Kleinert · 제10장 폴리머

$$ \mathcal{A}=\int_{t_{a}}^{t_{b}} d t \frac{M}{2}\left(\dot{x}^{i}\right)^{2}, \quad i=1,2,3 $$

$$ \boldsymbol{\nabla} \times[\boldsymbol{\nabla} \times \mathbf{A}(\mathbf{x})]=\mathbf{j}(\mathbf{x}) $$

$$ d x^{\alpha}=h_{\lambda}^{\alpha}(q) d q^{\lambda}, \quad \alpha=0,1,2,3 $$

$$ \mathcal{A}_{\mathrm{tot}}[q]=\int_{0}^{\beta} d \tau\left[\frac{1}{2} Z(\tau) \dot{q}^{2}(\tau)-\frac{1}{2} \delta(0) \log Z(\tau)\right] $$

$$ x^{i}=x^{i}(q) $$

$$ \mathbf{A}(\mathbf{x}) \rightarrow \mathbf{A}^{\prime}(\mathbf{x})=\mathbf{A}(\mathbf{x})+\nabla \Lambda(\mathbf{x}) $$

$$ g_{\alpha \beta}=h_{\alpha}^{\mu}(q) h_{\beta}^{\nu}(q) g_{\mu \nu}(q)=\eta_{\alpha \beta} $$

$$ J=e^{(1 / 2) \delta(0) \int_{0}^{\beta} d \tau \log Z(\tau)} $$

$$ \mathcal{A}=\int_{t_{a}}^{t_{b}} d t \frac{M}{2} g_{\mu \nu}(q) \dot{q}^{\mu} \dot{q}^{\nu} $$

$$ \left(\partial_{i} \partial_{j}-\partial_{j} \partial_{i}\right) \Lambda(\mathbf{x})=0 $$

$$ h_{\alpha}{ }^{\mu}(q) \equiv \eta_{\alpha \beta} g^{\mu \nu}(q) h^{\beta}{ }_{\nu}(q) $$

$$ \mathcal{A}^{(0)}=\int_{0}^{\beta} d \tau \frac{1}{2} \dot{q}^{2}(\tau), \quad \mathcal{A}^{\mathrm{int}}=\int_{0}^{\beta} d \tau \frac{1}{2}[Z(\tau)-1] \dot{q}^{2}(\tau) $$

$$ g_{\mu \nu}(q)=\partial_{\mu} x^{i}(q) \partial_{\nu} x^{i}(q) $$

$$ \left(\partial_{i} \partial_{j}-\partial_{j} \partial_{i}\right) \mathbf{A}(\mathbf{x})=0 $$

$$ h_{\alpha}{ }^{\mu} h^{\beta}{ }_{\mu}=\delta_{\alpha}{ }^{\beta}, \quad h^{\alpha}{ }_{\mu} h_{\alpha}{ }^{\nu}=\delta_{\mu}{ }^{\nu} . $$

$$ \begin{align*} (0 \beta \mid 00) & =J \int \mathcal{D} q(\tau) e^{-\mathcal{A}^{(0)}[q]-\mathcal{A}_{\mathrm{int}}[q]}=J \int \mathcal{D} q(\tau) e^{-\mathcal{A}^{(0)}[q]}\left(1-\mathcal{A}_{\mathrm{int}}+\frac{1}{2} \mathcal{A}_{\mathrm{int}}^{2}-\ldots\right) \\ & =(2 \pi \beta)^{-1 / 2} J\left[1-\left\langle\mathcal{A}_{\mathrm{int}}\right\rangle+\frac{1}{2}\left\langle\mathcal{A}_{\mathrm{int}}^{2}\right\rangle-\ldots\right] \\ & =(2 \pi \beta)^{-1 / 2} J e^{-\left\langle\mathcal{A}_{\mathrm{int}}\right\rangle_{c}+\frac{1}{2}\left\langle\mathcal{A}_{\mathrm{int}}^{2}\right\rangle_{c}-\ldots} \end{align*} $$

$$ l=\int_{t_{a}}^{t_{b}} d t \sqrt{g_{\mu \nu}(q) \dot{q}^{\mu} \dot{q}^{\nu}} $$

$$ \mathbf{A}(\mathbf{x})=\int d^{3} x^{\prime} \frac{\mathbf{j}\left(\mathbf{x}^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|} $$

$$ g_{\mu \nu}(q)=h^{\alpha}{ }_{\mu}(q) h^{\beta}{ }_{\nu}(q) \eta_{\alpha \beta} . $$

$$ \left\langle\mathcal{A}_{\mathrm{int}}\right\rangle_{c}=\int d \tau \frac{1}{2}[Z(\tau)-1] \cdot \Delta^{\cdot}(\tau, \tau)=-\frac{1}{2} \delta(0) \int d \tau[1-Z(\tau)] $$

$$ \partial_{t}\left(g_{\mu \nu} \dot{q}^{\nu}\right)-\frac{1}{2} \partial_{\mu} g_{\lambda \nu} \dot{q}^{\lambda} \dot{q}^{\nu}=g_{\mu \nu} \ddot{q}^{\nu}+\bar{\Gamma}_{\lambda \nu \mu} \dot{q}^{\lambda} \dot{q}^{\nu}=0 $$

$$ \mathbf{B}(\mathbf{x})=\int d^{3} x^{\prime} \frac{\mathbf{j}\left(\mathbf{x}^{\prime}\right) \times \mathbf{R}^{\prime}}{R^{\prime 3}}, \quad \mathbf{R}^{\prime} \equiv \mathbf{x}^{\prime}-\mathbf{x} $$

$$ {e^{a}}_{\mu}(q)=e^{a}{ }_{\alpha}(q) h^{\alpha}{ }_{\mu}(q), $$

$$ \begin{align*} \left\langle\mathcal{A}_{\mathrm{int}}^{2}\right\rangle_{c} & =\iint d \tau_{1} d \tau_{2} \frac{1}{2}(Z-1)_{1} \frac{1}{2}(Z-1)_{2} 2 \cdot \Delta^{\cdot}\left(\tau_{1}, \tau_{2}\right) \cdot \Delta^{\cdot}\left(\tau_{2}, \tau_{1}\right) \\ & \rightarrow \iint d^{d} x_{1} d^{d} x_{2} \frac{1}{2}(Z-1)_{1} \frac{1}{2}(Z-1)_{2} 2_{\mu} \Delta_{\nu}\left(x_{1}, x_{2}\right)_{\nu} \Delta_{\mu}\left(x_{2}, x_{1}\right) \\ & =\iint d^{d} x_{1} d^{d} x_{2} \frac{1}{2}(Z-1)_{1} \frac{1}{2}(Z-1)_{2} 2 \Delta_{\mu \mu}\left(x_{2}, x_{1}\right) \Delta_{\nu \nu}\left(x_{1}, x_{2}\right) \end{align*} $$

$$ \bar{\Gamma}_{\lambda \nu \mu} \equiv \frac{1}{2}\left(\partial_{\lambda} g_{\nu \mu}+\partial_{\nu} g_{\lambda \mu}-\partial_{\mu} g_{\lambda \nu}\right) $$

$$ \mathbf{j}(\mathbf{x})=I \boldsymbol{\delta}(\mathbf{x} ; L) $$

$$ \Lambda^{a}{ }_{\alpha}(q)=e^{a}{ }_{\alpha}(q), $$

$$ \left\langle\mathcal{A}_{\mathrm{int}}^{2}\right\rangle_{c}=\frac{1}{2} \iint d \tau_{1} d \tau_{2} z_{1} z_{2} \delta^{2}\left(\tau_{1}, \tau_{2}\right) $$

$$ \bar{\Gamma}_{\lambda \nu}^{\mu} \equiv g^{\mu \sigma} \bar{\Gamma}_{\lambda \nu \sigma}, $$

$$ \boldsymbol{\delta}(\mathbf{x} ; L)=\int_{L} d \mathbf{x}^{\prime} \delta^{(3)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right) $$

$$ \eta_{a b} \Lambda_{\alpha}^{a}(q) \Lambda^{b}{ }_{\beta}(q)=\eta_{\alpha \beta} . $$

$$ \begin{align*} & \left\langle\mathcal{A}_{\text {int }}^{3}\right\rangle_{c}=\iiint d \tau_{1} d \tau_{2} d \tau_{3} \frac{1}{2} z_{1} \frac{1}{2} z_{2} \frac{1}{2} z_{3} 8^{\cdot} \Delta^{\cdot}\left(\tau_{1}, \tau_{2}\right) \Delta^{\cdot}\left(\tau_{2}, \tau_{3}\right) \Delta^{\cdot}\left(\tau_{3}, \tau_{1}\right) \\ & \quad \rightarrow \iiint d^{d} x_{1} d^{d} x_{2} d^{d} x_{3} \frac{1}{2} z_{1} \frac{1}{2} z_{2} \frac{1}{2} z_{3} 8_{\mu} \Delta_{\nu}\left(x_{1}, x_{2}\right)_{\nu} \Delta_{\sigma}\left(x_{2}, x_{3}\right)_{\sigma} \Delta_{\mu}\left(x_{3}, x_{1}\right) \\ & \quad=-\iiint d^{d} x_{1} d^{d} x_{2} d^{d} x_{3} \frac{1}{2} z_{1} \frac{1}{2} z_{2} \frac{1}{2} z_{3} 8 \Delta_{\mu \mu}\left(x_{3}, x_{1}\right) \Delta_{\nu \nu}\left(x_{1}, x_{2}\right) \Delta_{\sigma \sigma}\left(x_{2}, x_{3}\right) \end{align*} $$

$$ \ddot{q}^{\mu}+\bar{\Gamma}_{\lambda \nu}^{\mu} \dot{q}^{\lambda} \dot{q}^{\nu}=0 . $$

$$ \boldsymbol{\nabla} \cdot \boldsymbol{\delta}(\mathbf{x} ; L)=0 $$

$$ \left(\partial_{\mu} \partial_{\nu}-\partial_{\nu} \partial_{\mu}\right) h_{\lambda}^{\alpha}(q)=0 $$

$$ \left\langle\mathcal{A}_{\mathrm{int}}^{3}\right\rangle_{c}=\iiint d \tau_{1} d \tau_{2} d \tau_{3} z_{1} z_{2} z_{3} \delta\left(\tau_{1}, \tau_{2}\right) \delta\left(\tau_{2}, \tau_{3}\right) \delta\left(\tau_{3}, \tau_{1}\right) $$

$$ \ddot{x}^{i}=0 $$

$$ \boldsymbol{\nabla} \cdot \boldsymbol{\delta}(\mathbf{x} ; L)=\delta\left(\mathbf{x}_{2}\right)-\delta\left(\mathbf{x}_{1}\right) . $$

$$ \Omega_{\alpha \beta}^{\gamma}(q)=\frac{1}{2} h_{\alpha}^{\mu}(q) h_{\beta}^{\nu}(q)\left[\partial_{\mu} h_{\nu}^{\gamma}(q)-\partial_{\nu} h_{\mu}^{\gamma}(q)\right] . $$

$$ -\left\langle\mathcal{A}_{\mathrm{int}}\right\rangle_{c}+\frac{1}{2}\left\langle\mathcal{A}_{\mathrm{int}}^{2}\right\rangle_{c}-\frac{1}{3!}\left\langle\mathcal{A}_{\mathrm{int}}^{3}\right\rangle_{c}+\ldots=\frac{1}{2} \sum_{1}^{\infty}(-1)^{n} \frac{c_{n}}{n}, $$

$$ \ddot{x}^{i}=\frac{\partial x^{i}}{\partial q^{\mu}} \ddot{q}^{\mu}+\frac{\partial^{2} x^{i}}{\partial q^{\lambda} \partial q^{\nu}} \dot{q}^{\lambda} \dot{q}^{\nu}=0 . $$

$$ \mathbf{A}(\mathbf{x})=I \int_{L} d \mathbf{x}^{\prime} \frac{1}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|} $$

$$ \stackrel{h}{K}_{\alpha \beta}^{\gamma}(q)=\Omega_{\alpha \beta}^{\gamma}(q)-\Omega_{\beta}^{\gamma}{ }_{\alpha}(q)+\Omega_{\alpha \beta}^{\gamma}(q), $$

$$ c_{n}=\int d \tau_{1} \ldots d \tau_{n} C\left(\tau_{1}, \tau_{2}\right) C\left(\tau_{2}, \tau_{3}\right) \ldots C\left(\tau_{n}, \tau_{1}\right) $$

$$ e_{\mu}^{i}(q) \equiv \frac{\partial x^{i}}{\partial q^{\mu}}, \quad e_{i}^{\mu}(q) \equiv \frac{\partial q^{\mu}}{\partial x^{i}} $$

$$ \mathbf{B}(\mathbf{x})=-I \int_{L} \frac{d \mathbf{x}^{\prime} \times \mathbf{R}^{\prime}}{R^{\prime 3}}, \quad \mathbf{R}^{\prime} \equiv \mathbf{x}^{\prime}-\mathbf{x} $$

$$ \Gamma_{\alpha \beta}{ }^{\gamma}=h^{\gamma}{ }_{\lambda} h_{\alpha}{ }^{\mu} h_{\beta}{ }^{\nu}\left(K_{\mu \nu}{ }^{\lambda}-\stackrel{h}{K}_{\mu \nu}{ }^{\lambda}\right), $$

$$ C\left(\tau, \tau^{\prime}\right)=[Z(\tau)-1] \delta\left(\tau, \tau^{\prime}\right) $$

$$ e_{i}{ }^{\mu} e^{i}{ }_{\nu}=\delta^{\mu}{ }_{\nu}, \quad e_{i}{ }^{\mu} e^{j}{ }_{\mu}=\delta_{i}{ }^{j} . $$

$$ \Omega(\mathbf{x} ; S)=\int_{S} \frac{d \mathbf{S}^{\prime} \cdot \mathbf{R}^{\prime}}{R^{\prime 3}} $$

$$ v_{\alpha}(q)=v_{\mu}(q) h_{\alpha}{ }^{\mu}(q), \quad v^{\alpha}(q)=v^{\mu}(q) h^{\alpha}{ }_{\mu}(q) . $$

$$ c_{n}=\iint d \tau_{1} d \tau_{n}\left[Z\left(\tau_{1}\right)-1\right]^{n} \delta^{2}\left(\tau_{1}-\tau_{n}\right)=\delta(0) \int d \tau[Z(\tau)-1]^{n} $$

$$ g_{\mu \nu}(q)=e_{\mu}^{i}(q) e_{\nu}^{i}(q) $$

$$ \mathbf{B}(\mathbf{x} ; S)=I \nabla \Omega(\mathbf{x} ; S) $$

$$ D_{\alpha} v_{\beta}(q)=\partial_{\alpha} v_{\beta}(q)-\Gamma_{\alpha \beta}^{\gamma}(q) v_{\gamma}(q), \quad D_{\alpha} v^{\beta}(q)=\partial_{\alpha} v^{\beta}(q)+\Gamma_{\alpha \gamma}{ }^{\beta}(q) v^{\gamma}(q) . $$

$$ \begin{align*} -\left\langle\mathcal{A}_{\text {int }}\right\rangle_{c}+\frac{1}{2}\left\langle\mathcal{A}_{\text {int }}^{2}\right\rangle_{c}-\frac{1}{3!}\left\langle\mathcal{A}_{\text {int }}^{3}\right\rangle_{c}+\ldots & =\frac{1}{2} \delta(0) \int d \tau \sum_{1}^{\infty}(-1)^{n} \frac{[Z(\tau)-1]^{n}}{n} \\ & =-\frac{1}{2} \delta(0) \int d \tau \log Z(\tau) \end{align*} $$

$$ \ddot{q}^{\mu}+e_{i}{ }^{\mu} \partial_{\lambda} e^{i}{ }_{\kappa} \dot{q}^{\kappa} \dot{q}^{\lambda}=0 . $$

$$ \nabla \Omega(\mathbf{x} ; S)=\int_{S} d S_{k}^{\prime} \nabla \frac{R_{k}^{\prime}}{R^{\prime 3}}=-\int_{S} d S_{k}^{\prime} \nabla^{\prime} \frac{R_{k}^{\prime}}{R^{\prime 3}} $$

$$ \Gamma_{\lambda \kappa}{ }^{\mu}=e_{i}{ }^{\mu} \partial_{\lambda} e^{i}{ }_{\kappa} . $$

$$ \nabla \Omega(\mathbf{x} ; S)=-\left[\int_{S}\left(d S_{k}^{\prime} \partial_{i}^{\prime} \frac{R_{k}^{\prime}}{R^{\prime 3}}-d S_{i}^{\prime} \partial_{k}^{\prime} \frac{R_{k}^{\prime}}{R^{\prime 3}}\right)+\int_{S} d S_{i}^{\prime} \partial_{k}^{\prime} \frac{R_{k}^{\prime}}{R^{\prime 3}}\right] $$

$$ \Gamma_{\lambda \kappa}{ }^{\mu}=-e^{i}{ }_{\kappa} \partial_{\lambda} e_{i}{ }^{\mu} . $$

$$ \int_{S}\left(d S_{k} \partial_{i}-d S_{i} \partial_{k}\right) f(\mathbf{x})=\epsilon_{k i l} \int_{L} d x_{l} f(\mathbf{x}) $$

$$ \ddot{q}^{\mu}+\Gamma_{\kappa \lambda}{ }^{\mu} \dot{q}^{\kappa} \dot{q}^{\lambda}=0 $$

$$ \boldsymbol{\nabla} \Omega(\mathbf{x} ; S)=-\left[\int_{L} \frac{d \mathbf{x}^{\prime} \times \mathbf{R}^{\prime}}{R^{\prime 3}}+4 \pi \int_{S} d \mathbf{S}^{\prime} \delta^{(3)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)\right] $$

$$ \left(\partial_{\lambda} \partial_{\kappa}-\partial_{\kappa} \partial_{\lambda}\right) x^{i}(q)=0 $$

$$ \boldsymbol{\delta}(\mathbf{x} ; S)=\int_{S} d \mathbf{S}^{\prime} \delta^{(3)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right) $$

$$ \partial_{\lambda} e_{\kappa}^{i}-\partial_{\kappa} e_{\lambda}^{i}=0 $$

$$ \nabla \times \delta(\mathbf{x} ; S)=\delta(\mathbf{x} ; L) $$

$$ d x^{i}=e^{i}{ }_{\mu}(q) d q^{\mu} . $$

$$ \nabla \Omega(\mathbf{x} ; S)=-\int d^{3} x^{\prime} \delta_{k}\left(\mathbf{x}^{\prime} ; S\right) \nabla^{\prime} \frac{R_{k}^{\prime}}{R^{\prime 3}} $$

$$ \partial_{\lambda} e_{\kappa}^{i}(q)-\partial_{\kappa} e_{\lambda}^{i}(q) \neq 0 $$

$$ \mathbf{B}_{i}(\mathbf{x} ; S)=-I\left[\int d^{3} x^{\prime}[\nabla \times \boldsymbol{\delta}(\mathbf{x} ; \mathbf{S})] \times \frac{\mathbf{R}^{\prime}}{R^{\prime 3}}+4 \pi \boldsymbol{\delta}\left(\mathbf{x}^{\prime} ; S\right)\right] $$

$$ \left(\partial_{\lambda} \partial_{\kappa}-\partial_{\kappa} \partial_{\lambda}\right) x^{i}(q) \neq 0 $$

$$ \boldsymbol{\delta}(\mathbf{x} ; S) \rightarrow \boldsymbol{\delta}\left(\mathbf{x} ; S^{\prime}\right)=\boldsymbol{\delta}(\mathbf{x} ; S)+\boldsymbol{\nabla} \delta(\mathbf{x} ; V) $$

$$ S_{\lambda \kappa}{ }^{\mu}=\frac{1}{2}\left(\Gamma_{\lambda \kappa}{ }^{\mu}-\Gamma_{\kappa \lambda}{ }^{\mu}\right)=\frac{1}{2} e_{i}{ }^{\mu}\left(\partial_{\lambda} e^{i}{ }_{\kappa}-\partial_{\kappa} e^{i}{ }_{\lambda}\right) . $$

$$ \delta(\mathbf{x} ; V) \equiv \int d^{3} x^{\prime} \delta^{(3)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right) $$

$$ S_{\mu} \equiv S_{\mu \lambda}{ }^{\lambda} $$

$$ \mathbf{B}(\mathbf{x})=I[\nabla \Omega(\mathbf{x} ; S)+4 \pi \delta(\mathbf{x} ; S)] $$

$$ \begin{align*} \Gamma_{\mu \nu \lambda} & =\frac{1}{2}\left\{\left[e_{i \lambda} \partial_{\mu} e_{\nu}^{i}+\partial_{\mu} e_{i \lambda} e_{\nu}^{i}\right]+\left[e_{i \mu} \partial_{\nu} e_{\lambda}^{i}+\partial_{\nu} e_{i \mu} e_{\lambda}^{i}\right]-\left[e_{i \mu} \partial_{\lambda} e_{\nu}^{i}+\partial_{\lambda} e_{i \mu} e_{\nu}^{i}\right]\right\} \\ & +\frac{1}{2}\left\{\left[e_{i \lambda} \partial_{\mu} e_{\nu}^{i}-e_{i \lambda} \partial_{\nu} e_{\mu}^{i}\right]-\left[e_{i \mu} \partial_{\nu} e_{\lambda}^{i}-e_{i \mu} \partial_{\lambda} e_{\nu}^{i}\right]+\left[e_{i \nu} \partial_{\lambda} e_{\mu}^{i}-e_{i \nu} \partial_{\mu} e_{\lambda}^{i}\right]\right\} \end{align*} $$

$$ \delta(V)=-\int_{-\infty}^{z} \delta_{z}(\mathbf{x} ; S) $$

$$ \Gamma_{\mu \nu}{ }^{\lambda}=\bar{\Gamma}_{\mu \nu}{ }^{\lambda}+K_{\mu \nu}{ }^{\lambda}, $$

$$ \nabla \Omega(\mathbf{x} ; L) \equiv \nabla \Omega(\mathbf{x} ; S)+4 \pi \delta(\mathbf{x} ; S) . $$

$$ K_{\mu \nu \lambda} \equiv S_{\mu \nu \lambda}-S_{\nu \lambda \mu}+S_{\lambda \mu \nu} $$

$$ \mathbf{B}(\mathbf{x})=I \nabla \Omega(\mathbf{x} ; L) . $$

$$ \Gamma_{\mu \nu}{ }^{\nu}=\bar{\Gamma}_{\mu \nu}{ }^{\nu} . $$

$$ \left(\partial_{i} \partial_{j}-\partial_{j} \partial_{i}\right) \Omega(\mathbf{x} ; L)=4 \pi \epsilon_{i j k} \delta_{k}(\mathbf{x} ; L) $$

$$ \eta_{a b}=\left(\begin{array}{cccc} 1 & & & \\ & -1 & & \\ & & -1 & \\ & & & -1 \end{array}\right)_{a b}, \quad a, b=0,1,2,3 . $$

$$ \varphi(\mathbf{x})=\arctan \frac{x^{2}}{x^{1}} $$

$$ R_{\mu \nu \lambda}{ }^{\kappa}=\partial_{\mu} \Gamma_{\nu \lambda}{ }^{\kappa}-\partial_{\nu} \Gamma_{\mu \lambda}{ }^{\kappa}-\left[\Gamma_{\mu}, \Gamma_{\nu}\right]_{\lambda}{ }^{\kappa}, \quad \mu, \nu, \ldots=0,1,2,3 . $$

$$ \partial_{i} \varphi(\mathbf{x})=-\epsilon_{i j} \frac{x_{j}}{\left(x^{1}\right)^{2}+\left(x^{2}\right)^{2}} $$

$$ \left[\Gamma_{\mu}, \Gamma_{\nu}\right]_{\lambda}{ }^{\kappa} \equiv\left(\Gamma_{\mu} \Gamma_{\nu}-\Gamma_{\nu} \Gamma_{\mu}\right)_{\lambda}{ }^{\kappa}=\Gamma_{\mu \lambda}{ }^{\sigma} \Gamma_{\nu \sigma}{ }^{\kappa}-\Gamma_{\nu \lambda}{ }^{\sigma} \Gamma_{\mu \sigma}{ }^{\kappa} . $$

$$ \int_{s} d^{2} x\left(\partial_{i} \partial_{j}-\partial_{j} \partial_{i}\right) \varphi(\mathbf{x})=\int_{c} d x_{i} \partial_{i} \varphi(\mathbf{x}) $$

$$ R_{\mu \nu \lambda}{ }^{\kappa}=e_{a}{ }^{\kappa}\left(\partial_{\mu} \partial_{\nu}-\partial_{\nu} \partial_{\mu}\right) e_{\lambda}^{a} . $$

$$ \left(\partial_{1} \partial_{2}-\partial_{2} \partial_{1}\right) \varphi(\mathbf{x})=2 \pi \delta^{(2)}(\mathbf{x}) $$

$$ \left(\partial_{\mu} \partial_{\nu}-\partial_{\nu} \partial_{\mu}\right) \partial_{\lambda} x^{a}(q) \neq 0 $$

$$ j(\mathbf{x})=\sum_{n} I_{n} \delta^{(2)}\left(\mathbf{x}-\mathbf{x}_{n}\right), $$

$$ \bar{R}_{\mu \nu \lambda}{ }^{\kappa}=\partial_{\mu} \bar{\Gamma}_{\nu \lambda}{ }^{\kappa}-\partial_{\nu} \bar{\Gamma}_{\mu \lambda}{ }^{\kappa}-\left[\bar{\Gamma}_{\mu}, \bar{\Gamma}_{\nu}\right]_{\lambda}{ }^{\kappa} . $$

$$ \left(\partial_{1} \partial_{2}-\partial_{2} \partial_{1}\right) f(\mathbf{x})=j(\mathbf{x}) $$

$$ R_{\mu \nu \lambda}{ }^{\kappa}=\bar{R}_{\mu \nu \lambda}{ }^{\kappa}+\bar{D}_{\mu} K_{\nu \lambda}{ }^{\kappa}-\bar{D}_{\nu} K_{\mu \lambda}{ }^{\kappa}-\left[K_{\mu}, K_{\nu}\right]_{\lambda}{ }^{\kappa} . $$

$$ G\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=\frac{1}{2 \pi} \varphi\left(\mathbf{x}-\mathbf{x}^{\prime}\right) $$

$$ \begin{align*} \bar{D}_{\mu} v_{\nu} & \equiv \partial_{\mu} v_{\nu}-\bar{\Gamma}_{\mu \nu}{ }^{\lambda} v_{\lambda} \\ \bar{D}_{\mu} v^{\nu} & \equiv \partial_{\mu} v^{\nu}+\bar{\Gamma}_{\mu \lambda}{ }^{\nu} v^{\lambda} \end{align*} $$

$$ \left(\partial_{1} \partial_{2}-\partial_{2} \partial_{1}\right) G\left(\mathbf{x}-\mathbf{x}^{\prime}\right)=\delta^{(2)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right) $$

$$ \Delta \sigma=g^{\mu \nu} \bar{D}_{\mu} \bar{D}_{\nu} \sigma . $$

$$ f(\mathbf{x})=\int d^{2} \mathbf{x}^{\prime} G\left(\mathbf{x}, \mathbf{x}^{\prime}\right) j(\mathbf{x}) $$

$$ \begin{align*} D_{\mu} v_{\nu} & \equiv \partial_{\mu} v_{\nu}-\Gamma_{\mu \nu}{ }^{\lambda} v_{\lambda} \\ D_{\mu} v^{\nu} & \equiv \partial_{\mu} v^{\nu}+\Gamma_{\mu \lambda}{ }^{\nu} v^{\lambda} \end{align*} $$

$$ \left(\partial_{1}^{2}+\partial_{2}^{2}\right) G_{\Delta}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)=\delta^{(2)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right) $$

$$ D_{\mu} e^{i}{ }_{\nu} \equiv \partial_{\mu} e^{i}{ }_{\nu}-\Gamma_{\mu \nu}{ }^{\lambda} e^{i}{ }_{\lambda}=0, \quad D_{\mu} e_{i}{ }^{\nu} \equiv \partial_{\mu} e_{i}{ }^{\nu}+\Gamma_{\mu \lambda}{ }^{\nu} e_{i}{ }^{\lambda}=0 . $$

$$ f(\mathbf{x})=\frac{1}{2 \pi} \sum_{n} I_{n} \arctan \frac{x^{2}-x_{n}^{2}}{x^{1}-x_{n}^{1}}, $$

$$ R_{\nu \lambda}=R_{\mu \nu \lambda}{ }^{\mu} $$

$$ \left(\partial_{i} \partial_{j}-\partial_{j} \partial_{i}\right) \Lambda(\mathbf{x})=0 $$

$$ R=g^{\nu \lambda} R_{\nu \lambda} $$

$$ \Delta B_{k}(\mathbf{x})=\epsilon_{k i j}\left(\partial_{i} \partial_{j}-\partial_{j} \partial_{i}\right) \Lambda(\mathbf{x}) $$

$$ G_{\mu \nu} \equiv R_{\mu \nu}-\frac{1}{2} g_{\mu \nu} R $$

$$ \mathbf{A}(\mathbf{x})=\boldsymbol{\nabla} \Lambda(\mathbf{x}) $$

$$ d x^{i}= \begin{cases}d q^{1} & \text { for } i=1 \\ d q^{2}+\epsilon \partial_{\mu} \phi(q) d q^{\mu} & \text { for } i=2\end{cases} $$

$$ \Lambda(\mathbf{x})=\Phi \Omega(\mathbf{x} ; L) $$

$$ \phi(q) \equiv \arctan \left(q^{2} / q^{1}\right) $$

$$ B_{k}(\mathbf{x})=\epsilon_{k i j}\left(\partial_{i} \partial_{j}-\partial_{j} \partial_{i}\right) \Lambda(\mathbf{x})=\Phi \delta_{k}(\mathbf{x} ; L) $$

$$ \begin{align*} e_{\mu}^{1} & =\delta_{\mu}^{1} \\ e_{\mu}^{2} & =\delta_{\mu}^{2}+\epsilon \partial_{\mu} \phi(q) \end{align*} $$

$$ \nabla \cdot \mathbf{B}(\mathbf{x})=4 \pi \rho_{\mathrm{m}}(\mathbf{x}) $$

$$ e^{1}{ }_{\lambda} S_{\mu \nu}{ }^{\lambda}=0, \quad e^{2}{ }_{\lambda} S_{\mu \nu}{ }^{\lambda}=\frac{\epsilon}{2}\left(\partial_{\mu} \partial_{\nu}-\partial_{\nu} \partial_{\mu}\right) \phi $$

$$ \frac{1}{2} \epsilon_{i j k}\left(\partial_{i} \partial_{j}-\partial_{j} \partial_{i}\right) A_{k}(\mathbf{x})=4 \pi \rho_{\mathrm{m}}(\mathbf{x}) $$

$$ \int d^{2} q\left(\partial_{1} \partial_{2}-\partial_{2} \partial_{1}\right) \phi=\oint d q^{\mu} \partial_{\mu} \phi=\oint d \phi=2 \pi $$

$$ \epsilon_{i j k} \partial_{i} \partial_{j} \partial_{k} \Lambda(\mathbf{x})=4 \pi \rho_{\mathrm{m}}(\mathbf{x}) $$

$$ e^{2}{ }_{\lambda} S_{12}{ }^{\lambda}=\frac{\epsilon}{2} 2 \pi \delta^{(2)}(q) . $$

$$ \mathbf{B}_{\text {inside }}(\mathbf{x} ; L)=4 \pi g \delta\left(\mathbf{x} ; L_{0}^{\uparrow}\right) $$

$$ b^{i} \equiv \oint_{C^{\prime}} d x^{i}=\oint_{C} d q^{\mu} e_{\mu}^{i} $$

$$ \boldsymbol{\delta}\left(\mathbf{x} ; L_{0}^{\uparrow}\right)=\int_{L_{0}^{\uparrow}, \mathbf{x}_{0}} d^{3} x^{\prime} \delta^{(3)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right) $$

$$ b^{i}=\oint_{S} d^{2} s^{\mu \nu} \partial_{\mu} e^{i}{ }_{\nu}=\oint_{S} d^{2} s^{\mu \nu} e^{i}{ }_{\lambda} S_{\mu \nu}{ }^{\lambda}, $$

$$ \boldsymbol{\nabla} \cdot \boldsymbol{\delta}\left(\mathbf{x} ; L_{0}^{\uparrow}\right)=\delta^{(3)}\left(\mathbf{x}-\mathbf{x}_{0}\right) $$

$$ b^{i}=(0, \epsilon) . $$

$$ \boldsymbol{\delta}\left(\mathbf{x} ; L_{0}^{\uparrow}\right) \rightarrow \boldsymbol{\delta}\left(\mathbf{x} ; L_{0}^{\prime \uparrow}\right)=\boldsymbol{\delta}\left(\mathbf{x} ; L_{0}^{\uparrow}\right)+\boldsymbol{\nabla} \times \boldsymbol{\delta}(\mathbf{x} ; S) $$

$$ b^{\mu} \equiv \oint_{C^{\prime}} d q^{\mu}=\oint_{C} d x^{i} e_{i}^{\mu} $$

$$ \mathbf{A}\left(\mathbf{x} ; L_{0}^{\uparrow}\right)=-g \int d^{3} x^{\prime} \frac{\boldsymbol{\nabla}^{\prime} \times \boldsymbol{\delta}\left(\mathbf{x}^{\prime} ; L_{0}^{\uparrow}\right)}{R^{\prime}}=g \int d^{3} x^{\prime} \boldsymbol{\delta}\left(\mathbf{x}^{\prime} ; L_{0}^{\uparrow}\right) \times \frac{\mathbf{R}^{\prime}}{R^{\prime 3}} $$

$$ \begin{align*} b^{\mu} & =\oint_{S} d^{2} s^{i j} \partial_{i} e_{j}^{\mu}=\oint_{S} d^{2} s^{i j} e_{i}^{\nu} \partial_{\nu} e_{j}^{\mu} \\ & =-\oint_{S} d^{2} s^{i j} e_{i}^{\nu} e_{j}{ }^{\lambda} S_{\nu \lambda}{ }^{\mu} \end{align*} $$

$$ \boldsymbol{\nabla} \times \mathbf{A}\left(\mathbf{x} ; L_{0}^{\uparrow}\right)=-g \int d^{3} x^{\prime} \frac{\nabla^{\prime} \times\left[\nabla^{\prime} \times \boldsymbol{\delta}\left(\mathbf{x}^{\prime} ; L_{0}^{\uparrow}\right)\right]}{R^{\prime}} $$

$$ x^{i}=\delta^{i}{ }_{\mu}\left[q^{\mu}+\frac{\Omega}{2 \pi} \epsilon^{\mu}{ }_{\nu} q^{\nu} \phi(q)\right], $$

$$ -g \int d^{3} x^{\prime} \frac{\boldsymbol{\nabla}^{\prime}\left[\boldsymbol{\nabla}^{\prime} \cdot \boldsymbol{\delta}\left(\mathbf{x}^{\prime} ; L_{0}^{\uparrow}\right)\right]}{R^{\prime}}+g \int d^{3} x^{\prime} \frac{\boldsymbol{\nabla}^{\prime 2} \boldsymbol{\delta}\left(\mathbf{x}^{\prime} ; L_{0}^{\uparrow}\right)}{R^{\prime}} $$

$$ g_{\mu \nu}=\delta_{\mu \nu}-\frac{\Omega}{\pi} \epsilon_{\mu \lambda} \epsilon_{\nu \kappa} \frac{q^{\lambda} q^{\kappa}}{q^{\sigma} q_{\sigma}} $$

$$ g \int d^{3} x^{\prime} \delta^{(3)}\left(\mathbf{x}-\mathbf{x}_{0}\right) \nabla^{\prime} \frac{1}{R^{\prime}}=g \frac{\mathbf{x}-\mathbf{x}_{0}}{\left|\mathbf{x}-\mathbf{x}_{0}\right|^{3}} $$

$$ \left(\partial_{1} \partial_{2}-\partial_{2} \partial_{1}\right) \omega(q)=-\Omega \delta^{(2)}(q) $$

$$ -4 \pi g \delta\left(\mathbf{x}^{\prime} ; L_{0}^{\uparrow}\right) $$

$$ R_{1212}=\left(\partial_{1} \partial_{2}-\partial_{2} \partial_{1}\right) \omega(q) $$

$$ \mathbf{B}(\mathbf{x})=\boldsymbol{\nabla} \times \mathbf{A}\left(\mathbf{x} ; L_{0}^{\uparrow}\right)+4 \pi g \boldsymbol{\delta}\left(\mathbf{x} ; L_{0}^{\uparrow}\right) . $$

$$ \dot{q}^{\mu}=e_{i}^{\mu}(q) \dot{x}^{i} $$

$$ \begin{align*} \mathbf{A}^{(g)}\left(\mathbf{x} ; L_{0}^{\uparrow}\right) & =-g \int_{0}^{\infty} d z^{\prime} \frac{\hat{\mathbf{z}} \times \mathbf{x}}{{\sqrt{x^{2}+y^{2}+\left(z^{\prime}-z\right)^{2}}}^{3 / 2}} \\ & =-g \frac{\hat{\mathbf{z}} \times \mathbf{x}}{r(r-z)}=g \frac{(y,-x, 0)}{r(r-z)} \end{align*} $$

$$ q^{\mu}(t)=q^{\mu}\left(t_{a}\right)+\int_{t_{a}}^{t} d t^{\prime} e_{i}^{\mu}\left(q\left(t^{\prime}\right)\right) \dot{x}^{i}\left(t^{\prime}\right) $$

$$ \begin{align*} \mathbf{A}^{(g)}\left(\mathbf{x} ; L_{0}^{\uparrow}\right) & =g \int_{-\infty}^{0} d z^{\prime} \frac{\hat{\mathbf{z}} \times \mathbf{x}}{{\sqrt{x^{2}+y^{2}+\left(z^{\prime}-z\right)^{2}}}^{3 / 2}} \\ & =g \frac{\hat{\mathbf{z}} \times \mathbf{x}}{r(r+z)}=-g \frac{(y,-x, 0)}{r(r+z)} \end{align*} $$

$$ \delta^{S} q^{\mu}(t)=\int_{t_{a}}^{t} d t^{\prime} \delta^{S}\left[e_{i}^{\mu}\left(q\left(t^{\prime}\right)\right) \dot{x}^{i}\left(t^{\prime}\right)\right] $$

$$ A_{\varphi}^{(g)}\left(\mathbf{x} ; L_{0}^{\uparrow}\right)=\frac{g}{r \sin \theta}(1-\cos \theta) \quad \text { or } \quad A_{\varphi}^{(g)}\left(\mathbf{x} ; L_{0}^{\uparrow}\right)=-\frac{g}{r \sin \theta}(1+\cos \theta) $$

$$ \delta^{S} \dot{q}^{\mu}(t)=\frac{d}{d t} \delta^{S} q^{\mu}(t) $$

$$ L=\frac{M}{2} \dot{\mathbf{x}}^{2} $$

$$ \delta \dot{x}^{i}(t)=\frac{d}{d t} \delta x^{i}(t) $$

$$ L \rightarrow L^{\prime}=L+\nabla \Lambda(\mathbf{x}) \dot{\mathbf{x}} $$

$$ \delta q^{\mu} \equiv e_{i}{ }^{\mu}(q) \delta x^{i} $$

$$ \mathcal{A}^{\prime} \rightarrow \mathcal{A}=\mathcal{A}+\Lambda\left(\mathbf{x}_{b}\right)-\Lambda\left(\mathbf{x}_{a}\right) $$

$$ \delta q\left(t_{a}\right)=\delta q\left(t_{b}\right)=0, $$

$$ L^{\prime}=\frac{M}{2} \dot{\mathbf{x}}^{2}+\mathbf{A}(\mathbf{x}) \dot{\mathbf{x}} $$

$$ \begin{align*} \frac{d}{d t} \delta^{S} q^{\mu}(t) & =\delta^{S} e_{i}^{\mu}(q(t)) \dot{x}^{i}(t)+e_{i}^{\mu}(q(t)) \frac{d}{d t} \delta x^{i}(t) \\ & =\delta^{S} e_{i}^{\mu}(q(t)) \dot{x}^{i}(t)+e_{i}^{\mu}(q(t)) \frac{d}{d t}\left[e_{\nu}^{i}(t) \delta q^{\nu}(t)\right] \end{align*} $$

$$ \mathbf{A}(\mathbf{x}) \rightarrow \mathbf{A}^{\prime}(\mathbf{x})=\mathbf{A}(\mathbf{x})+\nabla \Lambda(\mathbf{x}) $$

$$ \delta^{S} e_{i}^{\mu}(q)=-\Gamma_{\lambda \nu}^{\mu} \delta^{S} q^{\lambda} e_{i}^{\nu}, \quad \frac{d}{d t} e_{\nu}^{i}(q)=\Gamma_{\lambda \nu}^{\mu} \dot{q}^{\lambda} e_{\mu}^{i} $$

$$ L=\psi^{*}(\mathbf{x})\left(i \partial_{t}+\frac{1}{2 M} \nabla^{2}\right) \psi(\mathbf{x}) $$

$$ \frac{d}{d t} \delta^{S} q^{\mu}(t)=-\Gamma_{\lambda \nu}{ }^{\mu} \delta^{S} q^{\lambda} \dot{q}^{\nu}+\Gamma_{\lambda \nu}{ }^{\mu} \dot{q}^{\lambda} \delta q^{\nu}+\frac{d}{d t} \delta q^{\mu} $$

$$ \psi(\mathbf{x}, t) \rightarrow \psi^{\prime}(\mathbf{x})=e^{i \Lambda(\mathbf{x})} \psi(\mathbf{x}, t) $$

$$ \delta^{S} b^{\mu} \equiv \delta^{S} q^{\mu}-\delta q^{\mu} $$

$$ L=\psi^{*}(\mathbf{x}, t)\left(i \partial_{t}+\frac{1}{2 M} \mathbf{D}^{2}\right) \psi(\mathbf{x}, t) $$

$$ \frac{d}{d t} \delta^{S} b^{\mu}=-\Gamma_{\lambda \nu}^{\mu} \delta^{S} b^{\lambda} \dot{q}^{\nu}+2 S_{\lambda \nu}^{\mu} \dot{q}^{\lambda} \delta q^{\nu} $$

$$ \hat{\mathbf{P}}=-i \mathbf{D}=-i \boldsymbol{\nabla}-\mathbf{A}(\mathbf{x}) $$

$$ G_{\lambda}^{\mu}(t) \equiv \Gamma_{\lambda \nu}^{\mu}(q(t)) \dot{q}^{\nu}(t) $$

$$ \mathbf{D} \psi(\mathbf{x}, t) \rightarrow \mathbf{D} \psi^{\prime}(\mathbf{x}, t)=e^{i \Lambda(\mathbf{x})} \mathbf{D} \psi(\mathbf{x}, t) $$

$$ \Sigma^{\mu}{ }_{\nu}(t) \equiv 2 S_{\lambda \nu}{ }^{\mu}(q(t)) \dot{q}^{\lambda}(t) $$

$$ \mathcal{E}=\frac{1}{8 \pi} \int d^{3} x \mathbf{B}^{2}(\mathbf{x}) $$

$$ \frac{d}{d t} \delta^{S} b=-G \delta^{S} b+\Sigma(t) \delta q^{\nu}(t) $$

$$ \mathcal{E}=\frac{I^{2}}{8 \pi} \int d^{3} x[\nabla \Omega(\mathbf{x})]^{2} $$

$$ \delta^{S} b(t)=\int_{t_{a}}^{t} d t^{\prime} U\left(t, t^{\prime}\right) \Sigma\left(t^{\prime}\right) \delta q\left(t^{\prime}\right) $$

$$ \mathcal{E}=\frac{I^{2}}{8 \pi} \int d^{3} x[\nabla \Omega(\mathbf{x} ; S)+4 \pi \delta(\mathbf{x} ; S)]^{2} $$

$$ U\left(t, t^{\prime}\right)=T \exp \left[-\int_{t^{\prime}}^{t} d t^{\prime \prime} G\left(t^{\prime \prime}\right)\right] $$

$$ \mathcal{E}=\int d^{3} x\left\{-\frac{1}{8 \pi} \mathbf{B}^{2}(\mathbf{x})-\frac{I}{4 \pi} \mathbf{B}(\mathbf{x}) \cdot[\nabla \Omega(\mathbf{x} ; S)+4 \pi \delta(\mathbf{x} ; S)]\right\} $$

$$ \delta^{S} \mathcal{A}=M \int_{t_{a}}^{t_{b}} d t\left(g_{\mu \nu} \dot{q}^{\nu} \delta^{S} \dot{q}^{\mu}+\frac{1}{2} \partial_{\mu} g_{\lambda \kappa} \delta^{S} q^{\mu} \dot{q}^{\lambda} \dot{q}^{\kappa}\right) $$

$$ \nabla \cdot \mathbf{B}(\mathbf{x})=0 $$

$$ \delta^{S} \mathcal{A}=M \int_{t_{a}}^{t_{b}} d t\left[-g_{\mu \nu}\left(\ddot{q}^{\nu}+\bar{\Gamma}_{\lambda \kappa}{ }^{\nu} \dot{q}^{\lambda} \dot{q}^{\kappa}\right) \delta q^{\mu}+\left(g_{\mu \nu} \dot{q}^{\nu} \frac{d}{d t} \delta^{S} b^{\mu}+\Gamma_{\mu \lambda \kappa} \delta^{S} b^{\mu} \dot{q}^{\lambda} \dot{q}^{\kappa}\right)\right] . $$

$$ \mathbf{B}(\mathbf{x}) \equiv \boldsymbol{\nabla} \times \mathbf{A}(\mathbf{x}) $$

$$ \delta^{S} \mathcal{A}=-M \int_{t_{a}}^{t_{b}} d t g_{\mu \nu}\left(\ddot{q}^{\nu}+\bar{\Gamma}_{\lambda \kappa}{ }^{\nu} \dot{q}^{\lambda} \dot{q}^{\kappa}\right) \delta q^{\nu} $$

$$ \mathcal{E}=\int d^{3} x\left\{-\frac{1}{8 \pi}[\boldsymbol{\nabla} \times \mathbf{A}(\mathbf{x})]^{2}-I[\boldsymbol{\nabla} \times \mathbf{A}(\mathbf{x})] \cdot \boldsymbol{\delta}(\mathbf{x} ; S)\right\} $$

$$ \begin{align*} \delta^{S} \mathcal{A} & =-M \int_{t_{a}}^{t_{b}} d t g_{\mu \nu}\left[\ddot{q}^{\nu}+\left(\bar{\Gamma}_{\lambda \kappa}{ }^{\nu}+2 S_{\lambda \kappa}^{\nu}\right) \dot{q}^{\lambda} \dot{q}^{\kappa}\right] \delta q^{\mu} \\ & =-M \int_{t_{a}}^{t_{b}} d t g_{\mu \nu}\left(\ddot{q}^{\nu}+\Gamma_{\lambda \kappa}{ }^{\nu} \dot{q}^{\lambda} \dot{q}^{\kappa}\right) \delta q^{\mu} \end{align*} $$

$$ \mathcal{E}=\int d^{3} x\left\{-\frac{1}{8 \pi}[\boldsymbol{\nabla} \times \mathbf{A}(\mathbf{x})]^{2}-I \mathbf{A}(\mathbf{x}) \cdot[\boldsymbol{\nabla} \times \boldsymbol{\delta}(\mathbf{x} ; S)]\right\} $$

$$ d_{t} \delta q^{\mu}(t)=\partial_{\nu} e_{i}^{\mu}(q(t)) \dot{q}^{\nu}(t) \delta x^{i}(t)+e_{i}^{\mu}(q(t)) d_{t} \delta x^{i}(t) $$

$$ \mathbf{j}(\mathbf{x}) \equiv I \nabla \times \boldsymbol{\delta}(\mathbf{x} ; S)=I \boldsymbol{\delta}(\mathbf{x} ; L) $$

$$ d_{t} q^{\lambda}(t)=e_{i}{ }^{\lambda}(q(t)) d_{t} x^{i}(t) $$

$$ \mathcal{E}=\frac{1}{2} \int d^{3} x d^{3} x^{\prime} \mathbf{j}(\mathbf{x}) \frac{1}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|} \mathbf{j}\left(\mathbf{x}^{\prime}\right) $$

$$ \delta d_{t} q^{\mu}(t)=\partial_{\nu} e_{i}^{\mu}(q(t)) \delta q^{\nu} d_{t} x^{i}(t)+e_{i}^{\mu}(q(t)) \delta d_{t} x^{i} $$

$$ \mathcal{E}=\frac{1}{8 \pi} \int d^{3} x\left[\boldsymbol{\nabla} \times \mathbf{A}+4 \pi g \sum_{n} \boldsymbol{\delta}\left(\mathbf{x} ; L_{n}^{\uparrow}\right)\right]^{2} $$

$$ \delta d_{t} q^{\mu}(t)-d_{t} \delta q^{\mu}(t)=\partial_{\nu} e_{i}^{\mu}(q(t)) \delta q^{\nu} d_{t} x^{i}(t)-\partial_{\nu} e_{i}^{\mu}(q(t)) \dot{q}^{\nu}(t) \delta x^{i}(t) $$

$$ \mathcal{E}=\int d^{3} x\left\{-\frac{1}{8 \pi} \mathbf{B}^{2}(\mathbf{x})-\frac{1}{4 \pi} \mathbf{B}(\mathbf{x}) \cdot\left[\boldsymbol{\nabla} \times \mathbf{A}+g \sum_{n} \boldsymbol{\delta}\left(\mathbf{x} ; L_{n}^{\uparrow}\right)\right]\right\} $$

$$ \delta d_{t} q^{\mu}(t)-d_{t} \delta q^{\mu}(t)=2 S_{\nu \lambda}^{\mu} \dot{q}^{\nu}(t) \delta q^{\lambda}(t) $$

$$ \mathcal{E}=\int d^{3} x\left\{-\frac{1}{8 \pi}[\nabla \Lambda(\mathbf{x})]^{2}+g \Lambda(\mathbf{x}) \sum_{n} \nabla \cdot \delta\left(\mathbf{x} ; L_{n}^{\uparrow}\right)\right\} $$

$$ \mathcal{A}=\int_{t_{1}}^{t_{2}} d t L\left(q^{\mu}(t), \dot{q}^{\mu}(t)\right) $$

$$ \Lambda(\mathbf{x})=-\frac{4 \pi g}{\nabla^{2}} \sum_{n} \delta\left(\mathbf{x}-\mathbf{x}_{n}\right)=g \sum_{n} \frac{1}{\left|\mathbf{x}-\mathbf{x}_{n}\right|}, $$

$$ \delta \mathcal{A}=\int_{t_{1}}^{t_{2}} d t\left\{\frac{\partial L}{\partial q^{\mu}} \delta q^{\mu}+\frac{\partial L}{\partial \dot{q}^{\mu}} \frac{d}{d t} \delta q^{\mu}+2 S_{\nu \lambda}^{\mu} \frac{\partial L}{\partial \dot{q}^{\mu}} \dot{q}^{\nu} \delta q^{\lambda}\right\} $$

$$ \mathcal{E}=\frac{g^{2}}{2} \sum_{n, n^{\prime}} \frac{1}{\left|\mathbf{x}_{n}-\mathbf{x}_{n^{\prime}}\right|} $$

$$ \frac{\partial L}{\partial q^{\mu}}-\frac{d}{d t} \frac{\partial L}{\partial \dot{q}^{\mu}}=-2 S_{\mu \nu}^{\lambda} \dot{q}^{\nu} \frac{\partial L}{\partial \dot{q}^{\lambda}} $$

$$ M\left[\ddot{q}^{\mu}+g^{\mu \kappa}\left(\partial_{\nu} g_{\lambda \kappa}-\frac{1}{2} \partial_{\kappa} g_{\nu \lambda}\right)-2 S_{\nu \lambda}^{\mu}\right] \dot{q}^{\nu} \dot{q}^{\lambda}=0 $$

$$ \left(\mathbf{x} t \mid \mathbf{x}^{\prime} t^{\prime}\right)=\frac{1}{\sqrt{2 \pi i \epsilon \hbar / M}}{ }^{D} \prod_{n=1}^{N}\left[\int_{-\infty}^{\infty} d x_{n}\right] \prod_{n=1}^{N+1} K_{0}^{\epsilon}\left(\Delta \mathbf{x}_{n}\right) $$

$$ K_{0}^{\epsilon}\left(\Delta \mathbf{x}_{n}\right) \equiv\left\langle\mathbf{x}_{n}\right| \exp \left(-\frac{i}{\hbar} \epsilon \hat{H}\right)\left|\mathbf{x}_{n-1}\right\rangle=\frac{1}{\sqrt{2 \pi i \epsilon \hbar / M}^{D}} \exp \left[\frac{i}{\hbar} \frac{M}{2} \frac{\left(\Delta \mathbf{x}_{n}\right)^{2}}{\epsilon}\right] $$

$$ \Delta x^{i}(t)=\dot{x}^{i}\left(t_{0}\right) \Delta t $$

$$ \dot{x}_{n}^{i}=e^{i}{ }_{\mu}\left(q_{n}\right) \dot{q}_{n}^{\mu} $$

$$ \Delta q \equiv q^{\lambda}-q^{\prime \lambda}=\epsilon \dot{q}^{\lambda}-\frac{\epsilon^{2}}{2!} \ddot{q}^{\lambda}+\frac{\epsilon^{3}}{3!} \ddot{q}^{\lambda}+\ldots $$

$$ \ddot{q}^{\lambda}=-\Gamma_{\mu \nu}{ }^{\lambda} \dot{q}^{\mu} \dot{q}^{\nu} $$

$$ \ddot{q}^{\lambda}=-\left(\partial_{\sigma} \Gamma_{\mu \nu}^{\lambda}-2 \Gamma_{\mu \nu}^{\tau} \Gamma_{\{\sigma \tau\}}{ }^{\lambda}\right) \dot{q}^{\mu} \dot{q}^{\nu} \dot{q}^{\sigma} $$

$$ \begin{align*} \Delta x^{i} & =e_{\lambda}^{i} \dot{q}^{\lambda} \Delta t \\ & =e_{\lambda}^{i}\left[\Delta q^{\lambda}-\frac{1}{2!} \Gamma_{\mu \nu}{ }^{\lambda} \Delta q^{\mu} \Delta q^{\nu}+\frac{1}{3!}\left(\partial_{\sigma} \Gamma_{\mu \nu}{ }^{\lambda}+\Gamma_{\mu \nu}{ }^{\tau} \Gamma_{\{\sigma \tau\}}{ }^{\lambda}\right) \Delta q^{\mu} \Delta q^{\nu} \Delta q^{\sigma}+\ldots\right] \end{align*} $$

$$ \Delta \xi^{\mu} \equiv e_{i}^{\mu} \Delta x^{i} $$

$$ \Delta \xi^{\lambda}=\Delta q^{\lambda}+\frac{1}{2!} \Gamma_{\mu \nu}{ }^{\lambda} \Delta q^{\mu} \Delta q^{\nu}+\frac{1}{3!}\left(\partial_{\sigma} \Gamma_{\mu \nu}{ }^{\lambda}+\Gamma_{\mu \nu}{ }^{\tau} \Gamma_{\{\sigma \tau\}}{ }^{\lambda}\right) \Delta q^{\mu} \Delta q^{\nu} \Delta q^{\sigma}+\ldots $$

$$ F(x+\Delta x)=F(x)+\partial_{i} F(x) \Delta x^{i}+\frac{1}{2!} \partial_{i} \partial_{j} F(x) \Delta x^{i} \Delta x^{j}+\ldots, $$

$$ \begin{align*} \partial_{i} \partial_{j} f(q) & =e_{i}{ }^{\mu} \partial_{\mu} e_{j}{ }^{\nu} \partial_{\nu} f(q)=\left[e_{i}{ }^{\mu} e_{j}{ }^{\nu} \partial_{\mu} \partial_{\nu}+e_{i}{ }^{\mu}\left(\partial_{\mu} e_{j}{ }^{\nu}\right) \partial_{\nu}\right] f(q) \\ & =e_{i}{ }^{\mu} e_{j}{ }^{\nu}\left[\partial_{\mu} \partial_{\nu}-\Gamma_{\mu \nu}{ }^{\lambda} \partial_{\lambda}\right] f(q)=e_{i}{ }^{\mu} e_{j}{ }^{\nu} D_{\mu} \partial_{\nu} f(q)=e_{i}{ }^{\mu} e_{j}{ }^{\nu} D_{\mu} D_{\nu} f(q), \end{align*} $$

$$ f(q+\Delta q)=F(x)+D_{\mu} f(q) \Delta \xi^{i}+\frac{1}{2!} D_{\mu} D_{\nu} f(q) \Delta \xi^{\mu} \Delta \xi^{\nu}+\ldots $$

$$ \Delta x^{i}=x^{i}(q)-x^{i}(q-\Delta q)=e_{\lambda}^{i} \Delta q^{\lambda}-\frac{1}{2} e_{\nu, \mu}^{i} \Delta q^{\mu} \Delta q^{\nu}+\frac{1}{3!} e_{\nu, \mu \sigma}^{i} \Delta q^{\mu} \Delta q^{\nu} \Delta q^{\sigma}+\ldots, $$

$$ \Delta x^{i}=e_{\lambda}^{i}\left[\Delta q^{\lambda}-\frac{1}{2} e_{j}{ }^{\lambda} e_{\nu, \mu}^{j} \Delta q^{\mu} \Delta q^{\nu}+\frac{1}{3!} e_{j}{ }^{\lambda} e_{\nu, \mu \sigma}^{j} \Delta q^{\mu} \Delta q^{\nu} \Delta q^{\sigma}+\ldots\right] $$

$$ e_{i}{ }^{\lambda} e^{i}{ }_{\nu, \mu \sigma}=\partial_{\sigma}\left(e_{i}{ }^{\lambda} e^{i}{ }_{\nu, \mu}\right)-e^{i \tau} e^{i}{ }_{\nu, \mu} e^{j}{ }_{\tau} e^{j \lambda}{ }_{, \sigma}=\partial_{\sigma} \Gamma_{\mu \nu}{ }^{\lambda}+\Gamma_{\mu \nu}{ }^{\tau} \Gamma_{\sigma \tau}{ }^{\lambda} . $$

$$ \Delta x^{i}=e^{i}{ }_{\lambda}\left[\Delta q^{\lambda}-\frac{1}{2!} \Gamma_{\mu \nu}{ }^{\lambda} \Delta q^{\mu} \Delta q^{\nu}+\frac{1}{3!}\left(\partial_{\sigma} \Gamma_{\mu \nu}{ }^{\lambda}+\Gamma_{\mu \nu}{ }^{\tau} \Gamma_{\sigma \tau}{ }^{\lambda}\right) \Delta q^{\mu} \Delta q^{\nu} \Delta q^{\sigma}+\ldots\right] . $$

$$ K_{0}^{\epsilon}(\Delta \mathbf{x})=\langle\mathbf{x}|\left(-\frac{i}{\hbar} \epsilon \hat{H}\right)|\mathbf{x}-\Delta \mathbf{x}\rangle=\frac{1}{\sqrt{2 \pi i \epsilon \hbar / M}^{D}} e^{i \mathcal{A}_{>}^{\epsilon}(q, q-\Delta q) / \hbar} $$

$$ \begin{align*} & \mathcal{A}_{>}^{\epsilon}(q, q-\Delta q)=\left(\Delta x^{i}\right)^{2}=\epsilon \frac{M}{2} g_{\mu \nu} \dot{q}^{\mu} \dot{q}^{\nu} \\ & =\frac{M}{2 \epsilon}\left\{g_{\mu \nu} \Delta q^{\mu} \Delta q^{\nu}-\Gamma_{\mu \nu \lambda} \Delta q^{\mu} \Delta q^{\nu} \Delta q^{\lambda}\right. \\ & \left.\quad+\left[\frac{1}{3} g_{\mu \tau}\left(\partial_{\kappa} \Gamma_{\lambda \nu}{ }^{\tau}+\Gamma_{\lambda \nu}{ }^{\delta} \Gamma_{\{\kappa \delta\}}{ }^{\tau}\right)+\frac{1}{4} \Gamma_{\lambda \kappa}{ }^{\sigma} \Gamma_{\mu \nu \sigma}\right] \Delta q^{\mu} \Delta q^{\nu} \Delta q^{\lambda} \Delta q^{\kappa}+\ldots\right\} \end{align*} $$

$$ \begin{align*} & \mathcal{A}_{>}^{\epsilon}(q, q-\Delta q)=\frac{M}{2 \epsilon}\left\{g_{\mu \nu} \Delta q^{\mu} \Delta q^{\nu}-\bar{\Gamma}_{\mu \nu \lambda} \Delta q^{\mu} \Delta q^{\nu} \Delta q^{\lambda}\right. \\ & \left.\quad+\left[\frac{1}{3} g_{\mu \tau}\left(\partial_{\kappa} \bar{\Gamma}_{\lambda \nu}{ }^{\tau}+\bar{\Gamma}_{\lambda \nu}{ }^{\delta} \bar{\Gamma}_{\delta \kappa}{ }^{\tau}\right)+\frac{1}{4} \bar{\Gamma}_{\lambda \kappa}{ }^{\sigma} \bar{\Gamma}_{\mu \nu \sigma}+\frac{1}{3} S^{\sigma}{ }_{\lambda \kappa} S_{\sigma \mu \nu}\right] \Delta q^{\mu} \Delta q^{\nu} \Delta q^{\lambda} \Delta q^{\kappa}+\ldots\right\} \end{align*} $$

$$ \begin{align*} \dot{r} & =\frac{\Delta r}{\epsilon}+\frac{r(\Delta \phi)^{2}}{2 \epsilon}-\frac{\Delta r(\Delta \phi)^{2}}{\epsilon}+\ldots \\ \dot{\phi} & =\frac{\Delta \phi}{\epsilon}-\frac{\Delta r \Delta \phi}{\epsilon r}-\frac{(\Delta \phi)^{3}}{6 \epsilon}+\ldots \end{align*} $$

$$ \mathcal{A}^{\epsilon}=\frac{M}{2} \epsilon\left(\dot{r}^{2}+r^{2} \dot{\phi}^{2}\right) $$

$$ \mathcal{A}^{\epsilon}=\frac{M}{2 \epsilon}\left[\Delta r^{2}+r^{2}(\Delta \phi)^{2}-r \Delta r(\Delta \phi)^{2}-\frac{1}{12} r^{2}(\Delta \phi)^{4}+\ldots\right] $$

$$ r_{n}^{2}=r_{n}\left(r_{n-1}+\Delta r_{n}\right) $$

$$ \mathcal{A}^{\epsilon}=\frac{M}{2 \epsilon}\left[\Delta r_{n}^{2}+r_{n} r_{n-1}\left(\Delta \phi_{n}\right)^{2}-\frac{1}{12} r_{n} r_{n-1}\left(\Delta \phi_{n}\right)^{4}+\ldots\right] $$

$$ L(q, \dot{q})=\frac{M}{2} g_{\mu \nu}(q(t)) \dot{q}^{\mu}(t) \dot{q}^{\nu}(t) $$

$$ \mathcal{A}^{\epsilon}\left(q, q^{\prime}\right)=\frac{M}{2} \int_{t-\epsilon}^{t} d t g_{\mu \nu}(q(t)) \dot{q}^{\mu}(t) \dot{q}^{\nu}(t) $$

$$ \mathcal{A}^{\epsilon}=\frac{M}{2} \epsilon g_{\mu \nu}(q) \dot{q}^{\mu} \dot{q}^{\nu}=\frac{M}{2} \epsilon g_{\mu \nu}\left(q^{\prime}\right) \dot{q}^{\prime \mu} \dot{q}^{\prime \nu}=\frac{M}{2} \epsilon g_{\mu \nu}(\bar{q}) \dot{\bar{q}}^{\mu} \dot{\bar{q}}^{\nu} $$

$$ \begin{align*} & \overline{\mathcal{A}}^{\epsilon}\left(\bar{q}+\frac{\Delta q}{2}, \bar{q}-\frac{\Delta q}{2}\right)= \\ & \quad \frac{M}{2 \epsilon}\left[g_{\mu \nu}(\bar{q}) \Delta q^{\mu} \Delta q^{\nu}+\frac{1}{12} g_{\kappa \tau}\left(\partial_{\lambda} \Gamma_{\mu \nu}{ }^{\tau}+\Gamma_{\mu \nu}{ }^{\delta} \Gamma_{\{\lambda \delta\}}{ }^{\tau}\right) \Delta q^{\mu} \Delta q^{\nu} \Delta q^{\lambda} \Delta q^{\kappa}+\ldots\right] \end{align*} $$

$$ L(q, \dot{q}) \rightarrow L^{\epsilon}(q, \Delta q / \epsilon)=\frac{M}{2 \epsilon^{2}} g_{\mu \nu}(\bar{q}) \Delta q^{\mu}(t) \Delta q^{\nu}(t) $$

$$ \overline{\mathcal{A}}_{\mathrm{mpp}}^{\epsilon}=\epsilon L^{\epsilon}(q, \Delta q / \epsilon) $$

$$ \mathcal{A}^{\epsilon} \equiv \mathcal{A}_{>}^{\epsilon}(q, q-\Delta q) $$

$$ \prod_{n=2}^{N+1} \int d^{D} x_{n-1}^{i}=\prod_{n=2}^{N+1}\left\{\int d^{D} q_{n-1}^{\mu} \operatorname{det}\left[e_{\mu}^{i}\left(q_{n-1}\right)\right]\right\} $$

$$ \operatorname{det}\left(e^{i}{ }_{\mu}\right)=\sqrt{\operatorname{det} g_{\mu \nu}(q)} \equiv \sqrt{g(q)}, $$

$$ \prod_{n=2}^{N+1} \int d^{D} x_{n-1}^{i}=\prod_{n=2}^{N+1}\left[\int d^{D} q_{n-1}^{\mu} \sqrt{g\left(q_{n-1}\right)}\right] $$

$$ d x^{i}=e_{\mu}^{i}(q-\Delta q) d q^{\mu}=e_{\mu}^{i} d q^{\mu}-e_{\mu, \nu}^{i} d q^{\mu} \Delta q^{\nu}+\frac{1}{2} e_{\mu, \nu \lambda}^{i} d q^{\mu} \Delta q^{\nu} \Delta q^{\lambda}+\ldots $$

$$ J_{0}=\operatorname{det}\left(e^{i}{ }_{\kappa}\right) \operatorname{det}\left[\delta^{\kappa}{ }_{\mu}-e_{i}{ }^{\kappa} e_{\mu, \nu}^{i} \Delta q^{\nu}+\frac{1}{2} e_{i}{ }^{\kappa} e_{\mu, \nu \lambda}^{i} \Delta q^{\nu} \Delta q^{\lambda}\right] $$

$$ \prod_{n=2}^{N+1} \int d^{D} x_{n-1}^{i}=\prod_{n=2}^{N+1}\left\{\int d^{D} q_{n-1}^{\mu} J_{0 n}\right\} $$

$$ \operatorname{det}(1+B)=\exp \operatorname{tr} \log (1+B)=\exp \operatorname{tr}\left(B-B^{2} / 2+B^{3} / 3-\ldots\right) $$

$$ J_{0}=\operatorname{det}\left(e^{i}{ }_{\kappa}\right) \exp \left(\frac{i}{\hbar} \mathcal{A}_{J_{0}}^{\epsilon}\right) $$

$$ \frac{i}{\hbar} \mathcal{A}_{J_{0}}^{\epsilon}=-e_{i}{ }^{\kappa} e^{i}{ }_{\kappa, \mu} \Delta q^{\mu}+\frac{1}{2}\left[e_{i}{ }^{\mu} e_{\mu, \nu \lambda}^{i}-e_{i}{ }^{\mu} e^{i}{ }_{\kappa, \nu} e_{j}{ }^{\kappa} e^{j}{ }_{\mu, \lambda}\right] \Delta q^{\nu} \Delta q^{\lambda}+\ldots $$

$$ \begin{align*} e_{i \nu, \mu} e_{\kappa, \lambda}^{i} & =e_{i}^{\sigma} e_{\nu, \mu}^{i} e_{j \sigma} e_{\kappa, \lambda}^{j}=\Gamma_{\mu \nu}^{\sigma} \Gamma_{\lambda \kappa \sigma} \\ e_{i \mu} e_{\nu, \lambda \kappa}^{i} & =g_{\mu \tau}\left[\partial_{\kappa}\left(e_{i}^{\tau} e_{\nu, \lambda}^{i}\right)-e^{i \sigma} e_{\nu, \lambda}^{i} e_{\sigma}^{j} e^{j \tau}{ }_{, \kappa}\right] \\ & =g_{\mu \tau}\left(\partial_{\kappa} \Gamma_{\lambda \nu}{ }^{\tau}+\Gamma_{\lambda \nu}{ }^{\sigma} \Gamma_{\kappa \sigma}{ }^{\tau}\right) \end{align*} $$

$$ \frac{i}{\hbar} \mathcal{A}_{J_{0}}^{\epsilon}=-\Gamma_{\mu \nu}{ }^{\nu} \Delta q^{\mu}+\frac{1}{2} \partial_{\mu} \Gamma_{\nu \kappa}{ }^{\kappa} \Delta q^{\nu} \Delta q^{\mu}+\ldots $$

$$ J_{0}=\sqrt{g(q-\Delta q)} $$

$$ \exp \left(\frac{i}{\hbar} \mathcal{A}_{J_{0}}^{\epsilon}\right)=\frac{\sqrt{g(q-\Delta q)}}{\sqrt{g(q)}} $$

$$ \exp \left(\frac{i}{\hbar} \mathcal{A}_{\bar{J}_{0}}^{\epsilon}\right)=1-\frac{1}{\sqrt{g(q)}} \sqrt{g(q)}, \mu q^{\mu}+\frac{1}{2 \sqrt{g(q)}} \sqrt{g(q)}, \mu \nu, \Delta q^{\mu} \Delta q^{\nu}+\ldots $$

$$ \frac{1}{\sqrt{g}} \partial_{\mu} \sqrt{g}=\frac{1}{2} g^{\sigma \tau} \partial_{\mu} g_{\sigma \tau}=\bar{\Gamma}_{\mu \nu}^{\nu} $$

$$ \exp \left(\frac{i}{\hbar} \mathcal{A}_{\bar{J}_{0}}^{\epsilon}\right)=1-\bar{\Gamma}_{\mu \nu}{ }^{\nu} \Delta q^{\mu}+\frac{1}{2}\left(\partial_{\mu} \bar{\Gamma}_{\nu \lambda}{ }^{\lambda}+\bar{\Gamma}_{\mu \sigma}{ }^{\sigma} \bar{\Gamma}_{\nu \lambda}{ }^{\lambda}\right) \Delta q^{\mu} \Delta q^{\nu}+\ldots $$

$$ \frac{i}{\hbar} \mathcal{A}_{\bar{J}_{0}}^{\epsilon}=-\bar{\Gamma}_{\mu \nu}{ }^{\nu} \Delta q^{\mu}+\frac{1}{2} \partial_{\mu} \bar{\Gamma}_{\nu \lambda}{ }^{\lambda} \Delta q^{\mu} \Delta q^{\nu}+\ldots $$

$$ \prod_{n=2}^{N+1} \int d^{D} x_{n-1}^{i}=\prod_{n=2}^{N+1}\left\{\int d^{D} q_{n-1}^{\mu} \operatorname{det}\left[e_{\mu}^{i}\left(q_{n}\right)\right] \exp \left(\frac{i}{\hbar} \mathcal{A}_{J_{0 n}}^{\epsilon}\right)\right\} $$

$$ \left(\mathbf{x} t \mid \mathbf{x}^{\prime} t^{\prime}\right)=\frac{1}{\sqrt{2 \pi i \epsilon \hbar / M}}{ }^{D} \prod_{n=1}^{N}\left[\int_{-\infty}^{\infty} d \Delta x_{n}\right] \prod_{n=1}^{N+1} K_{0}^{\epsilon}\left(\Delta \mathbf{x}_{n}\right) $$

$$ \prod_{n=1}^{N} \int d^{D} x_{n} \equiv \prod_{n=2}^{N+1} \int d^{D} \Delta x_{n} $$

$$ \prod_{n=2}^{N+1} \int d^{D} \Delta x_{n} \rightarrow \prod_{n=2}^{N+1}\left[\int d^{D} \Delta q_{n} J_{n}\right] $$

$$ J=\frac{\partial(\Delta x)}{\partial(\Delta q)}=\operatorname{det}\left(e_{\kappa}^{i}\right) \operatorname{det}\left[\delta_{\mu}{ }^{\lambda}-\Gamma_{\{\mu \nu\}}{ }^{\lambda} \Delta q^{\nu}+\frac{1}{2}\left(\partial_{\{\sigma} \Gamma_{\mu \nu\}}{ }^{\lambda}+\Gamma_{\{\mu \nu}{ }^{\tau} \Gamma_{\{\tau \mid \sigma\}\}}{ }^{\lambda}\right) \Delta q^{\nu} \Delta q^{\sigma}+\ldots\right] . $$

$$ \frac{i}{\hbar} \mathcal{A}_{J}^{\epsilon}=-\Gamma_{\{\mu \nu\}}{ }^{\mu} \Delta q^{\nu}+\frac{1}{2}\left[\partial_{\{\mu} \Gamma_{\nu \kappa\}}{ }^{\kappa}+\Gamma_{\{\nu \kappa}{ }^{\sigma} \Gamma_{\{\sigma \mid \mu\}\}}{ }^{\kappa}-\Gamma_{\{\nu \kappa\}}{ }^{\sigma} \Gamma_{\{\sigma \mu\}}{ }^{\kappa}\right] \Delta q^{\nu} \Delta q^{\mu}+\ldots $$

$$ \langle q| e^{-i\left(t-t^{\prime}\right) \hat{H} / \hbar}\left|q^{\prime}\right\rangle=\frac{1}{\sqrt{2 \pi i \hbar \epsilon / M}^{D}} \prod_{n=2}^{N+1}\left[\int d^{D} \Delta q_{n} \frac{\sqrt{g\left(q_{n}\right)}}{\sqrt{2 \pi i \epsilon \hbar / M}^{D}}\right] e^{i \sum_{n=1}^{N+1}\left(\mathcal{A}^{\epsilon}+\mathcal{A}_{J}^{\epsilon}\right) / \hbar} $$

$$ \prod_{n=1}^{N} \int d^{D} x_{n}=\prod_{n=1}^{N}\left[\int d^{D} q_{n} \sqrt{g\left(q_{n}\right)}\right] $$

$$ \prod_{n=2}^{N+1}\left[\int d^{D} \Delta q_{n} \sqrt{g\left(q_{n}\right)}\right]=\prod_{n=1}^{N}\left[\int d^{D} q_{n} \sqrt{g\left(q_{n}\right)} e^{-i \mathcal{A}_{J_{0}}^{\epsilon} / \hbar}\right] . $$

$$ \langle q| e^{-i\left(t-t^{\prime}\right) \hat{H} / \hbar}\left|q^{\prime}\right\rangle=\frac{1}{\sqrt{2 \pi i \hbar \epsilon / M}^{D}} \prod_{n=1}^{N}\left[\int d^{D} q_{n} \frac{\sqrt{g\left(q_{n}\right)}}{\sqrt{2 \pi i \hbar \epsilon / M}^{D}}\right] e^{i \sum_{n=1}^{N+1}\left(\mathcal{A}^{\epsilon}+\Delta \mathcal{A}_{J}^{\epsilon}\right) / \hbar} $$

$$ \Delta \mathcal{A}_{J}^{\epsilon} \equiv \mathcal{A}_{J}^{\epsilon}-\mathcal{A}_{J_{0}}^{\epsilon} . $$

$$ \frac{i}{\hbar} \Delta \mathcal{A}_{J}^{\epsilon}=\frac{1}{6} \bar{R}_{\mu \nu} \Delta q^{\mu} \Delta q^{\nu} $$

$$ V_{\mathrm{eff}}(q)=-\frac{\hbar^{2}}{6 M} \bar{R}(q) $$

$$ \langle q| e^{-i\left(t-t^{\prime}\right) \hat{H} / \hbar}\left|q^{\prime}\right\rangle=\frac{1}{\sqrt{2 \pi i \hbar \epsilon / M}^{D}} \prod_{n=1}^{N}\left[\int d^{D} q_{n} \frac{\sqrt{g\left(q_{n}\right)}}{\sqrt{2 \pi i \epsilon \hbar / M}^{D}} e^{i \hbar R\left(q_{n}\right) / 6 M}\right] e^{i \sum_{n=1}^{N+1} \mathcal{A}^{\epsilon}\left[q_{n}\right] / \hbar} $$

$$ \langle q| e^{-i\left(t-t^{\prime}\right) \hat{H} / \hbar}\left|q^{\prime}\right\rangle=\int \mathcal{D}^{D} q \sqrt{g(q)} e^{i \int_{t_{a}}^{t_{b}} d t \mathcal{A}[q] / \hbar} $$

$$ \prod_{n=1}^{N+1}\left[\frac{d p_{n}}{2 \pi \hbar \sqrt{g\left(q_{n}\right)}}\right] e^{(i / \hbar) \sum_{n=1}^{N+1}\left[p_{n \mu} \Delta q^{\mu}-\epsilon \frac{1}{2 M} g^{\mu \nu}\left(q_{n}\right) p_{n \mu} p_{n \nu}\right]} $$

$$ \bar{R}=\frac{\left(D^{\prime}-1\right) D^{\prime}}{r^{2}} $$

$$ (d \mathbf{x})^{2}=\left(d x^{1}\right)^{2}+\left(d x^{2}\right)^{2}+\ldots+\left(d x^{D}\right)^{2} $$

$$ \left(x^{1}\right)^{2}+\left(x^{2}\right)^{2}+\ldots+\left(x^{D}\right)^{2}=r^{2} $$

$$ (d \mathbf{x})^{2}=\left(d x^{1}\right)^{2}+\left(d x^{2}\right)^{2}+\ldots+\left(d x^{D^{\prime}}\right)^{2}+\frac{\left(x^{1} d x^{1}+d x^{2}+\ldots+x^{D^{\prime}} d x^{D^{\prime}}\right)^{2}}{r^{2}-r^{\prime 2}} $$

$$ g_{\mu \nu}(x)=\delta_{\mu \nu}+\frac{x^{\mu} x^{\nu}}{r^{2}-r^{\prime 2}} $$

$$ \bar{R}_{\mu \nu \lambda \kappa} \approx \frac{1}{r^{2}}\left(\delta_{\mu \kappa} \delta_{\nu \lambda}-\delta_{\mu \lambda} \delta_{\nu \kappa}\right) $$

$$ \bar{R}_{\mu \nu \lambda \kappa}(x)=\frac{1}{r^{2}}\left[g_{\mu \kappa}(x) g_{\nu \lambda}(x)-g_{\mu \lambda}(x) g_{\nu \kappa}(x)\right], $$

$$ \bar{R}_{\nu \kappa}(x)=\bar{R}_{\mu \nu \kappa}^{\mu}(x)=\frac{D^{\prime}-1}{r^{2}} g_{\nu \kappa}(x) . $$

$$ V_{\mathrm{eff}}=-\frac{\hbar^{2}}{6 M r^{2}}(D-2)(D-1) $$

$$ E_{l}=\frac{\hbar^{2}}{2 M r^{2}} l(l+D-2) $$

$$ \left\langle q \mid q^{\prime}\right\rangle=\sqrt{g(q)}^{-1} \delta^{(D)}\left(q-q^{\prime}\right) $$

$$ \int d^{D} q \sqrt{g(q)}|q\rangle\langle q|=1 $$

$$ \mathcal{A}_{\mathrm{em}}=\int_{t_{a}}^{t_{b}} d t\left[\frac{e}{c} A_{\mu}(q(t)) \dot{q}^{\mu}-V(q(t))\right] $$

$$ \mathcal{A}^{\epsilon}=\frac{M}{2 \epsilon}\left(\Delta x^{i}\right)^{2}+\frac{e}{c} A_{i}(\mathbf{x}) \Delta x^{i}-\frac{e}{2 c} A_{i, j}(\mathbf{x}) \Delta x^{i} \Delta x^{j}-\epsilon V(\mathbf{x})+\ldots . $$

$$ \mathcal{A}^{\epsilon}=\int_{t-\epsilon}^{t} d t L(t) $$

$$ L(t)=\frac{M}{2} \dot{\mathbf{x}}^{2}(t)+\frac{e}{c} \mathbf{A}(\mathbf{x}(t)) \dot{\mathbf{x}}(t)-V(\mathbf{x}(t)) $$

$$ \frac{d}{d t} L=M \dot{\mathbf{x}} \ddot{\mathbf{x}}+\frac{e}{c} \mathbf{A}(\mathbf{x}) \ddot{\mathbf{x}}+\frac{e}{c} A_{i, j}(\mathbf{x}) \dot{x}^{i} \dot{x}^{j}-V_{i}(\mathbf{x}) \dot{x}^{i} $$

$$ \mathcal{A}^{\epsilon}=\int_{t-\epsilon}^{t} d t L(t)=\epsilon L(t)-\frac{1}{2} \epsilon^{2} \frac{d}{d t} L(t)+\ldots $$

$$ M \ddot{x}^{i}=-\frac{e}{c}\left(A_{i, j}(\mathbf{x})-A_{j, i}(\mathbf{x})\right) \dot{x}^{j}-V_{i}(\mathbf{x}) $$

$$ \begin{align*} \Delta x^{i} & =-\epsilon \dot{x}^{i}+\frac{1}{2} \epsilon^{2} \ddot{x}^{i}+\ldots \\ & =-\epsilon \dot{x}^{i}-\frac{e}{2 M c} \epsilon^{2}\left[\left(A_{i, j}-A_{j, i}\right) \dot{x}^{j}+V_{i}(\mathbf{x})\right]+\ldots \end{align*} $$

$$ \dot{x}^{i}=-\frac{\Delta x^{i}}{\epsilon}-\frac{e}{2 M c}\left(A_{i, j}-A_{j, i}\right) \Delta x^{j}+\ldots $$

$$ \Delta x^{i} \Delta x^{j} \rightarrow\left\langle\Delta x^{i} \Delta x^{j}\right\rangle=\delta_{i j} i \frac{\hbar \epsilon}{M}, $$

$$ \mathcal{A}^{\epsilon}=\frac{M}{2 \epsilon}\left(\Delta x^{i}\right)^{2}+\frac{e}{c} A_{i}(\mathbf{x}) \Delta x^{i}-i \epsilon \frac{\hbar e}{2 M c} A_{i, i}(\mathbf{x})-\epsilon V(\mathbf{x})+\ldots . $$

$$ \mathcal{A}^{\epsilon}=\frac{M}{2 \epsilon}\left(\Delta x^{i}\right)^{2}+\frac{e}{c} A_{i}(\overline{\mathbf{x}}) \Delta x^{i}-\epsilon V(\mathbf{x})+\ldots, $$

$$ \overline{\mathrm{x}}=\mathrm{x}-\frac{1}{2} \Delta \mathrm{x} $$

$$ \overline{\mathbf{x}}\left(t_{n}\right) \equiv \frac{1}{2}\left[\mathbf{x}\left(t_{n}\right)+\mathbf{x}\left(t_{n-1}\right)\right] $$

$$ \mathcal{A}_{\mathrm{em}}^{\epsilon}=\frac{e}{c} A_{\mu} \Delta q^{\mu}-\frac{e}{2 c} \partial_{\nu} A_{\mu} \Delta q^{\mu} \Delta q^{\nu}-\epsilon V(q)+\ldots $$

$$ \mathcal{A}_{\mathrm{em}}^{\epsilon}=\frac{e}{c} A_{\mu} \Delta q^{\mu}-i \epsilon \frac{\hbar e}{2 M c} \partial_{\mu} A^{\mu}-\epsilon V(q)+\ldots $$

$$ \mathcal{A}_{\mathrm{em}}^{N}=\sum_{n=1}^{N+1} \mathcal{A}_{\mathrm{em}}^{\epsilon} $$

$$ Z=\int \mathcal{D}^{D} q \sqrt{g} e^{-\mathcal{A}[q]} $$

$$ \mathcal{A}[q]=\int_{0}^{\beta} d \tau\left[\frac{1}{2} g_{\mu \nu}(q(\tau)) \dot{q}^{\mu}(\tau) \dot{q}^{\nu}(\tau)+V(q(\tau))\right] $$

$$ \mathcal{A}^{(0)}\left[q_{a} ; \delta q\right] \equiv \frac{1}{2} \int_{0}^{\beta} d \tau g_{\mu \nu}\left(q_{a}\right)\left[\delta \dot{q}^{\mu}(\tau) \delta \dot{q}^{\nu}(\tau)+\omega^{2} \delta q^{\mu}(\tau) \delta q^{\nu}(\tau)\right] $$

$$ \mathcal{A}^{\mathrm{int}}\left[q_{a} ; \delta q\right] \equiv \mathcal{A}[q]-\mathcal{A}^{(0)}\left[q_{a} ; \delta q\right] . $$

$$ \int \mathcal{D}^{D} q \sqrt{g} \equiv \prod_{\tau} \int d^{D} q(\tau) \sqrt{g(\tau)}=\left[\prod_{\tau} \int d^{D} q(\tau) \sqrt{g\left(q_{a}\right)}\right] \exp \left[\frac{1}{2} \sum_{\tau} \log \frac{g(q(\tau))}{g\left(q_{a}\right)}\right] . $$

$$ \int \mathcal{D}^{D} q \sqrt{g} \equiv\left[\prod_{\tau} \int d^{D} q(\tau) \sqrt{g\left(q_{a}\right)}\right] \exp \left[\frac{1}{2} \delta(0) \int_{0}^{\beta} d \tau \log \frac{g(q(\tau))}{g\left(q_{a}\right)}\right] $$

$$ \int \mathcal{D}^{D} q \sqrt{g\left(q_{a}\right)} e^{-\mathcal{A}^{g}[q]} $$

$$ \mathcal{A}_{g}[q]=-\frac{1}{2} \delta(0) \int_{0}^{\beta} d \tau \log \frac{g(q(\tau))}{g\left(q_{a}\right)} $$

$$ \mathcal{A}_{g}\left[q_{a}, \delta q\right]=-\frac{1}{2} \delta(0) \int_{0}^{\beta} d \tau\left[\log g\left(q_{a}+\delta q(\tau)\right)-\log g\left(q_{a}\right)\right] $$

$$ \mathcal{A}_{\mathrm{tot}}^{\mathrm{int}}\left[q_{a}, \delta q\right]=\mathcal{A}^{\mathrm{int}}\left[q_{a}, \delta q\right]+\mathcal{A}_{g}\left[q_{a}, \delta q\right] $$

$$ Z=\int \mathcal{D}^{D} q \sqrt{g\left(q_{a}\right)} e^{-\mathcal{A}^{(0)}[q]} e^{-\mathcal{A}_{\mathrm{tot}}^{\mathrm{int}}[q]} $$

$$ \begin{align*} Z & =\int \mathcal{D}^{D} q \sqrt{g\left(q_{a}\right)}\left(1-\mathcal{A}_{\mathrm{tot}}^{\mathrm{int}}+\frac{1}{2} \mathcal{A}_{\mathrm{tot}}^{\mathrm{int} 2}-\ldots\right) e^{-\mathcal{A}^{(0)}[q]} \\ & =Z_{\omega}\left[1-\left\langle\mathcal{A}_{\mathrm{tot}}^{\mathrm{int}}\right\rangle+\frac{1}{2!}\left\langle\mathcal{A}_{\mathrm{tot}}^{\mathrm{int} 2}\right\rangle-\ldots\right] \end{align*} $$

$$ Z_{\omega} \equiv e^{-\beta F_{\omega}}=\int \mathcal{D}^{D} q \sqrt{g\left(q_{a}\right)} e^{-\mathcal{A}^{(0)}[q]} $$

$$ \langle\ldots\rangle=Z_{\omega}^{-1} \int \mathcal{D} q \sqrt{g\left(q_{a}\right)}(\ldots) e^{-\mathcal{A}^{(0)}[q]} $$

$$ Z \equiv e^{-\beta F}=\exp \left[-\beta F_{\omega}-\left\langle\mathcal{A}_{\mathrm{tot}}^{\mathrm{int}}\right\rangle_{c}+\frac{1}{2!}\left\langle\mathcal{A}_{\mathrm{tot}}^{\mathrm{int} 2}\right\rangle_{c}-\ldots\right], $$

$$ Z_{\omega}=\int \mathcal{D} q e^{-\mathcal{A}_{\omega}[q]}=\exp \left[-\frac{D}{2} \operatorname{Tr} \log \left(-\partial^{2}+\omega^{2}\right)\right] \equiv e^{-\beta F_{\omega}} $$

$$ F_{\omega}=\frac{1}{\beta} \frac{D}{2} \operatorname{Tr} \log \left(-\partial^{2}+\omega^{2}\right) \underset{\beta \rightarrow \infty}{\rightarrow} \frac{D}{2} \int_{-\infty}^{\infty} \frac{d k}{2 \pi} \log \left(k^{2}+\omega^{2}\right)=\frac{D}{2} \omega . $$

$$ \begin{align*} G_{\mu \nu}^{(2)}\left(\tau, \tau^{\prime}\right) & \equiv\left\langle q_{\mu}(\tau) q_{\nu}\left(\tau^{\prime}\right)\right\rangle=-, \\ \partial_{\tau} G_{\mu \nu}^{(2)}\left(\tau, \tau^{\prime}\right) & \equiv\left\langle\dot{q}_{\mu}(\tau) q_{\nu}\left(\tau^{\prime}\right)\right\rangle=\cdots, \\ \partial_{\tau^{\prime}} G_{\mu \nu}^{(2)}\left(\tau, \tau^{\prime}\right) & \equiv\left\langle q_{\mu}(\tau) \dot{q}_{\nu}\left(\tau^{\prime}\right)\right\rangle=\ldots, \\ \partial_{\tau} \partial_{\tau^{\prime}} G_{\mu \nu}^{(2)}\left(\tau, \tau^{\prime}\right) & \equiv\left\langle\dot{q}_{\mu}(\tau) \dot{q}_{\nu}\left(\tau^{\prime}\right)\right\rangle=\ldots \ldots . \end{align*} $$

$$ G_{\mu \nu}^{(2)}\left(\tau, \tau^{\prime}\right)=\delta_{\mu \nu} \Delta\left(\tau-\tau^{\prime}\right), $$

$$ \Delta\left(\tau-\tau^{\prime}\right)=\int_{-\infty}^{\infty} \frac{d k}{2 \pi} \frac{e^{i k\left(\tau-\tau^{\prime}\right)}}{k^{2}+\omega^{2}}=\frac{1}{2 \omega} e^{-\omega\left|\tau-\tau^{\prime}\right|} $$

$$ \epsilon\left(\tau-\tau^{\prime}\right) \equiv-1+2 \int_{-\infty}^{\tau} d \tau^{\prime \prime} \delta\left(\tau^{\prime \prime}-\tau^{\prime}\right) $$

$$ \partial_{\tau^{\prime}} G_{\mu \nu}^{(2)}\left(\tau, \tau^{\prime}\right)=-\partial_{\tau} G_{\mu \nu}^{(2)}\left(\tau, \tau^{\prime}\right)=-\delta_{\mu \nu} \dot{\Delta}\left(\tau-\tau^{\prime}\right) . $$

$$ \begin{align*} \partial_{\tau} \partial_{\tau^{\prime}} G_{\mu \nu}^{(2)}\left(\tau, \tau^{\prime}\right) & =-\partial_{\tau}^{2} G_{\mu \nu}^{(2)}\left(\tau, \tau^{\prime}\right)=\delta_{\mu \nu} \int_{-\infty}^{\infty} \frac{d k}{2 \pi} \frac{e^{i k\left(\tau-\tau^{\prime}\right)} k^{2}}{k^{2}+\omega^{2}}=-\delta_{\mu \nu} \ddot{\Delta}\left(\tau-\tau^{\prime}\right) \\ & =\delta_{\mu \nu} \int_{-\infty}^{\infty} \frac{d k}{2 \pi} e^{i k\left(\tau-\tau^{\prime}\right)}\left(1-\frac{\omega^{2}}{k^{2}+\omega^{2}}\right)=\delta_{\mu \nu} \delta\left(\tau-\tau^{\prime}\right)-\omega^{2} G_{\mu \nu}^{(2)}\left(\tau, \tau^{\prime}\right) \end{align*} $$

$$ -\ddot{q}(\tau)+\omega^{2} q(\tau)=\delta\left(\tau-\tau^{\prime}\right) $$

$$ \ddot{\Delta}(\tau)=\omega^{2} \Delta(\tau)-\delta(\tau) . $$

$$ \mathcal{A}_{\omega}=\frac{1}{2} \int_{0}^{\beta} d \tau\left[\dot{x}^{2}(\tau)+\omega^{2} x^{2}(\tau)\right] $$

$$ Z_{\omega}=\int \mathcal{D} x e^{-\mathcal{A}_{\omega}[x]}=\exp \left[-\frac{D}{2} \operatorname{Tr} \log \left(-\partial^{2}+\omega^{2}\right)\right] \equiv e^{-\beta F_{\omega}} $$

$$ x(\tau)=f_{\eta}(\eta q(\tau)) \equiv \frac{1}{\eta} f(\eta q(\tau))=q-\frac{\eta}{3} q^{3}+a \frac{\eta^{2}}{5} q^{5}-\cdots $$

$$ Z=\int \mathcal{D} q(\tau) e^{-\mathcal{A}_{J}[q]} e^{-\mathcal{A}[q]} $$

$$ \mathcal{A}_{J}[q]=-\delta(0) \int d \tau \log \frac{\partial f(q(\tau))}{\partial q(\tau)} $$

$$ J=\prod_{\tau} \sqrt{\frac{\partial f(q(\tau))}{\partial q(\tau)}} $$

$$ \mathcal{A}_{\omega}[q]=\frac{1}{2} \int_{0}^{\beta} d \tau\left[\dot{q}^{2}(\tau)+\omega^{2} q^{2}(\tau)\right] $$

$$ \begin{align*} \mathcal{A}^{\mathrm{int}}[q]=\int_{0}^{\beta} d \tau & \left\{-\eta\left[q^{2}(\tau) \dot{q}^{2}(\tau)+\frac{\omega^{2}}{3} q^{4}(\tau)\right]\right. \\ & \left.+\eta^{2}\left[\left(\frac{1}{2}+a\right) q^{4}(\tau) \dot{q}^{2}(\tau)+\omega^{2}\left(\frac{1}{18}+\frac{2 a}{5}\right) q^{6}(\tau)\right]\right\} \end{align*} $$

$$ g_{00}(q)=g(q)=\left[f^{\prime}(\eta q)\right]^{2}=1-2 \eta q^{2}+(1+2 a) \eta^{2} q^{4}+\ldots $$

$$ \mathcal{A}_{J}[q]=-\delta(0) \int_{0}^{\beta} d \tau\left[-\eta q^{2}(\tau)+\eta^{2}\left(a-\frac{1}{2}\right) q^{4}(\tau)\right] $$

$$ \mathcal{A}_{\mathrm{tot}}^{\mathrm{int}}[q]=\mathcal{A}^{\mathrm{int}}[q]+\mathcal{A}_{J}[q] . $$

$$ \beta F=\beta F_{\omega}+\beta \sum_{n=1} \eta^{n} F_{n}=\beta F_{\omega}+\left\langle\mathcal{A}_{\mathrm{tot}}^{\mathrm{int}}\right\rangle_{c}-\frac{1}{2!}\left\langle\mathcal{A}_{\mathrm{tot}}^{\mathrm{int} 2}\right\rangle_{c}+\ldots $$

$$ \beta F_{1}=-\eta \circlearrowleft-\eta \omega^{2} \circlearrowleft+\eta \delta(0) \circlearrowleft . $$

$$ \beta F_{2}^{(1)}=\eta^{2}\left[3\left(\frac{1}{2}+a\right) \emptyset+15 \omega^{2}\left(\frac{1}{18}+\frac{a}{5}\right) \emptyset-3\left(a-\frac{1}{2}\right) \delta(0) \circlearrowleft\right] . $$

$$ \begin{align*} -\frac{1}{2!} \eta^{2} \int_{0}^{\beta} d \tau \int_{0}^{\beta} d \tau^{\prime}\langle & {\left[-q^{2}(\tau) \dot{q}^{2}(\tau)-\frac{\omega^{2}}{3} q^{4}(\tau)+\delta(0) q^{2}(\tau)\right] } \\ & \left.\times\left[-q^{2}\left(\tau^{\prime}\right) \dot{q}^{2}\left(\tau^{\prime}\right)-\frac{\omega^{2}}{3} q^{4}\left(\tau^{\prime}\right)+\delta(0) q^{2}\left(\tau^{\prime}\right)\right]\right\rangle_{c} \end{align*} $$

$$ \beta F_{2}^{(2)}=-\frac{\eta^{2}}{2!}\left\{2 \delta^{2}(0) \circlearrowleft-4 \delta(0)\left[\propto^{\prime} \oint+2 \omega^{2} \propto\right]\right\} . $$

$$ \beta F_{2}^{(3)}=-\frac{\eta^{2}}{2!}\left[4 \circlearrowleft x+2 \propto x+2 \cdots+8 \omega^{2} \cdots+8 \omega^{2} \propto x+8 \omega^{4} \propto \cap\right] \text {, } $$

$$ \beta F_{2}^{(4)}=-\frac{\eta^{2}}{2!} 4\left[\frac{2}{3} \omega^{4} \circlearrowleft+\Theta+4 \circlearrowleft+\Theta+4 \omega^{2} G\right] . $$

$$ \mathcal{A}_{\omega}=\frac{1}{2} \int d^{d} \tau\left[\partial_{\alpha} x(\tau) \partial_{\alpha} x(\tau)+\omega^{2} x^{2}(\tau)\right] $$

$$ \begin{align*} & G_{\alpha}^{(2)}\left(\tau, \tau^{\prime}\right)=\left\langle\partial_{\alpha} q(\tau) q\left(\tau^{\prime}\right)\right\rangle=\Delta_{\alpha}\left(\tau-\tau^{\prime}\right)=\int \frac{d^{d} k}{(2 \pi)^{d}} \frac{i k_{\alpha}}{k^{2}+\omega^{2}} e^{i k\left(\tau-\tau^{\prime}\right)} \\ & G_{\alpha \beta}^{(2)}\left(\tau, \tau^{\prime}\right)=\left\langle\partial_{\alpha} q(\tau) \partial_{\beta} q\left(\tau^{\prime}\right)\right\rangle=\Delta_{\alpha \beta}\left(\tau-\tau^{\prime}\right)=\int \frac{d^{d} k}{(2 \pi)^{d}} \frac{k_{\alpha} k_{\beta}}{k^{2}+\omega^{2}} e^{i k\left(\tau-\tau^{\prime}\right)} \end{align*} $$

$$ \left[=\frac{1}{p^{2}+\omega^{2}}, \quad \text { — } \cdots=i \frac{p_{\alpha}}{p^{2}+\omega^{2}}, \quad \cdots \cdots=\frac{p_{\alpha} p_{\beta}}{p^{2}+\omega^{2}}\right. $$

$$ \Omega^{-}=-\int d^{d} \tau_{1} \Delta\left(\tau_{1}-\tau_{2}\right) \Delta_{\alpha}\left(\tau_{1}-\tau_{2}\right) \Delta_{\beta}\left(\tau_{1}-\tau_{2}\right) \Delta_{\alpha \beta}\left(\tau_{1}-\tau_{2}\right) $$

$$ X=\int \frac{d k}{2 \pi} \frac{d p_{1}}{2 \pi} \frac{d p_{2}}{2 \pi} \frac{k^{2}\left(p_{1} p_{2}\right)}{\left(k^{2}+\omega^{2}\right)\left(p_{1}^{2}+\omega^{2}\right)\left(p_{2}^{2}+\omega^{2}\right)\left[\left(k+p_{1}+p_{2}\right)^{2}+\omega^{2}\right]} $$

$$ Y_{d}=\int \frac{d^{d} k}{(2 \pi)^{d}} \frac{d^{d} p_{1}}{(2 \pi)^{d}} \frac{d^{d} p_{2}}{(2 \pi)^{d}} \frac{k^{2}\left(p_{1} p_{2}\right)-\left(k p_{1}\right)\left(k p_{2}\right)}{\left(k^{2}+\omega^{2}\right)\left(p_{1}^{2}+\omega^{2}\right)\left(p_{2}^{2}+\omega^{2}\right)\left[\left(k+p_{1}+p_{2}\right)^{2}+\omega^{2}\right]}, $$

$$ I \equiv \int \frac{d^{d} k}{k^{2}+\omega^{2}}=\frac{\omega^{d-2}}{(4 \pi)^{d / 2}} \Gamma(1-d / 2) \underset{d=1}{=} \frac{1}{2 \omega} $$

$$ I_{\alpha}^{\beta} \equiv \int \frac{\partial^{d} k\left(k^{2}\right)^{\beta}}{\left(k^{2}+\omega^{2}\right)^{\alpha}}=\frac{\omega^{d+2 \beta-2 \alpha}}{(4 \pi)^{d / 2}} \frac{\Gamma(d / 2+\beta) \Gamma(\alpha-\beta-d / 2)}{\Gamma(d / 2) \Gamma(\alpha)} $$

$$ I_{0}^{\beta}=\int d^{d} k\left(k^{2}\right)^{\beta}=0 $$

$$ \begin{align*} & \bigcirc=\left\langle q^{2}\right\rangle \quad=\int \frac{d^{d} k}{k^{2}+\omega^{2}}=\frac{1}{d=1} \frac{1}{2 \omega}, \\ & \bigcirc=\left\langle q^{2}\right\rangle^{2}=\left(\int \frac{d^{d} k}{k^{2}+\omega^{2}}\right)^{2}=\frac{1}{d=1} \frac{1}{4 \omega^{2}}, \\ & \phi=\left\langle q^{2}\right\rangle^{3}=\left(\int \frac{d^{d} k}{k^{2}+\omega^{2}}\right)^{3}=\frac{1}{d=1} \frac{1}{8 \omega^{3}}, \\ & \phi=\left\langle q^{2}\right\rangle\langle\partial q \partial q\rangle=\int \frac{d^{d} k}{k^{2}+\omega^{2}} \int \frac{d^{d} p p^{2}}{p^{2}+\omega^{2}} \underset{d=1}{=}-\frac{1}{4}, \\ & \phi=\langle\partial q \partial q\rangle=\left(\int \frac{d^{d} k}{k^{2}+\omega^{2}}\right)^{2} \int \frac{d^{d} p p^{2}}{p^{2}+\omega^{2}}=-\frac{1}{d=1} . \end{align*} $$

$$ \begin{align*} \circlearrowleft & =\int d^{d} \tau_{1} \Delta^{2}\left(\tau_{1}-\tau_{2}\right) \int \frac{d^{d} p}{\left(p^{2}+\omega^{2}\right)^{2}} \underset{d=1}{=} \frac{1}{4 \omega^{3}} \\ \sigma^{\alpha} & =\int d^{d} \tau_{1} \Delta\left(\tau_{1}-\tau_{1}\right) \Delta_{\alpha}^{2}\left(\tau_{1}-\tau_{2}\right) \int \frac{d^{d} k}{k^{2}+\omega^{2}} \int \frac{d^{d} p p^{2}}{\left(p^{2}+\omega^{2}\right)^{2}}=\frac{1}{d=1} \frac{d^{d}}{8 \omega^{2}} \\ & =\int d^{d} \tau_{1} \Delta_{\alpha \alpha}\left(\tau_{1}-\tau_{1}\right) \Delta^{2}\left(\tau_{1}-\tau_{2}\right) \int \frac{d^{d} k k^{2}}{k^{2}+\omega^{2}} \int \frac{d^{d} p}{\left(p^{2}+\omega^{2}\right)^{2}} \underset{d=1}{=}-\frac{1}{8 \omega^{2}} \\ & =\int d^{d} \tau_{1} \Delta\left(\tau_{1}-\tau_{1}\right) \Delta^{2}\left(\tau_{1}-\tau_{2}\right) \int \frac{d^{d} k}{k^{2}+\omega^{2}} \int \frac{d^{d} p}{\left(p^{2}+\omega^{2}\right)^{2}}=\frac{1}{d=1} \end{align*} $$

$$ \begin{align*} \therefore & =\int d^{d} \tau_{1} \Delta\left(\tau_{1}-\tau_{1}\right) \Delta_{\alpha \alpha}^{2}\left(\tau_{1}-\tau_{2}\right) \Delta\left(\tau_{2}-\tau_{2}\right) \\ & =\left(\int \frac{d^{d} q}{q^{2}+\omega^{2}}\right)^{2} \int \frac{d^{d} p\left(p^{2}\right)^{2}}{\left(p^{2}+\omega^{2}\right)^{2}} \underset{d=1}{=}-\frac{3}{16 \omega} \\ & =\int d^{d} \tau_{1} \Delta_{\alpha \alpha}\left(\tau_{1}-\tau_{1}\right) \Delta^{2}\left(\tau_{1}-\tau_{2}\right) \Delta_{\beta \beta}\left(\tau_{2}-\tau_{2}\right) \\ & =\left[\int \frac{d^{d} q\left(q^{2}\right)^{2}}{q^{2}+\omega^{2}}\right]^{2} \int \frac{d^{d} k}{\left(k^{2}+\omega^{2}\right)^{2}} \underset{d=1}{=} \frac{1}{16 \omega} \\ & =\int d^{d} \tau_{1} \Delta\left(\tau_{1}-\tau_{1}\right) \Delta_{\alpha}^{2}\left(\tau_{1}-\tau_{2}\right) \Delta_{\beta \beta}\left(\tau_{2}-\tau_{2}\right) \\ & =\int \frac{d^{d} k}{k^{2}+\omega^{2}} \int \frac{d^{d} p p^{2}}{\left(p^{2}+\omega^{2}\right)^{2}} \int \frac{d^{d} q q^{2}}{q^{2}+\omega^{2}}=-\frac{1}{d=1} \\ & =\int d^{d} \tau_{1} \Delta_{\alpha \alpha}\left(\tau_{1}-\tau_{1}\right) \Delta^{2}\left(\tau_{1}-\tau_{2}\right) \Delta\left(\tau_{2}-\tau_{2}\right) \end{align*} $$

$$ \begin{align*} & =\int \frac{d^{d} k k^{2}}{k^{2}+\omega^{2}} \int \frac{d^{d} p}{\left(p^{2}+\omega^{2}\right)^{2}} \int \frac{d^{d} q}{q^{2}+\omega^{2}} \underset{d=1}{=}-\frac{1}{16 \omega^{3}}, \\ & =\int d^{d} \tau_{1} \Delta\left(\tau_{1}-\tau_{1}\right) \Delta_{\alpha}^{2}\left(\tau_{1}-\tau_{2}\right) \Delta\left(\tau_{2}-\tau_{2}\right) \\ & =\int \frac{d^{d} k}{k^{2}+\omega^{2}} \int \frac{d^{d} p p^{2}}{\left(p^{2}+\omega^{2}\right)^{2}} \int \frac{d^{d} q}{q^{2}+\omega^{2}} \underset{d=1}{=} \frac{1}{16 \omega^{3}}, \\ & =\int d^{d} \tau_{1} \Delta\left(\tau_{1}-\tau_{1}\right) \Delta^{2}\left(\tau_{1}-\tau_{2}\right) \Delta\left(\tau_{2}-\tau_{2}\right) \\ & =\int \frac{d^{d} k}{k^{2}+\omega^{2}} \int \frac{d^{d} p}{\left(p^{2}+\omega^{2}\right)^{2}} \int \frac{d^{d} q}{q^{2}+\omega^{2}} \underset{d=1}{=} \frac{1}{16 \omega^{5}} . \end{align*} $$

$$ \begin{align*} J\left(p^{2}\right) & =\int \frac{d^{d} k}{\left(k^{2}+\omega^{2}\right)\left[(k+p)^{2}+\omega^{2}\right]}=\int_{0}^{1} d x \int \frac{d^{d} k}{\left[k^{2}+p^{2} x(1-x)+\omega^{2}\right]^{2}} \\ & =\frac{\Gamma(2-d / 2)}{(4 \pi)^{d / 2}}\left(\frac{p^{2}+4 \omega^{2}}{4}\right)^{d / 2-2} F\left(2-\frac{d}{2}, \frac{1}{2} ; \frac{3}{2} ; \frac{p^{2}}{p^{2}+4 \omega^{2}}\right) \end{align*} $$

$$ J\left(p^{2}\right)=\frac{1}{\omega\left(p^{2}+4 \omega^{2}\right)} $$

$$ J_{\alpha_{1} \ldots \alpha_{n}}(p)=\int \frac{d^{d} k k_{\alpha_{1}} \cdots k_{\alpha_{n}}}{\left(k^{2}+\omega^{2}\right)\left[(k+p)^{2}+\omega^{2}\right]} $$

$$ J_{\alpha_{1} \ldots \alpha_{n}, \beta_{1} \ldots \beta_{m}}(p)=\int \frac{d^{d} k k_{\alpha_{1}} \cdots k_{\alpha_{n}}(k+p)_{\beta_{1}} \cdots(k+p)_{\beta_{m}}}{\left(k^{2}+\omega^{2}\right)\left[(k+p)^{2}+\omega^{2}\right]} . $$

$$ J_{\alpha, \beta}(p)=J_{\alpha}(p) p_{\beta}+J_{\alpha \beta}(p) $$

$$ J_{\alpha}(p)=\int \frac{d^{d} k k_{\alpha}}{\left(k^{2}+\omega^{2}\right)\left[(k+p)^{2}+\omega^{2}\right]}=-\frac{1}{2} p_{\alpha} J\left(p^{2}\right) $$

$$ \begin{align*} J_{\alpha \beta}(p)=\int & \frac{d^{d} k k_{\alpha} k_{\beta}}{\left(k^{2}+\omega^{2}\right)\left[(k+p)^{2}+\omega^{2}\right]}=\left[\delta_{\alpha \beta}+(d-2) \frac{p_{\alpha} p_{\beta}}{p^{2}}\right] \frac{I}{2(d-1)} \\ & +\left[-\delta_{\alpha \beta}\left(p^{2}+4 \omega^{2}\right)+\frac{p_{\alpha} p_{\beta}}{p^{2}}\left(d p^{2}+4 \omega^{2}\right)\right] \frac{J\left(p^{2}\right)}{4(d-1)} \end{align*} $$

$$ J_{\alpha \alpha}(p)=\int \frac{d^{d} k k^{2}}{\left(k^{2}+\omega^{2}\right)\left[(k+p)^{2}+\omega^{2}\right]}=I-\omega^{2} J\left(p^{2}\right) $$

$$ J_{\alpha \alpha \beta}(p)=\int \frac{d^{d} k k^{2} k_{\beta}}{\left(k^{2}+\omega^{2}\right)\left[(k+p)^{2}+\omega^{2}\right]}=\frac{1}{2} p_{\beta}\left[-I+\omega^{2} J\left(p^{2}\right)\right] $$

$$ K(a, b)=\int d^{d} p\left(p^{2}\right)^{a} J^{b}\left(p^{2}\right), \quad a \geq 0, \quad b \geq 1, \quad a \leq b $$

$$ J\left(p^{2}\right)=-\frac{\partial I}{\partial \omega^{2}}+\frac{1}{2} p^{2} \frac{\partial J\left(p^{2}\right)}{\partial \omega^{2}}-2 p^{2} \frac{\partial J\left(p^{2}\right)}{\partial p^{2}} $$

$$ K(a, b)=\frac{2 b(d / 2-1) I K(a-1, b-1)-2 \omega^{2}(2 a-2-b+d) K(a-1, b)}{(b+1) d / 2-2 b+a} $$

$$ \begin{align*} & K(0,0)=0, \quad K(0,1)=\int d^{d} p J\left(p^{2}\right)=I^{2} \\ & K(0,2)=\int d^{d} p J^{2}\left(p^{2}\right)=A, \ldots \end{align*} $$

$$ A \equiv \int \frac{d^{d} p_{1} d^{d} p_{2} d^{d} k}{\left(p_{1}^{2}+\omega^{2}\right)\left(p_{2}^{2}+\omega^{2}\right)\left(k^{2}+\omega^{2}\right)\left[\left(p_{1}+p_{2}+k\right)^{2}+\omega^{2}\right]} $$

$$ \mathscr{O}=\int d^{d} \tau_{1} \Delta\left(\tau_{1}-\tau_{2}\right) \Delta\left(\tau_{1}-\tau_{2}\right) \Delta\left(\tau_{1}-\tau_{2}\right) \Delta\left(\tau_{1}-\tau_{2}\right)=\int d^{d} k J^{2}(k)=A $$

$$ A=\int_{-\infty}^{\infty} d \tau \Delta^{4}(\tau, 0)=\int_{-\infty}^{\infty} d x\left(\frac{1}{2 \omega} e^{-\omega|x|}\right)^{4}=\frac{1}{32 \omega^{5}} $$

$$ K(a, 0) \equiv 0 $$

$$ \begin{align*} \int d^{d} p p^{2} J\left(p^{2}\right) & =K(1,1)=-2 \omega^{2} I^{2} \\ \int d^{d} p p^{2} J^{2}\left(p^{2}\right) & =K(1,2)=\frac{4}{3}\left(I^{3}-\omega^{2} A\right) \\ \int d^{d} p\left(p^{2}\right)^{2} J^{2}\left(p^{2}\right) & =K(2,2)=-8 \omega^{2} \frac{(6-5 d) I^{3}+2 d \omega^{2} A}{3(4-3 d)} \end{align*} $$

$$ \begin{align*} = & \int d^{d} \tau_{1} \Delta^{2}\left(\tau_{1}-\tau_{2}\right) \Delta_{\alpha \beta}^{2}\left(\tau_{1}-\tau_{2}\right) \\ = & \int d^{d} p d^{d} k d^{d} q \frac{(p k)^{2}}{\left(p^{2}+\omega^{2}\right)\left(k^{2}+\omega^{2}\right)\left(q^{2}+\omega^{2}\right)\left[(p+k+q)^{2}+\omega^{2}\right]} \\ \stackrel{=}{q \rightarrow q-p} \int d^{d} q J_{\alpha \beta}(q) J_{\alpha \beta}(q)=\int d^{d} k \frac{1}{16}\left(k^{2}\right)^{2} J^{2}(k) & \\ & +\int d^{d} k \frac{1}{4(d-1)}\left\{d I^{2}+\left[(d-2) k^{2}-4 \omega^{2}\right] I J(k)+\frac{1}{4}\left(k^{2}+4 \omega^{2}\right)^{2} J^{2}(k)\right\} \\ = & -\frac{\omega^{2}}{2} \frac{(6-5 d) I^{3}+2 d \omega^{2} A}{3(4-3 d)}-\frac{\omega^{2}}{6(4-3 d)}\left[(6-5 d) I^{3}+2 d \omega^{2} A\right] \\ = & -\frac{\omega^{2}}{3(4-3 d)}\left[(8-7 d) I^{3}+(d+4) \omega^{2} A\right]_{d=1}^{=}-\frac{\omega^{2}}{3}\left(I^{3}+5 \omega^{2} A\right)=-\frac{3}{32 \omega} \end{align*} $$

$$ \begin{align*} & =-\int d^{d} \tau_{1} \Delta\left(\tau_{1}-\tau_{2}\right) \Delta_{\alpha}\left(\tau_{1}-\tau_{2}\right) \Delta_{\beta}\left(\tau_{1}-\tau_{2}\right) \Delta_{\alpha \beta}\left(\tau_{1}-\tau_{2}\right) \\ & =\int d^{d} k d^{d} p_{1} d^{d} p_{2} \frac{\left(k p_{1}\right)\left(k p_{2}\right)}{\left(k^{2}+\omega^{2}\right)\left(p_{1}^{2}+\omega^{2}\right)\left(p_{2}^{2}+\omega^{2}\right)\left[\left(k+p_{1}+p_{2}\right)^{2}+\omega^{2}\right]} \\ & =\int d^{d} p\left[p_{\beta} J_{\alpha}(p) J_{\alpha \beta}(p)+J_{\alpha}(p) J_{\beta \alpha \beta}(p)\right] \\ & =-\frac{1}{8} \int d^{d} p p^{2} J\left(p^{2}\right)\left[\left(p^{2}+2 \omega^{2}\right) J\left(p^{2}\right)-2 I\right] \\ & =-\frac{\omega^{2}}{6(4-3 d)}\left[(8-5 d) I^{3}-2(4-d) \omega^{2} A\right]_{d=1}^{=}-\frac{\omega^{2}}{2}\left(I^{3}-2 \omega^{2} A\right)=-\frac{1}{32 \omega} \end{align*} $$

$$ =-\int d^{d} \tau_{1} \Delta_{\alpha}\left(\tau_{1}-\tau_{2}\right) \Delta_{\alpha}\left(\tau_{1}-\tau_{2}\right) \Delta_{\beta}\left(\tau_{1}-\tau_{2}\right) \Delta_{\beta}\left(\tau_{1}-\tau_{2}\right) $$

$$ \begin{align*} & =\int d^{d} \tau_{1} \Delta\left(\tau_{1}-\tau_{2}\right) \Delta_{\alpha \alpha}\left(\tau_{1}-\tau_{2}\right) \Delta_{\beta}\left(\tau_{1}-\tau_{2}\right) \Delta_{\beta}\left(\tau_{1}-\tau_{2}\right) \\ & +2 \int d^{d} \tau_{1} \Delta\left(\tau_{1}-\tau_{2}\right) \Delta_{\alpha}\left(\tau_{1}-\tau_{2}\right) \Delta_{\beta}\left(\tau_{1}-\tau_{2}\right) \Delta_{\alpha \beta}\left(\tau_{1}-\tau_{2}\right) \end{align*} $$

$$ \begin{align*} & \int d^{d} \tau_{1} \Delta\left(\tau_{1}-\tau_{2}\right) \Delta_{\alpha \alpha}\left(\tau_{1}-\tau_{2}\right) \Delta_{\beta}\left(\tau_{1}-\tau_{2}\right) \Delta_{\beta}\left(\tau_{1}-\tau_{2}\right) \\ & =\int d^{d} k d^{d} p_{1} d^{d} p_{2} \frac{k^{2}\left(p_{1} p_{2}\right)}{\left(k^{2}+\omega^{2}\right)\left(p_{1}^{2}+\omega^{2}\right)\left(p_{2}^{2}+\omega^{2}\right)\left[\left(k+p_{1}+p_{2}\right)^{2}+\omega^{2}\right]} \\ & =\int d^{d} p\left[p_{\alpha} J_{\alpha}(p) J_{\beta \beta}+J_{\alpha}(p) J_{\beta \alpha \beta}(p)\right]=\frac{\omega^{2}}{4} \int d^{d} p p^{2} J^{2}\left(p^{2}\right) \\ & =-\frac{\omega^{2}}{3}\left(I^{3}-\omega^{2} A\right) \underset{d=1}{=} \frac{1}{32 \omega} \end{align*} $$

$$ =\frac{1}{32 \omega} . $$

$$ \begin{align*} & =-\int d^{d} k d^{d} p_{1} d^{d} p_{2} \frac{p_{1} p_{2}}{\left(k^{2}+\omega^{2}\right)\left(p_{1}^{2}+\omega^{2}\right)\left(p_{2}^{2}+\omega^{2}\right)\left[\left(k+p_{1}+p_{2}\right)^{2}+\omega^{2}\right]} \\ \stackrel{=}{\overline{k \rightarrow k}-p_{2}}-\int d^{d} k d^{d} p_{1} d^{d} p_{2} \frac{p_{1} p_{2}}{\left[\left(k-p_{2}\right)^{2}+\omega^{2}\right]\left(p_{1}^{2}+\omega^{2}\right)\left(p_{2}^{2}+\omega^{2}\right)\left[\left(k+p_{1}\right)^{2}+\omega^{2}\right]} & \\ \stackrel{p_{2} \rightarrow-p_{2}}{=} \int d^{d} k d^{d} p_{1} \frac{p_{1 \alpha}}{\left(p_{1}^{2}+\omega^{2}\right)\left[\left(p_{1}+k\right)^{2}+\omega^{2}\right]} \int d^{d} p_{2} \frac{p_{2 \alpha}}{\left(p_{2}^{2}+\omega^{2}\right)\left[\left(p_{2}+k\right)^{2}+\omega^{2}\right]} & \\ & =\int d^{d} k J_{\alpha}^{2}(k)=\frac{1}{4} \int d^{d} k k^{2} J^{2}\left(k^{2}\right) \\ & =\frac{1}{4} \frac{4}{3}\left(I^{3}-\omega^{2} A\right) \underset{d=1}{=} \frac{1}{32 \omega^{3}} \end{align*} $$

$$ \delta^{(d)}(0)=\int \frac{d^{d} k}{(2 \pi)^{d}}=0 $$

$$ F_{2}^{(1)}=\eta^{2}\left[3\left(\frac{1}{2}+a\right)\left(-\frac{1}{8 \omega}\right)+15 \omega^{2}\left(\frac{1}{18}+\frac{a}{5}\right)\right]=-\frac{\eta^{2}}{12 \omega} $$

$$ \begin{align*} F_{2}^{(3)}=-\frac{\eta^{2}}{2!} & {\left[4\left(-\frac{1}{16 \omega}\right)+2\left(-\frac{3}{16 \omega}\right)+2\left(\frac{1}{16 \omega}\right)\right.} \\ & \left.+8 \omega^{2}\left(-\frac{1}{16 \omega^{3}}\right)+8 \omega^{2}\left(\frac{1}{16 \omega^{3}}\right)+8 \omega^{4}\left(\frac{1}{16 \omega^{5}}\right)\right]=0 . \end{align*} $$

$$ \beta F_{2}^{(4)}=-\frac{\eta^{2}}{2!}\left[-\frac{3}{32 \omega}+4\left(-\frac{1}{32 \omega}\right)+\frac{1}{32 \omega}+4 \omega^{2}\left(\frac{1}{32 \omega^{3}}\right)+\frac{2}{3} \omega^{4}\left(\frac{1}{32 \omega^{5}}\right)\right]=\frac{\eta^{2}}{12 \omega}, $$

$$ \delta_{\alpha_{1} \ldots \alpha_{n}}^{(d)}(\tau) \equiv \partial_{\alpha_{1} \ldots \alpha_{n}} \delta^{(d)}(\tau)=\int d^{d} k(i k)_{\alpha_{1}} \ldots(i k)_{\alpha_{n}} e^{i k x} $$

$$ \delta_{\alpha_{1} \ldots \alpha_{n}}^{(d)}(0)=\int d^{d} k(i k)_{\alpha_{1}} \ldots(i k)_{\alpha_{n}}=0 $$

$$ \Delta(0)=\int \frac{d^{d} k}{k^{2}+\omega^{2}}=\frac{\omega^{d-2}}{(4 \pi)^{d / 2}} \Gamma\left(1-\frac{d}{2}\right)=I \underset{d=1}{=} \frac{1}{2 \omega} $$

$$ \Delta_{\alpha}(\tau)=\int d^{d} k \frac{i k_{\alpha}}{k^{2}+\omega^{2}} e^{i k \tau} $$

$$ \Delta_{\alpha}(0)=0 $$

$$ \Delta_{\alpha \alpha}(\tau)=-\int d^{d} k \frac{k^{2}}{k^{2}+\omega^{2}} e^{i k x}=-\delta^{(d)}(\tau)+\omega^{2} \Delta(\tau) $$

$$ \left(-\partial_{\alpha}^{2}+\omega^{2}\right) q(\tau)=\delta^{(d)}(\tau) $$

$$ \Delta_{\alpha \alpha}(0)=\omega^{2} \Delta(0) \underset{d=1}{=} \frac{\omega}{2} $$

$$ F_{1}=-g \eta\left[-\Delta_{\alpha \alpha}(0)+\omega^{2} \Delta(0)\right] \Delta(0)=0 . $$

$$ \begin{align*} F_{2}^{(1)} & =-\eta^{2} 3 g^{2}\left[\left(\frac{1}{2}+a\right) \Delta_{\alpha \alpha}(0)-5\left(\frac{1}{18}+\frac{a}{5}\right) \omega^{2} \Delta(0)\right] \Delta^{2}(0) \\ & =-\eta^{2} \frac{2}{3} \omega^{2} \Delta^{3}(0) \underset{d=1}{=}-\frac{\eta^{2}}{12 \omega} \end{align*} $$

$$ \begin{align*} \int d^{d} \tau \Delta^{2}(\tau) & =\int d^{d} p d^{d} k \frac{\delta^{(d)}(k+p)}{\left(p^{2}+\omega^{2}\right)\left(k^{2}+\omega^{2}\right)} \\ & =\int \frac{d^{d} k}{\left(k^{2}+\omega^{2}\right)^{2}}=\frac{\omega^{d-4}}{(4 \pi)^{d / 2}} \Gamma\left(2-\frac{d}{2}\right)=\frac{(2-d)}{2 \omega^{2}} \Delta(0) \end{align*} $$

$$ \begin{align*} \int d^{d} \tau \Delta_{\alpha}^{2}(\tau) & =-\int d^{d} \tau \Delta(\tau)\left[-\delta^{(d)}(\tau)+\omega^{2} \Delta(\tau)\right]=\Delta(0)-\omega^{2} \int d^{d} \tau \Delta^{2}(\tau) \\ & =\frac{d}{2} \Delta(0) \end{align*} $$

$$ \begin{align*} \int d^{d} \tau \Delta_{\alpha \beta}^{2}(\tau) & =\int d^{d} p d^{d} k \frac{(k p)^{2} \delta^{(d)}(k+p)}{\left(k^{2}+\omega^{2}\right)\left(p^{2}+\omega^{2}\right)} \\ & =\int d^{d} k \frac{\left(k^{2}\right)^{2}}{\left(k^{2}+\omega^{2}\right)^{2}}=\int d^{d} \tau \Delta_{\alpha \alpha}^{2}(\tau) \end{align*} $$

$$ \begin{align*} \int d^{d} \tau \Delta_{\alpha \alpha}^{2}(\tau) & =\int d^{d} k \frac{\left(k^{2}\right)^{2}}{\left(k^{2}+\omega^{2}\right)^{2}}=-2 \omega^{2} \int \frac{d^{d} k}{\left(k^{2}+\omega^{2}\right)}+\omega^{4} \int \frac{z^{d} k}{\left(k^{2}+\omega^{2}\right)^{2}} \\ & =-2 \omega^{2} \Delta(0)+\omega^{4} \int d^{d} \tau \Delta^{2}(\tau) \end{align*} $$

$$ \begin{align*} \int d^{d} \tau \Delta_{\alpha \beta}^{2}(\tau) & =\int d^{d} \tau \Delta_{\alpha \alpha}^{2}(\tau)=-2 \omega^{2} \Delta(0)+\omega^{4} \int d^{d} \tau \Delta^{2}(\tau) \\ & =-(1+d / 2) \omega^{2} \Delta(0) \end{align*} $$

$$ \partial_{\alpha} \Delta_{\alpha \beta}(\tau)=\partial_{\beta} \Delta_{\alpha \alpha}(\tau) $$

$$ \int d^{d} \tau\left[\Delta_{\alpha \beta}^{2}(\tau)+2 \omega^{2} \Delta_{\alpha}^{2}(\tau)+\omega^{4} \Delta^{2}(\tau)\right]=0 $$

$$ F_{2}^{(3)}=-g^{2} \Delta^{2}(0) \int d^{d} \tau\left[\Delta_{\alpha \beta}^{2}(\tau)+2 \omega^{2} \Delta_{\alpha}^{2}(\tau)+\omega^{4} \Delta^{2}(\tau)\right]=0 $$

$$ \begin{align*} & =\int d^{d} \tau \Delta^{2}(\tau) \Delta_{\alpha \beta}^{2}(\tau) \\ 4 & =4 \int d^{d} \tau \Delta(\tau) \Delta_{\alpha}(\tau) \Delta_{\beta}(\tau) \Delta_{\alpha \beta}(\tau) \\ & =\int d^{d} \tau \Delta_{\alpha}(\tau) \Delta_{\alpha}(\tau) \Delta_{\beta}(\tau) \Delta_{\beta}(\tau) \end{align*} $$

$$ Y_{d}=\int d^{d} \tau \Delta^{2}(\tau)\left[\Delta_{\alpha \beta}^{2}(\tau)-\Delta_{\alpha \alpha}^{2}(\tau)\right] $$

$$ \int d^{d} \tau \Delta^{2}(\tau) \Delta_{\alpha \beta}^{2}(\tau)=\int d^{d} \tau \Delta^{2}(\tau) \Delta_{\alpha \alpha}^{2}(\tau)+Y_{d} $$

$$ -\int d^{d} \tau \Delta_{\alpha \alpha}(\tau) \Delta^{3}(\tau)=\Delta^{3}(0)-\omega^{2} \int d^{d} \tau \Delta^{4}(\tau) $$

$$ \int d^{d} \tau \Delta_{\alpha \alpha}(\tau) \Delta^{3}(\tau)=-3 \int d^{d} \tau \Delta_{\alpha}^{2}(\tau) \Delta^{2}(\tau) $$

$$ \int d^{d} \tau \Delta_{\alpha}^{2}(\tau) \Delta^{2}(\tau)=\frac{1}{3} \Delta^{3}(0)-\frac{1}{3} \omega^{2} \int d^{d} \tau \Delta^{4}(\tau) $$

$$ \int d^{d} \tau \Delta_{\alpha \alpha}^{2}(\tau) \Delta^{2}(\tau)-\omega^{4} \int d^{d} \tau \Delta^{4}(\tau)+2 \omega^{2} \Delta^{3}(0)=0 $$

$$ \int d^{d} \tau \Delta_{\alpha \alpha}(\tau) \Delta_{\beta}^{2}(\tau) \Delta(\tau)=\omega^{2} \int d^{d} \tau \Delta_{\beta}^{2}(\tau) \Delta^{2}(\tau) $$

$$ \int d^{d} \tau \Delta_{\alpha \alpha}(\tau) \Delta_{\beta}^{2}(\tau) \Delta(\tau)=\frac{1}{3} \omega^{2} \Delta^{3}(0)-\frac{1}{3} \omega^{4} \int d^{d} \tau \Delta^{4}(\tau) $$

$$ \begin{align*} \int d^{d} \tau \partial_{\beta} \Delta_{\alpha \alpha}(\tau) \Delta_{\beta}(\tau) \Delta^{2}(\tau) & =-\int d^{d} \tau \Delta_{\alpha \alpha}^{2}(\tau) \Delta^{2}(\tau)-2 \int d^{d} \tau \Delta_{\alpha \alpha}(\tau) \Delta_{\beta}^{2}(\tau) \Delta(\tau) \\ & =\frac{4}{3} \omega^{2} \Delta^{3}(0)-\frac{1}{3} \omega^{4} \int d^{d} \tau \Delta^{4}(\tau) \end{align*} $$

$$ 4 \int d^{d} \tau \Delta(\tau) \Delta_{\alpha}(\tau) \Delta_{\beta}(\tau) \Delta_{\alpha \beta}(\tau)=4 \omega^{2} \int d^{d} \tau \Delta^{2}(\tau) \Delta_{\alpha}^{2}(\tau)-2 Y_{d} $$

$$ \int d^{d} \tau \Delta_{\alpha}^{2}(\tau) \Delta_{\beta}^{2}(\tau)=-3 \omega^{2} \int d^{d} \tau \Delta^{2}(\tau) \Delta_{\alpha}^{2}(\tau)+Y_{d} $$

$$ +4 \longleftrightarrow+\underset{\sim}{\Theta}=\int d^{d} \tau \Delta^{2}(\tau) \Delta_{\alpha \alpha}^{2}(\tau)+\omega^{2} \int d^{d} \tau \Delta^{2}(\tau) \Delta_{\alpha}^{2}(\tau) . $$

$$ \frac{2}{3} \omega^{4} \bigcirc=\frac{2}{3} \omega^{4} \int d^{d} \tau \Delta^{4}(\tau) $$

$$ 4 \omega^{2} \circlearrowleft \quad=4 \omega^{2} \int d^{d} \tau \Delta^{2}(\tau) \Delta_{\alpha}^{2}(\tau) $$

$$ \begin{align*} F_{2}^{(4)} & =-2 \eta^{2} g^{2} \int d^{d} \tau \Delta^{2}(\tau)\left[\frac{2}{3} \omega^{4} \Delta^{2}(\tau)+\Delta_{\alpha \alpha}^{2}(\tau)+5 \omega^{2} \Delta_{\alpha}^{2}(\tau)\right] \\ & =\eta^{2} \frac{2}{3} \omega^{2} \Delta^{3}(0) \underset{d=1}{=} \frac{\eta^{2}}{12 \omega} \end{align*} $$

$$ \Delta(\tau)=c_{d} y^{1-d / 2} K_{1-d / 2}(y) $$

$$ c_{d}=\frac{\omega^{d-2}}{(2 \pi)^{d / 2}} $$

$$ K_{\beta}(y) \underset{y \approx 0}{\approx} \frac{1}{2} \Gamma(\beta)(y / 2)^{\mp \beta}, \quad \operatorname{Re} \beta \gtrless 0 $$

$$ \Delta_{\alpha}(\tau)=-c_{d} y^{1-d / 2} K_{d / 2}(y) \partial_{\alpha} y $$

$$ \Delta(\tau) \propto \text { const., } \quad \Delta_{\alpha}(\tau) \propto|\tau|^{\varepsilon} \partial_{\alpha}|\tau| . $$

$$ \Delta_{\alpha \beta}(\tau)=\Delta(\tau)\left(\partial_{\alpha} y\right)\left(\partial_{\beta} y\right)+\frac{c_{d}}{(d-2)} y^{d / 2} K_{d / 2}(y) \partial_{\alpha \beta} y^{2-d} $$

$$ \partial_{\alpha \beta} y^{2-d}=(2-d) \frac{\omega^{2-d}}{|y|^{d}}\left(\delta_{\alpha \beta}-d \frac{y_{\alpha} y_{\beta}}{y^{2}}\right) $$

$$ \partial^{2} y^{2-d}=(2-d) \omega^{2-d} S_{d} \delta^{(d)}(\tau) $$

$$ \begin{align*} \Delta_{\alpha \alpha}(\tau) & =\omega^{2} \Delta(\tau)-c_{d} m^{2-d} S_{d} \frac{1}{2} \Gamma(d / 2) 2^{d / 2} \delta^{(d)}(\tau) \\ & =\omega^{2} \Delta(\tau)-\delta^{(d)}(\tau) \end{align*} $$

$$ \partial_{\alpha} \Delta_{\alpha \beta}(\tau)=\partial_{\beta}\left[-\delta^{(d)}(\tau)+\omega^{2} \Delta(\tau)\right]+\omega S_{d}\left[\Delta(\tau)|y|^{d-1}\left(\partial_{\beta} y\right)\right] \delta^{(d)}(\tau)=\partial_{\beta} \Delta_{\lambda \lambda}(\tau) $$

$$ \int d^{d} \tau f(\tau)=S_{d} \int_{0}^{\infty} d r r^{d-1} f(r), \quad r \equiv|x| $$

$$ \int_{0}^{\infty} d y y K_{\beta}^{2}(y)=\frac{1}{2} \frac{\pi \beta}{\sin \pi \beta}=\frac{1}{2} \Gamma(1+\beta) \Gamma(1-\beta) $$

$$ \begin{align*} \int d^{d} \tau \Delta^{2}(\tau) & =\omega^{-d} c_{d}^{2} S_{d} \int_{0}^{\infty} d y y K_{1-d / 2}^{2}(y) \\ & =\omega^{-d} c_{d}^{2} S_{d} \frac{1}{2}(1-d / 2) \Gamma(1-d / 2) \Gamma(d / 2)=\frac{2-d}{2 \omega^{2}} \Delta(0) \end{align*} $$

$$ \begin{align*} \int d^{d} \tau \Delta_{\alpha}^{2}(\tau) & =\omega^{2-d} c_{d}^{2} S_{d} \int_{0}^{\infty} d y y K_{d / 2}^{2}(y) \\ & =\omega^{2-d} c_{d}^{2} S_{d} \frac{1}{2} \Gamma(1+d / 2) \Gamma(1-d / 2)=\frac{d}{2} \Delta(0) \end{align*} $$

$$ K_{d / 2}(y)=-y^{d / 2-1} \frac{d}{d y}\left[y^{1-d / 2} K_{1-d / 2}(y)\right] $$

$$ \begin{align*} \int d^{d} \tau \Delta_{\alpha}^{2}(\tau) & =-\left.\omega^{2-d} c_{d}^{2} S_{d}\left(y^{d / 2} K_{d / 2}\right)\left(y^{1-d / 2} K_{1-d / 2}\right)\right|_{0} ^{\infty}-\omega^{2} \int d^{d} \tau \Delta^{2}(\tau) \\ & =\Delta(0)-\omega^{2} \int d^{d} \tau \Delta^{2}(\tau) \end{align*} $$

$$ \begin{align*} & \int d^{d} \tau \Delta_{\alpha \alpha}^{2}(\tau)=\int d^{d} \tau \Delta_{\alpha \beta}^{2}(\tau)=\omega^{4} \int d^{d} \tau \Delta^{2}(\tau)-\omega^{4-d} c_{d}^{2} \Gamma(d / 2) \Gamma(1-d / 2) S_{d} \\ & =\omega^{4} \int d^{d} \tau \Delta^{2}(\tau)-2 \omega^{2} \Delta(0)=-(1+d / 2) \omega^{2} \Delta(0) \end{align*} $$

$$ (d-1)\left[\int_{0}^{\infty} d z K_{d / 2}(z) K_{1-d / 2}(z)+\frac{d}{2} \int_{0}^{\infty} d z z^{-1} K_{d / 2}^{2}(z)\right] $$

$$ \int d^{d} \tau\left[\delta^{(d)}(\tau)\right]^{2}=\omega^{4} \int d^{d} \tau \Delta^{2}(\tau)+2 \omega^{2} \int d^{d} \tau \Delta_{\alpha}^{2}(\tau)+\int d^{d} \tau \Delta_{\beta \beta}^{2}(\tau)=0 $$

$$ \int d^{d} \tau \delta^{(d)}(\tau) \delta^{(d)}(\tau)=\delta^{(d)}(0)=0 $$

$$ \int d^{d} \tau \delta^{(d)}(\tau) f(\tau)=f(0) $$

$$ \begin{align*} \int d^{d} \tau \Delta^{4}(\tau) & =c_{d}^{4} \omega^{-d} S_{d} \int_{0}^{\infty} d y y^{3-d} K_{1-d / 2}^{4}(y) \\ & =c_{1}^{4} \omega^{-1} S_{1} \frac{\pi^{2}}{2^{4}} \Gamma^{4}\left(\frac{3}{2}-\frac{d}{2}\right) \Gamma(d)=\frac{1}{32 \omega^{5}} \end{align*} $$

$$ y \equiv \omega \tau $$

$$ \begin{align*} \int d^{d} \tau \Delta^{2}(\tau) \Delta_{\alpha}^{2}(\tau) & =\omega^{2-d} c_{d}^{4} S_{d} \int_{0}^{\infty} d y y^{3-d} K_{d / 2}^{2}(y) K_{1-d / 2}^{2}(y) \\ & =\frac{1}{3} \omega^{2-d} c_{d}^{4} S_{d}\left[2^{-d-1} \Gamma(d / 2) \Gamma^{3}(1-d / 2)\right. \\ & \left.+\int_{0}^{\infty} d y\left(y^{1-d / 2} K_{1-d / 2}\right)^{3} \frac{d}{d y}\left(y^{d / 2} K_{d / 2}\right)\right] \\ & =\frac{1}{3}\left[\Delta^{3}(0)-\omega^{2} \int d^{d} \tau \Delta^{4}(\tau)\right] \underset{d=1}{=} \frac{1}{32 \omega} \end{align*} $$

$$ \int d^{d} \tau \Delta(\tau) \Delta_{\alpha}(\tau) \Delta_{\beta}(\tau) \Delta_{\alpha \beta}(\tau)=\omega^{2} \int d^{d} \tau \Delta^{2}(\tau) \Delta_{\alpha}^{2}(\tau)-\frac{1}{2} Y_{d} $$

$$ Y_{d}=-2(d-1) \omega^{4-d} c_{d}^{4} S_{d} \int_{0}^{\infty} d y y^{2-d} K_{1-d / 2}(y) K_{d / 2}^{3}(y) $$

$$ K_{\beta}(y)=\pi^{-1 / 2}(y / 2)^{-\beta} \Gamma\left(\frac{1}{2}+\beta\right) \int_{0}^{\infty} d t(\cosh t)^{-2 \beta} \cos (y \sinh t) $$

$$ \begin{align*} K_{(1 \mp \varepsilon) / 2}(y) & =\pi^{-1 / 2}(y / 2)^{-(1 \mp \varepsilon) / 2} \Gamma\left(1 \mp \frac{\varepsilon}{2}\right) \\ & \times\left[\frac{\pi}{2} e^{-y} \pm \varepsilon \int_{0}^{\infty} d t(\cosh t)^{-1} \ln (\cosh t) \cos (y \sinh t)\right] \end{align*} $$

$$ \begin{align*} Y_{d} & \underset{\varepsilon \approx 0}{\approx} 2\left(\omega^{4-d} c_{d}^{4} S_{d}\right) \varepsilon \frac{\pi^{2}}{4} \Gamma(1+\varepsilon / 2) \Gamma^{3}(1-\varepsilon / 2) \times 2^{-5 \varepsilon} \Gamma(2 \varepsilon) \\ & \underset{\varepsilon \rightarrow 0}{=}\left(\frac{1}{2 \omega \pi^{2}}\right) \frac{\pi^{2}}{4}=\frac{1}{8 \omega} \end{align*} $$

$$ \begin{align*} & \int d^{d} \tau \Delta^{2}(\tau) {\left[\Delta_{\alpha \beta}^{2}(\tau)-\Delta_{\alpha \alpha}^{2}(\tau)\right] } \\ &=-(d-1) \omega^{4-d} c_{d}^{4} S_{d} \int_{0}^{\infty} d y\left[y^{1-d / 2} K_{1-d / 2}(y)\right]^{2} \frac{d}{d y} K_{d / 2}^{2}(y) \end{align*} $$

$$ \int_{0}^{\infty} d y\left[y^{1-d / 2}(y) K_{1-d / 2}(y)\right]^{2} \frac{d}{d y} K_{d / 2}^{2}(y)=2 \int_{0}^{\infty} d y y^{2-d} K_{1-d / 2}(y) K_{d / 2}^{3}(y) $$

$$ \int d^{d} \tau \Delta_{\alpha}^{2}(\tau) \Delta_{\beta}^{2}(\tau)=-3 \omega^{2} \int d^{d} \tau \Delta^{2}(\tau) \Delta_{\alpha}^{2}(\tau)+Y_{d} $$

$$ \int d^{d} \tau \Delta^{2}(\tau) \Delta_{\lambda \lambda}^{2}(\tau)=\left[-2 \omega^{2} \Delta^{3}(0)+\omega^{4} \int d^{d} \tau \Delta^{4}(\tau)\right]_{d=1}=-\frac{7}{32 \omega} $$

$$ \begin{align*} & I_{1}=\int_{-\infty}^{\infty} d \tau \ddot{\Delta}^{2}(\tau) \Delta^{2}(\tau) \\ & I_{2}=\int_{-\infty}^{\infty} d \tau \ddot{\Delta}(\tau) \dot{\Delta}^{2}(\tau) \Delta(\tau) \end{align*} $$

$$ I_{1}=\int_{-\infty}^{\infty} d \tau \ddot{\Delta}^{2}(\tau) \Delta^{2}(\tau)=I_{1}^{\mathrm{div}}+I_{1}^{R} $$

$$ I_{1}^{\mathrm{div}}=\Delta^{2}(0) \int_{-\infty}^{\infty} d \tau \delta^{2}(\tau), \quad I_{1}^{R}=\int_{-\infty}^{\infty} d \tau \Delta^{2}(\tau)\left[\ddot{\Delta}^{2}(\tau)-\delta^{2}(\tau)\right] $$

$$ \begin{align*} \int_{-\infty}^{\infty} d \tau \Delta^{4}(\tau) & =\frac{1}{4 \omega^{2}} \Delta^{3}(0) \\ \int_{-\infty}^{\infty} d \tau \dot{\Delta}^{2}(\tau) \Delta^{2}(\tau) & =\frac{1}{4} \Delta^{3}(0) \\ \int_{-\infty}^{\infty} d \tau \dot{\Delta}^{4}(\tau) & =\frac{1}{4} \omega^{2} \Delta^{3}(0) \end{align*} $$

$$ \begin{gather*} -\frac{4}{2!} \int_{-\infty}^{\infty} d \tau\left[\Delta^{2}(\tau) \ddot{\Delta}^{2}(\tau)+4 \Delta(\tau) \dot{\Delta}^{2}(\tau) \ddot{\Delta}(\tau)+\dot{\Delta}^{4}(\tau)+4 \omega^{2} \Delta^{2}(\tau) \dot{\Delta}^{2}(\tau)+\frac{2}{3} \omega^{4} \Delta^{4}(\tau)\right] \\ =-2 \Delta^{2}(0) \int_{-\infty}^{\infty} d \tau \delta^{2}(\tau)-2\left(I_{1}^{R}+4 I_{2}\right)-\frac{17}{6} \omega^{2} \Delta^{3}(0) \end{gather*} $$

$$ \Sigma(\text { all })=3\left[\delta(0)-\int_{-\infty}^{\infty} d \tau \delta^{2}(\tau)\right] \Delta^{2}(0)-2\left(I_{1}^{R}+4 I_{2}\right)-\frac{7}{2} \omega^{2} \Delta^{3}(0) $$

$$ \int d \tau \delta^{2}(\tau) f(\tau) \equiv \delta(0) f(0) $$

$$ I_{1}^{R}+4 I_{2}=-\frac{7}{4} \omega^{2} \Delta^{3}(0)=-\frac{7}{32 \omega} $$

$$ \begin{align*} I_{1}^{R} & =\int_{-\infty}^{\infty} d \tau \Delta^{2}(\tau)\left[\ddot{\Delta}^{2}(\tau)-\delta^{2}(\tau)\right] \\ & =-2 \omega^{2} \Delta^{3}(0)+\omega^{4} \int_{-\infty}^{\infty} d \tau \Delta^{4}(\tau)=-\frac{7}{4} \omega^{2} \Delta^{3}(0)=-\frac{7}{32 \omega} \end{align*} $$

$$ \begin{align*} I_{2} & =-\int_{-\infty}^{\infty} d \tau \dot{\Delta}^{2}(\tau) \Delta(\tau) \delta(\tau)+\omega^{2} \int_{-\infty}^{\infty} d \tau \dot{\Delta}^{2}(\tau) \Delta^{2}(\tau) \\ & =-\frac{1}{8 \omega} \int_{-\infty}^{\infty} d \tau \epsilon^{2}(\tau) \delta(\tau)+\frac{1}{4} \omega^{2} \Delta^{3}(0)=\frac{1}{8 \omega}\left(-I_{\epsilon^{2} \delta}+\frac{1}{4}\right) \end{align*} $$

$$ I_{\epsilon^{2} \delta}=\int_{-\infty}^{\infty} d \tau \epsilon^{2}(\tau) \delta(\tau) $$

$$ I_{2}=\frac{1}{3} \int_{-\infty}^{\infty} d \tau \Delta(\tau) \frac{d}{d \tau}\left[\dot{\Delta}^{3}(\tau)\right]=-\frac{1}{3} \int_{-\infty}^{\infty} d \tau \dot{\Delta}^{4}(\tau)=-\frac{1}{12} \omega^{2} \Delta^{3}(0)=-\frac{1}{96 \omega} $$

$$ I_{\epsilon^{2} \delta}=\int_{-\infty}^{\infty} d \tau \epsilon^{2}(\tau) \delta(\tau)=0 $$

$$ \int d \tau \epsilon^{2 n+1}(\tau) \delta(\tau)=0, \quad n=\text { integer } $$

$$ \begin{align*} I_{2} & =\frac{1}{2} \int_{-\infty}^{\infty} d \tau \Delta(\tau) \dot{\Delta}(\tau) \frac{d}{d \tau}\left[\dot{\Delta}^{2}(\tau)\right] \\ & =-\frac{1}{2} \int_{-\infty}^{\infty} d \tau \dot{\Delta}^{4}(\tau)-\frac{1}{2} \int_{-\infty}^{\infty} d \tau \Delta(\tau) \dot{\Delta}^{2}(\tau) \ddot{\Delta}(\tau) \end{align*} $$

$$ \begin{align*} I_{2} & =-\frac{1}{2} \int_{-\infty}^{\infty} d \tau \dot{\Delta}^{4}(\tau)+\frac{1}{2} \int_{-\infty}^{\infty} d \tau \dot{\Delta}^{2}(\tau) \Delta(\tau) \delta(\tau)-\frac{1}{2} \omega^{2} \int_{-\infty}^{\infty} d \tau \dot{\Delta}^{2}(\tau) \Delta^{2}(\tau) \\ & =\frac{1}{16 \omega} I-\frac{1}{4} \omega^{2} \Delta^{3}(0)=-\frac{1}{32 \omega} \end{align*} $$

$$ \begin{align*} I_{1}^{R} & =\int_{-\infty}^{\infty} d \tau \Delta^{2}(\tau)\left[\ddot{\Delta}^{2}(\tau)-\delta^{2}(\tau)\right] \\ & =\int_{-\infty}^{\infty} d \tau\left[-\dddot{\Delta}(\tau) \dot{\Delta}(\tau) \Delta^{2}(\tau)-2 \ddot{\Delta}(\tau) \dot{\Delta}^{2}(\tau) \Delta(\tau)-\Delta^{2}(\tau) \delta^{2}(\tau)\right] \\ & =\int_{-\infty}^{\infty} d \tau\left[\dot{\Delta}(\tau) \Delta^{2}(\tau) \dot{\delta}(\tau)-\Delta^{2}(\tau) \delta^{2}(\tau)\right]-2 I_{2}-\omega^{2} \int_{-\infty}^{\infty} d \tau \dot{\Delta}^{2}(\tau) \Delta^{2}(\tau) \end{align*} $$

$$ \begin{gather*} \int_{-\infty}^{\infty} d \tau\left[\dot{\Delta}(\tau) \Delta^{2}(\tau) \dot{\delta}(\tau)-\Delta^{2}(\tau) \delta^{2}(\tau)\right]=-\int_{-\infty}^{\infty} d \tau\left[\ddot{\Delta}(\tau) \Delta^{2}(\tau)+2 \dot{\Delta}^{2}(\tau) \Delta(\tau)\right] \delta(\tau) \\ -\int_{-\infty}^{\infty} d \tau \Delta^{2}(\tau) \delta^{2}(\tau)=-\omega^{2} \Delta^{3}(0)-\frac{1}{4 \omega} I \end{gather*} $$

$$ I_{1}^{R}=\int_{-\infty}^{\infty} d \tau \Delta^{2}(\tau)\left[\ddot{\Delta}^{2}(\tau)-\delta^{2}(\tau)\right]=-2 I_{2}-\frac{5}{4} \omega^{2} \Delta^{3}(0)-\frac{1}{4 \omega} I=-\frac{3}{32 \omega} $$

$$ \left(-\partial_{\alpha}^{2}+\omega^{2}\right) \Delta(\tau)=\delta^{(d)}(\tau) $$

$$ \int d^{d} \tau\left\langle q_{\alpha}^{2}(\tau) q^{2}(\tau) q_{\beta}^{2}(0) q^{2}(0)\right\rangle $$

$$ \begin{align*} I_{1}^{d} & =\int d^{d} \tau\langle\sqrt[q_{\alpha}(\tau) q_{\alpha}(\tau) q(\tau) q(\tau) q_{\beta}(0) q_{\beta}(0) q(0) q(0)]{ } \\ & =\int d^{d} \tau \Delta^{2}(\tau) \Delta_{\alpha \beta}^{2}(\tau) \\ I_{2}^{d} & =\int d^{d} \tau\langle\sqrt[q_{\alpha}(\tau) q_{\alpha}(\tau) q(\tau) q(\tau) q_{\beta}(0) q_{\beta}(0) q(0) q(0)]{ } \\ & =\int d^{d} \tau \Delta(\tau) \Delta_{\alpha}(\tau) \Delta_{\beta}(\tau) \Delta_{\alpha \beta}(\tau) \end{align*} $$

$$ I_{2}^{d}=-\frac{1}{2} \int d^{d} \tau \Delta_{\beta}^{2}(\tau)\left[\Delta_{\alpha}^{2}(\tau)+\Delta(\tau) \Delta_{\alpha \alpha}(\tau)\right] $$

$$ I_{1}^{d}=-2 I_{2}^{d}+\int d^{d} \tau \Delta^{2}(\tau) \Delta_{\alpha \alpha}^{2}(\tau)+2 \int d^{d} \tau \Delta(\tau) \Delta_{\beta}^{2}(\tau) \Delta_{\alpha \alpha}(\tau) $$

$$ I_{\epsilon^{2} \delta}^{R}=\left[\int_{-\infty}^{\infty} d \tau \epsilon^{2}(\tau) \delta(\tau)\right]^{R}=8 \omega \int d^{d} \tau \Delta_{\beta}^{2}(\tau) \Delta(\tau) \delta^{(d)}(\tau)=0 $$

$$ \begin{align*} \beta F_{\omega} & =\frac{D}{2 \beta} \operatorname{Tr} \log \left(-\partial^{2}+\omega^{2}\right)=\frac{D}{2 \beta} \sum_{n} \log \left(\omega_{n}^{2}+\omega^{2}\right) \\ & =\frac{D}{\beta} \log [2 \sinh (\hbar \beta \omega / 2)] \end{align*} $$

$$ G_{\mu \nu}^{(2)}\left(0, \tau^{\prime}\right)=G_{\mu \nu}^{(2)}\left(\beta, \tau^{\prime}\right)=0, \quad G_{\mu \nu}^{(2)}(\tau, 0)=G_{\mu \nu}^{(2)}(\tau, \beta)=0 $$

$$ G_{\mu \nu}^{(2)}\left(\tau, \tau^{\prime}\right)=\delta_{\mu \nu} \Delta\left(\tau, \tau^{\prime}\right)= $$

$$ \Delta\left(\tau, \tau^{\prime}\right)=\Delta\left(\tau^{\prime}, \tau\right)=\frac{1}{\beta}\left(\beta-\tau_{>}\right) \tau_{<}=\frac{1}{2}\left[-\epsilon\left(\tau-\tau^{\prime}\right)\left(\tau-\tau^{\prime}\right)+\tau+\tau^{\prime}\right]-\frac{\tau \tau^{\prime}}{\beta}, $$

$$ \cdot \Delta\left(\tau, \tau^{\prime}\right)=\Delta^{\bullet}\left(\tau, \tau^{\prime}\right)=-\delta\left(\tau-\tau^{\prime}\right) $$

$$ \Delta^{\cdot}\left(\tau, \tau^{\prime}\right)=\delta\left(\tau-\tau^{\prime}\right)-1 / \beta $$

$$ \Delta\left(\tau, \tau^{\prime}\right) \equiv \frac{d}{d \tau} \Delta\left(\tau, \tau^{\prime}\right), \quad \Delta^{\cdot}\left(\tau, \tau^{\prime}\right) \equiv \frac{d}{d \tau^{\prime}} \Delta\left(\tau, \tau^{\prime}\right) $$

$$ \Delta\left(\tau, \tau^{\prime}\right)=-\frac{1}{2} \epsilon\left(\tau-\tau^{\prime}\right)+\frac{1}{2}-\frac{\tau^{\prime}}{\beta}, \quad \Delta^{\cdot}\left(\tau, \tau^{\prime}\right)=\frac{1}{2} \epsilon\left(\tau-\tau^{\prime}\right)+\frac{1}{2}-\frac{\tau}{\beta}={ }^{\cdot} \Delta\left(\tau^{\prime}, \tau\right) $$

$$ \begin{align*} \partial_{\tau} G_{\mu \nu}^{(2)}\left(\tau, \tau^{\prime}\right) & \equiv\left\langle\dot{q}_{\mu}(\tau) q_{\nu}\left(\tau^{\prime}\right)\right\rangle \\ \partial_{\tau^{\prime}} G_{\mu \nu}^{(2)}\left(\tau, \tau_{\mu \nu} \cdot \Delta\left(\tau, \tau^{\prime}\right)\right. & \equiv\left\langle q_{\mu}(\tau) \dot{q}_{\nu}\left(\tau^{\prime}\right)\right\rangle=\delta_{\mu \nu} \Delta^{\cdot}\left(\tau, \tau^{\prime}\right)=\ldots, \end{align*} $$

$$ \partial_{\tau} \partial_{\tau^{\prime}} G_{\mu \nu}^{(2)}\left(\tau, \tau^{\prime}\right)=\delta_{\mu \nu} \cdot \Delta^{\cdot}\left(\tau, \tau^{\prime}\right)=\cdots \cdots, $$

$$ Z=\int d^{D} x_{a}\left(\mathbf{x}_{a} \beta \mid \mathbf{x}_{a} 0\right) $$

$$ \left(\mathbf{x}_{a} \beta \mid \mathbf{x}_{a} 0\right)_{0}=\int \mathcal{D}^{D} x e^{-\mathcal{A}^{(0)}[\mathbf{x}]} $$

$$ \mathcal{A}^{(0)}[x]=\frac{1}{2} \int_{0}^{\beta} d \tau \dot{\mathbf{x}}^{2}(\tau) $$

$$ \left(\mathbf{x}_{a} \beta \mid \mathbf{x}_{a} 0\right)_{0}=e^{-(D / 2) \operatorname{Tr} \log \left(-\partial^{2}\right)}=[2 \pi \beta]^{-D / 2} $$

$$ \mathcal{A}[q]=\frac{1}{2} \int_{0}^{\beta} d \tau g_{\mu \nu}(q(\tau)) \dot{q}^{\mu}(\tau) \dot{q}^{\nu}(\tau), \text { with } g_{\mu \nu}(q) \equiv \frac{\partial x^{i}(q)}{\partial q^{\mu}} \frac{\partial x^{i}(q)}{\partial q^{\nu}} $$

$$ \int \mathcal{D}^{D} x(\tau) \equiv \prod_{\tau} \int d^{D} x(\tau)=J \prod_{\tau} \int d^{D} q(\tau) \equiv J \int \mathcal{D}^{D} q \sqrt{g\left(q_{a}\right)} $$

$$ J=\prod_{\tau}\left[\sqrt{\frac{\partial x^{i}(q(\tau))}{\delta q^{\mu}(\tau)}} / \sqrt{\frac{\partial x^{i}\left(q_{a}\right)}{\delta q_{0}^{\mu}}}\right]=\exp \left[\frac{1}{2} \delta(0) \int_{0}^{\beta} d \tau \log \frac{g(q(\tau))}{g\left(q_{a}\right)}\right] $$

$$ \left(\mathbf{x}_{a} \beta \mid \mathbf{x}_{a} 0\right)_{0} \equiv\left(q_{a} \beta \mid q_{a} 0\right)_{0}=\int \mathcal{D}^{D} q e^{-\mathcal{A}_{\mathrm{tot}}[x]} $$

$$ \mathcal{A}_{\mathrm{tot}}[q]=\int_{0}^{\beta} d \tau\left[\frac{1}{2} g_{\mu \nu}(q(\tau)) \dot{q}^{\mu}(\tau) \dot{q}^{\nu}(\tau)-\frac{1}{2} \delta(0) \log \frac{g(q(\tau))}{g\left(q_{a}\right)}\right] $$

$$ \mathcal{A}^{(0)}\left[q_{a}, \delta q\right]=\frac{1}{2} \int_{0}^{\beta} d \tau g_{\mu \nu}\left(q_{a}\right) \delta \dot{q}^{\mu}(\tau) \delta \dot{q}^{\nu}(\tau) $$

$$ \begin{align*} \mathcal{A}_{\mathrm{tot}}^{\mathrm{int}}\left[q_{a}, \delta q\right] & =\int_{0}^{\beta} d \tau \frac{1}{2}\left[g_{\mu \nu}(q)-g_{\mu \nu}\left(q_{a}\right)\right] \delta \dot{q}^{\mu} \delta \dot{q}^{\nu} \\ & -\int_{0}^{\beta} d \tau \frac{1}{2} \delta(0)\left\{\left[\frac{g\left(q_{a}+\delta q\right)}{g\left(q_{a}\right)}-1\right]-\frac{1}{2}\left[\frac{g\left(q_{a}+\delta q\right)}{g\left(q_{a}\right)}-1\right]^{2}+\ldots\right\} \end{align*} $$

$$ \begin{align*} \left(q_{a} \beta \mid q_{a} 0\right) & =\int \mathcal{D}^{D} q e^{\mathcal{A}^{(0)}[q]-\mathcal{A}_{\mathrm{tot}}^{\mathrm{int}}[q]}=\int \mathcal{D}^{D} q e^{-\mathcal{A}^{(0)}[q]}\left(1-\mathcal{A}_{\mathrm{tot}}^{\mathrm{int}}+\frac{1}{2} \mathcal{A}_{\mathrm{tot}}^{\mathrm{int} 2}-\ldots\right) \\ & =(2 \pi \beta)^{-D / 2}\left[1-\left\langle\mathcal{A}_{\mathrm{tot}}^{\mathrm{int}}\right\rangle+\frac{1}{2}\left\langle\mathcal{A}_{\mathrm{tot}}^{\mathrm{int} 2}\right\rangle-\ldots\right] \\ & =(2 \pi \beta)^{-D / 2} \exp \left\{-\left\langle\mathcal{A}_{\mathrm{tot}}^{\mathrm{int}}\right\rangle_{c}+\frac{1}{2}\left\langle\mathcal{A}_{\mathrm{tot}}^{\mathrm{int} 2}\right\rangle_{c}-\ldots\right\} \equiv e^{-\beta f(q)} \end{align*} $$

$$ \langle\ldots\rangle=(2 \pi \beta)^{D / 2} \int \mathcal{D}^{D} q(\tau)(\ldots) e^{-\mathcal{A}^{(0)}[q]} $$

$$ e^{-\beta f(q)}=\frac{1}{\sqrt{2 \pi \hbar^{2} / M k_{B} T}} e^{-\beta \tilde{V}_{\omega}^{\text {eff cl }}(q)} $$

$$ \Delta\left(\tau, \mathbf{z} ; \tau^{\prime}, \mathbf{z}^{\prime}\right)=\int \frac{d^{\varepsilon} k}{(2 \pi)^{\varepsilon}} e^{i \mathbf{k}\left(\mathbf{z}-\mathbf{z}^{\prime}\right)} \Delta_{\omega}\left(\tau, \tau^{\prime}\right), \quad \text { where } \quad \omega \equiv|\mathbf{k}| $$

$$ -{ }^{-} \Delta_{\omega}\left(\tau, \tau^{\prime}\right)+\omega^{2} \Delta_{\omega}\left(\tau, \tau^{\prime}\right)=\delta\left(\tau-\tau^{\prime}\right) $$

$$ \Delta_{\omega}(0, \tau)=\Delta_{\omega}(\beta, \tau)=0 $$

$$ \Delta_{\omega}\left(\tau, \tau^{\prime}\right)=\frac{\sinh \omega\left(\beta-\tau_{>}\right) \sinh \omega \tau_{<}}{\omega \sinh \omega \beta} $$

$$ \begin{align*} { }_{\mu \mu} \Delta\left(\tau, \mathbf{z} ; \tau^{\prime}, \mathbf{z}^{\prime}\right) & =\Delta_{\mu \mu}\left(\tau, \mathbf{z} ; \tau^{\prime}, \mathbf{z}^{\prime}\right)={ }^{-} \Delta\left(\tau, \mathbf{z} ; \tau^{\prime}, \mathbf{z}^{\prime}\right)+{ }_{\mathbf{z z}} \Delta\left(\tau, \mathbf{z} ; \tau^{\prime}, \mathbf{z}^{\prime}\right) \\ & =\int \frac{d^{\varepsilon} k}{(2 \pi)^{\varepsilon}} e^{i \mathbf{k}\left(\mathbf{z}-\mathbf{z}^{\prime}\right)}\left[\because \Delta_{\omega}\left(\tau, \tau^{\prime}\right)-\omega^{2} \Delta_{\omega}\left(\tau, \tau^{\prime}\right)\right]= \\ & =-\delta\left(\tau-\tau^{\prime}\right) \delta^{(\varepsilon)}\left(\mathbf{z}-\mathbf{z}^{\prime}\right) \equiv-\delta^{(d)}\left(z-z^{\prime}\right) \end{align*} $$

$$ { }_{\mu} \Delta_{\mu}(z, z)=\int \frac{d^{\varepsilon} k}{(2 \pi)^{\varepsilon}}\left[\Delta_{\omega}(\tau, \tau)+\omega^{2} \Delta_{\omega}(\tau, \tau)\right] $$

$$ \Delta_{\omega}(\tau, \tau)+\omega^{2} \Delta_{\omega}(\tau, \tau)=\delta(0)-\frac{\omega \cosh \omega(2 \tau-\beta)}{\sinh \omega \beta} $$

$$ { }_{\mu} \Delta_{\mu}(z, z)=\delta^{(d)}(z, z)-I^{\varepsilon} $$

$$ \begin{align*} I^{\varepsilon} & =\int \frac{d^{\varepsilon} k}{(2 \pi)^{\varepsilon}} \frac{\omega \cosh \omega(2 \tau-\beta)}{\sinh \omega \beta}=\frac{1}{\beta} \frac{S_{\varepsilon}}{(2 \pi \beta)^{\varepsilon}} \int_{0}^{\infty} d z z^{\varepsilon} \frac{\cosh z(1-2 \tau / \beta)}{\sinh z} \\ & =\frac{1}{\beta} \frac{S_{\varepsilon}}{(2 \pi \beta)^{\varepsilon}} \frac{\Gamma(\varepsilon+1)}{2^{\varepsilon+1}}[\zeta(\varepsilon+1,1-\tau / \beta)+\zeta(\varepsilon+1, \tau / \beta)] \end{align*} $$

$$ \mathcal{A}^{(0)}[q]=\frac{1}{2} \int_{0}^{\beta} d \tau \dot{q}^{2}(\tau) $$

$$ \mathcal{A}_{\mathrm{tot}}^{\mathrm{int}}=\int_{0}^{\beta} d \tau\left\{\left[-\eta q^{2}(\tau)+\frac{3}{2} \eta^{2} q^{4}(\tau)\right] \dot{q}^{2}(\tau)-\delta(0)\left[-\eta q^{2}(\tau)+\frac{1}{2} \eta^{2} q^{4}(\tau)\right]\right\} $$

$$ \beta f_{1}=\left\langle\mathcal{A}_{\mathrm{tot}}^{\mathrm{int}}\right\rangle_{c}=\eta \int_{0}^{\beta} d \tau\left\langle-q^{2}(\tau) \dot{q}^{2}(\tau)+\delta(0) q^{2}(\tau)\right\rangle+\mathcal{O}\left(\eta^{2}\right) $$

$$ \beta f_{1}=-\eta \circlearrowleft-2 \eta \circlearrowleft+\eta \delta(0) \circlearrowleft . $$

$$ \beta f_{2}^{(1)}=\eta^{2} \int_{0}^{\beta} d \tau\left\langle\frac{3}{2} q^{4}(\tau) \dot{q}^{2}(\tau)-\delta(0) \frac{1}{2} q^{4}(\tau)\right\rangle_{c} $$

$$ \beta f_{2}^{(1)}=\eta^{2}\left[\frac{9}{2} \bigcirc+18 \text { O }-\frac{3}{2} \delta(0) \bigcirc\right] . $$

$$ \beta f_{2}^{(2)}=-\frac{\eta^{2}}{2!}\left\{2 \delta^{2}(0) \circlearrowleft-4 \delta(0)[\circlearrowleft, 9+4 \circlearrowleft ?+Q]\right\} . $$

$$ \beta f_{2}^{(3)}=-\frac{\eta^{2}}{2!}[4 \cap, x+2 \cap-8 \hat{x}+4 \hat{x}+4 \hat{x}+2 \cap 0-8 \alpha \hat{x}] . $$

$$ \beta f_{2}^{(4)}=-\frac{\eta^{2}}{2!} 4[\circlearrowleft+4 \rightarrow+\square] . $$

$$ \beta f_{1}=-\eta \int_{0}^{\beta} d \tau\left[\Delta(\tau, \tau)^{\cdot} \Delta^{\cdot}(\tau, \tau)+2 \cdot \Delta^{2}(\tau, \tau)-\delta(0) \Delta(\tau, \tau)\right]=0 $$

$$ \int_{0}^{\beta} d \tau\left[-\frac{1}{\beta} \Delta(\tau, \tau)+2 \cdot \Delta^{2}(\tau, \tau)\right]=0 $$

$$ \Delta(\tau, \tau)=\tau-\frac{\tau^{2}}{\beta}, \quad \cdot \Delta^{2}(\tau, \tau)=\frac{1}{4}-\frac{\Delta(\tau, \tau)}{\beta} $$

$$ \frac{1}{2 \beta} \int_{0}^{\beta} d \tau \Delta(\tau, \tau)=\int_{0}^{\beta} d \tau \cdot \Delta^{2}(\tau, \tau)=\frac{\beta}{12} $$

$$ \beta f_{2}^{(1)}=\frac{3}{2} \eta^{2} \int_{0}^{\beta} d \tau\left[3 \Delta^{2}(\tau, \tau)^{\cdot} \Delta^{\cdot}(\tau, \tau)+12 \Delta(\tau, \tau) \cdot \Delta^{2}(\tau, \tau)-\delta(0) \Delta^{2}(\tau, \tau)\right] $$

$$ \beta f_{2}^{(1)}=\eta^{2}\left[3 \delta(0) \int_{0}^{\beta} d \tau \Delta^{2}(\tau, \tau)\right]=\eta^{2} \frac{\beta^{3}}{10} \delta(0) $$

$$ \begin{align*} & \beta f_{2}^{(2)}=-\frac{\eta^{2}}{2} \int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime}\left\{2 \delta^{2}(0) \Delta^{2}\left(\tau, \tau^{\prime}\right)\right. \\ & \left.\quad-4 \delta(0)\left[\Delta(\tau, \tau)^{\cdot} \Delta^{2}\left(\tau, \tau^{\prime}\right)+4 \cdot \Delta(\tau, \tau) \Delta\left(\tau, \tau^{\prime}\right) \cdot \Delta\left(\tau, \tau^{\prime}\right)+\Delta^{2}\left(\tau, \tau^{\prime}\right) \cdot \Delta^{\cdot}(\tau, \tau)\right]\right\} \end{align*} $$

$$ \begin{align*} \beta f_{2}^{(2)} & =-\frac{\eta^{2}}{2} \int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime}\left\{-2 \delta^{2}(0) \Delta^{2}\left(\tau, \tau^{\prime}\right)\right. \\ & \left.-4 \delta(0)\left[\Delta(\tau, \tau) \Delta^{2}\left(\tau, \tau^{\prime}\right)+4 \Delta(\tau, \tau) \Delta\left(\tau, \tau^{\prime}\right) \cdot \Delta\left(\tau, \tau^{\prime}\right)-\Delta^{2}\left(\tau, \tau^{\prime}\right) / \beta\right]\right\} \end{align*} $$

$$ \begin{align*} \beta f_{2}^{(2)} & =\frac{\eta^{2}}{2}\left\{2 \delta^{2}(0) \frac{\beta^{4}}{90}+4 \delta(0)\left[\frac{\beta^{3}}{45}+4 \frac{\beta^{3}}{180}-\frac{\beta^{3}}{90}\right]\right\} \\ & =\eta^{2}\left\{\frac{\beta^{4}}{90} \delta^{2}(0)+\frac{\beta^{3}}{15} \delta(0)\right\} \end{align*} $$

$$ \begin{align*} \left.\beta f_{2}^{(3)}\right|_{1,2,3} & =-\frac{\eta^{2}}{2} \int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime}\left\{4 \Delta(\tau, \tau) \cdot \Delta^{2}\left(\tau, \tau^{\prime}\right) \cdot \Delta^{\cdot}\left(\tau^{\prime}, \tau^{\prime}\right)\right. \\ & \left.+2 \cdot \Delta^{\cdot}(\tau, \tau) \Delta^{2}\left(\tau, \tau^{\prime}\right) \cdot \Delta^{\cdot}\left(\tau^{\prime}, \tau^{\prime}\right)+16 \cdot \Delta(\tau, \tau) \Delta\left(\tau, \tau^{\prime}\right) \cdot \Delta\left(\tau, \tau^{\prime}\right) \cdot \Delta^{\cdot}\left(\tau^{\prime}, \tau^{\prime}\right)\right\} \end{align*} $$

$$ \begin{align*} \beta f_{2}^{(2)}+\left.\beta f_{2}^{(3)}\right|_{1,2,3} & =-\frac{\eta^{2}}{2} \int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime}\left\{-\frac{4}{\beta} \Delta(\tau, \tau) \cdot \Delta^{2}\left(\tau, \tau^{\prime}\right)\right. \\ & \left.+\frac{2}{\beta^{2}} \Delta^{2}\left(\tau, \tau^{\prime}\right)-\frac{16}{\beta} \cdot \Delta(\tau, \tau) \Delta\left(\tau, \tau^{\prime}\right) \cdot \Delta\left(\tau, \tau^{\prime}\right)\right\} \end{align*} $$

$$ \beta f_{2}^{(2)}+\left.\beta f_{2}^{(3)}\right|_{1,2,3}=\frac{\eta^{2}}{2}\left(\frac{4}{\beta} \frac{\beta^{2}}{45}-\frac{2}{\beta^{2}} \frac{\beta^{4}}{90}+\frac{16}{\beta} \frac{\beta^{3}}{180}\right)=\frac{\eta^{2}}{2} \frac{7}{45} \beta^{2} . $$

$$ \text { 兑: } I_{5}=\int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \Delta(\tau, \tau)^{\cdot} \Delta\left(\tau, \tau^{\prime}\right) \Delta^{\cdot}\left(\tau, \tau^{\prime}\right) \Delta^{\cdot}\left(\tau^{\prime}, \tau^{\prime}\right)=-\frac{\beta^{2}}{720} . $$

$$ \text { 吕: } I_{4}=\int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \Delta(\tau, \tau) \Delta\left(\tau, \tau^{\prime}\right) \cdot \Delta^{\cdot}\left(\tau, \tau^{\prime}\right) \Delta^{\cdot}\left(\tau^{\prime}, \tau^{\prime}\right) $$

$$ I_{4}^{d}=\int_{0}^{\beta} \int_{0}^{\beta} d^{d} \tau d^{d} \tau_{\alpha}^{\prime} \Delta(\tau, \tau) \Delta\left(\tau, \tau^{\prime}\right)_{\alpha} \Delta_{\beta}\left(\tau, \tau^{\prime}\right) \Delta_{\beta}\left(\tau^{\prime}, \tau^{\prime}\right) $$

$$ \begin{align*} \left.\beta f_{2}^{(3)}\right|_{4,5}=-\frac{\eta^{2}}{2} 16\left(I_{4}+I_{5}\right) & =-\frac{\eta^{2}}{2} \int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \frac{16}{\beta} \cdot \Delta(\tau, \tau) \cdot \Delta\left(\tau, \tau^{\prime}\right) \Delta\left(\tau, \tau^{\prime}\right) \\ & =-\eta^{2} \frac{4}{45} \beta^{2} \end{align*} $$

$$ \beta f_{2}^{(2)}+\left.\beta f_{2}^{(3)}\right|_{6,7} ^{\prime}=\frac{\eta^{2}}{2} \frac{2}{15} \beta^{2} $$

$$ \begin{align*} a_{0}: I_{7} & =\int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \Delta(\tau, \tau)^{\cdot} \Delta\left(\tau, \tau^{\prime}\right)^{\cdot} \Delta^{\cdot}\left(\tau, \tau^{\prime}\right) \Delta^{\cdot}\left(\tau^{\prime}, \tau^{\prime}\right) \\ & \rightarrow \int_{0}^{\beta} \int_{0}^{\beta} d^{d} \tau d^{d} \tau^{\prime} \Delta(\tau, \tau)_{\alpha} \Delta\left(\tau, \tau^{\prime}\right)_{\alpha} \Delta_{\beta}\left(\tau, \tau^{\prime}\right) \Delta_{\beta}\left(\tau^{\prime}, \tau^{\prime}\right) \\ & =\frac{1}{2} \int_{0}^{\beta} \int_{0}^{\beta} d^{d} \tau d^{d} \tau^{\prime} \Delta(\tau, \tau) \Delta_{\beta}\left(\tau^{\prime}, \tau^{\prime}\right) \partial_{\beta}^{\prime}\left[{ }_{\alpha} \Delta\left(\tau, \tau^{\prime}\right)\right]^{2} \\ & \rightarrow \frac{1}{2} \int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \Delta(\tau, \tau)^{\cdot} \Delta\left(\tau^{\prime}, \tau^{\prime}\right) \frac{d}{d \tau^{\prime}}\left[\Delta^{2}\left(\tau, \tau^{\prime}\right)\right] \\ & =\frac{1}{2 \beta} \int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \Delta(\tau, \tau)^{\cdot} \Delta^{2}\left(\tau, \tau^{\prime}\right)=\frac{\beta^{2}}{90} \end{align*} $$

$$ \begin{align*} a: I_{6} & =\int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \Delta(\tau, \tau) \cdot \Delta^{\cdot 2}\left(\tau, \tau^{\prime}\right) \Delta\left(\tau^{\prime}, \tau^{\prime}\right) \\ & =\int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \Delta(\tau, \tau)\left[\Delta^{\cdot 2}\left(\tau, \tau^{\prime}\right)-\delta^{2}\left(\tau-\tau^{\prime}\right)\right] \Delta\left(\tau^{\prime}, \tau^{\prime}\right) \\ & +\int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \Delta^{2}(\tau, \tau) \delta^{2}\left(\tau-\tau^{\prime}\right) \end{align*} $$

$$ \begin{array}{r} I_{6}^{R} \rightarrow \int_{0}^{\beta} \int_{0}^{\beta} d^{d} \tau d^{d} \tau^{\prime} \Delta(\tau, \tau)\left[{ }_{\alpha} \Delta_{\beta}^{2}\left(\tau, \tau^{\prime}\right)-\Delta_{\gamma \gamma}^{2}\left(\tau, \tau^{\prime}\right)\right] \Delta\left(\tau^{\prime}, \tau^{\prime}\right) \\ =\int_{0}^{\beta} \int_{0}^{\beta} d^{d} \tau d^{d} \tau^{\prime}\left\{-\partial_{\alpha}[\Delta(\tau, \tau)] \Delta_{\beta}\left(\tau, \tau^{\prime}\right)_{\alpha} \Delta_{\beta}\left(\tau, \tau^{\prime}\right) \Delta\left(\tau^{\prime}, \tau^{\prime}\right)\right. \\ \left.+\Delta(\tau, \tau) \Delta_{\beta}\left(\tau, \tau^{\prime}\right) \Delta_{\gamma \gamma}\left(\tau, \tau^{\prime}\right) \partial_{\beta}^{\prime}\left[\Delta\left(\tau^{\prime}, \tau^{\prime}\right)\right]\right\} \\ \rightarrow \int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} 2\left\{-\Delta(\tau, \tau) \Delta^{\cdot}\left(\tau, \tau^{\prime}\right)^{\cdot} \Delta^{\cdot}\left(\tau, \tau^{\prime}\right) \Delta\left(\tau^{\prime}, \tau^{\prime}\right)+\right. \\ \left.\Delta(\tau, \tau) \Delta^{\cdot}\left(\tau, \tau^{\prime}\right)^{\cdot} \Delta\left(\tau^{\prime}, \tau^{\prime}\right) \Delta^{\cdot}\left(\tau, \tau^{\prime}\right)\right\} \end{array} $$

$$ I_{6}=I_{6}^{R}+I_{6}^{\mathrm{div}} $$

$$ \begin{align*} I_{6}^{R} & =2\left(-\frac{\beta^{2}}{90}-\frac{\beta^{2}}{120}\right)=\frac{1}{2}\left(-\frac{7 \beta^{2}}{90}\right) \\ I_{6}^{\mathrm{div}} & =\int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \Delta^{2}(\tau, \tau) \delta^{2}\left(\tau-\tau^{\prime}\right) \end{align*} $$

$$ I_{6}^{\mathrm{div}}=\delta(0) \int_{0}^{\beta} d \tau \Delta^{2}(\tau, \tau)=\delta(0) \frac{\beta^{3}}{30} $$

$$ \left.\beta f_{2}^{(3)}\right|_{6,7}=-\frac{\eta^{2}}{2}\left(2 I_{6}+16 I_{7}\right)=-\frac{\eta^{2}}{2}\left[2 \delta(0) \frac{\beta^{3}}{30}+\frac{\beta^{2}}{10}\right] $$

$$ \beta f_{2}^{(2)}+\beta f_{2}^{(3)}=-\frac{\eta^{2}}{2}\left[2 \delta(0) \frac{\beta^{3}}{30}+\frac{\beta^{2}}{30}\right] $$

$$ \begin{align*} \beta f_{2}^{(4)} & =-2 \eta^{2} \int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime}\left[\Delta^{2}\left(\tau, \tau^{\prime}\right) \Delta^{\cdot 2}\left(\tau, \tau^{\prime}\right)\right. \\ & \left.+4 \Delta\left(\tau, \tau^{\prime}\right) \cdot \Delta\left(\tau, \tau^{\prime}\right) \Delta^{\cdot}\left(\tau, \tau^{\prime}\right) \cdot \Delta^{\cdot}\left(\tau, \tau^{\prime}\right)+\Delta^{2}\left(\tau, \tau^{\prime}\right) \Delta^{\cdot 2}\left(\tau, \tau^{\prime}\right)\right] \end{align*} $$

$$ : I_{10}=\int_{0}^{\beta} d \tau \int_{0}^{\beta} d \tau^{\prime} \Delta^{2}\left(\tau, \tau^{\prime}\right) \Delta^{\cdot 2}\left(\tau, \tau^{\prime}\right)=\frac{\beta^{2}}{90} $$

$$ : I_{9}=\iint d^{d} \tau d^{d} \tau^{\prime} \Delta\left(\tau, \tau^{\prime}\right)_{\alpha} \Delta\left(\tau, \tau^{\prime}\right) \Delta_{\beta}\left(\tau, \tau^{\prime}\right)_{\alpha} \Delta_{\beta}\left(\tau, \tau^{\prime}\right) $$

$$ I_{9}=-\frac{1}{2} I_{10}+I_{9^{\prime}} \equiv-\frac{1}{2} I_{10}-\frac{1}{2} \iint d \tau d \tau^{\prime} \Delta\left(\tau, \tau^{\prime}\right) \Delta^{\cdot 2}\left(\tau, \tau^{\prime}\right)^{\bullet \Delta}\left(\tau, \tau^{\prime}\right) $$

$$ I_{9^{\prime}}=-\frac{1}{2} \iint d^{d} \tau d^{d} \tau^{\prime} \Delta\left(\tau, \tau^{\prime}\right) \Delta_{\beta}^{2}\left(\tau, \tau^{\prime}\right)_{\alpha \alpha} \Delta\left(\tau, \tau^{\prime}\right) $$

$$ I_{9^{\prime}}=\beta^{2}\left\{\frac{1}{48} \int d \tau \epsilon^{2}(\tau) \delta(\tau)+\frac{1}{240}\right\} $$

$$ I_{9}=-\frac{\beta^{2}}{720} $$

$$ \begin{array}{r} : I_{8}=\int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \Delta^{2}\left(\tau, \tau^{\prime}\right) \cdot \Delta^{\cdot 2}\left(\tau, \tau^{\prime}\right)=\int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \Delta^{2}(\tau, \tau) \delta^{2}\left(\tau-\tau^{\prime}\right) \\ +\int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \Delta^{2}\left(\tau, \tau^{\prime}\right)\left[\Delta^{\cdot 2}\left(\tau, \tau^{\prime}\right)-\delta^{2}\left(\tau-\tau^{\prime}\right)\right] \end{array} $$

$$ I_{8}^{R}=\iint d^{d} \tau d^{d} \tau^{\prime} \Delta^{2}\left(\tau, \tau^{\prime}\right)\left[{ }_{\alpha} \Delta_{\beta}^{2}\left(\tau, \tau^{\prime}\right)-\Delta_{\gamma \gamma}^{2}\left(\tau, \tau^{\prime}\right)\right] $$

$$ I_{8}^{R}=-2 I_{9}+2 I_{9^{\prime}}=-\frac{\beta^{2}}{72} $$

$$ I_{8}^{\mathrm{div}}=\int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \Delta^{2}(\tau, \tau) \delta^{2}\left(\tau-\tau^{\prime}\right)=I_{6}^{\mathrm{div}}=\delta(0) \frac{\beta^{3}}{30} $$

$$ \beta f_{2}^{(4)}=-2 \eta^{2}\left(I_{8}+4 I_{9}+I_{10}\right)=-\frac{\eta^{2}}{2}\left\{4 \delta(0) \frac{\beta^{2}}{30}-\frac{\beta^{2}}{30}\right\} $$

$$ \int d \tau \Delta(\tau, \tau) \delta(0)=\delta(0) \int d \tau \Delta(\tau, \tau) $$

$$ \int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \Delta^{2}(\tau, \tau) \delta^{2}\left(\tau-\tau^{\prime}\right)=\delta(0) \int d \tau \Delta^{2}(\tau, \tau) $$

$$ \int d \tau_{1} \ldots d \tau_{n} \Delta\left(\tau_{1}, \tau_{2}\right) \delta\left(\tau_{1}, \tau_{2}\right) \cdots \Delta\left(\tau_{n}, \tau_{1}\right) \delta\left(\tau_{n}, \tau_{1}\right)=\delta(0) \int d \tau \Delta^{n}(\tau, \tau) $$

$$ \iint d \tau_{1} d \tau_{n} \Delta^{n}\left(\tau_{1}, \tau_{1}\right) \delta^{2}\left(\tau_{1}-\tau_{n}\right)=\delta(0) \int d \tau \Delta^{n}(\tau, \tau) $$

$$ \mathcal{D}^{D} x \rightarrow \mathcal{D}^{D} q \sqrt{g} \exp \left(\int_{0}^{\beta} d \tau \bar{R} / 8\right) $$

$$ \partial_{\kappa} \bar{\Gamma}_{\tau \kappa}^{\mu}\left(q_{a}\right)=-\frac{1}{3}\left[\bar{R}_{\tau \kappa \sigma}^{\mu}\left(q_{a}\right)+\bar{R}_{\sigma \kappa \tau}^{\mu}\left(q_{a}\right)\right], \quad \text { for normal coordinates. } $$

$$ \begin{align*} g_{\mu \nu}(\xi) & =\delta_{\mu \nu}+\eta \frac{1}{3} \bar{R}_{\mu \lambda \nu \kappa} \xi^{\lambda} \xi^{\kappa}+\eta^{2} \frac{2}{45} \bar{R}_{\lambda \nu \kappa}{ }^{\delta} \bar{R}_{\sigma \mu \tau \delta} \xi^{\lambda} \xi^{\kappa} \xi^{\sigma} \xi^{\tau}+\ldots \\ g(\xi) & =1-\eta \frac{1}{3} \bar{R}_{\mu \nu} \xi^{\mu} \xi^{\nu}+\eta^{2} \frac{1}{18}\left(\bar{R}_{\mu \nu} \bar{R}_{\lambda \kappa}+\frac{1}{5} \bar{R}_{\mu \sigma \nu}{ }^{\tau} \bar{R}_{\lambda \tau \kappa}{ }^{\sigma}\right) \xi^{\mu} \xi^{\nu} \xi^{\lambda} \xi^{\kappa}+\ldots \end{align*} $$

$$ \begin{align*} \mathcal{A}_{\mathrm{tot}}^{\mathrm{int}}[\xi]=\int_{0}^{\beta} d \tau & \left\{\left[\eta \frac{1}{6} \bar{R}_{\mu \lambda \nu \kappa} \xi^{\lambda} \xi^{\kappa}+\eta^{2} \frac{1}{45} \bar{R}_{\lambda \mu \kappa} \delta \bar{R}_{\sigma \nu \tau \delta} \xi^{\lambda} \xi^{\kappa} \xi^{\sigma} \xi^{\tau}\right] \dot{\xi}^{\mu} \dot{\xi}^{\nu}\right. \\ & \left.+\eta \frac{1}{6} \delta(0) \bar{R}_{\mu \nu} \xi^{\mu} \xi^{\nu}+\eta^{2} \frac{1}{180} \delta(0) \bar{R}_{\mu \delta \nu}{ }^{\sigma} \bar{R}_{\lambda \sigma \kappa} \delta \xi^{\mu} \xi^{\nu} \xi^{\lambda} \xi^{\kappa}\right\} \end{align*} $$

$$ \begin{align*} \left\langle\xi^{\mu} \xi^{\nu}\right\rangle= & \delta^{\mu \nu}\langle\xi \xi\rangle \\ \left\langle\xi^{\lambda} \xi^{\kappa} \xi^{\mu} \xi^{\nu}\right\rangle= & \left(\delta^{\lambda \kappa} \delta^{\mu \nu}+\delta^{\lambda \mu} \delta^{\nu \kappa}+\delta^{\lambda \nu} \delta^{\kappa \mu}\right)\langle\xi \xi \xi\rangle \\ \left\langle\xi^{\lambda} \xi^{\kappa} \dot{\xi}^{\mu} \dot{\xi}^{\nu}\right\rangle= & \delta^{\lambda \kappa} \delta^{\mu \nu}\langle\xi \xi\rangle\langle\dot{\xi} \dot{\xi}\rangle+\left(\delta^{\lambda \mu} \delta^{\kappa \nu}+\delta^{\lambda \nu} \delta^{\kappa \mu}\right)\langle\xi \dot{\xi}\rangle\langle\xi \dot{\xi}\rangle \\ \left\langle\xi^{\lambda} \xi^{\kappa} \xi^{\sigma} \xi^{\tau} \dot{\xi}^{\mu} \dot{\xi}^{\nu}\right\rangle= & \left(\delta^{\lambda \kappa} \delta^{\sigma \tau}+\delta^{\lambda \sigma} \delta^{\kappa \tau}+\delta^{\lambda \tau} \delta^{\kappa \sigma}\right) \delta^{\mu \nu}\langle\xi \xi\rangle\langle\xi \xi\rangle\langle\dot{\xi} \dot{\xi}\rangle \\ + & {\left[\delta^{\lambda \mu}\left(\delta^{\kappa \nu} \delta^{\sigma \tau}+\delta^{\sigma \nu} \delta^{\tau \kappa}+\delta^{\tau \nu} \delta^{\kappa \sigma}\right)+\delta^{\kappa \mu}\left(\delta^{\sigma \nu} \delta^{\lambda \tau}+\delta^{\lambda \nu} \delta^{\tau \sigma}+\delta^{\tau \nu} \delta^{\sigma \lambda}\right)\right.} \\ + & \left.\delta^{\sigma \mu}\left(\delta^{\tau \nu} \delta^{\lambda \kappa}+\delta^{\lambda \nu} \delta^{\kappa \tau}+\delta^{\kappa \nu} \delta^{\tau \lambda}\right)+\delta^{\tau \mu}\left(\delta^{\lambda \nu} \delta^{\kappa \sigma}+\delta^{\kappa \nu} \delta^{\sigma \lambda}+\delta^{\sigma \nu} \delta^{\lambda \kappa}\right)\right] \\ & \times\langle\xi \xi\rangle\langle\xi \dot{\xi}\rangle\langle\xi \dot{\xi}\rangle \end{align*} $$

$$ \begin{align*} \left\langle\mathcal{A}_{\mathrm{tot}}^{\mathrm{int}}[\xi]\right\rangle & =\int_{0}^{\beta} d \tau\left\{\eta \frac{1}{6} \bar{R}[-\langle\xi \xi\rangle\langle\dot{\xi} \dot{\xi}\rangle+\langle\xi \dot{\xi}\rangle\langle\xi \dot{\xi}\rangle+\eta \delta(0)\langle\xi \xi\rangle]\right. \\ & +\eta^{2} \frac{1}{45}\left[\left(\bar{R}_{\mu \nu}^{2}+\bar{R}_{\mu \nu \lambda \kappa} \bar{R}^{\mu \nu \lambda \kappa}+\bar{R}_{\mu \nu \lambda \kappa} \bar{R}^{\lambda \nu \mu \kappa}\right)\left(\langle\xi \xi\rangle^{2}\langle\dot{\xi} \dot{\xi}\rangle-\langle\xi \xi\rangle\langle\xi \dot{\xi}\rangle^{2}\right)\right] \\ & \left.+\eta^{2} \frac{1}{180} \delta(0)\left(\bar{R}_{\mu \nu}^{2}+\bar{R}_{\mu \nu \lambda \kappa} \bar{R}^{\mu \nu \lambda \kappa}+\bar{R}_{\mu \nu \lambda \kappa} \bar{R}^{\lambda \nu \mu \kappa}\right)\langle\xi \xi\rangle\langle\xi \xi\rangle\right\} \end{align*} $$

$$ \begin{align*} \bar{R}_{\lambda \mu \kappa}{ }^{\delta} \bar{R}_{\sigma \nu \tau \delta} \delta^{\lambda \mu}\left(\delta^{\kappa \nu} \delta^{\sigma \tau}+\delta^{\sigma \nu} \delta^{\tau \kappa}+\delta^{\tau \nu} \delta^{\kappa \sigma}\right) & =0 \\ \bar{R}_{\lambda \mu \kappa}{ }^{\delta} \bar{R}_{\sigma \nu \tau \delta} \delta^{\kappa \mu}\left(\delta^{\sigma \nu} \delta^{\lambda \tau}+\delta^{\lambda \nu} \delta^{\tau \sigma}+\delta^{\tau \nu} \delta^{\sigma \lambda}\right) & =\bar{R}_{\lambda}{ }^{\delta}\left(-\bar{R}_{\delta}^{\lambda}+\bar{R}_{\delta}^{\lambda}\right)=0 \\ \bar{R}_{\lambda \mu \kappa}{ }^{\delta} \bar{R}_{\sigma \nu \tau \delta} \delta^{\sigma \mu}\left(\delta^{\tau \nu} \delta^{\lambda \kappa}+\delta^{\lambda \nu} \delta^{\kappa \tau}+\delta^{\kappa \nu} \delta^{\tau \lambda}\right) & =\bar{R}_{\lambda \mu \kappa}{ }^{\delta}\left(\bar{R}_{\delta} \delta^{\lambda \kappa}+\bar{R}_{\sigma \lambda \kappa \delta}+\bar{R}_{\sigma \kappa \lambda \delta}\right) \\ & =-\left(\bar{R}_{\mu \nu}^{2}+\bar{R}_{\mu \nu \lambda \kappa} \bar{R}^{\mu \nu \lambda \kappa}+\bar{R}_{\mu \nu \lambda \kappa} \bar{R}^{\lambda \nu \mu \kappa}\right) \\ \bar{R}_{\lambda \mu \kappa}{ }^{\delta} \bar{R}_{\sigma \nu \tau \delta} \delta^{\tau \mu}\left(\delta^{\lambda \nu} \delta^{\kappa \sigma}+\delta^{\kappa \nu} \delta^{\sigma \lambda}+\delta^{\sigma \nu} \delta^{\lambda \kappa}\right) & =\bar{R}_{\lambda \tau \kappa}{ }^{\delta}\left(R_{\kappa \lambda \tau \delta}+R_{\lambda \kappa \tau \delta}\right)=0 \end{align*} $$

$$ \bar{R}_{\mu \nu \lambda \kappa}+\bar{R}_{\mu \lambda \kappa \nu}+\bar{R}_{\mu \kappa \nu \lambda}=0 . $$

$$ R_{\mu \nu \lambda \kappa}=\frac{1}{2}\left(\partial_{\mu} \partial_{\lambda} g_{\nu \kappa}-\partial_{\mu} \partial_{\kappa} g_{\nu \lambda}-\partial_{\nu} \partial_{\lambda} g_{\mu \kappa}+\partial_{\mu} \partial_{\kappa} g_{\mu \lambda}\right)-\left[\bar{\Gamma}_{\mu}, \bar{\Gamma}_{\nu}\right]_{\lambda \kappa} $$

$$ R_{\mu \nu \lambda \kappa}=-R_{\mu \nu \kappa \lambda}, \quad R_{\mu \nu \lambda \kappa}=R_{\lambda \kappa \mu \nu} $$

$$ \bar{R}_{\mu \nu \lambda \kappa} \bar{R}^{\lambda \nu \mu \kappa}=\frac{1}{2} \bar{R}_{\mu \nu \lambda \kappa} \bar{R}^{\mu \nu \lambda \kappa} $$

$$ \beta f_{1}=\frac{1}{6} \bar{R}[-\eta \circlearrowleft+\eta \rightsquigarrow+\eta \delta(0) \circlearrowleft] . $$

$$ \beta f_{1}=-\eta \frac{1}{6} \bar{R} \int_{0}^{\beta} d \tau\left[\Delta(\tau, \tau)^{\cdot} \Delta^{\cdot}(\tau, \tau)-\Delta^{2}(\tau, \tau)-\delta(0) \Delta(\tau, \tau)\right] $$

$$ \beta f_{1}=-\frac{1}{6} \bar{R} \int_{0}^{\beta} d \tau\left[-\frac{1}{\beta} \Delta(\tau, \tau)-\Delta^{2}(\tau, \tau)\right] $$

$$ \beta f_{1}=\frac{1}{6} \bar{R} \int_{0}^{\beta} d \tau \frac{3}{2 \beta} \Delta(\tau, \tau)=\frac{\beta}{24} \bar{R} $$

$$ \left(q_{a} \beta \mid q_{a} 0\right)=\frac{1}{\sqrt{2 \pi \beta}^{D}} \exp \left[\frac{\beta}{12} \bar{R}\left(q_{a}\right)+\ldots\right] $$

$$ \beta f_{2}^{(1)}=\eta^{2} \frac{1}{45}\left(\bar{R}_{\mu \nu} \bar{R}^{\mu \nu}+\frac{3}{2} \bar{R}_{\mu \nu \lambda \kappa} \bar{R}^{\mu \nu \lambda \kappa}\right)\left[\varnothing-\frac{1}{4} \delta(0) \propto\right] . $$

$$ \int_{0}^{\beta} d \tau\left[\Delta^{2}(\tau, \tau)^{\cdot} \Delta^{\cdot}(\tau, \tau)-\Delta(\tau, \tau) \cdot \Delta^{2}(\tau, \tau)+\frac{1}{4} \delta(0) \Delta^{2}(\tau, \tau)\right] $$

$$ \int_{0}^{\beta} d \tau\left[-\frac{5}{4 \beta} \Delta^{2}(\tau, \tau)+\frac{5}{4} \delta(0) \Delta^{2}(\tau, \tau)\right]=\frac{5}{4} \frac{1}{30}[1-\delta(0)] $$

$$ \beta f_{2}^{(1)}=-\eta^{2} \frac{\beta^{2}}{1080}\left(\bar{R}_{\mu \nu}^{2}+\frac{3}{2} \bar{R}_{\mu \nu \kappa \lambda}^{2}\right)[1-\delta(0)] $$

$$ \begin{align*} -\frac{1}{2}\left\langle\mathcal{A}_{\mathrm{tot}}^{\mathrm{int} 2}\right\rangle_{c}= & -\frac{\eta^{2}}{2} \frac{1}{36} \int_{0}^{\beta} d \tau \int_{0}^{\beta} d \tau^{\prime}\left\langle\left[\delta^{2}(0) \bar{R}_{\mu \nu} \bar{R}_{\mu^{\prime} \nu^{\prime}} \xi^{\mu}(\tau) \xi^{\nu}(\tau) \xi^{\mu^{\prime}}\left(\tau^{\prime}\right) \xi^{\nu^{\prime}}\left(\tau^{\prime}\right)\right.\right. \\ & +2 \delta(0) \bar{R}_{\mu \lambda \nu \kappa} \bar{R}_{\mu^{\prime} \nu^{\prime}} \xi^{\lambda}(\tau) \xi^{\kappa}(\tau) \dot{\xi}^{\mu}(\tau) \dot{\xi}^{\nu}(\tau) \xi^{\mu^{\prime}}\left(\tau^{\prime}\right) \xi^{\nu^{\prime}}\left(\tau^{\prime}\right) \\ & \left.\left.+\bar{R}_{\mu \lambda \nu \kappa} \bar{R}_{\mu^{\prime} \lambda^{\prime} \nu^{\prime} \kappa^{\prime}} \xi^{\lambda}(\tau) \xi^{\kappa}(\tau) \dot{\xi}^{\mu}(\tau) \dot{\xi}^{\nu}(\tau) \xi^{\lambda^{\prime}}\left(\tau^{\prime}\right) \xi^{\kappa^{\prime}}\left(\tau^{\prime}\right) \dot{\xi}^{\mu^{\prime}}\left(\tau^{\prime}\right) \dot{\xi}^{\nu^{\prime}}\left(\tau^{\prime}\right)\right]\right\rangle_{c} \end{align*} $$

$$ \beta f_{2}^{(2)}=-\frac{\eta^{2}}{2} \frac{\bar{R}_{\mu \nu} \bar{R}^{\mu \nu}}{36}\left\{2 \delta^{2}(0) \circlearrowleft-4 \delta(0)\left[\alpha^{\prime} \oint-2 \circlearrowleft+\ldots\right]\right\}, $$

$$ \begin{align*} & \beta f_{2}^{(2)}=-\frac{\eta^{2}}{2} \frac{\bar{R}_{\mu \nu} \bar{R}^{\mu \nu}}{36} \int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime}\left\{2 \delta^{2}(0) \Delta^{2}\left(\tau, \tau^{\prime}\right)\right. \\ & \left.\quad-4 \delta(0)\left[\Delta(\tau, \tau)^{\cdot} \Delta^{2}\left(\tau, \tau^{\prime}\right)-2 \cdot \Delta(\tau, \tau) \Delta\left(\tau, \tau^{\prime}\right) \cdot \Delta\left(\tau, \tau^{\prime}\right)+\Delta^{2}\left(\tau, \tau^{\prime}\right)^{\cdot} \Delta^{\cdot}(\tau, \tau)\right]\right\} \end{align*} $$

$$ \beta f_{2}^{(3)}=-\frac{\eta^{2}}{2} \frac{\bar{R}_{\mu \nu} \bar{R}^{\mu \nu}}{36}[4 \cap \times+2 \cap \infty-8 \hat{x}+4 \hat{x}+4 \hat{x}+2 \cap \infty-8 \cap x] . $$

$$ \begin{align*} \left.\beta f_{2}^{(3)}\right|_{1,2,3} & =-\frac{\eta^{2}}{2} \frac{\bar{R}_{\mu \nu} \bar{R}^{\mu \nu}}{36} \int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime}\left\{4 \Delta(\tau, \tau)^{\cdot} \Delta^{2}\left(\tau, \tau^{\prime}\right) \Delta^{\cdot}\left(\tau^{\prime}, \tau^{\prime}\right)\right. \\ & \left.+2 \cdot \Delta^{\cdot}(\tau, \tau) \Delta^{2}\left(\tau, \tau^{\prime}\right) \cdot \Delta^{\cdot}\left(\tau^{\prime}, \tau^{\prime}\right)-8 \cdot \Delta(\tau, \tau) \Delta\left(\tau, \tau^{\prime}\right) \cdot \Delta\left(\tau, \tau^{\prime}\right) \cdot \Delta^{\cdot}\left(\tau^{\prime}, \tau^{\prime}\right)\right\} \end{align*} $$

$$ \begin{align*} \left.\beta f_{2}^{(3)}\right|_{4,5}=-\frac{\eta^{2}}{2} \frac{\bar{R}_{\mu \nu} \bar{R}^{\mu \nu}}{36}\left(I_{4}+I_{5}\right) & =-\frac{\eta^{2}}{2} \frac{\bar{R}_{\mu \nu} \bar{R}^{\mu \nu}}{36} \int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \frac{4}{\beta} \cdot \Delta(\tau, \tau) \cdot \Delta\left(\tau, \tau^{\prime}\right) \Delta\left(\tau, \tau^{\prime}\right) \\ & =-\frac{\eta^{2}}{2} \frac{\bar{R}_{\mu \nu} \bar{R}^{\mu \nu}}{36} \frac{1}{45} \beta^{2} \end{align*} $$

$$ \left.\beta f_{2}^{(3)}\right|_{6,7}=-\frac{\eta^{2}}{2} \frac{\bar{R}_{\mu \nu} \bar{R}^{\mu \nu}}{36}\left(2 I_{6}-8 I_{7}\right)=-\frac{\eta^{2}}{2} \frac{\bar{R}_{\mu \nu} \bar{R}^{\mu \nu}}{36}\left[2 \delta(0) \frac{\beta^{3}}{30}-\frac{\beta^{2}}{6}\right] . $$

$$ \beta f_{2}^{(2)}+\beta f_{2}^{(3)}=\eta^{2} \frac{\beta^{2}}{432} \bar{R}_{\mu \nu}^{2}-\eta^{2} \delta(0) \frac{\beta^{3}}{1080} \bar{R}_{\mu \nu}^{2} $$

$$ \beta f_{1}^{(2)}+\beta f_{2}^{(2)}+\beta f_{2}^{(3)}=\eta^{2} \frac{\beta^{2}}{720}\left(\bar{R}_{\mu \nu}^{2}-\bar{R}_{\mu \nu \kappa \lambda}^{2}\right)+\eta^{2} \delta(0) \frac{\beta^{3}}{1080} \bar{R}_{\mu \nu \kappa \lambda}^{2} $$

$$ \beta f_{2}^{(4)}=-\frac{\eta^{2}}{2} 2 \frac{1}{36}\left(\bar{R}_{\mu \nu \lambda \kappa} \bar{R}^{\mu \nu \lambda \kappa}+\bar{R}_{\mu \nu \lambda \kappa} \bar{R}^{\mu \lambda \nu \kappa}\right)[\circlearrowleft-2, \square+\square], $$

$$ \begin{align*} \beta f_{2}^{(3)}= & -\frac{\eta^{2}}{2} 2 \frac{3}{2} \frac{1}{36} \bar{R}_{\mu \nu \kappa \lambda}^{2} \int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime}\left[\Delta^{2}\left(\tau, \tau^{\prime}\right) \Delta^{\cdot 2}\left(\tau, \tau^{\prime}\right)\right. \\ & \left.-2 \Delta\left(\tau, \tau^{\prime}\right) \cdot \Delta\left(\tau, \tau^{\prime}\right) \Delta^{\cdot}\left(\tau, \tau^{\prime}\right) \cdot \Delta^{\cdot}\left(\tau, \tau^{\prime}\right)+\Delta^{2}\left(\tau, \tau^{\prime}\right) \Delta^{\cdot 2}\left(\tau, \tau^{\prime}\right)\right] \\ = & -\frac{\eta^{2}}{24} \bar{R}_{\mu \nu \kappa \lambda}^{2}\left(I_{8}-2 I_{9}+I_{10}\right) \end{align*} $$

$$ \beta f_{2}^{(3)}=-\frac{\eta^{2}}{24} \bar{R}_{\mu \nu \kappa \lambda}^{2} \int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \Delta^{2}(\tau, \tau) \delta^{2}\left(\tau-\tau^{\prime}\right)=-\eta^{2} \frac{\beta^{3}}{720} \bar{R}_{\mu \nu \kappa \lambda}^{2} \delta(0) $$

$$ \left(q_{a} \beta \mid q_{a} 0\right)=\frac{1}{\sqrt{2 \pi \beta}^{D}} \exp \left[\frac{\beta}{12} \bar{R}\left(q_{a}\right)+\frac{\beta^{2}}{720}\left(\bar{R}_{\mu \nu \kappa \lambda}^{2}-\bar{R}_{\mu \nu}^{2}\right)+\ldots\right] . $$

$$ \begin{align*} \left(q_{a} \beta \mid q_{a} 0\right) & \equiv\left\langle e^{\beta \Delta / 2}\right\rangle=\left(q_{a}\left|e^{\beta \Delta / 2}\right| q_{a}\right) \\ & =\frac{1}{\sqrt{2 \pi \beta}^{D}}\left\{1+\frac{\beta}{12} \bar{R}+\frac{\beta^{2}}{2}\left[\frac{1}{144} \bar{R}^{2}+\frac{1}{360}\left(\bar{R}^{\mu \nu \kappa \lambda} \bar{R}_{\mu \nu \kappa \lambda}-\bar{R}^{\mu \nu} \bar{R}_{\mu \nu}\right)\right]+\ldots\right\} \end{align*} $$

$$ \begin{align*} & I_{8}^{R}+4 I_{9}+I_{10}=-\frac{\beta^{2}}{120} \\ & I_{8}^{R}-2 I_{9}+I_{10}=0 \end{align*} $$

$$ \begin{align*} & I_{14}=\int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \Delta\left(\tau, \tau^{\prime}\right) \Delta^{\cdot}\left(\tau, \tau^{\prime}\right) \Delta^{\cdot}\left(\tau, \tau^{\prime}\right) \\ & I_{15}=\int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \Delta\left(\tau, \tau^{\prime}\right) \Delta^{\cdot 2}\left(\tau, \tau^{\prime}\right) \end{align*} $$

$$ I_{14}=\beta / 24, \quad I_{15}^{R}=-\beta / 8 $$

$$ \begin{align*} I_{14} & =\frac{1}{2} \int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \Delta\left(\tau, \tau^{\prime}\right) \frac{d}{d \tau}\left[\Delta^{\cdot 2}\left(\tau, \tau^{\prime}\right)\right]=-\frac{1}{2} \int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \Delta^{\cdot 2}\left(\tau \cdot \tau^{\prime}\right) \ddot{\Delta}\left(\tau, \tau^{\prime}\right) \\ & =-\frac{1}{6} \int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \frac{d}{d \tau^{\prime}}\left[\Delta^{\cdot 3}\left(\tau, \tau^{\prime}\right)\right]=\frac{1}{6} \int_{0}^{\beta} d \tau\left[\Delta^{\cdot 3}(\tau, 0)-\Delta^{\cdot 3}(\tau, \beta)\right]=\frac{\beta}{12} \end{align*} $$

$$ \begin{align*} I_{14} & =\int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \Delta\left(\tau, \tau^{\prime}\right) \Delta^{\cdot}\left(\tau, \tau^{\prime}\right) \delta\left(\tau-\tau^{\prime}\right)-\frac{1}{\beta} \int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \Delta\left(\tau, \tau^{\prime}\right) \Delta^{\cdot}\left(\tau, \tau^{\prime}\right) \\ & =\int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime}\left[-\frac{1}{4} \epsilon^{2}\left(\tau-\tau^{\prime}\right) \delta\left(\tau-\tau^{\prime}\right)\right]+\int_{0}^{\beta} d \tau \cdot \Delta^{2}(\tau, \tau)+\frac{\beta}{12} \\ & =\beta\left[-\frac{1}{4} \int d \tau \epsilon^{2}(\tau) \delta(\tau)+\frac{1}{6}\right] \end{align*} $$

$$ \begin{align*} I_{14} & =\frac{1}{2} \int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \Delta^{\cdot 2}\left(\tau, \tau^{\prime}\right) \delta\left(\tau-\tau^{\prime}\right)= \\ & =\frac{1}{8} \int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \epsilon^{2}\left(\tau-\tau^{\prime}\right) \delta\left(\tau-\tau^{\prime}\right)+\frac{1}{2} \int_{0}^{\beta} d \tau \cdot \Delta^{2}(\tau, \tau) \\ & =\beta\left[\frac{1}{8} \int d \tau \epsilon^{2}(\tau) \delta(\tau)+\frac{1}{24}\right] \end{align*} $$

$$ \int d \tau[\epsilon(\tau)]^{2} \delta(\tau) \equiv \frac{1}{3}, \quad \text { (false) } $$

$$ \int d \tau[\epsilon(\tau)]^{2} \delta(\tau)=\frac{1}{2} \int d \tau[\epsilon(\tau)]^{2} \dot{\epsilon}(\tau)=\frac{1}{6} \int d \tau \frac{d}{d \tau}[\epsilon(\tau)]^{3}=\frac{1}{3} $$

$$ I_{15}=\int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \Delta\left(\tau, \tau^{\prime}\right) \delta^{2}\left(\tau-\tau^{\prime}\right)-\frac{2}{\beta} \int_{0}^{\beta} d \tau \Delta(\tau, \tau)+\frac{1}{\beta^{2}} \int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \Delta\left(\tau, \tau^{\prime}\right) $$

$$ I_{15}^{\mathrm{div}}=\delta(0) \int_{0}^{\beta} d \tau \Delta(\tau, \tau)=\delta(0) \frac{\beta^{2}}{6} $$

$$ I_{15}=\int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \Delta\left(\tau, \tau^{\prime}\right)\left[\Delta^{\cdot 2}\left(\tau, \tau^{\prime}\right)-\delta^{2}\left(\tau-\tau^{\prime}\right)\right]+\delta(0) \frac{\beta^{2}}{6} $$

$$ \begin{align*} I_{15}^{R} & =\int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \Delta\left(\tau, \tau^{\prime}\right)\left[\cdot^{\cdot 2}\left(\tau, \tau^{\prime}\right)-\Delta^{\cdot 2}\left(\tau, \tau^{\prime}\right)\right] \\ & =\int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime}\left[-\cdot \Delta\left(\tau, \tau^{\prime}\right) \Delta^{\cdot}\left(\tau, \tau^{\prime}\right) \cdot \Delta^{\cdot}\left(\tau, \tau^{\prime}\right)-\Delta\left(\tau, \tau^{\prime}\right) \Delta^{\cdot}\left(\tau, \tau^{\prime}\right) \cdot \Delta d\left(\tau, \tau^{\prime}\right)\right] \\ & -\int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime}\left[-\Delta^{\cdot 2}\left(\tau, \tau^{\prime}\right) \Delta^{\cdot}\left(\tau, \tau^{\prime}\right)-\Delta\left(\tau, \tau^{\prime}\right) \Delta^{\cdot}\left(\tau, \tau^{\prime}\right) \Delta^{-}\left(\tau, \tau^{\prime}\right)\right] \\ & =-I_{14}+\int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \Delta^{\cdot 2}\left(\tau, \tau^{\prime}\right) \Delta^{\cdot}\left(\tau, \tau^{\prime}\right)=-I_{14}-\beta / 6 \end{align*} $$

$$ -\int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \Delta^{\cdot 2}\left(\tau, \tau^{\prime}\right) \delta\left(\tau-\tau^{\prime}\right)=-\beta\left[\frac{1}{4} \int d \tau \epsilon^{2}(\tau) \delta(\tau)+\frac{1}{12}\right] $$

$$ I_{14}^{d}=\iint d^{d} x d^{d} x^{\prime}{ }_{\mu} \Delta\left(x, x^{\prime}\right) \Delta_{\nu}\left(x, x^{\prime}\right)_{\mu} \Delta_{\nu}\left(x, x^{\prime}\right) $$

$$ I_{14}^{d}=-\frac{1}{2} \iint d^{d} x d^{d} x^{\prime} \Delta_{\nu}^{2}\left(x, x^{\prime}\right) \Delta_{\mu \mu}\left(x, x^{\prime}\right) $$

$$ I_{15}^{d}=\iint d^{d} x d^{d} x^{\prime} \Delta\left(x, x^{\prime}\right)\left[{ }_{\mu} \Delta_{\nu}\left(x, x^{\prime}\right)\right]^{2} $$

$$ I_{15}^{R}=-I_{14}+\int d^{d} x d^{d} x^{\prime} \Delta_{\nu}^{2}\left(x, x^{\prime}\right) \Delta_{\mu \mu}\left(x, x^{\prime}\right) $$

$$ g_{\mu \nu}(q)=\delta_{\mu \nu}+\sqrt{\eta}\left(\partial_{\lambda} g_{\mu \nu}\right) q^{\lambda}+\eta \frac{1}{2}\left(\partial_{\lambda} \partial_{\kappa} g_{\mu \nu}\right) q^{\lambda} q^{\kappa} $$

$$ \log g(q)=\sqrt{\eta} g^{\mu \nu}\left(\partial_{\lambda} g_{\mu \nu}\right) q^{\lambda}+\eta \frac{1}{2} g^{\mu \nu}\left[\left(\partial_{\lambda} \partial_{\kappa} g_{\mu \nu}\right)-g^{\sigma \tau}\left(\partial_{\lambda} g_{\mu \sigma}\right)\left(\partial_{\kappa} g_{\nu \tau}\right)\right] q^{\lambda} q^{\kappa} $$

$$ \begin{align*} \mathcal{A}_{\mathrm{tot}}^{\mathrm{int}}[q] & =\int_{0}^{\beta} d \tau\left\{\left[\frac{1}{2} \sqrt{\eta}\left(\partial_{\kappa} g_{\mu \nu}\right) q^{\kappa}+\frac{1}{4} \eta\left(\partial_{\lambda} \partial_{\kappa} g_{\mu \nu}\right) q^{\lambda} q^{\kappa}\right] \dot{q}^{\mu} \dot{q}^{\nu}\right. \\ & \left.-\frac{1}{2} \sqrt{\eta} \delta(0) g^{\mu \nu}\left(\partial_{\kappa} g_{\mu \nu}\right) q^{\kappa}-\frac{1}{4} \eta \delta(0) g^{\mu \nu}\left[\left(\partial_{\lambda} \partial_{\kappa} g_{\mu \nu}\right)-g^{\sigma \tau}\left(\partial_{\lambda} g_{\mu \sigma}\right)\left(\partial_{\kappa} g_{\nu \tau}\right)\right] q^{\lambda} q^{\kappa}\right\} \end{align*} $$

$$ \begin{align*} \partial_{\kappa} g_{\mu \nu} & =-g_{\mu \sigma} g_{\nu \tau} \partial_{\kappa} g^{\sigma \tau}=\Gamma_{\kappa \mu \nu}+\Gamma_{\kappa \nu \mu}=2 \Gamma_{\kappa\{\mu \nu\}}, \quad g^{\mu \nu}\left(\partial_{\kappa} g_{\mu \nu}\right)=2 \Gamma_{\kappa \mu}{ }^{\mu}, \\ \partial_{\lambda} \partial_{\kappa} g_{\mu \nu} & =\partial_{\lambda} \Gamma_{\kappa \mu \nu}+\partial_{\lambda} \Gamma_{\kappa \nu \mu}=\partial_{\lambda} \Gamma_{\kappa\{\mu \nu\}}, \\ g^{\mu \nu}\left(\partial_{\tau} \partial_{\lambda} g_{\mu \nu}\right) & =2 g^{\mu \nu} \partial_{\kappa} \Gamma_{\sigma \mu \nu}=2 \partial_{\kappa}\left(g^{\mu \nu} \Gamma_{\sigma \mu \nu}\right)-2 \Gamma_{\sigma \mu \nu} \partial_{\kappa} g^{\mu \nu} \\ & =2\left(\partial_{\kappa} \Gamma_{\sigma \mu}{ }^{\mu}+g^{\mu \nu} \Gamma_{\sigma \mu \tau} \Gamma_{\lambda \nu}{ }^{\tau}+\Gamma_{\sigma \mu}{ }^{\nu} \Gamma_{\lambda \nu}{ }^{\mu}\right), \end{align*} $$

$$ \begin{align*} \mathcal{A}_{\mathrm{tot}}^{\mathrm{int}}[q]= & \int_{0}^{\beta} d \tau\left\{\left[\sqrt{\eta} \Gamma_{\kappa \mu \nu} q^{\kappa}+\frac{\eta}{2} \partial_{\lambda} \Gamma_{\kappa \mu \nu} q^{\lambda} q^{\kappa}\right] \dot{q}^{\mu} \dot{q}^{\nu}\right. \\ & \left.-\delta(0)\left[\sqrt{\eta} \Gamma_{\mu \kappa}{ }^{\mu} q^{\kappa}+\frac{\eta}{2} \partial_{\lambda} \Gamma_{\tau \mu}{ }^{\mu} q^{\lambda} q^{\tau}\right]\right\} \end{align*} $$

$$ \begin{align*} \partial_{\lambda} \Gamma_{\tau \mu}{ }^{\mu} & =\partial_{\lambda} g^{\mu \nu} \Gamma_{\tau \mu \nu}=g^{\mu \nu} \partial_{\lambda} \Gamma_{\tau \mu \nu}=g^{\mu \nu} \partial_{\lambda} \Gamma_{\tau \mu \nu}-g^{\mu \sigma} g^{\nu \tau}\left(\partial_{\lambda} g_{\sigma \tau}\right) \Gamma_{\tau \mu \nu} \\ & =\partial_{\lambda} \Gamma_{\tau \mu \nu}-\left(\Gamma_{\lambda \mu \nu}+\Gamma_{\lambda \nu \mu}\right) \Gamma_{\tau}{ }^{\mu \nu} \end{align*} $$

$$ \beta f_{1}^{(1)}=\eta \int_{0}^{\beta} d \tau\left\langle\frac{1}{2} \partial_{\lambda} \Gamma_{\kappa \mu \nu} q^{\lambda} q^{\kappa} \dot{q}^{\mu} \dot{q}^{\nu}-\delta(0) \frac{1}{2} \partial_{\lambda} \Gamma_{\tau \mu}^{\mu} q^{\lambda} q^{\tau}\right\rangle_{c} $$

$$ \begin{align*} \beta f_{1}^{(1)} & =\frac{\eta}{2} \partial_{\lambda} \Gamma_{\kappa\{\mu \nu\}} \int_{0}^{\beta} d \tau\left\{g^{\mu \nu} g^{\kappa \lambda} \cdot \Delta^{\cdot}(\tau, \tau) \Delta(\tau, \tau)+2 g^{\mu \kappa} g^{\nu \lambda} \cdot \Delta^{2}(\tau, \tau)-\delta(0) g^{\mu \nu} g^{\kappa \lambda} \Delta(\tau, \tau)\right\} \\ & -\frac{\eta}{2} g^{\lambda \kappa}\left(\Gamma_{\lambda \nu}{ }^{\mu} \Gamma_{\mu \kappa}{ }^{\nu}+g^{\tau \mu} \Gamma_{\tau \kappa}{ }^{\nu} \Gamma_{\mu \lambda \nu}\right) \delta(0) \int_{0}^{\beta} d \tau \Delta(\tau, \tau) \end{align*} $$

$$ \beta f_{1}^{(1)}=-\beta \frac{\eta}{4}\left[\frac{g^{\mu \nu} g^{\kappa \lambda}}{6}\left(\partial_{\lambda} \Gamma_{\mu \kappa \nu}-\partial_{\mu} \Gamma_{\kappa \lambda \nu}\right)-\frac{g^{\lambda \kappa}}{3}\left(\Gamma_{\lambda \nu}{ }^{\mu} \Gamma_{\mu \kappa}{ }^{\nu}+g^{\tau \mu} \Gamma_{\tau \kappa}{ }^{\nu} \Gamma_{\mu \lambda \nu}\right) \delta(0)\right] $$

$$ \begin{align*} -\frac{\eta}{2!} \int_{0}^{\beta} d \tau \int_{0}^{\beta} d \tau^{\prime} & \left\langle\left[\Gamma_{\kappa \mu \nu} q^{\kappa}(\tau) \dot{q}^{\mu}(\tau) \dot{q}^{\nu}(\tau)+\delta(0) \Gamma_{\mu \kappa}{ }^{\mu} q^{\kappa}(\tau)\right]\right. \\ & \left.\times\left[\Gamma_{\kappa \mu \nu} q^{\kappa}\left(\tau^{\prime}\right) \dot{q}^{\mu}\left(\tau^{\prime}\right) \dot{q}^{\nu}\left(\tau^{\prime}\right)+\delta(0) \Gamma_{\mu \kappa}{ }^{\mu} q^{\kappa}\left(\tau^{\prime}\right)\right]\right\rangle_{c} \end{align*} $$

$$ \begin{align*} & \beta f_{1}^{(2)}=-\frac{\eta}{2} g^{\lambda \kappa} \Gamma_{\lambda \mu}{ }^{\mu} \Gamma_{\kappa \nu}{ }^{\nu}\left[1 \sigma^{\mu}-2 \delta(0)\right. \\ & \beta f_{1}^{(3)}=-\eta \Gamma_{\lambda \mu}{ }^{\mu}\left(g^{\lambda \kappa} \Gamma_{\kappa \nu}{ }^{\nu}+g^{\nu \kappa} \Gamma_{\nu \kappa}{ }^{\lambda}\right)\left[\delta^{2}(0)\right. \\ & \beta f_{1}^{(4)}=-\frac{\eta}{2}\left(g^{\mu \lambda} g^{\kappa \tau} \Gamma_{\mu \lambda}{ }^{\nu} \Gamma_{\kappa \tau \nu}+g^{\mu \nu} \Gamma_{\mu \kappa}{ }^{\kappa} \Gamma_{\nu \lambda}{ }^{\lambda}+2 g^{\mu \nu} \Gamma_{\mu \nu}{ }^{\kappa} \Gamma_{\kappa \lambda}{ }^{\lambda}\right) \\ & \beta f_{1}^{(5)}=-\frac{\eta}{2}\left(g^{\mu \kappa} g^{\nu \lambda} \Gamma_{\mu \lambda}{ }^{\tau} \Gamma_{\kappa \beta}+3 g^{\mu \kappa} \Gamma_{\mu \lambda}{ }^{\tau} \Gamma_{\tau \kappa}{ }^{\lambda}\right) \\ & \beta f_{1}^{(6)}=-\frac{\eta}{2} g^{\lambda \kappa}\left(\Gamma_{\lambda \nu}{ }^{\mu} \Gamma_{\mu \kappa}{ }^{\nu}+g^{\mu \tau} \Gamma_{\tau \kappa}{ }^{\nu} \Gamma_{\mu \lambda \nu}\right) \end{align*} $$

$$ I_{11}=\iint d \tau d \tau^{\prime}\left\{\Delta^{\cdot}(\tau, \tau) \Delta\left(\tau, \tau^{\prime}\right) \Delta^{\cdot}\left(\tau^{\prime}, \tau^{\prime}\right)-2 \delta(0) \Delta^{\cdot}(\tau, \tau) \Delta\left(\tau, \tau^{\prime}\right)+\delta^{2}(0) \Delta\left(\tau, \tau^{\prime}\right)\right\} $$

$$ I_{12}=\iint d \tau d \tau^{\prime}\left\{\cdot \Delta(\tau, \tau) \cdot \Delta\left(\tau, \tau^{\prime}\right)^{\cdot} \Delta^{\cdot}\left(\tau^{\prime}, \tau^{\prime}\right)-\delta(0) \cdot \Delta(\tau, \tau) \cdot \Delta\left(\tau, \tau^{\prime}\right)\right\} $$

$$ I_{11}=\frac{1}{\beta^{2}} \int_{0}^{\beta} d \tau \int_{0}^{\beta} d \tau^{\prime} \Delta\left(\tau, \tau^{\prime}\right)=\frac{\beta}{12} $$

$$ I_{12}=-\frac{1}{\beta} \int_{0}^{\beta} d \tau \int_{0}^{\beta} d \tau^{\prime} \Delta(\tau, \tau)^{\cdot} \Delta\left(\tau, \tau^{\prime}\right)=-\frac{\beta}{12} $$

$$ \begin{align*} I_{13} & =\iint d \tau d \tau^{\prime} \Delta(\tau, \tau) \Delta^{\cdot}\left(\tau^{\prime}, \tau^{\prime}\right) \Delta^{\cdot}\left(\tau, \tau^{\prime}\right) \\ & \rightarrow \iint d^{d} x d^{d} x^{\prime}{ }_{\mu} \Delta(x, x) \Delta_{\nu}\left(x^{\prime}, x^{\prime}\right)_{\mu} \Delta_{\nu}\left(x, x^{\prime}\right) \end{align*} $$

$$ I_{13}=\frac{1}{\beta} \iint d \tau d \tau^{\prime} \Delta^{\cdot}\left(\tau, \tau^{\prime}\right)^{\cdot} \Delta\left(\tau^{\prime}, \tau^{\prime}\right)=\frac{1}{\beta} \int_{0}^{\beta} d \tau \int_{0}^{\beta} d \tau^{\prime} \Delta(\tau, \tau) \cdot \Delta\left(\tau, \tau^{\prime}\right)=\frac{\beta}{12} $$

$$ \beta f_{1}^{(2)}+\beta f_{1}^{(3)}+\beta f_{1}^{(4)}=-\frac{\eta \beta}{24} g^{\mu \nu} g^{\kappa \lambda} \Gamma_{\mu \nu}^{\tau} \Gamma_{\kappa \lambda \tau} $$

$$ \begin{align*} \beta f_{1}^{(5)}+\beta f_{1}^{(6) R} & =-\frac{\eta}{2}\left\{g^{\mu \kappa} g^{\nu \lambda} \Gamma_{\mu \lambda}{ }^{\tau} \Gamma_{\kappa \nu \tau}\left(I_{14}+I_{15}^{R}\right)+g^{\lambda \kappa} \Gamma_{\lambda \nu}{ }^{\mu} \Gamma_{\mu \kappa}{ }^{\nu}\left(3 I_{14}+I_{15}^{R}\right)\right\} \\ & =\frac{\eta \beta}{24} g^{\mu \nu} g^{\kappa \lambda} \Gamma_{\mu \kappa}{ }^{\sigma} \Gamma_{\nu \lambda \sigma} . \end{align*} $$

$$ \sum_{i=2}^{6} \beta f_{1}^{(i)}=-\frac{\eta \beta}{24} g^{\mu \nu} g^{\kappa \lambda}\left(\Gamma_{\mu \nu}^{\sigma} \Gamma_{\kappa \lambda \sigma}-\Gamma_{\mu \kappa}^{\sigma} \Gamma_{j l, n}\right) . $$

$$ \sum_{i=1}^{6} \beta f_{1}^{(i)}=-\frac{\eta \beta}{24} g^{\mu \nu} g^{\kappa \lambda} R_{\lambda \mu \kappa \nu}=\frac{\eta \beta}{24} \bar{R} . $$

$$ \begin{align*} I_{14}+I_{15}^{R} & =-\frac{\beta}{12}, \\ 3 I_{14}+I_{15}^{R} & =0 \end{align*} $$

$$ Z=\oint \mathcal{D}^{D} q \sqrt{g(q)} e^{-\mathcal{A}[q]} $$

$$ q^{\mu}=q_{0}^{\mu}+\eta^{\mu}(\tau) $$

$$ Z=\int \frac{d^{D} q_{0}}{\sqrt{2 \pi \beta}^{D} \sqrt{g\left(q_{0}\right)}} e^{-\beta V^{\mathrm{effcl}\left(q_{0}\right)}} $$

$$ q^{\mu}(\tau)=q_{0}^{\mu}+\eta^{\mu}\left(q_{0}, \xi\right) $$

$$ \eta^{\mu}\left(q_{0}, \xi\right)=\xi^{\mu}-\frac{1}{2} \bar{\Gamma}_{\sigma \tau}{ }^{\mu}\left(q_{0}\right) \xi^{\sigma} \xi^{\tau}-\frac{1}{6} \bar{\Gamma}_{\sigma \tau \kappa}{ }^{\mu}\left(q_{0}\right) \xi^{\sigma} \xi^{\tau} \xi^{\kappa}-\ldots, $$

$$ \bar{\Gamma}_{\sigma \tau \kappa}{ }^{\mu}\left(q_{0}\right)=\nabla_{\kappa} \bar{\Gamma}_{\sigma \tau}{ }^{\mu}=\partial_{\kappa} \bar{\Gamma}_{\sigma \tau}{ }^{\mu}-2 \bar{\Gamma}_{\kappa \sigma}{ }^{\nu} \bar{\Gamma}_{\nu \tau}{ }^{\mu}, \ldots . $$

$$ \mathcal{A}^{(0)}\left[q_{0}, \xi\right]=g_{\mu \nu}\left(q_{0}\right) \int_{0}^{\beta} d \tau \frac{1}{2} \xi^{\mu}(\tau)\left(-\partial_{\tau}^{2}\right) \xi^{\nu}(\tau) $$

$$ \mathcal{A}_{\mathrm{tot}}^{\mathrm{int}}\left[q_{0} ; \xi\right]=\int_{0}^{\beta} d \tau\left[\frac{1}{6} \bar{R}_{\mu \lambda \nu \kappa} \xi^{\lambda} \xi^{\kappa} \dot{\xi}^{\mu} \dot{\xi}^{\nu}+\frac{1}{6} \delta(0) \bar{R}_{\mu \nu} \xi^{\mu} \xi^{\nu}\right] $$

$$ Z=\oint \mathcal{D}^{D} \xi(\tau) \sqrt{g\left(q_{0}\right)} e^{-\mathcal{A}^{(0)}\left[q_{0}, \xi\right]-\mathcal{A}_{\mathrm{tot}}^{\mathrm{int}}\left[q_{0}, \xi\right]} $$

$$ \xi_{0}^{\mu}=\bar{\xi}^{\mu} \equiv \beta^{-1} \int_{0}^{\beta} d \tau \xi^{\mu}(\tau) $$

$$ \left\langle\xi^{\mu}(\tau) \xi^{\nu}\left(\tau^{\prime}\right)\right\rangle^{q_{0}}=g^{\mu \nu}\left(q_{0}\right)\left(-\partial_{\tau}^{2}\right)^{-1} \delta\left(\tau-\tau^{\prime}\right)=g^{\mu \nu}\left(q_{0}\right) \bar{\Delta}\left(\tau, \tau^{\prime}\right) $$

$$ \bar{\Delta}\left(\tau, \tau^{\prime}\right)=\bar{\Delta}\left(\tau-\tau^{\prime}\right) \equiv \frac{\left(\tau-\tau^{\prime}\right)^{2}}{2 \beta}-\frac{\left|\tau-\tau^{\prime}\right|}{2}+\frac{\beta}{12}, \quad \tau, \tau^{\prime} \in[0, \hbar \beta] . $$

$$ \bar{\Delta}\left(\tau, \tau^{\prime}\right)=-\bar{\Delta} \cdot\left(\tau, \tau^{\prime}\right) \equiv \frac{\tau-\tau^{\prime}}{\beta}-\frac{\epsilon\left(\tau-\tau^{\prime}\right)}{2}, \quad \tau, \tau^{\prime} \in[0, \hbar \beta] $$

$$ -\cdots \bar{\Delta}\left(\tau, \tau^{\prime}\right)=-\bar{\Delta}^{\bullet}\left(\tau, \tau^{\prime}\right)=\cdot \bar{\Delta} \cdot\left(\tau, \tau^{\prime}\right)=\delta\left(\tau-\tau^{\prime}\right)-1 / \beta $$

$$ \frac{1}{\beta} \sum_{m \neq 0} e^{-i \omega_{m}\left(\tau-\tau^{\prime}\right)}=\delta\left(\tau-\tau^{\prime}\right)-\frac{1}{\beta} $$

$$ q_{0}^{\mu} \rightarrow q_{0 \varepsilon}^{\mu}=q_{0}^{\mu}+\varepsilon^{\mu}, \quad|\varepsilon| \ll 1 $$

$$ \xi^{\mu} \rightarrow \xi_{\varepsilon}^{\mu}=\xi^{\mu}-\varepsilon^{\nu} Q_{\nu}^{\mu}\left(q_{0}, \xi\right) $$

$$ \delta q^{\mu} \equiv q_{\varepsilon}^{\mu}-q^{\mu}=\varepsilon^{\nu} D_{\nu} q^{\mu}\left(q_{0}, \xi\right)=0 $$

$$ D_{\mu}=\frac{\partial}{\partial q_{0}^{\mu}}-Q_{\mu}^{\nu}\left(q_{0}, \xi\right) \frac{\partial}{\partial \xi^{\nu}} $$

$$ \delta_{\nu}^{\mu}+\frac{\partial \eta^{\mu}\left(q_{0}, \xi\right)}{\partial q_{0}^{\nu}}-Q_{\nu}^{\kappa}\left(q_{0}, \xi\right) \frac{\partial \eta^{\mu}\left(q_{0}, \xi\right)}{\partial \xi^{\kappa}}=0 $$

$$ \frac{\partial \eta^{\mu}\left(q_{0}, \xi\right)}{\partial q_{0}^{\nu}}=-\frac{1}{2} \partial_{\nu} \bar{\Gamma}_{(\sigma \tau)}^{\mu}\left(q_{0}\right) \xi^{\sigma} \xi^{\tau}-\ldots $$

$$ \begin{align*} & \frac{\partial \eta^{\mu}\left(q_{0}, \xi\right)}{\partial \xi^{\nu}}=\delta_{\nu}^{\mu}-\bar{\Gamma}_{(\nu \sigma)}^{\mu}\left(q_{0}\right) \xi^{\sigma}-\frac{1}{2} \bar{\Gamma}_{(\nu \sigma \tau)}^{\mu}\left(q_{0}\right) \xi^{\sigma} \xi^{\tau}-\ldots \\ & \quad=\delta_{\nu}^{\mu}-\bar{\Gamma}_{\nu \sigma}^{\mu} \xi^{\sigma}-\frac{1}{3}\left(\partial_{\sigma} \bar{\Gamma}_{\nu \tau}^{\mu}+\frac{1}{2} \partial_{\nu} \bar{\Gamma}_{\sigma \tau}^{\mu}-2 \bar{\Gamma}_{\tau \nu}{ }^{\kappa} \bar{\Gamma}_{\kappa \sigma}{ }^{\mu}-\bar{\Gamma}_{\tau \sigma}{ }^{\kappa} \bar{\Gamma}_{\kappa \nu}{ }^{\mu}\right) \xi^{\sigma} \xi^{\tau}-\ldots \end{align*} $$

$$ \begin{gather*} {\left[\left(\frac{\partial \eta\left(q_{0}, \xi\right)}{\partial \xi}\right)^{-1}\right]_{\nu}^{\mu}=\delta_{\nu}^{\mu}+\bar{\Gamma}_{\nu \sigma}^{\mu} \xi^{\sigma}+\frac{1}{3}\left(\partial_{\sigma} \bar{\Gamma}_{\nu \tau}^{\mu}+\frac{1}{2} \partial_{\nu} \bar{\Gamma}_{\sigma \tau}^{\mu}+\bar{\Gamma}_{\tau \nu}^{\kappa} \bar{\Gamma}_{\kappa \sigma}^{\mu}-\bar{\Gamma}_{\tau \sigma}^{\kappa} \bar{\Gamma}_{\kappa \nu}^{\mu}\right) \xi^{\sigma} \xi^{\tau}+\ldots} \\ =\left(\frac{\partial \xi^{\mu}\left(q_{0}, \eta\right)}{\partial \eta^{\nu}}\right)_{\eta=\eta\left(q_{0}, \xi\right)} \end{gather*} $$

$$ \xi^{\mu}\left(q_{0}, \eta\right)=\eta^{\mu}+\frac{1}{2} \tilde{\Gamma}_{\sigma \tau}{ }^{\mu}\left(q_{0}\right) \eta^{\sigma} \eta^{\tau}+\frac{1}{6} \tilde{\Gamma}_{\sigma \tau \kappa}{ }^{\mu}\left(q_{0}\right) \eta^{\sigma} \eta^{\tau} \eta^{\kappa}+\ldots $$

$$ \begin{align*} \tilde{\Gamma}_{\sigma \tau}{ }^{\mu}\left(q_{0}\right) & =\bar{\Gamma}_{\sigma \tau}{ }^{\mu}, \\ \tilde{\Gamma}_{\sigma \tau \kappa}{ }^{\mu}\left(q_{0}\right) & =\bar{\Gamma}_{\sigma \tau \kappa}{ }^{\mu}+3 \bar{\Gamma}_{\kappa \sigma}{ }^{\nu} \bar{\Gamma}_{\nu \tau}{ }^{\mu}=\partial_{\kappa} \bar{\Gamma}_{\sigma \tau}{ }^{\mu}+\bar{\Gamma}_{\kappa \sigma}{ }^{\nu} \bar{\Gamma}_{\nu \tau}{ }^{\mu}, \\ & \vdots \end{align*} $$

$$ Q_{\nu}^{\mu}\left(q_{0}, \xi\right)=\delta_{\nu}^{\mu}+\bar{\Gamma}_{\nu \sigma}^{\mu}\left(q_{0}\right) \xi^{\sigma}+\frac{1}{3} \bar{R}_{\sigma \nu \tau}^{\mu}\left(q_{0}\right) \xi^{\sigma} \xi^{\tau}+\ldots $$

$$ \eta^{\mu} \rightarrow \eta^{\prime \mu}=\eta^{\mu}-\varepsilon^{\nu} \bar{Q}_{\nu}^{\mu}\left(q_{0}, \eta\right), \quad \bar{Q}_{\nu}^{\mu}\left(q_{0}, 0\right)=\delta_{\nu}^{\mu} $$

$$ \bar{Q}_{\nu}^{\mu}\left(q_{0}, \eta\right)=\left[Q_{\nu}^{\kappa}\left(q_{0}, \xi\right) \frac{\partial \eta^{\mu}\left(q_{0}, \xi\right)}{\partial \xi^{\kappa}}-\frac{\partial \eta^{\mu}\left(q_{0}, \xi\right)}{\partial q_{0}^{\nu}}\right]_{\xi=\xi\left(q_{0}, \eta\right)} $$

$$ 1=\int d^{D} q_{0} \delta^{(D)}\left(q_{0 \varepsilon}-q_{0}\right)=\int d^{D} q_{0} \delta^{(D)}(\varepsilon) $$

$$ \oint \mathcal{D}^{D} \xi=\int \frac{d^{D} \xi_{0}}{\sqrt{2 \pi \beta}^{D}} \oint \mathcal{D}^{\prime D} \xi $$

$$ \bar{\xi}^{\mu} \rightarrow \bar{\xi}_{\varepsilon}^{\mu}=\bar{\xi}^{\mu}-\varepsilon^{\nu} \frac{1}{\beta} \int_{0}^{\beta} d \tau Q_{\nu}^{\mu}\left(q_{0}, \xi(\tau)\right) $$

$$ \int \frac{d^{D} \xi_{0}}{\sqrt{2 \pi \beta}^{D}} \rightarrow \int \frac{d^{D} \varepsilon}{\sqrt{2 \pi \beta}^{D}} \operatorname{det}\left[\frac{1}{\beta} \int_{0}^{\beta} d \tau Q_{\nu}^{\mu}\left(q_{0}, \xi(\tau)\right)\right] $$

$$ \begin{align*} \oint \mathcal{D}^{D} \xi & =\int d^{D} q_{0} \oint \frac{d^{D} \xi_{0}}{\sqrt{2 \pi \beta}^{D}} \delta^{(D)}(\varepsilon) \oint \mathcal{D}^{\prime D} \xi \\ & =\int \frac{d^{D} q_{0}}{\sqrt{2 \pi \beta}^{D}} \oint \mathcal{D}^{\prime D} \xi \operatorname{det}\left[\frac{1}{\beta} \int_{0}^{\beta} d \tau Q_{\nu}^{\mu}\left(q_{0}, \xi(\tau)\right)\right] \end{align*} $$

$$ \Delta\left[q_{0}, \xi\right]=\operatorname{det}\left[\frac{1}{\beta} \int_{0}^{\beta} d \tau Q_{\nu}^{\mu}\left(q_{0}, \xi\right)\right]=e^{-\mathcal{A}^{\mathrm{FP}}\left[q_{0}, \xi\right]} $$

$$ \mathcal{A}^{\mathrm{FP}}\left[q_{0}, \xi\right] \equiv-\operatorname{tr} \log \left[\frac{1}{\beta} \int_{0}^{\beta} \mathrm{d} \tau \mathrm{Q}_{\nu}^{\mu}\left(\mathrm{q}_{0}, \xi\right)\right] $$

$$ \begin{align*} \mathcal{A}^{\mathrm{FP}}\left[q_{0}, \xi\right] & =-\operatorname{tr} \log \left[\delta_{\nu}^{\mu}+(3 \beta)^{-1} \int_{0}^{\beta} d \tau \bar{R}_{\sigma \nu \tau}^{\mu}\left(q_{0}\right) \xi^{\sigma}(\tau) \xi^{\tau}(\tau)+\ldots\right] \\ & =\frac{1}{3 \beta} \int_{0}^{\beta} d \tau \bar{R}_{\mu \nu}\left(q_{0}\right) \xi^{\mu} \xi^{\nu}+\ldots \end{align*} $$

$$ \mathcal{A}_{\mathrm{tot}, \mathrm{FP}}^{\mathrm{int}}\left[q_{0}, \xi\right]=\mathcal{A}_{\mathrm{tot}}^{\mathrm{int}}\left[q_{0}, \xi\right]+\mathcal{A}^{\mathrm{FP}}\left[q_{0}, \xi\right] $$

$$ Z=\int \frac{d^{D} q_{0}}{\sqrt{2 \pi \beta}^{D}} \sqrt{g\left(q_{0}\right)} e^{-\beta V^{\mathrm{eff} \mathrm{cl}}\left(q_{0}\right)} $$

$$ B\left(q_{0}\right) \equiv e^{-\beta V^{\mathrm{eff} \text { cl }}\left(q_{0}\right)} $$

$$ B\left(q_{0}\right)=\oint \mathcal{D}^{\prime D} \xi \sqrt{g\left(q_{0}\right)} e^{-\mathcal{A}^{(0)}\left[q_{0}, \xi\right]-\mathcal{A}_{\mathrm{tot}, \mathrm{FP}}^{\mathrm{int}}\left[q_{0}, \xi\right]} $$

$$ B_{0}\left(q_{0}\right)=\oint \mathcal{D}^{\prime D} \xi \sqrt{g\left(q_{0}\right)} e^{-\int_{0}^{\beta} d \tau \frac{1}{2} g_{\mu \nu}\left(q_{0}\right) \dot{\xi}^{\mu} \dot{\xi}^{\nu}} $$

$$ B\left(q_{0}\right)=B_{0}\left(q_{0}\right)\left[1-\left\langle\mathcal{A}_{\mathrm{tot}, \mathrm{FP}}^{\mathrm{int}}\left[q_{0}, \xi\right]\right\rangle^{q_{0}}+\frac{1}{2}\left\langle\mathcal{A}_{\mathrm{tot}, \mathrm{FP}}^{\mathrm{int}}\left[q_{0}, \xi\right]^{2}\right\rangle^{q_{0}}-\ldots\right] $$

$$ \langle\ldots\rangle^{q_{0}}=B^{-1}\left(q_{0}\right) \oint \mathcal{D}^{\prime D} \xi[\ldots]^{q_{0}} e^{-\mathcal{A}^{(0)}\left[q_{0}, \xi\right]} $$

$$ \oint \mathcal{D}^{\prime D} \xi^{\mu} \sqrt{g\left(q_{0}\right)}=\oint \mathcal{D}^{\prime D} x^{i} $$

$$ B_{0}\left(q_{0}\right)=\oint \mathcal{D}^{\prime D} x^{i} e^{-\int_{0}^{\beta} d \tau \frac{1}{2} \dot{\xi}^{2}} $$

$$ x^{i}(\tau)=\sum_{m} x_{m}^{i} u_{m}(\tau)=x_{0}^{i}+\sum_{m \neq 0} x_{m}^{i} u_{m}(\tau), \quad x_{-m}^{i}=x_{m}^{i *}, \quad m>0, $$

$$ -\frac{1}{2} \int_{0}^{\beta} d \tau\left[x^{i}(\tau)\right]^{2}=-\frac{\beta}{2} \sum_{m \neq 0} \omega_{m}^{2} x_{-m}^{i} x_{m}^{i}=-\beta \sum_{m>0} \omega_{m}^{2} x_{m}^{i *} x_{m}^{i} $$

$$ B_{0}\left(q_{0}\right)=1 $$

$$ B\left(q_{0}\right)=1-\left\langle\mathcal{A}_{\mathrm{tot}, \mathrm{FP}}^{\mathrm{int}}\left[q_{0}, \xi\right]\right\rangle^{q_{0}}+\frac{1}{2}\left\langle\mathcal{A}_{\mathrm{tot}, \mathrm{FP}}^{\mathrm{int}}\left[q_{0}, \xi\right]^{2}\right\rangle^{q_{0}}-\ldots $$

$$ \left\langle x^{i}(\tau) x^{j}\left(\tau^{\prime}\right)\right\rangle^{q_{0}}=\delta^{i j} \bar{\Delta}\left(\tau, \tau^{\prime}\right) $$

$$ \begin{align*} \left\langle\mathcal{A}_{\mathrm{tot}}^{\mathrm{int}}\left[q_{0}, \xi\right]\right\rangle^{q_{0}} & =\int_{0}^{\beta} d \tau \frac{1}{6}\left[\bar{R}_{\mu \lambda \nu \kappa}\left(q_{0}\right)\left\langle\xi^{\lambda} \xi^{\kappa} \dot{\xi}^{\mu} \dot{\xi}^{\nu}\right\rangle^{q_{0}}+\delta(0) \bar{R}_{\mu \nu}\left(q_{0}\right)\left\langle\xi^{\mu} \xi^{\nu}\right\rangle^{q_{0}}\right] \\ & =\frac{1}{72} \bar{R}\left(q_{0}\right) \beta \end{align*} $$

$$ \left\langle\mathcal{A}^{\mathrm{FP}}\left[q_{0}, \xi\right]\right\rangle^{q_{0}}=\int_{0}^{\beta} d \tau \frac{1}{3 \beta} \bar{R}_{\mu \nu}\left(q_{0}\right)\left\langle\xi^{\mu} \xi^{\nu}\right\rangle^{q_{0}}=\frac{1}{36} \bar{R}\left(q_{0}\right) \beta $$

$$ B\left(q_{0}\right)=1-\left\langle\mathcal{A}_{\mathrm{tot}, \mathrm{FP}}^{\mathrm{int}}\left[q_{0}, \xi\right]\right\rangle^{q_{0}}+\ldots=1-\frac{1}{24} \bar{R}\left(q_{0}\right) \beta+\ldots $$

$$ Z^{\mathrm{P}}=\int \frac{d^{D} q_{0}}{\sqrt{2 \pi \beta}^{D}} \sqrt{g\left(q_{0}\right)} B\left(q_{0}\right) $$

$$ \mathcal{A}^{(0)}\left[q_{0}, \eta\right]=g_{\mu \nu}\left(q_{0}\right) \int_{0}^{\beta} d \tau \frac{1}{2} \eta^{\mu}(\tau)\left(-\partial_{\tau}^{2}\right) \eta^{\nu}(\tau) $$

$$ \begin{align*} \mathcal{A}_{\mathrm{tot}}^{\mathrm{int}}\left[q_{0} ; \eta\right]= & \int_{0}^{\beta} d \tau\left\{\left[\Gamma_{\kappa \mu \nu} \eta^{\kappa}+\frac{1}{2} \partial_{\lambda} \Gamma_{\kappa \mu \nu} \eta^{\lambda} \eta^{\kappa}\right] \dot{\eta}^{\mu} \dot{\eta}^{\nu}\right. \\ & \left.-\delta(0)\left[\Gamma_{\mu \kappa}{ }^{\mu} \eta^{\kappa}+\frac{1}{2} \partial_{\lambda} \Gamma_{\tau \mu}{ }^{\mu} \eta^{\lambda} \eta^{\tau}\right]\right\} \end{align*} $$

$$ \oint \mathcal{D}^{\prime D} \xi J\left(q_{0}, \xi\right) \Delta^{\mathrm{FP}}\left[q_{0}, \xi\right] \equiv \oint \mathcal{D}^{D} \xi(\tau) J\left(q_{0}, \xi\right) \delta^{(D)}\left(\xi_{0}\right) \Delta^{\mathrm{FP}}\left[q_{0}, \xi\right] $$

$$ \oint \mathcal{D}^{\prime D} \xi J\left(q_{0}, \xi\right) \Delta^{\mathrm{FP}}\left[q_{0}, \xi\right]=\oint \mathcal{D}^{\prime D} \eta \bar{\Delta}^{\mathrm{FP}}\left[q_{0}, \eta\right] $$

$$ \bar{\Delta}^{\mathrm{FP}}\left[q_{0}, \eta\right]=\Delta^{\mathrm{FP}}\left[q_{0}, \xi\left(q_{0}, \eta\right)\right] \times \operatorname{det}\left(\frac{\partial \bar{\eta}^{\mu}\left(q_{0}, \xi\right)}{\partial \xi^{\nu}}\right)_{\xi=\xi\left(q_{0}, \eta\right)} $$

$$ \operatorname{det}\left(\frac{\partial \bar{\eta}^{\mu}\left(q_{0}, \xi\right)}{\partial \xi^{\nu}}\right)_{\xi=\xi\left(q_{0}, \eta\right)}=\exp \left\{\operatorname{tr} \log \left[\frac{1}{\beta} \int_{0}^{\beta} d \tau\left(\frac{\partial \eta^{\mu}\left(q_{0}, \xi\right)}{\partial \xi^{\nu}}\right)_{\xi=\xi\left(q_{0}, \eta\right)}\right]\right\} $$

$$ \begin{align*} \left(\frac{\partial \eta^{\mu}\left(q_{0}, \xi\right)}{\partial \xi^{\nu}}\right)_{\xi=\xi\left(q_{0}, \eta\right)} & =\delta_{\nu}^{\mu}-\bar{\Gamma}_{\nu \sigma}{ }^{\mu} \eta^{\sigma} \\ & -\frac{1}{3}\left(\partial_{\sigma} \bar{\Gamma}_{\nu \tau}{ }^{\mu}+\frac{1}{2} \partial_{\nu} \bar{\Gamma}_{\sigma \tau}{ }^{\mu}-2 \bar{\Gamma}_{\tau \nu}{ }^{\kappa} \bar{\Gamma}_{\kappa \sigma}{ }^{\mu}+\frac{1}{2} \bar{\Gamma}_{\tau \sigma}{ }^{\kappa} \bar{\Gamma}_{\kappa \nu}{ }^{\mu}\right) \eta^{\sigma} \eta^{\tau}+\ldots \end{align*} $$

$$ \bar{\Delta}^{\mathrm{FP}}\left[q_{0}, \eta\right]=e^{-\overline{\mathcal{A}}^{\mathrm{FP}}\left[q_{0}, \eta\right]} $$

$$ \begin{align*} \overline{\mathcal{A}}^{\mathrm{FP}}\left[q_{0}, \eta\right] & =\mathcal{A}^{\mathrm{FP}}\left[q_{0}, \xi\left(q_{0}, \eta\right)\right]-\operatorname{tr} \log \left[\frac{1}{\beta} \int_{0}^{\beta} d \tau\left(\frac{\partial \eta^{\mu}\left(q_{0}, \xi\right)}{\partial \xi^{\nu}}\right)_{\xi=\xi\left(q_{0}, \eta\right)}\right] \\ & =\frac{1}{2 \beta} \int_{0}^{\beta} d \tau T_{\sigma \tau}\left(q_{0}\right) \eta^{\sigma} \eta^{\tau}+\ldots \end{align*} $$

$$ T_{\sigma \tau}\left(q_{0}\right)=\left(\partial_{\mu} \bar{\Gamma}_{\sigma \tau}{ }^{\mu}-2 \bar{\Gamma}_{\sigma \kappa}{ }^{\mu} \bar{\Gamma}_{\mu \tau}{ }^{\kappa}+\bar{\Gamma}_{\kappa \mu}{ }^{\mu} \bar{\Gamma}_{\sigma \tau}{ }^{\kappa}\right) . $$

$$ \left\langle\eta^{\mu}(\tau) \eta^{\nu}\left(\tau^{\prime}\right)\right\rangle^{q_{0}}=g^{\mu \nu}\left(q_{0}\right) \bar{\Delta}\left(\tau, \tau^{\prime}\right) $$

$$ B\left(q_{0}\right)=1-\left\langle\mathcal{A}_{\mathrm{tot}, \mathrm{FP}}^{\mathrm{int}}\left[q_{0}, \eta\right]\right\rangle^{q_{0}}+\frac{1}{2}\left\langle\left(\mathcal{A}_{\mathrm{tot}, \mathrm{FP}}^{\mathrm{int}}\left[q_{0}, \eta\right]\right)^{2}\right\rangle^{q_{0}}-\ldots, $$

$$ \mathcal{A}_{\mathrm{tot}, \mathrm{FP}}^{\mathrm{int}}\left[q_{0}, \eta\right]=\mathcal{A}_{\mathrm{tot}}^{\mathrm{int}}\left[q_{0}, \eta\right]+\overline{\mathcal{A}}^{\mathrm{FP}}\left[q_{0}, \eta\right] $$

$$ \bar{\Delta}(\tau, \tau)=\frac{1}{12}, \quad \cdot \bar{\Delta}(\tau, \tau)=\bar{\Delta} \cdot(\tau, \tau)=0 $$

$$ \begin{align*} \bar{f}_{1}^{(1)}=\left\langle\mathcal{A}_{\mathrm{tot}}^{\text {int }}\left[q_{0}, \eta\right]\right\rangle^{q_{0}} & =\frac{\beta}{24} g^{\sigma \tau}\left(\partial_{\sigma} \bar{\Gamma}_{\tau \mu}{ }^{\mu}+g^{\mu \nu} \bar{\Gamma}_{\tau \mu \kappa} \bar{\Gamma}_{\sigma \nu}{ }^{\kappa}+\bar{\Gamma}_{\tau \nu}{ }^{\mu} \bar{\Gamma}_{\sigma \mu}{ }^{\nu}\right) \\ & -\frac{\beta^{2}}{24} \delta(0) g^{\sigma \tau}\left(g^{\mu \nu} \bar{\Gamma}_{\tau \mu \kappa} \bar{\Gamma}_{\sigma \nu}{ }^{\kappa}+\bar{\Gamma}_{\tau \mu}{ }^{\nu} \bar{\Gamma}_{\sigma \nu}{ }^{\mu}\right) \end{align*} $$

$$ \left\langle\overline{\mathcal{A}}^{\mathrm{FP}}\left[q_{0}, \eta\right]\right\rangle^{q_{0}}=-\frac{\beta}{24} g^{\sigma \tau}\left(\partial_{\mu} \bar{\Gamma}_{\sigma \tau}{ }^{\mu}-2 \bar{\Gamma}_{\sigma \nu}{ }^{\mu} \bar{\Gamma}_{\mu \tau}{ }^{\nu}+\bar{\Gamma}_{\mu \kappa}{ }^{\mu} \bar{\Gamma}_{\sigma \tau}{ }^{\kappa}\right) $$

$$ \begin{align*} & : \bar{I}_{14}=\int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \bar{\Delta}\left(\tau, \tau^{\prime}\right) \bar{\Delta}^{\cdot}\left(\tau, \tau^{\prime}\right) \cdot \bar{\Delta} \cdot\left(\tau, \tau^{\prime}\right)=-\frac{\beta}{24}+\delta(0) \frac{\beta^{2}}{12} \\ & \bar{I}_{15}=\int_{0}^{\beta} \int_{0}^{\beta} d \tau d \tau^{\prime} \bar{\Delta}\left(\tau, \tau^{\prime}\right) \cdot \bar{\Delta}^{\cdot 2}\left(\tau, \tau^{\prime}\right)=-\frac{\beta}{24} . \end{align*} $$

$$ \begin{align*} \frac{1}{2}\left\langle\left(\mathcal{A}_{\mathrm{tot}, \mathrm{FP}}^{\mathrm{int}}\left[q_{0}, \eta\right]\right)^{2}\right\rangle_{c}^{q_{0}}= & -\frac{\beta}{24} g^{\sigma \tau}\left(g^{\mu \nu} \bar{\Gamma}_{\tau \mu \kappa} \bar{\Gamma}_{\sigma \nu}{ }^{\kappa}+2 \bar{\Gamma}_{\tau \nu}{ }^{\mu} \bar{\Gamma}_{\sigma \mu}{ }^{\nu}\right) \\ & +\frac{\beta^{2}}{24} \delta(0) g^{\sigma \tau}\left(g^{\mu \nu} \bar{\Gamma}_{\tau \mu \kappa} \bar{\Gamma}_{\sigma \nu}{ }^{\kappa}+\bar{\Gamma}_{\tau \mu}{ }^{\nu} \bar{\Gamma}_{\sigma \nu}{ }^{\mu}\right) \end{align*} $$

$$ g^{\sigma \tau} T_{\sigma \tau}=\nabla_{\mu} V^{\mu}, \quad V^{\mu}\left(q_{0}\right)=g^{\sigma \tau}\left(q_{0}\right) \bar{\Gamma}_{\sigma \tau}^{\mu}\left(q_{0}\right) . $$

$$ g_{\mu \nu}(q)=\delta_{\mu \nu}+\frac{q_{\mu} q_{\nu}}{1-q^{2}}, \quad g(q)=\frac{1}{1-q^{2}} $$

$$ q_{\varepsilon}^{\mu}=q^{\mu}+\varepsilon^{\mu} \sqrt{1-q^{2}}, \quad \varepsilon^{\mu}=\mathrm{const}, \mu=1, \ldots, D $$

$$ \Delta^{\mathrm{FP}}[q] \int d^{D} \varepsilon \delta^{(D)}\left(\bar{q}_{\varepsilon}\right)=\Delta^{\mathrm{FP}}[q] \int d^{D} \varepsilon \delta^{(D)}\left(\varepsilon^{\mu} \frac{1}{\beta} \int_{0}^{\beta} d \tau \sqrt{1-q^{2}}\right)=1 $$

$$ \Delta^{\mathrm{FP}}[q]=\left(\frac{1}{\beta} \int_{0}^{\beta} d \tau \sqrt{1-q^{2}}\right)^{D}=e^{-\mathcal{A}^{\mathrm{FP}}[q]} $$

$$ \mathcal{A}^{\mathrm{FP}}[q]=-D \log \left(\frac{1}{\beta} \int_{0}^{\beta} d \tau \sqrt{1-q^{2}}\right) $$

$$ \begin{align*} B & =\oint \prod_{\mu, \tau}\left[d q^{\mu}(\tau) \sqrt{g(q(\tau))}\right] \delta^{(D)}(\bar{q}) \Delta^{\mathrm{FP}}[q] e^{-\mathcal{A}[q]} \\ & =\oint \mathcal{D}^{\prime D} q \sqrt{g(q(\tau))} \Delta^{\mathrm{FP}}[q] e^{-\mathcal{A}[q]} \end{align*} $$

$$ B=\oint \mathcal{D}^{\prime D} q e^{-\mathcal{A}[q]-\mathcal{A}^{J}[q]-\mathcal{A}^{\mathrm{FP}}[q]} $$

$$ \prod_{\tau} \sqrt{g(q(\tau))} \equiv e^{-\mathcal{A}^{J}[q]} $$

$$ \mathcal{A}^{J}[q]=-\int_{0}^{\beta} d \tau \frac{1}{2} \delta(0) \log g(q)=\int_{0}^{\beta} d \tau \frac{1}{2} \delta(0) \log \left(1-q^{2}\right) $$

$$ g_{\mu \nu}(q)=\delta_{\mu \nu}+q_{\mu} q_{\nu}+\ldots, \quad g(q)=1+q^{2}+\ldots $$

$$ \mathcal{A}^{(0)}[q]=\int_{0}^{\beta} d \tau \frac{1}{2} \dot{q}^{2}(\tau) $$

$$ B_{0}=1 $$

$$ \left\langle q^{\mu}(\tau) q^{\nu}\left(\tau^{\prime}\right)\right\rangle=\delta^{\mu \nu} \bar{\Delta}\left(\tau, \tau^{\prime}\right) $$

$$ \mathcal{A}_{\mathrm{tot}}^{\mathrm{int}}[q]=\mathcal{A}^{\mathrm{int}}[q]+\mathcal{A}^{J}[q]=\int_{0}^{\beta} d \tau \frac{1}{2}\left[(q \dot{q})^{2}-\delta(0) q^{2}\right] $$

$$ \mathcal{A}^{\mathrm{FP}}[q]=\frac{D}{2 \beta} \int_{0}^{\beta} d \tau q^{2} $$

$$ B\left(q_{0}\right)=1-\left\langle\mathcal{A}_{\mathrm{tot}}^{\mathrm{int}}[q]\right\rangle^{q_{0}}-\left\langle\mathcal{A}^{\mathrm{FP}}[q]\right\rangle^{q_{0}}+\ldots=B(0) \equiv B . $$

$$ \begin{align*} \left\langle\mathcal{A}_{\mathrm{tot}}^{\mathrm{int}}[q]\right\rangle^{q_{0}} & =\frac{1}{2} \int_{0}^{\beta} d \tau\left\{D \cdot \bar{\Delta} \cdot(\tau, \tau) \bar{\Delta}(\tau, \tau)+D(D+1) \cdot \bar{\Delta}^{2}(\tau, \tau)-\delta(0) D \bar{\Delta}(\tau, \tau)\right\} \\ & =\frac{1}{2} \int_{0}^{\beta} d \tau\left\{-\frac{D}{\beta} \bar{\Delta}(\tau, \tau)+D(D+1) \cdot \bar{\Delta}^{2}(\tau, \tau)\right\}=-\frac{D}{24} \beta \end{align*} $$

$$ \left\langle\mathcal{A}^{\mathrm{FP}}[q]\right\rangle^{q_{0}}=\frac{D}{2 \beta} \int_{0}^{\beta} d \tau D \bar{\Delta}(\tau, \tau)=\frac{D^{2}}{24} \beta $$

$$ B=1-\frac{D(D-1)}{24} \beta+\ldots $$

$$ L(q, \dot{q})=\frac{1}{2} m(q) \dot{q}^{2}+V(q) $$

$$ V(q)=V(q(\widetilde{q})) \equiv \tilde{V}(\widetilde{q}), \quad m(q)=\widetilde{m}(\widetilde{q})[d \widetilde{q}(q) / d q]^{2} $$

$$ \mathcal{A}[q]=\int_{-\infty}^{\infty} d \tau L(q, \dot{q}) $$

$$ e^{-\Gamma[Q] / \hbar}=\int \mathcal{D} \mu(\delta q) e^{-(1 / \hbar)\left\{\mathcal{A}[Q+\delta q]-\int d \tau \delta q \delta \Gamma[Q] / \delta Q\right\}} $$

$$ \mathcal{D} \mu(\delta q)=Z^{-1} \prod_{\tau} d \delta q(\tau) \sqrt{m(Q)} e^{(1 / 2) \delta(0) \int d \tau \log [m(Q+\delta q) / m(Q)]} $$

$$ \Gamma[Q]=\mathcal{A}[Q]+\hbar \Gamma_{1}[Q]+\hbar^{2} \Gamma_{2}[Q]+\ldots $$

$$ \begin{align*} \mathcal{A}[Q+\delta q] & =\mathcal{A}[Q]+\int d \tau \frac{D \mathcal{A}}{\delta Q(\tau)} \delta x(\tau)+\frac{1}{2} \int d \tau \int d \tau^{\prime} \frac{D^{2} \mathcal{A}}{\delta Q(\tau) \delta Q\left(\tau^{\prime}\right)} \delta x(\tau) \delta x\left(\tau^{\prime}\right) \\ & +\frac{1}{6} \int d \tau \int d \tau^{\prime} \int d \tau^{\prime \prime} \frac{D^{3} \mathcal{A}}{\delta Q(\tau) \delta Q\left(\tau^{\prime}\right) \delta Q\left(\tau^{\prime \prime}\right)} \delta x(\tau) \delta x\left(\tau^{\prime}\right) \delta x\left(\tau^{\prime \prime}\right)+\ldots(10 \end{align*} $$

$$ \frac{D \mathcal{A}[Q]}{\delta Q(\tau)}=\frac{\delta \mathcal{A}[Q]}{\delta Q(\tau)}=V^{\prime}(Q)-\frac{1}{2} m^{\prime}(Q) \dot{Q}^{2}(\tau)-m(Q) \ddot{Q}(\tau) $$

$$ \frac{D^{2} \mathcal{A}[Q]}{\delta Q(\tau) \delta Q\left(\tau^{\prime}\right)}=\frac{\delta^{2} \mathcal{A}[Q]}{\delta Q(\tau) \delta Q\left(\tau^{\prime}\right)}-\Gamma(Q(\tau)) \frac{\delta \mathcal{A}[Q]}{\delta Q\left(\tau^{\prime}\right)} $$

$$ \frac{\delta^{2} \mathcal{A}[Q]}{\delta Q(\tau) \delta Q\left(\tau^{\prime}\right)}=-\left[m(Q) \partial_{\tau}^{2}+m^{\prime}(Q) \dot{Q} \partial_{\tau}+m^{\prime}(Q) \ddot{Q}+\frac{1}{2} m^{\prime \prime}(Q) \dot{Q}^{2}-V^{\prime \prime}(Q)\right] \delta\left(\tau-\tau^{\prime}\right) $$

$$ e^{-\Gamma_{1}[Q]}=\int \mathcal{D} \delta x \sqrt{m(Q)} e^{-\mathcal{A}^{(2)}[Q, \delta x]} $$

$$ \mathcal{A}^{(2)}[Q, \delta x]=\frac{1}{2} \int_{-\infty}^{\infty} d \tau d \tau^{\prime} \delta x(\tau) \frac{D^{2} \mathcal{A}[Q]}{\delta Q(\tau) \delta Q\left(\tau^{\prime}\right)} \delta x\left(\tau^{\prime}\right) $$

$$ \mathcal{A}^{(2)}[Q, \delta x]=\frac{1}{2} \int_{-\infty}^{\infty} d \tau d \tau^{\prime} \delta \tilde{x}(\tau) \frac{D^{2} \mathcal{A}[Q]}{\delta \tilde{Q}(\tau) \delta \tilde{Q}\left(\tau^{\prime}\right)} \delta \tilde{x}\left(\tau^{\prime}\right) $$

$$ \begin{align*} \omega^{2}(Q) & =e^{-1}(Q) D^{2} V(Q) e^{-1}(Q)=e^{-1}(Q) D V^{\prime}(Q) e^{-1}(Q) \\ & =\frac{1}{m(Q)}\left[V^{\prime \prime}(Q)-\Gamma(Q) V^{\prime}(Q)\right] \end{align*} $$

$$ \omega^{2}(Q)=\Delta V(Q)=\frac{1}{\sqrt{m(Q)}} \frac{d}{d Q}\left[\sqrt{m(Q)}\left(\frac{V^{\prime}(Q)}{m(Q)}\right)\right] $$

$$ \prod_{\tau} d \delta x(\tau) \sqrt{m(Q)}=\prod_{\tau} d \delta \tilde{x}(\tau) $$

$$ \Gamma_{1}[Q]=\frac{1}{2} \operatorname{Tr} \log \left[-\partial_{\tau}^{2}+\omega^{2}(Q(\tau))\right] $$

$$ \Gamma_{1}[Q]=\int_{-\infty}^{\infty} d \tau\left[V_{1}(Q)+\frac{1}{2} Z_{1}(Q) \dot{Q}^{2}+\cdots\right] $$

$$ \operatorname{Tr} \log \left[-\partial_{\tau}^{2}+\omega^{2}(\tau)\right] \equiv \int_{-\infty}^{\infty} d \tau\left\{\omega(\tau)+\frac{\left[\partial_{\tau} \omega^{2}(\tau)\right]^{2}}{32 \omega^{5}(\tau)}+\ldots\right\} $$

$$ V_{1}(Q)=\hbar \omega(Q) / 2, \quad Z_{1}(Q)=\frac{\left(D \omega^{2}\right)^{2}(Q)}{32 \omega^{5}(Q)} $$

$$ \Gamma^{\mathrm{eff}}[Q]=\int_{-\infty}^{\infty} d \tau\left[\frac{1}{2} m^{\mathrm{eff}}(Q) \dot{Q}^{2}+V^{\mathrm{eff}}(Q)\right] $$

$$ \begin{align*} m^{\mathrm{eff}}(Q) & =m(Q)+\hbar \frac{\left(D \omega^{2}\right)^{2}(Q)}{32 \omega^{5}(Q)} \\ V^{\mathrm{eff}}(Q) & =V(Q)+\hbar \frac{\omega(Q)}{2} \end{align*} $$