image
imagewidth (px) 3
1.29k
| latex
stringlengths 3
4.96k
| latex_norm
stringlengths 5
5.62k
| page_id
stringlengths 15
15
| expr_type
stringclasses 2
values |
|---|---|---|---|---|
$S(\phi )$
|
$ S ( \phi ) $
|
0001015_page002
|
embedded
|
|
$\phi $
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$ \phi $
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0001015_page002
|
embedded
|
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$\exp (-Ht)$
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$ e x p ( - H t ) $
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0001015_page002
|
embedded
|
|
$\psi (x)$
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$ \psi ( x ) $
|
0001015_page002
|
embedded
|
|
$d\mu $
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$ d \mu $
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0001015_page002
|
embedded
|
|
$x$
|
$ x $
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0001015_page002
|
embedded
|
|
$x(t)$
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$ x ( t ) $
|
0001015_page002
|
embedded
|
|
$V$
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$ V $
|
0001015_page002
|
embedded
|
|
$H= L +V(x)$
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$ H = L + V ( x ) $
|
0001015_page002
|
embedded
|
|
$L$
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$ L $
|
0001015_page002
|
embedded
|
|
$x(t)$
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$ x ( t ) $
|
0001015_page002
|
embedded
|
|
\begin {equation} H = \half p^2 + V(x) \end {equation}
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\begin{equation*} H = \frac { 1 } { 2 } p ^ { 2 } + V ( x ) \end{equation*}
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0001015_page002
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isolated
|
|
\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}
|
\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*}
|
0001015_page002
|
isolated
|
|
$x: I \to M$
|
$ x : I \rightarrow M $
|
0001015_page003
|
embedded
|
|
$I$
|
$ I $
|
0001015_page003
|
embedded
|
|
$[0,t]$
|
$ [ 0 , t ] $
|
0001015_page003
|
embedded
|
|
$M$
|
$ M $
|
0001015_page003
|
embedded
|
|
$n$
|
$ n $
|
0001015_page003
|
embedded
|
|
$g$
|
$ g $
|
0001015_page003
|
embedded
|
|
$\omega =dh$
|
$ \omega = d h $
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0001015_page003
|
embedded
|
|
$M$
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$ M $
|
0001015_page003
|
embedded
|
|
$\omega = \Omu (x) d\Xmu $
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$ \omega = \omega _ { \mu } ( x ) d x ^ { \mu } $
|
0001015_page003
|
embedded
|
|
$\omega = \Omu (x) d\Xmu $
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$ \omega = \omega _ { \mu } ( x ) d x ^ { \mu } $
|
0001015_page003
|
embedded
|
|
$\dot {x}^{\mu }(t')= \frac {d{x}^{\mu }}{d t'}$
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$ \dot { x } ^ { \mu } ( t ^ { \prime } ) = \frac { d x ^ { \mu } } { d t ^ { \prime } } $
|
0001015_page003
|
embedded
|
|
$S[x(.)] = i(h(x(t))-h(x(0)))$
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$ S [ x ( . ) ] = i ( h ( x ( t ) ) - h ( x ( 0 ) ) ) $
|
0001015_page003
|
embedded
|
|
$x(t)$
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$ x ( t ) $
|
0001015_page003
|
embedded
|
|
$\omega =0$
|
$ \omega = 0 $
|
0001015_page003
|
embedded
|
|
$h$
|
$ h $
|
0001015_page003
|
embedded
|
|
$M$
|
$ M $
|
0001015_page003
|
embedded
|
|
$\omega =dh$
|
$ \omega = d h $
|
0001015_page003
|
embedded
|
|
$\Pmu $
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$ p _ { \mu } $
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0001015_page003
|
embedded
|
|
$\Xmu $
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$ x ^ { \mu } $
|
0001015_page003
|
embedded
|
|
$n$
|
$ n $
|
0001015_page003
|
embedded
|
|
$H(p,x)= \Pmu \Xmu - \Lag (x,\dot {x})$
|
$ H ( p , x ) = p _ { \mu } x ^ { \mu } - L ( x , \dot { x } ) $
|
0001015_page003
|
embedded
|
|
$H(p,x)= \Pmu \Xmu - \Lag (x,\dot {x})$
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$ H ( p , x ) = p _ { \mu } x ^ { \mu } - L ( x , \dot { x } ) $
|
0001015_page003
|
embedded
|
|
$\omega $
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$ \omega $
|
0001015_page003
|
embedded
|
|
$\Pb {\Tmu }{T_{\nu }}=0$
|
$ \{ T _ { \mu } , T _ { \nu } \} = 0 $
|
0001015_page003
|
embedded
|
|
$\Pb {\Tmu }{H_c}=0$
|
$ \{ T _ { \mu } , H _ { c } \} = 0 $
|
0001015_page003
|
embedded
|
|
$\Tmu $
|
$ T _ { \mu } $
|
0001015_page003
|
embedded
|
|
$\psi (x)$
|
$ \psi ( x ) $
|
0001015_page003
|
embedded
|
|
$\Pmu =-i \Dmu $
|
$ p _ { \mu } = - i \partial _ { \mu } $
|
0001015_page003
|
embedded
|
|
$\Pmu $
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$ p _ { \mu } $
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0001015_page003
|
embedded
|
|
$-i\DDmu $
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$ - i \nabla _ { \mu } $
|
0001015_page003
|
embedded
|
|
$\XXmu \psi =0$
|
$ X ^ { \mu } \psi = 0 $
|
0001015_page003
|
embedded
|
|
$\XXmu =g^{\mu \nu } (p_{\nu } + i\omega _{\nu })$
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$ X ^ { \mu } = g ^ { \mu \nu } ( p _ { \nu } + i \omega _ { \nu } ) $
|
0001015_page003
|
embedded
|
|
\begin {equation}\label {ACeq} S[x(.)] = \Intot i\Omu (x(t'))\dot {x}^{\mu }(t') \, dt' \end {equation}
|
\begin{equation*} S [ x ( . ) ] = \int _ { 0 } ^ { t } i \omega _ { \mu } ( x ( t ^ { \prime } ) ) \dot { x } ^ { \mu } ( t ^ { \prime } ) \, d t ^ { \prime } \end{equation*}
|
0001015_page003
|
isolated
|
|
\begin {equation}\label {MOMeq} \Pmu = \frac {\delta \Lag }{\delta \dot {x}^\mu } = i\Omu , \end {equation}
|
\begin{equation*} p _ { \mu } = \frac { \delta L } { \delta \dot { x } ^ { \mu } } = i \omega _ { \mu } , \end{equation*}
|
0001015_page003
|
isolated
|
|
\begin {equation} \Tmu \equiv \Pmu - i\Omu . \end {equation}
|
\begin{equation*} T _ { \mu } \equiv p _ { \mu } - i \omega _ { \mu } . \end{equation*}
|
0001015_page003
|
isolated
|
|
\begin {equation}\label {GTeq} \delta _{\epsilon }\psi (x) =-i \epsilon (\Dmu \psi (x) + \Omu (x) \psi (x)) \end {equation}
|
\begin{equation*} \delta _ { \epsilon } \psi ( x ) = - i \epsilon ( \partial _ { \mu } \psi ( x ) + \omega _ { \mu } ( x ) \psi ( x ) ) \end{equation*}
|
0001015_page003
|
isolated
|
|
$\Etamu $
|
$ \eta ^ { \mu } $
|
0001015_page004
|
embedded
|
|
$\Pimu $
|
$ \pi _ { \mu } $
|
0001015_page004
|
embedded
|
|
$(2n,2n)$
|
$ ( 2 n , 2 n ) $
|
0001015_page004
|
embedded
|
|
$\Etamu ,\Pimu $
|
$ \eta ^ { \mu } , \pi _ { \mu } $
|
0001015_page004
|
embedded
|
|
$\nabla $
|
$ \nabla $
|
0001015_page004
|
embedded
|
|
$\psi (x,\eta )$
|
$ \psi ( x , \eta ) $
|
0001015_page004
|
embedded
|
|
$\Pmu =-i \DDmu $
|
$ p _ { \mu } = - i \nabla _ { \mu } $
|
0001015_page004
|
embedded
|
|
$\Pimu = -i\frac {\partial }{\partial \Etamu }$
|
$ \pi _ { \mu } = - i \frac { \partial } { \partial \eta ^ { \mu } } $
|
0001015_page004
|
embedded
|
|
$\psi (x,\eta )$
|
$ \psi ( x , \eta ) $
|
0001015_page004
|
embedded
|
|
$(n,n)$
|
$ ( n , n ) $
|
0001015_page004
|
embedded
|
|
$SM$
|
$ S M $
|
0001015_page004
|
embedded
|
|
$\Xmu ,\Etamu $
|
$ x ^ { \mu } , \eta ^ { \mu } $
|
0001015_page004
|
embedded
|
|
$Q$
|
$ Q $
|
0001015_page004
|
embedded
|
|
$Q=\Etamu \Tmu =-i\Etamu (\Dmu + \omega )$
|
$ Q = \eta ^ { \mu } T _ { \mu } = - i \eta ^ { \mu } ( \partial _ { \mu } + \omega ) $
|
0001015_page004
|
embedded
|
|
$Q=\Etamu \Tmu =-i\Etamu (\Dmu + \omega )$
|
$ Q = \eta ^ { \mu } T _ { \mu } = - i \eta ^ { \mu } ( \partial _ { \mu } + \omega ) $
|
0001015_page004
|
embedded
|
|
$\chi $
|
$ \chi $
|
0001015_page004
|
embedded
|
|
$\chi = \Pimu \XXmu = -ig^{\mu \nu } \Pimu (\nabla _{\nu }-\omega _{\nu })$
|
$ \chi = \pi _ { \mu } X ^ { \mu } = - i g ^ { \mu \nu } \pi _ { \mu } ( \nabla _ { \nu } - \omega _ { \nu } ) $
|
0001015_page004
|
embedded
|
|
$M$
|
$ M $
|
0001015_page004
|
embedded
|
|
$\psi (x,\eta )$
|
$ \psi ( x , \eta ) $
|
0001015_page004
|
embedded
|
|
$Q=-i\Emh d \Eph $
|
$ Q = - i e ^ { - h } d e ^ { h } $
|
0001015_page004
|
embedded
|
|
$\chi = \Eph \delta \Emh $
|
$ \chi = e ^ { h } \delta e ^ { - h } $
|
0001015_page004
|
embedded
|
|
$d$
|
$ d $
|
0001015_page004
|
embedded
|
|
$\delta = *d*$
|
$ \delta = \ast d \ast $
|
0001015_page004
|
embedded
|
|
$h$
|
$ h $
|
0001015_page004
|
embedded
|
|
$\chi =\Pimu \XXmu $
|
$ \chi = \pi _ { \mu } X ^ { \mu } $
|
0001015_page004
|
embedded
|
|
$h$
|
$ h $
|
0001015_page004
|
embedded
|
|
$Q$
|
$ Q $
|
0001015_page004
|
embedded
|
|
$h$
|
$ h $
|
0001015_page004
|
embedded
|
|
$H_g$
|
$ H _ { g } $
|
0001015_page004
|
embedded
|
|
\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{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*}
|
0001015_page004
|
isolated
|
|
\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{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*}
|
0001015_page004
|
isolated
|
|
\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}
|
\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*}
|
0001015_page004
|
isolated
|
|
$H_g$
|
$ H _ { g } $
|
0001015_page005
|
embedded
|
|
$x_t,\eta _t$
|
$ x _ { t } , \eta _ { t } $
|
0001015_page005
|
embedded
|
|
$SM$
|
$ S M $
|
0001015_page005
|
embedded
|
|
$\Xmu ,\Etamu $
|
$ x ^ { \mu } , \eta ^ { \mu } $
|
0001015_page005
|
embedded
|
|
$b_t$
|
$ b _ { t } $
|
0001015_page005
|
embedded
|
|
$(\theta _t,\rho _t)$
|
$ ( \theta _ { t } , \rho _ { t } ) $
|
0001015_page005
|
embedded
|
|
$(d+ \delta )^2$
|
$ ( d + \delta ) ^ { 2 } $
|
0001015_page005
|
embedded
|
|
$J$
|
$ J $
|
0001015_page005
|
embedded
|
|
$u:\Sigma \to M$
|
$ u : \Sigma \rightarrow M $
|
0001015_page005
|
embedded
|
|
$\Sigma $
|
$ \Sigma $
|
0001015_page005
|
embedded
|
|
$2m$
|
$ 2 m $
|
0001015_page005
|
embedded
|
|
$M$
|
$ M $
|
0001015_page005
|
embedded
|
|
$H^{\alpha }_{\mu }$
|
$ H _ { \mu } ^ { \alpha } $
|
0001015_page005
|
embedded
|
|
$\Etamu $
|
$ \eta ^ { \mu } $
|
0001015_page005
|
embedded
|
|
$\pi ^{\alpha }_{\mu }$
|
$ \pi _ { \mu } ^ { \alpha } $
|
0001015_page005
|
embedded
|
|
$\alpha =1,2$
|
$ \alpha = 1 , 2 $
|
0001015_page005
|
embedded
|
|
$\Sigma $
|
$ \Sigma $
|
0001015_page005
|
embedded
|
|
$\mu =1,\dots ,2 m$
|
$ \mu = 1 , \ldots , 2 m $
|
0001015_page005
|
embedded
|
|
$M$
|
$ M $
|
0001015_page005
|
embedded
|
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