Kleinert · 제11장 장론 입문

$$ \int \frac{d \Delta x}{\sqrt{2 \pi \epsilon}} \exp \left[-\frac{(\Delta x)^{2}}{2 \epsilon}\right] $$

$$ \mathcal{D}^{1 / 2}\left(q, q^{\prime}\right) e^{-i \mathcal{A}_{J}^{\epsilon} / \hbar} / \sqrt{g(q)} $$

$$ \psi(q, t)=\sqrt{g(q)} \int \frac{d^{D} \Delta q}{\sqrt{2 \pi i \epsilon \hbar / M}^{D}} \exp \left[\frac{i}{\hbar}\left(\mathcal{A}^{\epsilon}+\mathcal{A}_{J}^{\epsilon}\right)\right] \psi(q-\Delta q, t-\epsilon) $$

$$ \Delta x=\Delta q\left[1+a_{2} \Delta q+a_{3}(\Delta q)^{2}+\ldots\right] $$

$$ \psi(q, t)=\int d^{D} \Delta q K^{\epsilon}(q, \Delta q) \psi(q-\Delta q, t-\epsilon) $$

$$ \int \frac{d \Delta q}{\sqrt{2 \pi \epsilon}}\left[1+2 a_{2} \Delta q+3 a_{3}(\Delta q)^{2}+\ldots\right] \exp \left\{-\frac{(\Delta q)^{2}}{2 \epsilon}\left[1+2 a_{2} \Delta q+2 a_{3}(\Delta q)^{2}+a_{2}^{2}(\Delta q)^{2}+\ldots\right]\right\} $$

$$ K^{\epsilon}(q, \Delta q)=\frac{\sqrt{g(q)}}{\sqrt{2 \pi i \epsilon \hbar / M}^{D}} \exp \left[\frac{i}{\hbar}\left(\mathcal{A}^{\epsilon}+\mathcal{A}_{J}^{\epsilon}\right)\right] $$

$$ \begin{align*} \int \frac{d \Delta q}{\sqrt{2 \pi \epsilon}} \exp \left[-\frac{(\Delta q)^{2}}{2 \epsilon}\right][1 & -a_{2} \frac{(\Delta q)^{3}}{\epsilon}-a_{3} \frac{(\Delta q)^{4}}{\epsilon}-a_{2}^{2} \frac{(\Delta q)^{4}}{2 \epsilon} \\ & \left.+a_{2}^{2} \frac{(\Delta q)^{6}}{2 \epsilon}-2 a_{2}^{2} \frac{(\Delta q)^{4}}{\epsilon}+3 a_{3}(\Delta q)^{2}+\ldots\right] . \end{align*} $$

$$ K_{0}^{\epsilon}(q, \Delta \xi)=\frac{\sqrt{g(q)}}{\sqrt{2 \pi i \epsilon \hbar / M}} \exp \left[\frac{i}{\hbar} \frac{M}{2 \epsilon} g_{\mu \nu}(q) \Delta \xi^{\mu} \Delta \xi^{\nu}\right] $$

$$ \langle\mathcal{O}\rangle_{0} \equiv \int \frac{d \Delta q}{\sqrt{2 \pi i \epsilon}} \mathcal{O} \exp \left[-(\Delta q)^{2} / 2 \epsilon\right] $$

$$ \int d^{D} \Delta \xi K_{0}^{\epsilon}(q, \Delta \xi)=1 $$

$$ \left\langle(\Delta q)^{2}\right\rangle_{0}=\epsilon, \quad\left\langle(\Delta q)^{4}\right\rangle_{0}=3!\epsilon^{2}, \quad\left\langle(\Delta q)^{6}\right\rangle_{0}=5!\epsilon^{3}, \ldots . $$

$$ \psi(q-\Delta q, t-\epsilon)=\left(1-\Delta q^{\mu} \partial_{\mu}+\frac{1}{2} \Delta q^{\mu} \Delta q^{\nu} \partial_{\mu} \partial_{\nu}+\ldots\right) \psi(q, t-\epsilon) $$

$$ v_{3}^{\partial \Gamma}=-\frac{1}{2} g^{\mu \nu} \partial_{\{\mu} \Gamma_{\nu \lambda\}}{ }^{\lambda}+\frac{1}{6} g_{\mu \tau} \partial_{\kappa} \Gamma_{\lambda \nu}{ }^{\tau}\left(g^{\mu \nu} g^{\lambda \kappa}+g^{\mu \lambda} g^{\nu \kappa}+g^{\mu \kappa} g^{\nu \lambda}\right) . $$

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

$$ \begin{align*} v_{2}^{1}= & -\frac{1}{8}\left[\left(\Gamma_{1}+\Gamma_{2}\right)^{2}-\tilde{\Gamma}_{3}\left(\tilde{\Gamma}_{1}+\tilde{\Gamma}_{1}^{T}+\tilde{\Gamma}_{2}+\tilde{\Gamma}_{2}^{T}\right)\right] \\ v_{2}^{2}= & \frac{1}{8}\left[\Gamma_{3} \Gamma_{3}+\tilde{\Gamma}_{3}\left(\tilde{\Gamma}_{3}+\tilde{\Gamma}_{3}^{T}\right)\right] \\ v_{2}^{3}= & \frac{1}{4}\left[\left(\Gamma_{1}+\Gamma_{2}\right)^{2}+\Gamma_{3}\left(\Gamma_{1}+\Gamma_{2}\right)\right] \\ v_{2}^{4}= & -\frac{1}{8}\left[\Gamma_{1}^{2}+\Gamma_{2}^{2}+\Gamma_{3}^{2}+2\left(\Gamma_{1} \Gamma_{2}+\Gamma_{2} \Gamma_{3}+\Gamma_{3} \Gamma_{1}\right)\right. \\ & \left.+\tilde{\Gamma}_{3}\left(\tilde{\Gamma}_{1}+\tilde{\Gamma}_{1}^{T}+\tilde{\Gamma}_{2}+\tilde{\Gamma}_{2}^{T}+\Gamma_{3}+\tilde{\Gamma}_{3}^{T}\right)\right] \end{align*} $$

$$ \begin{align*} \psi(q, t) & =\int d^{D} \Delta \xi K_{0}^{\epsilon}(q, \Delta \xi) \\ & \times\left[1-\left(\Delta \xi^{\mu}+\frac{1}{2!} \Gamma_{\nu \lambda}^{\mu} \Delta \xi^{\nu} \Delta \xi^{\lambda}\right) \partial_{\mu}+\frac{1}{2} \Delta \xi^{\mu} \Delta \xi^{\nu} \partial_{\mu} \partial_{\nu}+\ldots\right] \psi(q, t-\epsilon) \end{align*} $$

$$ \begin{align*} \Delta v_{3}^{\partial \Gamma} & =\frac{1}{6} \bar{R}-\frac{2}{3} \partial_{\mu} S^{\mu}+\frac{1}{6}\left(\tilde{\Gamma}_{3} \tilde{\Gamma}_{2}^{T}-\Gamma_{3} \Gamma_{2}\right) \\ \Delta v_{3} \Gamma^{2} & =-\frac{1}{2} \tilde{\Gamma}_{3} \tilde{\Gamma}_{2}+\frac{1}{6}\left(\tilde{\Gamma}_{3} \tilde{\Gamma}_{2}+\tilde{\Gamma}_{3} \tilde{\Gamma}_{2}^{T}+\Gamma_{3} \Gamma_{2}\right) \end{align*} $$

$$ \left\langle\Delta \xi^{\mu} \Delta \xi^{\nu}\right\rangle=\int d^{D} \Delta \xi K_{0}^{\epsilon}(q, \Delta \xi) \Delta \xi^{\mu} \Delta \xi^{\nu}=\frac{i \hbar \epsilon}{M} g^{\mu \nu}(q) $$

$$ \psi(q, t)=\left[1+i \epsilon \frac{\hbar^{2}}{2 M}\left(g^{\mu \nu} \partial_{\mu} \partial_{\nu}-\Gamma_{\nu}^{\nu \mu} \partial_{\mu}\right)+\ldots\right] \psi(q, t-\epsilon) $$

$$ \Delta v_{3}=\frac{1}{6} \bar{R}-\frac{2}{3} \partial_{\mu} S^{\mu}+\frac{2}{3} \tilde{\Gamma}_{3} \tilde{S}_{1} $$

$$ D_{\mu} D^{\mu} \psi \equiv g^{\mu \nu} D_{\mu} D_{\nu} \psi=g^{\mu \nu} D_{\mu} \partial_{\nu} \psi=\left(g^{\mu \nu} \partial_{\mu} \partial_{\nu}-\Gamma_{\nu}{ }^{\nu \mu} \partial_{\mu}\right) \psi $$

$$ \tilde{S}_{3} \tilde{\Gamma}_{2}=-\tilde{\Gamma}_{3} \tilde{S}_{1} $$

$$ \Delta \psi=\frac{1}{\sqrt{g}} \partial_{\mu} \sqrt{g} g^{\mu \nu} \partial_{\nu} \psi $$

$$ \begin{gather*} \Delta v_{2}^{1}=-\frac{1}{2}\left(\Gamma_{1} \Gamma_{1}-\tilde{\Gamma}_{3} \tilde{\Gamma}_{2}\right)+\frac{1}{8}\left[\left(\Gamma_{1}+\Gamma_{2}\right)^{2}-\tilde{\Gamma}_{3}\left(\tilde{\Gamma}_{1}+\tilde{\Gamma}_{1}^{T}+\tilde{\Gamma}_{2}+\tilde{\Gamma}_{2}^{T}\right)\right] \\ \Delta v_{2}^{3}=\frac{1}{4}\left(\Gamma_{1}-\Gamma_{2}\right)\left(\Gamma_{1}+\Gamma_{2}+\Gamma_{3}\right) \end{gather*} $$

$$ \Delta=g^{\mu \nu} \partial_{\mu} \partial_{\nu}+\left(\frac{1}{\sqrt{g}} \partial_{\mu} \sqrt{g}\right) g^{\mu \nu} \partial_{\nu}+\left(\partial_{\mu} g^{\mu \nu}\right) \partial_{\nu} $$

$$ \begin{align*} \left(\frac{1}{\sqrt{g}} \partial_{\mu} \sqrt{g}\right) & =\frac{1}{2} g^{\sigma \tau} \partial_{\mu} g_{\sigma \tau}=\bar{\Gamma}_{\mu \nu}^{\nu} \\ \partial_{\mu} g^{\sigma \nu} & =-g^{\sigma \lambda} g^{\nu \kappa} \partial_{\mu} g_{\lambda \kappa} \\ \partial_{\mu} g^{\mu \nu} & =-\bar{\Gamma}_{\mu}{ }^{\mu \nu}-\bar{\Gamma}^{\nu}{ }_{\mu}{ }^{\mu} \end{align*} $$

$$ \Delta v_{2}=-\frac{1}{2} S_{1} S_{1}+\frac{1}{2} \Gamma_{3} S_{1}-\frac{1}{3} \tilde{\Gamma}_{3} \tilde{S}_{1}+\frac{1}{2} \tilde{S}_{1} \tilde{S}_{3}, $$

$$ \frac{1}{\sqrt{g}}\left(\partial_{\mu} g^{\mu \nu} \sqrt{g}\right)=-\bar{\Gamma}_{\mu}^{\mu \nu} $$

$$ \tilde{\Gamma}_{3} \tilde{\Gamma}_{2}^{T}=\tilde{\Gamma}_{3} \tilde{\Gamma}_{1} . $$

$$ \Delta \psi=\left(g^{\mu \nu} \partial_{\mu} \partial_{\nu}-\bar{\Gamma}_{\mu}^{\mu \nu} \partial_{\nu}\right) \psi=\bar{D}_{\mu} \bar{D}^{\mu} \psi $$

$$ v=\frac{1}{6} R-\frac{2}{3} g^{\mu \nu} \partial_{\mu} S_{\nu}-\frac{1}{2}\left(S_{1}^{2}-\Gamma_{3} S_{1}\right)+\frac{1}{2} \tilde{S}_{1} \tilde{S}_{3}-\frac{1}{3} \tilde{\Gamma}_{3} \tilde{S}_{1} $$

$$ D_{\mu} D^{\mu} \psi=\left(\bar{D}_{\mu} \bar{D}^{\mu}-K_{\mu}{ }^{\mu \nu} \partial_{\nu}\right) \psi=\left(\bar{D}_{\mu} \bar{D}^{\mu}-2 S^{\nu} \partial_{\nu}\right) \psi, $$

$$ -\frac{2}{3} D_{\mu} S^{\mu}-\frac{1}{6} \Gamma_{3} S_{1} $$

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

$$ \psi(q, t)=\left(1+\frac{i \epsilon \hbar}{2 M} D_{\mu} D^{\mu}\right) \psi(q, t-\epsilon)+\mathcal{O}\left(\epsilon^{2}\right) $$

$$ i \hbar \partial_{t} \psi(q, t)=\hat{H}_{0} \psi(q, t) $$

$$ \hat{H}_{0}=-\frac{\hbar^{2}}{2 M} D_{\mu} D^{\mu} $$

$$ \hat{H}_{0}=-\frac{\hbar^{2}}{2 M} \partial_{i}^{2} $$

$$ \partial_{i}=e_{i}^{\mu} \partial_{\mu} $$

$$ \partial_{i}^{2}=e_{i}^{\mu} \partial_{\mu} e^{i \nu} \partial_{\nu}=g^{\mu \nu} \partial_{\mu} \partial_{\nu}-\Gamma_{\mu}^{\mu \nu} \partial_{\nu}, $$

$$ \mathcal{A}_{0}^{\epsilon}=\frac{M}{2 \epsilon} g_{\mu \nu}(q) \Delta q^{\mu} \Delta q^{\nu} $$

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

$$ K_{0}^{\epsilon}(q, \Delta q)=\frac{\sqrt{g(q)}}{\sqrt{2 \pi i \epsilon \hbar / M}^{D}} \exp \left(\frac{i}{\hbar} \mathcal{A}_{0}^{\epsilon}\right) $$

$$ \int d^{D} \Delta q K_{0}^{\epsilon}(q, \Delta q)=1 $$

$$ K^{\epsilon}(q, \Delta q)=K_{0}^{\epsilon}(q, \Delta q)[1+C(\Delta q)] \equiv K_{0}^{\epsilon}(q, \Delta q)\left[1+\sum_{n=1}^{\infty} c_{n}(\Delta q)^{n}\right] . $$

$$ \begin{align*} \psi(q, t)= & \int d^{D} \Delta q K_{0}^{\epsilon}(q, \Delta q)\left[1+\sum_{n=1}^{\infty} c_{n}(\Delta q)^{n}\right] \\ & \times\left(1-\Delta q^{\mu} \partial_{\mu}+\frac{1}{2} \Delta q^{\mu} \Delta q^{\nu} \partial_{\mu} \partial_{\nu}+\ldots\right) \psi(q, t-\epsilon) \end{align*} $$

$$ C(\Delta q)=\exp \left[\frac{i}{\hbar}\left(\Delta \mathcal{A}^{\epsilon}+\mathcal{A}_{J}^{\epsilon}\right)\right]-1 $$

$$ C=C^{\mathrm{e}}+C^{\emptyset} $$

$$ C^{\emptyset}=-\Gamma_{\{\mu \nu\}}^{\nu} \Delta q^{\mu}-\frac{i}{\hbar} \frac{M}{2 \epsilon} \Gamma_{\mu \nu \lambda} \Delta q^{\mu} \Delta q^{\nu} \Delta q^{\lambda}+\ldots, $$

$$ C^{\mathrm{e}}=\sum_{a=1}^{4} C_{a}^{\mathrm{e}}+\ldots $$

$$ \begin{align*} C_{1}^{\mathrm{e}} & =\frac{1}{2}\left[\partial_{\{\mu} \Gamma_{\nu \lambda\}}^{\lambda}+\Gamma_{\{\nu \kappa}^{\sigma} \Gamma_{\{\sigma \mid \mu\}}{ }^{\kappa}+\Gamma_{\{\mu \sigma\}}^{\sigma} \Gamma_{\{\nu \lambda\}}^{\lambda}-\Gamma_{\{\nu \kappa\}}^{\sigma} \Gamma_{\{\mu \sigma\}}{ }^{\kappa}\right] \Delta q^{\mu} \Delta q^{\nu} \\ C_{2}^{\mathrm{e}} & =\frac{i M}{2 \hbar \epsilon} \Gamma_{\{\mu \nu}{ }^{\nu} \Gamma_{\sigma \lambda \kappa} \Delta q^{\mu} \Delta q^{\sigma} \Delta q^{\lambda} \Delta q^{\kappa} \\ C_{3}^{\mathrm{e}} & =\frac{i M}{2 \hbar \epsilon}\left[\frac{1}{3} g_{\kappa \tau}\left(\partial_{\lambda} \Gamma_{\mu \nu}^{\tau}+\Gamma_{\mu \nu}^{\sigma} \Gamma_{\{\lambda \sigma\}}^{\tau}\right)+\frac{1}{4} \Gamma_{\mu \nu}^{\sigma} \Gamma_{\lambda \kappa \sigma}\right] \Delta q^{\mu} \Delta q^{\nu} \Delta q^{\lambda} \Delta q^{\kappa} \\ C_{4}^{\mathrm{e}} & =-\frac{1}{2} \frac{M^{2}}{4 \hbar^{2} \mathrm{e}^{2}} \Gamma_{\mu \nu \lambda} \Gamma_{\sigma \tau \kappa} \Delta q^{\mu} \Delta q^{\nu} \Delta q^{\lambda} \Delta q^{\sigma} \Delta q^{\tau} \Delta q^{\kappa} \end{align*} $$

$$ \langle\mathcal{O}(\Delta q)\rangle_{0} \equiv \int d^{D} \Delta q K_{0}^{\epsilon}(q, \Delta q) \mathcal{O}(\Delta q) $$

$$ \begin{align*} \left\langle\Delta q^{\mu} \Delta q^{\nu}\right\rangle_{0} & =\frac{i \hbar \epsilon}{M} g^{\mu \nu} \\ \left\langle\Delta q^{\mu} \Delta q^{\nu} \Delta q^{\lambda} \Delta q^{\kappa}\right\rangle_{0} & =\left(\frac{i \hbar \epsilon}{M}\right)^{2} g^{\mu \nu \lambda \kappa} \\ \left\langle\Delta q^{\mu} \Delta q^{\nu} \Delta q^{\lambda} \Delta q^{\kappa} \Delta q^{\sigma} \Delta q^{\tau}\right\rangle_{0} & =\left(\frac{i \hbar \epsilon}{M}\right)^{3} g^{\mu \nu \lambda \kappa \sigma \tau} \end{align*} $$

$$ g^{\mu \nu \lambda \kappa} \equiv g^{\mu \nu} g^{\lambda \kappa}+g^{\mu \lambda} g^{\nu \kappa}+g^{\mu \kappa} g^{\nu \lambda} $$

$$ g^{\mu \nu \lambda \kappa \sigma \tau}=g^{\mu \nu} g^{\lambda \kappa \sigma \tau}+g^{\mu \lambda} g^{\nu \kappa \sigma \tau}+g^{\mu \kappa} g^{\nu \lambda \sigma \tau}+g^{\mu \sigma} g^{\nu \lambda \kappa \tau}+g^{\mu \tau} g^{\nu \lambda \kappa \sigma} $$

$$ i \epsilon \frac{\hbar^{2}}{2 M} g^{\mu \nu}(q) \partial_{\mu} \partial_{\nu} \psi(q, t-\epsilon) $$

$$ A^{\mu} \partial_{\mu} \psi(q, t-\epsilon), $$

$$ A^{\mu}=-\left\langle C^{\varnothing} \Delta q^{\mu}\right\rangle_{0} $$

$$ \begin{align*} A^{\mu} & =\left\langle\left(\Gamma_{\{\lambda \nu}{ }^{\nu} \Delta q^{\lambda}+\frac{i M}{2 \hbar \epsilon} \Gamma_{\sigma \tau \lambda} \Delta q^{\sigma} \Delta q^{\tau} \Delta q^{\lambda}\right) \Delta q^{\mu}+\ldots\right\rangle_{0} \\ & =i \epsilon \frac{\hbar}{M}\left[\Gamma^{\{\mu \nu\}}{ }_{\nu}-\frac{1}{2}\left(\Gamma_{\nu}^{\mu \nu}+\Gamma_{\nu}^{\nu \mu}+\Gamma_{\nu}{ }^{\nu \mu}\right)\right]+\ldots \\ & =-i \epsilon \frac{\hbar}{2 M} \Gamma_{\nu}{ }^{\nu \mu}+\ldots \end{align*} $$

$$ V_{\mathrm{eff}} \equiv \frac{i \hbar}{\epsilon}\langle C\rangle_{0}=\frac{i \hbar}{\epsilon}\left\langle C^{\mathrm{e}}\right\rangle_{0} $$

$$ V_{\mathrm{eff}}=\frac{\hbar^{2}}{M} v \equiv \frac{\hbar^{2}}{M} \sum_{A, B} v_{A}{ }^{B} $$

$$ \begin{align*} v_{2}^{1} & =-\frac{1}{2}\left(\Gamma_{\{\mu \sigma\}}{ }^{\sigma} \Gamma_{\{\nu \lambda\}}{ }^{\lambda}-\Gamma_{\{\nu \kappa\}}{ }^{\sigma} \Gamma_{\{\mu \sigma\}}{ }^{\kappa}\right) g^{\mu \nu}, \\ v_{2}^{2} & =\frac{1}{8} \Gamma_{\{\mu \nu\}}{ }^{\tau} \Gamma_{\lambda \sigma \tau} g^{\mu \nu \sigma \lambda}, \\ v_{2}^{3} & =\frac{1}{2} \Gamma_{\{\mu \kappa\}}{ }^{\kappa} \Gamma_{\nu \tau \lambda} g^{\mu \nu \tau \lambda}, \\ v_{2}^{4} & =-\frac{1}{8} \Gamma_{\mu \nu \lambda} \Gamma_{\sigma \tau \kappa} g^{\mu \nu \lambda \sigma \tau \kappa}, \\ v_{3}{ }^{1} & =-\frac{1}{2}\left(\partial_{\{\mu} \Gamma_{\nu \lambda\}}{ }^{\lambda}+\Gamma_{\{\nu \kappa}{ }^{\sigma} \Gamma_{\{\sigma \mid \mu\}\}}{ }^{\kappa}\right) g^{\mu \nu}, \\ v_{3}^{2} & =\frac{1}{6} g_{\mu \tau}\left(\partial_{\kappa} \Gamma_{\lambda \nu}{ }^{\tau}+\Gamma_{\lambda \nu}{ }^{\sigma} \Gamma_{\{\kappa \sigma\}}{ }^{\tau}\right) g^{\mu \nu \lambda \kappa} . \end{align*} $$

$$ \begin{align*} \psi(\mathbf{x}, t) & =\int \frac{d^{D} \Delta x}{\sqrt{2 \pi i \epsilon \hbar / M}^{D}} \exp \left[i \epsilon \frac{M}{2}\left(\Delta x^{i}\right)^{2}\right] \\ & \times\left(1-\Delta x^{i} \partial_{x^{i}}+\frac{1}{2} \Delta x^{i} \Delta x^{j} \partial_{x^{i}} \partial_{x^{j}}+\ldots\right) \psi(\mathbf{x}, t-\epsilon) \\ & =\left[1+\frac{i \epsilon \hbar}{2 M} \partial_{i}^{2}+\mathcal{O}\left(\epsilon^{2}\right)\right] \psi(\mathbf{x}, t-\epsilon) \end{align*} $$

$$ \Delta x^{i}=e_{\lambda}^{i}\left(\Delta q^{\lambda}-\frac{1}{2!} \Gamma_{\mu \nu}^{\lambda} \Delta q^{\mu} \Delta q^{\nu}\right) $$

$$ J=\frac{\partial(\Delta x)}{\partial(\Delta q)}=\operatorname{det}\left(e_{\kappa}^{i}\right) \operatorname{det}\left(\delta_{\mu}^{\kappa}-e_{i}{ }^{\kappa} e_{\{\mu, \nu\}}^{i} \Delta q^{\nu}\right) $$

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

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

$$ \begin{align*} \mathcal{A}^{\epsilon}=\frac{M}{2 \epsilon}\left(\Delta x^{i}\right)^{2}=\frac{M}{2 \epsilon} & \left(g_{\mu \nu} \Delta q^{\mu} \Delta q^{\nu}-\Gamma_{\mu \nu \lambda} \Delta q^{\mu} \Delta q^{\nu} \Delta q^{\lambda}\right. \\ & \left.+\frac{1}{4} \Gamma_{\lambda \kappa}{ }^{\sigma} \Gamma_{\mu \nu \sigma} \Delta q^{\mu} \Delta q^{\nu} \Delta q^{\lambda} \Delta q^{\kappa}+\ldots\right) \end{align*} $$

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

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

$$ \Delta^{\prime} x^{i}=\frac{1}{3!} e^{i}{ }_{[\tau, \sigma]} e_{k}{ }^{\tau} e^{k}{ }_{\nu, \mu} \Delta q^{\mu} \Delta q^{\nu} \Delta q^{\sigma}=\frac{1}{3!} e^{i}{ }_{\lambda} S_{\sigma \tau}{ }^{\lambda} \Gamma_{\mu \nu}{ }^{\tau} \Delta q^{\mu} \Delta q^{\nu} \Delta q^{\sigma} $$

$$ J=\frac{\partial(\Delta x)}{\partial(\Delta q)}=\operatorname{det}\left(e_{\kappa}^{i}\right) \operatorname{det}\left(\delta^{\kappa}{ }_{\mu}-e_{i}{ }^{\kappa} e_{\{\mu, \nu\}}^{i} \Delta q^{\nu}+\frac{1}{2} e_{i}^{\kappa} e_{\{\mu, \nu \lambda\}}^{i} \Delta q^{\nu} \Delta q^{\lambda}+\ldots\right) $$

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

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

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

$$ \begin{align*} & \int d^{D} \Delta q K^{\epsilon}(q, \Delta q)=1+\ldots \\ & \int d^{D} \Delta q K^{\epsilon}(q, \Delta q) \Delta q^{\nu}=-i \epsilon \frac{\hbar}{2 M} \Gamma_{\mu}^{\mu \nu}+\ldots \\ & \int d^{D} \Delta q K^{\epsilon}(q, \Delta q) \Delta q^{\mu} \Delta q^{\nu}=i \epsilon \frac{\hbar}{M} g^{\mu \nu}+\ldots \end{align*} $$

$$ K^{\epsilon}(q, \Delta q)=K_{0}^{\epsilon}(q, \Delta q)[1+C(\Delta q)] $$

$$ K_{0}^{\epsilon}(q, \Delta q)=\frac{\sqrt{g(q)}}{\sqrt{2 \pi i \epsilon \hbar / M}^{D}} \exp \left[\frac{i}{\hbar} g_{\mu \nu}(q) \Delta q^{\mu} \Delta q^{\nu}\right] $$

$$ \begin{align*} \langle C\rangle_{0} & =0+\ldots \\ \left\langle C \Delta q^{\mu}\right\rangle_{0} & =i \epsilon \frac{\hbar}{2 M} \Gamma_{\mu}^{\mu \nu}+\ldots \end{align*} $$

$$ \begin{align*} \left\langle C_{1}\right\rangle_{0}=\left\langle C_{2}\right\rangle_{0} & =\mathcal{O}\left(\epsilon^{2}\right) \\ \left\langle\left(C_{1}-C_{2}\right) \Delta q^{\mu}\right\rangle_{0} & =\mathcal{O}\left(\epsilon^{2}\right) \end{align*} $$

$$ K^{\epsilon}(q, \Delta q)=K_{0}^{\epsilon}(q, \Delta q)\left[1+\frac{1}{2} \Gamma_{\mu}^{\mu}{ }_{\nu} \Delta q^{\nu}\right] $$

$$ K^{\epsilon}(q, \Delta q)=K_{0}^{\epsilon}(q, \Delta q)\left[1-\frac{i}{D+2} \frac{M}{2 \hbar \epsilon} \Gamma_{\mu}^{\mu}{ }_{\nu} \Delta q^{\nu} g_{\lambda \kappa} \Delta q^{\lambda} \Delta q^{\kappa}\right] $$

$$ \begin{align*} \frac{i}{\hbar} \mathcal{A}_{J}^{\epsilon} & =\frac{1}{2} \Gamma_{\mu}{ }^{\mu}{ }_{\nu} \Delta q^{\nu}-\frac{1}{8}\left(\Gamma_{\mu}{ }^{\mu}{ }_{\nu} \Delta q^{\nu}\right)^{2} \\ \frac{i}{\hbar} \mathcal{A}_{J}^{\epsilon} & =-\frac{i}{D+2} \frac{M}{2 \hbar \epsilon} \Gamma_{\mu}{ }^{\mu}{ }_{\nu} \Delta q^{\nu} g_{\lambda \kappa} \Delta q^{\lambda} \Delta q^{\kappa} \end{align*} $$

$$ \begin{align*} -\frac{1}{8}\left(\Gamma_{\mu}{ }^{\mu}{ }_{\nu} \Delta q^{\nu}\right)^{2} & \rightarrow-\frac{1}{8} \Gamma_{\mu}{ }^{\mu}{ }_{\nu} \Gamma_{\lambda}{ }^{\lambda}{ }_{\kappa}\left\langle\Delta q^{\nu} \Delta q^{\kappa}\right\rangle_{0} \\ & =-i \epsilon \frac{\hbar}{8 M}\left(\Gamma_{\mu}{ }^{\mu}{ }_{\nu}\right)^{2} \end{align*} $$

$$ \frac{i}{\hbar} \mathcal{A}_{J}^{\epsilon}=\frac{1}{2} \Gamma_{\mu}{ }^{\mu}{ }_{\nu} \Delta q^{\nu}-i \epsilon \frac{\hbar}{8 M}\left(\Gamma_{\mu}{ }^{\mu}{ }_{\nu}\right)^{2} $$

$$ \Gamma_{\mu}{ }^{\mu \nu}=g^{\mu \lambda} \Gamma_{\mu \lambda}{ }^{\nu} $$

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

$$ \begin{align*} \psi(q, t)= & \int d^{D} \Delta q K_{0}^{\epsilon}(q, \Delta q)[1+C(\Delta q)] \\ & \times e^{i \mathcal{A}_{\mathrm{pot}}^{\epsilon} / \hbar}\left(1-\Delta q^{\mu} \partial_{\mu}+\frac{1}{2} \Delta q^{\mu} \Delta q^{\nu} \partial_{\mu} \partial_{\nu}\right) \psi(q, t-\epsilon)+\ldots \\ = & \int d^{D} \Delta q K_{0}^{\epsilon}(q, \Delta q)[1+C(\Delta q)]\left[1-\Delta q^{\mu}\left(\partial_{\mu}-i \frac{e}{\hbar c} A_{\mu}\right)\right. \\ & \left.+\frac{1}{2} \Delta q^{\mu} \Delta q^{\nu}\left(\partial_{\mu}-i \frac{e}{\hbar c} A_{\mu}\right)\left(\partial_{\nu}-i \frac{e}{\hbar c} A_{\nu}\right)-i \epsilon V(q)\right] \psi(q, t-\epsilon)+\ldots \end{align*} $$

$$ i \hbar \partial_{t} \psi(q, t)=\hat{H} \psi(q, t) $$

$$ \hat{H}_{0}=-\frac{\hbar^{2}}{2 M} D_{\mu} D^{\mu} $$

$$ D_{\mu}^{A} \equiv D_{\mu}-i \frac{e}{\hbar c} A_{\mu} $$

$$ \psi(q) \rightarrow e^{-i(e / \hbar c) \Lambda(q)} \psi(q) $$

$$ D_{\mu} \psi(q) \rightarrow e^{-i(e / \hbar c) \Lambda(q)} D_{\mu} \psi(q) $$

$$ A_{\mu} \rightarrow A_{\mu}+\partial_{\mu} \Lambda(q) $$

$$ \hat{H}=-\frac{\hbar^{2}}{2 M} D_{\mu}^{A} D^{A \mu}+V(q) $$

$$ -\frac{\hbar}{2 M c}\left(\hat{p}_{\mu} A^{\mu}+A^{\mu} \hat{p}_{\mu}\right) $$

$$ \left\langle\psi_{2} \mid \psi_{1}\right\rangle \equiv \int d^{D} q \sqrt{g(q)} \psi_{2}^{*}(q) \psi_{1}(q) $$

$$ \Delta=\frac{1}{\sqrt{g}} \partial_{\mu} \sqrt{g} g^{\mu \nu} \partial_{\nu} $$

$$ \left(D_{\mu} D^{\mu}-\Delta\right) \psi=-2 S^{\mu} \partial_{\mu} \psi . $$

$$ \left\langle\psi_{2} \mid \psi_{1}\right\rangle_{\mathrm{phys}} \equiv \int d^{D} q \sqrt{g} w(q) \psi_{2}^{*}(q) \psi_{1}(q) $$

$$ w(q)=e^{-2 \sigma(q)} $$

$$ \left\langle\psi_{2} \mid \psi_{1}\right\rangle_{\mathrm{phys}} \equiv \int d^{D} q \sqrt{g(q)} e^{-2 \sigma(q)} \psi_{2}^{*}(q) \psi_{1}(q) $$

$$ D_{\mu}^{*} \equiv\left(D_{\mu}+2 S_{\mu}\right) $$

$$ \int d^{D} q \sqrt{g} U^{\mu \nu_{1} \ldots \nu_{n}} D_{\mu} V_{\nu_{1} \ldots \nu_{n}} $$

$$ \begin{align*} & \text { surface term }-\int d^{D} d q\left[\left(\partial_{\mu} \sqrt{g} U^{\mu \nu_{1} \ldots \nu_{n}}\right) V_{\nu_{1} \ldots \nu_{n}}\right. \\ &\left.-\sum_{i} \sqrt{g} U^{\mu \nu_{1} \ldots \nu_{i} \ldots \nu_{n}} \Gamma_{\mu \nu_{i}}^{\lambda_{i}} V_{\nu_{1} \ldots \lambda_{i} \ldots \nu_{n}}\right] \end{align*} $$

$$ \partial_{\mu} \sqrt{g}=\sqrt{g} \bar{\Gamma}_{\mu \nu}{ }^{\nu}=\sqrt{g}\left(2 S_{\mu}+\Gamma_{\mu \nu}{ }^{\nu}\right), $$

$$ \begin{align*} \text { surface term } & -\int d^{D} q \sqrt{g}\left[\left(\partial_{\mu} U^{\mu \nu_{1} \ldots \nu_{n}}\right) V_{\nu_{1} \ldots \nu_{n}}\right. \\ & \left.-\sum_{i} \Gamma_{\mu \nu_{i}}^{\lambda_{i}} U^{\mu \nu_{1} \ldots \nu_{i} \ldots \nu_{n}} V_{\nu_{1} \ldots \lambda_{i} \ldots \nu_{n}}-2 S_{\mu} U^{\mu \nu_{1} \ldots \nu_{n}} V_{\nu_{1} \ldots \nu_{n}}\right] \end{align*} $$

$$ \text { surface term }-\int d^{D} q \sqrt{g}\left(D_{\mu}^{*} U^{\mu \nu_{1} \ldots \nu_{n}}\right) V_{\nu_{1} \ldots \nu_{n}} $$

$$ \begin{align*} \int d^{D} q \sqrt{g} & e^{-2 \sigma(q)} U^{\mu \nu_{1} \ldots \nu_{n}} D_{\mu} V_{\nu_{1} \ldots \nu_{n}}= \\ & =\text { surface term }-\int d^{D} q \sqrt{g}\left(D_{\mu}^{*} e^{-2 \sigma(q)} U^{\mu \nu_{1} \ldots \nu_{n}}\right) V_{\nu_{1} \ldots \nu_{n}} \\ & =\text { surface term }-\int d^{D} q \sqrt{g} e^{-2 \sigma(q)}\left(D_{\mu} \sqrt{g} U^{\mu \nu_{1} \ldots \nu_{n}}\right) V_{\nu_{1} \ldots \nu_{n}} \end{align*} $$

$$ \prod_{n=1}^{N} \int d q_{n} \sqrt{g\left(q_{n-1}\right)}=\prod_{n=1}^{N} \int d q_{n} \sqrt{g\left(q_{n}-\Delta q\right)} $$

$$ i \hbar \partial_{t} \psi(q, t)=\left(\hat{H}_{0}+V_{\mathrm{eff}}\right) \psi(q, t) $$

$$ \hat{H}_{0}=-\frac{\hbar^{2}}{2 M} \Delta $$

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

$$ \left(q \beta \mid q^{\prime} 0\right)=\left(q\left|e^{\beta \Delta / 2}\right| q^{\prime}\right)=\frac{1}{\sqrt{2 \pi \beta} e^{D}} e^{-g_{\mu \nu}(q) \Delta q^{\mu} \Delta q^{\nu} / 2 \beta} \sum_{k=0}^{\infty} \beta^{k} a_{k}\left(q, q^{\prime}\right) $$

$$ \begin{align*} a_{0}\left(q, q^{\prime}\right) & \equiv 1+\frac{1}{12} \bar{R}_{\mu \nu} \Delta q^{\mu} \Delta q^{\nu}+\left(\frac{1}{360} \bar{R}_{\kappa}^{\mu}{ }_{\lambda} \bar{R}_{\mu \sigma \nu \tau}+\frac{1}{288} \bar{R}_{\kappa \lambda} \bar{R}_{\sigma \tau}\right) \Delta q^{\kappa} \Delta q^{\lambda} \Delta q^{\sigma} \Delta q^{\tau} \\ a_{1}\left(q, q^{\prime}\right) & \equiv \frac{1}{12} \bar{R}+\left(\frac{1}{144} \bar{R} \bar{R}_{\mu \nu}+\frac{1}{360} \bar{R}^{\kappa \lambda} \bar{R}_{\kappa \mu \lambda \nu}+\frac{1}{360} \bar{R}_{\mu}^{\kappa \lambda \sigma} \bar{R}_{\kappa \lambda \sigma \nu}-\frac{1}{180} \bar{R}_{\mu}^{\kappa} \bar{R}_{\kappa \nu}\right) \Delta q^{\mu} \Delta q^{\nu} \\ a_{2}\left(q, q^{\prime}\right) & \equiv \frac{1}{288} \bar{R}^{2}+\frac{1}{720} \bar{R}^{\mu \nu \kappa \lambda} \bar{R}_{\mu \nu \kappa \lambda}-\frac{1}{720} \bar{R}^{\mu \nu} \bar{R}_{\mu \nu} \end{align*} $$

$$ (q \beta \mid q 0)=\frac{1}{\sqrt{2 \pi \beta}^{D}}\left\{1+\frac{\beta}{12} \bar{R}+\frac{\beta^{2}}{2}\left[\frac{1}{144} \bar{R}^{2}+\frac{1}{360}\left(\bar{R}^{\mu \nu \kappa \lambda} \bar{R}_{\mu \nu \kappa \lambda}-\bar{R}^{\mu \nu} \bar{R}_{\mu \nu}\right)\right]+\ldots\right\} . $$

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

$$ \Delta=\partial^{2}-\frac{1}{3} \bar{R}_{i k_{1} j k_{2}}\left(q_{0}\right)\left(q-q_{0}\right)^{k_{1}}\left(q-q_{0}\right)^{k_{2}} \partial_{\mu} \partial_{\nu}-\frac{2}{3} \bar{R}_{\mu \nu}\left(q_{0}\right)\left(q-q_{0}\right)^{\mu} \partial_{\nu} $$

$$ \begin{align*} \hat{H}_{0} & =-\frac{1}{2} \partial^{2} \\ \hat{H}_{\mathrm{int}} & =\frac{1}{6} \bar{R}_{i k_{1} j k_{2}}\left(q-q_{0}\right)^{k_{1}}\left(q-q_{0}\right)^{k_{2}} \partial_{\mu} \partial_{\nu}+\frac{1}{3} \bar{R}_{\mu \nu}\left(q-q_{0}\right)^{\mu} \partial_{\nu} \end{align*} $$

$$ \begin{align*} \left(q \beta \mid q^{\prime} 0\right) & =\langle q| e^{-\beta\left(\hat{H}_{0}+\hat{H}_{\mathrm{int}}\right)}\left|q^{\prime}\right\rangle=\langle q| e^{-\beta \hat{H}_{0}}\left[1-\int_{0}^{\beta} d \sigma e^{\sigma \hat{H}_{0}} \hat{H}_{\mathrm{int}} e^{-\sigma \hat{H}}\right]\left|q^{\prime}\right\rangle \\ & =\left(q \beta \mid q^{\prime} 0\right)_{0}-\int_{0}^{\beta} d \sigma \int d^{D} \bar{q}(q \beta-\sigma \mid \bar{q} 0)_{0} \hat{H}_{\mathrm{int}}(\bar{q})(\bar{q} \sigma \mid q 0) \end{align*} $$

$$ \left(q \beta \mid q^{\prime} 0\right)_{0}=\langle q| e^{-\beta \hat{H}_{0}}\left|q^{\prime}\right\rangle=\frac{1}{\sqrt{2 \pi \beta}^{n}} e^{-(\Delta q)^{2} / 2 \beta} $$

$$ \left(q \beta \mid q^{\prime} 0\right) \approx\left(q \beta \mid q^{\prime} 0\right)_{0}-\int_{0}^{\beta} d \sigma \int d^{D} \bar{q}(q \beta-\sigma \mid \bar{q} 0)_{0} \hat{H}_{\mathrm{int}}(\bar{q})(\bar{q} \sigma \mid q 0)_{0} $$

$$ \begin{align*} \left(q \beta \mid q^{\prime} 0\right) & =\left(q \beta \mid q^{\prime} 0\right)_{0}\left\{1+\int_{0}^{\beta} d \sigma \int \frac{d^{D}(\Delta \bar{q})}{\sqrt{2 \pi a}^{D}} e^{-[\Delta \bar{q}-(\sigma / \beta) \Delta q]^{2} / 2 a}\right. \\ & \left.\times\left[-\frac{1}{6} \bar{R}_{\mu \kappa \nu \lambda} \Delta \bar{q}^{\kappa} \Delta \bar{q}^{\lambda}\left(-\frac{\delta^{\mu \nu}}{\sigma}+\frac{\Delta \bar{q}^{\mu} \Delta \bar{q}^{\nu}}{\sigma^{2}}\right)+\frac{1}{3} \bar{R}_{\mu \nu} \frac{\Delta \bar{q}^{\mu} \Delta \bar{q}^{\nu}}{\sigma}\right]\right\} \end{align*} $$

$$ \begin{align*} \left(q \beta \mid q^{\prime} 0\right) & =\left(q \beta \mid q^{\prime} 0\right)_{0}\left\{1+\frac{1}{6} \int_{0}^{\beta} d \sigma\left[\frac{\sigma}{\beta^{2}} \bar{R}_{\mu \nu}\left(q^{\prime}\right) \Delta q^{\mu} \Delta q^{\nu}+\frac{a}{\sigma} \bar{R}\left(q^{\prime}\right)\right]\right\} \\ & =\left(q \beta \mid q^{\prime} 0\right)_{0}\left[1+\frac{1}{12} \bar{R}_{\mu \nu}\left(q^{\prime}\right) \Delta q^{\mu} \Delta q^{\nu}+\frac{\beta}{12} \bar{R}\left(q^{\prime}\right)\right] \end{align*} $$

$$ \begin{align*} & g_{\mu \nu}\left(q^{\prime}\right)=g_{\mu \nu}(q)+\frac{1}{3} \bar{R}_{i k_{1} j k_{2}}(q) \Delta q^{k_{1}} \Delta q^{k_{2}}+\ldots, \\ & g_{\mu \nu}\left(q^{\prime}\right) \Delta q^{\mu} \Delta q^{\nu}=g_{\mu \nu}(q) \Delta q^{\mu} \Delta q^{\nu} \end{align*} $$

$$ \left(q \beta \mid q^{\prime} 0\right) \simeq \frac{1}{\sqrt{2 \pi \beta}^{D}} e^{-g_{\mu \nu}(q) \Delta q^{\mu} \Delta q^{\nu} / 2 \beta}\left[1+\frac{1}{12} \bar{R}_{\mu \nu}(q) \Delta q^{\mu} \Delta q^{\mu}+\frac{\beta}{12} \bar{R}(q)\right] $$

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

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

$$ \bar{R}_{\mu \nu \kappa \lambda}^{2}=\frac{2(D-1)(D-2)}{r^{4}}, \quad \bar{R}_{\mu \nu}^{2}=\frac{(D-1)(D-2)^{2}}{r^{4}} $$

$$ \begin{align*} (q \beta \mid q 0)=\frac{1}{\sqrt{2 \pi \beta}^{D-1}}[1 & +(D-1)(D-2) \frac{\beta}{12 r^{2}} \\ & \left.+(D-1)(D-2)\left(5 D^{2}-17 D+18\right) \frac{\beta^{2}}{1440 r^{4}}+\ldots\right] \end{align*} $$

$$ Z(\beta)=\sum_{l=0}^{\infty} d_{l} \exp \left[-l(l+D-2) \beta / 2 r^{2}\right] $$

$$ (q \beta \mid q 0)=\frac{\Gamma(D / 2)}{2 \pi^{D / 2} r^{D-1}} Z(\beta) $$

$$ Z(\beta)=\sum_{l=0}^{\infty}(2 l+1) \exp \left[-l(l+1) \beta / 2 r^{2}\right] $$

$$ Z(\beta)=\int_{0}^{\infty} d[l(l+1)] \exp \left[-l(l+1) \beta / 2 r^{2}\right]+\sum_{l=0}^{\infty}(2 l+1)\left[1-l(l+1) \beta / 2 r^{2}+\ldots\right] $$

$$ \int_{0}^{\infty} d z \exp \left(-z \beta / 2 r^{2}\right)=\frac{2 r^{2}}{\beta} $$

$$ \begin{align*} \sum_{l=0}^{\infty}(2 l+1) & =1+\sum_{l=1}^{\infty}(2 l+1)=1+2 \zeta(-1)-\frac{1}{2}=\frac{1}{3} \\ -\frac{\beta}{2 r^{2}} \sum_{l=0}^{\infty}(2 l+1) l(l+1) & =-\frac{\beta}{2 r^{2}} \sum_{l=1}^{\infty}\left(2 l^{3}+l\right)=-\frac{\beta}{2 r^{2}}[2 \zeta(-3)+\zeta(-1)]=\frac{\beta}{30 r^{2}} \end{align*} $$

$$ Z(\beta)=\frac{2 r^{2}}{\beta}\left(1+\frac{\beta}{6 r^{2}}+\frac{\beta^{2}}{60 r^{4}}+\ldots\right) $$

$$ \left(q t \mid q^{\prime} 0\right)=\left(q\left|e^{i t \Delta / 2}\right| q^{\prime}\right) $$

$$ i \partial_{t}\left(q t \mid q^{\prime} 0\right)=-\frac{\Delta}{2}\left(q \beta \mid q^{\prime} 0\right) $$

$$ \left(q 0 \mid q^{\prime} 0\right)=\delta^{(D)}\left(q-q^{\prime}\right) $$

$$ \mathcal{A}=\frac{1}{2} \int_{0}^{t} d t^{\prime} g_{\mu \nu}\left(q\left(t^{\prime}\right)\right) \dot{q}^{\mu}\left(t^{\prime}\right) q^{\nu}\left(t^{\prime}\right) $$

$$ \sigma\left(q, q^{\prime}\right) \equiv t A\left(q, q^{\prime} ; t\right)=\sigma\left(q^{\prime}, q\right) $$

$$ p_{\mu}=\partial_{\mu} A\left(q, q^{\prime} ; t\right)=\partial_{\mu} \sigma\left(q, q^{\prime}\right) / t $$

$$ \partial_{t} \sigma\left(q, q^{\prime}\right)=\frac{1}{2} \partial_{\mu} \sigma\left(q ; q^{\prime}\right) \partial^{\mu} \sigma\left(q ; q^{\prime}\right) $$

$$ \left(q t \mid q^{\prime} 0\right)=\left(q\left|e^{i t \Delta / 2}\right| q^{\prime}\right)=\frac{1}{\sqrt{2 \pi i t}^{D}} \mathcal{D}^{1 / 2}\left(q, q^{\prime}\right) e^{i \sigma\left(q, q^{\prime}\right) / 2 t} $$

$$ \mathcal{D} \equiv \operatorname{det}_{D}\left[-\partial_{\mu} \partial_{\nu}^{\prime} A\left(q, q^{\prime} ; t\right)\right]=\frac{\operatorname{det}_{D}\left[-\partial_{\mu} \partial_{\nu} \sigma\left(q, q^{\prime}\right)\right]}{t^{D}} $$

$$ \left(q t \mid q^{\prime} 0\right)=\left(q\left|e^{i t \Delta / 2}\right| q^{\prime}\right)=\frac{1}{\sqrt{2 \pi i t}^{D}} \mathcal{D}^{1 / 2}\left(q, q^{\prime}\right) e^{i \sigma\left(q, q^{\prime}\right) / 2 t} \sum_{n=0}^{\infty} t^{n} a_{n}\left(q, q^{\prime} ; t\right) $$

$$ \begin{align*} \partial^{\mu} \sigma \partial_{\mu} a_{0} & =0 \\ \partial^{\mu} \sigma \partial_{\mu} a_{n+1}+(n+1) a_{n+1} & =\tilde{\mathcal{D}}^{-1 / 2} \bar{D}^{\mu} \partial_{\mu}\left(\tilde{\mathcal{D}}^{1 / 2} a_{n}\right), \quad n=0,1,2, \ldots \end{align*} $$

$$ \partial^{\mu} \sigma \partial_{\mu} I\left(q, q^{\prime}\right)=0, \quad \partial^{\mu^{\prime}} \sigma \partial_{\mu^{\prime}} I\left(q, q^{\prime}\right)=0, \quad I(q, q)=1 $$

$$ a_{n+1}\left(q, q^{\prime}\right)=\frac{1}{t^{n+1}} \int_{0}^{t} d t^{\prime \prime} t^{\prime \prime n} \tilde{\mathcal{D}}^{-1 / 2} \bar{D}^{\mu} \partial_{\mu}\left[\tilde{\mathcal{D}}^{1 / 2} a_{n}\left(q\left(\tau^{\prime \prime}\right), q^{\prime}\right)\right] $$