Kleinert · 제2장 경로적분 — 기본 성질

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\left\langle x_{b}\right| \hat{U}\left(t_{b}, t_{a}\right)\left|x_{a}\right\rangle, \quad t_{b}>t_{a} . $$

$$ e^{\hat{A}} e^{\hat{B}}=e^{\hat{C}} $$

$$ \hat{C}=\hat{B}+\int_{0}^{1} d t g\left(e^{\operatorname{ad} A t} e^{\operatorname{ad} B}\right)[\hat{A}] $$

$$ \int_{-\infty}^{\infty} d x_{n}\left|x_{n}\right\rangle\left\langle x_{n}\right|=1, \quad n=1, \ldots, N $$

$$ g(z) \equiv \frac{\log z}{z-1}=\sum_{n=0}^{\infty} \frac{(1-z)^{n}}{n+1} $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\prod_{n=1}^{N}\left[\int_{-\infty}^{\infty} d x_{n}\right] \prod_{n=1}^{N+1}\left(x_{n} t_{n} \mid x_{n-1} t_{n-1}\right) $$

$$ \operatorname{ad} B[\hat{A}] \equiv[\hat{B}, \hat{A}] $$

$$ \left(x_{n} t_{n} \mid x_{n-1} t_{n-1}\right)=\left\langle x_{n}\right| e^{-i \epsilon \hat{H}\left(t_{n}\right) / \hbar}\left|x_{n-1}\right\rangle, $$

$$ \begin{align*} \hat{C}=\hat{B}+\hat{A} & +\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n+1} \sum_{p_{i}, q_{i} ; p_{i}+q_{i} \geq 1} \frac{1}{1+\sum_{i=1}^{n} p_{i}} \\ & \times \frac{(\operatorname{ad} A)^{p_{1}}}{p_{1}!} \frac{(\operatorname{ad} B)^{q_{1}}}{q_{1}!} \cdots \frac{(\operatorname{ad} A)^{p_{n}}}{p_{n}!} \frac{(\operatorname{ad} B)^{q_{n}}}{q_{n}!}[\hat{A}] \end{align*} $$

$$ \hat{H}(t) \equiv H(\hat{p}, \hat{x}, t) $$

$$ \begin{align*} \hat{C}= & \hat{B}+\hat{A}-\frac{1}{2}\left[\frac{1}{2} \operatorname{ad} A+\operatorname{ad} B+\frac{1}{6}(\operatorname{ad} A)^{2}+\frac{1}{2} \operatorname{ad} A \operatorname{ad} B+\frac{1}{2}(\operatorname{ad} B)^{2}+\ldots\right][\hat{A}] \\ & +\frac{1}{3}\left[\frac{1}{3}(\operatorname{ad} A)^{2}+\frac{1}{2} \operatorname{ad} A \operatorname{ad} B+\frac{1}{2} \operatorname{ad} B \operatorname{ad} A+(\operatorname{ad} B)^{2}+\ldots\right][\hat{A}] \\ = & \hat{A}+\hat{B}+\frac{1}{2}[\hat{A}, \hat{B}]+\frac{1}{12}([\hat{A},[\hat{A}, \hat{B}]]+[\hat{B},[\hat{B}, \hat{A}]])+\frac{1}{24}[\hat{A},[[\hat{A}, \hat{B}], \hat{B}]] \ldots \end{align*} $$

$$ H(p, x, t)=T(p, t)+V(x, t) $$

$$ e^{\hat{A}+\hat{B}}=e^{\hat{A}} e^{\hat{B}} e^{\hat{Z}_{2}} e^{\hat{Z}_{3}} e^{\hat{Z}_{4}} \cdots $$

$$ e^{-i \epsilon \hat{H} / \hbar}=e^{-i \epsilon(\hat{T}+\hat{V}) / \hbar} $$

$$ \begin{align*} \hat{Z}_{2} & =\frac{1}{2}[\hat{B}, \hat{A}] \\ \hat{Z}_{3} & \left.=-\frac{1}{3}[\hat{B},[\hat{B}, \hat{A}]]-\frac{1}{6}[\hat{A},[\hat{B}, \hat{A}]]\right) \\ \hat{Z}_{4} & =\frac{1}{8}([[[\hat{B}, \hat{A}], \hat{B}], \hat{B}]+[[[\hat{B}, \hat{A}], \hat{A}], \hat{B}])+\frac{1}{24}[[[\hat{B}, \hat{A}], \hat{A}], \hat{A}] \\ & \vdots \end{align*} $$

$$ e^{-i \epsilon(\hat{T}+\hat{V}) / \hbar}=e^{-i \epsilon \hat{V} / \hbar} e^{-i \epsilon \hat{T} / \hbar} e^{-i \epsilon^{2} \hat{X} / \hbar^{2}} $$

$$ \hat{X} \equiv \frac{i}{2}[\hat{V}, \hat{T}]-\frac{\epsilon}{\hbar}\left(\frac{1}{6}[\hat{V},[\hat{V}, \hat{T}]]-\frac{1}{3}[[\hat{V}, \hat{T}], \hat{T}]\right)+\mathcal{O}\left(\epsilon^{2}\right) $$

$$ \begin{gather*} \left\langle x_{n}\right| e^{-i \epsilon H\left(\hat{p}, \hat{x}, t_{n}\right) / \hbar}\left|x_{n-1}\right\rangle \approx \int_{-\infty}^{\infty} d x\left\langle x_{n}\right| e^{-i \epsilon V\left(\hat{x}, t_{n}\right) / \hbar}|x\rangle\langle x| e^{-i \epsilon T\left(\hat{p}, t_{n}\right) / \hbar}\left|x_{n-1}\right\rangle \\ =\int_{-\infty}^{\infty} d x\left\langle x_{n}\right| e^{-i \epsilon V\left(\hat{x}, t_{n}\right) / \hbar}|x\rangle \int_{-\infty}^{\infty} \frac{d p_{n}}{2 \pi \hbar} e^{i p_{n}\left(x-x_{n-1}\right) / \hbar} e^{-i \epsilon T\left(p_{n}, t_{n}\right) / \hbar} \end{gather*} $$

$$ \hat{C}(t)=\log \left(e^{\hat{A} t} e^{\hat{B}}\right) $$

$$ \left\langle x_{n}\right| e^{-i \epsilon V\left(\hat{x}, t_{n}\right) / \hbar}|x\rangle=\delta\left(x_{n}-x\right) e^{-i \epsilon V\left(x_{n}, t_{n}\right) / \hbar} $$

$$ e^{\hat{C}(t)} \hat{M} e^{-\hat{C}(t)}=e^{\operatorname{ad} C(t)}[\hat{M}] $$

$$ \begin{align*} & \left\langle x_{n}\right| e^{-i \epsilon H\left(\hat{p}, \hat{x}, t_{n}\right) / \hbar}\left|x_{n-1}\right\rangle \approx \\ & \quad \int_{-\infty}^{\infty} \frac{d p_{n}}{2 \pi \hbar} \exp \left\{i p_{n}\left(x_{n}-x_{n-1}\right) / \hbar-i \epsilon\left[T\left(p_{n}, t_{n}\right)+V\left(x_{n}, t_{n}\right)\right] / \hbar\right\} \end{align*} $$

$$ e^{\operatorname{ad} C(t)}=e^{\operatorname{ad} A t} e^{\operatorname{ad} B} $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right) \approx \prod_{n=1}^{N}\left[\int_{-\infty}^{\infty} d x_{n}\right] \prod_{n=1}^{N+1}\left[\int_{-\infty}^{\infty} \frac{d p_{n}}{2 \pi \hbar}\right] \exp \left(\frac{i}{\hbar} \mathcal{A}^{N}\right) $$

$$ e^{\hat{C}(t)} \frac{d}{d t} e^{-\hat{C}(t)}=-\hat{A} $$

$$ \mathcal{A}^{N}=\sum_{n=1}^{N+1}\left[p_{n}\left(x_{n}-x_{n-1}\right)-\epsilon H\left(p_{n}, x_{n}, t_{n}\right)\right] $$

$$ e^{\hat{C}(t)} \frac{d}{d t} e^{-\hat{C}(t)}=-f(\operatorname{ad} C(t))[\dot{\hat{C}}(t)] $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right) \approx \prod_{n=1}^{N}\left[\int_{-\infty}^{\infty} d x_{n}\right] \prod_{n=1}^{N+1}\left[\int_{-\infty}^{\infty} \frac{d p_{n}}{2 \pi \hbar}\right] e^{i \sum_{n=1}^{N+1} p_{n}\left(x_{n}-x_{n-1}\right) / \hbar} $$

$$ f(z) \equiv \frac{e^{z}-1}{z} $$

$$ \begin{align*} \left(x_{b} t_{b} \mid x_{a} t_{a}\right) & \approx \prod_{n=1}^{N}\left[\int_{-\infty}^{\infty} d x_{n}\right] \prod_{n=1}^{N+1}\left\langle x_{n} \mid x_{n-1}\right\rangle=\prod_{n=1}^{N}\left[\int_{-\infty}^{\infty} d x_{n}\right] \prod_{n=1}^{N+1} \delta\left(x_{n}-x_{n-1}\right) \\ & =\delta\left(x_{b}-x_{a}\right) \end{align*} $$

$$ f(\operatorname{ad} C(t))[\dot{\hat{C}}(t)]=\hat{A} $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int \mathcal{D} x \int \frac{\mathcal{D} p}{2 \pi \hbar} e^{i \int d t p(t) \dot{x}(t) / \hbar}=\left\langle x_{b} \mid x_{a}\right\rangle=\delta\left(x_{b}-x_{a}\right) $$

$$ g\left(e^{z}\right) f(z) \equiv 1 $$

$$ [\hat{p}(t), x(t)]=-i \hbar $$

$$ \dot{\hat{C}}(t)=g\left(e^{\operatorname{ad} C(t)}\right) f(\operatorname{ad} C(t))[\dot{\hat{C}}(t)] . $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right) \approx \int_{-\infty}^{\infty} d x_{N}\left(x_{b} t_{b} \mid x_{N} t_{N}\right)\left(x_{N} t_{N} \mid x_{a} t_{a}\right) $$

$$ \dot{\hat{C}}(t)=g\left(e^{\operatorname{ad} C(t)}\right)[\hat{A}]=e^{\operatorname{ad} A t} e^{\operatorname{ad} B}[\hat{A}], $$

$$ \left(x_{b} t_{b} \mid x_{N} t_{N}\right) \approx \int_{-\infty}^{\infty} \frac{d p_{b}}{2 \pi \hbar} e^{(i / \hbar)\left[p_{b}\left(x_{b}-x_{N}\right)-\epsilon H\left(p_{b}, x_{b}, t_{b}\right)\right]} $$

$$ \hat{O}(s, t) \equiv e^{\hat{C}(t) s} \frac{d}{d t} e^{-\hat{C}(t) s} $$

$$ \left(x_{b} t_{b} \mid x_{N} t_{N}\right) \approx e^{-i \epsilon H\left(-i \hbar \partial_{x_{b}}, x_{b}, t_{b}\right) / \hbar} \int_{-\infty}^{\infty} \frac{d p_{b}}{2 \pi \hbar} e^{i p_{b}\left(x_{b}-x_{N}\right) / \hbar}=e^{-i \epsilon H\left(-i \hbar \partial_{x_{b}}, x_{b}, t_{b}\right) / \hbar} \delta\left(x_{b}-x_{N}\right) $$

$$ \begin{align*} \partial_{s} \hat{O}(s, t) & =e^{\hat{C}(t) s} \hat{C}(t) \frac{d}{d t}\left(e^{-\hat{C}(t) s}\right)-e^{\hat{C}(t) s} \frac{d}{d t}\left(\hat{C}(t) e^{-\hat{C}(t) s}\right) \\ & =-e^{\hat{C}(t) s} \dot{\hat{C}}(t) e^{-\hat{C}(t) s} \\ & =-e^{\operatorname{ad} C(t) s}[\dot{\hat{C}}(t)] \end{align*} $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right) \approx e^{-i \epsilon H\left(-i \hbar \partial_{x_{b}}, x_{b}, t_{b}\right) / \hbar}\left(x_{b} t_{b}-\epsilon \mid x_{a} t_{a}\right) $$

$$ \begin{align*} \hat{O}(s, t)-\hat{O}(0, t) & =\int_{0}^{s} d s^{\prime} \partial_{s^{\prime}} \hat{O}\left(s^{\prime}, t\right) \\ & =-\sum_{n=0}^{\infty} \frac{s^{n+1}}{(n+1)!}(\operatorname{ad} C(t))^{n}[\dot{\hat{C}}(t)] \end{align*} $$

$$ \frac{1}{\epsilon}\left[\left(x_{b} t_{b}+\epsilon \mid x_{a} t_{a}\right)-\left(x_{b} t_{b} \mid x_{a} t_{a}\right)\right] \approx \frac{1}{\epsilon}\left[e^{-i \epsilon H\left(-i \partial_{x_{b}}, x_{b}, t_{b}+\epsilon\right) / \hbar}-1\right]\left(x_{b} t_{b} \mid x_{a} t_{a}\right) $$

$$ \hat{O}(1, t)=e^{\hat{C}(t)} \frac{d}{d t} e^{-\hat{C}(t)}=-f(\operatorname{ad} C(t))[\dot{\hat{C}}(t)] $$

$$ i \hbar \partial_{t_{b}}\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=H\left(-i \hbar \partial_{x_{b}}, x_{b}, t_{b}\right)\left(x_{b} t_{b} \mid x_{a} t_{a}\right) $$

$$ \begin{align*} &+\frac{1}{4}\left(\frac{-i}{\hbar}\right)^{3}\{ \int_{t_{a}}^{t_{b}} d t_{3} \int_{t_{a}}^{t_{3}} d t_{2} \int_{t_{a}}^{t_{2}} d t_{1}\left[\hat{H}\left(t_{3}\right),\left[\hat{H}\left(t_{2}\right), \hat{H}\left(t_{1}\right)\right]\right] \\ &\left.+\frac{1}{3} \int_{t_{a}}^{t_{b}} d t_{3} \int_{t_{a}}^{t_{b}} d t_{2} \int_{t_{a}}^{t_{b}} d t_{1}\left[\left[\hat{H}\left(t_{3}\right), \hat{H}\left(t_{2}\right)\right], \hat{H}\left(t_{1}\right)\right]\right\}+\ldots \end{align*} $$

$$ e^{-i\left(t_{b}-t_{a}\right) \hat{H} / \hbar}=\lim _{N \rightarrow \infty}\left(e^{-i \epsilon \hat{V} / \hbar} e^{-i \epsilon \hat{T} / \hbar}\right)^{N+1} $$

$$ \left(x_{n} \epsilon \mid x_{n-1} 0\right)=\frac{1}{\sqrt{2 \pi \hbar i \epsilon / M}} \exp \left\{\frac{i}{\hbar} \frac{M}{2}\left[\frac{\left(x_{n}-x_{n-1}\right)^{2}}{\epsilon}-\epsilon \omega^{2} \frac{1}{2}\left(x_{n}^{2}+x_{n-1}^{2}\right)\right]\right\} . $$

$$ \mathcal{A}[p, x]=\int_{t_{a}}^{t_{b}} d t[p(t) \dot{x}(t)-H(p(t), x(t), t)] $$

$$ \left(x_{n} \epsilon \mid x_{n-1} 0\right)=\mathcal{N}_{1} \exp \left\{\frac{i}{\hbar}\left[a_{1}\left(x_{n}^{2}+x_{n-1}^{2}\right)-2 b_{1} x_{n} x_{n-1}\right]\right\}, $$

$$ \lim _{N \rightarrow \infty} \prod_{n=1}^{N}\left[\int_{-\infty}^{\infty} d x_{n}\right] \prod_{n=1}^{N+1}\left[\int_{-\infty}^{\infty} \frac{d p_{n}}{2 \pi \hbar}\right] \equiv \int_{x\left(t_{a}\right)=x_{a}}^{x\left(t_{b}\right)=x_{b}} \mathcal{D}^{\prime} x \int \frac{\mathcal{D} p}{2 \pi \hbar} $$

$$ \begin{align*} a_{1} & =\frac{M}{2 \epsilon}\left[1-2\left(\frac{\omega \epsilon}{2}\right)^{2}\right], \quad b_{1}=\frac{M}{2 \epsilon} \\ \mathcal{N}_{1} & =\frac{1}{\sqrt{2 \pi \hbar i \epsilon / M}} \end{align*} $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int_{x\left(t_{a}\right)=x_{a}}^{x\left(t_{b}\right)=x_{b}} \mathcal{D}^{\prime} x \int \frac{\mathcal{D} p}{2 \pi \hbar} e^{i \mathcal{A}[p, x] / \hbar} $$

$$ \left(x_{N} \epsilon \mid x_{N-1} 0\right)=\mathcal{N}_{N} \exp \left\{\frac{i}{\hbar}\left[a_{N}\left(x_{N}^{2}+x_{0}^{2}\right)-2 b_{N} x_{N} x_{0}\right]\right\} $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\sum_{\substack{\text { all histories } \\\left(x_{a}, t_{a}\right) \leadsto\left(x_{b}, t_{b}\right)}} e^{i \mathcal{A}[p, x] / \hbar} \text {, } $$

$$ \begin{align*} \mathcal{N}_{N+1} & =\mathcal{N}_{1} \mathcal{N}_{N} \sqrt{\frac{i \pi \hbar}{a_{N}+a_{1}}} \\ a_{N+1} & =\frac{a_{N}^{2}-b_{N}^{2}+a_{1} a_{N}}{a_{1}+a_{N}}=\frac{a_{1}^{2}-b_{1}^{2}+a_{1} a_{N}}{a_{1}+a_{N}} \\ b_{N+1} & =\frac{b_{1} b_{N}}{a_{1}+a_{N}} \end{align*} $$

$$ \left(p_{b} t_{b} \mid p_{a} t_{a}\right) \equiv\left\langle p_{b}\right| \hat{U}\left(t_{b}, t_{a}\right)\left|p_{a}\right\rangle $$

$$ \int_{-\infty}^{\infty} \frac{d p}{2 \pi \hbar}|p\rangle\langle p|=1 $$

$$ \left\langle p_{b} \mid p_{a}\right\rangle=2 \pi \hbar \delta\left(p_{b}-p_{a}\right) $$

$$ a_{N}^{2}=b_{N}^{2}+a_{1}^{2}-b_{1}^{2}, $$

$$ \begin{align*} \left(p_{b} t_{b} \mid p_{a} t_{a}\right) & \approx \prod_{n=1}^{N}\left[\int_{-\infty}^{\infty} \frac{d p_{n}}{2 \pi \hbar}\right] \prod_{n=0}^{N}\left[\int_{-\infty}^{\infty} d x_{n}\right] \\ & \times \exp \left\{\frac{i}{\hbar} \sum_{n=0}^{N}\left[-x_{n}\left(p_{n+1}-p_{n}\right)-\epsilon H\left(p_{n}, x_{n}, t_{n}\right)\right]\right\} \end{align*} $$

$$ b_{N+1}=\frac{b_{1} b_{N}}{a_{1}+\sqrt{b_{N}^{2}-\left(b_{1}^{2}-a_{1}^{2}\right)}} $$

$$ \begin{align*} \sum_{n=1}^{N+1} p_{n}\left(x_{n}-x_{n-1}\right)= & p_{N+1}\left(x_{N+1}-x_{N}\right)+p_{N}\left(x_{N}-x_{N-1}\right)+\ldots \\ & \ldots+p_{2}\left(x_{2}-x_{1}\right)+p_{1}\left(x_{1}-x_{0}\right) \\ = & p_{N+1} x_{N+1}-p_{1} x_{0} \\ & -\left(p_{N+1}-p_{N}\right) x_{N}-\left(p_{N}-p_{N-1}\right) x_{N-1}-\ldots-\left(p_{2}-p_{1}\right) x_{1} \\ = & p_{N+1} x_{N+1}-p_{1} x_{0}-\sum_{n=1}^{N}\left(p_{n+1}-p_{n}\right) x_{n} \end{align*} $$

$$ \frac{1}{b_{N+1}}=\frac{1}{b_{1}}\left(\frac{a_{1}}{b_{N}}+\sqrt{1-\frac{b_{1}^{2}-a_{1}^{2}}{b_{N}^{2}}}\right) . $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int \frac{d p_{b}}{2 \pi \hbar} e^{i p_{b} x_{b} / \hbar} \int \frac{d p_{a}}{2 \pi \hbar} e^{-i p_{a} x_{a} / \hbar}\left(p_{b} t_{b} \mid p_{a} t_{a}\right) $$

$$ a_{1}=\frac{M}{2 \epsilon} \cos \tilde{\omega} $$

$$ \left(p_{b} t_{b} \mid p_{a} t_{a}\right)=\int d x_{b} e^{-i p_{b} x_{b} / \hbar} \int d x_{a} e^{i p_{a} x_{a} / \hbar}\left(x_{b} t_{b} \mid x_{a} t_{a}\right) $$

$$ \frac{1}{b_{N+1}}=\frac{\cos \tilde{\omega} \epsilon}{b_{N}}+\frac{2 \epsilon}{M} \sqrt{1-\frac{M^{2}}{4 \epsilon^{2}} \frac{\sin ^{2} \tilde{\omega} \epsilon}{b_{N}^{2}}} $$

$$ \left(p_{b} t_{b} \mid p_{a} t_{a}\right)=\int_{p\left(t_{a}\right)=p_{a}}^{p\left(t_{b}\right)=p_{b}} \frac{\mathcal{D}^{\prime} p}{2 \pi \hbar} \int \mathcal{D} x e^{i \overline{\mathcal{A}}[p, x] / \hbar} $$

$$ \beta_{N} \equiv \frac{2 \epsilon}{M} b_{N} $$

$$ \overline{\mathcal{A}}[p, x]=\int_{t_{a}}^{t_{b}} d t[-\dot{p}(t) x(t)-H(p(t), x(t), t)]=\mathcal{A}[p, x]-p_{b} x_{b}+p_{a} x_{a} $$

$$ \beta_{1}=1, $$

$$ \left(p_{b} t_{b} \mid p_{a} t_{a}\right)=2 \pi \hbar \delta\left(p_{b}-p_{a}\right) e^{-i\left(t_{b}-t_{a}\right) H(p) / \hbar} $$

$$ \frac{1}{\beta_{N+1}}=\frac{\cos \tilde{\omega} \epsilon}{\beta_{N}}+\sqrt{1-\frac{\sin ^{2} \tilde{\omega} \epsilon}{\beta_{N}^{2}}} $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int \frac{d p}{2 \pi \hbar} e^{i p\left(x_{b}-x_{a}\right) / \hbar-i\left(t_{b}-t_{a}\right) H(p) / \hbar} $$

$$ \begin{align*} & \frac{1}{\beta_{2}}=\cos \tilde{\omega} \epsilon+\sqrt{1-\sin ^{2} \tilde{\omega} \epsilon}=\frac{\sin 2 \tilde{\omega} \epsilon}{\sin \tilde{\omega} \epsilon} \\ & \frac{1}{\beta_{3}}=\cos \tilde{\omega} \epsilon \frac{\sin 2 \tilde{\omega} \epsilon}{\sin \tilde{\omega} \epsilon}+\sqrt{1-\sin ^{2} \tilde{\omega} \epsilon \frac{\sin ^{2} 2 \tilde{\omega} \epsilon}{\sin ^{2} \tilde{\omega} \epsilon}}=\frac{\sin 3 \tilde{\omega} \epsilon}{\sin \tilde{\omega} \epsilon} \end{align*} $$

$$ Z_{\mathrm{QM}}\left(t_{b}, t_{a}\right)=\operatorname{Tr}\left(e^{-i\left(t_{b}-t_{a}\right) \hat{H} / \hbar}\right) $$

$$ \frac{1}{\beta_{N+1}}=\frac{\sin \tilde{\omega}(N+1) \epsilon}{\sin \tilde{\omega} \epsilon} $$

$$ Z_{\mathrm{QM}}\left(t_{b}, t_{a}\right)=\int_{-\infty}^{\infty} d x_{a}\left(x_{a} t_{b} \mid x_{a} t_{a}\right) $$

$$ b_{N+1}=\frac{M}{2 \epsilon} \frac{\sin \tilde{\omega} \epsilon}{\sin \tilde{\omega}(N+1) \epsilon} $$

$$ \int_{-\infty}^{\infty} d x_{N+1} \prod_{n=1}^{N}\left[\int_{-\infty}^{\infty} d x_{n}\right] \prod_{n=1}^{N+1}\left[\int_{-\infty}^{\infty} \frac{d p_{n}}{2 \pi \hbar}\right]=\prod_{n=1}^{N+1}\left[\iint_{-\infty}^{\infty} \frac{d x_{n} d p_{n}}{2 \pi \hbar}\right] $$

$$ \begin{align*} a_{N+1} & =\frac{M}{2 \epsilon} \sin \tilde{\omega} \epsilon \frac{\cos \tilde{\omega}(N+1) \epsilon}{\sin \tilde{\omega}(N+1) \epsilon} \\ \mathcal{N}_{N+1} & =\mathcal{N}_{1} \sqrt{\frac{\sin \tilde{\omega} \epsilon}{\sin \tilde{\omega}(N+1) \epsilon}} \end{align*} $$

$$ \lim _{N \rightarrow \infty} \prod_{n=1}^{N+1}\left[\iint_{-\infty}^{\infty} \frac{d x_{n} d p_{n}}{2 \pi \hbar}\right] \equiv \oint \mathcal{D} x \int \frac{\mathcal{D} p}{2 \pi \hbar} $$

$$ \int_{-\infty}^{\infty} d x_{a} \int_{x\left(t_{a}\right)=x_{a}}^{x\left(t_{b}\right)=x_{a}} \mathcal{D}^{\prime} x \int \frac{\mathcal{D} p}{2 \pi \hbar} \equiv \oint \mathcal{D} x \int \frac{\mathcal{D} p}{2 \pi \hbar} $$

$$ \tilde{F}\left(k, k^{\prime}\right)=\pi e^{-\left(k^{2}+k^{\prime 2}+a k k^{\prime}\right) / 2} $$

$$ \int_{-\infty}^{\infty} \frac{d p_{a}}{2 \pi \hbar} \int_{p\left(t_{a}\right)=p_{a}}^{p\left(t_{b}\right)=p_{a}} \frac{\mathcal{D}^{\prime} p}{2 \pi \hbar} \int \mathcal{D} x \equiv \oint \frac{\mathcal{D} p}{2 \pi \hbar} \int \mathcal{D} x $$

$$ F\left(x ; x^{\prime}\right)=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{d k}{2 \pi} \frac{d k^{\prime}}{2 \pi} e^{i k x+i k^{\prime} x} \tilde{F}\left(k, k^{\prime}\right) $$

$$ Z_{\mathrm{QM}}\left(t_{b}, t_{a}\right)=\oint \mathcal{D} x \int \frac{\mathcal{D} p}{2 \pi \hbar} e^{i \mathcal{A}[p, x] / \hbar}=\oint \frac{\mathcal{D} p}{2 \pi \hbar} \int \mathcal{D} x e^{i \overline{\mathcal{A}}[p, x] / \hbar} $$

$$ e^{k^{2} / 2-i k x}=\sum_{n=0}^{\infty} \frac{(-i k / 2)^{n}}{n!} H_{n}(x) $$

$$ \left.\sum_{n=1}^{N+1} p_{n}\left(x_{n}-x_{n-1}\right)\right|_{x_{N+1}=x_{0}}=-\left.\sum_{n=0}^{N}\left(p_{n+1}-p_{n}\right) x_{n}\right|_{p_{N+1}=p_{0}} . $$

$$ \begin{align*} \tilde{F}\left(k, k^{\prime}\right) & =\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} d x d x^{\prime} F\left(x, x^{\prime}\right) e^{-i k x-i k^{\prime} x}=e^{-\left(k^{2}+k^{\prime 2}\right) / 2} \\ & \times \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} d x d x^{\prime} F\left(x, x^{\prime}\right) \sum_{n=0}^{\infty} \sum_{n^{\prime}=0}^{\infty} \frac{(-i k / 2)^{n}}{n!} \frac{\left(-i k^{\prime} / 2\right)^{n^{\prime}}}{n^{\prime}!} H_{n}(x) H_{n^{\prime}}(x) \end{align*} $$

$$ H=\frac{p^{2}}{2 M}+V(x, t) $$

$$ \mathcal{A}^{N}=\sum_{n=1}^{N+1}\left[p_{n}\left(x_{n}-x_{n-1}\right)-\epsilon \frac{p_{n}^{2}}{2 M}-\epsilon V\left(x_{n}, t_{n}\right)\right] . $$

$$ \mathcal{A}^{N}=\sum_{n=1}^{N+1}\left[-\frac{\epsilon}{2 M}\left(p_{n}-\frac{x_{n}-x_{n-1}}{\epsilon} M\right)^{2}+\frac{M}{2} \epsilon\left(\frac{x_{n}-x_{n-1}}{\epsilon}\right)^{2}-\epsilon V\left(x_{n}, t_{n}\right)\right] . $$

$$ \int_{-\infty}^{\infty} \frac{d p_{n}}{2 \pi \hbar} \exp \left[-\frac{i}{\hbar} \frac{\epsilon}{2 M}\left(p_{n}-M \frac{x_{n}-x_{n-1}}{\epsilon}\right)^{2}\right]=\frac{1}{\sqrt{2 \pi \hbar i \epsilon / M}} $$

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

$$ \mathcal{A}^{N}=\epsilon \sum_{n=1}^{N+1}\left[\frac{M}{2}\left(\frac{x_{n}-x_{n-1}}{\epsilon}\right)^{2}-V\left(x_{n}, t_{n}\right)\right] $$

$$ \mathcal{A}[x]=\int_{t_{a}}^{t_{b}} d t L(x, \dot{x})=\int_{t_{a}}^{t_{b}} d t\left[\frac{M}{2} \dot{x}^{2}-V(x, t)\right] $$

$$ \begin{align*} \left(x_{b} t_{b} \mid x_{a} t_{a}\right) & \approx \int_{-\infty}^{\infty} d x_{N}\left(x_{b} t_{b} \mid x_{N} t_{N}\right)\left(x_{N} t_{N} \mid x_{a} t_{a}\right) \\ & =\int_{-\infty}^{\infty} d \Delta x\left(x_{b} t_{b} \mid x_{b}-\Delta x t_{b}-\epsilon\right)\left(x_{b}-\Delta x t_{b}-\epsilon \mid x_{a} t_{a}\right) \end{align*} $$

$$ \left(x_{b} t_{b} \mid x_{b}-\Delta x t_{b}-\epsilon\right) \approx \frac{1}{\sqrt{2 \pi \hbar i \epsilon / M}} \exp \left\{\epsilon \frac{i}{\hbar}\left[\frac{M}{2}\left(\frac{\Delta x}{\epsilon}\right)^{2}-V\left(x_{b}, t_{b}\right)\right]\right\} . $$

$$ \left(x_{b}-\Delta x t_{b}-\epsilon \mid x_{a} t_{a}\right)=\left[1-\Delta x \partial_{x_{b}}+\frac{1}{2}(\Delta x)^{2} \partial_{x_{b}}^{2}+\ldots\right]\left(x_{b}, t_{b}-\epsilon \mid x_{a} t_{a}\right) $$

$$ \int_{-\infty}^{\infty} \frac{d \Delta x}{\sqrt{2 \pi \hbar i \epsilon / M}}(\Delta x)^{2 n} \exp \left\{\epsilon \frac{i}{\hbar} \frac{M}{2}\left(\frac{\Delta x}{\epsilon}\right)^{2}\right\}=\left(i \frac{\hbar \epsilon}{M}\right)^{n} $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\left[1+\epsilon \frac{i \hbar}{2 M} \partial_{x_{b}}^{2}+\mathcal{O}\left(\epsilon^{2}\right)\right]\left[1-\epsilon \frac{i}{\hbar} V\left(x_{b}, t_{b}\right)+\mathcal{O}\left(\epsilon^{2}\right)\right]\left(x_{b}, t_{b}-\epsilon \mid x_{a} t_{a}\right) $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right) \equiv \int_{x\left(t_{a}\right)=x_{a}}^{x\left(t_{b}\right)=x_{b}} \mathcal{D} x e^{i \mathcal{A}[x] / \hbar} $$

$$ Z_{\mathrm{QM}}=\oint \mathcal{D} x e^{i \mathcal{A}[x] / \hbar} $$

$$ \oint \mathcal{D} x \approx \prod_{n=1}^{N+1} \int_{-\infty}^{\infty} \frac{d x_{n}}{\sqrt{2 \pi i \hbar \epsilon / M}} $$

$$ \epsilon_{n}=t_{n}-t_{n-1} $$

$$ \begin{align*} \left(x_{b} t_{b} \mid x_{a} t_{a}\right) & \approx \frac{1}{\sqrt{2 \pi \hbar i \epsilon_{b} / M}} \prod_{n=1}^{N}\left[\int_{-\infty}^{\infty} \frac{d x_{n}}{\sqrt{2 \pi i \hbar \epsilon_{n} / M}}\right] \\ & \times \exp \left\{\frac{i}{\hbar} \sum_{n=1}^{N+1}\left[\frac{M}{2} \frac{\left(x_{n}-x_{n-1}\right)^{2}}{\epsilon_{n}}-\epsilon_{n} V\left(x_{n}, t_{n}\right)\right]\right\} \end{align*} $$

$$ \left[\hat{H}\left(-i \hbar \partial_{x}, x\right)-i \hbar \partial_{t}\right]\left(x t \mid x_{a} t_{a}\right)=-i \hbar \delta\left(t-t_{a}\right) \delta\left(x-x_{a}\right) $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\Theta\left(t_{b}-t_{a}\right) \sum_{n} \psi_{n}\left(x_{b}\right) \psi_{n}^{*}\left(x_{a}\right) e^{-i E_{n}\left(t_{b}-t_{a}\right) / \hbar} $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int_{x\left(t_{a}\right)=x_{a}}^{x\left(t_{b}\right)=x_{b}} \mathcal{D}^{\prime} x \int \frac{\mathcal{D} p}{2 \pi \hbar} \exp \left[\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t\left(p \dot{x}-\frac{p^{2}}{2 M}\right)\right] $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int_{x\left(t_{a}\right)=x_{a}}^{x\left(t_{b}\right)=x_{b}} \mathcal{D} x \exp \left[\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t \frac{M}{2} \dot{x}^{2}\right] $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int \frac{d p}{2 \pi \hbar} e^{i p\left(x_{b}-x_{a}\right) / \hbar-i\left(t_{b}-t_{a}\right) p^{2} / 2 M \hbar} $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar\left(t_{b}-t_{a}\right) / M}} \exp \left[\frac{i}{\hbar} \frac{M}{2} \frac{\left(x_{b}-x_{a}\right)^{2}}{t_{b}-t_{a}}\right] $$

$$ \left(\mathbf{p}_{b} t_{b} \mid \mathbf{p}_{a} t_{a}\right)=(2 \pi \hbar)^{D} \delta^{(D)}\left(\mathbf{p}_{b}-\mathbf{p}_{a}\right) e^{-i\left(t_{b}-t_{a}\right) \mathbf{p}^{2} / 2 M \hbar} $$

$$ \left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)=\frac{1}{{\sqrt{2 \pi i \hbar\left(t_{b}-t_{a}\right) / M}}^{D}} \exp \left[\frac{i}{\hbar} \frac{M}{2} \frac{\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right)^{2}}{t_{b}-t_{a}}\right] $$

$$ \begin{gather*} \int d x^{\prime} \frac{1}{\sqrt{2 \pi i \hbar A \epsilon / M}} \exp \left[\frac{i}{\hbar} \frac{M}{2} \frac{\left(x^{\prime \prime}-x^{\prime}\right)^{2}}{A \epsilon}\right] \frac{1}{\sqrt{2 \pi i \hbar B \epsilon / M}} \exp \left[\frac{i}{\hbar} \frac{M}{2} \frac{\left(x^{\prime}-x\right)^{2}}{B \epsilon}\right] \\ =\frac{1}{\sqrt{2 \pi i \hbar(A+B) \epsilon / M}} \exp \left[\frac{i}{\hbar} \frac{M}{2} \frac{\left(x^{\prime \prime}-x\right)^{2}}{(A+B) \epsilon}\right] \end{gather*} $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar(N+1) \epsilon / M}} \exp \left[\frac{i}{\hbar} \frac{M}{2} \frac{\left(x_{b}-x_{a}\right)^{2}}{(N+1) \epsilon}\right] $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int_{x\left(t_{a}\right)=x_{a}}^{x\left(t_{b}\right)=x_{b}} \mathcal{D}^{\prime} x \int \frac{\mathcal{D} p}{2 \pi \hbar} \exp \left[\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t\left(p \dot{x}-\frac{p^{2}}{2 M g(t)}\right)\right] $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int_{x\left(t_{a}\right)=x_{a}}^{x\left(t_{b}\right)=x_{b}} \mathcal{D} x \sqrt{g} \exp \left[\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t \frac{M}{2} g(t) \dot{x}^{2}(t)\right] $$

$$ \int_{x\left(t_{a}\right)=x_{a}}^{x\left(t_{b}\right)=x_{b}} \mathcal{D} x \sqrt{g} \equiv \frac{1}{\sqrt{2 \pi \hbar i \epsilon / M g\left(t_{b}\right)}} \prod_{n=1}^{N}\left[\int_{-\infty}^{\infty} \frac{d x_{n}}{\sqrt{2 \pi \hbar i \epsilon / M g\left(t_{n}\right)}}\right] $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar M^{-1} \int_{t_{a}}^{t_{b}} g^{-1}(t)}} \exp \left[\frac{i}{\hbar} \frac{M}{2} \frac{\left(x_{b}-x_{a}\right)^{2}}{\int_{t_{a}}^{t_{b}} g^{-1}(t)}\right] $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int \frac{d p}{2 \pi \hbar} \exp \left\{\frac{i}{\hbar}\left[i p\left(x_{b}-x_{a}\right)-\frac{p^{2}}{2 M} \int_{t_{a}}^{t_{b}} g^{-1}(t)\right]\right\} $$

$$ x_{\mathrm{cl}}(t)=x_{a}+\frac{x_{b}-x_{a}}{t_{b}-t_{a}}\left(t-t_{a}\right) $$

$$ \ddot{x}_{\mathrm{cl}}(t)=0 $$

$$ x(t)=x_{\mathrm{cl}}(t)+\delta x(t) $$

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

$$ \left.\delta \mathcal{A}\right|_{x(t)=x_{\mathrm{cl}}(t)}=0 $$

$$ \mathcal{A}_{\mathrm{cl}} \equiv \mathcal{A}\left[x_{\mathrm{cl}}\right] $$

$$ \mathcal{A}=\mathcal{A}_{\mathrm{cl}}+\left.\frac{1}{2} \int_{t_{a}}^{t_{b}} d t \int_{t_{a}}^{t_{b}} d t^{\prime} \frac{\delta^{2} \mathcal{A}}{\delta x(t) \delta x\left(t^{\prime}\right)} \delta x(t) \delta x\left(t^{\prime}\right)\right|_{x(t)=x_{\mathrm{cl}}(t)}+\ldots $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int \mathcal{D} x e^{i \mathcal{A}[x] / \hbar}=e^{i \mathcal{A}_{\mathrm{cl}} / \hbar} F_{0}\left(t_{b}, t_{a}\right) $$

$$ \mathcal{A}_{\mathrm{cl}}=\int_{t_{a}}^{t_{b}} d t \frac{M}{2} \dot{x}_{\mathrm{cl}}^{2} $$

$$ F_{0}\left(t_{b}-t_{a}\right)=\int \mathcal{D} \delta x(t) \exp \left[\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t \frac{M}{2}(\delta \dot{x})^{2}\right] $$

$$ \mathcal{A}_{\mathrm{cl}}=\frac{M}{2} \frac{\left(x_{b}-x_{a}\right)^{2}}{t_{b}-t_{a}} $$

$$ F_{0}^{N}\left(t_{b}-t_{a}\right)=\frac{1}{\sqrt{2 \pi \hbar i \epsilon / M}} \prod_{n=1}^{N}\left[\int_{-\infty}^{\infty} \frac{d \delta x_{n}}{\sqrt{2 \pi \hbar i \epsilon / M}}\right] \exp \left(\frac{i}{\hbar} \mathcal{A}_{\mathrm{fl}}^{N}\right) $$

$$ \mathcal{A}_{\mathrm{fl}}^{N}=\frac{M}{2} \epsilon \sum_{n=1}^{N+1}\left(\frac{\delta x_{n}-\delta x_{n-1}}{\epsilon}\right)^{2} $$

$$ \nabla x(t) \equiv \frac{1}{\epsilon}[x(t+\epsilon)-x(t)], \quad \bar{\nabla} x(t) \equiv \frac{1}{\epsilon}[x(t)-x(t-\epsilon)] $$

$$ \nabla, \bar{\nabla} \xrightarrow{\epsilon \rightarrow 0} \partial_{t}, $$

$$ \begin{array}{ll} \nabla x_{n}=\frac{1}{\epsilon}\left(x_{n+1}-x_{n}\right), & N \geq n \geq 0 \\ \bar{\nabla} x_{n}=\frac{1}{\epsilon}\left(x_{n}-x_{n-1}\right), & N+1 \geq n \geq 1 \end{array} $$

$$ \mathcal{A}_{\mathrm{fl}}^{N}=\frac{M}{2} \epsilon \sum_{n=0}^{N}\left(\nabla x_{n}\right)^{2}=\frac{M}{2} \epsilon \sum_{n=1}^{N+1}\left(\bar{\nabla} x_{n}\right)^{2} . $$

$$ \mathcal{A}_{\mathrm{f}}^{N} \rightarrow \int_{t_{a}}^{t_{b}} d t \frac{M}{2} \dot{x}^{2} $$

$$ \int_{t_{a}}^{t_{b}} d t g(t) \dot{f}(t)=\left.g(t) f(t)\right|_{t_{a}} ^{t_{b}}-\int_{t_{a}}^{t_{b}} d t \dot{g}(t) f(t) $$

$$ \epsilon \sum_{n=1}^{N+1} p_{n} \bar{\nabla} x_{n}=\left.p_{n} x_{n}\right|_{0} ^{N+1}-\epsilon \sum_{n=0}^{N}\left(\nabla p_{n}\right) x_{n} $$

$$ \sum_{n=1}^{N+1} p_{n} \bar{\nabla} x_{n}=-\sum_{n=0}^{N}\left(\nabla p_{n}\right) x_{n}=-\sum_{n=1}^{N+1}\left(\nabla p_{n}\right) x_{n} $$

$$ \sum_{n=1}^{N+1} p_{n} \bar{\nabla} x_{n}=-\sum_{n=1}^{N+1}\left(\nabla p_{n}\right) x_{n} $$

$$ \sum_{n=1}^{N+1}\left(\bar{\nabla} x_{n}\right)^{2}=-\sum_{n=1}^{N} x_{n} \nabla \bar{\nabla} x_{n} $$

$$ \sum_{n=0}^{N}\left(\nabla x_{n}\right)^{2}=-\sum_{n=1}^{N+1} x_{n} \bar{\nabla} \nabla x_{n}=-\sum_{n=1}^{N} x_{n} \bar{\nabla} \nabla x_{n} $$

$$ \begin{align*} -\sum_{n=1}^{N} x_{n} \nabla \bar{\nabla} x_{n} & \equiv-\sum_{n, n^{\prime}=1}^{N} x_{n}(\nabla \bar{\nabla})_{n n^{\prime}} x_{n^{\prime}} \\ -\sum_{n=1}^{N} x_{n} \bar{\nabla} \nabla x_{n} & \equiv-\sum_{n, n^{\prime}=1}^{N} x_{n}(\bar{\nabla} \nabla)_{n n^{\prime}} x_{n^{\prime}} \end{align*} $$

$$ \nabla \bar{\nabla} \equiv \bar{\nabla} \nabla \equiv \frac{1}{\epsilon^{2}}\left(\begin{array}{rrrrrrr} -2 & 1 & 0 & \ldots & 0 & 0 & 0 \\ 1 & -2 & 1 & \ldots & 0 & 0 & 0 \\ \vdots & & & & & & \vdots \\ 0 & 0 & 0 & \ldots & 1 & -2 & 1 \\ 0 & 0 & 0 & \ldots & 0 & 1 & -2 \end{array}\right) $$

$$ x(t)=\int_{-\infty}^{\infty} d \omega e^{-i \omega t} x(\omega) $$

$$ \begin{align*} \nabla x\left(t_{n}\right) & =\int_{-\infty}^{\infty} d \omega \frac{1}{\epsilon}\left(e^{-i \omega\left(t_{n}+\epsilon\right)}-e^{-i \omega t_{n}}\right) x(\omega) \\ & =\int_{-\infty}^{\infty} d \omega e^{-i \omega t_{n}} \frac{1}{\epsilon}\left(e^{-i \omega \epsilon}-1\right) x(\omega) \end{align*} $$

$$ \frac{1}{\epsilon}\left(e^{-i \omega \epsilon}-1\right) $$

$$ (i \nabla x)(\omega)=\Omega x(\omega) \equiv \frac{i}{\epsilon}\left(e^{-i \omega \epsilon}-1\right) x(\omega) $$

$$ (i \bar{\nabla} x)(\omega)=\bar{\Omega} x(\omega) \equiv-\frac{i}{\epsilon}\left(e^{i \omega \epsilon}-1\right) x(\omega) $$

$$ \frac{i}{\epsilon}\left(e^{-i \omega \epsilon}-1\right) \frac{i}{\epsilon}\left(1-e^{i \omega \epsilon}\right)=\frac{1}{\epsilon^{2}}[2-2 \cos (\omega \epsilon)] \geq 0 $$

$$ x(t)=\int_{0}^{\infty} d \omega \sin \omega\left(t-t_{a}\right) x(\omega) $$

$$ \nu_{m}=\frac{\pi m}{t_{b}-t_{a}}=\frac{\pi m}{(N+1) \epsilon} $$

$$ x(t)=\sum_{m=1}^{\infty} \sqrt{\frac{2}{\left(t_{b}-t_{a}\right)}} \sin \nu_{m}\left(t-t_{a}\right) x\left(\nu_{m}\right) $$

$$ x\left(t_{n}\right)=\sum_{m=1}^{N} \sqrt{\frac{2}{N+1}} \sin \nu_{m}\left(t_{n}-t_{a}\right) x\left(\nu_{m}\right) $$

$$ \frac{2}{N+1} \sum_{n=1}^{N} \sin \nu_{m}\left(t_{n}-t_{a}\right) \sin \nu_{m^{\prime}}\left(t_{n}-t_{a}\right)=\delta_{m m^{\prime}} $$

$$ \frac{2}{N+1} \sum_{m=1}^{N} \sin \nu_{m}\left(t_{n}-t_{a}\right) \sin \nu_{m}\left(t_{n^{\prime}}-t_{a}\right)=\delta_{n n^{\prime}} $$

$$ \frac{2}{N+1} \frac{1}{2} \operatorname{Re} \sum_{n=0}^{N+1}\left\{\exp \left[\frac{i \pi\left(m-m^{\prime}\right)}{N+1} n\right]-\exp \left[\frac{i \pi\left(m+m^{\prime}\right)}{N+1} n\right]\right\}, $$

$$ \frac{2}{N+1} \frac{1}{2} \operatorname{Re}\left[\frac{1-e^{i \pi\left(m-m^{\prime}\right)} e^{i \pi\left(m-m^{\prime}\right) /(N+1)}}{1-e^{i \pi\left(m-m^{\prime}\right) /(N+1)}}-\left(m^{\prime} \rightarrow-m^{\prime}\right)\right] . $$

$$ \mathcal{A}_{\mathrm{fl}}^{N}=\frac{M}{2} \epsilon \sum_{n=0}^{N}\left(\bar{\nabla} x_{n}\right)^{2}=\frac{M}{2} \epsilon \sum_{m=1}^{N+1} x\left(\nu_{m}\right) \Omega_{m} \bar{\Omega}_{m} x\left(\nu_{m}\right) $$

$$ \Omega_{m} \bar{\Omega}_{m}=\frac{1}{\epsilon^{2}}\left[2-2 \cos \left(\nu_{m} \epsilon\right)\right]=\frac{1}{\epsilon^{2}}\left[2-2 \cos \left(\frac{\pi m}{N+1}\right)\right] $$

$$ \begin{align*} F_{0}^{N}\left(t_{b}-t_{a}\right)= & \frac{1}{\sqrt{2 \pi \hbar i \epsilon / M}} \prod_{n=1}^{N}\left[\int_{-\infty}^{\infty} \frac{d x_{n}}{\sqrt{2 \pi \hbar i \epsilon / M}}\right] \\ & \times \prod_{m=1}^{N} \exp \left\{\frac{i}{\hbar} \frac{M}{2} \epsilon \Omega_{m} \bar{\Omega}_{m}\left[x\left(\nu_{m}\right)\right]^{2}\right\} \end{align*} $$

$$ \prod_{n=1}^{N} d x_{n}=\prod_{m=1}^{N} d x\left(\nu_{m}\right) $$

$$ F_{0}^{N}\left(t_{b}-t_{a}\right)=\frac{1}{\sqrt{2 \pi \hbar i \epsilon / M}} \prod_{m=1}^{N} \frac{1}{\sqrt{\epsilon^{2} \Omega_{m} \bar{\Omega}_{m}}} $$

$$ \prod_{m=1}^{N}\left(1+x^{2}-2 x \cos \frac{m \pi}{N+1}\right)=\frac{x^{2(N+1)}-1}{x^{2}-1} . $$

$$ \prod_{m=1}^{N} \epsilon^{2} \Omega_{m} \bar{\Omega}_{m}=\prod_{m=1}^{N} 2\left(1-\cos \frac{m \pi}{N+1}\right)=N+1 $$

$$ F_{0}^{N}\left(t_{b}-t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar(N+1) \epsilon / M}} $$

$$ F_{0}\left(t_{b}-t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar\left(t_{b}-t_{a}\right) / M}} $$

$$ l\left(t_{b}-t_{a}\right) \equiv \sqrt{2 \pi \hbar\left(t_{b}-t_{a}\right) / M} $$

$$ F_{0}\left(t_{b}-t_{a}\right)=\frac{1}{\sqrt{i} l\left(t_{b}-t_{a}\right)} $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar\left(t_{b}-t_{a}\right) / M}} \exp \left[\frac{i}{\hbar} \frac{M}{2} \frac{\left(x_{b}-x_{a}\right)^{2}}{t_{b}-t_{a}}\right] $$

$$ \prod_{m=1}^{N} \epsilon^{2} \Omega_{m} \bar{\Omega}_{m} $$

$$ \prod_{m=1}^{N} \epsilon^{2} \Omega_{m} \bar{\Omega}_{m} \equiv \operatorname{det}_{N}\left(-\epsilon^{2} \nabla \bar{\nabla}\right) $$

$$ F_{0}^{N}\left(t_{b}-t_{b}\right)=\frac{1}{\sqrt{2 \pi \hbar i \epsilon / M}}\left[\operatorname{det}_{N}\left(-\epsilon^{2} \nabla \bar{\nabla}\right)\right]^{-1 / 2} $$

$$ \operatorname{det}_{N=1}\left(-\epsilon^{2} \nabla \bar{\nabla}\right)=|2|=2 . $$

$$ \operatorname{det}_{N=2}\left(-\epsilon^{2} \nabla \bar{\nabla}\right)=\left|\begin{array}{rr} 2 & -1 \\ -1 & 2 \end{array}\right|=3 . $$

$$ \operatorname{det}_{N}\left(-\epsilon^{2} \nabla \bar{\nabla}\right)=2 \operatorname{det}_{N-1}\left(-\epsilon^{2} \nabla \bar{\nabla}\right)-\operatorname{det}_{N-2}\left(-\epsilon^{2} \nabla \bar{\nabla}\right) . $$

$$ \operatorname{det}_{N}\left(-\epsilon^{2} \nabla \bar{\nabla}\right)=N+1 $$

$$ -\bar{\nabla} \nabla x(t)=0 $$

$$ x(t)=A t+B $$

$$ x_{\mathrm{cl}}\left(t_{n}\right)=x_{a}+\left(x_{b}-x_{a}\right) \frac{n}{N+1} $$

$$ \begin{align*} \mathcal{A}_{\mathrm{cl}} & =\epsilon \sum_{n=1}^{N+1} \frac{M}{2}\left(\bar{\nabla} x_{\mathrm{cl}}\right)^{2} \\ & =\frac{M}{2}\left(\left.x_{\mathrm{cl}} \nabla x_{\mathrm{cl}}\right|_{n=0} ^{N+1}-\epsilon \sum_{n=0}^{N} x_{\mathrm{cl}} \nabla \bar{\nabla} x_{\mathrm{cl}}\right) \\ & =\left.\frac{M}{2} x_{\mathrm{cl}} \nabla x_{\mathrm{cl}}\right|_{n=0} ^{N+1}=\frac{M}{2} \frac{\left(x_{b}-x_{a}\right)^{2}}{t_{b}-t_{a}} \end{align*} $$

$$ \begin{align*} \left(x_{b} t_{b} \mid x_{a} t_{a}\right) & =\int \mathcal{D}^{\prime} x \int \frac{\mathcal{D} p}{2 \pi \hbar} \exp \left\{\frac{i}{\hbar} \mathcal{A}[p, x]\right\} \\ & =\int \mathcal{D} x \exp \left\{\frac{i}{\hbar} \mathcal{A}[x]\right\} \end{align*} $$

$$ \mathcal{A}[p, x]=\int_{t_{a}}^{t_{b}} d t\left(p \dot{x}-\frac{1}{2 M} p^{2}-\frac{M \omega^{2}}{2} x^{2}\right) $$

$$ \mathcal{A}[x]=\int_{t_{a}}^{t_{b}} d t \frac{M}{2}\left(\dot{x}^{2}-\omega^{2} x^{2}\right) $$

$$ \mathcal{A}^{N}=\epsilon \frac{M}{2} \sum_{n=1}^{N+1}\left[\left(\bar{\nabla} x_{n}\right)^{2}-\omega^{2} x_{n}^{2}\right] $$

$$ \mathcal{A}_{\mathrm{cl}}=\int_{t_{a}}^{t_{b}} d t \frac{M}{2}\left(\dot{x}_{\mathrm{cl}}^{2}-\omega^{2} x_{\mathrm{cl}}^{2}\right) $$

$$ \mathcal{A}_{\mathrm{fl}}=\int_{t_{a}}^{t_{b}} d t \frac{M}{2}\left[(\delta \dot{x})^{2}-\omega^{2}(\delta x)^{2}\right] $$

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

$$ \ddot{x}_{\mathrm{cl}}=-\omega^{2} x_{\mathrm{cl}} $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int \mathcal{D} x e^{i \mathcal{A}[x] / \hbar}=e^{i \mathcal{A}_{\mathrm{cl}} / \hbar} F_{\omega}\left(t_{b}-t_{a}\right) $$

$$ x_{\mathrm{cl}}(t)=\frac{x_{b} \sin \omega\left(t-t_{a}\right)+x_{a} \sin \omega\left(t_{b}-t\right)}{\sin \omega\left(t_{b}-t_{a}\right)} . $$

$$ \mathcal{A}_{\mathrm{cl}}=\int_{t_{a}}^{t_{b}} d t \frac{M}{2}\left[x_{\mathrm{cl}}\left(-\ddot{x}_{\mathrm{cl}}-\omega^{2} x_{\mathrm{cl}}\right)\right]+\left.\frac{M}{2} x_{\mathrm{cl}} \dot{x}_{\mathrm{cl}}\right|_{t_{a}} ^{t_{b}} $$

$$ \mathcal{A}_{\mathrm{cl}}=\frac{M}{2}\left[x_{\mathrm{cl}}\left(t_{b}\right) \dot{x}_{\mathrm{cl}}\left(t_{b}\right)-x_{\mathrm{cl}}\left(t_{a}\right) \dot{x}_{\mathrm{cl}}\left(t_{a}\right)\right] $$

$$ \begin{align*} \dot{x}_{\mathrm{cl}}\left(t_{a}\right) & =\frac{\omega}{\sin \omega\left(t_{b}-t_{a}\right)}\left[x_{b}-x_{a} \cos \omega\left(t_{b}-t_{a}\right)\right] \\ \dot{x}_{\mathrm{cl}}\left(t_{b}\right) & =\frac{\omega}{\sin \omega\left(t_{b}-t_{a}\right)}\left[x_{b} \cos \omega\left(t_{b}-t_{a}\right)-x_{a}\right] \end{align*} $$

$$ \mathcal{A}_{\mathrm{cl}}=\frac{M \omega}{2 \sin \omega\left(t_{b}-t_{a}\right)}\left[\left(x_{b}^{2}+x_{a}^{2}\right) \cos \omega\left(t_{b}-t_{a}\right)-2 x_{b} x_{a}\right] . $$

$$ \begin{align*} F_{\omega}^{N}\left(t_{b}, t_{a}\right)= & \frac{1}{\sqrt{2 \pi \hbar i \epsilon / M}} \prod_{n=1}^{N}\left[\int_{-\infty}^{\infty} \frac{d \delta x_{n}}{\sqrt{2 \pi \hbar i \epsilon / M}}\right] \\ & \times \exp \left\{\frac{i}{\hbar} \frac{M}{2} \epsilon \sum_{n, n^{\prime}=1}^{N} \delta x_{n}\left[-\nabla \bar{\nabla}-\omega^{2}\right]_{n n^{\prime}} \delta x_{n^{\prime}}\right\} \end{align*} $$

$$ \Omega_{m} \bar{\Omega}_{m}-\omega^{2}=\frac{1}{\epsilon^{2}}\left[2-2 \cos \left(\nu_{m} \epsilon\right)\right]-\omega^{2} $$

$$ F_{\omega}^{N}\left(t_{b}, t_{a}\right)=\frac{1}{\sqrt{2 \pi \hbar i \epsilon / M}} \prod_{m=1}^{N} \frac{1}{\sqrt{\epsilon^{2} \Omega_{m} \bar{\Omega}_{m}-\epsilon^{2} \omega^{2}}} $$

$$ \sin \frac{\epsilon \tilde{\omega}}{2} \equiv \frac{\epsilon \omega}{2} $$

$$ \begin{align*} & \prod_{m=1}^{N}\left[\epsilon^{2} \Omega_{m} \bar{\Omega}_{m}-\epsilon^{2} \omega^{2}\right]=\prod_{m=1}^{N}\left[\epsilon^{2} \Omega_{m} \bar{\Omega}_{m}\right] \prod_{m=1}^{N}\left[\frac{\epsilon^{2} \Omega_{m} \bar{\Omega}_{m}-\epsilon^{2} \omega^{2}}{\epsilon^{2} \Omega_{m} \bar{\Omega}_{m}}\right] \\ & =\prod_{m=1}^{N}\left[\epsilon^{2} \Omega_{m} \bar{\Omega}_{m}\right]\left[\prod_{m=1}^{N}\left(1-\frac{\sin ^{2} \frac{\epsilon \tilde{\omega}}{2}}{\sin ^{2} \frac{m \pi}{2(N+1)}}\right)\right] \end{align*} $$

$$ \prod_{m=1}^{N}\left(1-\frac{\sin ^{2} x}{\sin ^{2} \frac{m \pi}{2(N+1)}}\right)=\frac{1}{\sin 2 x} \frac{\sin [2(N+1) x]}{(N+1)} $$

$$ \operatorname{det}_{N}\left(-\epsilon^{2} \nabla \bar{\nabla}-\epsilon^{2} \omega^{2}\right)=\prod_{m=1}^{N}\left[\epsilon^{2} \Omega_{m} \bar{\Omega}_{m}-\epsilon^{2} \omega^{2}\right]=\frac{\sin \tilde{\omega}\left(t_{b}-t_{a}\right)}{\sin \epsilon \tilde{\omega}} $$

$$ F_{\omega}^{N}\left(t_{b}, t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar / M}} \sqrt{\frac{\sin \tilde{\omega} \epsilon}{\epsilon \sin \tilde{\omega}\left(t_{b}-t_{a}\right)}}, \quad t_{b}-t_{a}<\pi / \tilde{\omega} $$

$$ t_{b}-t_{a}<\pi / \tilde{\omega} $$

$$ F_{\omega}^{N}\left(t_{b}, t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar / M}} \sqrt{\frac{\sin \tilde{\omega} \epsilon}{\epsilon\left|\sin \tilde{\omega}\left(t_{b}-t_{a}\right)\right|}} e^{-i \nu \pi / 2} $$

$$ \operatorname{det}_{N}\left(-\epsilon^{2} \nabla \bar{\nabla}-\epsilon^{2} \omega^{2}\right) \xrightarrow{\epsilon \rightarrow 0} \frac{\sin \omega\left(t_{b}-t_{a}\right)}{\omega \epsilon} . $$

$$ F_{\omega}\left(t_{b}-t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar / M}} \sqrt{\frac{\omega}{\sin \omega\left(t_{b}-t_{a}\right)}}, $$

$$ \begin{align*} \frac{\epsilon^{2} \Omega_{m} \bar{\Omega}_{m}-\epsilon^{2} \omega^{2}}{\epsilon^{2} \Omega_{m} \bar{\Omega}_{m}} & =1-\frac{\epsilon^{2} \omega^{2}}{2-2 \cos \left(\nu_{m} \epsilon\right)} \\ & \xrightarrow{\epsilon \rightarrow 0} 1-\frac{\omega^{2}\left(t_{b}-t_{a}\right)^{2}}{\pi^{2} m^{2}} \end{align*} $$

$$ \sin x=x \prod_{m=1}^{\infty}\left(1-\frac{x^{2}}{m^{2} \pi^{2}}\right) $$

$$ \prod_{m} \frac{\Omega_{m} \bar{\Omega}_{m}}{\Omega_{m} \bar{\Omega}_{m}-\omega^{2}} \xrightarrow{\epsilon \rightarrow 0} \prod_{m=1}^{\infty} \frac{\nu_{m}^{2}}{\nu_{m}^{2}-\omega^{2}}=\frac{\omega\left(t_{b}-t_{a}\right)}{\sin \omega\left(t_{b}-t_{a}\right)} $$

$$ \begin{align*} \left(x_{b} t_{b} \mid x_{a} t_{a}\right)= & \int \mathcal{D} x(t) \exp \left[\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t \frac{M}{2}\left(\dot{x}^{2}-\omega^{2} x^{2}\right)\right] \\ = & \frac{1}{\sqrt{2 \pi i \hbar / M}} \sqrt{\frac{\omega}{\sin \omega\left(t_{b}-t_{a}\right)}} \\ & \times \exp \left\{\frac{i}{2 \hbar} \frac{M \omega}{\sin \omega\left(t_{b}-t_{a}\right)}\left[\left(x_{b}^{2}+x_{a}^{2}\right) \cos \omega\left(t_{b}-t_{a}\right)-2 x_{b} x_{a}\right]\right\} \end{align*} $$

$$ \mathcal{A}=\int_{t_{a}}^{t_{b}} d t \frac{M}{2}\left(\dot{\mathbf{x}}^{2}-\omega^{2} \mathbf{x}^{2}\right) $$

$$ \begin{align*} \left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right) & =\prod_{i=1}^{D}\left(x_{b}^{i} t_{b} \mid x_{a}^{i} t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar / M}} \sqrt{\frac{\omega}{\sin \omega\left(t_{b}-t_{a}\right)}} \\ & \times \exp \left\{\frac{i}{2 \hbar} \frac{M \omega}{\sin \omega\left(t_{b}-t_{a}\right)}\left[\left(\mathbf{x}_{b}^{2}+\mathbf{x}_{a}^{2}\right) \cos \omega\left(t_{b}-t_{a}\right)-2 \mathbf{x}_{b} \mathbf{x}_{a}\right]\right\} \end{align*} $$

$$ \frac{\operatorname{det}\left(-\partial_{t}^{2}-\omega^{2}\right)}{\operatorname{det}\left(-\partial_{t}^{2}\right)} $$

$$ \nu_{m}^{2}=\left(\frac{\pi m}{t_{b}-t_{a}}\right)^{2} $$

$$ \frac{\operatorname{det}\left(-\partial_{t}^{2}-\omega^{2}\right)}{\operatorname{det}\left(-\partial_{t}^{2}\right)}=\prod_{m=1}^{\infty} \frac{\nu_{m}^{2}-\omega^{2}}{\nu_{m}^{2}} $$

$$ F_{\omega}\left(t_{b}-t_{a}\right)=\int \mathcal{D} \delta x \exp \left\{\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t \frac{M}{2}\left[(\delta \dot{x})^{2}-\omega^{2}(\delta x)^{2}\right]\right\} $$

$$ F_{\omega}\left(t_{b}, t_{a}\right) \xrightarrow{\epsilon \rightarrow 0} \frac{1}{\sqrt{2 \pi \hbar i \epsilon / M}} \frac{1}{\sqrt{\operatorname{det}\left(-\partial_{t}^{2}-\omega^{2}\right)}} \quad \text { (false). } $$

$$ \begin{align*} & \operatorname{det}\left(-\partial_{t}^{2}-\omega^{2}\right)=\prod_{m=1}^{\infty}\left(\nu_{m}^{2}-\omega^{2}\right) \\ & \quad=\prod_{m=1}^{\infty}\left(\nu_{m}^{2}\right) \prod_{m=1}^{\infty}\left[\frac{\nu_{m}^{2}-\omega^{2}}{\nu_{m}^{2}}\right]=\prod_{m=1}^{\infty}\left[\frac{\pi^{2} m^{2}}{\left(t_{b}-t_{a}\right)^{2}}\right] \times \frac{\sin \omega\left(t_{b}-t_{a}\right)}{\omega\left(t_{b}-t_{a}\right)} \end{align*} $$

$$ \Omega_{m} \bar{\Omega}_{m} \approx \nu_{m}^{2} \approx \frac{\pi^{2} m^{2}}{\left(t_{b}-t_{a}\right)^{2}} $$

$$ \begin{align*} F_{\omega}^{N}\left(t_{b}-t_{a}\right) & =\frac{1}{\sqrt{2 \pi \hbar i \epsilon / M}} \prod_{n=1}^{N}\left[\int \frac{d \delta x_{n}}{\sqrt{2 \pi \hbar i \epsilon / M}}\right] \exp \left[\frac{i}{\hbar} \frac{M}{2 \epsilon} \delta x^{T}\left(-\epsilon^{2} \nabla \bar{\nabla}-\epsilon^{2} \omega^{2}\right) \delta x\right] \\ & =\frac{1}{\sqrt{2 \pi \hbar i \epsilon / M}} \frac{1}{\sqrt{\operatorname{det}_{N}\left(-\epsilon^{2} \nabla \bar{\nabla}-\epsilon^{2} \omega^{2}\right)}} \end{align*} $$

$$ F_{\omega}^{N}\left(t_{b}-t_{a}\right)=\frac{1}{\sqrt{2 \pi \hbar i\left(t_{b}-t_{a}\right) / M}}\left[\frac{\operatorname{det}_{N}\left(-\epsilon^{2} \nabla \bar{\nabla}-\epsilon^{2} \omega^{2}\right)}{\operatorname{det}_{N}\left(-\epsilon^{2} \nabla \bar{\nabla}\right)}\right]^{-1 / 2} $$

$$ F_{\omega}^{N}\left(t_{b}-t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar / M}} \sqrt{\frac{\sin \tilde{\omega} \epsilon}{\epsilon \sin \tilde{\omega}\left(t_{b}-t_{a}\right)}}, $$

$$ \begin{align*} F_{\omega}\left(t_{b}-t_{a}\right) & =\frac{1}{\sqrt{2 \pi \hbar i\left(t_{b}-t_{a}\right) / M}}\left[\frac{\operatorname{det}\left(-\partial_{t}^{2}-\omega^{2}\right)}{\operatorname{det}\left(-\partial_{t}^{2}\right)}\right]^{-1 / 2} \\ & =\frac{1}{\sqrt{2 \pi \hbar i\left(t_{b}-t_{a}\right) / M}} \prod_{m=1}^{\infty}\left[\frac{\nu_{m}^{2}-\omega^{2}}{\nu_{m}^{2}}\right]^{-1 / 2} \\ & =\frac{1}{\sqrt{2 \pi \hbar i\left(t_{b}-t_{a}\right) / M}} \sqrt{\frac{\omega\left(t_{b}-t_{a}\right)}{\sin \omega\left(t_{b}-t_{a}\right)}} \end{align*} $$

$$ \begin{align*} & \left(\mathbf{p}_{b} t_{b} \mid \mathbf{p}_{a} t_{a}\right)=\int d^{D} x_{b} e^{-i \mathbf{p}_{b} \mathbf{x}_{b} / \hbar} \int d^{D} x_{a} e^{i \mathbf{p}_{a} \mathbf{x}_{a} / \hbar}\left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right) \\ & \quad=\frac{(2 \pi \hbar)^{D}}{\sqrt{2 \pi i \hbar}^{D}} \frac{1}{{\sqrt{M \omega \sin \omega\left(t_{b}-t_{a}\right)}}^{D}} \\ & \quad \times \exp \left\{\frac{i}{\hbar} \frac{1}{2 M \omega \sin \omega\left(t_{b}-t_{a}\right)}\left[\left(\mathbf{p}_{b}^{2}+\mathbf{p}_{a}^{2}\right) \cos \omega\left(t_{b}-t_{a}\right)-2 \mathbf{p}_{b} \mathbf{p}_{a}\right]\right\} \end{align*} $$

$$ \begin{align*} & \frac{1}{2 M \omega \sin \omega\left(t_{b}-t_{a}\right)}\left[\left(\mathbf{p}_{b}^{2}+\mathbf{p}_{a}\right) \cos \omega\left(t_{b}-t_{a}\right)-2 \mathbf{p}_{b} \mathbf{p}_{a}^{2}\right] \\ & \quad=\frac{1}{2 M \omega^{2}\left(t_{b}-t_{a}\right)}\left\{\left(\mathbf{p}_{b}-\mathbf{p}_{a}\right)^{2}-\frac{1}{2}\left(\mathbf{p}_{b}^{2}+\mathbf{p}_{a}^{2}\right)\left[\omega\left(t_{b}-t_{a}\right)\right]^{2}+\ldots\right\} \end{align*} $$

$$ \frac{(2 \pi)^{D}}{{\sqrt{2 \pi i \omega^{2}\left(t_{b}-t_{a}\right) \hbar M}}^{D}} \exp \left\{\frac{i}{\hbar} \frac{1}{2 M \omega^{2}\left(t_{b}-t_{a}\right)}\left(\mathbf{p}_{b}-\mathbf{p}_{a}\right)^{2}\right\} $$

$$ \mathcal{A}^{N}=\epsilon \frac{M}{2} \sum_{n=1}^{N+1}\left[\left(\bar{\nabla} x_{n}\right)^{2}-\omega^{2}\left(x_{n}^{2}+x_{n-1}^{2}\right) / 2\right] $$

$$ \mathcal{A}^{N}=\epsilon \frac{M}{2} \sum_{n=0}^{N}\left[\left(\nabla x_{n}\right)^{2}-\omega^{2}\left(x_{n+1}^{2}+x_{n}^{2}\right) / 2\right] $$

$$ \epsilon \sum_{n=1}^{N+1}\left(\bar{\nabla} x_{n}\right)^{2}=\epsilon \sum_{n=0}^{N}\left(\nabla x_{n}\right)^{2}=\left[x_{b} \bar{\nabla} x_{b}-x_{a} \nabla x_{a}\right]-\epsilon \sum_{n=1}^{N} x_{n} \nabla \bar{\nabla} x_{n} $$

$$ \mathcal{A}^{N}=\frac{M}{2}\left(x_{b} \bar{\nabla} x_{b}-x_{a} \nabla x_{a}\right)-\epsilon \frac{M}{4} \omega^{2}\left(x_{b}^{2}+x_{a}^{2}\right)-\epsilon \frac{M}{2} \sum_{n=1}^{N} x_{n}\left(\nabla \bar{\nabla}+\omega^{2}\right) x_{n} $$

$$ \left(\nabla \bar{\nabla}+\omega^{2}\right) x_{\mathrm{cl}}(t)=0 $$

$$ x_{\mathrm{cl}}(t)=\frac{1}{\sin \tilde{\omega}\left(t_{b}-t_{a}\right)}\left[x_{b} \sin \tilde{\omega}\left(t-t_{a}\right)+x_{a} \sin \tilde{\omega}\left(t_{b}-t\right)\right] $$

$$ \mathcal{A}_{\mathrm{cl}}^{N}=\frac{M}{2 \epsilon} \frac{\sin \tilde{\omega} \epsilon}{\sin \tilde{\omega}\left(t_{b}-t_{a}\right)}\left[\left(x_{b}^{2}+x_{a}^{2}\right) \cos \tilde{\omega}\left(t_{b}-t_{a}\right)-2 x_{b} x_{a}\right] $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=e^{i \mathcal{A}_{\mathrm{cl}}^{N} / \hbar} F_{\omega}^{N}\left(t_{b}-t_{a}\right) $$

$$ F\left(t_{b}, t_{a}\right)=\int \mathcal{D} \delta x(t) \exp \left(\frac{i}{\hbar} \mathcal{A}\right) $$

$$ \mathcal{A}=\int_{t_{a}}^{t_{b}} d t \frac{M}{2}\left[(\delta \dot{x})^{2}-\Omega^{2}(t)(\delta x)^{2}\right] $$

$$ F^{N}\left(t_{b}, t_{a}\right)=\frac{1}{\sqrt{2 \pi \hbar i\left(t_{b}-t_{a}\right) / M}}\left[\frac{\operatorname{det}_{N}\left(-\epsilon^{2} \nabla \bar{\nabla}-\epsilon^{2} \Omega^{2}\right)}{\operatorname{det}_{N}\left(-\epsilon^{2} \nabla \bar{\nabla}\right)}\right]^{-1 / 2} $$

$$ \Omega^{2}(t)=\left(\begin{array}{lll} \Omega_{N}^{2} & & \\ & \ddots & \\ & & \Omega_{1}^{2} \end{array}\right) $$

$$ \begin{align*} D_{N} & \equiv \operatorname{det}_{N}\left(-\epsilon^{2} \bar{\nabla} \nabla-\epsilon^{2} \Omega^{2}\right) \\ & \equiv\left|\begin{array}{ccccccc} 2-\epsilon^{2} \Omega_{N}^{2} & -1 & 0 & \ldots & 0 & 0 & 0 \\ -1 & 2-\epsilon^{2} \Omega_{N-1}^{2} & -1 & \ldots & 0 & 0 & 0 \\ \vdots & & & & & & \vdots \\ 0 & 0 & 0 & \ldots & -1 & 2-\epsilon^{2} \Omega_{2}^{2} & -1 \\ 0 & 0 & 0 & \ldots & 0 & -1 & 2-\epsilon^{2} \Omega_{1}^{2} \end{array}\right| . \end{align*} $$

$$ D_{N}=\left(2-\epsilon^{2} \Omega_{N}^{2}\right) D_{N-1}-D_{N-2} $$

$$ \epsilon^{2}\left[\frac{1}{\epsilon}\left(\frac{D_{N}-D_{N-1}}{\epsilon}-\frac{D_{N-1}-D_{N-2}}{\epsilon}\right)+\Omega_{N}^{2} D_{N-1}\right]=0 $$

$$ \left(\nabla \bar{\nabla}+\Omega_{N+1}^{2}\right) D_{N}=0 $$

$$ \begin{align*} D_{1} & =\left(2-\epsilon^{2} \Omega_{1}^{2}\right) \\ D_{2} & =\left(2-\epsilon^{2} \Omega_{1}^{2}\right)\left(2-\epsilon^{2} \Omega_{2}^{2}\right)-1 \end{align*} $$

$$ \left(\nabla \bar{\nabla}+\omega^{2}\right) D_{N}=0 $$

$$ D_{N}=\frac{\sin (N+1) \epsilon \tilde{\omega}}{\sin \epsilon \tilde{\omega}} $$

$$ \begin{align*} & D_{1}=2 \cos \epsilon \tilde{\omega}, \\ & D_{2}=4 \cos ^{2} \epsilon \tilde{\omega}-1, \end{align*} $$

$$ D_{\mathrm{ren}}\left(t_{N}\right)=\epsilon D_{N} $$

$$ \begin{align*} (\epsilon D)_{1} & =D_{\text {ren }}\left(t_{a}\right)=0 \\ \frac{\epsilon D_{2}-\epsilon D_{1}}{\epsilon}=(\nabla \epsilon D)_{1} \xrightarrow{\epsilon \rightarrow 0} \dot{D}_{\text {ren }}\left(t_{a}\right) & =1 \end{align*} $$

$$ \left[\partial_{t}^{2}+\Omega^{2}(t)\right] D_{\mathrm{ren}}(t)=0 $$

$$ D_{\mathrm{ren}}(t)=\frac{1}{\omega} \sin \omega\left(t-t_{a}\right) $$

$$ \operatorname{det}\left(-\epsilon^{2} \nabla \bar{\nabla}-\epsilon^{2} \omega^{2}\right) \xrightarrow{\epsilon \rightarrow 0} \frac{1}{\epsilon} \frac{\sin \omega\left(t_{b}-t_{a}\right)}{\omega} $$

$$ \left[\partial_{t}^{2}+\Omega^{2}(t)\right] x(t)=0 $$

$$ D_{\mathrm{ren}}(t)=\alpha \xi(t)+\beta \eta(t) $$

$$ \begin{align*} & \alpha \xi\left(t_{a}\right)+\beta \eta\left(t_{a}\right)=0 \\ & \alpha \dot{\xi}\left(t_{a}\right)+\beta \dot{\eta}\left(t_{a}\right)=1 \end{align*} $$

$$ D_{\mathrm{ren}}(t)=\frac{\xi(t) \eta\left(t_{a}\right)-\xi\left(t_{a}\right) \eta(t)}{\dot{\xi}\left(t_{a}\right) \eta\left(t_{a}\right)-\xi\left(t_{a}\right) \dot{\eta}\left(t_{a}\right)} $$

$$ W \equiv \xi(t) \stackrel{\leftrightarrow}{\partial}_{t} \eta(t) \equiv \xi(t) \dot{\eta}(t)-\dot{\xi}(t) \eta(t) $$

$$ \frac{d}{d t}\left[a(t) \frac{d y(t)}{d t}\right]+b(t) y(t)=0 $$

$$ D_{\mathrm{ren}}(t)=-\frac{1}{W}\left[\xi(t) \eta\left(t_{a}\right)-\xi\left(t_{a}\right) \eta(t)\right] $$

$$ D_{\mathrm{ren}}=-\frac{1}{W}\left[\xi\left(t_{b}\right) \eta\left(t_{a}\right)-\xi\left(t_{a}\right) \eta\left(t_{b}\right)\right] $$

$$ \tilde{D}_{\mathrm{ren}}(t)=-\frac{1}{W}\left[\xi\left(t_{b}\right) \eta(t)-\xi(t) \eta\left(t_{b}\right)\right] $$

$$ \tilde{D}_{\text {ren }}\left(t_{b}\right)=0, \quad \dot{\tilde{D}}_{\text {ren }}\left(t_{b}\right)=-1 $$

$$ \begin{array}{lc} {\left[\partial_{t}^{2}+\Omega^{2}(t)\right] D_{a}(t)=0 ;} & D_{a}\left(t_{a}\right)=0, \\ {\left[\partial_{t}^{2}+\Omega^{2}(t)\right] D_{b}(t)=0 ;} & D_{b}\left(t_{b}\right)=0, \end{array} $$

$$ D_{\mathrm{ren}}=D_{a}\left(t_{b}\right)=D_{b}\left(t_{a}\right) $$

$$ \dot{D}_{a}\left(t_{b}\right)=-\dot{D}_{b}\left(t_{a}\right), \quad \text { for } \quad \Omega(t)=\Omega(-t) . $$

$$ \dot{D}_{a}\left(t_{b}\right)+\dot{D}_{b}\left(t_{a}\right)=-2 \int_{t_{a}}^{t_{b}} d t \Omega(t) \dot{\Omega}(t) D_{a}(t) D_{b}(t) $$

$$ \xi(t)=\cos \omega t, \quad \eta(t)=\sin \omega t . $$

$$ W=\omega $$

$$ D_{\mathrm{ren}}=-\frac{1}{\omega}\left(\cos \omega t_{b} \sin \omega t_{a}-\cos \omega t_{a} \sin \omega t_{b}\right)=\frac{1}{\omega} \sin \omega\left(t_{b}-t_{a}\right) . $$

$$ \eta(t)=w \xi(t) \int^{t} \frac{d t^{\prime}}{\xi^{2}\left(t^{\prime}\right)} $$

$$ \dot{\eta}=\frac{\dot{\xi} \eta}{\xi}+\frac{w}{\xi}, \quad \ddot{\eta}=\frac{\ddot{\xi} \eta}{\xi} . $$

$$ W=\xi(t) \dot{\eta}(t)-\dot{\xi}(t) \eta(t)=w $$

$$ D_{\mathrm{ren}}(t)=D_{a}(t)=\xi(t) \xi\left(t_{a}\right) \int_{t_{a}}^{t} \frac{d t^{\prime}}{\xi^{2}\left(t^{\prime}\right)}, \quad \tilde{D}_{\mathrm{ren}}(t)=D_{b}(t)=\xi\left(t_{b}\right) \xi(t) \int_{t}^{t_{b}} \frac{d t^{\prime}}{\xi^{2}\left(t^{\prime}\right)} $$

$$ D_{\mathrm{ren}}=\xi\left(t_{b}\right) \xi\left(t_{a}\right) \int_{t_{a}}^{t_{b}} \frac{d t^{\prime}}{\xi^{2}\left(t^{\prime}\right)} $$

$$ x\left(x_{a}, \dot{x}_{a} ; t\right)=\frac{1}{D_{b}\left(t_{a}\right)}\left[D_{b}(t)-D_{a}(t) \dot{D}_{b}\left(t_{a}\right)\right] x_{a}+D_{a}(t) \dot{x}_{a} $$

$$ D_{\mathrm{ren}}(t)=\frac{\partial x\left(x_{a}, \dot{x}_{a} ; t\right)}{\partial \dot{x}_{a}} $$

$$ D_{\mathrm{ren}}=\frac{\partial x_{b}}{\partial \dot{x}_{a}}, $$

$$ x\left(x_{b}, \dot{x}_{b} ; t\right)=\frac{1}{D_{a}\left(t_{b}\right)}\left[D_{a}(t)+D_{b}(t) \dot{D}_{a}\left(t_{b}\right)\right] x_{b}-D_{b}(t) \dot{x}_{b} $$

$$ D_{\mathrm{ren}}=-\frac{\partial x_{a}}{\partial \dot{x}_{b}} $$

$$ D_{\mathrm{ren}}=\operatorname{Det}\left[-\partial_{t}^{2} \delta_{i j}-\Omega_{i j}^{2}(t)\right]=\operatorname{det} D_{i j}\left(t_{b}\right) $$

$$ D_{\mathrm{ren}}=\operatorname{det} \frac{\partial x_{b}^{i}}{\partial \dot{x}_{a}^{j}}=\operatorname{det}\left(-\frac{\partial x_{a}^{i}}{\partial \dot{x}_{b}^{j}}\right) . $$

$$ x\left(x_{b}, x_{a} ; t\right)=\frac{D_{b}(t)}{D_{b}\left(t_{a}\right)} x_{a}+\frac{D_{a}(t)}{D_{a}\left(t_{b}\right)} x_{b} $$

$$ \frac{D_{a}(t)}{D_{a}\left(t_{b}\right)}=\frac{\partial x\left(x_{b}, x_{a} ; t\right)}{\partial x_{b}}, \quad \frac{D_{b}(t)}{D_{b}\left(t_{a}\right)}=\frac{\partial x\left(x_{b}, x_{a} ; t\right)}{\partial x_{a}} . $$

$$ \begin{align*} \dot{x}_{a} & =\frac{\dot{D}_{b}\left(t_{a}\right)}{D_{b}\left(t_{a}\right)} x_{a}+\frac{1}{D_{a}\left(t_{b}\right)} x_{b} \\ \dot{x}_{b} & =-\frac{1}{D_{b}\left(t_{a}\right)} x_{a}+\frac{\dot{D}_{a}\left(t_{b}\right)}{D_{a}\left(t_{b}\right)} x_{b} \end{align*} $$

$$ D_{\mathrm{ren}}=\left(\frac{\partial \dot{x}_{a}}{\partial x_{b}}\right)^{-1}=-\left(\frac{\partial \dot{x}_{b}}{\partial x_{a}}\right)^{-1}, $$

$$ \left.\frac{\partial x_{b}}{\partial \dot{x}_{a}}\right|_{x_{a}}=\left(\left.\frac{\partial \dot{x}_{a}}{\partial x_{b}}\right|_{x_{a}}\right)^{-1} . $$

$$ \mathcal{A}=\int_{t_{a}}^{t_{b}} d t \frac{M}{2}\left[(\delta \dot{\mathbf{x}})^{2}-\delta \mathbf{x}^{T} \mathbf{\Omega}^{2}(t) \delta \mathbf{x}\right] $$

$$ F^{N}\left(t_{b}, t_{a}\right)=\frac{1}{{\sqrt{2 \pi \hbar i\left(t_{b}-t_{a}\right) / M}}^{D}}\left[\frac{\operatorname{det}_{N}\left(-\epsilon^{2} \nabla \bar{\nabla}-\epsilon^{2} \boldsymbol{\Omega}^{2}\right)}{\operatorname{det}_{N}\left(-\epsilon^{2} \nabla \bar{\nabla}\right)}\right]^{-1 / 2} . $$

$$ D_{\mathrm{ren}}=\operatorname{det} \mathbf{D}_{a}\left(t_{b}\right)=\operatorname{det} \mathbf{D}_{b}\left(t_{a}\right) $$

$$ \begin{array}{lll} {\left[\partial_{t}^{2}+\boldsymbol{\Omega}^{2}(t)\right] \mathbf{D}_{a}(t)=0 ;} & \mathbf{D}_{a}\left(t_{a}\right)=0, & \dot{\mathbf{D}}_{a}\left(t_{a}\right)=\mathbf{1} \\ {\left[\partial_{t}^{2}+\boldsymbol{\Omega}^{2}(t)\right] \mathbf{D}_{b}(t)=0 ;} & \mathbf{D}_{b}\left(t_{b}\right)=0, & \dot{\mathbf{D}}_{b}\left(t_{b}\right)=-\mathbf{1} \end{array} $$

$$ D_{\mathrm{ren}}=\left(\operatorname{det} \frac{\partial \dot{x}_{a}^{i}}{\partial x_{b}^{j}}\right)^{-1}=\left[\operatorname{det}\left(-\frac{\partial \dot{x}_{b}^{i}}{\partial x_{a}^{j}}\right)\right]^{-1} . $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int \mathcal{D} x \exp \left\{\frac{i}{\hbar} \mathcal{A}[x]\right\}, $$

$$ \mathcal{A}[x]=\frac{M}{2} \int_{t_{a}}^{t_{b}} d t\left[\dot{x}^{2}(t)-\Omega^{2}(t) x^{2}(t)\right] $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int \mathcal{D} x e^{i \mathcal{A}[x] / \hbar}=F_{\Omega}\left(t_{b}, t_{a}\right) e^{i \mathcal{A}_{\mathrm{cl}} / \hbar} $$

$$ F_{\Omega}\left(t_{b}, t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar / M}} \frac{1}{\sqrt{D_{a}\left(t_{b}\right)}} $$

$$ F_{\Omega}\left(t_{b}, t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar / M}}\left(\frac{\partial x_{b}}{\partial \dot{x}_{a}}\right)^{-1 / 2}=\frac{1}{\sqrt{2 \pi i \hbar / M}}\left(\frac{\partial \dot{x}_{a}}{\partial x_{b}}\right)^{1 / 2}, $$

$$ F_{\Omega}\left(t_{b}, t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar / M}}\left(-\frac{\partial x_{a}}{\partial \dot{x}_{b}}\right)^{-1 / 2}=\frac{1}{\sqrt{2 \pi i \hbar / M}}\left(-\frac{\partial \dot{x}_{b}}{\partial x_{a}}\right)^{1 / 2} . $$

$$ \mathcal{A}_{\mathrm{cl}}=\frac{M}{2}\left(x_{b} \dot{x}_{b}-x_{a} \dot{x}_{a}\right) $$

$$ \mathcal{A}_{\mathrm{cl}}=\frac{M}{2}\left(x_{b} \frac{\partial \dot{x}_{b}}{\partial x_{b}} x_{b}-x_{a} \frac{\partial \dot{x}_{a}}{\partial x_{a}} x_{a}+x_{b} \frac{\partial \dot{x}_{b}}{\partial x_{a}} x_{a}-x_{a} \frac{\partial \dot{x}_{a}}{\partial x_{b}} x_{b}\right) . $$

$$ \mathcal{A}_{\mathrm{cl}}=\frac{M}{2 D_{a}\left(t_{b}\right)}\left[x_{b}^{2} \dot{D}_{a}\left(t_{b}\right)-x_{a}^{2} \dot{D}_{b}\left(t_{a}\right)-2 x_{b} x_{a}\right] $$

$$ D_{\mathrm{ren}}=D_{a}\left(t_{b}\right)=D_{b}\left(t_{a}\right)=-M\left(\frac{\partial^{2}}{\partial x_{b} \partial x_{a}} \mathcal{A}_{\mathrm{cl}}\right)^{-1} . $$

$$ F_{\Omega}\left(t_{b}, t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar / M}^{D}}\left[\operatorname{det} \frac{\partial x_{b}^{i}}{\partial \dot{x}_{a}^{j}}\right]^{-1 / 2}=\frac{1}{\sqrt{2 \pi i \hbar / M}^{D}}\left[\operatorname{det} \frac{\partial \dot{x}_{a}^{i}}{\partial x_{b}^{j}}\right]^{1 / 2} . $$

$$ F_{\Omega}\left(t_{b}, t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar / M}^{D}}\left|\operatorname{det} \frac{\partial x_{b}^{i}}{\partial \dot{x}_{a}^{j}}\right|^{-1 / 2} e^{-i \nu \pi / 2}=\frac{1}{\sqrt{2 \pi i \hbar / M}^{D}}\left|\operatorname{det} \frac{\partial \dot{x}_{a}^{i}}{\partial x_{b}^{j}}\right|^{1 / 2} e^{-i \nu \pi / 2}, $$

$$ \mathbf{x}_{b} \approx\left(t_{b}-t_{a}\right) \dot{\mathbf{x}}_{a}+\mathbf{x}_{a}, \quad \mathbf{x}_{a} \approx-\left(t_{b}-t_{a}\right) \dot{\mathbf{x}}_{b}+\mathbf{x}_{b}, $$

$$ \frac{\partial x_{b}^{i}}{\partial \dot{x}_{a}^{j}}=\delta_{i j}\left(t_{b}-t_{a}\right), \quad \frac{\partial x_{a}}{\partial \dot{x}_{b}^{j}}=-\delta_{i j}\left(t_{b}-t_{a}\right) $$

$$ \dot{\mathbf{x}}_{b} \approx \dot{\mathbf{x}}_{a} \approx \frac{\mathbf{x}_{b}-\mathbf{x}_{a}}{t_{b}-t_{a}} $$

$$ \frac{\partial \dot{x}_{b}^{i}}{\partial x_{a}^{j}}=-\delta_{i j} \frac{1}{t_{b}-t_{a}}, \quad \frac{\partial \dot{x}_{a}^{i}}{\partial x_{b}^{j}}=\delta_{i j} \frac{1}{t_{b}-t_{a}} . $$

$$ \mathcal{A}_{\mathrm{cl}} \approx \frac{M}{2} \frac{\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right)^{2}}{t_{b}-t_{a}} $$

$$ \mathcal{A}_{\mathrm{cl}}=\frac{M}{2}\left(x_{b}, x_{a}\right) A\binom{x_{b}}{x_{a}} $$

$$ A=\left(\begin{array}{rr} \frac{\partial \dot{x}_{b}}{\partial x_{b}} & \frac{\partial \dot{x}_{b}}{\partial x_{a}} \\ -\frac{\partial \dot{x}_{a}}{\partial x_{b}} & -\frac{\partial \dot{x}_{a}}{\partial x_{a}} \end{array}\right) . $$

$$ A^{-1}=\left(\begin{array}{cc} \frac{\partial x_{b}}{\partial \dot{x}_{b}} & -\frac{\partial x_{b}}{\partial \dot{x}_{a}} \\ \frac{\partial x_{a}}{\partial \dot{x}_{b}} & -\frac{\partial x_{a}}{\partial \dot{x}_{a}} \end{array}\right) $$

$$ \begin{align*} x\left(\dot{x}_{b}, \dot{x}_{a} ; t\right) & =\frac{1}{\dot{D}_{a}\left(t_{b}\right) \dot{D}_{b}\left(t_{a}\right)+1} \\ & \times\left\{\left[D_{a}(t)+D_{b}(t) \dot{D}_{a}\left(t_{b}\right)\right] \dot{x}_{a}+\left[-D_{b}(t)+D_{a}(t) \dot{D}_{b}\left(t_{a}\right)\right] \dot{x}_{b}\right\} \end{align*} $$

$$ \begin{align*} x_{a} & =\frac{1}{\dot{D}_{a}\left(t_{b}\right) \dot{D}_{b}\left(t_{a}\right)+1}\left[D_{b}\left(t_{a}\right) \dot{D}_{a}\left(t_{a}\right) \dot{x}_{b}-D_{b}\left(t_{a}\right) \dot{x}_{b}\right] \\ x_{b} & =\frac{1}{\dot{D}_{a}\left(t_{b}\right) \dot{D}_{b}\left(t_{a}\right)+1}\left[D_{a}\left(t_{b}\right) \dot{x}_{a}+D_{a}\left(t_{b}\right) \dot{D}_{b}\left(t_{a}\right) \dot{x}_{b}\right] \end{align*} $$

$$ A^{-1}=\frac{D_{a}\left(t_{b}\right)}{\dot{D}_{a}\left(t_{b}\right) \dot{D}_{b}\left(t_{a}\right)+1}\left(\begin{array}{cc} \dot{D}_{b}\left(t_{a}\right) & -1 \\ -1 & -\dot{D}_{a}\left(t_{b}\right) \end{array}\right) . $$

$$ \operatorname{det} A=-\frac{\partial\left(\dot{x}_{b}, \dot{x}_{a}\right)}{\partial\left(x_{b}, x_{a}\right)}=-\frac{\dot{D}_{a}\left(t_{b}\right) \dot{D}_{b}\left(t_{a}\right)+1}{D_{a}\left(t_{b}\right) D_{b}\left(t_{a}\right)} $$

$$ \begin{align*} \left(p_{b} t_{b} \mid p_{a} t_{a}\right) & =\int d x_{b} e^{-i p_{b} x_{b} / \hbar} \int d x_{a} e^{i p_{a} x_{a} / \hbar}\left(x_{b} t_{b} \mid x_{a} t_{a}\right) \\ = & \sqrt{\frac{2 \pi \hbar}{i M}} \sqrt{\frac{D_{a}\left(t_{b}\right)}{\dot{D}_{a}\left(t_{b}\right) \dot{D}_{b}\left(t_{a}\right)+1}} \\ \times & \exp \left\{\frac{i}{\hbar} \frac{1}{2 M} \frac{D_{a}\left(t_{b}\right)}{\dot{D}_{a}\left(t_{b}\right) \dot{D}_{b}\left(t_{a}\right)+1}\left[-\dot{D}_{b}\left(t_{a}\right) p_{b}^{2}+\dot{D}_{a}\left(t_{b}\right) p_{a}^{2}-2 p_{b} p_{a}\right]\right\} \end{align*} $$

$$ \mathcal{A}_{\mathrm{cl}}=\frac{M}{2}\left(\mathbf{x}_{b}^{T}, \mathbf{x}_{a}^{T}\right) \mathbf{A}\binom{\mathbf{x}_{b}}{\mathbf{x}_{a}} $$

$$ \mathbf{A}=\left(\begin{array}{cc} \frac{\partial \dot{\mathbf{x}}_{b}}{\partial \mathbf{x}_{b}} & \frac{\partial \dot{\mathbf{x}}_{b}}{\partial \mathbf{x}_{a}} \\ -\frac{\partial \dot{\mathbf{x}}_{a}}{\partial \mathbf{x}_{b}} & -\frac{\partial \dot{\mathbf{x}}_{a}}{\partial \mathbf{x}_{a}} \end{array}\right), \quad \mathbf{A}^{-1}=\left(\begin{array}{cc} \frac{\partial \mathbf{x}_{b}}{\partial \dot{\mathbf{x}}_{b}} & -\frac{\partial \mathbf{x}_{b}}{\partial \dot{\mathbf{x}}_{a}} \\ \frac{\partial \mathbf{x}_{a}}{\partial \dot{\mathbf{x}}_{b}} & -\frac{\partial \mathbf{x}_{a}}{\partial \dot{\mathbf{x}}_{a}} \end{array}\right) . $$

$$ \mathbf{A}=\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) $$

$$ \mathbf{A}=\left(\begin{array}{ll} a & b \\ c & d \end{array}\right)=\left(\begin{array}{cc} a & 0 \\ c & 1 \end{array}\right)\left(\begin{array}{cc} 1 & a^{-1} b \\ 0 & d-c a^{-1} b \end{array}\right)=\left(\begin{array}{cc} 1 & b \\ 0 & d \end{array}\right)\left(\begin{array}{cc} a-b d^{-1} c & 0 \\ d^{-1} c & 1 \end{array}\right) $$

$$ \operatorname{det}\left(\begin{array}{ll} a & b \\ c & d \end{array}\right)=\operatorname{det} a \cdot \operatorname{det}\left(d-c a^{-1} b\right)=\operatorname{det}\left(a-b d^{-1} c\right) \cdot \operatorname{det} d $$

$$ \mathbf{A}=\left(\begin{array}{cc} 1 & -a^{-1} b x \\ 0 & x \end{array}\right)\left(\begin{array}{cc} a^{-1} & 0 \\ -c a^{-1} & 1 \end{array}\right)=\left(\begin{array}{cc} a^{-1}+a^{-1} b x c a^{-1}-a^{-1} b x \\ -x c a^{-1} & x \end{array}\right), x \equiv\left(d-c a^{-1} b\right)^{-1} $$

$$ \begin{align*} \left(\mathbf{p}_{b} t_{b} \mid \mathbf{p}_{a} t_{a}\right) & =\int d x_{b} e^{-i \mathbf{p}_{b} \mathbf{x}_{b} / \hbar} \int d x_{a} e^{i \mathbf{p}_{a} \mathbf{x}_{a} / \hbar}\left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right) \\ = & \frac{2 \pi}{\sqrt{2 \pi i \hbar M}} \frac{1}{\sqrt{D_{\mathrm{ren}} \operatorname{det} \mathbf{A}}} \exp \left\{\frac{i}{\hbar} \frac{1}{2 M}\left[\left(\mathbf{p}_{b}^{T}, \mathbf{p}_{a}^{T}\right) \mathbf{A}^{-1}\binom{\mathbf{p}_{b}}{\mathbf{p}_{a}}\right]\right\} \end{align*} $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int \frac{d p}{(2 \pi \hbar)} e^{i p\left(x_{b}-x_{a}\right) / \hbar} e^{-i p^{2}\left(t_{b}-t_{a}\right) / 2 M \hbar} $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\sum_{n=0}^{\infty} \psi_{n}\left(x_{b}\right) \psi_{n}^{*}\left(x_{a}\right) e^{-i E_{n}\left(t_{b}-t_{a}\right) / \hbar} $$

$$ \psi_{p}(x)=\frac{1}{\sqrt{2 \pi \hbar}} e^{i p x} $$

$$ \begin{align*} \left(x_{b} t_{b} \mid x_{a} t_{a}\right)= & \frac{1}{\sqrt{2 \pi i \hbar \sin \left[\omega\left(t_{b}-t_{a}\right)\right] / M \omega}} \\ & \times \exp \left\{\frac{i M \omega}{2 \hbar \sin \left[\omega\left(t_{b}-t_{a}\right)\right]}\left[\left(x_{b}^{2}+x_{a}^{2}\right) \cos \omega\left(t_{b}-t_{a}\right)-2 x_{b} x_{a}\right]\right\} \end{align*} $$

$$ \begin{align*} & \frac{1}{\sqrt{1-a^{2}}} \exp \left\{-\frac{1}{2\left(1-a^{2}\right)}\left[\left(x^{2}+x^{\prime 2}\right)\left(1+a^{2}\right)-4 x x^{\prime} a\right]\right\} \\ & \quad=\exp \left(-x^{2} / 2-x^{\prime 2} / 2\right) \sum_{n=0}^{\infty} \frac{a^{n}}{2^{n} n!} H_{n}(x) H_{n}\left(x^{\prime}\right) \end{align*} $$

$$ x \equiv \sqrt{M \omega / \hbar} x_{b}, \quad x^{\prime} \equiv \sqrt{M \omega / \hbar} x_{a}, \quad a \equiv e^{-i \omega\left(t_{b}-t_{a}\right)} $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\sum_{n=0}^{\infty} \psi_{n}\left(x_{b}\right) \psi_{n}\left(x_{a}\right) e^{-i(n+1 / 2) \omega\left(t_{b}-t_{a}\right)} $$

$$ E_{n}=\hbar \omega(n+1 / 2) $$

$$ \psi_{n}(x)=N_{n} \lambda_{\omega}^{-1 / 2} e^{-x^{2} / 2 \lambda_{\omega}^{2}} H_{n}\left(x / \lambda_{\omega}\right) $$

$$ \lambda_{\omega} \equiv \sqrt{\frac{\hbar}{M \omega}} $$

$$ N_{n}=\left(1 / 2^{n} n!\sqrt{\pi}\right)^{1 / 2} $$

$$ \int_{-\infty}^{\infty} d x \psi_{n}(x) \psi_{n^{\prime}}(x)^{*}=\delta_{n n^{\prime}} $$

$$ \frac{1}{2^{n} n!\sqrt{\pi}} \int_{-\infty}^{\infty} d x e^{-x^{2}} H_{n}(x) H_{n^{\prime}}(x)=\delta_{n, n^{\prime}} $$

$$ \mathcal{A}[x]=\int_{t_{a}}^{t_{b}} d t \frac{M}{2}\left[g(t) \dot{x}^{2}(t)-\Omega^{2}(t) x^{2}(t)\right] $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int_{x\left(t_{a}\right)=x_{a}}^{x\left(t_{b}\right)=x_{b}} \mathcal{D}^{\prime} x \int \frac{\mathcal{D} p}{2 \pi \hbar} e^{i \mathcal{A}[p, x] / \hbar} $$

$$ \mathcal{A}[p, x]=\int_{t_{a}}^{t_{b}} d t\left[p \dot{x}-\frac{p^{2}}{2 M g(t)}-\frac{M}{2} \Omega^{2}(t) x^{2}(t)\right] . $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right) \approx \frac{1}{\sqrt{2 \pi \hbar i \epsilon / M g\left(t_{N+1}\right)}} \prod_{n=1}^{N}\left[\int_{-\infty}^{\infty} \frac{d x_{n}}{\sqrt{2 \pi \hbar i \epsilon / M g\left(t_{n}\right)}}\right] \exp \left(\frac{i}{\hbar} \mathcal{A}^{N}\right) $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int \mathcal{D} x \sqrt{g} \exp \left\{\frac{i}{\hbar} \mathcal{A}[x]\right\}, $$

$$ \left[-\partial_{t} g(t) \partial_{t}-\Omega^{2}(t)\right] x(t)=0 $$

$$ \tilde{x}(t)=\sqrt{g(t)} x(t), \quad \tilde{\Omega}^{2}(t)=\frac{1}{g(t)}\left[\Omega^{2}(t)+\frac{\dot{g}^{2}(t)}{4 g(t)}-\frac{\ddot{g}(t)}{2}\right], $$

$$ \sqrt{g(t)}\left[-\partial_{t}^{2}-\tilde{\Omega}^{2}(t)\right] \tilde{x}(t)=0 $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int \mathcal{D} x \sqrt{g} e^{i \mathcal{A}[x] / \hbar}=F\left(x_{b}, t_{b} ; x_{a}, t_{a}\right) e^{i \mathcal{A}_{\mathrm{cl}} / \hbar} $$

$$ F\left(x_{b}, t_{b} ; x_{a}, t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar / M}} \frac{1}{\sqrt{D_{a}\left(t_{b}\right)}}, $$

$$ \mathcal{A}_{\mathrm{cl}}=\frac{M}{2}\left(g_{b} x_{b} \dot{x}_{b}-g_{a} x_{a} \dot{x}_{a}\right), $$

$$ \begin{array}{lll} {\left[\partial_{t} g(t) \partial_{t}+\Omega^{2}(t)\right] D_{a}(t)=0 ;} & D_{a}\left(t_{a}\right)=0, & \dot{D}_{a}\left(t_{a}\right)=1 / g_{a}, \\ {\left[\partial_{t} g(t) \partial_{t}+\Omega^{2}(t)\right] D_{b}(t)=0 ;} & D_{b}\left(t_{b}\right)=0, & \dot{D}_{b}\left(t_{b}\right)=-1 / g_{b}, \end{array} $$

$$ x\left(x_{b}, x_{a} ; t\right)=\frac{D_{b}(t)}{D_{b}\left(t_{a}\right)} x_{a}+\frac{D_{a}(t)}{D_{a}\left(t_{b}\right)} x_{b} $$

$$ \mathcal{A}_{\mathrm{cl}}=\frac{M}{2 D_{a}\left(t_{b}\right)}\left[g_{b} x_{b}^{2} \dot{D}_{a}\left(t_{b}\right)-g_{a} x_{a}^{2} \dot{D}_{b}\left(t_{a}\right)-2 x_{b} x_{a}\right] $$

$$ D_{\mathrm{ren}}=D_{a}\left(t_{b}\right)=D_{b}\left(t_{a}\right)=-M\left(\frac{\partial^{2} \mathcal{A}_{\mathrm{cl}}}{\partial x_{b} \partial x_{a}}\right)^{-1}, $$

$$ F\left(x_{b}, t_{b} ; x_{a}, t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar}} \sqrt{-\frac{\partial^{2} \mathcal{A}_{\mathrm{cl}}}{\partial x_{b} \partial x_{a}}} . $$

$$ D_{a}(t)=\int_{t_{a}}^{t} d t^{\prime} g^{-1}\left(t^{\prime}\right), \quad D_{b}(t)=\int_{t}^{t_{b}} d t^{\prime} g^{-1}\left(t^{\prime}\right), \quad D_{\mathrm{ren}}=D_{a}\left(t_{b}\right)=D_{b}\left(t_{a}\right)=\int_{t_{a}}^{t_{b}} d t^{\prime} g^{-1}\left(t^{\prime}\right) $$

$$ \mathcal{A}_{\mathrm{cl}}=\frac{M}{2} \frac{\left(x_{b}-x_{a}\right)^{2}}{D_{a}\left(t_{b}\right)} $$

$$ \mathcal{A}=\int_{t_{a}}^{t_{b}} d t \frac{M}{2}\left[g(t) \dot{x}^{2}+2 b(t) x \dot{x}-\Omega^{2}(t) x^{2}\right] $$

$$ \left[\partial_{t} g(t) \partial_{t}+\dot{b}(t)+\Omega^{2}(t)\right] x=0 $$

$$ Z=\operatorname{Tr}\left(e^{-\hat{H} / k_{B} T}\right)=\sum_{n} e^{-E_{n} / k_{B} T} $$

$$ Z_{\mathrm{QM}}=\operatorname{Tr}\left(e^{-i\left(t_{b}-t_{a}\right) \hat{H} / \hbar}\right) $$

$$ t_{b}-t_{a}=-\frac{i \hbar}{k_{B} T} \equiv-i \hbar \beta $$

$$ Z \equiv \int_{-\infty}^{\infty} d x z(x)=\int_{-\infty}^{\infty} d x\langle x| e^{-\beta \hat{H}}|x\rangle=\left.\int_{-\infty}^{\infty} d x\left(x t_{b} \mid x t_{a}\right)\right|_{t_{b}-t_{a}=-i \hbar \beta} $$

$$ z(x) \equiv\langle x| e^{-\beta \hat{H}}|x\rangle=\left.\left(x t_{b} \mid x t_{a}\right)\right|_{t_{b}-t_{a}=-i \hbar \beta} $$

$$ z_{\omega}(x)=\frac{1}{\sqrt{2 \pi \hbar / M}} \sqrt{\frac{\omega}{\sinh \hbar \beta \omega}} \exp \left(-\frac{M \omega}{\hbar} \tanh \frac{\hbar \beta \omega}{2} x^{2}\right) . $$

$$ \begin{align*} Z & \equiv \prod_{n=1}^{N+1}\left[\int_{-\infty}^{\infty} d x_{n}\right] \\ & \times\left\langle x_{N+1}\right| e^{-\epsilon \hat{H} / \hbar}\left|x_{N}\right\rangle\left\langle x_{N}\right| e^{-\epsilon \hat{H} / \hbar}\left|x_{N-1}\right\rangle \times \ldots \times\left\langle x_{2}\right| e^{-\epsilon \hat{H} / \hbar}\left|x_{1}\right\rangle\left\langle x_{1}\right| e^{-\epsilon \hat{H} / \hbar}\left|x_{N+1}\right\rangle \end{align*} $$

$$ \left\langle x_{n}\right| e^{-\epsilon \hat{H} / \hbar}\left|x_{n-1}\right\rangle \approx \int_{-\infty}^{\infty} \frac{d p_{n}}{2 \pi \hbar} e^{i p_{n}\left(x_{n}-x_{n-1}\right) / \hbar-\epsilon H\left(p_{n}, x_{n}\right) / \hbar} $$

$$ Z \approx \prod_{n=1}^{N+1}\left[\int_{-\infty}^{\infty} d x_{n} \int_{-\infty}^{\infty} \frac{d p_{n}}{2 \pi \hbar}\right] \exp \left(-\frac{1}{\hbar} \mathcal{A}_{\mathrm{e}}^{N}\right) $$

$$ \mathcal{A}_{\mathrm{e}}^{N}=\sum_{n=1}^{N+1}\left[-i p_{n}\left(x_{n}-x_{n-1}\right)+\epsilon H\left(p_{n}, x_{n}\right)\right] $$

$$ \mathcal{A}_{\mathrm{e}}[p, x]=\int_{0}^{\hbar \beta} d \tau[-i p(\tau) \dot{x}(\tau)+H(p(\tau), x(\tau))] $$

$$ Z=\int \mathcal{D} x \int \frac{\mathcal{D} p}{2 \pi \hbar} e^{-\mathcal{A}_{\mathrm{e}}[p, x] / \hbar} $$

$$ H=L_{\mathrm{e}}+i \frac{\partial L_{\mathrm{e}}}{\partial \dot{x}} \dot{x}=L_{\mathrm{e}}+i p \dot{x} $$

$$ \oint \mathcal{D} x \int \frac{\mathcal{D} p}{2 \pi \hbar}=\oint \frac{\mathcal{D} p}{2 \pi \hbar} \int \mathcal{D} x=\prod_{n=1}^{N+1} \int_{-\infty}^{\infty} d x_{n} \int_{-\infty}^{\infty} \frac{d p_{n}}{2 \pi \hbar} $$

$$ \rho\left(x_{a}\right) \equiv Z^{-1}\left\langle x_{a}\right| e^{-\hat{H} / k_{B} T}\left|x_{a}\right\rangle . $$

$$ \int_{-\infty}^{\infty} d x \rho(x)=1 $$

$$ \rho\left(x_{a}\right)=\sum_{n}\left|\psi_{n}\left(x_{a}\right)\right|^{2} e^{-\beta E_{n}} / \sum_{n} e^{-\beta E_{n}} . $$

$$ \rho\left(x_{a}\right) \xrightarrow{T \rightarrow 0}\left|\psi_{0}\left(x_{a}\right)\right|^{2} . $$

$$ Z \xrightarrow{T \rightarrow \infty} Z_{\mathrm{cl}}=\int_{-\infty}^{\infty} d x \int_{-\infty}^{\infty} \frac{d p}{2 \pi \hbar} e^{-H(p, x) / k_{B} T} $$

$$ \rho(x) \xrightarrow{T \rightarrow \infty} \rho_{\mathrm{cl}}(x)=Z_{\mathrm{cl}}^{-1} \int_{-\infty}^{\infty} \frac{d p}{2 \pi \hbar} e^{-H(p, x) / k_{B} T} $$

$$ Z \approx\left[\int_{-\infty}^{\infty} d x\right]\langle x| e^{-\epsilon \hat{H} / \hbar}|x\rangle $$

$$ \langle x| e^{-\epsilon \hat{H}}|x\rangle \approx \int_{-\infty}^{\infty} \frac{d p_{n}}{2 \pi \hbar} e^{-\epsilon H\left(p_{n}, x\right) / \hbar} $$

$$ H(p, x)=\frac{p^{2}}{2 M}+V(x) $$

$$ \int_{-\infty}^{\infty} \frac{d p}{2 \pi \hbar} e^{-a p^{2} / 2 \hbar}=\frac{1}{\sqrt{2 \pi \hbar a}} $$

$$ Z_{\mathrm{cl}}=\int_{-\infty}^{\infty} \frac{d x}{\sqrt{2 \pi \hbar^{2} / M k_{B} T}} e^{-V(x) / k_{B} T}=\int_{-\infty}^{\infty} \frac{d x}{l_{\mathrm{e}}(\hbar \beta)} e^{-\beta V(x)} $$

$$ l_{\mathrm{e}}(\hbar \beta) \equiv \sqrt{2 \pi \hbar^{2} \beta / M} $$

$$ \rho(x) \xrightarrow{T \rightarrow \infty} \rho_{\mathrm{cl}}(x)=Z_{\mathrm{cl}}^{-1} \frac{1}{l_{\mathrm{e}}(\hbar \beta)} e^{-\bar{V}(x)} $$

$$ Z_{\mathrm{cl}}=\frac{L}{l_{\mathrm{e}}(\hbar \beta)} $$

$$ Z_{\mathrm{cl}}=\frac{V_{D}}{l_{\mathrm{e}}^{D}(\hbar \beta)} $$

$$ Z_{\mathrm{cl}}=\frac{l_{\omega}^{D}}{l^{D}(\hbar \beta)} $$

$$ l_{\omega} \equiv \sqrt{\frac{2 \pi}{\beta M \omega^{2}}} $$

$$ l_{\omega} l_{\mathrm{e}}(\hbar \beta)=2 \pi \lambda_{\omega}^{2} $$

$$ l_{\omega} \xrightarrow[\omega \rightarrow 0]{ } L $$

$$ \frac{1}{\omega} \underset{\omega \rightarrow 0}{\longrightarrow} \sqrt{\frac{\beta M}{2 \pi}} L $$

$$ \frac{1}{\omega} \underset{\omega \rightarrow 0}{\longrightarrow} \sqrt{\frac{\left(t_{b}-t_{a}\right) M}{2 \pi \hbar}} L $$

$$ \begin{align*} \rho\left(x_{a}\right) & =Z^{-1} \int_{x(0)=x_{a}}^{x(\hbar \beta)=x_{b}} \mathcal{D}^{\prime} x \int \frac{\mathcal{D} p}{2 \pi \hbar} e^{-\mathcal{A}_{\mathrm{e}}[p, x] / \hbar} \\ & =Z^{-1} \int_{x(0)=x_{a}}^{x(\hbar \beta)=x_{b}} \mathcal{D} x e^{-\mathcal{A}_{\mathrm{e}}[x] / \hbar} \end{align*} $$

$$ \langle\hat{O}\rangle_{T} \equiv Z^{-1} \sum_{n} e^{-\beta E_{n}}\langle n| \hat{O}|n\rangle $$

$$ \langle\hat{O}\rangle_{T}=Z^{-1} \iint_{-\infty}^{\infty} d x_{b} d x_{a}\left\langle x_{b}\right| e^{-\beta \hat{H}}\left|x_{a}\right\rangle\left\langle x_{a}\right| \hat{O}\left|x_{b}\right\rangle $$

$$ \langle f(\hat{x})\rangle_{T}=Z^{-1} \iint_{-\infty}^{\infty} d x_{b} d x_{a}\left\langle x_{b}\right| e^{-\beta \hat{H}}\left|x_{a}\right\rangle \delta\left(x_{b}-x_{a}\right) f\left(x_{a}\right)=\int d x \rho(x) f(x) $$

$$ \rho\left(x_{b}, x_{a}\right) \equiv Z^{-1}\left\langle x_{b}\right| e^{-\beta \hat{H}}\left|x_{a}\right\rangle, $$

$$ \hat{U}_{\mathrm{e}}\left(\tau_{b}, \tau_{a}\right) \equiv e^{-\left(\tau_{b}-\tau_{a}\right) \hat{H} / \hbar}, \quad \tau_{b}>\tau_{a}, $$

$$ \left(x_{b} \tau_{b} \mid x_{a} \tau_{a}\right) \equiv\left\langle x_{b}\right| \hat{U}_{\mathrm{e}}\left(\tau_{b}, \tau_{a}\right)\left|x_{a}\right\rangle, \quad \tau_{b}>\tau_{a} $$

$$ Z=\int_{-\infty}^{\infty} d x(x \hbar \beta \mid x 0) $$

$$ \rho\left(x_{b}, x_{a}\right)=Z^{-1}\left(x_{b} \hbar \beta \mid x_{a} 0\right) . $$

$$ \hat{U}\left(\tau_{b}, \tau_{a}\right)=T_{\tau} \exp \left[-\frac{1}{\hbar} \int_{\tau_{a}}^{\tau_{b}} d \tau \hat{H}(-i \tau)\right] $$

$$ \left(x_{b} \tau_{b} \mid x_{a} \tau_{a}\right) \approx \prod_{n=1}^{N}\left[\int_{-\infty}^{\infty} d x_{n}\right] \prod_{n=1}^{N+1}\left[\int_{-\infty}^{\infty} \frac{d p_{n}}{2 \pi \hbar}\right] \exp \left(-\mathcal{A}_{\mathrm{e}}^{N} / \hbar\right) $$

$$ \mathcal{A}_{\mathrm{e}}^{N}=\sum_{n=1}^{N+1}\left[-i p_{n}\left(x_{n}-x_{n-1}\right)+\epsilon H\left(p_{n}, x_{n}, \tau_{n}\right)\right] $$

$$ \left(x_{b} \tau_{b} \mid x_{a} \tau_{a}\right)=\int \mathcal{D}^{\prime} x \int \frac{\mathcal{D} p}{2 \pi \hbar} \exp \left\{-\frac{1}{\hbar} \mathcal{A}_{\mathrm{e}}[p, x]\right\} $$

$$ \begin{align*} \left(x_{b} \tau_{b} \mid x_{a} \tau_{a}\right)= & \int \mathcal{D} x \exp \left\{-\frac{1}{\hbar} \int_{0}^{\hbar \beta} d \tau\left[\frac{M}{2}\left(\partial_{\tau} x\right)^{2}+V(x, \tau)\right]\right\} \\ \approx & \frac{1}{\sqrt{2 \pi \hbar \epsilon / M}} \prod_{n=1}^{N}\left[\int_{-\infty}^{\infty} \frac{d x_{n}}{\sqrt{2 \pi \beta / M}}\right] \\ & \times \exp \left\{-\frac{1}{\hbar} \epsilon \sum_{n=1}^{N+1}\left[\frac{M}{2}\left(\frac{x_{n}-x_{n-1}}{\epsilon}\right)^{2}+V\left(x_{n}, \tau_{n}\right)\right]\right\} \end{align*} $$

$$ \begin{align*} Z & =\int_{-\infty}^{\infty} d x(x \hbar \beta \mid x 0) \\ & =\int d x \int_{x(0)=x}^{x(\hbar \beta)=x} \mathcal{D} x e^{-\mathcal{A}_{\mathrm{e}}[x] / \hbar}=\oint \mathcal{D} x e^{-\mathcal{A}_{\mathrm{e}}[x] / \hbar} \end{align*} $$

$$ \mathcal{A}_{\mathrm{e}}[x]=\int_{\tau_{a}}^{\tau_{b}} d \tau\left[\frac{M}{2} x^{\prime 2}+V(x, \tau)\right] $$

$$ \oint \mathcal{D} x \approx \prod_{n=1}^{N+1} \int_{-\infty}^{\infty} \frac{d x_{n}}{\sqrt{2 \pi \hbar \epsilon / M}} $$

$$ x(\tau)=\sum_{m=-\infty}^{\infty} \frac{1}{\sqrt{N+1}} e^{-i \omega_{m} \tau} x_{m} $$

$$ \omega_{m} \equiv 2 \pi m k_{B} T / \hbar=\frac{2 \pi m}{\hbar \beta}, \quad m=0, \pm 1, \pm 2, \ldots $$

$$ x(\tau)=x(\tau+\hbar \beta) $$

$$ x_{m}=x_{-m}^{*} \quad(\text { modulo } N+1) $$

$$ Z_{\omega}^{N}=\prod_{n=0}^{N}\left[\int_{-\infty}^{\infty} \frac{d x_{n}}{\sqrt{2 \pi \hbar \epsilon / M}}\right] \exp \left(-\mathcal{A}_{\mathrm{e}}^{N} / \hbar\right) $$

$$ \mathcal{A}_{\mathrm{e}}^{N}=\frac{M}{2 \epsilon} \sum_{n=1}^{N+1} x_{n}\left(-\epsilon^{2} \nabla \bar{\nabla}+\epsilon^{2} \omega^{2}\right) x_{n} $$

$$ Z_{\omega}^{N}=\frac{1}{\sqrt{\operatorname{det}_{N+1}\left(-\epsilon^{2} \nabla \bar{\nabla}+\epsilon^{2} \omega^{2}\right)}} $$

$$ \epsilon^{2} \Omega_{m} \bar{\Omega}_{m}+\epsilon^{2} \omega^{2}=2-2 \cos \omega_{m} \epsilon+\epsilon^{2} \omega^{2}, $$

$$ \begin{align*} T_{m n} x_{n}= & \left(T_{m}\right)_{n} x_{n} \\ = & \sqrt{\frac{2}{N+1}}\left(\frac{1}{\sqrt{2}}, \cos \frac{m}{N+1} 2 \pi \cdot 1, \sin \frac{m}{N+1} 2 \pi \cdot 1,\right. \\ & \cos \frac{m}{N+1} 2 \pi \cdot 2, \sin \frac{m}{N+1} 2 \pi \cdot 2, \ldots \\ & \left.\ldots, \cos \frac{m}{N+1} 2 \pi \cdot n, \sin \frac{m}{N+1} 2 \pi \cdot n, \ldots\right)_{n} x_{n} . \end{align*} $$

$$ \mathcal{A}_{\mathrm{e}}^{N}=\frac{M}{2} \epsilon\left\{\begin{array}{ccc} {\left[\omega^{2} x_{0}^{2}+2 \sum_{m=1}^{N / 2}\left(\Omega_{m} \bar{\Omega}_{m}+\omega^{2}\right)\left|x_{m}\right|^{2}\right]} & \text { for } & N=\text { even }, \\ {\left[\omega^{2} x_{0}^{2}+\left(\Omega_{(N+1) / 2} \bar{\Omega}_{(N+1) / 2}+\omega^{2}\right) x_{N+1}^{2}\right.} & & \\ \left.+2 \sum_{m=1}^{(N-1) / 2}\left(\Omega_{m} \bar{\Omega}_{m}+\omega^{2}\right)\left|x_{m}\right|^{2}\right] & \text { for } & N=\text { odd } . \end{array}\right. $$

$$ \int_{-\infty}^{\infty} d x_{0} \int_{-\infty}^{\infty} d x_{(N+1) / 2} \prod_{m=1}^{(N-1) / 2} \int_{-\infty}^{\infty} d \operatorname{Re} x_{m} \int_{-\infty}^{\infty} d \operatorname{Im} x_{m} \quad \text { for } \quad N=\text { odd } $$

$$ \begin{align*} Z_{\omega}^{N} & =\left[\operatorname{det}_{N+1}\left(-\epsilon^{2} \nabla \bar{\nabla}+\epsilon^{2} \omega^{2}\right)\right]^{-1 / 2}=\left[\prod_{m=0}^{N}\left(\epsilon^{2} \Omega_{m} \bar{\Omega}_{m}+\epsilon^{2} \omega^{2}\right)\right]^{-1 / 2} \\ & =\left\{\prod_{m=0}^{N}\left[2\left(1-\cos \omega_{m} \epsilon\right)+\epsilon^{2} \omega^{2}\right]\right\}^{-1 / 2}=\left[\prod_{m=0}^{N}\left(4 \sin ^{2} \frac{\omega_{m} \epsilon}{2}+\epsilon^{2} \omega^{2}\right)\right]^{-1 / 2} \end{align*} $$

$$ \sin ^{2} \frac{\omega_{m} \epsilon}{2}=\left(1+\cos \frac{\omega_{m} \epsilon}{2}\right)\left(1-\cos \frac{\omega_{m} \epsilon}{2}\right), $$

$$ 1+\cos \frac{\omega_{m} \epsilon}{2} \equiv 1+\cos \frac{\pi m}{N+1} $$

$$ 1-\cos \frac{\omega_{m} \epsilon}{2}=1-\cos \frac{\pi m}{N+1} \equiv 1+\cos \pi \frac{N+1-m}{N+1} $$

$$ Z_{\omega}^{N}=\frac{1}{\epsilon \omega}\left[\prod_{m=1}^{N} 2\left(1-\cos \frac{\omega_{m} \epsilon}{2}\right)\right]^{-1}\left[\prod_{m=1}^{N}\left(1+\frac{\epsilon^{2} \omega^{2}}{4 \sin ^{2} \frac{\omega_{m} \epsilon}{2}}\right)\right]^{-1 / 2} $$

$$ Z_{\omega}^{N}=\frac{k_{B} T}{\hbar \omega}\left[\prod_{m=1}^{N}\left(1+\frac{\epsilon^{2} \omega^{2}}{4 \sin ^{2} \frac{\omega_{m} \epsilon}{2}}\right)\right]^{-1 / 2} $$

$$ Z_{\omega}^{N}=\frac{k_{B} T}{\hbar \omega}\left[\prod_{m=1}^{N / 2}\left(1+\frac{\epsilon^{2} \omega^{2}}{4 \sin ^{2} \frac{m \pi}{N+1}}\right)\right]^{-1} $$

$$ Z_{\omega}^{N}=\frac{k_{B} T}{\hbar \omega}\left[\left(1+\frac{\epsilon^{2} \omega^{2}}{4}\right)^{1 / 2} \prod_{m=1}^{(N-1) / 2}\left(1+\frac{\epsilon^{2} \omega^{2}}{4 \sin ^{2} \frac{\pi m}{N+1}}\right)\right]^{-1} . $$

$$ \sin i \frac{\tilde{\omega}_{\mathrm{e}} \epsilon}{2} \equiv i \frac{\omega \epsilon}{2}, \quad \sinh \frac{\tilde{\omega}_{\mathrm{e}} \epsilon}{2} \equiv \frac{\omega \epsilon}{2} $$

$$ \prod_{m=1}^{(N-1) / 2}\left[1-\frac{\sin ^{2} x}{\sin ^{2} \frac{m \pi}{(N+1)}}\right]=\frac{2}{\sin 2 x} \frac{\sin [(N+1) x]}{(N+1)} $$

$$ Z_{\omega}^{N}=\frac{k_{B} T}{\hbar \omega}\left[\frac{1}{\sinh \left(\tilde{\omega}_{\mathrm{e}} \epsilon / 2\right)} \frac{\sinh \left[(N+1) \tilde{\omega}_{\mathrm{e}} \epsilon / 2\right]}{N+1}\right]^{-1} $$

$$ \prod_{m=1}^{N / 2}\left[1-\frac{\sin ^{2} x}{\sin ^{2} \frac{m \pi}{(N+1)}}\right]=\frac{1}{\sin x} \frac{\sin [(N+1) x]}{(N+1)} $$

$$ Z_{\omega}^{N}=\frac{1}{2 \sinh \left(\hbar \tilde{\omega}_{\mathrm{e}} \beta / 2\right)} $$

$$ Z_{\omega}^{N}=e^{-\hbar \tilde{\omega}_{\mathrm{e}} / 2 k_{B} T}+e^{-3 \hbar \tilde{\omega}_{\mathrm{e}} / 2 k_{B} T}+e^{-5 \hbar \tilde{\omega}_{\mathrm{e}} / 2 k_{B} T}+\ldots $$

$$ E_{n}=\left(n+\frac{1}{2}\right) \hbar \tilde{\omega}_{\mathrm{e}} $$

$$ \tilde{\omega}_{\mathrm{e}}=\frac{2}{\epsilon} \operatorname{arsinh} \frac{\omega \epsilon}{2} $$

$$ Z_{\omega}=\frac{1}{2 \sinh (\beta \hbar \omega / 2)} $$

$$ Z_{\omega}=\frac{k_{B} T}{\hbar \omega}\left[\prod_{m=1}^{\infty}\left(1+\frac{\omega^{2}}{\omega_{m}^{2}}\right)\right]^{-1} $$

$$ Z_{\omega}=\frac{k_{B} T}{\hbar \omega} \frac{\hbar \omega / 2 k_{B} T}{\sinh \left(\hbar \omega / 2 k_{B} T\right)}=\frac{1}{2 \sinh (\beta \hbar \omega / 2)} $$

$$ \begin{align*} Z_{\omega}^{N} & =\left[\operatorname{det}_{N+1}\left(-\epsilon^{2} \nabla \bar{\nabla}+\epsilon^{2} \omega^{2}\right)\right]^{-1 / 2} \\ & =\left[\operatorname{det}_{N+1}^{\prime}\left(-\epsilon^{2} \nabla \bar{\nabla}\right)\right]^{-1 / 2}\left[\frac{\operatorname{det}_{N+1}\left(-\epsilon^{2} \nabla \bar{\nabla}+\epsilon^{2} \omega^{2}\right)}{\operatorname{det}_{N+1}^{\prime}\left(-\epsilon^{2} \nabla \bar{\nabla}\right)}\right]^{-1 / 2} \\ & \xrightarrow{\epsilon \rightarrow 0} \frac{k_{B} T}{\hbar}\left[\frac{\operatorname{det}\left(-\partial_{\tau}^{2}+\omega^{2}\right)}{\operatorname{det}^{\prime}\left(-\partial_{\tau}^{2}\right)}\right]^{-1 / 2}=\frac{k_{B} T}{\hbar \omega} \prod_{m=1}^{\infty}\left[\frac{\omega_{m}^{2}+\omega^{2}}{\omega_{m}^{2}}\right]^{-1} \end{align*} $$

$$ \begin{align*} \left(x_{b} \tau_{b} \mid x_{a} \tau_{a}\right) & =\frac{1}{\sqrt{2 \pi \hbar / M}} \sqrt{\frac{\omega}{\sinh \omega\left(\tau_{b}-\tau_{a}\right)}} \\ & \times \exp \left\{-\frac{1}{2 \hbar} \frac{M \omega}{\sinh \omega\left(\tau_{b}-\tau_{a}\right)}\left[\left(x_{b}^{2}+x_{a}^{2}\right) \cosh \omega\left(\tau_{b}-\tau_{a}\right)-2 x_{b} x_{a}\right]\right\} \end{align*} $$

$$ \begin{align*} Z_{\omega}=\int_{-\infty}^{\infty} d x\left(x \tau_{b} \mid x \tau_{a}\right)= & \frac{1}{\sqrt{2 \pi \hbar\left(\tau_{b}-\tau_{a}\right) / M}} \sqrt{\frac{\omega\left(\tau_{b}-\tau_{a}\right)}{\sinh \left[\omega\left(\tau_{b}-\tau_{a}\right)\right]}} \\ & \times \frac{\sqrt{2 \pi \hbar \sinh \left[\omega\left(\tau_{b}-\tau_{a}\right)\right] / \omega M}}{2 \sinh \left[\omega\left(\tau_{b}-\tau_{a}\right) / 2\right]}=\frac{1}{2 \sinh \left[\omega\left(\tau_{b}-\tau_{a}\right) / 2\right]} \end{align*} $$

$$ \begin{align*} Z_{\omega}^{\text {open }} & =\int_{-\infty}^{\infty} d x_{b} \int_{-\infty}^{\infty} d x_{a}\left(x_{b} \tau_{b} \mid x_{a} \tau_{a}\right)=\frac{1}{\sqrt{2 \pi \hbar\left(\tau_{b}-\tau_{a}\right) / M}} \sqrt{\frac{\omega\left(\tau_{b}-\tau_{a}\right)}{\sinh \left[\omega\left(\tau_{b}-\tau_{a}\right)\right]}} \frac{2 \pi \hbar}{M \omega} \\ & =\sqrt{\frac{2 \pi \hbar}{M \omega}} \frac{1}{\sqrt{\sinh \left[\omega\left(\tau_{b}-\tau_{a}\right)\right]}} \end{align*} $$

$$ \begin{align*} \left(x_{b} \tau_{b} \mid x_{a} \tau_{a}\right) & =\int \mathcal{D}^{\prime} x \int \frac{\mathcal{D} p}{2 \pi \hbar} e^{-\int_{\tau_{a}}^{\tau_{b}} d \tau\left[-i p \dot{x}+p^{2} / 2 M+M \Omega^{2}(\tau) x^{2} / 2\right] / \hbar} \\ & =\int \mathcal{D} x e^{-\int_{\tau_{a}}^{\tau_{b}} d \tau\left[M \dot{x}^{2}+\Omega^{2}(\tau) x^{2}\right] / 2 \hbar} \end{align*} $$

$$ F^{N}\left(\tau_{a}-\tau_{b}\right)=\operatorname{det}_{N+1}\left[-\epsilon^{2} \nabla \bar{\nabla}+\epsilon \Omega^{2}(\tau)\right]^{-1 / 2} $$

$$ F\left(\tau_{a}-\tau_{b}\right)=\frac{k_{B} T}{\hbar}\left[\frac{\operatorname{det}\left(-\partial_{\tau}^{2}+\Omega^{2}(\tau)\right)}{\operatorname{det}^{\prime}\left(-\partial_{\tau}^{2}\right)}\right]^{-1 / 2} $$

$$ F\left(\tau_{b}, \tau_{a}\right)=\frac{1}{2 \sinh (\beta \hbar \omega / 2)}\left[\frac{\operatorname{det}\left(-\partial_{\tau}^{2}+\Omega^{2}(\tau)\right)}{\operatorname{det}\left(-\partial_{\tau}^{2}+\omega^{2}\right)}\right]^{-1 / 2} $$

$$ -\epsilon^{2} \nabla \bar{\nabla}=\left(\begin{array}{cccccccc} 2 & -1 & 0 & & \ldots & 0 & 0 & -1 \\ -1 & 2 & -1 & & \ldots & 0 & 0 & 0 \\ \vdots & & & & & & & \vdots \\ 0 & 0 & & 0 & \ldots & -1 & 2 & -1 \\ -1 & 0 & 0 & & \ldots & 0 & -1 & 2 \end{array}\right) . $$

$$ -\epsilon^{2} \nabla \bar{\nabla}+\epsilon^{2} \Omega^{2}=\left(\begin{array}{cccccc} 2+\epsilon^{2} \Omega_{N+1}^{2} & -1 & 0 & \ldots & 0 & -\alpha \\ -1 & 2+\epsilon^{2} \Omega_{N}^{2} & -1 & \ldots & 0 & 0 \\ \vdots & & & & & \vdots \\ -\alpha & 0 & 0 & \ldots & -1 & 2+\epsilon^{2} \Omega_{1}^{2} \end{array}\right) . $$

$$ \begin{align*} \tilde{D}_{N+1}= & \left(2+\epsilon^{2} \Omega_{N+1}^{2}\right) \\ & \times \operatorname{det}_{N}\left(\begin{array}{cccccc} 2+\epsilon^{2} \Omega_{N}^{2} & -1 & 0 & \ldots & 0 & 0 \\ \vdots & & & & & \vdots \\ 0 & 0 & 0 & \ldots & -1 & 2+\epsilon^{2} \Omega_{1}^{2} \end{array}\right) \end{align*} $$

$$ \left(-\epsilon^{2} \bar{\nabla} \nabla+\epsilon^{2} \Omega_{N+1}^{2}\right) D_{N}=0 $$

$$ \begin{align*} D_{1} & =2+\epsilon^{2} \Omega_{1}^{2} \\ D_{2} & =\left(2+\epsilon^{2} \Omega_{1}^{2}\right)\left(2+\epsilon^{2} \Omega_{2}^{2}\right)-1 \end{align*} $$

$$ -D_{N-1}-\alpha $$

$$ (-1)^{N}\left[1+\left(2+\epsilon^{2} \Omega_{N}^{2}\right) H_{N-1}-H_{N-2}\right], $$

$$ \begin{align*} & H_{N-1} \equiv(-1)^{N-1} \\ & \quad \times \operatorname{det}_{N-1}\left(\begin{array}{ccccccc} 0 & 0 & 0 & \ldots & 0 & 0 & -\alpha \\ 2+\epsilon^{2} \Omega_{N-1}^{2} & -1 & 0 & \ldots & 0 & 0 & 0 \\ -1 & 2+\epsilon^{2} \Omega_{N-2}^{2} & -1 & \ldots & 0 & 0 & 0 \\ \vdots & & & & & & \vdots \\ 0 & 0 & 0 & \ldots & -1 & 2+\epsilon^{2} \Omega_{2}^{2} & -1 \end{array}\right) . \end{align*} $$

$$ \left(-\epsilon^{2} \bar{\nabla} \nabla+\epsilon^{2} \Omega_{N+1}^{2}\right) H_{N}=0 $$

$$ \begin{align*} H_{2} & =\left|\begin{array}{cc} 0 & -\alpha \\ 2+\epsilon^{2} \Omega_{2}^{2} & -1 \end{array}\right|=\alpha\left(2+\epsilon^{2} \Omega_{2}^{2}\right), \\ H_{3} & =-\left|\begin{array}{ccc} 0 & 0 & -\alpha \\ 2+\epsilon^{2} \Omega_{3}^{2} & -1 & 0 \\ -1 & 2+\epsilon^{2} \Omega_{2}^{2} & -1 \end{array}\right| \\ & =\alpha\left[\left(2+\epsilon^{2} \Omega_{2}^{2}\right)\left(2+\epsilon^{2} \Omega_{3}^{2}\right)-1\right] . \end{align*} $$

$$ H_{N}=\alpha D_{N-1}^{+} $$

$$ \begin{align*} \tilde{D}_{N+1}= & \left(2+\epsilon^{2} \Omega_{N}^{2}\right) D_{N}-D_{N-1}-\alpha \\ & -\alpha\left[1+\left(2+\epsilon^{2} \Omega_{N}^{2}\right) \alpha D_{N-2}^{+}-\alpha D_{N-3}^{+}\right] \end{align*} $$

$$ \tilde{D}_{N+1}=D_{N+1}-\alpha^{2} D_{N-1}^{+}-2 \alpha $$

$$ \tilde{D}_{N+1}=D_{N+1}-D_{N-1}^{+}-2 $$

$$ \left[-\partial_{\tau}^{2}+\Omega^{2}(\tau)\right] D_{\text {ren }}(\tau)=0, \quad D_{\text {ren }}(0)=0, \quad \dot{D}_{\text {ren }}(0)=1 . $$

$$ \operatorname{det}\left(-\epsilon^{2} \bar{\nabla} \nabla+\epsilon \Omega^{2}\right)_{T} \xrightarrow{\epsilon \rightarrow 0} 2\left[\dot{D}_{\mathrm{ren}}(\hbar \beta)-1\right] $$

$$ Z_{\Omega}=\frac{1}{\sqrt{2\left[\dot{D}_{\mathrm{ren}}(\hbar \beta)-1\right]}} $$

$$ Z_{\Omega}=\frac{1}{2 \sqrt{\dot{D}_{a}\left(t_{b}\right)-1}}, \quad t_{b}=i \hbar \beta $$

$$ D_{\mathrm{ren}}(\tau)=\frac{1}{\omega} \sinh \omega \tau $$

$$ 2\left[\dot{D}_{\mathrm{ren}}(\tau)-1\right]=2(\cosh \beta \hbar \omega-1)=4 \sinh ^{2}(\beta \hbar \omega / 2) $$

$$ \begin{align*} Z_{\omega} & =\left.\left\{2\left[\dot{D}_{\mathrm{ren}}(\tau)-1\right]\right\}^{-1 / 2}\right|_{\tau=\hbar \beta} \\ & =\frac{1}{2 \sinh (\beta \hbar \omega / 2)} \end{align*} $$

$$ D_{N}=\frac{\sinh (N+1) \tilde{\omega_{\mathrm{e}}} \epsilon}{\sinh \tilde{\omega_{\mathrm{e}}} \epsilon} $$

$$ \begin{align*} \tilde{D}_{N+1} & =\frac{1}{\sinh \tilde{\omega_{\mathrm{e}}} \epsilon}\left[\sinh (N+2) \tilde{\omega_{\mathrm{e}}} \epsilon-\sinh N \tilde{\omega_{\mathrm{e}}} \epsilon\right]-2 \\ & =2\left[\cosh (N+1) \tilde{\omega_{\mathrm{e}}} \epsilon-1\right]=4 \sinh ^{2}\left[(N+1) \tilde{\omega_{\mathrm{e}}} \epsilon / 2\right] \end{align*} $$

$$ Z_{\omega}=\frac{1}{\sqrt{\tilde{D}_{N+1}}}=\frac{1}{2 \sinh \left(\hbar \tilde{\omega_{\mathrm{e}}} \beta / 2\right)} $$

$$ x(\tau)=x_{0}+\eta(\tau) \equiv x_{0}+\sum_{m=1}^{\infty}\left(x_{m} e^{i \omega_{m} \tau}+\text { c.c. }\right), \quad x_{0}=\text { real, } \quad x_{-m} \equiv x_{m}^{*} . $$

$$ x_{0}=\bar{x} \equiv \frac{k_{B} T}{\hbar} \int_{0}^{\hbar / k_{B} T} d \tau x(\tau) $$

$$ \begin{align*} \mathcal{A}_{\mathrm{e}} & =\frac{M}{2} \int_{0}^{\hbar / k_{B} T} d \tau\left(\dot{x}^{2}+\omega^{2} x^{2}\right) \\ & =\frac{M \hbar}{k_{B} T}\left[\frac{\omega^{2}}{2} x_{0}^{2}+\sum_{m=1}^{\infty}\left(\omega_{m}^{2}+\omega^{2}\right)\left|x_{m}\right|^{2}\right] . \end{align*} $$

$$ \int_{-\infty}^{\infty} d x_{0} \prod_{m=1}^{\infty} \int_{-\infty}^{\infty} d \operatorname{Re} x_{m} \int_{-\infty}^{\infty} d \operatorname{Im} x_{m} $$

$$ \oint \mathcal{D} x \equiv \int_{-\infty}^{\infty} \frac{d x_{0}}{l_{\mathrm{e}}(\hbar \beta)} \prod_{m=1}^{\infty}\left[\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{d \operatorname{Re} x_{m} d \operatorname{Im} x_{m}}{\pi k_{B} T / M \omega_{m}^{2}}\right] $$

$$ \oint \mathcal{D} x \equiv \int_{-\infty}^{\infty} \frac{d x_{0}}{l_{\mathrm{e}}(\hbar \beta)} \oint \mathcal{D}^{\prime} x $$

$$ \begin{align*} Z_{\omega}^{x_{0}} \equiv \oint \mathcal{D}^{\prime} x e^{-\mathcal{A}_{\mathrm{e}} / \hbar} & =\prod_{m=1}^{\infty}\left[\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{d \operatorname{Re} x_{m} d \operatorname{Im} x_{m}}{\pi k_{B} T / M \omega_{m}^{2}}\right] e^{-M \hbar\left[\omega^{2} x_{0}^{2} / 2+\sum_{m=1}^{\infty}\left(\omega_{m}^{2}+\omega^{2}\right)\left|x_{m}\right|^{2}\right] / k_{B} T} \\ & =e^{-M \omega^{2} x_{0}^{2} / 2 k_{B} T} \prod_{m=1}^{\infty}\left[\frac{\omega_{m}^{2}+\omega^{2}}{\omega_{m}^{2}}\right]^{-1} \end{align*} $$

$$ Z_{\omega}=\oint \mathcal{D} x e^{-\mathcal{A}_{\mathrm{e}} / \hbar}=\int_{-\infty}^{\infty} \frac{d x_{0}}{l_{\mathrm{e}}(\hbar \beta)} Z_{\omega}^{x_{0}}=\frac{k_{B} T}{\hbar \omega} \prod_{m=1}^{\infty}\left[\frac{\omega_{m}^{2}+\omega^{2}}{\omega_{m}^{2}}\right]^{-1} $$

$$ \dot{x}\left(\tau_{a}\right)=v_{a}=0, \quad \dot{x}\left(\tau_{b}\right)=v_{b}=0 $$

$$ x(\tau)=x_{0}+\eta(\tau)=x_{0}+\sum_{n=1}^{\infty} x_{n} \cos \nu_{n}\left(\tau-\tau_{a}\right), \quad \nu_{n}=n \pi / \beta $$

$$ \mathcal{A}_{\mathrm{e}}=\frac{M}{2} \int_{0}^{\hbar / k_{B} T} d \tau\left(\dot{x}^{2}+\omega^{2} x^{2}\right)=\frac{M \hbar}{k_{B} T}\left[\frac{\omega^{2}}{2} x_{0}^{2}+\frac{1}{2} \sum_{n=1}^{\infty}\left(\nu_{n}^{2}+\omega^{2}\right) x_{n}^{2}\right] $$

$$ \begin{align*} \oint \mathcal{D} x & \equiv \int_{-\infty}^{\infty} \frac{d x_{0}}{l_{\mathrm{e}}(\hbar \beta)} \prod_{n=1}^{\infty}\left[\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{d x_{n}}{\pi k_{B} T / 2 M \nu_{n}^{2}}\right] \\ & \equiv \int_{-\infty}^{\infty} \frac{d x_{0}}{l_{\mathrm{e}}(\hbar \beta)} \oint \mathcal{D}^{\prime} x \end{align*} $$

$$ \begin{align*} Z_{\omega}^{\mathrm{N}, x_{0}} \equiv \oint \mathcal{D}^{\prime} x e^{-\mathcal{A}_{\mathrm{e}} / \hbar} & =\prod_{n=1}^{\infty}\left[\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{d x_{n}}{\pi k_{B} T / 2 M \nu_{n}^{2}}\right] e^{-M \hbar\left[\omega^{2} x_{0}^{2} / 2+\sum_{n=1}^{\infty}\left(\nu_{n}^{2}+\omega^{2}\right)\left|x_{n}\right|^{2}\right] / k_{B} T} \\ & =e^{-M \omega^{2} x_{0}^{2} / 2 k_{B} T} \prod_{n=1}^{\infty}\left[\frac{\nu_{n}^{2}+\omega^{2}}{\nu_{n}^{2}}\right]^{-1} \end{align*} $$

$$ Z_{\omega}^{\mathrm{N}, x_{0}}=\sqrt{\frac{\omega \hbar \beta}{\sinh \omega \hbar \beta}} \exp \left(-\beta \frac{M}{2} \omega^{2} x_{0}^{2}\right) . $$

$$ Z_{\omega}^{\mathrm{N}}=\frac{1}{l_{\mathrm{e}}(\hbar \beta)} \sqrt{\frac{2 \pi \hbar}{M \omega}} \frac{1}{\sqrt{\sinh \omega \hbar \beta}} $$

$$ Z_{\omega}^{\text {open }}=\left.\left(p_{b} \hbar \beta \mid p_{a} 0\right)\right|_{p_{b}=p_{a}=0} $$

$$ \int_{0}^{\hbar \beta} d \tau \frac{M}{2} \dot{x}^{2}=\frac{M \hbar}{k_{B} T} \sum_{m=1}^{\infty} \omega_{m}^{2}\left|x_{m}\right|^{2} $$

$$ Z=\oint \mathcal{D} x \exp \left[-\frac{M}{k_{B} T} \sum_{m=1}^{\infty} \omega_{m}^{2}\left|x_{m}\right|^{2}-\frac{1}{\hbar} \int_{0}^{\hbar / k_{B} T} d \tau V\left(x_{0}+\sum_{m=-\infty}^{\infty}{ }^{\prime} x_{m} e^{-i \omega_{m} \tau}\right)\right] $$

$$ Z \xrightarrow{T \rightarrow \infty} \oint \mathcal{D} x \exp \left[-\frac{M}{k_{B} T} \sum_{m=1}^{\infty} \omega_{m}^{2}\left|x_{m}\right|^{2}-\frac{1}{k_{B} T} V\left(x_{0}\right)\right] . $$

$$ Z \xrightarrow{T \rightarrow \infty} Z_{\mathrm{cl}}=\int_{-\infty}^{\infty} \frac{d x_{0}}{l_{\mathrm{e}}(\hbar \beta)} e^{-V\left(x_{0}\right) / k_{B} T} $$

$$ \rho(x) \xrightarrow{T \rightarrow \infty} Z_{\mathrm{cl}}^{-1} e^{-V(x) / k_{B} T} . $$

$$ Z=\prod_{m=0}^{N}\left[2\left(1-\cos \omega_{m} \epsilon\right)+\epsilon^{2} \omega^{2}\right]^{-1 / 2} $$

$$ F=-k_{B} T \log Z=\frac{1}{2} k_{B} T \sum_{m=0}^{N} \log \left[2\left(1-\cos \omega_{m} \epsilon\right)+\epsilon^{2} \omega^{2}\right] . $$

$$ F=\frac{1}{2} k_{B} T(N+1) \sum_{n=-\infty}^{\infty} \int_{0}^{2 \pi} \frac{d \lambda}{2 \pi} e^{i \lambda n(N+1)} \log \left[2(1-\cos \lambda)+\epsilon^{2} \omega^{2}\right] $$

$$ \int_{0}^{2 \pi} \frac{d \lambda}{2 \pi} e^{i \lambda n(N+1)} \log \left[2(1-\cos \lambda)+\epsilon^{2} \omega^{2}\right] $$

$$ \log a=\lim _{\delta \rightarrow 0}\left[-\int_{\delta}^{\infty} \frac{d \tau}{\tau} e^{-\tau a / 2}\right]+\log (2 \delta)+\gamma $$

$$ \gamma \equiv-\Gamma^{\prime}(1) / \Gamma(1)=\lim _{N \rightarrow \infty}\left(\sum_{n=1}^{N} \frac{1}{n}-\log N\right) \approx 0.5773156649 \ldots $$

$$ E_{1}(x)=\int_{x}^{\infty} \frac{d t}{t} e^{-t} $$

$$ E_{1}(x)=-\gamma-\log x-\sum_{k=1}^{\infty} \frac{(-x)^{k}}{k k!} $$

$$ F=\frac{1}{2 \epsilon} \sum_{n=-\infty}^{\infty} \lim _{\delta \rightarrow 0}\left\{-\int_{\delta}^{\infty} \frac{d \tau}{\tau} \int_{0}^{2 \pi} \frac{d \lambda}{2 \pi} e^{i \lambda n(N+1)-\tau\left[2(1-\cos \lambda)+\epsilon^{2} \omega^{2}\right] / 2}-\delta_{n 0}[\log (2 \delta)+\gamma]\right\} $$

$$ F=\frac{1}{2 \epsilon} \sum_{n=-\infty}^{\infty} \lim _{\delta \rightarrow 0}\left\{-\int_{\delta}^{\infty} \frac{d \tau}{\tau} I_{n(N+1)}(\tau) e^{-\tau\left(2+\epsilon^{2} \omega^{2}\right) / 2}-\delta_{n 0}[\log (2 \delta)+\gamma]\right\} $$

$$ \frac{\partial F}{\partial m^{2}}=\frac{1}{4 \epsilon} \sum_{n=-\infty}^{\infty} \int_{0}^{\infty} d \tau I_{n(N+1)}(\tau) e^{-\tau\left(2+m^{2}\right) / 2} $$

$$ \int_{0}^{\infty} d \tau I_{\nu}(\mu \tau) e^{-\tau \alpha}=\mu^{\nu} \frac{\left(\alpha-\sqrt{\alpha^{2}-\mu^{2}}\right)^{-\nu},}{\sqrt{\alpha^{2}-\mu^{2}}}=\mu^{-\nu} \frac{\left(\alpha-\sqrt{\alpha^{2}-\mu^{2}}\right)^{\nu}}{\sqrt{\alpha^{2}-\mu^{2}}} $$

$$ \frac{\partial F}{\partial m^{2}}=\frac{1}{2 \epsilon} \sum_{n=-\infty}^{\infty} \frac{1}{\sqrt{\left(m^{2}+2\right)^{2}-4}}\left[\frac{m^{2}+2-\sqrt{\left(m^{2}+2\right)^{2}-4}}{2}\right]^{|n|(N+1)} $$

$$ \log \left[\left(m^{2}+2+\sqrt{\left(m^{2}+2\right)^{2}-4}\right) / 2\right]+\mathrm{const} $$

$$ -\frac{1}{|n|(N+1)}\left[\left(m^{2}+2+\sqrt{\left(m^{2}+2\right)^{2}-4}\right) / 2\right]^{-|n|(N+1)}+\mathrm{const} $$

$$ I_{\alpha}(z) \sim \frac{1}{|\alpha|!}\left(\frac{z}{2}\right)^{\alpha}\left[1+O\left(z^{2}\right)\right] $$

$$ \begin{align*} -\frac{1}{(|n|(N+1))!} \int_{\delta}^{\infty} & \frac{d \tau}{\tau}\left(\frac{\tau}{2}\right)^{|n|(N+1)} e^{-\tau m^{2} / 2} \\ & \approx\left\{\begin{array}{ll} \log m^{2}+\gamma+\log (2 \delta) & n=0 \\ -\left(m^{2}\right)^{-|n|(N+1)} /|n|(N+1) & n \neq 0 \end{array}\right\} \end{align*} $$

$$ \begin{align*} F & =\frac{1}{2 \beta} \sum_{m=0}^{N} \log \left[2\left(1-\cos \left(\omega_{m} \epsilon\right)\right)+\epsilon^{2} \omega^{2}\right] \\ & =\frac{1}{2 \epsilon}\left\{\log \left[\left(\epsilon^{2} \omega^{2}+2+\sqrt{\left(\epsilon^{2} \omega^{2}+2\right)^{2}-4}\right) / 2\right]\right. \\ & \left.-\frac{2}{N+1} \sum_{n=1}^{\infty} \frac{1}{n}\left[\left(\epsilon^{2} \omega^{2}+2+\sqrt{\left(\epsilon^{2} \omega^{2}+2\right)^{2}-4}\right) / 2\right]^{-|n|(N+1)}\right\} \end{align*} $$

$$ \epsilon \tilde{\omega}_{\mathrm{e}} \equiv \log \left\{\left[\epsilon^{2} \omega^{2}+2+\sqrt{\left(\epsilon^{2} \omega^{2}+2\right)^{2}-4}\right] / 2\right\} $$

$$ \cosh \left(\epsilon \tilde{\omega}_{\mathrm{e}}\right)=\left(\epsilon^{2} \omega^{2}+2\right) / 2, \quad \sinh \left(\epsilon \tilde{\omega}_{\mathrm{e}}\right)=\sqrt{\left(\epsilon^{2} \omega^{2}+2\right)^{2}-4} / 2 $$

$$ \begin{align*} F & =\frac{\hbar}{2}\left[\tilde{\omega}_{\mathrm{e}}-\frac{2}{\epsilon(N+1)} \sum_{n=1}^{\infty} \frac{1}{n} e^{-\epsilon \tilde{\omega}_{\mathrm{e}} n(N+1)}\right] \\ & =\frac{1}{2}\left[\hbar \tilde{\omega}_{\mathrm{e}}+2 k_{B} T \log \left(1-e^{-\beta \hbar \tilde{\omega}_{\mathrm{e}}}\right)\right] \\ & =\frac{1}{\beta} \log \left[2 \sinh \left(\beta \hbar \tilde{\omega}_{\mathrm{e}} / 2\right)\right] \end{align*} $$

$$ F^{\epsilon} \stackrel{\epsilon \rightarrow 0}{=} \frac{1}{\beta} \log [2 \sinh (\beta \hbar \omega / 2)]=\frac{\hbar \omega}{2}+\frac{1}{\beta} \log \left(1-e^{-\beta \hbar \omega}\right) $$

$$ Z_{\omega}=\oint \mathcal{D} x e^{-\int_{0}^{\hbar \beta} M\left[\dot{x}^{2}(\tau)+\omega^{2} x^{2}(\tau)\right] / 2}=\oint \mathcal{D} x e^{-\int_{0}^{\hbar \beta} M x(\tau)\left[-\partial_{\tau}^{2}+\omega^{2}\right] x(\tau) / 2} $$

$$ Z_{\omega}=\frac{1}{\sqrt{\operatorname{Det}\left(-\partial_{\tau}^{2}+\omega^{2}\right)}}=e^{-\frac{1}{2} \operatorname{Tr} \log \left(-\partial_{\tau}^{2}+\omega^{2}\right)} $$

$$ Z_{\omega}=\prod_{\omega^{\prime}} \frac{1}{\sqrt{\omega^{\prime 2}+\omega^{2}}} $$

$$ Z_{\omega} \equiv e^{-F_{\omega} / k_{B} T}=e^{-\frac{1}{2} \sum_{\omega^{\prime}} \log \left(\omega^{\prime 2}+\omega^{2}\right)} $$

$$ Z_{\omega}=\exp \left[-\frac{1}{2} \sum_{m=-\infty}^{\infty} \log \left(\omega_{m}^{2}+\omega^{2}\right)\right] . $$

$$ F_{\omega}=\left.\frac{1}{2 \beta} \operatorname{Tr} \log \left(-\partial_{\tau}^{2}+\omega^{2}\right)\right|_{\text {per }}=\frac{1}{2 \beta} \sum_{m=-\infty}^{\infty} \log \left(\omega_{m}^{2}+\omega^{2}\right) $$

$$ \left.F_{\omega} \equiv \frac{1}{2 \beta} \operatorname{Tr} \log \left(-\partial_{\tau}^{2}+\omega^{2}\right)\right|_{ \pm \infty}=\frac{\hbar}{2} \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \log \left(\omega^{\prime 2}+\omega^{2}\right) $$

$$ \sum_{\omega^{\prime}} \underset{T \rightarrow 0}{\longrightarrow} \hbar \beta \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} $$

$$ \log \left(\omega^{\prime 2}+\omega^{2}\right)=-\left.\frac{d}{d \epsilon}\left(\omega^{\prime 2}+\omega^{2}\right)^{-\epsilon}\right|_{\epsilon=0} $$

$$ l_{\mathrm{MS}}(\epsilon)=-\frac{1}{\epsilon}\left(\omega^{\prime 2}+\omega^{2}\right)^{-\epsilon}+\frac{1}{\epsilon} $$

$$ l_{\mathrm{MS}}(\epsilon)=-\left.\frac{1}{\epsilon}\left(\omega^{\prime 2}+\omega^{2}\right)^{-\epsilon}\right|_{\mathrm{MS}, \epsilon \rightarrow 0} $$

$$ \frac{1}{\hbar \beta} \operatorname{Tr} \log \left(-\partial_{\tau}^{2}+\omega^{2}\right)=-\left.\frac{d}{d \epsilon} \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi}\left(\omega^{\prime 2}+\omega^{2}\right)^{-\epsilon}\right|_{\epsilon=0} $$

$$ \int_{0}^{\infty} \frac{d \tau}{\tau} \tau^{\mu} e^{-\tau \omega^{2}}=\omega^{-\mu / 2} \Gamma(\mu) $$

$$ a^{-\epsilon}=\frac{1}{\Gamma(\epsilon)} \int_{0}^{\infty} \frac{d \tau}{\tau} \tau^{\epsilon} e^{-\tau a} $$

$$ \frac{1}{\hbar \beta} \operatorname{Tr} \log \left(-\partial_{\tau}^{2}+\omega^{2}\right)=-\left.\frac{d}{d \epsilon} \frac{1}{\Gamma(\epsilon)} \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \int_{0}^{\infty} \frac{d \tau}{\tau} \tau^{\epsilon} e^{-\tau\left(\omega^{\prime 2}+\omega^{2}\right)}\right|_{\epsilon=0} $$

$$ \frac{1}{\hbar \beta} \operatorname{Tr} \log \left(-\partial_{\tau}^{2}+\omega^{2}\right)=-\left.\frac{d}{d \epsilon} \frac{1}{\Gamma(\epsilon)} \int_{0}^{\infty} \frac{d \tau}{\tau} \tau^{\epsilon} \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} e^{-\tau\left(\omega^{\prime 2}+\omega^{2}\right)}\right|_{\epsilon=0} $$

$$ \frac{1}{\hbar \beta} \operatorname{Tr} \log \left(-\partial_{\tau}^{2}+\omega^{2}\right)=-\left.\frac{d}{d \epsilon} \frac{1}{\Gamma(\epsilon)} \int_{0}^{\infty} \frac{d \tau}{\tau} \tau^{\epsilon} \frac{1}{2 \sqrt{\tau \pi}} e^{-\tau \omega^{2}}\right|_{\epsilon=0} $$

$$ \frac{1}{\hbar \beta} \operatorname{Tr} \log \left(-\partial_{\tau}^{2}+\omega^{2}\right)=-\left.\frac{1}{2 \sqrt{\pi}} \omega^{1-2 \epsilon} \frac{d}{d \epsilon} \frac{1}{\Gamma(\epsilon)} \Gamma(\epsilon-1 / 2)\right|_{\epsilon=0} $$

$$ \frac{1}{\hbar \beta} \operatorname{Tr} \log \left(-\partial_{\tau}^{2}+\omega^{2}\right)=\int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \log \left(\omega^{\prime 2}+\omega^{2}\right)=\omega $$

$$ F_{\omega}=\frac{\hbar \omega}{2} . $$

$$ \log a=-\int_{0}^{\infty} \frac{d \tau}{\tau} e^{-\tau a} $$

$$ \frac{1}{\hbar \beta} \operatorname{Tr} \log \left(-\partial_{\tau}^{2}+\omega^{2}\right)=-\frac{1}{\epsilon} \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi}\left[\frac{1}{\epsilon}\left(\omega^{\prime 2}+\omega^{2}\right)^{-\epsilon}-\frac{1}{\epsilon}\right]_{\epsilon \rightarrow 0} . $$

$$ \int_{0}^{\infty} d \omega^{\prime}\left(\omega^{\prime}\right)^{\alpha}=0 \quad \text { for all } \alpha $$

$$ \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \frac{\left(\omega^{\prime 2}\right)^{\gamma}}{\left(\omega^{\prime 2}+\omega^{2}\right)^{\epsilon}}=\frac{\Gamma(\gamma+1 / 2)}{2 \pi \Gamma(\epsilon)}\left(\omega^{2}\right)^{\gamma+1 / 2-\epsilon} $$

$$ \frac{1}{\Gamma(\epsilon)} \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi}\left(\omega^{\prime 2}\right)^{\gamma} \int_{0}^{\infty} \frac{d \tau}{\tau} \tau^{\epsilon} e^{-\tau\left(\omega^{\prime 2}+\omega^{2}\right)} $$

$$ \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi}\left(\omega^{\prime 2}\right)^{\gamma} e^{-\tau\left(\omega^{\prime 2}+\omega^{2}\right)}=\frac{1}{2 \pi} \int_{0}^{\infty} \frac{d \omega^{\prime 2}}{\omega^{\prime 2}}\left(\omega^{\prime 2}\right)^{\gamma+1 / 2} e^{-\tau\left(\omega^{\prime 2}+\omega^{2}\right)}=\frac{\tau^{-\gamma-1 / 2}}{2 \pi} \Gamma(\gamma+1 / 2) $$

$$ \int_{0}^{\infty} \frac{d \tau}{\tau} \tau^{\epsilon-\gamma-1 / 2} e^{-\tau \omega^{2}}=\left(\omega^{2}\right)^{\gamma+1 / 2+\epsilon} $$

$$ k_{B} T \sum_{m=-\infty}^{\infty} c=\int_{-\infty}^{\infty} \frac{d \omega_{m}}{2 \pi} c $$

$$ k_{B} T \sum_{m=-\infty}^{\infty} \log \omega^{2}=0 $$

$$ \Delta F_{\omega}=\frac{k_{B} T}{2} \sum_{m=-\infty}^{\infty} \log \left(\frac{\omega_{m}^{2}}{\omega^{2}}+1\right)-\frac{\hbar}{2} \int_{-\infty}^{\infty} \frac{d \omega_{m}}{2 \pi} \log \left(\frac{\omega_{m}^{2}}{\omega^{2}}+1\right) $$

$$ \Delta_{1} F_{\omega}=k_{B} T \sum_{m=1}^{\infty}\left[\log \left(\frac{\omega_{m}^{2}}{\omega^{2}}+1\right)-\log \frac{\omega_{m}^{2}}{\omega^{2}}\right]=k_{B} T \sum_{m=1}^{\infty} \log \left(1+\frac{\omega^{2}}{\omega_{m}^{2}}\right), $$

$$ \Delta_{2} F_{\omega}=k_{B} T \sum_{m=1}^{\infty} \log \frac{\omega_{m}^{2}}{\omega^{2}} $$

$$ \prod_{m=1}^{\infty}\left(1+\frac{\omega^{2}}{\omega_{m}^{2}}\right)=\frac{\sinh (\beta \hbar \omega / 2)}{\beta \hbar \omega / 2} $$

$$ \Delta F_{1}=\frac{1}{\beta} \log \frac{\sinh (\beta \hbar \omega / 2)}{\beta \hbar \omega / 2} $$

$$ \sum_{m=1}^{\infty} \log \frac{\omega_{m}^{2}}{\omega^{2}}=-\left[2 \frac{d}{d \epsilon} \sum_{m=1}^{\infty}\left(\frac{\omega_{m}}{\omega}\right)^{-\epsilon}\right]_{\epsilon \rightarrow 0}=-\left[2 \frac{d}{d \epsilon}\left(\frac{2 \pi}{\beta \hbar \omega}\right)^{-\epsilon} \sum_{m=1}^{\infty} m^{-\epsilon}\right]_{\epsilon \rightarrow 0} $$

$$ \zeta(z)=\sum_{m=1}^{\infty} m^{-z} $$

$$ \zeta(0)=-1 / 2, \quad \zeta^{\prime}(0)=-\frac{1}{2} \log 2 \pi $$

$$ \zeta(z) \approx-\frac{1}{2}(2 \pi)^{z}, \quad z \approx 0 $$

$$ \sum_{m=1}^{\infty} \log \frac{\omega_{m}^{2}}{\omega^{2}}=-\left[2 \frac{d}{d \epsilon}\left(\frac{2 \pi}{\beta \hbar \omega}\right)^{-\epsilon} \zeta(\epsilon)\right]_{\epsilon \rightarrow 0}=\left.\frac{d}{d \epsilon}(\beta \hbar \omega)^{\epsilon}\right|_{\epsilon \rightarrow 0}=\log \hbar \omega \beta $$

$$ \Delta F_{\omega}=\frac{1}{\beta} \log \left(1-e^{-\hbar \beta \omega}\right) $$

$$ \begin{align*} F_{\omega} & =\frac{1}{2 \beta} \operatorname{Tr} \log \left(-\partial_{\tau}^{2}+\omega^{2}\right)=\frac{1}{2 \beta} \sum_{m=-\infty}^{\infty} \log \left(\omega_{m}^{2}+\omega^{2}\right)=\frac{\hbar \omega}{2}+\frac{1}{\beta} \log \left(1-e^{-\hbar \omega / k_{B} T}\right) \\ & =\frac{1}{\beta} \log \left(2 \sinh \frac{\hbar \omega \beta}{2}\right) \end{align*} $$

$$ \sum_{m=-\infty}^{\infty} c=\sum_{m=-\infty}^{-1} c+c+\sum_{m=1}^{\infty} c=0 $$

$$ \sum_{m=1}^{\infty} 1=\sum_{m=-1}^{-\infty} 1=\zeta(0)=-1 / 2 $$

$$ \frac{1}{2 \beta} \sum_{m=-\infty}^{\infty} \log \left(\frac{\omega_{m}^{2}}{\omega^{2}}+1\right)=\frac{1}{\beta} \sum_{m=1}^{\infty} \log \left(\frac{\omega_{m}^{2}}{\omega^{2}}+1\right) $$

$$ F_{0}(\Delta t)=\int \mathcal{D} \delta x(t) \exp \left[\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t \frac{M}{2} \delta x\left(-\partial_{t}^{2}\right) \delta x\right]=\frac{1}{\sqrt{2 \pi \hbar i \Delta t / M}} $$

$$ F_{\omega}(\Delta t)=F_{0}(\Delta t)\left[\frac{\operatorname{Det}\left(-\partial_{t}^{2}-\omega^{2}\right)}{\operatorname{Det}\left(-\partial_{t}^{2}\right)}\right]^{-1 / 2} . $$

$$ \frac{\operatorname{Det}\left(-\partial_{t}^{2}-\omega^{2}\right)}{\operatorname{Det}\left(-\partial_{t}^{2}\right)}=\exp \left\{\sum_{n=1}^{\infty}\left[\log \left(\nu_{n}^{2}-\omega^{2}\right)-\log \nu_{n}^{2}\right]\right\} $$

$$ \frac{\operatorname{Det}\left(-\partial_{t}^{2}-\omega^{2}\right)}{\operatorname{Det}\left(-\partial_{t}^{2}\right)}=\frac{\sin \omega \Delta t}{\omega \Delta t} $$

$$ \begin{align*} \operatorname{Det}\left(-\partial_{t}^{2}-\omega^{2}\right) & =\left.\sum_{n=1}^{\infty} \log \left(\nu_{n}^{2}+\omega^{2}\right)\right|_{\omega \rightarrow i \omega}=\sum_{n=1}^{\infty}\left[\log \left(\frac{\nu_{n}^{2}}{\omega^{2}}+1\right)+\log \omega^{2}\right]_{\omega \rightarrow i \omega} \\ & =\left[\sum_{n=1}^{\infty} \log \left(\frac{\nu_{n}^{2}}{\omega^{2}}+1\right)-\frac{1}{2} \log \omega^{2}\right]_{\omega \rightarrow i \omega}=\log \left(2 \frac{\sin \omega \Delta t}{\omega}\right) \end{align*} $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\frac{1}{\sqrt{\pi i / M}} \operatorname{Det}^{-1 / 2}\left(-\partial_{t}^{2}-\omega^{2}\right) e^{i \mathcal{A}_{\mathrm{cl}} / \hbar}=\frac{1}{\sqrt{\pi i / M}} \sqrt{\frac{\omega}{2 \sin \omega \Delta t}} e^{i \mathcal{A}_{\mathrm{cl}} / \hbar} $$

$$ \operatorname{Tr} \log \left(-\partial_{\tau}^{2}+\omega^{2}\right)=\operatorname{Tr} \log \left(\partial_{\tau}+\omega\right)+\operatorname{Tr} \log \left(-\partial_{\tau}+\omega\right) $$

$$ \begin{align*} \operatorname{Tr} \log \left(\partial_{\tau}+\omega\right)=\operatorname{Tr} \log \left(-\partial_{\tau}+\omega\right) & =\hbar \beta \int_{-\infty}^{\infty} \frac{d \omega}{2 \pi} \log \left(-i \omega^{\prime}+\omega\right)=\hbar \beta \int_{-\infty}^{\infty} \frac{d \omega}{2 \pi} \log \left(i \omega^{\prime}+\omega\right) \\ & =\frac{\hbar \beta \omega}{2} \end{align*} $$

$$ \operatorname{Tr} \log \left(\partial_{\tau}+\omega\right)=\operatorname{Tr} \log \left(-\partial_{\tau}+\omega\right)=\frac{1}{2} \operatorname{Tr} \log \left(-\partial_{\tau}^{2}+\omega^{2}\right)=\log \left(2 \sinh \frac{\beta \hbar \omega}{2}\right) $$

$$ \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \log \left(-\omega^{\prime 2}+\omega^{2}-i \eta\right)=\omega, \quad \omega \geq 0 $$

$$ \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \log \left[\omega^{\prime} \pm(\omega-i \eta)\right]=i \frac{\omega}{2}, \quad \omega \geq 0 $$

$$ \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \log \left(\omega^{\prime} \pm i \omega\right)=\mp \epsilon\left(\omega_{R}\right) \frac{\omega}{2} $$

$$ \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \log \left(\omega^{\prime} \pm \omega\right)=-i \epsilon\left(\omega_{I}\right) \frac{\omega}{2} $$

$$ \frac{k_{B} T}{\hbar} \sum_{m=-\infty}^{\infty} \log \left(\omega_{m} \pm i \omega\right)=\frac{k_{B} T}{\hbar} \log \left[2 \epsilon\left(\omega_{R}\right) \sinh \frac{\hbar \omega}{2 k_{B} T}\right] $$

$$ \frac{k_{B} T}{\hbar} \sum_{m=-\infty}^{\infty} \log \left(\omega_{m} \pm \omega\right)=\frac{k_{B} T}{\hbar} \log \left[-2 i \epsilon\left(\omega_{I}\right) \sin \frac{\hbar \omega}{2 k_{B} T}\right] $$

$$ \begin{align*} \operatorname{Tr} \log \left[ \pm \partial_{\tau}+\Omega(\tau)\right] & =\log \left\{2 \sinh \left[\frac{1}{2} \int_{0}^{\hbar \beta} d \tau^{\prime \prime} \Omega\left(\tau^{\prime \prime}\right)\right]\right\} \\ & =\frac{1}{2} \int_{0}^{\hbar \beta} d \tau^{\prime \prime} \Omega\left(\tau^{\prime \prime}\right)+\log \left[1-e^{-\int_{0}^{\hbar \beta} d \tau^{\prime \prime} \Omega\left(\tau^{\prime \prime}\right)}\right] \end{align*} $$

$$ \operatorname{Det}\left[-\hbar^{2} \partial_{\tau}^{2}+w^{2}(\tau)\right]=\operatorname{Det}\left[-\hbar \partial_{\tau}-\bar{w}(\tau)\right] \times \operatorname{Det}\left[\hbar \partial_{\tau}-\bar{w}(\tau)\right] $$

$$ \hbar \partial_{\tau} \bar{w}(\tau)+\bar{w}^{2}(\tau)=w^{2}(\tau) $$

$$ \operatorname{Tr} \log \left[-\hbar^{2} \partial_{\tau}^{2}+w^{2}(\tau)\right]=\log \left\{4 \sinh ^{2}\left[\frac{1}{2 \hbar} \int_{0}^{\hbar \beta} d \tau^{\prime} \bar{w}\left(\tau^{\prime}\right)\right]\right\} $$

$$ \bar{w}(\tau)=2 \hbar \partial_{\tau} \operatorname{arsinh} \sqrt{\left[\dot{D}_{\mathrm{ren}}(\tau)-1\right] / 2} $$

$$ \bar{w}(\tau)=\sum_{n=0}^{\infty} \bar{w}_{n}(\tau) \hbar^{n} $$

$$ \bar{w}_{n}(\tau)=-\frac{1}{2 w(\tau)}\left(\dot{\bar{w}}_{n-1}(\tau)+\sum_{k=1}^{n-1} \bar{w}_{n-k}(\tau) \bar{w}_{k}(\tau)\right), \quad n \geq 1 $$

$$ \begin{align*} \{\sqrt{v(\tau)}, & -\frac{v^{\prime}(\tau)}{4 v(\tau)}, \quad-\frac{5 v^{\prime}(\tau)^{2}}{32 v(\tau)^{5 / 2}}+\frac{v^{\prime \prime}(\tau)}{8 v(\tau)^{3 / 2}}, \quad-\frac{15 v^{\prime}(\tau)^{3}}{64 v(\tau)^{4}}+\frac{9 v^{\prime}(\tau) v^{\prime \prime}(\tau)}{32 v(\tau)^{3}}-\frac{v^{(3)}(\tau)}{16 v(\tau)^{2}} \\ & \left.-\frac{1105 v^{\prime}(\tau)^{4}}{2048 v(\tau)^{11 / 2}}+\frac{221 v^{\prime}(\tau)^{2} v^{\prime \prime}(\tau)}{256 v(\tau)^{9 / 2}}-\frac{19 v^{\prime \prime}(\tau)^{2}}{128 v(\tau)^{7 / 2}}-\frac{7 v^{\prime}(\tau) v^{(3)}(\tau)}{32 v(\tau)^{7 / 2}}+\frac{v^{(4)}(\tau)}{32 v(\tau)^{5 / 2}}\right\} \end{align*} $$

$$ \Delta F_{\omega}=\frac{k_{B} T}{2}\left(\sum_{m=-\infty}^{\infty}-\frac{\hbar}{k_{B} T} \int_{-\infty}^{\infty} \frac{d \omega_{m}}{2 \pi}\right) \log \left(\omega_{m}^{2}+\omega^{2}\right) $$

$$ \Delta F_{\omega}=\frac{k_{B} T}{2}\left(\sum_{m=-\infty}^{\infty}-\int_{-\infty}^{\infty} d m\right) \log \left[\left(\frac{2 \pi k_{B} T}{\hbar}\right)^{2} m^{2}+\omega^{2}\right] $$

$$ \Delta F_{\omega}=-\frac{k_{B} T}{2} \int_{0}^{\infty} \frac{d \tau}{\tau}\left(\sum_{m=-\infty}^{\infty}-\int_{-\infty}^{\infty} d m\right) e^{-\tau\left[\left(2 \pi k_{B} T / \hbar\right)^{2} m^{2}+\omega^{2}\right]} $$

$$ \Delta F_{\omega}=-\frac{1}{2 \beta} \int_{0}^{\infty} \frac{d \tau}{\tau} \int_{-\infty}^{\infty} d \mu\left(\sum_{n=-\infty}^{\infty} e^{2 \pi \mu n i}-1\right) e^{-\tau\left[(2 \pi / \hbar \beta)^{2} \mu^{2}+\omega^{2}\right]} $$

$$ 2 \pi \mu n i-\tau\left(\frac{2 \pi}{\hbar \beta}\right)^{2} \mu^{2}=-\tau\left(\frac{2 \pi}{\hbar \beta}\right)^{2}\left[\mu-i \frac{n \hbar^{2} \beta^{2}}{4 \pi \tau}\right]^{2}-\frac{1}{4 \tau}(\hbar \beta n)^{2}, $$

$$ \Delta F_{\omega}=-\frac{\hbar}{2 \sqrt{\pi}} \int_{0}^{\infty} \frac{d \tau}{\tau} \tau^{-1 / 2} \sum_{n=1}^{\infty} e^{-(n \hbar \beta)^{2} / 4 \tau-\tau \omega^{2}} $$

$$ \int_{0}^{\infty} \frac{d \tau}{\tau} \tau^{\nu} e^{-a^{2} / \tau-b^{2} \tau}=2\left(\frac{a}{b}\right)^{\nu} K_{\nu}(2 a b), \quad K_{\nu}(2 a b)=K_{-\nu}(2 a b) $$

$$ \Delta F_{\omega}=-\frac{\hbar \omega}{2 \sqrt{\pi}} \sum_{n=1}^{\infty} 2(n \beta \hbar \omega)^{-1 / 2} \sqrt{2} K_{1 / 2}(n \beta \hbar \omega) $$

$$ K_{1 / 2}(z)=\sqrt{\frac{\pi}{2 z}} e^{-z} $$

$$ \Delta F_{\omega}=-\frac{1}{\beta} \sum_{n=1} \frac{1}{n} e^{-\beta \hbar \omega n}=\frac{1}{\beta} \log \left(1-e^{-\beta \hbar \omega}\right) $$

$$ S(\beta \hbar \omega)=k_{B} T \sum_{m=1}^{\infty} \log \left(1+\frac{\omega^{2}}{\omega_{m}^{2}}\right) $$

$$ S(\beta \hbar \omega)=\frac{\beta \hbar \omega}{2}-\log \beta \hbar \omega-\sum_{n=1}^{\infty} \frac{1}{n} e^{-n \beta \hbar \omega} $$

$$ S(\beta \hbar \omega)=-\sum_{k=1}^{\infty} \frac{(-1)^{k}}{k}\left(\sum_{m=1}^{\infty} \frac{1}{m^{2 k}}\right)\left[\left(\frac{\beta \hbar \omega}{2 \pi}\right)^{2}\right]^{k} $$

$$ S(\beta \hbar \omega)=-\sum_{m=1}^{\infty} \frac{(-1)^{k}}{k} \zeta(2 k)\left(\frac{\beta \hbar \omega}{2 \pi}\right)^{2 k} $$

$$ \zeta(2 n)=\frac{(2 \pi)^{2 n}}{2(2 n)!}\left|B_{2 n}\right| $$

$$ \frac{t}{e^{t}-1}=\sum_{n=0}^{\infty} B_{n} \frac{t^{n}}{n!} $$

$$ \zeta(1-2 n)=-\frac{B_{2 n}}{2 n}, $$

$$ \zeta(z)=2^{z} \pi^{z-1} \sin (\pi z / 2) \Gamma(1-z) \zeta(1-z)=2^{z-1} \pi^{z} \zeta(1-z) / \Gamma(z) \cos \frac{z \pi}{2} . $$

$$ \zeta(2)=\frac{\pi^{2}}{6}, \quad \zeta(4)=\frac{\pi^{4}}{90}, \quad \zeta(6)=\frac{\pi^{6}}{945}, \quad \ldots \quad, \quad \zeta(\infty)=1 . $$

$$ \sum_{n=1}^{\infty} \frac{1}{n} e^{-n \beta \hbar \omega} \approx \sum_{n=1}^{N-1} \frac{1}{n}+\sum_{n=N}^{\infty} \frac{1}{n} e^{-n \beta \hbar \omega} $$

$$ \psi(z) \equiv \frac{\Gamma^{\prime}(z)}{\Gamma(z)} $$

$$ \psi(z)=-\gamma-\sum_{n=0}^{\infty}\left(\frac{1}{n+z}-\frac{1}{n+1}\right) $$

$$ \psi(N)=-\gamma+\sum_{n=1}^{N-1} \frac{1}{n} $$

$$ \psi(z) \approx \log z-\frac{1}{2 z}-\sum_{n=1}^{\infty} \frac{B_{2 n}}{2 n z^{2 n}} $$

$$ \sum_{n=1}^{\infty} \frac{1}{n} e^{-n \beta \hbar \omega} \underset{T \rightarrow \infty}{\approx}-\log \beta \hbar \omega+\mathcal{O}(\beta) $$

$$ g(\beta \hbar \omega) \equiv \sum_{n=1}^{\infty} \frac{1}{n} e^{-n \beta \hbar \omega} $$

$$ \zeta_{\nu}\left(e^{\beta \hbar \omega}\right) \equiv \sum_{n=1}^{\infty} \frac{1}{n^{\nu}} e^{-n \beta \hbar \omega} $$

$$ \zeta_{\nu}\left(e^{\beta \hbar \omega}\right)=\int_{0}^{\infty} d n \frac{1}{n^{\nu}} e^{-n \beta \hbar \omega}+\left(\sum_{n=1}^{\infty}-\int_{0}^{\infty}\right) \frac{1}{n^{\nu}} e^{-n \beta \hbar \omega} $$

$$ \zeta_{\nu}\left(e^{\beta \hbar \omega}\right)=\int_{0}^{\infty} d n \frac{1}{n^{\nu}} e^{-n \beta \hbar \omega}+\left(\sum_{n=1}^{\infty}-\int_{0}^{\infty}\right) \frac{1}{n^{\nu}}+\sum_{k=1}^{\infty}\left[\left(\sum_{n=1}^{\infty}-\int_{0}^{\infty}\right) n^{k-\nu}\right] \frac{(-1)^{k}}{k!}(\beta \hbar \omega)^{k} $$

$$ \left(\sum_{n=1}^{\infty}-\int_{0}^{\infty}\right) \frac{1}{\nu^{k}}=\zeta(\nu) $$

$$ \zeta_{\nu}\left(e^{\beta \hbar \omega}\right)=\Gamma(1-\nu)(\beta \hbar \omega)^{\nu-1}+\zeta(\nu)+\sum_{k=1}^{\infty} \frac{1}{k!}(-\beta \hbar \omega)^{k} \zeta(\nu-k) . $$

$$ \left(\sum_{n=1}^{\infty}-\int_{0}^{\infty}\right) \frac{e^{n \beta \hbar \omega}}{n^{\nu}}=\sum_{k=1}^{\infty} \frac{1}{k!}(-\beta \hbar \omega)^{k} \zeta(\nu-k) \equiv \bar{\zeta}_{\nu}\left(e^{\beta \hbar \omega}\right) $$

$$ 2^{z} \Gamma(1-z) \zeta(1-z) \sin \frac{\pi z}{2}=\pi^{1-z} \zeta(z) $$

$$ \zeta(\nu)=\frac{1}{\nu-1}+\gamma+\mathcal{O}(\nu-1)=-\Gamma(1-\nu)+\mathcal{O}(\nu-1) $$

$$ \zeta(-2 p)=0, \quad \zeta(1-2 p)=\frac{1}{p}(-1)^{p} \frac{(2 p)!}{(2 \pi)^{2 p}} \zeta(2 p), \quad p=1,2,3, \ldots . $$

$$ g(\beta \hbar \omega)=\zeta_{1}\left(e^{\beta \hbar \omega}\right)=-\log \beta \hbar \omega+\frac{\beta \hbar \omega}{2}+\sum_{k=1}^{\infty} \zeta(2 k) \frac{(-1)^{k}}{k!}(\beta \hbar \omega)^{2 k} . $$

$$ \zeta_{1}\left(e^{\beta \hbar \omega}\right)=\sum_{p=0}^{\infty}\left(\sum_{n=1}^{\infty} n^{p-1}\right) \frac{(-1)^{p}}{p!}(\beta \hbar \omega)^{p}=-\zeta(1)+\sum_{p=1}^{\infty} \zeta(1-p) \frac{(-1)^{p}}{p!}(\beta \hbar \omega)^{p}, $$

$$ \zeta(1) \rightarrow \zeta_{\mathrm{reg}}(1)=-\log \beta \hbar \omega . $$

$$ \begin{align*} \zeta_{\nu}\left(e^{\beta \hbar \omega}\right) & \equiv \sum_{m=-\infty}^{\infty} \int_{0}^{\infty} d n e^{(2 \pi i m+\beta \hbar \omega) n} \frac{1}{n^{\nu}}=\Gamma(1-\nu)(-\beta \hbar \omega)^{\nu-1} \\ & +\Gamma(1-\nu) 2 \operatorname{Re} \sum_{m=1}^{\infty}(-\beta \hbar \omega-2 \pi i m)^{\nu-1} \end{align*} $$

$$ \begin{align*} & 2 \operatorname{Re} \sum_{m=1}^{\infty}(-2 \pi i m)^{\nu-1}\left(1+\frac{\beta \hbar \omega}{2 \pi i m}\right)^{\nu-1} \\ & \quad=2 \sum_{k=0}^{\infty}\binom{\nu-1}{k} \cos [(1-\nu-k) \pi / 2](2 \pi)^{\nu-1-k} \zeta(1-\nu+k)(\beta \hbar \omega)^{k} \end{align*} $$

$$ \begin{align*} \zeta_{\nu}\left(e^{\beta \hbar \omega}\right) & \equiv \sum_{m=-\infty}^{\infty} \int_{0}^{\infty} d n e^{\left(i \omega_{m}+\hbar \omega\right) \beta n} \frac{1}{n^{\nu}} \\ & =\Gamma(1-\nu)(-\beta \hbar \omega)^{\nu-1}\left[1+2 \operatorname{Re} \sum_{m=1}^{\infty}\left(1+i \omega_{m} / \hbar \omega\right)^{\nu-1}\right] \end{align*} $$

$$ \begin{align*} \sum_{k=0}^{K} F(a+k \Delta) & =\frac{1}{\Delta} \int_{a}^{b} d t F(t)+\frac{1}{2}[F(a)+F(b)] \\ & +\sum_{p=1}^{\infty} \frac{\Delta^{2 p-1}}{(2 p)!} B_{2 p}\left[F^{(2 p-1)}(b)-F^{(2 p-1)}(a)\right] \end{align*} $$

$$ \sum_{k=0}^{K-1} F(a+(k+\kappa) \Delta)=\frac{1}{\Delta} \int_{a}^{b} d t F(t)+\sum_{p=1}^{\infty} \frac{\Delta^{p-1}}{p!} B_{p}(\kappa)\left[F^{(p-1)}(b)-F^{(p-1)}(a)\right] $$

$$ \frac{t e^{\kappa t}}{e^{t}-1}=\sum_{n=0}^{\infty} B_{n}(\kappa) \frac{t^{n}}{n!} $$

$$ \sum_{k=0}^{K-1} F(a+(k+\kappa) \Delta)=\frac{1}{\Delta} \int_{a}^{b} d t\left[1+\sum_{p=0}^{\infty} \frac{\Delta^{p}}{p!} B_{p}(\kappa) \partial_{t}^{p}\right] F(t) $$

$$ \begin{align*} \sum_{m=0}^{M}\left[\log \left(\omega_{m}^{2}+\omega^{2}\right)-\log \left(\omega_{m}^{2}\right)\right] & =\left.\left\{\pi \frac{\omega}{\omega_{1}}+\frac{\omega_{m}}{\omega_{1}}\left[\log \left(\omega_{m}^{2}+\omega^{2}\right)-2\right]\right\}\right|_{m=1} ^{m=M}-\left.\{\omega=0\}\right|_{m=1} ^{m=M} \\ & +\frac{1}{2}\left\{\log \left(\omega_{1}^{2}+\omega^{2}\right)+\log \left(\omega_{M}^{2}+\omega^{2}\right)\right\}-\{\omega=0\} . \end{align*} $$

$$ \pi \frac{\omega}{\omega_{1}}-\frac{1}{2} \log \frac{\omega^{2}}{\omega_{1}^{2}}, $$

$$ \begin{align*} & \frac{\partial\left(\beta \tilde{\omega}_{\mathrm{e}}\right)}{\partial \beta}=\frac{\omega}{\cosh \left(\epsilon \tilde{\omega}_{\mathrm{e}} / 2\right)} \\ & \frac{\partial\left(\epsilon \tilde{\omega}_{\mathrm{e}}\right)}{\partial \beta}=\frac{2}{\beta} \tanh \left(\epsilon \tilde{\omega}_{\mathrm{e}} / 2\right) \end{align*} $$

$$ \begin{align*} E & =\frac{\partial}{\partial \beta}(\beta F)=\frac{\hbar}{2} \operatorname{coth}\left(\beta \hbar \tilde{\omega}_{\mathrm{e}} / 2\right) \frac{\partial\left(\beta \tilde{\omega}_{\mathrm{e}}\right)}{\partial \beta} \\ & =\frac{\hbar \omega}{2} \frac{\operatorname{coth}\left(\beta \hbar \tilde{\omega}_{\mathrm{e}} / 2\right)}{\cosh \left(\epsilon \tilde{\omega}_{\mathrm{e}} / 2\right)} \end{align*} $$

$$ \begin{align*} \frac{1}{k_{B}} C & =-\beta^{2} \frac{\partial^{2}}{\partial \beta^{2}}(\beta F)=-\beta^{2} \frac{\partial}{\partial \beta} E \\ & =\frac{1}{4} \beta^{2} \hbar^{2} \omega^{2}\left[\frac{1}{\sinh ^{2}\left(\beta \hbar \tilde{\omega}_{\mathrm{e}} / 2\right)}+\operatorname{coth}\left(\beta \hbar \tilde{\omega}_{\mathrm{e}} / 2\right) \tanh \left(\epsilon \tilde{\omega}_{\mathrm{e}} / 2\right) \frac{\epsilon}{\hbar \beta}\right] \frac{1}{\cosh ^{2}\left(\epsilon \tilde{\omega}_{\mathrm{e}} / 2\right)} \end{align*} $$

$$ \begin{align*} F & \rightarrow \frac{1}{\beta} \log \beta, \\ E & \rightarrow \frac{1}{\beta}=T, \\ C & \rightarrow 1 . \end{align*} $$

$$ \begin{align*} \tilde{\omega}_{\mathrm{e}} & \rightarrow \frac{1}{\epsilon} \log \left(\epsilon^{2} \omega^{2}\right) \\ \cosh \left(\epsilon \tilde{\omega}_{\mathrm{e}} / 2\right) & \rightarrow \epsilon \omega / 2 \end{align*} $$

$$ E \xrightarrow{T \rightarrow 0} \frac{1}{\beta} \operatorname{coth}[(N+1) \log (\epsilon \omega)] \xrightarrow{T \rightarrow 0} 0, $$

$$ C \xrightarrow{T \rightarrow 0} N+1 \text {. } $$

$$ \tilde{\omega}_{\mathrm{e}}=\omega\left(1-\frac{1}{24} \epsilon^{2} \omega^{2}+\ldots\right) $$

$$ \tilde{\omega}_{\mathrm{e}}=\omega\left[1-\frac{1}{24} \frac{\hbar^{2} \omega^{2}}{k_{b}^{2} T^{2}(N+1)^{2}}+\ldots\right] $$

$$ V(\mathbf{x})=V_{0}+M \mathbf{g} \cdot \mathbf{x} $$

$$ \ddot{\mathbf{x}}=-\mathbf{g}, $$

$$ \mathbf{x}=\mathbf{x}_{a}+\mathbf{v}_{a}\left(t-t_{a}\right)+\frac{\mathbf{g}}{2}\left(t-t_{a}\right)^{2} $$

$$ \mathbf{v}_{a}=\frac{\mathbf{x}_{b}-\mathbf{x}_{a}}{t_{b}-t_{a}}-\frac{\mathbf{g}}{2}\left(t_{b}-t_{a}\right) $$

$$ \mathcal{A}=\int_{t_{a}}^{t_{b}} d t\left(\frac{M}{2} \dot{\mathbf{x}}^{2}-V_{0}-\mathbf{g} \cdot \mathbf{x}\right) $$

$$ \mathcal{A}_{\mathrm{cl}}=-V_{0}\left(t_{b}-t_{a}\right)+\frac{M}{2} \frac{\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right)^{2}}{t_{b}-t_{a}}-\frac{1}{2}\left(t_{b}-t_{a}\right) \mathbf{g} \cdot\left(\mathbf{x}_{b}+\mathbf{x}_{a}\right)-\frac{1}{24}\left(t_{b}-t_{a}\right)^{3} \mathbf{g}^{2} . $$

$$ \begin{align*} & \left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)=\frac{1}{{\sqrt{2 \pi i \hbar\left(t_{b}-t_{a}\right) / M}}^{3}} e^{-\frac{i}{\hbar} V_{0}\left(t_{b}-t_{a}\right)} \\ & \quad \times \exp \left\{\frac{i M}{2 \hbar}\left[\frac{\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right)^{2}}{t_{b}-t_{a}}-\left(t_{b}-t_{a}\right) \mathbf{g} \cdot\left(\mathbf{x}_{b}+\mathbf{x}_{a}\right)-\frac{1}{12}\left(t_{b}-t_{a}\right)^{3} \mathbf{g}^{2}\right]\right\} \end{align*} $$

$$ V(\mathbf{x})=V_{0}+\frac{M}{2} \omega^{2}\left(\mathbf{x}-\mathbf{x}_{0}\right)^{2} $$

$$ \omega \rightarrow 0, \quad \mathbf{x}_{0}=-\mathbf{g} / \omega^{2} \rightarrow-\infty \approx \hat{\mathbf{g}}, \quad V_{0}=-M \mathbf{x}_{0}^{2} / 2=-M g^{2} / 2 \omega^{4} \rightarrow-\infty $$

$$ \mathbf{g}=-M \omega^{2} \mathbf{x}_{0} $$

$$ v_{0}=V_{0}+\frac{M}{2} \omega^{2} \mathbf{x}_{0}^{2} $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int d E A_{E}\left(x_{b}\right) A_{E}^{*}\left(x_{a}\right) e^{-i\left(E-v_{0}\right)\left(t_{b}-t_{a}\right) / \hbar} $$

$$ A_{E}(x)=\frac{1}{\sqrt{l \varepsilon}} \mathrm{Ai}\left(\frac{x}{l}-\frac{E}{\varepsilon}\right) $$

$$ \operatorname{Ai}^{\prime \prime}(z)=z \operatorname{Ai}(z) $$

$$ \operatorname{Ai}(z)=\frac{\sqrt{z}}{2}\left[I_{-1 / 3}\left(2 z^{3 / 2} / 3\right)-I_{1 / 3}\left(2 z^{3 / 2} / 3\right)\right]=\frac{1}{\pi} \sqrt{\frac{z}{3}} K_{1 / 3}\left(2 z^{3 / 2} / 3\right) $$

$$ \operatorname{Ai}(z) \rightarrow \frac{1}{2 \sqrt{\pi} z^{1 / 4}} e^{-2 z^{3 / 2} / 3}, \quad z \rightarrow \infty $$

$$ \begin{align*} I_{\nu}(\xi)=e^{-\pi \nu i / 2} J\left(e^{\pi i / 2} \xi\right), \quad-\pi<\arg \xi \leq \pi / 2 \\ I_{\nu}(\xi)=e^{-\pi \nu i / 2} J\left(e^{\pi i / 2} \xi\right), \quad \pi / 2<\arg \xi \leq \pi \end{align*} $$

$$ \operatorname{Ai}(z)=\frac{1}{3} \sqrt{z}\left[J_{-1 / 3}\left(2(-z)^{3 / 2} / 3\right)+J_{1 / 3}\left(2(-z)^{3 / 2} / 3\right)\right] $$

$$ J_{\nu}(\xi) \rightarrow \sqrt{\frac{2}{\pi \xi}} \cos (\xi-\pi \nu / 2-\pi / 4)+\mathcal{O}\left(\xi^{-1}\right) $$

$$ \operatorname{Ai}(z) \rightarrow \frac{1}{\sqrt{\pi} z^{1 / 4}} \sin \left[2(-z)^{3 / 2} / 3+\pi / 4\right], \quad z \rightarrow-\infty $$

$$ \operatorname{Ai}(x)=\int_{-\infty}^{\infty} \frac{d k}{2 \pi} e^{i\left(x k+k^{3} / 3\right)} $$

$$ \langle p \mid E\rangle=\sqrt{\frac{l}{\varepsilon}} e^{-i\left(p E-p^{3} / 6 M\right) l / \varepsilon \hbar} $$

$$ \int \frac{d p}{2 \pi \hbar}\left\langle E^{\prime} \mid p\right\rangle\langle p \mid E\rangle=\delta\left(E^{\prime}-E\right), \quad \int d E\left\langle p^{\prime} \mid E\right\rangle\langle E \mid p\rangle=2 \pi \hbar \delta\left(p^{\prime}-p\right) $$

$$ \mathcal{A}_{\mathrm{mag}}=\frac{e}{c} \int_{t_{a}}^{t_{b}} d t \dot{\mathbf{x}}(t) \cdot \mathbf{A}(\mathbf{x}(t)) $$

$$ \mathcal{A}[\mathbf{x}]=\int_{t_{a}}^{t_{b}} d t\left[\frac{M}{2} \dot{\mathbf{x}}^{2}(t)+\frac{e}{c} \dot{\mathbf{x}}(t) \cdot \mathbf{A}(\mathbf{x}(t))\right] $$

$$ \mathbf{A}(\mathbf{x})=(0, B x, 0) $$

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

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

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

$$ \tilde{\mathbf{A}}(\mathbf{x})=\frac{1}{2} \mathbf{B} \times \mathbf{x} $$

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

$$ \Lambda(\mathbf{x})=-\frac{1}{2} B x y $$

$$ \mathcal{A}[\mathbf{p}, \mathbf{x}]=\int_{t_{a}}^{t_{b}} d t\left\{\mathbf{p} \cdot \dot{\mathbf{x}}-\frac{1}{2 M}\left[\mathbf{p}-\frac{e}{c} \mathbf{A}(\mathbf{x})\right]^{2}\right\} $$

$$ \mathbf{p} \rightarrow \mathbf{P} \equiv \mathbf{p}-\frac{e}{c} \mathbf{A}(\mathbf{x}) $$

$$ \mathcal{A}[\mathbf{p}, \mathbf{x}]=\int_{t_{a}}^{t_{b}} d t[\mathbf{p} \cdot \dot{\mathbf{x}}-H(\mathbf{p}, \mathbf{x})] $$

$$ H(\mathbf{p}, \mathbf{x})=\frac{\mathbf{p}^{2}}{2 M}+\frac{1}{8} M \omega_{L}^{2} \mathbf{x}^{2}-\frac{1}{2} \omega_{L} l_{z}(\mathbf{p}, \mathbf{x}), $$

$$ l_{z}(\mathbf{p}, \mathbf{x})=(\mathbf{x} \times \mathbf{p})_{z}=x p_{y}-y p_{x} $$

$$ \omega_{L}=\frac{e}{M c} B $$

$$ \mu_{B} \equiv \frac{\hbar e}{M c}, $$

$$ \omega_{L}=\mu_{B} B / \hbar $$

$$ \omega_{B} \equiv \frac{\omega_{L}}{2} $$

$$ H(\mathbf{p}, \mathbf{x})=\frac{\mathbf{p}^{2}}{2 M}+\frac{1}{2} M \omega_{L}^{2} x^{2}-\omega_{L} x p_{y} $$

$$ \mathcal{A}_{\mathrm{e}}^{N}=\sum_{n=1}^{N+1}\left\{\mathbf{p}_{n}\left(\mathbf{x}_{n}-\mathbf{x}_{n-1}\right)-\frac{1}{2 M}\left[p_{x n}^{2}+\left(p_{y n}-B x_{n}\right)^{2}+p_{z n}^{2}\right]\right\}, $$

$$ \left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)=\prod_{n=1}^{N}\left[\int d^{3} x_{n}\right] \prod_{n=1}^{N+1}\left[\int \frac{d^{3} p_{n}}{(2 \pi \hbar)^{3}}\right] \exp \left(\frac{i}{\hbar} \mathcal{A}_{\mathrm{e}}^{N}\right) $$

$$ \mathcal{A}^{N}=\sum_{n=1}^{N+1}\left\{\mathbf{p}_{n}\left(\mathbf{x}_{n}-\mathbf{x}_{n-1}\right)-\frac{1}{2 M}\left[p_{x n}^{2}+\left(p_{y n}-B x_{n}\right)^{2}+p_{z n}^{2}\right]\right\} $$

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

$$ \Delta \mathcal{A}=\frac{e}{c} \int_{t_{a}}^{t_{b}} d t \dot{\mathbf{x}} \cdot \nabla \Lambda(\mathbf{x})=\frac{e}{c}\left[\Lambda\left(\mathbf{x}_{b}\right)-\Lambda\left(\mathbf{x}_{a}\right)\right] $$

$$ \left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)_{A} \rightarrow\left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)_{A^{\prime}}=e^{i e \Lambda\left(\mathbf{x}_{b}\right) / c \hbar}\left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)_{A} e^{-i e \Lambda\left(\mathbf{x}_{a}\right) / c \hbar} $$

$$ (2 \pi \hbar)^{2} \delta^{(2)}\left(\mathbf{p}_{N+1}^{\prime}-\mathbf{p}_{N}^{\prime}\right) \cdots(2 \pi \hbar)^{2} \delta^{(2)}\left(\mathbf{p}_{2}^{\prime}-\mathbf{p}_{1}^{\prime}\right) $$

$$ \begin{align*} \left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right) & =\int_{-\infty}^{\infty} \frac{d p_{y} d p_{z}}{(2 \pi \hbar)^{2}} \prod_{n=1}^{N}\left[\int_{-\infty}^{\infty} d x_{n}\right]_{n=1}^{N+1}\left[\int \frac{d p_{x n}}{2 \pi \hbar}\right] \\ & \times \exp \left\{\frac{i}{\hbar}\left[p_{y}\left(y_{b}-y_{a}\right)+p_{z}\left(z_{b}-z_{a}\right)-\left(t_{b}-t_{a}\right) \frac{p_{z}^{2}}{2 M}\right]\right\} \exp \left(\frac{i}{\hbar} \mathcal{A}_{x}^{N}\right) \end{align*} $$

$$ \mathcal{A}_{x}^{N}=\sum_{n=1}^{N+1}\left[p_{x n}\left(x_{n}-x_{n-1}\right)-\frac{p_{x n}^{2}}{2 M}-\frac{1}{2 M}\left(p_{y}-\frac{e}{c} B x_{n}\right)^{2}\right] $$

$$ x_{0}=p_{y} / M \omega_{L} $$

$$ \begin{align*} & \left(x_{b} t_{b} \mid x_{a} t_{a}\right)_{x_{0}}=\sqrt{\frac{M \omega_{L}}{2 \pi i \hbar \sin \omega_{L}\left(t_{b}-t_{a}\right)}} \\ & \quad \times \exp \left(\frac { i } { \hbar } \frac { M \omega _ { L } } { 2 \operatorname { s i n } \omega _ { L } ( t _ { b } - t _ { a } ) } \left\{\left[\left(x_{b}-x_{0}\right)^{2}+\left(x_{a}-x_{0}\right)^{2}\right] \cos \omega_{L}\left(t_{b}-t_{a}\right)\right.\right. \\ & \left.\left.\quad-2\left(x_{b}-x_{0}\right)\left(x_{a}-x_{0}\right)\right\}\right) \end{align*} $$

$$ \left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar\left(t_{b}-t_{a}\right) / M}} e^{i \frac{M}{2 \hbar} \frac{\left(z_{b}-z_{a}\right)^{2}}{t_{b}-t_{a}}}\left(\mathbf{x}_{b}^{\perp} t_{b} \mid \mathbf{x}_{a}^{\perp} t_{a}\right) $$

$$ \left(\mathbf{x}_{b}^{\perp} t_{b} \mid \mathbf{x}_{a}^{\perp} t_{a}\right) \equiv \frac{M \omega_{L}}{2 \pi \hbar} \int_{-\infty}^{\infty} d x_{0} e^{i M \omega_{L} x_{0}\left(y_{b}-y_{a}\right) / \hbar}\left(x_{b} t_{b} \mid x_{a} t_{a}\right)_{x_{0}} $$

$$ \begin{align*} & \frac{i M \omega_{L}}{2 \hbar}\left[-\left(x_{b}^{2}+x_{a}^{2}\right) \tan \left[\omega_{L}\left(t_{b}-t_{a}\right) / 2\right]+\left(x_{b}-x_{a}\right)^{2} \frac{1}{\sin \omega_{L}\left(t_{b}-t_{a}\right)}\right] \\ - & i \frac{M \omega_{L}}{\hbar} \tan \left[\omega_{L}\left(t_{b}-t_{a}\right) / 2\right]\left(x_{0}-\frac{x_{b}+x_{a}}{2}-\frac{y_{b}-y_{a}}{2 \tan \left[\omega_{L}\left(t_{b}-t_{a}\right) / 2\right]}\right)^{2} \\ + & i \frac{M \omega_{L}}{2 \hbar}\left[\frac{\left(x_{b}+x_{a}\right)^{2}}{2} \tan \left[\omega_{L}\left(t_{b}-t_{a}\right) / 2\right]+\frac{\left(y_{b}-y_{a}\right)^{2}}{2 \tan \left[\omega_{L}\left(t_{b}-t_{a}\right) / 2\right]}\right] \\ + & i \frac{M \omega_{L}}{2 \hbar}\left(x_{b}+x_{a}\right)\left(y_{b}-y_{a}\right) \end{align*} $$

$$ \frac{M \omega_{L}}{2 \pi \hbar} \sqrt{\frac{\pi \hbar}{i M \omega_{L} \tan \left[\omega_{L}\left(t_{b}-t_{a}\right) / 2\right]}} $$

$$ \left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)={\sqrt{\frac{M}{2 \pi i \hbar\left(t_{b}-t_{a}\right)}}}^{3} \frac{\omega_{L}\left(t_{b}-t_{a}\right) / 2}{\sin \left[\omega_{L}\left(t_{b}-t_{a}\right) / 2\right]} \exp \left[\frac{i}{\hbar}\left(\mathcal{A}_{\mathrm{cl}}+\mathcal{A}_{\mathrm{sf}}\right)\right] $$

$$ \begin{align*} \mathcal{A}_{\mathrm{cl}}=\frac{M}{2}\left\{\frac{\left(z_{b}-z_{a}\right)^{2}}{t_{b}-t_{a}}+\frac{\omega_{L}}{2} \cot \left[\omega_{L}\left(t_{b}-t_{a}\right) / 2\right]\left[\left(x_{b}\right.\right.\right. & \left.\left.-x_{a}\right)^{2}+\left(y_{b}-y_{a}\right)^{2}\right] \\ & \left.+\omega_{L}\left(x_{a} y_{b}-x_{b} y_{a}\right)\right\} \end{align*} $$

$$ \mathcal{A}_{\mathrm{sf}}=\frac{M \omega_{L}}{2}\left(x_{b} y_{b}-x_{a} y_{a}\right)=\left.\frac{e}{2 c} B x y\right|_{a} ^{b} $$

$$ \mathcal{A}_{\mathrm{cl}}^{\perp}=\int_{t_{a}}^{t_{b}} d t\left\{\frac{M}{2} \frac{d}{d t}(x \dot{x}+y \dot{y})+\frac{M}{2}\left[x\left(-\ddot{x}+\omega_{L} \dot{y}\right)+y\left(-\ddot{y}-\omega_{L} \dot{x}\right)\right]\right\} $$

$$ \ddot{x}=\omega_{L} \dot{y}, \quad \ddot{y}=-\omega_{L} \dot{x} $$

$$ \mathcal{A}_{\mathrm{cl}}^{\perp}=\left.\frac{M}{2}(x \dot{x}+y \dot{y})\right|_{t_{a}} ^{t_{b}}=\frac{M}{2}\left(\left[x_{b} \dot{x}_{b}-x_{a} \dot{x}_{a}\right]+\left[y_{b} \dot{y}_{b}-y_{a} \dot{y}_{a}\right]\right) . $$

$$ \dot{\ddot{x}}+\omega_{L}^{2} \dot{x}=0, \quad \dot{\ddot{y}}+\omega_{L}^{2} \dot{y}=0 . $$

$$ \begin{align*} & x=\frac{1}{\sin \omega_{L}\left(t_{b}-t_{a}\right)}\left[\left(x_{b}-x_{0}\right) \sin \omega_{L}\left(t-t_{a}\right)-\left(x_{a}-x_{0}\right) \sin \omega_{L}\left(t-t_{b}\right)\right]+x_{0} \\ & y=\frac{1}{\sin \omega_{L}\left(t_{b}-t_{a}\right)}\left[\left(y_{b}-y_{0}\right) \sin \omega_{L}\left(t-t_{a}\right)-\left(y_{a}-y_{0}\right) \sin \omega_{L}\left(t-t_{b}\right)\right]+y_{0} \end{align*} $$

$$ \begin{align*} & x_{0}=\frac{1}{2}\left[\left(x_{b}+x_{a}\right)+\left(y_{b}-y_{a}\right) \cot \frac{\omega_{L}}{2}\left(t_{b}-t_{a}\right)\right] \\ & y_{0}=\frac{1}{2}\left[\left(y_{b}+y_{a}\right)-\left(x_{b}-x_{a}\right) \cot \frac{\omega_{L}}{2}\left(t_{b}-t_{a}\right)\right] \end{align*} $$

$$ \begin{align*} x_{b} \dot{x}_{b} & =\frac{\omega_{L}}{\sin \omega_{L}\left(t_{b}-t_{a}\right)} x_{b}\left[\left(x_{0}-x_{a}\right)+\left(x_{b}-x_{0}\right) \cos \omega_{L}\left(t_{b}-t_{a}\right)\right] \\ x_{a} \dot{x}_{a} & =\frac{\omega_{L}}{\sin \omega_{L}\left(t_{b}-t_{a}\right)} x_{a}\left[\left(x_{0}-x_{a}\right) \cos \omega_{L}\left(t_{b}-t_{a}\right)+\left(x_{b}-x_{0}\right)\right] \end{align*} $$

$$ \begin{align*} x_{b} \dot{x}_{b}-x_{a} \dot{x}_{a}= & \omega_{L} x_{0}\left(x_{b}+x_{a}\right) \tan \frac{\omega_{L}}{2}\left(t_{b}-t_{a}\right) \\ & +\frac{\omega_{L}}{\sin \omega_{L}\left(t_{b}-t_{a}\right)}\left[\left(x_{b}^{2}+x_{a}^{2}\right) \cos \omega_{L}\left(t_{b}-t_{a}\right)-2 x_{b} x_{a}\right] \\ = & \frac{\omega_{L}}{2}\left[\left(x_{b}-x_{a}\right)^{2} \cot \frac{\omega_{L}}{2}\left(t_{b}-t_{a}\right)+\left(x_{b}+x_{a}\right)\left(y_{b}-y_{a}\right)\right] \end{align*} $$

$$ \begin{align*} y_{b} \dot{y}_{b}-y_{a} \dot{y}_{a}= & \omega_{L} y_{0}\left(y_{b}+y_{a}\right) \tan \frac{\omega_{L}}{2}\left(t_{b}-t_{a}\right) \\ & +\frac{\omega_{L}}{\sin \omega_{L}\left(t_{b}-t_{a}\right)}\left[\left(y_{b}^{2}+y_{a}^{2}\right) \cos \omega_{L}\left(t_{b}-t_{a}\right)-2 y_{b} y_{a}\right] \\ = & \frac{\omega_{L}}{2}\left[\left(y_{b}-y_{a}\right)^{2} \cot \frac{\omega_{L}}{2}\left(t_{b}-t_{a}\right)-\left(x_{b}-x_{a}\right)\left(y_{b}+y_{a}\right)\right] \end{align*} $$

$$ \mathcal{A}_{\mathrm{cl}}^{\perp}=\frac{M}{2}\left\{\frac{\omega_{L}}{2} \cot \left[\omega_{L}\left(t_{b}-t_{a}\right) / 2\right]\left[\left(x_{b}-x_{a}\right)^{2}+\left(y_{b}-y_{a}\right)^{2}\right]+\omega_{L}\left(x_{a} y_{b}-x_{b} y_{a}\right)\right\}, $$

$$ \Delta \mathcal{A}=\frac{M \omega_{L}}{2}\left(x_{a} y_{b}-x_{b} y_{a}\right) $$

$$ \mathbf{x} \rightarrow \mathbf{x}+\mathbf{d} $$

$$ \frac{M \omega_{L}}{2}\left[d_{x}\left(y_{b}-y_{a}\right)+d_{y}\left(x_{a}-x_{b}\right)\right]=\frac{M \omega_{L}}{2}\left[(\mathbf{d} \times \mathbf{x})_{b}-(\mathbf{d} \times \mathbf{x})_{a}\right]_{z} $$

$$ \left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right) \rightarrow e^{i e \Lambda\left(\mathbf{x}_{b}\right) / c \hbar}\left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right) e^{-i e \Lambda\left(\mathbf{x}_{a}\right) / c \hbar} $$

$$ \Lambda(\mathbf{x})=-\frac{M \omega_{L} \hbar c}{2 e}[\mathbf{d} \times \mathbf{x}]_{z} $$

$$ H(\mathbf{p}, \mathbf{x})=\frac{\mathbf{p}^{2}}{2 M}+\frac{1}{2} M \omega^{2} \mathbf{x}^{2}-\omega_{B} l_{z}(\mathbf{p}, \mathbf{x}) $$

$$ \mathcal{A}_{\mathrm{e}}[\mathbf{p}, \mathbf{x}]=\int_{\tau_{a}}^{\tau_{b}} d \tau[-i \mathbf{p} \cdot \dot{\mathbf{x}}+H(\mathbf{p}, \mathbf{x})] $$

$$ \mathcal{A}_{\mathrm{e}}[\mathbf{x}]=\int_{0}^{\hbar \beta} d \tau\left\{\frac{M}{2} \dot{\mathbf{x}}^{2}(\tau)+\frac{1}{2} M\left(\omega^{2}-\omega_{B}^{2}\right) \mathbf{x}^{2}(\tau)-i M \omega_{B}[\mathbf{x}(\tau) \times \dot{\mathbf{x}}(\tau)]_{z}\right\} $$

$$ \mathcal{A}_{\mathrm{cl}}=\int_{0}^{\hbar \beta} d \tau\left[\frac{M}{2} \frac{d}{d \tau}(\mathbf{x} \dot{\mathbf{x}})+\frac{M}{2} \mathbf{x}^{T} \mathbf{D}_{\omega^{2}, B} \mathbf{x}\right] $$

$$ \mathbf{D}_{\omega^{2}, B}\left(\tau, \tau^{\prime}\right) \equiv\left(\begin{array}{cc} -\partial_{\tau}^{2}+\omega^{2}-\omega_{B}^{2} & -2 i \omega_{B} \partial_{\tau} \\ 2 i \omega_{B} \partial_{\tau} & -\partial_{\tau}^{2}+\omega^{2}-\omega_{B}^{2} \end{array}\right) \delta\left(\tau-\tau^{\prime}\right) $$

$$ Z=\frac{1}{(2 \pi \hbar / M)^{2}} \operatorname{det} \mathbf{D}_{\omega^{2}, B}^{-1 / 2} $$

$$ \mathbf{D}_{\omega^{2}, B}\left(\tau, \tau^{\prime}\right)=\frac{1}{\hbar \beta} \sum_{m=-\infty}^{\infty} \tilde{\mathbf{D}}_{\omega^{2}, B}\left(\omega_{m}\right) e^{-i \omega_{m}\left(\tau-\tau^{\prime}\right)} $$

$$ \tilde{\mathbf{D}}_{\omega^{2}, B}\left(\omega_{m}\right)=\left(\begin{array}{cc} \omega_{m}^{2}+\omega^{2}-\omega_{B}^{2} & -2 \omega_{B} \omega_{m} \\ 2 \omega_{B} \omega_{m} & \omega_{m}^{2}+\omega^{2}-\omega_{B}^{2} \end{array}\right) $$

$$ \operatorname{det} \tilde{\mathbf{D}}_{\omega^{2}, B}\left(\omega_{m}\right)=\left(\omega_{m}^{2}+\omega^{2}-\omega_{B}^{2}\right)^{2}+4 \omega_{B}^{2} \omega_{m}^{2} . $$

$$ \operatorname{det} \tilde{\mathbf{D}}_{\omega^{2}, B}\left(\omega_{m}\right)=\left(\omega_{m}^{2}+\omega_{+}^{2}\right)\left(\omega_{m}^{2}+\omega_{-}^{2}\right) $$

$$ \omega_{ \pm} \equiv \omega \pm \omega_{B} $$

$$ \mathbf{e}_{+}=\frac{1}{\sqrt{2}}\binom{1}{i}, \quad \mathbf{e}_{-}=-\frac{1}{\sqrt{2}}\binom{1}{-i}, $$

$$ d_{ \pm}=\omega_{m}^{2}+\omega^{2} \pm 2 i \omega_{m} \omega_{B}=\left(\omega_{m}+i \omega_{ \pm}\right)\left(\omega_{m}-i \omega_{\mp}\right) $$

$$ \begin{align*} \mathcal{A}_{\mathrm{cl}} & =\int_{0}^{\hbar \beta} d \tau\left\{\frac{M}{2} \frac{d}{d \tau}\left(x_{+}^{*} \dot{x}_{+}+x_{-}^{*} \dot{x}_{-}\right)\right. \\ & \left.+\frac{M}{2}\left[x_{+}^{*}\left(-\partial_{\tau}-\omega_{+}\right)\left(-\partial_{\tau}+\omega_{-}\right) x_{+}+x_{-}^{*}\left(-\partial_{\tau}-\omega_{-}\right)\left(-\partial_{\tau}+\omega_{+}\right) x_{-}\right]\right\} \end{align*} $$

$$ Z=\frac{1}{2 \sinh \left(\hbar \beta \omega_{+} / 2\right)} \frac{1}{2 \sinh \left(\hbar \beta \omega_{-} / 2\right)} $$

$$ \frac{1}{\omega^{2}-\omega_{B}^{2}} \underset{\omega \rightarrow \omega_{B}}{\longrightarrow} \frac{1}{2 \omega \omega_{-}} \underset{\omega_{-} \rightarrow 0}{\longrightarrow} \frac{\beta}{2 \pi / M} V_{2} $$

$$ Z \underset{\omega_{-} \rightarrow 0}{\longrightarrow} \frac{1}{2 \sinh (\hbar \beta \omega)} \frac{V_{2}}{\lambda_{\omega}^{2}}, $$

$$ \begin{gather*} \mathcal{A}[\mathbf{x}]=\int_{t_{a}}^{t_{b}} d t\left\{\frac{M}{2} \dot{\mathbf{x}}^{2}+\frac{e}{c} \dot{\mathbf{x}}(t)[\mathbf{A}(\mathbf{x}, t)+\boldsymbol{\nabla} \Lambda(\mathbf{x}, t)]-V(\mathbf{x}, t)+\frac{e}{c} \partial_{t} \Lambda(\mathbf{x}, t)\right\} \\ -\frac{e}{c}\left[\Lambda\left(\mathbf{x}_{b}, t_{b}\right)-\Lambda\left(\mathbf{x}_{a}, t_{a}\right)\right] \end{gather*} $$

$$ \frac{1}{2 M}\left[\nabla A(\mathbf{x}, t)-\frac{e}{c} \mathbf{A}(\mathbf{x}, t)\right]^{2}+\partial_{t} A(\mathbf{x}, t)+V(\mathbf{x}, t)=0 $$

$$ \begin{array}{r} \mathcal{A}[\mathbf{x}]=\int_{t_{a}}^{t_{b}} d t \frac{1}{2 M}\left[M \dot{\mathbf{x}}-\nabla A(\mathbf{x}, t)+\frac{e}{c} \mathbf{A}(\mathbf{x}, t)\right]^{2} \\ +A\left(\mathbf{x}_{b}, t_{b}\right)-A\left(\mathbf{x}_{a}, t_{a}\right) \end{array} $$

$$ \mathbf{v} \cdot \nabla \delta A+\partial_{t} \delta A=0 $$

$$ \mathbf{v}(\mathbf{x}, t) \equiv(1 / M)\left[\nabla A(\mathbf{x}, t)-\frac{e}{c} \mathbf{A}(\mathbf{x}, t)\right] $$

$$ \begin{align*} \left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)= & e^{i A\left(\mathbf{x}_{b}, t_{b} ; \mathbf{x}_{a}, t_{a}\right) / \hbar} \int_{\mathbf{x}\left(t_{a}\right)=\mathbf{x}_{a}}^{\mathbf{x}\left(t_{b}\right)=\mathbf{x}_{b}} \mathcal{D} \mathbf{x} \\ & \times \exp \left\{\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t \frac{1}{2 M}\left[M \dot{\mathbf{x}}-\nabla A\left(\mathbf{x}, t ; \mathbf{x}_{a}, t_{a}\right)+\frac{e}{c} \mathbf{A}(\mathbf{x}, t)\right]^{2}\right\} \end{align*} $$

$$ \left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)=e^{i A\left(\mathbf{x}_{b}, t_{b} ; \mathbf{x}_{a}, t_{a}\right) / \hbar} \int_{\mathbf{x}\left(t_{a}\right)=\mathbf{x}_{a}}^{\mathbf{x}\left(t_{b}\right)=\mathbf{x}_{b}} \mathcal{D} \mathbf{x} \exp \left[\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t \frac{M}{2}(\dot{\mathbf{x}}-\mathbf{v})^{2}\right] $$

$$ \begin{align*} \left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right) & =e^{i A\left(\mathbf{x}_{b}, t_{b} ; \mathbf{x}_{a}, t_{a}\right) / \hbar} \int_{\mathbf{x}\left(t_{a}\right)=\mathbf{x}_{a}}^{\mathbf{x}\left(t_{b}\right)=\mathbf{x}_{b}} \mathcal{D}^{\prime} \mathbf{x} \int \frac{\mathcal{D} \mathbf{p}}{2 \pi \hbar} \\ & \times \exp \left(\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t\left\{\mathbf{p}(t)[\dot{\mathbf{x}}(t)-\mathbf{v}(\mathbf{x}(t), t)]-\frac{1}{2 M} \mathbf{p}^{2}(t)\right\}\right) \end{align*} $$

$$ \left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)=\int \mathcal{D}^{3} x \delta\left(\mathbf{x}_{b}-\mathbf{x}_{a}-\int_{t_{a}}^{t_{b}} d t \dot{\mathbf{x}}(t)\right) \exp \left\{\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t\left[\frac{M}{2} \dot{\mathbf{x}}^{2}-V(\mathbf{x})\right]\right\} $$

$$ \begin{align*} \mathbf{x}(t) & =\mathbf{x}_{b}-\int_{t}^{t_{b}} d t \mathbf{v}(t) \\ \mathbf{x}(t) & =\mathbf{x}_{a}+\int_{t_{a}}^{t} d t \mathbf{v}(t) \\ \mathbf{x}(t) & =\mathbf{X}+\frac{1}{2} \int_{t_{a}}^{t_{b}} d t^{\prime} \mathbf{v}\left(t^{\prime}\right) \epsilon\left(t^{\prime}-t\right) \end{align*} $$

$$ \mathbf{X} \equiv \frac{\mathbf{x}_{b}+\mathbf{x}_{a}}{2} $$

$$ \int \mathcal{D}^{3} v \exp \left\{\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t\left[\frac{M}{2} \mathbf{v}^{2}\right]\right\}=1 $$

$$ \delta\left(\mathbf{x}_{b}-\mathbf{x}_{a}-\int_{t_{a}}^{t_{b}} d t \mathbf{v}(t)\right)=\int \frac{d^{3} p}{(2 \pi i)^{3}} \exp \left[\frac{i}{\hbar} \mathbf{p}\left(\mathbf{x}_{b}-\mathbf{x}_{a}-\int_{t_{a}}^{t_{b}} d t \mathbf{v}(t)\right)\right] $$

$$ \left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)=\int \frac{d^{3} p}{(2 \pi i)^{3}} \exp \left\{\frac{i}{\hbar}\left[\mathbf{p}\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right)-\frac{\mathbf{p}^{2}}{2 M}\left(t_{b}-t_{a}\right)\right]\right\} $$

$$ \begin{align*} \left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right) & =\int \mathcal{D}^{3} v \delta\left(\Delta \mathbf{x}-\int_{t_{a}}^{t_{b}} d t \mathbf{v}(t)\right) \exp \left\{\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t \frac{M}{2} \mathbf{v}^{2}\right\} \\ & \times \exp \left\{-\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t V\left(\mathbf{X}+\frac{1}{2} \int_{t_{a}}^{t_{b}} d t^{\prime} \mathbf{v}\left(t^{\prime}\right) \epsilon\left(t^{\prime}-t\right)\right)\right\} \end{align*} $$

$$ \begin{align*} \int d^{3} x_{a}\left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right) & =\int \mathcal{D}^{3} v \exp \left\{\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t \frac{M}{2} \mathbf{v}^{2}\right\} \\ & \times \exp \left\{-\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t V\left(\mathbf{x}_{b}-\frac{1}{2} \int_{t}^{t_{b}} d t^{\prime} \mathbf{v}\left(t^{\prime}\right)\right)\right\} \end{align*} $$

$$ \left\langle\mathbf{p}_{b}\right| \hat{S}\left|\mathbf{p}_{a}\right\rangle \equiv \lim _{t_{b}-t_{a} \rightarrow \infty} e^{i\left(E_{b} t_{b}-E_{a} t_{a}\right) / \hbar} \int d^{3} x_{b} \int d^{3} x_{a} e^{-i\left(\mathbf{p}_{b} \mathbf{x}_{b}-\mathbf{p}_{a} \mathbf{x}_{a}\right) / \hbar}\left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right) $$

$$ e^{-i \mathbf{p}_{a}\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right) / \hbar}\left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)=\int \mathcal{D}^{3} x \exp \left\{\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t\left[\frac{M}{2} \dot{\mathbf{x}}^{2}-\mathbf{p}_{a} \dot{\mathbf{x}}-V(\mathbf{x})\right]\right\} $$

$$ \mathbf{y}(t)=\mathbf{x}(t)-\frac{\mathbf{p}_{a}}{M} t $$

$$ e^{-i \mathbf{p}_{a}\left(\mathbf{x}_{b}-\mathbf{x}_{b}\right) / \hbar}\left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)=e^{-i \mathbf{p}_{a}^{2}\left(t_{b}-t_{a}\right) / 2 M \hbar} \int \mathcal{D}^{3} y \exp \left\{\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t\left[\frac{M}{2} \dot{\mathbf{y}}^{2}-V\left(\mathbf{y}+\frac{\mathbf{P}}{M} t\right)\right]\right\} $$

$$ \begin{align*} \left\langle\mathbf{p}_{b}\right| \hat{S}\left|\mathbf{p}_{a}\right\rangle & \equiv \lim _{t_{b}-t_{a} \rightarrow \infty} e^{i \mathbf{q}^{2} t_{b} / 2 M \hbar} \int d^{3} y_{b} e^{-i \mathbf{q} \mathbf{y}_{b} / \hbar} \int d^{3} y_{a} \\ & \times \int \mathcal{D}^{3} y \exp \left\{\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t\left[\frac{M}{2} \dot{\mathbf{y}}^{2}-V\left(\mathbf{y}+\frac{\mathbf{p}_{a}}{M} t\right)\right]\right\} \end{align*} $$

$$ \int d^{3} y_{a} \frac{1}{\sqrt{2 \pi \hbar i\left(t_{b}-t_{a}\right) / M}} \exp \left[\frac{i}{\hbar} \frac{M}{2} \frac{\left(\mathbf{y}_{b}-\mathbf{y}_{a}\right)^{2}}{\left.t_{b}-t_{a}\right)}\right]=1 $$

$$ \left.\left\langle\mathbf{p}_{b}\right| \hat{S}\left|\mathbf{p}_{a}\right\rangle\right|_{V \equiv 0}=\lim _{t_{b}-t_{a} \rightarrow \infty} e^{i \mathbf{q}^{2}\left(t_{b}-t_{a}\right) / 8 M \hbar}(2 \pi \hbar)^{3} \delta^{(3)}(\mathbf{q})=(2 \pi \hbar)^{3} \delta^{(3)}\left(\mathbf{p}_{b}-\mathbf{p}_{a}\right) $$

$$ \begin{align*} \left\langle\mathbf{p}_{b}\right| \hat{S}_{1}\left|\mathbf{p}_{a}\right\rangle & =-\frac{i}{\hbar} \lim _{t_{b}-t_{a} \rightarrow \infty} e^{i \mathbf{q}^{2} t_{b} / 2 M \hbar} \int d^{3} y_{b} e^{-i \mathbf{q}_{b} / \hbar} \int \frac{d^{3} Q}{(2 \pi \hbar)^{3}} V(\mathbf{Q}) \int d^{3} y_{a} \\ & \times \int_{t_{a}}^{t_{b}} d t^{\prime} \exp \left(\frac{i}{\hbar} \frac{\mathbf{p}_{a} \mathbf{Q}}{M} t^{\prime}\right) \int \mathcal{D}^{3} y \exp \left\{\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t\left[\frac{M}{2} \dot{\mathbf{y}}^{2}+\delta\left(t^{\prime}-t\right) \mathbf{Q} \mathbf{y}\right]\right\} \end{align*} $$

$$ \begin{align*} & \frac{1}{{\sqrt{2 \pi i \hbar\left(t_{b}-t_{a}\right) / M}}^{3}} \exp \left[\frac{i}{\hbar} \frac{M}{2} \frac{\left(\mathbf{y}_{b}-\mathbf{y}_{a}\right)^{2}}{t_{b}-t_{a}}\right] \\ & \times \exp \left(\frac{i}{\hbar} \frac{1}{t_{b}-t_{a}}\left\{\left[\mathbf{y}_{b}\left(t^{\prime}-t_{a}\right)+\mathbf{y}_{a}\left(t_{b}-t^{\prime}\right)\right] \mathbf{Q}-\frac{1}{2 M}\left(t_{b}-t^{\prime}\right)\left(t^{\prime}-t_{a}\right) \mathbf{Q}^{2}\right\}\right) \end{align*} $$

$$ \exp \left\{\frac{i}{\hbar} \mathbf{Q} \mathbf{y}_{b}\right\} \exp \left\{-\frac{i}{\hbar} \frac{1}{2 M}\left(t_{b}-t^{\prime}\right) \mathbf{Q}^{2}\right\} $$

$$ \left\langle\mathbf{p}_{b}\right| \hat{S}_{1}\left|\mathbf{p}_{a}\right\rangle=-2 \pi i \delta\left(E_{b}-E_{a}\right) V(\mathbf{q}) $$

$$ \begin{align*} & 2 \pi \hbar i \delta\left(E_{b}-E_{a}\right)\left\langle\mathbf{p}_{b}\right| \hat{T}\left|\mathbf{p}_{a}\right\rangle \equiv-\lim _{t_{b}-t_{a} \rightarrow \infty} e^{i \mathbf{q}^{2} t_{b} / 2 M \hbar} \int d^{3} y_{b} e^{-i \mathbf{q y}_{b} / \hbar} \int d^{3} y_{a} \\ & \quad \times \int \mathcal{D}^{3} y \exp \left(\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t \frac{M}{2} \dot{\mathbf{y}}^{2}\right)\left\{\exp \left[-\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t V\left(\mathbf{y}+\frac{\mathbf{p}_{a}}{M} t\right)\right]-1\right\} \end{align*} $$

$$ 1=\frac{\left|\mathbf{p}_{a}\right|}{M} \int_{-\infty}^{\infty} d t_{0} \delta\left(\hat{\mathbf{p}}_{a}\left(\mathbf{y}_{b}+\mathbf{p}_{a} t_{0} / M\right)\right) $$

$$ \mathbf{p} \equiv \mathbf{p}_{a}, \quad p \equiv\left|\mathbf{p}_{a}\right|=\left|\mathbf{p}_{b}\right| $$

$$ \begin{align*} \left\langle\mathbf{p}_{b}\right| \hat{T}\left|\mathbf{p}_{a}\right\rangle & \equiv i \frac{p}{M} \lim _{t_{b}-t_{a} \rightarrow \infty} e^{i \mathbf{q}^{2}\left(t_{b}-t_{a}\right) / 8 M \hbar} \int d^{3} y_{b} \delta\left(\hat{\mathbf{p}}_{a} \mathbf{y}_{b}\right) e^{-i \mathbf{q} \mathbf{y}_{b} / \hbar} \int d^{3} y_{a} \\ & \times \int \mathcal{D}^{3} y \exp \left(\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t \frac{M}{2} \dot{\mathbf{y}}^{2}\right)\left\{\exp \left[-\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t V\left(\mathbf{y}+\frac{\mathbf{P}}{M} t\right)\right]-1\right\} \end{align*} $$

$$ \mathbf{b} \equiv \mathbf{y}_{b}-\left(\hat{\mathbf{p}}_{a} \mathbf{y}_{b}\right) \hat{\mathbf{p}} / a $$

$$ \begin{align*} \left\langle\mathbf{p}_{b}\right| \hat{T}\left|\mathbf{p}_{a}\right\rangle \equiv i \frac{p}{M} \lim _{t_{b}-t_{a} \rightarrow \infty} & e^{i \mathbf{q}^{2} t_{b} / 2 M \hbar} \int d^{2} b e^{-i \mathbf{q} / \hbar} \\ & \times \int \mathcal{D}^{3} v \exp \left(\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t \frac{M}{2} \mathbf{v}^{2}\right)\left[e^{i \chi_{\mathbf{b}, \mathbf{p}}[\mathbf{v}]}-1\right] \end{align*} $$

$$ \chi_{\mathbf{b}, \mathbf{p}}[\mathbf{v}] \equiv-\frac{1}{\hbar} \int_{t_{a}}^{t_{b}} d t V\left(\mathbf{b}+\frac{\mathbf{p}}{M} t-\int_{t}^{t_{b}} d t^{\prime} \mathbf{v}\left(t^{\prime}\right)\right) $$

$$ \int \mathcal{D}^{3} v \exp \left\{\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t\left[\frac{M}{2} \mathbf{v}^{2}-\Theta\left(t_{b}-t^{\prime}\right) \mathbf{Q} \mathbf{v}\right]\right\}=e^{-\frac{i}{2 M \hbar} \int_{t_{a}}^{t_{b}} d t \Theta^{2}\left(t_{b}-t^{\prime}\right) \mathbf{Q}^{2}}=e^{-\frac{i}{2 M \hbar}\left(t_{b}-t^{\prime}\right) \mathbf{Q}^{2}} $$

$$ e^{i \mathbf{q}^{2} t_{b} / 2 M \hbar}=\int \mathcal{D}^{3} w \exp \left[-\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t \frac{M}{2} \mathbf{w}^{2}(t)\right] e^{i \int_{t_{a}}^{t_{b}} d t \Theta(t) \mathbf{w}(t) \mathbf{q} / \hbar} $$

$$ \begin{align*} f_{\mathbf{p}_{b} \mathbf{p}_{a}} & =\lim _{t_{b}-t_{a} \rightarrow \infty} \frac{p}{2 \pi i \hbar} \int d^{2} b e^{-i \mathbf{q} \mathbf{b} / \hbar} \int \mathcal{D}^{3} w \\ & \times \int \mathcal{D}^{3} v \exp \left[\frac{i}{\hbar} \int_{-\infty}^{\infty} d t \frac{M}{2}\left(\mathbf{v}^{2}-\mathbf{w}^{2}\right)\right]\left[\exp \left(i \chi_{\mathbf{b}_{\mathbf{w}}, \mathbf{p}}\right)-1\right] \end{align*} $$

$$ \chi_{\mathbf{b}_{\mathbf{w}}, \mathbf{p}}[\mathbf{v}, \mathbf{w}]=-\frac{1}{\hbar} \int_{-\infty}^{\infty} d t V\left(\mathbf{b}+\frac{\mathbf{p}}{M} t-\int_{t_{a}}^{t_{b}} d t^{\prime}\left[\Theta\left(t^{\prime}-t\right) \mathbf{v}\left(t^{\prime}\right)-\Theta\left(t^{\prime}\right) \mathbf{w}\left(t^{\prime}\right)\right]\right) $$

$$ \begin{align*} f_{\mathbf{p}_{b} \mathbf{p}_{a}} & =\lim _{t_{b}-t_{a} \rightarrow \infty} \frac{p}{2 \pi i \hbar} \int d^{2} b e^{-i \mathbf{q} / \hbar} \int d^{3} y_{a} \int d^{3} z_{a} \\ & \times \int \mathcal{D}^{3} y \int \mathcal{D}^{3} z \exp \left[\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t \frac{M}{2}\left(\dot{\mathbf{y}}^{2}-\dot{\mathbf{z}}^{2}\right)\right]\left[e^{i \chi_{\mathbf{b}_{\mathbf{z}}, \mathbf{p}}[\mathbf{y}]}-1\right] \end{align*} $$

$$ \chi_{\mathbf{b}_{\mathbf{z}}, \mathbf{p}}[\mathbf{y}] \equiv-\frac{1}{\hbar} \int_{t_{a}}^{t_{b}} d t V\left(\mathbf{b}+\frac{\mathbf{p}}{M} t+\mathbf{y}(t)-\mathbf{z}(0)\right) $$

$$ \int d^{3} y_{a} \int d^{3} z_{a} \int \mathcal{D}^{3} y \int \mathcal{D}^{3} z \exp \left[\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t \frac{M}{2}\left(\dot{\mathbf{y}}^{2}-\dot{\mathbf{z}}^{2}\right)\right] $$

$$ f_{\mathbf{p}_{b} \mathbf{p}_{a}}^{\mathrm{ei}} \equiv \frac{p}{2 \pi i \hbar} \int d^{2} b e^{-i \mathbf{q b} / \hbar}\left[\exp \left(i \chi_{\mathbf{b}, \mathbf{p}}^{\mathrm{ei}}\right)-1\right] $$

$$ \chi_{\mathbf{b}, \mathbf{p}}^{\mathrm{ei}} \equiv-\frac{1}{\hbar} \int_{-\infty}^{\infty} d t V\left(\mathbf{b}+\frac{\mathbf{p}}{M} t\right) $$

$$ \chi_{\mathbf{b}, \mathbf{p}}^{\mathrm{ei}} \equiv-\frac{M}{p} \frac{1}{\hbar} \int_{-\infty}^{\infty} d z V(\mathbf{b}+\hat{\mathbf{p}} z) $$

$$ \chi_{\mathbf{b}, \mathbf{p}}^{\mathrm{ei}} \equiv-\frac{M}{p} \frac{1}{\hbar} \int_{-\infty}^{\infty} d z V\left(\sqrt{b^{2}+z^{2}}\right) $$

$$ \frac{1}{2 \pi} \int_{-\pi}^{\pi} d \theta \exp \left(\frac{i}{\hbar} q b \cos \theta\right)=J_{0}(q b) $$

$$ f_{\mathbf{p}_{b} \mathbf{p}_{a}}^{\mathrm{ei}}=\frac{p}{i \hbar} \int d b b J_{0}(q b)\left[\exp \left(i \chi_{\mathbf{b}, \mathbf{p}}^{\mathrm{ei}}\right)-1\right] $$

$$ \chi_{\mathbf{b}, \mathbf{p}}^{\mathrm{ei}}=2 i \delta_{p b / \hbar}(p) $$

$$ \left(\mathbf{x} t \mid \mathbf{x}^{\prime} 0\right)=\langle\mathbf{x}| e^{-i \hat{H} t / \hbar}\left|\mathbf{x}^{\prime}\right\rangle $$

$$ \hat{H}=H(\hat{\mathbf{p}})=\frac{\hat{\mathbf{p}}^{2}}{2 M} $$

$$ \begin{align*} i \hbar \partial_{t}\left\langle\mathbf{x} t \mid \mathbf{x}^{\prime} 0\right\rangle & \equiv\langle\mathbf{x}| \hat{H} e^{-i \hat{H} t / \hbar}\left|\mathbf{x}^{\prime}\right\rangle=\langle\mathbf{x}| e^{-i \hat{H} t / \hbar}\left[e^{i \hat{H} t / \hbar} \hat{H} e^{-i \hat{H} t / \hbar}\right]\left|\mathbf{x}^{\prime}\right\rangle \\ & =\langle\mathbf{x} t| H(\hat{\mathbf{p}}(t))\left|\mathbf{x}^{\prime} 0\right\rangle \end{align*} $$

$$ \hat{H}=H(\hat{\mathbf{x}}(t), \hat{\mathbf{x}}(0) ; t) $$

$$ \langle\mathbf{x} t| \hat{\mathbf{x}}(t)=\mathbf{x}\langle\mathbf{x} t|, \quad \hat{\mathbf{x}}(0)\left|\mathbf{x}^{\prime} 0\right\rangle=\mathbf{x}^{\prime}\left|\mathbf{x}^{\prime} 0\right\rangle $$

$$ \langle\mathbf{x} t| H(\hat{\mathbf{x}}(t), \hat{\mathbf{x}}(0) ; t)|\hat{\mathbf{x}} 0\rangle=H\left(\mathbf{x}, \mathbf{x}^{\prime} ; t\right)\left\langle\mathbf{x} t \mid \mathbf{x}^{\prime} 0\right\rangle $$

$$ i \hbar \partial_{t}\left\langle\mathbf{x} t \mid \mathbf{x}^{\prime} 0\right\rangle \equiv H\left(\mathbf{x}, \mathbf{x}^{\prime} ; t\right)\left\langle\mathbf{x} t \mid \mathbf{x}^{\prime} 0\right\rangle $$

$$ \left\langle\mathbf{x} t \mid \mathbf{x}^{\prime} 0\right\rangle=C\left(\mathbf{x}, \mathbf{x}^{\prime}\right) E\left(\mathbf{x}, \mathbf{x}^{\prime} ; t\right) \equiv C\left(\mathbf{x}, \mathbf{x}^{\prime}\right) e^{-i \int_{t} d t^{\prime} H\left(\mathbf{x}, \mathbf{x}^{\prime} ; t^{\prime}\right) / \hbar} $$

$$ \begin{align*} \frac{d \hat{\mathbf{x}}(t)}{d t} & =\frac{i}{\hbar}[\hat{H}, \hat{\mathbf{x}}(t)]=\frac{\hat{\mathbf{p}}(t)}{M}, \\ \frac{d \hat{\mathbf{p}}(t)}{d t} & =\frac{i}{\hbar}[\hat{H}, \hat{\mathbf{p}}(t)]=0 . \end{align*} $$

$$ \hat{\mathbf{p}}(t)=\hat{\mathbf{p}}(0) $$

$$ \hat{\mathbf{x}}(t)-\hat{\mathbf{x}}(0)=t \frac{\hat{\mathbf{p}}(t)}{M} $$

$$ \hat{H}=\frac{M}{2 t^{2}}[\hat{\mathbf{x}}(t)-\hat{\mathbf{x}}(0)]^{2} $$

$$ \hat{H}=\frac{M}{2 t^{2}}\left\{\hat{\mathbf{x}}^{2}(t)-2 \hat{\mathbf{x}}(t) \hat{\mathbf{x}}(0)+\hat{\mathbf{x}}^{2}(0)+[\hat{\mathbf{x}}(t), \hat{\mathbf{x}}(0)]\right\} $$

$$ [\hat{\mathbf{x}}(t), \hat{\mathbf{x}}(0)]=-\frac{i \hbar}{M} D t, $$

$$ \hat{H}=H(\hat{\mathbf{x}}(t), \hat{\mathbf{x}}(0) ; t)=\frac{M}{2 t^{2}}\left[\hat{\mathbf{x}}^{2}(t)-2 \hat{\mathbf{x}}(t) \hat{\mathbf{x}}(0)+\hat{\mathbf{x}}^{2}(0)\right]-i \hbar \frac{D}{2 t} $$

$$ H\left(\mathbf{x}, \mathbf{x}^{\prime} ; t\right)=\frac{M}{2 t^{2}}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)^{2}-i \hbar \frac{D}{2 t} . $$

$$ E\left(\mathbf{x}, \mathbf{x}^{\prime} ; t\right)=e^{-i \int d t H\left(\mathbf{x}, \mathbf{x}^{\prime} ; t\right) / \hbar}=\exp \left[\frac{i}{\hbar} \frac{M}{2 t}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)^{2}-\frac{D}{2} \log t\right] $$

$$ \left\langle\mathbf{x} t \mid \mathbf{x}^{\prime} 0\right\rangle=C\left(\mathbf{x}, \mathbf{x}^{\prime}\right) \frac{1}{t^{D / 2}} \exp \left[\frac{i}{\hbar} \frac{M}{2 t}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)^{2}\right] $$

$$ \begin{align*} -i \hbar \nabla\left\langle\mathbf{x} t \mid \mathbf{x}^{\prime} 0\right\rangle & =\langle\mathbf{x}| \hat{\mathbf{p}} e^{-i \hat{H} t / \hbar}\left|\mathbf{x}^{\prime}\right\rangle=\langle\mathbf{x}| e^{-i \hat{H} t}\left[e^{i \hat{H} t / \hbar} \hat{\mathbf{p}} e^{-i \hat{H} t / \hbar}\right]\left|\mathbf{x}^{\prime}\right\rangle=\langle\mathbf{x} t| \hat{\mathbf{p}}(t)\left|\mathbf{x}^{\prime} 0\right\rangle \\ i \hbar \nabla^{\prime}\left\langle\mathbf{x} t \mid \mathbf{x}^{\prime} 0\right\rangle & =\langle\mathbf{x}| e^{-i \hat{H} t / \hbar} \hat{\mathbf{p}}\left|\mathbf{x}^{\prime}\right\rangle\langle\mathbf{x} t| \hat{\mathbf{p}}(0)\left|\mathbf{x}^{\prime} 0\right\rangle \end{align*} $$

$$ \begin{align*} -i \hbar \nabla\left\langle\mathbf{x} t \mid \mathbf{x}^{\prime} 0\right\rangle & =\frac{M}{t}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)\left\langle\mathbf{x} t \mid \mathbf{x}^{\prime} 0\right\rangle, \\ i \hbar \nabla^{\prime}\left\langle\mathbf{x} t \mid \mathbf{x}^{\prime} 0\right\rangle & =\frac{M}{t}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)\left\langle\mathbf{x} t \mid \mathbf{x}^{\prime} 0\right\rangle . \end{align*} $$

$$ -i \nabla C\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=0, \quad i \nabla^{\prime} C\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=0, $$

$$ \lim _{t \rightarrow 0}\left\langle\mathbf{x} t \mid \mathbf{x}^{\prime} 0\right\rangle=\delta^{(D)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right) $$

$$ C={\frac{M}{2 \pi i \hbar}^{D}}^{D}, $$

$$ \left\langle\mathbf{x} t \mid \mathbf{x}^{\prime} 0\right\rangle \equiv{\frac{M}{2 \pi i \hbar t}^{D}}^{D} \exp \left[\frac{i}{\hbar} \frac{M}{2 t}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)^{2}\right] $$

$$ \hat{H}=H(\hat{\mathbf{p}}, \hat{\mathbf{x}})=\frac{\hat{\mathbf{p}}^{2}}{2 M}+\frac{M \omega^{2}}{2} \mathbf{x}^{2} $$

$$ \begin{align*} \frac{d \hat{\mathbf{x}}(t)}{d t} & =\frac{i}{\hbar}[\hat{H}, \hat{\mathbf{x}}(t)]=\frac{\hat{\mathbf{p}}(t)}{M} \\ \frac{d \hat{\mathbf{p}}(t)}{d t} & =\frac{i}{\hbar}[\hat{H}, \hat{\mathbf{p}}(t)]=-M \omega^{2} \hat{\mathbf{x}}(t) \end{align*} $$

$$ \hat{\mathbf{p}}(t)=M \frac{\omega}{\sin \omega t}[\hat{\mathbf{x}}(t) \cos \omega t-\hat{\mathbf{x}}(0)] $$

$$ \hat{H}=\frac{M \omega^{2}}{2 \sin ^{2} \omega t}\left\{[\hat{\mathbf{x}}(t) \cos \omega t-\hat{\mathbf{x}}(0)]^{2}+\sin ^{2} \omega t \hat{\mathbf{x}}^{2}(t)\right\} $$

$$ \hat{H}=\frac{M \omega^{2}}{2 \sin ^{2} \omega t}\left\{\hat{\mathbf{x}}^{2}(t)+\hat{\mathbf{x}}^{2}(0)-2 \cos \omega t \hat{\mathbf{x}}(t) \hat{\mathbf{x}}(0)+\cos \omega t[\hat{\mathbf{x}}(t), \hat{\mathbf{x}}(0)]\right\} $$

$$ [\hat{\mathbf{x}}(t), \hat{\mathbf{x}}(0)]=-\frac{i \hbar}{M} D \frac{\sin \omega t}{\omega}, $$

$$ H\left(\mathbf{x}, \mathbf{x}^{\prime} ; t\right)=\frac{M \omega^{2}}{2 \sin ^{2} \omega t}\left(\mathbf{x}^{2}+\mathbf{x}^{\prime 2}-2 \cos \omega t \mathbf{x} \mathbf{x}^{\prime}\right)-i \hbar \frac{D}{2} \omega \cot \omega t $$

$$ \int d t H\left(\mathbf{x}, \mathbf{x}^{\prime} ; t\right)=-\frac{M \omega}{2 \sin \omega t}\left[\left(\mathbf{x}^{2}+\mathbf{x}^{\prime 2}\right) \cos \omega t-2 \mathbf{x} \mathbf{x}^{\prime}\right]-i \hbar \frac{D}{2} \log \frac{\sin \omega t}{\omega} . $$

$$ \hat{\mathbf{P}} \equiv \hat{\mathbf{p}}-\frac{e}{c} \mathbf{A}(\hat{\mathbf{x}}) $$

$$ \hat{H}=H(\hat{\mathbf{p}}, \hat{\mathbf{x}})=\frac{\hat{\mathbf{P}}^{2}}{2 M} $$

$$ \left[\hat{P}_{i}, \hat{P}_{j}\right]=-\frac{e}{c}\left[\hat{p}_{i}, \hat{A}_{j}\right]-\frac{e}{c}\left[\hat{A}_{i}, \hat{p}_{j}\right]=i \frac{e \hbar}{c}\left(\nabla_{i} A_{j}-\nabla_{j} A_{i}\right)=i \frac{e \hbar}{c} B_{i j} $$

$$ \begin{align*} \frac{d \hat{\mathbf{x}}(t)}{d t} & =\frac{i}{\hbar}[\hat{H}, \hat{\mathbf{x}}(t)]=\frac{\hat{\mathbf{P}}(t)}{M} \\ \frac{d \hat{\mathbf{P}}(t)}{d t} & =\frac{i}{\hbar}[\hat{H}, \hat{\mathbf{P}}(t)]=\frac{e}{M c} B(\hat{\mathbf{x}}(t)) \hat{\mathbf{P}}(t)+i \frac{e \hbar}{M c} \nabla_{j} B_{j i}(\hat{\mathbf{x}}(t)) \end{align*} $$

$$ \hat{\mathbf{P}}(t)=e^{\Omega_{L} t} \hat{\mathbf{P}}(0) $$

$$ \Omega_{L i j} \equiv \frac{e}{M c} B_{i j} $$

$$ \boldsymbol{\omega}_{L} \equiv \frac{e}{M c} \mathbf{B} $$

$$ \left(L_{k}\right)_{i j} \equiv-i \epsilon_{k i j} $$

$$ \Omega_{L}=i \mathbf{L} \cdot \boldsymbol{\omega}_{L} $$

$$ \hat{\mathbf{x}}(t)=\hat{\mathbf{x}}(0)+\frac{e^{\Omega_{L} t}-1}{\Omega_{L}} \frac{\hat{\mathbf{P}}(0)}{M}, $$

$$ \frac{e^{\Omega_{L} t}-1}{\Omega_{L}}=t+\Omega_{L} \frac{t^{2}}{2}+\Omega_{L}^{2} \frac{t^{3}}{3!}+\ldots $$

$$ \frac{\hat{\mathbf{P}}(0)}{M}=\frac{\Omega_{L} / 2}{\sinh \Omega_{L} t / 2} e^{-\Omega_{L} t / 2}[\hat{\mathbf{x}}(t)-\hat{\mathbf{x}}(0)] $$

$$ \hat{\mathbf{P}}(t)=M N\left(\Omega_{L} t\right)[\hat{\mathbf{x}}(t)-\hat{\mathbf{x}}(0)] $$

$$ N\left(\Omega_{L} t\right) \equiv \frac{\Omega_{L} / 2}{\sinh \Omega_{L} t / 2} e^{\Omega_{L} t / 2} $$

$$ \frac{\hat{\mathbf{P}}^{2}(t)}{2 M}=\frac{M}{2}[\hat{\mathbf{x}}(t)-\hat{\mathbf{x}}(0)]^{T} K\left(\Omega_{L} t\right)[\hat{\mathbf{x}}(t)-\hat{\mathbf{x}}(0)], $$

$$ K\left(\Omega_{L} t\right)=N^{T}\left(\Omega_{L} t\right) N\left(\Omega_{L} t\right) . $$

$$ K\left(\Omega_{L} t\right)=N\left(-\Omega_{L} t\right) N\left(\Omega_{L} t\right)=\frac{\Omega_{L}^{2} / 4}{\sinh ^{2} \Omega_{L} t / 2} $$

$$ \left[\hat{\mathbf{x}}_{i}(t), \hat{\mathbf{x}}_{j}(0)\right]=-\frac{i}{M}\left(\frac{e^{\Omega_{L} t}-1}{\Omega_{L}}\right)_{i j}, $$

$$ \begin{align*} & {\left[\hat{\mathbf{x}}_{i}(t), \hat{\mathbf{x}}_{j}(0)\right]+\left[\hat{\mathbf{x}}_{j}(t), \hat{\mathbf{x}}_{i}(0)\right]=-\frac{i}{M}\left(\frac{e^{\Omega_{L} t}-1}{\Omega_{L}}+\frac{e^{\Omega_{L}^{T} t}-1}{\Omega_{L}^{T}}\right)_{i j}} \\ & =-\frac{i}{M}\left(\frac{e^{\Omega_{L} t}-e^{-\Omega_{L} t}}{\Omega_{L}}\right)_{i j}=-2 \frac{i}{M}\left[\frac{\sinh \Omega_{L} t}{\Omega_{L}}\right]_{i j} . \end{align*} $$

$$ \begin{align*} H(\hat{\mathbf{x}}(t), \hat{\mathbf{x}}(0)) & =\frac{M}{2}\left[\hat{\mathbf{x}}^{T}(t) K\left(\Omega_{L} t\right) \hat{\mathbf{x}}(t)-2 \hat{\mathbf{x}}^{T} K\left(\Omega_{L} t\right) \hat{\mathbf{x}}(0)+\hat{\mathbf{x}}^{T} K\left(\Omega_{L} t\right) \hat{\mathbf{x}}(0)\right] \\ & -\frac{i \hbar}{2} \operatorname{tr}\left[\frac{\Omega_{L}}{2} \operatorname{coth} \frac{\Omega_{L} t}{2}\right] \end{align*} $$

$$ \int d t K\left(\Omega_{L} t\right)=\int d t \frac{\Omega_{L}^{2} / 2}{\sinh ^{2} \Omega_{L} t / 2}=-\frac{\Omega_{L}}{2} \operatorname{coth} \frac{\Omega_{L} t}{2} $$

$$ \int d t \frac{1}{2} \operatorname{tr}\left[\frac{\Omega_{L}}{2} \operatorname{coth} \frac{\Omega_{L} t}{2}\right]=\operatorname{tr} \log \frac{\sinh \Omega_{L} t / 2}{\Omega_{L} / 2}=\operatorname{tr} \log \frac{\sinh \Omega_{L} t / 2}{\Omega_{L} t / 2}+3 \log t $$

$$ E\left(\mathbf{x}, \mathbf{x}^{\prime} ; t\right)=\frac{1}{t^{3 / 2}} \exp \left\{\frac{i}{\hbar} \frac{M}{2}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)^{T}\left(\frac{\Omega_{L}}{2} \operatorname{coth} \frac{\Omega_{L} t}{2}\right)\left(\mathbf{x}-\mathbf{x}^{\prime}\right)-\frac{1}{2} \operatorname{tr} \log \frac{\sinh \Omega_{L} t / 2}{\Omega_{L} t / 2}\right\} . $$

$$ \left[\operatorname{det} \frac{\sinh \Omega_{L} t / 2}{\Omega_{L} t / 2}\right]^{-1 / 2} . $$

$$ \begin{align*} {\left[-i \hbar \nabla-\frac{e}{c} \mathbf{A}(\mathbf{x})\right]\left\langle\mathbf{x} t \mid \mathbf{x}^{\prime} 0\right\rangle } & =\langle\mathbf{x}| \hat{\mathbf{P}} e^{-i \hat{H} t / \hbar}\left|\mathbf{x}^{\prime}\right\rangle=\langle\mathbf{x}| e^{-i \hat{H} t / \hbar}\left[e^{i \hat{H} t / \hbar} \hat{\mathbf{P}} e^{-i \hat{H} t / \hbar}\right]\left|\mathbf{x}^{\prime}\right\rangle \\ & =\langle\mathbf{x} t| \hat{\mathbf{P}}(t)\left|\mathbf{x}^{\prime} 0\right\rangle=L\left(\Omega_{L} t\right)\left(\mathbf{x}-\mathbf{x}^{\prime}\right)\left\langle\mathbf{x} t \mid \mathbf{x}^{\prime} 0\right\rangle, \\ {\left[i \hbar \nabla^{\prime}-\frac{e}{c} \mathbf{A}(\mathbf{x})\right]\left\langle\mathbf{x} t \mid \mathbf{x}^{\prime} 0\right\rangle } & =\langle\mathbf{x}| e^{-i \hat{H} t / \hbar} \hat{\mathbf{P}}\left|\mathbf{x}^{\prime}\right\rangle \\ & =\langle\mathbf{x} t| \hat{\mathbf{P}}(0)\left|\mathbf{x}^{\prime} 0\right\rangle=L\left(\Omega_{L} t\right)\left(\mathbf{x}-\mathbf{x}^{\prime}\right)\left\langle\mathbf{x} t \mid \mathbf{x}^{\prime} 0\right\rangle . \end{align*} $$

$$ M\left(\frac{\Omega_{L}}{2} \operatorname{coth} \frac{\Omega_{L} t}{2}\right)\left(\mathbf{x}-\mathbf{x}^{\prime}\right)-M L\left(\Omega_{L} t\right)\left(\mathbf{x}-\mathbf{x}^{\prime}\right)=-\frac{M}{2} \Omega_{L}\left(\mathbf{x}-\mathbf{x}^{\prime}\right), $$

$$ \left[-i \hbar \nabla-\frac{e}{c} A(\mathbf{x})-\frac{M}{2} \Omega_{L}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)\right] C\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=0 . $$

$$ \left[i \hbar \nabla^{\prime}-\frac{e}{c} A(\mathbf{x})-\frac{M}{2} \Omega_{L}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)\right] C\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=0 . $$

$$ C\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=C \exp \left\{\frac{i}{\hbar} \int_{\mathbf{x}^{\prime}}^{\mathbf{x}} d \boldsymbol{\xi}\left[\frac{e}{c} \mathbf{A}(\boldsymbol{\xi})+\frac{M}{2} \Omega_{L}\left(\boldsymbol{\xi}-\mathbf{x}^{\prime}\right)\right]\right\} $$

$$ \frac{e}{c} \mathbf{A}^{\prime}(\boldsymbol{\xi}) \equiv \frac{e}{c} \mathbf{A}(\boldsymbol{\xi})+\frac{\Omega_{L}}{2}\left(\boldsymbol{\xi}-\mathbf{x}^{\prime}\right)=\frac{e}{c}\left[\mathbf{A}(\boldsymbol{\xi})-\frac{1}{2} \mathbf{B} \times\left(\boldsymbol{\xi}-\mathbf{x}^{\prime}\right)\right] $$

$$ C\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=C \exp \left\{i \frac{e}{c} \int_{\mathbf{x}^{\prime}}^{\mathbf{x}} d \boldsymbol{\xi} \mathbf{A}(\boldsymbol{\xi})\right\} $$

$$ \begin{align*} \left\langle\mathbf{x} t \mid \mathbf{x}^{\prime} 0\right\rangle & =\frac{1}{{\sqrt{2 \pi i \hbar^{2} t / M}}^{3}}\left[\operatorname{det} \frac{\sinh \Omega_{L} t / 2}{\Omega_{L} t / 2}\right]^{-1 / 2} \exp \left\{i \frac{e}{c} \int_{\mathbf{x}^{\prime}}^{\mathbf{x}} d \boldsymbol{\xi} \mathbf{A}(\boldsymbol{\xi})\right\} \\ & \times \exp \left\{\frac{i}{\hbar} \frac{M}{2}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)^{T}\left(\frac{\Omega_{L}}{2} \operatorname{coth} \frac{\Omega_{L} t}{2}\right)\left(\mathbf{x}-\mathbf{x}^{\prime}\right)\right\} \end{align*} $$

$$ \Omega_{L}=\left(\begin{array}{ccc} 0 & \omega_{L} & 0 \\ -\omega_{L} & 0 & 0 \\ 0 & 0 & 0 \end{array}\right), $$

$$ \cos \frac{\Omega_{L} t}{2}=\left(\begin{array}{ccc} \cos \omega_{L} t / 2 & 0 & 0 \\ 0 & \cos \omega_{L} t / 2 & 0 \\ 0 & 0 & 1 \end{array}\right) $$

$$ \frac{\sinh \Omega_{L} t / 2}{\Omega_{L} t / 2}=\left(\begin{array}{ccc} 0 & \sin \omega_{L} t / 2 & 0 \\ -\sin \omega_{L} t / 2 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right) $$

$$ \operatorname{det} \frac{\sinh \Omega_{L} t / 2}{\Omega_{L} t / 2}=\left(\frac{\sinh \omega_{L} t / 2}{\omega_{L} t / 2}\right)^{2} . $$

$$ \boldsymbol{\xi}=\mathbf{x}^{\prime}+s\left(\mathbf{x}-\mathbf{x}^{\prime}\right), \quad s \in[0,1] . $$

$$ \begin{align*} \int_{\mathbf{x}^{\prime}}^{\mathbf{x}} d \boldsymbol{\xi} \mathbf{A}(\boldsymbol{\xi}) & =B\left(y-y^{\prime}\right) \int_{0}^{1} d s\left[x^{\prime}+s\left(x-x^{\prime}\right)\right]=B\left(y-y^{\prime}\right)\left(x+x^{\prime}\right) \\ & =B\left(x y-x^{\prime} y^{\prime}\right)+B\left(x^{\prime} y-x y^{\prime}\right) \end{align*} $$