\\begin{equation*} W _ { n } \\, \\overset { d e f } { = } \\, \\{ t \\in D ^ { \\prime } ( R ^ { 4 n } ) _ { s y m m } \\> , \\> s u p p \\> t ~ c o m p a c t ~ , \\> W F ( t ) \\> \\cap \\> ( R ^ { 4 n } \\times \\overline { V _ { + } ^ { n } \\cup V _ { - } ^ { n } } ) = \\emptyset \\} \\end{equation*}"
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0001129_page10 | {
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"\\begin {equation} (\\varphi ^{\\otimes n})(t) \\=d\\int :\\varphi (x_1)...\\varphi (x_n): t(x_1,...,x_n)\\,dx_1...dx_n,\\quad (\\varphi ^{\\otimes 0})\\=d {\\bf 1},\\label {W3} \\end {equation}",
"\\begin {equation} :\\varphi ^n(f):=(\\varphi ^{\\otimes n})(t)\\ ,\\ f\\in {\\cal D}(\\RR ^4)\\ ,\\ t(x_{1},\\ldots ,x_{n})=f(x_{1})\\prod _{i=2}^n \\delta (x_{i}-x_{1}) \\label {E:Wick powers} \\end {equation}",
"\\begin {equation} (\\varphi ^{\\otimes n})(t)\\times _\\hbar (\\varphi ^{\\otimes m})(s)= \\sum _{k=0}^{{\\rm min}\\{n,m\\}}\\hbar ^k(\\varphi ^{\\otimes (n+m-2k)}) (t\\otimes _{k} s)\\label {W6} \\end {equation}",
"\\begin {eqnarray} (t\\otimes _{k} s)(x_1,...,x_{n+m-2k})={\\cal S}\\frac {n!m!i^k} {k!(n-k)!(m-k)!}\\int dy_1...dy_{2k}\\Delta _+(y_1-y_2)...\\\\ \\Delta _+(y_{2k-1}-y_{2k})t(x_1,...,x_{n-k},y_1,y_3,...,y_{2k-1})\\\\ s(x_{n-k+1},...,x_{n+m-2k},y_2,y_4,...,y_{2k})\\end {eqnarray}",
"\\begin {equation} (t\\times _{\\hbar }s)_{n}=\\sum _{m+l-2k=n}\\hbar ^k t_{m}\\otimes _{k}s_{l}. \\label {product on W} \\end {equation}",
"\\begin {eqnarray} \\lim _{\\hbar \\to 0}\\phi (t)\\times _\\hbar \\phi (s)&=& \\lim _{\\hbar \\to 0}\\phi (\\sum _{n}\\hbar ^n t\\otimes _n s)\\\\ &=& \\phi (t\\otimes _0 s)=\\phi (t)\\cdot \\phi (s) \\end {eqnarray}",
"\\begin {equation} \\lim _{\\hbar \\to 0}\\frac {1}{i\\hbar }[\\phi (t),\\phi (s)]_\\hbar = \\phi (t\\otimes _1 s-s\\otimes _1 t)=\\{\\phi (t),\\phi (s)\\} \\end {equation}"
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"\\begin{equation*} ( \\varphi ^ { \\otimes n } ) ( t ) \\, \\overset { d e f } { = } \\, \\int : \\varphi ( x _ { 1 } ) . . . \\varphi ( x _ { n } ) : t ( x _ { 1 } , . . . , x _ { n } ) \\, d x _ { 1 } . . . d x _ { n } , \\quad ( \\varphi ^ { \\otimes 0 } ) \\, \\overset { d e f } { = } \\, 1 , \\end{equation*}",
"\\begin{equation*} : \\varphi ^ { n } ( f ) : = ( \\varphi ^ { \\otimes n } ) ( t ) ~ , ~ f \\in D ( R ^ { 4 } ) ~ , ~ t ( x _ { 1 } , \\ldots , x _ { n } ) = f ( x _ { 1 } ) \\prod _ { i = 2 } ^ { n } \\delta ( x _ { i } - x _ { 1 } ) \\end{equation*}",
"\\begin{equation*} ( \\varphi ^ { \\otimes n } ) ( t ) \\times _ { \\hbar } ( \\varphi ^ { \\otimes m } ) ( s ) = \\sum _ { k = 0 } ^ { m i n \\{ n , m \\} } \\hbar ^ { k } ( \\varphi ^ { \\otimes ( n + m - 2 k ) } ) ( t \\otimes _ { k } s ) \\end{equation*}",
"\\begin{align*} ( t \\otimes _ { k } s ) ( x _ { 1 } , . . . , x _ { n + m - 2 k } ) = S \\frac { n ! m ! i ^ { k } } { k ! ( n - k ) ! ( m - k ) ! } \\int d y _ { 1 } . . . d y _ { 2 k } \\Delta _ { + } ( y _ { 1 } - y _ { 2 } ) . . . \\\\ \\Delta _ { + } ( y _ { 2 k - 1 } - y _ { 2 k } ) t ( x _ { 1 } , . . . , x _ { n - k } , y _ { 1 } , y _ { 3 } , . . . , y _ { 2 k - 1 } ) \\\\ s ( x _ { n - k + 1 } , . . . , x _ { n + m - 2 k } , y _ { 2 } , y _ { 4 } , . . . , y _ { 2 k } ) \\end{align*}",
"\\begin{equation*} ( t \\times _ { \\hbar } s ) _ { n } = \\sum _ { m + l - 2 k = n } \\hbar ^ { k } t _ { m } \\otimes _ { k } s _ { l } . \\end{equation*}",
"\\begin{align*} \\underset { \\hbar \\rightarrow 0 } { \\operatorname { l i m } } \\phi ( t ) \\times _ { \\hbar } \\phi ( s ) & = & \\underset { \\hbar \\rightarrow 0 } { \\operatorname { l i m } } \\phi ( \\sum _ { n } \\hbar ^ { n } t \\otimes _ { n } s ) \\\\ & = & \\phi ( t \\otimes _ { 0 } s ) = \\phi ( t ) \\cdot \\phi ( s ) \\end{align*}",
"\\begin{equation*} \\underset { \\hbar \\rightarrow 0 } { \\operatorname { l i m } } \\frac { 1 } { i \\hbar } [ \\phi ( t ) , \\phi ( s ) ] _ { \\hbar } = \\phi ( t \\otimes _ { 1 } s - s \\otimes _ { 1 } t ) = \\{ \\phi ( t ) , \\phi ( s ) \\} \\end{equation*}"
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"\\begin{equation*} : \\mitvarphi ^ { \\mitn } ( \\mitf ) : = ( \\mitvarphi ^ { \\otimes \\mitn } ) ( \\mitt ) ~ , ~ \\mitf \\in \\mitD ( \\BbbR ^ { 4 } ) ~ , ~ \\mitt ( \\mitx _ { 1 } , \\ldots , \\mitx _ { \\mitn } ) = \\mitf ( \\mitx _ { 1 } ) \\prod _ { \\miti = 2 } ^ { \\mitn } \\mitdelta ( \\mitx _ { \\miti } - \\mitx _ { 1 } ) \\end{equation*}",
"\\begin{equation*} ( \\mitvarphi ^ { \\otimes \\mitn } ) ( \\mitt ) \\times _ { \\hslash } ( \\mitvarphi ^ { \\otimes \\mitm } ) ( \\mits ) = \\sum _ { \\mitk = 0 } ^ { \\mathrm { m i n } \\{ \\mitn , \\mitm \\} } \\hslash ^ { \\mitk } ( \\mitvarphi ^ { \\otimes ( \\mitn + \\mitm - 2 \\mitk ) } ) ( \\mitt \\otimes _ { \\mitk } \\mits ) \\end{equation*}",
"\\begin{align*} ( \\mitt \\otimes _ { \\mitk } \\mits ) ( \\mitx _ { 1 } , . . . , \\mitx _ { \\mitn + \\mitm - 2 \\mitk } ) = \\mitS \\frac { \\mitn ! \\mitm ! \\miti ^ { \\mitk } } { \\mitk ! ( \\mitn - \\mitk ) ! ( \\mitm - \\mitk ) ! } \\int \\mitd \\mity _ { 1 } . . . \\mitd \\mity _ { 2 \\mitk } \\mupDelta _ { + } ( \\mity _ { 1 } - \\mity _ { 2 } ) . . . \\\\ \\mupDelta _ { + } ( \\mity _ { 2 \\mitk - 1 } - \\mity _ { 2 \\mitk } ) \\mitt ( \\mitx _ { 1 } , . . . , \\mitx _ { \\mitn - \\mitk } , \\mity _ { 1 } , \\mity _ { 3 } , . . . , \\mity _ { 2 \\mitk - 1 } ) \\\\ \\mits ( \\mitx _ { \\mitn - \\mitk + 1 } , . . . , \\mitx _ { \\mitn + \\mitm - 2 \\mitk } , \\mity _ { 2 } , \\mity _ { 4 } , . . . , \\mity _ { 2 \\mitk } ) \\end{align*}",
"\\begin{equation*} ( \\mitt \\times _ { \\hslash } \\mits ) _ { \\mitn } = \\sum _ { \\mitm + \\mitl - 2 \\mitk = \\mitn } \\hslash ^ { \\mitk } \\mitt _ { \\mitm } \\otimes _ { \\mitk } \\mits _ { \\mitl } . \\end{equation*}",
"\\begin{align*} \\underset { \\hslash \\rightarrow 0 } { \\operatorname { l i m } } \\mitphi ( \\mitt ) \\times _ { \\hslash } \\mitphi ( \\mits ) & = & \\underset { \\hslash \\rightarrow 0 } { \\operatorname { l i m } } \\mitphi ( \\sum _ { \\mitn } \\hslash ^ { \\mitn } \\mitt \\otimes _ { \\mitn } \\mits ) \\\\ & = & \\mitphi ( \\mitt \\otimes _ { 0 } \\mits ) = \\mitphi ( \\mitt ) \\cdot \\mitphi ( \\mits ) \\end{align*}",
"\\begin{equation*} \\underset { \\hslash \\rightarrow 0 } { \\operatorname { l i m } } \\frac { 1 } { \\miti \\hslash } [ \\mitphi ( \\mitt ) , \\mitphi ( \\mits ) ] _ { \\hslash } = \\mitphi ( \\mitt \\otimes _ { 1 } \\mits - \\mits \\otimes _ { 1 } \\mitt ) = \\{ \\mitphi ( \\mitt ) , \\mitphi ( \\mits ) \\} \\end{equation*}"
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0001129_page11 | {
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"$(\\bar {\\cal W},\\times _\\hbar )$",
"$\\varphi _{\\rm class}$",
"$\\omega _0(t)=t_{0}$",
"$\\hbar \\ne 0$",
"$\\hbar =0$",
"$n$",
"$T$",
"$T_n\\vert _{{\\cal U}_n}$",
"$\\Delta _n=\\{(x_1,...,x_n)\\in \\RR ^{4n}|x_1=x_2=...=x_n\\}$",
"$\\Delta _n$",
"${\\cal L}$",
"$\\phi $",
"$\\phi $",
"$T(A_1(x_1)...A_n(x_n))^*=\\bar T(A_1^*(x_1)... A_n^*(x_n))$",
"$\\quad [T(A_1(x_1)...A_n(x_n)),\\phi (x)]=$",
"$\\quad \\quad \\quad \\quad =i\\hbar \\sum _{k=1}^n T(A_1(x_1)... \\frac {\\d A_k}{\\d \\phi }(x_k)...A_n(x_n))\\Delta (x_k-x)$",
"$\\quad (\\w _x+m^2)T(A_1(x_1)...A_n(x_n)\\phi (x))=$",
"$\\quad \\quad \\quad \\quad \\quad \\quad =-i\\hbar \\sum _{k=1}^nT(A_1(x_1)... \\frac {\\d A_k}{\\d \\phi }(x_k)...A_n(x_n))\\delta (x_k-x)$",
"$[\\phi (x),\\phi (y)]=i\\hbar \\Delta (x-y)$",
"$*$",
"$T$",
"$\\Delta _n$",
"$t_0\\in {\\cal D}'(\\RR ^{4(n-1)}\\setminus \\{0\\})$",
"$t\\in {\\cal D}'(\\RR ^{4(n-1)})$",
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"$t_0(y)$",
"$t(y)$",
"$y=0$",
"${\\rm sd}(t_0)\\leq {\\rm sd}(t)$",
"${\\rm sd}(t_0)<4(n-1)$",
"$\\delta $",
"\\begin {equation} {\\rm sd}(t)\\=d {\\rm inf}\\{\\delta \\in \\RR \\>,\\>\\lim _{\\lambda \\to 0} \\lambda ^\\delta t(\\lambda x)=0\\}.\\label {4.3a} \\end {equation}",
"\\begin {equation} {\\rm sd}(t_0)={\\rm sd}(t).\\label {4.3b} \\end {equation}"
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"$ \\hbar = 0 $",
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"$ \\quad \\quad \\quad \\quad = i \\hbar \\sum _ { k = 1 } ^ { n } T ( A _ { 1 } ( x _ { 1 } ) . . . \\frac { \\partial A _ { k } } { \\partial \\phi } ( x _ { k } ) . . . A _ { n } ( x _ { n } ) ) \\Delta ( x _ { k } - x ) $",
"$ \\quad ( \\square { } _ { x } + m ^ { 2 } ) T ( A _ { 1 } ( x _ { 1 } ) . . . A _ { n } ( x _ { n } ) \\phi ( x ) ) = $",
"$ \\quad \\quad \\quad \\quad \\quad \\quad = - i \\hbar \\sum _ { k = 1 } ^ { n } T ( A _ { 1 } ( x _ { 1 } ) . . . \\frac { \\partial A _ { k } } { \\partial \\phi } ( x _ { k } ) . . . A _ { n } ( x _ { n } ) ) \\delta ( x _ { k } - x ) $",
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0001129_page12 | {
"latex": [
"$({\\rm sd}(t_0)-4(n-1))$",
"$t$",
"$C_a$",
"${\\rm dim}({\\cal L})\\leq 4$",
"$C_a$",
"$\\phi $",
"$g,f\\in {\\cal D}(\\RR ^4,{\\cal V})$",
"$R$",
"$n$",
"$m$",
"$R_{n,m}$",
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"$(a_1\\times _\\hbar ...\\times _\\hbar a_n)^c$",
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"\\begin {equation} t(y)+\\sum _{|a|\\leq {\\rm sd}(t_0)-4(n-1)}C_a\\d ^a\\delta (y)\\label {4.3c} \\end {equation}",
"\\begin {equation} S_{g}(f)=\\sum _{n,m}\\frac {i^{n+m}}{n!m!}R_{n,m} (g^{\\otimes n};f^{\\otimes m}), \\label {E:retarded products} \\end {equation}",
"\\begin {equation} R_{n,m}(g^{\\otimes n};f^{\\otimes m})= \\sum _{k=0}^{n}(-1)^k\\frac {n!}{k!(n-k)!} \\bar {T}_{k}(g^{\\otimes k})\\times _\\hbar T_{n-k+m}(g^{\\otimes (n-k)} \\otimes f^{\\otimes m}).\\label {R=T} \\end {equation}",
"\\begin {equation} {\\rm supp}\\>R_{n,m}\\bigl (...\\bigr )\\subset \\{(y_1,...y_n,x_1,...,x_m)\\>,\\>\\{y_1,...y_n\\}\\subset (\\{x_1,...,x_m\\} +\\bar V_-)\\} \\ .\\label {L15a} \\end {equation}",
"\\begin {equation} a\\times _\\hbar b=\\sum _{n\\geq 0}\\hbar ^n M_n(a,b),\\label {W2a} \\end {equation}",
"\\begin {equation} :\\prod _{i\\in I}\\varphi (x_i):\\cdot :\\prod _{j\\in J}\\varphi (x_j): = :\\prod _{i\\in I}\\varphi (x_i)\\prod _{j\\in J}\\varphi (x_j):\\label {C} \\end {equation}",
"\\begin {equation} (a_1\\times _\\hbar ...\\times _\\hbar a_n)^c\\=d (a_1\\times _\\hbar ... \\times _\\hbar a_n) -\\sum _{|P|\\geq 2}\\prod _{J\\in P}(a_{j_1}\\times _\\hbar ...\\times _\\hbar a_{j_{|J|}})^c,\\label {conn} \\end {equation}"
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"\\begin{equation*} t ( y ) + \\sum _ { \\vert a \\vert \\leq s d ( t _ { 0 } ) - 4 ( n - 1 ) } C _ { a } \\partial ^ { a } \\delta ( y ) \\end{equation*}",
"\\begin{equation*} S _ { g } ( f ) = \\sum _ { n , m } \\frac { i ^ { n + m } } { n ! m ! } R _ { n , m } ( g ^ { \\otimes n } ; f ^ { \\otimes m } ) , \\end{equation*}",
"\\begin{equation*} R _ { n , m } ( g ^ { \\otimes n } ; f ^ { \\otimes m } ) = \\sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \\frac { n ! } { k ! ( n - k ) ! } \\bar { T } _ { k } ( g ^ { \\otimes k } ) \\times _ { \\hbar } T _ { n - k + m } ( g ^ { \\otimes ( n - k ) } \\otimes f ^ { \\otimes m } ) . \\end{equation*}",
"\\begin{equation*} s u p p \\> R _ { n , m } ( . . . ) \\subset \\{ ( y _ { 1 } , . . . y _ { n } , x _ { 1 } , . . . , x _ { m } ) \\> , \\> \\{ y _ { 1 } , . . . y _ { n } \\} \\subset ( \\{ x _ { 1 } , . . . , x _ { m } \\} + \\bar { V } _ { - } ) \\} ~ . \\end{equation*}",
"\\begin{equation*} a \\times _ { \\hbar } b = \\sum _ { n \\geq 0 } \\hbar ^ { n } M _ { n } ( a , b ) , \\end{equation*}",
"\\begin{equation*} : \\prod _ { i \\in I } \\varphi ( x _ { i } ) : \\cdot : \\prod _ { j \\in J } \\varphi ( x _ { j } ) : = : \\prod _ { i \\in I } \\varphi ( x _ { i } ) \\prod _ { j \\in J } \\varphi ( x _ { j } ) : \\end{equation*}",
"\\begin{equation*} ( a _ { 1 } \\times _ { \\hbar } . . . \\times _ { \\hbar } a _ { n } ) ^ { c } \\, \\overset { d e f } { = } \\, ( a _ { 1 } \\times _ { \\hbar } . . . \\times _ { \\hbar } a _ { n } ) - \\sum _ { \\vert P \\vert \\geq 2 } \\prod _ { J \\in P } ( a _ { j _ { 1 } } \\times _ { \\hbar } . . . \\times _ { \\hbar } a _ { j _ { \\vert J \\vert } } ) ^ { c } , \\end{equation*}"
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"\\begin{equation*} \\mitS _ { \\mitg } ( \\mitf ) = \\sum _ { \\mitn , \\mitm } \\frac { \\miti ^ { \\mitn + \\mitm } } { \\mitn ! \\mitm ! } \\mitR _ { \\mitn , \\mitm } ( \\mitg ^ { \\otimes \\mitn } ; \\mitf ^ { \\otimes \\mitm } ) , \\end{equation*}",
"\\begin{equation*} \\mitR _ { \\mitn , \\mitm } ( \\mitg ^ { \\otimes \\mitn } ; \\mitf ^ { \\otimes \\mitm } ) = \\sum _ { \\mitk = 0 } ^ { \\mitn } ( - 1 ) ^ { \\mitk } \\frac { \\mitn ! } { \\mitk ! ( \\mitn - \\mitk ) ! } \\bar { \\mitT } _ { \\mitk } ( \\mitg ^ { \\otimes \\mitk } ) \\times _ { \\hslash } \\mitT _ { \\mitn - \\mitk + \\mitm } ( \\mitg ^ { \\otimes ( \\mitn - \\mitk ) } \\otimes \\mitf ^ { \\otimes \\mitm } ) . \\end{equation*}",
"\\begin{equation*} \\mathrm { s u p p } \\> \\mitR _ { \\mitn , \\mitm } \\big ( . . . \\big ) \\subset \\{ ( \\mity _ { 1 } , . . . \\mity _ { \\mitn } , \\mitx _ { 1 } , . . . , \\mitx _ { \\mitm } ) \\> , \\> \\{ \\mity _ { 1 } , . . . \\mity _ { \\mitn } \\} \\subset ( \\{ \\mitx _ { 1 } , . . . , \\mitx _ { \\mitm } \\} + \\bar { \\mitV } _ { - } ) \\} ~ . \\end{equation*}",
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"\\begin{equation*} : \\prod _ { \\miti \\in \\mitI } \\mitvarphi ( \\mitx _ { \\miti } ) : \\cdot : \\prod _ { \\mitj \\in \\mitJ } \\mitvarphi ( \\mitx _ { \\mitj } ) : = : \\prod _ { \\miti \\in \\mitI } \\mitvarphi ( \\mitx _ { \\miti } ) \\prod _ { \\mitj \\in \\mitJ } \\mitvarphi ( \\mitx _ { \\mitj } ) : \\end{equation*}",
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0001129_page13 | {
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"$\\{j_1,...,j_{|J|}\\}=J$",
"$j_1<...<j_{|J|}$",
"$P$",
"$\\{1,...,n\\}$",
"$\\prod $",
"$T_n^c$",
"$\\bar T_n^c\\equiv (\\bar T_n)^c$",
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"${\\cal O} (\\hbar ^0)$",
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"$a_1\\times _\\hbar ...\\times _\\hbar a_n$",
"$\\Delta _+$",
"$\\hbar $",
"$\\sim \\hbar ^0$",
"$a_1,...,a_n$",
"$n$",
"$1$",
"$0$",
"$(n-1)$",
"${\\cal O}(\\hbar ^{n-1})$",
"$\\quad \\w $",
"${\\cal B}\\ni A_1,...,A_n={\\cal O}(\\hbar ^0)$",
"$x_i\\not = x_j,\\> \\forall 1\\leq i<j\\leq n$",
"$\\pi \\in {\\cal S}_n$",
"$n$",
"$T^c$",
"$n$",
"$\\Delta _n$",
"$T(J):=T(\\prod _{j\\in J}A_j(x_j)),\\> J\\subset \\{1,... ,n\\}$",
"$I\\subset \\{1,...,n\\},\\> I\\not =\\emptyset ,\\>I^c\\not =\\emptyset $",
"$\\sqcup $",
"$k=0$",
"$(r+s)$",
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"$T^c(I_l)$",
"$T^c(J_m)$",
"$\\sum _{l=1}^r (|I_l|-1)+\\sum _{m=1}^r (|J_m|-1)+(r+s-1)=n-1$",
"$\\hbar $",
"$\\hbar $",
"${\\cal D}(\\RR ^{4n}\\setminus \\Delta _n)$",
"$(T_n-T_n^c)$",
"$<n$",
"$\\Delta _n$",
"$T_n$",
"$\\Delta _n$",
"\\begin {equation} T_n^c(f_1\\otimes ...\\otimes f_n)\\=d T_n(f_1\\otimes ...\\otimes f_n) -\\sum _{|P|\\geq 2}\\prod _{p\\in P}T_{|p|}^c(\\otimes _{j\\in p}f_j),\\label {T^c} \\end {equation}",
"\\begin {equation} (a_1\\times _\\hbar ...\\times _\\hbar a_n)^c={\\cal O}(\\hbar ^{n-1}). \\end {equation}",
"\\begin {equation} T^c\\bigl (A_1(x_1)...A_n(x_n)\\bigr )=(A_{\\pi 1}(x_{\\pi 1})\\times _\\hbar ... \\times _\\hbar A_{\\pi n}(x_{\\pi n}))^c={\\cal O}(\\hbar ^{n-1}). \\label {ordnung:hbar} \\end {equation}",
"\\begin {equation} T^c\\bigl (A_1(x_1)...A_n(x_n)\\bigr )={\\cal O}(\\hbar ^{n-1})\\quad \\quad {\\rm on}\\quad {\\cal D}(\\RR ^{4n}).\\label {T^c:hbar} \\end {equation}",
"\\begin {eqnarray} T\\bigl (A_1(x_1)...A_n(x_n)\\bigr )= T(I)\\times _\\hbar T(I^c)= \\sum _{r=1}^{{}|I|}\\sum _{s=1}^{{}|I^c|}\\sum _{I_1\\sqcup ...\\sqcup I_r=I} \\sum _{J_1\\sqcup ...\\sqcup J_s=I^c}\\\\ \\sum _{k\\geq 0}\\hbar ^k M_k\\Bigl (T^c(I_1)\\cdot ...\\cdot T^c(I_r), T^c(J_1)\\cdot ...\\cdot T^c(J_s)\\Bigr ),\\end {eqnarray}"
],
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"$ j _ { 1 } < . . . < j _ { \\vert J \\vert } $",
"$ P $",
"$ \\{ 1 , . . . , n \\} $",
"$ \\prod $",
"$ T _ { n } ^ { c } $",
"$ \\bar { T } _ { n } ^ { c } \\equiv ( \\bar { T } _ { n } ) ^ { c } $",
"$ a _ { 1 } , . . . , a _ { n } $",
"$ O ( \\hbar ^ { 0 } ) $",
"$ a _ { i } $",
"$ a _ { 1 } \\times _ { \\hbar } . . . \\times _ { \\hbar } a _ { n } $",
"$ \\Delta _ { + } $",
"$ \\hbar $",
"$ \\sim \\hbar ^ { 0 } $",
"$ a _ { 1 } , . . . , a _ { n } $",
"$ n $",
"$ 1 $",
"$ 0 $",
"$ ( n - 1 ) $",
"$ O ( \\hbar ^ { n - 1 } ) $",
"$ \\quad \\square $",
"$ B \\ni A _ { 1 } , . . . , A _ { n } = O ( \\hbar ^ { 0 } ) $",
"$ x _ { i } \\ne x _ { j } , \\> \\forall 1 \\leq i < j \\leq n $",
"$ \\pi \\in S _ { n } $",
"$ n $",
"$ T ^ { c } $",
"$ n $",
"$ \\Delta _ { n } $",
"$ T ( J ) : = T ( \\prod _ { j \\in J } A _ { j } ( x _ { j } ) ) , \\> J \\subset \\{ 1 , . . . , n \\} $",
"$ I \\subset \\{ 1 , . . . , n \\} , \\> I \\ne \\emptyset , \\> I ^ { c } \\ne \\emptyset $",
"$ \\sqcup $",
"$ k = 0 $",
"$ ( r + s ) $",
"$ k \\geq ( r + s - 1 ) $",
"$ T ^ { c } ( I _ { l } ) $",
"$ T ^ { c } ( J _ { m } ) $",
"$ \\sum _ { l = 1 } ^ { r } ( \\vert I _ { l } \\vert - 1 ) + \\sum _ { m = 1 } ^ { r } ( \\vert J _ { m } \\vert - 1 ) + ( r + s - 1 ) = n - 1 $",
"$ \\hbar $",
"$ \\hbar $",
"$ D ( R ^ { 4 n } \\setminus \\Delta _ { n } ) $",
"$ ( T _ { n } - T _ { n } ^ { c } ) $",
"$ < n $",
"$ \\Delta _ { n } $",
"$ T _ { n } $",
"$ \\Delta _ { n } $",
"\\begin{equation*} T _ { n } ^ { c } ( f _ { 1 } \\otimes . . . \\otimes f _ { n } ) \\, \\overset { d e f } { = } \\, T _ { n } ( f _ { 1 } \\otimes . . . \\otimes f _ { n } ) - \\sum _ { \\vert P \\vert \\geq 2 } \\prod _ { p \\in P } T _ { \\vert p \\vert } ^ { c } ( \\otimes _ { j \\in p } f _ { j } ) , \\end{equation*}",
"\\begin{equation*} ( a _ { 1 } \\times _ { \\hbar } . . . \\times _ { \\hbar } a _ { n } ) ^ { c } = O ( \\hbar ^ { n - 1 } ) . \\end{equation*}",
"\\begin{equation*} T ^ { c } ( A _ { 1 } ( x _ { 1 } ) . . . A _ { n } ( x _ { n } ) ) = ( A _ { \\pi 1 } ( x _ { \\pi 1 } ) \\times _ { \\hbar } . . . \\times _ { \\hbar } A _ { \\pi n } ( x _ { \\pi n } ) ) ^ { c } = O ( \\hbar ^ { n - 1 } ) . \\end{equation*}",
"\\begin{equation*} T ^ { c } ( A _ { 1 } ( x _ { 1 } ) . . . A _ { n } ( x _ { n } ) ) = O ( \\hbar ^ { n - 1 } ) \\quad \\quad o n \\quad D ( R ^ { 4 n } ) . \\end{equation*}",
"\\begin{align*} T ( A _ { 1 } ( x _ { 1 } ) . . . A _ { n } ( x _ { n } ) ) = T ( I ) \\times _ { \\hbar } T ( I ^ { c } ) = \\sum _ { r = 1 } ^ { \\vert I \\vert } \\sum _ { s = 1 } ^ { \\vert I ^ { c } \\vert } \\sum _ { I _ { 1 } \\sqcup . . . \\sqcup I _ { r } = I } \\sum _ { J _ { 1 } \\sqcup . . . \\sqcup J _ { s } = I ^ { c } } \\\\ \\sum _ { k \\geq 0 } \\hbar ^ { k } M _ { k } ( T ^ { c } ( I _ { 1 } ) \\cdot . . . \\cdot T ^ { c } ( I _ { r } ) , T ^ { c } ( J _ { 1 } ) \\cdot . . . \\cdot T ^ { c } ( J _ { s } ) ) , \\end{align*}"
],
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"$ \\{ \\mitj _ { 1 } , . . . , \\mitj _ { \\vert \\mitJ \\vert } \\} = \\mitJ $",
"$ \\mitj _ { 1 } < . . . < \\mitj _ { \\vert \\mitJ \\vert } $",
"$ \\mitP $",
"$ \\{ 1 , . . . , \\mitn \\} $",
"$ \\prod $",
"$ \\mitT _ { \\mitn } ^ { \\mitc } $",
"$ \\bar { \\mitT } _ { \\mitn } ^ { \\mitc } \\equiv ( \\bar { \\mitT } _ { \\mitn } ) ^ { \\mitc } $",
"$ \\mita _ { 1 } , . . . , \\mita _ { \\mitn } $",
"$ \\mitO ( \\hslash ^ { 0 } ) $",
"$ \\mita _ { \\miti } $",
"$ \\mita _ { 1 } \\times _ { \\hslash } . . . \\times _ { \\hslash } \\mita _ { \\mitn } $",
"$ \\mupDelta _ { + } $",
"$ \\hslash $",
"$ \\sim \\hslash ^ { 0 } $",
"$ \\mita _ { 1 } , . . . , \\mita _ { \\mitn } $",
"$ \\mitn $",
"$ 1 $",
"$ 0 $",
"$ ( \\mitn - 1 ) $",
"$ \\mitO ( \\hslash ^ { \\mitn - 1 } ) $",
"$ \\quad \\square $",
"$ \\mitB \\ni \\mitA _ { 1 } , . . . , \\mitA _ { \\mitn } = \\mitO ( \\hslash ^ { 0 } ) $",
"$ \\mitx _ { \\miti } \\ne \\mitx _ { \\mitj } , \\> \\forall 1 \\leq \\miti < \\mitj \\leq \\mitn $",
"$ \\mitpi \\in \\mitS _ { \\mitn } $",
"$ \\mitn $",
"$ \\mitT ^ { \\mitc } $",
"$ \\mitn $",
"$ \\mupDelta _ { \\mitn } $",
"$ \\mitT ( \\mitJ ) : = \\mitT ( \\prod _ { \\mitj \\in \\mitJ } \\mitA _ { \\mitj } ( \\mitx _ { \\mitj } ) ) , \\> \\mitJ \\subset \\{ 1 , . . . , \\mitn \\} $",
"$ \\mitI \\subset \\{ 1 , . . . , \\mitn \\} , \\> \\mitI \\ne \\varnothing , \\> \\mitI ^ { \\mitc } \\ne \\varnothing $",
"$ \\sqcup $",
"$ \\mitk = 0 $",
"$ ( \\mitr + \\mits ) $",
"$ \\mitk \\geq ( \\mitr + \\mits - 1 ) $",
"$ \\mitT ^ { \\mitc } ( \\mitI _ { \\mitl } ) $",
"$ \\mitT ^ { \\mitc } ( \\mitJ _ { \\mitm } ) $",
"$ \\sum _ { \\mitl = 1 } ^ { \\mitr } ( \\vert \\mitI _ { \\mitl } \\vert - 1 ) + \\sum _ { \\mitm = 1 } ^ { \\mitr } ( \\vert \\mitJ _ { \\mitm } \\vert - 1 ) + ( \\mitr + \\mits - 1 ) = \\mitn - 1 $",
"$ \\hslash $",
"$ \\hslash $",
"$ \\mitD ( \\BbbR ^ { 4 \\mitn } \\setminus \\mupDelta _ { \\mitn } ) $",
"$ ( \\mitT _ { \\mitn } - \\mitT _ { \\mitn } ^ { \\mitc } ) $",
"$ < \\mitn $",
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"$ \\mitT _ { \\mitn } $",
"$ \\mupDelta _ { \\mitn } $",
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"\\begin{equation*} ( \\mita _ { 1 } \\times _ { \\hslash } . . . \\times _ { \\hslash } \\mita _ { \\mitn } ) ^ { \\mitc } = \\mitO ( \\hslash ^ { \\mitn - 1 } ) . \\end{equation*}",
"\\begin{equation*} \\mitT ^ { \\mitc } \\big ( \\mitA _ { 1 } ( \\mitx _ { 1 } ) . . . \\mitA _ { \\mitn } ( \\mitx _ { \\mitn } ) \\big ) = ( \\mitA _ { \\mitpi 1 } ( \\mitx _ { \\mitpi 1 } ) \\times _ { \\hslash } . . . \\times _ { \\hslash } \\mitA _ { \\mitpi \\mitn } ( \\mitx _ { \\mitpi \\mitn } ) ) ^ { \\mitc } = \\mitO ( \\hslash ^ { \\mitn - 1 } ) . \\end{equation*}",
"\\begin{equation*} \\mitT ^ { \\mitc } \\big ( \\mitA _ { 1 } ( \\mitx _ { 1 } ) . . . \\mitA _ { \\mitn } ( \\mitx _ { \\mitn } ) \\big ) = \\mitO ( \\hslash ^ { \\mitn - 1 } ) \\quad \\quad \\mathrm { o n } \\quad \\mitD ( \\BbbR ^ { 4 \\mitn } ) . \\end{equation*}",
"\\begin{align*} \\mitT \\big ( \\mitA _ { 1 } ( \\mitx _ { 1 } ) . . . \\mitA _ { \\mitn } ( \\mitx _ { \\mitn } ) \\big ) = \\mitT ( \\mitI ) \\times _ { \\hslash } \\mitT ( \\mitI ^ { \\mitc } ) = \\sum _ { \\mitr = 1 } ^ { \\vert \\mitI \\vert } \\sum _ { \\mits = 1 } ^ { \\vert \\mitI ^ { \\mitc } \\vert } \\sum _ { \\mitI _ { 1 } \\sqcup . . . \\sqcup \\mitI _ { \\mitr } = \\mitI } \\sum _ { \\mitJ _ { 1 } \\sqcup . . . \\sqcup \\mitJ _ { \\mits } = \\mitI ^ { \\mitc } } \\\\ \\sum _ { \\mitk \\geq 0 } \\hslash ^ { \\mitk } \\mitM _ { \\mitk } \\Big ( \\mitT ^ { \\mitc } ( \\mitI _ { 1 } ) \\cdot . . . \\cdot \\mitT ^ { \\mitc } ( \\mitI _ { \\mitr } ) , \\mitT ^ { \\mitc } ( \\mitJ _ { 1 } ) \\cdot . . . \\cdot \\mitT ^ { \\mitc } ( \\mitJ _ { \\mits } ) \\Big ) , \\end{align*}"
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0001129_page14 | {
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"$T_n^c$",
"$T_n^c$",
"$\\bar T_n^c$",
"$T_n$",
"$\\bar T_n$",
"$T_n^c$",
"$\\bar T_n^c$",
"$<\\Omega ,T^c(A_1...A_n)\\Omega >$",
"$A_j$",
"$\\Omega $",
"$T_n$",
"$\\bar T_n^c$",
"${\\cal D}(\\RR ^4,{\\cal V})\\ni f_j,g_k= {\\cal O}(\\hbar ^0)$",
"$R_{n,m}(f_1\\otimes ...\\otimes f_n; g_1\\otimes ...\\otimes g_m)$",
"$f_j$",
"$g_k$",
"$R_{n,m}(f_1\\otimes ...\\otimes f_n;g_1\\otimes ...\\otimes g_m) ={\\cal O}(\\hbar ^n)$",
"$R_{n,m}(Y;X),\\>Y\\equiv \\{y_1,...,y_n\\},\\>X\\equiv \\{x_1,...,x_m\\}$",
"$R_{n,m}(Y;X),\\>Y\\equiv \\{y_1,...,y_n\\},\\>X\\equiv \\{x_1,...,x_m\\}$",
"$J\\subset Y$",
"$(Y\\setminus J)\\cup X$",
"$R_{n,m}(Y;X)$",
"$\\sum _{P\\subset J}(-1)^{|P|} \\bar T(P) \\times _\\hbar T(J\\setminus P)=0$",
"$S^{-1}S={\\bf 1}$",
"$J$",
"$R$",
"$T$",
"$\\bar T$",
"$\\prod $",
"$\\sqcup $",
"$\\hbar $",
"$m=1$",
"$n\\geq |P|+|Q|-1$",
"$m\\geq 1$",
"$\\hbar $",
"$m$",
"$g_j$",
"$m=1$",
"$n\\geq |P|+|Q|-m$",
"$(|I|-|P|)+(|I^c|+m-|Q|)+(|P|+|Q|-m)=n$",
"$\\quad \\w $",
"\\begin {equation} \\sum _{I\\subset Y}(-1)^{|I|}\\Bigl (\\bar T(I\\cap J^c)\\bar T(I\\cap J)\\Bigr ) \\times _\\hbar \\Bigl (T(I^c\\cap J)T(I^c\\cap J^c,X)\\Bigr ). \\end {equation}",
"\\begin {eqnarray} R_{n,m}(f_1\\otimes ...\\otimes f_n;g_1\\otimes ...\\otimes g_m)= \\sum _{I\\subset \\{1,...,n\\}}(-1)^{|I|}\\sum _{P\\in {\\rm Part}(I)} \\sum _{Q\\in {\\rm Part}(I^c\\sqcup \\{1,...,m\\})}\\\\ \\Bigl (\\prod _{p\\in P}\\bar T_{|p|}^c(\\otimes _{i\\in p}f_i)\\Bigr )\\times _\\hbar \\Bigl (\\prod _{q\\in Q}T_{|q|}^c(\\otimes _{i\\in q}f_i\\otimes \\otimes _{j\\in q}g_j)\\Bigr ) \\end {eqnarray}",
"\\begin {equation} \\prod _{p\\in P}\\bar T_{|p|}^c(\\otimes _{i\\in p}f_i)={\\cal O}(\\hbar ^{|I|-|P|}), \\quad \\prod _{q\\in Q}T_{|q|}^c(\\otimes _{i\\in q}f_i\\otimes \\otimes _{j\\in q}g_j)={\\cal O}(\\hbar ^{|I^c|+m-|Q|}).\\label {prod:hbar} \\end {equation}",
"\\begin {equation} \\Bigl (\\prod _{p\\in P}\\bar T_{|p|}^c(...)\\Bigr )\\times _\\hbar \\Bigl (\\prod _{q\\in Q}T_{|q|}^c(...)\\Bigr )=\\sum _{n\\geq 0}\\hbar ^n M_n\\Bigl (\\prod _{p\\in P}\\bar T_{|p|}^c(...),\\prod _{q\\in Q}T_{|q|}^c(...)\\Bigr ) \\end {equation}"
],
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"$ T _ { n } ^ { c } $",
"$ T _ { n } ^ { c } $",
"$ \\bar { T } _ { n } ^ { c } $",
"$ T _ { n } $",
"$ \\bar { T } _ { n } $",
"$ T _ { n } ^ { c } $",
"$ \\bar { T } _ { n } ^ { c } $",
"$ < \\Omega , T ^ { c } ( A _ { 1 } . . . A _ { n } ) \\Omega > $",
"$ A _ { j } $",
"$ \\Omega $",
"$ T _ { n } $",
"$ \\bar { T } _ { n } ^ { c } $",
"$ D ( R ^ { 4 } , V ) \\ni f _ { j } , g _ { k } = O ( \\hbar ^ { 0 } ) $",
"$ R _ { n , m } ( f _ { 1 } \\otimes . . . \\otimes f _ { n } ; g _ { 1 } \\otimes . . . \\otimes g _ { m } ) $",
"$ f _ { j } $",
"$ g _ { k } $",
"$ R _ { n , m } ( f _ { 1 } \\otimes . . . \\otimes f _ { n } ; g _ { 1 } \\otimes . . . \\otimes g _ { m } ) = O ( \\hbar ^ { n } ) $",
"$ R _ { n , m } ( Y ; X ) , \\> Y \\equiv \\{ y _ { 1 } , . . . , y _ { n } \\} , \\> X \\equiv \\{ x _ { 1 } , . . . , x _ { m } \\} $",
"$ R _ { n , m } ( Y ; X ) , \\> Y \\equiv \\{ y _ { 1 } , . . . , y _ { n } \\} , \\> X \\equiv \\{ x _ { 1 } , . . . , x _ { m } \\} $",
"$ J \\subset Y $",
"$ ( Y \\setminus J ) \\cup X $",
"$ R _ { n , m } ( Y ; X ) $",
"$ \\sum _ { P \\subset J } ( - 1 ) ^ { \\vert P \\vert } \\bar { T } ( P ) \\times _ { \\hbar } T ( J \\setminus P ) = 0 $",
"$ S ^ { - 1 } S = 1 $",
"$ J $",
"$ R $",
"$ T $",
"$ \\bar { T } $",
"$ \\prod $",
"$ \\sqcup $",
"$ \\hbar $",
"$ m = 1 $",
"$ n \\geq \\vert P \\vert + \\vert Q \\vert - 1 $",
"$ m \\geq 1 $",
"$ \\hbar $",
"$ m $",
"$ g _ { j } $",
"$ m = 1 $",
"$ n \\geq \\vert P \\vert + \\vert Q \\vert - m $",
"$ ( \\vert I \\vert - \\vert P \\vert ) + ( \\vert I ^ { c } \\vert + m - \\vert Q \\vert ) + ( \\vert P \\vert + \\vert Q \\vert - m ) = n $",
"$ \\quad \\square $",
"\\begin{equation*} \\sum _ { I \\subset Y } ( - 1 ) ^ { \\vert I \\vert } ( \\bar { T } ( I \\cap J ^ { c } ) \\bar { T } ( I \\cap J ) ) \\times _ { \\hbar } ( T ( I ^ { c } \\cap J ) T ( I ^ { c } \\cap J ^ { c } , X ) ) . \\end{equation*}",
"\\begin{align*} R _ { n , m } ( f _ { 1 } \\otimes . . . \\otimes f _ { n } ; g _ { 1 } \\otimes . . . \\otimes g _ { m } ) = \\sum _ { I \\subset \\{ 1 , . . . , n \\} } ( - 1 ) ^ { \\vert I \\vert } \\sum _ { P \\in P a r t ( I ) } \\sum _ { Q \\in P a r t ( I ^ { c } \\sqcup \\{ 1 , . . . , m \\} ) } \\\\ ( \\prod _ { p \\in P } \\bar { T } _ { \\vert p \\vert } ^ { c } ( \\otimes _ { i \\in p } f _ { i } ) ) \\times _ { \\hbar } ( \\prod _ { q \\in Q } T _ { \\vert q \\vert } ^ { c } ( \\otimes _ { i \\in q } f _ { i } \\otimes \\otimes _ { j \\in q } g _ { j } ) ) \\end{align*}",
"\\begin{equation*} \\prod _ { p \\in P } \\bar { T } _ { \\vert p \\vert } ^ { c } ( \\otimes _ { i \\in p } f _ { i } ) = O ( \\hbar ^ { \\vert I \\vert - \\vert P \\vert } ) , \\quad \\prod _ { q \\in Q } T _ { \\vert q \\vert } ^ { c } ( \\otimes _ { i \\in q } f _ { i } \\otimes \\otimes _ { j \\in q } g _ { j } ) = O ( \\hbar ^ { \\vert I ^ { c } \\vert + m - \\vert Q \\vert } ) . \\end{equation*}",
"\\begin{equation*} ( \\prod _ { p \\in P } \\bar { T } _ { \\vert p \\vert } ^ { c } ( . . . ) ) \\times _ { \\hbar } ( \\prod _ { q \\in Q } T _ { \\vert q \\vert } ^ { c } ( . . . ) ) = \\sum _ { n \\geq 0 } \\hbar ^ { n } M _ { n } ( \\prod _ { p \\in P } \\bar { T } _ { \\vert p \\vert } ^ { c } ( . . . ) , \\prod _ { q \\in Q } T _ { \\vert q \\vert } ^ { c } ( . . . ) ) \\end{equation*}"
],
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"$ \\mitT _ { \\mitn } ^ { \\mitc } $",
"$ \\mitT _ { \\mitn } ^ { \\mitc } $",
"$ \\bar { \\mitT } _ { \\mitn } ^ { \\mitc } $",
"$ \\mitT _ { \\mitn } $",
"$ \\bar { \\mitT } _ { \\mitn } $",
"$ \\mitT _ { \\mitn } ^ { \\mitc } $",
"$ \\bar { \\mitT } _ { \\mitn } ^ { \\mitc } $",
"$ < \\mupOmega , \\mitT ^ { \\mitc } ( \\mitA _ { 1 } . . . \\mitA _ { \\mitn } ) \\mupOmega > $",
"$ \\mitA _ { \\mitj } $",
"$ \\mupOmega $",
"$ \\mitT _ { \\mitn } $",
"$ \\bar { \\mitT } _ { \\mitn } ^ { \\mitc } $",
"$ \\mitD ( \\BbbR ^ { 4 } , \\mitV ) \\ni \\mitf _ { \\mitj } , \\mitg _ { \\mitk } = \\mitO ( \\hslash ^ { 0 } ) $",
"$ \\mitR _ { \\mitn , \\mitm } ( \\mitf _ { 1 } \\otimes . . . \\otimes \\mitf _ { \\mitn } ; \\mitg _ { 1 } \\otimes . . . \\otimes \\mitg _ { \\mitm } ) $",
"$ \\mitf _ { \\mitj } $",
"$ \\mitg _ { \\mitk } $",
"$ \\mitR _ { \\mitn , \\mitm } ( \\mitf _ { 1 } \\otimes . . . \\otimes \\mitf _ { \\mitn } ; \\mitg _ { 1 } \\otimes . . . \\otimes \\mitg _ { \\mitm } ) = \\mitO ( \\hslash ^ { \\mitn } ) $",
"$ \\mitR _ { \\mitn , \\mitm } ( \\mitY ; \\mitX ) , \\> \\mitY \\equiv \\{ \\mity _ { 1 } , . . . , \\mity _ { \\mitn } \\} , \\> \\mitX \\equiv \\{ \\mitx _ { 1 } , . . . , \\mitx _ { \\mitm } \\} $",
"$ \\mitR _ { \\mitn , \\mitm } ( \\mitY ; \\mitX ) , \\> \\mitY \\equiv \\{ \\mity _ { 1 } , . . . , \\mity _ { \\mitn } \\} , \\> \\mitX \\equiv \\{ \\mitx _ { 1 } , . . . , \\mitx _ { \\mitm } \\} $",
"$ \\mitJ \\subset \\mitY $",
"$ ( \\mitY \\setminus \\mitJ ) \\cup \\mitX $",
"$ \\mitR _ { \\mitn , \\mitm } ( \\mitY ; \\mitX ) $",
"$ \\sum _ { \\mitP \\subset \\mitJ } ( - 1 ) ^ { \\vert \\mitP \\vert } \\bar { \\mitT } ( \\mitP ) \\times _ { \\hslash } \\mitT ( \\mitJ \\setminus \\mitP ) = 0 $",
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"$ \\hslash $",
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"$ \\mitm = 1 $",
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"$ \\quad \\square $",
"\\begin{equation*} \\sum _ { \\mitI \\subset \\mitY } ( - 1 ) ^ { \\vert \\mitI \\vert } \\Big ( \\bar { \\mitT } ( \\mitI \\cap \\mitJ ^ { \\mitc } ) \\bar { \\mitT } ( \\mitI \\cap \\mitJ ) \\Big ) \\times _ { \\hslash } \\Big ( \\mitT ( \\mitI ^ { \\mitc } \\cap \\mitJ ) \\mitT ( \\mitI ^ { \\mitc } \\cap \\mitJ ^ { \\mitc } , \\mitX ) \\Big ) . \\end{equation*}",
"\\begin{align*} \\mitR _ { \\mitn , \\mitm } ( \\mitf _ { 1 } \\otimes . . . \\otimes \\mitf _ { \\mitn } ; \\mitg _ { 1 } \\otimes . . . \\otimes \\mitg _ { \\mitm } ) = \\sum _ { \\mitI \\subset \\{ 1 , . . . , \\mitn \\} } ( - 1 ) ^ { \\vert \\mitI \\vert } \\sum _ { \\mitP \\in \\mathrm { P a r t } ( \\mitI ) } \\sum _ { \\mitQ \\in \\mathrm { P a r t } ( \\mitI ^ { \\mitc } \\sqcup \\{ 1 , . . . , \\mitm \\} ) } \\\\ \\Big ( \\prod _ { \\mitp \\in \\mitP } \\bar { \\mitT } _ { \\vert \\mitp \\vert } ^ { \\mitc } ( \\otimes _ { \\miti \\in \\mitp } \\mitf _ { \\miti } ) \\Big ) \\times _ { \\hslash } \\Big ( \\prod _ { \\mitq \\in \\mitQ } \\mitT _ { \\vert \\mitq \\vert } ^ { \\mitc } ( \\otimes _ { \\miti \\in \\mitq } \\mitf _ { \\miti } \\otimes \\otimes _ { \\mitj \\in \\mitq } \\mitg _ { \\mitj } ) \\Big ) \\end{align*}",
"\\begin{equation*} \\prod _ { \\mitp \\in \\mitP } \\bar { \\mitT } _ { \\vert \\mitp \\vert } ^ { \\mitc } ( \\otimes _ { \\miti \\in \\mitp } \\mitf _ { \\miti } ) = \\mitO ( \\hslash ^ { \\vert \\mitI \\vert - \\vert \\mitP \\vert } ) , \\quad \\prod _ { \\mitq \\in \\mitQ } \\mitT _ { \\vert \\mitq \\vert } ^ { \\mitc } ( \\otimes _ { \\miti \\in \\mitq } \\mitf _ { \\miti } \\otimes \\otimes _ { \\mitj \\in \\mitq } \\mitg _ { \\mitj } ) = \\mitO ( \\hslash ^ { \\vert \\mitI ^ { \\mitc } \\vert + \\mitm - \\vert \\mitQ \\vert } ) . \\end{equation*}",
"\\begin{equation*} \\Big ( \\prod _ { \\mitp \\in \\mitP } \\bar { \\mitT } _ { \\vert \\mitp \\vert } ^ { \\mitc } ( . . . ) \\Big ) \\times _ { \\hslash } \\Big ( \\prod _ { \\mitq \\in \\mitQ } \\mitT _ { \\vert \\mitq \\vert } ^ { \\mitc } ( . . . ) \\Big ) = \\sum _ { \\mitn \\geq 0 } \\hslash ^ { \\mitn } \\mitM _ { \\mitn } \\Big ( \\prod _ { \\mitp \\in \\mitP } \\bar { \\mitT } _ { \\vert \\mitp \\vert } ^ { \\mitc } ( . . . ) , \\prod _ { \\mitq \\in \\mitQ } \\mitT _ { \\vert \\mitq \\vert } ^ { \\mitc } ( . . . ) \\Big ) \\end{equation*}"
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0001129_page15 | {
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"$\\theta {\\cal L}$",
"$\\theta \\in {\\cal D}(\\RR ^4)$",
"${\\cal O}$",
"${\\cal O}$",
"$\\theta ,\\theta '\\in \\Theta ({\\cal O})$",
"$v$",
"$v(\\varphi _{\\theta \\cal L}(x))= \\varphi _{\\theta '\\cal L}(x) $",
"$x\\in {\\cal O}$",
"$\\varphi _{\\cal L}$",
"$\\times _{\\hbar }$",
"$k\\geq 2$",
"$S$",
"\\begin {equation} (\\w +m^2)\\varphi _{\\cal L}(x) =-\\Bigl (\\frac {\\d {\\cal L}}{\\d \\varphi }\\Bigr )_{\\cal L}(x), \\label {E:field equation} \\end {equation}",
"\\begin {equation} \\begin {split} \\{\\varphi _{\\cal L}(0,{\\bf x}),\\varphi _{\\cal L}(0,{\\bf y})\\}=&0= \\{\\dot {\\varphi }_{\\cal L}(0,{\\bf x}),\\dot {\\varphi }_{\\cal L} (0,{\\bf y})\\}\\\\ \\{\\varphi _{\\cal L}(0,{\\bf x}),\\dot {\\varphi }_{\\cal L}(0,{\\bf y})\\}&= \\delta ({\\bf x}-{\\bf y})\\ . \\end {split} \\label {E:canonical Poisson brackets} \\end {equation}",
"\\begin {eqnarray} \\varphi _{\\theta {\\cal L}}(x)=\\sum _{n=0}^{\\infty } \\int _{y_1^0\\leq y_2^0 \\leq ...y_n^0\\leq x^0}dy_1dy_2...dy_n\\,\\theta (y_1)...\\theta (y_n)\\\\ \\{{\\cal L}(y_1),\\{{\\cal L}(y_2),... \\{{\\cal L}(y_n),\\varphi (x)\\}...\\}\\}\\end {eqnarray}",
"\\begin {equation} \\{\\cdot ,\\cdot \\}\\to \\frac {1}{i\\hbar }[\\cdot ,\\cdot ]_{\\hbar }\\label {coPb} \\end {equation}",
"\\begin {eqnarray} \\frac {1}{i\\hbar }[:\\varphi ^n(x):&,&:\\varphi ^m(y):]_{\\hbar }=\\sum _{k=1}^{{}{\\rm min} \\>\\{n,m\\}} (i\\hbar )^{k-1}\\frac {n!m!}{(n-k)!(m-k)!}\\\\ &&\\Bigl (\\Delta _+(x-y)^k-\\Delta _+(y-x)^k\\Bigr ) :\\varphi ^{(n-k)}(x)\\varphi ^{(m-k)}(y): \\end {eqnarray}"
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"$ x \\in O $",
"$ \\varphi _ { L } $",
"$ \\times _ { \\hbar } $",
"$ k \\geq 2 $",
"$ S $",
"\\begin{equation*} ( \\square + m ^ { 2 } ) \\varphi _ { L } ( x ) = - ( \\frac { \\partial L } { \\partial \\varphi } ) _ { L } ( x ) , \\end{equation*}",
"\\begin{align*} \\begin{array}{rl} \\{ \\varphi _ { L } ( 0 , x ) , \\varphi _ { L } ( 0 , y ) \\} = & 0 = \\{ \\dot { \\varphi } _ { L } ( 0 , x ) , \\dot { \\varphi } _ { L } ( 0 , y ) \\} \\\\ \\{ \\varphi _ { L } ( 0 , x ) , \\dot { \\varphi } _ { L } ( 0 , y ) \\} & = \\delta ( x - y ) ~ . \\end{array} \\end{align*}",
"\\begin{align*} \\varphi _ { \\theta L } ( x ) = \\sum _ { n = 0 } ^ { \\infty } \\int _ { y _ { 1 } ^ { 0 } \\leq y _ { 2 } ^ { 0 } \\leq . . . y _ { n } ^ { 0 } \\leq x ^ { 0 } } d y _ { 1 } d y _ { 2 } . . . d y _ { n } \\, \\theta ( y _ { 1 } ) . . . \\theta ( y _ { n } ) \\\\ \\{ L ( y _ { 1 } ) , \\{ L ( y _ { 2 } ) , . . . \\{ L ( y _ { n } ) , \\varphi ( x ) \\} . . . \\} \\} \\end{align*}",
"\\begin{equation*} \\{ \\cdot , \\cdot \\} \\rightarrow \\frac { 1 } { i \\hbar } [ \\cdot , \\cdot ] _ { \\hbar } \\end{equation*}",
"\\begin{align*} \\frac { 1 } { i \\hbar } [ : \\varphi ^ { n } ( x ) : & , & : \\varphi ^ { m } ( y ) : ] _ { \\hbar } = \\sum _ { k = 1 } ^ { m i n \\> \\{ n , m \\} } ( i \\hbar ) ^ { k - 1 } \\frac { n ! m ! } { ( n - k ) ! ( m - k ) ! } \\\\ & & ( \\Delta _ { + } ( x - y ) ^ { k } - \\Delta _ { + } ( y - x ) ^ { k } ) : \\varphi ^ { ( n - k ) } ( x ) \\varphi ^ { ( m - k ) } ( y ) : \\end{align*}"
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"\\begin{align*} \\begin{array}{rl} \\{ \\mitvarphi _ { \\mitL } ( 0 , \\mitx ) , \\mitvarphi _ { \\mitL } ( 0 , \\mity ) \\} = & 0 = \\{ \\dot { \\mitvarphi } _ { \\mitL } ( 0 , \\mitx ) , \\dot { \\mitvarphi } _ { \\mitL } ( 0 , \\mity ) \\} \\\\ \\{ \\mitvarphi _ { \\mitL } ( 0 , \\mitx ) , \\dot { \\mitvarphi } _ { \\mitL } ( 0 , \\mity ) \\} & = \\mitdelta ( \\mitx - \\mity ) ~ . \\end{array} \\end{align*}",
"\\begin{align*} \\mitvarphi _ { \\mittheta \\mitL } ( \\mitx ) = \\sum _ { \\mitn = 0 } ^ { \\infty } \\int _ { \\mity _ { 1 } ^ { 0 } \\leq \\mity _ { 2 } ^ { 0 } \\leq . . . \\mity _ { \\mitn } ^ { 0 } \\leq \\mitx ^ { 0 } } \\mitd \\mity _ { 1 } \\mitd \\mity _ { 2 } . . . \\mitd \\mity _ { \\mitn } \\, \\mittheta ( \\mity _ { 1 } ) . . . \\mittheta ( \\mity _ { \\mitn } ) \\\\ \\{ \\mitL ( \\mity _ { 1 } ) , \\{ \\mitL ( \\mity _ { 2 } ) , . . . \\{ \\mitL ( \\mity _ { \\mitn } ) , \\mitvarphi ( \\mitx ) \\} . . . \\} \\} \\end{align*}",
"\\begin{equation*} \\{ \\cdot , \\cdot \\} \\rightarrow \\frac { 1 } { \\miti \\hslash } [ \\cdot , \\cdot ] _ { \\hslash } \\end{equation*}",
"\\begin{align*} \\frac { 1 } { \\miti \\hslash } [ : \\mitvarphi ^ { \\mitn } ( \\mitx ) : & , & : \\mitvarphi ^ { \\mitm } ( \\mity ) : ] _ { \\hslash } = \\sum _ { \\mitk = 1 } ^ { \\mathrm { m i n } \\> \\{ \\mitn , \\mitm \\} } ( \\miti \\hslash ) ^ { \\mitk - 1 } \\frac { \\mitn ! \\mitm ! } { ( \\mitn - \\mitk ) ! ( \\mitm - \\mitk ) ! } \\\\ & & \\Big ( \\mupDelta _ { + } ( \\mitx - \\mity ) ^ { \\mitk } - \\mupDelta _ { + } ( \\mity - \\mitx ) ^ { \\mitk } \\Big ) : \\mitvarphi ^ { ( \\mitn - \\mitk ) } ( \\mitx ) \\mitvarphi ^ { ( \\mitm - \\mitk ) } ( \\mity ) : \\end{align*}"
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|
0001129_page16 | {
"latex": [
"$\\supp g$",
"$\\supp f, \\supp h$",
"$\\bar {T}_{1}=T_{1}$",
"$m=1$",
"$y_i\\not = y_j\\>\\forall i\\not = j$",
"$R_{n,1}(y_1,...,y_n;x)$",
"${\\cal L}$",
"$y_i$",
"$\\hbar \\rightarrow 0$",
"$R\\bigl (\\hbar ^{-1}{\\cal L}(y_1)...\\hbar ^{-1}{\\cal L} (y_n);\\varphi (x)\\bigr )$",
"$\\hbar $",
"$\\hbar ^{-1}$",
"${\\cal L}$",
"$R_{n,1}((\\theta {\\cal L})^{\\otimes n};f\\varphi )$",
"$R_{n,1}$",
"$\\varphi _{\\cal L}$",
"$\\Bigl ( \\frac {\\d {\\cal L}}{\\d \\varphi } \\Bigr )_{\\cal L}$",
"$\\varphi _{\\cal L}$",
"$S$",
"$S_{\\hbar ^{-1}\\theta {\\cal L}}(f)\\>(f\\in {\\cal D} (\\RR ^4,{\\cal V}))$",
"${\\cal A}_{\\hbar ^{-1}\\theta \\cal L}$",
"$\\hbar $",
"$V\\in {\\cal U}(\\theta ,\\theta ^\\prime )$",
"$S$",
"$\\theta _-\\in {\\cal D}(\\RR ^4)$",
"$(\\theta -\\theta ^\\prime )$",
"$\\theta -\\theta ^\\prime =\\theta _+ +\\theta _-$",
"$\\theta -\\theta ^\\prime =\\theta _+ +\\theta _-$",
"${\\rm supp}\\>\\theta _+ \\cap (C({\\cal O}) +\\bar V_-)=\\emptyset $",
"${\\rm supp}\\>\\theta _- \\cap (C({\\cal O}) +\\bar V_+)=\\emptyset $",
"\\begin {equation} R_{n+1,m}(g\\otimes h^{\\otimes n};f^{\\otimes m}) = -[T_{1}(g),R_{n,m}(h^{\\otimes n};f^{\\otimes m})]_\\hbar \\label {recursionR} \\end {equation}",
"\\begin {eqnarray} R\\bigl ({\\cal L}(y_1)...{\\cal L}(y_n);\\varphi (x)\\bigr )=(-1)^n \\sum _{\\pi \\in {\\cal S}_n}\\Theta (x^0-y_{\\pi n}^0)\\Theta (y_{\\pi n}^0- y_{\\pi (n-1)}^0)...\\\\ \\Theta (y_{\\pi 2}^0-y_{\\pi 1}^0) [{\\cal L}(y_{\\pi 1}),[{\\cal L}(y_{\\pi 2})... [{\\cal L}(y_{\\pi n}),\\varphi (x)]_\\hbar ...]_\\hbar ]_\\hbar .\\end {eqnarray}",
"\\begin {equation} \\varphi _{\\theta {\\cal L}}(h)=\\sum _{n=0}^\\infty \\frac {i^n}{n!\\hbar ^n} R_{n,1}((\\theta {\\cal L})^{\\otimes n};h\\varphi ),\\quad \\quad h\\in {\\cal D}(\\RR ^4),\\label {intfield} \\end {equation}",
"\\begin {equation} (\\w +m^2)\\varphi _{\\cal L}(x)=-\\Bigl ( \\frac {\\d {\\cal L}}{\\d \\varphi } \\Bigr )_{\\cal L}(x).\\label {4.4} \\end {equation}",
"\\begin {equation} V=S_{\\hbar ^{-1}\\theta {\\cal L}}(\\hbar ^{-1}\\theta _-{\\cal L})^{-1} \\in {\\cal U}(\\theta ,\\theta ^\\prime )\\label {E:V=S} \\end {equation}"
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"$ L $",
"$ y _ { i } $",
"$ \\hbar \\rightarrow 0 $",
"$ R ( \\hbar ^ { - 1 } L ( y _ { 1 } ) . . . \\hbar ^ { - 1 } L ( y _ { n } ) ; \\varphi ( x ) ) $",
"$ \\hbar $",
"$ \\hbar ^ { - 1 } $",
"$ L $",
"$ R _ { n , 1 } ( ( \\theta L ) ^ { \\otimes n } ; f \\varphi ) $",
"$ R _ { n , 1 } $",
"$ \\varphi _ { L } $",
"$ ( \\frac { \\partial L } { \\partial \\varphi } ) _ { L } $",
"$ \\varphi _ { L } $",
"$ S $",
"$ S _ { \\hbar ^ { - 1 } \\theta L } ( f ) \\> ( f \\in D ( R ^ { 4 } , V ) ) $",
"$ A _ { \\hbar ^ { - 1 } \\theta L } $",
"$ \\hbar $",
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"$ S $",
"$ \\theta _ { - } \\in D ( R ^ { 4 } ) $",
"$ ( \\theta - \\theta ^ { \\prime } ) $",
"$ \\theta - \\theta ^ { \\prime } = \\theta _ { + } + \\theta _ { - } $",
"$ \\theta - \\theta ^ { \\prime } = \\theta _ { + } + \\theta _ { - } $",
"$ s u p p \\> \\theta _ { + } \\cap ( C ( O ) + \\bar { V } _ { - } ) = \\emptyset $",
"$ s u p p \\> \\theta _ { - } \\cap ( C ( O ) + \\bar { V } _ { + } ) = \\emptyset $",
"\\begin{equation*} R _ { n + 1 , m } ( g \\otimes h ^ { \\otimes n } ; f ^ { \\otimes m } ) = - [ T _ { 1 } ( g ) , R _ { n , m } ( h ^ { \\otimes n } ; f ^ { \\otimes m } ) ] _ { \\hbar } \\end{equation*}",
"\\begin{align*} R ( L ( y _ { 1 } ) . . . L ( y _ { n } ) ; \\varphi ( x ) ) = ( - 1 ) ^ { n } \\sum _ { \\pi \\in S _ { n } } \\Theta ( x ^ { 0 } - y _ { \\pi n } ^ { 0 } ) \\Theta ( y _ { \\pi n } ^ { 0 } - y _ { \\pi ( n - 1 ) } ^ { 0 } ) . . . \\\\ \\Theta ( y _ { \\pi 2 } ^ { 0 } - y _ { \\pi 1 } ^ { 0 } ) [ L ( y _ { \\pi 1 } ) , [ L ( y _ { \\pi 2 } ) . . . [ L ( y _ { \\pi n } ) , \\varphi ( x ) ] _ { \\hbar } . . . ] _ { \\hbar } ] _ { \\hbar } . \\end{align*}",
"\\begin{equation*} \\varphi _ { \\theta L } ( h ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { i ^ { n } } { n ! \\hbar ^ { n } } R _ { n , 1 } ( ( \\theta L ) ^ { \\otimes n } ; h \\varphi ) , \\quad \\quad h \\in D ( R ^ { 4 } ) , \\end{equation*}",
"\\begin{equation*} ( \\square + m ^ { 2 } ) \\varphi _ { L } ( x ) = - ( \\frac { \\partial L } { \\partial \\varphi } ) _ { L } ( x ) . \\end{equation*}",
"\\begin{equation*} V = S _ { \\hbar ^ { - 1 } \\theta L } ( \\hbar ^ { - 1 } \\theta _ { - } L ) ^ { - 1 } \\in U ( \\theta , \\theta ^ { \\prime } ) \\end{equation*}"
],
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"$ \\mathrm { s u p p } \\> \\mitg $",
"$ \\mathrm { s u p p } \\> \\mitf , \\mathrm { s u p p } \\> \\Planckconst $",
"$ \\bar { \\mitT } _ { 1 } = \\mitT _ { 1 } $",
"$ \\mitm = 1 $",
"$ \\mity _ { \\miti } \\ne \\mity _ { \\mitj } \\> \\forall \\miti \\ne \\mitj $",
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"$ \\mitL $",
"$ \\mity _ { \\miti } $",
"$ \\hslash \\rightarrow 0 $",
"$ \\mitR \\big ( \\hslash ^ { - 1 } \\mitL ( \\mity _ { 1 } ) . . . \\hslash ^ { - 1 } \\mitL ( \\mity _ { \\mitn } ) ; \\mitvarphi ( \\mitx ) \\big ) $",
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"$ \\hslash ^ { - 1 } $",
"$ \\mitL $",
"$ \\mitR _ { \\mitn , 1 } ( ( \\mittheta \\mitL ) ^ { \\otimes \\mitn } ; \\mitf \\mitvarphi ) $",
"$ \\mitR _ { \\mitn , 1 } $",
"$ \\mitvarphi _ { \\mitL } $",
"$ \\Big ( \\frac { \\mitpartial \\mitL } { \\mitpartial \\mitvarphi } \\Big ) _ { \\mitL } $",
"$ \\mitvarphi _ { \\mitL } $",
"$ \\mitS $",
"$ \\mitS _ { \\hslash ^ { - 1 } \\mittheta \\mitL } ( \\mitf ) \\> ( \\mitf \\in \\mitD ( \\BbbR ^ { 4 } , \\mitV ) ) $",
"$ \\mitA _ { \\hslash ^ { - 1 } \\mittheta \\mitL } $",
"$ \\hslash $",
"$ \\mitV \\in \\mitU ( \\mittheta , \\mittheta ^ { \\prime } ) $",
"$ \\mitS $",
"$ \\mittheta _ { - } \\in \\mitD ( \\BbbR ^ { 4 } ) $",
"$ ( \\mittheta - \\mittheta ^ { \\prime } ) $",
"$ \\mittheta - \\mittheta ^ { \\prime } = \\mittheta _ { + } + \\mittheta _ { - } $",
"$ \\mittheta - \\mittheta ^ { \\prime } = \\mittheta _ { + } + \\mittheta _ { - } $",
"$ \\mathrm { s u p p } \\> \\mittheta _ { + } \\cap ( \\mitC ( \\mitO ) + \\bar { \\mitV } _ { - } ) = \\varnothing $",
"$ \\mathrm { s u p p } \\> \\mittheta _ { - } \\cap ( \\mitC ( \\mitO ) + \\bar { \\mitV } _ { + } ) = \\varnothing $",
"\\begin{equation*} \\mitR _ { \\mitn + 1 , \\mitm } ( \\mitg \\otimes \\Planckconst ^ { \\otimes \\mitn } ; \\mitf ^ { \\otimes \\mitm } ) = - [ \\mitT _ { 1 } ( \\mitg ) , \\mitR _ { \\mitn , \\mitm } ( \\Planckconst ^ { \\otimes \\mitn } ; \\mitf ^ { \\otimes \\mitm } ) ] _ { \\hslash } \\end{equation*}",
"\\begin{align*} \\mitR \\big ( \\mitL ( \\mity _ { 1 } ) . . . \\mitL ( \\mity _ { \\mitn } ) ; \\mitvarphi ( \\mitx ) \\big ) = ( - 1 ) ^ { \\mitn } \\sum _ { \\mitpi \\in \\mitS _ { \\mitn } } \\mupTheta ( \\mitx ^ { 0 } - \\mity _ { \\mitpi \\mitn } ^ { 0 } ) \\mupTheta ( \\mity _ { \\mitpi \\mitn } ^ { 0 } - \\mity _ { \\mitpi ( \\mitn - 1 ) } ^ { 0 } ) . . . \\\\ \\mupTheta ( \\mity _ { \\mitpi 2 } ^ { 0 } - \\mity _ { \\mitpi 1 } ^ { 0 } ) [ \\mitL ( \\mity _ { \\mitpi 1 } ) , [ \\mitL ( \\mity _ { \\mitpi 2 } ) . . . [ \\mitL ( \\mity _ { \\mitpi \\mitn } ) , \\mitvarphi ( \\mitx ) ] _ { \\hslash } . . . ] _ { \\hslash } ] _ { \\hslash } . \\end{align*}",
"\\begin{equation*} \\mitvarphi _ { \\mittheta \\mitL } ( \\Planckconst ) = \\sum _ { \\mitn = 0 } ^ { \\infty } \\frac { \\miti ^ { \\mitn } } { \\mitn ! \\hslash ^ { \\mitn } } \\mitR _ { \\mitn , 1 } ( ( \\mittheta \\mitL ) ^ { \\otimes \\mitn } ; \\Planckconst \\mitvarphi ) , \\quad \\quad \\Planckconst \\in \\mitD ( \\BbbR ^ { 4 } ) , \\end{equation*}",
"\\begin{equation*} ( \\square + \\mitm ^ { 2 } ) \\mitvarphi _ { \\mitL } ( \\mitx ) = - \\Big ( \\frac { \\mitpartial \\mitL } { \\mitpartial \\mitvarphi } \\Big ) _ { \\mitL } ( \\mitx ) . \\end{equation*}",
"\\begin{equation*} \\mitV = \\mitS _ { \\hslash ^ { - 1 } \\mittheta \\mitL } ( \\hslash ^ { - 1 } \\mittheta _ { - } \\mitL ) ^ { - 1 } \\in \\mitU ( \\mittheta , \\mittheta ^ { \\prime } ) \\end{equation*}"
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|
0001129_page17 | {
"latex": [
"$C({\\cal O})$",
"${\\cal O}$",
"$\\theta $",
"$\\theta ^\\prime $",
"$V$",
"$\\hbar $",
"${\\cal A}$",
"${\\cal A} ({\\cal O})$",
"$\\bigvee _{n\\in \\NN _0}\\hbar ^n {\\cal A}$",
"$\\bigvee _{n\\in \\NN _0}\\hbar ^n {\\cal A}({\\cal O})$",
"${\\cal A}$",
"$\\hbar $",
"$R_{n,m}(...;...)=\\sum _{a=1}^m R_{n,m}^{(a)} (...;...)$",
"$R_{n,m}^{(a)}(...;...)$",
"$a$",
"$a$",
"$\\times _\\hbar $",
"${\\cal O}(\\hbar ^{-1})$",
"$A\\in {\\cal A}({\\cal O})$",
"$V\\times _\\hbar A\\times _\\hbar V^{-1}$",
"$V$",
"$V$",
"$V^{-1}$",
"$A$",
"$V$",
"$A$",
"$V^{-1}$",
"$A$",
"$A$",
"$n$",
"$\\hbar $",
"$V\\times _\\hbar A\\times _\\hbar V^{-1}$",
"$\\hbar $",
"$\\hbar ^{n-1}$",
"$V\\times _\\hbar A\\times _\\hbar V^{-1}$",
"$V$",
"$V^{-1}$",
"$r$",
"$s$",
"$(r+s)$",
"$\\hbar ^{(r+s)}$",
"$Y_1\\sqcup Y_2=Y$",
"$X_1\\sqcup X_2=X$",
"$R(Y,X)$",
"$(Y_1,X_1)$",
"$(Y_2,X_2)$",
"${\\bf 1}=VV^{-1}=VV^*$",
"\\begin {equation} R_{n,m}^{(a)}((\\hbar ^{-1}\\theta {\\cal L})^{\\otimes n}; (\\hbar ^{-1}\\theta _-{\\cal L})^{\\otimes m})={\\cal O}(\\hbar ^{-a}). \\label {R^a} \\end {equation}",
"\\begin {equation} {\\cal A}({\\cal O})\\ni A={\\cal O}(\\hbar ^n)\\quad \\Longrightarrow \\quad V\\times _\\hbar A\\times _\\hbar V^{-1}={\\cal O}(\\hbar ^n)\\label {E:Ad(V)A} \\end {equation}",
"\\begin {eqnarray} \\sum _{I\\subset Y}(-1)^{|I\\cap Y_1|}[\\bar T(I\\cap Y_1)\\times _\\hbar T(I^c\\cap Y_1,X_1)]\\cdot \\\\ (-1)^{|I\\cap Y_2|}[\\bar T(I\\cap Y_2)\\times _\\hbar T(I^c\\cap Y_2,X_2)]\\\\ =R(Y_1,X_1)\\cdot R(Y_2,X_2).\\end {eqnarray}",
"\\begin {equation} \\sum _{Y_1\\sqcup Y_2=Y,\\>X_1\\sqcup X_2=X}(-1)^{(|Y_1|+|X_1|)} R^*(Y_1,X_1)\\times _\\hbar R(Y_2,X_2)=0\\label {V^*V} \\end {equation}"
],
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"$ O $",
"$ \\theta $",
"$ \\theta ^ { \\prime } $",
"$ V $",
"$ \\hbar $",
"$ A $",
"$ A ( O ) $",
"$ \\vee _ { n \\in N _ { 0 } } \\hbar ^ { n } A $",
"$ \\vee _ { n \\in N _ { 0 } } \\hbar ^ { n } A ( O ) $",
"$ A $",
"$ \\hbar $",
"$ R _ { n , m } ( . . . ; . . . ) = \\sum _ { a = 1 } ^ { m } R _ { n , m } ^ { ( a ) } ( . . . ; . . . ) $",
"$ R _ { n , m } ^ { ( a ) } ( . . . ; . . . ) $",
"$ a $",
"$ a $",
"$ \\times _ { \\hbar } $",
"$ O ( \\hbar ^ { - 1 } ) $",
"$ A \\in A ( O ) $",
"$ V \\times _ { \\hbar } A \\times _ { \\hbar } V ^ { - 1 } $",
"$ V $",
"$ V $",
"$ V ^ { - 1 } $",
"$ A $",
"$ V $",
"$ A $",
"$ V ^ { - 1 } $",
"$ A $",
"$ A $",
"$ n $",
"$ \\hbar $",
"$ V \\times _ { \\hbar } A \\times _ { \\hbar } V ^ { - 1 } $",
"$ \\hbar $",
"$ \\hbar ^ { n - 1 } $",
"$ V \\times _ { \\hbar } A \\times _ { \\hbar } V ^ { - 1 } $",
"$ V $",
"$ V ^ { - 1 } $",
"$ r $",
"$ s $",
"$ ( r + s ) $",
"$ \\hbar ^ { ( r + s ) } $",
"$ Y _ { 1 } \\sqcup Y _ { 2 } = Y $",
"$ X _ { 1 } \\sqcup X _ { 2 } = X $",
"$ R ( Y , X ) $",
"$ ( Y _ { 1 } , X _ { 1 } ) $",
"$ ( Y _ { 2 } , X _ { 2 } ) $",
"$ 1 = V V ^ { - 1 } = V V ^ { \\ast } $",
"\\begin{equation*} R _ { n , m } ^ { ( a ) } ( ( \\hbar ^ { - 1 } \\theta L ) ^ { \\otimes n } ; ( \\hbar ^ { - 1 } \\theta _ { - } L ) ^ { \\otimes m } ) = O ( \\hbar ^ { - a } ) . \\end{equation*}",
"\\begin{equation*} A ( O ) \\ni A = O ( \\hbar ^ { n } ) \\quad \\Longrightarrow \\quad V \\times _ { \\hbar } A \\times _ { \\hbar } V ^ { - 1 } = O ( \\hbar ^ { n } ) \\end{equation*}",
"\\begin{align*} \\sum _ { I \\subset Y } ( - 1 ) ^ { \\vert I \\cap Y _ { 1 } \\vert } [ \\bar { T } ( I \\cap Y _ { 1 } ) \\times _ { \\hbar } T ( I ^ { c } \\cap Y _ { 1 } , X _ { 1 } ) ] \\cdot \\\\ ( - 1 ) ^ { \\vert I \\cap Y _ { 2 } \\vert } [ \\bar { T } ( I \\cap Y _ { 2 } ) \\times _ { \\hbar } T ( I ^ { c } \\cap Y _ { 2 } , X _ { 2 } ) ] \\\\ = R ( Y _ { 1 } , X _ { 1 } ) \\cdot R ( Y _ { 2 } , X _ { 2 } ) . \\end{align*}",
"\\begin{equation*} \\sum _ { Y _ { 1 } \\sqcup Y _ { 2 } = Y , \\> X _ { 1 } \\sqcup X _ { 2 } = X } ( - 1 ) ^ { ( \\vert Y _ { 1 } \\vert + \\vert X _ { 1 } \\vert ) } R ^ { \\ast } ( Y _ { 1 } , X _ { 1 } ) \\times _ { \\hbar } R ( Y _ { 2 } , X _ { 2 } ) = 0 \\end{equation*}"
],
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"$ \\mitO $",
"$ \\mittheta $",
"$ \\mittheta ^ { \\prime } $",
"$ \\mitV $",
"$ \\hslash $",
"$ \\mitA $",
"$ \\mitA ( \\mitO ) $",
"$ \\bigvee _ { \\mitn \\in \\BbbN _ { 0 } } \\hslash ^ { \\mitn } \\mitA $",
"$ \\bigvee _ { \\mitn \\in \\BbbN _ { 0 } } \\hslash ^ { \\mitn } \\mitA ( \\mitO ) $",
"$ \\mitA $",
"$ \\hslash $",
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"$ \\mitR _ { \\mitn , \\mitm } ^ { ( \\mita ) } ( . . . ; . . . ) $",
"$ \\mita $",
"$ \\mita $",
"$ \\times _ { \\hslash } $",
"$ \\mitO ( \\hslash ^ { - 1 } ) $",
"$ \\mitA \\in \\mitA ( \\mitO ) $",
"$ \\mitV \\times _ { \\hslash } \\mitA \\times _ { \\hslash } \\mitV ^ { - 1 } $",
"$ \\mitV $",
"$ \\mitV $",
"$ \\mitV ^ { - 1 } $",
"$ \\mitA $",
"$ \\mitV $",
"$ \\mitA $",
"$ \\mitV ^ { - 1 } $",
"$ \\mitA $",
"$ \\mitA $",
"$ \\mitn $",
"$ \\hslash $",
"$ \\mitV \\times _ { \\hslash } \\mitA \\times _ { \\hslash } \\mitV ^ { - 1 } $",
"$ \\hslash $",
"$ \\hslash ^ { \\mitn - 1 } $",
"$ \\mitV \\times _ { \\hslash } \\mitA \\times _ { \\hslash } \\mitV ^ { - 1 } $",
"$ \\mitV $",
"$ \\mitV ^ { - 1 } $",
"$ \\mitr $",
"$ \\mits $",
"$ ( \\mitr + \\mits ) $",
"$ \\hslash ^ { ( \\mitr + \\mits ) } $",
"$ \\mitY _ { 1 } \\sqcup \\mitY _ { 2 } = \\mitY $",
"$ \\mitX _ { 1 } \\sqcup \\mitX _ { 2 } = \\mitX $",
"$ \\mitR ( \\mitY , \\mitX ) $",
"$ ( \\mitY _ { 1 } , \\mitX _ { 1 } ) $",
"$ ( \\mitY _ { 2 } , \\mitX _ { 2 } ) $",
"$ 1 = \\mitV \\mitV ^ { - 1 } = \\mitV \\mitV ^ { \\ast } $",
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"\\begin{equation*} \\mitA ( \\mitO ) \\ni \\mitA = \\mitO ( \\hslash ^ { \\mitn } ) \\quad \\Longrightarrow \\quad \\mitV \\times _ { \\hslash } \\mitA \\times _ { \\hslash } \\mitV ^ { - 1 } = \\mitO ( \\hslash ^ { \\mitn } ) \\end{equation*}",
"\\begin{align*} \\sum _ { \\mitI \\subset \\mitY } ( - 1 ) ^ { \\vert \\mitI \\cap \\mitY _ { 1 } \\vert } [ \\bar { \\mitT } ( \\mitI \\cap \\mitY _ { 1 } ) \\times _ { \\hslash } \\mitT ( \\mitI ^ { \\mitc } \\cap \\mitY _ { 1 } , \\mitX _ { 1 } ) ] \\cdot \\\\ ( - 1 ) ^ { \\vert \\mitI \\cap \\mitY _ { 2 } \\vert } [ \\bar { \\mitT } ( \\mitI \\cap \\mitY _ { 2 } ) \\times _ { \\hslash } \\mitT ( \\mitI ^ { \\mitc } \\cap \\mitY _ { 2 } , \\mitX _ { 2 } ) ] \\\\ = \\mitR ( \\mitY _ { 1 } , \\mitX _ { 1 } ) \\cdot \\mitR ( \\mitY _ { 2 } , \\mitX _ { 2 } ) . \\end{align*}",
"\\begin{equation*} \\sum _ { \\mitY _ { 1 } \\sqcup \\mitY _ { 2 } = \\mitY , \\> \\mitX _ { 1 } \\sqcup \\mitX _ { 2 } = \\mitX } ( - 1 ) ^ { ( \\vert \\mitY _ { 1 } \\vert + \\vert \\mitX _ { 1 } \\vert ) } \\mitR ^ { \\ast } ( \\mitY _ { 1 } , \\mitX _ { 1 } ) \\times _ { \\hslash } \\mitR ( \\mitY _ { 2 } , \\mitX _ { 2 } ) = 0 \\end{equation*}"
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|
0001129_page18 | {
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"\\begin {eqnarray} V\\times _\\hbar A\\times _\\hbar V^{-1}=\\sum _{n,m}\\frac {1}{n!m!}\\int dy_1...dy_ndx_1...dx_m\\,\\theta (y_1)...\\theta (y_n)\\theta _-(x_1)... \\theta _-(x_m)\\\\ \\sum _{Y_1\\sqcup Y_2=Y,\\>X_1\\sqcup X_2=X} (-i)^{(|Y_1|+|X_1|)}i^{(|Y_2|+|X_2|)}R^*(Y_1,X_1)\\times _\\hbar A \\times _\\hbar R(Y_2,X_2),\\end {eqnarray}",
"\\begin {eqnarray} \\sum _{Y_1\\sqcup Y_2=Y,\\>X_1\\sqcup X_2=X}(-1)^{(|Y_1\\cap Y_4|+|X_1\\cap X_4|)} [R^*(Y_1\\cap Y_4,X_1\\cap X_4) \\times _\\hbar \\\\ A\\times _\\hbar R(Y_2\\cap Y_4,X_2\\cap X_4)]\\cdot \\\\ (-1)^{(|Y_1\\cap Y_3|+|X_1\\cap X_3|)} [R^*(Y_1\\cap Y_3,X_1\\cap X_3) \\times _\\hbar R(Y_2\\cap Y_3,X_2\\cap X_3)]=0.\\quad \\w \\end {eqnarray}",
"\\begin {equation} {\\cal A}_{\\cal L}^{(n)}\\=d\\frac {{\\cal A}_{\\cal L}}{I_{n+1}},\\quad \\quad {\\cal A}_{\\cal L}^{(n)}({\\cal O})\\=d\\frac {{\\cal A}_{\\cal L} ({\\cal O})}{I_{n+1}\\cap {\\cal A}_{\\cal L}({\\cal O})}.\\label {L16a} \\end {equation}"
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"\\begin{align*} V \\times _ { \\hbar } A \\times _ { \\hbar } V ^ { - 1 } = \\sum _ { n , m } \\frac { 1 } { n ! m ! } \\int d y _ { 1 } . . . d y _ { n } d x _ { 1 } . . . d x _ { m } \\, \\theta ( y _ { 1 } ) . . . \\theta ( y _ { n } ) \\theta _ { - } ( x _ { 1 } ) . . . \\theta _ { - } ( x _ { m } ) \\\\ \\sum _ { Y _ { 1 } \\sqcup Y _ { 2 } = Y , \\> X _ { 1 } \\sqcup X _ { 2 } = X } ( - i ) ^ { ( \\vert Y _ { 1 } \\vert + \\vert X _ { 1 } \\vert ) } i ^ { ( \\vert Y _ { 2 } \\vert + \\vert X _ { 2 } \\vert ) } R ^ { \\ast } ( Y _ { 1 } , X _ { 1 } ) \\times _ { \\hbar } A \\times _ { \\hbar } R ( Y _ { 2 } , X _ { 2 } ) , \\end{align*}",
"\\begin{align*} \\sum _ { Y _ { 1 } \\sqcup Y _ { 2 } = Y , \\> X _ { 1 } \\sqcup X _ { 2 } = X } ( - 1 ) ^ { ( \\vert Y _ { 1 } \\cap Y _ { 4 } \\vert + \\vert X _ { 1 } \\cap X _ { 4 } \\vert ) } [ R ^ { \\ast } ( Y _ { 1 } \\cap Y _ { 4 } , X _ { 1 } \\cap X _ { 4 } ) \\times _ { \\hbar } \\\\ A \\times _ { \\hbar } R ( Y _ { 2 } \\cap Y _ { 4 } , X _ { 2 } \\cap X _ { 4 } ) ] \\cdot \\\\ ( - 1 ) ^ { ( \\vert Y _ { 1 } \\cap Y _ { 3 } \\vert + \\vert X _ { 1 } \\cap X _ { 3 } \\vert ) } [ R ^ { \\ast } ( Y _ { 1 } \\cap Y _ { 3 } , X _ { 1 } \\cap X _ { 3 } ) \\times _ { \\hbar } R ( Y _ { 2 } \\cap Y _ { 3 } , X _ { 2 } \\cap X _ { 3 } ) ] = 0 . \\quad \\square \\end{align*}",
"\\begin{equation*} A _ { L } ^ { ( n ) } \\, \\overset { d e f } { = } \\, \\frac { A _ { L } } { I _ { n + 1 } } , \\quad \\quad A _ { L } ^ { ( n ) } ( O ) \\, \\overset { d e f } { = } \\, \\frac { A _ { L } ( O ) } { I _ { n + 1 } \\cap A _ { L } ( O ) } . \\end{equation*}"
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0001129_page19 | {
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"${\\cal G}_{\\cal L}({\\cal O})$",
"$g\\rightarrow 1$",
"$\\lambda :\\varphi ^{2n}:$",
"${\\cal O}$",
"$x_1,...,x_k\\not \\in ((\\bar {\\cal O}\\cup \\{x_{k+l+1},...,x_n\\})+\\bar V_-),\\>x_{k+1},...,x_{k+l} \\in {\\cal O}$",
"$x_{k+l+1},...,x_n\\not \\in (\\bar {\\cal O}+\\bar V_+)$",
"$\\theta \\in \\Theta ({\\cal O})$",
"$\\{x_1,...,x_k\\}\\cap ({\\rm supp}\\>\\theta +\\bar V_-)=\\emptyset $",
"$\\{x_{k+l+1},...,x_n\\}\\cap ({\\rm supp}\\>\\theta +\\bar V_+)=\\emptyset $",
"$R$",
"$T_{\\theta {\\cal L}}\\bigl (\\varphi (x_{k+l+1})...\\varphi (x_n)\\bigr )$",
"$T_0\\bigl (\\varphi (x_{k+l+1})... \\varphi (x_n)\\bigr )$",
"$S_{\\theta {\\cal L}}(f\\varphi )=S(\\theta {\\cal L})^{-1}S(f\\varphi ) S(\\theta {\\cal L})$",
"${\\rm supp}\\>f\\cap ({\\rm supp}\\>\\theta +\\bar V_-)=\\emptyset $",
"$T_0\\bigl ( \\varphi (x_{k+l+1})...\\varphi (x_n)\\bigr )\\Omega $",
"$S(\\theta {\\cal L})^{-1}T_0 \\bigl (\\varphi (x_1)...\\varphi (x_k)\\bigr )^*S(\\theta {\\cal L})\\Omega $",
"$g$",
"$g_0\\in {\\cal D}(\\RR ^4),\\>g_0(0)=1$",
"$\\epsilon \\rightarrow 0\\>\\>(\\epsilon >0)$",
"$g_\\epsilon (x)\\equiv g_0(\\epsilon x)$",
"$g_0$",
"\\begin {eqnarray} \\Bigl (\\Omega ,T_{\\theta {\\cal L}}\\bigl (\\varphi (x_1)...\\varphi (x_n)\\bigr ) \\Omega \\Bigr )= \\Bigl (T_{\\theta {\\cal L}}\\bigl (\\varphi (x_1)...\\varphi (x_k)\\bigr )^*\\Omega , \\\\ T_{\\theta {\\cal L}}\\bigl (\\varphi (x_{k+1})...\\varphi (x_{k+l})\\bigr ) T_{\\theta {\\cal L}}\\Bigl (\\varphi (x_{k+l+1})...\\varphi (x_n)\\bigr )\\Omega \\Bigr ). \\end {eqnarray}",
"\\begin {equation} T_{\\theta {\\cal L}}\\bigl (\\varphi (x_1)...\\varphi (x_k)\\bigr )^*=S(\\theta {\\cal L})^{-1}T_0\\bigl (\\varphi (x_1)...\\varphi (x_k)\\bigr )^* S(\\theta {\\cal L}). \\end {equation}"
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"$ x _ { k + l + 1 } , . . . , x _ { n } \\notin ( \\bar { O } + \\bar { V } _ { + } ) $",
"$ \\theta \\in \\Theta ( O ) $",
"$ \\{ x _ { 1 } , . . . , x _ { k } \\} \\cap ( s u p p \\> \\theta + \\bar { V } _ { - } ) = \\emptyset $",
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"$ T _ { 0 } ( \\varphi ( x _ { k + l + 1 } ) . . . \\varphi ( x _ { n } ) ) $",
"$ S _ { \\theta L } ( f \\varphi ) = S ( \\theta L ) ^ { - 1 } S ( f \\varphi ) S ( \\theta L ) $",
"$ s u p p \\> f \\cap ( s u p p \\> \\theta + \\bar { V } _ { - } ) = \\emptyset $",
"$ T _ { 0 } ( \\varphi ( x _ { k + l + 1 } ) . . . \\varphi ( x _ { n } ) ) \\Omega $",
"$ S ( \\theta L ) ^ { - 1 } T _ { 0 } ( \\varphi ( x _ { 1 } ) . . . \\varphi ( x _ { k } ) ) ^ { \\ast } S ( \\theta L ) \\Omega $",
"$ g $",
"$ g _ { 0 } \\in D ( R ^ { 4 } ) , \\> g _ { 0 } ( 0 ) = 1 $",
"$ \\epsilon \\rightarrow 0 \\> \\> ( \\epsilon > 0 ) $",
"$ g _ { \\epsilon } ( x ) \\equiv g _ { 0 } ( \\epsilon x ) $",
"$ g _ { 0 } $",
"\\begin{align*} ( \\Omega , T _ { \\theta L } ( \\varphi ( x _ { 1 } ) . . . \\varphi ( x _ { n } ) ) \\Omega ) = ( T _ { \\theta L } ( \\varphi ( x _ { 1 } ) . . . \\varphi ( x _ { k } ) ) ^ { \\ast } \\Omega , \\\\ T _ { \\theta L } ( \\varphi ( x _ { k + 1 } ) . . . \\varphi ( x _ { k + l } ) ) T _ { \\theta L } ( \\varphi ( x _ { k + l + 1 } ) . . . \\varphi ( x _ { n } ) ) \\Omega ) . \\end{align*}",
"\\begin{equation*} T _ { \\theta L } ( \\varphi ( x _ { 1 } ) . . . \\varphi ( x _ { k } ) ) ^ { \\ast } = S ( \\theta L ) ^ { - 1 } T _ { 0 } ( \\varphi ( x _ { 1 } ) . . . \\varphi ( x _ { k } ) ) ^ { \\ast } S ( \\theta L ) . \\end{equation*}"
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"$ \\{ \\mitx _ { \\mitk + \\mitl + 1 } , . . . , \\mitx _ { \\mitn } \\} \\cap ( \\mathrm { s u p p } \\> \\mittheta + \\bar { \\mitV } _ { + } ) = \\varnothing $",
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"$ \\mitT _ { \\mittheta \\mitL } \\big ( \\mitvarphi ( \\mitx _ { \\mitk + \\mitl + 1 } ) . . . \\mitvarphi ( \\mitx _ { \\mitn } ) \\big ) $",
"$ \\mitT _ { 0 } \\big ( \\mitvarphi ( \\mitx _ { \\mitk + \\mitl + 1 } ) . . . \\mitvarphi ( \\mitx _ { \\mitn } ) \\big ) $",
"$ \\mitS _ { \\mittheta \\mitL } ( \\mitf \\mitvarphi ) = \\mitS ( \\mittheta \\mitL ) ^ { - 1 } \\mitS ( \\mitf \\mitvarphi ) \\mitS ( \\mittheta \\mitL ) $",
"$ \\mathrm { s u p p } \\> \\mitf \\cap ( \\mathrm { s u p p } \\> \\mittheta + \\bar { \\mitV } _ { - } ) = \\varnothing $",
"$ \\mitT _ { 0 } \\big ( \\mitvarphi ( \\mitx _ { \\mitk + \\mitl + 1 } ) . . . \\mitvarphi ( \\mitx _ { \\mitn } ) \\big ) \\mupOmega $",
"$ \\mitS ( \\mittheta \\mitL ) ^ { - 1 } \\mitT _ { 0 } \\big ( \\mitvarphi ( \\mitx _ { 1 } ) . . . \\mitvarphi ( \\mitx _ { \\mitk } ) \\big ) ^ { \\ast } \\mitS ( \\mittheta \\mitL ) \\mupOmega $",
"$ \\mitg $",
"$ \\mitg _ { 0 } \\in \\mitD ( \\BbbR ^ { 4 } ) , \\> \\mitg _ { 0 } ( 0 ) = 1 $",
"$ \\mitepsilon \\rightarrow 0 \\> \\> ( \\mitepsilon > 0 ) $",
"$ \\mitg _ { \\mitepsilon } ( \\mitx ) \\equiv \\mitg _ { 0 } ( \\mitepsilon \\mitx ) $",
"$ \\mitg _ { 0 } $",
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"\\begin{equation*} \\mitT _ { \\mittheta \\mitL } \\big ( \\mitvarphi ( \\mitx _ { 1 } ) . . . \\mitvarphi ( \\mitx _ { \\mitk } ) \\big ) ^ { \\ast } = \\mitS ( \\mittheta \\mitL ) ^ { - 1 } \\mitT _ { 0 } \\big ( \\mitvarphi ( \\mitx _ { 1 } ) . . . \\mitvarphi ( \\mitx _ { \\mitk } ) \\big ) ^ { \\ast } \\mitS ( \\mittheta \\mitL ) . \\end{equation*}"
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0001129_page20 | {
"latex": [
"$x_1,...,x_k$",
"$(\\bar {\\cal O}\\cup \\{x_{k+l+1},...,x_n\\})$",
"$S$",
"$\\hat l$",
"$\\hat j$",
"$S$",
"$g\\equiv \\theta \\in \\Theta ({\\cal O})$",
"$x\\in {\\cal O}$",
"$\\rho $",
"${\\rm supp}\\>\\rho \\subset \\{y|\\theta (y)=1\\}$",
"$Z(f)$",
"$<\\Omega | T\\bigl (\\phi _{\\cal L}(x_1)...\\phi _{\\cal L}(x_m)\\bigr )|\\Omega >$",
"$S$",
"$\\Omega $",
"$\\Delta (x)$",
"$3$",
"$\\frac {\\delta S}{\\delta \\phi (x)}$",
"$S=\\int d^4x\\, [\\frac {1}{2}(\\d _\\mu \\phi (x)\\d ^\\mu \\phi (x)-m^2\\phi ^2(x))+g(x){\\cal L}(x)]$",
"$S=\\int d^4x\\, [\\frac {1}{2}(\\d _\\mu \\phi (x)\\d ^\\mu \\phi (x)-m^2\\phi ^2(x))+g(x){\\cal L}(x)]$",
"${\\cal L}$",
"$4$",
"\\begin {eqnarray} (\\w _x+m^2)R\\bigl ({\\cal L}(y_1)...{\\cal L}(y_n);\\phi (x)\\phi (x_1)... \\phi (x_m)\\bigr )=\\\\ -i\\sum _{l=1}^{n}\\delta (x-y_l)R\\bigl ({\\cal L}(y_1)...\\hat l...{\\cal L}(y_n); \\frac {\\d {\\cal L}}{\\d \\phi }(x)\\phi (x_1)...\\phi (x_m)\\bigr )\\\\ -i\\sum _{j=1}^{m}\\delta (x-x_j)R\\bigl ({\\cal L}(y_1)...{\\cal L}(y_n);\\phi (x_1)... \\hat j...\\phi (x_m)\\bigr ),\\end {eqnarray}",
"\\begin {equation} f(x)S_{g{\\cal L}}(f\\phi )=(\\w _x+m^2)\\frac {\\delta }{i\\delta f(x)} S_{g{\\cal L}}(f\\phi ) -\\frac {\\delta }{i\\delta \\rho (x)}\\vert _{\\rho =0}S_{g{\\cal L}}(f\\phi +\\rho g\\frac {\\d {\\cal L}}{\\d \\phi }).\\label {4.5a} \\end {equation}",
"\\begin {equation} (\\w _x+m^2)\\frac {\\delta }{i\\delta f(x)} S_{\\cal L}(f\\phi )=f(x)S_{\\cal L}(f\\phi )+ \\frac {\\delta }{i\\delta \\rho (x)}\\vert _{\\rho =0}S_{\\cal L}(f\\phi +\\rho \\frac {\\d {\\cal L}}{\\d \\phi }),\\quad x\\in {\\cal O}.\\label {4.5aa} \\end {equation}",
"\\begin {equation} Z(f)=\\lim _{g\\to 1}(\\Omega ,S_{g{\\cal L}}(f\\phi )\\Omega ),\\label {4.5b} \\end {equation}",
"\\begin {equation} f(x)Z(f)=-\\Delta (x)\\cdot Z(f),\\label {4.5c} \\end {equation}"
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"$ S = \\int d ^ { 4 } x \\, [ \\frac { 1 } { 2 } ( \\partial _ { \\mu } \\phi ( x ) \\partial ^ { \\mu } \\phi ( x ) - m ^ { 2 } \\phi ^ { 2 } ( x ) ) + g ( x ) L ( x ) ] $",
"$ S = \\int d ^ { 4 } x \\, [ \\frac { 1 } { 2 } ( \\partial _ { \\mu } \\phi ( x ) \\partial ^ { \\mu } \\phi ( x ) - m ^ { 2 } \\phi ^ { 2 } ( x ) ) + g ( x ) L ( x ) ] $",
"$ L $",
"$ 4 $",
"\\begin{align*} ( \\square _ { x } + m ^ { 2 } ) R ( L ( y _ { 1 } ) . . . L ( y _ { n } ) ; \\phi ( x ) \\phi ( x _ { 1 } ) . . . \\phi ( x _ { m } ) ) = \\\\ - i \\sum _ { l = 1 } ^ { n } \\delta ( x - y _ { l } ) R ( L ( y _ { 1 } ) . . . \\hat { l } . . . L ( y _ { n } ) ; \\frac { \\partial L } { \\partial \\phi } ( x ) \\phi ( x _ { 1 } ) . . . \\phi ( x _ { m } ) ) \\\\ - i \\sum _ { j = 1 } ^ { m } \\delta ( x - x _ { j } ) R ( L ( y _ { 1 } ) . . . L ( y _ { n } ) ; \\phi ( x _ { 1 } ) . . . \\hat { j } . . . \\phi ( x _ { m } ) ) , \\end{align*}",
"\\begin{equation*} f ( x ) S _ { g L } ( f \\phi ) = ( \\square _ { x } + m ^ { 2 } ) \\frac { \\delta } { i \\delta f ( x ) } S _ { g L } ( f \\phi ) - \\frac { \\delta } { i \\delta \\rho ( x ) } \\vert _ { \\rho = 0 } S _ { g L } ( f \\phi + \\rho g \\frac { \\partial L } { \\partial \\phi } ) . \\end{equation*}",
"\\begin{equation*} ( \\square _ { x } + m ^ { 2 } ) \\frac { \\delta } { i \\delta f ( x ) } S _ { L } ( f \\phi ) = f ( x ) S _ { L } ( f \\phi ) + \\frac { \\delta } { i \\delta \\rho ( x ) } \\vert _ { \\rho = 0 } S _ { L } ( f \\phi + \\rho \\frac { \\partial L } { \\partial \\phi } ) , \\quad x \\in O . \\end{equation*}",
"\\begin{equation*} Z ( f ) = \\underset { g \\rightarrow 1 } { \\operatorname { l i m } } ( \\Omega , S _ { g L } ( f \\phi ) \\Omega ) , \\end{equation*}",
"\\begin{equation*} f ( x ) Z ( f ) = - \\Delta ( x ) \\cdot Z ( f ) , \\end{equation*}"
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"$ 3 $",
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"$ \\mitS = \\int \\nolimits \\mitd ^ { 4 } \\mitx \\, [ \\frac { 1 } { 2 } ( \\mitpartial _ { \\mitmu } \\mitphi ( \\mitx ) \\mitpartial ^ { \\mitmu } \\mitphi ( \\mitx ) - \\mitm ^ { 2 } \\mitphi ^ { 2 } ( \\mitx ) ) + \\mitg ( \\mitx ) \\mitL ( \\mitx ) ] $",
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"\\begin{equation*} \\mitf ( \\mitx ) \\mitS _ { \\mitg \\mitL } ( \\mitf \\mitphi ) = ( \\square _ { \\mitx } + \\mitm ^ { 2 } ) \\frac { \\mitdelta } { \\miti \\mitdelta \\mitf ( \\mitx ) } \\mitS _ { \\mitg \\mitL } ( \\mitf \\mitphi ) - \\frac { \\mitdelta } { \\miti \\mitdelta \\mitrho ( \\mitx ) } \\vert _ { \\mitrho = 0 } \\mitS _ { \\mitg \\mitL } ( \\mitf \\mitphi + \\mitrho \\mitg \\frac { \\mitpartial \\mitL } { \\mitpartial \\mitphi } ) . \\end{equation*}",
"\\begin{equation*} ( \\square _ { \\mitx } + \\mitm ^ { 2 } ) \\frac { \\mitdelta } { \\miti \\mitdelta \\mitf ( \\mitx ) } \\mitS _ { \\mitL } ( \\mitf \\mitphi ) = \\mitf ( \\mitx ) \\mitS _ { \\mitL } ( \\mitf \\mitphi ) + \\frac { \\mitdelta } { \\miti \\mitdelta \\mitrho ( \\mitx ) } \\vert _ { \\mitrho = 0 } \\mitS _ { \\mitL } ( \\mitf \\mitphi + \\mitrho \\frac { \\mitpartial \\mitL } { \\mitpartial \\mitphi } ) , \\quad \\mitx \\in \\mitO . \\end{equation*}",
"\\begin{equation*} \\mitZ ( \\mitf ) = \\underset { \\mitg \\rightarrow 1 } { \\operatorname { l i m } } ( \\mupOmega , \\mitS _ { \\mitg \\mitL } ( \\mitf \\mitphi ) \\mupOmega ) , \\end{equation*}",
"\\begin{equation*} \\mitf ( \\mitx ) \\mitZ ( \\mitf ) = - \\mupDelta ( \\mitx ) \\cdot \\mitZ ( \\mitf ) , \\end{equation*}"
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0001129_page21 | {
"latex": [
"$R_{\\cal L}(g^{\\otimes n};f^{\\otimes m})$",
"${\\cal L},g,f\\in {\\cal D}(\\RR ^4,{\\cal V})$",
"$T$",
"$\\bar T$",
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"\\begin {equation} S_{{\\cal L}+g}(f)=S_{\\cal L}(g)^{-1}S_{\\cal L}(g+f)\\=d \\sum _{n,m=0}^\\infty \\frac {i^{n+m}}{n!m!}R_{\\cal L} (g^{\\otimes n};f^{\\otimes m}) \\label {Q2b} \\end {equation}",
"\\begin {equation} R_{\\cal L}(g^{\\otimes n};f^{\\otimes m})= \\sum _{k=0}^{n}(-1)^k\\frac {n!}{k!(n-k)!} \\bar {T}_{\\cal L}(g^{\\otimes k}) T_{\\cal L}(g^{\\otimes (n-k)} \\otimes f^{\\otimes m}).\\label {R=Tww} \\end {equation}",
"\\begin {equation} \\bar T_{\\cal L}(f^{\\otimes m})= \\frac {d^m}{(-i)^m d\\lambda ^m}\\vert _{\\lambda =0}S_{\\cal L} (\\lambda f)^{-1},\\label {Q2a} \\end {equation}",
"\\begin {equation} {\\cal L}_0\\rightarrow {\\cal L}_0+\\epsilon {\\cal L}_1\\label {Q1} \\end {equation}",
"\\begin {eqnarray} \\frac {d^m}{d\\epsilon ^m}\\vert _{\\epsilon =0} T_{\\theta ({\\cal L}_0+\\epsilon {\\cal L}_1)}(f^{\\otimes l})&=& \\frac {\\d ^m}{\\d \\epsilon ^m}\\vert _{\\epsilon =0}\\frac {\\d ^l}{i^l\\d \\lambda ^l} \\vert _{\\lambda =0}S_{\\theta ({\\cal L}_0+\\epsilon {\\cal L}_1)}(\\lambda f) \\\\ &=&i^m R_{\\theta {\\cal L}_0}((\\theta {\\cal L}_1)^{\\otimes m};f^{\\otimes l}). \\end {eqnarray}",
"\\begin {equation} \\Theta _0({\\cal O})\\=d\\{\\theta \\in \\Theta ({\\cal O})\\quad |\\quad \\theta \\vert _{{\\rm supp}\\> {\\cal L}_1}\\equiv 1\\}. \\end {equation}",
"\\begin {equation} \\frac {d^m}{d\\epsilon ^m}\\vert _{\\epsilon =0} T_{{\\cal L}_0+\\epsilon {\\cal L}_1}(f^{\\otimes l})= i^m R_{{\\cal L}_0}({\\cal L}_1^{\\otimes m};f^{\\otimes l}).\\label {Q2c} \\end {equation}"
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"\\begin{equation*} \\Theta _ { 0 } ( O ) \\, \\overset { d e f } { = } \\, \\{ \\theta \\in \\Theta ( O ) \\quad \\vert \\quad \\theta \\vert _ { s u p p \\> L _ { 1 } } \\equiv 1 \\} . \\end{equation*}",
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"\\begin{equation*} \\mitR _ { \\mitL } ( \\mitg ^ { \\otimes \\mitn } ; \\mitf ^ { \\otimes \\mitm } ) = \\sum _ { \\mitk = 0 } ^ { \\mitn } ( - 1 ) ^ { \\mitk } \\frac { \\mitn ! } { \\mitk ! ( \\mitn - \\mitk ) ! } \\bar { \\mitT } _ { \\mitL } ( \\mitg ^ { \\otimes \\mitk } ) \\mitT _ { \\mitL } ( \\mitg ^ { \\otimes ( \\mitn - \\mitk ) } \\otimes \\mitf ^ { \\otimes \\mitm } ) . \\end{equation*}",
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"\\begin{equation*} \\mupTheta _ { 0 } ( \\mitO ) \\, \\overset { \\mathrm { d e f } } { = } \\, \\{ \\mittheta \\in \\mupTheta ( \\mitO ) \\quad \\vert \\quad \\mittheta \\vert _ { \\mathrm { s u p p } \\> \\mitL _ { 1 } } \\equiv 1 \\} . \\end{equation*}",
"\\begin{equation*} \\frac { \\mitd ^ { \\mitm } } { \\mitd \\mitepsilon ^ { \\mitm } } \\vert _ { \\mitepsilon = 0 } \\mitT _ { \\mitL _ { 0 } + \\mitepsilon \\mitL _ { 1 } } ( \\mitf ^ { \\otimes \\mitl } ) = \\miti ^ { \\mitm } \\mitR _ { \\mitL _ { 0 } } ( \\mitL _ { 1 } ^ { \\otimes \\mitm } ; \\mitf ^ { \\otimes \\mitl } ) . \\end{equation*}"
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0001129_page22 | {
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"$m$",
"$\\epsilon $",
"$(\\theta {\\cal L}_1)^{\\otimes m}$",
"$(\\theta {\\cal L}_1)^{\\otimes m}$",
"${\\cal L}_0$",
"${\\cal L}_1$",
"${\\rm dim}({\\cal L}_j)=4$",
"${\\rm dim}({\\cal L}_j)<4$",
"${\\cal L}_j$",
"${\\rm dim}({\\cal L}_j) =4$",
"${\\rm sd}(t_0)\\leq {\\rm sd}(t)\\leq 4n-b$",
"$b$",
"${\\cal L}_j$",
"$4$",
"$m,l\\in \\NN _0$",
"$\\frac {\\d ^m}{\\d \\epsilon ^m}$",
"$\\frac {\\d ^l}{\\d \\lambda ^l}$",
"$\\theta \\rightarrow 1$",
"$P_\\Omega $",
"$\\Omega $",
"$P_\\Omega ^\\bot \\=d 1-P_\\Omega $",
"$S(\\theta {\\cal L})^* =S(\\theta {\\cal L})^{-1}$",
"$N_\\delta \\{\\prod _{j=1}^l\\varphi _{i_j\\> {\\cal L}}(x)\\},\\>\\delta \\geq d\\equiv \\sum _{j=1}^l d(\\varphi _{i_j\\>{\\cal L}})$",
"$\\delta =d$",
"$N_\\delta \\{\\prod _{j=1}^l\\varphi _{i_j\\>{\\cal L}}(x)\\}$",
"$(:\\prod _{j=1}^l\\varphi _{i_j}(x):)_{g{\\cal L}}$",
"\\begin {equation} i^m T_{\\theta {\\cal L}_0}((\\theta {\\cal L}_1)^{\\otimes m} \\otimes f^{\\otimes l})=\\frac {\\d ^m}{\\d \\epsilon ^m}\\vert _{\\epsilon =0} \\frac {\\d ^l}{i^l\\d \\lambda ^l}\\vert _{\\lambda =0}S_{\\theta {\\cal L}_0} (\\theta \\epsilon {\\cal L}_1+\\lambda f).\\label {Q3} \\end {equation}",
"\\begin {equation} \\frac {d^m}{d\\epsilon ^m}\\vert _{\\epsilon =0}\\lim _{\\theta \\to 1} \\Bigl (\\Omega ,T_{\\theta ({\\cal L}_0+\\epsilon {\\cal L}_1)}(f^{\\otimes l}) \\Omega \\Bigr )=i^m\\lim _{\\theta \\to 1}\\Bigl (\\Omega ,T_{\\theta {\\cal L}_0} ((\\theta {\\cal L}_1)^{\\otimes m}\\otimes f^{\\otimes l})\\Omega \\Bigr )\\label {Q5} \\end {equation}",
"\\begin {equation} \\lim _{\\theta \\to 1}\\Bigl (\\Omega ,S_{\\theta ({\\cal L}_0+\\epsilon {\\cal L}_1)} (\\lambda f)\\Omega \\Bigr )=\\lim _{\\theta \\to 1}\\Bigl (\\Omega ,S_{\\theta {\\cal L}_0} (\\theta \\epsilon {\\cal L}_1+\\lambda f)\\Omega \\Bigr ).\\label {Q6} \\end {equation}",
"\\begin {equation} \\lim _{\\theta \\to 1}(\\Omega ,S_{\\theta {\\cal L}}(f)\\Omega )= \\lim _{\\theta \\to 1}\\frac {(\\Omega ,S(\\theta {\\cal L}+f)\\Omega )} {(\\Omega ,S(\\theta {\\cal L})\\Omega )},\\label {Q4} \\end {equation}",
"\\begin {eqnarray} (\\Omega ,S_{\\theta {\\cal L}}(f)\\Omega )&=&(S(\\theta {\\cal L})\\Omega , (P_\\Omega +P_\\Omega ^\\bot )S(\\theta {\\cal L}+f)\\Omega )\\\\ &=&\\frac {(\\Omega ,S(\\theta {\\cal L}+f)\\Omega )}{(\\Omega ,S(\\theta {\\cal L})\\Omega )} \\cdot |(\\Omega ,S(\\theta {\\cal L})\\Omega )|^2\\\\ &+&(\\Omega ,S(\\theta {\\cal L})^{-1} P_\\Omega ^\\bot S(\\theta {\\cal L}+f)\\Omega )\\end {eqnarray}"
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"$ ( \\theta L _ { 1 } ) ^ { \\otimes m } $",
"$ ( \\theta L _ { 1 } ) ^ { \\otimes m } $",
"$ L _ { 0 } $",
"$ L _ { 1 } $",
"$ d i m ( L _ { j } ) = 4 $",
"$ d i m ( L _ { j } ) < 4 $",
"$ L _ { j } $",
"$ d i m ( L _ { j } ) = 4 $",
"$ s d ( t _ { 0 } ) \\leq s d ( t ) \\leq 4 n - b $",
"$ b $",
"$ L _ { j } $",
"$ 4 $",
"$ m , l \\in N _ { 0 } $",
"$ \\frac { \\partial ^ { m } } { \\partial \\epsilon ^ { m } } $",
"$ \\frac { \\partial ^ { l } } { \\partial \\lambda ^ { l } } $",
"$ \\theta \\rightarrow 1 $",
"$ P _ { \\Omega } $",
"$ \\Omega $",
"$ P _ { \\Omega } ^ { \\bot } \\, \\overset { d e f } { = } \\, 1 - P _ { \\Omega } $",
"$ S ( \\theta L ) ^ { \\ast } = S ( \\theta L ) ^ { - 1 } $",
"$ N _ { \\delta } \\{ \\prod _ { j = 1 } ^ { l } \\varphi _ { i _ { j } \\> L } ( x ) \\} , \\> \\delta \\geq d \\equiv \\sum _ { j = 1 } ^ { l } d ( \\varphi _ { i _ { j } \\> L } ) $",
"$ \\delta = d $",
"$ N _ { \\delta } \\{ \\prod _ { j = 1 } ^ { l } \\varphi _ { i _ { j } \\> L } ( x ) \\} $",
"$ ( : \\prod _ { j = 1 } ^ { l } \\varphi _ { i _ { j } } ( x ) : ) _ { g L } $",
"\\begin{equation*} i ^ { m } T _ { \\theta L _ { 0 } } ( ( \\theta L _ { 1 } ) ^ { \\otimes m } \\otimes f ^ { \\otimes l } ) = \\frac { \\partial ^ { m } } { \\partial \\epsilon ^ { m } } \\vert _ { \\epsilon = 0 } \\frac { \\partial ^ { l } } { i ^ { l } \\partial \\lambda ^ { l } } \\vert _ { \\lambda = 0 } S _ { \\theta L _ { 0 } } ( \\theta \\epsilon L _ { 1 } + \\lambda f ) . \\end{equation*}",
"\\begin{equation*} \\frac { d ^ { m } } { d \\epsilon ^ { m } } \\vert _ { \\epsilon = 0 } \\underset { \\theta \\rightarrow 1 } { \\operatorname { l i m } } ( \\Omega , T _ { \\theta ( L _ { 0 } + \\epsilon L _ { 1 } ) } ( f ^ { \\otimes l } ) \\Omega ) = i ^ { m } \\underset { \\theta \\rightarrow 1 } { \\operatorname { l i m } } ( \\Omega , T _ { \\theta L _ { 0 } } ( ( \\theta L _ { 1 } ) ^ { \\otimes m } \\otimes f ^ { \\otimes l } ) \\Omega ) \\end{equation*}",
"\\begin{equation*} \\underset { \\theta \\rightarrow 1 } { \\operatorname { l i m } } ( \\Omega , S _ { \\theta ( L _ { 0 } + \\epsilon L _ { 1 } ) } ( \\lambda f ) \\Omega ) = \\underset { \\theta \\rightarrow 1 } { \\operatorname { l i m } } ( \\Omega , S _ { \\theta L _ { 0 } } ( \\theta \\epsilon L _ { 1 } + \\lambda f ) \\Omega ) . \\end{equation*}",
"\\begin{equation*} \\underset { \\theta \\rightarrow 1 } { \\operatorname { l i m } } ( \\Omega , S _ { \\theta L } ( f ) \\Omega ) = \\underset { \\theta \\rightarrow 1 } { \\operatorname { l i m } } \\frac { ( \\Omega , S ( \\theta L + f ) \\Omega ) } { ( \\Omega , S ( \\theta L ) \\Omega ) } , \\end{equation*}",
"\\begin{align*} ( \\Omega , S _ { \\theta L } ( f ) \\Omega ) & = & ( S ( \\theta L ) \\Omega , ( P _ { \\Omega } + P _ { \\Omega } ^ { \\bot } ) S ( \\theta L + f ) \\Omega ) \\\\ & = & \\frac { ( \\Omega , S ( \\theta L + f ) \\Omega ) } { ( \\Omega , S ( \\theta L ) \\Omega ) } \\cdot \\vert ( \\Omega , S ( \\theta L ) \\Omega ) \\vert ^ { 2 } \\\\ & + & ( \\Omega , S ( \\theta L ) ^ { - 1 } P _ { \\Omega } ^ { \\bot } S ( \\theta L + f ) \\Omega ) \\end{align*}"
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"$ \\mitepsilon $",
"$ ( \\mittheta \\mitL _ { 1 } ) ^ { \\otimes \\mitm } $",
"$ ( \\mittheta \\mitL _ { 1 } ) ^ { \\otimes \\mitm } $",
"$ \\mitL _ { 0 } $",
"$ \\mitL _ { 1 } $",
"$ \\mathrm { d i m } ( \\mitL _ { \\mitj } ) = 4 $",
"$ \\mathrm { d i m } ( \\mitL _ { \\mitj } ) < 4 $",
"$ \\mitL _ { \\mitj } $",
"$ \\mathrm { d i m } ( \\mitL _ { \\mitj } ) = 4 $",
"$ \\mathrm { s d } ( \\mitt _ { 0 } ) \\leq \\mathrm { s d } ( \\mitt ) \\leq 4 \\mitn - \\mitb $",
"$ \\mitb $",
"$ \\mitL _ { \\mitj } $",
"$ 4 $",
"$ \\mitm , \\mitl \\in \\BbbN _ { 0 } $",
"$ \\frac { \\mitpartial ^ { \\mitm } } { \\mitpartial \\mitepsilon ^ { \\mitm } } $",
"$ \\frac { \\mitpartial ^ { \\mitl } } { \\mitpartial \\mitlambda ^ { \\mitl } } $",
"$ \\mittheta \\rightarrow 1 $",
"$ \\mitP _ { \\mupOmega } $",
"$ \\mupOmega $",
"$ \\mitP _ { \\mupOmega } ^ { \\bot } \\, \\overset { \\mathrm { d e f } } { = } \\, 1 - \\mitP _ { \\mupOmega } $",
"$ \\mitS ( \\mittheta \\mitL ) ^ { \\ast } = \\mitS ( \\mittheta \\mitL ) ^ { - 1 } $",
"$ \\mitN _ { \\mitdelta } \\{ \\prod _ { \\mitj = 1 } ^ { \\mitl } \\mitvarphi _ { \\miti _ { \\mitj } \\> \\mitL } ( \\mitx ) \\} , \\> \\mitdelta \\geq \\mitd \\equiv \\sum _ { \\mitj = 1 } ^ { \\mitl } \\mitd ( \\mitvarphi _ { \\miti _ { \\mitj } \\> \\mitL } ) $",
"$ \\mitdelta = \\mitd $",
"$ \\mitN _ { \\mitdelta } \\{ \\prod _ { \\mitj = 1 } ^ { \\mitl } \\mitvarphi _ { \\miti _ { \\mitj } \\> \\mitL } ( \\mitx ) \\} $",
"$ ( : \\prod _ { \\mitj = 1 } ^ { \\mitl } \\mitvarphi _ { \\miti _ { \\mitj } } ( \\mitx ) : ) _ { \\mitg \\mitL } $",
"\\begin{equation*} \\miti ^ { \\mitm } \\mitT _ { \\mittheta \\mitL _ { 0 } } ( ( \\mittheta \\mitL _ { 1 } ) ^ { \\otimes \\mitm } \\otimes \\mitf ^ { \\otimes \\mitl } ) = \\frac { \\mitpartial ^ { \\mitm } } { \\mitpartial \\mitepsilon ^ { \\mitm } } \\vert _ { \\mitepsilon = 0 } \\frac { \\mitpartial ^ { \\mitl } } { \\miti ^ { \\mitl } \\mitpartial \\mitlambda ^ { \\mitl } } \\vert _ { \\mitlambda = 0 } \\mitS _ { \\mittheta \\mitL _ { 0 } } ( \\mittheta \\mitepsilon \\mitL _ { 1 } + \\mitlambda \\mitf ) . \\end{equation*}",
"\\begin{equation*} \\frac { \\mitd ^ { \\mitm } } { \\mitd \\mitepsilon ^ { \\mitm } } \\vert _ { \\mitepsilon = 0 } \\underset { \\mittheta \\rightarrow 1 } { \\operatorname { l i m } } \\Big ( \\mupOmega , \\mitT _ { \\mittheta ( \\mitL _ { 0 } + \\mitepsilon \\mitL _ { 1 } ) } ( \\mitf ^ { \\otimes \\mitl } ) \\mupOmega \\Big ) = \\miti ^ { \\mitm } \\underset { \\mittheta \\rightarrow 1 } { \\operatorname { l i m } } \\Big ( \\mupOmega , \\mitT _ { \\mittheta \\mitL _ { 0 } } ( ( \\mittheta \\mitL _ { 1 } ) ^ { \\otimes \\mitm } \\otimes \\mitf ^ { \\otimes \\mitl } ) \\mupOmega \\Big ) \\end{equation*}",
"\\begin{equation*} \\underset { \\mittheta \\rightarrow 1 } { \\operatorname { l i m } } \\Big ( \\mupOmega , \\mitS _ { \\mittheta ( \\mitL _ { 0 } + \\mitepsilon \\mitL _ { 1 } ) } ( \\mitlambda \\mitf ) \\mupOmega \\Big ) = \\underset { \\mittheta \\rightarrow 1 } { \\operatorname { l i m } } \\Big ( \\mupOmega , \\mitS _ { \\mittheta \\mitL _ { 0 } } ( \\mittheta \\mitepsilon \\mitL _ { 1 } + \\mitlambda \\mitf ) \\mupOmega \\Big ) . \\end{equation*}",
"\\begin{equation*} \\underset { \\mittheta \\rightarrow 1 } { \\operatorname { l i m } } ( \\mupOmega , \\mitS _ { \\mittheta \\mitL } ( \\mitf ) \\mupOmega ) = \\underset { \\mittheta \\rightarrow 1 } { \\operatorname { l i m } } \\frac { ( \\mupOmega , \\mitS ( \\mittheta \\mitL + \\mitf ) \\mupOmega ) } { ( \\mupOmega , \\mitS ( \\mittheta \\mitL ) \\mupOmega ) } , \\end{equation*}",
"\\begin{align*} ( \\mupOmega , \\mitS _ { \\mittheta \\mitL } ( \\mitf ) \\mupOmega ) & = & ( \\mitS ( \\mittheta \\mitL ) \\mupOmega , ( \\mitP _ { \\mupOmega } + \\mitP _ { \\mupOmega } ^ { \\bot } ) \\mitS ( \\mittheta \\mitL + \\mitf ) \\mupOmega ) \\\\ & = & \\frac { ( \\mupOmega , \\mitS ( \\mittheta \\mitL + \\mitf ) \\mupOmega ) } { ( \\mupOmega , \\mitS ( \\mittheta \\mitL ) \\mupOmega ) } \\cdot \\vert ( \\mupOmega , \\mitS ( \\mittheta \\mitL ) \\mupOmega ) \\vert ^ { 2 } \\\\ & + & ( \\mupOmega , \\mitS ( \\mittheta \\mitL ) ^ { - 1 } \\mitP _ { \\mupOmega } ^ { \\bot } \\mitS ( \\mittheta \\mitL + \\mitf ) \\mupOmega ) \\end{align*}"
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0001129_page23 | {
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"$(\\Omega ,S(\\theta {\\cal L})^{-1}P_\\Omega ^\\bot S(\\theta {\\cal L}+f)\\Omega )$",
"$S(\\theta {\\cal L})^{-1}$",
"$S(\\theta {\\cal L}+f)$",
"$(\\Omega ,S(\\theta {\\cal L})^{-1}\\Omega )\\break (\\Omega , S(\\theta {\\cal L}+f)\\Omega )$",
"$(\\Omega ,S(\\theta {\\cal L})^{-1}\\Omega )\\break (\\Omega , S(\\theta {\\cal L}+f)\\Omega )$",
"$f=0$",
"${\\cal L}_0$",
"${\\cal L}_1$",
"${\\rm dim}({\\cal L}_j)\\leq 4$",
"${\\rm dim}({\\cal L}_j)\\leq 4$",
"$4$",
"$\\quad \\w $",
"${\\cal L}_0$",
"${\\cal L}_1$",
"$\\lambda _1,...,\\lambda _s$",
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"${\\cal D}(\\RR ^4,{\\cal V})$",
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"\\begin {eqnarray} 1&=&(\\Omega ,S(\\theta {\\cal L})^{-1}(P_\\Omega +P_\\Omega ^\\bot )S(\\theta {\\cal L}) \\Omega )\\\\ &=&|(\\Omega ,S(\\theta {\\cal L})\\Omega )|^2+ (\\Omega ,S(\\theta {\\cal L})^{-1}P_\\Omega ^\\bot S(\\theta {\\cal L})\\Omega ). \\end {eqnarray}",
"\\begin {equation} \\lim _{\\theta \\to 1}(\\Omega ,S(\\theta {\\cal L})^{-1}P_\\Omega ^\\bot S(\\theta {\\cal L}+f)\\Omega )=0.\\label {gml3} \\end {equation}",
"\\begin {equation} \\lim _{\\theta \\to 1}\\frac {(\\Omega ,S(\\theta ({\\cal L}_0+\\epsilon {\\cal L}_1) +\\lambda f)\\Omega )} {(\\Omega ,S(\\theta ({\\cal L}_0+\\epsilon {\\cal L}_1))\\Omega )} =\\lim _{\\theta \\to 1}\\frac {(\\Omega ,S(\\theta ({\\cal L}_0+\\epsilon {\\cal L}_1) +\\lambda f)\\Omega )} {(\\Omega ,S(\\theta {\\cal L}_0)\\Omega )}.\\label {gml4} \\end {equation}",
"\\begin {equation} \\lim _{\\theta \\to 1}\\frac {(\\Omega ,S(\\theta ({\\cal L}_0+\\epsilon {\\cal L}_1)) \\Omega )}{(\\Omega ,S(\\theta {\\cal L}_0)\\Omega )}=1.\\label {gml5} \\end {equation}",
"\\begin {equation} \\lim _{\\theta \\to 1}(\\Omega ,S(\\theta {\\cal L}_0)\\Omega )=1,\\quad \\quad \\lim _{\\theta \\to 1}(\\Omega ,S(\\theta ({\\cal L}_0+\\epsilon {\\cal L}_1))\\Omega )=1. \\end {equation}",
"\\begin {equation} \\lim _{\\theta \\to 1}\\Bigl (\\Omega ,S_{\\theta ({\\cal L}_0+\\epsilon {\\cal L}_1)} (\\lambda f)\\Omega \\Bigr )=\\lim _{\\theta \\to 1}\\frac {\\Bigl (\\Omega ,S_{\\theta {\\cal L}_0}(\\theta \\epsilon {\\cal L}_1+\\lambda f) \\Omega \\Bigr )}{\\Bigl (\\Omega ,S_{\\theta {\\cal L}_0}(\\theta \\epsilon {\\cal L}_1) \\Omega \\Bigr )}\\label {Q7} \\end {equation}"
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"\\begin{equation*} \\underset { \\theta \\rightarrow 1 } { \\operatorname { l i m } } ( \\Omega , S _ { \\theta ( L _ { 0 } + \\epsilon L _ { 1 } ) } ( \\lambda f ) \\Omega ) = \\underset { \\theta \\rightarrow 1 } { \\operatorname { l i m } } \\frac { ( \\Omega , S _ { \\theta L _ { 0 } } ( \\theta \\epsilon L _ { 1 } + \\lambda f ) \\Omega ) } { ( \\Omega , S _ { \\theta L _ { 0 } } ( \\theta \\epsilon L _ { 1 } ) \\Omega ) } \\end{equation*}"
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"\\begin{equation*} \\underset { \\mittheta \\rightarrow 1 } { \\operatorname { l i m } } \\frac { ( \\mupOmega , \\mitS ( \\mittheta ( \\mitL _ { 0 } + \\mitepsilon \\mitL _ { 1 } ) ) \\mupOmega ) } { ( \\mupOmega , \\mitS ( \\mittheta \\mitL _ { 0 } ) \\mupOmega ) } = 1 . \\end{equation*}",
"\\begin{equation*} \\underset { \\mittheta \\rightarrow 1 } { \\operatorname { l i m } } ( \\mupOmega , \\mitS ( \\mittheta \\mitL _ { 0 } ) \\mupOmega ) = 1 , \\quad \\quad \\underset { \\mittheta \\rightarrow 1 } { \\operatorname { l i m } } ( \\mupOmega , \\mitS ( \\mittheta ( \\mitL _ { 0 } + \\mitepsilon \\mitL _ { 1 } ) ) \\mupOmega ) = 1 . \\end{equation*}",
"\\begin{equation*} \\underset { \\mittheta \\rightarrow 1 } { \\operatorname { l i m } } \\Big ( \\mupOmega , \\mitS _ { \\mittheta ( \\mitL _ { 0 } + \\mitepsilon \\mitL _ { 1 } ) } ( \\mitlambda \\mitf ) \\mupOmega \\Big ) = \\underset { \\mittheta \\rightarrow 1 } { \\operatorname { l i m } } \\frac { \\Big ( \\mupOmega , \\mitS _ { \\mittheta \\mitL _ { 0 } } ( \\mittheta \\mitepsilon \\mitL _ { 1 } + \\mitlambda \\mitf ) \\mupOmega \\Big ) } { \\Big ( \\mupOmega , \\mitS _ { \\mittheta \\mitL _ { 0 } } ( \\mittheta \\mitepsilon \\mitL _ { 1 } ) \\mupOmega \\Big ) } \\end{equation*}"
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0001129_page24 | {
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"$ \\mitP $",
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"$ \\mscrO _ { \\mitP } ( 1 ) $",
"$ \\mitg _ { 2 } $",
"$ \\mitg _ { 3 } $",
"$ \\mscrL ^ { 4 } $",
"$ \\mscrL ^ { 6 } $",
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"$ \\mbfY $",
"$ \\mbfZ $",
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"$ \\mitz = \\mbfZ ^ { 3 } $",
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"$ \\mbfZ $",
"$ \\mscrO ( \\mitsigma ) $",
"$ \\mitz = \\mbfZ ^ { 3 } $",
"\\begin{equation*} \\mitz \\mity ^ { 2 } = 4 \\mitx ^ { 3 } - \\mitg _ { 2 } \\mitx \\mitz ^ { 2 } - \\mitg _ { 3 } \\mitz ^ { 3 } \\end{equation*}",
"\\begin{equation*} \\mbfX \\sim \\mscrO ( 2 \\mitsigma ) \\otimes \\mscrL ^ { 2 } , \\qquad \\mbfY \\sim \\mscrO ( 3 \\mitsigma ) \\otimes \\mscrL ^ { 3 } , \\qquad \\mbfZ \\sim \\mscrO ( \\mitsigma ) \\end{equation*}"
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0001133_page07 | {
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"$s$",
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"\\begin {equation} D(a_{0}) \\text { is an irreducible divisor in } X \\label {eq:13} \\end {equation}",
"\\begin {equation} D(a_{n}) \\text { is an effective class in } H_{4}(X, {\\mathbb Z}). \\label {eq:14} \\end {equation}"
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|
0001133_page09 | {
"latex": [
"$a=12$",
"$b=15$",
"$n=5$",
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"\\begin {equation} \\eta = 24\\cS +36\\cE \\label {eq:25} \\end {equation}",
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"\\begin {equation} n \\quad \\mbox {is even}, \\qquad \\lambda =m, \\qquad \\eta =c_{1}(B) \\quad \\mod 2 \\label {eq:29} \\end {equation}"
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0001133_page11 | {
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0001133_page14 | {
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0001133_page20 | {
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0001133_page23 | {
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0001133_page24 | {
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0001133_page25 | {
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0001133_page26 | {
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|
0001133_page27 | {
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0001133_page31 | {
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0001133_page36 | {
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0001178_page02 | {
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0001178_page03 | {
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0001178_page06 | {
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"\\begin{equation*} \\square \\phi ^ { i } = 0 \\; , \\quad \\partial ^ { \\alpha \\beta } \\psi _ { \\beta } ^ { [ i j ] } = \\partial ^ { \\alpha \\beta } F _ { ( \\beta \\gamma ) } ^ { i } = \\partial ^ { \\alpha \\beta } \\chi _ { ( \\beta \\gamma \\delta ) } = 0 \\end{equation*}"
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"\\begin{equation*} \\square \\mitphi ^ { \\miti } = 0 \\; , \\quad \\mitpartial ^ { \\mitalpha \\mitbeta } \\mitpsi _ { \\mitbeta } = 0 \\; . \\end{equation*}",
"\\begin{align*} \\mitN = ( 2 , 0 ) : \\qquad \\mitW ^ { \\miti } & = & \\mitphi ^ { \\miti } + \\mittheta _ { \\mitj } ^ { \\mitalpha } \\mitpsi _ { \\mitalpha } ^ { [ \\miti \\mitj ] } + \\mittheta _ { \\mitk } ^ { \\mitalpha } \\mittheta _ { \\mitl } ^ { \\mitbeta } \\mitepsilon ^ { \\mitk \\mitl \\miti \\mitj } \\mitF _ { ( \\mitalpha \\mitbeta ) \\mitj } \\\\ & & + \\mittheta _ { \\mitj } ^ { \\mitalpha } \\mittheta _ { \\mitk } ^ { \\mitbeta } \\mittheta _ { \\mitl } ^ { \\mitgamma } \\mitepsilon ^ { \\miti \\mitj \\mitk \\mitl } \\mitchi _ { ( \\mitalpha \\mitbeta \\mitgamma ) } + \\mathrm { d } . ~ \\mathrm { t } . ~ \\end{align*}",
"\\begin{equation*} \\square \\mitphi ^ { \\miti } = 0 \\; , \\quad \\mitpartial ^ { \\mitalpha \\mitbeta } \\mitpsi _ { \\mitbeta } ^ { [ \\miti \\mitj ] } = \\mitpartial ^ { \\mitalpha \\mitbeta } \\mitF _ { ( \\mitbeta \\mitgamma ) } ^ { \\miti } = \\mitpartial ^ { \\mitalpha \\mitbeta } \\mitchi _ { ( \\mitbeta \\mitgamma \\mitdelta ) } = 0 \\end{equation*}"
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0001178_page10 | {
"latex": [
"$D^{11}$",
"$U\\hskip -2pt Sp(4)$",
"$D^{11}$",
"$D^{12}$",
"$D^{13}$",
"$D^{22}$",
"$D^{11}=2[D^{12},D^{13}]$",
"$D^{11}=2[D^{12},D^{13}]$",
"$D^{12}=[D^{22},D^{13}]$",
"$U\\hskip -2pt Sp(4)$",
"$N=(2,0)$",
"$D^{1,2}_\\alpha = D^i_\\alpha u^{1,2}_i$",
"$W^{12}=W^{\\{ij\\}}u^1_i u^2_j$",
"$W^{12}$",
"$U\\hskip -2pt Sp(4)$",
"$u^1_{[k} u^1_{i]} = u^2_{[k} u^2_{i]}=0$",
"$\\Omega ^{ij}u^1_iu^2_j=0$",
"$W^{12}$",
"$\\theta ^3=-\\theta _2$",
"$\\theta ^4=-\\theta _1$",
"$N=(2,0)$",
"$W^{12}$",
"\\begin {eqnarray} U\\hskip -2pt Sp(2): &\\quad & D^{11}f^1(u) = 0 \\\\ &\\quad & \\Rightarrow \\ f^1(u) = f^iu^1_i \\;; \\\\ U\\hskip -2pt Sp(4): &\\quad & D^{11}f^1(u) = D^{12}f^1(u) = D^{13}f^1(u) = D^{22}f^1(u) = 0 \\\\ &\\quad & \\Rightarrow \\ f^1(u) = f^iu^1_i\\;; \\\\ &\\quad & D^{11}f^{12}(u) = D^{12}f^{12}(u) = D^{13}f^{12}(u) = D^{22}f^{12}(u) = 0 \\\\ &\\quad & \\Rightarrow \\ f^{12}(u) = f^{\\{ij\\}}u^1_iu^2_j\\;. \\end {eqnarray}",
"\\begin {equation}\\label {34} D^{(k}_\\alpha W^{\\{i)j\\}}=0 \\ \\times \\left \\{\\begin {array}{lll} u^1_k u^1_i u^2_j & \\Rightarrow & D^1_\\alpha W^{12} =0 \\\\ u^2_k u^2_i u^1_j & \\Rightarrow & D^2_\\alpha W^{12} =0 \\end {array} \\right .\\;. \\end {equation}",
"\\begin {equation}\\label {35} D^1_\\alpha W^{12} = D^2_\\alpha W^{12} =0\\;. \\end {equation}",
"\\begin {equation}\\label {36} D^1_\\alpha W^{12} = D^2_\\alpha W^{12} =0 \\ \\Rightarrow \\^^MW^{12} = W^{12}(x_A,\\theta ^1,\\theta ^2,u) \\end {equation}",
"\\begin {equation}\\label {37} x^{\\alpha \\beta }_A = x^{\\alpha \\beta } - i\\theta ^{\\alpha (i}\\theta ^{\\beta j)} (u^1_iu^4_j + u^2_iu^3_j)\\;, \\quad \\theta ^{1\\alpha } = \\theta ^\\alpha _4 = \\theta ^{\\alpha }_i u^i_4\\;, \\ \\theta ^{2\\alpha } = \\theta ^\\alpha _3 = \\theta ^{\\alpha }_i u^i_3\\;. \\end {equation}"
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"$ U \\hspace{-2.0pt} S p ( 4 ) $",
"$ D ^ { 1 1 } $",
"$ D ^ { 1 2 } $",
"$ D ^ { 1 3 } $",
"$ D ^ { 2 2 } $",
"$ D ^ { 1 1 } = 2 [ D ^ { 1 2 } , D ^ { 1 3 } ] $",
"$ D ^ { 1 1 } = 2 [ D ^ { 1 2 } , D ^ { 1 3 } ] $",
"$ D ^ { 1 2 } = [ D ^ { 2 2 } , D ^ { 1 3 } ] $",
"$ U \\hspace{-2.0pt} S p ( 4 ) $",
"$ N = ( 2 , 0 ) $",
"$ D _ { \\alpha } ^ { 1 , 2 } = D _ { \\alpha } ^ { i } u _ { i } ^ { 1 , 2 } $",
"$ W ^ { 1 2 } = W ^ { \\{ i j \\} } u _ { i } ^ { 1 } u _ { j } ^ { 2 } $",
"$ W ^ { 1 2 } $",
"$ U \\hspace{-2.0pt} S p ( 4 ) $",
"$ u _ { [ k } ^ { 1 } u _ { i ] } ^ { 1 } = u _ { [ k } ^ { 2 } u _ { i ] } ^ { 2 } = 0 $",
"$ \\Omega ^ { i j } u _ { i } ^ { 1 } u _ { j } ^ { 2 } = 0 $",
"$ W ^ { 1 2 } $",
"$ \\theta ^ { 3 } = - \\theta _ { 2 } $",
"$ \\theta ^ { 4 } = - \\theta _ { 1 } $",
"$ N = ( 2 , 0 ) $",
"$ W ^ { 1 2 } $",
"\\begin{align*} U \\hspace{-2.0pt} S p ( 2 ) : & & D ^ { 1 1 } f ^ { 1 } ( u ) = 0 \\\\ & & \\Rightarrow ~ f ^ { 1 } ( u ) = f ^ { i } u _ { i } ^ { 1 } \\; ; \\\\ U \\hspace{-2.0pt} S p ( 4 ) : & & D ^ { 1 1 } f ^ { 1 } ( u ) = D ^ { 1 2 } f ^ { 1 } ( u ) = D ^ { 1 3 } f ^ { 1 } ( u ) = D ^ { 2 2 } f ^ { 1 } ( u ) = 0 \\\\ & & \\Rightarrow ~ f ^ { 1 } ( u ) = f ^ { i } u _ { i } ^ { 1 } \\; ; \\\\ & & D ^ { 1 1 } f ^ { 1 2 } ( u ) = D ^ { 1 2 } f ^ { 1 2 } ( u ) = D ^ { 1 3 } f ^ { 1 2 } ( u ) = D ^ { 2 2 } f ^ { 1 2 } ( u ) = 0 \\\\ & & \\Rightarrow ~ f ^ { 1 2 } ( u ) = f ^ { \\{ i j \\} } u _ { i } ^ { 1 } u _ { j } ^ { 2 } \\; . \\end{align*}",
"\\begin{align*} D _ { \\alpha } ^ { ( k } W ^ { \\{ i ) j \\} } = 0 ~ \\times \\{ \\begin{array}{ccc} u _ { k } ^ { 1 } u _ { i } ^ { 1 } u _ { j } ^ { 2 } & \\Rightarrow & D _ { \\alpha } ^ { 1 } W ^ { 1 2 } = 0 \\\\ u _ { k } ^ { 2 } u _ { i } ^ { 2 } u _ { j } ^ { 1 } & \\Rightarrow & D _ { \\alpha } ^ { 2 } W ^ { 1 2 } = 0 \\end{array} \\; . \\end{align*}",
"\\begin{equation*} D _ { \\alpha } ^ { 1 } W ^ { 1 2 } = D _ { \\alpha } ^ { 2 } W ^ { 1 2 } = 0 \\; . \\end{equation*}",
"\\begin{equation*} D _ { \\alpha } ^ { 1 } W ^ { 1 2 } = D _ { \\alpha } ^ { 2 } W ^ { 1 2 } = 0 ~ \\Rightarrow ~ W ^ { 1 2 } = W ^ { 1 2 } ( x _ { A } , \\theta ^ { 1 } , \\theta ^ { 2 } , u ) \\end{equation*}",
"\\begin{equation*} x _ { A } ^ { \\alpha \\beta } = x ^ { \\alpha \\beta } - i \\theta ^ { \\alpha ( i } \\theta ^ { \\beta j ) } ( u _ { i } ^ { 1 } u _ { j } ^ { 4 } + u _ { i } ^ { 2 } u _ { j } ^ { 3 } ) \\; , \\quad \\theta ^ { 1 \\alpha } = \\theta _ { 4 } ^ { \\alpha } = \\theta _ { i } ^ { \\alpha } u _ { 4 } ^ { i } \\; , ~ \\theta ^ { 2 \\alpha } = \\theta _ { 3 } ^ { \\alpha } = \\theta _ { i } ^ { \\alpha } u _ { 3 } ^ { i } \\; . \\end{equation*}"
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"$ \\mitU \\hspace{-2.0pt} \\mitS \\mitp ( 4 ) $",
"$ \\mitD ^ { 1 1 } $",
"$ \\mitD ^ { 1 2 } $",
"$ \\mitD ^ { 1 3 } $",
"$ \\mitD ^ { 2 2 } $",
"$ \\mitD ^ { 1 1 } = 2 [ \\mitD ^ { 1 2 } , \\mitD ^ { 1 3 } ] $",
"$ \\mitD ^ { 1 1 } = 2 [ \\mitD ^ { 1 2 } , \\mitD ^ { 1 3 } ] $",
"$ \\mitD ^ { 1 2 } = [ \\mitD ^ { 2 2 } , \\mitD ^ { 1 3 } ] $",
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"$ \\mitN = ( 2 , 0 ) $",
"$ \\mitD _ { \\mitalpha } ^ { 1 , 2 } = \\mitD _ { \\mitalpha } ^ { \\miti } \\mitu _ { \\miti } ^ { 1 , 2 } $",
"$ \\mitW ^ { 1 2 } = \\mitW ^ { \\{ \\miti \\mitj \\} } \\mitu _ { \\miti } ^ { 1 } \\mitu _ { \\mitj } ^ { 2 } $",
"$ \\mitW ^ { 1 2 } $",
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"$ \\mitu _ { [ \\mitk } ^ { 1 } \\mitu _ { \\miti ] } ^ { 1 } = \\mitu _ { [ \\mitk } ^ { 2 } \\mitu _ { \\miti ] } ^ { 2 } = 0 $",
"$ \\mupOmega ^ { \\miti \\mitj } \\mitu _ { \\miti } ^ { 1 } \\mitu _ { \\mitj } ^ { 2 } = 0 $",
"$ \\mitW ^ { 1 2 } $",
"$ \\mittheta ^ { 3 } = - \\mittheta _ { 2 } $",
"$ \\mittheta ^ { 4 } = - \\mittheta _ { 1 } $",
"$ \\mitN = ( 2 , 0 ) $",
"$ \\mitW ^ { 1 2 } $",
"\\begin{align*} \\mitU \\hspace{-2.0pt} \\mitS \\mitp ( 2 ) : & & \\mitD ^ { 1 1 } \\mitf ^ { 1 } ( \\mitu ) = 0 \\\\ & & \\Rightarrow ~ \\mitf ^ { 1 } ( \\mitu ) = \\mitf ^ { \\miti } \\mitu _ { \\miti } ^ { 1 } \\; ; \\\\ \\mitU \\hspace{-2.0pt} \\mitS \\mitp ( 4 ) : & & \\mitD ^ { 1 1 } \\mitf ^ { 1 } ( \\mitu ) = \\mitD ^ { 1 2 } \\mitf ^ { 1 } ( \\mitu ) = \\mitD ^ { 1 3 } \\mitf ^ { 1 } ( \\mitu ) = \\mitD ^ { 2 2 } \\mitf ^ { 1 } ( \\mitu ) = 0 \\\\ & & \\Rightarrow ~ \\mitf ^ { 1 } ( \\mitu ) = \\mitf ^ { \\miti } \\mitu _ { \\miti } ^ { 1 } \\; ; \\\\ & & \\mitD ^ { 1 1 } \\mitf ^ { 1 2 } ( \\mitu ) = \\mitD ^ { 1 2 } \\mitf ^ { 1 2 } ( \\mitu ) = \\mitD ^ { 1 3 } \\mitf ^ { 1 2 } ( \\mitu ) = \\mitD ^ { 2 2 } \\mitf ^ { 1 2 } ( \\mitu ) = 0 \\\\ & & \\Rightarrow ~ \\mitf ^ { 1 2 } ( \\mitu ) = \\mitf ^ { \\{ \\miti \\mitj \\} } \\mitu _ { \\miti } ^ { 1 } \\mitu _ { \\mitj } ^ { 2 } \\; . \\end{align*}",
"\\begin{align*} \\mitD _ { \\mitalpha } ^ { ( \\mitk } \\mitW ^ { \\{ \\miti ) \\mitj \\} } = 0 ~ \\times \\left\\{ \\begin{array}{ccc} \\mitu _ { \\mitk } ^ { 1 } \\mitu _ { \\miti } ^ { 1 } \\mitu _ { \\mitj } ^ { 2 } & \\Rightarrow & \\mitD _ { \\mitalpha } ^ { 1 } \\mitW ^ { 1 2 } = 0 \\\\ \\mitu _ { \\mitk } ^ { 2 } \\mitu _ { \\miti } ^ { 2 } \\mitu _ { \\mitj } ^ { 1 } & \\Rightarrow & \\mitD _ { \\mitalpha } ^ { 2 } \\mitW ^ { 1 2 } = 0 \\end{array} \\right. \\; . \\end{align*}",
"\\begin{equation*} \\mitD _ { \\mitalpha } ^ { 1 } \\mitW ^ { 1 2 } = \\mitD _ { \\mitalpha } ^ { 2 } \\mitW ^ { 1 2 } = 0 \\; . \\end{equation*}",
"\\begin{equation*} \\mitD _ { \\mitalpha } ^ { 1 } \\mitW ^ { 1 2 } = \\mitD _ { \\mitalpha } ^ { 2 } \\mitW ^ { 1 2 } = 0 ~ \\Rightarrow ~ \\mitW ^ { 1 2 } = \\mitW ^ { 1 2 } ( \\mitx _ { \\mitA } , \\mittheta ^ { 1 } , \\mittheta ^ { 2 } , \\mitu ) \\end{equation*}",
"\\begin{equation*} \\mitx _ { \\mitA } ^ { \\mitalpha \\mitbeta } = \\mitx ^ { \\mitalpha \\mitbeta } - \\miti \\mittheta ^ { \\mitalpha ( \\miti } \\mittheta ^ { \\mitbeta \\mitj ) } ( \\mitu _ { \\miti } ^ { 1 } \\mitu _ { \\mitj } ^ { 4 } + \\mitu _ { \\miti } ^ { 2 } \\mitu _ { \\mitj } ^ { 3 } ) \\; , \\quad \\mittheta ^ { 1 \\mitalpha } = \\mittheta _ { 4 } ^ { \\mitalpha } = \\mittheta _ { \\miti } ^ { \\mitalpha } \\mitu _ { 4 } ^ { \\miti } \\; , ~ \\mittheta ^ { 2 \\mitalpha } = \\mittheta _ { 3 } ^ { \\mitalpha } = \\mittheta _ { \\miti } ^ { \\mitalpha } \\mitu _ { 3 } ^ { \\miti } \\; . \\end{equation*}"
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0001178_page12 | {
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"$N=(1,0)$",
"$N=(2,0)$",
"$U\\hskip -2pt Sp(2n)$",
"$\\phi ^1=\\phi ^i(x)u^1_i$",
"$\\psi _\\alpha =\\psi _\\alpha (x)$",
"$\\psi ^{23}_\\alpha = \\psi ^{\\{ij\\}}_\\alpha (x) u^2_i u^3_j$",
"$F^3_{(\\alpha \\beta )} = F^i_{(\\alpha \\beta )}(x)u^3_i$",
"$u^I_iu^I_j$",
"$I$",
"$I=1,2$",
"$N=(1,0)$",
"$I=1,2,3,4$",
"$N=(2,0)$",
"$\\theta ^I$",
"$W$",
"$U\\hskip -2pt Sp(2n)$",
"$W$",
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"$W^{12}$",
"$W^1$",
"\\begin {equation}\\label {41} D^1_\\alpha W^1 = 0 \\quad \\Rightarrow \\quad \\left \\{ \\begin {array}{ll} W^1 = W^1(\\theta ^1)\\;, & N=(1,0) \\\\ W^1 = W^1(\\theta ^1,\\theta ^2,\\theta ^3)\\;, & N=(2,0) \\end {array} \\right . \\;. \\end {equation}",
"\\begin {eqnarray} N=(1,0): &\\ & D^{11}W^1=0\\;;\\\\ N=(2,0): &\\ & D^{11}W^1=D^{12}W^1=D^{13}W^1=D^{22}W^1=0\\;.\\end {eqnarray}",
"\\begin {equation}\\label {44} N=(1,0): \\quad W^1 = \\phi ^1 + \\theta ^{1\\alpha }\\psi _\\alpha + \\mbox {\\small d.t.} \\end {equation}",
"\\begin {eqnarray} N=(2,0): && W^1 = \\phi ^1 + \\theta ^{1\\alpha }\\psi _\\alpha - (\\theta ^{1\\alpha }\\psi ^{23}_\\alpha + \\mbox {\\small cycle 123}) \\\\ && - (\\theta ^{1\\alpha }\\theta ^{2\\beta }F^3_{(\\alpha \\beta )} + \\mbox {\\small cycle 123}) + 6 \\theta ^{1\\alpha }\\theta ^{2\\beta } \\theta ^{3\\gamma }\\chi _{(\\alpha \\beta \\gamma )} + \\mbox {\\small d.t.} \\end {eqnarray}",
"\\begin {equation}\\label {47} D^I_\\alpha D^I_\\beta W = 0 \\end {equation}",
"\\begin {equation}\\label {47'} N=(1,0): \\qquad W = \\phi + {1\\over 2}(\\theta ^{1\\alpha }\\psi ^2_\\alpha - \\theta ^{2\\alpha }\\psi ^1_\\alpha ) + \\theta ^{1\\alpha }\\theta ^{2\\beta } F_{(\\alpha \\beta )} + \\mbox {\\small d.t.} \\end {equation}"
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"$ F _ { ( \\alpha \\beta ) } ^ { 3 } = F _ { ( \\alpha \\beta ) } ^ { i } ( x ) u _ { i } ^ { 3 } $",
"$ u _ { i } ^ { I } u _ { j } ^ { I } $",
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"\\begin{align*} D _ { \\alpha } ^ { 1 } W ^ { 1 } = 0 \\quad \\Rightarrow \\quad \\{ \\begin{array}{ll} W ^ { 1 } = W ^ { 1 } ( \\theta ^ { 1 } ) \\; , & N = ( 1 , 0 ) \\\\ W ^ { 1 } = W ^ { 1 } ( \\theta ^ { 1 } , \\theta ^ { 2 } , \\theta ^ { 3 } ) \\; , & N = ( 2 , 0 ) \\end{array} \\; . \\end{align*}",
"\\begin{align*} N = ( 1 , 0 ) : & & D ^ { 1 1 } W ^ { 1 } = 0 \\; ; \\\\ N = ( 2 , 0 ) : & & D ^ { 1 1 } W ^ { 1 } = D ^ { 1 2 } W ^ { 1 } = D ^ { 1 3 } W ^ { 1 } = D ^ { 2 2 } W ^ { 1 } = 0 \\; . \\end{align*}",
"\\begin{equation*} N = ( 1 , 0 ) : \\quad W ^ { 1 } = \\phi ^ { 1 } + \\theta ^ { 1 \\alpha } \\psi _ { \\alpha } + d . t . \\end{equation*}",
"\\begin{align*} N = ( 2 , 0 ) : & & W ^ { 1 } = \\phi ^ { 1 } + \\theta ^ { 1 \\alpha } \\psi _ { \\alpha } - ( \\theta ^ { 1 \\alpha } \\psi _ { \\alpha } ^ { 2 3 } + c y c l e ~ 1 2 3 ) \\\\ & & - ( \\theta ^ { 1 \\alpha } \\theta ^ { 2 \\beta } F _ { ( \\alpha \\beta ) } ^ { 3 } + c y c l e ~ 1 2 3 ) + 6 \\theta ^ { 1 \\alpha } \\theta ^ { 2 \\beta } \\theta ^ { 3 \\gamma } \\chi _ { ( \\alpha \\beta \\gamma ) } + d . t . \\end{align*}",
"\\begin{equation*} D _ { \\alpha } ^ { I } D _ { \\beta } ^ { I } W = 0 \\end{equation*}",
"\\begin{equation*} N = ( 1 , 0 ) : \\qquad W = \\phi + \\frac { 1 } { 2 } ( \\theta ^ { 1 \\alpha } \\psi _ { \\alpha } ^ { 2 } - \\theta ^ { 2 \\alpha } \\psi _ { \\alpha } ^ { 1 } ) + \\theta ^ { 1 \\alpha } \\theta ^ { 2 \\beta } F _ { ( \\alpha \\beta ) } + d . t . \\end{equation*}"
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"$ \\mitu _ { \\miti } ^ { \\mitI } \\mitu _ { \\mitj } ^ { \\mitI } $",
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"$ \\mitN = ( 2 , 0 ) $",
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"$ \\mitW ^ { 1 } $",
"\\begin{align*} \\mitD _ { \\mitalpha } ^ { 1 } \\mitW ^ { 1 } = 0 \\quad \\Rightarrow \\quad \\left\\{ \\begin{array}{ll} \\mitW ^ { 1 } = \\mitW ^ { 1 } ( \\mittheta ^ { 1 } ) \\; , & \\mitN = ( 1 , 0 ) \\\\ \\mitW ^ { 1 } = \\mitW ^ { 1 } ( \\mittheta ^ { 1 } , \\mittheta ^ { 2 } , \\mittheta ^ { 3 } ) \\; , & \\mitN = ( 2 , 0 ) \\end{array} \\right. \\; . \\end{align*}",
"\\begin{align*} \\mitN = ( 1 , 0 ) : & & \\mitD ^ { 1 1 } \\mitW ^ { 1 } = 0 \\; ; \\\\ \\mitN = ( 2 , 0 ) : & & \\mitD ^ { 1 1 } \\mitW ^ { 1 } = \\mitD ^ { 1 2 } \\mitW ^ { 1 } = \\mitD ^ { 1 3 } \\mitW ^ { 1 } = \\mitD ^ { 2 2 } \\mitW ^ { 1 } = 0 \\; . \\end{align*}",
"\\begin{equation*} \\mitN = ( 1 , 0 ) : \\quad \\mitW ^ { 1 } = \\mitphi ^ { 1 } + \\mittheta ^ { 1 \\mitalpha } \\mitpsi _ { \\mitalpha } + \\mathrm { d } . \\mathrm { t } . \\end{equation*}",
"\\begin{align*} \\mitN = ( 2 , 0 ) : & & \\mitW ^ { 1 } = \\mitphi ^ { 1 } + \\mittheta ^ { 1 \\mitalpha } \\mitpsi _ { \\mitalpha } - ( \\mittheta ^ { 1 \\mitalpha } \\mitpsi _ { \\mitalpha } ^ { 2 3 } + \\mathrm { c y c l e } ~ 1 2 3 ) \\\\ & & - ( \\mittheta ^ { 1 \\mitalpha } \\mittheta ^ { 2 \\mitbeta } \\mitF _ { ( \\mitalpha \\mitbeta ) } ^ { 3 } + \\mathrm { c y c l e } ~ 1 2 3 ) + 6 \\mittheta ^ { 1 \\mitalpha } \\mittheta ^ { 2 \\mitbeta } \\mittheta ^ { 3 \\mitgamma } \\mitchi _ { ( \\mitalpha \\mitbeta \\mitgamma ) } + \\mathrm { d } . \\mathrm { t } . \\end{align*}",
"\\begin{equation*} \\mitD _ { \\mitalpha } ^ { \\mitI } \\mitD _ { \\mitbeta } ^ { \\mitI } \\mitW = 0 \\end{equation*}",
"\\begin{equation*} \\mitN = ( 1 , 0 ) : \\qquad \\mitW = \\mitphi + \\frac { 1 } { 2 } ( \\mittheta ^ { 1 \\mitalpha } \\mitpsi _ { \\mitalpha } ^ { 2 } - \\mittheta ^ { 2 \\mitalpha } \\mitpsi _ { \\mitalpha } ^ { 1 } ) + \\mittheta ^ { 1 \\mitalpha } \\mittheta ^ { 2 \\mitbeta } \\mitF _ { ( \\mitalpha \\mitbeta ) } + \\mathrm { d } . \\mathrm { t } . \\end{equation*}"
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|
0001178_page14 | {
"latex": [
"$N=(1,0)$",
"$(W)^2$",
"$(W)^2$",
"$W$",
"$W$",
"$W$",
"$W^1$",
"$W(W^1)^p$",
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"\\begin {equation}\\label {52} \\partial ^{\\alpha \\beta } A_{[\\alpha \\beta ][\\gamma \\delta ]}=0\\;, \\end {equation}",
"\\begin {equation}\\label {53} D^1_\\alpha D^1_\\beta (W(W^1)^p) =0 \\end {equation}",
"\\begin {equation}\\label {54} D^{11} (W(W^1)^p) =0\\;. \\end {equation}",
"\\begin {equation}\\label {55} \\partial ^{\\alpha \\beta }\\psi _{[\\alpha \\beta ]\\gamma } = 0\\;. \\end {equation}",
"\\begin {equation}\\label {56} (W^{12})^2 = \\phi ^{1122} + \\ldots + \\theta ^{1\\alpha }\\theta ^{1\\beta }\\theta ^{2\\gamma }\\theta ^{2\\delta } A_{[\\alpha \\beta ][\\gamma \\delta ]} + \\mbox {\\small d.t.} \\end {equation}",
"\\begin {equation}\\label {series1} (W^{12})^p = \\phi ^{\\stackrel {\\underbrace {\\mbox {\\scriptsize 1\\ldots 1}}}{p} \\stackrel {\\underbrace {\\mbox {\\scriptsize 2\\ldots 2}}}{p}} + \\ldots + \\theta ^{1\\alpha }\\theta ^{1\\beta }\\theta ^{2\\gamma }\\theta ^{2\\delta } A^{\\stackrel {\\underbrace {\\mbox {\\scriptsize 1\\ldots 1}}}{p-2} \\stackrel {\\underbrace {\\mbox {\\scriptsize 2\\ldots 2}}}{p-2}}_{[\\alpha \\beta ][\\gamma \\delta ]} + \\mbox {\\small d.t.} \\end {equation}"
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"\\begin{equation*} D _ { \\alpha } ^ { 1 } D _ { \\beta } ^ { 1 } ( W ( W ^ { 1 } ) ^ { p } ) = 0 \\end{equation*}",
"\\begin{equation*} D ^ { 1 1 } ( W ( W ^ { 1 } ) ^ { p } ) = 0 \\; . \\end{equation*}",
"\\begin{equation*} \\partial ^ { \\alpha \\beta } \\psi _ { [ \\alpha \\beta ] \\gamma } = 0 \\; . \\end{equation*}",
"\\begin{equation*} ( W ^ { 1 2 } ) ^ { 2 } = \\phi ^ { 1 1 2 2 } + \\ldots + \\theta ^ { 1 \\alpha } \\theta ^ { 1 \\beta } \\theta ^ { 2 \\gamma } \\theta ^ { 2 \\delta } A _ { [ \\alpha \\beta ] [ \\gamma \\delta ] } + d . t . \\end{equation*}",
"\\begin{equation*} ( W ^ { 1 2 } ) ^ { p } = \\phi ^ { \\overset { \\underbrace { 1 \\ldots 1 } } { p } \\overset { \\underbrace { 2 \\ldots 2 } } { p } } + \\ldots + \\theta ^ { 1 \\alpha } \\theta ^ { 1 \\beta } \\theta ^ { 2 \\gamma } \\theta ^ { 2 \\delta } A _ { [ \\alpha \\beta ] [ \\gamma \\delta ] } ^ { \\overset { \\underbrace { 1 \\ldots 1 } } { p - 2 } \\overset { \\underbrace { 2 \\ldots 2 } } { p - 2 } } + d . t . \\end{equation*}"
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"$ \\mitW ^ { 1 2 } $",
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"\\begin{equation*} \\mitpartial ^ { \\mitalpha \\mitbeta } \\mitA _ { [ \\mitalpha \\mitbeta ] [ \\mitgamma \\mitdelta ] } = 0 \\; , \\end{equation*}",
"\\begin{equation*} \\mitD _ { \\mitalpha } ^ { 1 } \\mitD _ { \\mitbeta } ^ { 1 } ( \\mitW ( \\mitW ^ { 1 } ) ^ { \\mitp } ) = 0 \\end{equation*}",
"\\begin{equation*} \\mitD ^ { 1 1 } ( \\mitW ( \\mitW ^ { 1 } ) ^ { \\mitp } ) = 0 \\; . \\end{equation*}",
"\\begin{equation*} \\mitpartial ^ { \\mitalpha \\mitbeta } \\mitpsi _ { [ \\mitalpha \\mitbeta ] \\mitgamma } = 0 \\; . \\end{equation*}",
"\\begin{equation*} ( \\mitW ^ { 1 2 } ) ^ { 2 } = \\mitphi ^ { 1 1 2 2 } + \\ldots + \\mittheta ^ { 1 \\mitalpha } \\mittheta ^ { 1 \\mitbeta } \\mittheta ^ { 2 \\mitgamma } \\mittheta ^ { 2 \\mitdelta } \\mitA _ { [ \\mitalpha \\mitbeta ] [ \\mitgamma \\mitdelta ] } + \\mathrm { d } . \\mathrm { t } . \\end{equation*}",
"\\begin{equation*} ( \\mitW ^ { 1 2 } ) ^ { \\mitp } = \\mitphi ^ { \\overset { \\underbrace { 1 \\ldots 1 } } { \\mitp } \\overset { \\underbrace { 2 \\ldots 2 } } { \\mitp } } + \\ldots + \\mittheta ^ { 1 \\mitalpha } \\mittheta ^ { 1 \\mitbeta } \\mittheta ^ { 2 \\mitgamma } \\mittheta ^ { 2 \\mitdelta } \\mitA _ { [ \\mitalpha \\mitbeta ] [ \\mitgamma \\mitdelta ] } ^ { \\overset { \\underbrace { 1 \\ldots 1 } } { \\mitp - 2 } \\overset { \\underbrace { 2 \\ldots 2 } } { \\mitp - 2 } } + \\mathrm { d } . \\mathrm { t } . \\end{equation*}"
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0001213_page04 | {
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"\\begin {eqnarray} && ds^2 = H^{-1/2}[-f dt^2 +dx_1^2 +\\cdots +dx_{p-2}^2 +h(dx_{p-1}^2 +dx_p^2)] +H^{1/2}(f^{-1}dr^2 +r^2 d\\Omega _{8-p}^2), \\\\ && e^{2\\phi } = g^2 H^{\\frac {3-p}{2}}h, \\ \\ \\ B_{p-1,p}=\\tan \\theta H^{-1}h, \\\\ && A^{p}_{012\\cdots p} = g^{-1}(H^{-1}-1)h\\cos \\theta \\coth \\alpha ,\\ \\ \\^^MA^{p-2}_{012\\cdots (p-2)}=g^{-1}(H^{-1}-1)\\sin \\theta \\coth \\alpha , \\end {eqnarray}",
"\\begin {equation} \\label {2e2} H=1+\\frac {r_0^{7-p}\\sinh ^2\\alpha }{r^{7-p}}, \\ \\ \\ f=1-\\left (\\frac {r_0} {r}\\right )^{7-p}, \\ \\ \\ h^{-1}=\\cos ^2\\theta +H^{-1}\\sin ^2\\theta . \\end {equation}",
"\\begin {equation} \\label {2e3} Q_p =\\frac {1}{2\\kappa ^2}\\int _{\\Omega _{8-p}}*F_{p+2} =\\frac {(7-p)\\Omega _{8-p} \\cos \\theta }{2\\kappa ^2 g}r_0^{7-p} \\sinh \\alpha \\cosh \\alpha , \\end {equation}"
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"\\begin{equation*} \\mitH = 1 + \\frac { \\mitr _ { 0 } ^ { 7 - \\mitp } { \\operatorname { s i n h } } ^ { 2 } \\mitalpha } { \\mitr ^ { 7 - \\mitp } } , ~ ~ ~ \\mitf = 1 - { \\left( \\frac { \\mitr _ { 0 } } { \\mitr } \\right) } ^ { 7 - \\mitp } , ~ ~ ~ \\Planckconst ^ { - 1 } = { \\operatorname { c o s } } ^ { 2 } \\mittheta + \\mitH ^ { - 1 } { \\operatorname { s i n } } ^ { 2 } \\mittheta . \\end{equation*}",
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|
0001213_page05 | {
"latex": [
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"\\begin {equation} \\tilde {Q}_{p-2}=\\frac {Q_{p-2}}{V_2}. \\end {equation}",
"\\begin {equation} \\tilde {R}^{7-p} = N_p\\frac {2\\kappa ^2 gT_p}{(7-p)\\Omega _{8-p}\\cos \\theta } =N_{p-2}\\frac {2\\kappa ^2 gT_{p-2}}{(7-p)\\Omega _{8-p}V_2\\sin \\theta }, \\end {equation}",
"\\begin {equation} \\label {2e7} \\tan \\theta =\\frac {\\tilde {Q}_{p-2}}{Q_p}=\\frac {1}{V_2}\\frac {Q_{p-2}}{Q_p} =\\frac {1}{V_2}\\frac {T_{p-2}}{T_p}\\frac {N_{p-2}}{N_p}. \\end {equation}",
"\\begin {eqnarray} && M=\\frac {(8-p)\\Omega _{8-p}V_p r_0^{7-p}}{2\\kappa ^2g^2}\\left (1 +\\frac {7-p}{8-p}\\sinh ^2\\alpha \\right ), \\\\ && T=\\frac {7-p}{4\\pi r_0\\cosh \\alpha }, \\\\ && S=\\frac {4\\pi \\Omega _{8-p}V_p}{2\\kappa ^2g^2}r_0^{8-p}\\cosh \\alpha . \\end {eqnarray}"
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"\\begin{equation*} Q _ { p - 2 } = \\frac { 1 } { 2 \\kappa ^ { 2 } } \\int _ { V _ { 2 } \\times \\Omega _ { 8 - p } } \\ast F _ { p } = \\frac { ( 7 - p ) \\Omega _ { 8 - p } V _ { 2 } \\operatorname { s i n } \\theta } { 2 \\kappa ^ { 2 } g } r _ { 0 } ^ { 7 - p } \\operatorname { s i n h } \\alpha \\operatorname { c o s h } \\alpha . \\end{equation*}",
"\\begin{equation*} \\widetilde { Q } _ { p - 2 } = \\frac { Q _ { p - 2 } } { V _ { 2 } } . \\end{equation*}",
"\\begin{equation*} \\widetilde { R } ^ { 7 - p } = N _ { p } \\frac { 2 \\kappa ^ { 2 } g T _ { p } } { ( 7 - p ) \\Omega _ { 8 - p } \\operatorname { c o s } \\theta } = N _ { p - 2 } \\frac { 2 \\kappa ^ { 2 } g T _ { p - 2 } } { ( 7 - p ) \\Omega _ { 8 - p } V _ { 2 } \\operatorname { s i n } \\theta } , \\end{equation*}",
"\\begin{equation*} \\operatorname { t a n } \\theta = \\frac { \\widetilde { Q } _ { p - 2 } } { Q _ { p } } = \\frac { 1 } { V _ { 2 } } \\frac { Q _ { p - 2 } } { Q _ { p } } = \\frac { 1 } { V _ { 2 } } \\frac { T _ { p - 2 } } { T _ { p } } \\frac { N _ { p - 2 } } { N _ { p } } . \\end{equation*}",
"\\begin{align*} & & M = \\frac { ( 8 - p ) \\Omega _ { 8 - p } V _ { p } r _ { 0 } ^ { 7 - p } } { 2 \\kappa ^ { 2 } g ^ { 2 } } ( 1 + \\frac { 7 - p } { 8 - p } { \\operatorname { s i n h } } ^ { 2 } \\alpha ) , \\\\ & & T = \\frac { 7 - p } { 4 \\pi r _ { 0 } \\operatorname { c o s h } \\alpha } , \\\\ & & S = \\frac { 4 \\pi \\Omega _ { 8 - p } V _ { p } } { 2 \\kappa ^ { 2 } g ^ { 2 } } r _ { 0 } ^ { 8 - p } \\operatorname { c o s h } \\alpha . \\end{align*}"
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"$ 2 \\mitkappa ^ { 2 } = ( 2 \\mitpi ) ^ { 7 } \\mitalpha ^ { \\prime 4 } $",
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"\\begin{equation*} \\mitQ _ { \\mitp - 2 } = \\frac { 1 } { 2 \\mitkappa ^ { 2 } } \\int _ { \\mitV _ { 2 } \\times \\mupOmega _ { 8 - \\mitp } } \\ast \\mitF _ { \\mitp } = \\frac { ( 7 - \\mitp ) \\mupOmega _ { 8 - \\mitp } \\mitV _ { 2 } \\operatorname { s i n } \\mittheta } { 2 \\mitkappa ^ { 2 } \\mitg } \\mitr _ { 0 } ^ { 7 - \\mitp } \\operatorname { s i n h } \\mitalpha \\operatorname { c o s h } \\mitalpha . \\end{equation*}",
"\\begin{equation*} \\tilde { \\mitQ } _ { \\mitp - 2 } = \\frac { \\mitQ _ { \\mitp - 2 } } { \\mitV _ { 2 } } . \\end{equation*}",
"\\begin{equation*} \\tilde { \\mitR } ^ { 7 - \\mitp } = \\mitN _ { \\mitp } \\frac { 2 \\mitkappa ^ { 2 } \\mitg \\mitT _ { \\mitp } } { ( 7 - \\mitp ) \\mupOmega _ { 8 - \\mitp } \\operatorname { c o s } \\mittheta } = \\mitN _ { \\mitp - 2 } \\frac { 2 \\mitkappa ^ { 2 } \\mitg \\mitT _ { \\mitp - 2 } } { ( 7 - \\mitp ) \\mupOmega _ { 8 - \\mitp } \\mitV _ { 2 } \\operatorname { s i n } \\mittheta } , \\end{equation*}",
"\\begin{equation*} \\operatorname { t a n } \\mittheta = \\frac { \\tilde { \\mitQ } _ { \\mitp - 2 } } { \\mitQ _ { \\mitp } } = \\frac { 1 } { \\mitV _ { 2 } } \\frac { \\mitQ _ { \\mitp - 2 } } { \\mitQ _ { \\mitp } } = \\frac { 1 } { \\mitV _ { 2 } } \\frac { \\mitT _ { \\mitp - 2 } } { \\mitT _ { \\mitp } } \\frac { \\mitN _ { \\mitp - 2 } } { \\mitN _ { \\mitp } } . \\end{equation*}",
"\\begin{align*} & & \\mitM = \\frac { ( 8 - \\mitp ) \\mupOmega _ { 8 - \\mitp } \\mitV _ { \\mitp } \\mitr _ { 0 } ^ { 7 - \\mitp } } { 2 \\mitkappa ^ { 2 } \\mitg ^ { 2 } } \\left( 1 + \\frac { 7 - \\mitp } { 8 - \\mitp } { \\operatorname { s i n h } } ^ { 2 } \\mitalpha \\right) , \\\\ & & \\mitT = \\frac { 7 - \\mitp } { 4 \\mitpi \\mitr _ { 0 } \\operatorname { c o s h } \\mitalpha } , \\\\ & & \\mitS = \\frac { 4 \\mitpi \\mupOmega _ { 8 - \\mitp } \\mitV _ { \\mitp } } { 2 \\mitkappa ^ { 2 } \\mitg ^ { 2 } } \\mitr _ { 0 } ^ { 8 - \\mitp } \\operatorname { c o s h } \\mitalpha . \\end{align*}"
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0001213_page06 | {
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"$\\theta $",
"$\\theta $",
"$p$",
"$B$",
"$(p-2)$",
"$B$",
"$B$",
"$(p-2)$",
"$(p-2)$",
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"$\\mu _{p-2}$",
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"$q_p= Q_pV_p$",
"$q_{p-2}=V_{p-2}Q_{p-2}$",
"$r_0 \\rightarrow 0$",
"$\\alpha \\rightarrow \\infty $",
"$\\tilde {R}^{7-p}$",
"$(p-2)$",
"$p$",
"$p$",
"\\begin {eqnarray} dM &=& TdS + \\mu _pdq_p +\\mu _{p-2}dq_{p-2} \\\\ &=& TdS +\\mu _pV_pT_pdN_p +\\mu _{p-2}V_{p-2}T_{p-2}dN_{p-2}, \\end {eqnarray}",
"\\begin {equation} \\mu _p= \\cos \\theta \\tanh \\alpha /g, \\ \\ \\ \\mu _{p-2}=\\sin \\theta \\tanh \\alpha /g. \\end {equation}",
"\\begin {equation} M^2_{\\rm ext.}= q_p^2 +q_{p-2}^2, \\end {equation}",
"\\begin {eqnarray} \\alpha ' \\rightarrow 0:&& \\tan \\theta =\\frac {\\tilde {b}}{\\alpha '}, \\ \\ \\ r=\\alpha ' u, \\ \\ \\ r_0=\\alpha 'u_0, \\\\ && g=\\tilde {g}\\alpha '^{(5-p)/2},\\ \\ \\ x_{0,1,\\cdots ,p-2} =\\tilde {x}_{0,1,\\cdots ,p-2},\\ \\ \\^^Mx_{p-1,p}= \\frac {\\alpha '}{\\tilde {b}}\\tilde {x}_{p-1,p}, \\end {eqnarray}"
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"$ \\mu _ { p - 2 } $",
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"$ q _ { p } = Q _ { p } V _ { p } $",
"$ q _ { p - 2 } = V _ { p - 2 } Q _ { p - 2 } $",
"$ r _ { 0 } \\rightarrow 0 $",
"$ \\alpha \\rightarrow \\infty $",
"$ \\widetilde { R } ^ { 7 - p } $",
"$ ( p - 2 ) $",
"$ p $",
"$ p $",
"\\begin{align*} d M & = & T d S + \\mu _ { p } d q _ { p } + \\mu _ { p - 2 } d q _ { p - 2 } \\\\ & = & T d S + \\mu _ { p } V _ { p } T _ { p } d N _ { p } + \\mu _ { p - 2 } V _ { p - 2 } T _ { p - 2 } d N _ { p - 2 } , \\end{align*}",
"\\begin{equation*} \\mu _ { p } = \\operatorname { c o s } \\theta \\operatorname { t a n h } \\alpha \\slash g , ~ ~ ~ \\mu _ { p - 2 } = \\operatorname { s i n } \\theta \\operatorname { t a n h } \\alpha \\slash g . \\end{equation*}",
"\\begin{equation*} M _ { e x t . } ^ { 2 } = q _ { p } ^ { 2 } + q _ { p - 2 } ^ { 2 } , \\end{equation*}",
"\\begin{align*} \\alpha ^ { \\prime } \\rightarrow 0 : & & \\operatorname { t a n } \\theta = \\frac { \\widetilde { b } } { \\alpha ^ { \\prime } } , ~ ~ ~ r = \\alpha ^ { \\prime } u , ~ ~ ~ r _ { 0 } = \\alpha ^ { \\prime } u _ { 0 } , \\\\ & & g = \\widetilde { g } \\alpha ^ { \\prime ( 5 - p ) \\slash 2 } , ~ ~ ~ x _ { 0 , 1 , \\cdots , p - 2 } = \\widetilde { x } _ { 0 , 1 , \\cdots , p - 2 } , ~ ~ ~ x _ { p - 1 , p } = \\frac { \\alpha ^ { \\prime } } { \\widetilde { b } } \\widetilde { x } _ { p - 1 , p } , \\end{align*}"
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"$ \\mittheta $",
"$ \\mittheta $",
"$ \\mitp $",
"$ \\mitB $",
"$ ( \\mitp - 2 ) $",
"$ \\mitB $",
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"$ \\mitmu _ { \\mitp - 2 } $",
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"$ \\mitq _ { \\mitp } = \\mitQ _ { \\mitp } \\mitV _ { \\mitp } $",
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"\\begin{align*} \\mitd \\mitM & = & \\mitT \\mitd \\mitS + \\mitmu _ { \\mitp } \\mitd \\mitq _ { \\mitp } + \\mitmu _ { \\mitp - 2 } \\mitd \\mitq _ { \\mitp - 2 } \\\\ & = & \\mitT \\mitd \\mitS + \\mitmu _ { \\mitp } \\mitV _ { \\mitp } \\mitT _ { \\mitp } \\mitd \\mitN _ { \\mitp } + \\mitmu _ { \\mitp - 2 } \\mitV _ { \\mitp - 2 } \\mitT _ { \\mitp - 2 } \\mitd \\mitN _ { \\mitp - 2 } , \\end{align*}",
"\\begin{equation*} \\mitmu _ { \\mitp } = \\operatorname { c o s } \\mittheta \\operatorname { t a n h } \\mitalpha \\slash \\mitg , ~ ~ ~ \\mitmu _ { \\mitp - 2 } = \\operatorname { s i n } \\mittheta \\operatorname { t a n h } \\mitalpha \\slash \\mitg . \\end{equation*}",
"\\begin{equation*} \\mitM _ { \\mathrm { e x t } . } ^ { 2 } = \\mitq _ { \\mitp } ^ { 2 } + \\mitq _ { \\mitp - 2 } ^ { 2 } , \\end{equation*}",
"\\begin{align*} \\mitalpha ^ { \\prime } \\rightarrow 0 : & & \\operatorname { t a n } \\mittheta = \\frac { \\tilde { \\mitb } } { \\mitalpha ^ { \\prime } } , ~ ~ ~ \\mitr = \\mitalpha ^ { \\prime } \\mitu , ~ ~ ~ \\mitr _ { 0 } = \\mitalpha ^ { \\prime } \\mitu _ { 0 } , \\\\ & & \\mitg = \\tilde { \\mitg } \\mitalpha ^ { \\prime ( 5 - \\mitp ) \\slash 2 } , ~ ~ ~ \\mitx _ { 0 , 1 , \\cdots , \\mitp - 2 } = \\tilde { \\mitx } _ { 0 , 1 , \\cdots , \\mitp - 2 } , ~ ~ ~ \\mitx _ { \\mitp - 1 , \\mitp } = \\frac { \\mitalpha ^ { \\prime } } { \\tilde { \\mitb } } \\tilde { \\mitx } _ { \\mitp - 1 , \\mitp } , \\end{align*}"
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0001213_page07 | {
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"\\begin {eqnarray} && ds^2=\\alpha '\\left [\\left (\\frac {u}{R}\\right )^{(7-p)/2}\\left (-\\tilde {f} dt^2 +d\\tilde {x}_1^2 +\\cdots +d\\tilde {x}_{p-2}^2 +\\tilde {h}(d\\tilde {x}_{p-1}^2 +d\\tilde {x}_p^2)\\right ) \\right . \\\\ &&~~~~~~~~\\left . +\\left (\\frac {R}{u}\\right )^{(7-p)/2} \\left (\\tilde {f}^{-1}du^2 +u^2 d\\Omega ^2_{8-p}\\right )\\right ], \\\\ && e^{2\\phi } = \\tilde {g}^2\\tilde {b}^2 \\tilde {h}\\left (\\frac {R}{u} \\right )^{(7-p)(3-p)/2}, \\ \\ \\ B_{p-1,p} = \\frac {\\alpha '}{\\tilde {b}}\\frac {(au)^{7-p}}{1+(au)^{7-p}}, \\end {eqnarray}",
"\\begin {equation} \\tilde {f} = 1 - \\left (\\frac {u_0}{u}\\right )^{7-p},\\ \\ \\tilde {h} = \\frac {1} {1+(au)^{7-p}}, \\ \\ \\ a^{7-p} = \\tilde {b}^2/R^{7-p}, \\label {fha} \\end {equation}",
"\\begin {equation} \\label {R} R^{7-p} = \\frac {1}{2}(2\\pi )^{6-p}\\pi ^{-(7-p)/2}\\Gamma [(7-p)/2]\\tilde {g} \\tilde {b} N_p. \\end {equation}",
"\\begin {eqnarray} && E = \\frac {(9-p)\\Omega _{8-p}\\tilde {V}_p}{2(2\\pi )^7 (\\tilde {g}\\tilde {b})^2} u_0^{7-p}, \\\\ && T = \\frac {7-p}{4\\pi }R^{-\\frac {7-p}{2}}u_0^{\\frac {5-p}{2}}, \\\\ && S = \\frac {2\\Omega _{8-p}\\tilde {V}_p}{(2\\pi )^6(\\tilde {g}\\tilde {b})^2} R^{\\frac {7-p}{2}}u_0^{(9-p)/2}. \\end {eqnarray}",
"\\begin {eqnarray} F &=& - \\frac {(5-p)\\Omega _{8-p}\\tilde {V}_p}{2(2\\pi )^7(\\tilde {g}\\tilde {b})^2} u_0^{7-p} \\\\ &=& -\\frac {\\Omega _{8-p}V_{p-2}\\tilde {V}_2}{(2\\pi )^7 \\tilde {g}^2 \\tilde {b}^2} \\frac {5-p}{2}\\left (\\frac {4\\pi }{7-p}\\right )^{\\frac {2(7-p)} {5-p}} R^{\\frac {(7-p)^2}{5-p}}T^{\\frac {2(7-p)}{5-p}}, \\end {eqnarray}"
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"\\begin{equation*} \\widetilde { f } = 1 - { ( \\frac { u _ { 0 } } { u } ) } ^ { 7 - p } , ~ ~ \\widetilde { h } = \\frac { 1 } { 1 + ( a u ) ^ { 7 - p } } , ~ ~ ~ a ^ { 7 - p } = \\widetilde { b } ^ { 2 } \\slash R ^ { 7 - p } , \\end{equation*}",
"\\begin{equation*} R ^ { 7 - p } = \\frac { 1 } { 2 } ( 2 \\pi ) ^ { 6 - p } \\pi ^ { - ( 7 - p ) \\slash 2 } \\Gamma [ ( 7 - p ) \\slash 2 ] \\widetilde { g } \\widetilde { b } N _ { p } . \\end{equation*}",
"\\begin{align*} & & E = \\frac { ( 9 - p ) \\Omega _ { 8 - p } \\widetilde { V } _ { p } } { 2 ( 2 \\pi ) ^ { 7 } ( \\widetilde { g } \\widetilde { b } ) ^ { 2 } } u _ { 0 } ^ { 7 - p } , \\\\ & & T = \\frac { 7 - p } { 4 \\pi } R ^ { - \\frac { 7 - p } { 2 } } u _ { 0 } ^ { \\frac { 5 - p } { 2 } } , \\\\ & & S = \\frac { 2 \\Omega _ { 8 - p } \\widetilde { V } _ { p } } { ( 2 \\pi ) ^ { 6 } ( \\widetilde { g } \\widetilde { b } ) ^ { 2 } } R ^ { \\frac { 7 - p } { 2 } } u _ { 0 } ^ { ( 9 - p ) \\slash 2 } . \\end{align*}",
"\\begin{align*} F & = & - \\frac { ( 5 - p ) \\Omega _ { 8 - p } \\widetilde { V } _ { p } } { 2 ( 2 \\pi ) ^ { 7 } ( \\widetilde { g } \\widetilde { b } ) ^ { 2 } } u _ { 0 } ^ { 7 - p } \\\\ & = & - \\frac { \\Omega _ { 8 - p } V _ { p - 2 } \\widetilde { V } _ { 2 } } { ( 2 \\pi ) ^ { 7 } \\widetilde { g } ^ { 2 } \\widetilde { b } ^ { 2 } } \\frac { 5 - p } { 2 } { ( \\frac { 4 \\pi } { 7 - p } ) } ^ { \\frac { 2 ( 7 - p ) } { 5 - p } } R ^ { \\frac { ( 7 - p ) ^ { 2 } } { 5 - p } } T ^ { \\frac { 2 ( 7 - p ) } { 5 - p } } , \\end{align*}"
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"\\begin{align*} & & \\mitE = \\frac { ( 9 - \\mitp ) \\mupOmega _ { 8 - \\mitp } \\tilde { \\mitV } _ { \\mitp } } { 2 ( 2 \\mitpi ) ^ { 7 } ( \\tilde { \\mitg } \\tilde { \\mitb } ) ^ { 2 } } \\mitu _ { 0 } ^ { 7 - \\mitp } , \\\\ & & \\mitT = \\frac { 7 - \\mitp } { 4 \\mitpi } \\mitR ^ { - \\frac { 7 - \\mitp } { 2 } } \\mitu _ { 0 } ^ { \\frac { 5 - \\mitp } { 2 } } , \\\\ & & \\mitS = \\frac { 2 \\mupOmega _ { 8 - \\mitp } \\tilde { \\mitV } _ { \\mitp } } { ( 2 \\mitpi ) ^ { 6 } ( \\tilde { \\mitg } \\tilde { \\mitb } ) ^ { 2 } } \\mitR ^ { \\frac { 7 - \\mitp } { 2 } } \\mitu _ { 0 } ^ { ( 9 - \\mitp ) \\slash 2 } . \\end{align*}",
"\\begin{align*} \\mitF & = & - \\frac { ( 5 - \\mitp ) \\mupOmega _ { 8 - \\mitp } \\tilde { \\mitV } _ { \\mitp } } { 2 ( 2 \\mitpi ) ^ { 7 } ( \\tilde { \\mitg } \\tilde { \\mitb } ) ^ { 2 } } \\mitu _ { 0 } ^ { 7 - \\mitp } \\\\ & = & - \\frac { \\mupOmega _ { 8 - \\mitp } \\mitV _ { \\mitp - 2 } \\tilde { \\mitV } _ { 2 } } { ( 2 \\mitpi ) ^ { 7 } \\tilde { \\mitg } ^ { 2 } \\tilde { \\mitb } ^ { 2 } } \\frac { 5 - \\mitp } { 2 } { \\left( \\frac { 4 \\mitpi } { 7 - \\mitp } \\right) } ^ { \\frac { 2 ( 7 - \\mitp ) } { 5 - \\mitp } } \\mitR ^ { \\frac { ( 7 - \\mitp ) ^ { 2 } } { 5 - \\mitp } } \\mitT ^ { \\frac { 2 ( 7 - \\mitp ) } { 5 - \\mitp } } , \\end{align*}"
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0001213_page08 | {
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"\\begin {equation} \\label {number} \\tan \\theta = \\frac {\\tilde {b}}{\\alpha '} = \\frac {(2\\pi )^2\\tilde {b}^2} {\\alpha '\\tilde {V}_2}\\frac {N_{p-2}}{N_p}\\ \\ \\ \\^^M\\Longrightarrow \\ \\ \\frac {N_{p-2}}{N_p} = \\frac {\\tilde {V}_2}{(2\\pi )^2\\tilde {b}}. \\end {equation}"
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0001213_page09 | {
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0001213_page10 | {
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"\\begin {eqnarray} ds^2 &=&\\alpha ' \\left [\\left (\\frac {u}{R}\\right )^{(7-p)/2}\\left (-\\tilde {f}dt^2 +d\\tilde {x}_1^2 +\\cdots +d\\tilde {x}_{p-2}^2 +dx_{p-1}^2 +dx_p^2\\right ) \\right . \\\\ && ~~~ \\left . +\\left (\\frac {R}{u}\\right )^{(7-p)/2}\\left (\\tilde {f}^{-1}du^2 + u^2d\\Omega ^2_{8-p}\\right ) \\right ], \\\\ e^{2\\phi } &=& \\frac {(2\\pi )^4\\tilde {g}^2 \\tilde {b}^4}{\\tilde {V}_2^2} \\left (\\frac {u}{R}\\right )^{(7-p)(p-3)/2}, \\ \\ \\ \\tilde {B}_{p-1,p}=0. \\end {eqnarray}",
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"\\begin{equation*} \\mitg _ { \\mathrm { Y M } } ^ { 2 } = ( 2 \\mitpi ) ^ { \\mitp - 4 } \\tilde { \\mitg } , \\end{equation*}"
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0001213_page11 | {
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"\\begin {eqnarray} && ds^2 = \\alpha ' \\left [ \\left (\\frac {u}{R}\\right )^{(7-p)/2} \\left ( -\\tilde {f} dt^2 + d\\tilde {x}_1^2 +\\cdots +d\\tilde {x}^2_{p-2}+ dx_{p-1}^2 +dx_p^2 \\right ) \\right . \\\\ &&~~~~~~ \\left . +\\left (\\frac {R}{u}\\right )^{(7-p)/2}\\left ({\\tilde f}^{-1} du^2 +u^2 d\\Omega _{8-p}^2 \\right ) \\right ], \\\\ && e^{2\\phi } = \\frac {(2\\pi )^4\\tilde {g}^2 \\tilde {b}^4}{\\tilde {V}_2^2} \\left (\\frac {u}{R}\\right )^{(7-p)(p-3)/2}, \\ \\ \\ \\tilde {B}_{p-1,p}= \\frac {\\alpha '}{\\tilde {b}}, \\end {eqnarray}"
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0001213_page12 | {
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0001213_page13 | {
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0001213_page14 | {
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"\\begin {equation} \\label {3e7} S_p=-\\frac {V_{p-2}\\tilde {V}_2}{(2\\pi )^p\\tilde {g}\\tilde {b}}\\int d\\tau \\left (\\frac {u}{R}\\right )^{7-p}\\left [\\sqrt {\\tilde {f}}-1+ \\frac {u_0^{7-p}}{2u^{7-p}}\\right ]. \\end {equation}",
"\\begin {equation} \\label {3e8} F_p=\\frac {V_{p-2}\\tilde {V}_2}{(2\\pi )^p\\tilde {g}\\tilde {b}} \\left (\\frac {u}{R}\\right )^{7-p}\\left [\\sqrt {\\tilde {f}}-1+ \\frac {u_0^{7-p}}{2u^{7-p}}\\right ]. \\end {equation}"
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"\\begin{equation*} \\mitF _ { \\mitp } = \\frac { \\mitV _ { \\mitp - 2 } \\tilde { \\mitV } _ { 2 } } { ( 2 \\mitpi ) ^ { \\mitp } \\tilde { \\mitg } \\tilde { \\mitb } } { \\left( \\frac { \\mitu } { \\mitR } \\right) } ^ { 7 - \\mitp } \\left[ \\sqrt { \\tilde { \\mitf } } - 1 + \\frac { \\mitu _ { 0 } ^ { 7 - \\mitp } } { 2 \\mitu ^ { 7 - \\mitp } } \\right] . \\end{equation*}"
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|
0001213_page15 | {
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"\\begin {equation} F_p|_{u=u_0}=\\frac {dF}{dN_p}\\delta N_p, \\label {fe} \\end {equation}",
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|
0001213_page16 | {
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"\\begin {equation} \\label {dp-2action} S_{p-2}=-T_{p-2}\\int d^{p-1}x e^{-\\phi }\\sqrt {-\\det G_{ab}}+T_{p-2} \\int A^{p-2}. \\end {equation}",
"\\begin {equation} \\label {3e13} S_{p-2}=-\\frac {T_{p-2}V_{p-2}}{g}\\int d\\tau H^{-1}\\left [H^{1/2}h^{-1/2} \\sqrt {f} -(1-H_0)\\sin \\theta -H \\right ], \\end {equation}",
"\\begin {equation} \\label {3e14} U_{p-2}|_{r=r_0} =\\frac {V_{p-2}T_{p-2}}{g}\\left (1 -\\sin \\theta \\tanh \\alpha \\right ). \\end {equation}"
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