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f ^ { + + i } = e ^ { -- } f _ { -- } ^ { ~ + + i } ~ , \qquad f ^ { -- i } = e ^ { + + } f _ { + + } ^ { ~ -- i } | 14e67a6db6.png | tokenized |
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E = E _ 3 T _ 3 \ , \ \ \ E _ 3 = 2 \pi \ . | 6df3686585.png | tokenized |
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\zeta \left( \begin{array} { r c r } { s _ { 1 } } & { \ldots } & { s _ { k } } \\ { \sigma _ { 1 } } & { \ldots } & { \sigma _ { k } } \\ \end{array} \right) : = \sum _ { n _ { j } > n _ { j + 1 } > 0 } \quad \prod _ { j = 1 } ^ { k } \frac { \sigma _ { j } ^ { n _ { j } } } { n _ { j } ^ { s _ { j } } } | 45506327a9.png | normalized |
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\label{ssftmotion}\eta_0 \Bigl( {\rm e}^{-\Phi} Q_B {\rm e}^{\Phi}\Bigr)=0. | 70bdf1298e.png | none |
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\partial _ { \mu } T ^ { \mu \nu } = 0 \, . | 7b6b415f2d.png | tokenized |
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\displaystyle{\prod_{i<j} |x_{i} - x_{j} |^{4}} | 105684eb6d.png | none |
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E ( t _ 1 , t _ 3 ) = \int _ { c } { \rm d } t _ 2 \, B ( t _ 1 , t _ 2 ) C ( t _ 2 , t _ 3 ) . | 33858e12ba.png | tokenized |
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\alpha = - \left ( 1 - \frac { 3 b ^ 2 } { 4 \kappa ^ 2 } \right ) ^ { - 1 } \: \ln \biggl | \frac { z } { l } \biggr | , | 74bfe88079.png | tokenized |
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t ( x ) \sim \left \{ \begin {array} { c c c } \overline { t } & \mathrm { n e a r } & x = \pi R \\ - \overline { t } & \mathrm { n e a r } & x = - \pi R \end {array} \right . | 4b32a1a77d.png | tokenized |
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G ^ { \mu } ( \rho ) = { \pm } { \epsilon } ^ { { \mu } { \nu } } { \partial } _ { \nu } { \rho } { \equiv } { \pm } { \epsilon } ^ { { \mu } { \nu } } F _ { \nu } ( \rho ) , | 48e2f625ce.png | tokenized |
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\sqrt { { \bf p } ^ { 2 } + { { \tilde { \mu } } } ^ { 2 } } { \tilde { \psi } ( { \bf x } ) } - { \frac { \alpha } { r } } { \tilde { \psi } } ( { \bf x } ) = ( { \tilde { E } } + { \tilde { \mu } } ) { \tilde { \psi } } ( { \bf x } ) , | 5a35a450b3.png | normalized |
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\frac{dT}{ds}= \frac{C_0}{\alpha (X)}\ \ \ ; \ \ \frac{dX}{ds}=\epsilon_1(C_0^2-\epsilon \alpha(X))^{\frac{1}{2}}\label{1.2.1} | 5a9037ae25.png | none |
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K : x + \sqrt { 2 } y + \sqrt { 3 } = 0 \ , \ u = 0 \ . | 1ab73f1c3c.png | normalized |
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f(r,x_3)=\frac 2{a(a+b)}=\frac 1{r(r-x_3)}\label{s3-13} | 69fc429019.png | none |
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{ \cal H } _ { \rm t o p } = \sum _ j V _ j \otimes V _ j , | 6810242ccd.png | tokenized |
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A_{\mu}= \frac {1}{2q} \frac{ i(\overline{\psi} \gamma_{\mu} \slash{\partial} \psi - \overline{\psi}\overleftarrow{\slash{\partial}} \gamma_{\mu}\psi) -2 m j_\mu} {\overline{\psi}\psi}. \label{eq:4DDiracSoln} | 5a97c95d42.png | none |
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\kappa ( \Lambda , T ) = \frac { m ^ 2 ( \Lambda , T ) } { \Lambda ^ 2 } | 328375bb94.png | tokenized |
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\mathcal{L}_X(f\psi) \;=\; (Xf)\psi + f\mathcal{L}_X\psi\,. | 3775246f21.png | none |
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2 C = B = A \, , \quad a = b = D = E = 0 \, , | b550726da6.png | tokenized |
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{ \sf A } = \frac { e ^ a } { 2 l } \Gamma _ a + \frac { 1 } { 2 } \omega ^ { a b } \Gamma _ { a b } . | 510fe5850f.png | tokenized |
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S ( \{ p _ { m } \} ) = - \sum _ { m = 1 } ^ { M } p _ { m } \operatorname { l o g } p _ { m } . | 6b27180c35.png | normalized |
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\hat { \hat { T } } _ { \hat { \hat { \mu } } \hat { \hat { \nu } } } { } ^ { \hat { \hat { \rho } } } = - \left ( i _ { \hat { \hat { k } } _ { ( n ) } } \hat { \hat { C } } \right ) _ { \hat { \hat { \mu } } \hat { \hat { \nu } } } Q ^ { n m } \hat { \hat { k } } _ { ( m ) } { } ^ { \hat { \hat { \rho } } } \, . | 3f28904075.png | tokenized |
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\Omega _ { \cal I } ^ { ( 1 ) } = \Omega _ { \cal I } ^ { \tau } Y _ { \cal I } + \Omega _ { \cal I } ^ { \theta } \Gamma _ { \cal I } . | 4844da87f5.png | tokenized |
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\delta_{\epsilon , \epsilon'} \theta ^{\alpha'} = - A ^{\alpha'}{}_\beta[(1+C)^{-1}] ^\beta{}_ \alpha \epsilon ^\alpha + \epsilon^{\alpha'} | 18ee819ec0.png | none |
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e ^ { - 2 \chi } \left( 2 f ( X ( e ^ { \chi } ) ) - e ^ { \chi } X ( e ^ { \chi } ) \right) = 0 . | 2acf5fbc4d.png | normalized |
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m ^ 2 = k ( k - 1 ) , ( k ^ 2 + 3 k + 2 ) | 6345d41dfe.png | tokenized |
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{ \cal L } = i \hat { \bar { \psi } } \gamma ^ { \mu } * ( \partial _ { \mu } + i g \hat { A } _ { \mu } ) * \hat { \psi } . | 5abd355991.png | normalized |
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P d P = V ( V ^ { \dagger } V ) ^ { - 1 } d V ^ { \dagger } \{ { \bf 1 } - V ( V ^ { \dagger } V ) ^ { - 1 } V ^ { \dagger } \} | 4c1959423a.png | normalized |
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{ \chi } _ { m } ^ { F } ( g h , . ) = { \chi } _ { m ^ { \prime } } ^ { F } ( g , . ) \overline { { \rho } } ( h ) _ { m ^ { \prime } m } . | 34033027aa.png | normalized |
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( 1 - \gamma ^ { 1 } ) k _ { n } ( 0 , x ^ { \prime } ) = \left( \begin{array} { c c } { 1 } & { i } \\ { - i } & { 1 } \\ \end{array} \right) k _ { n } ( 0 , x ^ { \prime } ) = 0 . | 688672b558.png | normalized |
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J ^ { \mu \nu } \overline { { \epsilon } } ^ { I } \gamma _ { \nu } \gamma _ { \mu } \eta ^ { J } \epsilon _ { I J } = J ^ { \mu \nu } g _ { \mu \nu } \overline { { \epsilon } } ^ { I } \eta ^ { J } \epsilon _ { I J } = 0 \, . | 3a9742e95f.png | normalized |
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T ^ { P P } | _ { A T } = T ^ { S S } | _ { A T } = \frac { - i \pi ^ { 2 } } { ( 2 \pi ) ^ { 4 } } s \cdot ( s + p ) , | 261bec155b.png | tokenized |
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\phi ^ { a } = F \ \frac { r ^ { a } } { r } \quad ( r \rightarrow \infty ) \ . | 750703697e.png | normalized |
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2\pi \chi (\mathcal{M})=\int _{\partial {\cal M}}d\tau k | 4baaa2c0be.png | none |
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G ( \beta , \tilde { \mu } ) = \left( \frac { \pi \mu \Gamma ( \frac { \lambda } { \lambda + 1 } ) } { 2 \Gamma ( \frac { 1 } { \lambda + 1 } ) } \right) ^ { \frac { 1 } { 2 \lambda } } g _ { 0 } ( \beta ) g _ { S } ( \beta , Z ) \, , | 1ac7762f5c.png | normalized |
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\int R \sqrt { g } d ^ { 4 } x = 4 \pi \left( 1 - { \frac { \beta } { 8 \pi m } } \right) A | 285c6149d8.png | normalized |
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\label{cothexpansion}\pi\,y\coth (\pi y) = 1 + 2y^2 \, \sum_{n=1}^{\infty} 1/(y^2 + n^2)\,, | 55d100739c.png | none |
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\mathrm { T r } \, [ \phi _ { 1 } ( \hat { x } ) \phi _ { 2 } ( \hat { x } ) ] = \int d x \, ( \phi _ { 1 } * \phi _ { 2 } ) ( x ) . | d0553c62cf.png | normalized |
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E _ { ( m ) } ^ { \mu \nu \lambda \rho } = \hat { X } ^ { \mu \nu \lambda \rho } L _ { ( m ) } ~ ~ ~ . | 44ca780f24.png | normalized |
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P _ { N } = \frac { x ^ { N } } { \sum _ { M = 0 } ^ { + \infty } x ^ { M } } = \left( 1 - x ^ { \prime } \right) x ^ { N } \; , | 36d8d9f944.png | normalized |
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\exp \left [ \frac { i } { \hbar } S ^ \prime \right ] = K \exp \left [ \frac { i } { \hbar } S \right ] . | 1305d83577.png | tokenized |
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\sum_{i=1}^{4} N_i = \mbox{even\,(odd)}\label{chi87} | 72ed3c9d05.png | none |
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\sigma = \pi^2 \omega R^2 \frac{e^{\frac{\omega}{T_{H}}} -1}{(e^{\frac{\omega}{2T_{L}}} - 1)(e^{\frac{\omega}{2T_{R}}} -1)}, | 3b5d3d9d7f.png | none |
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\label{quantrel}e^{iZ(\lambda _{j})}=-(-1)^{\delta }. | 426c855270.png | none |
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\vec { T } ^ { ( 1 , s , 1 ) } ( \lambda - u ) \; = \; { \vec { T } ^ { ( 1 , s , 1 ) } } ( u ) ^ { \dagger } | 3a7352a84f.png | normalized |
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\label{bc2} V_- e^{-\frac{4}{3} d_1} = V_+ e^{\frac{4}{3} d_2} = -2 . | 115d76f13f.png | none |
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e ^ { 3 C } = { { c ( x ^ { 1 1 } ) } \over { - \alpha \sqrt { 2 } D + a } } \ , | 572d65a068.png | tokenized |
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\varepsilon _ { i j } p _ { a } ^ { i } p _ { b } ^ { j } = \star ( \pi _ { a } \wedge \pi _ { b } ) . | 4b23c4c5e1.png | normalized |
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M _ { \hat A \hat B } = \Lambda _ { ( \hat A } \Lambda _ { \hat B ] } , | 3468c467b6.png | tokenized |
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N \equiv - M , \ \ \ \Delta \equiv - \frac 1 2 \tilde { G } ^ { 3 } , \ \ \ \Lambda _ { \pm } \equiv \pm \frac i 2 \Xi _ { \mp } | c36c899123.png | normalized |
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H_1 |\sigma\rangle = \sum_{\rho\in Y, \tau \in ``2"}{2\over \dim(\tilde \rho)}\chi_{\tilde\rho}(\sigma) \chi_{\tilde\rho}(\tau) \chi_\rho . | 3f2017e165.png | none |
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g | P _ { L } , P _ { R } \rangle = e ^ { 2 \pi i ( P _ { L } \cdot a _ { L } - P _ { R } \cdot a _ { R } ) } | \theta _ { L } P _ { L } , \theta _ { R } P _ { R } \rangle . | 1f61ab144c.png | normalized |
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F = STU + f(T,U)+ f^{non-pert}.\label{partik3} | 6516a2853c.png | none |
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\label{schrodinger-eq}\frac{\psi''(z)}{\psi(z)}=\frac{\chi''(z)}{\chi(z)}=\gamma^2 (T_g(z)-J_2'(z)). | 1b3c504701.png | none |
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d s _ { 5 } ^ { 2 } = - N ^ { 2 } ( \tau , w ) d \tau ^ { 2 } + r ^ { 2 } ( \tau , w ) d \Sigma _ { K } ^ { 2 } + d w ^ { 2 } , | 1bbffb96b6.png | normalized |
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\langle \ X ^ \mu ( 0 ) \ \partial _ \parallel X ^ \nu ( e ^ { i \theta } ) \ \rangle = 0 . | 53b8af46b3.png | tokenized |
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\hat { a } \rightarrow \sum _ { a = 1 } ^ { m ^ { 2 } } \hat { a } ^ { a } \otimes T ^ { a } , | 3e793f0e4d.png | tokenized |
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X^{i}=i\hat{\partial}_{i}\,,~~~i=1,2,3,4\,.\label{LM5sol} | 1397e1b6fc.png | none |
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\vec{X} \:\Psi(\vec{x}) \;=\; \vec{x} \:\Psi(\vec{x}) \;. | 287fbe1c74.png | none |
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\begin{array} { c c c c c c c c c } { p ^ { \prime } } & { 5 } & { 7 } & { 9 } & { 1 1 } & { 1 3 } & { 1 5 } & { \cdots } \\ { \hline \Delta } & { 7 } & { 1 1 } & { 1 5 } & { 1 9 } & { 2 3 } & { 2 7 } & { \cdots } \\ \end{array} | 265d6dc511.png | normalized |
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\mathcal { L } _ { Q E D } = \bar { \psi } ( i \partial \! \! \! / - e A \! \! \! / ) \psi - \frac { 1 } { 4 } | 736ff12d92.png | tokenized |
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q _ { i } = r ^ { 2 } \partial _ { i } h \partial _ { j } \partial _ { j } h - 2 x _ { j } \partial _ { j } h \partial _ { i } h + x _ { i } \partial _ { j } h \partial _ { j } h . | 6647a06b19.png | normalized |
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\partial _ { \nu } j ^ { \nu } ( x ) = \int d ^ { 4 } y \ \ K _ { a b } ( x , y ) \varphi ^ { a } ( y ) \tau _ { c } ^ { b } \varphi ^ { c } ( x ) \ . | 9949fbfcd3.png | normalized |
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[\gamma _3]=[\gamma_1][\gamma _2][\gamma_1]^{-1}[\gamma _2]^{-1} | 55d40d4e86.png | none |
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{ \cal { L } } = \frac { 1 } { 2 } ( \partial _ { \mu } \phi ) ^ { 2 } - U ( \phi ) , | 1d6665cc7a.png | normalized |
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\Sigma _ { A j } ^ { \, \, \, \, i } = \frac { i } { 2 } \sigma _ { A j } ^ { \, \, \, \, i } | 4946a53331.png | tokenized |
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Z_{\rm flat\, space} = {1\over |Im\tau|^2|\eta |^8}\sum_i{1\over 2}\left|{\theta_i\over\eta}\right|^4 | 1213c38222.png | none |
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\S _ { k } { q } _ { k p } { \bar { q } } _ { k p } = \S _ { k } { q ^ { \prime } } _ { k p } { \bar { q ^ { \prime } } } _ { k p } = \cdots = i n v , | 14acbdd059.png | normalized |
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G\left( D\right) =\sum\limits_{n=0}^{\infty }g_{2n+1}D^{2n+1} . | 6230d53383.png | none |
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W ( x ) = \left \{ \begin {array} { c l l } - \omega x & { \rm f o r } & x > 0 \\ \infty & { \rm f o r } & x < 0 \end {array} \right . | 7f16c8c2c9.png | tokenized |
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[ \eta _ i , \eta _ j ] = f _ { i j } ^ { \, \, \, \, k } \eta _ k . | 484050b817.png | tokenized |
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{ \cal P } _ { a } = p _ { a } \qquad { \cal J } _ { a } = \epsilon _ { a b c } x ^ { b } p ^ { c } - s \frac { p _ { a } } { ( - p ^ { 2 } ) \lefteqn { { } ^ { 1 / 2 } } } \qquad , | 4482baf132.png | normalized |
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\label{gl}{\cal L}_{GL} = (D_{\mu} \varphi)(D_{\mu} \varphi)^{\ast}- m^2 |\varphi | ^2-\lambda | \varphi |^4 | 719d091e73.png | none |
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{ \cal L } ( a , B ) = - \frac { 1 } { 2 } \frac { B ^ 2 } { \mu ( a m ) } \; , | 2ff0ab502d.png | tokenized |
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q _ { 1 c l } ( \tau ) = q _ { 1 } ( \tau ; q _ { 1 a } , p _ { 1 } ( 0 ) ) | 5e84b929d8.png | normalized |
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Z _ 1 = 1 + \frac { r _ 1 ^ 2 } { r ^ 2 } \ , \ Z _ 5 = 1 + \frac { r _ 5 ^ 2 } { r ^ 2 } , | 5ea1ce5c7c.png | tokenized |
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\langle c _ { l , m } ^ { \pm } \rangle \; = \; 1 | 393b3285dc.png | normalized |
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D _ \pm \eta = \partial _ \pm \eta + [ A _ \pm , \eta ] = \pm { 1 \over t } \partial _ \pm \eta . | 5e93d6f491.png | tokenized |
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s = { 4 \pi \over g ^ 2 } \int ^ \infty _ 0 d r \left [ \left ( { d K \over d r } \right ) ^ 2 + \left ( { K ^ 2 - 1 \over 2 r ^ 2 } \right ) \right ] | 4e7f9af031.png | tokenized |
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\lambda _ { a } : x _ { i } \rightarrow \lambda _ { a } ^ { Q _ { i } ^ { a } } x _ { i } | 3f13ee9b4c.png | normalized |
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\left[ \partial _ { 0 } ^ { 2 } + L ( i \partial _ { 0 } ) \right] \phi = 0 . | 5cdbc6e4d7.png | normalized |
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\sum_{l=1}^{L-1} \sin^{-4} {\pi l\over L}={(L^4 - 1) \over 45}+ {2 (L^2 - 1) \over 9}\label{k4} | 2e3e52f21c.png | none |
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\{ x _ i , x _ j \} _ D = 0 = \{ p ^ i , p ^ j \} _ D \, \, \, \, \{ x _ i , p ^ j \} _ D = \delta ^ j _ i \, \, . | 53b7b69733.png | tokenized |
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\lambda = \frac { 8 \pi } { \beta ^ { 2 } } - 1 \quad ; \quad M = m ^ { \frac { 8 \pi } { 8 \pi - \beta ^ { 2 } } } \kappa ( \beta ) \quad , | 38b3859b49.png | normalized |
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< A _ z \vert \equiv < 0 \vert \exp \left ( \sqrt { \frac { \kappa } { 2 } } \int d ^ 2 { \bf x } A ^ a _ { z } { \cal A } ^ { a } \right ) . | 1177dae397.png | tokenized |
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c_{\rm BH} \sim N^2 (T/\lambda^{1/3})^{9/5} \approx S_{\rm BH} | a9faac05f6.png | none |
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\left[ \sigma _ { 1 } , U \right] = 0 \; , \qquad \left[ \sigma _ { 1 } , L \right] = 0 \; . | 2368129962.png | normalized |
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\{ \Phi ^ { i } ( x ) , \Phi ^ { j } ( y ) \} = \omega ^ { i j } ( x , y ) , | 558d96d6c4.png | normalized |
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\begin {array} { l l } s g _ { \mu \nu } = \hat { g } _ { \mu \nu } ~ & ~ s \hat { g } _ { \mu \nu } = 0 . \end {array} | 71af947958.png | tokenized |
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I ( k ) = \int _ { \frac { \Lambda } { s } < | p | < \Lambda } \frac { d ^ { D } p } { ( 2 \pi ) ^ { D } } \frac { e ^ { i k \wedge p } } { p ^ { 2 } } . | 5bd8ffcf80.png | normalized |
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P ( y , x ) \; = \; P ( x , y ) ^ * \; = \; \chi _ L \: \overline { g _ j ( x , y ) } \: \gamma ^ j \; . | 552892c213.png | tokenized |
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| \Phi \rangle = { \cal O } | \Omega \rangle \; . | 7655f857db.png | tokenized |
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R^{\mu }U\psi =R^{\mu }\psi +\omega ^{\mu }{}_{\alpha }R^{\alpha }\psi ,\label{eq18-7} | 21eccc7e70.png | none |
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f _ { k } ( \tau ) \to \alpha _ { k } f _ { k } ( \tau ) + \beta _ { k } f _ { k } ^ { * } ( \tau ) , \hspace { 1 e m } | \alpha _ { k } | ^ { 2 } - | \beta _ { k } | ^ { 2 } , | 1baba03963.png | normalized |
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{\bf a}^n=s_ns_{n+1}{\bf n}_n\times {\bf n}_{n+1},\;\;\;{\bf v}^n=s_nc_{n+1}%{\bf n}_n-s_{n+1}c_n{\bf n}_{n+1}. | 40b6f80e52.png | none |
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d x _ { \perp } ^ { 2 } = - d t ^ { 2 } + d \vec { x } ^ { 2 } = d \rho ^ { 2 } + \rho ^ { 2 } d s _ { d S _ { 5 } } ^ { 2 } \; , | 274e57c195.png | normalized |
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S^{\mu} \rightarrow S^{\mu} \, \, ; \, \, A^{\mu}\rightarrow A^{\mu} , | 5ccf2d85b2.png | none |
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P ^ r = { \prod } { P _ { 0 } } ^ { r _ { 0 } } { P _ { 1 } } ^ { r _ { 1 } } \ldots { P _ { d } } ^ { r _ { d } } { P _ { d + 1 } } ^ { r _ { d + 1 } } | 797dc9b578.png | tokenized |
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\Psi ( { \bf u } , \lambda ) = M ( { \bf u } , \lambda ) M ^ { - 1 } ( { \bf u ^ { \prime } } , \lambda ) | 385b8c4593.png | normalized |
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1 = A \frac { ( c - d ) ^ { 2 } } { 1 6 } + \sqrt { ( 1 / 2 - c ) ( 1 / 2 - d ) } . | 63f0e7058a.png | normalized |