harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
e^{-1}{\calL}=\frac{1}{2}R-{\frac{1}{4}}G_{IJ}F_{\mu\nu}{}^{I}F^{\mu\nuJ}-\frac{1}{2}{\calG}_{ij}\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j}+{\frac{e^{-1}}{48}}\epsilon^{\mu\nu\rho\sigma\lambda}C_{IJK}F_{\mu\nu}^{I}F_{\rho\sigma}^{J}A_{\lambda}^{K}.
V_{0}=\prod_{a}\beta_{a}^{-J_{a}}\prod_{\sigmaa}(1+q_{\sigma}\zeta_{a\sigma})^{2j_{a\sigma}}\prod_{\sigma}\Bigr(\sum_{a}\frac{\beta_{a}\zeta_{a\sigma}}{1+q_{\sigma}\zeta_{a\sigma}}\Bigl)^{{\bfJ}_{\sigma}-\frac{k}{2}}\,.
Q^{ab}=(1+\nu)_{ab}^{-1}+(1+\nu)_{ac}^{-1}\left\{\nu^{cd}Q^{db}-\mathrm{e}^{-i\mathbf{q.x}}C(-\Box)\frac{\delta^{2}\Gamma}{\delta\phi^{c}\delta\phi^{d}}\left(\mathrm{e}^{i\mathbf{q.x}}Q^{db}\right)\right\}
\vartheta_{\sigma}^{(n)}(w|\zeta)=\Theta_{x^{2}}\left(-\sigmax\prod_{a\inA}w_{a}^{-1}\prod_{j=1}^{2n}\zeta_{j}\right)\prod_{j=1}^{2n}\zeta_{j}^{-(n-\frac{1}{2})(1-\frac{1}{r})}\prod_{a,b\inA\atopa<b}x^{v_{b}-v_{a}}\Theta_{x^{2}}(w_{a}/w_{b}).
[{\calD}\mu]={\calD}A^{\mua}{\calD}\phi^{\dagger}{\calD}\phi{\calD}\lambda^{a}{\calD}\xi^{a}\prod_{\beta=0}^{3}\delta(G_{\alpha}^{a}[A_{0}^{a}+\xi^{a},A_{i}^{a},\lambda^{a}])\middet\{\Gamma_{\alpha}^{a},G_{\beta}^{b}\}\mid(detM_{ab})^{-1/2},
P_{F}(q)_{{\bfs}^{\prime}{\bfs}}^{{\hat{\mu}}{\hat{\nu}}}=\sum_{N_{X}=0}^{\infty}d(N_{X})\cdot{\bar{\psi}}_{{\bfs}^{\prime}}q_{\hat{\rho}}\Gamma^{\hat{\rho}}\psi_{\bfs}{\frac{\eta^{{\hat{\mu}}{\hat{\nu}}}}{q^{2}+M_{N_{X}}^{2}}}.
|\tilde{Z}_{1}|^{2}={\frac{1}{(4G)^{2}}}e^{-\tilde{\eta}_{0}}[(\tilde{Q}_{R}{}^{1})^{2}+(\tilde{Q}_{R}{}^{2})^{2}]\qquad|\tilde{Z}_{2}|^{2}={\frac{1}{(4G)^{2}}}e^{-\tilde{\eta}_{0}}[(\tilde{Q}_{L}{}^{1})^{2}+(\tilde{Q}_{L}{}^{2})^{2}]\.
\int_{ext}\left<T_{00}(\vec{x})\right>_{B}=-\frac1{16\pi^{2}}\left(a\int_{ext}\zeta_{B}(2|0,x_{2})+b\int_{ext}\zeta_{B}(2|x_{1},0)\right)+\frac1{32\pi^{2}}\int_{ext}\zeta_{B}(2|x_{1},x_{2})
m_{\varphi{\bar{\varphi}}}^{2}\simG_{\varphi{\bar{\varphi}}}^{-1}\,\left(G^{i{\bar{\jmath}}}R_{i{\bar{\jmath}}\varphi{\bar{\varphi}}}-G_{\varphi{\bar{\varphi}}}\right){\frac{m_{3/2}^{4}}{M_{p}^{2}}}{\calJ}\,
\widehatT^{\mu\nu}(q)=-\,{\frac{e^{2}}{q^{2}}}\,\left(\eta^{\mu\nu}q^{2}-q^{\mu}q^{\nu}\right)\left[\left.\hat{h}\right|_{q^{2}=0}+\left(\hat{h}(q^{2})-\left.\hat{h}\right|_{q^{2}=0}\right)\right].
\widehat{D}_{0}\pi_{a}+i\varsigma{\frac{\partialE_{a}}{\partial\phi_{i}}}\tau_{i}-i\overline{{p}}^{\prime}{\frac{\partial\overline{{J}}^{a}}{\partial\overline{{\phi}}_{i}}}\overline{{\tau}}_{i}+iE_{a}\beta-i\overline{{J}}^{a}\overline{{\gamma}}+{\calO}(1/e^{2})=0
\widehat{\Psi}_{R}^{c}(x)=\frac{1}{2^{\frac{1}{4}}\sqrt{2L}}\sum_{k=0}^{\infty}\{\widehat{a}_{k}^{c}e^{-\frac{i\pi}{L}(k+\frac{1}{2})x^{-}}+\widehat{d}_{k}^{c\dagger}e^{\frac{i\pi}{L}(k+\frac{1}{2})x^{-}}\},
{\calA}_{b}^{<}=-{\frac{e\rhoZ_{-}^{2}}{2\sqrt{1-e^{2}\rho^{2}}}}+N\int_{1}^{{\frac{1}{e\rho}}}{\frac{dx}{\sqrt{x^{2}-1}}}+{\frac{1}{2}}\int_{1}^{{\frac{1}{e\rho}}}{\frac{dx}{(x^{2}-1)^{3/2}}}\left(xZ_{-}^{2}-W_{-}^{2}(\rhox)\right)\cdotp
V(x)=E[1-{\frac{1+{\frac{b^{n}}{2(x^{2}+4Q_{p}z)^{\frac{n}{2}}}}}{(1+{\frac{b^{n}}{(x^{2}+4Q_{p}z)^{\frac{n}{2}}}})^{2}}}]+{\frac{J_{\parallel}^{2}}{2m_{f}x^{2}}}{\frac{1}{(1+{\frac{b^{n}}{(x^{2}+4Q_{p}z)^{\frac{n}{2}}}})^{2}}},
\chi_{2}^{i}(F,G;{\bfp^{\prime},q})=-\frac{q^{i}}{12\pi^{2}}\tilde{f}{\bf(p^{\prime}})\tilde{g}({\bf-p^{\prime}-q})(b-\frac{3}{2})\xi^{b-3/2}U(b-1/2,b-1/2,\xi),
{\calL}_{mass}=-\frac{1}{R^{2}}\sum_{n=-\infty}^{\infty}\left(\begin{matrix}{\phi^{(n)\,\dagger}}&{\phi^{c\,(n)\,\dagger}}\\\end{matrix}\right)\left(\begin{matrix}{(n+\alpha)^{2}+F_{T}^{2}/4}&{i(n+\alpha)F_{T}}\\{-i(n+\alpha)F_{T}}&{(n+\alpha)^{2}+F_{T}^{2}/4}\\\end{matrix}\right)\left(\begin{matrix}{\phi^{(n)}}\\{\phi^{c\,(n)}}\\\end{matrix}\right)\,,
H=\sum_{j=1}^{N-1}H_{j,j+1}+\frac{1}{2}\stackrel{1}{K}_{-}^{\prime}(0)+\frac{\mathrm{Tr}_{0}\stackrel{0}{K}_{+}(0)H_{N0}}{\mathrm{Tr}K_{+}(0)}\nonumber\,
0=2\sum_{a=1}^{\tilde{n}_{c}}\frac{\delta_{\ell}-\phi_{a}}{\Lambda\omega^{2\ell-1}}+2\sum_{k=1}^{n_{c}-\tilde{n}_{c}}\frac{\delta_{\ell}-\gamma_{k}}{\Lambda\omega^{2\ell-1}-\Lambda\omega^{2k}}-\sum_{i=1}^{n_{f}}\frac{\delta_{\ell}+m_{i}}{\Lambda\omega^{2\ell-1}}.
(QN)_{i_{1}i_{2}}^{i}=-\frac{1}{6}N_{[j_{1}j_{2}]}^{i}\Gamma^{j_{1}j_{2}}{}_{i_{1}i_{2}}-\frac{1}{3}\left(N_{[i_{1}j]}^{i}\Gamma^{j}{}_{i_{2}}-N_{[i_{2}j]}^{i}\Gamma^{j}{}_{i_{1}}\right),
\rho(N)\simeq\frac{c\;(ap)^{\frac{2b+1}{2(p+1)}}}{\sqrt{2\pi(p+1)}}\;N^{-\frac{2b+p+2}{2(p+1)}}\;e^{AN^{\frac{p}{p+1}}}\;,\;\;\;A=\frac{p+1}{p}\;(ap)^{\frac{1}{p+1}}\;.
\left[A,L_{1},L_{2},L_{3},R_{1},R_{2}\right]=\frac{3}{2}\left(i\hbar\right)^{2}\left[\left\{A,R_{3}\right\},I_{L}\right]+\frac{1}{2}\left(i\hbar\right)^{2}\sum_{i}\left[\,\left[\left[A,L_{i}\right],L_{i}\right]\,,\,R_{3}\,\right]\;,
{\calS}_{ij}=a_{1}R+a_{2}\left(\widehat{R}_{i}+\widehat{R}_{j}\right)+a_{3}\widetilde{R}+a_{4}R_{\mu\nu}\left(n_{i}^{\mu}\widehat{n}_{i}^{\nu}+n_{j}^{\mu}\widehat{n}_{j}^{\nu}\right)+\mathrm{extrinsic~terms}
\begin{array}{lclcl}{T^{(I)}(0)}&{=}&{T_{0}^{(I)}(0)}&{=}&{0}\\{}&{}&{}&{}&{}\\{{\dot{T}}^{(I)}(0)}&{=}&{{\dot{T}}_{0}^{(I)}(0)}&{=}&{M}\\\end{array}
G[\sigma]=\intd^{4}x\;\Bigg[A_{0}-\frac{1}{4A_{1}}\sigma^{2}+\frac{A_{2}}{4A_{1}^{2}}\sigma\Box\sigma+\frac{A_{3}}{16A_{1}^{4}}\sigma^{4}+\left(\frac{A_{4}}{4A_{1}^{2}}-\frac{A_{2}^{2}}{4A_{1}^{3}}\right)\sigma\Box\Box\sigma+\cdots\Bigg]\;.
\begin{aligned}{J_{mn}^{1}}&{{}=-\bar{\epsilon}_{(2)}\gamma_{mn}\epsilon_{(3)},}\\{J_{mn}^{2}}&{{}=-\bar{\epsilon}_{(3)}\gamma_{mn}\epsilon_{(1)},}\\{J_{mn}^{3}}&{{}=-\bar{\epsilon}_{(1)}\gamma_{mn}\epsilon_{(2)}.}\\\end{aligned}
\int\mathrm{d}xP_{\pm}(x)={\calP}_{\pm}+{\calP}_{\pm}^{(0)}x+\sum_{n\ne0}\frac{{\calP}_{\pm}^{(n)}}{in}\mathrm{e}^{inx},{}~~~~{\calP}_{\pm}\equiv-\sum_{n\ne0}\frac{{\calP}_{\pm}^{(n)}}{in}.
\begin{array}{rclrcl}{Solv_{3}}&{\equiv}&{Solv\left(SO^{\star}(12)/U(6)\right)}&{Solv_{4}}&{\equiv}&{Solv\left(E_{6(4)}/SU(2)\timesSU(6)\right)}\\{\mathrm{rank~}\,Solv_{3}}&{=}&{3}&{\mathrm{rank~}\,Solv_{4}}&{=}&{4}\\{\mathrm{dim~}\,Solv_{3}}&{=}&{30}&{\mathrm{dim~}\,Solv_{4}}&{=}&{40}\\\end{array}
\tilde{\alpha}_{\omega^{\prime}\omega}^{R}=\frac{1}{\kappa}\,\sqrt{\frac{\omega}{\omega^{\prime}}}\,e^{-\,i\omega^{\prime}x_{H}}\,(\xi\omega^{\prime})^{i\frac{\omega}{\kappa}}\:e^{\frac{\pi\omega}{2\kappa}}\:\Gamma(-i\,\frac{\omega}{\kappa})\,,
{\calL}_{F}=-\sqrt{-h}\left(\bar{\psi}^{1},\,\bar{\psi}^{2}\right)\left(\begin{array}{cc}{i\gamma^{+}\nabla_{+}}&{2}\\{-2}&{i\gamma^{-}\nabla_{-}}\\\end{array}\right)\left(\begin{array}{c}{\psi^{1}}\\{\psi^{2}}\\\end{array}\right),
(M_{0,\pi})_{A}^{\;B}=\frac{1}{2}\alpha_{0,\pi}(\sigma_{1})_{A}^{\;B}\,\quad(\bar{M}_{0,\pi})_{A}^{\;B}=\frac{1}{2}\alpha_{0,\pi}\mathrm{i}(\sigma_{2})_{A}^{\;B}\.
K^{\mu}=(1/3!)\,\varepsilon^{\mu\nu\rho\sigma}\,C_{\nu\rho\sigma}\,=-\,\frac{1}{16\pi^{2}}\,\varepsilon^{\mu\alpha\beta\gamma}\mathrm{tr}\left(F_{\alpha\beta}\,A_{\gamma}-\frac{2}{3}\,A_{\alpha}A_{\beta}A_{\gamma}\right)\,.
H_{CTM}=-2\mathrm{i}\sum_{j=1}^{\infty}j\left\{(1+\Gamma)\tau_{j}^{y}\tau_{j+1}^{x}-(1-\Gamma)\tau_{j}^{x}\tau_{j+1}^{y}+h(\tau_{j}^{x}\tau_{j}^{y}+\tau_{j+1}^{x}\tau_{j+1}^{y})\right\}\nonumber
\{A,B\}_{D}^{\ast}=\{A,B\}^{\ast}+i\sum_{i=1}^{N}\{A,\chi_{i}^{\mu}(\tau)\}^{\ast}\eta_{\mu\nu}\{\chi_{i}^{\nu}(\tau),B\}^{\ast}-\frac{i}{p_{s}^{2}}\sum_{i=1}^{N}\{A,\phi_{i}(\tau)\}^{\ast}\{\phi_{i}(\tau),B\}^{\ast}.
Tr(f_{1}(\hat{P})f_{2}(\hat{P})\cdotsf_{n}(\hat{P}))=\int\frac{d^{p+1}P}{(2\pi)^{(p+1)/2}\sqrt{|\mathrm{det}\theta|}}f_{1}(P)*f_{2}(P)\cdotsf_{n}(P).
a_{1,\mu}=\frac{1}{R}z_{\mu}^{(3)}-\frac{a_{1}}{R}\dot{z}_{\mu}+\{a_{2}+\frac{1}{R}(\ddot{z}^{\alpha}\ddot{z}_{\alpha})-a_{1}a+\epsilon\,\frac{a_{1}}{R}\}\lambda_{\mu}
\Omega=2i\left(1-\frac{1}{2}\frac{1+z\bar{z}}{\zeta}\theta\bar{\theta}\right)\frac{\mathrm{d}z\wedge\mathrm{d}\bar{z}}{\zeta^{2}}+i\left(\frac{\mathrm{d}\theta\wedge\mathrm{d}\bar{\theta}}{\zeta}-\frac{\bar{z}\theta}{\zeta^{2}}\mathrm{d}z\wedge\mathrm{d}\bar{\theta}-\frac{z\bar{\theta}}{\zeta^{2}}\mathrm{d}\bar{z}\wedge\mathrm{d}\theta\right).
K_{3}v_{\lambda,l,m}^{(\circ)}=-i{\frac{(l+1+i\Lambda)(l+1-i\Lambda)}{2l+3}}C_{l+1,m}v_{\lambda,l+1,m}^{\circ}+i(2l+1)C_{l,m}v_{\lambda,l-1,m}^{\circ},
V^{\sigma}=-\frac{m}{4\pi}g^{2}\{(1-\frac{m^{2}}{8M^{2}})\phi+\frac{m^{2}}{2M^{2}}(\frac{1}{x}+\frac{1}{x^{2}})\phi{\bfL}\cdot{\bfS}+\frac{3m^{4}}{16M^{4}}\frac{1}{x^{2}}\chiQ_{12}\}
\deltag_{\mu\nu}(x,y)=a^{2}\left[h_{\mu\nu}(x)-\frac{2y}{l}\eta_{\mu\nu}\delta\phi(x)+\stackrel{(1)}{\deltag_{\mu\nu}}(x,y)+\stackrel{(2)}{\deltag_{\mu\nu}}(x,y)+\cdots\right].\quad
\int_{m_{e}}^{\infty}dk\(k^{2}-m_{e}^{2})^{1-s}\partial_{k}\frac{t^{j}}{m^{n}}=-\frac{m_{e}^{2-2s}}{2}\frac{\Gamma(2-s)\Gamma\left(\frac{1+j-n}{2}\right)\Gamma(s+\frac{n-3}{2})}{(Rm_{e})^{n-1}\Gamma(j/2)}\,
{\cal{A}}={\mathrm{\bf{j}}}\Big(iA^{i}T_{i}+\frac{i}{2\sqrt{3}}BY\Big)+{\mathrm{\bf{k}}}\Big(\frac{i}{\sqrt{2}}(\phi_{0}\Sigma_{+}+\phi_{+}\Sigma_{-}+\phi_{0}^{*}\Sigma_{-}^{\prime}+\phi_{+}^{*}\Sigma_{+}^{\prime})\Big),
\kappa^{2}={\frac{z_{\beta}-z_{\alpha}}{z_{i}-z_{\alpha}}},\,\lambda^{2}={\frac{z_{\beta}-z_{\alpha}}{z_{j}-z_{\alpha}}},\,\mu^{2}={\frac{z_{\beta}-z_{\alpha}}{z_{k}-z_{\alpha}}}.
\left(\begin{matrix}{\sigma_{0}^{(\mathrm{op)}}}\\{\pi_{0}^{(\mathrm{op)}}}\\\end{matrix}\right)=-{\frac{\mu^{2}}{N}}\left(m_{\mathrm{ZM}}^{2}-\partial_{\bot}^{2}\right)^{-1}\left[:\bar{\Psi}_{M}^{a}\left(\begin{matrix}{1}\\{i\gamma_{5}}\\\end{matrix}\right)\Psi_{M}^{a}:\right]_{0},
\begin{matrix}{{N_{A}}^{B}={M^{B}}_{A}(-1)^{\varepsilon_{A}(\varepsilon_{B}+1)},}&{{(N^{-1})_{A}}^{B}={(M^{-1})^{B}}_{A}(-1)^{\varepsilon_{A}(\varepsilon_{B}+1)}.}\\\end{matrix}
\left(\begin{array}{c}{\vec{\alpha}}\\{\vec{\beta}}\\\end{array}\right)=\left(\begin{array}{cc}{A_{2}\,{\mathbfL}_{1}}&{A_{2}\,{\mathbfL}_{2}}\\{-A_{1}\,{\mathbfL}_{2}}&{A_{1}\,{\mathbfL}_{1}}\\\end{array}\right)\left(\begin{array}{c}{\vec{Q}}\\{\vec{P}}\\\end{array}\right)
exp({\frac{i}{\hbar}}S^{\prime}(\Phi^{A},{\frac{\partial\psi}{\partial\phi^{A}}})))=\intD\tilde{\phi}^{A}\,D\pi^{A}\,DB^{A}exp\,\Big({\frac{i}{\hbar}}\,\,{\underline{S}}_{col.}\Big)
\mathrm{tr}(T^{A}T^{B}T^{C}T^{D})={\frac{1}{12}}\delta_{AB}\delta_{CD}-{\frac{1}{8}}f_{ABE}f_{CDE}+{\frac{1}{8}}d_{ABE}d_{CDE}+{\frac{i}{8}}(f_{ABE}d_{CDE}+f_{CDE}d_{ABE}).
\left(\begin{matrix}{X_{1}}\\{X_{2}}\\{X_{3}}\\{X_{4}}\\\end{matrix}\right)=\frac{1}{2r}\left(\begin{matrix}{3}&{1}&{1}\\{1}&{-1}&{-3}\\{-1}&{-1}&{3}\\{-3}&{1}&{-1}\\\end{matrix}\right)\left(\begin{matrix}{\varphi^{(1)}}\\{\varphi^{(2)}}\\{\varphi^{(3)}}\\\end{matrix}\right),
{\calL}=\frac{1}{2}(\partial_{\mu}\sigma)^{2}+\frac{1}{2}(\partial_{\mu}\pi)^{2}-\frac{1}{2}\mu^{2}(\sigma^{2}+\pi^{2})-\frac{\lambda}{4}(\sigma^{2}+\pi^{2})^{2}+c\sigma,
\partial_{k}S_{k}^{I}[\phi]={\frac{1}{2}}{\frac{\deltaS_{k}^{I}[\phi]}{\delta\phi}}\,\partial_{k}G_{k}\,{\frac{\deltaS_{k}^{I}[\phi]}{\delta\phi}}-{\frac{1}{2}}{\mathrm{Tr}}\left[\partial_{k}G_{k}{\frac{\delta^{2}S_{k}^{I}[\phi]}{\delta\phi\delta\phi}}\right].
\{{\calJ}_{\mu},x_{\nu}\}=-\epsilon_{\mu\nu\lambda}x^{\lambda}-\eta_{\mu\nu}\epsilon_{\lambda\rho\sigma}p^{\lambda}\partial^{\rho}A^{\sigma}+\epsilon_{\mu\rho\sigma}(p_{\nu}\partial^{\rho}-\partial_{\nu}p^{\rho})A^{\sigma}
H=\int{dx\left\{{-\frac{1}{2}\Pi_{1}\left({1+\frac{{g^{2}}}{\pi}\frac{a}{{a-1}}\frac{1}{{\partial^{2}}}}\right)^{-1}\Pi^{1}+c^{\prime}(x)\left({\partial_{1}\Pi^{1}-J^{0}}\right)}\right\}},
\Pi(t,\Theta)=\frac{\partial}{\partial\frac{d\phi}{dt}}L=\frac{i}{2}\frac{\partial}{\partial\frac{d\phi}{dt}}\intd\Theta(D_{\Theta}\phi)\dot{\phi}=\frac{1}{2}\intd\Theta\{\Theta\dot{\phi}+iD_{\Theta}\phi\},\qquad\dot{\phi}\equiv\frac{d\phi}{dt},
\frac{\partialS}{\partialt}=-H\bigg(q,\psi,\frac{\partialS}{\partialq},\frac{\partialS}{\partial\psi},\lambda\big(q,\psi,\frac{\partialS}{\partialq},\frac{\partialS}{\partial\psi}\big),\lambda^{\alpha},t\bigg).
c_{j_{q}k_{{s}_{1}}}^{h_{{t}_{1}}}c_{h_{{t}_{1}}k_{{s}_{2}}}^{h_{{t}_{2}}}\ldotsc_{h_{{t}_{n-2}}k_{{t}_{n-1}}}^{h_{{t}_{n-1}}}c_{h_{{t}_{n-1}}k_{{s}_{n}}}^{i_{p}}g^{k_{{s}_{1}}}g^{k_{{s}_{2}}}\ldotsg^{k_{{s}_{n-1}}}g^{k_{{s}_{n}}}dg^{j_{q}}\;,
S_{l.c.}=\frac{1}{2}\!\int\!\!d^{d}x(\phi^{IJ}\Box\phi^{IJ}-\frac{d(d-2)}{4z^{2}}|\phi_{2}|^{2}-\frac{(d-2)(d-4)}{4z^{2}}|\phi_{1}|^{2}-\frac{(d-4)(d-6)}{4z^{2}}|\phi_{0}|^{2})
\begin{array}{lll}{\alpha_{ik}}&{=}&{1\otimes\cdots\otimes1\otimes\alpha_{i}\otimes1\otimes\cdots\otimes1\inH^{1}((M)^{N},{\bf{Z}})}\\{\beta_{k}}&{=}&{1\otimes\cdots\otimes1\otimes\beta\otimes1\otimes\cdots\otimes1\inH^{2}((M)^{N},{\bf{Z}}),}\\\end{array}
C_{+}=C_{-}=\left[{\frac{\frac{1}{2}}{\int_{\tilde{a}}^{\tilde{a^{\prime}}}\frac{d\phi}{\sqrt{2(E-V)}}}}\right]^{\frac{1}{2}}=\left[{\frac{\mu\sqrt{1+u}}{2K(k^{\prime})}}\right]^{\frac{1}{2}}\equivC
e^{-\frac{1}{\hbar}\left(\tilde{L}^{\Lambda,\Lambda_{0}}(\varrho\,;\varphi)+\tilde{I}^{\Lambda,\Lambda_{0}}(\varrho)\right)}=e^{-\frac{1}{2}\langle\varphi,(\hbarC^{\Lambda,\Lambda_{0}})^{-1}\varphi\rangle}\tilde{Z}^{\Lambda,\Lambda_{0}}(\varrho\,;(C^{\Lambda,\Lambda_{0}})^{-1}\varphi).
L=P_{\mu}\Pi_{\tau}^{\mu}+Z_{\alpha\beta}\Pi_{\tau}^{\alpha\beta}+\bar{Z}_{\dot{\alpha}\dot{\beta}}\bar{\Pi}_{\tau}^{\dot{\alpha}\dot{\beta}}-\lambda_{v}h_{v}-\bar{\lambda}_{v}\bar{h}_{v}-\lambda_{u}h_{u}-\bar{\lambda}_{u}\bar{h}_{u}\,.
\begin{array}{lcl}{T_{\mu\nu}}&{=-}&{\nabla_{\mu}\Phi\nabla_{\nu}\Phi+2\nabla_{\mu}\nabla_{\nu}\Phi+\frac{1}{2}g_{\mu\nu}\nabla^{\alpha}\Phi\nabla_{\alpha}\Phi}\\{}&{}&{-g_{\mu\nu}(2R+\frac{1}{2}\Lambda),}\\\end{array}
\frac{\partialt^{\prime}}{\partialx^{\nu}}=\eta_{\nu\nu}(x^{\nu}-y^{\nu}(t^{\prime}))\left(\sum_{\alpha=0}^{3}\eta_{\alpha\alpha}(x^{\alpha}-y^{\alpha}(t^{\prime}))\frac{dy^{\alpha}(t^{\prime})}{dt^{\prime}}\right)^{-1}
[\omega]=\left(\begin{matrix}{0}&{a}&{0}&{b}&{0}&{c}\\{-a}&{0}&{-{b^{\prime}}}&{0}&{-{c^{\prime}}}&{0}\\{0}&{{b^{\prime}}}&{0}&{d}&{0}&{f}\\{-{b^{\prime}}}&{0}&{-d}&{0}&{-{f^{\prime}}}&{0}\\{0}&{{c^{\prime}}}&{0}&{{f^{\prime}}}&{0}&{g}\\{-c}&{0}&{-f}&{0}&{-g}&{0}\\\end{matrix}\right)
\begin{array}{l}{A^{1}(0,0)=5F_{(6)},}\\{A^{1}(n,0)=A^{1}(0,n^{\prime})=F_{(5,1)}+F_{(4,2)}+F_{(3,3)}+F_{(2,4)}+F_{(1,5)}=0,}\\{A^{1}(n,n)=2F_{(0,0,6)}+F_{(0,2,4)}+F_{(2,0,4)}+F_{(1,1,4)},}\\{A^{1}(n,-n)=F_{(1,0,5)}+3F_{(2,2,2)}+F_{(5,0,1)},}\\\end{array}
\frac{\lambda}{4!}\Phi^{4}(\vec{r},t)=\frac{\lambda}{4}\langle\Phi^{2}(\vec{r},t)\rangle\Phi^{2}(\vec{r},t)+\left(\frac{\lambda}{4!}\Phi^{4}(\vec{r},t)-\frac{\lambda}{4}\langle\Phi^{2}(\vec{r},t)\rangle\Phi^{2}(\vec{r},t)\right)
E^{2}=\frac{1}{p^{2}+\left(\frac{4\pi^{2}\alpha^{\prime}}{L_{1}L_{2}}\right)^{2}q^{2}}\left(\left(\frac{2\pi}{L_{1}}n_{1}\right)^{2}+\left(\frac{2\pi}{L_{2}}n_{2}\right)^{2}\right).
{\calL}=\intd^{4}\theta\left(Q^{i\dagger}e^{2V}Q^{i}+\frac{1}{2}\mathrm{Tr}S^{\dagger}S\right)+\intd^{2}\theta\left(\frac{1}{4g_{h}^{2}}W^{a}W^{a}+\frac{1}{2}\lambdaS_{ij}Q^{i}Q^{j}\right)+h.c.
\begin{array}{l}{\lbrack[B,C],\varphi_{o}A]-[[A,C],\eta_{o}]+(B\varphi_{o})[A,C]-A(B\varphi_{o})C-(AB^{2}\varphi_{o})A}\\{\qquad\qquad\qquad\qquad\qquad-(BAB\varphi_{o})A=0.}\\\end{array}\tag{3.14e}
i\partial_{t}\psi_{\alpha\beta}=[\sqrt{(m_{\alpha}+\frac{1}{2}V_{s})^{2}+{\bfp}^{2}}+\sqrt{(m_{\beta}+\frac{1}{2}V_{s})^{2}+{\bfp}^{2}}-\frac{4}{3}\frac{\alpha_{s}}{r}+k\frac{{\bfs}_{\alpha}\cdot{\bfs}_{\beta}}{m_{\alpha}m_{\beta}}]\psi_{\alpha\beta}
\rho=rM_{\mathrm{P}}\;\sqrt{\frac{3\lambda}{8\pi}}\,\qquadA=RM_{\mathrm{P}}\;\sqrt{\frac{3\lambda}{8\pi}}\,\qquadF={\frac{f}{M_{\mathrm{P}}}}\sqrt{\frac{8\pi}{3}}\,\qquadU(F)\equiv\lambda^{-1}V(\phi)\.
u(\mathbf{p,}i)=\left[\begin{array}{c}{\left(1+\frac{\mathbf{p}^{2}}{8m^{2}}\right)}\\{\frac{(\overrightarrow{\mathbf{\sigma}}\cdot\mathbf{p})}{2m}}\\\end{array}\right]\varphi_{i},\;\\\\\\\v(\mathbf{p,}i)=\left[\begin{array}{c}{\frac{(\overrightarrow{\mathbf{\sigma}}\mathbf{\cdotp})}{2m}}\\{\left(1+\frac{\mathbf{p}^{2}}{8m^{2}}\right)}\\\end{array}\right]\chi_{i},
B_{4}(\triangle)={\itTr}\Biggl(\frac{1}{180}\left(R_{\mu\nu\sigma\lambda}^{2}-R_{\mu\nu}^{2}\right)+\frac{1}{2}\left(Z+\frac{R}{6}\right)+\frac{1}{12}Y_{\mu\nu}Y^{\mu\nu}\Biggr)
\begin{array}{rcl}{\left({\calV}_{1,2}\otimes{\calR}_{1,1}\right)_{\mathrmf}}&{=}&{{\calV}_{1,2}\oplus{\calV}_{1,2}\oplus{\calV}_{2,2}\,,}\\{\left({\calR}_{1,1}\otimes{\calR}_{1,1}\right)_{\mathrmf}}&{=}&{{\calR}_{1,1}\oplus{\calR}_{1,1}\oplus{\calR}_{2,1}\,.}\\\end{array}
A_{N}=\frac{{\calN}_{N}}{2\pi^{2}\pi^{3}}{\pi^{3}}\int_{0}^{1}\frac{dz\,z}{\sqrt{z}\sqrt{1-z}}\int_{0}^{\infty}\frac{dt}{t^{2}}\left(\frac{1}{2\pi^{2}t}\right)^{{\textstyle\frac{k+1-D}{2}}}e^{-tY^{2}}
\langleT_{\\mu}^{\mu}\rangle=\frac{1}{48\pi}\Biggl[\left(\partial_{\mu}\partial_{\nu}h^{\mu\nu}-\partial_{\mu}\partial^{\mu}h_{\\nu}^{\nu}\right)\mp\frac{1}{2}\varepsilon_{\mu\lambda}\partial^{\lambda}\partial_{\nu}h^{\mu\nu}\Biggr].
xy^{2}=\Lambda^{4(n_{c}-\tilde{n}_{c})}\left[x\prod_{a=1}^{\tilde{n}_{c}-r}(x-\phi_{a}^{2})\prod_{i=1}^{r}m_{i}^{2}+\frac{1}{\Lambda^{n_{c}-\tilde{n}_{c}}}\prod_{i=1}^{n_{f}}m_{i}\right]^{2}-\frac{1}{\Lambda^{2(n_{c}-\tilde{n}_{c})}}\prod_{i=1}^{n_{f}}m_{i}^{2},
q^{\frac{3}{2}(\Lambda,\Lambda+2\rho)}=\sum_{\nu\inM(\Lambda)}q^{\frac{1}{2}(\nu,\nu+2(\Lambda+\rho))}\frac{D_{q}[\Lambda+\nu]}{D_{q}[\Lambda]}\sum_{s=0}^{\infty}(m_{\nu-s\delta}^{+}-m_{\nu-s\delta}^{-})\,q^{-s(3g+2\kappa_{\Lambda})}\;.
\sigma^{1}=\left(\begin{array}{ll}{0}&{1}\\{1}&{0}\\\end{array}\right),\quad\sigma^{2}=\left(\begin{array}{ll}{0}&{i}\\{-i}&{0}\\\end{array}\right),\quad\sigma^{3}=\left(\begin{array}{ll}{1}&{0}\\{0}&{-1}\\\end{array}\right).
H=-\frac{e}{2}(p^{2}-m^{2})+i(p_{\mu}\psi_{a}^{\mu}+m\psi_{a}^{5})\chi_{a}-\frac{1}{2}f_{ab}\left(i\left[\psi_{an},\;\psi_{b}^{n}\right]_{-}+\kappa_{ab}\right).
I_{int}=\intd^{n}x\{\sum_{a}\big({\frac{-1}{2(p_{a}+1)!}}H_{\mu_{1}\ldots\mu_{p_{a}+1}}^{a}H^{a\mu_{1}\ldots\mu_{p_{a}+1}}+g_{a}S^{a\mu_{1}\ldots\mu_{p_{a}}}B_{\mu_{1}\ldots\mu_{p_{a}}}^{a}\big)\},
P_{\mu}\bar{\varepsilon}_{0}\gamma^{\mu}\varepsilon_{0}=\int_{\Sigma}G_{\mu\nu}\bar{\varepsilon}\gamma^{\mu}\varepsilond\Sigma^{\nu}+\int_{\Sigma}\nabla_{\mu}\bar{\varepsilon}(\gamma^{\nu}\sigma^{\mu\rho}+\sigma^{\mu\rho}\gamma^{\nu})\nabla_{\rho}\varepsilond\Sigma_{\nu},
\int_{C_{b}^{(s)}}f_{s}(t)\frac{d\thetadz}{2\pii}\varphi_{s}(t)=-1,\quad\int_{C_{b}^{(s)}}f_{s}(t)\frac{d\thetadz}{2\pii}=0\quad\mathrm{and}\quad\int_{C_{b}^{(s)}}K_{s}^{(1)}(t,t^{\prime})\frac{d\theta^{\prime}dz^{\prime}}{2\pii}=0.
\alpha=\left(\begin{array}{ccc}{\omega_{k}^{2}}&{0}&{0}\\{0}&{\omega_{k}^{-1}}&{0}\\{0}&{0}&{\omega_{k}^{-1}}\\\end{array}\right)~~~~~~~~\beta=\left(\begin{array}{ccc}{1}&{0}&{0}\\{0}&{\omega_{2k^{\prime}}}&{0}\\{0}&{0}&{\omega_{2k^{\prime}}^{-1}}\\\end{array}\right)~~~~~~~\gamma=\left(\begin{array}{ccc}{1}&{0}&{0}\\{0}&{0}&{i}\\{0}&{i}&{0}\\\end{array}\right)
U_{\underline{{m}}}^{~\underline{{a}}}=\left(U_{\underline{{m}}}^{0},U_{\underline{{m}}}^{~I},U_{\underline{{m}}}^{9}\right)=\left({\frac{1}{2}}(U_{\underline{{m}}}^{++}+U_{\underline{{m}}}^{--}),U_{\underline{{m}}}^{~I},{\frac{1}{2}}(U_{\underline{{m}}}^{++}-U_{\underline{{m}}}^{--})\right)~\in~SO(1,9)
N_{i}(m^{2},t)=\int_{-\infty}^{\infty}d\mu(k)\,\,k^{-2}\frac{1}{\left[m^{2}+k^{2}(1-\xi^{-1})\right]^{2}}e^{-k^{2}t}e^{-m_{i}^{2}t}
\begin{array}{l}{\ell_{3}^{-}=-\ell^{-};\;\ell^{+}=0}\\{\hat{h}=\frac{k^{+}}{k}\ell^{-}=-\frac{\hat{q}}{2}}\\{\hat{h}=\hat{h}^{\prime},\;\hat{q}=-\hat{q}^{\prime}}\\\end{array}
I_{1}(\rho,g)=\frac{1}{2\pii}\int_{C_{0}}ds\frac{e^{\rho/s}}{s}F_{1}(s^{})=\frac{1}{2\pii}\int_{C_{0}^{\prime}}ds^{\prime}\frac{e^{\rho^{\prime}/s^{\prime}}}{s^{\prime}}F_{1}({s^{\prime}})=\tilde{I}_{1}(\rho^{\prime})
i\hbar\frac{\partial}{\partialt}\left[\begin{array}{cc}{\Psi_{e}(t)}\\{\Psi_{\mu}(t)}\\\end{array}\right]=H_{\mathrm{flavor}}\left[\begin{array}{cc}{\Psi_{e}(t)}\\{\Psi_{\mu}(t)}\\\end{array}\right]\,,
Z_{DD}={\frac{1}{N}}\sum_{p=0}^{N-1}\chi_{p}^{f}(q){\frac{1}{\eta(q)}}\int_{-\infty}^{\infty}dxq^{x^{2}/2N}\sum_{m=-\infty}^{\infty}|a_{m}|^{2}e^{2\piimx}
\left(\lambda^{\alpha,A}\right)^{\ast}=-\lambda^{\beta,B}\epsilon_{\beta\alpha}\eta_{BA}^{1}\,\qquad\left(\bar{\lambda}_{\alpha^{\prime},A}\right)^{\ast}=-\bar{\lambda}_{\beta^{\prime},B}\epsilon^{\beta^{\prime}\alpha^{\prime}}\eta^{1,BA}\.
S_{int}=\frac{g}{4!(2\pi)^{d}}\intdp_{1}dp_{2}dp_{3}dp_{4}\,e^{-ip_{1}\wedgep_{2}-ip_{3}\wedgep_{4}}\varphi(p_{1})\varphi(p_{2})\varphi(p_{3})\varphi(p_{4})\delta(p_{1}+p_{2}+p_{3}+p_{4}).
E_{1+2}=E_{1}+E_{2}\,,\quad\overrightarrow{p}_{1+2}=\overrightarrow{p}_{1}+\overrightarrow{p}_{2}+\frac{1}{\kappa}(E_{1}\overrightarrow{p}_{2}-E_{2}\overrightarrow{p}_{1})\,.
f^{(1)}=\left(\begin{array}{cccc}{0}&{-\delta_{ij}}&{0}&{0}\\{\delta_{ji}}&{0}&{0}&{\partial_{i}^{y}}\\{0}&{0}&{0}&{m^{2}}\\{0}&{-\partial_{j}^{x}}&{-m^{2}}&{0}\\\end{array}\right)\,\delta^{(3)}({\vec{x}}-{\vec{y}}),
{\calU}_{j}\,{\calU}_{k}=e^{2\pii\theta_{jk}}{\calU}_{k}\,{\calU}_{j},\qquad\left[\nabla_{j},{\calU}_{k}\right]=\delta_{jk}\,{\calU}_{k},\qquad\left[\nabla_{j},\nabla_{k}\right]=iF_{jk}\,{\bfI},
S_{\mathrm{GF}}^{H}=\int\mathrm{d}^{3}x\left[\frac{\xi_{0}}{2}b^{a}b^{a}-b^{a}H(\partial/\Lambda)(\partial^{\mu}A_{\mu})^{a}+\overline{c}^{a}H(\partial/\Lambda)(\partial_{\mu}D^{\mu}c)^{a}\right].
S_{l}=[s_{l}(x^{\prime})e_{l}^{\prime}(x)-s_{l}^{\prime}(x^{\prime})e_{l}(x)]^{2}-\xi^{2}[s_{l}(x^{\prime})e_{l}^{\prime}(x)+s_{l}^{\prime}(x^{\prime})e_{l}(x)]^{2},
[4(\vec{k}\,^{2}+m^{2})-M^{2}]\psi(\vec{k},\hat{n})=-{\frac{m^{2}}{2\pi^{3}}}\int{\frac{d^{3}k^{\prime}}{\varepsilon_{k^{\prime}}}}V(\vec{k},\vec{k^{\prime}},\hat{n},M^{2})\psi(\vec{k^{\prime}},\hat{n}),
S_{0}={\frac{1}{2\pi}}\intd^{2}x\sqrt{-g}\left\{e^{-2\phi}\left[R+4\left(\nabla\phi\right)^{2}-4\Lambda\right]-\frac{1}{2}\sum_{i=1}^{N}\left(\nablaf_{i}\right)^{2}\right\}\\,
\mathbf{S}_{T}(S)\equiv\mathbf{S}_{Q}(S)+\rho^{\mu}\mathbf{\bar{Q}}_{\mu}S-i\xi^{\mu}\mathbf{P}_{\mu}S+\bar{\xi}\mathbf{Z}S+\xi\mathbf{\bar{Z}}S-\rho^{\mu}\frac{\partialS}{\partial\xi^{\mu}}+\frac{\partialS}{\partial\bar{\xi}}-\rho^{\mu}\rho_{\mu}\frac{\partialS}{\partial\xi},
\begin{array}{l}{p^{00}=0,}\\{p^{0k}=-g^{ik}\partial_{i}(ae^{-\Phi}),}\\{p^{ik}=\frac{1}{2}ae^{-\Phi}\left[(\dot{h}_{lm}g^{lm}-2\dot{\varphi})g^{ik}+\dot{h}^{ik}\right],}\\{p_{\varphi}=ae^{-\Phi}(2\dot{\varphi}-\dot{h}_{lm}g^{lm}).}\\\end{array}
\gamma^{a\underline{b}}=\left(\begin{matrix}{\gamma^{z\underline{z}}}&{\gamma^{z\underline{\zeta}}}\\{\gamma^{\zeta\underline{z}}}&{\gamma^{\zeta\underline{\zeta}}}\\\end{matrix}\right)=\left(\begin{matrix}{0}&{(1+{\frac{1}{2}}z\overline{z})^{2}}\\{(1+{\frac{1}{2}}z\overline{z})^{2}}&{(1+{\frac{1}{2}}z\overline{z})(z\overline{\zeta}+\overline{z}\zeta)}\\\end{matrix}\right).
|I\rangle={\frac{1}{4i}}b^{+}\left({\frac{\pi}{2}}\right)b^{-}\left({\frac{\pi}{2}}\right)|I\rangle_{M}|I^{r}\rangle_{G}=[b^{\dagger}]_{\calO}(b_{0}+2[b^{\dagger}]_{\calE})|I\rangle_{M}|I^{r}\rangle_{G},
\tau_{1}=\left(\begin{matrix}{0}&{-i}&{0}\\{-i}&{0}&{0}\\{0}&{0}&{0}\\\end{matrix}\right)\qquad\tau_{2}=\left(\begin{matrix}{0}&{-1}&{0}\\{1}&{0}&{0}\\{0}&{0}&{0}\\\end{matrix}\right)\qquad\tau_{3}=\left(\begin{matrix}{-i}&{0}&{0}\\{0}&{i}&{0}\\{0}&{0}&{0}\\\end{matrix}\right)~,
{\frac{(4\pi)^{2}V}{2a\mu^{4}}}={\frac{x_{\ast}^{2}}{2}}\left[{\frac{1}{g_{4R}(\mu)}}-{\frac{1}{g_{4R}^{\ast}}}\right]+{\frac{x_{\ast}^{4/(2-w)}}{4}}\left[{\frac{3}{2}}+{\frac{\alpha_{c}}{\alpha}}\right]+{\frac{Rx_{\ast}^{2/(2-w)}}{12\mu^{2}}}\left[{\frac{1}{2}}+{\frac{\alpha_{c}}{\alpha}}\right].
K_{N}(\lambda,\lambda^{\prime})~=~\sqrt{\lambda\lambda^{\prime}}\prod_{f=1}^{\alpha}\sqrt{\lambda^{2}+m_{f}^{2}}\sqrt{{\lambda^{\prime}}^{2}+m_{f}^{2}}e^{-\frac{N}{2}(V(\lambda^{2})+V({\lambda^{\prime}}^{2}))}\frac{1}{N}\sum_{j=0}^{N-1}\frac{Q_{j}(\lambda)Q_{j}(\lambda^{\prime})}{r_{j}}~,
f(r,t)=-i\frac{\omega^{3}}{105\Omega_{8}}\frac{1}{r^{4}}e^{i\omegar}\left(1+\frac{6i}{\omegar}-\frac{15}{(\omegar)^{2}}-\frac{15i}{(\omegar)^{3}}\right)e^{-i\omegat}
\zeta^{\prime}\left(0|A\right)=\frac{1}{4\pi}\int\left[d^{2}z\right]\,tr^{\prime}[k_{1}\left(z\right)]\qquad,\qquad\left[\frac{d}{ds}\zeta^{\prime}\left(s|A\right)\right]_{s=0}=\gamma\zeta^{\prime}\left(0|A\right)
\int_{C_{h}^{-}}{f(z)dz}-\int_{C_{h}^{+}}{f(z)dz}=2\int_{a}^{b}{f(x)dx}+2\pii\left[\sum_{k}{\mathrm{Res}}_{{\mathrm{Im}}z_{f,k}<0}f(z)-\sum_{k}{\mathrm{Res}}_{{\mathrm{Im}}z_{f,k}>0}f(z)\right].
S_{4}(k_{1},k_{2},k_{3},k_{4})=i[g_{YM}^{0}]^{2}\int{\frac{d^{4}k}{(2\pi)^{4}}}{\frac{S_{3}(k_{1},k_{2};k)S_{3}(k_{3},k_{4};-k)}{-k^{2}+i\epsilon}}\,.