かろん / Karonの投稿

$$ \boxed{ \begin{aligned} \mathcal{Z}[J] &= \frac{1}{\mathcal{N}} \int \exp\Bigg\{ i \int_{\mathbb{R}^{4}} \Bigg[ -\frac14F_{\mu\nu}F^{\mu\nu} + \bar{\psi} \left( i\gamma^\mu D_\mu -m \right) \psi + \frac12 \partial_\mu\phi\, \partial^\mu\phi - \frac{m^2}{2}\phi^2 - \frac{\lambda}{4!}\phi^4 + J(x)\phi(x) \Bigg] \,d^4x \Bigg\} \mathcal{D}\phi\, \mathcal{D}\psi\, \mathcal{D}\bar{\psi} \\[1.2em] \langle \phi(x_1)\cdots\phi(x_n) \rangle &= \left. \frac{1}{\mathcal{Z}[0]} \frac{\delta^n\mathcal{Z}[J]} {\delta J(x_1)\cdots\delta J(x_n)} \right|_{J=0} \\[1.2em] R_{\mu\nu} - \frac12 Rg_{\mu\nu} + \Lambda g_{\mu\nu} &= \frac{8\pi G}{c^4} T_{\mu\nu} \\[1.2em] \frac{\partial u}{\partial t} &= \sum_{i,j=1}^{n} a_{ij}(x) \frac{\partial^2u} {\partial x_i\partial x_j} + \sum_{i=1}^{n} b_i(x) \frac{\partial u}{\partial x_i} + c(x)u + f(x,t) \\[1.2em] \mathcal{L}(\theta) &= -\sum_{i=1}^{N} \log \left( \frac{ e^{z_{y_i}} }{ \sum_{k=1}^{K} e^{z_k} } \right) + \lambda \lVert \theta \rVert_2^2 \\[1.2em] \hat{f}(\xi) &= \int_{\mathbb{R}^{n}} f(x) e^{-2\pi i x\cdot\xi} \,dx = \sum_{k=-\infty}^{\infty} c_ke^{ikx} = \zeta(s) = \sum_{n=1}^{\infty} \frac1{n^s} \end{aligned} } $$