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\rho^2 \ln(\Lambda l)^2 \ll \frac{1}{l^2} \ln\frac{1}{(\rho l)^2}\label{eq20} | expression | |
S^{\rm W}(p) = -i\gamma\cdot p \sigma_{V}^{\rm W}(p) \label{Csym} | expression | |
{}\label{hamac}S^G [h_{ij}, \pi^{ij}, N, N^i] = \int_{\cal M} dt \, d^3x \left( {\pi}^{ij} {\dot h}_{ij} - N H^G - N^i H^G_i \right), | expression | |
X^{A{A^\prime}}_{0}:=i{\sqrt2}(\sigma^{A} \beta^{A^\prime}- \varsigma^{A} \alpha^{A^\prime})=-i{\sqrt2}({\bar \sigma}^{A^\prime}{\bar \beta}^{A} -{\bar \varsigma}^{A^\prime}{\bar \alpha}^{A}), | expression | |
P[R>s]\ \ge\ \lim_{m\to\infty} P[\cap_{n\ge m}A_n(t^{2n})^c]\ . | expression | |
\Gamma(b_1 \to \gamma + \gamma)= \zeta \left({M_1\over2m_P}\right)^4 \left({A\over\ell_P^2}\right)^2{M_1\over2\hbar}. | expression | |
{\widehat \Gamma}_{\; \; \; BC}^{A'} \equiv \Gamma_{\; \; \; BC}^{A'}+\nabla_{\; \; \; B}^{A'} \; \nu_{C} \; , | expression | |
\lambda^2 - \mbox{Tr} Z \, \lambda +\mbox{det}Z =0 \label{simple} | expression | |
\label{5} [x_0, x_i] = -\frac{i}{\kappa}\, x_i, \quad [x_i, x_j] = 0., | expression | |
{\cal L} = - \frac{1}{4} F^a_{\mu\nu} F^{\mu\nu,a} + \frac{1}{2}m^2A_\mu^aA^{\mu,a} \label{4.1} | expression | |
\bar{Z}(j) = 1 + \bar{X}(j\bar{Z}(j)), \;\;\;\;\;\; \bar{Z}(j) = \sum_{w}j^{\bar{w}}<\phi^{w}> \label{ff} | expression | |
D_i=\sum_j (X_{ji}^\dagger X_{ji}-X_{ij}X_{ij}^\dagger )=\theta_i,\label{eq:D-flatness} | expression | |
x(t)=x_{r}(t)={1\over t},\quad y(t)=y_{r}(t)=-{1\over{t^2}}, \quad z(t)=z_{r}(t)=-{1\over{t^2}}. \label{eq:functions} | expression | |
\int_0^{\Delta t(r_P)} dt = \int_{r_P}^c e^{-r} dr. | expression | |
N^{AB}=\bar{\epsilon}_1\Gamma^{ABC}D_C\epsilon_2+\textstyle{1\over 8}\bar{\epsilon_1}\Gamma^{C_1C_2}\epsilon_2F^{AB}_{\ \ C_1C_2}+\textstyle{1\over{96}}\bar{\epsilon}_1\Gamma^{ABC_1\cdots C_4}\epsilon_2F_{C_1\cdots C_4}.\label{11dsgnester} | expression | |
\overline{g}=\sqrt{g^{2}+g^{\prime 2}} = \frac{1}{2\sqrt{\omega}}\sqrt{\tilde{g}^{2}+\tilde{g}^{\prime 2}} \label{VII34} | expression | |
\{w,z\} = \{w,u\}~(\frac{du}{dz})^{2} + \{u,z\} | expression | |
K_{i}E_{\alpha}=q^{(\alpha,\alpha_{i})}E_{\alpha}K_{i},K_{i}F_{\alpha}=q^{-(\alpha,\alpha_{i})}F_{\alpha}K_{i}, | expression | |
\label{obh:matter}{\delta L\over \delta\Psi} = {\partial L\over \partial\Psi} -(-1)^{p}D{\partial L \over \partial D\Psi}=0. | expression | |
%F_{2,a} - F_{2,-a}, F_{2,b} - F_{2,-b}%\label{5.16}% | expression | |
{\cal P} : T \mapsto -T, \qquad Q_\pm \mapsto\bar{Q}_\mp , \qquad \bar{Q}_\pm \mapsto Q_\mp\,. | expression | |
F(g)= -\frac{3}{8} -\frac{5 A^2}{48}-\frac{A^4}{384} -\frac{1}{2} \log\frac{A^2}{4} + \frac{C_1}{g^2},\label{sol1} | expression | |
\Psi_D =\left( \begin{array}{c}A\\B\\C\\D\end{array} \right). | expression | |
\label{quater.constr.}\phi_2^4=\frac{\sqrt{g}}{\alpha}\left[p_1^2-m^2+\sqrt{g}~\frac{\alpha m+\beta\sigma\sqrt{m^2-p_1^2}}{(q_2^2)^{3/2}}\right],~~~~~~\sigma=\mbox{sign}~\varepsilon(p_1q_2q_3), | expression | |
E_{eff(3+1)}=-\frac{1}{2\pi^{2}}\int_{0}^{+\infty}k\ln\left(\frac{k^{2}+m^{2}_{f}}{m_{f}^{2}}\right) (\Delta(k)-c) dk | expression | |
[\partial_A-\Phi_A,\partial_B-\Phi_B]=\partial_B\Phi_A-\partial_A\Phi_B+[\Phi_A,\Phi_B]=0 | expression | |
{\sum_{j=1}^{m+1} \sum_{k=0}^{\infty} E^{(-2k-1)}_{jj}u^{(-2k-1)}_j} | expression | |
\{\gamma ^{\mu },\gamma ^{\nu }\}=2g^{\mu \upsilon }, \label{Dirac} | expression | |
\label{tension}T_{Dp}= \frac{1}{l_s^{p+1} g_s} | expression | |
%V(x) = \frac{3}{4}\frac{1}{\cos^2 x}-\frac{1}{4}.%\label{potential}% | expression | |
S_{ij}(i\pi -\theta )=S_{ij}(h\theta _{h}+H\theta _{H}-\theta)=\prod\limits_{x=1}^{h}\prod\limits_{y=1}^{H}\left\{ x+h,y+H\right\}_{\theta }^{-\mu _{ij}(x+h,y+H)}\,\,. \label{133} | expression | |
C(r)=\ln(2\sqrt{2\pi}e^{\gamma +1} r)\quad . \label{l3} | expression | |
\omega_\xi(X,Y)=\langle[X,Y],\xi\rangle=-\langle Y,[X,\xi]\rangle | expression | |
\bar\delta S^{(2)}=\int_{t_0}^tdt\int dx\,\epsilon^{\mu\nu}\frac{\epsilon^{\rho\lambda}}{\sqrt{-g}}\,D_\rho\partial_\mu\phi\,\bar\delta\bigl(D_\lambda\partial_\nu\phi\bigr)\,,\label{4.2} | expression | |
\sum_{\pi(1,2,...n)} \mbox{Tr}\Big[\tau^{(l_0)} \tau^{(l_{\pi(1)})} .....\tau^{(l_{\pi(n)})} \Big]\frac{1}{\tilde{x}_0 - \tilde{x}_{\pi(1)}}\frac{1}{\tilde{x}_{\pi(1)} - \tilde{x}_{\pi(2)}} .....\frac{1}{\tilde{x}_{\pi(n)} - \tilde{x}_0} \; , | expression | |
\beta F = - \frac{N}{\beta^3 } \int d \theta d \phi \int_{r_+ + h}^L dr \frac{\sqrt{g_4}}{ ( - g^{'}_{tt } )^2 } = - N \int_0^\beta d \tau \int d \theta d \phi \int_{r_+ + h}^L dr \sqrt{g_4} \frac{1}{ \beta_{local}^4}, \label{free} | expression | |
\delta \phi = \delta \phi_0 + \delta \phi_{-} t^{-1}, | expression | |
\begin{array}{rcl}\omega(\lambda)\wedge\omega(\lambda) & = & 0 \,, \\d \omega(\lambda) & = & 0 \,.\end{array} | expression | |
\label{ctildeocho}\begin{array}{rcl}{\tilde C}^{(8)\prime}_{\mu_1\dots\mu_7 z}&=&(i_k N^{(8)})_{\mu_1\dots\mu_7}+7(i_k N^{(7)})_{[\mu_1\dots\mu_6}(C^{(1)}_{\mu_7]}-C^{(1)}_z \frac{g_{z\mu_7]}}{g_{zz}})\\& & \\& & +35(C^{(3)}_{[\mu_1\dots\mu_3}-3C^{(3)}_{[\mu_1\mu_2 z}\frac{g_{z\mu_3}}{g_{zz}})C^{(3)}_{\mu_4\mu_5 z}C^{(... | expression | |
\frac{\delta ^{P}I[\phi +\psi ]}{\delta \eta } | expression | |
E_C^{(2)\;reg}(R\to\infty,\Lambda)=-\frac{\Lambda}{2\pi}\ln\left[\sinh\left(R\sqrt{\Lambda^2+\omega_0^2}\right)\right]+\frac{1}{2\pi}\int\limits_{0}^{\Lambda}dy\left(R\sqrt{y^2+\omega_0^2}-\ln 2\right)-\frac{\omega_0^2}{4}.\label{b3.17} | expression | |
S= \int dt d^2x\left\{\mbox{ Tr }(2Kg^{-1}\dot g)-\frac{\kappa}{2}\epsilon^{ij}A^A_i \dot A^A_j-{\cal H}+A_0^AG^A\right\} | expression | |
\lambda tK_{\left| n\right| }(\mu t)I_{\left| n\right| }(\mu t)q_{-i}q_i.\label{dh} | expression | |
I_{\mu}^{\ \nu}\equiv \left[ \delta_\mu^\nu + \phi^{-1}\phi_{\mu\alpha}h^{\alpha\nu}\right] \;,\quad I^{-1}\ _\mu^{\ \alpha} I_\alpha^{\ \nu} =\delta_\mu^\nu \;,\quad I\equiv detI_\mu^{\ \nu}\; . \label{defI} | expression | |
\label{hyper1}(1+z)(1-z) \frac{d^2}{dz^2} u(z) + \left( (c-2)z + c -2a -2b \right)\frac{d}{dz}u(z) + \frac{2ab}{1+z} = 0, | expression | |
\eta_{11}= - {4\Gamma(d-2) \over \Gamma(2-{d \over 2})\Gamma({d \over 2}-1)\Gamma({d \over 2}-2)\Gamma({d \over 2}+1)}, | expression | |
f_{\rm em}=-{\pi^2\over 240}{1\over a^4}.\label{casimirclassic} | expression | |
\Psi(\theta)= \left( \begin{array}{lr} \psi_{n,m-1}(\theta) \\ \psi_{n,m}(\theta) \\ \psi_{n,m-1}(\theta) \\ \psi_{n,m}(\theta) \\ \,\,\,\,\,\,\,\vdots\\ \psi_{n,m-1}(\theta) \\ \end{array} \right)_{(2k+1) \times 1} | expression | |
N=T^a \mbox{\tiny $\wedge$} T_a - R_{ab}\mbox{\tiny $\wedge$}e^a\mbox{\tiny $\wedge$} e^b.\label{NY} | expression | |
{\cal L}_{\rm hyp.kin.} \rightarrow {\cal L}_{\rm hyp.kin.} - \frac32 \sigma'' \phi^*_i \phi^i. | expression | |
\left\{ A,B\right\} _{{\footnotesize PB}}=\sum_{i}\frac{\partial\left(A,B\right) }{\partial\left( x_{i},p_{i}\right) }=-\left\{ B,A\right\}_{{\footnotesize PB}}\;. | expression | |
\{C^a,g,\Theta_{\mu},\partial_{\mu}\Theta_{\nu},\ldots\}, | expression | |
t_2 \frac{t_0^4}{4!}+ 2 \frac{t_1^2}{2!} \frac{t_0^3}{3!} | expression | |
[\frac{1}{u^{3}}\partial _{u}(u^{3}\partial _{u})-\frac{Nk^{2}}{u^{2}}-k'^{2}-\frac{l(l+2)}{u^{2}}]\tilde{\varphi }(u)=0 | expression | |
\hat{K}=\left(\begin{array}{ccc} \frac{B}{4} \delta^{\mu \nu \alpha \beta} & 0 & 0\\ 0 & -\frac{B}{16} & -\frac{B_1}{4} \\ 0 & -\frac{B_1}{4} & \frac{B_1^2}{2B} -A\end{array}\right) | expression | |
\pmatrix{w&x\cr y&z\cr} \label{eq:soq-lie} | expression | |
[\tilde{E}_{r},P_{+}]=0,\quad [\tilde{E}_{r},P_{s}]=\delta_{rs}P_{+},\quad [\tilde{E}_{3},P_{+}]=P_{+},\quad [\tilde{E}_{3},P_{s}]=0 | expression | |
\label{fl1}F(H*)=\int_{C} d\lambda \tilde{F} (\lambda) \exp(\lambda H*) | expression | |
G_0(\tau):=\left(\frac{\theta^2_{D_r}(\tau)}{\eta(\tau)^{r+1}}\right)^{24}=\left(\frac{\eta(\frac{2\tau}{h})}{\eta(\frac{\tau}{2})\eta(\frac{\tau}{h})}\right)^{24}. | expression | |
F_{\rm cl.}=-\frac{1}{2}\sum_{i,j}e_ie_j\frac{1}{2\pi}\ln\vert\vec x_i- \vec x_j \vert | expression | |
V(x-y)=\frac{1}{N}\sum^{N-1}_{n=1} e^{i 2\pi n (x-y)/N}\frac{1}{4\sin^2\frac{\pi n}{N}}\label{cpo} | expression | |
Z = \int\! {\cal D}A_{\mu}\, {\cal D}B_{\mu}\, e^{iS}\, , | expression | |
{\cal W}(M_{\alpha})=\int_{M_{\alpha}}H_{\alpha}^2dA_{\alpha}+\int_{\partial M_{\alpha}}\kappa^{\alpha} ds=\bar{\cal W}(\bar{M}_{\alpha})=\int_{\bar{M}_{\alpha}}\left(\bar{H}_{\alpha}^2+\bar{R}_{\alpha}\right) d\bar{A}_{\alpha}+\int_{\partial\bar{M}_{\alpha}}\bar{\kappa}^{\alpha}d\bar{s}, | expression | |
\label{solution1}ds^2 = e^{- 2 k |y|} \eta_{\mu \nu} dx^{\mu} dx^{\nu} + d y^2, | expression | |
M^2\equiv\left.\frac{ \langle \psi |H|\psi \rangle}{\langle\psi |\psi \rangle}\right|_{min}= 1 +\frac{2\pi}{\sqrt{3}}\sqrt{ m^2 - \frac{3}{4\pi^2}\theta_{eff}^2},\label{eq:m2var} | expression | |
S = - \; \frac{1}{4}\; \int d^{4}x (G_{\mu\nu}(A), G^{\mu\nu}(A))\label{eq:action} | expression | |
\label{psej}{\cal J}^m_n{}^\ddag={\cal J}^m_n. | expression | |
W_\alpha \left( y,\theta \right) =i\lambda _\alpha -\theta_\alpha D -\frac{i}{2}\left( \theta\sigma ^\mu\bar \sigma ^\nu \right) _\alpha F_{\mu\nu} -\theta^2 (\not\!\nabla \bar\lambda)_\alpha \label{6} | expression | |
[\hat{x}^{\prime\mu},\hat{x}^{\prime\nu}] =i(1+2\varepsilon)\sigma^{\mu\nu}+\mathcal{O}(\varepsilon^2). | expression | |
h^{0}(z, V|_{z}(-1)) = \sum_{i=1}^{16} m_i.\label{3.31} | expression | |
\label{gl76}f^c(B,a) = (D_i(B) + 2D_0^{-1}(B) \hat{F}_{0i} (B))^{cd} a_i^d\equiv (N_i(B) a_i)^c. | expression | |
\lambda_{6,H_1} = \frac1{\lambda_{6,H_2}}. | expression | |
B^{\mu\nu}(s)=-\frac{1}{32\pi^{2}}\Lambda^{s/4\pi}(g^{\mu\nu}-\frac{q^{\mu}q^{\nu}}{q^{2}})\int_{0}^{1}dx\int_{0}^{\Lambda}dm\; m^{-1-s/4\pi}\ln{(1+\frac{x(1-x)q^{2}}{m})} | expression | |
{\cal L}_{inter}=-\frac{\lambda_3}{4}\left(\phi\phi^*-\eta^2_1\right)\left(\chi\chi^*-\eta^2_2\right) | expression | |
\label{Axial_Anomaly}\bar\theta\theta\,\langle \partial_\mu \Big(\bar\Psi \gamma^\mu\gamma_5\Psi\Big)\rangle= - \bar\theta\theta\,\frac{1}{8\pi^2} F_{\mu\nu}\tilde F^{\mu\nu}. | expression | |
dM = TdS + \Omega_HdJ + \Phi_HdQ + \Psi_HdP. | expression | |
\exp\{\frac{i}{\hbar} n \hat{\rm T}_b \} \tilde{\psi}_{\pm}\exp\{- \frac{i}{\hbar} n \hat{\rm T}_b \} =\tilde{\psi}_{\pm} | expression | |
\left({\partial^2\over\partial t^2}-\nabla^2\right)G(x,x')=-\delta(x-x'),\label{gfneqn} | expression | |
S^{\alpha } = \pm C \, {}^{-}\eta ^{\alpha }_{\ ab}e^a \wedge e^b \label S | expression | |
{\cal H}_{\mu\nu} = c_1 F_{\mu\nu} +c_2 \tilde F_{\mu\nu} +{ c_3\over 2} \epsilon_{\mu\nu\alpha\beta} F_{\alpha\beta}+{c_4 \over 2} \epsilon_{\mu\nu\alpha\beta} \tilde F_{\alpha\beta}\,; | expression | |
\eta=\sum\limits_{\omega \in P}\frac{2n_\omega }{(\omega ,\omega)}\omega\ ,\ \ \ \ n_\omega \in {\bf Z} | expression | |
\overline{(\Delta n)^2}=\frac{\sinh\alpha}{\alpha} \, \, [n]\,[1+\, \sigma \, n] \ \ . | expression | |
\frac{1}{2}s_n=t_{n,P}~,~~~~~~~~~~~~~n=0,1,2,\cdots.\label{eq:11} | expression | |
\sigma_l = \frac{2\pi^{\frac{d-1}{2}}\Gamma \left( l+d-2 \right) \left( l+d/2 - 1\right)}{\omega^{d-1}\Gamma \left( d/2 -1/2\right) \Gamma \left( l+1 \right)} P(l)\,, | expression | |
\label{4.19} \hat{\rho}^H_{\epsilon}(\beta)=\hat{\rho}^R_{\mu\epsilon_x} (2\pi\mu\alpha)~~~ . | expression | |
c_{1}(B)= 3l- \sum_{r=1}^{8} E_{i}, \qquad c_{2}(B)= 11\label{eq:x31} | expression | |
\label{eq:massmatrix} m \equiv \left(\partial_{\underline{y}}\Lambda \right) \Lambda^{-1}= {\textstyle\frac{1}{2}}m^{i}T_{i} ={\textstyle\frac{1}{2}}\left(\begin{array}{cc}m^{1} & m^{2}+m^{3} \\& \\m^{2}-m^{3} & -m_{1} \\\end{array}\right)\, . | expression | |
{\det}_{(\pi)}(\exp Q):={\det}_{(\pi)}(A\exp Q)\big/{\det}_{(\pi)}(A),\label{B280} | expression | |
N + \frac{1}{2} \tilde N = 16 - 2(h_1 f_2 - h_2 f_1)\leq 16. | expression | |
\left\{\begin{array}{rcl}\tilde{ds}^{2} & = & W^{-\frac{2}{1+a^{2}}} dt^{2}-W^{+\frac{2}{1+a^{2}}}d\vec{x}^{2}\, ,\\& &\\e^{\varphi} & = & W^{+\frac{a}{1+a^{2}}}\, ,\\& &\\F_{\underline{i}\underline{j}} & = & \mp\sqrt{\frac{2}{1+a^{2}}}\\epsilon_{ijk}\partial_{\underline{k}}W\, .\end{array}\right. | expression | |
\delta \phi^i = Q^i_\alpha \epsilon^\alpha, | expression | |
\left( - \frac{\partial}{{\partial}z_{+}} + z_{+} \right) u_2^{+} =i \sqrt{\frac{2}{\Delta}} e^{i\frac{\pi}{4}} u_1^{+}, | expression | |
{\partial V_i^a \over \partial t} = {1\over 2} \epsilon_{ijk}[V_j,V_k]^a,\label{ajs} | expression | |
\label{e25}-{\bf I}\partial\sb\tau\sp2+ge\sb0F\partial\sb\tau=\exp\left(\frac {ge\sb0}{2}F\tau\right)\left(-{\bf I}\partial\sb\tau\sp2+\frac{g\sp2e\sb0\sp2}{4}F\sp2\right)\exp\left(-\frac{ge\sb0}{2}F\tau\right)\,, | expression | |
S_{E}[X]\ -\ S_{E}[UX]\ =\ i2\pi n\quad\mbox{ with\ $n \in{\bf Z}$}\ .\label{CS} | expression | |
%\theta = {\bf p} d{\bf X} \pm {\alpha^2\over m}%(p_0 + p_1) d \left({p_2\over%p_0 +p_1}\right).% | expression | |
\bar{\partial} u^\psi = -{\lambda\over\pi} h_{\psi } + F\left[{\bf h},{\bf u}, u^\psi\right] , | expression | |
q_s^2{\partial^2 H\over\partial q_i\partial q_{i + 2}} =-(cq_s/(2\Delta x))^2 \neq 0\quad\hbox{and}\quad{\partial^2 H\over\partial p_i\partial p_{i + 2}} = 0,\label{Eq:KGHamrl} | expression | |
G_c = G_R \cdot ( 1 + 2 f') - ( 1 + 2 f') \cdot G_A - 2 G_R \cdot \Sigma_{off} \cdot G_A , \label{mom1} | expression | |
\phi_b(r)=\left\{\begin{array}{ll} \phi_t & \mbox{$0<r<R-\Delta R$} \\\phi_{\rm wall}(r-R) & \mbox{$R-\Delta R<r<R+\Delta R$} \\\phi_f & \mbox{$R+\Delta R<r<\infty$}\end{array} \right. \;,\label{bounce} | expression |
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