Kleinert · 제8장 위상공간

$$ \mathbf{x}=r(\cos \varphi, \sin \varphi), $$

$$ \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)=\int \mathcal{D}^{2} x(\tau) \exp \left\{-\frac{1}{\hbar} \int_{\tau_{a}}^{\tau_{b}} d \tau\left[\frac{M}{2} \dot{\mathbf{x}}^{2}+V(r)\right]\right\} $$

$$ \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right) \approx \prod_{n=1}^{N}\left[\int \frac{d^{2} x_{n}}{2 \pi \hbar \epsilon / M}\right] \exp \left\{-\frac{\epsilon}{\hbar} \sum_{n=1}^{N+1}\left[\frac{M}{2 \epsilon^{2}}\left(\mathbf{x}_{n}-\mathbf{x}_{n-1}\right)^{2}+V\left(r_{n}\right)\right]\right\} . $$

$$ \exp \left[-\frac{1}{\hbar} \frac{M}{2 \epsilon}\left(\mathbf{x}_{n}-\mathbf{x}_{n-1}\right)^{2}\right]=\exp \left\{-\frac{1}{\hbar} \frac{M}{2 \epsilon}\left[r_{n}^{2}+r_{n-1}^{2}-2 r_{n} r_{n-1} \cos \left(\varphi_{n}-\varphi_{n-1}\right)\right]\right\} $$

$$ e^{a \cos \varphi}=\sum_{m=-\infty}^{\infty} I_{m}(a) e^{i m \varphi} $$

$$ \begin{align*} & \exp \left[-\frac{1}{\hbar} \frac{M}{2 \epsilon}\left(\mathbf{x}_{n}-\mathbf{x}_{n-1}\right)^{2}\right] \\ & \quad=\exp \left[-\frac{1}{\hbar} \frac{M}{2 \epsilon}\left(r_{n}^{2}+r_{n-1}^{2}\right)\right] \sum_{m=-\infty}^{\infty} I_{m}\left(\frac{M}{\hbar \epsilon} r_{n} r_{n-1}\right) e^{i m\left(\varphi_{n}-\varphi_{n-1}\right)} \end{align*} $$

$$ \prod_{n=1}^{N} 2 \pi \delta_{m_{n}, m_{n-1}} $$

$$ \begin{align*} & \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right) \approx \frac{2 \pi}{2 \pi \hbar \epsilon / M} \prod_{n=1}^{N}\left[\int_{0}^{\infty} \frac{d r_{n} r_{n} 2 \pi}{2 \pi \hbar \epsilon / M}\right] \sum_{m=-\infty}^{\infty} \frac{1}{2 \pi} e^{i m\left(\varphi_{b}-\varphi_{a}\right)} \\ & \quad \times \prod_{n=1}^{N+1}\left\{\exp \left[-\frac{1}{\hbar} \frac{M}{2 \epsilon}\left(r_{n}^{2}+r_{n-1}^{2}\right)\right] I_{m}\left(\frac{M}{\hbar \epsilon} r_{n} r_{n-1}\right)\right\} \exp \left[-\frac{\epsilon}{\hbar} \sum_{n=1}^{N+1} V\left(r_{n}\right)\right] \end{align*} $$

$$ \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)=\sum_{m=-\infty}^{\infty} \frac{1}{\sqrt{r_{b} r_{a}}}\left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{m} \frac{1}{2 \pi} e^{i m\left(\varphi_{b}-\varphi_{a}\right)} $$

$$ \begin{align*} & \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{m} \approx \frac{1}{\sqrt{2 \pi \hbar \epsilon / M}} \prod_{n=1}^{N}\left[\int_{0}^{\infty} \frac{d r_{n}}{\sqrt{2 \pi \hbar \epsilon / M}}\right] \\ & \quad \times \prod_{n=1}^{N+1}\left[\exp \left\{-\frac{M}{2 \epsilon \hbar}\left(r_{n}-r_{n-1}\right)^{2}\right\} \tilde{I}_{m}\left(\frac{M r_{n} r_{n-1}}{\hbar \epsilon}\right)\right] \exp \left\{-\frac{\epsilon}{\hbar} \sum_{n=1}^{N+1} V\left(r_{n}\right)\right\} \end{align*} $$

$$ \tilde{I}_{m}(z) \equiv \sqrt{2 \pi z} e^{-z} I_{m}(z) $$

$$ \begin{align*} & \tilde{I}_{m}(z) \xrightarrow{z \rightarrow \infty} 1-\frac{m^{2}-1 / 4}{2 z}+\ldots=e^{-\frac{m^{2}-1 / 4}{2 z}}+\ldots \\ & \tilde{I}_{m}(z) \xrightarrow{z \rightarrow 0} 2 \sqrt{\pi}(z / 2)^{m+1 / 2}+\ldots \end{align*} $$

$$ \begin{gather*} \int_{0}^{\infty} d r^{\prime} \exp \left(-\frac{r^{\prime \prime 2}}{2 \epsilon_{2}}-\frac{r^{2}}{2 \epsilon_{1}}\right) \exp \left[-r^{\prime 2}\left(\frac{1}{2 \epsilon_{2}}+\frac{1}{2 \epsilon_{1}}\right)\right] \frac{\sqrt{r^{\prime \prime} r^{\prime}}}{\epsilon_{2}} I_{m}\left(\frac{r^{\prime \prime} r^{\prime}}{\epsilon_{2}}\right) \frac{\sqrt{r^{\prime} r}}{\epsilon_{1}} I_{m}\left(\frac{r^{\prime} r}{\epsilon_{1}}\right) \\ =\exp \left[-\frac{r^{\prime \prime 2}+r^{2}}{2\left(\epsilon_{1}+\epsilon_{2}\right)}\right] \frac{\sqrt{r^{\prime \prime} r}}{\epsilon_{1}+\epsilon_{2}} I_{m}\left(\frac{r^{\prime \prime} r}{\epsilon_{1}+\epsilon_{2}}\right) \end{gather*} $$

$$ \int_{0}^{\infty} d r r e^{-r^{2} / \epsilon} I_{\nu}(\beta r) I_{\nu}(\alpha r)=\frac{\epsilon}{2} e^{\left(\alpha^{2}+\beta^{2}\right) \epsilon / 4} I_{\nu}(\epsilon \alpha \beta / 2) $$

$$ \begin{align*} \epsilon & =2 \epsilon_{1} \epsilon_{2} /\left(\epsilon_{1}+\epsilon_{2}\right) \\ \alpha & =r / \epsilon_{1} \\ \beta & =r^{\prime \prime} / \epsilon_{2} \end{align*} $$

$$ \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{m}=\frac{M}{\hbar} \frac{\sqrt{r_{b} r_{a}}}{\tau_{b}-\tau_{a}} \exp \left[-\frac{M}{2 \hbar} \frac{r_{b}^{2}+r_{a}^{2}}{\left(\tau_{b}-\tau_{a}\right)}\right] I_{m}\left(\frac{M}{\hbar} \frac{r_{b} r_{a}}{\tau_{b}-\tau_{a}}\right) $$

$$ \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)=\frac{M}{2 \pi \hbar\left(\tau_{b}-\tau_{a}\right)} \exp \left[-\frac{M}{2 \hbar} \frac{\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right)^{2}}{\tau_{b}-\tau_{a}}\right], $$

$$ \sum_{m=-\infty}^{\infty} I_{m}\left(\frac{M}{\hbar} \frac{r_{b} r_{a}}{\tau_{b}-\tau_{a}}\right) e^{i m\left(\varphi_{b}-\varphi_{a}\right)} $$

$$ \int_{0}^{\infty} d r r\left(r_{b} \tau_{b} \mid r \tau\right)_{m}\left(r \tau \mid r_{a} \tau_{a}\right)_{m}=\left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{m} $$

$$ \mathcal{A}_{m}=\int_{\tau_{a}}^{\tau_{b}} d \tau\left(\frac{M}{2} \dot{r}^{2}+\frac{\hbar^{2}}{2 M} \frac{m^{2}-1 / 4}{r^{2}}+V(r)\right) $$

$$ \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{m}=\int_{0}^{\infty} \mathcal{D} r \exp \left(-\frac{1}{\hbar} \mathcal{A}_{m}\right) $$

$$ \begin{align*} \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{m} & \approx \frac{1}{\sqrt{2 \pi \hbar \epsilon / M}} \prod_{n=1}^{N}\left[\int_{0}^{\infty} \frac{d r_{n}}{\sqrt{2 \pi \hbar \epsilon / M}}\right] \\ & \times \exp \left\{-\frac{1}{\hbar} \sum_{n=1}^{N+1}\left[\frac{M}{2 \epsilon}\left(r_{n}-r_{n-1}\right)^{2}+\epsilon \frac{\hbar^{2}}{2 M} \frac{m^{2}-1 / 4}{r_{n} r_{n-1}}+\epsilon V\left(r_{n}\right)\right]\right\} \end{align*} $$

$$ \tilde{I}_{m}\left(\frac{M r_{n} r_{n-1}}{\hbar \epsilon}\right) \xrightarrow{\epsilon \rightarrow 0} \exp \left(-\epsilon \frac{\hbar}{2 M} \frac{m^{2}-1 / 4}{r_{n} r_{n-1}}\right) $$

$$ \tilde{I}_{m}(z)=1-\frac{m^{2}-1 / 4}{2 z}+\frac{\left(m^{2}-1 / 4\right)\left(m^{2}-1 / 9\right)}{2!(2 z)^{2}}-\ldots $$

$$ \left(r_{a} \tau_{a}+\epsilon \mid r_{a} \tau_{a}\right)=\frac{1}{\sqrt{2 \pi \hbar \epsilon / M}} \exp \left(-\epsilon \frac{\hbar}{2 M} \frac{m^{2}-1 / 4}{r_{a}^{2}}\right) $$

$$ Z_{\mathrm{cl}}=\int_{0}^{\infty} \frac{d r}{\sqrt{2 \pi a}}\left[\exp \left(-a \frac{m^{2}-1 / 4}{2 r^{2}}\right)-1\right]=-\frac{1}{2} \sqrt{m^{2}-\frac{1}{4}} $$

$$ Z=\frac{1}{2} \int_{0}^{\infty} d z e^{-z}\left[I_{m}(z)-I_{1 / 2}(z)\right] $$

$$ \int_{0}^{\infty} d z e^{-\alpha z} I_{\mu}(z)=\left(\alpha^{2}-1\right)^{-1 / 2}\left(\alpha+\sqrt{\alpha^{2}-1}\right)^{-\mu} $$

$$ Z=-\frac{1}{2}(m-1 / 2) $$

$$ \tilde{\mathcal{A}}_{m}^{N}=\sum_{n=1}^{N+1}\left[\frac{M}{2 \epsilon}\left(r_{n}-r_{n-1}\right)^{2}-\hbar \log \tilde{I}_{m}\left(\frac{M}{\hbar \epsilon} r_{n} r_{n-1}\right)\right] $$

$$ e^{-\beta(\hat{T}+\hat{V})}=\lim _{N \rightarrow \infty}\left(e^{-\epsilon \hat{T}} e^{-\epsilon \hat{V}}\right)^{N+1}, \quad \epsilon \equiv \beta /(N+1) $$

$$ e^{-\beta H}=\lim _{N \rightarrow \infty}\left(e^{-\epsilon\left(\hat{T}+\hat{V}_{\mathrm{cb}}\right)} e^{-\epsilon \hat{V}}\right)^{N+1}, \quad \epsilon \equiv \beta /(N+1) $$

$$ \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{m} \approx \prod_{n=1}^{N}\left[\int_{0}^{\infty} d r_{n}\right] \prod_{n=1}^{N+1}\left[\int \frac{d p_{n}}{2 \pi \hbar}\right] \exp \left\{-\frac{1}{\hbar} \tilde{\mathcal{A}}_{m}^{N}[p, r]\right\} $$

$$ \tilde{\mathcal{A}}_{m}^{N}[p, r]=\sum_{n=1}^{N+1}\left[-i p_{n}\left(r_{n}-r_{n-1}\right)+\epsilon \frac{p_{n}^{2}}{2 M}-\hbar \log \tilde{I}_{m}\left(\frac{M}{\hbar \epsilon} r_{n} r_{n-1}\right)+\epsilon V\left(r_{n}\right)\right] $$

$$ \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{m} \approx \frac{1}{\sqrt{2 \pi \hbar \epsilon / M}} \prod_{n=1}^{N}\left[\int_{0}^{\infty} \frac{d r_{n}}{\sqrt{2 \pi \hbar \epsilon / M}}\right] \exp \left\{-\frac{1}{\hbar} \tilde{\mathcal{A}}_{m}^{N}[r]\right\} $$

$$ \tilde{A}_{m}^{N}[r, \dot{r}]=\sum_{n=1}^{N+1}\left[\frac{M}{2 \epsilon}\left(r_{n}-r_{n-1}\right)^{2}-\hbar \log \tilde{I}_{m}\left(\frac{M r_{n} r_{n-1}}{\hbar \epsilon}\right)+\epsilon V\left(r_{n}\right)\right] $$

$$ \begin{align*} \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)=\frac{1}{2 \pi} & \frac{M \omega}{\hbar \sinh \left[\omega\left(\tau_{b}-\tau_{a}\right)\right]} e^{-\frac{M \omega}{2 \hbar} \operatorname{coth}\left[\omega\left(\tau_{b}-\tau_{a}\right)\right]\left(r_{b}^{2}+r_{a}^{2}\right)} \\ & \times \sum_{m=-\infty}^{\infty} I_{m}\left(\frac{M \omega r_{b} r_{a}}{\hbar \sinh \left[\omega\left(\tau_{b}-\tau_{a}\right)\right]}\right) e^{i m\left(\varphi_{b}-\varphi_{a}\right)} \end{align*} $$

$$ \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{m}=\frac{M}{\hbar} \frac{\omega \sqrt{r_{b} r_{a}}}{\operatorname{coth}\left[\omega\left(\tau_{b}-\tau_{a}\right)\right]} I_{m}\left(\frac{M \omega r_{b} r_{a}}{\hbar \sinh \left[\omega\left(\tau_{b}-\tau_{a}\right)\right]}\right) $$

$$ \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{m}=\frac{M}{\hbar} \frac{\sqrt{r_{b} r_{a}}}{\tau_{b}-\tau_{a}} I_{m}\left(\frac{M}{\hbar} \frac{r_{b} r_{a}}{\tau_{b}-\tau_{a}}\right) $$

$$ \begin{align*} & \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)=\sum_{l=-\infty, \infty} \int_{0}^{\infty} \mathcal{D} r r \int_{-\infty}^{\infty} \mathcal{D} \varphi \\ & \quad \times\left.\exp \left\{\frac{1}{\hbar} \int_{\tau_{a}}^{\tau_{b}} d \tau\left[\frac{M}{2}\left(\dot{r}^{2}+r^{2} \dot{\varphi}^{2}\right)+V(r)\right]\right\}\right|_{\varphi\left(\tau_{b}\right)=\varphi_{b}+2 \pi l} \end{align*} $$

$$ \begin{align*} \sum_{\varphi_{N+1}=\varphi_{b}+2 \pi l, l=-\infty, \infty} & \frac{1}{2 \pi \hbar \epsilon / M} \prod_{n=1}^{N}\left[\int_{0}^{\infty} \frac{d r_{n} r_{n}}{\sqrt{2 \pi \hbar \epsilon / M}} \int_{-\infty}^{\infty} \frac{d \varphi_{n}}{\sqrt{2 \pi \hbar \epsilon / M}}\right] \\ \quad \times \exp & \left\{-\frac{1}{\hbar} \epsilon \sum_{n=1}^{N+1}\left[\frac{M}{2 \epsilon^{2}}\left[\left(r_{n}-r_{n-1}\right)^{2}+r_{n}^{2}\left(\varphi_{n}-\varphi_{n-1}\right)^{2}\right]+V\left(r_{n}\right)\right]\right\} \end{align*} $$

$$ \begin{align*} \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right) & \approx \frac{1}{2 \pi \hbar \epsilon / M} \frac{1}{r_{b}} \sum_{l-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{d p}{2 \pi \hbar} e^{(i / \hbar) p\left(\varphi_{b}-\varphi_{a}+2 \pi l\right)} \prod_{n=1}^{N}\left[\int_{0}^{\infty} \frac{d r_{n}}{2 \pi \hbar \epsilon / M}\right] \\ & \times \exp \left\{-\frac{\epsilon}{\hbar} \sum_{n=1}^{N+1}\left[\frac{M}{2} \frac{\left(r_{n}-r_{n-1}\right)^{2}}{\epsilon^{2}}+\frac{p^{2}}{2 M r_{n}^{2}}+V\left(r_{n}\right)\right]\right\} \end{align*} $$

$$ \begin{align*} \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{m} & \approx \sqrt{r_{b} r_{a}} \frac{1}{2 \pi \hbar \epsilon / M} \frac{1}{r_{b}} \prod_{n=1}^{N}\left[\int_{0}^{\infty} \frac{d r_{n}}{2 \pi \hbar \epsilon / M}\right] \\ & \times \exp \left\{-\frac{\epsilon}{\hbar} \sum_{n=1}^{N+1}\left[\frac{M}{2} \frac{\left(r_{n}-r_{n-1}\right)^{2}}{\epsilon^{2}}+\frac{\hbar^{2}}{2 M} \frac{m^{2}}{r_{n}^{2}}+V\left(r_{n}\right)\right]\right\} \end{align*} $$

$$ =\exp \left(-\frac{1}{2} \sum_{n=1}^{N+1} \frac{r_{n}-r_{n-1}}{\sqrt{r_{n} r_{n-1}}}+\ldots\right) $$

$$ \exp \left\{-\frac{\epsilon}{\hbar} \sum_{n=1}^{N+1}\left[\frac{M}{2}\left(\frac{r_{n}-r_{n-1}}{\epsilon}+i \frac{\hbar}{2 M \sqrt{r_{n} r_{n-1}}}\right)^{2}+\frac{\hbar^{2}}{2 M} \frac{m^{2}-1 / 4}{r_{n}^{2}}\right]\right\} $$

$$ \exp \left\{-i p_{n}\left(r_{n}-r_{n-1}\right)+\frac{\epsilon}{\hbar}\left[\frac{p_{n}^{2}}{2 M}-i \hbar \frac{p_{n}}{2 M r_{n}}\right]\right\} $$

$$ \epsilon \frac{\hbar^{2}}{2 M} \frac{m^{2}-1 / 4}{r_{n} r_{n-1}} " \equiv-\hbar \log \tilde{I}_{m}\left(\frac{M r_{n} r_{n-1}}{\hbar \epsilon}\right) $$

$$ \tilde{A}_{m}^{N}[r, \dot{r}]=\sum_{n=1}^{N+1}\left[\frac{M}{2 \epsilon}\left(r_{n}-r_{n-1}\right)^{2}+\epsilon \frac{\hbar^{2}}{2 \mu} \frac{m^{2}-1 / 4 "}{r_{n} r_{n-1}}+\epsilon V\left(r_{n}\right)\right] $$

$$ \mathcal{A}_{m}=\int_{\tau_{a}}^{\tau_{b}} d \tau\left[\frac{M}{2} \dot{r}^{2}+\frac{" \hbar^{2}}{2 M} \frac{m^{2}-1 / 4 "}{r^{2}}+V(r)\right] $$

$$ \left(\mathbf{x}_{n}-\mathbf{x}_{n-1}\right)^{2}=\left(r_{n}-r_{n-1}\right)^{2}+2 r_{n} r_{n-1}\left[1-\cos \left(\varphi_{n}-\varphi_{n-1}\right)\right], $$

$$ \begin{align*} \mathcal{A}^{N}=\frac{M}{2 \epsilon} \sum_{n=1}^{N+1}\left(\mathbf{x}_{n}-\mathbf{x}_{n-1}\right)^{2}= & \frac{M}{2 \epsilon} \sum_{n=1}^{N+1}\left\{\left(r_{n}-r_{n-1}\right)^{2}\right. \\ + & \left.2 r_{n} r_{n-1}\left[\frac{1}{2!}\left(\varphi_{n}-\varphi_{n-1}\right)^{2}-\frac{1}{4!}\left(\varphi_{n}-\varphi_{n-1}\right)^{4}+\ldots\right]\right\} \cdot \end{align*} $$

$$ \int \frac{d \varphi_{n-1}}{\sqrt{2 \pi \epsilon / a}} e^{-(a / 2 \epsilon)\left[\left(\varphi_{n}-\varphi_{n-1}\right)^{2}+a_{4}\left(\varphi_{n}-\varphi_{n-1}\right)^{4}+\ldots\right]} $$

$$ \left\langle\left(\varphi_{n}-\varphi_{n-1}\right)^{2}\right\rangle_{0}=\frac{\epsilon}{a} $$

$$ \int \frac{d u_{n-1}}{\sqrt{2 \pi / a}} e^{-(a / 2)\left[\left(u_{n}-u_{n-1}\right)^{2}+\epsilon a_{4}\left(u_{n}-u_{n-1}\right)^{4}+\ldots\right]} $$

$$ \int \frac{d u}{\sqrt{2 \pi / a}} e^{-(a / 2) u^{2}}\left\{\begin{array}{c} u^{2} \\ u^{4} \\ \vdots \\ u^{2 n} \end{array}\right\}=\left\{\begin{array}{r} a^{-1} \\ 3 a^{-2} \\ \vdots \\ (2 n-1)!!a^{-n} \end{array}\right\} $$

$$ 1-\epsilon \frac{a}{2} a_{4} 3 a^{-2}+\mathcal{O}\left(\epsilon^{2}\right) $$

$$ \int \frac{d \varphi_{n-1}}{\sqrt{2 \pi \epsilon / a}} e^{-(a / 2 \epsilon)\left(\varphi_{n}-\varphi_{n-1}\right)^{2}-3 a_{4} \epsilon / 2 a+\ldots} $$

$$ \Delta A=\frac{a}{2 \epsilon} a_{4}\left(\varphi_{n}-\varphi_{n-1}\right)^{4} $$

$$ A_{\mathrm{eff}}=\epsilon \frac{3 a_{4}}{2 a} $$

$$ A_{\mathrm{eff}}=\langle\Delta A\rangle_{0} $$

$$ \int \frac{d^{D} u}{\sqrt{2 \pi / a}^{D}} e^{-(a / 2) \mathbf{u}^{2}}\left\{\begin{array}{c} u_{i} u_{j} \\ u_{i} u_{j} u_{k} u_{l} \\ \vdots \\ u_{i_{1}} \cdot \ldots \cdot u_{i_{2 n}} \end{array}\right\}=\left\{\begin{array}{l} a^{-1} \delta_{i j} \\ a^{-2}\left(\delta_{i j} \delta_{k l}+\delta_{i k} \delta_{j l}+\delta_{i l} \delta_{j k}\right) \\ \vdots \\ a^{-n} \delta_{i_{1} \ldots i_{2 n}} \end{array}\right\} $$

$$ \delta_{i_{1} \ldots i_{2 n}}=\delta_{i_{1} i_{2}} \delta_{i_{3} i_{4} \ldots i_{2 n}}+\delta_{i_{1} i_{3}} \delta_{i_{2} i_{4} \ldots i_{2 n}}+\ldots+\delta_{i_{1} i_{2 n}} \delta_{i_{2} i_{3} \ldots i_{2 n-1}} $$

$$ \Delta A=\frac{a}{2 \epsilon}\left(a_{4}\right)_{i j k l}\left(\varphi_{n}-\varphi_{n-1}\right)_{i}\left(\varphi_{n}-\varphi_{n-1}\right)_{j}\left(\varphi_{n}-\varphi_{n-1}\right)_{k}\left(\varphi_{n}-\varphi_{n-1}\right)_{l} $$

$$ \int \frac{d \delta}{\sqrt{2 \pi / a}} e^{-a \delta^{2}} e^{-a \Delta^{2} / 4}\left\{1-\frac{a}{2} \epsilon a_{4}\left[(-\delta+\Delta / 2)^{4}+(\delta+\Delta / 2)^{4}\right]+\ldots\right\} . $$

$$ \left\langle\left(u_{n-1}-\bar{u}_{n-1}\right)^{2}\right\rangle_{0}=\frac{1}{2 a} $$

$$ \frac{1}{\sqrt{2 \pi \epsilon / a}} e^{-(a / 2 \epsilon)\left[\left(\varphi_{n}-\varphi_{n-1}\right)^{2}+a_{4}\left(\varphi_{n}-\varphi_{n-1}\right)^{4}+\ldots\right]} $$

$$ \int_{-\infty}^{\infty} \frac{d \varphi_{n-1}}{\sqrt{2 \pi \epsilon / a}} e^{-(a / 2 \epsilon)\left[\left(\varphi_{n}-\varphi_{n-1}\right)^{2}+a_{4}\left(\varphi_{n}-\varphi_{n-1}\right)^{4}+\ldots\right]} \psi\left(\varphi_{n-1}\right) $$

$$ \psi\left(\varphi_{n-1}\right)=\psi\left(\varphi_{n}\right)-\left(\varphi_{n}-\varphi_{n-1}\right) \psi^{\prime}\left(\varphi_{n}\right)+\frac{1}{2}\left(\varphi_{n}-\varphi_{n-1}\right)^{2} \psi^{\prime \prime}\left(\varphi_{n}\right)+\ldots, $$

$$ \left(1-A_{\mathrm{eff}}\right) \psi\left(\varphi_{n}\right)+\frac{\epsilon}{2 a} \psi^{\prime \prime}\left(\varphi_{n}\right)+\ldots $$

$$ \begin{align*} & \frac{1}{\sqrt{2 \pi \epsilon / a}} e^{-(a / 2 \epsilon)\left[\left(\varphi_{n}-\varphi_{n-1}\right)^{2}+a_{4}\left(\varphi_{n}-\varphi_{n-1}\right)^{4}+\ldots\right]} \\ & \quad=\left(1-A_{\text {eff }}\right) \delta\left(\varphi_{n}-\varphi_{n-1}\right)+\frac{\epsilon}{2 a} \delta^{\prime \prime}\left(\varphi_{n}-\varphi_{n-1}\right)+\ldots \end{align*} $$

$$ \frac{1}{\sqrt{2 \pi \epsilon / a}} e^{-(a / 2 \epsilon)\left(\varphi_{n}-\varphi_{n-1}\right)^{2}} e^{-A_{\mathrm{eff}}} $$

$$ \begin{align*} & \frac{1}{\sqrt{2 \pi \epsilon / a}} e^{-(a / 2 \epsilon)\left[\left(\varphi_{n}-\varphi_{n-1}\right)^{2}+a_{3}\left(\varphi_{n}-\varphi_{n-1}\right)^{3}+a_{4}\left(\varphi_{n}-\varphi_{n-1}\right)^{4}+\ldots\right]} \\ & \quad=\left(1-A_{\text {eff }}\right) \delta\left(\varphi_{n}-\varphi_{n-1}\right)+3 a_{3} \frac{\epsilon}{2 a} \delta^{\prime}\left(\varphi_{n}-\varphi_{n-1}\right)+\frac{\epsilon}{2 a} \delta^{\prime \prime}\left(\varphi_{n}-\varphi_{n-1}\right)+\ldots \end{align*} $$

$$ A_{\mathrm{eff}}=\frac{\epsilon}{2 a}\left[3 a_{4}-\frac{15}{4} a_{3}^{2}\right] . $$

$$ \frac{1}{\sqrt{2 \pi \epsilon / a}} e^{-(a / 2 \epsilon)\left(\varphi_{n}-\varphi_{n-1}\right)^{2}} e^{-A_{\mathrm{eff}}}\left[1-\frac{3 a_{3}}{2}\left(\varphi_{n}-\varphi_{n-1}\right)+\ldots\right] $$

$$ \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)=\int \mathcal{D}^{D} x(\tau) \exp \left\{-\frac{1}{\hbar} \int_{\tau_{a}}^{\tau_{b}}\left[\frac{M}{2} \dot{\mathbf{x}}^{2}+V(r)\right]\right\} $$

$$ \exp \left[-\frac{1}{\hbar} \frac{M}{2 \epsilon} \sum_{n=1}^{N+1}\left(r_{n}^{2}+r_{n-1}^{2}-2 r_{n} r_{n-1} \cos \Delta \vartheta_{n}\right)\right] $$

$$ \mathbf{x}=r(\cos \theta \cos \varphi, \cos \theta \sin \varphi, \sin \theta) $$

$$ \cos \Delta \vartheta_{n}=\cos \theta_{n} \cos \theta_{n-1}+\sin \theta_{n} \sin \theta_{n-1} \cos \left(\varphi_{n}-\varphi_{n-1}\right) $$

$$ \frac{1}{\sqrt{2 \pi \hbar \epsilon / M}^{3}} \prod_{n=1}^{N} \int \frac{d^{3} x_{n}}{\sqrt{2 \pi \hbar \epsilon / M}^{3}} $$

$$ \frac{1}{\sqrt{2 \pi \hbar \epsilon / M}^{3}} \prod_{n=1}^{N} \int \frac{d r_{n} r_{n}^{2} d \cos \theta_{n} d \varphi_{n}}{\sqrt{2 \pi \hbar \epsilon / M}^{3}} $$

$$ e^{h \cos \Delta \vartheta_{n}}=\sqrt{\frac{\pi}{2 h}} \sum_{l=0}^{\infty} I_{l+1 / 2}(h)(2 l+1) P_{l}\left(\cos \Delta \vartheta_{n}\right), $$

$$ Y_{l m}(\theta, \varphi)=(-1)^{m}\left[\frac{2 l+1}{4 \pi} \frac{(l-m)!}{(l+m)!}\right]^{1 / 2} P_{l}^{m}(\cos \theta) e^{i m \varphi} $$

$$ \frac{2 l+1}{4 \pi} P_{l}\left(\cos \Delta \vartheta_{n}\right)=\sum_{m=-l}^{l} Y_{l m}\left(\theta_{n}, \varphi_{n}\right) Y_{l m}^{*}\left(\theta_{n-1}, \varphi_{n-1}\right), $$

$$ P_{l}^{m}(z)=\frac{\left(1-z^{2}\right)^{m / 2}}{2^{l} l!} \frac{(l-m)!}{(l+m)!} \frac{d^{l+m}}{d z^{l+m}}\left(z^{2}-1\right)^{l} $$

$$ \left[-\frac{1}{\sin \theta} \frac{d}{d \theta}\left(\sin \theta \frac{d}{d \theta}\right)+\frac{m^{2}}{\sin ^{2} \theta}\right] P_{l}^{m}(\cos \theta)=l(l+1) P_{l}^{m}(\cos \theta) $$

$$ e^{h \cos \Delta \vartheta_{n}}=\sqrt{\frac{\pi}{2 h}} 4 \pi \sum_{l=0}^{\infty} I_{l+1 / 2}(h) \sum_{m=-l}^{l} Y_{l m}\left(\theta_{n}, \varphi_{n}\right) Y_{l m}^{*}\left(\theta_{n-1}, \varphi_{n-1}\right) . $$

$$ \begin{align*} & \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right) \approx \frac{4 \pi}{\sqrt{2 \pi \hbar \epsilon / M}} \prod_{n=1}^{N}\left[\int_{0}^{\infty} \frac{d r_{n} r_{n}^{2} d \cos \theta_{n} d \varphi_{n} 4 \pi}{\sqrt{2 \pi \hbar \epsilon / M}}\right] \\ & \quad \times \prod_{n=1}^{N+1}\left[\left(\frac{\hbar \epsilon \pi}{2 M r_{n} r_{n-1}}\right)^{1 / 2} \sum_{l_{n}=0}^{\infty} \sum_{m_{n}=-l_{n}}^{l_{n}} I_{l_{n}+1 / 2}\left(\frac{M}{\hbar \epsilon} r_{n} r_{n-1}\right)\right. \\ & \left.\quad \times Y_{l_{n} m_{n}}\left(\theta_{n}, \varphi_{n}\right) Y_{l_{n} m_{n}}^{*}\left(\theta_{n-1}, \varphi_{n-1}\right)\right] \exp \left\{-\frac{\epsilon}{\hbar} \sum_{n=1}^{N+1}\left[\frac{M}{2} \frac{r_{n}^{2}+r_{n-1}^{2}}{\epsilon^{2}}+V\left(r_{n}\right)\right]\right\} \end{align*} $$

$$ \int_{-1}^{1} d \cos \theta \int_{-\pi}^{\pi} d \varphi Y_{l m}^{*}(\theta, \varphi) Y_{l^{\prime} m^{\prime}}(\theta, \varphi)=\delta_{l l^{\prime}} \delta_{m m^{\prime}} $$

$$ \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)=\sum_{l=0}^{\infty} \sum_{m=-l}^{l} \frac{1}{r_{b} r_{a}}\left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{l} Y_{l m}\left(\theta_{b}, \varphi_{b}\right) Y_{l m}^{*}\left(\theta_{a}, \varphi_{a}\right), $$

$$ \begin{align*} \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{l} & \approx \frac{4 \pi r_{b} r_{a}}{\sqrt{2 \pi \hbar \epsilon / M}^{3}} \prod_{n=1}^{N}\left[\int \frac{d r_{n} r_{n}^{2} 4 \pi}{\sqrt{2 \pi \hbar \epsilon / M}^{3}}\right] \prod_{n=1}^{N+1}\left[\frac{\hbar \epsilon}{2 M r_{n} r_{n-1}}\right] \\ & \times \prod_{n=1}^{N+1}\left[\tilde{I}_{l+1 / 2}\left(\frac{M}{\hbar \epsilon} r_{n} r_{n-1}\right)\right] \exp \left\{-\frac{\epsilon}{\hbar} \sum_{n=1}^{N+1}\left[\frac{M}{2 \epsilon^{2}}\left(r_{n}-r_{n-1}\right)^{2}+V\left(r_{n}\right)\right]\right\} \end{align*} $$

$$ \frac{1}{\sqrt{2 \pi \hbar \epsilon / M}} \prod_{n=1}^{N}\left[\int_{0}^{\infty} \frac{d r_{n}}{\sqrt{2 \pi \hbar \epsilon / M}}\right] $$

$$ \mathcal{A}_{l}^{N}=\epsilon \sum_{n=1}^{N+1}\left[\frac{M}{2 \epsilon^{2}}\left(r_{n}-r_{n-1}\right)^{2}+\frac{\hbar^{2}}{2 M} \frac{l(l+1)}{r_{n} r_{n-1}}+V\left(r_{n}\right)\right] $$

$$ \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{l} \approx \frac{1}{\sqrt{2 \pi \epsilon \hbar / M}} \prod_{n=1}^{N}\left[\int_{0}^{\infty} \frac{d r_{n}}{\sqrt{2 \pi \hbar \epsilon / M}}\right] \exp \left(-\frac{1}{\hbar} \mathcal{A}_{l}^{N}\right) $$

$$ \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{l}=\int_{0}^{\infty} \mathcal{D} r(\tau) e^{-\frac{1}{\hbar} \mathcal{A}_{l}[r]} $$

$$ \mathcal{A}_{l}[r]=\int_{\tau_{a}}^{\tau_{b}} d \tau\left[\frac{M}{2} \dot{r}^{2}+\frac{\hbar^{2}}{2 M} \frac{l(l+1)}{r^{2}}+V(r)\right] $$

$$ V_{\mathrm{cf}}=\frac{\hbar^{2}}{2 M} \frac{l(l+1)}{r^{2}} $$

$$ e^{h \cos \Delta \vartheta_{n}}=e^{-\frac{M}{\hbar \epsilon} r_{n} r_{n-1} \cos \Delta \vartheta_{n}} $$

$$ e^{h \cos \Delta \vartheta_{n}}=\sum_{l=0}^{\infty} a_{l}(h) \frac{l+D / 2-1}{D / 2-1} \frac{1}{S_{D}} C_{l}^{(D / 2-1)}\left(\cos \Delta \vartheta_{n}\right), $$

$$ \begin{align*} a_{l}(h) & \equiv(2 \pi)^{D / 2} h^{1-D / 2} I_{l+D / 2-1}(h) \\ & \equiv e^{h} \tilde{a}_{l}(h)=e^{h}\left(\frac{2 \pi}{h}\right)^{(D-1) / 2} \tilde{I}_{l+D / 2-1}(h) \end{align*} $$

$$ \frac{1}{\left(1-2 t \alpha+\alpha^{2}\right)^{\lambda}}=\sum_{n=0}^{\infty} C_{n}^{(\lambda)} \alpha^{n} $$

$$ \begin{gather*} \int_{0}^{\pi} d \vartheta \sin ^{\nu} \vartheta e^{h \cos \vartheta} C_{l}^{(\nu)}(\cos \vartheta)=\pi \frac{2^{1-\nu} \Gamma(2 \nu+l)}{l!\Gamma(\nu)} h^{-\nu} I_{\nu+l}(h) \\ \int_{0}^{\pi} d \vartheta \sin ^{\nu} \vartheta C_{l}^{(\nu)}(\cos \vartheta) C_{l^{\prime}}^{(\nu)}(\cos \vartheta)=\pi \frac{2^{1-2 \nu} \Gamma(2 \nu+l)}{l!(l+\nu) \Gamma(\nu)^{2}} \delta_{l l^{\prime}} \end{gather*} $$

$$ P_{l}^{(\alpha, \beta)}(z) \equiv \frac{1}{l!} \frac{\Gamma(l+1+\beta)}{\Gamma(1+\beta)} F(-l, l+1+\alpha+\beta ; 1+\beta ;(1-z) / 2) . $$

$$ C_{l}^{(\nu)}(z)=\frac{\Gamma(2 \nu+l) \Gamma(\nu+1 / 2)}{\Gamma(2 \nu) \Gamma(\nu+l+1 / 2)} P_{l}^{(\nu-1 / 2, \nu-1 / 2)}(z) $$

$$ C_{l}^{(\nu)}(z)=\frac{1}{l!} \frac{\Gamma(l+2 \nu)}{\Gamma(2 \nu)} F(-l, l+2 \nu ; 1 / 2+\nu ;(1-z) / 2) $$

$$ \begin{gather*} \lim _{\nu \rightarrow 0} \frac{1}{\nu} C_{l}^{(\nu)}(\cos \vartheta)=\frac{1}{2 l} \cos l \vartheta, \\ C_{l}^{(1 / 2)}(\cos \vartheta)=P_{l}^{(0,0)}(\cos \vartheta)=P_{l}(\cos \vartheta), \end{gather*} $$

$$ C_{l}^{(1)}(\cos \vartheta)=\frac{\sin (l+1) \beta}{\sin \beta} $$

$$ \begin{align*} \hat{x}^{1} & =\sin \varphi_{D-1} \cdots \sin \varphi_{1} \\ \hat{x}^{2} & =\sin \varphi_{D-1} \cdots \cos \varphi_{1} \\ & \vdots \\ \hat{x}^{D} & =\cos \varphi_{D-1} \end{align*} $$

$$ \begin{align*} & 0 \leq \varphi_{1}<2 \pi \\ & 0 \leq \varphi_{i}<\pi, \quad i \neq 1 \end{align*} $$

$$ d_{l}=\frac{(2 l+D-2)(l+D-3)!}{l!(D-2)!} $$

$$ \int d \hat{\mathbf{x}} Y_{l \mathbf{m}}^{*}(\hat{\mathbf{x}}) Y_{l^{\prime} \mathbf{m}^{\prime}}(\hat{\mathbf{x}})=\delta_{l l^{\prime}} \delta_{\mathbf{m m}^{\prime}} $$

$$ \int d \hat{\mathbf{x}}=\int d \varphi_{D-1} \sin ^{D-2} \varphi_{D-1} \int d \varphi_{D-2} \sin ^{D-3} \varphi_{D-2} \cdots \int d \varphi_{2} \sin \varphi_{2} \int d \varphi_{1} $$

$$ \int d^{D-1} \hat{\mathbf{x}}=\int d \varphi_{D-1} \sin ^{D-2} \varphi_{D-1} \int d^{D-2} \hat{\mathbf{x}}_{\perp} $$

$$ S_{D}=\frac{\sqrt{\pi} \Gamma((D-1) / 2)}{\Gamma(D / 2)} \times S_{D-1} $$

$$ \hat{\mathbf{x}}=(\cos \theta, \sin \theta \cos \psi, \sin \theta \sin \psi \cos \varphi, \sin \theta \sin \psi \sin \varphi) $$

$$ d \hat{\mathbf{x}}=d \theta \sin ^{2} \theta d \psi \sin \psi d \varphi $$

$$ \begin{align*} \hat{x}^{1} & =\cos (\theta / 2) \cos [(\varphi+\gamma) / 2], \\ \hat{x}^{2} & =-\cos (\theta / 2) \sin [(\varphi+\gamma) / 2], \\ \hat{x}^{3} & =\sin (\theta / 2) \cos [(\varphi-\gamma) / 2], \\ \hat{x}^{4} & =\sin (\theta / 2) \sin [(\varphi-\gamma) / 2], \end{align*} $$

$$ \theta \in[0, \pi), \quad \varphi \in[0,2 \pi), \quad \gamma \in[-2 \pi, 2 \pi) . $$

$$ g(\varphi, \theta, \gamma)=\exp \left(i \varphi \sigma_{3} / 2\right) \exp \left(i \theta \sigma_{2} / 2\right) \exp \left(i \gamma \sigma_{3} / 2\right) $$

$$ d \hat{\mathbf{x}}=\frac{1}{8} d \theta \sin \theta d \varphi d \gamma $$

$$ Y_{l, m_{1}, m_{2}}(\hat{\mathbf{x}})=\sqrt{\frac{l+1}{2 \pi^{2}}} \mathcal{D}_{m_{1} m_{2}}^{l / 2}(\varphi, \theta, \gamma)=\sqrt{\frac{l+1}{2 \pi^{2}}} d_{m_{1} m_{2}}^{l / 2}(\theta) e^{i\left(m_{1} \varphi+m_{2} \gamma\right)} $$

$$ \frac{2 l+D-2}{D-2} \frac{1}{S_{D}} C_{l}^{(D / 2-1)}\left(\cos \Delta \vartheta_{n}\right)=\sum_{\mathbf{m}} Y_{l \mathbf{m}}\left(\hat{\mathbf{x}}_{n}\right) Y_{l \mathbf{m}}^{*}\left(\hat{\mathbf{x}}_{n-1}\right) . $$

$$ \frac{1}{4 \pi}(2 l+1) P_{l}\left(\cos \Delta \vartheta_{n}\right)=\sum_{m=-l}^{l} Y_{l m}\left(\hat{\mathbf{x}}_{n}\right) Y_{l m}^{*}\left(\hat{\mathbf{x}}_{n-1}\right) $$

$$ \frac{l+1}{2 \pi^{2}} C_{l}^{(1)}\left(\cos \Delta \vartheta_{n}\right)=\frac{l+1}{2 \pi^{2}} \sum_{m_{1}, m_{2}=-l / 2}^{l / 2} \mathcal{D}_{m_{1} m_{2}}^{l / 2}\left(\varphi_{n}, \theta_{n}, \gamma_{n}\right) \mathcal{D}_{m_{1} m_{2}}^{l / 2 *}\left(\varphi_{n-1}, \theta_{n-1}, \gamma_{n-1}\right), $$

$$ \begin{align*} \cos \Delta \vartheta_{n}= & \cos \left(\theta_{n} / 2\right) \cos \left(\theta_{n-1} / 2\right) \cos \left[\left(\varphi_{n}-\varphi_{n-1}+\gamma_{n}-\gamma_{n-1}\right) / 2\right] \\ & +\sin \left(\theta_{n} / 2\right) \sin \left(\theta_{n-1} / 2\right) \cos \left[\left(\varphi_{n}-\varphi_{n-1}-\gamma_{n}+\gamma_{n-1}\right) / 2\right] \end{align*} $$

$$ e^{h\left(\cos \Delta \vartheta_{n}-1\right)}=\sum_{l=0}^{\infty} \tilde{a}_{l}(h) \sum_{\mathbf{m}} Y_{l \mathbf{m}}\left(\hat{\mathbf{x}}_{n}\right) Y_{l \mathbf{m}}^{*}\left(\hat{\mathbf{x}}_{n-1}\right) $$

$$ e^{h\left(\cos \Delta \vartheta_{n}-1\right)}=\sum_{l=0}^{\infty} \tilde{a}_{l}(h) \frac{l+1}{2 \pi^{2}} \sum_{m_{1}, m_{2}=-l / 2}^{l / 2} \mathcal{D}_{m_{1} m_{2}}^{l / 2}\left(\varphi_{n}, \theta_{n}, \gamma_{n}\right) \mathcal{D}_{m_{1} m_{2}}^{l / 2 *}\left(\varphi_{n-1}, \theta_{n-1}, \gamma_{n-1}\right) $$

$$ \begin{align*} \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right) & \approx \frac{1}{\sqrt{2 \pi \hbar \epsilon / M}} \prod_{n=1}^{N}\left[\int_{0}^{\infty} \frac{d r_{n} r_{n}^{D-1} d \hat{\mathbf{x}}_{n}}{\sqrt{2 \pi \hbar \epsilon / M}^{D}}\right] \\ & \times \prod_{n=1}^{N+1}\left[\left(\frac{2 \pi \hbar \epsilon}{M r_{n} r_{n-1}}\right)^{(D-1) / 2} \sum_{l_{n}=0}^{\infty} \tilde{I}_{D / 2-1+l_{n}}\left(\frac{M}{\hbar \epsilon} r_{n} r_{n-1}\right)\right. \\ & \left.\times \sum_{\mathbf{m}_{n}} Y_{l_{n} \mathbf{m}_{n}}\left(\hat{\mathbf{x}}_{n}\right) Y_{l_{n} \mathbf{m}_{n}}^{*}\left(\hat{\mathbf{x}}_{n-1}\right)\right] \exp \left\{-\frac{\epsilon}{\hbar} \sum_{n=1}^{N+1}\left[\frac{M}{2 \epsilon^{2}}\left(r_{n}-r_{n-1}\right)^{2}+V\left(r_{n}\right)\right]\right\} \end{align*} $$

$$ \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)=\frac{1}{\left(r_{b} r_{a}\right)^{(D-1) / 2}} \sum_{l=0}^{\infty}\left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{l} \sum_{\mathbf{m}} Y_{l \mathbf{m}}\left(\hat{\mathbf{x}}_{b}\right) Y_{l \mathbf{m}}^{*}\left(\hat{\mathbf{x}}_{a}\right) $$

$$ \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{l} \approx \frac{1}{\sqrt{2 \pi \hbar \epsilon / M}} \prod_{n=1}^{N}\left[\int_{0}^{\infty} \frac{d r_{n}}{\sqrt{2 \pi \hbar \epsilon / M}}\right] \exp \left\{-\frac{1}{\hbar} \mathcal{A}_{l}^{N}[r]\right\} $$

$$ \mathcal{A}_{l}^{N}[r]=\epsilon \sum_{n=1}^{N+1}\left[\frac{M}{2 \epsilon^{2}}\left(r_{n}-r_{n-1}\right)^{2}-\frac{\hbar}{\epsilon} \log \tilde{I}_{l+D / 2-1}\left(\frac{M}{\hbar \epsilon} r_{n} r_{n-1}\right)+V\left(r_{n}\right)\right] $$

$$ \mathcal{A}_{l}^{N}[r, \bar{r}] \approx \epsilon \sum_{n=1}^{N+1}\left[\frac{M}{2 \epsilon^{2}}\left(r_{n}-r_{n-1}\right)^{2}+\frac{\hbar^{2}}{2 M} \frac{(l+D / 2-1)^{2}-1 / 4}{r_{n} r_{n-1}}+V\left(r_{n}\right)\right] $$

$$ \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{l}=\int \mathcal{D} r(\tau) \exp \left\{-\frac{1}{\hbar} \mathcal{A}_{l}[r]\right\} $$

$$ \mathcal{A}_{l}[r]=\int_{\tau_{a}}^{\tau_{b}} d \tau\left[\frac{M}{2} \dot{r}^{2}+\frac{" \hbar^{2}}{2 M} \frac{(l+D / 2-1)^{2}-1 / 4 "}{r^{2}}+V(r)\right] $$

$$ \frac{\hbar^{2}}{2 M r^{2}}\left[(l+D / 2-1)^{2}-1 / 4\right] \text { ", } $$

$$ \left." \frac{\epsilon \hbar^{2}}{2 M r_{n} r_{n-1}}\left[(l+D / 2-1)^{2}-1 / 4\right)\right] " \equiv-\hbar \log \tilde{I}_{l+D / 2-1}\left(\frac{M}{\hbar \epsilon} r_{n} r_{n-1}\right) $$

$$ \begin{align*} \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right) & =\frac{1}{\sqrt{2 \pi \hbar / M}^{D}} \sqrt{\frac{\omega}{\sinh \left[\omega\left(\tau_{b}-\tau_{a}\right)\right]}} \\ & \times \exp \left\{-\frac{1}{\hbar} \frac{M \omega}{\sinh \left[\omega\left(\tau_{b}-\tau_{a}\right)\right]}\left(r_{b}^{2}+r_{a}^{2}\right) \cosh \left[\omega\left(\tau_{b}-\tau_{a}\right)\right]\right\} \\ & \times \sum_{l=0}^{\infty} a_{l}\left(\frac{M \omega r_{b} r_{a}}{\hbar \sinh \left[\omega\left(\tau_{b}-\tau_{a}\right)\right]}\right) \sum_{\mathbf{m}} Y_{l \mathbf{m}}\left(\hat{\mathbf{x}}_{b}\right) Y_{l \mathbf{m}}^{*}\left(\hat{\mathbf{x}}_{a}\right) \end{align*} $$

$$ \begin{align*} \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{l} & =\frac{M}{\hbar} \frac{\omega{\sqrt{r_{b} r_{a}}}^{D-1}}{\sinh \left[\omega\left(\tau_{b}-\tau_{a}\right)\right]} \\ & \times e^{-(M \omega / 2 \hbar) \operatorname{coth}\left[\omega\left(\tau_{b}-\tau_{a}\right)\right]\left(r_{b}^{2}+r_{a}^{2}\right)} I_{l+D / 2-1}\left(\frac{M \omega r_{b} r_{a}}{\hbar \sinh \left[\omega\left(\tau_{b}-\tau_{a}\right)\right]}\right) \end{align*} $$

$$ \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{l}=\frac{M}{\hbar} \frac{{\sqrt{r_{b} r_{a}}}^{D-1}}{\left(\tau_{b}-\tau_{a}\right)} e^{-M\left(r_{b}^{2}+r_{a}^{2}\right) / 2 \hbar\left(\tau_{b}-\tau_{a}\right)} I_{l+D / 2-1}\left(\frac{M r_{b} r_{a}}{\hbar\left(\tau_{b}-\tau_{a}\right)}\right) $$

$$ \begin{align*} \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{l} & =\int_{0}^{\infty} \mathcal{D} r(\tau) \exp \left[-\frac{1}{\hbar} \int_{\tau_{a}}^{\tau_{b}} d \tau\left(\frac{M}{2} \dot{r}^{2}+\frac{" \hbar^{2}}{2 M} \frac{\mu^{2}-1 / 4 "}{r^{2}}+\frac{M}{2} \omega^{2} r^{2}\right)\right] \\ & =\frac{M}{\hbar} \frac{\omega \sqrt{r_{b} r_{a}} D-1}{\sinh \left[\omega\left(\tau_{b}-\tau_{a}\right)\right]} e^{-(M \omega / 2 \hbar) \operatorname{coth}\left[\omega\left(\tau_{b}-\tau_{a}\right)\right]\left(r_{b}^{2}+r_{a}^{2}\right)} I_{\mu}\left(\frac{M \omega r_{b} r_{a}}{\hbar \sinh \left[\omega\left(\tau_{b}-\tau_{a}\right)\right]}\right) \end{align*} $$

$$ V_{\text {extra }}(r)=\hbar^{2} \frac{l_{\text {extra }}^{2}}{2 M r^{2}} $$

$$ \mu=\sqrt{(l+D / 2-1)^{2}+l_{\mathrm{extra}}^{2}} $$

$$ \mathcal{A}=\frac{M}{2} r^{2} \int_{\tau_{a}}^{\tau_{b}} d \tau \dot{\mathbf{u}}^{2}(\tau) $$

$$ \frac{M}{2 \epsilon} \sum_{n=1}^{N+1}\left(r_{n}^{2}+r_{n-1}^{2}-2 r_{n} r_{n-1} \cos \Delta \vartheta_{n}\right) $$

$$ -\frac{M}{2 \epsilon} \sum_{n=1}^{N+1}\left(r_{n}^{2}+r_{n-1}^{2}-2 r_{n} r_{n-1}\right)+\frac{M}{2 \epsilon} \sum_{n=1}^{N+1} 2 r_{n} r_{n-1}\left(1-\cos \Delta \vartheta_{n}\right) $$

$$ -\frac{M}{2 \epsilon} \sum_{n=1}^{N+1} r_{n} r_{n-1}\left(\hat{\mathbf{x}}_{n}-\hat{\mathbf{x}}_{n-1}\right)^{2} $$

$$ \left(\mathbf{u}_{b} \tau_{b} \mid \mathbf{u}_{a} \tau_{a}\right) \approx \frac{1}{\sqrt{2 \pi \hbar \epsilon / M r}^{D-1}} \prod_{n=1}^{N}\left[\int \frac{d \mathbf{u}_{n}}{\sqrt{2 \pi \hbar \epsilon / M r}^{D-1}}\right] \exp \left(-\frac{1}{\hbar} \mathcal{A}^{N}\right) $$

$$ \mathcal{A}^{N}=\frac{M}{2 \epsilon} r^{2} \sum_{n=1}^{N+1}\left(\mathbf{u}_{n}-\mathbf{u}_{n-1}\right)^{2} $$

$$ \exp \left[-\frac{M r^{2}}{2 \hbar \epsilon}\left(\mathbf{u}_{n}-\mathbf{u}_{n-1}\right)^{2}\right]=\exp \left[-\frac{M r^{2}}{\hbar \epsilon}\left(1-\cos \Delta \vartheta_{n}\right)\right] $$

$$ \begin{align*} \exp \left[-\frac{M r^{2}}{2 \hbar \epsilon}\left(\mathbf{u}_{n}-\mathbf{u}_{n-1}\right)^{2}\right]= & \sum_{l=0}^{\infty} \tilde{a}_{l}(h) \frac{l+D / 2-1}{D / 2-1} \frac{1}{S_{D}} C_{l}^{(D / 2-1)}\left(\cos \Delta \vartheta_{n}\right) \\ & =\sum_{l=0}^{\infty} \tilde{a}_{l}(h) \sum_{\mathbf{m}} Y_{l \mathbf{m}}\left(\mathbf{u}_{n}\right) Y_{l \mathbf{m}}^{*}\left(\mathbf{u}_{n-1}\right) \end{align*} $$

$$ \tilde{a}_{l}(h)=\left(\frac{2 \pi}{h}\right)^{(D-1) / 2} \tilde{I}_{l+D / 2-1}(h), \quad h=\frac{M r^{2}}{\hbar \epsilon} . $$

$$ \left(\mathbf{u}_{b} \tau_{b} \mid \mathbf{u}_{a} \tau_{a}\right)=\left(\frac{h}{2 \pi}\right)^{(N+1)(D-1) / 2} \sum_{l=0}^{\infty} \tilde{a}_{l}(h)^{N+1} \sum_{\mathbf{m}} Y_{l \mathbf{m}}\left(\mathbf{u}_{b}\right) Y_{l \mathbf{m}}^{*}\left(\mathbf{u}_{a}\right) $$

$$ \begin{align*} \left(\frac{h}{2 \pi}\right)^{(N+1)(D-1) / 2} & \tilde{a}_{l}(h)^{N+1}=\left[\tilde{I}_{l+D / 2-1}\left(\frac{M r^{2}}{\hbar \epsilon}\right)\right]^{N+1} \\ & \xrightarrow{\epsilon \rightarrow 0} \exp \left\{-\left(\tau_{b}-\tau_{a}\right) \hbar \frac{(l+D / 2-1)^{2}-1 / 4}{2 M r^{2}}\right\} \end{align*} $$

$$ \left(\mathbf{u}_{b} \tau_{b} \mid \mathbf{u}_{a} \tau_{a}\right)=\sum_{l=0}^{\infty} \exp \left[-\frac{\hbar L_{2}}{2 M r^{2}}\left(\tau_{b}-\tau_{a}\right)\right] \sum_{\mathbf{m}} Y_{l \mathbf{m}}\left(\mathbf{u}_{b}\right) Y_{l \mathbf{m}}^{*}\left(\mathbf{u}_{a}\right) $$

$$ L_{2} \equiv(l+D / 2-1)^{2}-1 / 4 $$

$$ \begin{align*} \left(\mathbf{u}_{b} \tau_{b} \mid \mathbf{u}_{a} \tau_{a}\right)= & \sum_{l=0}^{\infty} \frac{2 l+1}{4 \pi} \exp \left\{-\frac{\hbar L_{2}}{2 M r^{2}}\left(\tau_{b}-\tau_{a}\right)\right\} \\ & \times \sum_{m=-l}^{l} \frac{(l-m)!}{(l+m)!} P_{l}^{m}\left(\cos \theta_{b}\right) P_{l}^{m}\left(\cos \theta_{a}\right) e^{i m\left(\varphi_{b}-\varphi_{a}\right)} \end{align*} $$

$$ \left(\mathbf{u}_{b} \tau_{b} \mid \hat{\mathbf{z}}_{a} \tau_{a}\right)=\sum_{l=0}^{\infty} \frac{2 l+1}{4 \pi} \exp \left[-\frac{\hbar L_{2}}{2 M r^{2}}\left(\tau_{b}-\tau_{a}\right)\right] P_{l}\left(\cos \theta_{b}\right) P_{l}(1) $$

$$ \begin{align*} & \left(\mathbf{u}_{b} \tau_{b} \mid \mathbf{u}_{a} \tau_{a}\right)=\sum_{l=0}^{\infty} \exp \left[-\frac{\hbar L_{2}}{2 M r^{2}}\left(\tau_{b}-\tau_{a}\right)\right] \\ & \quad \times \frac{l+1}{2 \pi^{2}} \sum_{m_{1}, m_{2}=-l / 2}^{l / 2} \mathcal{D}_{m_{1} m_{2}}^{l / 2}\left(\varphi_{b}, \theta_{b}, \gamma_{b}\right) \mathcal{D}_{m_{1} m_{2}}^{l / 2 *}\left(\varphi_{a}, \theta_{a}, \gamma_{a}\right) \end{align*} $$

$$ \left(\mathbf{u}_{b} \tau_{b} \mid \mathbf{u}_{a} \tau_{a}\right)=\sum_{m}\left(\sin \theta_{b} \tau_{b} \mid \sin \theta_{a} \tau_{a}\right)_{m} \frac{1}{2 \pi} e^{i m\left(\varphi_{b}-\varphi_{a}\right)} . $$

$$ \left(\theta_{b} \tau_{b} \mid \theta_{a} \tau_{a}\right)_{m} \equiv \sqrt{\sin \theta_{b} \sin \theta_{a}}\left(\sin \theta_{b} \tau_{b} \mid \sin \theta_{a} \tau_{a}\right)_{m} $$

$$ \left(\mathbf{u}_{b} \tau_{b} \mid \mathbf{u}_{a} \tau_{a}\right)=\sum_{m} \frac{1}{\sqrt{\sin \theta_{b} \sin \theta_{a}}}\left(\theta_{b} \tau_{b} \mid \theta_{a} \tau_{a}\right)_{m} \frac{1}{2 \pi} e^{i m\left(\varphi_{b}-\varphi_{a}\right)} $$

$$ \left(\sin \theta_{b} \tau \mid \sin \theta_{a} \tau\right)_{m}=\frac{1}{\sin \theta_{a}} \delta\left(\theta_{b}-\theta_{a}\right) $$

$$ \left(\theta_{b} \tau \mid \theta_{a} \tau\right)_{m}=\delta\left(\theta_{b}-\theta_{a}\right) $$

$$ \begin{align*} \left(\theta_{b} \tau_{b} \mid \theta_{a} \tau_{a}\right)_{m} & =\sqrt{\sin \theta_{b} \sin \theta_{a}} \\ & \times \sum_{l=m}^{\infty} \exp \left[-\frac{\hbar l(l+1)}{2 M r^{2}}\left(\tau_{b}-\tau_{a}\right)\right] 2 \pi Y_{l m}\left(\theta_{b}, 0\right) Y_{l m}^{*}\left(\theta_{a}, 0\right) \end{align*} $$

$$ \begin{align*} \left(\theta_{b} \tau_{b} \mid \theta_{a} \tau_{a}\right)_{m}=\sqrt{\sin \theta_{b} \sin \theta_{a}} & \sum_{l=m}^{\infty} \exp \left\{-\frac{\hbar l(l+1)}{2 M r^{2}}\left(\tau_{b}-\tau_{a}\right)\right\} \\ & \times \frac{(2 l+1)}{2} \frac{(l-m)!}{(l+m)!} P_{l}^{m}\left(\cos \theta_{b}\right) P_{l}^{m}\left(\cos \theta_{a}\right) \end{align*} $$

$$ \left(\mathbf{u}_{b} \tau_{b} \mid \mathbf{u}_{a} \tau_{a}\right) \approx \frac{1}{2 \pi \hbar \epsilon / M r^{2}} \prod_{n=1}^{N}\left[\int \frac{d \cos \theta_{n} d \varphi_{n}}{2 \pi \hbar \epsilon / M r^{2}}\right] \exp \left(-\frac{1}{\hbar} \mathcal{A}^{N}\right) $$

$$ \cos \Delta \vartheta_{n}=\cos \theta_{n} \cos \theta_{n-1}+\sin \theta_{n} \sin \theta_{n-1} \cos \left(\varphi_{n}-\varphi_{n-1}\right) $$

$$ \begin{align*} & \exp \left[-\frac{M r^{2}}{2 \hbar \epsilon}\left(\mathbf{u}_{n}-\mathbf{u}_{n-1}\right)^{2}\right]=\exp \left[-\frac{M r^{2}}{\hbar \epsilon}\left(1-\cos \Delta \vartheta_{n}\right)\right] \\ & \quad=\exp \left[-\frac{M r^{2}}{\hbar \epsilon}\left(1-\cos \theta_{n} \cos \theta_{n-1}-\sin \theta_{n} \sin \theta_{n-1}\right)\right] \\ & \quad \times \frac{1}{\sqrt{2 \pi h_{n}}} \sum_{m=-\infty}^{\infty} \tilde{I}_{m}\left(h_{n}\right) e^{i m\left(\varphi_{n}-\varphi_{n-1}\right)} \end{align*} $$

$$ h_{n} \equiv \frac{M r^{2}}{\hbar \epsilon} \sin \theta_{n} \sin \theta_{n-1} $$

$$ \left(\theta_{b} \tau_{b} \mid \theta_{a} \tau_{a}\right)_{m} \approx \frac{1}{\sqrt{2 \pi \epsilon \hbar / M r^{2}}} \prod_{n=1}^{N}\left[\int_{0}^{\pi} \frac{d \theta_{n}}{\sqrt{2 \pi \epsilon \hbar / M r^{2}}}\right] \exp \left(-\frac{1}{\hbar} \mathcal{A}_{m}^{N}\right) $$

$$ \mathcal{A}_{m}^{N}=\sum_{n=1}^{N+1}\left\{\frac{M r^{2}}{\epsilon}\left[1-\cos \left(\theta_{n}-\theta_{n-1}\right)\right]-\hbar \log \tilde{I}_{m}\left(h_{n}\right)\right\} $$

$$ \mathcal{A}_{m}^{N} \approx \epsilon \sum_{n=1}^{N+1}\left\{\frac{M r^{2}}{2 \epsilon^{2}}\left[\left(\Delta \theta_{n}\right)^{2}-\frac{1}{12}\left(\Delta \theta_{n}\right)^{4}+\ldots\right]+\frac{\hbar^{2}}{2 M r^{2}} \frac{m^{2}-1 / 4}{\sin \theta_{n} \sin \theta_{n-1}}\right\}, $$

$$ \mathcal{A}_{m}=\int_{\tau_{a}}^{\tau_{b}} d \tau\left(\frac{M r^{2}}{2} \dot{\theta}^{2}-\frac{\hbar^{2}}{8 M r^{2}}+\frac{\hbar^{2}}{2 M r^{2}} \frac{m^{2}-1 / 4}{\sin ^{2} \theta}\right) $$

$$ \begin{align*} & \sqrt{\frac{2 \zeta}{\pi}} e^{\zeta \cos \theta_{n} \cos \theta_{n-1}} I_{m}\left(\zeta \sin \theta_{n} \sin \theta_{n-1}\right) \\ & \quad=\sum_{l=m}^{\infty} I_{l+1 / 2}(\zeta)(2 l+1) \frac{(l-m)!}{(l+m)!} P_{l}^{m}\left(\cos \theta_{n}\right) P_{l}^{m}\left(\cos \theta_{n-1}\right) \end{align*} $$

$$ \begin{align*} e^{-\zeta\left(1-\cos \Delta \theta_{n}\right)} & =e^{-\zeta\left[1-\cos \theta_{n} \cos \theta_{n-1}-\sin \theta_{n} \sin \theta_{n-1} \cos \left(\varphi_{n}-\varphi_{n-1}\right)\right]} \\ & \times \frac{1}{\sqrt{2 \pi \zeta \sin \theta_{n} \sin \theta_{n-1}}} \sum_{m=-\infty}^{\infty} \tilde{I}_{m}\left(\zeta \sin \theta_{n} \sin \theta_{n-1}\right) e^{i m\left(\varphi_{n}-\varphi_{n-1}\right)} \\ e^{-\zeta\left(1-\cos \Delta \theta_{n}\right)} & =e^{-\zeta} \sqrt{\frac{\pi}{2 \zeta}} \sum_{l=0}^{\infty}(2 l+1) I_{l+1 / 2}(\zeta) P_{l}\left(\cos \Delta \theta_{n}\right) \end{align*} $$

$$ P_{l}\left(\cos \Delta \theta_{n}\right)=\sum_{m=-l}^{l} \frac{(l-m)!}{(l+m)!} P_{l}^{m}\left(\theta_{n}\right) P_{l}^{m}\left(\theta_{n-1}\right) e^{i m\left(\varphi_{n}-\varphi_{n-1}\right)} $$

$$ \int_{-1}^{1} \frac{d \cos \theta}{\sin ^{2} \theta} P_{l}^{m}(\cos \theta) P_{l^{\prime}}^{m}(\cos \theta)=\frac{(l+m)!}{(l-m)!} \frac{2}{2 l+1} \delta_{l l^{\prime}} $$

$$ \begin{align*} \left(\theta_{b} \tau_{b} \mid \theta_{a} \tau_{a}\right)_{m}= & \sqrt{\sin \theta_{b} \sin \theta_{a}} \sum_{l=m}^{\infty}\left[\tilde{I}_{m+l+1 / 2}(\zeta)\right]^{N+1} \\ & \times \frac{(2 l+1)}{2} \frac{(l-m)!}{(l+m)!} P_{l}^{m}\left(\cos \theta_{b}\right) P_{l}^{m}\left(\cos \theta_{a}\right) \end{align*} $$

$$ \left[\tilde{I}_{m+l+1 / 2}(\zeta)\right]^{N+1} \approx \exp \left[-\frac{\hbar}{2 M r^{2}} L_{2}\left(\tau_{b}-\tau_{a}\right)\right] $$

$$ P_{l}^{m}(z)=(-)^{m} P_{l}^{-m} \frac{(l+m)!}{(l-m)!} $$

$$ \begin{align*} \left(\theta_{b} \tau_{b} \mid \theta_{a} \tau_{a}\right)_{\mu} & =\sqrt{\sin \theta_{b} \sin \theta_{a}} \sum_{n=0}^{\infty}\left[\tilde{I}_{n+\mu+1 / 2}(\zeta)\right]^{N+1} \\ & \times \frac{(2 n+2 \mu+1)}{2} \frac{(n+2 \mu)!}{n!} P_{n+\mu}^{-\mu}\left(\cos \theta_{b}\right) P_{n+\mu}^{-\mu}\left(\cos \theta_{a}\right) \end{align*} $$

$$ \begin{align*} \left(\theta_{b} \tau_{b} \mid \theta_{a} \tau_{a}\right)_{\mu} & =\sqrt{\sin \theta_{b} \sin \theta_{a}} \sum_{n=0}^{\infty} \exp \left[-\frac{\hbar(n+\mu)(n+\mu+1)}{2 M r^{2}}\left(\tau_{b}-\tau_{a}\right)\right] \\ & \times \frac{(2 n+2 \mu+1)}{2} \frac{(n+2 \mu)!}{n!} P_{n+\mu}^{-\mu}\left(\cos \theta_{b}\right) P_{n+\mu}^{-\mu}\left(\cos \theta_{a}\right) \end{align*} $$

$$ \begin{align*} & (\sin \alpha \sin \beta)^{-\mu} J_{\mu}(z \sin \alpha \sin \beta) e^{i z \cos \alpha \cos \beta}=\frac{2^{2 \mu+1} \Gamma^{2}(\mu+1 / 2)}{\sqrt{2 \pi z}} \\ & \quad \times \sum_{n=0}^{\infty} \frac{i^{n} n!(n+\mu+1 / 2)}{\Gamma(n+2 \mu+1)} J_{n+\mu+1 / 2}(z) C_{n}^{(\mu+1 / 2)}(\cos \alpha) C_{n}^{(\mu+1 / 2)}(\cos \beta) \end{align*} $$

$$ \begin{align*} & (\sin \alpha \sin \beta)^{-\mu} I_{\mu}(\zeta \sin \alpha \sin \beta) \exp (\zeta \cos \alpha \cos \beta)=\frac{2^{2 \mu+1} \Gamma^{2}(\mu+1 / 2)}{\sqrt{2 \pi \zeta}} \\ & \quad \times \sum_{n=0}^{\infty} \frac{n!(n+\mu+1 / 2)}{\Gamma(n+2 \mu+1)} I_{n+\mu+1 / 2}(\zeta) C_{n}^{(\mu+1 / 2)}(\cos \alpha) C_{n}^{(\mu+1 / 2)}(\cos \beta) \end{align*} $$

$$ P_{n}^{(\mu, \mu)}(z)=(-2)^{\mu} \frac{(n+\mu)!}{n!}\left(1-z^{2}\right)^{-\mu / 2} P_{n+\mu}^{-\mu}(z) $$

$$ C_{n}^{(\mu+1 / 2)}(z)=\frac{\Gamma(n+2 \mu+1) \Gamma(\mu+1)}{\Gamma(2 \mu+1) n!}\left[\frac{1-z^{2}}{4}\right]^{-\mu / 2} P_{n+\mu}^{-\mu}(z) $$

$$ \int_{-1}^{1} \frac{d \cos \theta}{\sin ^{2} \theta} P_{n+\mu}^{-\mu}(\cos \theta) P_{n^{\prime}+\mu}^{-\mu}(\cos \theta)=\frac{n!}{(n+2 \mu)!} \frac{2}{2 n+1 \mu+1} \delta_{n n^{\prime}} $$

$$ P_{\nu}^{\mu}(z)=\frac{1}{\Gamma(1-\mu)}\left(\frac{1+z}{1-z}\right)^{\mu / 2} F(-\nu, \nu+1 ; 1-\mu ;(1-z) / 2) $$

$$ \int_{-1}^{1} d z\left(1-z^{2}\right)^{\mu} C_{n}^{(\mu+1 / 2)}(z) C_{n^{\prime}}^{(\mu+1 / 2)}(z)=\delta_{n n^{\prime}} \frac{\pi 2^{-2 \mu} \Gamma(2 \mu+2+n)}{n!(n+\mu)[\Gamma(\mu+1 / 2)]^{2}} $$

$$ V(\theta)=\frac{\hbar^{2}}{2 M r^{2}} \frac{s(s+1)}{\sin ^{2} \theta} $$

$$ \begin{gather*} {\left[\frac{\hbar^{2}}{M r^{2}}\left(-\frac{1}{2} \frac{1}{\sin \theta} \frac{d}{d \theta} \sin \theta \frac{d}{d \theta}+\frac{m^{2}}{2 \sin ^{2} \theta}\right)+\hbar \partial_{\tau}\right]\left(\sin \theta \tau \mid \sin \theta_{a} \tau_{a}\right)_{m}} \\ =\hbar \delta\left(\tau-\tau_{a}\right) \delta\left(\cos \theta-\cos \theta_{a}\right) \end{gather*} $$

$$ \left[\frac{\hbar^{2}}{M r^{2}}\left(-\frac{1}{2} \frac{d}{d \theta^{2}}-\frac{1}{8}+\frac{m^{2}-1 / 4}{2 \sin ^{2} \theta}\right)+\hbar \partial_{\tau}\right]\left(\theta \tau \mid \theta_{a} \tau_{a}\right)_{m}=\hbar \delta\left(\tau-\tau_{a}\right) \delta\left(\theta-\theta_{a}\right) $$

$$ \begin{align*} \left(\mathbf{u}_{b} \tau_{b} \mid \mathbf{u}_{a} \tau_{a}\right) & =\sum_{l=0}^{\infty} \exp \left\{-\frac{\hbar L_{2}}{2 M r^{2}}\left(\tau_{b}-\tau_{a}\right)\right\} \\ & \times \frac{l+1}{2 \pi^{2}} \sum_{m_{1} m_{2}=-l / 2}^{l / 2} d_{m_{1} m_{2}}^{l / 2}\left(\theta_{b}\right) d_{m_{1} m_{2}}^{l / 2}\left(\theta_{a}\right) e^{i m_{1}\left(\varphi_{b}-\varphi_{a}\right)+i m_{2}\left(\gamma_{b}-\gamma_{a}\right)} \end{align*} $$

$$ L_{2} \equiv(l+1)^{2}-1 / 4=4(l / 2)(l / 2+1)+3 / 4 $$

$$ \left(\mathbf{u}_{b} \tau_{b} \mid \mathbf{u}_{a} \tau_{a}\right)=8 \sum_{m_{1} m_{2}}\left(\sin \theta_{b} \tau_{b} \mid \sin \theta_{a} \tau_{a}\right)_{m_{1} m_{2}} \frac{1}{2 \pi} e^{i m_{1}\left(\varphi_{b}-\varphi_{a}\right)} \frac{1}{4 \pi} e^{i m_{2}\left(\gamma_{b}-\gamma_{a}\right)} . $$

$$ \left(\theta_{b} \tau_{b} \mid \theta_{a} \tau_{a}\right)_{m_{1} m_{2}} \equiv \sqrt{\sin \theta_{b} \sin \theta_{a}}\left(\sin \theta_{b} \tau_{b} \mid \sin \theta_{a} \tau_{a}\right)_{m_{1} m_{2}} $$

$$ \left(\mathbf{u}_{b} \tau_{b} \mid \mathbf{u}_{a} \tau_{a}\right)=\sum_{m} \frac{8}{\sqrt{\sin \theta_{b} \sin \theta_{a}}}\left(\theta_{b} \tau_{b} \mid \theta_{a} \tau_{a}\right)_{m_{1} m_{2}} \frac{1}{2 \pi} e^{i m\left(\varphi_{b}-\varphi_{a}\right)} \frac{1}{4 \pi} e^{i m\left(\gamma_{b}-\gamma_{a}\right)} . $$

$$ \begin{align*} & \left(\theta_{b} \tau_{b} \mid \theta_{a} \tau_{a}\right)_{m_{1} m_{2}}=\sqrt{\sin \theta_{b} \sin \theta_{a}} \\ & \quad \times \sum_{l}^{\infty} \exp \left\{-\frac{\hbar\left[(l+1)^{2}-1 / 4\right]}{2 M r^{2}}\left(\tau_{b}-\tau_{a}\right)\right\} \frac{l+1}{2} d_{m_{1} m_{2}}^{l / 2}\left(\theta_{b}\right) d_{m_{1} m_{2}}^{l / 2}\left(\theta_{a}\right) \end{align*} $$

$$ \left(\mathbf{u}_{b} \tau_{b} \mid \mathbf{u}_{a} \tau_{a}\right) \approx \frac{1}{\sqrt{2 \pi \hbar \epsilon / M r^{2}}} \prod_{n=1}^{N}\left[\int_{0}^{\pi} \int_{0}^{2 \pi} \int_{0}^{4 \pi} \frac{d \theta_{n} \sin \theta_{n} d \varphi_{n} d \gamma_{n}}{8 \sqrt{2 \pi \hbar \epsilon / M r}^{3}}\right] \exp \left(-\frac{1}{\hbar} \mathcal{A}^{N}\right) $$

$$ \begin{align*} & \exp \left[-\frac{M r^{2}}{2 \hbar \epsilon}\left(\mathbf{u}_{n}-\mathbf{u}_{n-1}\right)^{2}\right]=\exp \left[-\frac{M r^{2}}{\hbar \epsilon}\left(1-\cos \Delta \vartheta_{n}\right)\right] \\ & =\exp \left\{-\frac{M r^{2}}{2 \hbar \epsilon}\left[1-\cos \left(\theta_{n} / 2\right) \cos \left(\theta_{n-1} / 2\right)-\sin \left(\theta_{n} / 2\right) \sin \left(\theta_{n-1} / 2\right)\right]\right\} \\ & \quad \times \frac{1}{\sqrt{2 \pi h_{n}^{c}}} \frac{1}{\sqrt{4 \pi h_{n}^{s}}} \sum_{m_{1}, m_{2}=-\infty}^{\infty} \tilde{I}_{\left|m_{1}+m_{2}\right|}\left(h_{n}^{c}\right) \tilde{I}_{\left|m_{1}-m_{2}\right|}\left(h_{n}^{s}\right) \\ & \quad \times \exp \left\{i m_{1}\left(\varphi_{n}-\varphi_{n-1}\right)+i m_{2}\left(\gamma_{n}-\gamma_{n-1}\right)\right\} \end{align*} $$

$$ h_{n}^{c}=\frac{M r^{2}}{\hbar \epsilon} \cos \left(\theta_{n} / 2\right) \cos \left(\theta_{n-1} / 2\right), \quad h_{n}^{s}=\frac{M r^{2}}{\hbar \epsilon} \sin \left(\theta_{n} / 2\right) \sin \left(\theta_{n-1} / 2\right) $$

$$ \left(\theta_{b} \tau_{b} \mid \theta_{a} \tau_{a}\right)_{m_{1} m_{2}} \approx \frac{1}{\sqrt{2 \pi \epsilon \hbar / 4 M r^{2}}} \prod_{n=1}^{N}\left[\int_{0}^{\pi} \frac{d \theta_{n}}{\sqrt{2 \pi \epsilon \hbar / 4 M r^{2}}}\right] \exp \left\{-\frac{1}{\hbar} \mathcal{A}_{m_{1} m 2}^{N}\right\} $$

$$ \begin{align*} \mathcal{A}_{m_{1} m_{2}}^{N}=\sum_{n=1}^{N+1} & \left\{\frac { M r ^ { 2 } } { \epsilon } \left[1-\cos \left[\left(\theta_{n}-\theta_{n-1}\right) / 2\right]\right.\right. \\ & \left.-\hbar \log \tilde{I}_{\left|m_{1}+m_{2}\right|}\left(h_{n}^{c}\right)-\hbar \log \tilde{I}_{\left|m_{1}-m_{2}\right|}\left(h_{n}^{s}\right)\right\} \end{align*} $$

$$ \begin{align*} \mathcal{A}_{m_{1} m_{2}}^{N} \rightarrow \epsilon & \sum_{n=1}^{N+1}\left\{\frac{M r^{2}}{2 \epsilon^{2}}\left[\left(\Delta \theta_{n} / 2\right)^{2}-\frac{1}{12}\left(\Delta \theta_{n} / 2\right)^{4}+\ldots\right]\right. \\ & \left.+\frac{\hbar^{2}}{2 M r^{2}} \frac{\left(m_{1}+m_{2}\right)^{2}-1 / 4}{\cos \left(\theta_{n} / 2\right) \cos \left(\theta_{n-1} / 2\right)}+\frac{\hbar^{2}}{2 M r^{2}} \frac{\left(m_{1}-m_{2}\right)^{2}-1 / 4}{\sin \left(\theta_{n} / 2\right) \sin \left(\theta_{n-1} / 2\right)}\right\} \end{align*} $$

$$ \begin{align*} \mathcal{A}_{m_{1} m_{2}}=\int_{\tau_{a}}^{\tau_{b}} d \tau\left(\frac{M r^{2}}{8} \dot{\theta}^{2}-\frac{\hbar^{2}}{8 M r^{2}}\right. & +\frac{\hbar^{2}}{2 M r^{2}} \frac{\left|m_{1}+m_{2}\right|^{2}-1 / 4}{\cos ^{2}(\theta / 2)} \\ & \left.+\frac{\hbar^{2}}{2 M r^{2}} \frac{\left|m_{1}-m_{2}\right|^{2}-1 / 4}{\sin ^{2}(\theta / 2)}\right) \end{align*} $$

$$ \mu=M / 4 $$

$$ \mathcal{A}_{m_{1} m_{2}}=\int_{\tau_{a}}^{\tau_{b}} d \tau\left(\frac{\mu r^{2}}{2} \dot{\theta}^{2}-\frac{\hbar^{2}}{32 \mu r^{2}}+\frac{\hbar^{2}}{2 \mu r^{2}} \frac{m_{1}^{2}+m_{2}^{2}-1 / 4-2 m_{1} m_{2} \cos \theta}{\sin ^{2} \theta}\right) $$

$$ \begin{align*} & \left(\theta_{b} \tau_{b} \mid \theta_{a} \tau_{a}\right)_{\mu_{1} \mu_{2}}=\sqrt{\sin \theta_{b} \sin \theta_{a}} \\ & \times \sum_{n=0}^{\infty} \exp \left\{\frac{\hbar\left[\left(n+\mu_{1}+1\right)^{2}-1 / 4\right]}{2 M r^{2}}\left(\tau_{b}-\tau_{a}\right)\right\} \frac{n+\mu_{1}+1}{2} d_{\mu_{1} \mu_{2}}^{n+\mu_{1}}\left(\theta_{b}\right) d_{\mu_{1} \mu_{2}}^{n+\mu_{1}}\left(\theta_{a}\right) \end{align*} $$

$$ \begin{align*} & \frac{z}{2} J_{\mu_{+}}(z \cos \alpha \cos \beta) J_{\mu_{-}}(z \sin \alpha \sin \beta)=\cos ^{\mu_{+}} \alpha \cos ^{\mu_{+}} \beta \sin ^{\mu_{-}} \alpha \sin ^{\mu_{-}} \beta \\ & \quad \times \sum_{n=0}^{\infty}(-1)^{n}\left(\mu_{+}+\mu_{-}+2 n+1\right) J_{\mu_{+}+\mu_{-}+2 n+1}(z) \\ & \quad \times \frac{\Gamma\left(n+\mu_{+}+\mu_{-}+1\right) \Gamma\left(n+\mu_{-}+1\right)}{n!\Gamma\left(n+\mu_{+}+1\right)\left[\Gamma\left(\mu_{-}+1\right)\right]^{2}} \\ & \quad \times F\left(-n, n+\mu_{+}+\mu_{-}+1 ; \mu_{-}+1 ; \sin ^{2} \alpha\right) \\ & \quad \times F\left(-n, n+\mu_{+}+\mu_{-}+1 ; \mu_{-}+1 ; \sin ^{2} \beta\right) \end{align*} $$

$$ P_{n}^{\left(\mu_{-}, \mu_{+}\right)}(x)=\frac{1}{n!} \frac{\Gamma\left(n+\mu_{-}+1\right)}{\Gamma\left(\mu_{-}+1\right)} F\left(-n, n+\mu_{+}+\mu_{-}+1 ; 1+\mu_{-} ;(1-x) / 2\right) $$

$$ \begin{align*} & I_{\mu_{+}}\left(\zeta \cos \frac{\theta_{n}}{2} \cos \frac{\theta_{n-1}}{2}\right) I_{\mu_{-}}\left(\zeta \sin \frac{\theta_{n}}{2} \sin \frac{\theta_{n-1}}{2}\right) \\ & =\frac{4}{\zeta} \sum_{n=0}^{\infty} I_{2 n+\mu_{+}+\mu_{-}+1}(\zeta) \frac{\left(n+\mu_{1}+\mu_{2}\right)!\left(n+\mu_{1}-\mu_{2}\right)!}{\left(n+2 \mu_{1}\right)!n!} d_{\mu_{1} \mu_{2}}^{n+\mu_{1}}\left(\theta_{n}\right) d_{\mu_{1} \mu_{2}}^{n+\mu_{1}}\left(\theta_{n-1}\right) \end{align*} $$

$$ \int_{-1}^{1} d \cos \theta d_{\mu_{1} \mu_{2}}^{n+\mu_{1}}(\theta) d_{\mu_{1} \mu_{2}}^{n^{\prime}+\mu_{1}}(\theta)=\delta_{n n^{\prime}} \frac{2}{2 n+1} $$

$$ \begin{align*} \int_{-1}^{1} d x(1-x)^{\mu_{-}}(1+x)^{\mu_{+}} P_{n}^{\left(\mu_{-}, \mu_{+}\right)}(x) P_{n^{\prime}}^{\left(\mu_{-}, \mu_{+}\right)}(x) & \\ \quad= & \delta_{n n^{\prime}} \frac{2^{\mu_{+}+\mu_{-}+1} \Gamma\left(\mu_{+}+n+1\right) \Gamma\left(\mu_{-}+n+1\right)}{n!\left(\mu_{+}+\mu_{-}+1+2 n\right) \Gamma\left(\mu_{+}+\mu_{-}+n+1\right)} \end{align*} $$

$$ \left(\theta_{b} \tau_{b} \mid \theta_{a} \tau_{a}\right)_{\mu_{1} \mu_{2}}=\sqrt{\sin \theta_{b} \sin \theta_{a}} \sum_{n=0}^{\infty}\left[\tilde{I}_{2 n+\mu_{+}+\mu_{-}+1}(\zeta)\right]^{N+1} d_{\mu_{1} \mu_{2}}^{n+\mu_{1}}\left(\theta_{b}\right) d_{\mu_{1} \mu_{2}}^{n+\mu_{1}}\left(\theta_{a}\right) $$

$$ \left(\theta_{b} \tau_{b} \mid \theta_{a} \tau_{a}\right)_{\mu_{1} \mu_{2}}=\sqrt{\sin \theta_{b} \sin \theta_{a}} \sum_{n=0}^{\infty} e^{-E_{n}\left(\tau_{b}-\tau_{a}\right) / \hbar} d_{\mu_{1} \mu_{2}}^{n+\mu_{1}}\left(\theta_{b}\right) d_{\mu_{1} \mu_{2}}^{n+\mu_{1}}\left(\theta_{a}\right) $$

$$ E_{n}=\frac{\hbar}{2 M r^{2}}\left[\left(2 n+\mu_{+}+\mu_{-}+1\right)^{2}-1 / 4\right] $$

$$ V_{\mathcal{P} \mathcal{T}^{\prime}}(\theta)=\frac{\hbar^{2}}{2 M r^{2}}\left[\frac{s_{1}\left(s_{1}+1\right)}{\sin ^{2}(\theta / 2)}+\frac{s_{2}\left(s_{2}+1\right)}{\cos ^{2}(\theta / 2)}\right] $$

$$ \begin{gather*} {\left[\frac{\hbar^{2}}{2 \mu r^{2}}\left(-\frac{1}{\sin \theta} \frac{d}{d \theta} \sin \theta \frac{d}{d \theta}+\frac{3}{16}+\frac{m_{1}^{2}+m_{2}^{2}-2 m_{1} m_{2} \cos \theta}{\sin ^{2} \theta}\right)+\hbar \partial_{\tau}\right]} \\ \times\left(\sin \theta \tau \mid \sin \theta_{a} \tau_{a}\right)_{m_{1} m_{2}}=\hbar \delta\left(\tau-\tau_{a}\right) \delta\left(\cos \theta-\cos \theta_{a}\right) \end{gather*} $$

$$ \begin{gather*} {\left[\frac{\hbar^{2}}{2 \mu r^{2}}\left(-\frac{d^{2}}{d \theta^{2}}-\frac{1}{16}+\frac{m_{1}^{2}+m_{2}^{2}-1 / 4-2 m_{1} m_{2} \cos \theta}{\sin ^{2} \theta}\right)+\hbar \partial_{\tau}\right]} \\ \times\left(\theta \tau \mid \theta_{a} \tau_{a}\right)_{m_{1} m_{2}}=\hbar \delta\left(\tau-\tau_{a}\right) \delta\left(\theta-\theta_{a}\right) \end{gather*} $$

$$ E_{l}=\frac{\hbar^{2}}{2 M r^{2}}\left(L_{2}^{2}\right)_{l} $$

$$ \left(L_{2}^{2}\right)_{l}=(l+D / 2-1)^{2}-1 / 4, \quad l=0,1,2, \ldots $$

$$ E_{l}=\frac{\hbar^{2}}{2 M r^{2}}\left(\hat{L}^{2}\right)_{l} $$

$$ \left(\hat{L}^{2}\right)_{l}=l(l+D-2), \quad l=0,1,2, \ldots $$

$$ \Delta\left(L_{2}^{2}\right)_{l} \equiv \hat{L}^{2}-L_{2}^{2}=\frac{1}{4}-\left(\frac{D}{2}-1\right)^{2}=-\frac{(D-1)(D-3)}{4} $$

$$ \mathcal{A}_{\text {on sphere }}^{N}=\frac{M}{\epsilon} r^{2} \sum_{n=1}^{N+1} \frac{\left(\Delta \vartheta_{n}\right)^{2}}{2} $$

$$ \mathcal{A}^{N}=\frac{M}{\epsilon} r^{2} \sum_{n=1}^{N+1}\left(1-\cos \Delta \vartheta_{n}\right) . $$

$$ \mathcal{A}_{\text {on sphere }}^{N}=\frac{M}{\epsilon} r^{2} \sum_{n=1}^{N+1}\left[\left(1-\cos \Delta \vartheta_{n}\right)+\frac{1}{24}\left(\Delta \vartheta_{n}\right)^{4}-\ldots\right] $$

$$ \frac{1}{\sqrt{2 \pi \hbar \epsilon / M r^{2}}} \prod_{n=1}^{N+1} \int_{-\pi / 2}^{\pi / 2} \frac{d \varphi_{n}}{\sqrt{2 \pi \hbar \epsilon / M r^{2}}} $$

$$ \left\langle\left(\Delta \vartheta_{n}\right)^{4}\right\rangle_{0}=3 \frac{\epsilon \hbar}{M r^{2}} $$

$$ \Delta_{\mathrm{qu}} \mathcal{A}^{N}=\frac{M}{\epsilon} r^{2} \sum_{n=1}^{N+1} \frac{1}{24}\left(\Delta \vartheta_{n}\right)^{4} $$

$$ \left\langle\Delta_{\mathrm{qu}} \mathcal{A}^{N}\right\rangle_{0}=(N+1) \epsilon \frac{\hbar^{2} / 4}{2 M r^{2}} $$

$$ \Delta_{\mathrm{qu}} \mathcal{A}^{N} \approx \frac{M}{\epsilon} \sum_{n=1}^{N+1} \frac{1}{24 r^{2}}\left(\Delta \mathbf{x}_{n}\right)^{4} $$

$$ \left\langle\left(\Delta \mathbf{x}_{n}\right)_{i}\left(\Delta \mathbf{x}_{n}\right)_{j}\right\rangle_{0}=\frac{\hbar \epsilon}{M} \delta_{i j} $$

$$ \left\langle\Delta \mathcal{A}^{N}\right\rangle_{0}=(N+1) \epsilon \frac{\hbar^{2}}{2 M r^{2}} \Delta_{\mathrm{qu}} L_{2}^{2} $$

$$ \Delta_{\mathrm{qu}} L_{2}^{2}=\frac{D^{2}-1}{12} . $$

$$ \left\langle\Delta x_{i} \Delta x_{j} \Delta x_{k} \Delta x_{l}\right\rangle_{0}=\left(\frac{\epsilon \hbar}{M}\right)^{2}\left(\delta_{i j} \delta_{k l}+\delta_{i k} \delta_{j l}+\delta_{i l} \delta_{j k}\right) $$

$$ \Delta_{\mathrm{f}} L_{2}^{2}=\Delta L_{2}^{2}-\Delta_{\mathrm{qu}} L_{2}^{2}=-\frac{1}{3}(D-1)(D-2) . $$

$$ \prod_{n=1}^{N}\left[\int \frac{d^{D-1} \mathbf{u}_{n}}{\sqrt{2 \pi \hbar \epsilon / M r}^{D-1}}\right] $$

$$ \prod_{n=1}^{N}\left[1+\frac{D-2}{6}\left(\Delta \vartheta_{n}\right)^{2}\right] $$

$$ \prod_{n=1}^{N}\left[1+\frac{(D-2)(D-1)}{6 r^{2}} \frac{\epsilon \hbar}{M}\right] $$

$$ \left\langle\Delta \mathcal{A}_{f}^{N}\right\rangle_{0}=(N+1) \epsilon \frac{\hbar^{2}}{2 M r^{2}} \Delta_{f} L_{2}^{2} $$

$$ \left(\mathbf{u}_{b} \tau_{b} \mid \mathbf{u}_{a} \tau_{a}\right)=\sum_{l=0}^{\infty} \exp \left[-\frac{\hbar \hat{L}^{2}}{2 M r^{2}}\left(\tau_{b}-\tau_{a}\right)\right] \sum_{\mathbf{m}} Y_{l \mathbf{m}}\left(\mathbf{u}_{b}\right) Y_{l \mathbf{m}}^{*}\left(\mathbf{u}_{a}\right) $$

$$ \hat{L}^{2}=l(l+D-2), $$

$$ \int d^{D-1} \mathbf{u}_{b}\left(\mathbf{u}_{b} \tau_{b} \mid \mathbf{u}_{a} \tau_{a}\right)=1 $$

$$ \begin{align*} \int d^{D-1} \mathbf{u}_{b} \sum_{\mathbf{m}} Y_{l \mathbf{m}}\left(\mathbf{u}_{b}\right) Y_{l \mathbf{m}}^{*}\left(\mathbf{u}_{a}\right) & =\delta_{l 0} \int d^{D-1} \mathbf{u}_{b} Y_{0 \mathbf{0}}\left(\mathbf{u}_{b}\right) Y_{0 \mathbf{0}}^{*}\left(\mathbf{u}_{a}\right) \\ & =\delta_{l 0} \int d^{D-1} \mathbf{u}_{b} 1 / S_{D}=\delta_{l 0} \end{align*} $$

$$ \int d^{D-1} \mathbf{u}_{b}\left(\mathbf{u}_{b} \tau_{b} \mid \mathbf{u}_{a} \tau_{a}\right)=\exp \left[-\frac{(D / 2-1)^{2}-1 / 4}{2 \mu r^{2}}\left(\tau_{b}-\tau_{a}\right)\right] $$

$$ \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)=\int_{-i \infty}^{i \infty} \int \mathcal{D}^{2} x(\tau) \frac{\mathcal{D} \lambda(\tau)}{2 \pi i \hbar} \exp \left(-\frac{1}{\hbar} \int_{\tau_{a}}^{\tau_{b}} d \tau\left\{\frac{M}{2} \dot{\mathbf{x}}^{2}+\frac{\lambda(\tau)}{2 r}\left[x^{2}(\tau)-r^{2}\right]\right\}\right) $$

$$ \left(\mathbf{u}_{b} t_{b} \mid \mathbf{u}_{a} t_{a}\right) \approx \prod_{n=1}^{N}\left[\int d^{D} x_{n}\right] \prod_{n=1}^{N}\left[\int \frac{d \lambda_{n}}{2 \pi i \hbar / \epsilon}\right] \exp \left(\frac{i}{\hbar} \mathcal{A}^{N}\right) $$

$$ \mathcal{A}^{N}=\sum_{n=1}^{N+1}\left[\frac{M}{2 \epsilon}\left(\mathbf{x}_{n}-\mathbf{x}_{n-1}\right)^{2}+\epsilon \frac{\lambda_{n}}{2 r}\left(\mathbf{x}_{n}^{2}-r^{2}\right)\right] . $$

$$ \mathcal{A}^{N}=\frac{M}{2 \epsilon} r^{2} \sum_{n=1}^{N+1}\left(\mathbf{u}_{n}-\mathbf{u}_{n-1}\right)^{2}=\frac{M r^{2}}{\epsilon} \sum_{n=1}^{N+1}\left(1-\cos \Delta \vartheta_{n}\right) $$

$$ \mathcal{A}^{N}=\frac{M}{\epsilon} r^{2} \sum_{n=1}^{N+1}\left[1-\frac{1}{2} \operatorname{tr}\left(g_{n} g_{n-1}^{-1}\right)\right], $$

$$ g_{n}=g\left(\varphi_{n}, \theta_{n}, \gamma_{n}\right) $$

$$ \begin{align*} g(\varphi, \theta, \gamma) & =\exp \left(i \varphi \sigma_{3} / 2\right) \exp \left(i \theta \sigma_{2} / 2\right) \exp \left(i \gamma \sigma_{3} / 2\right) \\ & =\left(\begin{array}{cc} e^{i \varphi / 2} & 0 \\ 0 & e^{-i \varphi / 2} \end{array}\right)\left(\begin{array}{rr} \cos (\theta / 2) & \sin (\theta / 2) \\ -\sin (\theta / 2) & \cos (\theta / 2) \end{array}\right)\left(\begin{array}{cc} e^{i \gamma / 2} & 0 \\ 0 & e^{-i \gamma / 2} \end{array}\right) \end{align*} $$

$$ \begin{align*} \frac{1}{2} \operatorname{tr}\left(g_{n} g_{n-1}^{-1}\right) & =\cos \left(\theta_{n} / 2\right) \cos \left(\theta_{n-1} / 2\right) \cos \left[\left(\varphi_{n}-\varphi_{n-1}+\gamma_{n}-\gamma_{n-1}\right) / 2\right] \\ & +\sin \left(\theta_{n} / 2\right) \sin \left(\theta_{n-1} / 2\right) \cos \left[\left(\varphi_{n}-\varphi_{n-1}-\gamma_{n}+\gamma_{n-1}\right) / 2\right] \end{align*} $$

$$ \int d g \equiv \frac{1}{16 \pi^{2}} \int_{0}^{\pi} \int_{0}^{2 \pi} \int_{0}^{4 \pi} d \theta \sin \theta d \varphi d \gamma=\frac{1}{2 \pi^{2}} \int d^{3} \mathbf{u}=1 $$

$$ \left(\mathbf{u}_{b} \tau_{b} \mid \mathbf{u}_{a} \tau_{a}\right) \equiv \frac{1}{2 \pi^{2}}\left(g_{b} \tau_{b} \mid g_{a} \tau_{a}\right) $$

$$ \left(g_{b} \tau_{b} \mid g_{a} \tau_{a}\right) \approx \frac{2 \pi^{2}}{\sqrt{2 \pi \hbar \epsilon / M r}^{2}} \prod_{n=1}^{N}\left[\int \frac{2 \pi^{2} d g_{n}}{\sqrt{2 \pi \hbar \epsilon / M r}^{3}}\right] \exp \left(-\frac{1}{\hbar} \mathcal{A}^{N}\right) $$

$$ \begin{align*} & \exp \left\{-\frac{M r^{2}}{\hbar \epsilon}\left[1-\frac{1}{2} \operatorname{tr}\left(g_{n} g_{n-1}^{-1}\right)\right]\right\}=\sum_{l=0}^{\infty} \tilde{a}_{l}(h) \frac{l+1}{2 \pi^{2}} C_{l}^{(1)}\left(\cos \Delta \vartheta_{n}\right) \\ & \quad=\sum_{l=0}^{\infty} \tilde{a}_{l}(h) \frac{l+1}{2 \pi^{2}} \sum_{m_{1}, m_{2}=-l / 2}^{l / 2} \mathcal{D}_{m_{1} m_{2}}^{l / 2}\left(\varphi_{n}, \theta_{n}, \gamma_{n}\right) \mathcal{D}_{m_{1} m_{2}}^{l / 2 *}\left(\varphi_{n-1}, \theta_{n-1}, \gamma_{n-1}\right) \end{align*} $$

$$ \exp \left[\frac{h}{2} \operatorname{tr}\left(g_{n} g_{n-1}^{-1}\right)\right]=\frac{1}{h} \sum_{l=0}^{\infty}(l+1) I_{l+1}(h) \chi^{(l / 2)}\left(g_{n} g_{n-1}^{-1}\right) $$

$$ \chi^{(l / 2)}(g)=\mathcal{D}_{m m}^{l / 2}(g) $$

$$ \chi^{(l / 2)}\left(g_{n} g_{n-1}^{-1}\right)=\mathcal{D}_{m m^{\prime}}^{l / 2}\left(g_{n}\right) \mathcal{D}_{m m^{\prime}}^{l / 2 *}\left(g_{n-1}\right) $$

$$ \int d g \chi^{(L)}\left(g_{1} g^{-1}\right) \chi^{\left(L^{\prime}\right)}\left(g g_{2}^{-1}\right)=\delta_{L L^{\prime}} \frac{1}{d_{L}} \chi^{(L)}\left(g_{1} g_{2}^{-1}\right) $$

$$ \begin{align*} \left(g_{b} \tau_{b} \mid g_{a} \tau_{a}\right) & =\sum_{l=0}^{\infty} \exp \left[-\frac{\hbar L_{2}}{2 M r^{2}}\left(\tau_{b}-\tau_{a}\right)\right] \\ & \times(l+1) \sum_{m_{1}, m_{2}=-l / 2}^{l / 2} \mathcal{D}_{m_{1} m_{2}}^{l / 2}\left(\varphi_{n}, \theta_{n}, \gamma_{n}\right) \mathcal{D}_{m_{1} m_{2}}^{l / 2 *}\left(\varphi_{n-1}, \theta_{n-1}, \gamma_{n-1}\right) \end{align*} $$

$$ \Delta E=-\frac{3 \hbar^{2}}{8 M} $$

$$ \begin{align*} & \left(\varphi_{b}, \theta_{b}, \gamma_{b} \tau_{b} \mid \varphi_{b}, \theta_{b}, \gamma_{b} \tau_{a}\right)_{\mathrm{top}} \\ & \quad=\left(\varphi_{b}, \theta_{b}, \gamma_{b} \tau_{b} \mid \varphi_{b}, \theta_{b}, \gamma_{b} \tau_{a}\right)+\left(\varphi_{b}, \theta_{b}, \gamma_{b}+2 \pi \tau_{b} \mid \varphi_{b}, \theta_{b}, \gamma_{b} \tau_{a}\right) \end{align*} $$

$$ \begin{align*} & \left(\varphi_{b}, \theta_{b}, \gamma_{b} \tau_{b} \mid \varphi_{b}, \theta_{b}, \gamma_{b} \tau_{a}\right)_{\text {fermionic }} \\ & \quad=\left(\varphi_{b}, \theta_{b}, \gamma_{b} \tau_{b} \mid \varphi_{b}, \theta_{b}, \gamma_{b} \tau_{a}\right)-\left(\varphi_{b}, \theta_{b}, \gamma_{b}+2 \pi \tau_{b} \mid \varphi_{b}, \theta_{b}, \gamma_{b} \tau_{a}\right) \end{align*} $$

$$ \frac{1}{\epsilon^{2}}\left[1-\frac{1}{2} \operatorname{tr}\left(g_{n} g_{n-1}^{-1}\right)\right], $$

$$ \omega_{a}=-i \operatorname{tr}\left(\dot{\sigma}_{a} g^{-1}\right), \quad a=\xi, \eta, \zeta . $$

$$ \begin{align*} \frac{1}{\epsilon^{2}}\left\{I_{\xi}\left[1-\frac{1}{2} \operatorname{tr}\left(g_{n} \sigma_{\xi} g_{n-1}^{-1}\right)\right]\right. & +I_{\eta}\left[1-\frac{1}{2} \operatorname{tr}\left(g_{n} \sigma_{\eta} g_{n-1}^{-1}\right)\right] \\ & \left.+I_{\zeta}\left[1-\frac{1}{2} \operatorname{tr}\left(g_{n} \sigma_{\zeta} g_{n-1}^{-1}\right)\right]\right\} \end{align*} $$

$$ |\theta \varphi\rangle \equiv R^{s}(\theta, \varphi)|s s\rangle \equiv e^{-i S_{3} \varphi} e^{-i S_{2} \theta}|s s\rangle, $$

$$ |\theta \varphi\rangle=\sum_{s_{3}=-s}^{s}\left|s s_{3}\right\rangle\left\langle s s_{3}\right| \hat{R}(\theta, \varphi)|s s\rangle=\sum_{s_{3}=-s}^{s}\left|s s_{3}\right\rangle e^{-i s_{3} \varphi} d_{s_{3} s}^{s}(\theta) $$

$$ |\theta \varphi\rangle=e^{i \varphi / 2} \cos \theta / 2\left|\frac{1}{2} \frac{1}{2}\right\rangle-e^{-i \varphi / 2} \sin \theta / 2\left|\frac{1}{2}-\frac{1}{2}\right\rangle . $$

$$ Y_{m q}^{j}(\theta, \varphi) \equiv \sqrt{\frac{2 j+1}{4 \pi}} e^{i m \varphi} d_{m q}^{j}(\theta) $$

$$ \int_{-1}^{1} d \cos \theta \int_{0}^{2 \pi} d \varphi Y_{m q}^{j *}(\theta, \varphi) Y_{m^{\prime} q}^{j^{\prime}}(\theta, \varphi)=\delta_{j j^{\prime}} \delta_{m m^{\prime}} $$

$$ \sum_{j, m} Y_{m q}^{j}(\theta, \varphi) Y_{m q}^{j *}\left(\theta^{\prime}, \varphi^{\prime}\right)=\delta\left(\cos \theta-\cos \theta^{\prime}\right) \delta\left(\varphi-\varphi^{\prime}\right) $$

$$ |\mathbf{u}\rangle \equiv \sqrt{\frac{2 j+1}{4 \pi}}|\theta \varphi\rangle=\sum_{s_{3}=-s}^{s}\left|s s_{3}\right\rangle Y_{s_{3} s}^{s *}(\theta, \varphi) $$

$$ \int_{-1}^{1} d \cos \theta \int_{0}^{2 \pi} d \varphi=\int d^{3} u \delta\left(\mathbf{u}^{2}-1\right) \equiv \int d \mathbf{u} $$

$$ \begin{align*} \int d \mathbf{u}|\mathbf{u}\rangle\langle\mathbf{u}| & =\sum_{s_{3}=-s}^{s} \sum_{s_{3}^{\prime}=-s}^{s}\left|s s_{3}\right\rangle \int_{-1}^{1} d \cos \theta \int_{0}^{2 \pi} d \varphi Y_{s_{3} s}^{s *}(\theta, \varphi) Y_{s_{3}^{\prime} s}^{s}(\theta, \varphi)\left\langle s s_{3}^{\prime}\right| \\ & =\sum_{s_{3}=-s}^{s}\left|s s_{3}\right\rangle\left\langle s s_{3}\right|=1^{s} \end{align*} $$

$$ \left\langle\mathbf{u} \mid \mathbf{u}^{\prime}\right\rangle=\sum_{s_{3}=-s}^{s} \sum_{s_{3}^{\prime}=-s}^{s} Y_{s_{3} s}^{s}(\mathbf{u})\left\langle s s_{3} \mid s s_{3}^{\prime}\right\rangle Y_{s_{3}^{\prime} s}^{s}\left(\mathbf{u}^{\prime}\right)=\sum_{s_{3}=-s}^{s} Y_{s_{3} s}^{s}(\mathbf{u}) Y_{s_{3} s}^{s *}\left(\mathbf{u}^{\prime}\right) $$

$$ \begin{gather*} \frac{2 j+1}{4 \pi}\langle s s| e^{i \theta S_{2}} e^{i \varphi S_{3}} e^{-i \varphi^{\prime} S_{3}} e^{-i \theta^{\prime} S_{2}}|s s\rangle=\frac{2 j+1}{4 \pi}\langle s s| e^{-i s A S_{3}} e^{-i \beta S_{2}}|s s\rangle \\ \quad=\frac{2 j+1}{4 \pi} e^{-i s A S_{3}} d_{s s}^{s}(\beta)=\frac{2 j+1}{4 \pi} e^{-i s A S_{3}}\left(\frac{1+\mathbf{u} \cdot \mathbf{u}^{\prime}}{2}\right)^{s} \end{gather*} $$

$$ A=\sqrt{s(s-a)(s-b)(s-c)}, \quad s=(a+b+c) / 2=\text { semiperimeter } $$

$$ \tan \frac{E}{4}=\sqrt{\tan \frac{\phi_{s}}{2} \tan \frac{\phi_{s}-\phi_{a}}{2} \tan \frac{\phi_{s}-\phi_{b}}{2} \tan \frac{\phi_{s}-\phi_{c}}{2}}, $$

$$ \phi_{s}=\left(\phi_{a}+\phi_{b}+\phi_{c}\right) / 2 $$

$$ \left\langle\mathbf{u}_{b} \mid \mathbf{u}_{a}\right\rangle=\left[\prod_{n=1}^{N} \int d \mathbf{u}_{n}\right]\left\langle\mathbf{u}_{b} \mid \mathbf{u}_{N}\right\rangle\left\langle\mathbf{u}_{N} \mid \mathbf{u}_{N-1}\right\rangle\left\langle\mathbf{u}_{N-1}\right| \cdots\left|\mathbf{u}_{1}\right\rangle\left\langle\mathbf{u}_{1} \mid \mathbf{u}_{a}\right\rangle . $$

$$ \left\langle\mathbf{u}_{n} \mid \mathbf{u}_{n-1}\right\rangle \approx \frac{2 s+1}{4 \pi} e^{i \Delta A_{n}}\left[1+\frac{1}{2} \mathbf{u}_{n}\left(\mathbf{u}_{n}-\mathbf{u}_{n-1}\right)\right]^{s} \approx \frac{2 s+1}{4 \pi} e^{i \Delta A_{n}+\frac{s}{2} \mathbf{u}_{n}\left(\mathbf{u}_{n}-\mathbf{u}_{n-1}\right)} . $$

$$ \frac{2 s+1}{4 \pi} \exp \left[i \epsilon s \cos \theta \dot{\varphi}-\frac{s}{4} \epsilon^{2}\left(\dot{\theta}^{2}+\sin ^{2} \theta \dot{\varphi}^{2}\right)+\ldots\right] $$

$$ \begin{align*} \langle\theta \varphi| i \partial_{t}|\theta \varphi\rangle & =\langle s s| e^{i S_{2} \theta} e^{i S_{3} \varphi} i \partial_{t} e^{-i S_{3} \varphi} e^{-i S_{2} \theta}|s s\rangle \\ & =\langle s s| e^{i S_{2} \theta} e^{i S_{3} \varphi}\left(\dot{\varphi} S_{3} e^{-i S_{3} \varphi} e^{-i S_{2} \theta}+e^{-i S_{3} \varphi} \dot{\theta} S_{2} e^{-i S_{2} \theta}\right)|s s\rangle \\ & =\langle s s|\left(\cos \theta S_{3}-\sin \theta S_{1}\right) \dot{\varphi}+\dot{\theta} S_{2}|s s\rangle=s \cos \theta \dot{\varphi} \end{align*} $$

$$ \left\langle\mathbf{u}_{b} \mid \mathbf{u}_{a}\right\rangle=\int \mathcal{D} \mathbf{u} e^{\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t \hbar s \cos \theta \dot{\varphi}} $$

$$ \int \mathcal{D} \mathbf{u} \equiv \lim _{N \rightarrow \infty}\left[\frac{2 s+1}{4 \pi} \prod_{n=1}^{N} \int d \mathbf{u}_{n}\right] $$

$$ \left(\mathbf{u}_{b} t_{b} \mid \mathbf{u}_{a} t_{a}\right)=\int \mathcal{D} \mathbf{u} e^{i \int_{t_{a}}^{t_{b}} d t[\hbar s \cos \theta \dot{\varphi}-H(\theta, \varphi)] / \hbar} $$

$$ \langle\mathbf{u}| \hat{\mathbf{S}}|\mathbf{u}\rangle=\langle s s| \hat{R}^{-1}(\theta, \varphi) \hat{\mathbf{S}} \hat{R}(\theta, \varphi)|s s\rangle=R_{i j}(\theta, \varphi)\langle s s| S_{j}|s s\rangle=s \mathbf{u} $$

$$ \mathcal{A}_{0}=\frac{e}{c} \int_{t_{a}}^{t_{b}} \mathbf{A}^{(g)}(\mathbf{u}) \cdot \dot{\mathbf{u}}=\hbar s \int_{t_{a}}^{t_{b}} d t \cos \theta \dot{\varphi} $$

$$ |\theta \varphi\rangle^{\prime} \equiv \hat{R}(\theta, \varphi)|s s\rangle=e^{-i S_{3} \varphi} e^{-i S_{2} \theta} e^{i S_{3} \varphi}|s s\rangle=|\theta \varphi\rangle e^{i s \varphi}, $$

$$ \mathcal{A}_{0}=\hbar s \int d t \frac{\hat{\mathbf{z}} \times \mathbf{u}(\mathbf{t})}{1-u_{z}(t)} \cdot \dot{\mathbf{u}}(t) $$

$$ \mathcal{A}_{0}=\hbar s \int d t \frac{\mathbf{n} \times \mathbf{u}(\mathbf{t})}{1-\mathbf{n} \cdot \mathbf{u}(t)} \cdot \dot{\mathbf{u}}(t) $$

$$ \mathbf{B}^{(g)}=\boldsymbol{\nabla} \times \mathbf{A}^{(g)}=g \frac{\mathbf{u}}{|\mathbf{u}|}-4 \pi g \int_{0}^{\infty} d s \hat{\mathbf{n}} \delta^{(3)}(\mathbf{u}-s \hat{\mathbf{n}}) $$

$$ \mathcal{A}_{0}=\frac{e}{c} \int d t \mathbf{A}^{(g)}(t) \cdot \mathbf{u}(t)=\frac{e}{c} \int d \mathbf{S} \cdot\left[\boldsymbol{\nabla} \times \mathbf{A}^{(g)}\right]=\frac{e}{c} \int d \mathbf{S} \cdot \mathbf{B}^{(g)} $$

$$ Z_{\mathrm{QM}}=\oint d \mathbf{u} e^{i\left(\mathcal{A}_{0}+\mathcal{A}\right) / \hbar} $$

$$ \frac{g e}{\hbar c}=s, $$

$$ \boldsymbol{\nabla} \times \mathbf{B}(\mathbf{u})=4 \pi g \delta^{(3)}(\mathbf{u}) $$

$$ \mathcal{A}=\frac{M}{2} \int d t\left[\dot{\mathbf{u}}^{2}(t)-V\left(\mathbf{u}^{2}(t)\right)\right], \quad \mathbf{u}^{2}(t) \equiv 1 $$

$$ \mathcal{A}_{0}=\int d t \psi^{*}(t)\left[i \hbar \partial_{t}+\gamma \mathbf{u}(t) \cdot \boldsymbol{\sigma}\right] \psi(t) $$

$$ \psi(\mathbf{u}(t))=\left(\begin{array}{cc} e^{-i \phi(t) / 2} & \cos \theta(t) / 2 \\ e^{i \phi(t) / 2} & \sin \theta(t) / 2 \end{array}\right) $$

$$ \mathcal{A}_{0}=\int d t \psi(\mathbf{u}(t)) i \hbar \partial_{t} \psi(\mathbf{u}(t))=\hbar \frac{1}{2} \cos \theta(t) \dot{\phi}(t) \equiv \hbar \beta(t) $$

$$ M\left(\ddot{\mathbf{u}}+\dot{\mathbf{u}}^{2} \mathbf{u}\right)=-\partial_{\mathbf{u}} V(\mathbf{u})-\frac{\delta}{\delta \mathbf{u}(t)} \mathcal{A}_{0} $$

$$ \begin{align*} \frac{\delta}{\delta u_{i}(t)} \hbar \int_{t_{a}}^{t_{b}} d t \mathbf{A}^{(g)} \cdot \dot{\mathbf{u}} & =\hbar\left(\partial_{u_{i}} A_{j}^{(g)} \dot{u}_{j}-\partial_{t} A_{i}^{(g)}\right)=\hbar\left[\left(\partial_{u_{i}} A_{j}^{(g)}-\partial_{u_{j}} A_{i}^{(g)}\right) \dot{u}_{j}\right] \\ & =\hbar\left[\left(\boldsymbol{\nabla} \times \mathbf{A}^{(g)}\right) \times \mathbf{u}\right]_{i} \end{align*} $$

$$ M\left(\ddot{\mathbf{u}}+\dot{\mathbf{u}}^{2} \mathbf{u}\right)=-\partial_{\mathbf{u}} V(\mathbf{u})-\hbar g \mathbf{u} \times \dot{\mathbf{u}} $$

$$ M \mathbf{u} \times \ddot{\mathbf{u}}=-\mathbf{u} \times \partial_{\mathbf{u}} V(\mathbf{u})-\hbar g\left[\mathbf{u}(\mathbf{u} \cdot \dot{\mathbf{u}})-\dot{\mathbf{u}} \mathbf{u}^{2}\right] . $$

$$ M \mathbf{u} \times \ddot{\mathbf{u}}-\hbar g \dot{\mathbf{u}}=-\mathbf{u} \times \partial_{\mathbf{u}} V(\mathbf{u}(t)) . $$

$$ \hbar s \dot{\mathbf{u}}=-\mathbf{u} \times \partial_{\mathbf{u}} V(\mathbf{u}) $$

$$ H_{\mathrm{int}}=-\boldsymbol{\mu} \cdot \mathbf{B}, \quad \boldsymbol{\mu}=-\gamma \mathbf{S}, $$

$$ \boldsymbol{\mu}=\gamma s \hbar \mathbf{u} $$

$$ H_{\mathrm{int}}=-\gamma s \hbar \mathbf{u} \cdot \mathbf{B} $$

$$ \dot{\mathbf{u}}=-\gamma \mathbf{B} \times \mathbf{u} $$

$$ \hbar \dot{\mathbf{S}}=i[\hat{H}, \mathbf{S}] $$

$$ \dot{\mathbf{S}}=-\gamma \mathbf{B} \times \mathbf{S}, $$