Kleinert · 제13장 게이지장

$$ H=\frac{p^{2}}{2 M}-\frac{e^{2}}{r} $$

$$ \hat{a} \equiv\binom{\hat{a}_{1}}{\hat{a}_{2}}, \quad \hat{b} \equiv\binom{\hat{b}_{1}}{\hat{b}_{2}} $$

$$ \left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)=\int \mathcal{D}^{3} x(t) \exp \left[\frac{i}{\hbar} \int_{t_{b}}^{t_{a}} d t(\mathbf{p} \dot{\mathbf{x}}-H)\right] $$

$$ \begin{align*} \hat{L}_{i j} & =\frac{1}{2}\left(\hat{a}^{\dagger} \sigma_{k} \hat{a}+\hat{b}^{\dagger} \sigma_{k} \hat{b}\right) \quad i, j, k=1,2,3 \text { cyclic, } \\ \hat{L}_{i 4} & =\frac{1}{2}\left(\hat{a}^{\dagger} \sigma_{i} \hat{a}-\hat{b}^{\dagger} \sigma_{i} \hat{b}\right), \\ \hat{L}_{i 5} & =\frac{1}{2}\left(\hat{a}^{\dagger} \sigma_{i} c \hat{b}^{\dagger}-\hat{a} c \sigma_{i} \hat{b}\right), \\ \hat{L}_{i 6} & =\frac{i}{2}\left(\hat{a}^{\dagger} \sigma_{i} c \hat{b}^{\dagger}+\hat{a} c \sigma_{i} \hat{b}\right), \\ \hat{L}_{45} & =\frac{1}{2 i}\left(\hat{a}^{\dagger} c \hat{b}^{\dagger}-\hat{a} c \hat{b}\right), \\ \hat{L}_{46} & =\frac{1}{2}\left(\hat{a}^{\dagger} c \hat{b}^{\dagger}+\hat{a} c \hat{b}\right), \\ \hat{L}_{56} & =\frac{1}{2}\left(\hat{a}^{\dagger} \hat{a}+\hat{b}^{\dagger} \hat{b}+2\right) . \end{align*} $$

$$ f_{l}(\mathbf{x})=f(\mathbf{x})^{1-\lambda}, \quad f_{r}(\mathbf{x})=f(\mathbf{x})^{\lambda} $$

$$ \frac{1}{2}\left(n_{1}^{a}+n_{2}^{a}+n_{1}^{b}+n_{2}^{b}+2\right)=n . $$

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=\int_{0}^{\infty} d S\left\langle\mathbf{x}_{b}\right| \hat{\mathcal{U}}_{E}(S)\left|\mathbf{x}_{a}\right\rangle $$

$$ \left[\hat{L}_{A B}, \hat{L}_{A C}\right]=i g_{A A} \hat{L}_{B C} $$

$$ \left\langle\mathbf{x}_{b}\right| \hat{\mathcal{U}}_{E}(S)\left|\mathbf{x}_{a}\right\rangle=r_{b}^{\lambda} r_{a}^{1-\lambda} \int \mathcal{D}^{D} x(s) \int \frac{\mathcal{D}^{D} p(s)}{(2 \pi \hbar)^{D}} \exp \left\{\frac{i}{\hbar} \int_{0}^{S} d s\left[\mathbf{p x}^{\prime}-r^{1-\lambda}(H-E) r^{\lambda}\right]\right\} $$

$$ \begin{align*} r & =\hat{L}_{56}-\hat{L}_{46}, \\ x^{i} & =\hat{L}_{i 5}-\hat{L}_{i 4}, \\ -i\left(\mathbf{x} \partial_{\mathbf{x}}+1\right) & =\hat{L}_{45}, \\ -i r \partial_{x^{i}} & =\hat{L}_{i 6} . \end{align*} $$

$$ \left\langle\mathbf{x}_{b}\right| \hat{\mathcal{U}}_{E}(S)\left|\mathbf{x}_{a}\right\rangle \approx r_{b}^{\lambda} r_{a}^{1-\lambda} \prod_{n=2}^{N+1}\left[\int_{-\infty}^{\infty} d^{D} \Delta x_{n}\right] \prod_{n=1}^{N+1}\left[\int_{-\infty}^{\infty} \frac{d^{D} p_{n}}{(2 \pi \hbar)^{D}}\right] e^{i \mathcal{A}_{E}^{N} / \hbar}, $$

$$ \partial_{x^{i}}=\frac{1}{2 \vec{u}^{2}} e_{\mu}^{i} \partial_{\mu} $$

$$ \mathcal{A}_{E}^{N}[\mathbf{p}, \mathbf{x}]=\sum_{n=1}^{N+1}\left[\mathbf{p}_{n} \Delta \mathbf{x}_{n}-\epsilon_{s} r_{n}^{1-\lambda} r_{n-1}^{\lambda}\left(\frac{\mathbf{p}_{n}{ }^{2}}{2 M}-E\right)+\epsilon_{s} e^{2}\right] . $$

$$ \begin{array}{ll} u^{1}=\frac{1}{2}\left(z_{1}+z_{1}^{*}\right), & u^{2}=\frac{1}{2 i}\left(z_{1}-z_{1}^{*}\right) \\ u^{3}=\frac{1}{2}\left(z_{2}+z_{2}^{*}\right), & u^{4}=\frac{1}{2 i}\left(z_{2}-z_{2}^{*}\right) \end{array} $$

$$ \left\langle\mathbf{x}_{b}\right| \hat{\mathcal{U}}_{E}(S)\left|\mathbf{x}_{a}\right\rangle \approx \frac{r_{b}^{\lambda} r_{a}^{1-\lambda}}{\sqrt{2 \pi i \epsilon_{s} \hbar r_{b}^{1-\lambda} r_{a}^{\lambda} / M}}{ }^{D} \prod_{n=2}^{N+1}\left[\int \frac{d^{D} \Delta \mathbf{x}_{n}}{{\sqrt{2 \pi i \epsilon_{s} \hbar r_{n-1} / M}}^{D}}\right] e^{i \mathcal{A}_{E}^{N}\left[\mathbf{x}, \mathbf{x}^{\prime}\right] / \hbar} $$

$$ \begin{array}{ll} \partial_{1}=\left(\partial_{z_{1}}+\partial_{z_{1}^{*}}\right), & \partial_{2}=i\left(\partial_{z_{1}}-\partial_{z_{1}^{*}}\right), \\ \partial_{3}=\left(\partial_{z_{2}}+\partial_{z_{2}^{*}}\right), & \partial_{4}=i\left(\partial_{z_{2}}-\partial_{z_{2}^{*}}\right) \end{array} $$

$$ \mathcal{A}_{E}^{N}\left[\mathbf{x}, \mathbf{x}^{\prime}\right]=(N+1) \epsilon_{s} e^{2}+\sum_{n=1}^{N+1}\left[\frac{M}{2} \frac{\left(\Delta \mathbf{x}_{n}\right)^{2}}{\epsilon_{s} r_{n}^{1-\lambda} r_{n-1}^{\lambda}}+\epsilon_{s} r_{n} E\right] . $$

$$ -i r \partial_{x^{i}}=-\frac{i}{2}\left(\bar{z} \sigma_{i} \partial_{\bar{z}}+\partial_{z} \sigma_{i} z\right) $$

$$ \mathcal{A}_{E}\left[\mathbf{x}, \mathbf{x}^{\prime}\right]=e^{2} S+\int_{0}^{S} d s\left(\frac{M}{2 r} \mathbf{x}^{\prime 2}+E r\right) $$

$$ \begin{align*} \hat{L}_{i j} & =\frac{1}{2}\left(\bar{z} \sigma_{k} \partial_{\bar{z}}-\partial_{z} \sigma_{k} z\right), \\ \hat{L}_{i 4} & =-\frac{1}{2}\left(\bar{z} \sigma_{i} z-\partial_{z} \sigma_{i} \partial_{\bar{z}}\right), \\ \hat{L}_{i 5} & =\frac{1}{2}\left(\bar{z} \sigma_{i} z+\partial_{z} \sigma_{i} \partial_{\bar{z}}\right), \\ \hat{L}_{i 6} & =-\frac{i}{2}\left(\bar{z} \sigma_{i} \partial_{\bar{z}}+\partial_{z} \sigma_{i} z\right), \\ \hat{L}_{45} & =-\frac{i}{2}\left(\bar{z} \partial_{\bar{z}}+\partial_{z} z\right), \\ \hat{L}_{46} & =-\frac{1}{2}\left(\bar{z} z+\partial_{z} \partial_{\bar{z}}\right), \\ \hat{L}_{56} & =\frac{1}{2}\left(\bar{z} z-\partial_{z} \partial_{\bar{z}}\right) . \end{align*} $$

$$ \begin{align*} x^{1} & =\left(u^{1}\right)^{2}-\left(u^{2}\right)^{2} \\ x^{2} & =2 u^{1} u^{2} \end{align*} $$

$$ \begin{align*} \hat{L}_{i j} & =-i\left(x_{i} \partial_{x^{j}}-x_{j} \partial_{x^{i}}\right), \\ \hat{L}_{i 4} & =\frac{1}{2}\left(-x^{i} \partial_{\mathbf{x}}^{2}-x^{i}+2 \partial_{x^{i}} \mathbf{x} \partial_{\mathbf{x}}\right), \\ \hat{L}_{i 5} & =\frac{1}{2}\left(-x^{i} \partial_{\mathbf{x}}^{2}+x^{i}+2 \partial_{x^{i}} \mathbf{x} \partial_{\mathbf{x}}\right), \\ \hat{L}_{i 6} & =-i r \partial_{x^{i}}, \\ \hat{L}_{45} & =-i\left(x^{i} \partial_{x^{i}}+1\right), \\ \hat{L}_{46} & =\frac{1}{2}\left(-r \partial_{\mathbf{x}}^{2}-r\right), \\ \hat{L}_{56} & =\frac{1}{2}\left(-r \partial_{\mathbf{x}}^{2}+r\right), \end{align*} $$

$$ u=\sqrt{x} $$

$$ \begin{align*} {\left[P_{\mu}, P_{\nu}\right] } & =0 \\ {\left[L_{\mu \nu}, P_{\lambda}\right] } & =-i\left(g_{\mu \lambda} P_{\nu}-g_{\nu \lambda} P_{\mu}\right) \\ {\left[L_{\mu \nu}, L_{\lambda \kappa}\right] } & =-i\left(g_{\mu \lambda} L_{\nu \kappa}-g_{\nu \lambda} L_{\mu \kappa}-g_{\mu \kappa} L_{\nu \lambda}-g_{\nu \kappa} L_{\mu \lambda}\right) \end{align*} $$

$$ A(\mathbf{u})=\left(\begin{array}{rr} u^{1} & -u^{2} \\ u^{2} & u^{1} \end{array}\right) $$

$$ \mathbf{x}=A(\mathbf{u}) \mathbf{u} . $$

$$ e_{\mu}^{i}(\mathbf{u})=\frac{\partial x^{i}}{\partial u^{\mu}}(\mathbf{u})=2 A_{\mu}^{i}(\mathbf{u}) $$

$$ x^{\mu} \rightarrow \frac{x^{\mu}-c^{\mu} x^{2}}{1-2 c x+c^{2} x^{2}} $$

$$ e_{i}{ }^{\mu}(\mathbf{u})=\frac{1}{2}\left(A^{-1}\right)^{T}{ }_{i}{ }^{\mu}(\mathbf{u})=\frac{1}{2 \mathbf{u}^{2}} A^{i}{ }_{\mu}(\mathbf{u}) . $$

$$ \begin{align*} {\left[D, P_{\mu}\right] } & =-i P_{\mu}, \quad\left[D, K_{\mu}\right]=i K_{\mu},\left[D, L_{\mu \nu}\right]=0 \\ {\left[K_{\mu}, K_{\nu}\right] } & =0, \quad\left[K_{\mu}, P_{\nu}\right]=-2 i\left(g_{\mu \nu} D+L_{\mu \nu}\right), \quad\left[K_{\mu}, L_{\nu \lambda}\right]=i\left(g_{\mu \nu} K_{\lambda}-g_{\mu \lambda} K_{\nu}\right) \end{align*} $$

$$ \Gamma_{\mu \nu}^{\lambda}=e_{i}^{\lambda} \partial_{\mu} e_{\nu}^{i}=\frac{1}{\mathbf{u}^{2}}\left[\left(\partial_{\mu} A\right)^{T} A\right]_{\nu \lambda} $$

$$ \begin{align*} \left(\Gamma_{1}\right)_{\mu}{ }^{\nu} & =\frac{1}{\mathbf{u}^{2}}\left(\begin{array}{rr} u^{1} & -u^{2} \\ u^{2} & u^{1} \end{array}\right)_{\mu}^{\nu}=\frac{1}{2 \mathbf{u}^{2}} A(\mathbf{u})_{\nu}^{\mu} \\ \left(\Gamma_{2}\right)_{\mu}{ }^{\nu} & =\frac{1}{\mathbf{u}^{2}}\left(\begin{array}{rr} u^{2} & u^{1} \\ -u^{1} & u^{2} \end{array}\right)_{\mu}^{\nu} . \end{align*} $$

$$ \begin{align*} \hat{P}_{\mu} & =i \partial_{\mu}, \quad \hat{M}_{\mu \nu}=i\left(x_{\mu} \partial_{\nu}-x_{\nu} \partial_{\mu}\right), \quad \hat{D}=i x^{\mu} \partial_{\mu} \\ \hat{K}_{\mu} & =i\left(2 x_{\mu} x^{\nu} \partial_{\nu}-x^{2} \partial_{\mu}\right) \end{align*} $$

$$ \Gamma_{\mu}{ }^{\mu \lambda} \equiv 0, $$

$$ \Gamma_{\mu}{ }^{\mu \lambda} \equiv g^{\mu \nu} e_{i}{ }^{\lambda} \partial_{\mu} e^{i}{ }_{\nu}, $$

$$ J_{\mu \nu} \equiv L_{\mu \nu}, J_{\mu 5} \equiv \frac{1}{2}\left(P_{\mu}-K_{\mu}\right), J_{\mu 6} \equiv \frac{1}{2}\left(P_{\mu}+K_{\mu}\right), J_{56} \equiv D $$

$$ \partial_{\mu} e^{i}{ }_{\mu}=\partial_{\mathbf{u}}^{2} x^{i}(\mathbf{u})=0 $$

$$ \left[J_{A B}, J_{C D}\right]=-i\left(\bar{g}_{A C} J_{B D}-\bar{g}_{B C} J_{A D}+\bar{g}_{B D} J_{A C}-\bar{g}_{B C} J_{A D}\right), $$

$$ \begin{align*} e_{i}{ }^{\lambda}\left(\partial_{\mu} e^{i}{ }_{\nu}-\partial_{\nu} e^{i}{ }_{\mu}\right) & \equiv 0, \\ e_{i}{ }^{\kappa}\left(\partial_{\mu} \partial_{\nu}-\partial_{\nu} \partial_{\mu}\right) e^{i}{ }_{\lambda} & \equiv 0, \end{align*} $$

$$ \begin{align*} & \hat{L}_{12}=i\left(u^{1} \partial_{2}-u^{2} \partial_{1}-u^{3} \partial_{4}+u^{4} \partial_{3}\right) / 2 \\ & \hat{L}_{13}=i\left(u^{1} \partial_{3}+u^{2} \partial_{4}-u^{3} \partial_{1}-u^{4} \partial_{2}\right) / 2 \\ & \hat{L}_{14}=-\left(u^{1} u^{3}+u^{2} u^{4}\right)+\left(\partial_{1} \partial_{3}+\partial_{2} \partial_{4}\right) / 4 \\ & \hat{L}_{15}=\left(u^{1} u^{3}+u^{2} u^{4}\right)+\left(\partial_{1} \partial_{3}+\partial_{2} \partial_{4}\right) / 4 \\ & \hat{L}_{16}=-i\left(u^{1} \partial_{3}+u^{2} \partial_{4}+u^{3} \partial_{1}+u^{4} \partial_{2}\right) / 2, \\ & \hat{L}_{23}=i\left(u^{1} \partial_{4}-u^{2} \partial_{3}+u^{3} \partial_{2}-u^{4} \partial_{1}\right) / 2, \\ & \hat{L}_{24}=-\left(u^{1} u^{4}-u^{2} u^{3}\right)+\left(\partial_{1} \partial_{4}-\partial_{2} \partial_{3}\right) / 4, \\ & \hat{L}_{25}=\left(u^{1} u^{4}+u^{2} u^{3}\right)+\left(\partial_{1} \partial_{4}-\partial_{2} \partial_{3}\right) / 4, \\ & \hat{L}_{26}=-i\left(u^{1} \partial_{4}-u^{2} \partial_{3}-u^{3} \partial_{2}+u^{4} \partial_{1}\right) / 2, \\ & \hat{L}_{34}=\left[\left(u^{1}\right)^{2}+\left(u^{2}\right)^{2}-\left(u^{3}\right)^{2}-\left(u^{4}\right)^{2}\right] / 2+\left(\partial_{1}^{2}+\partial_{2}^{2}-\partial_{3}^{2}-\partial_{4}^{2}\right) / 8, \\ & \hat{L}_{35}=-\left[\left(u^{1}\right)^{2}+\left(u^{2}\right)^{2}-\left(u^{3}\right)^{2}-\left(u^{4}\right)^{2}\right] / 2+\left(\partial_{1}^{2}+\partial_{2}^{2}-\partial_{3}^{2}-\partial_{4}^{2}\right) / 8, \\ & \hat{L}_{36}=-i\left(u^{1} \partial_{1}+u^{2} \partial_{2}-u^{3} \partial_{3}-u^{4} \partial_{4}\right) / 2, \\ & \hat{L}_{45}=-i\left(u^{1} \partial_{1}+u^{2} \partial_{2}+u^{3} \partial_{3}+u^{4} \partial_{4}+2\right) / 2, \\ & \hat{L}_{46}=-\left(u^{\mu}\right)^{2} / 2-\partial_{\mu}^{2} / 8, \\ & \hat{L}_{56}=\left(u^{\mu}\right)^{2} / 2-\partial_{\mu}^{2} / 8 . \end{align*} $$

$$ \left(-\frac{a_{H}}{2} r \nabla^{2}-\frac{E}{E_{H}} \frac{r}{a_{H}}-1\right) \psi(\mathbf{x})=0, $$

$$ \mathbf{x}^{\prime 2}=4 \mathbf{u}^{2} \mathbf{u}^{\prime 2}=4 r \mathbf{u}^{\prime 2} $$

$$ \left[\frac{1}{2}\left(\hat{L}_{56}+\hat{L}_{46}\right)-E\left(\hat{L}_{56}-\hat{L}_{46}\right)-1\right] \psi=0 $$

$$ \mathcal{A}[\mathbf{x}]=e^{2} S+\int_{0}^{S} d s\left(\frac{4 M}{2} \mathbf{u}^{\prime 2}+E \mathbf{u}^{2}\right) $$

$$ \left[e^{i \vartheta \hat{L}_{45}} \hat{L}_{56} e^{-i \vartheta \hat{L}_{45}}-1\right] \psi=0 $$

$$ \mathcal{A}_{\mathrm{oS}}[\mathbf{u}]=\int_{0}^{S} d s \frac{\mu}{2}\left(\mathbf{u}^{\prime 2}-\omega^{2} \mathbf{u}^{2}\right) $$

$$ \vartheta=\frac{1}{2} \log (-2 E) $$

$$ \mu=4 M $$

$$ \vartheta=\vartheta_{n}=-\log n $$

$$ \omega=\sqrt{-E / 2 M} $$

$$ \left\langle\psi_{n^{\prime}}^{\prime H} \mid \psi_{n}^{H}\right\rangle_{\mathrm{phys}} \equiv\left\langle\psi_{n^{\prime}}^{\prime s}\right|\left(\hat{L}_{56}-\hat{L}_{46}\right)\left|\psi_{n}^{s}\right\rangle=\delta_{n^{\prime} n} $$

$$ d \mathbf{x}=2 A(\mathbf{u}) d \mathbf{u} $$

$$ \psi_{n}^{H}(\mathbf{x})=\frac{1}{\sqrt{n}} e^{i \vartheta_{n} \hat{D}} \psi_{n}^{s}\left(u^{\mu}\right)=\frac{1}{\sqrt{n}} \psi_{n}^{s}\left(u^{\mu} / \sqrt{n}\right) $$

$$ d^{2} x_{n}=4 \mathbf{u}_{n}^{2} d^{2} u_{n} $$

$$ \left\langle\mathbf{x}_{b}\right| \hat{\mathcal{U}}_{E}(S)\left|\mathbf{x}_{a}\right\rangle=\frac{1}{4} e^{i e^{2} S / \hbar}\left[\left(\mathbf{u}_{b} S \mid \mathbf{u}_{a} 0\right)+\left(-\mathbf{u}_{b} S \mid \mathbf{u}_{a} 0\right)\right], $$

$$ F_{n l}(p)=\sqrt{\frac{2}{\pi}} \sqrt{\frac{(n-l-1)!}{(n-l)!}} n^{2} 2^{2(l+1)} l!\frac{n^{l} p^{l}}{\left(n^{2} p^{2}+1\right)^{l+2}} C_{n-l-1}^{(l+1)}\left(\frac{n^{2} p^{2}-1}{n^{2} p^{2}+1}\right) $$

$$ \begin{align*} \left(\mathbf{u}_{b} S \mid \mathbf{u}_{a} 0\right) & \approx \frac{1}{2 \pi i \hbar \epsilon_{s} / \mu} \prod_{n=1}^{N}\left[\int \frac{d^{2} u_{n}}{2 \pi i \hbar \epsilon_{s} / \mu}\right] \\ & \times \exp \left\{\frac{i}{\hbar} \sum_{n=1}^{N} \frac{\mu}{2}\left(\frac{1}{\epsilon_{s}} \Delta \mathbf{u}_{n}{ }^{2}-\epsilon_{s} \omega^{2} \mathbf{u}_{n}{ }^{2}\right)\right\} \end{align*} $$

$$ C_{0}^{(\lambda)}(z)=1, \quad C_{1}^{(\lambda)}(z)=2 \lambda z, \quad C_{2}^{(\lambda)}(z)=2 \lambda(\lambda+1) z^{2}-\lambda, \ldots $$

$$ \left(\mathbf{u}_{b} S \mid \mathbf{u}_{a} 0\right)=\frac{\mu \omega}{2 \pi i \hbar \sin \omega S} \exp \left\{\frac{i}{2 \hbar} \frac{\mu \omega}{\sin \omega S}\left[\left(\mathbf{u}_{b}^{2}+\mathbf{u}_{a}^{2}\right) \cos \omega S-2 \mathbf{u}_{b} \mathbf{u}_{a}\right]\right\} . $$

$$ F_{10}=4 \sqrt{\frac{2}{\pi}} \frac{1}{\left(p^{2}+1\right)^{2}}, \quad F_{20}=\frac{32}{\sqrt{\pi}} \frac{4 p^{2}-1}{\left(4 p^{2}+1\right)^{3}}, \quad F_{21}=\frac{128}{\sqrt{3 \pi}} \frac{p}{\left(4 p^{2}+1\right)^{3}} $$

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=\int_{0}^{\infty} d S e^{i e^{2} S / \hbar} \frac{1}{4}\left[\left(\mathbf{u}_{b} S \mid \mathbf{u}_{a} 0\right)+\left(-\mathbf{u}_{b} S \mid \mathbf{u}_{a} 0\right)\right] $$

$$ \begin{align*} \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E} & =\frac{1}{2} \int_{0}^{\infty} d S \exp \left(i e^{2} S / \hbar\right) F^{2}(S) \\ & \times \exp \left[-\pi F^{2}(S)\left(\mathbf{u}_{b}^{2}+\mathbf{u}_{a}^{2}\right) \cos \omega S\right] \cosh \left[2 \pi F^{2}(S) \mathbf{u}_{b} \mathbf{u}_{a}\right] \end{align*} $$

$$ F(S)=\sqrt{\mu \omega / 2 \pi i \hbar \sin \omega S} $$

$$ \mathbf{u}_{a, b}^{2}=r_{a, b}, \quad \mathbf{u}_{b} \mathbf{u}_{a}=\sqrt{\left(r_{b} r_{a}+\mathbf{x}_{b} \mathbf{x}_{a}\right) / 2} $$

$$ \begin{align*} \varrho & \equiv e^{-2 i \omega S}=e^{-2 \omega \sigma} \\ \kappa & \equiv \frac{\mu \omega}{2 \hbar}=\frac{2 M \omega}{\hbar}=\sqrt{-2 M E / \hbar^{2}} \\ \nu & \equiv \frac{e^{2}}{2 \omega \hbar}=\sqrt{\frac{e^{4} M}{-2 \hbar^{2} E}} \end{align*} $$

$$ \begin{align*} \pi F^{2}(S) & =\kappa \frac{2 \sqrt{\varrho}}{1-\varrho} \\ e^{i e^{2} s / \hbar} F^{2}(S) & =\frac{2}{\pi} \kappa \frac{\varrho^{1 / 2-\nu}}{1-\varrho}, \end{align*} $$

$$ \begin{gather*} \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=-i \frac{M}{\pi \hbar} \int_{0}^{1} d \varrho \frac{\varrho^{-1 / 2-\nu}}{1-\varrho} \cos \left[2 \kappa \frac{2 \sqrt{\varrho}}{1-\varrho} \sqrt{\left(r_{b} r_{a}+\mathbf{x}_{b} \mathbf{x}_{a}\right) / 2}\right] \\ \times \exp \left[-\kappa \frac{1+\varrho}{1-\varrho}\left(r_{b}+r_{a}\right)\right] \end{gather*} $$

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=\sum_{n=1}^{\infty} \frac{i \hbar}{E-E_{n}} \sum_{m=0}^{n-1}\left[\psi_{n_{r}, m}\left(\mathbf{x}_{b}\right) \psi_{n_{r}, m}^{*}\left(\mathbf{x}_{b}\right)+\psi_{n_{r},-m}\left(\mathbf{x}_{b}\right) \psi_{n_{r},-m}^{*}\left(\mathbf{x}_{b}\right)\right], $$

$$ \psi_{n_{r}, m}(\mathbf{x})=\frac{1}{\sqrt{r}} R_{n_{r},|m|}(r) \frac{1}{\sqrt{2 \pi}} e^{i m \phi}, \quad r \equiv $$

$$ E_{n}=-\frac{M e^{4}}{\hbar^{4}} \frac{1}{\left(n-\frac{1}{2}\right)^{2}} $$

$$ \frac{1}{\sqrt{r}} R_{n_{r},|m|}(r)=N_{n_{r},|m|}\left(\frac{2 r}{r_{n}}\right)^{|m|} e^{-r / r_{n}}{ }_{1} F_{1}\left(-n+|m|+1,2|m|+1,2 r / r_{n}\right), $$

$$ N_{n_{r},|m|} \equiv \frac{2}{r_{n}} \frac{1}{2|m|!} \sqrt{\frac{(n+|m|-1)!}{(2 n-1)(n-|m|-1)!}} $$

$$ \zeta \equiv \frac{1+\varrho}{1-\varrho}, $$

$$ \frac{d \varrho}{(1-\varrho)^{2}}=\frac{1}{2} d \zeta, \quad \varrho=\frac{\zeta-1}{\zeta+1} $$

$$ \begin{align*} \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E} & =-i \frac{M}{\pi \hbar} \frac{1}{2} \int_{1}^{\infty} d \zeta(\zeta-1)^{-\nu-1 / 2}(\zeta+1)^{\nu-1 / 2} \\ \times & \cos \left\{2 \kappa \sqrt{\zeta^{2}-1} \sqrt{\left(r_{b} r_{a}+\mathbf{x}_{b} \mathbf{x}_{a}\right) / 2}\right\} e^{-\kappa \zeta\left(r_{b}+r_{a}\right)} \end{align*} $$

$$ \int_{1}^{\infty} d \zeta(\zeta-1)^{-\nu-1 / 2} \ldots \rightarrow \frac{\pi e^{i \pi(\nu+1 / 2)}}{\sin [\pi(\nu+1 / 2)]} \frac{1}{2 \pi i} \int_{C} d \zeta(\zeta-1)^{-\nu-1 / 2} \ldots $$

$$ \begin{align*} \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}= & -i \frac{M}{\pi \hbar} \frac{1}{2} \frac{\pi e^{i \pi(\nu+1 / 2)}}{\sin [\pi(\nu+1 / 2)]} \int_{C} \frac{d \zeta}{2 \pi i}(\zeta-1)^{-\nu-1 / 2}(\zeta+1)^{\nu-1 / 2} \\ & \times \cos \left[2 \kappa \sqrt{\zeta^{2}-1} \sqrt{\left(r_{b} r_{a}+\mathbf{x}_{b} \mathbf{x}_{a}\right) / 2}\right] e^{-\kappa \zeta\left(r_{b}+r_{a}\right)} \end{align*} $$

$$ \mathbf{x}_{n}=A\left(\mathbf{u}_{n}\right) \mathbf{u}_{n} $$

$$ \Delta x^{i}=2 A_{\mu}^{i}(\mathbf{u}) \Delta u^{\mu}-\partial_{\nu} A_{\mu}^{i}(\mathbf{u}) \Delta u^{\mu} \Delta u^{\nu} . $$

$$ \Delta \mathbf{x}_{n}=A\left(\mathbf{u}_{n}\right) \mathbf{u}_{n}-A\left(\mathbf{u}_{n-1}\right) \mathbf{u}_{n-1}=2 A\left(\mathbf{u}_{n}-\frac{1}{2} \Delta \mathbf{u}_{n}\right) \Delta \mathbf{u}_{n}=2 A\left(\overline{\mathbf{u}}_{n}\right) \Delta \mathbf{u}_{n}, $$

$$ \begin{align*} \left(\Delta \mathbf{x}_{n}\right)^{2} & =4 \overline{\mathbf{u}}_{n}^{2}\left(\Delta \mathbf{u}_{n}\right)^{2} \\ \overline{\mathbf{u}}_{n} & \equiv\left(\mathbf{u}_{n}+\mathbf{u}_{n-1}\right) / 2 \end{align*} $$

$$ \mathcal{A}^{\epsilon}=\frac{M}{2 \epsilon_{s}} \frac{\left(\Delta \mathbf{x}_{n}\right)^{2}}{r_{n}^{1-\lambda} r_{n-1}^{\lambda}}=\frac{M}{2 \epsilon_{s}} \frac{4 \overline{\mathbf{u}}_{n}^{2}}{\left(\mathbf{u}_{n}^{2}\right)^{1-\lambda}\left(\mathbf{u}_{n-1}^{2}\right)^{\lambda}}\left(\Delta \mathbf{u}_{n}\right)^{2} $$

$$ \begin{align*} \overline{\mathbf{u}}_{n}= & \mathbf{u}_{n}-\frac{1}{2} \Delta \mathbf{u}_{n} \\ \mathbf{u}_{n-1}= & \mathbf{u}_{n}-\Delta \mathbf{u}_{n} \\ \frac{\overline{\mathbf{u}}_{n}^{2}}{\left(\mathbf{u}_{n}^{2}\right)^{1-\lambda}\left(\mathbf{u}_{n-1}^{2}\right)^{\lambda}}= & 1+(2 \lambda-1) \frac{\mathbf{u}_{n} \Delta \mathbf{u}_{n}}{\mathbf{u}_{n}^{2}}+\left(\frac{1}{4}-\lambda\right) \frac{\Delta \mathbf{u}_{n}^{2}}{\mathbf{u}_{n}^{2}} \\ & +2 \lambda^{2}\left(\frac{\mathbf{u}_{n} \Delta \mathbf{u}_{n}}{\mathbf{u}_{n}^{2}}\right)^{2} \end{align*} $$

$$ \mathcal{A}_{0}^{\epsilon}\left(\Delta \mathbf{u}_{n}\right)=4 M \frac{\left(\Delta \mathbf{u}_{n}\right)^{2}}{2 \epsilon_{s}} $$

$$ \begin{align*} \Delta \mathcal{A}^{\epsilon} & =4 M \frac{\left(\Delta \mathbf{u}_{n}\right)^{2}}{2 \epsilon_{s}} \\ & \times\left[(2 \lambda-1) \frac{\mathbf{u}_{n} \Delta \mathbf{u}_{n}}{\mathbf{u}_{n}^{2}}+\left(\frac{1}{4}-\lambda\right) \frac{\Delta \mathbf{u}_{n}^{2}}{\mathbf{u}_{n}^{2}}+2 \lambda^{2}\left(\frac{\mathbf{u}_{n} \Delta \mathbf{u}_{n}}{\mathbf{u}_{n}^{2}}\right)^{2}\right] \end{align*} $$

$$ \begin{align*} \Delta x^{i} & =2 A^{i}{ }_{\mu}\left(\mathbf{u}-\frac{1}{2} \Delta \mathbf{u}\right) \Delta u^{\mu} \\ & =2 A^{i}{ }_{\mu}(\mathbf{u}) \Delta u^{\mu}-\partial_{\nu} A^{i}{ }_{\mu}(\mathbf{u}) \Delta u^{\mu} \Delta u^{\nu} . \end{align*} $$

$$ e^{i}{ }_{\mu}(\mathbf{u})=2 A^{i}{ }_{\mu}(\mathbf{u}) . $$

$$ \begin{align*} \frac{i}{\hbar} \mathcal{A}_{J}^{\epsilon} & =-e_{i}{ }^{\mu} e^{i}{ }_{\{\mu, \nu\}} \Delta u^{\nu}-\frac{1}{2} e_{i}{ }^{\mu} e^{i}{ }_{\{\kappa, \nu\}} e_{j}{ }^{\kappa} e^{j}{ }_{\mu, \lambda} \Delta u^{\nu} \Delta u^{\lambda} \\ & =-\Gamma_{\{\nu \mu\}}{ }^{\mu} \Delta u^{\nu}-\frac{1}{2} \Gamma_{\{\nu \kappa\}}{ }^{\mu} \Gamma_{\{\mu \lambda\}}{ }^{\kappa} \Delta u^{\nu} \Delta u^{\lambda} \end{align*} $$

$$ e_{i}{ }^{\kappa}=\frac{1}{2 \mathbf{u}^{2}} e^{i}{ }_{\kappa}, $$

$$ \begin{align*} \Gamma_{\nu \mu}{ }^{\mu} & =e_{i}{ }^{\mu} \partial_{\nu} e^{i}{ }_{\mu}=\frac{2 u^{\nu}}{\mathbf{u}^{2}}, \\ \Gamma_{\mu \nu}{ }^{\mu} & =-e^{i}{ }_{\nu} \partial_{\mu} e_{i}{ }^{\mu}=\frac{2 u^{\nu}}{\mathbf{u}^{2}}, \\ \Gamma_{\nu \kappa}{ }^{\mu} \Gamma_{\lambda \mu}{ }^{\kappa} & =-\partial_{\lambda} e_{i}{ }^{\kappa} \partial_{\nu} e^{i}{ }_{\kappa}=-\frac{2}{\mathbf{u}^{4}}\left(\delta^{\nu \lambda} \mathbf{u}^{2}-2 u^{\nu} u^{\lambda}\right) . \end{align*} $$

$$ -\partial_{\mu} e_{i}^{\mu}=-\partial_{\mu}\left(2 \mathbf{u}^{2}\right)^{-1} e_{\mu}^{i}=e_{i}^{\mu} 2 u^{\mu} \mathbf{u}^{-2}, $$

$$ \frac{i}{\hbar} \mathcal{A}_{J}^{\epsilon}=-\left[2 \frac{\mathbf{u}_{n} \Delta \mathbf{u}_{n}}{\mathbf{u}_{n}^{2}}-\frac{\Delta \mathbf{u}_{n}^{2}}{\mathbf{u}_{n}^{2}}+2\left(\frac{\mathbf{u} \Delta \mathbf{u}_{n}}{\mathbf{u}_{n}^{2}}\right)^{2}+\ldots\right] $$

$$ \begin{align*} \frac{\left(r_{b} / r_{a}\right)^{2 \lambda-1}}{2 \pi i \epsilon_{s} \hbar} \prod_{n=1}^{N}\left[\int \frac{d^{2} \Delta x_{n}}{2 \pi i \epsilon_{s} r_{n-1} / M}\right] & \approx \frac{1}{2 \pi i \epsilon_{s} \hbar} \prod_{n=1}^{N}\left[\int \frac{d^{2} \Delta x_{n}}{2 \pi i \epsilon_{s} \hbar r_{n} / M}\right] \prod_{n=1}^{N+1}\left(\frac{r_{n}}{r_{n-1}}\right)^{2 \lambda} \\ & =\frac{1}{2 \pi i \epsilon_{s} \hbar} \prod_{n=2}^{N+1}\left[\int \frac{d^{2} \Delta x_{n}}{2 \pi i \epsilon_{s} \hbar r_{n} / M}\right] e^{i \mathcal{A}_{f}^{N} / \hbar} \end{align*} $$

$$ \mathcal{A}_{f}^{N} \equiv \sum_{n=1}^{N+1} \mathcal{A}_{f}^{\epsilon} $$

$$ \frac{i}{\hbar} \mathcal{A}_{f}^{\epsilon}=2 \lambda \log \frac{r_{n}^{2}}{r_{n-1}^{2}}=2 \lambda \log \frac{\mathbf{u}_{n}^{2}}{\mathbf{u}_{n-1}^{2}} $$

$$ d^{2} \Delta x=4 \mathbf{u}^{2} d^{2} \Delta u \exp \left(\frac{i}{\hbar} \mathcal{A}_{J}^{\epsilon}\right) $$

$$ \frac{1}{2} \times \frac{4}{2 \cdot 2 \pi i \epsilon_{s} \hbar} \prod_{n=1}^{N}\left[\int \frac{4 d^{2} \Delta u_{n}}{2 \cdot 2 \pi i \epsilon_{s} \hbar / M}\right] \exp \left[\frac{i}{\hbar}\left(\mathcal{A}_{J}^{N}+\mathcal{A}_{f}^{N}\right)\right] $$

$$ \mathcal{A}_{J}^{N} \equiv \sum_{n=1}^{N+1} \mathcal{A}_{J}^{\epsilon} $$

$$ \frac{i}{\hbar} \mathcal{A}_{f}^{\epsilon}=2 \lambda \log \left(\frac{\mathbf{u}_{n}^{2}}{\mathbf{u}_{n-1}^{2}}\right)=2 \lambda\left[2 \frac{\mathbf{u}_{n} \Delta \mathbf{u}_{n}}{\mathbf{u}_{n}^{2}}-\frac{\Delta \mathbf{u}_{n}^{2}}{\mathbf{u}_{n}^{2}}+2\left(\frac{\mathbf{u} \Delta \mathbf{u}_{n}}{\mathbf{u}_{n}^{2}}\right)^{2}+\ldots\right] $$

$$ \mathcal{A}^{\epsilon}=\mathcal{A}_{0}^{\epsilon}+\Delta_{\mathrm{corr}} \mathcal{A}^{\epsilon} $$

$$ \mathcal{A}_{0}^{\epsilon}\left(\Delta \mathbf{u}_{n}\right)=4 M \frac{\left(\Delta \mathbf{u}_{n}\right)^{2}}{2 \epsilon_{s}} $$

$$ \begin{align*} & \frac{i}{\hbar} \Delta_{\mathrm{corr}} \mathcal{A}^{\epsilon} \equiv \frac{i}{\hbar}\left(\Delta \mathcal{A}^{\epsilon}+\mathcal{A}_{J}^{\epsilon}+\mathcal{A}_{f}^{\epsilon}\right) \\ & =\frac{i}{\hbar} 4 M \frac{\Delta \mathbf{u}_{n}^{2}}{2 \epsilon_{s}}\left[(2 \lambda-1) \frac{\mathbf{u}_{n} \Delta \mathbf{u}_{n}}{\mathbf{u}_{n}^{2}}+\left(\frac{1}{4}-\lambda\right) \frac{\Delta \mathbf{u}_{n}^{2}}{\mathbf{u}_{n}^{2}}+2 \lambda^{2}\left(\frac{\mathbf{u}_{n} \Delta \mathbf{u}_{n}}{\mathbf{u}_{n}^{2}}\right)^{2}\right] \\ & \quad+(2 \lambda-1)\left[2 \frac{\mathbf{u}_{n} \Delta \mathbf{u}_{n}}{\mathbf{u}_{n}^{2}}-\frac{\Delta \mathbf{u}_{n}^{2}}{\mathbf{u}_{n}^{2}}+2\left(\frac{\mathbf{u}_{n} \Delta \mathbf{u}_{n}}{\mathbf{u}_{n}^{2}}\right)^{2}\right]+\ldots \end{align*} $$

$$ K^{\epsilon}(\Delta \mathbf{u})=\frac{4}{2 \cdot 2 \pi i \epsilon_{s} \hbar / M} \exp \left[\frac{i}{\hbar}\left(\mathcal{A}_{0}^{\epsilon}+\Delta_{\mathrm{corr}} \mathcal{A}^{\epsilon}\right)\right] $$

$$ K_{0}^{\epsilon}(\Delta \mathbf{u})=\frac{4}{2 \cdot 2 \pi i \epsilon_{s} \hbar / M} \exp \left[\frac{i}{\hbar} \mathcal{A}_{0}^{\epsilon}\right] . $$

$$ C_{1}=C \equiv \exp \left(\frac{i}{\hbar} \Delta_{\mathrm{corr}} \mathcal{A}^{\epsilon}\right)-1 $$

$$ C_{2}=0 $$

$$ \begin{array}{r} \langle C\rangle_{0}=0 \\ \langle C(\mathbf{p} \Delta \mathbf{u})\rangle_{0}=0 \end{array} $$

$$ \begin{align*} \left\langle\Delta u^{\mu} \Delta u^{\nu}\right\rangle_{0} & \equiv \frac{i \hbar \epsilon_{s}}{4 M} \delta^{\mu \nu} \\ \left\langle\Delta u^{\mu_{1}} \cdots \Delta u^{\mu_{2 n}}\right\rangle_{0} & =\left(\frac{i \hbar \epsilon_{s}}{4 M}\right)^{n} \delta^{\mu_{1} \ldots \mu_{2 n}}, \quad n>1 \end{align*} $$

$$ \delta^{\mu_{1} \ldots \mu_{2 n}} \equiv \delta^{\mu_{1} \mu_{2}} \delta^{\mu_{3} \mu_{4} \ldots \mu_{2 n}}+\delta^{\mu_{1} \mu_{3}} \delta^{\mu_{2} \mu_{4} \ldots \mu_{2 n}}+\ldots+\delta^{\mu_{1} \mu_{2 n}} \delta^{\mu_{2} \mu_{3} \ldots \mu_{2 n-1}} $$

$$ \left\langle(\Delta \mathbf{u})^{2 k}(\mathbf{u} \Delta \mathbf{u})^{2 l}\right\rangle_{0}=\left(\frac{i \hbar \epsilon_{s}}{4 M}\right)^{k+l} \frac{[D+2(k+l-1)]!!}{(D+2 l-2)!!}(2 l-1)!!\left(\mathbf{u}^{2}\right)^{l}, $$

$$ \left\langle(\Delta \mathbf{u})^{2 k}(\mathbf{u} \Delta \mathbf{u})^{2 l}(\mathbf{u} \Delta \mathbf{u})(\mathbf{p} \Delta \mathbf{u})\right\rangle_{0}=\left(\frac{i \hbar \epsilon_{s}}{4 M}\right)^{k+l+1} \frac{[D+2(k+l)]!!}{(D+2 l)!!}(2 l-1)!!(\mathbf{u p}), $$

$$ \begin{array}{r} \langle C\rangle_{0}=\frac{i}{\hbar}\left\langle\Delta_{\mathrm{corr}} \mathcal{A}^{\epsilon}\right\rangle_{0}+\frac{1}{2!}\left(\frac{i}{\hbar}\right)^{2}\left\langle\left(\Delta_{\mathrm{corr}} \mathcal{A}^{\epsilon}\right)^{2}\right\rangle_{0}=0 \\ \langle C(\mathbf{p} \Delta \mathbf{u})\rangle_{0}=\frac{i}{\hbar}\left\langle\Delta_{\mathrm{corr}} \mathcal{A}^{\epsilon}(\mathbf{p} \Delta \mathbf{u})\right\rangle_{0}=0 \end{array} $$

$$ \frac{i}{\hbar}\left\langle\Delta_{\mathrm{corr}} \mathcal{A}^{\epsilon}\right\rangle_{0}=2 i \frac{\hbar \epsilon_{s}}{M}\left[-\left(\frac{1}{4}-\lambda\right) \frac{(D+2) D}{16}-2 \lambda^{2} \frac{D+2}{16}\right] $$

$$ \frac{i}{\hbar}\left\langle\Delta_{\mathrm{corr}} \mathcal{A}^{\epsilon}\right\rangle_{0}=-i \frac{\hbar \epsilon_{s}}{M}\left(\lambda-\frac{1}{2}\right)^{2} $$

$$ \frac{1}{2!}\left(\frac{i}{\hbar}\right)^{2}\left\langle\left(\Delta_{\mathrm{corr}} \mathcal{A}^{\epsilon}\right)^{2}\right\rangle_{0}=\frac{i}{2} \frac{\hbar \epsilon_{s}}{M}\left[4(2 \lambda-1)^{2} \frac{(D+4)(D+2)}{64}+4(2 \lambda-1)^{2} \frac{1}{4}-8(2 \lambda-1)^{2} \frac{D+2}{16}\right] $$

$$ \frac{1}{2!}\left(\frac{i}{\hbar}\right)^{2}\left\langle\left(\Delta_{\mathrm{corr}} \mathcal{A}^{\epsilon}\right)^{2}\right\rangle_{0}=i \frac{\hbar \epsilon_{s}}{M}\left(\lambda-\frac{1}{2}\right)^{2} $$

$$ \left\langle\Delta_{\mathrm{corr}} \mathcal{A}^{\epsilon}(\mathbf{p} \Delta \mathbf{u})\right\rangle_{0}=-\frac{\hbar^{2} \epsilon_{s}}{4 M}[(2 \lambda-1)(D+2) / 4-(2 \lambda-1)] $$

$$ x^{i}=\bar{z} \sigma^{i} z, \quad r=\bar{z} z $$

$$ z=\binom{z_{1}}{z_{2}}, \quad \bar{z}=\left(z_{1}^{*}, z_{2}^{*}\right) $$

$$ z_{1}=\left(u^{1}+i u^{2}\right), \quad z_{2}=\left(u^{3}+i u^{4}\right) $$

$$ \left(\begin{array}{c} x^{1} \\ x^{2} \\ x^{3} \end{array}\right)=A(\vec{u})\left(\begin{array}{c} u^{1} \\ u^{2} \\ u^{3} \\ u^{4} \end{array}\right) $$

$$ A(\vec{u})=\left(\begin{array}{rrrr} u^{3} & u^{4} & u^{1} & u^{2} \\ u^{4} & -u^{3} & -u^{2} & u^{1} \\ u^{1} & u^{2} & -u^{3} & -u^{4} \end{array}\right) . $$

$$ r=\left(u^{1}\right)^{2}+\left(u^{2}\right)^{2}+\left(u^{3}\right)^{2}+\left(u^{4}\right)^{2} \equiv(\vec{u})^{2}, $$

$$ \left(\begin{array}{l} d x^{1} \\ d x^{2} \\ d x^{3} \end{array}\right)=2\left(\begin{array}{rrrr} u^{3} & u^{4} & u^{1} & u^{2} \\ u^{4} & -u^{3} & -u^{2} & u^{1} \\ u^{1} & u^{2} & -u^{3} & -u^{4} \end{array}\right)\left(\begin{array}{l} d u^{1} \\ d u^{2} \\ d u^{3} \\ d u^{4} \end{array}\right) $$

$$ d \vec{x}=2 A(\vec{u}) d \vec{u} $$

$$ A(\vec{u})=\left(\begin{array}{rrrr} u^{3} & u^{4} & u^{1} & u^{2} \\ u^{4} & -u^{3} & -u^{2} & u^{1} \\ u^{1} & u^{2} & -u^{3} & -u^{4} \\ u^{2} & -u^{1} & u^{4} & -u^{3} \end{array}\right) . $$

$$ \begin{align*} d x^{4} & =2\left(u^{2} d u^{1}-u^{1} d u^{2}+u^{4} d u^{3}-u^{3} d u^{4}\right) \\ & =r(\cos \theta d \varphi+d \gamma) \end{align*} $$

$$ \left(\partial_{u^{1}} \partial_{u^{2}}-\partial_{u^{2}} \partial_{u^{1}}\right) x^{4}\left(u^{\mu}\right)=-4, \quad\left(\partial_{u^{3}} \partial_{u^{4}}-\partial_{u^{4}} \partial_{u^{3}}\right) x^{4}\left(u^{\mu}\right)=-4 $$

$$ \mathcal{A}_{\mathrm{kin}}^{N} \equiv \sum_{n=1}^{N+1} \frac{M}{2} \frac{\left(\vec{x}_{n}-\vec{x}_{n-1}\right)^{2}}{\epsilon_{s} r_{n}^{1-\lambda} r_{n-1}^{\lambda}} $$

$$ \prod_{n=1}^{N+1} \int_{-\infty}^{\infty} \frac{d\left(\Delta x^{4}\right)_{n}}{\sqrt{2 \pi i \epsilon_{s} \hbar r_{n}^{1-\lambda} r_{n-1}^{\lambda} / M}} $$

$$ \prod_{n=1}^{N+1}\left[\int_{-\infty}^{\infty} \frac{d\left(\Delta x^{4}\right)_{n}}{\sqrt{2 \pi i \epsilon_{s} \hbar r_{n}^{1-\lambda} r_{n-1}^{\lambda} / M}}\right] \exp \left[\frac{i}{\hbar} \sum_{n=1}^{N+1} \frac{M}{2} \frac{\left(\Delta x_{n}^{4}\right)^{2}}{\epsilon_{s} r_{n}^{1-\lambda} r_{n-1}^{\lambda}}\right]=1 $$

$$ \begin{align*} \left\langle\mathbf{x}_{b}\right| \hat{\mathcal{U}}_{E}(S)\left|\mathbf{x}_{a}\right\rangle & =\int d x_{a}^{4} \frac{r_{b}^{\lambda} r_{a}^{1-\lambda}}{\left(2 \pi i \epsilon_{s} \hbar r_{b}^{1-\lambda} r_{a}^{\lambda} / M\right)^{2}} \\ & \times \prod_{n=2}^{N+1}\left[\int_{-\infty}^{\infty} \frac{d^{4} \Delta x_{n}}{\left(2 \pi i \epsilon_{s} \hbar r_{n-1} / M\right)^{2}}\right] \exp \left(\frac{i}{\hbar} \mathcal{A}_{E}^{N}\right) \end{align*} $$

$$ \begin{align*} & \left\langle\mathbf{x}_{b}\right| \hat{\mathcal{U}}_{E}(S)\left|\mathbf{x}_{a}\right\rangle=\frac{1}{\left(2 \pi i \epsilon_{s} \hbar / M\right)^{2}} \int_{-\infty}^{\infty} \frac{d x_{a}^{4}}{r_{a}} \\ & \quad \times \prod_{n=2}^{N+1}\left[\int \frac{d^{4} \Delta \vec{x}_{n}}{\left(2 \pi i \epsilon_{s} \hbar r_{n} / M\right)^{2}}\right] \exp \left[\frac{i}{\hbar}\left(\mathcal{A}_{E}^{N}+\mathcal{A}_{f}^{N}\right)\right] \end{align*} $$

$$ \mathcal{A}_{E}^{N}\left[\vec{x}, \vec{x}^{\prime}\right]=(N+1) \epsilon_{s} e^{2}+\sum_{n=1}^{N+1}\left[\frac{M}{2} \frac{\left(\Delta \vec{x}_{n}\right)^{2}}{\epsilon_{s} r_{n}^{1-\lambda} r_{n-1}^{\lambda}}+\epsilon_{s} r_{n}^{1-\lambda} r_{n-1}^{\lambda} E\right] . $$

$$ \frac{i}{\hbar} \mathcal{A}_{f}^{N}=3 \lambda \sum_{n=1}^{N+1} \log \left(\frac{\vec{u}_{n}^{2}}{\vec{u}_{n-1}^{2}}\right) $$

$$ \begin{align*} A^{T} & =\vec{u}^{2} A^{-1} \\ \operatorname{det} A & =\sqrt{\operatorname{det}\left(A A^{T}\right)}=r^{2} \end{align*} $$

$$ \begin{align*} \vec{x}^{\prime 2} & =4 \vec{u}^{2} \vec{u}^{\prime 2}=4 r \vec{u}^{\prime 2} \\ d^{4} x & =16 r^{2} d^{4} u \end{align*} $$

$$ \left\langle\mathbf{x}_{b}\right| \hat{\mathcal{U}}_{E}(S)\left|\mathbf{x}_{a}\right\rangle=e^{i e^{2} S / \hbar} \frac{1}{16} \int \frac{d x_{a}^{4}}{r_{a}}\left(\vec{u}_{b} S \mid \vec{u}_{a} 0\right) $$

$$ \left(\vec{u}_{b} S \mid \vec{u}_{a} 0\right)=\int \mathcal{D}^{4} u(s) \exp \left(\frac{i}{\hbar} \mathcal{A}_{\mathrm{os}}\right) $$

$$ \mathcal{A}_{\mathrm{os}}=\int_{0}^{S} d s \frac{\mu}{2}\left(\vec{u}^{\prime 2}-\omega^{2} \vec{u}^{2}\right) $$

$$ \mu=4 M, \quad \omega=\sqrt{-E / 2 M} $$

$$ \left\langle\mathbf{x}_{b}\right| \hat{\mathcal{U}}_{E}(S)\left|\mathbf{x}_{a}\right\rangle=e^{i e^{2} S / \hbar} \frac{1}{16} \int_{0}^{4 \pi} d \gamma_{a}\left(\vec{u}_{b} S \mid \vec{u}_{a} 0\right) $$

$$ \left(\vec{u}_{n}+\vec{u}_{n-1}\right)^{2}\left(\vec{u}_{n}-\vec{u}_{n-1}\right)^{2} $$

$$ \begin{align*} \left(\vec{u}_{b} S \mid \vec{u}_{a} 0\right) & =\frac{1}{\left(2 \pi i \hbar \epsilon_{s} / \mu\right)^{2}} \prod_{n=1}^{N}\left[\int \frac{d^{4} \Delta u_{n}}{2 \pi i \hbar \epsilon_{s} / \mu}\right] \exp \left[\frac{i}{\hbar} \sum_{n=1}^{N} \frac{\mu}{2}\left(\frac{1}{\epsilon_{s}} \Delta \vec{u}_{n}^{2}-\epsilon_{s} \omega^{2} \vec{u}_{n}^{2}\right)\right] \\ & =\frac{\omega^{2}}{(2 \pi i \hbar \sin \omega S / \mu)^{2}} \exp \left\{\frac{i}{2 \hbar} \frac{\mu \omega}{\sin \omega S}\left[\left(\vec{u}_{b}^{2}+\vec{u}_{a}^{2}\right) \cos \omega S-2 \vec{u}_{b} \vec{u}_{a}\right]\right\} . \end{align*} $$

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=\int_{0}^{\infty} d S e^{i e^{2} S / \hbar} \frac{1}{16} \int_{0}^{4 \pi} d \gamma_{a}\left(\vec{u}_{b} S \mid \vec{u}_{a} 0\right) $$

$$ \begin{align*} \vec{u}_{b} \vec{u}_{a}=\sqrt{r_{b} r_{a}} & \left\{\cos \left(\theta_{b} / 2\right) \cos \left(\theta_{a} / 2\right) \cos \left[\left(\varphi_{b}-\varphi_{a}+\gamma_{b}-\gamma_{a}\right) / 2\right]\right. \\ & \left.+\sin \left(\theta_{b} / 2\right) \sin \left(\theta_{a} / 2\right) \cos \left[\left(\varphi_{b}-\varphi_{a}-\gamma_{b}+\gamma_{a}\right) / 2\right]\right\} . \end{align*} $$

$$ \begin{align*} \vec{u}_{b} \vec{u}_{a}=\sqrt{r_{b} r_{a}} & \left\{\cos \left[\left(\theta_{b}-\theta_{a}\right) / 2\right] \cos \left[\left(\varphi_{b}-\varphi_{a}\right) / 2\right] \cos \left[\left(\gamma_{b}-\gamma_{a}\right) / 2\right]\right. \\ & \left.-\cos \left[\left(\theta_{b}+\theta_{a}\right) / 2\right] \sin \left[\left(\varphi_{b}-\varphi_{a}\right) / 2\right] \sin \left[\left(\gamma_{b}-\gamma_{a}\right) / 2\right]\right\} \end{align*} $$

$$ \vec{u}_{b} \vec{u}_{a}=\sqrt{\left(r_{b} r_{a}+\mathbf{x}_{b} \mathbf{x}_{a}\right) / 2} \cos \left[\left(\gamma_{b}-\gamma_{a}+\beta\right) / 2\right] $$

$$ \tan \frac{\beta}{2}=\frac{\cos \left[\left(\theta_{b}+\theta_{a}\right) / 2\right] \sin \left[\left(\varphi_{b}-\varphi_{a}\right) / 2\right]}{\cos \left[\left(\theta_{b}-\theta_{a}\right) / 2\right] \cos \left[\left(\varphi_{b}-\varphi_{a}\right) / 2\right]} $$

$$ \cos \frac{\beta}{2}=\cos \frac{\theta_{b}-\theta_{a}}{2} \cos \frac{\varphi_{b}-\varphi_{a}}{2} \sqrt{\frac{r_{b} r_{a}}{\left(r_{b} r_{a}+\mathbf{x}_{b} \mathbf{x}_{a}\right) / 2}} . $$

$$ \begin{align*} \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E} & =-i \frac{M \kappa}{\pi \hbar} \int_{0}^{1} d \varrho \frac{\varrho^{-\nu}}{(1-\varrho)^{2}} I_{0}\left(2 \kappa \frac{2 \sqrt{\varrho}}{1-\varrho} \sqrt{\left(r_{b} r_{a}+\mathbf{x}_{b} \mathbf{x}_{a}\right) / 2}\right) \\ & \times \exp \left[-\kappa \frac{1+\varrho}{1-\varrho}\left(r_{b}+r_{a}\right)\right] \end{align*} $$

$$ \int_{1}^{\infty} d \zeta(\zeta-1)^{-\nu} \ldots \rightarrow \frac{\pi e^{i \pi \nu}}{\sin \pi \nu} \int_{C} \frac{d \zeta}{2 \pi i}(\zeta-1)^{-\nu} \ldots $$

$$ \begin{align*} \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=- & i \frac{M}{\pi \hbar} \frac{\kappa}{2} \frac{\pi e^{i \pi \nu}}{\sin \pi \nu} \int_{C} \frac{d \zeta}{2 \pi i}(\zeta-1)^{-\nu}(\zeta+1)^{\nu} \\ & \times I_{0}\left(2 \kappa \sqrt{\zeta^{2}-1} \sqrt{\left(r_{b} r_{a}+\mathbf{x}_{b} \mathbf{x}_{a}\right) / 2}\right) e^{-\kappa \zeta\left(r_{b}+r_{a}\right)} \end{align*} $$

$$ \begin{align*} \frac{i}{\hbar} \mathcal{A}_{J}^{\epsilon} & =-e^{\mu} e_{\{\mu, \nu\}}^{i} \Delta u^{\nu}-e_{i}{ }^{\mu} e^{i}{ }_{\{\kappa, \nu\}} e_{j}{ }^{\kappa} e^{i}{ }_{\{\mu, \lambda\}} \Delta u^{\nu} \Delta u^{\lambda} \\ & =-\Gamma_{\{\nu \mu\}}{ }^{\mu} \Delta u^{\nu}-\frac{1}{2} \Gamma_{\{\nu \kappa\}}{ }^{\sigma} \Gamma_{\{\lambda \sigma\}}{ }^{\kappa} \Delta u^{\nu} \Delta u^{\lambda} \end{align*} $$

$$ e_{\mu}^{i}=\partial x^{i} / \partial u^{\mu}=2 A_{\mu}^{i}(\vec{u}), \quad i=1,2,3,4, $$

$$ e_{i}{ }^{\mu}=\frac{1}{2 \vec{u}^{2}} e^{i}{ }_{\mu} $$

$$ \begin{align*} \left(\Gamma_{1}\right)_{\mu}^{\nu} & =\frac{1}{\vec{u}^{2}}\left(\begin{array}{rrrr} u^{1} & u^{2} & -u^{3} & -u^{4} \\ -u^{2} & u^{1} & -u^{4} & u^{3} \\ u^{3} & u^{4} & u^{1} & u^{2} \\ u^{4} & -u^{3} & -u^{2} & u^{1} \end{array}\right)_{\mu}^{\nu} \\ \left(\Gamma_{2}\right)_{\mu}^{\nu} & =\frac{1}{\vec{u}^{2}}\left(\begin{array}{rrrr} u^{2} & -u^{1} & u^{4} & -u^{3} \\ u^{1} & u^{2} & -u^{3} & -u^{4} \\ -u^{4} & u^{3} & u^{2} & -u^{1} \\ u^{3} & u^{4} & u^{1} & u^{2} \end{array}\right)_{\mu}^{\nu} \\ \left(\Gamma_{3}\right)_{\mu}^{\nu} & =\frac{1}{\vec{u}^{2}}\left(\begin{array}{rrrr} u^{3} & u^{4} & u^{1} & u^{2} \\ -u^{4} & u^{3} & u^{2} & -u^{1} \\ -u^{1} & -u^{2} & u^{3} & u^{4} \\ -u^{2} & u^{1} & -u^{4} & u^{3} \end{array}\right)_{\mu}^{\nu} \\ \left(\Gamma_{4}\right)_{\mu}^{\nu} & =\frac{1}{\vec{u}^{2}}\left(\begin{array}{rrrr} u^{4} & -u^{3} & -u^{2} & u^{1} \\ u^{3} & u^{4} & u^{1} & u^{2} \\ u^{2} & -u^{1} & u^{4} & -u^{3} \\ -u^{1} & -u^{2} & u^{3} & u^{4} \end{array}\right)_{\mu}^{\nu} \end{align*} $$

$$ \Gamma_{\mu}{ }^{\mu \nu} \equiv 0 $$

$$ \partial_{\mu} e^{i}{ }_{\mu}=0 $$

$$ S_{12}^{\lambda}=S_{34}^{\lambda}=\frac{1}{\vec{u}^{2}}\left(-u^{2}, u^{1},-u^{4}, u^{3}\right)^{\lambda} $$

$$ S_{\mu}=S_{\mu \nu}^{\nu}=\frac{u^{\mu}}{\vec{u}^{2}} $$

$$ \begin{align*} \Gamma_{\nu \mu}{ }^{\mu} & =e_{i}{ }^{\mu} \partial_{\nu} e^{i}{ }_{\mu}=\frac{4 u^{\nu}}{\vec{u}^{2}}, \\ \Gamma_{\mu \nu}{ }^{\mu} & =-e^{i}{ }_{\nu} \partial_{\mu} e_{i}{ }^{\mu}=\frac{2 u^{\nu}}{\vec{u}^{2}} \end{align*} $$

$$ \Gamma_{\{\nu \mu\}}^{\mu}=\frac{3 u^{\nu}}{\vec{u}^{2}} $$

$$ \Gamma_{\nu \kappa}{ }^{\sigma} \Gamma_{\lambda \sigma}{ }^{\kappa}=-\frac{4}{\vec{u}^{4}}\left(\delta^{\nu \lambda} \vec{u}^{2}-2 u^{\nu} u^{\lambda}\right) $$

$$ \begin{align*} \Gamma_{\{\nu \kappa\}}{ }^{\sigma} \Gamma_{\{\lambda \sigma\}}{ }^{\kappa} & =\Gamma_{\nu \kappa}{ }^{\sigma} \Gamma_{\lambda \sigma}{ }^{\kappa}-2 \Gamma_{\nu \kappa}{ }^{\sigma} S_{\lambda \sigma}{ }^{\kappa}+S_{\nu \kappa}{ }^{\sigma} S_{\lambda \sigma}{ }^{\kappa} \\ & =\Gamma_{\nu \kappa}{ }^{\sigma} \Gamma_{\lambda \sigma}{ }^{\kappa}-2\left(-\delta_{\nu \lambda} \vec{u}^{2}+2 u_{\nu} u_{\lambda}\right) / \vec{u}^{4}+u_{\nu} u_{\lambda} / \vec{u}^{4} \end{align*} $$

$$ \frac{i}{\hbar} \mathcal{A}_{J}^{\epsilon}=-\left[3 \frac{\vec{u}_{n} \Delta \vec{u}_{n}}{\vec{u}_{n}^{2}}-\frac{\Delta \vec{u}_{n}^{2}}{\vec{u}_{n}^{2}}+\frac{5}{2}\left(\frac{\vec{u}_{n} \Delta \vec{u}_{n}}{\vec{u}_{n}^{2}}\right)^{2}+\ldots\right] $$

$$ \begin{align*} \frac{i}{\hbar} \mathcal{A}_{f}^{\epsilon} & =3 \lambda \log \left(\frac{\vec{u}_{n}^{2}}{\vec{u}_{n-1}^{2}}\right) \\ & =3 \lambda\left[2 \frac{\vec{u}_{n} \Delta \vec{u}_{n}}{\vec{u}_{n}^{2}}-\frac{\Delta \vec{u}_{n}^{2}}{\vec{u}_{n}^{2}}+2\left(\frac{\vec{u} \Delta \vec{u}_{n}}{\vec{u}_{n}^{2}}\right)^{2}+\ldots\right] \end{align*} $$

$$ \begin{align*} \frac{i}{\hbar} \mathcal{A}_{J}^{\epsilon}= & -2 \log \left[\frac{\vec{u}^{2}}{(\vec{u}-\Delta \vec{u})^{2}}\right]+\frac{\vec{u} \Delta \vec{u}}{\vec{u}^{2}} \\ & -\frac{\Delta \vec{u}^{2}}{\vec{u}^{2}}+\frac{3}{2}\left(\frac{\vec{u} \Delta \vec{u}}{\vec{u}^{2}}\right)^{2}+\ldots, \end{align*} $$

$$ \begin{align*} \frac{i}{\hbar} \Delta_{\mathrm{corr}} \mathcal{A}^{\epsilon}= & \frac{i}{\hbar} 4 M \frac{\Delta \vec{u}^{2}}{2 \epsilon}\left[(2 \lambda-1) \frac{\vec{u} \Delta \vec{u}}{\vec{u}^{2}}+\left(\frac{1}{4}-\lambda\right) \frac{\Delta \vec{u}^{2}}{\vec{u}^{2}}+2 \lambda^{2}\left(\frac{\vec{u} \Delta \vec{u}}{\vec{u}^{2}}\right)^{2}\right] \\ & +(3 \lambda-2)\left[2 \frac{\vec{u} \Delta \vec{u}}{\vec{u}^{2}}-\frac{\Delta \vec{u}^{2}}{\vec{u}^{2}}+2\left(\frac{\vec{u} \Delta \vec{u}}{\vec{u}^{2}}\right)^{2}\right] \\ & +\frac{\vec{u} \Delta \vec{u}}{\vec{u}^{2}}-\frac{(\Delta \vec{u})^{2}}{\vec{u}^{2}}+\frac{3}{2}\left(\frac{\vec{u} \Delta \vec{u}}{\vec{u}^{2}}\right)^{2}+\ldots \end{align*} $$

$$ C=\exp \left(\frac{i}{\hbar} \Delta_{\mathrm{corr}} \mathcal{A}^{\epsilon}\right)-1 $$

$$ \begin{align*} \langle C\rangle_{0} & =0 \\ \langle C(\vec{p} \Delta \vec{u})\rangle_{0} & =0 \end{align*} $$

$$ \begin{align*} \frac{i}{\hbar}\left\langle\Delta_{\mathrm{corr}} \mathcal{A}^{\epsilon}\right\rangle_{0}+\frac{1}{2}\left(\frac{i}{\hbar}\right)^{2}\left\langle\left(\Delta_{\mathrm{corr}} \mathcal{A}^{\epsilon}\right)^{2}\right\rangle_{0} & =0 \\ \frac{i}{\hbar}\left\langle\Delta_{\mathrm{corr}} \mathcal{A}(\vec{p} \Delta \vec{u})\right\rangle_{0} & =0 \end{align*} $$

$$ i\left[-2(2 \lambda-1) \frac{D+2}{16}+2(3 \lambda-2) \frac{1}{4}+\frac{1}{4}\right] $$

$$ i\left[-2\left(\frac{1}{4}-\lambda\right) \frac{(D+2) D}{16}-4 \lambda^{2} \frac{D+2}{16}-(3 \lambda-2)\left(\frac{D}{4}-\frac{2}{4}\right)-\left(\frac{D}{4}-\frac{3}{8}\right)\right], $$

$$ -i \frac{3}{8}(2 \lambda-1)^{2} $$

$$ i \frac{1}{2}\left[4(2 \lambda-1)^{2} \frac{(D+4)(D+2)}{64}+9(2 \lambda-1)^{2} \frac{1}{4}-12(2 \lambda-1)^{2} \frac{D+2}{16}\right] $$

$$ \Gamma_{\mu}{ }^{\mu \lambda}=g^{\mu \nu} e_{i}{ }^{\lambda} \partial_{\mu} e^{i}{ }_{\nu}=0, $$

$$ f_{l}=f(\mathbf{x}), \quad f_{r} \equiv 1 $$

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

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

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

$$ \Delta \mathcal{A}^{\epsilon}=4 M \frac{(\Delta \mathbf{u})^{2}}{2 \epsilon_{s}} $$

$$ \left(-\frac{1}{2 M} \hbar^{2} \nabla^{2}-E\right) \psi(\mathbf{x})=\frac{e^{2}}{r} \psi(\mathbf{x}) $$

$$ \left(-\frac{1}{2 M} \hbar^{2} r \boldsymbol{\nabla}^{2}-E r\right) \psi(\mathbf{x})=e^{2} \psi(\mathbf{x}) . $$

$$ \left[-\frac{1}{8 M} \hbar^{2} \partial_{\mu}^{2}-E\left(u^{\mu}\right)^{2}\right] \psi\left(u^{\mu}\right)=e^{2} \psi\left(u^{\mu}\right) . $$

$$ \left\langle\psi^{\prime} \mid \psi\right\rangle=\int d^{4} u \psi^{\prime}\left(u^{\mu}\right)\left(u^{\mu}\right)^{2} \psi(u) $$

$$ S_{\mu}=\partial_{\mu} \sigma(\vec{u}), \quad \sigma(\vec{u})=\frac{1}{2} \log \vec{u}^{2} $$

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

$$ \sqrt{g}=16 \vec{u}^{4} $$

$$ \left\langle\psi_{2} \mid \psi_{1}\right\rangle_{\mathrm{phys}}=\int d^{4} u \sqrt{g} e^{-2 \sigma} \psi_{2}^{*}(\vec{u}) \psi_{1}(\vec{u})=\int d^{4} u 16 \vec{u}^{2} \psi_{2}^{*}(\vec{u}) \psi_{1}(\vec{u}) $$

$$ \left\langle\psi_{2} \mid \psi_{1}\right\rangle_{\mathrm{phys}}=\int d^{2} u \sqrt{g} \psi_{2}^{*}(\mathbf{u}) \psi_{1}(\mathbf{u})=\int d^{2} u 4 \mathbf{u}^{2} \psi_{2}^{*}(\mathbf{u}) \psi_{1}(\mathbf{u}) $$

$$ \left[-\frac{1}{2 \mu} \hbar^{2} \partial_{\mu}^{2}+\frac{\mu}{2} \omega^{2}\left(u^{\mu}\right)^{2}\right] \psi\left(u^{\mu}\right)=\mathcal{E} \psi\left(u^{\mu}\right) $$

$$ \mathcal{E}_{N}=\hbar \omega\left(N+D_{u} / 2\right) $$

$$ N=\sum_{i=1}^{D_{u}} n_{i} $$

$$ \mathcal{E}_{n}=\hbar \omega 2\left(n+D_{u} / 4-1\right), \quad n=1,2,3, \ldots $$

$$ \mathcal{E}_{n}=e^{2} $$

$$ \omega=\omega_{n} \equiv \frac{e^{2}}{2\left(n+D_{u} / 4-1\right)}, \quad n=1,2,3 \ldots $$

$$ E_{n}=-2 M \omega_{n}^{2}=-\frac{M e^{4}}{\hbar^{2}} \frac{1}{2 n^{2}}=-M c^{2} \frac{\alpha^{2}}{2 n^{2}} $$

$$ \begin{align*} -i r \partial_{x^{4}} \psi(\mathbf{x}) & =-i r e_{4}{ }^{\mu} \partial_{\mu} \psi\left(u^{\mu}\right)=-i \frac{1}{2}\left[\left(u^{2} \partial_{1}-u^{1} \partial_{2}\right)+\left(u^{4} \partial_{3}-u^{3} \partial_{4}\right)\right] \psi\left(u^{\mu}\right) \\ & =-i \partial_{\gamma} \psi\left(u^{\mu}\right)=0 \end{align*} $$

$$ \frac{1}{2}\left[\bar{z} \partial_{\bar{z}}-z \partial_{z}\right] \psi\left(z, z^{*}\right)=0 $$

$$ \hat{h} \psi\left(u^{\mu}\right) \equiv \frac{1}{2}\left[-\partial_{\mu}^{2}+16 \omega^{2}\left(u^{\mu}\right)^{2}\right] \psi\left(u^{\mu}\right)=4 \psi\left(u^{\mu}\right) . $$

$$ \hat{h}^{s}=\frac{1}{2}\left[-\partial_{\mu}^{2}+4\left(u^{\mu}\right)^{2}\right], $$

$$ \hat{h}=4 \omega e^{i \vartheta \hat{D}} \hat{h}^{s} e^{-i \vartheta \hat{D}} $$

$$ \hat{D} \equiv-\frac{1}{2} i u^{\mu} \partial_{\mu}, $$

$$ \vartheta=\log (2 \omega) $$

$$ \psi\left(u^{\mu}\right)=e^{i \vartheta \hat{D}} \psi^{s}\left(u^{\mu}\right)=\psi^{s}\left(\sqrt{2 \omega} u^{\mu}\right) $$

$$ \psi_{n}\left(u^{\mu}\right)=\psi_{n}^{s}\left(u^{\mu} / \sqrt{n}\right) $$

$$ c=i \sigma^{2}=\left(\begin{array}{rr} 0 & 1 \\ -1 & 0 \end{array}\right), $$

$$ \begin{array}{ll} \hat{a}_{1}^{\dagger} \equiv-\frac{1}{\sqrt{2}}\left(-\partial_{z_{2}^{*}}+z_{2}\right), & \hat{b}_{1}^{\dagger} \equiv \frac{1}{\sqrt{2}}\left(-\partial_{z_{1}}+z_{1}^{*}\right) \\ \hat{a}_{2}^{\dagger} \equiv \frac{1}{\sqrt{2}}\left(-\partial_{z_{1}^{*}}+z_{1}\right), & \hat{b}_{2}^{\dagger} \equiv \frac{1}{\sqrt{2}}\left(-\partial_{z_{2}}+z_{2}^{*}\right) \end{array} $$

$$ \begin{array}{ll} \hat{a}_{1} \equiv-\frac{1}{\sqrt{2}}\left(\partial_{z_{2}}+z_{2}^{*}\right), & \hat{b}_{1} \equiv \frac{1}{\sqrt{2}}\left(\partial_{z_{1}^{*}}+z_{1}\right) \\ \hat{a}_{2} \equiv \frac{1}{\sqrt{2}}\left(\partial_{z_{1}}+z_{1}^{*}\right), & \hat{b}_{2} \equiv \frac{1}{\sqrt{2}}\left(\partial_{z_{2}^{*}}+z_{2}\right) \end{array} $$

$$ \hat{h}^{s}=2\left(\hat{a}^{\dagger} \hat{a}+\hat{b}^{\dagger} \hat{b}+2\right) $$

$$ \left\langle z, z^{*} \mid 0\right\rangle=\psi_{s, 0000}\left(z, z^{*}\right)=\frac{1}{\sqrt{\pi}} e^{-z_{1} z_{1}^{*}-z_{2} z_{2}^{*}}=\frac{1}{\sqrt{\pi}} e^{-\left(u^{\mu}\right)^{2}} . $$

$$ \left|n_{1}^{a}, n_{2}^{a}, n_{1}^{b}, n_{2}^{b}\right\rangle=N_{n_{1}^{a}, n_{2}^{a}, n_{1}^{b}, n_{2}^{b}} \hat{a}_{1}^{\dagger n_{1}^{a}} \hat{a}_{2}^{\dagger n_{2}^{a}} \hat{b}_{1}^{\dagger n_{1}^{b}} \hat{b}_{2}^{\dagger n_{2}^{b}}|0\rangle $$

$$ N_{n_{1}^{a}, n_{2}^{a}, n_{1}^{b}, n_{2}^{b}}=\frac{1}{\sqrt{n_{1}^{a}!n_{2}^{a}!n_{1}^{b}!n_{2}^{b}!}} $$

$$ 2\left(n_{1}^{a}+n_{2}^{a}+n_{1}^{b}+n_{2}^{b}+2\right)=2(N+2)=4 n . $$

$$ \hat{L}_{05}=-\frac{1}{2}\left(\hat{a}^{\dagger} \hat{a}-\hat{b}^{\dagger} \hat{b}\right) \psi^{s}=0 $$

$$ \hat{L}_{i}^{a} \equiv \frac{1}{2} \hat{a}^{\dagger} \sigma_{i} \hat{a}, \quad \hat{L}_{i}^{a} \equiv \frac{1}{2} \hat{b}^{\dagger} \sigma_{i} \hat{b} $$

$$ \begin{array}{ll} l^{a}=\left(n_{1}^{a}+n_{2}^{a}\right) / 2, & m^{a}=\left(n_{1}^{a}-n_{2}^{a}\right) / 2 \\ l^{b}=\left(n_{1}^{b}+n_{2}^{b}\right) / 2, & m^{b}=\left(n_{1}^{b}-n_{2}^{b}\right) / 2 \end{array} $$

$$ \begin{array}{ll} n_{1}^{a} \equiv n_{1}+m, \quad n_{2}^{a} \equiv n_{2}, \quad n_{1}^{b}=n_{2}+m, \quad n_{2}^{b}=n_{1}, & \text { for } m \geq 0, \\ n_{1}^{a} \equiv n_{1}, \quad n_{2}^{a} \equiv n_{2}-m, \quad n_{1}^{b}=n_{2}, \quad n_{2}^{b}=n_{1}-m, & \text { for } m \leq 0, \end{array} $$

$$ \begin{align*} \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=-i & \frac{M \kappa}{\pi \hbar} \int_{0}^{1} d \varrho \frac{\varrho^{-\nu}}{(1-\varrho)^{2}} I_{0}\left(2 \kappa \frac{2 \sqrt{\varrho}}{1-\varrho} \sqrt{\left(r_{b} r_{a}+\mathbf{x}_{b} \mathbf{x}_{a}\right) / 2}\right) \\ & \times \exp \left\{-\kappa \frac{1+\varrho}{1-\varrho}\left(r_{b}+r_{a}\right)\right\} \end{align*} $$

$$ z \equiv 2 \kappa \sqrt{r_{b} r_{a}} \frac{2 \sqrt{\varrho}}{1-\varrho} $$

$$ \left(\frac{1}{2} k z\right)^{\mu-\nu} I_{\nu}(k z)=k^{\mu} \sum_{l=0}^{\infty} \frac{1}{l!} \frac{\Gamma(l+\mu)}{\Gamma(1+\nu)}(2 l+\mu) F\left(-l, l+\mu ; 1+\nu ; k^{2}\right)(-)^{l} I_{2 l+\mu}(z) $$

$$ I_{2 q}(z \cos (\theta / 2))=\frac{2}{z} \sum_{l=|q|}^{\infty}(2 l+1) d_{q q}^{l}(\theta) I_{2 l+1}(z) $$

$$ I_{0}(z \cos (\theta / 2))=\frac{2}{z} \sum_{l=0}^{\infty}(2 l+1) P_{l}(\cos \theta) I_{2 l+1}(z) $$

$$ \varrho=e^{-2 y}, \quad z=2 \kappa \sqrt{r_{b} r_{a}} \frac{1}{\sinh y} $$

$$ \begin{align*} \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E} & =\frac{1}{r_{b} r_{a}} \sum_{l=0}^{\infty}\left(r_{b} \mid r_{a}\right)_{E, l} \frac{2 l+1}{4 \pi} P_{l}(\cos \theta) \\ & =\frac{1}{r_{b} r_{a}} \sum_{l=0}^{\infty}\left(r_{b} \mid r_{a}\right)_{E, l} \sum_{m=-l}^{l} Y_{l m}\left(\hat{\mathbf{x}}_{b}\right) Y_{l m}^{*}\left(\hat{\mathbf{x}}_{a}\right) \end{align*} $$

$$ \begin{align*} \left(r_{b} \mid r_{a}\right)_{E, l}= & -i \sqrt{r_{b} r_{a}} \frac{2 M}{\hbar} \int_{0}^{\infty} d y \frac{1}{\sinh y} e^{2 \nu y} \\ & \times \exp \left[-\kappa \operatorname{coth} y\left(r_{b}+r_{a}\right)\right] I_{2 l+1}\left(2 \kappa \sqrt{r_{b} r_{a}} \frac{1}{\sinh y}\right) \end{align*} $$

$$ \left(r_{b} \mid r_{a}\right)_{E, l}=-i \frac{M}{\hbar \kappa} \frac{\Gamma(-\nu+l+1)}{(2 l+1)!} W_{\nu, l+1 / 2}\left(2 \kappa r_{b}\right) M_{\nu, l+1 / 2}\left(2 \kappa r_{a}\right) . $$

$$ E_{n}=-\frac{M e^{4}}{\hbar^{2}} \frac{1}{2 n^{2}}=-M c^{2} \frac{\alpha^{2}}{2 n^{2}} $$

$$ \kappa=\frac{1}{a_{H}} \frac{1}{\nu}, $$

$$ a_{H} \equiv \frac{\hbar^{2}}{M e^{2}} $$

$$ \begin{align*} \Gamma(-\nu+l+1) & \approx-\frac{(-)^{n_{r}}}{n_{r}!} \frac{1}{\nu-n} \\ \frac{1}{\nu-n} & \approx \frac{2}{n} \frac{\hbar^{2} \kappa^{2}}{2 M} \frac{1}{E-E_{n}} \\ \kappa & \approx \frac{1}{a_{H}} \frac{1}{n} \end{align*} $$

$$ -i \Gamma(-\nu+l+1) \frac{M}{\hbar \kappa} \approx \frac{(-)^{n_{r}}}{n^{2} n_{r}!} \frac{1}{a_{H}} \frac{i \hbar}{E-E_{n}} . $$

$$ \left(r_{b} \mid r_{a}\right)_{E, l}=\sum_{n=l+1}^{\infty} \frac{i \hbar}{E-E_{n}} R_{n l}\left(r_{b}\right) R_{n l}\left(r_{a}\right)+\ldots $$

$$ \psi_{n l m}(\mathbf{x})=\frac{1}{r} R_{n, l}(r) Y_{l m}(\hat{\mathbf{x}}) $$

$$ \begin{align*} R_{n l}(r) & =\frac{1}{a_{H}^{1 / 2} n} \frac{1}{(2 l+1)!} \sqrt{\frac{(n+l)!}{(n-l-1)!}} \\ & \times\left(2 r / n a_{H}\right)^{l+1} e^{-r / n a_{H}} M\left(-n+l+1,2 l+2,2 r / n a_{H}\right) \\ & =\frac{1}{a_{H}^{1 / 2} n} \sqrt{\frac{(n-l-1)!}{(n+l)!}} e^{-r / n a_{H}}\left(2 r / n a_{H}\right)^{l+1} L_{n-l-1}^{2 l+1}\left(2 r / n a_{H}\right) \end{align*} $$

$$ z L_{n}^{\mu}(z)=(2 n+\mu+1) L_{n}^{\mu}(z)-(n+\mu) L_{n-1}^{\mu}(z)-(n+1) L_{n+1}^{\mu}(z) $$

$$ \begin{align*} & \operatorname{disc}\left(r_{b} \mid r_{a}\right)_{E, l}=\left(r_{b} \mid r_{a}\right)_{E+i \eta, l}-\left(r_{b} \mid r_{a}\right)_{E-i \eta, l} \\ & \quad=\frac{M}{\hbar k}\left[\frac{\Gamma\left(-i \nu^{\prime}+l+1\right)}{(2 l+1)!} W_{i \nu^{\prime}, l+1 / 2}\left(-2 i k r_{b}\right) M_{i \nu^{\prime}, l+1 / 2}\left(-2 i k r_{a}\right)+\left(\nu^{\prime} \rightarrow-\nu^{\prime}\right)\right] . \end{align*} $$

$$ M_{i \nu^{\prime}, l+1 / 2}(-2 i k r)=e^{-i \pi(l+1)} M_{-i \nu^{\prime}, l+1 / 2}(2 i k r) $$

$$ \operatorname{disc}\left(r_{b} \mid r_{a}\right)_{E, l}=\frac{M}{\hbar k} \frac{\left|\Gamma\left(-i \nu^{\prime}+l+1\right)\right|^{2}}{(2 l+1)!^{2}} e^{\pi \nu^{\prime}} M_{i \nu^{\prime}, l+1 / 2}\left(-2 i k r_{b}\right) M_{-i \nu^{\prime}, l+1 / 2}\left(2 i k r_{a}\right) . $$

$$ \int_{0}^{\infty} \frac{d E}{2 \pi \hbar} \operatorname{disc}\left(r_{b} \mid r_{a}\right)_{E, l}+\sum_{n=l+1}^{\infty} R_{n l}\left(r_{b}\right) R_{n l}^{*}\left(r_{a}\right)=\delta\left(r_{b}-r_{a}\right) $$

$$ \int_{-\infty}^{\infty} d k R_{k l}\left(r_{b}\right) R_{k l}^{*}\left(r_{a}\right) $$

$$ R_{k l}(r)=\sqrt{\frac{1}{2 \pi}} \frac{\left|\Gamma\left(-i \nu^{\prime}+l+1\right)\right|}{(2 l+1)!} e^{\pi \nu^{\prime} / 2} M_{i \nu^{\prime}, l+1 / 2}(-2 i k r) . $$

$$ M_{\lambda, \mu}(z)=z^{\mu+1 / 2} e^{-z / 2} M(\mu-\lambda+1 / 2,2 \mu+1, z) $$

$$ R_{k l}(r)=\sqrt{\frac{1}{2 \pi}} \frac{\left|\Gamma\left(-i \nu^{\prime}+l+1\right)\right|}{(2 l+1)!} e^{\pi \nu^{\prime} / 2} e^{i k r}(-2 i k r)^{l+1} M\left(-i \nu^{\prime}+l+1,2 l+2,-2 i k r\right) . $$

$$ \left(\partial_{\mu} \partial_{\nu}-\partial_{\nu} \partial_{\mu}\right) e_{\lambda}^{i}(\vec{u})=0 $$

$$ \begin{align*} \bar{R}_{\nu \lambda} & =\bar{R}_{\mu \nu \lambda}^{\mu} \\ & =-\frac{3}{2 \vec{u}^{6}}\left(\delta_{\nu \lambda} \vec{u}^{2}-\vec{u}_{\nu} \vec{u}_{\lambda}\right) \end{align*} $$

$$ \bar{R}=g^{\nu \lambda} \bar{R}_{\nu \lambda}=-\frac{9}{2 \vec{u}^{4}} $$

$$ g_{\mu \nu}(q)=\Omega^{2}(q) \delta_{\mu \nu} $$

$$ g_{\mu \nu}(q) \rightarrow \Omega^{2}(q) g_{\mu \nu}(q) $$

$$ \bar{\Gamma}_{\mu \nu}{ }^{\lambda} \rightarrow \bar{\Gamma}_{\mu \nu}{ }^{\lambda}+\Omega_{, \mu} \delta_{\nu}{ }^{\lambda}+\Omega_{, \nu} \delta_{\mu}{ }^{\lambda}-g_{\mu \nu} g^{\lambda \kappa} \Omega_{, \kappa} $$

$$ \begin{align*} \bar{R}_{\mu \nu} \rightarrow & \Omega^{-2} \bar{R}_{\mu \nu}-(D-2)\left(\Omega^{-3} \Omega_{; \mu \nu}-2 \Omega^{-4} \Omega_{, \mu} \Omega_{, \nu}\right) \\ & -g_{\mu \nu} g^{\lambda \kappa}\left[(D-3) \Omega^{-4} \Omega_{, \lambda} \Omega_{, \kappa}+\Omega^{-3} \Omega_{; \lambda \kappa}\right] \\ = & \Omega^{-2} \bar{R}_{\mu \nu}+(D-2) \Omega^{-1}\left(\Omega^{-1}\right)_{; \mu \nu}-g_{\mu \nu}(D-2)^{-1} \Omega^{-D}\left(\Omega^{D-2}\right)_{; \lambda \kappa} g^{\lambda \kappa} \end{align*} $$

$$ \Omega_{; \mu \nu}=D_{\nu} \Omega_{, \mu}=\Omega_{\mu \nu}-\bar{\Gamma}_{\mu \nu}^{\lambda} \Omega_{, \lambda} $$

$$ \bar{R} \rightarrow \bar{R}^{\Omega}=\Omega^{-2}\left[\bar{R}-2(D-1) \Omega^{-1} \Omega_{; \mu \nu} g^{\mu \nu}-(D-1)(D-4) \Omega^{-2} \Omega_{, \mu} \Omega_{, \nu} g^{\mu \nu}\right] $$

$$ \Omega_{, \mu}=2 \frac{u^{\mu}}{|\vec{u}|}, \quad \Omega_{, \mu \nu}=\frac{2}{|\vec{u}|^{3}}\left(\delta^{\mu \nu} \vec{u}^{2}-u^{\mu} u^{\nu}\right) $$

$$ \begin{align*} \bar{R}_{\mu \nu} & =-3(D-2) \frac{1}{4 \vec{u}^{6}}\left(\delta_{\mu \nu} \vec{u}^{2}-\vec{u}_{\mu} \vec{u}_{\nu}\right) \\ \bar{R} & =-3(D-1)(D-2) \frac{1}{4 \vec{u}^{4}} \end{align*} $$

$$ S_{\mu}(q) \equiv S_{\mu \nu^{\lambda}}(q)=\frac{1}{2 \Omega^{2}(q)} \partial_{\mu} \Omega^{2}(q) $$

$$ S_{\mu \nu}{ }^{\lambda}(q)=\frac{1}{2}\left[\delta_{\mu}{ }^{\lambda} \partial_{\nu} s(q)-\delta_{\nu}{ }^{\lambda} \partial_{\mu} s(q)\right] $$

$$ \phi(q) \rightarrow \Omega^{1-D / 2}(q) \phi(q) $$

$$ \Delta^{\Omega}=\Omega^{-2}\left[\Delta-\frac{1}{2}(D-2) \Omega^{-1} \Omega_{; \mu \nu} g^{\mu \nu}-\frac{1}{4}(D-2)(D-4) \Omega^{-2} \Omega_{, \mu} \Omega_{, \nu} g^{\mu \nu}\right] . $$

$$ \Delta-\frac{1}{4} \frac{D-2}{D-1} \bar{R} $$

$$ \left(\Delta-\frac{1}{4} \frac{D-2}{D-1} \bar{R}\right) \phi(q) \longrightarrow \Omega^{-1-D / 2}\left(\Delta-\frac{1}{4} \frac{D-2}{D-1} \bar{R}\right) \phi(q) . $$

$$ \hat{\mathbf{M}}=\frac{M}{\hat{p}_{E}}\left[\frac{1}{2 M}(\hat{\mathbf{p}} \times \hat{\mathbf{L}}-\hat{\mathbf{L}} \times \hat{\mathbf{p}})-e^{2} \frac{\mathbf{r}}{r}\right] $$

$$ \left[\hat{L}_{i}, \hat{L}_{j}\right]=i \epsilon_{i j k} \hat{L}_{k}, \quad\left[\hat{L}_{i}, \hat{M}_{j}\right]=i \epsilon_{i j k} \hat{M}_{k}, \quad\left[\hat{M}_{i}, \hat{M}_{j}\right]=i \epsilon_{i j k} \hat{M}_{k} $$

$$ \hat{\mathbf{M}} \cdot \hat{\mathbf{L}}=\hat{\mathbf{L}} \cdot \hat{\mathbf{M}}=0 $$

$$ \hat{\mathbf{L}}^{2}+\hat{\mathbf{M}}^{2}+\hbar^{2}=-e^{4} \frac{M}{2 \hat{H}}=e^{4} \frac{M^{2}}{\hat{p}_{E}^{2}} $$

$$ \hat{\mathbf{J}}^{(1,2)} \equiv \frac{1}{2}(\hat{\mathbf{L}} \pm \hat{\mathbf{M}}) $$

$$ \hat{\mathbf{L}}^{2}+\hat{\mathbf{M}}^{2}+\hbar^{2}=4(\hat{\mathbf{L}} \pm \hat{\mathbf{M}})^{2}+\hbar^{2}=[4 j(j+1)+1] \hbar^{2}=e^{4} \frac{M^{2}}{\hat{p}_{E}^{2}}=-\hbar^{2} \alpha^{2} \frac{M c^{2}}{2 \hat{H}} $$

$$ \left|n m_{1} m_{2}\right\rangle=\left|j m_{1}\right\rangle^{(1)} \otimes\left|j m_{2}\right\rangle^{(2)} $$

$$ |n l m\rangle=\sum_{m_{1}, m_{2}=-j, \ldots, j}\left|j m_{1}\right\rangle^{(1)} \otimes\left|j m_{2}\right\rangle^{(2)}\left(j, m_{1} ; j, m_{2} \mid l, m\right) $$

$$ \hat{R}=\frac{i}{\hat{f}(E-\hat{H})} \hat{f} $$

$$ \begin{align*} \left\langle\mathbf{p}_{b}\right| \hat{\mathcal{U}}_{E}(S)\left|\mathbf{p}_{a}\right\rangle & =\int \mathcal{D}^{3} x(s) \int \frac{\mathcal{D}^{3} p(s)}{2 \pi \hbar} \\ & \times \exp \left\{i \int_{0}^{S} d s\left[-\mathbf{p}^{\prime} \cdot \mathbf{x}-f\left(\frac{\mathbf{p}^{2}}{2}-E\right)+f \frac{\alpha}{r}\right]\right\} f_{a} \end{align*} $$

$$ \left(\mathbf{p}_{b} \mid \mathbf{p}_{a}\right)_{E}^{f}=\int_{0}^{\infty} d S\left\langle\mathbf{p}_{b}\right| \hat{\mathcal{U}}_{E}(S)\left|\mathbf{p}_{a}\right\rangle $$

$$ \left(\mathbf{p}_{b} \mid \mathbf{p}_{a}\right)_{E}=\int \mathcal{D} f \Phi[f]\left(\mathbf{p}_{b} \mid \mathbf{p}_{a}\right)_{E}^{f} $$

$$ \Phi[f]=\prod_{s} \frac{1}{r} \exp \left\{-\frac{i}{2 r^{2}}\left[f-r^{2}\left(\frac{\mathbf{p}^{2}}{2}-E\right)\right]^{2}\right\} $$

$$ \mathcal{A}[\mathbf{p}, \mathbf{x}, f]=\int_{0}^{S} d s\left[-\mathbf{p}^{\prime} \cdot \mathbf{x}-\frac{r^{2}}{2}\left(\frac{\mathbf{p}^{2}}{2}-E\right)^{2}-\frac{1}{2 r^{2}} f^{2}+\frac{f}{r} \alpha\right] $$

$$ \mathcal{A}[\mathbf{p}]=\frac{1}{2} \int_{0}^{S} d s\left[\frac{4 \mathbf{p}^{\prime 2}}{\left(\mathbf{p}^{2}+p_{E}^{2}\right)^{2}}+\alpha^{2}\right] $$

$$ \boldsymbol{\pi} \equiv \frac{2 p_{E} \mathbf{p}}{\mathbf{p}^{2}+p_{E}^{2}}, \quad \pi_{4} \equiv \frac{\mathbf{p}^{2}-p_{E}^{2}}{\mathbf{p}^{2}+p_{E}^{2}} $$

$$ \mathcal{A}[\vec{\pi}]=\frac{1}{2} \int_{0}^{S} d s\left(\frac{1}{p_{E}^{2}} \vec{\pi}^{2}+\alpha^{2}\right) $$

$$ \left(\vec{\pi}_{b} S \mid \vec{\pi}_{a} 0\right)=\int \frac{\mathcal{D} \vec{\pi}}{(2 \pi)^{3 / 2} p_{E}^{3}} e^{i \mathcal{A}[\vec{\pi}]} $$

$$ \left(\vec{\pi}_{b} S \mid \vec{\pi}_{a} 0\right)=e^{-i S p_{E}^{2} / 2} \int \frac{\mathcal{D} \vec{\pi}}{(2 \pi)^{3 / 2} p_{E}^{3}} e^{i \mathcal{A}[\vec{\pi}]} $$

$$ Y_{n l m}(\vec{\pi})=\sum_{m_{1}, m_{2}=-j, \ldots, j} Y_{2 j, m_{1}, m_{2}}(\vec{\pi})\left(j, m_{1} ; j, m_{2} \mid l, m\right) $$

$$ \int d \vec{\pi} Y_{n^{\prime} l^{\prime} m^{\prime}}^{*}(\vec{\pi}) Y_{n l m}(\vec{\pi})=\delta_{n n^{\prime}} \delta_{l l^{\prime}} \delta_{m m^{\prime}}, \quad \sum_{n, l, m} Y_{n l m}\left(\vec{\pi}^{\prime}\right) Y_{n l m}(\vec{\pi})=\delta^{(4)}\left(\vec{\pi}^{\prime}-\vec{\pi}\right) $$

$$ \sum_{l, m} Y_{n l m}\left(\vec{\pi}^{\prime}\right) Y_{n l m}(\vec{\pi})=\frac{n^{2}}{2 \pi^{2}} P_{n}(\cos \vartheta), \quad P_{n}(\cos \vartheta)=\frac{\sin n \vartheta}{n \sin \vartheta} $$

$$ \cos \vartheta=\vec{\pi}_{b} \vec{\pi}_{a}=\frac{\left(\mathbf{p}_{b}^{2}-p_{E}^{2}\right)\left(\mathbf{p}_{a}^{2}-p_{E}^{2}\right)+4 p_{E}^{2} \mathbf{p}_{b} \cdot \mathbf{p}_{a}}{\left(\mathbf{p}_{b}^{2}+p_{E}^{2}\right)\left(\mathbf{p}_{a}^{2}+p_{E}^{2}\right)} $$

$$ \left(\vec{\pi}_{b} S \mid \vec{\pi}_{a} 0\right)=(2 \pi)^{3 / 2} p_{E}^{3} \sum_{n=1}^{\infty} \frac{n^{2}}{2 \pi^{2}} P_{n}(\cos \vartheta) \exp \left\{\left[-i\left(p_{E}^{2} n^{2}-\alpha^{2}\right)\right] \frac{S}{2}\right\} . $$

$$ \left(\vec{\pi}_{b} \mid \vec{\pi}_{a}\right)_{0}=(2 \pi)^{3 / 2} p_{E}^{3} \sum_{n=1}^{\infty} \frac{n^{2}}{2 \pi^{2}} P_{n}(\cos \vartheta) \frac{2 i}{2 E n^{2}+\alpha^{2}} $$

$$ E_{n}=-\frac{\alpha^{2}}{2 n^{2}}, \quad n=1,2,3, \ldots $$

$$ \mathcal{A}[\mathbf{p}]=\frac{1}{2} \int_{0}^{S} d s\left[\frac{1}{h} \frac{4 \mathbf{p}^{\prime 2}}{\left(\mathbf{p}^{2}+p_{E}^{2}\right)^{2}}+\alpha^{2} h\right] $$

$$ \mathcal{A}[\mathbf{p}]=2 \alpha \int_{\tau_{a}}^{\tau_{b}} d \tau \sqrt{\frac{\dot{\mathbf{p}}^{2}}{\left(\mathbf{p}^{2}+p_{E}^{2}\right)^{2}}} $$

$$ S\left(\mathbf{p}_{b}, \mathbf{p}_{a} ; E\right)=-\int_{\mathbf{p}_{a}}^{\mathbf{p}_{b}} d \tau \dot{\mathbf{p}} \cdot \mathbf{x} $$