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U _ { \Lambda } ( g ) = U _ { \Lambda } ^ { * } ( U ( g ) \bar { \otimes } \underline { 1 } ) U _ { \Lambda } | 27769840a7.png | tokenized |
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\pi ( d _ { f } p _ { i } ) = i [ D , \pi ( p _ { i } ) ] . | 56ee898b7d.png | tokenized |
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i \int \frac{d^{\, 4} P}{(2 \pi)^4} \,\left[ G_c^{(0)} (X; P) + G_{c 2}^{(1)} (X; P) \right] {\cal F} (P) , \label{kiss} | 78f28b2e42.png | none |
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<0|\lambda^2|0>=\Lambda^3. \label{vacuum} | 4c3435e629.png | none |
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\overrightarrow{C_x}\equiv \frac{{}_1}{{}^2}\beta ^{-1}\left[ -\hbar^2\partial _x^2+(\beta ^2-\alpha ^2)x^2\right] \label{op2} | 1431213364.png | none |
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V _ { \mathrm { e f f } } \approx \frac { m ^ { 2 } } { 2 \upsilon ^ { 2 } } \Phi ^ { 2 } ( \Phi - \upsilon ) ^ { 2 } , | 28bacc0ebb.png | normalized |
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W_\alpha =\frac 14\bar D_{\dot\beta}\bar D^{\dot\beta} D_\alpha V \label{9} | 3267149fa1.png | none |
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I(x) = - 1 - \pi x \int_1^\infty\frac{z dz}{\sqrt{z^2-1}}\frac{1}{\sinh^2(\pi x z)}. | 4165be74c6.png | none |
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G = 4 \gamma e ^ { - { \cal G } } \; , \; \; \psi ^ { \prime } = \phi ^ { \prime } - { \cal G } ^ { \prime } \; , | a85997bd4e.png | normalized |
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ds^2 = \frac{u - \beta m}{u^{2 \beta + 1}} \left(\frac{(u^\prime dr)^2}{4 h^2 (u+1)} - (u+1) dt^2 \right)\label{gmet2a} | 5c24ed2aa5.png | none |
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\left\{ \begin{array}{l}\psi^{\downarrow}_{+\frac{1}{2}} (x,{\vec k}_{\perp})=\frac{(+k^1-{\mathrm i} k^2)}{x }\,\varphi \ ,\\\psi^{\downarrow}_{-\frac{1}{2}} (x,{\vec k}_{\perp})=(M+\frac{m}{x})\,\varphi \ .\end{array}\right.\label{sn2a} | 2fd80bf7b1.png | none |
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ds^2 = \frac{l^2}{t^2}dt^2 - \left(\frac tl - \frac{T_B(\sigma)l}{2\eta_0 t}\right)^2 d\sigma^2\,, \label{genmetrdS} | 7a62259469.png | none |
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\displaystyle \left . \quad \quad \quad + \frac { 2 7 } { 8 9 6 } \zeta ( 6 ) + \frac { 3 } { 4 4 8 0 } \zeta ( 3 ) \zeta ( 4 ) + \frac { 7 } { 1 9 2 0 } \zeta ( 7 ) \right ) | 614e8ddae9.png | tokenized |
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\left.\begin{array}{ccl} D^AF_{AB}^3 & = & 0 \\ [.05in]D^AF_{AB}^i & = & M^2A_B^i \end{array}\right\}\label{eqnsmot} | 63e07b25b9.png | none |
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p ^ { \mu } \partial _ { \mu } \phi ( q , p ) = \lambda \frac { \partial V ( \phi , \phi ^ { \ast } ) } { \partial \phi ^ { \ast } } . | 6ba6487dcb.png | tokenized |
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\frac { 1 } { S _ { q + 1 } } \left ( S _ q X _ j ( \lambda ) + X _ j ( - q \lambda ) \right ) \: = \: \vec { I } \ | 75f36df367.png | tokenized |
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ds^2_{4} = (d\tau + 2q \cos\theta d\psi)^2 + dr^2 + r^2d\Omega_{2}^{2}, | 7c00df8944.png | none |
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\frac { \partial H ^ { \upsilon } } { \partial V ^ { i } } = 0 | b1d365eedb.png | normalized |
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\label{BC}\lim_{s\rightarrow0}K_{\beta}(x,x';s)=\delta^{(4)}(x,x') , | 2d9133872e.png | none |
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D \overline { { D } } = 1 , \hspace { 0 . 3 i n } \overline { { D } } ( x ) = \gamma _ { 4 } D ^ { + } \gamma _ { 4 } . | 79bf28cc3c.png | normalized |
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\left ( - \partial _ z ^ 2 + V _ { Q M } - k ^ 2 \right ) \psi = 0 | 610b71d275.png | tokenized |
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\frac { \kappa ( f _ { + } ) - 1 } { \kappa ( f _ { + } ) } = 1 - \frac { 2 M } { R } \ , | 29e80f24c1.png | normalized |
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\left . - i \vec \gamma \cdot \hat r \eta = \eta \right | _ { \rm r = R } \ . | 36f84d499b.png | tokenized |
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: \! \varphi ^ { N } \! : \, \leftrightarrow p _ { n } ^ { ( N ) } ( \underline { { l } } | 3c2c7b2079.png | normalized |
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L _ { x ^ { \lambda } } = F ^ { \mu \lambda } \partial _ { \mu } . | 6fc3d34a82.png | normalized |
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4 m ^ 4 q ( \tau ) = \Lambda ^ 4 , \quad m \rightarrow \infty , \quad q \rightarrow 0 , | 1ceb655f5b.png | tokenized |
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H _ { B } = - \frac { 1 } { 2 } \frac { \partial ^ { 2 } } { \partial \phi _ { I } ^ { a } \partial \phi _ { I } ^ { a } } + \frac { 1 } { 2 } \Omega ^ { 2 } _ { I J } \phi _ { I } ^ { a } \phi _ { J } ^ { a } | 6840645a8c.png | tokenized |
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A ^ { \prime } = \eta { \frac { 2 } { D - 2 } } { \frac { 1 } { \sqrt { D - 1 } } } \phi ^ { \prime } \ \ \ \ \ \ \ ( \eta \equiv \pm 1 ) . | 63cb699b8c.png | normalized |
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{\cal B}= exp(\sum {\lambda}_{i}(1){\lambda}_{i}(2))\label{CLB} | 562e4d4154.png | none |
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\omega _ { S _ 2 } = \lambda _ + \omega _ { F _ + } + \lambda _ - \omega _ { F _ - } | 99d0906926.png | tokenized |
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Q = \left( \frac { d - 3 } { 2 } \right) ^ { 2 } + \xi ^ { 2 } , | 38c49362c4.png | normalized |
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|q\rangle\to D^{(R)}(a(C,x_0))~|q\rangle~, | 1d290eb954.png | none |
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R e ( \varepsilon ) \nabla ^ { 2 } \varepsilon = \nabla \varepsilon \cdot \nabla \varepsilon | 13d59cd1f6.png | normalized |
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\sum \limits _ { k = 1 } ^ r \, { \cal P } _ { i 1 k } ^ { [ n - m ] } ( x ) A { \bf e } _ k = 0 | 73323b6390.png | tokenized |
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\mathbf { \omega } _ { b } ^ { a } = - \eta ^ { a c } \eta _ { b d } \mathbf { \omega } _ { c } ^ { d } | 5f8ef0e613.png | normalized |
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2 [ 1 - e ^ { - i p \xi } - e ^ { - i q \xi } + e ^ { i \xi ( p + q ) } ] \rho ( \xi ) \; + O ( m ^ { 3 } ) \; . | 28c37f044e.png | normalized |
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{ \cal D } _ \mu e _ \nu ^ b = \partial _ \mu e _ \nu ^ b + \omega _ \mu ^ { b c } e _ { \nu c } \; . | 7abf8b6c6c.png | tokenized |
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G_\xi=\int d\sigma [\frac{1}{2}\alpha\mu\epsilon^{ij}\partial_j A_i-\mu J^0]\delta\xi. | 6278739609.png | none |
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{ \cal Z } ( A ) = \sum _ { R } ( d _ { R } ) ^ { 2 } \operatorname { e x p } \left[ - { \frac { g ^ { 2 } A } { 4 } } C _ { 2 } ( R ) \right] , | 60a4baaecf.png | normalized |
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S _ { E } = - \frac { 1 } { 1 6 \pi G } \int d ^ { 3 } x \sqrt { g } ( R - 2 \Lambda ) | 744970012f.png | normalized |
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\psi^{L}_{f_i} = \psi^{L}_{e_i} ,\psi^{L}_{u_i} ,\psi^{L}_{d_i}\/. | 13c0f9b8fa.png | none |
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\delta B^{v(2)}_{i} = \delta \beta^{v(2)}_{i} +\frac{1}{2}\bar{\nabla}_{i} \delta S^{0}, | 3302461243.png | none |
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\prod _ { \lambda \ne \nu } \tilde { D } _ { \lambda } D _ { \nu } F ^ { \mu \nu } = j ^ { \mu } , | 6ec448c4bd.png | normalized |
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\tilde { H } = K _ { a } ^ { \prime } + H _ { 0 } | 3e6275b3dd.png | normalized |
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\mathrm { r e s } | _ { \lambda _ { 0 } } \Phi ( - \partial _ { \eta } + \frac { V _ { ( 0 ) } } { 1 + \lambda } ) \Phi ^ { - 1 } = 0 . | 1adaacb2c8.png | normalized |
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{ \cal L } _ \Phi = \frac { 1 } { 2 } \partial _ \mu \Phi \partial ^ \mu \Phi + \frac { 1 } { m } \Phi \partial _ \mu J ^ \mu \, , | 3b9430e71a.png | tokenized |
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(T_a)_{bc} \equiv \frac{1}{2} (D_{abc} - i F_{abc})\label{A.10} | 33a3a429e5.png | none |
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\operatorname* { l i m } _ { \epsilon \rightarrow 0 } \widetilde { \mathcal V } _ { \epsilon } ( z ) = \left\{ \begin{array} { l l } { 0 } & { \mathrm { f o r } \; z < 1 } \\ { - \sigma \infty } & { \mathrm { f o r } \; z > 1 } \\ \end{array} \right. \; ; | 66221541c5.png | normalized |
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\left ( \alpha _ { \rm m a x } ( \gamma ) \right ) ^ 2 + \gamma ^ 2 \approx \rm c o n s t \ . | 359ea4595d.png | tokenized |
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\begin{array} { l l } { { \cal L } _ { \chi } } & { = ( 2 \pi R ) { \cal L } _ { 5 } , _ { \chi } = \tilde { g } ^ { M N } ( \partial _ { M } \chi ) ^ { * } ( \partial _ { N } \chi ) - V } \\ \end{array} | 40c54025f9.png | normalized |
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\lambda _ { n c } = f \lambda + 4 ( u ^ { 2 } - \Lambda ^ { 4 } ) f ^ { \prime } \lambda ^ { \prime } \ \ , | 30cbefd128.png | normalized |
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D_2=\frac{1}{\sqrt{\frac{2}{k^2}(K-E)}},\label{30} | 213f19daf3.png | none |
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R \; ( v \otimes w ) = \mu \; v ^ { \prime } \otimes w ^ { \prime } , | 6987d26928.png | normalized |
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\theta\rightarrow {p\pi+\delta\over (N+1)} \ , | f432d1a5cc.png | none |
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{\cal H}={1\over2}p^2-{g^2\over2}\sum_{\alpha\in\Delta} x(\alpha\cdot q)x(-\alpha\cdot q), \label{eq:hamiltonian} | 100dbfc451.png | none |
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E = \sum _ { n = - \infty } ^ { \infty } \bar { E } _ { n } + \frac { c \xi ^ { 2 } } { 1 6 \pi a ^ { 2 } } { , } | 6ddc33ea64.png | normalized |
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\epsilon ^ { \beta \gamma \delta } V _ \alpha C _ { \beta \gamma } ^ \alpha = 0 , | 1f60424bbe.png | tokenized |
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B = \frac { 1 } { 2 \pi } \frac { \beta ^ { 2 } } { 1 + \beta ^ { 2 } / 4 \pi } | 6dcf00d6ac.png | tokenized |
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V ^ 2 = - \Lambda r ^ 2 - m - 2 q ^ 2 \ln \left ( \frac { r } { r _ 0 } \right ) . | 26747da53c.png | tokenized |
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H_{ua} = (R_{a})_{*}H_{u},\,\,\forall u\in P\,\mbox{ and } \forall a\in G, | 394132ae7f.png | none |
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\left . f _ l ( r ) \right | _ { r = R } = 0 , \qquad \left . \partial _ r \left ( r \, g _ l ( r ) \right ) \right | _ { r = R } = 0 . | 6c715df93e.png | tokenized |
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E _ { \mathrm { b } } = 2 \left( { \frac { \gamma } { 2 } } \right) \sp { ( n - 1 ) / 2 } \; , | 24880a7553.png | normalized |
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E_0(m,\lambda)=<-{1\over2}{d^2\over d\phi^2}+{1\over2} m^2 \phi^2+\frac{\lambda}{4} \phi^4>, | 37f229d740.png | none |
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Q _ { 1 } = \partial _ { \theta ^ { i } } d x ^ { i } + \theta \partial _ { x ^ { i } } d x ^ { i } + M | 58cdbbddc2.png | tokenized |
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\phi ^ { \prime } = \frac { a b } { 2 } \frac { \partial U _ { B } } { \partial \phi } . | 1d8312b99f.png | normalized |
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S_{E}=\int dt dz\, (\,\frac{\theta}{2\pi}E+ \frac{R^2}{2\lambda^2 e^{2\phi}}E^2). | 33ce82e3a8.png | none |
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\label{9}\varsigma =4\pi N\sqrt{\lambda_2\lambda_0} (1-\cos f)^{\frac{k}{2}} \sin f\frac{df}{d\rho} . | 3d8bfaa08a.png | none |
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W _ { i } = ( 0 ( 0 ~ W _ { i } ^ { 1 } ) ( 0 ~ W _ { i } ^ { 2 } ) ( 0 ~ W _ { i } ^ { 2 } ) \vert \vert W _ { i } ^ { 4 } ~ . . . ~ W _ { i } ^ { 1 4 } ) ~ . | 7521f45876.png | normalized |
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\label{resistance}R_{12}={1\over r_3}({r_1r_2 + r_1r_3 + r_2r_3}), | 4ca48143b8.png | none |
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\gamma _ { \star } : = \gamma ^ { 0 } \gamma ^ { 1 } = - \sigma _ { 1 } | 719437e5c6.png | normalized |
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\frac {r^9}{l_p^9} \sim p_{11}R,\label{eq:t7} | 77f5ba3038.png | none |
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\lambda_a^{N+M}=\mu^{N+M}, \hskip10mm a=1,2,\cdots ,N.\label{grring} | 3ac9f97099.png | none |
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0 \rightarrow S _ { t } ^ { n , n _ { c } + l } \rightarrow S _ { t } ^ { n , n _ { c } - l } | 7d8a0867cf.png | normalized |
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f \longrightarrow \hat { f } , \quad g \longrightarrow \hat { g } , | 2300befa23.png | tokenized |
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\label{eq:local--nonlocal}\begin{array}{rcl}\frac{\partial^n}{\partial \lambda^{n}} \psi&=&(-1)^n\,n!\,L(\lambda)^{-n}\,\psi\,,\\\frac{\partial^n}{\partial \lambda^{n}} \bar{\psi}&=&(-1)^n\,n!\,L(\lambda)^{-n}\,\bar{\psi}\,,\quad n\geq 1\,,\end{array} | 5e6ebf36b9.png | none |
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X^{-}=\frac{1}{2p^{+}}(l_{1}l_{2}R_{1}R_{2})^{2}\tau \;. | 3f13d75fab.png | none |
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d s ^ { 2 } = ( d X ^ { 0 } ) ^ { 2 } - R ( X ^ { 0 } ) ^ { 2 } \sum _ { i = 1 } ^ { D - 1 } ( d X ^ { i } ) ^ { 2 } | 48369efe93.png | normalized |
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d \bar { l } ^ { 2 } = \bar { h } _ { i j } d x ^ { i } d x ^ { j } = e ^ { 2 \sigma } d l ^ { 2 } . | 1989b8764c.png | normalized |
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U _ { 1 } = e ^ { - i l \hat { y _ { 2 } } } , ~ ~ ~ ~ ~ ~ ~ ~ U _ { 2 } = e ^ { i l ( \tau _ { 2 } \hat { y _ { 1 } } - \tau _ { 1 } \hat { y _ { 2 } } ) } | 54f65ed091.png | normalized |
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c _ { 4 } \approx - \frac { 9 } { 2 } \int _ { 0 } ^ { 1 } d x \; \left[ \frac { 6 8 } { 2 4 3 } + \frac { 6 5 0 } { 1 9 6 8 3 } x ^ { 2 } + \cdots \right] \, . | 3cbe554584.png | normalized |
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Q _ { A } ( u , \bar { u } ) = \langle P , u | \hat { A } | P , u \rangle \; . | 36d8e76922.png | normalized |
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I(R) \; \simeq \; l_{sq} (R) \label{lsq1} | 5f2f5d67a1.png | none |
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e^{-2A_+-\alpha^{(0)}_+}\bar{\tau}_{+(T)}^{[0]} + e^{-2A_--\alpha^{(0)}_-}\bar{\tau}_{-(T)}^{[0]} = 0. \label{eqn:relation-tauT} | 42a6a3a737.png | none |
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{\cal I}_n = Tr(\Phi^n), (n = 1,\ldots) | 32c379c91f.png | none |
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\bar{\vartheta}_{a}{\cal Q}^{a}+\bar{{\cal Q}}_{a}\vartheta^{a}=\bar{Q}_{i}\theta^{i}+\bar{\tilde{Q}}_{i}\tilde{\theta}^{i} | 1d81e9b435.png | none |
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X _ { + 1 } X _ { + 2 } X _ { - 2 } = 2 \frac { - m _ 1 + m _ 2 + 3 m _ 3 } { m _ 1 + m _ 2 + m _ 3 } X _ { + 1 } . | 3fa616152e.png | tokenized |
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\frac { 1 } { A B } \frac { d } { d r } ( A \sqrt { 1 + B ^ { 2 } \dot { r } ^ { 2 } } ) = - \frac { \kappa _ { 5 } ^ { 2 } } { 6 } ( 2 \rho + 3 p ) , | 41a3fe0a16.png | normalized |
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P _ { I } ^ { 1 } = P _ { I } ^ { 2 } = 0 \, , \quad P _ { I } ^ { 3 } = | 5cb1e39fe8.png | normalized |
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H ( x , x ^ { \prime } ) = \int \frac { d ^ { n } k } { ( 2 \pi ) ^ { n } } e ^ { i k ( x - x ^ { \prime } ) } H ( k , \overline { x } ) . | 7703c4ed4a.png | normalized |
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S ^ M _ { \rm g h } = - \int d ^ 4 x { \bar c } \biggl [ ( 1 - \kappa ) M + \kappa \tilde M \biggr ] c . | 2739099991.png | tokenized |
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q _ I = \mu \sinh ^ 2 \beta _ I , \quad \tilde q _ I = \mu \sinh \beta _ I \cosh \beta _ I . | be6a7c18cb.png | tokenized |
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[ ( E - V ) ^ { 2 } + \hbar ^ { 2 } c ^ { 2 } \triangle - m ^ { 2 } c ^ { 2 } ] \Psi \: = \: 0 | 7476ad1c33.png | normalized |
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{ \cal F } = - \sqrt { a } \, \phi ^ { * } \sigma , | 3b6b0f8f2c.png | normalized |
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\Phi ^ { \Lambda \vert { A B } } \equiv \left( \Phi _ { A B } ^ { \Lambda } \right) ^ { \star } = - \epsilon ^ { A B C D } \, \Phi _ { C D } ^ { \Lambda } ~ . | 5063dd55d9.png | normalized |
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\Gamma = b^{(3)} + {e \over \mu c}{\sqrt{g(x)}\over 4\pi}(Q-N)\nonumber | 3e51c6772c.png | none |
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\Bigl ( \omega _ { A B } ^ { a b } ( x , y ) \Bigr ) = \left( \begin{array} { c c } { 0 } & { - \, 1 } \\ { 1 } & { 0 } \\ \end{array} \right) \, \delta ^ { a b } \delta ( x - y ) \, , | 578b48db34.png | normalized |
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\dot { \varphi } = \partial _ { x } \varphi | 3a6aea68ec.png | normalized |
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Z [ \phi ( x ) ] = \int D \phi ( x ) e ^ { \{ - N ^ { 2 } S ( \phi ) \} } , | 3f8db068d7.png | normalized |
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\left\langle 0 \left | H \right| 0 \right\rangle = 0. | 67c68bf693.png | none |