Kleinert · 제3장 외부 소스·상관함수·섭동

$$ (p-n) A_{k}^{p}=\frac{1}{2}(p+2)(p+1) A_{k}^{p+2}-\sum_{k^{\prime}=1}^{k}(-1)^{k^{\prime}} v_{k^{\prime}+2} A_{k-k^{\prime}}^{p-j-2}+\sum_{k^{\prime}=1}^{k}(-1)^{k^{\prime}} E_{k^{\prime}}^{(n)} A_{k-k^{\prime}}^{p} . $$

$$ G_{\omega^{2}}\left(t, t^{\prime}\right)=C \sin \omega\left(t_{b}-t_{>}\right) \sin \omega\left(t_{<}-t_{a}\right) $$

$$ V(x)=\frac{M}{2} \omega^{2} x^{2}+g v_{3} x^{3}+g^{2} v_{4} x^{4} $$

$$ \partial_{t} G_{\omega^{2}}\left(t, t^{\prime}\right)=-C \omega \cos \omega\left(t_{b}-t\right) \sin \omega\left(t^{\prime}-t_{a}\right) $$

$$ \psi^{(n)}(x)=\left(\frac{M \omega}{\pi \hbar}\right)^{1 / 4} \exp \left[-\frac{M \omega}{2 \hbar} x^{2}+\phi^{(n)}(x)\right] $$

$$ \partial_{t} G_{\omega^{2}}\left(t, t^{\prime}\right)=C \omega \sin \omega\left(t_{b}-t^{\prime}\right) \cos \omega\left(t-t_{a}\right) . $$

$$ \phi^{(n)}(x)=\sum_{k=1}^{\infty} g^{k} \phi_{k}^{(n)}(x) $$

$$ \left.\partial_{t} G_{\omega^{2}}\left(t, t^{\prime}\right)\right|_{t=t^{\prime}+\epsilon}-\left.\partial_{t} G_{\omega^{2}}\left(t, t^{\prime}\right)\right|_{t=t^{\prime}-\epsilon}=-C \omega \sin \omega\left(t_{b}-t_{a}\right) . $$

$$ \left[-\frac{\hbar^{2}}{2 M} \frac{d^{2}}{d x^{2}}+\left(\frac{M}{2} \omega^{2} x^{2}+g v_{3} x^{3}+g^{2} v_{4} x^{4}\right)-E^{(n)}\right] \psi^{(n)}(x)=0 $$

$$ -\partial_{t}^{2} G_{\omega^{2}}\left(t, t^{\prime}\right)=C \omega \sin \omega\left(t_{b}-t_{a}\right) \delta\left(t-t^{\prime}\right) $$

$$ -\frac{\hbar^{2}}{2 M} \phi^{\prime \prime}(x)+\hbar \omega x \phi^{\prime}(x)-\frac{\hbar^{2}}{2 M}\left[\phi^{\prime}(x)\right]^{2}+g v_{3} x^{3}+g^{2} v_{4} x^{4}=n \hbar \omega+\epsilon $$

$$ G_{\omega^{2}}\left(t, t^{\prime}\right)=\frac{\sin \omega\left(t_{b}-t_{>}\right) \sin \omega\left(t_{<}-t_{a}\right)}{\omega \sin \omega\left(t_{b}-t_{a}\right)} . $$

$$ E=\hbar \omega\left(n+\frac{1}{2}\right)+\epsilon $$

$$ W[\xi(t), \eta(t)] \equiv \xi(t) \dot{\eta}(t)-\dot{\xi}(t) \eta(t) $$

$$ \epsilon=\sum_{k=1}^{\infty} g^{k} \epsilon_{k} $$

$$ G_{\omega^{2}}\left(t, t^{\prime}\right)=\frac{-\cos \omega\left(t_{b}-t_{a}-\left|t-t^{\prime}\right|\right)+\cos \omega\left(t_{b}+t_{a}-t-t^{\prime}\right)}{2 \omega \sin \omega\left(t_{b}-t_{a}\right)} . $$

$$ -\frac{\hbar^{2}}{2 M} \phi_{k}^{\prime \prime}(x)+\hbar \omega x \phi_{k}^{\prime}(x)-\frac{\hbar^{2}}{2 M} \sum_{l=1}^{k-1} \phi_{k-l}^{\prime}(x) \phi_{l}^{\prime}(x)+\delta_{k, 1} v_{3} x^{3}+\delta_{k, 2} v_{4} x^{4}=\epsilon_{k} . $$

$$ \begin{align*} G_{0}\left(t, t^{\prime}\right) & =\frac{1}{\left(t_{b}-t_{a}\right)}\left(t_{b}-t_{>}\right)\left(t_{<}-t_{a}\right) \\ & =\frac{1}{t_{b}-t_{a}}\left[-t t^{\prime}-\frac{1}{2}\left(t_{b}-t_{a}\right)\left|t-t^{\prime}\right|+\frac{1}{2}\left(t_{a}+t_{b}\right)\left(t+t^{\prime}\right)-t_{a} t_{b}\right] \end{align*} $$

$$ \phi_{k}(x)=\sum_{m=1}^{\infty} c_{m}^{(k)} x^{m}, \quad \text { with } c_{m}^{(k)} \equiv 0 \quad \text { for } m>k+2 $$

$$ G_{\Omega^{2}}\left(t, t^{\prime}\right)=\Theta\left(t-t^{\prime}\right) \Delta\left(t, t^{\prime}\right) $$

$$ c_{1}^{(1)}=-\frac{v_{3}}{M \omega^{2}}, \quad c_{2}^{(1)}=0, \quad c_{3}^{(1)}=-\frac{v_{3}}{3 \hbar \omega}, \quad \epsilon_{1}=0 $$

$$ \begin{align*} {\left[-\partial_{t}^{2}-\Omega^{2}(t)\right] G_{\Omega^{2}}\left(t, t^{\prime}\right) } & =\Theta\left(t-t^{\prime}\right)\left[-\partial_{t}^{2}-\Omega^{2}(t)\right] \Delta\left(t, t^{\prime}\right) \\ & -\dot{\delta}\left(t-t^{\prime}\right)-2 \partial_{t} \Delta\left(t, t^{\prime}\right) \delta\left(t-t^{\prime}\right) \end{align*} $$

$$ \begin{align*} c_{1}^{(2)} & =0, \quad c_{2}^{(2)}=\frac{7 v_{3}^{2}}{8 M^{2} \omega^{4}}-\frac{3 v_{4}}{4 M \omega^{2}}, \quad c_{3}^{(2)}=0, \quad c_{4}^{(2)}=\frac{v_{3}^{2}}{8 M \hbar \omega^{3}}-\frac{v_{4}}{4 \hbar \omega} \\ \epsilon_{2} & =-\frac{11 v_{3}^{2} \hbar^{2}}{8 M^{3} \omega^{4}}+\frac{3 v_{4} \hbar^{2}}{4 M^{2} \omega^{2}} \end{align*} $$

$$ \Delta\left(t, t^{\prime}\right)=\Delta(t, t)+\left[\partial_{t} \Delta\left(t, t^{\prime}\right)\right]_{t=t^{\prime}}\left(t-t^{\prime}\right)+\frac{1}{2}\left[\partial_{t}^{2} \Delta\left(t, t^{\prime}\right)\right]_{t=t^{\prime}}\left(t-t^{\prime}\right)^{2}+\ldots $$

$$ \left(t-t^{\prime}\right) \dot{\delta}\left(t-t^{\prime}\right)=-\delta\left(t-t^{\prime}\right), \quad\left(t-t^{\prime}\right)^{n} \dot{\delta}\left(t-t^{\prime}\right)=0 \quad \text { for } \quad n>1 $$

$$ \begin{align*} c_{m}^{(k)} & =\frac{(m+2)(m+1) \hbar}{2 m M \omega} c_{m+2}^{(k)}+\frac{\hbar}{2 m M \omega} \sum_{l=1}^{k-1} \sum_{n=1}^{m+1} n(m+2-n) c_{n}^{(l)} c_{m+2-n}^{(k-l)} \\ \epsilon_{k} & =-\frac{\hbar^{2}}{M} c_{2}^{(k)}-\frac{\hbar^{2}}{2 M} \sum_{l=1}^{k-1} c_{1}^{(l)} c_{1}^{(k-l)} \end{align*} $$

$$ -\dot{\delta}\left(t-t^{\prime}\right) \Delta\left(t, t^{\prime}\right)-\delta\left(t-t^{\prime}\right) \partial_{t} \Delta\left(t, t^{\prime}\right) $$

$$ \Delta(t, t)=0,\left.\quad \dot{\Delta}\left(t, t^{\prime}\right)\right|_{t^{\prime}=t}=-1 $$

$$ \begin{align*} c_{1}^{(3)}=-\frac{5 v_{3}^{3} \hbar}{M^{4} \omega^{7}}+\frac{6 v_{3} v_{4} \hbar}{M^{3} \omega^{5}}, c_{2}^{(3)} & =0, \quad c_{3}^{(3)}=-\frac{13 v_{3}^{3}}{12 M^{3} \omega^{6}}+\frac{3 v_{3} v_{4}}{2 M^{2} \omega^{4}}, \\ c_{4}^{(3)} & =0, \quad c_{5}^{(3)}=-\frac{v_{3}^{3}}{10 M^{2} \hbar \omega^{5}}+\frac{v_{3} v_{4}}{5 M \hbar \omega^{3}}, \quad \epsilon_{3}=0, \end{align*} $$

$$ \left[-\partial_{t}^{2}-\Omega^{2}(t)\right] \Delta\left(t, t^{\prime}\right)=0, \quad \text { for } t>t^{\prime} $$

$$ \begin{align*} c_{1}^{(4)} & =0, c_{2}^{(4)}=\frac{305 v_{3}^{4} \hbar}{32 M^{5} \omega^{9}}-\frac{123 v_{3}^{2} v_{4} \hbar}{8 M^{4} \omega^{7}}+\frac{21 v_{4}^{2} \hbar}{8 M^{3} \omega^{5}}, c_{3}^{(4)}=0, c_{4}^{(4)}=\frac{99 v_{3}^{4}}{64 M^{4} \omega^{8}}-\frac{47 v_{3}^{2} v_{4}}{16 M \omega^{6}}+\frac{11 v_{4}^{2}}{16 M^{2} \omega^{4}} \\ c_{5}^{(4)} & =0, c_{6}^{(4)}=\frac{5 v_{3}^{4}}{48 M^{3} \hbar \omega^{7}}-\frac{v_{3}^{2} v_{4}}{4 M^{2} \hbar \omega^{5}}+\frac{v_{4}^{2}}{12 M \hbar \omega^{3}} \\ \epsilon_{4} & =-\frac{465 v_{3}^{4} \hbar^{3}}{32 M^{6} \omega^{9}}+\frac{171 v_{3}^{2} v_{4} \hbar^{3}}{8 M^{5} \omega^{7}}-\frac{21 v_{4}^{2} \hbar^{3}}{8 M^{4} \omega^{5}} \end{align*} $$

$$ \Delta\left(t, t^{\prime}\right)=\alpha\left(t^{\prime}\right) \xi(t)+\beta\left(t^{\prime}\right) \eta(t) $$

$$ \left[-\partial_{t}^{2}-\Omega^{2}(t)\right] \xi(t)=0, \quad\left[-\partial_{t}^{2}-\Omega^{2}(t)\right] \eta(t)=0 . $$

$$ \begin{align*} c_{m}^{(k)} & =\sum_{\lambda=0}^{\lfloor k / 2\rfloor} \frac{v_{3}^{k-2 \lambda} v_{4}^{\lambda}}{\omega^{5 k / 2-m / 2-2 \lambda}} c_{m, \lambda}^{(k)}, \quad \text { with } c_{m, \lambda}^{(k)} \equiv 0 \text { for } m>k+2, \text { or } \lambda>\left\lfloor\frac{k}{2}\right\rfloor \\ \epsilon_{k} & =\sum_{\lambda=0}^{\lfloor k / 2\rfloor} \frac{v_{3}^{k-2 \lambda} v_{4}^{\lambda}}{\omega^{5 k / 2-1-2 \lambda}} \epsilon_{k, \lambda} . \end{align*} $$

$$ \Delta\left(t, t^{\prime}\right)=\frac{1}{W}\left[\xi(t) \eta\left(t^{\prime}\right)-\xi\left(t^{\prime}\right) \eta(t)\right] $$

$$ \begin{gather*} \Delta\left(t, t^{\prime}\right)=\frac{\Delta\left(t_{b}, t\right) \Delta\left(t^{\prime}, t_{a}\right)-\Delta\left(t, t_{a}\right) \Delta\left(t_{b}, t^{\prime}\right)}{\Delta\left(t_{b}, t_{a}\right)} \\ \Delta\left(t_{b}, t\right) \partial_{t_{b}} \Delta\left(t_{b}, t_{a}\right)-\Delta\left(t, t_{a}\right)=\Delta\left(t_{b}, t_{a}\right) \partial_{t} \Delta\left(t_{b}, t\right) \\ \Delta\left(t, t_{a}\right) \partial_{t_{b}} \Delta\left(t_{b}, t_{a}\right)-\Delta\left(t_{b}, t\right)=\Delta\left(t_{b}, t_{a}\right) \partial_{t} \Delta\left(t, t_{a}\right) \end{gather*} $$

$$ c_{m, \lambda}^{(k)}=\frac{(m+2)(m+1)}{2 m} c_{m+2, \lambda}^{(k)}+\frac{1}{2 m} \sum_{l=1}^{k-1} \sum_{n=1}^{m+1} \sum_{\lambda^{\prime}=0}^{\lambda} n(m+2-n) c_{n, \lambda-\lambda^{\prime}}^{(l)} c_{m+2-n, \lambda^{\prime}}^{(k-l)} $$

$$ \begin{array}{lll} c_{1,0}^{(1)}=-1, & c_{2,0}^{(1)}=0, & c_{3,0}^{(1)}=-\frac{1}{3}, \\ c_{1,0}^{(2)}=0, & c_{1,1}^{(2)}=0, & c_{2,0}^{(2)}=\frac{7}{8}, \\ c_{3,0}^{(2)}=0, & c_{3,1}^{(2)}=0, & c_{4,0}^{(2)}=\frac{1}{8}, \end{array}, c_{4,1}^{(2)}=-\frac{3}{4} . $$

$$ G_{\Omega^{2}}\left(t, t^{\prime}\right)=\Theta\left(t-t^{\prime}\right) \Delta\left(t, t^{\prime}\right)+a\left(t^{\prime}\right) \xi(t)+b\left(t^{\prime}\right) \eta(t) $$

$$ \epsilon_{k, \lambda}=-c_{2, \lambda}^{(k)}-\frac{1}{2} \sum_{l=1}^{k-1} \sum_{\lambda^{\prime}=0}^{\lambda} c_{1, \lambda-\lambda^{\prime}}^{(l)} c_{1, \lambda^{\prime}}^{(k-l)} $$

$$ \begin{align*} & G_{\Omega^{2}}\left(t_{b}, t\right)=0, \quad t_{b} \neq t \\ & G_{\Omega^{2}}\left(t, t_{a}\right)=0, \quad t \neq t_{a} \end{align*} $$

$$ -\frac{\hbar^{2}}{2 M} \psi^{\prime \prime}(x)+\left(\frac{M}{2} \omega^{2} x^{2}+g v_{3} x^{3}+g^{2} v_{4} x^{4}-j x\right) \psi(x)=E \psi(x) . $$

$$ \begin{align*} a(t) \xi\left(t_{a}\right)+b(t) \eta\left(t_{a}\right) & =0 \\ a(t) \xi\left(t_{b}\right)+b(t) \eta\left(t_{b}\right) & =\Delta\left(t, t_{b}\right) \end{align*} $$

$$ x^{\prime}=x-\frac{j}{M \omega^{2}} \text { and } E^{\prime}=E+\frac{j^{2}}{2 M \omega^{2}} $$

$$ \psi(x) \propto e^{\phi(x)}, \quad \text { with } \quad \phi(x)=\frac{j}{\hbar \omega} x-\frac{M \omega}{2 \hbar} x^{2}+\sum_{k=1}^{\infty} g^{k} \phi_{k}(x), $$

$$ \Lambda=\left(\begin{array}{ll} \xi\left(t_{a}\right) & \eta\left(t_{a}\right) \\ \xi\left(t_{b}\right) & \eta\left(t_{b}\right) \end{array}\right) $$

$$ E(j)=\frac{\hbar \omega}{2}-\frac{j^{2}}{2 M \omega^{2}}+\sum_{k=1}^{\infty} g^{k} \epsilon_{k} . $$

$$ \operatorname{det} \Lambda=W \Delta\left(t_{a}, t_{b}\right) \neq 0 $$

$$ -\frac{\hbar^{2}}{2 M} \phi_{k}^{\prime \prime}(x)-\frac{\hbar^{2}}{2 M} \sum_{l=1}^{k-1} \phi_{k-l}^{\prime}(x) \phi_{l}^{\prime}(x)+\left(\hbar \omega x-\frac{j \hbar}{M \omega}\right) \phi_{k}^{\prime}(x)+\delta_{k, 1} v_{3} x^{3}+\delta_{k, 2} v_{4} x^{4}=\epsilon_{k} $$

$$ G_{\Omega^{2}}\left(t, t^{\prime}\right)=\frac{\Theta\left(t-t^{\prime}\right) \Delta\left(t_{b}, t\right) \Delta\left(t^{\prime}, t_{a}\right)+\Theta\left(t^{\prime}-t\right) \Delta\left(t, t_{a}\right) \Delta\left(t_{b}, t^{\prime}\right)}{\Delta\left(t_{a}, t_{b}\right)} . $$

$$ c_{1}^{(1)}=-\frac{v_{3}}{M \omega^{2}}-\frac{j^{2} v_{3}}{M^{2} \hbar \omega^{5}}, c_{2}^{(1)}=-\frac{j v_{3}}{2 M \hbar \omega^{3}}, c_{3}^{(1)}=-\frac{v_{3}}{3 \hbar \omega}, \epsilon_{1}=\frac{3 \hbar j v_{3}}{2 M \omega^{3}}+\frac{j^{3} v_{3}}{M^{3} \omega^{6}}, $$

$$ D_{a}(t) \equiv \Delta\left(t, t_{a}\right), \quad D_{b}(t) \equiv \Delta\left(t_{b}, t\right) $$

$$ \begin{align*} c_{1}^{(2)} & =\frac{17 j v_{3}^{2}}{4 M^{3} \omega^{6}}+\frac{4 j^{3} v_{3}^{2}}{M^{4} \hbar \omega^{9}}-\frac{5 j v_{4}}{2 M^{2} \omega^{4}}-\frac{j^{3} v_{4}}{M^{3} \hbar \omega^{7}}, \\ c_{2}^{(2)} & =\frac{7 v_{3}^{2}}{8 M^{2} \omega^{4}}+\frac{3 j^{2} v_{3}^{2}}{2 M^{3} \hbar \omega^{7}}-\frac{3 v_{4}}{4 M \omega^{2}}-\frac{j^{2} v_{4}}{2 M^{2} \hbar \omega^{5}}, \\ c_{3}^{(2)} & =\frac{j v_{3}^{2}}{2 M^{2} \hbar \omega^{5}}-\frac{j v_{4}}{3 M \hbar \omega^{3}}, \quad c_{4}^{(2)}=\frac{v_{3}^{2}}{8 M \hbar \omega^{3}}-\frac{v_{4}}{4 \hbar \omega}, \\ \epsilon_{2} & =-\frac{11 \hbar^{2} v_{3}^{2}}{8 M^{3} \omega^{4}}-\frac{27 \hbar j^{2} v_{3}^{2}}{4 M^{4} \omega^{7}}-\frac{9 j^{4} v_{3}^{2}}{2 M^{5} \omega^{10}}+\frac{3 \hbar^{2} v_{4}}{4 M^{2} \omega^{2}}+\frac{3 \hbar j^{2} v_{4}}{M^{3} \omega^{5}}+\frac{j^{4} v_{4}}{M^{4} \omega^{8}} . \end{align*} $$

$$ G_{\Omega^{2}}\left(t, t^{\prime}\right)=\frac{\Theta\left(t-t^{\prime}\right) D_{b}(t) D_{a}\left(t^{\prime}\right)+\Theta\left(t^{\prime}-t\right) D_{a}(t) D_{b}\left(t^{\prime}\right)}{D_{a}\left(t_{b}\right)} $$

$$ \begin{align*} c_{m}^{(k)}= & \frac{(m+2)(m+1) \hbar}{2 m M \omega} c_{m+2}^{(k)}+\frac{\hbar}{2 m M \omega} \sum_{l=1}^{k-1} \sum_{n=1}^{m+1} n(m+2-n) c_{n}^{(l)} c_{m+2-n}^{(k-l)} \\ & +\frac{j(m+1)}{M m \omega^{2}} c_{m+1}^{(k)} \end{align*} $$

$$ \left(-\partial_{t}^{2}-\omega^{2}\right) \delta x(t)=0 $$

$$ \epsilon_{k}=-\frac{j \hbar}{M \omega} c_{1}^{(k)}-\frac{\hbar^{2}}{M} c_{2}^{(k)}-\frac{\hbar^{2}}{2 M} \sum_{l=1}^{k-1} c_{1}^{(l)} c_{1}^{(k-l)} $$

$$ x_{n}(t)=\sqrt{\frac{2}{t_{b}-t_{a}}} \sin \nu_{n}\left(t-t_{a}\right) $$

$$ e^{-\beta V_{\mathrm{eff}}^{\mathrm{fl}}(X)}=\int \mathcal{D} \delta x \exp \left(-\frac{1}{\hbar}\left\{\mathcal{A}[X+\delta x]-\mathcal{A}[X]-\mathcal{A}_{X}[X] \delta x-V_{\mathrm{eff} X}^{\mathrm{f}}(X) \delta x\right\}\right) $$

$$ \nu_{n}=\frac{\pi n}{t_{b}-t_{a}} $$

$$ \begin{align*} e^{-\beta\left[V_{\text {eff }}(X)-V(X)\right]} & =\oint \mathcal{D} \delta x \exp \left\{-\frac{1}{\hbar} \int_{0}^{\hbar \beta} d \tau\right. \\ & \left.\times\left[\frac{M}{2} \delta \dot{x}^{2}(\tau)+V(X+\delta x)-V(X)-V_{\text {eff }}^{\prime}(X) \delta x\right]\right\} \end{align*} $$

$$ \int_{t_{a}}^{t_{b}} d t x_{n}(t) x_{n^{\prime}}(t)=\delta_{n n^{\prime}} $$

$$ e^{-\beta\left[V_{\mathrm{eff}}(X)-V_{\mathrm{eff}}^{\prime}(X) X\right]}=\oint \mathcal{D} x \exp \left\{-\frac{1}{\hbar} \int_{0}^{\hbar \beta} d \tau\left[\frac{M}{2} \dot{x}^{2}(\tau)+V(x)-V_{\mathrm{eff}}^{\prime}(X) x\right]\right\} $$

$$ \int_{t_{a}}^{t_{b}} d t G_{\omega^{2}}\left(t, t^{\prime}\right) x_{n}\left(t^{\prime}\right)=G_{n} x_{n}(t) $$

$$ -\frac{\hbar^{2}}{2 M} \psi^{\prime \prime}(x)+\left[V(x)-V_{\mathrm{eff}}^{\prime}(X) x\right] \psi(x)=\left[V_{\mathrm{eff}}(X)-V_{\mathrm{eff}}^{\prime}(X) X\right] \psi(x) $$

$$ G_{\omega^{2}}\left(t, t^{\prime}\right)=\sum_{n=1}^{\infty} G_{n} x_{n}(t) x_{n}\left(t^{\prime}\right) $$

$$ \begin{array}{r} -\frac{\hbar^{2}}{2 M} \psi^{\prime \prime}(x)+\left[\frac{M}{2} \omega^{2} x^{2}+g v_{3} x^{3}+g^{2} v_{4} x^{4}-V_{\mathrm{eff}}^{\prime}(X) x\right] \psi(x) \\ =\left[V_{\mathrm{eff}}(X)-V_{\mathrm{eff}}^{\prime}(X) X\right] \psi(x), \end{array} $$

$$ G_{n}=\left(\nu_{n}^{2}-\omega^{2}\right)^{-1} $$

$$ V_{\mathrm{eff}}(X)=\sum_{k=0}^{\infty} g^{k} V_{k}(X) $$

$$ G_{\omega^{2}}\left(t, t^{\prime}\right)=\frac{2}{t_{b}-t_{a}} \sum_{n=1}^{\infty} \frac{\sin \nu_{n}\left(t-t_{a}\right) \sin \nu_{n}\left(t^{\prime}-t_{a}\right)}{\nu_{n}^{2}-\omega^{2}} . $$

$$ V_{k}(X)=\sum_{m=0}^{k+2} C_{m}^{(k)} X^{m} $$

$$ G_{\omega^{2}}\left(t, t^{\prime}\right)=\frac{2}{t_{b}-t_{a}} \sum_{n=1}^{\infty}(-1)^{n+1} \frac{\sin \nu_{n}\left(t_{b}-t\right) \sin \nu_{n}\left(t^{\prime}-t_{a}\right)}{\nu_{n}^{2}-\omega^{2}} . $$

$$ \phi(x)=\frac{M \omega X}{\hbar} x-\frac{M \omega}{2 \hbar} x^{2}+\sum_{k=1}^{\infty} g^{k} \phi_{k}(x), $$

$$ K(t) y_{n}(t)=\lambda_{n} y_{n}(t) $$

$$ V_{\mathrm{eff}}(X)=\frac{\hbar \omega}{2}+\frac{M}{2} \omega^{2} X^{2}+\sum_{k=1}^{\infty} g^{k} V_{k}(X), $$

$$ \begin{align*} \int_{t_{a}}^{t_{b}} d t y_{n}(t) y_{n^{\prime}}(t) & =\delta_{n n^{\prime}} \\ \sum_{n} y_{n}(t) y_{n}\left(t^{\prime}\right) & =\delta\left(t-t^{\prime}\right) \end{align*} $$

$$ \begin{align*} -\frac{\hbar^{2}}{2 M} \phi_{k}^{\prime \prime}(x)-\frac{\hbar^{2}}{2 M} \sum_{l=1}^{k-1} \phi_{k-l}^{\prime}(x) \phi_{l}^{\prime}(x)+\hbar \omega(x-X) \phi_{k}^{\prime}(x) & -x V_{k}^{\prime}(X)+\delta_{k, 1} v_{3} x^{3}+\delta_{k, 2} v_{4} x^{4} \\ & =V_{k}(X)-V_{k}^{\prime}(X) X \end{align*} $$

$$ c_{1}^{(1)}=\frac{v_{3}}{2 M \omega^{2}}+\frac{2 v_{3} X^{2}}{\hbar \omega}, c_{2}^{(1)}=-\frac{v_{3} X}{2 \hbar \omega}, c_{3}^{(1)}=-\frac{v_{3}}{3 \hbar \omega}, V_{1}(X)=\frac{3 v_{3} \hbar}{2 M \omega}+v_{3} X^{3}, $$

$$ G_{\Omega^{2}}\left(t, t^{\prime}\right)=\sum_{n} \frac{y_{n}(t) y_{n}\left(t^{\prime}\right)}{\lambda_{n}} $$

$$ \begin{align*} c_{1}^{(2)} & =-\frac{13 v_{3}^{2} X}{4 M^{2} \omega^{4}}-\frac{2 v_{3}^{2} X^{3}}{M \hbar \omega^{3}}+\frac{7 v_{4} X}{2 M \omega^{2}}+\frac{3 v_{4} X^{3}}{\hbar \omega}, \quad c_{2}^{(2)}=\frac{v_{3}^{2}}{8 M^{2} \omega^{4}}-\frac{3 v_{4}}{4 M \omega^{2}}-\frac{v_{4} X^{2}}{2 \hbar \omega}, \\ c_{3}^{(2)} & =\frac{v_{3}^{2} X}{2 M \hbar \omega^{3}}-\frac{v_{4} X}{3 \hbar \omega} c_{4}^{(2)}=\frac{v_{3}^{2}}{8 M \hbar \omega^{3}}-\frac{v_{4}}{4 \hbar \omega}, \\ V_{2}(X) & =-\frac{\hbar^{2} v_{3}^{2}}{4 M^{3} \omega^{4}}-\frac{9 \hbar v_{3}^{2} X^{2}}{4 M^{2} \omega^{3}}+\frac{3 \hbar^{2} v_{4}}{4 M^{2} \omega^{2}}+\frac{3 \hbar v_{4} X^{2}}{M \omega}+v_{4} X^{4} . \end{align*} $$

$$ \left[i \partial_{t}-\Omega(t)\right] G_{\Omega}\left(t, t^{\prime}\right)=i \delta\left(t-t^{\prime}\right), \quad t-t^{\prime} \in\left[0, t_{b}-t_{a}\right) $$

$$ \left[\partial_{\tau}-\Omega(\tau)\right] G_{\Omega, \mathrm{e}}\left(\tau, \tau^{\prime}\right)=\delta\left(\tau-\tau^{\prime}\right), \quad \tau-\tau^{\prime} \in[0, \hbar \beta) . $$

$$ \begin{align*} c_{m}^{(k)}= & \frac{(m+2)(m+1) \hbar}{2 m M \omega} c_{m+2}^{(k)}+\frac{\hbar}{2 m M \omega} \sum_{l=1}^{k-1} \sum_{n=1}^{m+1} n(m+2-n) c_{n}^{(l)} c_{m+2-n}^{(k-l)} \\ & +\frac{X(m+1)}{m} c_{m+1}^{(k)} \text { for } m \geq 2 \text { and with } c_{m}^{(k)} \equiv 0 \text { for } m>k+2, \\ c_{1}^{(k)}= & \frac{3 \hbar}{M \omega} c_{3}^{(k)}+2 X c_{2}^{(k)}+\frac{\hbar}{M \omega} \sum_{l=1}^{k-1}\left(c_{2}^{(k-l)} c_{1}^{(l)}+c_{1}^{(k-l)} c_{2}^{(l)}\right)+\frac{1}{\hbar \omega} V_{k}^{\prime}(X), \end{align*} $$

$$ \left(i \partial_{t}-\omega\right) G_{\omega}^{\mathrm{p}}\left(t, t^{\prime}\right)=i \delta\left(t-t^{\prime}\right), \quad t-t^{\prime} \in\left[0, t_{b}-t_{a}\right) $$

$$ G_{\omega}^{\mathrm{p}}\left(t, t^{\prime}\right) \equiv G_{\omega}^{\mathrm{p}}\left(t-t^{\prime}\right)=G_{\omega}^{\mathrm{p}}\left(t-t^{\prime}+t_{b}-t_{a}\right) $$

$$ \begin{align*} V_{k}(X)= & -\frac{\hbar^{2}}{M} c_{2}^{(k)}-\frac{3 \hbar^{2}}{M} X c_{3}^{(k)}-2 \hbar \omega X^{2} c_{2}^{(k)}-\frac{\hbar^{2}}{M} X \sum_{l=1}^{k-1}\left(c_{2}^{(k-l)} c_{1}^{(l)}+c_{1}^{(k-l)} c_{2}^{(l)}\right) \\ & -\frac{\hbar^{2}}{2 M} \sum_{l=1}^{k-1} c_{1}^{(l)} c_{1}^{(k-l)} . \end{align*} $$

$$ G_{\omega}^{\mathrm{p}}\left(t-t^{\prime}\right)=\frac{1}{t_{b}-t_{a}} \sum_{m=-\infty}^{\infty} e^{-i \omega_{m}\left(t-t^{\prime}\right)} \frac{i}{\omega_{m}-\omega} $$

$$ \left[-\frac{1}{2} \frac{d^{2}}{d r^{2}}-\frac{1}{2} \frac{D-1}{r} \frac{d}{d r}+\frac{l(l+D-2)}{2 r^{2}}+\frac{1}{2} r^{2}+\frac{g}{4} r^{4}\right] R_{n}(r)=E^{(n)} R_{n}(r) . $$

$$ \omega_{m} \equiv \frac{2 \pi m}{t_{b}-t_{a}}, \quad m=0, \pm 1, \pm 2, \pm 3, \ldots $$

$$ E^{(n)}=2 n^{\prime}+l+D / 2=n+D / 2, \quad n=0,1,2,3, \ldots \quad, l=0,1,2,3, \ldots . $$

$$ \sum_{m=-\infty}^{\infty} f(m)=\int_{-\infty}^{\infty} d \mu \sum_{n=-\infty}^{\infty} e^{2 \pi i \mu n} f(\mu) $$

$$ \left(-\frac{1}{2} \frac{d^{2}}{d r^{2}}-\frac{1}{2} \frac{2 l+D-1}{r} \frac{d}{d r}+\frac{1}{2} r^{2}+\frac{g}{4} r^{4}\right) w_{n}(r)=E^{(n)} w_{n}(r) . $$

$$ G_{\omega}^{\mathrm{p}}(t)=\sum_{n=-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} e^{-i \omega^{\prime}\left[t-\left(t_{b}-t_{a}\right) n\right]} \frac{i}{\omega^{\prime}-\omega} $$

$$ r \Phi_{k}^{\prime}(r)-2 n^{\prime} \Phi_{k}(r)=\frac{1}{2} \Phi_{k}^{\prime \prime}(r)+\frac{(2 l+D-1)}{2 r} \Phi_{k}^{\prime}(r)+r^{4} \Phi_{k-1}(r)+\sum_{k^{\prime}=1}^{k}(-1)^{k^{\prime}} E_{k^{\prime}}^{(n)} \Phi_{k-k^{\prime}}(r) . $$

$$ G_{\omega}(t)=\int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} e^{-i \omega^{\prime} t} \frac{i}{\omega^{\prime}-\omega} $$

$$ \left(p-2 n^{\prime}\right) A_{k}^{p}=\frac{1}{2}[(p+2)(p+1)+(p+2)(2 l+D-1)] A_{k}^{p+2}+A_{k-1}^{p-4}+\sum_{k^{\prime}=1}^{k}(-1)^{k^{\prime}} E_{k^{\prime}}^{(n)} A_{k-k^{\prime}}^{p} $$

$$ G_{\omega}^{\mathrm{p}}(t)=\sum_{n=-\infty}^{\infty} G_{\omega}\left(t-\left(t_{b}-t_{a}\right) n\right) . $$

$$ C_{k}^{0}=(2 l+D) \delta_{0 k} $$

$$ \begin{align*} G_{\omega}(t) & =\int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} e^{-i \omega^{\prime} t} \frac{i}{\omega^{\prime}-\omega+i \eta} \\ & =e^{-i \omega t} \times\left\{\begin{array}{lll} 1 & \text { for } & t>0 \\ \frac{1}{2} & \text { for } & t=0 \\ 0 & \text { for } & t<0 \end{array}\right. \end{align*} $$

$$ 2\left(p^{\prime}-n^{\prime}\right) C_{k}^{p^{\prime}}=\left[\left(2 p^{\prime}+1\right)\left(p^{\prime}+1\right)+\left(p^{\prime}+1\right)(l+D / 2-1 / 2)\right] C_{k}^{p^{\prime}+1}+C_{k-1}^{p^{\prime}-2}-\sum_{k^{\prime}=1}^{k} C_{k^{\prime}}^{1} C_{k-k^{\prime}}^{p^{\prime}}, $$

$$ G_{\omega}(t)=e^{-i \omega t} \bar{\Theta}(t) . $$

$$ C_{k}^{1}=3(2 l+D) \delta_{0 k} $$

$$ G_{\omega}^{\mathrm{p}}(t)=\sum_{n=-\infty}^{\infty} e^{-i \omega\left[t-\left(t_{b}-t_{a}\right) n\right]} \bar{\Theta}\left(t-\left(t_{b}-t_{a}\right) n\right) $$

$$ 2\left(p^{\prime}-n^{\prime}\right) C_{k}^{p^{\prime}}=\left[\left(2 p^{\prime}+3\right)\left(p^{\prime}+1\right)+\left(p^{\prime}+3 / 2\right)(l+D / 2-1 / 2)\right] C_{k}^{p^{\prime}+1}+C_{k-1}^{p^{\prime}-2}-\sum_{k^{\prime}=1}^{k} C_{k^{\prime}}^{1} C_{k-k^{\prime}}^{p^{\prime}} $$

$$ t \in\left[0, t_{b}-t_{a}\right) $$

$$ E_{k}^{(n)}=-\frac{(-1)^{k}}{2} \frac{2 l+D+1}{2 l+D} C_{k}^{1} $$

$$ \begin{align*} G_{\omega}^{\mathrm{p}}(t) & =\sum_{n=-\infty}^{0} e^{-i \omega\left[t-\left(t_{b}-t_{a}\right) n\right]}=\frac{e^{-i \omega t}}{1-e^{-i \omega\left(t_{b}-t_{a}\right)}} \\ & =-i \frac{e^{-i \omega\left[t-\left(t_{b}-t_{a}\right) / 2\right]}}{2 \sin \left[\omega\left(t_{b}-t_{a}\right) / 2\right]}, \quad t \in\left(0, t_{b}-t_{a}\right) \end{align*} $$

$$ \begin{align*} & {\left[-\frac{1}{2} \frac{d^{2}}{d r^{2}}-\frac{1}{2} \frac{D-1}{r} \frac{d}{d r}+\frac{l(l+D-2)}{2 r^{2}}+\frac{1}{2} r^{2}\right.} \\ & \left.\quad+\frac{g}{4}\left(a_{4} r^{4}+a_{6} r^{6}+\ldots+a_{2 q} x^{2 q}\right)\right] R_{n}(r)=E^{(n)} R_{n}(r), \end{align*} $$

$$ G_{\omega}^{\mathrm{p}}(0)=G_{\omega}^{\mathrm{p}}(0+)-\frac{1}{2} $$

$$ C_{k-1}^{p^{\prime}-2} \rightarrow a_{4} C_{k-1}^{p^{\prime}-2}+a_{6} C_{k-1}^{p^{\prime}-3}+\ldots+a_{2 q} C_{k-1}^{p^{\prime}-q} $$

$$ G_{\omega}^{\mathrm{p}}(t)=-i \frac{e^{-i \omega\left[t+\left(t_{b}-t_{a}\right) / 2\right]}}{2 \sin \left[\omega\left(t_{b}-t_{a}\right) / 2\right]}, \quad t \in\left(-\left(t_{b}-t_{a}\right), 0\right) $$

$$ G_{\omega, \mathrm{e}}^{\mathrm{p}}(\tau)=\frac{1}{1-e^{-\hbar \omega / k_{B} T}} e^{-\omega \tau}, \quad \tau \in(0, \hbar \beta) $$

$$ n_{\omega}^{\mathrm{b}}=\frac{1}{e^{\hbar \omega / k_{B} T}-1} $$

$$ G_{\omega, \mathrm{e}}^{\mathrm{p}}(\tau)=\left(1+n_{\omega}^{\mathrm{b}}\right) e^{-\omega \tau}, \quad \tau \in(0, \hbar \beta) . $$

$$ G_{\omega^{2}}^{\mathrm{p}}\left(t, t^{\prime}\right)=\left(-\partial_{t}^{2}-\omega^{2}\right)^{-1} \delta\left(t-t^{\prime}\right), \quad t-t^{\prime} \in\left[t_{a}, t_{b}\right) $$

$$ G_{\omega^{2}}^{\mathrm{p}}\left(t, t^{\prime}\right) \equiv G_{\omega^{2}}^{\mathrm{p}}\left(t-t^{\prime}\right)=G_{\omega^{2}}^{\mathrm{p}}\left(t-t^{\prime}+t_{b}-t_{a}\right) $$

$$ G_{\omega^{2}}^{\mathrm{p}}(t)=\frac{1}{t_{b}-t_{a}} \sum_{m=-\infty}^{\infty} e^{-i \omega_{m} t} \frac{1}{\omega_{m}^{2}-\omega^{2}} $$

$$ \frac{1}{\omega^{\prime 2}-\omega^{2}+i \eta}=\frac{1}{2 i \omega}\left(\frac{i}{\omega^{\prime}-\omega+i \eta}-\frac{i}{\omega^{\prime}+\omega-i \eta}\right) $$

$$ \begin{align*} G_{\omega^{2}}^{\mathrm{p}}(t) & =\frac{1}{2 \omega i}\left[G_{\omega}^{\mathrm{p}}(t)-G_{-\omega}^{\mathrm{p}}(t)\right] \\ & =-\frac{1}{2 \omega} \frac{\cos \omega\left[t-\left(t_{b}-t_{a}\right) / 2\right]}{\sin \left[\omega\left(t_{b}-t_{a}\right) / 2\right]}, \quad t \in\left[0, t_{b}-t_{a}\right) \end{align*} $$

$$ G_{-\omega}(t)=-e^{-i \omega t} \bar{\Theta}(-t) $$

$$ \left(i \partial_{t}-\omega\right) G_{\omega}^{\mathrm{a}}\left(t, t^{\prime}\right)=i \delta\left(t-t^{\prime}\right), \quad t-t^{\prime} \in\left[0, t_{b}-t_{a}\right) $$

$$ G_{\omega}^{\mathrm{a}}\left(t, t^{\prime}\right) \equiv G_{\omega}^{\mathrm{a}}\left(t-t^{\prime}\right)=-G_{\omega}^{\mathrm{a}}\left(t-t^{\prime}+t_{b}-t_{a}\right) $$

$$ G_{\omega}^{\mathrm{a}}(t)=\frac{1}{t_{b}-t_{a}} \sum_{m=-\infty}^{\infty} e^{-i \omega_{m}^{\mathrm{f}} t} \frac{i}{\omega_{m}^{\mathrm{f}}-\omega} $$

$$ \omega_{m}^{\mathrm{f}}=\frac{\pi(2 m+1)}{t_{b}-t_{a}} $$

$$ \sum_{m=-\infty}^{\infty} f(m+1 / 2)=\int_{-\infty}^{\infty} d \mu \sum_{n=-\infty}^{\infty}(-)^{n} e^{2 \pi i \mu n} f(\mu) $$

$$ \begin{align*} G_{\omega}^{\mathrm{a}}(t) & =\sum_{n=-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi}(-)^{n} e^{-i \omega^{\prime}\left[t-\left(t_{b}-t_{a}\right) n\right]} \frac{i}{\omega^{\prime}-\omega+i \eta} \\ & =\sum_{n=-\infty}^{\infty}(-)^{n} G_{\omega}\left(t-\left(t_{b}-t_{a}\right) n\right) \end{align*} $$

$$ G_{\omega}^{\mathrm{a}}(t)=\sum_{n=-\infty}^{\infty} e^{-i \omega\left[t-\left(t_{b}-t_{a}\right) n\right]}(-)^{n} \bar{\Theta}\left(t-\left(t_{b}-t_{a}\right) n\right) $$

$$ \begin{align*} G_{\omega}^{\mathrm{a}}(t) & =\sum_{n=-\infty}^{0} e^{-i \omega\left[t-\left(t_{b}-t_{a}\right) n\right]}(-)^{n}=\frac{e^{i \omega t}}{1+e^{-i \omega\left(t_{b}-t_{a}\right)}} \\ & =\frac{e^{-i \omega\left[t-\left(t_{b}-t_{a}\right) / 2\right]}}{2 \cos \left[\omega\left(t_{b}-t_{a}\right) / 2\right]}, \quad t \in\left[0, t_{b}-t_{a}\right) \end{align*} $$

$$ G_{0}^{\mathrm{a}}(t)=\frac{1}{2} \epsilon(t), \quad t \in\left[-\left(t_{b}-t_{a}\right),\left(t_{b}-t_{a}\right)\right] $$

$$ G_{\omega, \mathrm{e}}^{\mathrm{a}}(\tau)=\frac{1}{1+e^{-\hbar \omega / k_{B} T}} e^{-\omega \tau}, \quad \tau \in[0, \hbar \beta) . $$

$$ n_{\omega}^{\mathrm{f}}=\frac{1}{e^{\hbar \omega / k_{B} T}+1} $$

$$ G_{\omega, \mathrm{e}}^{\mathrm{a}}(\tau)=\left(1-n_{\omega}^{\mathrm{f}}\right) e^{-\omega \tau}, \quad \tau \in[0, \hbar \beta) . $$

$$ \begin{align*} G_{\omega^{2}}^{\mathrm{a}}(t) & =\frac{1}{t_{b}-t_{a}} \sum_{m=0}^{\infty} e^{-i \omega_{m}^{\mathrm{f}} t} \frac{1}{\omega_{m}^{\mathrm{f} 2}-\omega^{2}} \\ & =\frac{1}{2 \omega i}\left[G_{\omega}^{\mathrm{a}}(t)-G_{-\omega}^{\mathrm{a}}(t)\right] \\ & =-\frac{1}{2 \omega} \frac{\sin \omega\left[t-\left(t_{b}-t_{a}\right) / 2\right]}{\cos \left[\omega\left(t_{b}-t_{a}\right) / 2\right]}, \quad t \in\left[0, t_{b}-t_{a}\right] \end{align*} $$

$$ G_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}(0)=\frac{1}{\hbar \beta} \sum_{m=-\infty}^{\infty} \frac{1}{\omega_{m}^{2}+\omega^{2}}, \quad G_{\omega, \mathrm{e}}^{\mathrm{p}}(0)=\frac{1}{\hbar \beta} \sum_{m=-\infty}^{\infty} \frac{1}{\omega_{m}^{\mathrm{f} 2}+\omega^{2}}, $$

$$ \begin{align*} & =\frac{1}{2 \omega} \operatorname{coth} \frac{\hbar \omega \beta}{2}, \\ \frac{1}{2 \omega}\left[G_{\omega, \mathrm{e}}^{\mathrm{a}}(\eta)+G_{\omega, \mathrm{e}}^{\mathrm{a}}(-\eta)\right] & =\frac{1}{2 \omega}\left[G_{\omega, \mathrm{e}}^{\mathrm{a}}(\eta)-G_{\omega, \mathrm{e}}^{\mathrm{a}}(\hbar \beta-\eta)\right]=\frac{1}{2 \omega}\left(1-n_{\omega}^{\mathrm{f}}\right)\left(1-e^{-\beta \omega}\right) \\ & =\frac{1}{2 \omega} \tanh \frac{\hbar \omega \beta}{2}, \end{align*} $$

$$ G_{\Omega}\left(t, t^{\prime}\right)=\bar{\Theta}\left(t-t^{\prime}\right) g\left(t, t^{\prime}\right) . $$

$$ \left[i \partial_{t}-\Omega(t)\right] g\left(t, t^{\prime}\right)=0 $$

$$ g\left(t, t^{\prime}\right)=K\left(t^{\prime}\right) e^{-i \int_{c}^{t} d t^{\prime \prime} \Omega\left(t^{\prime \prime}\right)} $$

$$ G_{\Omega}\left(t, t^{\prime}\right)=\bar{\Theta}\left(t-t^{\prime}\right) e^{-i \int_{t^{\prime}}^{t} d t^{\prime \prime} \Omega\left(t^{\prime \prime}\right)} $$

$$ G_{\Omega}\left(t, t^{\prime}\right)=\left[\bar{\Theta}\left(t-t^{\prime}\right)+C\left(t^{\prime}\right)\right] e^{-i \int_{t^{\prime}}^{t} d t^{\prime \prime} \Omega\left(t^{\prime \prime}\right)} $$

$$ C\left(t^{\prime}\right) e^{-i \int_{t^{\prime}}^{t_{a}} d t^{\prime \prime} \Omega\left(t^{\prime \prime}\right)}=\left[1+C\left(t^{\prime}\right)\right] e^{-i \int_{t^{\prime}}^{t_{b}} d t^{\prime \prime} \Omega\left(t^{\prime \prime}\right)} $$

$$ C=n_{\Omega}^{\mathrm{p}} \equiv \frac{1}{e^{i \int_{t_{a}}^{t_{b}} d t^{\prime \prime} \Omega\left(t^{\prime \prime}\right)}-1} $$

$$ G_{\Omega}^{\mathrm{p}}\left(t, t^{\prime}\right)=\left[\bar{\Theta}\left(t-t^{\prime}\right)+n_{\Omega}^{\mathrm{p}}\right] e^{-i \int_{t^{\prime}}^{t} d t^{\prime \prime} \Omega\left(t^{\prime \prime}\right)} $$

$$ n_{\Omega}^{\mathrm{a}} \equiv \frac{1}{e^{i \int_{t_{a}}^{t_{b}} d t \Omega(t)}+1} $$

$$ \left[-i \partial_{t}-\Omega(t)\right] G_{\Omega}\left(t, t^{\prime}\right)=i \delta\left(t-t^{\prime}\right) $$

$$ e^{i \int_{t_{a}}^{t_{b}} d t \Omega(t)} \rightarrow \hat{T} e^{i \int_{t_{a}}^{t_{b}} d t \Omega(t)} $$

$$ G_{\Omega}\left(\tau, \tau^{\prime}\right)=\bar{\Theta}\left(\tau-\tau^{\prime}\right) e^{-\int_{\tau^{\prime}}^{\tau} d \tau^{\prime \prime} \Omega\left(\tau^{\prime \prime}\right)} $$

$$ G_{\Omega}\left(\tau, \tau^{\prime}\right)=\left[\bar{\Theta}\left(\tau-\tau^{\prime}\right)+n^{\mathrm{b}}\right] e^{-\int_{\tau^{\prime}}^{\tau} d \tau^{\prime \prime} \Omega\left(\tau^{\prime \prime}\right)} $$

$$ n^{\mathrm{b}} \equiv \frac{1}{e^{\int_{0}^{\hbar \beta} d \tau^{\prime \prime} \Omega\left(\tau^{\prime \prime}\right)}-1} $$

$$ n^{\mathrm{f}} \equiv \frac{1}{e^{\int_{0}^{\hbar \beta} d \tau^{\prime \prime} \Omega\left(\tau^{\prime \prime}\right)}+1} $$

$$ \operatorname{Tr} \log \left[\partial_{\tau}+g \Omega(\tau)\right]=\int_{0}^{g} d g^{\prime} G_{g^{\prime} \Omega}(\tau, \tau) $$

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

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

$$ \begin{align*} \operatorname{Tr} \log \left[ \pm \partial_{\tau}+\Omega(\tau)\right] & =\operatorname{Tr} \log \left[ \pm \partial_{\tau}+\eta\right]+\operatorname{Tr} \log \left[1+\left( \pm \partial_{\tau}+\eta\right)^{-1} \Omega(\tau)\right] \\ & =\operatorname{Tr} \log \left[ \pm \partial_{\tau}+\eta\right]+\operatorname{Tr} \log \left[1+\left( \pm \partial_{\tau}+\eta\right)^{-1} \Omega(\tau)\right] \end{align*} $$

$$ \left[ \pm \partial_{\tau}+\eta\right]^{-1}\left(\tau, \tau^{\prime}\right)=\left\{\begin{array}{l} \bar{\Theta}\left(\tau-\tau^{\prime}\right) \\ \bar{\Theta}\left(\tau^{\prime}-\tau\right) \end{array}\right\} $$

$$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \int d \tau_{1} \cdots \mathrm{~d} \tau_{n} \Omega\left(\tau_{1}\right) \bar{\Theta}\left(\tau_{1}-\tau_{2}\right) \Omega\left(\tau_{2}\right) \bar{\Theta}\left(\tau_{2}-\tau_{3}\right) \cdots \Omega\left(\tau_{n}\right) \bar{\Theta}\left(\tau_{n}-\tau_{1}\right) $$

$$ \int d \tau_{1} \Omega\left(\tau_{1}\right) \bar{\Theta}\left(\tau_{1}-\tau_{1}\right)=\frac{1}{2} \int d \tau \Omega(\tau) $$

$$ \operatorname{Tr} \log \left[\hbar \partial_{\tau}+\hat{H}(\tau)\right]=\frac{1}{2 \hbar} \operatorname{Tr}\left[\int_{0}^{\hbar \beta} d \tau \hat{H}(\tau)\right]-\sum_{n=1}^{\infty} \frac{1}{n} \operatorname{Tr}\left[\hat{T} e^{-n \int_{0}^{\hbar \beta} d \tau^{\prime \prime} \hat{H}\left(\tau^{\prime \prime}\right) / \hbar}\right] $$

$$ \begin{align*} G_{\omega^{2}}\left(t, t^{\prime}\right) & =\frac{2}{t_{b}-t_{a}} \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{(2 i)^{2}} \frac{\left(e^{i \nu_{n} t_{2}}-e^{-i \nu_{n} t_{2}}\right)\left(e^{i \nu_{n} t_{1}}-e^{-i \nu_{n} t_{1}}\right)}{\nu_{n}^{2}-\omega^{2}} \\ & =\frac{1}{2} \frac{1}{t_{b}-t_{a}} \sum_{n=1}^{\infty}(-1)^{n} \frac{\left[\left(e^{-i \nu_{n}\left(t_{2}+t_{1}\right)}-e^{-i \nu_{n}\left(t_{2}-t_{1}\right)}\right)+\text { c.c. }\right]}{\nu_{n}^{2}-\omega^{2}} \\ & =\frac{1}{2} \frac{1}{t_{b}-t_{a}} \sum_{n=-\infty}^{\infty}(-1)^{n} \frac{e^{-i \nu_{n}\left(t_{2}+t_{1}\right)}-e^{-i \nu_{n}\left(t_{2}-t_{1}\right)}}{\nu_{n}^{2}-\omega^{2}} \end{align*} $$

$$ \begin{align*} G_{\omega^{2}}\left(t, t^{\prime}\right)=\frac{1}{2} & \left\{\frac{1}{t_{b}-t_{a}} \sum_{m=-\infty}^{\infty} \frac{e^{-i \omega_{m}\left(t_{2}+t_{1}\right)}}{\omega_{m}^{2}-\omega^{2}}-\frac{1}{t_{b}-t_{a}} \sum_{m=-\infty}^{\infty} \frac{e^{-i \omega_{m}^{\mathrm{f}}\left(t_{2}+t_{1}\right)}}{\omega_{m}^{\mathrm{f}} 2-\omega^{2}}\right. \\ & \left.-\frac{1}{t_{b}-t_{a}} \sum_{m=-\infty}^{\infty} \frac{e^{-i \omega_{m}\left(t_{2}-t_{1}\right)}}{\omega_{m}^{2}-\omega^{2}}+\frac{1}{t_{b}-t_{a}} \sum_{m=-\infty}^{\infty} \frac{e^{-i \omega_{m}^{\mathrm{f}}\left(t_{2}-t_{1}\right)}}{\omega_{m}^{\mathrm{f}} 2-\omega^{2}}\right\} . \end{align*} $$

$$ G_{\omega^{2}}\left(t, t^{\prime}\right)=\frac{1}{2}\left[G_{\omega}^{\mathrm{p}}\left(t_{2}+t_{1}\right)-G_{\omega}^{\mathrm{a}}\left(t_{2}+t_{1}\right)-G_{\omega}^{\mathrm{p}}\left(t_{2}-t_{1}\right)+G_{\omega}^{\mathrm{a}}\left(t_{2}-t_{1}\right)\right] $$

$$ \begin{align*} G_{\omega}^{\mathrm{p}}\left(t_{2}+t_{1}\right)-G_{\omega}^{\mathrm{p}}\left(t_{2}-t_{1}\right) & =\frac{\sin \omega\left[t_{2}-\left(t_{b}-t_{a}\right) / 2\right] \sin \omega t_{1}}{\omega \sin \left[\omega\left(t_{b}-t_{a}\right) / 2\right]} \\ G_{\omega}^{\mathrm{a}}\left(t_{2}+t_{1}\right)-G_{\omega}^{\mathrm{a}}\left(t_{2}-t_{1}\right) & =-\frac{\cos \omega\left[t_{2}-\left(t_{b}-t_{a}\right) / 2\right] \sin \omega t_{1}}{\omega \cos \left[\omega\left(t_{b}-t_{a}\right) / 2\right]} \end{align*} $$

$$ G_{\omega^{2}}\left(t, t^{\prime}\right)=\frac{1}{\omega \sin \omega\left(t_{b}-t_{a}\right)} \sin \omega t_{2} \sin \omega t_{1} $$

$$ t_{a} \rightarrow-\infty, \quad t_{b} \rightarrow \infty . $$

$$ G_{\omega^{2}}\left(t, t^{\prime}\right)=-\frac{i}{2 \omega} e^{-i \omega\left|t-t^{\prime}\right|}, $$

$$ \left(-\partial_{t}^{2}-\omega^{2}\right) G_{\omega^{2}}\left(t, t^{\prime}\right)=\delta\left(t-t^{\prime}\right) $$

$$ \begin{align*} G_{\omega^{2}}^{\mathrm{p}}\left(t, t^{\prime}\right) & =\sum_{n=-\infty}^{\infty} G\left(t+n\left(t_{b}-t_{a}\right), t^{\prime}\right) \\ G_{\omega^{2}}^{\mathrm{a}}\left(t, t^{\prime}\right) & =\sum_{n=-\infty}^{\infty}(-1)^{n} G_{\omega^{2}}\left(t+n\left(t_{b}-t_{a}\right), t^{\prime}\right) \end{align*} $$

$$ x_{0}(t)=\sqrt{\frac{1}{t_{b}-t_{a}}}, \quad x_{n}(t)=\sqrt{\frac{2}{t_{b}-t_{a}}} \cos \nu_{n}\left(t-t_{a}\right), $$

$$ G_{\omega^{2}}^{\mathrm{N}}\left(t, t^{\prime}\right)=\frac{2}{t_{b}-t_{a}}\left[-\frac{1}{2 \omega^{2}}+\sum_{n=1}^{\infty} \frac{\cos \nu_{n}\left(t-t_{a}\right) \cos \nu_{n}\left(t^{\prime}-t_{a}\right)}{\nu_{n}^{2}-\omega^{2}}\right] . $$

$$ \left.\partial_{t} G_{\omega^{2}}^{\mathrm{N}}\left(t, t^{\prime}\right)\right|_{t=t_{b}}=0,\left.\quad \partial_{t^{\prime}} G_{\omega^{2}}^{\mathrm{N}}\left(t, t^{\prime}\right)\right|_{t^{\prime}=t_{a}}=0 . $$

$$ G_{\omega^{2}}^{\mathrm{N}}\left(t, t^{\prime}\right)=\frac{1}{2}\left[G_{\omega}^{\mathrm{p}}\left(t_{2}+t_{1}\right)-G_{\omega}^{\mathrm{a}}\left(t_{2}+t_{1}\right)+G_{\omega}^{\mathrm{p}}\left(t_{2}-t_{1}\right)-G_{\omega}^{\mathrm{a}}\left(t_{2}-t_{1}\right)\right] . $$

$$ \begin{align*} G_{\omega}^{\mathrm{p}}\left(t_{2}+t_{1}\right)+G_{\omega}^{\mathrm{p}}\left(t_{2}-t_{1}\right) & =-\frac{\cos \omega\left[t_{2}-\left(t_{b}-t_{a}\right) / 2\right] \cos \omega t_{1}}{\omega \sin \left[\omega\left(t_{b}-t_{a}\right) / 2\right]} \\ G_{\omega}^{\mathrm{a}}\left(t_{2}+t_{1}\right)+G_{\omega}^{\mathrm{a}}\left(t_{2}-t_{1}\right) & =-\frac{\sin \omega\left[t_{2}-\left(t_{b}-t_{a}\right) / 2\right] \cos \omega t_{1}}{\omega \cos \left[\omega\left(t_{b}-t_{a}\right) / 2\right]} \end{align*} $$

$$ G_{\omega^{2}}^{\mathrm{N}}\left(t, t^{\prime}\right)=-\frac{1}{\omega \sin \omega\left(t_{b}-t_{a}\right)} \cos \omega\left(t_{b}-t_{>}\right) \cos \omega\left(t_{<}-t_{a}\right) $$

$$ G_{\omega^{2}}^{\mathrm{N}}\left(t, t^{\prime}\right) \underset{\omega^{2} \approx 0}{\approx}-\frac{1}{\left(t_{b}-t_{a}\right) \omega^{2}}+\frac{t_{b}-t_{a}}{3}-\frac{1}{2}\left|t-t^{\prime}\right|-\frac{1}{2}\left(t+t^{\prime}\right)+\frac{1}{2\left(t_{b}-t_{a}\right)}\left(t^{2}+t^{\prime 2}\right) . $$

$$ \left[-\partial_{t}^{2}-\Omega^{2}(t)\right] G_{\Omega^{2}}\left(t, t^{\prime}\right)=\delta\left(t-t^{\prime}\right) $$

$$ G_{\Omega^{2}}^{\mathrm{p}, \mathrm{a}}\left(t, t^{\prime}\right)=\bar{\Theta}\left(t-t^{\prime}\right) \Delta\left(t, t^{\prime}\right)+a\left(t^{\prime}\right) \xi(t)+b\left(t^{\prime}\right) \eta(t) $$

$$ \delta^{\mathrm{p}, \mathrm{a}}\left(t-t^{\prime}\right) \equiv \sum_{n=-\infty}^{\infty} \delta\left(t-t^{\prime}-n \hbar \beta\right)\left\{\begin{array}{c} 1 \\ (-1)^{n} \end{array}\right\} $$

$$ \left[-\partial_{t}^{2}-\Omega^{2}(t)\right] \Delta\left(t, t^{\prime}\right)=0 ; \quad \Delta(t, t)=0,\left.\quad \partial_{t} \Delta\left(t, t^{\prime}\right)\right|_{t^{\prime}=t}=-1 $$

$$ \begin{align*} a(t)\left[\xi\left(t_{b}\right) \mp \xi\left(t_{a}\right)\right]+b(t)\left[\eta\left(t_{b}\right) \mp \eta\left(t_{a}\right)\right] & =-\Delta\left(t_{b}, t\right) \\ a(t)\left[\dot{\xi}\left(t_{b}\right) \mp \dot{\xi}\left(t_{a}\right)\right]+b(t)\left[\dot{\eta}\left(t_{b}\right) \mp \dot{\eta}\left(t_{a}\right)\right] & =-\partial_{t} \Delta\left(t_{b}, t\right) \end{align*} $$

$$ \bar{\Lambda}^{\mathrm{p}, \mathrm{a}}\left(t_{a}, t_{b}\right)=\left(\begin{array}{ll} \xi\left(t_{b}\right) \mp \xi\left(t_{a}\right) & \eta\left(t_{b}\right) \mp \eta\left(t_{a}\right) \\ \dot{\xi}\left(t_{b}\right) \mp \dot{\xi}\left(t_{a}\right) & \dot{\eta}\left(t_{b}\right) \mp \dot{\eta}\left(t_{a}\right) \end{array}\right) $$

$$ \operatorname{det} \bar{\Lambda}^{\mathrm{p}, \mathrm{a}}\left(t_{a}, t_{b}\right)=W \bar{\Delta}^{\mathrm{p}, \mathrm{a}}\left(t_{a}, t_{b}\right) \neq 0 $$

$$ \bar{\Delta}^{\mathrm{p}, \mathrm{a}}\left(t_{a}, t_{b}\right)=2 \pm \partial_{t} \Delta\left(t_{a}, t_{b}\right) \pm \partial_{t} \Delta\left(t_{b}, t_{a}\right) $$

$$ G_{\Omega^{2}}^{\mathrm{p}, \mathrm{a}}\left(t, t^{\prime}\right)=G_{\Omega^{2}}\left(t, t^{\prime}\right) \mp \frac{\left[\Delta\left(t, t_{a}\right) \pm \Delta\left(t_{b}, t\right)\right]\left[\Delta\left(t^{\prime}, t_{a}\right) \pm \Delta\left(t_{b}, t^{\prime}\right)\right]}{\bar{\Delta}^{\mathrm{p}, \mathrm{a}}\left(t_{a}, t_{b}\right) \Delta\left(t_{a}, t_{b}\right)} $$

$$ \begin{align*} \mathcal{A}_{j, \mathrm{~A}} & =-\frac{1}{2 M} \int_{t_{a}}^{t_{b}} d t \int_{t_{a}}^{t_{b}} d t^{\prime} G_{\omega^{2}}\left(t, t^{\prime}\right) j(t) j\left(t^{\prime}\right) \\ & =-\frac{1}{M} \frac{1}{\omega \sin \omega\left(t_{b}-t_{a}\right)} \int_{t_{a}}^{t_{b}} d t \int_{t_{a}}^{t} d t^{\prime} \sin \omega\left(t_{b}-t\right) \sin \omega\left(t^{\prime}-t_{a}\right) j(t) j\left(t^{\prime}\right) \end{align*} $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)_{\omega}^{j}=\int \mathcal{D} x \exp \left\{\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t\left[\frac{M}{2}\left(\dot{x}^{2}-\omega^{2} x^{2}\right)+j x\right]\right\}=e^{(i / \hbar) \mathcal{A}_{j, \mathrm{cl}}} F_{\omega, j}\left(t_{b}, t_{a}\right) $$

$$ \begin{align*} \mathcal{A}_{j, \mathrm{cl}} & =\frac{1}{2} \frac{M \omega}{\sin \omega\left(t_{b}-t_{a}\right)}\left[\left(x_{b}^{2}+x_{a}^{2}\right) \cos \omega\left(t_{b}-t_{a}\right)-2 x_{b} x_{a}\right] \\ & +\frac{1}{\sin \omega\left(t_{b}-t_{a}\right)} \int_{t_{a}}^{t_{b}} d t\left[x_{a} \sin \omega\left(t_{b}-t\right)+x_{b} \sin \omega\left(t-t_{a}\right)\right] j(t) \end{align*} $$

$$ \begin{align*} & F_{\omega, j}\left(t_{b}, t_{a}\right)=F_{\omega}\left(t_{b}, t_{a}\right) e^{i \mathcal{A}_{j, \mathrm{fl}} / \hbar}=\frac{1}{\sqrt{2 \pi i \hbar / M}} \sqrt{\frac{\omega}{\sin \omega\left(t_{b}-t_{a}\right)}} \\ & \quad \times \exp \left\{-\frac{i}{\hbar M \omega \sin \omega\left(t_{b}-t_{a}\right)} \int_{t_{a}}^{t_{b}} d t \int_{t_{a}}^{t} d t^{\prime} \sin \omega\left(t_{b}-t\right) \sin \omega\left(t^{\prime}-t_{a}\right) j(t) j\left(t^{\prime}\right)\right\} \end{align*} $$

$$ \mathcal{A}_{j, \mathrm{cl}}=\frac{M}{2 D_{a}\left(t_{b}\right)}\left[x_{b}^{2} \dot{D}_{a}\left(t_{b}\right)-x_{a}^{2} \dot{D}_{b}\left(t_{a}\right)-2 x_{b} x_{a}\right]+\frac{1}{D_{a}\left(t_{b}\right)} \int_{t_{a}}^{t_{b}} d t\left[x_{b} D_{a}(t)+x_{a} D_{b}(t)\right] j(t) $$

$$ \begin{align*} & F_{\omega, j}\left(t_{b}, t_{a}\right)=F_{\omega}\left(t_{b}, t_{a}\right) e^{i \mathcal{A}_{j, \mathrm{~A}} / \hbar}=\frac{1}{\sqrt{2 \pi i \hbar / M}} \frac{1}{\sqrt{D_{a}\left(t_{b}\right)}} \exp \left\{-\frac{i}{2 \hbar M D_{a}\left(t_{b}\right)}\right. \\ & \left.\quad \times \int_{t_{a}}^{t_{b}} d t \int_{t_{a}}^{t} d t^{\prime} j(t)\left[\bar{\Theta}\left(t-t^{\prime}\right) D_{b}(t) D_{a}\left(t^{\prime}\right)+\bar{\Theta}\left(t^{\prime}-t\right) D_{a}(t) D_{b}\left(t^{\prime}\right)\right] j\left(t^{\prime}\right)\right\} \end{align*} $$

$$ \begin{align*} A(\omega) & \equiv \frac{1}{M \omega} \int_{t_{a}}^{t_{b}} d t e^{-i \omega\left(t-t_{a}\right)} j(t) \\ B(\omega) & \equiv \frac{1}{M \omega} \int_{t_{a}}^{t_{b}} d t e^{-i \omega\left(t_{b}-t\right)} j(t)=-e^{-i \omega\left(t_{b}-t_{a}\right)} A(-\omega) \end{align*} $$

$$ \mathcal{A}_{j, \mathrm{cl}}=-i \frac{M \omega}{\sin \omega\left(t_{b}-t_{a}\right)}\left\{\left[x_{b}\left(e^{i \omega\left(t_{b}-t_{a}\right)} A-B\right)\right]+x_{a}\left(e^{i \omega\left(t_{b}-t_{a}\right)} B-A\right)\right\} $$

$$ \mathcal{A}_{j, \mathrm{fl}}=\frac{i}{4 M \omega} \int_{t_{a}}^{t_{b}} d t \int_{t_{b}}^{t_{b}} d t^{\prime} e^{-i \omega\left|t-t^{\prime}\right|} j(t) j\left(t^{\prime}\right)-\frac{M \omega}{2 \sin \omega\left(t_{b}-t_{a}\right)}\left[e^{i \omega\left(t_{b}-t_{a}\right)}\left(A^{2}+B^{2}\right)-2 A B\right] $$

$$ \begin{align*} &-\left[\sin \omega\left(t_{b}-t\right) \sin \omega\left(t^{\prime}-t_{a}\right) \bar{\Theta}\left(t-t^{\prime}\right)+\sin \omega\left(t_{b}-t^{\prime}\right) \sin \omega\left(t-t_{a}\right) \bar{\Theta}\left(t^{\prime}-t\right)\right] \\ &=\frac{1}{4}\left[\left(e^{i \omega\left(t_{b}-t_{a}\right)} e^{-i \omega\left(t-t^{\prime}\right)}+\text { c.c. }\right)\right.\left.-\left(e^{i \omega\left(t_{b}+t_{a}\right)} e^{-i \omega\left(t+t^{\prime}\right)}+\text { c.c. }\right)\right] \bar{\Theta}\left(t-t^{\prime}\right) \\ &+\left\{t \leftrightarrow t^{\prime}\right\} \end{align*} $$

$$ \begin{align*} \frac{1}{4}\{ & -\left(e^{i \omega\left(t_{b}+t_{a}\right)} e^{-i \omega\left(t^{\prime}+t\right)}+\text { c.c. }\right) \\ & +e^{i \omega\left(t_{b}-t_{a}\right)}\left(e^{-i \omega\left(t-t^{\prime}\right)} \bar{\Theta}\left(t-t^{\prime}\right)+e^{-i \omega\left(t^{\prime}-t\right)} \bar{\Theta}\left(t^{\prime}-t\right)\right) \\ & \left.+e^{-i \omega\left(t_{b}-t_{a}\right)}\left[e^{i \omega\left(t-t^{\prime}\right)}\left(1-\bar{\Theta}\left(t^{\prime}-t\right)\right)+e^{i \omega\left(t^{\prime}-t\right)}\left(1-\bar{\Theta}\left(t-t^{\prime}\right)\right)\right]\right\} \end{align*} $$

$$ \begin{align*} & \frac{1}{4}\left[-e^{i \omega\left(t_{b}-t_{a}\right)} 4 M^{2} \omega^{2}\left(B^{2}+A^{2}\right)\right. \\ & \left.\quad+\left(e^{i \omega\left(t_{b}-t_{a}\right)}-e^{-i \omega\left(t_{b}-t_{a}\right)}\right) \int_{t_{a}}^{t_{b}} d t \int_{t_{b}}^{t_{b}} d t^{\prime} e^{-i \omega\left|t-t^{\prime}\right|} j(t) j\left(t^{\prime}\right)+4 M^{2} \omega^{2} 2 A B\right] \end{align*} $$

$$ \begin{align*} \frac{i}{\hbar} \mathcal{A}_{j}=\frac{i}{\hbar}\left(\mathcal{A}_{j, \mathrm{cl}}+\mathcal{A}_{j, \mathrm{fl}}\right)= & \frac{i}{\hbar}\left\{\frac{1}{\omega \sin \omega\left(t_{b}-t_{a}\right)}\left[1-\cos \omega\left(t_{b}-t_{a}\right)\right]\left(x_{b}+x_{a}\right) j\right. \\ & \left.+\frac{1}{2 M \omega^{3}}\left[\omega\left(t_{b}-t_{a}\right)+2 \frac{\cos \omega\left(t_{b}-t_{a}\right)-1}{\sin \omega\left(t_{b}-t_{a}\right)}\right] j^{2}\right\} \end{align*} $$

$$ \mathcal{A}_{j}=\frac{1}{\omega} \tan \frac{\omega\left(t_{b}-t_{a}\right)}{2}\left(x_{b}+x_{a}\right) j+\frac{1}{2 M \omega^{3}}\left[\omega\left(t_{b}-t_{a}\right)-2 \tan \frac{\omega\left(t_{b}-t_{a}\right)}{2}\right] j^{2} $$

$$ -\int_{t_{a}}^{t_{b}} d t\left(\frac{M}{2} \omega^{2} x^{2}-x j\right) $$

$$ -\int_{t_{a}}^{t_{b}} d t \frac{M}{2} \omega^{2}\left(x-\frac{j}{M \omega^{2}}\right)^{2}+\frac{t_{b}-t_{a}}{2 M \omega^{2}} j^{2} $$

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

$$ \begin{align*} \left(x_{b} t_{a} \mid x_{a} t_{a}\right)_{0}^{j=\mathrm{const}} & =\frac{1}{\sqrt{2 \pi i \hbar\left(t_{b}-t_{a}\right) / M}} \exp \left[\frac{i}{\hbar} \frac{M}{2} \frac{\left(x_{b}-x_{a}\right)^{2}}{t_{b}-t_{a}}\right] \\ & \times \exp \left\{\frac{i}{\hbar}\left[\frac{1}{2}\left(x_{b}+x_{a}\right)\left(t_{b}-t_{a}\right) j-\frac{1}{24 M}\left(t_{b}-t_{a}\right)^{3} j^{2}\right]\right\} \end{align*} $$

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

$$ \ddot{x}_{j, \mathrm{cl}}=j / M $$

$$ x_{j, \mathrm{cl}}(t)=x_{a}+\left[x_{b}-x_{a}-\frac{j}{2 M}\left(t_{b}-t_{a}\right)^{2}\right] \frac{t-t_{a}}{t_{b}-t_{a}}+\frac{j}{2 M}\left(t-t_{a}\right)^{2} . $$

$$ \mathcal{A}_{j, \mathrm{cl}}=\frac{M}{2} \frac{\left(x_{b}-x_{a}\right)^{2}}{t_{b}-t_{a}}+\frac{1}{2}\left(x_{b}+x_{a}\right)\left(t_{b}-t_{a}\right) j-\frac{\left(t_{b}-t_{a}\right)^{3}}{24} \frac{j^{2}}{M}, $$

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

$$ \ddot{x}_{j, \mathrm{cl}}(t)+\omega^{2} x_{j, \mathrm{cl}}(t)=j(t) / M $$

$$ x_{j, \mathrm{cl}}(t)=x_{a} \frac{\sin \omega\left(t_{b}-t\right)}{\sin \omega\left(t_{b}-t_{a}\right)}+x_{b} \frac{\sin \omega\left(t-t_{a}\right)}{\sin \omega\left(t_{b}-t_{a}\right)}+\frac{1}{M} \int_{t_{a}}^{t_{b}} d t^{\prime} G_{\omega^{2}}\left(t, t^{\prime}\right) j\left(t^{\prime}\right) $$

$$ \begin{align*} \mathcal{A}_{\mathrm{cl}} & =\int_{t_{a}}^{t_{b}} d t\left[\frac{M}{2}\left(\dot{x}_{\mathrm{cl}, j}^{2}-\omega^{2} x_{j, \mathrm{cl}}^{2}\right)+j x_{j, \mathrm{cl}}\right]=\left.\frac{M}{2} x_{j, \mathrm{cl}} \dot{x}_{j, \mathrm{cl}}\right|_{t_{a}} ^{t_{b}} \\ & +\int_{t_{a}}^{t_{b}} d t\left[\frac{M}{2} x_{j, \mathrm{cl}}\left(-\ddot{x}_{j, \mathrm{cl}}-\omega^{2} x_{j, \mathrm{cl}}+\frac{j}{M}\right)\right]+\frac{1}{2} \int_{t_{a}}^{t_{b}} d t x_{j, \mathrm{cl}} j \\ & =\left.\frac{M}{2}\left(x_{b} \dot{x}_{b}-x_{a} \dot{x}_{a}\right)\right|_{x=x_{j, \mathrm{cl}}}+\frac{1}{2} \int_{t_{a}}^{t_{b}} d t x_{j, \mathrm{cl}}(t) j(t) \end{align*} $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)_{\omega}^{x_{0}}=\left(t_{b}-t_{a}\right) \int_{-\infty}^{\infty} \frac{d j}{2 \pi \hbar} e^{-i j\left(t_{b}-t_{a}\right) x_{0} / \hbar}\left(x_{b} t_{b} \mid x_{a} t_{a}\right)_{\omega}^{j} $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)_{\omega}^{x_{0}}=\int \mathcal{D} x \delta\left(x_{0}-\bar{x}\right) \exp \left\{\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t \frac{M}{2}\left(\dot{x}^{2}-\omega^{2} x^{2}\right)\right\} $$

$$ \mathcal{A}_{j}-j\left(t_{b}-t_{a}\right) x_{0}=\frac{1}{2 M \omega^{3}}\left[\omega\left(t_{b}-t_{a}\right)-2 \tan \frac{\omega\left(t_{b}-t_{a}\right)}{2}\right]\left(j-j_{0}\right)^{2}+\mathcal{A}^{x_{0}} $$

$$ j_{0}=\frac{M \omega^{2}}{\omega\left(t_{b}-t_{a}\right)-2 \tan \frac{\omega\left(t_{b}-t_{a}\right)}{2}}\left[\omega\left(t_{b}-t_{a}\right) x_{0}-\tan \frac{\omega\left(t_{b}-t_{a}\right)}{2}\left(x_{b}+x_{a}\right)\right] $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)_{\omega}^{x_{0}}=\left(x_{b} t_{b} \mid x_{a} t_{a}\right) \frac{t_{b}-t_{a}}{\sqrt{2 \pi \hbar}} \sqrt{\frac{i M \omega^{3}}{\omega\left(t_{b}-t_{a}\right)-2 \tan \frac{\omega\left(t_{b}-t_{a}\right)}{2}}} \exp \left(\frac{i}{\hbar} \mathcal{A}^{x_{0}}\right) . $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)_{\omega}^{x_{0}}=\frac{\sqrt{3} M}{\pi \hbar i\left(t_{b}-t_{a}\right)} \exp \left\{\frac{M i}{2 \hbar\left(t_{b}-t_{a}\right)}\left[\left(x_{b}-x_{a}\right)^{2}+12\left(x_{0}-\frac{x_{b}+x_{a}}{2}\right)^{2}\right]\right\} . $$

$$ Z_{\omega}^{x_{0}}=\frac{1}{\sqrt{2 \pi \hbar\left(t_{b}-t_{a}\right) / M i}} \frac{\omega\left(t_{b}-t_{a}\right) / 2}{\sin \left[\omega\left(t_{b}-t_{a}\right) / 2\right]} \exp \left[-\frac{i}{2 \hbar}\left(t_{b}-t_{a}\right) M \omega^{2} x_{0}^{2}\right] $$

$$ Z_{\omega}^{\text {open }, x_{0}}=\sqrt{\frac{\omega\left(t_{b}-t_{a}\right)}{\sin \omega\left(t_{b}-t_{a}\right)}} \exp \left[-\frac{i}{2 \hbar}\left(t_{b}-t_{a}\right) M \omega^{2} x_{0}^{2}\right] $$

$$ \left(x_{b} \hbar \beta \mid x_{a} 0\right)_{\omega}^{j}=\int \mathcal{D} x(\tau) \exp \left\{-\frac{1}{\hbar} \int_{0}^{\hbar \beta} d \tau\left[\frac{M}{2}\left(\dot{x}^{2}+\omega^{2} x^{2}\right)-j(\tau) x(\tau)\right]\right\} $$

$$ \left(x_{b} \hbar \beta \mid x_{a} 0\right)_{\omega}^{j}=\sqrt{\frac{M}{2 \pi \hbar^{2} \beta}} \sqrt{\frac{\omega \hbar \beta}{\sinh \omega \hbar \beta}} \exp \left\{-\frac{1}{\hbar} \mathcal{A}_{\mathrm{e}}^{\mathrm{ext}}[j]\right\} $$

$$ \mathcal{A}_{\mathrm{e}}^{\mathrm{ext}}[j]=\mathcal{A}_{\mathrm{e}}+\mathcal{A}_{\mathrm{e}}^{j}=\mathcal{A}_{\mathrm{e}}+\mathcal{A}_{1, \mathrm{e}}^{j}+\mathcal{A}_{2, \mathrm{e}}^{j} $$

$$ \mathcal{A}_{\mathrm{e}}=\frac{M \omega}{2 \sinh \beta \hbar \omega}\left[\left(x_{b}^{2}+x_{a}^{2}\right) \cosh \omega \hbar \beta-2 x_{b} x_{a}\right] $$

$$ \mathcal{A}_{1, \mathrm{e}}^{j}=-\frac{1}{\sinh \omega \hbar \beta} \int_{\tau_{a}}^{\tau_{b}} d \tau\left[x_{a} \sinh \omega(\hbar \beta-\tau)+x_{b} \sinh \omega \tau\right] j(\tau) $$

$$ \mathcal{A}_{2, \mathrm{e}}^{j}=-\frac{1}{M} \int_{0}^{\hbar \beta} d \tau \int_{0}^{\tau} d \tau^{\prime} j(\tau) G_{\omega^{2}, \mathrm{e}}\left(\tau, \tau^{\prime}\right) j\left(\tau^{\prime}\right) $$

$$ \begin{align*} G_{\omega^{2}, \mathrm{e}}\left(\tau, \tau^{\prime}\right) & =\frac{\sinh \omega\left(\hbar \beta-\tau_{>}\right) \sinh \omega \tau_{<}}{\omega \sinh \omega \hbar \beta} \\ & =\frac{\cosh \omega\left(\hbar \beta-\left|\tau-\tau^{\prime}\right|\right)-\cosh \omega\left(\hbar \beta-\tau-\tau^{\prime}\right)}{2 \omega \sinh \omega \hbar \beta} \end{align*} $$

$$ \left(-\partial_{\tau}^{2}+\omega^{2}\right) G_{\omega^{2}, \mathrm{e}}\left(\tau, \tau^{\prime}\right)=\delta\left(\tau-\tau^{\prime}\right) $$

$$ G_{\omega^{2}, \mathrm{e}}\left(\tau, \tau^{\prime}\right)=i G_{\omega^{2}}\left(-i \tau,-i \tau^{\prime}\right) $$

$$ \mathcal{A}_{1, \mathrm{e}}^{j}=-\frac{M \omega}{\sinh \omega \hbar \beta}\left\{\left[x_{b}\left(e^{-\beta \hbar \omega} A_{\mathrm{e}}-B_{\mathrm{e}}\right)\right] x_{a}\left(e^{-\beta \hbar \omega} B_{\mathrm{e}}-A_{\mathrm{e}}\right)\right\} $$

$$ \begin{align*} \mathcal{A}_{2, \mathrm{e}}^{j}=-\frac{1}{4 M \omega} \int_{0}^{\hbar \beta} d \tau & \int_{0}^{\hbar \beta} d \tau^{\prime} e^{-\omega\left|\tau-\tau^{\prime}\right|} j(\tau) j\left(\tau^{\prime}\right) \\ & +\frac{M \omega}{2 \sinh \omega \hbar \beta}\left[e^{\beta \hbar \omega}\left(A_{\mathrm{e}}^{2}+B_{\mathrm{e}}^{2}\right)-2 A_{\mathrm{e}} B_{\mathrm{e}}\right] \end{align*} $$

$$ \begin{align*} \left.A_{\mathrm{e}}(\omega) \equiv i A(\omega)\right|_{t_{b}-t_{a}=-i \hbar \beta} & =\frac{1}{M \omega} \int_{0}^{\hbar \beta} d \tau e^{-\omega \tau} j(\tau) \\ \left.B_{\mathrm{e}}(\omega) \equiv i B(\omega)\right|_{t_{b}-t_{a}=-i \hbar \beta} & =\frac{1}{M \omega} \int_{0}^{\hbar \beta} d \tau e^{-\omega(\hbar \beta-\tau)} j(\tau)=-e^{-\beta \hbar \omega} A_{\mathrm{e}}(-\omega) \end{align*} $$

$$ \mathcal{A}_{\mathrm{e}}=\frac{M \omega}{\sinh \beta \hbar \omega} 2 \sinh ^{2}(\omega \hbar \beta / 2) x^{2} . $$

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

$$ \mathcal{A}_{\mathrm{e}}^{j}=\mathcal{A}_{\mathrm{fl}, \mathrm{e}}^{j}+\mathcal{A}_{\mathrm{r}, \mathrm{e}}^{j}, $$

$$ \mathcal{A}_{\mathrm{r}, \mathrm{e}}^{j}=-\frac{M \omega}{2 \sinh \omega \beta} e^{\beta \hbar \omega}\left(A_{\mathrm{e}}+B_{\mathrm{e}}\right)^{2} $$

$$ \mathcal{A}_{\mathrm{fl}, \mathrm{e}}^{j}+\mathcal{A}_{\mathrm{r}, \mathrm{e}}^{j}=-\frac{1}{4 M \omega} \int_{0}^{\hbar \beta} d \tau \int_{0}^{\hbar \beta} d \tau^{\prime} e^{-\omega\left|\tau-\tau^{\prime}\right|} j(\tau) j\left(\tau^{\prime}\right)-\frac{M \omega}{\sinh (\beta \hbar \omega / 2)} e^{\beta \hbar \omega / 2} A_{\mathrm{e}} B_{\mathrm{e}} $$

$$ \mathcal{A}_{\mathrm{e}}^{j}=-\frac{1}{4 M \omega} \int_{0}^{\hbar \beta} d \tau \int_{0}^{\hbar \beta} d \tau^{\prime} \frac{\cosh \omega\left(\left|\tau-\tau^{\prime}\right|-\hbar \beta / 2\right)}{\sinh (\beta \hbar \omega / 2)} j(\tau) j\left(\tau^{\prime}\right) $$

$$ \begin{align*} G_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}(\tau) & \left.\equiv i G_{\omega^{2}}^{\mathrm{p}}(-i \tau)\right|_{t_{b}-t_{a}=-i \hbar \beta} \\ & =\frac{1}{2 \omega} \frac{\cosh \omega(\tau-\hbar \beta / 2)}{\sinh (\beta \hbar \omega / 2)}, \quad \tau \in[0, \hbar \beta] \end{align*} $$

$$ Z_{\omega}[j]=Z_{\omega} \exp \left(-\frac{1}{\hbar} \mathcal{A}_{\mathrm{e}}^{j}\right) $$

$$ Z_{\omega}^{\mathrm{open}}[j]=\sqrt{\frac{2 \pi \hbar}{M \omega}} \frac{1}{\sqrt{\sinh \left[\omega\left(\tau_{b}-\tau_{a}\right)\right]}} e^{-\left(\mathcal{A}_{2, \mathrm{e}}^{j}+\tilde{\mathcal{A}}_{2, \mathrm{e}}^{j}\right) / \hbar} $$

$$ \tilde{\mathcal{A}}_{2, \mathrm{e}}^{j}=-\frac{1}{M} \int_{0}^{\hbar \beta} d \tau \int_{0}^{\tau} d \tau^{\prime} j(\tau) \tilde{G}_{\omega^{2}}\left(\tau, \tau^{\prime}\right) j\left(\tau^{\prime}\right) $$

$$ \begin{array}{r} \tilde{G}_{\omega^{2}}\left(\tau, \tau^{\prime}\right)=\frac{1}{2 \omega \sinh ^{3} \omega \hbar \beta}\left\{\cosh \omega \hbar \beta\left[\sinh \omega(\hbar \beta-\tau) \sinh \omega\left(\hbar \beta-\tau^{\prime}\right)+\sinh \omega \tau \sinh \omega \tau^{\prime}\right]\right. \\ \left.+\sinh \omega(\hbar \beta-\tau) \sinh \omega \tau^{\prime}+\sinh \omega\left(\hbar \beta-\tau^{\prime}\right) \sinh \omega \tau\right\} .(3.226) \end{array} $$

$$ \tilde{G}_{\omega^{2}}\left(\tau, \tau^{\prime}\right)=\frac{1}{\omega} \frac{\cosh \omega\left(\hbar \beta-\tau-\tau^{\prime}\right)}{\sinh \omega \hbar \beta} $$

$$ \begin{align*} & \sinh \omega \tau\left[\cosh \omega \hbar \beta \sinh \omega \tau^{\prime}+\sinh \omega\left(\hbar \beta-\tau^{\prime}\right)\right] \\ + & \sinh \omega\left(\hbar \beta-\tau^{\prime}\right)[\cosh \omega \hbar \beta \sinh \omega(\hbar \beta-\tau)+\sinh \omega(\hbar \beta-((\hbar \beta-\tau))] \end{align*} $$

$$ \sinh \omega \hbar \beta\left[\sinh \omega \tau \cosh \omega \tau^{\prime}+\sinh \omega(\hbar \beta-\tau) \cosh \omega\left(\hbar \beta-\tau^{\prime}\right)\right] $$

$$ \frac{1}{2}\left[\sinh \omega\left(\tau+\tau^{\prime}\right)+\sinh \omega\left(\tau-\tau^{\prime}\right)+\sinh \omega\left(2 \hbar \beta-\tau-\tau^{\prime}\right)+\sinh \omega\left(\tau^{\prime}-\tau\right)\right] $$

$$ \frac{1}{2}\left[\sinh \omega\left(\hbar \beta+\tau+\tau^{\prime}-\hbar \beta\right)+\sinh \omega\left(\hbar \beta+\hbar \beta-\tau-\tau^{\prime}\right)\right] $$

$$ \frac{1}{2}\left[2 \sinh \omega \hbar \beta \cosh \omega\left(\hbar \beta-\tau-\tau^{\prime}\right)\right] $$

$$ \left(\mathcal{A}_{2, \mathrm{e}}^{j}+\tilde{\mathcal{A}}_{2, \mathrm{e}}^{j}\right)=-\frac{1}{M} \int_{0}^{\hbar \beta} d \tau \int_{0}^{\tau} d \tau^{\prime} j(\tau) G_{\omega^{2}, \mathrm{e}}^{\mathrm{open}}\left(\tau, \tau^{\prime}\right) j\left(\tau^{\prime}\right) $$

$$ \begin{align*} G_{\omega^{2}, \mathrm{e}}^{\mathrm{open}}\left(\tau, \tau^{\prime}\right) & =\frac{\cosh \omega\left(\hbar \beta-\left|\tau-\tau^{\prime}\right|\right)+\cosh \omega\left(\hbar \beta-\tau-\tau^{\prime}\right)}{2 \omega \sinh \omega \hbar \beta} \\ & =\frac{\cosh \omega\left(\hbar \beta-\tau_{>}\right) \cosh \omega \tau_{<}}{\omega \sinh \omega \hbar \beta} \end{align*} $$

$$ G_{\omega^{2}, \mathrm{e}}^{\mathrm{open}}\left(\tau, \tau^{\prime}\right) \underset{\omega^{2} \approx 0}{\approx} \frac{1}{\beta \omega^{2}}+\frac{\beta}{3}-\frac{1}{2}\left|\tau-\tau^{\prime}\right|-\frac{1}{2}\left(\tau+\tau^{\prime}\right)+\frac{1}{2 \beta}\left(\tau^{2}+\tau^{\prime 2}\right), $$

$$ Z_{\omega}[j]=\int \mathcal{D} x(\tau) e^{-\mathcal{A}_{\mathrm{e}}[j] / \hbar} $$

$$ \mathcal{A}_{\mathrm{e}}[j]=\int_{0}^{\hbar \beta} d \tau\left[\frac{M}{2}\left(\dot{x}^{2}+\omega^{2} x^{2}\right)-j(\tau) x(\tau)\right] $$

$$ \mathcal{A}_{\mathrm{e}}[j]=\int_{0}^{\hbar \beta} d \tau\left[\frac{M}{2} x(\tau)\left(-\partial_{\tau}^{2}+\omega^{2}\right) x(\tau)-j(\tau) x(\tau)\right] $$

$$ D_{\omega^{2}, \mathrm{e}}\left(\tau, \tau^{\prime}\right) \equiv\left(-\partial_{\tau}^{2}+\omega^{2}\right) \delta\left(\tau-\tau^{\prime}\right), \quad \tau-\tau^{\prime} \in[0, \hbar \beta] . $$

$$ G_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}\left(\tau, \tau^{\prime}\right)=G_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}\left(\tau-\tau^{\prime}\right)=D_{\omega^{2}, \mathrm{e}}^{-1}\left(\tau, \tau^{\prime}\right)=\left(-\partial_{\tau}^{2}+\omega^{2}\right)^{-1} \delta\left(\tau-\tau^{\prime}\right) $$

$$ x \rightarrow x^{\prime}=x-\frac{1}{M} G_{\omega^{2}, \mathrm{e}}^{P} j $$

$$ \mathcal{A}_{\mathrm{e}}[j]=\int_{0}^{\hbar \beta} d \tau \frac{M}{2} x^{\prime}\left(-\partial_{\tau}^{2}+\omega^{2}\right) x^{\prime}-\frac{1}{2 M} \int_{0}^{\hbar \beta} d \tau \int_{0}^{\hbar \beta} d \tau^{\prime} j(\tau) G_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}\left(\tau-\tau^{\prime}\right) j\left(\tau^{\prime}\right) $$

$$ Z_{\omega}=\operatorname{Det} D_{\omega^{2}, \mathrm{e}}^{-1 / 2} $$

$$ \operatorname{Det} D_{\omega^{2}, \mathrm{e}}=\prod_{m=-\infty}^{\infty}\left(\omega_{m}^{2}+\omega^{2}\right)=\exp \left[\sum_{m=-\infty}^{\infty} \log \left(\omega_{m}^{2}+\omega^{2}\right)\right] $$

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

$$ Z[j]=Z_{\omega} \exp \left\{-\frac{1}{\hbar} \mathcal{A}_{\mathrm{e}}^{j}[j]\right\} $$

$$ \mathcal{A}_{\mathrm{e}}^{j}[j]=-\frac{1}{2 M} \int_{0}^{\hbar \beta} d \tau \int_{0}^{\hbar \beta} d \tau^{\prime} j(\tau) G_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}\left(\tau-\tau^{\prime}\right) j\left(\tau^{\prime}\right) $$

$$ G_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}(\tau)=\frac{1}{\hbar \beta} \sum_{m=-\infty}^{\infty} \frac{1}{\omega_{m}^{2}+\omega^{2}} e^{-i \omega_{m} \tau} $$

$$ G_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}(\tau) \underset{T=0}{=} \int \frac{d \omega_{m}}{2 \pi} \frac{1}{\omega_{m}^{2}+\omega^{2}} e^{-i \omega_{m} \tau}=\frac{1}{2 \omega} e^{-\omega|\tau|} $$

$$ \sum_{\bar{m}=-\infty}^{\infty} \delta(m-\bar{m})=\sum_{n=-\infty}^{\infty} e^{i 2 \pi n m}=\sum_{n=-\infty}^{\infty} e^{i n \omega_{m} \hbar \beta} $$

$$ \begin{align*} G_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}(\tau) & =\sum_{n=-\infty}^{\infty} \frac{1}{2 \omega} e^{-\omega|\tau+n \hbar \beta|} \\ & =\frac{1}{2 \omega} \frac{\cosh \omega(\tau-\hbar \beta / 2)}{\sinh (\beta \hbar \omega / 2)}, \quad \tau \in[0, \hbar \beta] \end{align*} $$

$$ G_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}(\tau)=\frac{1}{\hbar \beta \omega^{2}}+\frac{\tau^{2}}{2 \hbar \beta}-\frac{\tau}{2}+\frac{\hbar \beta}{12}+\ldots $$

$$ G_{\omega^{2}, \mathrm{e}}^{\mathrm{p} \prime}(\tau) \equiv G_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}(\tau)-\frac{1}{\hbar \beta \omega^{2}} $$

$$ G_{0, \mathrm{e}}^{\mathrm{p} /}(\tau)=\frac{1}{\hbar \beta} \sum_{m= \pm 1, \pm 2, \ldots} \frac{1}{\omega_{m}^{2}} e^{-i \omega_{m} \tau}=\frac{\tau^{2}}{2 \hbar \beta}-\frac{\tau}{2}+\frac{\hbar \beta}{12}, $$

$$ \frac{1}{\hbar \beta} \sum_{m= \pm 1, \pm 2, \ldots} \frac{(-1)^{m}}{\omega_{m}^{2}} e^{-i \omega_{m}(\tau-\hbar \beta / 2)} $$

$$ -\frac{2}{\hbar \beta}\left(\frac{\hbar \beta}{2 \pi}\right)^{2} \sum_{n=0,2,4, \ldots} \frac{1}{n!}\left[-i \frac{2 \pi}{\hbar \beta}(\tau-\hbar \beta / 2)\right]^{n} \sum_{m=1}^{\infty} \frac{(-1)^{m-1}}{m^{2-n}} . $$

$$ \eta(z) \equiv \sum_{m=1}^{\infty} \frac{(-1)^{m-1}}{m^{z}} $$

$$ \eta(z)=\left(1-2^{1-z}\right) \zeta(z) $$

$$ \eta(0)=-\zeta(0)=1 / 2, \quad \eta(2)=\zeta(2) / 2=\pi^{2} / 12 $$

$$ -\frac{2}{\hbar \beta}\left(\frac{\hbar \beta}{2 \pi}\right)^{2}\left[\frac{\pi^{2}}{12}-\frac{1}{4}\left(\frac{2 \pi}{\hbar \beta}\right)^{2}(\tau-\hbar \beta / 2)^{2}\right]=\frac{\tau^{2}}{2 \hbar \beta}-\frac{\tau}{2}+\frac{\hbar \beta}{12} $$

$$ G_{0, \mathrm{e}}^{\mathrm{p} \prime}(\tau)=\frac{\hbar \beta}{2} B_{2}(\tau / \hbar \beta) $$

$$ B_{n}(x)=\sum_{k=0}^{n}\binom{n}{k} B_{k} z^{n-k} $$

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

$$ B_{2 n}(z)=(-1)^{n-1} \frac{2(2 n)!}{(2 \pi)^{2 n}} \sum_{k=0}^{\infty} \frac{\cos (2 \pi k z)}{k^{2 n}} $$

$$ B_{1}(z)=z-1 / 2, \quad B_{2}(z)=z^{2}-z+1 / 6, \ldots $$

$$ \begin{align*} G_{\omega^{2}, \mathrm{e}}^{\mathrm{a}}(\tau) & =\sum_{n=-\infty}^{\infty} \frac{(-1)^{n}}{2 \omega} e^{-\omega|\tau+n \hbar \beta|} \\ & =\frac{1}{2 \omega} \frac{\sinh \omega(\tau-\hbar \beta / 2)}{\cosh (\beta \hbar \omega / 2)}, \quad \tau \in[0, \hbar \beta] \end{align*} $$

$$ G_{\omega^{2}, \mathrm{e}}^{\mathrm{a}}(\tau)=\frac{\tau}{2}-\frac{\hbar \beta}{4}, \quad \tau \in[0, \hbar \beta] $$

$$ G_{\omega^{2}, \mathrm{e}}^{\mathrm{a}}(\tau) \equiv \frac{1}{\hbar \beta} \sum_{m=-\infty}^{\infty} \frac{1}{\omega_{m}^{\mathrm{f} 2}} e^{-i \omega_{m}^{\mathrm{f}} \tau} $$

$$ \frac{1}{\hbar \beta} \sum_{m=-\infty}^{\infty} \frac{1}{\omega_{m}^{\mathrm{f} 2}} \cos \left(\omega_{m}^{\mathrm{f}} \tau\right)=\frac{1}{\hbar \beta} \sum_{m=-\infty}^{\infty} \frac{(-1)^{m}}{\omega_{m}^{\mathrm{f} 2}} \sin \left[\omega_{m}^{\mathrm{f}}(\tau-\hbar \beta / 2)\right] . $$

$$ \frac{2}{\hbar \beta}\left(\frac{\hbar \beta}{2 \pi}\right)^{2} \sum_{n=1,3,5, \ldots} \frac{(-1)^{(n-1) / 2}}{n!}\left[\frac{2 \pi}{\hbar \beta}(\tau-\hbar \beta / 2)\right]^{n} \sum_{m=0}^{\infty} \frac{(-1)^{m}}{\left(m+\frac{1}{2}\right)^{2-n}} $$

$$ \beta(z) \equiv \frac{1}{2^{z}} \sum_{m=0}^{\infty} \frac{(-1)^{m}}{\left(m+\frac{1}{2}\right)^{z}} $$

$$ \zeta(z, q) \equiv \sum_{m=0}^{\infty} \frac{1}{(m+q)^{z}} $$

$$ \sum_{m=0}^{\infty} \frac{(-1)^{m}}{(m+q)^{z}}=\zeta(z, q)-2^{2-z} \zeta(z,(q+1) / 2) $$

$$ \beta(z) \equiv \frac{1}{2^{z}}\left[\zeta(z, 1 / 2)-2^{2-z} \zeta(z, 3 / 4)\right] $$

$$ \zeta(z, q)=\frac{1}{z-1}-\psi(q)+\mathcal{O}(z-1) $$

$$ \beta(1)=\lim _{z \rightarrow 1} \frac{1}{2}\left[\zeta(z, 1 / 2)-2^{1-z} \zeta(z, 3 / 4)\right]=\frac{1}{2}[-\psi(1 / 2)+\psi(3 / 4)+\log 2]=\frac{\pi}{4} $$

$$ \psi(1 / 2)=-\gamma-2 \log 2, \quad \psi(3 / 4)=-\gamma-3 \log 2+\frac{\pi}{2} $$

$$ \beta(1)=\frac{1}{4}[\psi(3 / 2)-\psi(1 / 4)] $$

$$ \mathcal{A}_{\mathrm{e}}^{j}[j]=-\frac{1}{4 M \omega} \int_{0}^{\hbar \beta} d \tau \int_{-\infty}^{\infty} d \tau^{\prime} e^{-\omega\left|\tau-\tau^{\prime}\right|} j(\tau) j\left(\tau^{\prime}\right) $$

$$ G_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}(\tau)=\frac{1}{2 \omega} \sum_{n=-\infty}^{\infty} e^{-\omega|\tau+n \hbar \beta|} $$

$$ \left[\partial_{\tau}^{2}-\Omega^{2}(\tau)\right] G_{\omega^{2}, \mathrm{e}}^{\mathrm{p}, \mathrm{a}}\left(\tau, \tau^{\prime}\right)=\delta^{\mathrm{p}, \mathrm{a}}\left(\tau-\tau^{\prime}\right) $$

$$ \delta^{\mathrm{p}, \mathrm{a}}\left(\tau-\tau^{\prime}\right)=\sum_{n=-\infty}^{\infty} \delta\left(\tau-\tau^{\prime}-n \hbar \beta\right)\left\{\begin{array}{c} 1 \\ (-1)^{n} \end{array}\right\} $$

$$ G_{\omega^{2}, \mathrm{e}}^{\mathrm{p}, \mathrm{a}}\left(\tau, \tau^{\prime}\right)=G_{\omega^{2}, \mathrm{e}}\left(\tau, \tau^{\prime}\right) \mp \frac{\left[\Delta\left(\tau, \tau_{a}\right) \pm \Delta\left(\tau_{b}, \tau\right)\right]\left[\Delta\left(\tau^{\prime}, \tau_{a}\right) \pm \Delta\left(\tau_{b}, \tau^{\prime}\right)\right]}{\bar{\Delta}^{\mathrm{p}, \mathrm{a}}\left(\tau_{a}, \tau_{b}\right) \Delta\left(\tau_{a}, \tau_{b}\right)} $$

$$ G_{\omega^{2}, \mathrm{e}}\left(\tau, \tau^{\prime}\right)=\frac{\bar{\Theta}\left(\tau-\tau^{\prime}\right) \Delta\left(\tau_{b}, \tau\right) \Delta\left(\tau^{\prime}, \tau_{a}\right)+\bar{\Theta}\left(\tau-\tau^{\prime}\right) \Delta\left(\tau_{b}, \tau^{\prime}\right) \Delta\left(\tau, \tau_{a}\right)}{\Delta\left(\tau_{a}, \tau_{b}\right)} $$

$$ \Delta\left(\tau, \tau^{\prime}\right)=\frac{1}{W}\left[\xi(\tau) \eta\left(\tau^{\prime}\right)-\xi\left(\tau^{\prime}\right) \eta(\tau)\right], \quad W=\xi(\tau) \dot{\eta}(\tau)-\dot{\xi}(\tau) \eta(\tau), $$

$$ \bar{\Delta}^{\mathrm{p}, \mathrm{a}}\left(\tau_{a}, \tau_{b}\right)=2 \pm \partial_{\tau} \Delta\left(\tau_{a}, \tau_{b}\right) \pm \partial_{\tau} \Delta\left(\tau_{b}, \tau_{a}\right) $$

$$ \begin{align*} G_{\omega, \mathrm{e}}^{\mathrm{p}}(\tau) & =\frac{1}{\hbar \beta} \sum_{m} e^{-i \omega_{m} \tau} \frac{-1}{i \omega_{m}-\omega}=e^{-\omega(\tau-\hbar \beta / 2)} \frac{1}{2 \sinh (\beta \hbar \omega / 2)} \\ & =\left(1+n_{\omega}^{\mathrm{b}}\right) e^{-\omega \tau} \end{align*} $$

$$ \begin{align*} G_{\omega, \mathrm{e}}^{\mathrm{a}}(\tau) & =\frac{1}{\hbar \beta} \sum_{m} e^{-i \omega_{m}^{\mathrm{f}} \tau} \frac{-1}{i \omega_{m}^{\mathrm{f}}-\omega}=e^{-\omega(\tau-\hbar \beta / 2)} \frac{1}{2 \cosh (\beta \hbar \omega / 2)} \\ & =\left(1-n_{\omega}^{\mathrm{f}}\right) e^{-\omega \tau} \end{align*} $$

$$ G_{\omega, \mathrm{e}}^{\mathrm{p}, \mathrm{a}}(\tau)= \pm G_{\omega, \mathrm{e}}^{\mathrm{p}, \mathrm{a}}(\tau+\hbar \beta) $$

$$ \left[-\partial_{\tau}-\Omega(\tau)\right] G_{\Omega, \mathrm{e}}^{\mathrm{p}, \mathrm{a}}\left(\tau, \tau^{\prime}\right)=\delta^{\mathrm{p}, \mathrm{a}}\left(\tau-\tau^{\prime}\right) $$

$$ G_{\Omega, \mathrm{e}}^{\mathrm{p}, \mathrm{a}}\left(\tau, \tau^{\prime}\right)=\bar{\Theta}\left(\tau-\tau^{\prime}\right) e^{-\int_{0}^{\tau} d \tau^{\prime} \Omega\left(\tau^{\prime}\right)} $$

$$ G_{\Omega, \mathrm{e}}^{\mathrm{p}, \mathrm{a}}\left(\tau, \tau^{\prime}\right)=\sum_{n=0}^{\infty} e^{-\int_{0}^{\tau+n \hbar \beta} d \tau^{\prime} \Omega\left(\tau^{\prime}\right)}\left\{\begin{array}{c} 1 \\ (-1)^{n} \end{array}\right\}, \quad \hbar \beta>\tau>\tau^{\prime}>0 $$

$$ G_{\omega^{2}}\left(\tau, \tau^{\prime}\right)=\frac{\epsilon}{2 \sinh \epsilon \tilde{\omega}_{\mathrm{e}}} e^{-\tilde{\omega}_{\mathrm{e}}\left|\tau-\tau^{\prime}\right|}=\frac{1}{2 \omega} \frac{1}{\cosh \left(\epsilon \tilde{\omega}_{\mathrm{e}} / 2\right)} e^{-\tilde{\omega}_{\mathrm{e}}\left|\tau-\tau^{\prime}\right|} $$

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

$$ G_{\omega^{2}}\left(\tau, \tau^{\prime}\right)=G_{\omega^{2}}\left(\tau-\tau^{\prime}\right)=\int \frac{d \omega^{\prime}}{2 \pi} e^{-i \omega^{\prime}\left(\tau-\tau^{\prime}\right)} \frac{\epsilon^{2}}{2\left(1-\cos \epsilon \omega^{\prime}\right)+\epsilon^{2} \omega^{2}} $$

$$ G_{\omega^{2}}\left(\tau, \tau^{\prime}\right)=\int_{0}^{\infty} d s \int \frac{d \omega^{\prime}}{2 \pi} e^{-i \omega^{\prime} \epsilon n} e^{-s\left[2\left(1-\cos \epsilon \omega^{\prime}\right)+\epsilon^{2} \omega^{2}\right] / \epsilon^{2}} $$

$$ \begin{align*} G_{\mathrm{e}}^{\mathrm{p}}(\tau) & =\frac{1}{\hbar \beta} \sum_{m=-\infty}^{\infty} \frac{\epsilon^{2}}{2\left(1-\cos \epsilon \omega_{m}\right)+\epsilon^{2} \omega^{2}} e^{-i \omega_{m} \tau} \\ & =\frac{1}{2 \omega} \frac{1}{\cosh (\epsilon \tilde{\omega} / 2)} \frac{\cosh \tilde{\omega}(\tau-\hbar \beta / 2)}{\sinh (\hbar \tilde{\omega} \beta / 2)}, \quad \tau \in[0, \hbar \beta] \end{align*} $$

$$ \begin{align*} G_{\omega^{2}}^{(n)}\left(\tau_{1}, \ldots, \tau_{n}\right) & \equiv\left\langle x\left(\tau_{1}\right) x\left(\tau_{2}\right) \cdots x\left(\tau_{n}\right)\right\rangle \\ & \equiv Z^{-1} \int \mathcal{D} x x\left(\tau_{1}\right) x\left(\tau_{2}\right) \cdots x\left(\tau_{n}\right) \exp \left(-\frac{1}{\hbar} \mathcal{A}_{\mathrm{e}}\right) \end{align*} $$

$$ G_{\omega^{2}}^{(n)}\left(\tau_{1}, \ldots, \tau_{n}\right)=Z^{-1} \operatorname{Tr}\left\{\hat{T}_{\tau}\left[\hat{x}_{H}\left(\tau_{1}\right) \hat{x}_{H}\left(\tau_{2}\right) \cdots \hat{x}_{H}\left(\tau_{n}\right) e^{-\hat{H} / k_{B} T}\right]\right\}, $$

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

$$ \begin{align*} & \left.G_{\omega^{2}}^{(n)}\left(\tau_{1}, \ldots, \tau_{n}\right)=\prod_{i=1}^{n+1}\left[\int_{-\infty}^{\infty} d x_{\tau_{t(i)}}\right]\left(x_{t(n+1)} \tau_{b}\right) \mid x_{t(n)} \tau_{t(n)}\right) \cdot x\left(\tau_{t(n)}\right) \cdot \ldots \\ & \quad \cdot\left(x_{t(i+1)} \tau_{t(i+1)} \mid x_{t(i)} \tau_{t(i)}\right) \cdot x\left(\tau_{t(i)}\right) \cdot\left(x_{t(i)} \tau_{t(i)} \mid x_{t(i-1)} \tau_{t(i-1)}\right) \cdot x\left(\tau_{t(i-1)}\right) \\ & \quad \cdot \ldots \cdot\left(x_{t(2)} \tau_{t(2)} \mid x_{t(1)} \tau_{t(1)}\right) \cdot x\left(\tau_{t(1)}\right) \cdot\left(x_{t(1)} \tau_{t(1)} \mid x_{t(0)} \tau_{a}\right) \end{align*} $$

$$ G_{\omega^{2}}^{(n)}\left(\tau_{1}, \ldots, \tau_{n}\right)=\left[Z[j]^{-1} \hbar \frac{\delta}{\delta j\left(\tau_{1}\right)} \cdots \hbar \frac{\delta}{\delta j\left(\tau_{n}\right)} Z[j]\right]_{j=0} $$

$$ G_{\omega^{2}}^{(n)}\left(\tau_{1}, \ldots, \tau_{n}\right)=\left[\hbar \frac{\delta}{\delta j\left(\tau_{1}\right)} \cdots \hbar \frac{\delta}{\delta j\left(\tau_{n}\right)}\right. $$

$$ G_{\omega^{2}}^{(2)}\left(\tau, \tau^{\prime}\right)=\left\langle x(\tau) x\left(\tau^{\prime}\right)\right\rangle=\frac{\hbar}{M} G_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}\left(\tau-\tau^{\prime}\right) $$

$$ G_{\omega^{2}}^{(n)}\left(\tau_{1}, \ldots, \tau_{n}\right)=\sum_{\text {pairs }} G_{\omega^{2}}^{(2)}\left(\tau_{p(1)}, \tau_{p(2)}\right) \cdots G_{\omega^{2}}^{(2)}\left(\tau_{p(n-1)}, \tau_{p(n)}\right) . $$

$$ \dot{\tau}_{1} \dot{\tau}_{2} \tau_{3} \tau_{4} \ldots \tau_{n}+\dot{\tau}_{1} \tau_{2} \dot{\tau}_{3} \tau_{4} \ldots \tau_{n}+\dot{\tau}_{1} \tau_{2} \tau_{3} \dot{\tau}_{4} \ldots \tau_{n}+\ldots+\dot{\tau}_{1} \tau_{2} \tau_{3} \tau_{4} \ldots \dot{\tau}_{n}, $$

$$ \left\langle e^{K x}\right\rangle=e^{K^{2}\left\langle x^{2}\right\rangle / 2} $$

$$ Z_{\omega}[j]=Z_{\omega} \times\left\langle e^{\int d \tau j(\tau) x(\tau) / \hbar}\right\rangle $$

$$ \left\langle e^{\int d \tau j(\tau) x(\tau) / \hbar}\right\rangle=e^{(1 / 2 M \hbar) \int d \tau \int d \tau^{\prime} j(\tau) G_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}\left(\tau, \tau^{\prime}\right) j\left(\tau^{\prime}\right)} $$

$$ \left\langle e^{\int d \tau j(\tau) x(\tau) / \hbar}\right\rangle=e^{\int d \tau \int d \tau^{\prime} j(\tau)\left\langle x(\tau) x\left(\tau^{\prime}\right)\right\rangle j\left(\tau^{\prime}\right) / 2 \hbar^{2}} $$

$$ e^{-W} \equiv\left\langle e^{-\nabla \cdot \mathbf{u}(\mathbf{x})}\right\rangle=e^{\left.\left.-\Sigma_{\mathbf{k}}\langle | \mathbf{k} \cdot \mathbf{u}(\mathbf{k})\right]^{2}\right\rangle / 2} $$

$$ \left\langle e^{P x}\right\rangle=e^{P\langle x(\tau)\rangle+P^{2}\langle x-\langle x(\tau)\rangle\rangle^{2} / 2} $$

$$ G_{\omega^{2}}^{(2)}\left(t, t^{\prime}\right)=\left\langle x(t) x\left(t^{\prime}\right)\right\rangle=i \frac{\hbar}{M} G_{\omega^{2}}\left(t-t^{\prime}\right) $$

$$ \left\langle\dot{x}(t) \dot{x}\left(t^{\prime}\right)\right\rangle=i \frac{\hbar}{M} \frac{\cos \omega\left(t_{b}-t_{>}\right) \cos \omega\left(t_{<}-t_{a}\right)}{\omega \sin \omega\left(t_{b}-t_{a}\right)} $$

$$ \left\langle\dot{x}\left(t_{b}\right) \dot{x}\left(t_{b}\right)\right\rangle=i \frac{\hbar}{M} \cot \omega\left(t_{b}-t_{a}\right) $$

$$ i \hbar \partial_{t_{b}}\left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)=-\int \mathcal{D}^{D} x L\left(\mathbf{x}_{b}, \dot{\mathbf{x}}_{b}\right) e^{i \int_{t_{a}}^{t_{b}} d t L(\mathbf{x}, \dot{\mathbf{x}}) / \hbar}=-\left\langle L\left(\mathbf{x}_{b}, \dot{\mathbf{x}}_{b}\right)\right\rangle\left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right) $$

$$ \left\langle L_{\mathrm{fl}}\left(\mathbf{x}_{b}, \dot{\mathbf{x}}_{b}\right)\right\rangle=\frac{M}{2}\left\langle\delta \dot{\mathbf{x}}_{b}^{2}\right\rangle . $$

$$ \left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)=C\left(\mathbf{x}_{b}, \mathbf{x}_{a}\right) e^{i A\left(\mathbf{x}_{b}, \mathbf{x}_{a} ; t_{b}-t_{a}\right) / \hbar} \exp \left(\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t_{b^{\prime}} \frac{M}{2}\left\langle\delta \dot{\mathbf{x}}_{b^{\prime}}^{2}\right\rangle\right) $$

$$ -i \hbar \nabla_{b}\left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)=\left\langle\mathbf{p}_{b}\right\rangle\left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)=\mathbf{p}_{\mathrm{cl}}\left(t_{b}\right)\left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right), $$

$$ \left\langle L_{\mathrm{fl}}\left(\mathbf{x}_{b}, \dot{\mathbf{x}}_{b}\right)\right\rangle=\frac{M}{2}\left\langle\delta \dot{\mathbf{x}}_{b}^{2}\right\rangle=i \frac{\hbar \omega}{2} D \cot \omega\left(t_{b}-t_{a}\right) $$

$$ \mathbf{G}_{\omega^{2}, B}^{(2)}\left(\tau, \tau^{\prime}\right)=\frac{\hbar}{M} \mathbf{D}_{\omega^{2}, B}^{-1}\left(\tau, \tau^{\prime}\right) $$

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

$$ \tilde{\mathbf{G}}_{\omega^{2}, B}^{(2)}\left(\omega_{m}\right)=\frac{\hbar}{M} \frac{1}{\left(\omega_{m}^{2}+\omega_{+}^{2}\right)\left(\omega_{m}^{2}+\omega_{-}^{2}\right)}\left(\begin{array}{cc} \omega_{m}^{2}+\omega^{2}-\omega_{B}^{2} & 2 \omega_{B} \omega_{m} \\ -2 \omega_{B} \omega_{m} & \omega_{m}^{2}+\omega^{2}-\omega_{B}^{2} \end{array}\right) $$

$$ \begin{align*} & \frac{1}{2\left(\omega_{m}^{2}+\omega_{+}^{2}\right)\left(\omega_{m}^{2}+\omega_{-}^{2}\right)}\left[\left(\omega_{m}^{2}+\omega_{+}^{2}\right)+\left(\omega_{m}^{2}+\omega_{-}^{2}\right)-4 \omega_{B}^{2}\right] \\ & \quad=\frac{1}{2}\left\{\left[\frac{1}{\omega_{m}^{2}+\omega_{+}^{2}}+\frac{1}{\omega_{m}^{2}+\omega_{-}^{2}}\right]+\frac{\omega_{B}}{\omega}\left[\frac{1}{\omega_{m}^{2}+\omega_{+}^{2}}-\frac{1}{\omega_{m}^{2}+\omega_{-}^{2}}\right]\right\} \end{align*} $$

$$ G_{\omega^{2}, B, x x}^{(2)}=\frac{\hbar}{4 M \omega}\left[\frac{\cosh \omega_{+}\left(\left|\tau-\tau^{\prime}\right|-\hbar \beta / 2\right)}{\sinh \left(\omega_{+} \hbar \beta / 2\right)}+\frac{\cosh \omega_{-}\left(\left|\tau-\tau^{\prime}\right|-\hbar \beta / 2\right)}{\sinh \left(\omega_{-} \hbar \beta / 2\right)}\right] $$

$$ \frac{2 \omega_{B} \omega_{m}}{\left(\omega_{m}^{2}+\omega_{+}^{2}\right)\left(\omega_{m}^{2}+\omega_{-}^{2}\right)}=\frac{\omega_{m}}{2 \omega}\left[\frac{1}{\omega_{m}^{2}+\omega_{+}^{2}}-\frac{1}{\omega_{m}^{2}+\omega_{-}^{2}}\right] $$

$$ \begin{align*} G_{\omega^{2}, B, x y}^{(2)}\left(\tau, \tau^{\prime}\right)=-G_{\omega^{2}, B, y x}^{(2)}\left(\tau, \tau^{\prime}\right)=\frac{\hbar}{2 M} i \partial_{\tau} & {\left[\frac{1}{2 \omega_{+}} \frac{\cosh \omega_{+}\left(\left|\tau-\tau^{\prime}\right|-\hbar \beta / 2\right)}{\sinh \left(\omega_{+} \hbar \beta / 2\right)}\right.} \\ & \left.-\frac{1}{2 \omega_{-}} \frac{\cosh \omega_{-}\left(\left|\tau-\tau^{\prime}\right|-\hbar \beta / 2\right)}{\sinh \left(\omega_{-} \hbar \beta / 2\right)}\right] \end{align*} $$

$$ \begin{align*} G_{\omega^{2}, B, x y}^{(2)}\left(\tau, \tau^{\prime}\right)=G_{\omega^{2}, B, y x}^{(2)}\left(\tau, \tau^{\prime}\right)=\frac{\hbar \epsilon\left(\tau-\tau^{\prime}\right)}{2 M i} & {\left[\frac{1}{2 \omega_{+}} \frac{\sinh \omega_{+}\left(\left|\tau-\tau^{\prime}\right|-\hbar \beta / 2\right)}{\sinh \left(\omega_{+} \hbar \beta / 2\right)}\right.} \\ & \left.-\frac{1}{2 \omega_{-}} \frac{\sinh \omega_{-}\left(\left|\tau-\tau^{\prime}\right|-\hbar \beta / 2\right)}{\sinh \left(\omega_{-} \hbar \beta / 2\right)}\right] \end{align*} $$

$$ G_{\omega^{2}, B, x x}^{(2) \prime}\left(\tau, \tau^{\prime}\right)=G_{\omega^{2}, B, y y}^{(2) \prime}\left(\tau, \tau^{\prime}\right)=G_{\omega^{2}, B, x x}^{(2)}-\frac{1}{\beta M \omega_{+} \omega_{-}}, $$

$$ G_{\omega^{2}, B, x x}^{(2) \prime}\left(\tau, \tau^{\prime}\right)=G_{\omega^{2}, B, y y}^{(2) \prime}\left(\tau, \tau^{\prime}\right)=\frac{\hbar}{4 M \omega}\left[\frac{\cosh 2 \omega\left(\left|\tau-\tau^{\prime}\right|-\hbar \beta / 2\right)}{\sinh (\beta \hbar \omega)}-\frac{1}{\omega \hbar \beta}\right] . $$

$$ G_{\omega^{2}, B, x y}^{(2) \prime}\left(\tau, \tau^{\prime}\right)=-G_{\omega^{2}, B, y x}^{(2) \prime}\left(\tau, \tau^{\prime}\right)=G_{\omega^{2}, B, x y}^{(2)}+\frac{\hbar \omega_{B}}{2 M i \omega_{+} \omega_{-}} \epsilon\left(\tau-\tau^{\prime}\right) $$

$$ \begin{align*} & G_{\omega^{2}}^{(m, n)}\left(\tau_{1}, \ldots, \tau_{m} ; \tau_{1}, \ldots, \tau_{n}\right) \equiv\left\langle x\left(\tau_{1}\right) x\left(\tau_{2}\right) \cdots x\left(\tau_{m}\right) p\left(\tau_{1}\right) p\left(\tau_{2}\right) \cdots p\left(\tau_{n}\right)\right\rangle \\ & \quad \equiv Z^{-1} \int \mathcal{D} x(\tau) \int \frac{\mathcal{D} p(\tau)}{2 \pi} x\left(\tau_{1}\right) x\left(\tau_{2}\right) \cdots x\left(\tau_{m}\right) p(\tau) p\left(\tau_{1}\right) p\left(\tau_{2}\right) \cdots p\left(\tau_{n}\right) \exp \left(-\frac{1}{\hbar} \mathcal{A}_{\mathrm{e}}\right) \end{align*} $$

$$ Z[j, k]=\int \mathcal{D} x(\tau) e^{-\mathcal{A}_{\mathrm{e}}[j, k] / \hbar} $$

$$ \mathcal{A}_{\mathrm{e}}[j, k]=\int_{0}^{\hbar \beta} d \tau\left[-i p(\tau) \dot{x}(\tau)+\frac{1}{2 M} p^{2}+\frac{M}{2} \omega^{2} x^{2}-j(\tau) x(\tau)-k(\tau) p(\tau)\right] $$

$$ \mathcal{A}_{\mathrm{e}}[\mathbf{J}]=\int_{0}^{\hbar \beta} d \tau\left(\frac{1}{2} \mathbf{V}^{T} \mathbf{D}_{\omega^{2}, \mathrm{e}} \mathbf{V}-\mathbf{V}^{T} \mathbf{J}\right) $$

$$ \mathbf{D}_{\omega^{2}, \mathrm{e}}\left(\tau, \tau^{\prime}\right) \equiv\left(\begin{array}{cc} M \omega^{2} & i \partial_{\tau} \\ -i \partial_{\tau} & M^{-1} \end{array}\right) \delta\left(\tau-\tau^{\prime}\right), \quad \tau-\tau^{\prime} \in[0, \hbar \beta] $$

$$ \begin{align*} \mathbf{G}_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}\left(\tau, \tau^{\prime}\right) & =\mathbf{G}_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}\left(\tau-\tau^{\prime}\right)=\mathbf{D}_{\omega^{2}, \mathrm{e}}^{-1}\left(\tau, \tau^{\prime}\right) \\ & =\left(\begin{array}{cc} M^{-1} & -i \partial_{\tau} \\ i \partial_{\tau} & M \omega^{2} \end{array}\right)\left(-\partial_{\tau}^{2}+\omega^{2}\right)^{-1} \delta\left(\tau-\tau^{\prime}\right) \end{align*} $$

$$ \mathbf{V} \rightarrow \mathbf{V}^{\prime}=\mathbf{V}+\mathbf{G}_{\omega^{2}, \mathrm{e}}^{\mathrm{p}} \mathbf{J} $$

$$ \mathcal{A}_{\mathrm{e}}[\mathbf{J}]=\int_{0}^{\hbar \beta} d \tau \frac{1}{2} \mathbf{V}^{\prime T} \mathbf{D}_{\omega^{2}, \mathrm{e}} \mathbf{V}^{\prime}-\frac{1}{2} \int_{0}^{\hbar \beta} d \tau \int_{0}^{\hbar \beta} d \tau^{\prime} \mathbf{J}^{T}\left(\tau^{\prime}\right) \mathbf{G}_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}\left(\tau-\tau^{\prime}\right) \mathbf{J}\left(\tau^{\prime}\right) $$

$$ Z_{\omega}=\operatorname{Det} \mathbf{D}_{\omega^{2}, \mathrm{e}}^{-1 / 2} $$

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

$$ \mathbf{D}_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}\left(\omega_{m}\right)=\left(\begin{array}{cc} M^{-1} & \omega_{m} \\ -\omega_{m} & M \omega^{2} \end{array}\right) $$

$$ \operatorname{det} \mathbf{D}_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}\left(\omega_{m}\right)=\omega_{m}^{2}+\omega^{2} $$

$$ \mathbf{G}_{\mathrm{e}}^{\mathrm{p}}\left(\omega_{m}\right)=\left[\mathbf{D}_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}\left(\omega_{m}\right)\right]^{-1}=\left(\begin{array}{cc} M \omega^{2} & -\omega_{m} \\ \omega_{m} & M^{-1} \end{array}\right) \frac{1}{\omega_{m}^{2}+\omega^{2}} $$

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

$$ Z[\mathbf{J}]=Z_{\omega} \exp \left\{-\frac{1}{\hbar} \mathcal{A}_{\mathrm{e}}^{\mathbf{J}}[\mathbf{J}]\right\} $$

$$ \mathcal{A}_{\mathrm{e}}^{\mathbf{J}}[\mathbf{J}]=-\frac{1}{2} \int_{0}^{\hbar \beta} d \tau \int_{0}^{\hbar \beta} d \tau^{\prime} \mathbf{J}^{T}(\tau) \mathbf{G}_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}\left(\tau, \tau^{\prime}\right) \mathbf{J}\left(\tau^{\prime}\right) $$

$$ \mathbf{G}_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}\left(\tau, \tau^{\prime}\right)=\mathbf{G}_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}\left(\tau-\tau^{\prime}\right)=\mathbf{D}_{\omega^{2}, \mathrm{e}}^{-1}\left(\tau, \tau^{\prime}\right)=\left(\begin{array}{cc} M^{-1} & -i \partial_{\tau} \\ i \partial_{\tau} & M \omega^{2} \end{array}\right) G_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}\left(\tau-\tau^{\prime}\right), $$

$$ \begin{align*} G_{\omega^{2}, \mathrm{e}, x x}^{(2)}\left(\tau, \tau^{\prime}\right) & \equiv\left\langle x(\tau) x\left(\tau^{\prime}\right)\right\rangle=\frac{\hbar}{M} G_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}\left(\tau-\tau^{\prime}\right) \\ G_{\omega^{2}, \mathrm{e}, x p}^{(2)}\left(\tau, \tau^{\prime}\right) & \equiv\left\langle x(\tau) p\left(\tau^{\prime}\right)\right\rangle=-i \hbar \dot{G}_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}\left(\tau-\tau^{\prime}\right) \\ G_{\omega^{2}, \mathrm{e}, p x}^{(2)}\left(\tau, \tau^{\prime}\right) & \equiv\left\langle p(\tau) x\left(\tau^{\prime}\right)\right\rangle=i \hbar \dot{G}_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}\left(\tau-\tau^{\prime}\right) \\ G_{\omega^{2}, \mathrm{e}, p p}^{(2)}\left(\tau, \tau^{\prime}\right) & \equiv\left\langle p(\tau) p\left(\tau^{\prime}\right)\right\rangle=\hbar M \omega^{2} G_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}\left(\tau-\tau^{\prime}\right) \end{align*} $$

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

$$ \left\langle\dot{x}(\tau) \dot{x}\left(\tau^{\prime}\right)\right\rangle=-\hbar M \partial_{\tau}^{2} G_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}\left(\tau-\tau^{\prime}\right) $$

$$ \begin{align*} \left\langle p(\tau) p\left(\tau^{\prime}\right)\right\rangle & =\left\langle\dot{x}(\tau) \dot{x}\left(\tau^{\prime}\right)\right\rangle+\frac{\hbar}{M}\left(-\partial_{\tau}^{2}+\omega^{2}\right) G_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}\left(\tau-\tau^{\prime}\right) \\ & =\left\langle\dot{x}(\tau) \dot{x}\left(\tau^{\prime}\right)\right\rangle+\frac{\hbar}{M} \delta\left(\tau-\tau^{\prime}\right) \end{align*} $$

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

$$ \begin{align*} G_{\omega^{2}, B, x p_{x}}^{(2)}\left(\tau, \tau^{\prime}\right) \equiv & \left\langle x(\tau) p_{x}\left(\tau^{\prime}\right)\right\rangle= \\ G_{\omega^{2}, B, x p_{y}}^{(2)}\left(\tau, \tau^{\prime}\right) \equiv\left\langle x(\tau) p_{y}\left(\tau^{\prime}\right)\right\rangle= & i M \partial_{\tau^{\prime}} G_{\omega^{2}, B, x y}^{(2)}\left(\tau, \tau^{\prime}\right)+M \omega_{B} G_{\omega^{2}, B, x x}^{(2)}\left(\tau, \tau^{\prime}\right), \\ G_{\omega^{2}, B, z p_{z}}^{(2)}\left(\tau, \tau^{\prime}\right) \equiv & \left\langle z(\tau) p_{z}\left(\tau^{\prime}\right)\right\rangle= \\ G_{\omega^{2}, B, p_{x} p_{x}}^{(2)}\left(\tau, \tau^{\prime}\right) \equiv\left\langle p_{x}(\tau) p_{x}\left(\tau^{\prime}\right)\right\rangle= & -M_{\omega^{2}, B, z z}^{(2)} \partial_{\tau} \partial_{\tau^{\prime}} G_{\omega^{2}, B, x x}^{(2)}\left(\tau, \tau^{\prime}\right), \\ & +M^{2} \omega_{B}^{2} G_{\omega^{2}, B, x x}^{(2)}\left(\tau, \tau^{\prime}\right)+\hbar M \delta\left(\tau-\tau^{\prime}\right), \\ G_{\omega^{2}, B, p_{x} p_{y}}^{(2)}\left(\tau, \tau^{\prime}\right) \equiv\left\langle p_{x}(\tau) p_{y}\left(\tau^{\prime}\right)\right\rangle= & -M^{2} \partial_{\tau} \partial_{\tau^{\prime}} G_{\omega^{2}, B, x y}^{(2)}\left(\tau, \tau^{\prime}\right)+i M^{2} \partial_{\tau} G_{\omega^{2}, B, x x}^{(2)}\left(\tau, \tau^{\prime}\right) \\ & +M^{2} \omega_{B}^{2} G_{\omega^{2}, B, x y}^{(2)}\left(\tau, \tau^{\prime}\right), \\ G_{\omega^{2}, B, p_{z} p_{z}}^{(2)}\left(\tau, \tau^{\prime}\right) \equiv\left\langle p_{z}(\tau) p_{z}\left(\tau^{\prime}\right)\right\rangle= & -M^{2} \partial_{\tau} \partial_{\tau^{\prime}} G_{\omega^{2}, B, z z}^{(2)}\left(\tau, \tau^{\prime}\right)+\hbar M \delta\left(\tau-\tau^{\prime}\right) . \end{align*} $$

$$ \left(x_{b} \hbar \beta \mid x_{a} 0\right)[j, k]=\int_{x(0)=x_{a}}^{x(\hbar \beta)=x_{b}} D x \frac{D p}{2 \pi \hbar} \exp \left\{-\frac{1}{\hbar} \mathcal{A}_{\mathrm{e}}[j, k]\right\} $$

$$ \left(p_{b} \hbar \beta \mid p_{a} 0\right)[j, k]=\int_{p(0)=p_{a}}^{p(\hbar \beta)=p_{b}} D x \frac{D p}{2 \pi \hbar} \exp \left\{-\frac{1}{\hbar} \mathcal{A}_{\mathrm{e}}[j, k]\right\} $$

$$ \left(p_{b} \hbar \beta \mid p_{a} 0\right)[j, k]=\int_{-\infty}^{+\infty} d x_{a} \int_{-\infty}^{+\infty} d x_{b} e^{-i\left(p_{b} x_{b}-p_{a} x_{a}\right) / \hbar}\left(x_{b} \hbar \beta \mid x_{a} 0\right)[j, k] $$

$$ \begin{align*} \lim _{\tau_{b} \uparrow \hbar \beta} \lim _{\tau_{a} \downarrow 0}\left(p_{b}\right. & \left.\hbar \beta \mid p_{a} 0\right)\left[j(\tau), k(\tau)+i x_{b} \delta\left(\tau_{b}-\tau\right)-i x_{a} \delta\left(\tau-\tau_{a}\right)\right] \\ & =\exp \left\{\frac{i}{\hbar}\left(p_{b} x_{b}-p_{a} x_{a}\right)\right\}\left(p_{b} \hbar \beta \mid p_{a} 0\right)[j(\tau), k(\tau)] \end{align*} $$

$$ \left(x_{b} \hbar \beta \mid x_{a} 0\right)[j, k]=\int_{-\infty}^{+\infty} \frac{d p_{a}}{2 \pi \hbar} \int_{-\infty}^{+\infty} \frac{d p_{b}}{2 \pi \hbar} e^{i\left(p_{b} x_{b}-p_{a} x_{a}\right) / \hbar}\left(p_{b} \hbar \beta \mid p_{a} 0\right)[j, k], $$

$$ \left(x_{b} \hbar \beta \mid x_{a} 0\right)[j, k]=\lim _{\tau_{b} \uparrow \hbar \beta} \lim _{\tau_{a} \downarrow 0}(0 \hbar \beta \mid 00)\left[j(\tau), k(\tau)+i x_{b} \delta\left(\tau_{b}-\tau\right)-i x_{a} \delta\left(\tau-\tau_{a}\right)\right] . $$

$$ \int_{x(0)=x}^{x(\hbar \beta)=x} \frac{\mathcal{D} x \mathcal{D} p}{2 \pi \hbar}=\oint \frac{\mathcal{D} x \mathcal{D} p}{2 \pi \hbar} \delta(x(0)-x) $$

$$ \int_{x(0)=x}^{x(\hbar \beta)=x} \frac{\mathcal{D} x \mathcal{D} p}{2 \pi \hbar}=\lim _{\tau_{a}^{\prime} \downarrow 0} \int_{-\infty}^{+\infty} \frac{d p_{a}}{2 \pi \hbar} e^{i p_{a} x / \hbar} \oint \frac{\mathcal{D} x \mathcal{D} p}{2 \pi \hbar} e^{-i \int_{0}^{\hbar \beta} d \tau p_{a} \delta\left(\tau-\tau_{a}^{\prime}\right) x(\tau) / \hbar} $$

$$ \begin{align*} \left(x_{b} \hbar \beta \mid x_{a} 0\right)[k, j] & =\lim _{\tau_{b} \uparrow \hbar \beta} \lim _{\tau_{a} \downarrow 0} \lim _{\tau_{a}^{\prime} \downarrow 0} \int_{-\infty}^{+\infty} \frac{d p_{a}}{2 \pi \hbar} \\ \times & Z\left[j(\tau)-i p_{a} \delta\left(\tau-\tau_{a}^{\prime}\right), k(\tau)+i x_{b} \delta\left(\tau_{b}-\tau\right)-i x_{a} \delta\left(\tau-\tau_{a}\right)\right] \end{align*} $$

$$ \tilde{k}(\tau)=k(\tau)+i x_{b} \delta\left(\tau_{b}-\tau\right)-i x_{a} \delta\left(\tau-\tau_{a}\right), \quad \tilde{j}(\tau)=j(\tau)-i p \delta\left(\tau-\tau_{a}^{\prime}\right), $$

$$ Z_{\omega}[\tilde{k}, \tilde{j}]=Z_{\omega}^{(0)}[0,0] Z_{\omega}^{(1)}[k, j] Z_{\omega}^{\mathrm{p}}[k, j] . $$

$$ \begin{align*} & \left.\left.-x_{a}^{2} G_{p p}^{\mathrm{p}}\left(\tau_{a}, \tau_{a}\right)-x_{b}^{2} G_{p p}^{\mathrm{p}}\left(\tau_{b}, \tau_{b}\right)+2 x_{a} x_{b} G_{p p}^{\mathrm{p}}\left(\tau_{a}, \tau_{b}\right)\right\}\right) \\ Z_{\omega}^{(1)}[k, j]= & \exp \left(\frac { 1 } { \hbar ^ { 2 } } \int _ { 0 } ^ { \hbar \beta } d \tau \left\{j(\tau)\left[-i p G_{x x}^{\mathrm{p}}\left(\tau, \tau_{a}^{\prime}\right)+i x_{b} G_{x p}^{\mathrm{p}}\left(\tau, \tau_{b}\right)-i x_{a} G_{x p}^{\mathrm{p}}\left(\tau, \tau_{a}\right)\right]\right.\right. \\ & \left.\left.+k(\tau)\left[-i p G_{x p}^{\mathrm{p}}\left(\tau, \tau_{a}^{\prime}\right)+i x_{b} G_{p p}^{\mathrm{p}}\left(\tau, \tau_{b}\right)-i x_{a} G_{p p}^{\mathrm{p}}\left(\tau, \tau_{a}\right)\right]\right\}\right) \\ Z_{\omega}^{\mathrm{p}}[k, j]= & \exp \left\{\frac { 1 } { 2 \hbar ^ { 2 } } \int _ { 0 } ^ { \hbar \beta } d \tau _ { 1 } \int _ { 0 } ^ { \hbar \beta } d \tau _ { 2 } \left[\left(j\left(\tau_{1}\right), k\left(\tau_{2}\right)\right)\right.\right. \\ & \left.\left.\quad \times\left(\begin{array}{ll} G_{x x}^{\mathrm{p}}\left(\tau_{1}, \tau_{1}\right) & G_{x p}^{\mathrm{p}}\left(\tau_{1}, \tau_{2}\right) \\ G_{p x}^{\mathrm{p}}\left(\tau_{1}, \tau_{2}\right) & G_{p p}^{\mathrm{p}}\left(\tau_{1}, \tau_{2}\right) \end{array}\right)\binom{j\left(\tau_{2}\right)}{k\left(\tau_{2}\right)}\right]\right\} \end{align*} $$

$$ \begin{align*} & \left(x_{b} \hbar \beta \mid x_{a} 0\right)[k, j]=\left(x_{b} \hbar \beta \mid x_{a} 0\right)[0,0] \times \exp \left\{\frac{1}{\hbar} \int_{0}^{\hbar \beta} d \tau\left[x_{\mathrm{cl}}(\tau) j(\tau)+p_{\mathrm{cl}}(\tau) k(\tau)\right]\right\} \\ & \times \exp \left\{\frac{1}{2 \hbar^{2}} \int_{0}^{\hbar \beta} d \tau_{1} \int_{0}^{\hbar \beta} d \tau_{2}\left[\left(j\left(\tau_{1}\right), k\left(\tau_{2}\right)\right)\left(\begin{array}{ll} G_{x x}^{(\mathrm{D})}\left(\tau_{1}, \tau_{2}\right) & G_{x p}^{(\mathrm{D})}\left(\tau_{1}, \tau_{2}\right) \\ G_{p x}^{(\mathrm{D})}\left(\tau_{1}, \tau_{2}\right) & G_{p p}^{(\mathrm{D})}\left(\tau_{1}, \tau_{2}\right) \end{array}\right)\binom{j\left(\tau_{2}\right)}{k\left(\tau_{2}\right)}\right]\right\}, \end{align*} $$

$$ \begin{gather*} \left(x_{b} \hbar \beta \mid x_{a} 0\right)[0,0]=\lim _{\tau_{b} \uparrow \hbar \beta} \lim _{\tau_{a} \downarrow 0} \lim _{\tau^{\prime} \downarrow 0} \frac{Z_{\omega}}{2 \pi \hbar} \sqrt{\frac{2 \pi \hbar^{2}}{G_{x x}^{\mathrm{p}}\left(\tau_{a}^{\prime}, \tau_{a}^{\prime}\right)}} \\ \quad \times \exp \left[\frac { 1 } { 2 \hbar ^ { 2 } } \left(x_{a}^{2}\left\{\frac{G_{x p}^{\mathrm{p}}{ }^{2}\left(\tau_{a}^{\prime}, \tau_{a}\right)}{G_{x x}^{\mathrm{p}}\left(\tau_{a}^{\prime}, \tau_{a}^{\prime}\right)}-G_{p p}^{\mathrm{p}}\left(\tau_{a}, \tau_{a}\right)\right\}+x_{b}^{2}\left\{\frac{G_{x p}^{\mathrm{p}}{ }^{2}\left(\tau_{a}^{\prime}, \tau_{b}\right)}{G_{x x}^{\mathrm{p}}\left(\tau_{a}^{\prime}, \tau_{a}^{\prime}\right)}-G_{p p}^{\mathrm{p}}\left(\tau_{b}, \tau_{b}\right)\right\}\right.\right. \\ \left.\left.\quad-2 x_{a} x_{b}\left\{\frac{G_{x p}^{\mathrm{p}}\left(\tau_{a}^{\prime}, \tau_{a}\right) G_{x p}^{\mathrm{p}}\left(\tau_{a}^{\prime}, \tau_{b}\right)}{G_{x x}^{\mathrm{p}}\left(\tau_{a}^{\prime}, \tau_{a}^{\prime}\right)}-G_{p p}^{\mathrm{p}}\left(\tau_{a}, \tau_{b}\right)\right\}\right)\right] \end{gather*} $$

$$ \lim _{\tau_{a} \downarrow 0} \lim _{\tau_{a}^{\prime} \downarrow 0} G_{x p}^{\mathrm{p}}\left(\tau_{a}^{\prime}, \tau_{a}\right)=-i \frac{\hbar}{2}, $$

$$ \begin{align*} \left(x_{b} \hbar \beta \mid x_{a} 0\right)[0,0] & =\sqrt{\frac{M \omega}{2 \pi \hbar \sinh \hbar \beta \omega}} \\ & \times \exp \left\{-\frac{M \omega}{2 \hbar \sinh \hbar \beta \omega}\left[\left(x_{a}^{2}+x_{b}^{2}\right) \cosh \hbar \beta \omega-2 x_{a} x_{b}\right]\right\} \end{align*} $$

$$ \left.-x_{b}\left[\frac{G_{x p}^{\mathrm{p}}\left(\tau_{a}^{\prime}, \tau_{b}\right) G_{x x}^{\mathrm{p}}\left(\tau, \tau_{a}^{\prime}\right)}{G_{x x}^{\mathrm{p}}\left(\tau_{a}^{\prime}, \tau_{a}^{\prime}\right)}+G_{x p}^{\mathrm{p}}\left(\tau_{b}, \tau\right)\right]\right\} $$

$$ \begin{align*} p_{\mathrm{cl}}(\tau)=\lim _{\tau_{b} \uparrow \hbar \beta} \lim _{\tau_{a} \downarrow 0} \lim _{\tau_{a}^{\prime} \downarrow 0} \frac{i}{\hbar}\{ & x_{a}\left[\frac{G_{x p}^{\mathrm{p}}\left(\tau_{a}^{\prime}, \tau_{a}\right) G_{x p}^{\mathrm{p}}\left(\tau_{a}^{\prime}, \tau\right)}{G_{x x}^{\mathrm{p}}\left(\tau_{a}^{\prime}, \tau_{a}^{\prime}\right)}-G_{p p}^{\mathrm{p}}\left(\tau_{a}, \tau\right)\right] \\ & \left.-x_{b}\left[\frac{G_{x p}^{\mathrm{p}}\left(\tau_{a}^{\prime}, \tau_{b}\right) G_{x p}^{\mathrm{p}}\left(\tau_{a}^{\prime}, \tau\right)}{G_{x x}^{\mathrm{p}}\left(\tau_{a}^{\prime}, \tau_{a}^{\prime}\right)}-G_{p p}^{\mathrm{p}}\left(\tau_{b}, \tau\right)\right]\right\} . \end{align*} $$

$$ x_{\mathrm{cl}}(\tau)=\frac{x_{a} \sinh \omega(\hbar \beta-\tau)+x_{b} \sinh \omega \tau}{\sinh \hbar \beta \omega} $$

$$ p_{\mathrm{cl}}(\tau)=i M \omega \frac{-x_{a} \cosh \omega(\hbar \beta-\tau)+x_{b} \cosh \omega \tau}{\sinh \hbar \beta \omega}, $$

$$ \begin{align*} & G_{x x}^{(\mathrm{D})}\left(\tau_{1}, \tau_{2}\right)=G_{x x}^{\mathrm{p}}\left(\tau_{1}, \tau_{2}\right)-\frac{G_{x x}^{\mathrm{p}}\left(\tau_{1}, 0\right) G_{x x}^{\mathrm{p}}\left(\tau_{2}, 0\right)}{G_{x x}^{\mathrm{p}}\left(\tau_{1}, \tau_{1}\right)} \\ & G_{x p}^{(\mathrm{D})}\left(\tau_{1}, \tau_{2}\right)=G_{x p}^{\mathrm{p}}\left(\tau_{1}, \tau_{2}\right)+\frac{G_{x x}^{\mathrm{p}}\left(\tau_{1}, 0\right) G_{x p}^{\mathrm{p}}\left(\tau_{2}, 0\right)}{G_{x x}^{\mathrm{p}}\left(\tau_{1}, \tau_{1}\right)} \\ & G_{p x}^{(\mathrm{D})}\left(\tau_{1}, \tau_{2}\right)=G_{p x}^{\mathrm{p}}\left(\tau_{1}, \tau_{2}\right)+\frac{G_{x p}^{\mathrm{p}}\left(\tau_{1}, 0\right) G_{x x}^{\mathrm{p}}\left(\tau_{2}, 0\right)}{G_{x x}^{\mathrm{p}}\left(\tau_{1}, \tau_{1}\right)} \\ & G_{p p}^{(\mathrm{D})}\left(\tau_{1}, \tau_{2}\right)=G_{p p}^{\mathrm{p}}\left(\tau_{1}, \tau_{2}\right)-\frac{G_{x p}^{\mathrm{p}}\left(\tau_{1}, 0\right) G_{x p}^{\mathrm{p}}\left(\tau_{2}, 0\right)}{G_{x x}^{\mathrm{p}}\left(\tau_{1}, \tau_{1}\right)} \end{align*} $$

$$ \begin{align*} G_{x x}^{(\mathrm{D})}\left(\tau_{1}, \tau_{2}\right)= & \frac{\hbar}{2 M \omega \sinh \hbar \beta \omega}\left[\cosh \omega\left(\hbar \beta-\left|\tau_{1}-\tau_{2}\right|\right)-\cosh \omega\left(\hbar \beta-\tau_{1}-\tau_{2}\right)\right], \\ G_{x p}^{(\mathrm{D})}\left(\tau_{1}, \tau_{2}\right)= & \frac{i \hbar}{2 \sinh \hbar \beta \omega}\left\{\theta\left(\tau_{1}-\tau_{2}\right) \sinh \omega\left(\hbar \beta-\left|\tau_{1}-\tau_{2}\right|\right)\right. \\ & \left.-\theta\left(\tau_{2}-\tau_{1}\right) \sinh \omega\left(\hbar \beta-\left|\tau_{2}-\tau_{1}\right|\right)+\sinh \omega\left(\hbar \beta-\tau_{1}-\tau_{2}\right)\right\}, \\ G_{p x}^{(\mathrm{D})}\left(\tau_{1}, \tau_{2}\right)= & -\frac{i \hbar}{2 \sinh \hbar \beta \omega}\left\{\theta\left(\tau_{1}-\tau_{2}\right) \sinh \omega\left(\hbar \beta-\left|\tau_{1}-\tau_{2}\right|\right)\right. \\ & \left.-\theta\left(\tau_{2}-\tau_{1}\right) \sinh \omega\left(\hbar \beta-\left|\tau_{2}-\tau_{1}\right|\right)-\sinh \omega\left(\hbar \beta-\tau_{1}-\tau_{2}\right)\right\}, \\ G_{p p}^{(\mathrm{D})}\left(\tau_{1}, \tau_{2}\right)= & \frac{M \hbar \omega}{2 \sinh \hbar \beta \omega}\left[\cosh \omega\left(\hbar \beta-\left|\tau_{1}-\tau_{2}\right|\right)+\cosh \omega\left(\hbar \beta-\tau_{1}-\tau_{2}\right)\right] . \end{align*} $$

$$ \begin{array}{lc} G_{x x}^{(\mathrm{D})}\left(\tau_{1}, \tau_{2}\right)=G_{x x}^{(\mathrm{D})}\left(\tau_{2}, \tau_{1}\right), & G_{x p}^{(\mathrm{D})}\left(\tau_{1}, \tau_{2}\right)=-G_{x p}^{(\mathrm{D})}\left(\tau_{2}, \tau_{1}\right), \\ G_{p x}^{(\mathrm{D})}\left(\tau_{1}, \tau_{2}\right)=-G_{p x}^{(\mathrm{D})}\left(\tau_{2}, \tau_{1}\right), & G_{p p}^{(\mathrm{D})}\left(\tau_{1}, \tau_{2}\right)=G_{p p}^{(\mathrm{D})}\left(\tau_{2}, \tau_{1}\right), \end{array} $$

$$ G_{x p}^{(\mathrm{D})}\left(\tau_{1}, \tau_{2}\right)=G_{p x}^{(\mathrm{D})}\left(\tau_{2}, \tau_{1}\right) $$

$$ \begin{align*} G_{x p}^{(\mathrm{D})}\left(\tau_{1}, \tau_{2}\right) & =-i M \frac{\partial}{\partial \tau_{1}} G_{x x}^{(\mathrm{D})}\left(\tau_{1}, \tau_{2}\right)=i M \frac{\partial}{\partial \tau_{2}} G_{x x}^{(\mathrm{D})}\left(\tau_{1}, \tau_{2}\right) \\ G_{p x}^{(\mathrm{D})}\left(\tau_{1}, \tau_{2}\right) & =i M \frac{\partial}{\partial \tau_{1}} G_{x x}^{(\mathrm{D})}\left(\tau_{1}, \tau_{2}\right)=-i M \frac{\partial}{\partial \tau_{2}} G_{x x}^{(\mathrm{D})}\left(\tau_{1}, \tau_{2}\right) \\ G_{p p}^{(\mathrm{D})}\left(\tau_{1}, \tau_{2}\right) & =\hbar M \delta\left(\tau_{1}-\tau_{2}\right)-M^{2} \frac{\partial^{2}}{\partial \tau_{1} \partial \tau_{2}} G_{x x}^{(\mathrm{D})}\left(\tau_{1}-\tau_{2}\right) \end{align*} $$

$$ \begin{align*} \left(x_{b} \hbar \beta \mid x_{a} 0\right) & =\prod_{i} \oint \mathcal{D} X_{i}(\tau) \int_{x(0)=x_{a}}^{x(\hbar \beta)=x_{b}} \mathcal{D} x(\tau) \\ & \times \exp \left\{-\frac{1}{\hbar} \int_{0}^{\hbar \beta} d \tau \sum_{i}\left[\frac{M_{i}}{2}\left(\dot{X}_{i}^{2}+\Omega_{i}^{2} X_{i}^{2}\right)\right]\right\} \\ & \times \exp \left\{-\frac{1}{\hbar} \int_{0}^{\hbar \beta} d \tau\left[\frac{M}{2} \dot{x}^{2}+V(x(\tau))-\sum_{i} c_{i} X_{i}(\tau) x(\tau)\right]\right\} \times \frac{1}{\prod_{i} Z_{i}} \end{align*} $$

$$ \begin{align*} Z_{i} & \equiv \oint \mathcal{D} X_{i}(\tau) \exp \left\{-\frac{1}{\hbar} \int_{0}^{\hbar \beta} d \tau\left[\frac{M_{i}}{2}\left(\dot{X}_{i}^{2}+\Omega_{i}^{2} X_{i}^{2}\right)\right]\right\} \\ & =\frac{1}{2 \sinh \left(\hbar \beta \Omega_{i} / 2\right)} \end{align*} $$

$$ \left(x_{b} \hbar \beta \mid x_{a} 0\right)=\int_{x(0)=x_{a}}^{x(\hbar \beta)=x_{b}} \mathcal{D} x(\tau) \exp \left\{-\frac{1}{\hbar} \int_{0}^{\hbar \beta} d \tau\left[\frac{M}{2} \dot{x}^{2}+V(x(\tau))\right]-\frac{1}{\hbar} \mathcal{A}_{\mathrm{bath}}[x]\right\} $$

$$ \mathcal{A}_{\mathrm{bath}}[x]=-\frac{1}{2} \int_{0}^{\hbar \beta} d \tau \int_{0}^{\hbar \beta} d \tau^{\prime} x(\tau) \alpha\left(\tau-\tau^{\prime}\right) x\left(\tau^{\prime}\right) $$

$$ \begin{align*} \alpha\left(\tau-\tau^{\prime}\right) & =\sum_{i} c_{i}^{2} \frac{1}{M_{i}} G_{\Omega_{i}^{2}, \mathrm{e}}^{\mathrm{p}}\left(\tau-\tau^{\prime}\right) \\ & =\sum_{i} \frac{c_{i}^{2}}{2 M_{i} \Omega_{i}} \frac{\cosh \Omega_{i}\left(\left|\tau-\tau^{\prime}\right|-\hbar \beta / 2\right)}{\sinh \left(\Omega_{i} \hbar \beta / 2\right)} \end{align*} $$

$$ \alpha\left(\tau-\tau^{\prime}\right)=\frac{1}{\hbar \beta} \sum_{m=-\infty}^{\infty} \alpha_{m} e^{-i \omega_{m}\left(\tau-\tau^{\prime}\right)} $$

$$ \alpha_{m}=\sum_{i} \frac{c_{i}^{2}}{M_{i}} \frac{1}{\omega_{m}^{2}+\omega_{i}^{2}} $$

$$ \mathcal{A}_{\mathrm{bath}}[x]=-\frac{1}{2} \int_{0}^{\hbar \beta} d \tau \int_{-\infty}^{\infty} d \tau^{\prime} x(\tau) \alpha_{0}\left(\tau-\tau^{\prime}\right) x\left(\tau^{\prime}\right) $$

$$ \alpha_{0}\left(\tau-\tau^{\prime}\right)=\sum_{i} \frac{c_{i}^{2}}{2 M_{i} \Omega_{i}} e^{-\Omega_{i}\left|\tau-\tau^{\prime}\right|} $$

$$ \rho_{\mathrm{b}}\left(\omega^{\prime}\right) \equiv 2 \pi \sum_{i} \frac{c_{i}^{2}}{2 M_{i} \Omega_{i}} \delta\left(\omega^{\prime}-\Omega_{i}\right) $$

$$ \alpha_{0}\left(\tau-\tau^{\prime}\right)=\int_{0}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \rho_{\mathrm{b}}\left(\omega^{\prime}\right) e^{-\omega^{\prime}\left|\tau-\tau^{\prime}\right|} $$

$$ \alpha\left(\tau-\tau^{\prime}\right)=\int_{0}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \rho_{\mathrm{b}}\left(\omega^{\prime}\right) \frac{\cosh \omega^{\prime}\left(\left|\tau-\tau^{\prime}\right|-\hbar \beta / 2\right)}{\sinh \left(\omega^{\prime} \hbar \beta / 2\right)} $$

$$ \alpha_{m}=\int_{0}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \rho_{\mathrm{b}}\left(\omega^{\prime}\right) \frac{2 \omega^{\prime}}{\omega_{m}^{2}+\omega^{2}} $$

$$ \alpha_{m}=2 \int_{0}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \frac{\rho_{\mathrm{b}}\left(\omega^{\prime}\right)}{\omega^{\prime}}\left(1-\frac{\omega_{m}^{2}}{\omega_{m}^{2}+\omega^{\prime 2}}\right)=\alpha_{0}-g_{m} $$

$$ \alpha\left(\tau-\tau^{\prime}\right)=\alpha_{0} \delta^{\mathrm{p}}\left(\tau-\tau^{\prime}\right)-g\left(\tau-\tau^{\prime}\right) $$

$$ \delta^{\mathrm{p}}\left(\tau-\tau^{\prime}\right)=\frac{1}{\hbar \beta} \sum_{m=-\infty}^{\infty} e^{-i \omega_{m}\left(\tau-\tau^{\prime}\right)}=\sum_{n=-\infty}^{\infty} \delta\left(\tau-\tau^{\prime}-n \hbar \beta\right) $$

$$ g\left(\tau-\tau^{\prime}\right)=\frac{1}{\hbar \beta} \sum_{m=-\infty}^{\infty} g\left(\omega_{m}\right) e^{-i \omega_{m}\left(\tau-\tau^{\prime}\right)}, $$

$$ g_{m}=\sum_{i} \frac{c_{i}^{2}}{M_{i}} \frac{\omega_{m}^{2}}{\omega_{m}^{2}+\Omega_{i}^{2}}=\int_{0}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \frac{\rho_{\mathrm{b}}\left(\omega^{\prime}\right)}{\omega^{\prime}} \frac{2 \omega_{m}^{2}}{\omega_{m}^{2}+\omega^{\prime 2}} $$

$$ \mathcal{A}_{\mathrm{bath}}[x]=\mathcal{A}_{\mathrm{loc}}+\mathcal{A}_{\mathrm{bath}}^{\prime}[x] $$

$$ \mathcal{A}_{\mathrm{bath}}^{\prime}[x]=\frac{1}{2} \int_{0}^{\hbar \beta} d \tau \int_{0}^{\hbar \beta} d \tau^{\prime} x(\tau) g\left(\tau-\tau^{\prime}\right) x\left(\tau^{\prime}\right) $$

$$ \mathcal{A}_{\mathrm{loc}}=-\frac{\alpha_{0}}{2} \int_{0}^{\hbar \beta} d \tau x^{2}(\tau) $$

$$ M \Delta \omega^{2} \equiv-\alpha_{0}=-2 \int_{0}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \frac{\rho_{\mathrm{b}}\left(\omega^{\prime}\right)}{\omega^{\prime}}=-\sum_{i} \frac{c_{i}^{2}}{M_{i} \Omega_{i}^{2}} $$

$$ \mathcal{A}_{\mathrm{loc}}=\frac{M}{2} \Delta \omega^{2} \int_{0}^{\hbar \beta} d \tau x^{2}(\tau) $$

$$ V_{\mathrm{ren}}(x)=V(x)+\frac{M}{2} \Delta \omega^{2} x^{2} $$

$$ \left(x_{b} \hbar \beta \mid x_{a} 0\right)=\int_{x(0)=x_{a}}^{x(\hbar \beta)=x_{b}} \mathcal{D} x(\tau) \exp \left\{-\frac{1}{\hbar} \int_{0}^{\hbar \beta} d \tau\left[\frac{M}{2} \dot{x}^{2}+V_{\mathrm{ren}}(x(\tau))\right]-\frac{1}{\hbar} \mathcal{A}_{\mathrm{bath}}^{\prime}[x]\right\} $$

$$ \int_{0}^{\hbar \beta} d \tau g\left(\tau-\tau^{\prime}\right)=0 $$

$$ x(\tau) x\left(\tau^{\prime}\right)=\frac{1}{2}\left\{x^{2}(\tau)+x^{2}\left(\tau^{\prime}\right)-\left[x(\tau)-x\left(\tau^{\prime}\right)\right]^{2}\right\} $$

$$ \mathcal{A}_{\mathrm{bath}}^{\prime}[x]=-\frac{1}{4} \int_{0}^{\hbar \beta} d \tau \int_{0}^{\hbar \beta} d \tau^{\prime} g\left(\tau-\tau^{\prime}\right)\left[x(\tau)-x\left(\tau^{\prime}\right)\right]^{2} $$

$$ \rho_{\mathrm{b}}\left(\omega^{\prime}\right) \approx 2 M \gamma \omega^{\prime} $$

$$ \rho_{\mathrm{b}}\left(\omega^{\prime}\right) \approx 2 M \gamma \omega^{\prime} \frac{\omega_{D}^{2}}{\omega_{D}^{2}+\omega^{\prime 2}} $$

$$ g_{m}=2 M \gamma \omega_{D}^{2} \int_{0}^{\infty} \frac{d \omega}{2 \pi} \frac{1}{\omega_{D}^{2}+\omega^{2}} \frac{2 \omega_{m}^{2}}{\omega_{m}^{2}+\omega^{2}}=M\left|\omega_{m}\right| \gamma \frac{\omega_{D}}{\left|\omega_{m}\right|+\omega_{D}} $$

$$ g_{m} \equiv M\left|\omega_{m}\right| \gamma_{m} $$

$$ \gamma_{m}=\gamma \frac{\omega_{D}}{\left|\omega_{m}\right|+\omega_{D}}, $$

$$ \Delta \omega^{2}=-\gamma \omega_{D} $$

$$ \mathbf{A}(\mathbf{x}, t)=\sum_{\mathbf{k}} c_{\mathbf{k}}(\mathbf{x}) \mathbf{X}_{\mathbf{k}}(t), \quad c_{\mathbf{k}}=\frac{e^{i \mathbf{k x}}}{\sqrt{2 \Omega_{\mathbf{k}} V}}, \quad \sum_{\mathbf{k}}=\int \frac{d^{3} k V}{(2 \pi)^{3}} $$

$$ \mathcal{A}_{\mathrm{bath}}[x]=-\frac{1}{2} \int_{0}^{\hbar \beta} d \tau \int_{0}^{\hbar \beta} d \tau^{\prime} \dot{x}^{i}(\tau) \alpha^{i j}\left(\mathbf{x}(\tau), \tau ; \mathbf{x}\left(\tau^{\prime}\right), \tau^{\prime}\right) \dot{x}^{j}\left(\tau^{\prime}\right) $$

$$ \alpha^{i j}\left(\mathbf{x}, \tau ; \mathbf{x}^{\prime}, \tau^{\prime}\right)=\frac{e^{2}}{\hbar c^{2}} \sum_{\mathbf{k}} c_{-\mathbf{k}}(\mathbf{x}) c_{\mathbf{k}}\left(\mathbf{x}^{\prime}\right)\left\langle X_{-\mathbf{k}}^{i}(\tau) X_{\mathbf{k}}^{j}\left(\tau^{\prime}\right)\right\rangle $$

$$ { }^{T} \delta_{\mathbf{k}}^{i j} \equiv\left(\delta^{i j}-k^{i} k^{j} / \mathbf{k}^{2}\right) $$

$$ G_{-\mathbf{k}^{\prime} \mathbf{k}}^{i j}\left(\tau-\tau^{\prime}\right)=\left\langle\hat{X}_{-\mathbf{k}^{\prime}}^{i}(\tau) \hat{X}_{\mathbf{k}^{\prime}}^{j}\left(\tau^{\prime}\right)\right\rangle=\hbar^{T} \delta_{\mathbf{k}}^{i j} \delta_{\mathbf{k k}^{\prime}} G_{\omega^{2}, \mathrm{e} \mathbf{k}}^{\mathrm{p}}\left(\tau-\tau^{\prime}\right) $$

$$ G_{\omega^{2}, \mathrm{e} \mathbf{k}}^{\mathrm{p}}\left(\tau-\tau^{\prime}\right) \equiv \frac{1}{2 \Omega_{\mathbf{k}}} \frac{\cosh \Omega_{\mathbf{k}}\left(\left|\tau-\tau^{\prime}\right|-\hbar \beta / 2\right)}{\sinh \left(\Omega_{\mathbf{k}} \hbar \beta / 2\right)} $$

$$ \alpha^{i j}\left(\mathbf{x}, \tau ; \mathbf{x}^{\prime}, \tau^{\prime}\right)=\frac{e^{2}}{c^{2}} \int \frac{d^{3} k}{(2 \pi)^{3}}{ }^{T} \delta_{\mathbf{k}}^{i j} \frac{e^{i \mathbf{k}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)}}{2 \Omega_{\mathbf{k}}} \frac{\cosh \Omega_{\mathbf{k}}\left(\left|\tau-\tau^{\prime}\right|-\hbar \beta / 2\right)}{\sinh \left(\Omega_{\mathbf{k}} \hbar \beta / 2\right)} $$

$$ \alpha^{i j}\left(\mathbf{x}, \tau ; \mathbf{x}^{\prime}, \tau^{\prime}\right)=\frac{e^{2}}{c^{3}} \int \frac{d^{3} k}{(2 \pi)^{3}}{ }^{T} \delta_{\mathbf{k}}^{i j} \frac{e^{i \mathbf{k}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)-c|\mathbf{k}|\left|\tau-\tau^{\prime}\right|}}{2|\mathbf{k}|} $$

$$ G_{\mathrm{e}}^{R}\left(\mathbf{x}, \tau ; \mathbf{x}^{\prime}, \tau^{\prime}\right)=\frac{1}{4 \pi^{2} c^{2}} \frac{1}{\left(\tau-\tau^{\prime}\right)^{2}+\left(\mathbf{x}-\mathbf{x}^{\prime}\right)^{2} / c^{2}} $$

$$ \alpha^{i j}\left(\mathbf{x}, \tau ; \mathbf{x}^{\prime}, \tau^{\prime}\right)=\frac{2 e^{2}}{3 c^{2}} \delta^{i j} \frac{1}{2 \pi c^{2}} \int \frac{d \omega}{2 \pi} \omega \frac{\cosh \omega\left(\left|\tau-\tau^{\prime}\right|-\hbar \beta / 2\right)}{\sinh (\omega \hbar \beta / 2)} . $$

$$ \rho_{\mathrm{pb}}\left(\omega^{\prime}\right)=\frac{e^{2}}{3 c^{2} \pi} \omega^{\prime} $$

$$ \rho_{\mathrm{pb}}\left(\omega^{\prime}\right) \approx 2 M \gamma \omega^{\prime 3}, \quad \gamma=\frac{e^{2}}{6 c^{2} \pi M} $$

$$ V_{\mathrm{ren}}(x)=\frac{M}{2} \omega^{2} x^{2} $$

$$ \mathcal{A}_{\mathrm{e}}=\frac{M \hbar}{k_{B} T}\left\{\frac{\omega^{2}}{2} x_{0}^{2}+\sum_{m=1}^{\infty}\left[\omega_{m}^{2}+\omega^{2}+\omega_{m} \gamma_{m}\right]\left|x_{m}\right|^{2}\right\} $$

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

$$ Z_{\omega}^{\mathrm{damp}}=\frac{k_{B} T}{\hbar \omega} \prod_{m=1}^{\infty} \frac{\omega_{m}^{2}\left(\omega_{m}+\omega_{D}\right)}{\omega_{m}^{3}+\omega_{m}^{2} \omega_{D}+\omega_{m}\left(\omega^{2}+\gamma \omega_{D}\right)+\omega_{D} \omega^{2}} $$

$$ w^{3}-w^{2} \omega_{D}+w\left(\omega^{2}+\gamma \omega_{D}\right)-\omega^{2} \omega_{D}=0 $$

$$ Z_{\omega}^{\mathrm{damp}}=\frac{k_{B} T}{\hbar \omega} \prod_{m=1}^{\infty} \frac{\omega_{m}}{\omega_{m}+w_{1}} \frac{\omega_{m}}{\omega_{m}+w_{2}} \frac{\omega_{m}}{\omega_{m}+w_{3}} \frac{\omega_{m}+\omega_{D}}{\omega_{m}} . $$

$$ \Gamma(z)=\lim _{n \rightarrow \infty} \frac{n^{z}}{z} \prod_{m=1}^{n} \frac{m}{m+z} $$

$$ w_{1}+w_{2}+w_{3}-\omega_{D}=0, \quad w_{1} w_{2} w_{3}=\omega^{2} \omega_{D} $$

$$ Z_{\omega}^{\mathrm{damp}}=\frac{1}{2 \pi} \frac{\omega}{\omega_{1}} \frac{\Gamma\left(w_{1} / \omega_{1}\right) \Gamma\left(w_{2} / \omega_{1}\right) \Gamma\left(w_{3} / \omega_{1}\right)}{\Gamma\left(\omega_{D} / \omega_{1}\right)} $$

$$ w_{1}=\gamma / 2+i \delta, \quad w_{1}=\gamma / 2-i \delta, \quad w_{3}=\omega_{D}-\gamma $$

$$ \delta \equiv \sqrt{\omega^{2}-\gamma^{2} / 4} $$

$$ Z_{\omega}^{\mathrm{damp}}=\frac{1}{2 \pi} \frac{\omega}{\omega_{1}} \Gamma\left(w_{1} / \omega_{1}\right) \Gamma\left(w_{2} / \omega_{1}\right) $$

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

$$ \Gamma\left(i \omega / \omega_{1}\right) \Gamma\left(-i \omega / \omega_{1}\right)=\frac{\omega_{1}}{\omega} \frac{\pi}{\sinh \left(\pi \omega / \omega_{1}\right)}=\frac{\omega_{1}}{\omega} \frac{\pi}{\sinh \left(\omega \hbar / 2 k_{B} T\right)} $$

$$ \begin{align*} F(T)=-k_{B} T & {\left[\log \left(\omega / 2 \pi \omega_{1}\right)-\log \Gamma\left(\omega_{D} / \omega_{1}\right)\right.} \\ & \left.+\log \Gamma\left(w_{1} / \omega_{1}\right)+\log \Gamma\left(w_{2} / \omega_{1}\right)+\log \Gamma\left(w_{3} / \omega_{1}\right)\right] \end{align*} $$

$$ \log \Gamma(z)=\left(z-\frac{1}{2}\right) \log z-z+\frac{1}{2} \log 2 \pi+\frac{1}{12 z}-\frac{1}{360 z^{3}}-\mathcal{O}\left(1 / z^{5}\right) $$

$$ F(T) \sim E_{0}-\left(\frac{1}{w_{1}}+\frac{1}{w_{2}}+\frac{1}{w_{1}}-\frac{\omega^{2}}{w_{1} w_{2} w_{3}}\right) \frac{\pi}{6 \hbar}\left(k_{B} T\right)^{2}=E_{0}-\frac{\gamma \pi}{6 \omega^{2} \hbar}\left(k_{B} T\right)^{2}, $$

$$ E_{0}=-\frac{\hbar}{2 \pi}\left[w_{1} \log \left(w_{1} / \omega_{D}\right)+w_{2} \log \left(w_{2} / \omega_{D}\right)+w_{3} \log \left(w_{3} / \omega_{D}\right)\right] $$

$$ E_{0}=\frac{\hbar \omega}{2}+\frac{\gamma}{2 \pi} \log \frac{\omega_{D}}{\omega}-\frac{\gamma^{2}}{16 \omega}\left(1+\frac{4 \omega}{\pi \omega_{D}}\right)+\mathcal{O}\left(\gamma^{3}\right) $$

$$ \frac{1}{\hbar \beta} \sum_{m} \underset{T \rightarrow 0}{\longrightarrow} \int_{0}^{\infty} \frac{d \omega_{m}}{2 \pi} $$

$$ E_{0}=\frac{\hbar}{2 \pi} \int_{0}^{\infty} d \omega_{m} \log \left[\frac{\omega_{m}^{3}+\omega_{m}^{2} \omega_{D}+\omega_{m}\left(\omega^{2}+\gamma \omega_{D}\right)+\omega_{D} \omega^{2}}{\omega_{m}^{2}\left(\omega_{m}+\omega_{D}\right)}\right] $$

$$ \rho(\varepsilon)=\frac{1}{2 \pi i} \int_{\eta-i \infty}^{\eta+i \infty} d \beta e^{i \varepsilon \beta} Z_{\omega}^{\mathrm{damp}}(\beta) $$

$$ \rho(\varepsilon)=\sum_{n=0}^{\infty} \delta(\varepsilon-(n+1 / 2) \hbar \omega) $$

$$ \Gamma(z)=\int_{1}^{\infty} d t t^{z-1} e^{-t}+\sum_{n=0}^{\infty} \frac{(-1)^{n}}{n!(z+n)} $$

$$ \begin{gather*} \rho(\varepsilon)=\frac{1}{\omega} \sum_{n=1}^{\infty} \sum_{i=1}^{3} R_{n, i} e^{-2 \pi n \varepsilon / w_{i}} \\ R_{n, 1}=\frac{\omega}{w_{1}^{2}} \frac{(-1)^{n-1}}{(n-1)!} \frac{\Gamma\left(-n w_{2} / w_{1}\right) \Gamma\left(-n w_{3} / w_{1}\right)}{\Gamma\left(-n \omega_{D} / w_{1}\right)}, \end{gather*} $$

$$ Z_{\omega}^{\mathrm{damp}}=\frac{k_{B} T}{\hbar \omega}\left\{\prod_{m=1}^{\infty}\left[\frac{\omega_{m}^{2}+\omega^{2}+\omega_{m}^{3} \gamma}{\omega_{m}^{2}}\right]\right\}^{-1}, $$

$$ Z_{\omega}^{\mathrm{damp}}=\frac{k_{B} T}{\hbar \omega} \prod_{m=1}^{\infty} \frac{\omega_{m}^{2}\left(\omega_{m}+\omega_{D}\right)\left(1+\gamma \omega_{D}\right)}{\omega_{m}^{3}\left(1+\gamma \omega_{D}\right)+\omega_{m}^{2} \omega_{D}+\omega_{m} \omega^{2}+\omega_{D} \omega^{2}} . $$

$$ w^{3}\left(1+\gamma \omega_{D}\right)-w^{2} \omega_{D}+w \omega^{2}-\omega^{2} \omega_{D}=0 . $$

$$ w_{1} \approx \gamma_{\mathrm{pb}}^{\mathrm{eff}} / 2+i \omega, \quad w_{1} \approx \gamma_{\mathrm{pb}}^{\mathrm{eff}} / 2-i \omega, \quad w_{3} \approx \omega_{D} /\left(1+\gamma_{\mathrm{pb}}^{\mathrm{eff}} \omega_{D} / \omega^{2}\right), $$

$$ \gamma_{\mathrm{pb}}^{\mathrm{eff}}=\frac{e^{2}}{6 c^{2} \pi M} \omega^{2}, $$

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

$$ \begin{align*} \left(x_{b} t_{b} \mid x_{a} t_{a}\right)= & \int_{x\left(t_{a}\right)=x_{a}}^{x\left(t_{b}\right)=x_{b}} \mathcal{D} x\left[1-\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t V(x(t))\right. \\ & -\frac{1}{2!\hbar^{2}} \int_{t_{a}}^{t_{b}} d t_{2} V\left(x\left(t_{2}\right)\right) \int_{t_{a}}^{t_{b}} d t_{1} V\left(x\left(t_{1}\right)\right) \\ & \left.+\frac{i}{3!\hbar^{3}} \int_{t_{a}}^{t_{b}} d t_{3} V\left(x\left(t_{3}\right)\right) \int_{t_{a}}^{t_{b}} d t_{2} V\left(x\left(t_{2}\right)\right) \int_{t_{a}}^{t_{b}} d t_{1} V\left(x\left(t_{1}\right)\right)+\ldots\right] \\ & \times \exp \left[\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t \frac{M}{2}\left(\dot{x}^{2}-\omega^{2} x^{2}\right)\right] \end{align*} $$

$$ \begin{align*} & \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\left(x_{b} t_{b} \mid x_{a} t_{a}\right)-\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t_{1} \int d x_{1}\left(x_{b} t_{b} \mid x_{1} t_{1}\right) V\left(x_{1}\right)\left(x_{b} t_{b} \mid x_{a} t_{a}\right) \\ & \quad-\frac{1}{2!\hbar^{2}} \int_{t_{a}}^{t_{b}} d t_{2} \int_{t_{a}}^{t_{b}} d t_{1} \int d x_{1} d x_{2}\left(x_{b} t_{b} \mid x_{2} t_{2}\right) V\left(x_{2}\right)\left(x_{2} t_{2} \mid x_{1} t_{1}\right) V\left(x_{1}\right)\left(x_{1} t_{1} \mid x_{a} t_{a}\right)+\ldots \end{align*} $$

$$ Z=\oint \mathcal{D} x \exp \left\{-\frac{1}{\hbar} \int_{0}^{\hbar \beta} d \tau\left[\frac{M}{2}\left(\dot{x}^{2}+\omega^{2} x^{2}\right)+V(x)\right]\right\} $$

$$ \begin{align*} Z=\int \mathcal{D} x[1- & \frac{1}{\hbar} \int_{0}^{\hbar \beta} d \tau V(x(\tau))+\frac{1}{2!\hbar^{2}} \int_{0}^{\hbar \beta} d \tau_{2} V\left(x\left(\tau_{2}\right)\right) \int_{0}^{\hbar \beta} d \tau_{1} V\left(x\left(\tau_{1}\right)\right) \\ & \left.-\frac{1}{3!\hbar^{3}} \int_{0}^{\hbar \beta} d \tau_{3} V\left(x\left(\tau_{3}\right)\right) \int_{0}^{\hbar \beta} d \tau_{2} V\left(x\left(\tau_{2}\right)\right) \int_{0}^{\hbar \beta} d \tau_{1} V\left(x\left(\tau_{1}\right)\right)+\ldots\right] \\ & \times \exp \left\{-\frac{1}{\hbar} \int_{0}^{\hbar \beta} d t\left[\frac{M}{2} \dot{x}^{2}+M \frac{\omega^{2}}{2} x^{2}\right]\right\} \end{align*} $$

$$ \mathcal{A}_{\mathrm{int}, \mathrm{e}} \equiv \int_{0}^{\hbar \beta} d \tau V(x(\tau)) $$

$$ \langle\ldots\rangle_{\omega} \equiv Z_{\omega}^{-1} \int \mathcal{D} x \ldots \exp \left\{-\frac{1}{\hbar} \int_{0}^{\hbar \beta} d \tau\left[\frac{M}{2}\left(\dot{x}^{2}+\omega^{2} x^{2}\right)\right]\right\} $$

$$ Z=\left(1-\frac{1}{\hbar}\left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}\right\rangle_{\omega}+\frac{1}{2!\hbar^{2}}\left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}^{2}\right\rangle_{\omega}-\frac{1}{3!\hbar^{3}}\left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}^{3}\right\rangle_{\omega}+\ldots\right) Z_{\omega} $$

$$ \begin{align*} & 1-\frac{1}{\hbar}\left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}\right\rangle_{\omega}+\frac{1}{2!\hbar^{2}}\left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}^{2}\right\rangle_{\omega}-\frac{1}{3!\hbar^{3}}\left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}^{3}\right\rangle_{\omega}+\ldots \\ & =\exp \left\{-\frac{1}{\hbar}\left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}\right\rangle_{\omega}+\frac{1}{2!\hbar^{2}}\left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}^{2}\right\rangle_{\omega, c}-\frac{1}{3!\hbar^{3}}\left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}^{3}\right\rangle_{\omega, c}+\ldots\right\} \end{align*} $$

$$ \begin{align*} \left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}^{2}\right\rangle_{\omega, c} & \equiv\left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}^{2}\right\rangle_{\omega}-\left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}\right\rangle_{\omega}^{2} \\ & =\left\langle\left[\mathcal{A}_{\mathrm{int}, \mathrm{e}}-\left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}\right\rangle_{\omega}\right]^{2}\right\rangle_{\omega} \\ \left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}^{3}\right\rangle_{\omega, \mathrm{c}} & \equiv\left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}^{3}\right\rangle_{\omega}-3\left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}^{2}\right\rangle_{\omega}\left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}\right\rangle_{\omega}+2\left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}\right\rangle_{\omega}^{3} \\ & =\left\langle\left[\mathcal{A}_{\mathrm{int}, \mathrm{e}}-\left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}\right\rangle_{\omega}\right]^{3}\right\rangle_{\omega} \\ & \vdots \end{align*} $$

$$ \Delta F=\frac{1}{\beta}\left(\frac{1}{\hbar}\left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}\right\rangle_{\omega}-\frac{1}{2!\hbar^{2}}\left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}^{2}\right\rangle_{\omega, \mathrm{c}}+\frac{1}{3!\hbar^{3}}\left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}^{3}\right\rangle_{\omega, \mathrm{c}}+\ldots\right) $$

$$ \Delta E_{0}=\lim _{\beta \rightarrow \infty} \frac{1}{\beta}\left(\frac{1}{\hbar}\left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}\right\rangle_{\omega}-\frac{1}{2!\hbar^{2}}\left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}^{2}\right\rangle_{\omega, \mathrm{c}}+\frac{1}{3!\hbar^{3}}\left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}^{3}\right\rangle_{\omega, \mathrm{c}}+\ldots\right) $$

$$ Z[j]=\oint \mathcal{D} x \exp \left\{-\frac{1}{\hbar} \int_{0}^{\hbar \beta} d \tau\left[\frac{M}{2}\left(\dot{x}^{2}+\omega^{2} x^{2}\right)+V(x)-j x\right]\right\} $$

$$ Z[j]=e^{-\frac{1}{\hbar} \int_{0}^{\hbar \beta} d \tau V(\delta / \delta j(\tau))} Z_{\omega}[j] $$

$$ Z=Z[0] . $$

$$ Z_{\omega}[j]=\int \mathcal{D} x \exp \left\{-\frac{1}{\hbar} \int_{0}^{\hbar \beta} d \tau\left[\frac{M}{2}\left(\dot{x}^{2}+\omega^{2} x^{2}\right)-j x\right]\right\} $$

$$ \begin{align*} & e^{-\frac{1}{\hbar} \int_{0}^{\hbar \beta} d \tau V(\delta / \delta j(\tau))}=1-\frac{1}{\hbar} \int_{0}^{\hbar \beta} d \tau V(\delta / \delta j(\tau)) \\ & \quad+\frac{1}{2!\hbar^{2}} \int_{0}^{\hbar \beta} d \tau_{2} V\left(\delta / \delta j\left(\tau_{2}\right)\right) \int_{0}^{\hbar \beta} d \tau_{1} V\left(\delta / \delta j\left(\tau_{1}\right)\right) \\ & \quad-\frac{1}{3!\hbar^{3}} \int_{0}^{\hbar \beta} d \tau_{3} V\left(\delta / \delta j\left(\tau_{3}\right)\right) \int_{0}^{\hbar \beta} d \tau_{2} V\left(\delta / \delta\left(\tau_{2}\right)\right) \int_{0}^{\hbar \beta} d \tau_{1} V\left(\delta / \delta\left(\tau_{1}\right)\right)+\ldots \end{align*} $$

$$ \left\langle\mathcal{A}^{n}\right\rangle=\left.Z^{-1} \frac{\partial^{n}}{\partial \hbar^{-1^{n}}} Z_{\omega}[j]\right|_{\hbar^{-1}=0} $$

$$ \langle\mathcal{A}\rangle=\lim _{\hbar \rightarrow \infty} Z_{\omega}^{-1} \hbar^{2} \frac{\partial}{\partial \hbar} Z_{\omega}=\lim _{\hbar \rightarrow \infty} \hbar \frac{\hbar \omega \beta}{2 \sinh \hbar \omega \beta / 2}=0 $$

$$ \left\langle\dot{x}^{2}(\tau)\right\rangle_{\omega}+\omega^{2}\left\langle x^{2}(\tau)\right\rangle_{\omega}=\int \frac{d \omega^{\prime}}{2 \pi} \frac{\omega^{\prime 2}}{\omega^{\prime 2}+\omega^{2}}+\int \frac{d \omega^{\prime}}{2 \pi} \frac{\omega^{2}}{\omega^{\prime 2}+\omega^{2}}=\int \frac{d \omega^{\prime}}{2 \pi}=0 . $$

$$ \left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}\right\rangle_{\omega} \equiv Z_{\omega}^{-1} \int_{0}^{\hbar \beta} d \tau_{1} \int d x d x_{1}\left(x \hbar \beta \mid x_{1} \tau_{1}\right)_{\omega} V\left(x_{1}\right)\left(x_{1} \tau_{1} \mid x 0\right)_{\omega} $$

$$ \begin{array}{r} \frac{1}{2}\left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}^{2}\right\rangle_{\omega} \equiv Z_{\omega}^{-1} \int_{0}^{\hbar \beta} d \tau_{2} \int_{0}^{\hbar \beta} d \tau_{1} \int d x d x_{2} d x_{1}\left(x \hbar \beta \mid x_{2} \tau_{2}\right)_{\omega} V\left(x_{2}\right) \\ \times\left(x_{2} \tau_{2} \mid x_{1} \tau_{1}\right)_{\omega} V\left(x_{1}\right)\left(x_{1} \tau_{1} \mid x 0\right)_{\omega} \end{array} $$

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

$$ \begin{align*} & \int d x_{b} d x_{a} \psi_{n}^{*}\left(x_{b}\right)\left(x_{b} t_{b} \mid x_{a} t_{a}\right) \psi_{n}\left(x_{a}\right)=\int d x_{b} d x_{a} \psi_{n}^{*}\left(x_{b}\right)\left(x_{b} t_{b} \mid x_{a} t_{a}\right)_{\omega} \psi_{n}\left(x_{a}\right) \\ & \quad \times\left(1+\frac{i}{\hbar}\langle n| \mathcal{A}_{\mathrm{int}}|n\rangle_{\omega}-\frac{1}{2!\hbar^{2}}\langle n| \mathcal{A}_{\mathrm{int}}^{2}|n\rangle_{\omega}-\frac{i}{3!\hbar^{3}}\langle n| \mathcal{A}_{\mathrm{int}}^{3}|n\rangle_{\omega}+\ldots\right) \end{align*} $$

$$ \mathcal{A}_{\mathrm{int}} \equiv-\int_{t_{a}}^{t_{b}} d t V(x(t)) $$

$$ \langle n| \ldots|n\rangle_{\omega} \equiv Z_{\mathrm{QM}, \omega, n}^{-1} \int d x_{b} d x_{a} \psi_{n}^{*}\left(x_{b}\right)\left(\int_{x\left(t_{a}\right)=x_{a}}^{x\left(t_{b}\right)=x_{b}} \mathcal{D} x \ldots e^{i \mathcal{A}_{\omega} / \hbar}\right) \psi_{n}\left(x_{a}\right) $$

$$ Z_{\mathrm{QM}, \omega, n} \equiv e^{-i \omega(n+1 / 2)\left(t_{b}-t_{a}\right)} $$

$$ \begin{align*} \langle n| \mathcal{A}_{\mathrm{int}}|n\rangle_{\omega} \equiv-Z_{\mathrm{QM}, \omega, n}^{-1} \int_{t_{a}}^{t_{b}} d t_{1} \int & d x_{b} d x_{a} d x_{1} \psi_{n}^{*}\left(x_{b}\right)\left(x_{b} t_{b} \mid x_{1} t_{1}\right)_{\omega} \\ & \times V\left(x_{1}\right)\left(x_{1} t_{1} \mid x_{a} t_{a}\right)_{\omega} \psi_{n}\left(x_{a}\right) \end{align*} $$

$$ \begin{align*} & \frac{1}{2}\langle n| \mathcal{A}_{\mathrm{int}}^{2}|n\rangle_{\omega} \equiv Z_{\mathrm{QM}, \omega, n}^{-1} \int_{t_{a}}^{t_{b}} d t_{2} \int_{t_{a}}^{t_{2}} d t_{1} \int d x_{b} d x_{a} d x_{2} d x_{1} \\ & \quad \times \psi_{n}^{*}\left(x_{b}\right)\left(x_{b} t_{b} \mid x_{2} t_{2}\right)_{\omega} V\left(x_{2}\right)\left(x_{2} t_{2} \mid x_{1} t_{1}\right)_{\omega} V\left(x_{1}\right)\left(x_{1} t_{1} \mid x_{a} t_{a}\right)_{\omega} \psi_{n}\left(x_{a}\right) \end{align*} $$

$$ \begin{align*} 1+\frac{i}{\hbar} & \langle n| \mathcal{A}_{\text {int }}|n\rangle_{\omega}-\frac{1}{2!\hbar^{2}}\langle n| \mathcal{A}_{\text {int }}^{2}|n\rangle_{\omega}-\frac{i}{3!\hbar^{3}}\langle n| \mathcal{A}_{\text {int }}^{3}|n\rangle_{\omega}+\ldots \\ & =\exp \left\{\frac{i}{\hbar}\langle n| \mathcal{A}_{\text {int }}|n\rangle_{\omega}-\frac{1}{2!\hbar^{2}}\langle n| \mathcal{A}_{\text {int }}^{2}|n\rangle_{\omega, \mathrm{c}}-\frac{i}{3!\hbar^{3}}\langle n| \mathcal{A}_{\text {int }}^{3}|n\rangle_{\omega, \mathrm{c}}+\ldots\right\} \end{align*} $$

$$ \begin{align*} \langle n| \mathcal{A}_{\text {int }}^{2}|n\rangle_{\omega, \mathrm{c}} & \equiv\langle n| \mathcal{A}_{\text {int }}^{2}|n\rangle_{\omega}-\langle n| \mathcal{A}_{\text {int }}|n\rangle_{\omega}^{2} \\ & =\langle n|\left[\mathcal{A}_{\text {int }}-\langle n| \mathcal{A}_{\text {int }}|n\rangle_{\omega}\right]^{2}|n\rangle_{\omega} \\ \langle n| \mathcal{A}_{\text {int }}^{3}|n\rangle_{\omega, \mathrm{c}} & \equiv\langle n| \mathcal{A}_{\text {int }}^{3}|n\rangle_{\omega}-3\langle n| \mathcal{A}_{\text {int }}^{2}|n\rangle_{\omega}\langle n| \mathcal{A}_{\text {int }}|n\rangle_{\omega}+2\langle n| \mathcal{A}_{\text {int }}|n\rangle_{\omega}^{3} \\ & =\langle n|\left[\mathcal{A}_{\text {int }}-\langle n| \mathcal{A}_{\text {int }}|n\rangle_{\omega}\right]^{3}|n\rangle_{\omega} \\ & \vdots \\ & \cdot \end{align*} $$

$$ \begin{align*} \Delta E_{n}=\lim _{t_{b}-t_{a} \rightarrow \infty} \frac{i \hbar}{t_{b}-t_{a}}\left\{\frac{i}{\hbar}\langle n| \mathcal{A}_{\mathrm{int}}|n\rangle_{\omega}\right. & -\frac{1}{2!\hbar^{2}}\langle n| \mathcal{A}_{\mathrm{int}}^{2}|n\rangle_{\omega, \mathrm{c}} \\ & \left.-\frac{i}{3!\hbar^{3}}\langle n| \mathcal{A}_{\mathrm{int}}^{3}|n\rangle_{\omega, \mathrm{c}}+\ldots\right\}, \end{align*} $$

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

$$ \langle n| \mathcal{A}_{\mathrm{int}}|n\rangle_{\omega} \equiv-\int_{t_{a}}^{t_{b}} d t \int d x \psi_{n}^{*}(x) V(x) \psi_{n}(x) \equiv-\left(t_{b}-t_{a}\right) V_{n n} $$

$$ \begin{align*} \frac{1}{2}\langle n| \mathcal{A}_{\mathrm{int}}^{2}|n\rangle_{\omega} & \equiv Z_{\mathrm{QM}, \omega, n}^{-1} \int_{t_{a}}^{t_{b}} d t_{2} \int_{t_{a}}^{t_{2}} d t_{1} \\ & \times \sum_{k} e^{-i E_{n}\left(t_{b}-t_{2}\right) / \hbar-i E_{k}\left(t_{2}-t_{1}\right) / \hbar-i E_{n}\left(t_{1}-t_{a}\right) / \hbar} V_{n k} V_{k n} \end{align*} $$

$$ \int_{t_{a}}^{t_{b}} d t_{2} \int_{t_{a}}^{t_{2}} d t_{1} \sum_{k} e^{i\left(E_{n}-E_{k}\right) t_{2} / \hbar+i\left(E_{k}-E_{n}\right) t_{1} / \hbar} V_{n k} V_{k n} $$

$$ -\sum_{k} \frac{V_{n k} V_{k n}}{E_{k}-E_{n}}\left\{i \hbar\left(t_{b}-t_{a}\right)-\frac{\hbar^{2}}{E_{n}-E_{k}}\left[e^{i\left(E_{n}-E_{k}\right)\left(t_{b}-t_{a}\right) / \hbar}-1\right]\right\} $$

$$ V_{n n} V_{n n} \frac{\left(t_{b}-t_{a}\right)^{2}}{2}, $$

$$ \frac{1}{2}\langle n| \mathcal{A}_{\mathrm{int}}^{2}|n\rangle_{\omega}=-\sum_{m \neq n} \frac{V_{n m} V_{m n}}{E_{m}-E_{n}} i \hbar\left(t_{b}-t_{a}\right)+V_{n n} V_{n n} \frac{\left(t_{b}-t_{a}\right)^{2}}{2} $$

$$ \frac{1}{2}\langle n| \mathcal{A}_{\mathrm{int}}^{2}|n\rangle_{\omega, \mathrm{c}}=-\sum_{k \neq n} \frac{V_{n k} V_{k n}}{E_{k}-E_{n}} i \hbar\left(t_{b}-t_{a}\right) . $$

$$ \Delta_{1} E_{n}+\Delta_{2} E_{n}=V_{n n}-\sum_{k \neq n} \frac{V_{n k} V_{k n}}{E_{k}-E_{n}} $$

$$ \Delta_{3} E_{n}=\sum_{k \neq n} \sum_{l \neq n} \frac{V_{n k} V_{k l} V_{l n}}{\left(E_{k}-E_{n}\right)\left(E_{l}-E_{n}\right)}-V_{n n} \sum_{k \neq n} \frac{V_{n k} V_{k n}}{\left(E_{k}-E_{n}\right)^{2}} . $$

$$ \Delta E_{n}=\bar{R}_{n n}\left(E_{n}+\Delta E_{n}\right), $$

$$ \hat{\bar{R}}(E)=\hat{V}+\hat{V} \frac{1-\hat{P}_{n}}{E-\hat{H}_{\omega}} \hat{\bar{R}}(E) $$

$$ \hat{\bar{R}}(E)=\hat{V}+\hat{V} \frac{1-\hat{P}_{n}}{E-\hat{H}_{\omega}} \hat{V}+\hat{V} \frac{1-\hat{P}_{n}}{E-\hat{H}_{\omega}} \hat{V} \frac{1-\hat{P}_{n}}{E-\hat{H}_{\omega}} \hat{V}+\ldots $$

$$ E-E_{n}=R_{n n}(E)=V_{n n}+\sum_{k \neq n} \frac{V_{n k} V_{k n}}{E-E_{k}}+\sum_{k \neq n} \sum_{l \neq n} \frac{V_{n k} V_{k l} V_{l n}}{\left(E-E_{k}\right)\left(E-E_{l}\right)}+\ldots $$

$$ \begin{align*} & \Delta E_{n}=R_{n n}\left(E_{n}\right)+R_{n n}\left(E_{n}\right) R_{n n}^{\prime}\left(E_{n}\right)+\left[R_{n n}\left(E_{n}\right) R_{n n}^{\prime}\left(E_{n}\right)^{2}+\frac{1}{2} R_{n n}^{2}\left(E_{n}\right) R_{n n}^{\prime \prime}\left(E_{n}\right)\right] \\ & +\left[R_{n n}\left(E_{n}\right) R_{n n}^{\prime}\left(E_{n}\right)^{3}+\frac{3}{2} R_{n n}^{2}\left(E_{n}\right) R_{n n}^{\prime}\left(E_{n}\right) R_{n n}^{\prime \prime}\left(E_{n}\right)+\frac{1}{6} R_{n n}^{3}\left(E_{n}\right) R_{n n}^{\prime \prime \prime}\left(E_{n}\right)\right]+\ldots .(3 . \end{align*} $$

$$ \begin{align*} \Delta E_{n} & =\frac{\hbar \omega}{2}(2 n+1)+\frac{g}{4} 3\left(2 n^{2}+2 n+1\right) a^{4} \\ & -\left(\frac{g}{4}\right)^{2} 2\left(34 n^{3}+51 n^{2}+59 n+21\right) a^{8} \frac{1}{\hbar \omega} \\ & +\left(\frac{g}{4}\right)^{3} 4 \cdot 3\left(125 n^{4}+250 n^{3}+472 n^{2}+347 n+111\right) a^{12} \frac{1}{\hbar^{2} \omega^{2}} \end{align*} $$

$$ \hat{H}=\hat{H}_{0}+\hat{V} . $$

$$ \hat{H}_{0}|n\rangle=E_{0}^{(n)}|n\rangle, \quad \hat{H}\left|\psi^{(n)}\right\rangle=E^{(n)}\left|\psi^{(n)}\right\rangle . $$

$$ a_{n}^{(n)} \equiv\left\langle n \mid \psi^{(n)}\right\rangle=1 . $$

$$ \left|\psi^{(n)}\right\rangle=|n\rangle+\sum_{m \neq n} a_{m}^{(n)}|m\rangle $$

$$ a_{m}^{(n)} \equiv\left\langle m \mid \psi^{(n)}\right\rangle $$

$$ E_{0}^{(m)} a_{m}^{(n)}+\langle m| \hat{V}\left|\psi^{(n)}\right\rangle=E^{(n)} a_{m}^{(n)} . $$

$$ E_{0}^{(m)} a_{m}^{(n)}+\langle m| \hat{V}|n\rangle+\sum_{k \neq n} a_{k}^{(n)}\langle m| \hat{V}|k\rangle=E^{(n)} a_{m}^{(n)}, $$

$$ E_{0}^{(n)}+\langle n| \hat{V}|n\rangle+\sum_{k \neq n} a_{k}^{(n)}\langle n| \hat{V}|k\rangle=E^{(n)} $$

$$ a_{m}^{(n)}=\frac{1}{E_{0}^{(n)}-E_{0}^{(m)}}\left[\left\langle m-a_{m}^{(n)} n\right| \hat{V}|n\rangle+\sum_{k \neq n} a_{k}^{(n)}\left\langle m-a_{m}^{(n)} n\right| \hat{V}|k\rangle\right], $$

$$ a_{m}^{(n)}(g)=\sum_{l=1}^{\infty} a_{m, l}^{(n)}(-g)^{l} \quad(m \neq n) $$

$$ E^{(n)}=E_{0}^{(n)}-\sum_{l=1}^{\infty}(-g)^{l} E_{l}^{(n)} . $$

$$ \begin{align*} E_{1}^{(n)} & =\langle n| \hat{V}|n\rangle \\ E_{l}^{(n)} & =\sum_{k \neq n} a_{k, l-1}^{(n)}\langle n| \hat{V}|k\rangle \quad l>1 \end{align*} $$

$$ a_{m, 1}^{(n)}=\frac{\langle m| \hat{V}|n\rangle}{E_{0}^{(m)}-E_{0}^{(n)}} $$

$$ a_{m, l}^{(n)}=\frac{1}{E_{0}^{(m)}-E_{0}^{(n)}}\left[-a_{m, l-1}^{(n)}\langle n| \hat{V}|n\rangle+\sum_{k \neq n} a_{k, l-1}^{(n)}\langle m| \hat{V}|k\rangle-\sum_{l^{\prime}=1}^{l-2} a_{m, l^{\prime}}^{(n)} \sum_{k \neq n} a_{k, l-1-l^{\prime}}^{(n)}\langle n| \hat{V}|k\rangle\right] . $$

$$ a_{m, l}^{(n)}=\frac{1}{E_{0}^{(m)}-E_{0}^{(n)}}\left[\sum_{k \neq n} a_{k, l-1}^{(n)}\langle m| \hat{V}|k\rangle-\sum_{l^{\prime}=1}^{l-1} a_{m, l^{\prime}}^{(n)} E_{l-l^{\prime}}^{(n)}\right] . $$

$$ E_{2}^{(n)}=\sum_{k \neq n} a_{k, 1}^{(n)}\langle n| \hat{V}|k\rangle=\sum_{k \neq n} \frac{\langle k| \hat{V}|n\rangle\langle n| \hat{V}|k\rangle}{E_{0}^{(k)}-E_{0}^{(n)}} $$

$$ V(x)=\frac{g}{4} x^{4} $$

$$ \left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}\right\rangle_{\omega}=\frac{g}{4} \int_{0}^{\hbar \beta} d \tau\left\langle x^{4}(\tau)\right\rangle_{\omega} $$

$$ \left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}\right\rangle_{\omega}=3 \frac{g}{4} \int_{0}^{\hbar \beta} d \tau G_{\omega^{2}}^{(2)}(\tau, \tau)^{2} $$

$$ \left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}^{2}\right\rangle_{\omega}=\left(\frac{g}{4}\right)^{2} \int_{0}^{\hbar \beta} d \tau_{2} \int_{0}^{\hbar \beta} d \tau_{1}\left\langle x^{4}\left(\tau_{2}\right) x^{4}\left(\tau_{1}\right)\right\rangle_{\omega} $$

$$ \begin{align*} \left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}^{2}\right\rangle_{\omega}=\left(\frac{g}{4}\right)^{2} \int_{0}^{\hbar \beta} & d \tau_{2} \int_{0}^{\hbar \beta} d \tau_{1}\left[72 G_{\omega^{2}}^{(2)}\left(\tau_{2}, \tau_{2}\right) G_{\omega^{2}}^{(2)}\left(\tau_{2}, \tau_{1}\right)^{2} G_{\omega^{2}}^{(2)}\left(\tau_{1}, \tau_{1}\right)\right. \\ & \left.+24 G_{\omega^{2}}^{(2)}\left(\tau_{2}, \tau_{1}\right)^{4}+9 G_{\omega^{2}}^{(2)}\left(\tau_{2}, \tau_{2}\right)^{2} G_{\omega^{2}}^{(2)}\left(\tau_{1}, \tau_{1}\right)^{2}\right] \end{align*} $$

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

$$ \beta F_{\omega}=-\frac{1}{2} \operatorname{Tr} \log G_{\omega^{2}}^{(2)}=-\frac{1}{2 \hbar \beta} \int_{0}^{\hbar \beta} d \tau\left[\log G_{\omega^{2}}^{(2)}\right](\tau, \tau)=-\frac{1}{2} \bigcirc . $$

$$ \begin{align*} 0 & =\int_{0}^{\hbar \beta} d \tau G_{\omega^{2}}^{(2)}(\tau, \tau)=\hbar \beta a^{2} \\ & =\int_{0}^{\hbar \beta} \int_{0}^{\hbar \beta} d \tau_{1} d \tau_{2} G_{\omega^{2}}^{(2)}\left(\tau_{1}, \tau_{2}\right)^{2} \equiv \hbar \beta \frac{1}{\omega} a_{2}^{4} \\ & =\int_{0}^{\hbar \beta} \int_{0}^{\hbar \beta} \int_{0}^{\hbar \beta} d \tau_{1} d \tau_{2} d \tau_{3} G_{\omega^{2}}^{(2)}\left(\tau_{1}, \tau_{2}\right) G_{\omega^{2}}^{(2)}\left(\tau_{2}, \tau_{3}\right) G_{\omega^{2}}^{(2)}\left(\tau_{3}, \tau_{1}\right) \\ & \equiv \hbar \beta\left(\frac{1}{\omega}\right)^{2} a_{3}^{6} \\ & =\int_{0}^{\hbar \beta} \int_{0}^{\hbar \beta} d \tau_{1} d \tau_{2} G_{\omega^{2}}^{(2)}\left(\tau_{1}, \tau_{2}\right)^{4} \equiv \hbar \beta \frac{1}{\omega} a_{2}^{8} \\ & =\int_{0}^{\hbar \beta} \int_{0}^{\hbar \beta} \int_{0}^{\hbar \beta} d \tau_{1} d \tau_{2} d \tau_{3} G_{\omega^{2}}^{(2)}\left(\tau_{1}, \tau_{2}\right) G_{\omega^{2}}^{(2)}\left(\tau_{2}, \tau_{3}\right) G_{\omega^{2}}^{(2)}\left(\tau_{3}, \tau_{1}\right)^{3} \\ & \equiv \hbar \beta\left(\frac{1}{\omega}\right)^{2} a_{3}^{10} \\ & =\int_{0}^{\hbar \beta} \int_{0}^{\hbar \beta} \int_{0}^{\hbar \beta} d \tau_{1} d \tau_{2} d \tau_{3} G_{\omega^{2}}^{(2)}\left(\tau_{1}, \tau_{2}\right)^{2} G_{\omega^{2}}^{(2)}\left(\tau_{2}, \tau_{3}\right)^{2} G_{\omega^{2}}^{(2)}\left(\tau_{3}, \tau_{1}\right)^{2} \\ & \equiv \hbar \beta\left(\frac{1}{\omega}\right)^{2} a_{3}^{12} . \end{align*} $$

$$ a_{V}^{2 L}=\left(\frac{\hbar}{M \omega}\right)^{L} \alpha_{V}^{2 L}(x) $$

$$ \begin{align*} F & =F_{\omega}+\frac{g}{4} 3 a^{4}-\frac{1}{2!\hbar \omega}\left(\frac{g}{4}\right)^{2}\left(72 a^{2} a_{2}^{4} a^{2}+24 a_{2}^{8}\right) \\ & +\frac{1}{3!\hbar^{2} \omega^{2}}\left(\frac{g}{4}\right)^{3}\left[2592 a^{2}\left(a_{2}^{4}\right)^{2} a^{2}+1728 a_{3}^{6}\left(a^{2}\right)^{3}+3456 a_{3}^{10} a^{2}+1728 a_{3}^{12}\right]+\ldots \end{align*} $$

$$ \begin{align*} & a_{2}^{4} \rightarrow a^{4}, \\ & a_{2}^{8} \rightarrow \frac{1}{2} a^{8}, \\ & a_{3}^{12} \rightarrow \frac{3}{8} a^{12}, \end{align*} $$

$$ F=\frac{\hbar \omega}{2}+\frac{g}{4} 3 a^{4}-\left(\frac{g}{4}\right)^{2} 42 a^{8} \frac{1}{\hbar \omega}+\left(\frac{g}{4}\right)^{3} 4 \cdot 333 a^{12}\left(\frac{1}{\hbar \omega}\right)^{2}+\ldots . $$

$$ Z_{\omega}[j]=\oint \mathcal{D} x \exp \left\{-\frac{1}{\hbar} \int_{0}^{\hbar \beta} d \tau\left[\frac{M}{2}\left(\dot{x}^{2}+\omega^{2} x^{2}\right)-j x\right]\right\} $$

$$ Z_{\omega}[j]=\frac{1}{2 \sin (\omega \hbar \beta / 2)} \exp \left\{\frac{1}{2 M \hbar} \int_{0}^{\hbar \beta} d \tau \int_{0}^{\hbar \beta} d \tau^{\prime} j(\tau) G_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}\left(\tau-\tau^{\prime}\right) j\left(\tau^{\prime}\right)\right\} $$

$$ G_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}(\tau)=\frac{1}{2 \omega} \frac{\cosh \omega(\tau-\hbar \beta / 2)}{\sinh (\beta \hbar \omega / 2)}, \quad \tau \in[0, \hbar \beta] $$

$$ Z[j]=e^{-\frac{1}{\hbar} \int_{0}^{\hbar \beta} d \tau V(\hbar \delta / \delta j(\tau))} Z_{\omega}[j] $$

$$ W[j]=\log Z[j] $$

$$ G_{\mathrm{c}}^{(n)}\left(\tau_{1}, \ldots, \tau_{n}\right)=\frac{\delta}{\delta j\left(\tau_{1}\right)} \cdots \frac{\delta}{\delta j\left(\tau_{n}\right)} W[j] $$

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

$$ \mathcal{A}_{\mathrm{e}}[x, j]=\int_{0}^{\hbar \beta} d \tau\left[\frac{M}{2}\left(\dot{x}^{2}+\omega^{2} x^{2}\right)+V(x)-j(\tau) x(\tau)\right] $$

$$ \int \mathcal{D} x \frac{\delta}{\delta x(\tau)} e^{-\mathcal{A}_{\mathrm{e}}[x, j]}=0 $$

$$ \int \mathcal{D} x \frac{\delta \mathcal{A}_{\mathrm{e}}[x, j]}{\delta x(\tau)} e^{-\mathcal{A}_{\mathrm{e}}[x, j]}=0 $$

$$ \frac{\delta \mathcal{A}_{\mathrm{e}}[x, j]}{\delta x(\tau)}=M\left(-\ddot{x}+\omega^{2} x\right)+\frac{\lambda}{3!} x^{3}-j=0, $$

$$ \frac{\delta \mathcal{A}_{\mathrm{e}}[x, j]}{\delta x(\tau)}=G_{0}^{-1} x+\frac{\lambda}{3!} x^{3}-j=0 $$

$$ \int \mathcal{D} x\left\{G_{0}^{-1} x(\tau)+\frac{\lambda}{3!} x^{3}(\tau)-j(\tau)\right\} e^{-\mathcal{A}_{\mathrm{e}}[x, j]}=0 $$

$$ \left\{G_{0}^{-1} \frac{\delta}{\delta j(\tau)}+\frac{\lambda}{3!}\left[\frac{\delta}{\delta j(\tau)}\right]^{3}-j(\tau)\right\} Z[j]=0 $$

$$ Z_{j\left(\tau_{1}\right) j\left(\tau_{2}\right) \ldots j\left(\tau_{n}\right)}[j] \equiv \frac{\delta}{\delta j\left(\tau_{1}\right)} \frac{\delta}{\delta j\left(\tau_{2}\right)} \cdots \frac{\delta}{\delta j\left(\tau_{n}\right)} Z[j] $$

$$ G_{0}^{-1} Z_{j(\tau)}+\frac{\lambda}{3!} Z_{j(\tau) j(\tau) j(\tau)}-j(\tau)=0 $$

$$ G_{0}^{-1} W_{j}+\frac{\lambda}{3!}\left(W_{j j j}+3 W_{j j} W_{j}+W_{j}^{3}\right)-j=0 . $$

$$ W_{j\left(\tau_{1}\right) j\left(\tau_{2}\right) \ldots j\left(\tau_{n}\right)}[j] \equiv \frac{\delta}{\delta j\left(\tau_{1}\right)} \frac{\delta}{\delta j\left(\tau_{2}\right)} \cdots \frac{\delta}{\delta j\left(\tau_{n}\right)} W[j] $$

$$ W_{j}=-\frac{\lambda}{3!} G_{0}\left(W_{j j j}+3 W_{j j} W_{j}+W_{j}^{3}\right)+G_{0} j $$

$$ G_{\mathrm{c}}^{(1)}=W_{j} $$

$$ G_{\mathrm{c}}^{(1)}=-\frac{\lambda}{3!} G_{0}\left\{G_{\mathrm{c} j j}^{(1)}+3 G_{\mathrm{c} j}^{(1)} G_{\mathrm{c}}^{(1)}+\left[G_{\mathrm{c}}^{(1)}\right]^{3}\right\}+G_{0} j $$

$$ G_{\mathrm{c}}^{(1)}=G_{0} j . $$

$$ W_{0}[j]=\int \mathcal{D} j G_{\mathrm{c}}^{(1)}=\frac{1}{2} j G_{0} j $$

$$ G_{\mathrm{c}}^{(1)}=-G_{0} \frac{\lambda}{3!}\left[3 G_{0} G_{0} j+\left(G_{0} j\right)^{3}\right]+G_{0} j $$

$$ G^{(1)}(\tau)=Z^{-1}[j] \frac{\delta}{\delta j(\tau)} Z[j]=\frac{\delta}{\delta j(\tau)} W[j]=G_{\mathrm{c}}^{(1)}(\tau) $$

$$ \langle x(\tau)\rangle \equiv G^{(1)}(\tau)=G_{\mathrm{c}}^{(1)}(\tau)=X $$

$$ \begin{align*} G^{(2)}\left(\tau_{1}, \tau_{2}\right) & =Z^{-1}[j] \frac{\delta}{\delta j\left(\tau_{1}\right)} \frac{\delta}{\delta j\left(\tau_{2}\right)} Z[j] \\ & =Z^{-1}[j] \frac{\delta}{\delta j\left(\tau_{1}\right)}\left\{\left(\frac{\delta}{\delta j\left(\tau_{2}\right)} W[j]\right) Z[j]\right\} \\ & =Z^{-1}[j]\left\{W_{j\left(\tau_{1}\right) j\left(\tau_{2}\right)}+W_{j\left(\tau_{1}\right)} W_{j\left(\tau_{2}\right)}\right\} Z[j] \\ & =G_{\mathrm{c}}^{(2)}\left(\tau_{1}, \tau_{2}\right)+G_{\mathrm{c}}^{(1)}\left(\tau_{1}\right) G_{\mathrm{c}}^{(1)}\left(\tau_{2}\right) \end{align*} $$

$$ \begin{align*} & G^{(3)}\left(\tau_{1}, \tau_{2}, \tau_{3}\right)=Z^{-1}[j] \frac{\delta}{\delta j\left(\tau_{1}\right)} \frac{\delta}{\delta j\left(\tau_{2}\right)} \frac{\delta}{\delta j\left(\tau_{3}\right)} Z[j] \\ & \quad=Z^{-1}[j] \frac{\delta}{\delta j\left(\tau_{1}\right)} \frac{\delta}{\delta j\left(\tau_{2}\right)}\left\{\left[\frac{\delta}{\delta j\left(\tau_{3}\right)} W[j]\right] Z[j]\right\} \\ & \quad=Z^{-1}[j] \frac{\delta}{\delta j\left(\tau_{1}\right)}\left\{\left[W_{j\left(\tau_{3}\right) j\left(\tau_{2}\right)}+W_{j\left(\tau_{2}\right)} W_{j\left(\tau_{3}\right)}\right] Z[j]\right\} \\ & \quad=Z^{-1}[j]\left\{W_{j\left(\tau_{1}\right) j\left(\tau_{2}\right) j\left(\tau_{3}\right)}+\left(W_{j\left(\tau_{1}\right)} W_{j\left(\tau_{2}\right) j\left(\tau_{3}\right)}+W_{j\left(\tau_{2}\right)} W_{j\left(\tau_{1}\right) j\left(\tau_{3}\right)}\right.\right. \\ & \left.\left.\quad+W_{j\left(\tau_{3}\right)} W_{j\left(\tau_{1}\right) j\left(\tau_{2}\right)}\right)+W_{j\left(\tau_{1}\right)} W_{j\left(\tau_{2}\right)} W_{j\left(\tau_{3}\right)}\right\} Z[j] \\ & =G_{\mathrm{c}}^{(3)}\left(\tau_{1}, \tau_{2}, \tau_{3}\right)+\left[G_{\mathrm{c}}^{(1)}\left(\tau_{1}\right) G_{\mathrm{c}}^{(2)}\left(\tau_{2}, \tau_{3}\right)+2 \text { perm }\right]+G_{\mathrm{c}}^{(1)}\left(\tau_{1}\right) G_{\mathrm{c}}^{(1)}\left(\tau_{2}\right) G_{\mathrm{c}}^{(1)}\left(\tau_{3}\right) \end{align*} $$

$$ \begin{align*} G^{(4)}\left(\tau_{1}, \ldots, \tau_{4}\right)= & G_{\mathrm{c}}^{(4)}\left(\tau_{1}, \ldots, \tau_{4}\right)+\left[G_{\mathrm{c}}^{(3)}\left(\tau_{1}, \tau_{2}, \tau_{3}\right) G_{\mathrm{c}}^{(1)}\left(\tau_{4}\right)+3 \text { perm }\right] \\ & +\left[G_{\mathrm{c}}^{(2)}\left(\tau_{1}, \tau_{2}\right) G_{\mathrm{c}}^{(2)}\left(\tau_{3}, \tau_{4}\right)+2 \text { perm }\right] \\ & +\left[G_{\mathrm{c}}^{(2)}\left(\tau_{1}, \tau_{2}\right) G_{\mathrm{c}}^{(1)}\left(\tau_{3}\right) G_{\mathrm{c}}^{(1)}\left(\tau_{4}\right)+5 \text { perm }\right] \\ & +G_{\mathrm{c}}^{(1)}\left(\tau_{1}\right) \cdots G_{\mathrm{c}}^{(1)}\left(\tau_{4}\right) \end{align*} $$

$$ \begin{align*} G^{(1)} & =e^{-W}\left(e^{W}\right)_{j}=W_{j}=G_{\mathrm{c}}^{(1)} \\ G^{(2)}=e^{-W}\left(e^{W}\right)_{j j} & =W_{j j}+W_{j}^{2}=G_{\mathrm{c}}^{(2)}+G_{\mathrm{c}}^{(1) 2} \\ G^{(3)}=e^{-W}\left(e^{W}\right)_{j j j} & =W_{j j j}+3 W_{j j} W_{j}+W_{j}^{3}=G_{\mathrm{c}}^{(3)}+3 G_{\mathrm{c}}^{(2)} G_{\mathrm{c}}^{(1)}+G_{\mathrm{c}}^{(1) 3} \\ G^{(4)}=e^{-W}\left(e^{W}\right)_{j j j j} & =W_{j j j j}+4 W_{j j j} W_{j}+3 W_{j j}^{2}+6 W_{j j} W_{j}^{2}+W_{j}^{4} \\ & =G_{\mathrm{c}}^{(4)}+4 G_{\mathrm{c}}^{(3)} G_{\mathrm{c}}^{(1)}+3 G_{\mathrm{c}}^{(2) 2}+6 G_{\mathrm{c}}^{(2)} G_{\mathrm{c}}^{(1) 2}+G_{\mathrm{c}}^{(1) 4} \end{align*} $$

$$ G^{(n)}=G_{j}^{(n-1)}+G^{(n-1)} G_{\mathrm{c}}^{(1)}, \quad n \geq 2 $$

$$ \begin{align*} G_{\mathrm{c}}^{(1)} & =G^{(1)} \\ G_{\mathrm{c}}^{(2)} & =G^{(2)}-G^{(1)} G^{(1)} \\ G_{\mathrm{c}}^{(3)} & =G^{(3)}-3 G^{(2)} G^{(1)}+2 G^{(1) 3} \\ G_{\mathrm{c}}^{(4)} & =G^{(4)}-4 G^{(3)} G^{(1)}+12 G^{(2)} G^{(1) 2}-3 G^{(2) 2}-6 G^{(1) 4} \end{align*} $$

$$ G_{j}^{(n)}=G^{(n+1)}-G^{(n)} G^{(1)} $$

$$ Z[K]=\int \mathcal{D} x(\tau) e^{-\mathcal{A}_{\mathrm{e}}[x, K]} $$

$$ \mathcal{A}_{\mathrm{e}}[x, K] \equiv \mathcal{A}_{0}[x]+\mathcal{A}^{\mathrm{int}}[x]+\frac{1}{2} \int d \tau \int d \tau^{\prime} x(\tau) K\left(\tau, \tau^{\prime}\right) x\left(\tau^{\prime}\right) $$

$$ G^{(2)}\left(\tau, \tau^{\prime}\right)=-2 Z^{-1}[K] \frac{\delta Z}{\delta K\left(\tau, \tau^{\prime}\right)} $$

$$ G^{(4)}\left(\tau_{1}, \tau_{2}, \tau_{3}, \tau_{4}\right)=4 Z^{-1}[K] \frac{\delta^{2} Z}{\delta K\left(\tau_{1}, \tau_{2}\right) \delta K\left(\tau_{3}, \tau_{4}\right)} . $$

$$ \begin{align*} G^{(2)}\left(\tau, \tau^{\prime}\right) & =2 \frac{\delta W}{\delta K\left(\tau, \tau^{\prime}\right)} \\ G^{(4)}\left(\tau_{1}, \tau_{2}, \tau_{3}, \tau_{4}\right) & =4\left[\frac{\delta^{2} W}{\delta K\left(\tau_{1}, \tau_{2}\right) \delta K\left(\tau_{3}, \tau_{4}\right)}+\frac{\delta W}{\delta K\left(\tau_{1}, \tau_{2}\right)} \frac{\delta W}{\delta K\left(\tau_{3}, \tau_{4}\right)}\right] . \end{align*} $$

$$ G^{(2)}=2 W_{K}, \quad G^{(4)}=4\left[W_{K K}+W_{K} W_{K}\right]=4 W_{K K}+G^{(2)} G^{(2)} . $$

$$ G^{(4)}=G_{\mathrm{c}}^{(4)}+3 G_{\mathrm{c}}^{(2)} G_{\mathrm{c}}^{(2)} $$

$$ 4 W_{K K}=G_{\mathrm{c}}^{(4)}+2 G_{\mathrm{c}}^{(2)} G_{\mathrm{c}}^{(2)} $$

$$ \begin{align*} & \frac{4 \delta^{2} W}{\delta K\left(\tau_{1}, \tau_{2}\right) \delta K\left(\tau_{3}, \tau_{4}\right)} \\ & \quad=G_{\mathrm{c}}^{(4)}\left(\tau_{1}, \tau_{2}, \tau_{3}, \tau_{4}\right)+G_{\mathrm{c}}^{(2)}\left(\tau_{1}, \tau_{3}\right) G_{\mathrm{c}}^{(2)}\left(\tau_{2}, \tau_{4}\right)+G_{\mathrm{c}}^{(2)}\left(\tau_{1}, \tau_{4}\right) G_{\mathrm{c}}^{(2)}\left(\tau_{2}, \tau_{3}\right) . \end{align*} $$

$$ \int \mathcal{D} x x(\tau) \frac{\delta}{\delta x\left(\tau^{\prime}\right)} e^{-\mathcal{A}_{\mathrm{e}}[x, K]}=-\delta\left(\tau-\tau^{\prime}\right) Z[K] $$

$$ \int \mathcal{D} x x(\tau) \frac{\delta \mathcal{A}_{\mathrm{e}}[x, K]}{\delta x\left(\tau^{\prime}\right)} e^{-\mathcal{A}_{\mathrm{e}}[x, K]}=\delta\left(\tau-\tau^{\prime}\right) Z[K] $$

$$ \int \mathcal{D} x \int d \tau \int d \tau^{\prime}\left\{x(\tau) G_{0}^{-1}\left(\tau, \tau^{\prime}\right) x\left(\tau^{\prime}\right)+\frac{\lambda}{3!} x(\tau) x^{3}\left(\tau^{\prime}\right)\right\} e^{-\mathcal{A}_{\mathrm{e}}[x, K]}=\delta\left(\tau-\tau^{\prime}\right) Z[K] $$

$$ G_{0} \rightarrow\left[G_{0}^{-1}-K\right]^{-1} $$

$$ G_{0}^{-1} Z_{K}+\frac{\lambda}{3} Z_{K K}=\frac{1}{2} Z $$

$$ G_{0}^{-1} W_{K}+\frac{\lambda}{3}\left(W_{K K}+W_{K} W_{K}\right)=\frac{1}{2} $$

$$ W_{K}=G_{0}^{2} W_{G_{0}}, \quad W_{K K}=2 G_{0}^{3} W_{G_{0}}+G_{0}^{4} W_{G_{0} G_{0}}, $$

$$ G_{0} W_{G_{0}}+\frac{\lambda}{3}\left(G_{0}^{4} W_{G_{0} G_{0}}+2 G_{0}^{3} W_{G_{0}}+G_{0}^{4} W_{G_{0}} W_{G_{0}}\right)=\frac{1}{2} $$

$$ W^{(0)}\left[G_{0}\right]=\frac{1}{2} \operatorname{Tr} \log \left(G_{0}\right) $$

$$ W\left[G_{0}\right]=W^{(0)}\left[G_{0}\right]+W^{\mathrm{int}}\left[G_{0}\right] $$

$$ G_{0} W_{G_{0}}^{\mathrm{int}}+\frac{\lambda}{3}\left(G_{0}^{4} W_{G_{0} G_{0}}^{\mathrm{int}}+3 G_{0}^{3} W_{G_{0}}^{\mathrm{int}}+G_{0}^{4} W_{G_{0}}^{\mathrm{int}} W_{G_{0}}^{\mathrm{int}}\right)=6 \frac{-\lambda}{4!} G_{0}^{2} . $$

$$ W^{\mathrm{int}}\left[G_{0}\right]=3 \frac{-\lambda}{4!} G_{0}^{2} $$

$$ W^{\mathrm{int}}\left[G_{0}\right]=\sum_{p=1}^{\infty} \frac{1}{p!} W_{p}\left(\frac{-\lambda}{4!}\right)^{p}\left(G_{0}\right)^{2 p} $$

$$ W_{p+1}=4\left\{[2 p(2 p-1)+3(2 p)] W_{p}+\sum_{q=1}^{p-1}\binom{p}{q} 2 q W_{q} \times 2(p-q) W_{p-q}\right\} $$

$$ \begin{align*} Z\left[G_{0}\right] & =\exp \left[\frac{1}{2} \operatorname{Tr} \log G_{0}+\sum_{p=1}^{\infty} \frac{1}{p!} W_{p}\left(\frac{-\lambda}{4!}\right)^{p}\left(G_{0}\right)^{2 p}\right] \\ & =\operatorname{Det}^{1 / 2}\left[G_{0}\right]\left[1+\sum_{p=1}^{\infty} \frac{1}{p!} z_{p}\left(\frac{-\lambda}{4!}\right)^{p}\left(G_{0}\right)^{2 p}\right] \end{align*} $$

$$ W_{p}+3\binom{p-1}{1} W_{p-1}+7 \cdot 5 \cdot 3\binom{p-1}{2}+\ldots+(4 p-5)!!\binom{p-1}{p-1}=(4 p-1)!!,( $$

$$ W_{p+1}=4\left[G_{0}^{4} \frac{d^{2}}{d \cap^{2}} W_{p}+3 \cdot G_{0}^{3} \frac{d}{d \cap} W_{p}+\sum_{q=1}^{p-1}\binom{p}{q}\left(\frac{d}{d \cap} W_{q}\right) G_{0}^{2} \cdot G_{0}^{2}\left(\frac{d}{d \cap} W_{p-q}\right)\right] $$

$$ W_{1}[0]=\frac{1}{8} \bigcirc \quad G_{1}^{(2)}\left(\tau_{1}, \tau_{2}\right)=2 \times \frac{1}{8} 2 \underset{\tau_{1}}{\Omega_{\tau_{2}}} . $$

$$ W_{2}[0]=\frac{1}{16} \bigcirc \quad+\frac{1}{48} \text {. } $$

$$ G^{(4)}=4 \times\left(2 \cdot 1 \cdot \frac{1}{16}+4 \cdot 3 \cdot \frac{1}{48}\right) \times \circlearrowleft . $$

$$ W_{2}[0]=\frac{1}{48} \circlearrowleft . $$

$$ G^{(4)}=4!\times \frac{1}{48} 3>\circlearrowleft $$

$$ -\Gamma[X] \equiv W[j]-W_{j} j $$

$$ X(\tau) \equiv \frac{\delta W[j]}{\delta j(\tau)} \equiv W_{j(\tau)}=\langle x\rangle_{j(\tau)} $$

$$ -\Gamma[X] \equiv W[j]-X j $$

$$ \Gamma_{X}[X]=j $$

$$ \Gamma_{X}[X]=0 . $$

$$ X_{0}=\left.\langle x\rangle\right|_{j=0} $$

$$ \Gamma^{(n)}\left(\tau_{1}, \ldots, \tau_{n}\right) \equiv \frac{\delta}{\delta X\left(\tau_{1}\right)} \ldots \frac{\delta}{\delta X\left(\tau_{n}\right)} \Gamma[X] $$

$$ \begin{align*} \Gamma^{(2)}\left(\omega_{1}, \omega_{2}\right) & =2 \pi \delta\left(\omega_{1}+\omega_{2}\right) \bar{\Gamma}^{(2)}\left(\omega_{1}\right) \\ \Gamma^{(4)}\left(\omega_{1}, \omega_{2}, \omega_{3}, \omega_{4}\right) & =2 \pi \delta\left(\sum_{i=1}^{4} \omega_{i}\right) \bar{\Gamma}^{(4)}\left(\omega_{1}, \omega_{2}, \omega_{3}, \omega_{4}\right) \end{align*} $$

$$ \Gamma_{X\left(\tau_{1}\right) \ldots X\left(\tau_{n}\right)} \equiv \frac{\delta}{\delta X\left(\tau_{1}\right)} \ldots \frac{\delta}{\delta X\left(\tau_{n}\right)} \Gamma[X] $$

$$ G_{\mathrm{c}}^{(1)}=X . $$

$$ G_{\mathrm{c}}^{(2)}=G_{j}^{(1)}=W_{j j}=\frac{\delta X}{\delta j}=\left(\frac{\delta j}{\delta X}\right)^{-1}=\Gamma_{X X}^{-1} $$

$$ \Gamma_{X(\tau) X(\mathbf{y})}^{-1} \equiv\left[\frac{\delta^{2} \Gamma}{\delta X(\tau) \delta X\left(\tau^{\prime}\right)}\right]^{-1} $$

$$ \int d \tau^{\prime} \Gamma_{X(\tau) X\left(\tau^{\prime}\right)}^{-1} \Gamma_{X\left(\tau^{\prime}\right) X\left(\tau^{\prime \prime}\right.}=\delta\left(\tau-\tau^{\prime \prime}\right) $$

$$ \left.G_{\mathrm{c}}^{(2)}\right|_{j=0}=\left.\Gamma_{X X}^{-1}\right|_{X=X_{0}} $$

$$ G_{\omega^{2}}(\omega) \equiv \bar{G}^{(2)}(\mathrm{k})=\frac{1}{\bar{\Gamma}^{(2)}(\omega)} $$

$$ W_{j j j}=-\Gamma_{X X}^{-2} \Gamma_{X X X} \frac{\delta X}{\delta j}=-\Gamma_{X X}^{-3} \Gamma_{X X X}=-G_{c}^{(2)^{3}} \Gamma_{X X X} $$

$$ G_{\mathrm{c}}^{(3)}=W_{j j j}=-G_{c}^{(2)^{3}} \Gamma_{X X X} $$

$$ G_{c}^{(2)}{ }_{j}=W_{j j j}=G_{c}^{(3)}=-G_{c}^{(2)^{3}} \Gamma_{X X X} . $$

$$ G_{\mathrm{c}}^{(n)}\left(\tau_{1}, \ldots, \tau_{n}\right)=\frac{\delta}{\delta j\left(\tau_{n}\right)} G_{\mathrm{c}}^{(n-1)}\left(\tau_{1}, \ldots, \tau_{n-1}\right), $$

$$ \Gamma_{X \ldots X j}=\Gamma_{X \ldots X X} \frac{\delta X}{\delta j}=\Gamma_{X \ldots X X} G_{c}^{(2)}, $$

$$ \bar{\Gamma}^{(2)}=G_{0}^{-1}+\bar{\Gamma}_{X X}^{\mathrm{int}}, $$

$$ G=\left(1+G_{0} \bar{\Gamma}_{X X}^{\mathrm{int}}\right)^{-1} G_{0} $$

$$ G=G_{0}-G_{0} \bar{\Gamma}_{X X}^{\mathrm{int}} G_{0}+G_{0} \bar{\Gamma}_{X X}^{\mathrm{int}} G_{0} \bar{\Gamma}_{X X}^{\mathrm{int}} G_{0}-\ldots $$

$$ \Sigma \equiv-\bar{\Gamma}_{X X}^{\mathrm{int}} $$

$$ G \equiv\left[G_{0}^{-1}-\Sigma\right]^{-1} $$

$$ G=G_{0}+G_{0} \Sigma G $$

$$ \Gamma[X]=\sum_{n=0}^{\infty} \frac{1}{n!} \int d \tau_{1} \ldots d \tau_{n} \Gamma^{(n)}\left(\tau_{1}, \ldots, \tau_{n}\right) X\left(\tau_{1}\right) \ldots X\left(\tau_{n}\right) $$

$$ \begin{align*} \Gamma_{0}^{(2)}\left(\tau_{1}, \tau_{2}\right) & =M\left(-\partial_{\tau_{1}}^{2}+\omega^{2}\right) \delta\left(\tau_{1}-\tau_{2}\right), \\ \Gamma_{0}^{(4)}\left(\tau_{1}, \tau_{2}, \tau_{3}, \tau_{4}\right) & =\lambda \delta\left(\tau_{1}-\tau_{2}\right) \delta\left(\tau_{1}-\tau_{3}\right) \delta\left(\tau_{1}-\tau_{4}\right) . \end{align*} $$

$$ \Gamma_{0}[X]=\frac{M}{2!} \int d \tau\left[\left(\partial_{\tau} X\right)^{2}+\omega^{2} X^{2}\right]+\frac{\lambda}{4!} \int d \tau X^{4} $$

$$ \Gamma[\mathbf{M}]=\int d^{3} x\left[\frac{1}{2} \sum_{i=1}^{3}\left(\partial_{i} \mathbf{M}\right)^{2}+\frac{m^{2}}{2!} \mathbf{M}^{2}+\frac{\lambda}{4!} \mathbf{M}^{4}\right] . $$

$$ G^{(1, n)}\left(\tau, \tau_{1}, \ldots, \tau_{n}\right)=\frac{1}{2}\left\langle x^{2}(\tau) x\left(\tau_{1}\right) \cdots x\left(\tau_{n}\right)\right\rangle . $$

$$ \int d \tau G^{(1, n)}\left(\tau, \tau_{1}, \ldots, \tau_{n}\right)=-\left.Z^{-1} \frac{\partial}{M \partial \omega^{2}} \frac{\delta}{\delta j\left(\tau_{1}\right)} \cdots \frac{\delta}{\delta j\left(\tau_{n}\right)} Z[j]\right|_{j=0} . $$

$$ \int d \tau G_{\mathrm{c}}^{(1, n)}\left(\tau, \tau_{1}, \ldots, \tau_{n}\right)=-\left.\frac{\partial}{M \partial \omega^{2}} \frac{\delta}{\delta j\left(\tau_{1}\right)} \cdots \frac{\delta}{\delta j\left(\tau_{n}\right)} W[j]\right|_{j=0} . $$

$$ \int d \tau G_{\mathrm{c}}^{(1, n)}\left(\tau, \tau_{1}, \ldots, \tau_{n}\right)=-\frac{\partial}{M \partial \omega^{2}} G_{\mathrm{c}}^{(n)}\left(\tau_{1}, \ldots, \tau_{n}\right) $$

$$ \int d \tau \Gamma^{(1, n)}\left(\tau, \tau_{1}, \ldots, \tau_{n}\right)=-\left.\frac{\partial}{M \partial \omega^{2}} \frac{\delta}{\delta X\left(\tau_{1}\right)} \cdots \frac{\delta}{\delta X\left(\tau_{n}\right)} \Gamma[X]\right|_{X_{0}} $$

$$ \int d \tau \Gamma^{(1, n)}\left(\tau, \tau_{1}, \ldots, \tau_{n}\right)=-\frac{\partial}{M \partial \omega^{2}} \Gamma^{(n)}\left(\tau_{1}, \ldots, \tau_{n}\right) $$

$$ Z[j]=e^{i W[j] / \hbar} $$

$$ X(t) \equiv\langle x(t)\rangle $$

$$ X(t)=\delta W[j] / \delta j(t) $$

$$ j(t)=j[X](t) $$

$$ \Gamma[X] \equiv W[j]-\int d t j(t) X(t) $$

$$ V^{\mathrm{eff}}(X) \equiv-\frac{1}{t_{b}-t_{a}} \Gamma[X] $$

$$ \frac{\delta \Gamma[X]}{\delta X(t)}=-j(t) . $$

$$ W[j]=\Gamma[X]+\int d t j(t) X(t) $$

$$ Z[j]=\int \mathcal{D} x(t) e^{(i / \hbar)\left\{\mathcal{A}[x]+\int d t j(t) x(t)\right\}} $$

$$ e^{\frac{i}{\hbar}\left\{\Gamma[X]+\int d t j(t) X(t)\right\}}=\int \mathcal{D} x(t) e^{(i / \hbar)\left\{\mathcal{A}[x]+\int d t j(t) x(t)\right\}} $$

$$ \left.\frac{\delta \mathcal{A}[x]}{\delta x(t)}\right|_{x=x_{\mathrm{cl}}(t)}=-j(t), $$

$$ W[j]=\Gamma[X]+\int d t j(t) X(t) \approx \mathcal{A}\left[x_{\mathrm{cl}}[j]\right]+\int d t j(t) x_{\mathrm{cl}}(t)[j] $$

$$ X=\frac{\delta W}{\delta j}=\frac{\delta \Gamma}{\delta X} \frac{\delta X}{\delta j}+X+j \frac{\delta X}{\delta j} $$

$$ X=\frac{\delta \mathcal{A}}{\delta x_{\mathrm{cl}}} \frac{\delta x_{\mathrm{cl}}}{\delta j}+x_{\mathrm{cl}}+j \frac{\delta x_{\mathrm{cl}}}{\delta j}=x_{\mathrm{cl}} $$

$$ \Gamma_{0}[X]=\mathcal{A}[X] . $$

$$ \Gamma_{0}[\mathbf{X}]=\int d t\left[\frac{1}{2}\left(\dot{X}_{a}^{2}-\omega^{2} X_{a}^{2}\right)-\frac{g}{4!}\left(X_{a}^{2}\right)^{2}\right] $$

$$ V_{0}^{\mathrm{eff}}(\mathbf{X})=V(\mathbf{X})=\frac{\omega^{2}}{2} X_{a}^{2}+\frac{g}{4!}\left(X_{a}^{2}\right)^{2} . $$

$$ \begin{align*} \Gamma^{(2)}\left(t_{1}, t_{2}\right)_{a b} & \left.\equiv \frac{\delta^{2} \Gamma}{\delta X_{a}\left(t_{1}\right) \delta X_{b}\left(t_{2}\right)}\right|_{X_{a}=0}=\left.\frac{\delta^{2} \mathcal{A}}{x_{a}\left(t_{1}\right) x_{b}\left(t_{2}\right)}\right|_{x_{a}=X_{a}=0} \\ & =\left(-\partial_{t}^{2}-\omega^{2}\right) \delta_{a b} \delta\left(t_{1}-t_{2}\right) \end{align*} $$

$$ \Gamma^{(2)}\left(t_{1}, t_{2}\right)_{a b}=\left[i \hbar G^{-1}\right]_{a b}\left(t_{1}, t_{2}\right) $$

$$ G_{a b}\left(t_{1}, t_{2}\right)=G_{0 a b}\left(t_{1}, t_{2}\right) . $$

$$ \Gamma^{(4)}\left(t_{1}, t_{2}, t_{3}, t_{4}\right)_{a b c d} \equiv \frac{\delta^{4} \Gamma}{\delta X_{a}\left(t_{1}\right) \delta X_{b}\left(t_{2}\right) \delta X_{c}\left(t_{3}\right) \delta X_{d}\left(t_{4}\right)}=g T_{a b c d} $$

$$ T_{a b c d}=\frac{1}{3}\left(\delta_{a b} \delta_{c d}+\delta_{a c} \delta_{b d}+\delta_{a d} \delta_{b c}\right) $$

$$ \left|\mathbf{X}_{0}\right|=\sqrt{-6 \omega^{2} / g} . $$

$$ \begin{align*} \Gamma^{(2)}\left(t_{1}, t_{2}\right)_{a b} & \left.\equiv \frac{\delta^{2} \Gamma}{\delta X_{a}\left(t_{1}\right) \delta X_{b}\left(t_{2}\right)}\right|_{X_{a} \neq 0}=\left.\frac{\delta^{2} \mathcal{A}}{x_{a}\left(t_{1}\right) x_{b}\left(t_{2}\right)}\right|_{x_{a}=X_{a} \neq 0} \\ & =\left[-\partial_{t}^{2}-\omega^{2}-\frac{g}{6}\left(\delta_{a b} X_{c}^{2}+2 X_{a} X_{b}\right)\right] \delta\left(t_{1}-t_{2}\right) . \end{align*} $$

$$ P_{L a b}(\hat{\mathbf{X}})=\hat{X}_{a} \hat{X}_{b}, \quad P_{T a b}(\hat{\mathbf{X}})=\delta_{a b}-\hat{X}_{a} \hat{X}_{b} $$

$$ \Gamma^{(2)}\left(t_{1}, t_{2}\right)_{a b}=\Gamma_{L}^{(2)}\left(t_{1}, t_{2}\right)_{a b} P_{L a b}(\hat{\mathbf{X}})+\Gamma_{T}^{(2)}\left(t_{1}, t_{2}\right)_{a b} P_{T a b}(\hat{\mathbf{X}}) $$

$$ \Gamma_{T}^{(2)}\left(t_{1}, t_{2}\right)_{a b}=\left[-\partial_{t}^{2}-\left(\omega^{2}+\frac{g}{6} \mathbf{X}^{2}\right)\right] \delta\left(t_{1}-t_{2}\right) $$

$$ \Gamma_{L}^{(2)}\left(t_{1}, t_{2}\right)_{a b}=\left[-\partial_{t}^{2}-\left(\omega^{2}+3 \frac{g}{6} \mathbf{X}^{2}\right)\right] \delta\left(t_{1}-t_{2}\right) $$

$$ \begin{align*} \mathcal{G}_{L}\left(t_{1}, t_{2}\right)_{a b} & =\frac{i \hbar}{\Gamma_{L}\left(t_{1}, t_{2}\right)}=\frac{i \hbar}{-\partial_{t}^{2}-\omega_{L}^{2}(\mathbf{X})} \\ \mathcal{G}_{T}\left(t_{1}, t_{2}\right)_{a b} & =\frac{i \hbar}{\Gamma_{T}^{(2)}\left(t_{1}, t_{2}\right)}=\frac{i \hbar}{-\partial_{t}^{2}-\omega_{T}^{2}(\mathbf{X})} \end{align*} $$

$$ \omega_{L}^{2}(\mathbf{X}) \equiv \omega^{2}+3 \frac{g}{6} \mathbf{X}^{2}, \quad \omega_{T}^{2}(\mathbf{X}) \equiv \omega^{2}+\frac{g}{6} \mathbf{X}^{2} $$

$$ \left.\mathcal{G}_{L}\left(t_{1}, t_{2}\right)_{a b}\right|_{\mathbf{X}=\mathbf{0}}=\left.\mathcal{G}_{T}\left(t_{1}, t_{2}\right)_{a b}\right|_{\mathbf{X}=\mathbf{0}}=\left.\mathcal{G}\left(t_{1}, t_{2}\right)_{a b}\right|_{\mathbf{X}=\mathbf{0}}=\frac{i \hbar}{-\partial_{t}^{2}-\omega^{2}} $$

$$ \left.\mathcal{G}_{L}\left(t_{1}, t_{2}\right)_{a b}\right|_{\mathbf{X}=\mathbf{x}_{0}}=\frac{i \hbar}{-\partial_{t}^{2}+2 \omega^{2}},\left.\quad \mathcal{G}_{T}\left(t_{1}, t_{2}\right)_{a b}\right|_{\mathbf{X}=\mathbf{x}_{0}}=\frac{i \hbar}{-\partial_{t}^{2}} . $$

$$ \delta x(t) \equiv x(t)-x_{\mathrm{cl}}(t) $$

$$ \begin{align*} \mathcal{A}\left[x_{\mathrm{cl}}\right. & +\delta x]+\int d t j(t)\left[x_{\mathrm{cl}}(t)+\delta x(t)\right] \\ & =\mathcal{A}\left[x_{\mathrm{cl}}\right]+\int d t j(t) x_{\mathrm{cl}}(t)+\int d t\left\{j(t)+\left.\frac{\delta \mathcal{A}}{\delta x(t)}\right|_{x=x_{\mathrm{cl}}}\right\} \delta x(t) \\ & +\left.\frac{1}{2} \int d t d t^{\prime} \delta x(t) \frac{\delta^{2} \mathcal{A}}{\delta x(t) \delta x\left(t^{\prime}\right)}\right|_{x=x_{\mathrm{cl}}} \delta x\left(t^{\prime}\right)+\mathcal{O}\left((\delta x)^{3}\right) \end{align*} $$

$$ Z[j] \approx e^{(i / \hbar)\left\{\mathcal{A}\left[x_{\mathrm{cl}}\right]+\int d t j(t) x_{\mathrm{cl}}(t)\right\}} \int \mathcal{D} \delta x \exp \left\{\left.\frac{i}{\hbar} \int d t d t^{\prime} \delta x(t) \frac{\delta^{2} \mathcal{A}}{\delta x(t) \delta x\left(t^{\prime}\right)}\right|_{x=x_{\mathrm{cl}}} \delta x\left(t^{\prime}\right)\right\} $$

$$ \begin{align*} & e^{(i / \hbar)\left\{\mathcal{A}\left[x_{\mathrm{cl}}\right]+\int d t j(t) x_{\mathrm{cl}}(t)\right\}\left[\operatorname{det} \frac{\delta^{2} \mathcal{A}}{\delta x(t) \delta x\left(t^{\prime}\right)}\right]_{x=x_{\mathrm{cl}}}^{-1 / 2}} \\ & \quad=e^{(i / \hbar)\left\{\mathcal{A}\left[x_{\mathrm{cl}}\right]+\int d t j(t) x_{\mathrm{cl}}(t)+i(\hbar / 2) \operatorname{Tr} \log \left[\delta^{2} \mathcal{A} /\left.\delta x(t) \delta x\left(t^{\prime}\right)\right|_{x=x_{\mathrm{cl}}}\right\}\right.} \end{align*} $$

$$ \Gamma[X]+\int d t j(t) X(t)=\mathcal{A}\left[x_{\mathrm{cl}}[j]\right]+\int d t j(t) x_{\mathrm{cl}}(t)[j]+\frac{i \hbar}{2} \operatorname{Tr} \log \frac{\delta^{2} \mathcal{A}\left[x_{\mathrm{cl}}[j]\right]}{\delta x(t) \delta x\left(t^{\prime}\right)} $$

$$ W[j]=W_{0}[j]+\hbar W_{1}[j]+\mathcal{O}\left(\hbar^{2}\right) . $$

$$ X=x_{\mathrm{cl}}+\hbar X_{1}+\mathcal{O}\left(\hbar^{2}\right) $$

$$ \begin{align*} \Gamma[X]+\int d t j X= & \mathcal{A}\left[X-\hbar X_{1}\right]+\int d t j X-\hbar \int d t j X_{1} \\ & +\left.\frac{i}{2} \hbar \operatorname{Tr} \log \frac{\delta^{2} \mathcal{A}}{\delta x_{a} \delta x_{b}}\right|_{x=X-\hbar X_{1}}+\mathcal{O}\left(\hbar^{2}\right) \end{align*} $$

$$ \Gamma[X]=\mathcal{A}[X]-\hbar \int d t\left\{\frac{\delta \mathcal{A}[X]}{\delta X}+j\right\} X_{1}+\left.\frac{i}{2} \hbar \operatorname{Tr} \log \frac{\delta^{2} \mathcal{A}}{\delta x_{a} \delta x_{b}}\right|_{x=X}+\mathcal{O}\left(\hbar^{2}\right) $$

$$ \begin{align*} \Gamma[X]=\Gamma_{0}[X]+\hbar \Gamma_{1}[X] & =\int d t\left[\frac{1}{2} \dot{X}^{2}-\frac{\omega^{2}}{2} X_{a}^{2}-\frac{g}{4!}\left(X_{a}^{2}\right)^{2}\right] \\ & +\frac{i}{2} \hbar \operatorname{Tr} \log \left[-\partial_{t}^{2}-\omega^{2}-\frac{g}{6}\left(\delta_{a b} X_{c}^{2}+2 X_{a} X_{b}\right)\right] \end{align*} $$

$$ \begin{align*} \hbar \Gamma_{1}[X] & =\frac{i}{2} \hbar \operatorname{Tr} \log \Gamma_{L}^{(2)}\left(t_{1}, t_{2}\right)_{a b}+\frac{i}{2}(N-1) \hbar \operatorname{Tr} \log \Gamma_{T}^{(2)}\left(t_{1}, t_{2}\right)_{a b} \\ & =\frac{i}{2} \hbar \operatorname{Tr} \log \left(-\partial_{t}^{2}-\omega_{L}^{2}(\mathbf{X})\right)+\frac{i}{2}(N-1) \hbar \operatorname{Tr} \log \left(-\partial_{t}^{2}-\omega_{T}^{2}(\mathbf{X})\right) \end{align*} $$

$$ \begin{align*} & \frac{i}{2} \hbar \operatorname{Tr} \log \left(-\partial_{t}^{2}-\omega^{2}-\frac{g}{2} X^{2}\right)=\frac{i}{2} \hbar \operatorname{Tr} \log \left(-\partial_{t}^{2}-\omega^{2}\right)+\frac{i}{2} \hbar \operatorname{Tr} \log \left(1+\frac{i}{-\partial_{t}^{2}-\omega^{2}} i g \frac{X^{2}}{2}\right) \\ & \quad=i \frac{\hbar}{2} \operatorname{Tr} \log \left(-\partial_{t}^{2}-\omega^{2}\right)-i \frac{\hbar}{2} \sum_{n=1}^{\infty}\left(-i \frac{g}{2}\right)^{n} \frac{1}{n} \operatorname{Tr}\left(\frac{i}{-\partial_{t}^{2}-\omega^{2}} X^{2}\right)^{n} \end{align*} $$

$$ G_{0}=\frac{i}{-\partial_{t}^{2}-\omega^{2}} $$

$$ i \frac{\hbar}{2} \operatorname{Tr} \log \left(-\partial_{t}^{2}-\omega^{2}\right)-i \frac{\hbar}{2} \sum_{n=1}^{\infty}\left(-i \frac{g}{2}\right)^{n} \frac{1}{n} \operatorname{Tr}\left(G_{0} X^{2}\right)^{n} . $$

$$ \begin{align*} & -\frac{\hbar}{4} g \int d t d t^{\prime} \delta\left(t-t^{\prime}\right) G_{0}\left(t, t^{\prime}\right) X^{2}\left(t^{\prime}\right) \\ & +i \hbar \frac{g^{2}}{16} \int d t d t^{\prime} d t^{\prime \prime} \delta^{4}\left(t-t^{\prime \prime}\right) G_{0}\left(t, t^{\prime}\right) X^{2}\left(t^{\prime}\right) G_{0}\left(t^{\prime}, t^{\prime \prime}\right) X^{2}\left(t^{\prime \prime}\right)+\ldots \end{align*} $$

$$ i \frac{\hbar}{2} \operatorname{Tr} \log \left(-\partial_{t}^{2}-\omega^{2}\right)=\frac{1}{2} \bigcirc $$

$$ \begin{align*} \Gamma^{(2)}(q) & =q^{2}-\omega^{2}-\hbar \frac{g}{2} \int \frac{d k}{2 \pi} \frac{i}{k^{2}-\omega^{2}+i \eta} \\ \Gamma^{(4)}\left(q_{i}\right) & =g-i \frac{g^{2}}{2}\left[\int \frac{d k}{2 \pi} \frac{i}{k^{2}-\omega^{2}+i \eta} \frac{i}{\left(q_{1}+q_{2}-k\right)^{2}-\omega^{2}+i \eta}+2 \text { perm }\right] . \end{align*} $$

$$ \begin{align*} \Gamma^{(2)}(q) & =-q^{2}-\omega^{2}-\hbar \frac{g}{2} \int \frac{d k}{2 \pi} \frac{1}{k^{2}+\omega^{2}} \\ & =-\left(q^{2}+\omega^{2}+\hbar \frac{g}{2} \frac{1}{2 \omega}\right) \\ \Gamma^{(4)}\left(q_{i}\right) & =g-\hbar \frac{g^{2}}{2}\left[I\left(q_{1}+q_{2}\right)+2 \text { perm }\right] \end{align*} $$

$$ I\left(q_{1}+q_{2}\right)=\int \frac{d k}{2 \pi} \frac{1}{k^{2}+\omega^{2}} \frac{i}{\left(q_{1}+q_{2}-k\right)^{2}+\omega^{2}} $$

$$ e^{(i / \hbar)\{\Gamma[X]+j X\}}=e^{i(\hbar / 2) W[j]}=e^{(i / \hbar)\left\{\left(\mathcal{A}\left[x_{\mathrm{cl}}\right]+j x_{\mathrm{cl}}\right)+(i \hbar / 2) \operatorname{Tr} \log \mathcal{A}_{x x}\left[x_{\mathrm{cl}}\right]\right\}} e^{(i / \hbar) \hbar^{2} W_{2}\left[x_{\mathrm{cl}}\right]} $$

$$ e^{(i / \hbar) \hbar^{2} W_{2}\left[x_{\mathrm{cl}}\right]}=\frac{\int \mathcal{D} x \exp \frac{i}{\hbar}\left\{\frac{1}{2} \delta x \mathcal{D}\left[x_{\mathrm{cl}}\right] \delta x+\mathcal{R}\left[x_{\mathrm{cl}}, \delta x\right]\right\}}{\int \mathcal{D} \delta x \exp \frac{i}{\hbar}\left\{\frac{1}{2} \delta x \mathcal{A}_{x x}\left[x_{\mathrm{cl}}\right] \delta x\right\}} $$

$$ \mathcal{D}\left[x_{\mathrm{cl}}\right] \equiv \mathcal{A}_{x x}\left[x_{\mathrm{cl}}\right]=-\partial_{t}^{2}-\omega^{2}-\frac{g}{2} x_{\mathrm{cl}}^{2} $$

$$ \begin{align*} \mathcal{R}\left[x_{\mathrm{cl}}, \delta x\right]=\mathcal{A}\left[x_{\mathrm{cl}}+\delta x\right]-\mathcal{A}\left[x_{\mathrm{cl}}\right] & -\int d t \mathcal{A}_{x}\left[x_{\mathrm{cl}}\right](t) \delta x(t) \\ & -\frac{1}{2} \int d t d t^{\prime} \delta x(t) \mathcal{A}_{x x}\left[x_{\mathrm{cl}}\right]\left(t, t^{\prime}\right) \delta x\left(t^{\prime}\right) \end{align*} $$

$$ \frac{1}{2} \int d t d t^{\prime} \delta x(t) \mathcal{A}_{x x}\left[x_{\mathrm{cl}}\right]\left(t, t^{\prime}\right) \delta x\left(t^{\prime}\right) \rightarrow \frac{1}{2} \delta x \mathcal{A}_{x x}\left[x_{\mathrm{cl}}\right] \delta x $$

$$ W[j]=\mathcal{A}\left[x_{\mathrm{cl}}\right]+x_{\mathrm{cl}} j+\hbar \Delta_{1}\left[x_{\mathrm{cl}}\right] $$

$$ \Delta_{1}\left[x_{\mathrm{cl}}\right]=\frac{i}{2} \operatorname{Tr} \log \mathcal{D}\left[x_{\mathrm{cl}}\right]+\hbar W_{2}\left[x_{\mathrm{cl}}\right] $$

$$ X=\frac{\delta W[j]}{\delta j}=x_{\mathrm{cl}}+\hbar \Delta_{1 x_{\mathrm{cl}}}\left[x_{\mathrm{cl}}\right] \frac{\delta x_{\mathrm{cl}}}{\delta j} $$

$$ X_{1}=\Delta_{1 x_{\mathrm{cl}}}\left[x_{\mathrm{cl}}\right] \frac{\delta x_{\mathrm{cl}}}{\delta j} $$

$$ W\left[x_{\mathrm{cl}}\right]=\mathcal{A}\left[x_{\mathrm{cl}}\right]+x_{\mathrm{cl}} j\left[x_{\mathrm{cl}}\right]+\hbar \Delta_{1}\left[x_{\mathrm{cl}}\right] $$

$$ \begin{gather*} \Gamma[X]=\mathcal{A}[X]-\hbar \mathcal{A}_{X}[X] X_{1}-\hbar X_{1} j[X]+\hbar^{2} X_{1} j_{X}[X] X_{1}+\frac{1}{2} \hbar^{2} X_{1} \mathcal{D}[X] X_{1} \\ +\hbar \Delta_{1}[X]-\hbar^{2} \Delta_{1 X}[X] X_{1}+\mathcal{O}\left(\hbar^{3}\right) \end{gather*} $$

$$ \mathcal{A}_{X}\left[X-\hbar X_{1}\right]=-j[X]+\mathcal{O}\left(\hbar^{2}\right) $$

$$ \mathcal{A}_{X}[X]=-j[X]+\hbar \mathcal{A}_{X X}[X] X_{1}+\mathcal{O}\left(\hbar^{2}\right)=-j[X]+\hbar \mathcal{D}[X] X_{1}+\mathcal{O}\left(\hbar^{2}\right) $$

$$ \Gamma[X]=\mathcal{A}[X]+\hbar \Delta_{1}[X]+\hbar^{2}\left\{-\frac{1}{2} X_{1} \mathcal{D}[X] X_{1}+X_{1} j_{X}[X] X_{1}-\Delta_{1 X} X_{1}\right\} . $$

$$ \frac{\delta j}{\delta x_{\mathrm{cl}}} X_{1}=\Delta_{1 x_{\mathrm{cl}}}\left[x_{\mathrm{cl}}\right] $$

$$ \frac{\delta j}{\delta X} X=\Delta_{1 X}[X]+\mathcal{O}(\hbar) $$

$$ -\frac{\hbar^{2}}{2} X_{1} \mathcal{D}[X] X_{1}+\hbar^{2} W_{2}[X]+\mathcal{O}\left(\hbar^{3}\right) $$

$$ \frac{\delta j}{\delta x_{\mathrm{cl}}}=-\mathcal{A}_{x x}\left[x_{\mathrm{cl}}\right]=-\mathcal{D}\left[x_{\mathrm{cl}}\right] $$

$$ X_{1}=-\mathcal{D}^{-1}[X] \Delta_{1 X}[X]+\mathcal{O}(\hbar) $$

$$ \Delta_{1 X}[X]=\frac{i}{2} \operatorname{Tr}\left(\mathcal{D}^{-1}[X] \frac{\delta}{\delta X} \mathcal{D}[X]\right)+\hbar W_{2 X}[X]+\mathcal{O}\left(\hbar^{2}\right) $$

$$ \begin{align*} \Gamma[X] & =\mathcal{A}[X]+\hbar \Gamma_{1}[X]+\hbar^{2} \Gamma_{2}[X] \\ & =\mathcal{A}[X]+i \frac{\hbar}{2} \operatorname{Tr} \log \mathcal{D}[X]+\hbar^{2} W_{2}[X] \\ & +\frac{\hbar^{2}}{2} \frac{1}{2} \operatorname{Tr}\left(\mathcal{D}^{-1}[X] \frac{\delta}{\delta X} \mathcal{D}[X]\right) \mathcal{D}^{-1}[X] \frac{1}{2} \operatorname{Tr}\left(\mathcal{D}^{-1}[X] \frac{\delta}{\delta X} \mathcal{D}[X]\right) \end{align*} $$

$$ \mathcal{R}[X ; \delta x]=\frac{1}{3!} \mathcal{A}_{X X X}[X] \delta x \delta x \delta x+\frac{1}{4!} \mathcal{A}_{X X X X}[X] \delta x \delta x \delta x \delta x+\ldots $$

$$ (i / \hbar) \mathcal{A}_{X X X X}[X]=(i / \hbar) \mathcal{D}_{X X}[X] $$

$$ (i / \hbar) \mathcal{A}_{X X X}[X]=(i / \hbar) \mathcal{D}_{X}[X] $$

$$ \frac{\hbar^{2}}{8} \mathcal{D}_{X_{1} X_{2}}^{-1} \mathcal{A}_{X_{1} X_{2} X_{3}} \mathcal{D}_{X_{3} X_{3^{\prime}}}^{-1} \mathcal{A}_{X_{3^{\prime}} X_{1^{\prime}} X_{2^{\prime}}} \mathcal{D}_{X_{1^{\prime}} X_{2^{\prime}}}^{-1} $$

$$ i \Gamma_{2}[X]=i \frac{3}{4!} \mathcal{D}_{12}^{-1} \mathcal{A}_{X_{1} X_{2} X_{3} X_{4}} \mathcal{D}_{34}^{-1}+i \frac{1}{4!^{2}} \mathcal{A}_{X_{1} X_{2} X_{3}} \mathcal{D}_{X_{1} X_{1^{\prime}}}^{-1} \mathcal{D}_{X_{2} X_{2^{\prime}}}^{-1} \mathcal{D}_{X_{3} X_{3^{\prime}}}^{-1} \mathcal{A}_{X_{1} X_{2} X_{3}} $$

$$ \mathcal{G}_{L}\left(\tau_{1}, \tau_{2}\right)=\frac{\hbar}{2 M \omega_{L}} \frac{\cosh \left(\omega_{L}\left|\tau_{1}-\tau_{2}\right|-\hbar \beta \omega_{L} / 2\right)}{\sinh \left(\hbar \beta \omega_{L} / 2\right)} $$

$$ \mathcal{G}_{T}\left(\tau_{1}, \tau_{2}\right)=\frac{\hbar}{2 M \omega_{T}} \frac{\cosh \left(\omega_{T}\left|\tau_{1}-\tau_{2}\right|-\hbar \beta \omega_{T} / 2\right)}{\sinh \left(\hbar \beta \omega_{T} / 2\right)}, $$

$$ \omega_{L}^{2}(\mathbf{X}) \equiv \frac{1}{M} v^{\prime \prime}(X), \quad \omega_{T}^{2}(\mathbf{X}) \equiv \frac{1}{M X} v^{\prime}(X) $$

$$ \frac{\partial^{3} v(X)}{\partial X_{i} \partial X_{j} \partial X_{k}}=P_{i j k}^{L} v^{\prime \prime \prime}(X)+P_{i j k}^{T}\left[\frac{v^{\prime \prime}(X)}{X}-\frac{v^{\prime}(X)}{X^{2}}\right], $$

$$ P_{i j k}^{L} \equiv \frac{X_{i} X_{j} X_{k}}{X^{3}} \quad \text { and } \quad P_{i j k}^{T} \equiv \delta_{i j} \frac{X_{k}}{X}+\delta_{i k} \frac{X_{j}}{X}+\delta_{j k} \frac{X_{i}}{X}-3 P_{i j k}^{L} . $$

$$ \frac{\partial^{4} v(X)}{\partial X_{i} \partial X_{j} \partial X_{k} \partial X_{l}}=P_{i j k l}^{L} v^{(4)}(X)+P_{i j k l}^{T} \frac{v^{\prime \prime \prime}(X)}{X}+P_{i j k l}^{S}\left[\frac{v^{\prime \prime}(X)}{X^{2}}-\frac{v^{\prime}(X)}{X^{3}}\right], $$

$$ \begin{align*} P_{i j k l}^{L} & =\frac{X_{i} X_{j} X_{k} X_{l}}{X^{4}} \\ P_{i j k l}^{T} & =\delta_{i j} \frac{X_{k} X_{l}}{X^{2}}+\delta_{i k} \frac{X_{j} X_{l}}{X^{2}}+\delta_{i l} \frac{X_{j} X_{k}}{X^{2}}+\delta_{j k} \frac{X_{i} X_{l}}{X^{2}}+\delta_{j l} \frac{X_{i} X_{k}}{X^{2}}+\delta_{k l} \frac{X_{i} X_{k}}{X^{2}}-6 P_{i j k l}^{L}, \\ P_{i j k l}^{S} & =\delta_{i j} \delta_{k l}+\delta_{i k} \delta_{j l}+\delta_{i l} \delta_{j k}-3 P_{i j k l}^{L}-3 P_{i j k l}^{T} . \end{align*} $$

$$ \begin{gather*} \frac{X_{i}}{X} P_{i j k}^{L}=P_{j k}^{L}, \quad \frac{X_{i}}{X} P_{i j k}^{T}=P_{j k}^{T}, \\ P_{i j}^{L} P_{i k l}^{L}=P_{j k l}^{L}, \quad P_{i j}^{T} P_{i k l}^{T}=\frac{X_{k}}{X} P_{j l}^{T}+\frac{X_{l}}{X} P_{j k}^{T}, P_{i j}^{L} P_{i k l}^{T}=\frac{X_{j}}{X} P_{k l}^{T}, P_{i j}^{T} P_{i k l}^{L}=0, \\ P_{h i j}^{L} P_{h k l}^{L}=P_{i j k l}^{L}, \quad P_{h i j}^{T} P_{h k l}^{T}=P_{i j}^{T} P_{k l}^{T}+P_{i k}^{L} P_{j l}^{T}+P_{i l}^{L} P_{j k}^{T}+P_{j k}^{L} P_{i l}^{T}+P_{j l}^{L} P_{i k}^{T}, \\ P_{h i j}^{L} P_{h k l}^{T}=P_{i j}^{L} P_{k l}^{T}, \quad P_{h i j}^{T} P_{h k l}^{L}=P_{i j}^{T} P_{k l}^{L}, P_{i j}^{L} P_{i j k l}^{L}=P_{k l}^{L}, P_{i j}^{T} P_{i j k l}^{T}=(D-1) P_{k l}^{L}, \\ P_{i j}^{L} P_{i j k l}^{T}=P_{k l}^{T}, \quad P_{i j}^{T} P_{i j k l}^{L}=0, \\ P_{i j}^{L} P_{i j k l}^{S}=-2 P_{k l}^{T}, \quad P_{i j}^{T} P_{i j k l}^{S}=(D+1) P_{k l}^{T}-2(D-1) P_{k l}^{L} . \end{gather*} $$

$$ (i / \hbar) \Gamma[\mathbf{X}] \rightarrow-\beta F(\mathbf{X}) $$

$$ -\beta F_{\mathrm{MF}}=-\int_{0}^{\hbar \beta} d \tau\left[\frac{M}{2} \dot{\mathbf{X}}^{2}+v(X)\right] $$

$$ -\beta F_{1-\text { loop }}=-\log \left[2 \sinh \left(\hbar \beta \omega_{L} / 2\right)\right]-(D-1) \log \left[2 \sinh \left(\hbar \beta \omega_{T} / 2\right)\right] $$

$$ \begin{align*} -\beta \Delta_{1} F_{2-\text { loop }} & =-\beta\left\{\mathcal{G}_{L}^{2}(\tau, \tau) v^{(4)}(X)+\left(D^{2}-1\right) \mathcal{G}_{T}^{2}(\tau, \tau)\left[\frac{v^{\prime \prime}(X)}{X^{2}}-\frac{v^{\prime}(X)}{X^{3}}\right]\right. \\ & \left.+2(D-1) \mathcal{G}_{L}(\tau, \tau) \mathcal{G}_{T}(\tau, \tau)\left[\frac{v^{\prime \prime \prime}(X)}{X}-\frac{2 v^{\prime \prime}(X)}{X^{2}}+\frac{2 v^{\prime}(X)}{X^{3}}\right]\right\} \end{align*} $$

$$ \begin{align*} -\beta \Delta_{2} F_{2-\text { loop }} & =\frac{1}{\hbar^{2}} \int_{0}^{\hbar \beta} d \tau_{1} \int_{0}^{\hbar \beta} d \tau_{2}\left\{\mathcal{G}_{L}^{3}\left(\tau_{1}, \tau_{2}\right)\left[v^{\prime \prime \prime}(X)\right]^{2}\right. \\ & \left.+3(D-1) \mathcal{G}_{L}\left(\tau_{1}, \tau_{2}\right) \mathcal{G}_{T}^{2}\left(\tau_{1}, \tau_{2}\right)\left[\frac{v^{\prime \prime}(X)}{X}-\frac{v^{\prime}(X)}{X^{2}}\right]^{2}\right\} \end{align*} $$

$$ \begin{align*} -\beta \Delta_{1} F_{2-\text { loop }}= & -\frac{\hbar^{2} \beta}{(2 M)^{2}}\left\{\frac{1}{\omega_{L}^{2}} \operatorname{coth}^{2}\left(\hbar \beta \omega_{L} / 2\right) v^{(4)}(X)\right. \\ & +\frac{D^{2}-1}{\omega_{T}^{2}} \operatorname{coth}^{2}\left(\hbar \beta \omega_{T} / 2\right)\left[\frac{v^{\prime \prime}(X)}{X^{2}}-\frac{v^{\prime}(X)}{X^{3}}\right] \\ +\frac{2(D-1)}{\omega_{L} \omega_{T}} & \left.\operatorname{coth}\left(\hbar \beta \omega_{L} / 2\right) \operatorname{coth}\left(\hbar \beta \omega_{T} / 2\right)\left[\frac{v^{\prime \prime \prime}(X)}{X}-\frac{2 v^{\prime \prime}(X)}{X^{2}}+\frac{2 v^{\prime}(X)}{X^{3}}\right]\right\} \end{align*} $$

$$ \begin{align*} &-\beta \Delta_{2} F_{2-\text { loop }}=\frac{2 \hbar^{2} \beta}{\omega_{L}} \frac{1}{\left(2 M \omega_{L}\right)^{3}}\left[v^{\prime \prime \prime}(X)\right]^{2}\left[\frac{1}{3}+\frac{1}{\sinh ^{2}\left(\hbar \beta \omega_{L} / 2\right)}\right] \\ &+\frac{6 \hbar^{2} \beta(D-1)}{2 \omega_{T}+\omega_{L}} \frac{1}{2 M \omega_{L}} \frac{1}{\left(2 M \omega_{T}\right)^{2}}\left[\frac{v^{\prime \prime}(X)}{X}-\frac{v^{\prime}(X)}{X^{2}}\right]^{2} \\ & \times\left[\operatorname{coth}^{2}\left(\hbar \beta \omega_{T} / 2\right)+\frac{\omega_{T}}{\omega_{L}} \frac{1}{\sinh ^{2}\left(\hbar \beta \omega_{T} / 2\right)}+\frac{\omega_{T}}{2 \omega_{T}-\omega_{L}} \frac{\sinh \left[\hbar \beta\left(2 \omega_{T}-\omega_{L}\right) / 2\right]}{\sinh \left(\hbar \beta \omega_{L} / 2\right) \sinh ^{2}\left(\hbar \beta \omega_{T} / 2\right)}\right] \end{align*} $$

$$ \begin{align*} & V_{\mathrm{eff}}(X) \underset{T \rightarrow 0}{=} v(X)+\frac{\hbar \omega_{L}}{2}+(D-1) \frac{\hbar \omega_{T}}{2}+\frac{\hbar^{2}}{8(2 M)^{2}}\left\{\frac{1}{\omega_{L}^{2}} v^{(4)}(X)\right. \\ & \left.+\frac{D^{2}-1}{\omega_{T}^{2}}\left[\frac{v^{\prime \prime}(X)}{X^{2}}-\frac{v^{\prime}(X)}{X^{3}}\right]+\frac{2(D-1)}{\omega_{L} \omega_{T}}\left[\frac{v^{\prime \prime \prime}(X)}{X}-\frac{2 v^{\prime \prime}(X)}{X^{2}}+\frac{2 v^{\prime}(X)}{X^{3}}\right]\right\} \\ & -\frac{\hbar^{2}}{6(2 M)^{3}}\left\{\frac{1}{3 \omega_{L}^{4}}\left[v^{\prime \prime \prime}(X)\right]^{2}+\frac{3(D-1)}{2 \omega_{T}+\omega_{L}} \frac{1}{\omega_{L} \omega_{T}^{2}}\left[\frac{v^{\prime \prime}(X)}{X}-\frac{v^{\prime}(X)}{X^{2}}\right]^{2}\right\}+\mathcal{O}\left(\hbar^{3}\right) \end{align*} $$

$$ V(x)=\frac{M}{2} \omega^{2} x^{2}+\frac{g_{3}}{3!} x^{3}+\frac{g_{4}}{4!} x^{4}, $$

$$ \begin{align*} V_{\mathrm{eff}}(X)= & \frac{M}{2} \omega^{2} X^{2}+g_{3} X^{3}+g_{4} X^{4}+\frac{1}{\beta} \log (2 \sinh \hbar \beta \omega / 2)+\hbar^{2} \frac{g_{4}}{8(2 M \omega)^{2}} \frac{1}{\tanh ^{2}(\hbar \beta \omega / 2)} \\ & -\frac{\hbar^{2}}{6 \omega} \frac{\left(g_{3}+g_{4} X\right)^{2}}{(2 M \omega)^{3}}\left[\frac{1}{3}+\frac{1}{\sinh ^{2}(\hbar \beta \omega / 2)}\right]+\mathcal{O}\left(\hbar^{3}\right) \end{align*} $$

$$ \begin{align*} V_{\mathrm{eff}}(X) \underset{T \rightarrow 0}{=} & \frac{M}{2} \omega^{2} X^{2}+\frac{g_{3}}{3!} X^{3}+\frac{g_{4}}{4!} X^{4}+\frac{\hbar \omega}{2}+\hbar^{2} \frac{g_{4}}{8(2 M \omega)^{2}} \\ & -\frac{\hbar^{2}}{18 \omega} \frac{\left(g_{3}+g_{4} X\right)^{2}}{(2 M \omega)^{3}}+\mathcal{O}\left(\hbar^{3}\right) \end{align*} $$

$$ \Gamma[\mathbf{X}]=\mathcal{A}[\mathbf{X}]+\Gamma^{\mathrm{fl}}[\mathbf{X}] . $$

$$ \mathbf{x}(t)=\mathbf{X}(t)+\delta \mathbf{x}(t) $$

$$ \exp \left\{\frac{i}{\hbar} W[\mathbf{j}]\right\}=\int \mathcal{D} \delta \mathbf{x} \exp \left\{\frac{i}{\hbar}(\mathcal{A}[\mathbf{X}+\delta \mathbf{x}]+\mathbf{j}[\mathbf{X}](\mathbf{X}+\delta \mathbf{x}))\right\} $$

$$ \exp \left\{\frac{i}{\hbar}\left(\Gamma\left[\mathbf{X}^{\mathbf{j}}\right]+\mathbf{j}\left[\mathbf{X}^{\mathbf{j}}\right] \mathbf{X}^{\mathbf{j}}\right)\right\}=\int \mathcal{D} \delta \mathbf{x} \exp \left\{\frac{i}{\hbar}(\mathcal{A}[\mathbf{X}+\delta \mathbf{x}]+\mathbf{j}[\mathbf{X}](\mathbf{X}+\delta \mathbf{x}))\right\} $$

$$ \mathbf{j}=-\left.\frac{\delta \Gamma[\mathbf{X}]}{\partial \mathbf{X}}\right|_{\mathbf{X}=\mathbf{X}^{\mathbf{j}}}=-\Gamma_{\mathbf{X}}\left[\mathbf{X}^{\mathbf{j}}\right] . $$

$$ \exp \left\{\frac{i}{\hbar} \Gamma[\mathbf{X}]\right\}=\int \mathcal{D} \delta \mathbf{x} \exp \left(\frac{i}{\hbar}\left\{\mathcal{A}[\mathbf{X}+\delta \mathbf{x}]-\Gamma_{\mathbf{X}}[\mathbf{X}] \delta \mathbf{x}\right\}\right) $$

$$ =\mathcal{G}_{a b}[\mathbf{X}] \equiv i \hbar\left[\frac{\delta^{2} \mathcal{A}[\mathbf{X}]}{\delta X_{a} \delta X_{b}}\right]_{a b}^{-1} $$

$$ 1 \overbrace{4}^{n} 5=\frac{\delta^{n} \mathcal{A}[\mathbf{X}]}{\delta X_{a_{1}} \delta X_{a_{2}} \ldots \delta X_{a_{n}}} $$

$$ \exp \{i \tilde{W}[\mathbf{X}, \tilde{\mathbf{j}}] / \hbar\} \equiv \int \mathcal{D} \delta \mathbf{x} \exp \left(\frac{i}{\hbar}\left\{\tilde{\mathcal{A}}[\mathbf{X}, \delta \mathbf{x}]+\int d t \tilde{\mathbf{j}}(t) \delta \mathbf{x}(t)\right\}\right) $$

$$ \tilde{\mathcal{A}}[\mathbf{X}, \delta \mathbf{x}]=\mathcal{A}[\mathbf{X}+\delta \mathbf{x}]-\mathcal{A}[\mathbf{X}]-\mathcal{A}_{\mathbf{X}}[\mathbf{X}] \delta \mathbf{x} $$

$$ \tilde{\mathbf{j}}=-\Gamma_{\mathbf{X}}[\mathbf{X}]+\mathcal{A}_{\mathbf{X}}[\mathbf{X}]=-\tilde{\Gamma}_{\mathbf{X}}[\mathbf{X}], $$

$$ \tilde{\Gamma}[\mathbf{X}, \tilde{\mathbf{X}}] \equiv \tilde{W}[\mathbf{X}, \tilde{\mathbf{j}}]-\int d t \tilde{\mathbf{j}} \tilde{\mathbf{X}} $$

$$ \tilde{\mathbf{X}}=\frac{\delta \tilde{W}[\mathbf{X}, \tilde{\mathbf{j}}]}{\delta \tilde{\mathbf{j}}}=\tilde{\mathbf{X}}[\mathbf{X}, \tilde{\mathbf{j}}] $$

$$ \begin{align*} \frac{i}{\hbar} \sum_{n \geq 3} i \hbar^{n} \Gamma_{n}[\mathbf{X}]= & \frac{1}{8} \bigcirc+\frac{1}{12} \bigcirc+\frac{1}{48} \bigcirc+\frac{1}{16} \bigcirc \\ & +\frac{1}{8} \bigcirc+\frac{1}{8} \text { \& } \frac{1}{24} \text { \& } \frac{1}{16} \text { Q. } \end{align*} $$

$$ \begin{align*} \Gamma[X] & =\mathcal{A}[X]+\frac{i}{2} \hbar \operatorname{Tr} \log \Gamma^{(2)}\left(t_{b}, t_{a}\right) \\ & =\int_{t_{a}}^{t_{b}} d t\left[\frac{M}{2} \dot{X}^{2}-\frac{M \omega^{2}}{2} X^{2}\right]+\frac{i}{2} \hbar \operatorname{Tr} \log \left(-\partial_{t}^{2}-\omega^{2}\right) \end{align*} $$

$$ V^{\mathrm{eff}}(X)=V(X)-\frac{i}{2\left(t_{b}-t_{a}\right)} \log \left\{2 \pi i \sin \left[\omega\left(t_{b}-t_{a}\right)\right] / M \omega\right\} . $$

$$ V^{\mathrm{eff}}(X)=V(X)-\frac{i}{\left(t_{b}-t_{a}\right)} \log \left\{2 i \sin \left[\omega\left(t_{b}-t_{a}\right) / 2\right]\right\} $$

$$ j_{a} \rightarrow j_{a}-i \epsilon_{c d}\left(L_{c d}\right)_{a b} j_{b} $$

$$ \left(L_{c d}\right)_{a b}=i\left(\delta_{c a} \delta_{d b}-\delta_{d a} \delta_{c b}\right) $$

$$ \delta W[\mathbf{j}]=0=\int d t \frac{\delta W[\mathbf{j}]}{\delta j_{a}(x)} i\left(L_{c d}\right)_{a b} j_{b} \epsilon_{c d}=0 $$

$$ \int d t X_{a}(t) i\left(L_{c d}\right)_{a b} \frac{\delta \Gamma[\mathbf{X}]}{\delta X_{b}(t)} \epsilon_{c d}=0 $$

$$ \begin{align*} \left(L_{c d}\right)_{a b} j_{b}(t) & =\left(L_{c d}\right)_{a b} \frac{\delta \Gamma[\mathbf{X}]}{\delta X(t)_{b}} \\ & =-\int d t^{\prime} X_{a^{\prime}}\left(t^{\prime}\right)\left(L_{c d}\right)_{a^{\prime} b} \frac{\delta^{2} \Gamma[\mathbf{X}]}{\delta X_{b}\left(t^{\prime}\right) \delta X_{n}(t)} \end{align*} $$

$$ \left.\int d t^{\prime} \bar{X}_{a^{\prime}}\left(t^{\prime}\right)\left(L_{c d}\right)_{a^{\prime} b} \frac{\delta^{2} \Gamma[\mathbf{X}]}{\delta X_{b}\left(t^{\prime}\right) \delta X_{a}(t)}\right|_{\mathbf{X}(t)=\overline{\mathbf{X}}}=0 $$

$$ \tilde{\Gamma}^{(2)}\left(\omega^{\prime}\right) \equiv \int d t^{\prime} e^{i \omega^{\prime} t} \Gamma^{(2)}\left(t^{\prime}, t\right) $$

$$ X_{a^{\prime}}^{0}\left(L_{c d}\right)_{a^{\prime} b} \tilde{G}_{b a}^{-1}\left(\omega^{\prime}=0\right)=0 $$

$$ V^{\mathrm{eff}}(X)=V(X)+\frac{1}{\beta} \log \left(2 \sinh \frac{\beta \hbar \omega}{2}\right) . $$

$$ Z=\exp \left[-\left.\beta V(X)\right|_{\min }\right] $$

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

$$ \begin{align*} \left\langle\delta x(\tau) \delta x\left(\tau^{\prime}\right)\right\rangle & =G_{\Omega^{2}(X)}^{(2)}\left(\tau, \tau^{\prime}\right)=\frac{\hbar}{M} G_{\Omega^{2}(X), \mathrm{e}}^{\mathrm{p}}\left(\tau-\tau^{\prime}\right) \\ & =\frac{\hbar}{M} \frac{1}{2 \Omega(X)} \frac{\cosh \Omega(X)\left(\left|\tau-\tau^{\prime}\right|-\hbar \beta / 2\right)}{\sinh [\Omega(X) \hbar \beta / 2]}, \quad\left|\tau-\tau^{\prime}\right| \in[0, \hbar \beta] \end{align*} $$

$$ \Omega^{2}(X)=\omega^{2}+3 \frac{g}{6} X^{2} $$

$$ \left\langle[\delta x(\tau)]^{2}\right\rangle=\frac{\hbar}{M} \frac{1}{2 \Omega(X)} \operatorname{coth} \frac{\Omega(X) \hbar \beta}{2} $$

$$ \left\langle[\delta x(\tau)]^{2}\right\rangle \xrightarrow{T \rightarrow \infty} \frac{k_{B} T}{M \Omega^{2}} $$

$$ \frac{M \Omega^{2}}{2}\left\langle x^{2}\right\rangle=\frac{k_{B} T}{2} $$

$$ G_{\Omega^{2}, \mathrm{e}}^{\mathrm{p}}\left(\tau-\tau^{\prime}\right)=\frac{1}{\hbar \beta} \sum_{m=-\infty}^{\infty} \frac{1}{\omega_{m}^{2}+\Omega^{2}} e^{-i \omega_{m}\left(\tau-\tau^{\prime}\right)} $$

$$ G_{\Omega^{2}, \mathrm{e}}^{\mathrm{p} \prime}(\tau) \equiv G_{\Omega^{2}, \mathrm{e}}^{\mathrm{p}}(\tau)-\frac{1}{\hbar \beta \Omega^{2}}=\frac{1}{2 \Omega} \frac{\cosh \Omega(|\tau|-\hbar \beta / 2)}{\sinh [\Omega \hbar \beta / 2]}-\frac{1}{\hbar \beta \Omega^{2}} $$

$$ a_{\Omega}^{2} \equiv G_{\Omega^{2}, \mathrm{e}}^{\mathrm{p} \prime}(0)=\frac{1}{2 \Omega} \operatorname{coth} \frac{\Omega \hbar \beta}{2}-\frac{1}{\hbar \beta \Omega^{2}} $$

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

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

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

$$ B\left(x_{0}\right) \equiv e^{-V^{\text {eff cl }}\left(x_{0}\right) / k_{B} T} $$

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

$$ V^{\mathrm{eff} \mathrm{cl}}\left(x_{0}\right) \xrightarrow{T \rightarrow \infty} V\left(x_{0}\right) . $$

$$ B\left(x_{0}\right) \equiv \oint \mathcal{D}^{\prime} x e^{-\mathcal{A}_{\mathrm{e}} / \hbar} $$

$$ \bar{x} \equiv \frac{1}{\hbar \beta} \int_{0}^{\hbar \beta} d \tau x(\tau) $$

$$ \tilde{\delta}\left(\bar{x}-x_{0}\right) \equiv l_{\mathrm{e}}(\hbar \beta) \delta\left(\bar{x}-x_{0}\right)=\sqrt{\frac{2 \pi \hbar^{2} \beta}{M}} \delta\left(\bar{x}-x_{0}\right) $$

$$ \begin{align*} B\left(x_{0}\right) \equiv e^{-V^{\mathrm{eff} \mathrm{cl}}\left(x_{0}\right) / k_{B} T} & =\oint \mathcal{D}^{\prime} x e^{-\mathcal{A}_{\mathrm{e}} / \hbar}=\oint \mathcal{D} x \tilde{\delta}\left(\bar{x}-x_{0}\right) e^{-\mathcal{A}_{\mathrm{e}} / \hbar} \\ & =\oint \mathcal{D} \eta \tilde{\delta}(\bar{\eta}) e^{-\mathcal{A}_{\mathrm{e}} / \hbar} \end{align*} $$

$$ \mathcal{A}_{\mathrm{e}}\left[x_{0}+\eta\right]=\hbar \beta \frac{M \omega^{2}}{2} x_{0}^{2}+\frac{M}{2} \int_{0}^{\hbar \beta} d \tau\left[\dot{\eta}^{2}(\tau)+\omega^{2} \eta^{2}(\tau)\right] $$

$$ \tilde{\delta}(\bar{\eta})=l_{\mathrm{e}}(\hbar \beta) \int_{-i \infty}^{i \infty} \frac{d \lambda}{2 \pi i} \exp \left(\lambda \frac{1}{\hbar \beta} \int d \tau \eta(\tau)\right) $$

$$ \begin{align*} B_{\omega}\left(x_{0}\right)=\oint \mathcal{D} \eta \tilde{\delta}(\bar{\eta}) e^{-\mathcal{A}_{\mathrm{e}} / \hbar} & =e^{-\beta M \omega^{2} x_{0}^{2} / 2} l_{\mathrm{e}}(\hbar \beta) \int_{-i \infty}^{i \infty} \frac{d \lambda}{2 \pi i} \\ & \times \oint \mathcal{D} \eta \exp \left\{-\frac{1}{\hbar} \int_{0}^{\hbar \beta} d \tau\left[\frac{M}{2} \dot{\eta}^{2}(\tau)-\frac{\lambda}{\beta} \eta(\tau)\right]\right\} \end{align*} $$

$$ \frac{1}{2 \sinh (\beta \hbar \omega / 2)} \exp \left\{\frac{\lambda^{2}}{2 M \hbar \beta^{2}} \int_{0}^{\hbar \beta} d \tau \int_{0}^{\hbar \beta} d \tau^{\prime} G_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}\left(\tau-\tau^{\prime}\right)\right\} $$

$$ \int_{0}^{\hbar \beta} d \tau \int_{0}^{\hbar \beta} d \tau^{\prime} G_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}\left(\tau-\tau^{\prime}\right)=\int_{0}^{\hbar \beta} d \tau \int_{0}^{\hbar \beta} d \tau^{\prime} \frac{1}{\hbar \beta} \sum_{m=-\infty}^{\infty} \frac{1}{\omega_{m}^{2}+\omega^{2}} e^{-i \omega_{m}\left(\tau-\tau^{\prime}\right)}=\frac{\hbar \beta}{\omega^{2}} $$

$$ \frac{1}{2 \sinh (\beta \hbar \omega / 2)} \int_{-i \infty}^{i \infty} \frac{d \lambda}{2 \pi i} \exp \left(\frac{\lambda^{2}}{2 M \omega^{2} \beta}\right)=\frac{1}{2 \sinh (\beta \hbar \omega / 2)} \frac{1}{l_{\mathrm{e}}(\hbar \beta)} \omega \hbar \beta . $$

$$ B_{\omega}\left(x_{0}\right) \equiv e^{-V_{\omega}^{\text {eff cl }}\left(x_{0}\right) / k_{B} T}=\oint \mathcal{D} \eta \tilde{\delta}(\bar{\eta}) e^{-\mathcal{A}_{\mathrm{e}} / \hbar}=\frac{\beta \hbar \omega / 2}{\sinh (\beta \hbar \omega / 2)} e^{-\beta M \omega^{2} x_{0}^{2}} $$

$$ B\left(p_{0}, x_{0}\right) \equiv \exp \left[-\beta H^{\mathrm{eff} \mathrm{cl}}\left(p_{0}, x_{0}\right)\right] \equiv \oint \mathcal{D} x \oint \frac{\mathcal{D} p}{2 \pi \hbar} \delta\left(x_{0}-\bar{x}\right) 2 \pi \hbar \delta\left(p_{0}-\bar{p}\right) e^{-\mathcal{A}_{\mathrm{e}}[p, x] / \hbar} $$

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

$$ Z=\int_{-\infty}^{\infty} d x_{0} \int_{-\infty}^{\infty} \frac{d p_{0}}{2 \pi \hbar} e^{-\beta H^{\mathrm{eff} \mathrm{cl}}\left(p_{0}, x_{0}\right)} $$

$$ B_{\omega}\left(p_{0}, x_{0}\right) \equiv e^{-H_{\omega}^{\mathrm{eff} \mathrm{cl}}\left(p_{0}, x_{0}\right) / k_{B} T}=l_{\mathrm{e}}(\hbar \beta) \frac{\beta \hbar \omega / 2}{\sinh (\beta \hbar \omega / 2)} e^{-\beta\left(p_{0}^{2} / 2 M+M \omega^{2} x_{0}^{2}\right)} . $$

$$ H(\mathbf{p}, \mathbf{x})=\frac{1}{2 M} \mathbf{p}^{2}+\frac{M}{2} \omega_{\perp}^{2} \mathbf{x}_{\perp}^{2}(\tau)+\frac{M}{2} \omega_{\|}^{2} z^{2}(\tau)+\omega_{B} l_{z}(\mathbf{p}(\tau), \mathbf{x}(\tau)) $$

$$ B\left(\mathbf{p}_{0}, \mathbf{x}_{0}\right)=e^{-\beta H^{\mathrm{eff} \mathrm{cl}}\left(\mathbf{p}_{0}, \mathbf{x}_{0}\right)}=l_{\mathrm{e}}^{3}(\hbar \beta) \frac{\hbar \beta \omega_{+} / 2}{\sinh \hbar \beta \omega_{+} / 2} \frac{\hbar \beta \omega_{-} / 2}{\sinh \hbar \beta \omega_{-} / 2} \frac{\hbar \beta \omega_{\|} / 2}{\sinh \hbar \beta \omega_{\|} / 2} e^{-\beta H\left(\mathbf{p}_{0}, \mathbf{x}_{0}\right)}, $$

$$ \begin{align*} V_{\omega}^{\mathrm{eff} \mathrm{cl}}\left(x_{0}\right) & =k_{B} T \log \frac{\sinh \left(\hbar \omega / 2 k_{B} T\right)}{\hbar \omega / 2 k_{B} T}+\frac{M}{2} \omega^{2} x_{0}^{2} \\ & =\frac{M}{2} \omega^{2} x_{0}^{2}+\frac{\hbar \omega}{2}+k_{B} T\left[\log \left(1-e^{-\hbar \omega / k_{B} T}\right)-\log \frac{\hbar \omega}{k_{B} T}\right] \end{align*} $$

$$ V_{\omega}^{\mathrm{eff} \mathrm{cl}}\left(x_{0}\right)=\frac{M}{2} \omega^{2} x_{0}^{2}+\hbar \omega\left[\frac{1}{24} \frac{\hbar \omega}{k_{B} T}-\frac{1}{2880}\left(\frac{\hbar \omega}{k_{B} T}\right)^{3}+\ldots\right] . $$

$$ \begin{align*} V_{\omega}^{\mathrm{eff}}\left(x_{0}\right) & =k_{B} T \log \left[2 \sinh \left(\hbar \omega / 2 k_{B} T\right)\right]+\frac{M}{2} \omega^{2} x_{0}^{2} \\ & =\frac{M}{2} \omega^{2} x_{0}^{2}+\frac{\hbar \omega}{2}+k_{B} T \log \left(1-e^{-\hbar \omega / k_{B} T}\right), \end{align*} $$

$$ V^{\mathrm{eff} \mathrm{cl}}\left(x_{0}\right) \underset{T \rightarrow 0}{\rightarrow} V^{\mathrm{eff}}\left(x_{0}\right) \equiv \Gamma_{\mathrm{e}}[X] /\left.\beta\right|_{X=x_{0}}, $$

$$ V_{\omega}^{\mathrm{eff} \mathrm{cl}}\left(x_{0}\right) \xrightarrow{T \rightarrow 0} \frac{\hbar \omega}{2}+\frac{M}{2} \omega^{2} x_{0}^{2}-k_{B} T \log \frac{\hbar \omega}{k_{B} T}, $$

$$ \begin{align*} Z_{\omega} & \xrightarrow{T \rightarrow 0} e^{-\hbar \omega / 2 k_{B} T} \frac{\hbar \omega}{k_{B} T} \int_{-\infty}^{\infty} \frac{d x_{0}}{l_{\mathrm{e}}(\hbar \beta)} e^{-M \omega^{2} x_{0}^{2} / 2 k_{B} T} \\ & =e^{-\hbar \omega / 2 k_{B} T} \end{align*} $$

$$ \tilde{V}_{\omega}^{\mathrm{eff} \mathrm{cl}}(x) \equiv k_{B} T \log \left[l_{\mathrm{e}}(\hbar \beta) z(x)\right] $$

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

$$ \tilde{V}_{\omega}^{\mathrm{eff} \mathrm{cl}}(x)=-\frac{k_{B} T}{2} \log \frac{2 \hbar \omega}{k_{B} T}+\frac{\hbar \omega}{2}+k_{B} T\left[\log \left(1-e^{-2 \hbar \omega / k_{B} T}\right)+\frac{M \omega}{\hbar} \tanh \frac{\hbar \omega}{k_{B} T} x^{2}\right] . $$

$$ \tilde{V}^{\mathrm{eff} \mathrm{cl}}\left(x_{0}\right) \xrightarrow{T \rightarrow 0} \frac{\hbar \omega}{2}+k_{B} T \frac{M \omega}{\hbar} x^{2}-\frac{k_{B} T}{2} \log \frac{2 \hbar \omega}{k_{B} T}, $$

$$ \begin{align*} Z_{\omega} \xrightarrow{T \rightarrow 0} & e^{-\hbar \omega / 2 k_{B} T} \sqrt{\frac{2 \hbar \omega}{k_{B} T}} \int_{-\infty}^{\infty} \frac{d x}{l_{\mathrm{e}}(\hbar \beta)} e^{-M \omega x^{2} / \hbar} \\ & =e^{-\hbar \omega / 2 k_{B} T} \end{align*} $$

$$ \tilde{B}(x) \equiv l_{\mathrm{e}}(\hbar \beta) z(x)=e^{-\tilde{V}^{\mathrm{eff} \mathrm{cl}}(x) / k_{B} T} $$

$$ \left\langle\eta(\tau) \eta\left(\tau^{\prime}\right)\right\rangle_{\omega}=\frac{\hbar}{M} G_{\omega^{2}, \mathrm{e}}^{\mathrm{p} \prime}\left(\tau-\tau^{\prime}\right)=\frac{\hbar}{2 M \omega} \frac{\cosh \omega\left(\left|\tau-\tau^{\prime}\right|-\hbar \beta / 2\right)}{\sinh (\beta \hbar \omega / 2)}-\frac{1}{\hbar \beta \omega^{2}}, $$

$$ \left\langle\eta^{2}(\tau)\right\rangle_{\omega} \equiv a_{\omega}^{2}=G_{\omega^{2}, \mathrm{e}}^{\mathrm{p} \prime}(0)=\frac{1}{2 \omega} \operatorname{coth} \frac{\beta \hbar \omega}{2}-\frac{1}{\hbar \beta \omega^{2}}, $$

$$ \begin{align*} & \frac{1}{2 M \hbar \beta^{2}}\left\{\int_{0}^{\hbar \beta} d \tau \int_{0}^{\hbar \beta} d \tau^{\prime}\left[\lambda^{2}+\lambda \beta j(\tau)+\lambda \beta j\left(\tau^{\prime}\right)\right] G_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}\left(\tau-\tau^{\prime}\right)\right\} \\ & \quad \times \exp \left\{\frac{1}{2 M \hbar} \int_{0}^{\hbar \beta} d \tau \int_{0}^{\hbar \beta} d \tau^{\prime} j(\tau) G_{\omega^{2}, \mathrm{e}}^{\mathrm{p}}\left(\tau-\tau^{\prime}\right) j\left(\tau^{\prime}\right)\right\} \end{align*} $$

$$ \frac{1}{2 M \hbar \beta^{2}}\left\{\lambda^{2} \frac{\hbar \beta}{\omega^{2}}+2 \frac{\lambda \beta}{\omega^{2}} \int_{0}^{\hbar \beta} d \tau j(\tau)\right\} $$

$$ \exp \left\{-\frac{1}{2 M \beta \hbar^{2} \omega^{2}} \int_{0}^{\hbar \beta} d \tau \int_{0}^{\hbar \beta} d \tau^{\prime} j(\tau) j\left(\tau^{\prime}\right)\right\} $$

$$ Z_{\omega}^{x_{0}}[j]=\frac{\beta \hbar \omega / 2}{\sin (\beta \hbar \omega / 2)} e^{-\beta M \omega^{2} x_{0}^{2} / 2} \exp \left\{\frac{1}{2 M \hbar} \int_{0}^{\hbar \beta} d \tau \int_{0}^{\hbar \beta} d \tau^{\prime} j(\tau) G_{\omega^{2}, \mathrm{e}}^{\mathrm{p} \prime}\left(\tau-\tau^{\prime}\right) j\left(\tau^{\prime}\right)\right\} $$

$$ \mathcal{A}_{\mathrm{e}}[x]=\int_{0}^{\hbar \beta} d \tau\left[\frac{M}{2} \dot{x}^{2}+V(x)\right] $$

$$ \mathcal{A}_{\mathrm{e}}=\hbar \beta V\left(x_{0}\right)+\mathcal{A}_{\mathrm{e}}^{(0)}[\eta]+\mathcal{A}_{\text {int }, \mathrm{e}}\left[x_{0} ; \eta\right] $$

$$ \mathcal{A}_{\mathrm{e}}^{(0)}[\eta]=\int_{0}^{\hbar \beta} d \tau\left[\frac{M}{2} \dot{\eta}^{2}(\tau)+\frac{M}{2} \Omega^{2}\left(x_{0}\right) \eta^{2}(\tau)\right], \quad \Omega^{2}\left(x_{0}\right) \equiv V^{\prime \prime}\left(x_{0}\right) / M $$

$$ \mathcal{A}_{\mathrm{int}, \mathrm{e}}\left[x_{0} ; \eta\right]=\int_{0}^{\hbar \beta} d \tau V^{\mathrm{int}}\left(x_{0} ; \eta(\tau)\right) $$

$$ V^{\mathrm{int}}\left(x_{0} ; \eta(\tau)\right)=V\left(x_{0}+\eta(\tau)\right)-V\left(x_{0}\right)-V^{\prime}\left(x_{0}\right) \eta(\tau)-\frac{1}{2} V^{\prime \prime}\left(x_{0}\right) \eta^{2}(\tau) $$

$$ V^{\mathrm{int}}\left(x_{0} ; \eta\right)=\frac{1}{3!} V^{\prime \prime \prime}\left(x_{0}\right) \eta^{3}+\frac{1}{4!} V^{(4)}\left(x_{0}\right) \eta^{4}+\ldots $$

$$ B\left(x_{0}\right)=\left(1-\frac{1}{\hbar}\left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}\right\rangle_{\Omega}^{x_{0}}+\frac{1}{2!\hbar^{2}}\left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}^{2}\right\rangle_{\Omega}^{x_{0}}-\frac{1}{3!\hbar^{3}}\left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}^{3}\right\rangle_{\Omega}^{x_{0}}+\ldots\right) B_{\Omega}\left(x_{0}\right) $$

$$ B_{\Omega}\left(x_{0}\right)=\int \mathcal{D} \eta \tilde{\delta}(\bar{\eta}) e^{-\mathcal{A}_{\mathrm{e}}^{(0)}[\eta] / / \hbar} $$

$$ \langle F[x]\rangle_{\Omega}^{x_{0}}=B_{\Omega}^{-1}\left(x_{0}\right) \int \mathcal{D} \eta \tilde{\delta}(\bar{\eta}) F[x] e^{-\mathcal{A}_{\mathrm{e}}^{(0)}[\eta] / \hbar} $$

$$ \langle F[x]\rangle_{\Omega}^{x_{0}}=\left[Z_{\Omega}^{x_{0}}\right]^{-1} \prod_{m=1}^{\infty}\left[\int \frac{d x_{m}^{\mathrm{re}} d x_{m}^{\mathrm{im}}}{\pi k_{B} T / M \omega_{m}^{2}}\right] e^{-\frac{M}{k_{B} T} \Sigma_{m=1}^{\infty}\left[\omega_{m}^{2}+\Omega^{2}\left(x_{0}\right)\right]\left|x_{m}\right|^{2}} F[x] $$

$$ \left\langle x_{m} x_{m^{\prime}}^{*}\right\rangle_{\Omega}^{x_{0}}=\delta_{m m^{\prime}} \frac{k_{B} T}{M} \frac{1}{\omega_{m}^{2}+\Omega^{2}\left(x_{0}\right)} $$

$$ \left\langle\eta(\tau) \eta\left(\tau^{\prime}\right)\right\rangle_{\Omega}^{x_{0}}=\left\langle\sum_{m, m^{\prime} \neq 0}^{\infty} x_{m} x_{m^{\prime}}^{*} e^{-i\left(\omega_{m}-\omega_{m^{\prime}}\right) \tau}\right\rangle_{\Omega}^{x_{0}}=2 \frac{1}{M \beta} \sum_{m=1} \frac{1}{\omega_{m}^{2}+\Omega^{2}\left(x_{0}\right)} $$

$$ Z(j)=\int \mathcal{D} x(\tau) \exp \left\{-\int_{0}^{\beta} d \tau\left[\frac{1}{2} \dot{x}^{2}+V(x(\tau))\right]+\beta j \bar{x}\right\} $$

$$ Z(j)=\int_{-\infty}^{\infty} \frac{d x_{0}}{\sqrt{2 \pi \beta}} e^{-\beta\left[V^{\mathrm{eff} \mathrm{cl}}\left(x_{0}\right)-j x_{0}\right]} $$

$$ V^{\mathrm{eff}}(X)=-\frac{1}{\beta} W(j)+X j $$

$$ X=X(j)=\frac{1}{\beta} \frac{d}{d j} W(j) . $$

$$ X=Z(j)^{-1} \int_{-\infty}^{\infty} \frac{d x_{0}}{\sqrt{2 \pi \beta}} x_{0} \exp \left\{-\beta\left[V^{\text {eff cl }}\left(x_{0}\right)-j x_{0}\right]\right\} $$

$$ j(X)=\frac{d V^{\mathrm{eff}}(X)}{d X} . $$

$$ j=d V(X) / d X . $$

$$ j=-X+g X^{3} . $$

$$ B\left(x_{0}\right)=\left(1-\frac{1}{\hbar}\left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}\right\rangle_{\Omega}^{x_{0}}+\ldots\right) B_{\Omega}\left(x_{0}\right), $$

$$ V^{\mathrm{int}}\left(x_{0} ; \eta(\tau)\right)=\int_{-\infty}^{\infty} \frac{d k}{2 \pi} e^{i k\left(x_{0}+\eta(\tau)\right)} \tilde{V}^{\mathrm{int}}(k) $$

$$ \left\langle\mathcal{A}_{\mathrm{int}, \mathrm{e}}\left[x_{0} ; \eta\right]\right\rangle_{\Omega}^{x_{0}}=\int_{0}^{\hbar \beta} d \tau \int_{-\infty}^{\infty} \frac{d k}{2 \pi} \tilde{V}^{\mathrm{int}}(k) e^{i k x_{0}}\left\langle e^{i k \eta(\tau)}\right\rangle_{\Omega}^{x_{0}} $$

$$ \left\langle e^{i k \eta(\tau)}\right\rangle_{\Omega}^{x_{0}}=e^{-k^{2}\left\langle\eta^{2}(\tau)\right\rangle_{\Omega}^{x_{0}} / 2} $$

$$ \left\langle e^{i k \eta(\tau)}\right\rangle_{\Omega}^{x_{0}}=e^{-k^{2} a_{\Omega\left(x_{0}\right)}^{2} / 2} $$

$$ \left\langle\mathcal{A}_{\text {int }, \mathrm{e}}\left[x_{0} ; \eta\right]\right\rangle_{\Omega}^{x_{0}}=\int_{0}^{\hbar \beta} d \tau \int_{-\infty}^{\infty} \frac{d k}{2 \pi} \tilde{V}^{\text {int }}(k) e^{i k x_{0}-k^{2} a_{\Omega\left(x_{0}\right)}^{2} / 2} $$

$$ \tilde{V}^{\mathrm{int}}(k)=\int_{-\infty}^{\infty} d x V^{\mathrm{int}}\left(x_{0} ; \eta\right) e^{-i k\left(x_{0}+\eta\right)} $$

$$ \left\langle V^{\mathrm{int}}(x(\tau))\right\rangle_{\Omega}^{x_{0}} \equiv V_{a_{\Omega}^{2}}^{\mathrm{int}}\left(x_{0}\right)=\int_{-\infty}^{\infty} \frac{d x_{0}^{\prime}}{\sqrt{2 \pi a_{\Omega\left(x_{0}\right)}^{2}}} e^{-\eta^{2} / 2 a_{\Omega\left(x_{0}\right)}^{2}} V^{\mathrm{int}}\left(x_{0} ; \eta\right) $$

$$ B\left(x_{0}\right) \approx \frac{\Omega\left(x_{0}\right) \hbar \beta}{2 \sin \left[\Omega\left(x_{0}\right) \hbar \beta / 2\right]} \exp \left\{-\beta M \Omega\left(x_{0}\right)^{2} x_{0}^{2} / 2-\beta V_{a_{\Omega}^{2}}^{\mathrm{int}}\left(x_{0}\right)\right\} $$

$$ V^{\mathrm{eff} \mathrm{cl}}\left(x_{0}\right) \approx V_{\Omega\left(x_{0}\right)}^{\mathrm{eff} \mathrm{cl}}\left(x_{0}\right)+V_{a_{\Omega}^{2}}^{\mathrm{int}}\left(x_{0}\right) . $$

$$ V^{\mathrm{int}}\left(x_{0} ; \eta\right)=\sum_{k=3}^{\infty} \frac{1}{k!} V^{(k)}\left(x_{0}\right) \hbar^{k} $$

$$ \int_{-\infty}^{\infty} \frac{d \eta}{\sqrt{2 \pi a^{2}}} e^{-\eta^{2} / 2 a^{2}} \eta^{k}=\left\{\begin{array}{c} (k-1)!!a^{k} \\ 0 \end{array}\right\} \text { for } k=\left\{\begin{array}{c} \text { even } \\ \text { odd } \end{array}\right\} $$

$$ V_{a^{2}}^{\mathrm{int}}\left(x_{0}\right)=\sum_{k=4,6, \ldots}^{\infty} \frac{(k-1)!!}{k!} V^{(k)}\left(x_{0}\right) a^{k}\left(x_{0}\right) $$

$$ \int d^{3} y_{a} \int d^{3} z_{a} \int \mathcal{D}^{3} y \int \mathcal{D}^{3} z \exp \left[\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t \frac{M}{2}\left(\dot{\mathbf{y}}^{2}-\dot{\mathbf{z}}^{2}\right)\right] F[\mathbf{y}(t)-\mathbf{z}(0)] $$

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

$$ \begin{align*} Z\left[\mathbf{j}_{\mathbf{y}}\right] & =\int d^{3} y_{a}\left(\mathbf{y}_{b} t_{b} \mid \mathbf{y}_{a} t_{a}\right)_{\omega}^{\mathbf{j}_{\mathbf{y}}} \\ & =\exp \left\{\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t \frac{1}{\sin \omega\left(t_{b}-t_{a}\right)}\left[\mathbf{y}_{b}\left(\sin \left[\omega\left(t-t_{a}\right)\right]+\sin \left[\omega\left(t_{b}-t\right)\right]\right) \mathbf{j}_{\mathbf{y}}\right]\right\} \\ & \times \exp \left\{-\frac{i}{\hbar^{2}} \frac{\hbar}{M} \int_{t_{a}}^{t_{b}} d t \int_{t_{a}}^{t} d t^{\prime} \mathbf{j}_{\mathbf{y}}(t) \bar{G}_{\omega^{2}}\left(t, t^{\prime}\right) \mathbf{j}_{\mathbf{y}}\left(t^{\prime}\right)\right\} \end{align*} $$

$$ \bar{G}_{\omega^{2}}\left(t, t^{\prime}\right)=\frac{1}{\omega \sin \omega\left(t_{b}-t_{a}\right)} \sin \omega\left(t_{b}-t_{>}\right)\left[\sin \omega\left(t_{<}-t_{a}\right)+\sin \omega\left(t_{b}-t_{<}\right)\right] $$

$$ \bar{G}_{\omega^{2}}\left(t, t^{\prime}\right)=t_{b}-t_{>} . $$

$$ \begin{align*} Z[\mathbf{j}] & \equiv \int d^{3} y_{a} \int d^{3} z_{a} \int \mathcal{D}^{3} y \int \mathcal{D}^{3} z \\ & \times \exp \left(\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t\left\{\frac{M}{2}\left[\dot{\mathbf{y}}^{2}-\dot{\mathbf{z}}^{2}-\omega^{2}\left(\mathbf{y}^{2}-\mathbf{z}^{2}\right)-\mathbf{j} \mathbf{y}_{\mathbf{z}}\right]\right\}\right) \end{align*} $$

$$ \mathbf{y}_{\mathbf{z}}(t) \equiv \mathbf{y}(t)-\mathbf{z}(0) $$

$$ Z[\mathbf{j}]=\frac{1}{D_{\omega}} \exp \left\{-\frac{i}{\hbar^{2}} \frac{\hbar}{2 M} \int_{t_{a}}^{t_{b}} d t \int_{t_{a}}^{t_{b}} d t^{\prime} \mathbf{j}(t) \bar{G}_{\omega^{2}}^{\prime}\left(t, t^{\prime}\right) \mathbf{j}\left(t^{\prime}\right)\right\} $$

$$ D_{\omega}=\frac{1}{\cos ^{2}\left[\omega\left(t_{b}-t_{a}\right)\right]} \exp \left[\int_{0}^{t_{b}-t_{a}} \frac{d t}{t}(\cos \omega t-1)\right] $$

$$ \bar{G}_{\omega^{2}}^{\prime}\left(t, t^{\prime}\right) \equiv \bar{G}_{\omega^{2}}\left(t, t^{\prime}\right)-\bar{G}_{\omega^{2}}(0,0) . $$

$$ \bar{G}_{0}^{\prime}\left(t, t^{\prime}\right) \equiv-t_{>}, $$

$$ \begin{align*} f_{\mathbf{p}_{b} \mathbf{p}_{a}} & =\frac{p}{2 \pi i \hbar} \int d^{2} b e^{-i \mathbf{q} \mathbf{b} / \hbar} \\ & \times \int \mathcal{D}^{3} y_{\mathbf{z}} \exp \left\{\frac{i}{\hbar} \int_{-\infty}^{\infty} d t \frac{M}{2} \mathbf{y}_{\mathbf{z}}\left[\bar{G}_{0}^{\prime}\left(t, t^{\prime}\right)\right]^{-1} \mathbf{y}_{\mathbf{z}}\right\}\left[e^{i \chi_{\mathbf{b}, \mathbf{p}}\left[\mathbf{y}_{\mathbf{z}}\right]}-1\right] \end{align*} $$

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

$$ \chi_{\mathbf{b}, \mathbf{p}}[\mathbf{y}]=\chi_{\mathbf{b}, \mathbf{p}}^{\mathrm{ei}}-\frac{1}{\hbar} \int_{-\infty}^{\infty} d t \nabla V\left(\mathbf{b}+\frac{\mathbf{p}}{M} t\right) \mathbf{y}_{\mathbf{z}}(t) $$

$$ -\frac{1}{\hbar} \int_{-\infty}^{\infty} d t \mathbf{y}_{\mathbf{z}}(t) \mathbf{j}(t) $$

$$ \mathbf{j}(t)=\nabla V\left(\mathbf{b}+\frac{\mathbf{p}}{M} t\right) . $$

$$ \Delta_{1} \chi_{\mathbf{b}, \mathbf{p}}^{\mathrm{ei}}=\frac{1}{2 M \hbar} \int_{-\infty}^{\infty} d t_{1} \int_{-\infty}^{\infty} d t_{2} \nabla V\left(\mathbf{b}+\frac{\mathbf{p}}{M} t_{1}\right) \nabla V\left(\mathbf{b}+\frac{\mathbf{p}}{M} t_{2}\right) t_{>} $$

$$ \boldsymbol{\nabla}_{\|} V=z V^{\prime} / r, \quad \boldsymbol{\nabla}_{\perp} V=\mathbf{b} V^{\prime} / r $$

$$ \Delta_{1} \chi_{\mathbf{b}, \mathbf{p}}^{\mathrm{ei}}=\frac{M^{2}}{2 \hbar p^{3}} \int_{-\infty}^{\infty} d z_{1} \int_{-\infty}^{\infty} d z_{2} \frac{V^{\prime}\left(r_{1}\right)}{r_{1}} \frac{V^{\prime}\left(r_{2}\right)}{r_{2}}\left(b^{2}+z_{1} z_{2}\right) z_{1} $$

$$ \Delta_{1} \chi_{\mathbf{b}, \mathbf{p}}^{\mathrm{ei}}=\frac{M^{2}}{\hbar p^{3}} \int_{-\infty}^{\infty} d z_{1} z_{1} \frac{V^{\prime}\left(r_{1}\right)}{r_{1}} \int_{-\infty}^{\infty} d z_{2} \frac{V^{\prime}\left(r_{2}\right)}{r_{2}}\left(b^{2}-z_{2}^{2}\right) $$

$$ z V^{\prime} / r=\partial_{z} V, \quad b V^{\prime} / r=\partial_{b} V, $$

$$ \Delta_{1} \chi_{\mathbf{b}, \mathbf{p}}^{\mathrm{ei}}=-\frac{M^{2}}{\hbar p^{3}}\left(1+b \partial_{b}\right) \int_{-\infty}^{\infty} d z V^{2}\left(\sqrt{b^{2}+z^{2}}\right) $$

$$ \left\langle\mathbf{p}_{b}\right| \hat{S}\left|\mathbf{p}_{a}\right\rangle \equiv \lim _{t_{b}-t_{a} \rightarrow \infty} e^{i\left(E_{b}-E_{a}\right) t_{b} / \hbar}\left(\mathbf{p}_{b} 0 \mid \mathbf{p}_{a} t_{a}\right) e^{-i E_{a} t_{a} / \hbar} $$

$$ \left(\mathbf{p}_{b} 0 \mid \mathbf{p}_{a} t_{a}\right)=\int d^{3} x_{b} d^{3} x_{a} e^{-i \mathbf{p}_{b} \mathbf{x}_{b}}\left(\mathbf{x}_{b} 0 \mid \mathbf{x}_{a} t_{a}\right) e^{i \mathbf{p}_{a} \mathbf{x}_{a}} $$

$$ \begin{align*} &\left(\mathbf{p}_{b} 0 \mid \mathbf{p}_{a} t_{a}\right)=\left(\mathbf{p}_{b} 0 \mid \mathbf{p}_{a} t_{a}\right)_{0} \\ &+\frac{i}{\hbar}\left\langle\mathbf{p}_{b}\right| \mathcal{A}_{\mathrm{int}}\left|\mathbf{p}_{a}\right\rangle_{0}-\frac{1}{2!\hbar^{2}}\left\langle\mathbf{p}_{b}\right| \mathcal{A}_{\mathrm{int}}^{2}\left|\mathbf{p}_{a}\right\rangle_{0}-\frac{i}{3!\hbar^{3}}\left\langle\mathbf{p}_{b}\right| \mathcal{A}_{\mathrm{int}}^{3}\left|\mathbf{p}_{a}\right\rangle_{0}+\ldots \end{align*} $$

$$ \left(\mathbf{p}_{b} 0 \mid \mathbf{p}_{a} t_{a}\right)_{0}=(2 \pi \hbar)^{3} \delta^{(3)}\left(\mathbf{p}_{b}-\mathbf{p}_{a}\right) e^{i \mathbf{p}_{b}^{2} t_{a} / 2 M \hbar} $$

$$ \left\langle\mathbf{p}_{b}\right| \ldots\left|\mathbf{p}_{a}\right\rangle_{0} \equiv \int d^{3} x_{b} d^{3} x_{a} e^{-i \mathbf{p}_{b} \mathbf{x}_{b}}\left(\int \mathcal{D}^{3} x \ldots e^{i \mathcal{A}_{0} / \hbar}\right) e^{i \mathbf{p}_{a} \mathbf{x}_{a}} $$

$$ \begin{align*} \left\langle\mathbf{p}_{b}\right| \mathcal{A}_{\mathrm{int}}\left|\mathbf{p}_{a}\right\rangle_{0}=-\int_{t_{a}}^{0} d t_{1} \int d^{3} x_{b} d^{3} x_{a} d^{3} x_{1} e^{-i \mathbf{p}_{b} \mathbf{x}_{b}}\left(\mathbf{x}_{b} 0 \mid \mathbf{x}_{1} t_{1}\right)_{0} & \\ & \times V\left(\mathbf{x}_{1}\right)\left(\mathbf{x}_{1} t_{1} \mid \mathbf{x}_{a} t_{a}\right)_{0} e^{i \mathbf{p}_{a} \mathbf{x}_{a}} \end{align*} $$

$$ \begin{align*} \int d^{3} x_{b} e^{-i \mathbf{p}_{b} \mathbf{x}_{b}}\left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{1} t_{1}\right)_{0} & =e^{-i \mathbf{p}_{b} \mathbf{x}_{1}} e^{-i \mathbf{p}_{b}^{2}\left(t_{b}-t_{1}\right) / 2 M \hbar} \\ \int d^{3} x_{a}\left(\mathbf{x}_{1} t_{1} \mid \mathbf{x}_{a} t_{a}\right)_{0} e^{-i \mathbf{p}_{b} \mathbf{x}_{b}} & =e^{-i \mathbf{p}_{a} \mathbf{x}_{1}} e^{i \mathbf{p}_{a}^{2}\left(t_{1}-t_{a}\right) / 2 M \hbar} \end{align*} $$

$$ \left\langle\mathbf{p}_{b}\right| \mathcal{A}_{\mathrm{int}}\left|\mathbf{p}_{a}\right\rangle_{0}=-\int_{t_{a}}^{0} d t_{1} e^{i\left(\mathbf{p}_{b}^{2}-\mathbf{p}_{a}^{2}\right) t_{1} / 2 M \hbar} V_{\mathbf{p}_{b} \mathbf{p}_{a}} e^{i \mathbf{p}_{a}^{2} t_{a} / 2 M \hbar} $$

$$ V_{\mathbf{p}_{b} \mathbf{p}_{a}} \equiv\left\langle\mathbf{p}_{b}\right| \hat{V}\left|\mathbf{p}_{a}\right\rangle=\int d^{3} x e^{i\left(\mathbf{p}_{b}-\mathbf{p}_{a}\right) \mathbf{x} / \hbar} V(\mathbf{x})=\tilde{V}\left(\mathbf{p}_{b}-\mathbf{p}_{a}\right) $$

$$ \frac{i}{\hbar}\left\langle\mathbf{p}_{b}\right| \mathcal{A}_{\mathrm{int}}\left|\mathbf{p}_{a}\right\rangle_{0}=-\frac{1}{E_{b}-E_{a}-i \eta} V_{\mathbf{p}_{b} \mathbf{p}_{a}} e^{i E_{a} t_{a}} $$

$$ \left\langle\mathbf{p}_{b}\right| \hat{S}\left|\mathbf{p}_{a}\right\rangle \equiv \lim _{t_{b}-t_{a} \rightarrow \infty} e^{i\left(E_{b}-E_{a}\right) t_{b} / \hbar}\left[(2 \pi \hbar)^{3} \delta^{(3)}\left(\mathbf{p}_{b}-\mathbf{p}_{a}\right)-\frac{1}{E_{b}-E_{a}-i \eta} V_{\mathbf{p}_{b} \mathbf{p}_{a}}\right] $$

$$ \begin{align*} T_{\mathbf{p}_{b} \mathbf{p}_{a}} & =V_{\mathbf{p}_{b} \mathbf{p}_{a}}-\int \frac{d^{3} p_{c}}{(2 \pi \hbar)^{3}} V_{\mathbf{p}_{b} \mathbf{p}_{c}} \frac{1}{E_{c}-E_{a}-i \eta} V_{\mathbf{p}_{c} \mathbf{p}_{a}} \\ & +\int \frac{d^{3} p_{c}}{(2 \pi \hbar)^{3}} \int \frac{d^{3} p_{d}}{(2 \pi \hbar)^{3}} V_{\mathbf{p}_{b} \mathbf{p}_{c}} \frac{1}{E_{c}-E_{a}-i \eta} V_{\mathbf{p}_{c} \mathbf{p}_{d}} \frac{1}{E_{d}-E_{a}-i \eta} V_{\mathbf{p}_{d} \mathbf{p}_{a}}+\ldots \end{align*} $$

$$ T_{\mathbf{p}_{b} \mathbf{p}_{a}}=V_{\mathbf{p}_{b} \mathbf{p}_{a}}-\int \frac{d^{3} p_{c}}{(2 \pi \hbar)^{3}} V_{\mathbf{p}_{b} \mathbf{p}_{c}} \frac{1}{E_{c}-E_{a}-i \eta} T_{\mathbf{p}_{c} \mathbf{p}_{a}} $$

$$ \mathcal{O}(t) G_{\omega^{2}}\left(t, t^{\prime}\right) \equiv\left[-\partial_{t}^{2}-\Omega^{2}(t)\right] G_{\omega^{2}}\left(t, t^{\prime}\right)=\delta\left(t-t^{\prime}\right) $$

$$ \operatorname{Det} \mathcal{O}=e^{\operatorname{Tr} \log \mathcal{O}}, $$

$$ \begin{align*} \operatorname{Tr}\left\{\int_{0}^{1} d g \Omega^{2}(t)\left[-\partial_{t}^{2}-g \Omega^{2}(t)\right]^{-1} \delta\left(t-t^{\prime}\right)\right\}= & -\operatorname{Tr}\left\{\log \left[-\partial_{t}^{2}-\Omega^{2}(t)\right] \delta\left(t-t^{\prime}\right)\right\} \\ & +\operatorname{Tr}\left\{\log \left[-\partial_{t}^{2}\right] \delta\left(t-t^{\prime}\right)\right\} \end{align*} $$

$$ \mathcal{O}_{g}(t) G_{g}\left(t, t^{\prime}\right) \equiv\left[-\partial_{t}^{2}-g \Omega^{2}(t)\right] G_{g}\left(t, t^{\prime}\right)=\delta\left(t-t^{\prime}\right) $$

$$ \operatorname{Det}\left(\mathcal{O}_{0}^{-1} \mathcal{O}_{1}\right)=e^{-\int_{0}^{1} d g \operatorname{Tr}\left[\Omega^{2}(t) G_{g}\left(t, t^{\prime}\right)\right]} $$

$$ \mathcal{O}_{g}(t) D_{g}(t)=0 ; \quad D_{g}\left(t_{a}\right)=0, \quad \dot{D}_{g}\left(t_{a}\right)=1 $$

$$ \mathcal{O}_{g}(t) D_{g}^{\prime}(t)=\Omega^{2}(t) D_{g}(t) ; \quad D_{g}^{\prime}\left(t_{a}\right)=0, \quad \dot{D}_{g}^{\prime}\left(t_{a}\right)=0 $$

$$ D_{g}(t)=\frac{\xi_{g}\left(t_{a}\right) \eta_{g}(t)-\xi_{g}(t) \eta_{g}\left(t_{a}\right)}{W_{g}}=\Delta_{g}\left(t, t_{a}\right) $$

$$ W_{g}=\xi_{g}(t) \dot{\eta}_{g}(t)-\eta_{g}(t) \dot{\xi}_{g}(t) $$

$$ D_{g}\left(t_{b}\right)=\frac{\operatorname{Det} \Lambda_{g}}{W_{g}}=\Delta_{g}\left(t_{b}, t_{a}\right) $$

$$ \Lambda_{g}=\left(\begin{array}{ll} \xi_{g}\left(t_{a}\right) & \eta_{g}\left(t_{a}\right) \\ \xi_{g}\left(t_{b}\right) & \eta_{g}\left(t_{b}\right) \end{array}\right) $$

$$ D_{g}^{\prime}(t)=\int_{t_{a}}^{t} d t^{\prime} \Omega^{2}\left(t^{\prime}\right) \Delta_{g}\left(t, t^{\prime}\right) \Delta_{g}\left(t^{\prime}, t_{a}\right) $$

$$ D_{g}^{\prime}\left(t_{b}\right)=\Delta_{g}\left(t_{b}, t_{a}\right) \int_{t_{a}}^{t_{b}} d t^{\prime} \Omega^{2}\left(t^{\prime}\right) G_{g}\left(t^{\prime}, t^{\prime}\right) $$

$$ \operatorname{Tr}\left[\Omega^{2}(t) G_{g}\left(t, t^{\prime}\right)\right]=-\partial_{g} \log \left(\frac{\operatorname{det} \Lambda_{g}}{W_{g}}\right)=-\partial_{g} \log D_{g}\left(t_{b}\right) $$

$$ \operatorname{Det}\left(\mathcal{O}_{0}^{-1} \mathcal{O}_{g}\right)=C\left(t_{b}, t_{a}\right) D_{g}\left(t_{b}\right) $$

$$ D_{0}(t)=t-t_{a} $$

$$ \operatorname{Det}\left(\mathcal{O}_{0}^{-1} \mathcal{O}_{1}\right)=\frac{\operatorname{det} \Lambda_{1}}{W_{1}} / \frac{\operatorname{Det} \Lambda_{0}}{W_{0}}=\frac{D_{1}\left(t_{b}\right)}{t_{b}-t_{a}}, $$

$$ \mathcal{O}_{g}(\tau) G_{g}^{\mathrm{p}, \mathrm{a}}\left(\tau, \tau^{\prime}\right) \equiv\left[\partial_{\tau}^{2}-g \Omega^{2}(\tau)\right] G_{g}^{\mathrm{p}, \mathrm{a}}\left(\tau, \tau^{\prime}\right)=\delta^{\mathrm{p}, \mathrm{a}}\left(\tau-\tau^{\prime}\right) $$

$$ \operatorname{Det}\left(\mathcal{O}_{0}^{-1} \mathcal{O}_{1}\right)=e^{-\int_{0}^{1} d g \operatorname{Tr}\left[\Omega^{2}(\tau) G_{g}\left(\tau, \tau^{\prime}\right)\right]} $$

$$ \begin{align*} & G_{g}^{\mathrm{p}, \mathrm{a}}\left(\tau_{b}, \tau^{\prime}\right)= \pm G_{g}^{\mathrm{p}, \mathrm{a}}\left(\tau_{a}, \tau^{\prime}\right), \\ & \dot{G}_{g}^{\mathrm{p}, \mathrm{a}}\left(\tau_{b}, \tau^{\prime}\right)= \pm \dot{G}_{g}^{\mathrm{p}, \mathrm{a}}\left(\tau_{a}, \tau^{\prime}\right) . \end{align*} $$

$$ G_{g}^{\mathrm{p}, \mathrm{a}}\left(\tau, \tau^{\prime}\right)=G_{g}\left(\tau, \tau^{\prime}\right) \mp \frac{\left[\Delta_{g}\left(\tau, \tau_{a}\right) \pm \Delta_{g}\left(\tau_{b}, \tau\right)\right]\left[\Delta_{g}\left(\tau^{\prime}, \tau_{a}\right) \pm \Delta_{g}\left(\tau_{b}, \tau^{\prime}\right)\right]}{\bar{\Delta}_{g}^{\mathrm{p}, \mathrm{a}}\left(\tau_{a}, \tau_{b}\right) \cdot \Delta_{g}\left(\tau_{a}, \tau_{b}\right)} $$

$$ \Delta\left(\tau, \tau^{\prime}\right)=\frac{1}{W}\left[\xi(\tau) \eta\left(\tau^{\prime}\right)-\xi\left(\tau^{\prime}\right) \eta(\tau)\right] $$

$$ \bar{\Delta}_{g}^{\mathrm{p}, \mathrm{a}}\left(\tau_{a}, \tau_{b}\right)=2 \pm \partial_{\tau} \Delta_{g}\left(\tau_{a}, \tau_{b}\right) \pm \partial_{\tau} \Delta_{g}\left(\tau_{b}, \tau_{a}\right) $$

$$ \operatorname{det} \bar{\Lambda}_{g}^{\mathrm{p}, \mathrm{a}}=W_{g} \bar{\Delta}_{g}^{\mathrm{p}, \mathrm{a}}\left(\tau_{a}, \tau_{b}\right) \neq 0 $$

$$ \mathcal{O}_{g}(\tau) \bar{D}_{g}(\tau)=0 ; \quad \bar{D}_{g}\left(\tau_{a}\right)=1, \quad \dot{\bar{D}}_{g}\left(\tau_{a}\right)=0 $$

$$ \bar{D}_{g}(\tau)=\frac{\xi_{g}(\tau) \dot{\eta}_{g}\left(\tau_{a}\right)-\dot{\xi}_{g}\left(\tau_{a}\right) \eta_{g}(\tau)}{W_{g}} $$

$$ \dot{D}_{g}\left(\tau_{b}\right)+\bar{D}_{g}\left(\tau_{b}\right)= \pm\left[2-\bar{\Delta}_{g}^{\mathrm{p}, \mathrm{a}}\left(\tau_{a}, \tau_{b}\right)\right] $$

$$ \mathcal{O}_{g}(\tau) \bar{D}_{g}^{\prime}(\tau)=\Omega^{2}(\tau) \bar{D}_{g}^{\prime}(\tau) ; \quad \bar{D}_{g}^{\prime}\left(\tau_{a}\right)=1, \dot{\bar{D}}_{g}^{\prime}\left(\tau_{a}\right)=0 $$

$$ \bar{D}_{g}^{\prime}(\tau)=-\int_{\tau_{a}}^{\tau} d \tau^{\prime} \Omega^{2}\left(\tau^{\prime}\right) \Delta_{g}\left(\tau, \tau^{\prime}\right) \dot{\Delta}_{g}\left(\tau_{a}, \tau^{\prime}\right) $$

$$ \dot{D}_{g}^{\prime}\left(\tau_{b}\right)+\bar{D}_{g}^{\prime}\left(\tau_{b}\right)= \pm \bar{\Delta}_{g}^{\mathrm{p}, \mathrm{a}}\left(\tau_{a}, \tau_{b}\right) \int_{\tau_{a}}^{\tau_{b}} d \tau \Omega^{2}(\tau) G_{g}^{\mathrm{p}, \mathrm{a}}(\tau, \tau) $$

$$ \begin{align*} \operatorname{Tr}\left[\Omega^{2}(\tau) G_{g}^{\mathrm{p}, \mathrm{a}}\left(\tau, \tau^{\prime}\right)\right] & =-\partial_{g} \log \left(\frac{\operatorname{det} \bar{\Lambda}_{g}^{\mathrm{p}, \mathrm{a}}}{W g}\right) \\ & =-\partial_{g} \log \left[2 \mp \dot{D}_{g}\left(\tau_{b}\right) \mp \bar{D}_{g}\left(\tau_{b}\right)\right] \end{align*} $$

$$ \operatorname{Det}\left(\tilde{\mathcal{O}}^{-1} \mathcal{O}_{g}\right)=C\left(t_{b}, t_{a}\right)\left[2 \mp \dot{D}_{g}\left(\tau_{b}\right) \mp \bar{D}_{g}\left(\tau_{b}\right)\right] $$

$$ D_{1}^{\omega}(\tau)=\frac{1}{\omega} \sin \omega\left(\tau-\tau_{a}\right), \quad \bar{D}_{1}^{\omega}(\tau)=\cos \omega\left(\tau-\tau_{a}\right) $$

$$ 1=C\left(t_{b}, t_{a}\right) \begin{cases}4 \sin ^{2}\left[\omega\left(\tau_{b}-\tau_{a}\right) / 2\right] & \text { periodic case } \\ 4 \cos ^{2}\left[\omega\left(\tau_{b}-\tau_{a}\right) / 2\right] & \text { antiperiodic case. }\end{cases} $$

$$ \operatorname{Det}\left(\tilde{\mathcal{O}}^{-1} \mathcal{O}_{1}\right)=\frac{\operatorname{det} \bar{\Lambda}_{1}^{\mathrm{p}}}{W_{1}} / \frac{\operatorname{Det} \bar{\Lambda}_{1}^{\omega \mathrm{p}}}{W_{1}^{\omega}}=\frac{2-\dot{D}_{1}\left(\tau_{b}\right)-\bar{D}_{1}\left(\tau_{b}\right)}{4 \sin ^{2}\left[\omega\left(\tau_{b}-\tau_{a}\right) / 2\right]} $$

$$ \operatorname{Det}\left(\tilde{\mathcal{O}}^{-1} \mathcal{O}_{1}\right)=\frac{\operatorname{det} \bar{\Lambda}_{1}^{\mathrm{a}}}{W_{1}} / \frac{\operatorname{Det} \bar{\Lambda}_{1}^{\omega \mathrm{a}}}{W_{1}^{\omega}}=\frac{2+\dot{D}_{1}\left(\tau_{b}\right)+\bar{D}_{1}\left(\tau_{b}\right)}{4 \cos ^{2}\left[\omega\left(\tau_{b}-\tau_{a}\right) / 2\right]} $$

$$ \xi(t)=q(t) \cos \phi(t), \quad \eta(t)=q(t) \sin \phi(t) $$

$$ \dot{\phi}(t) q^{2}(t)=W $$

$$ \ddot{q}+\Omega^{2}(t) q-W^{2} q^{-3}=0 . $$

$$ \operatorname{Det}\left(\mathcal{O}_{0}^{-1} \mathcal{O}_{1}\right)=\frac{1}{W} \frac{q\left(t_{a}\right) q\left(t_{b}\right) \sin \left[\phi\left(t_{b}\right)-\phi\left(t_{a}\right)\right]}{t_{b}-t_{a}} $$

$$ \operatorname{Det}\left(\tilde{\mathcal{O}}^{-1} \mathcal{O}_{1}\right)=4 \sin ^{2} \frac{\phi\left(t_{b}\right)}{2} / 4 \sin ^{2} \frac{\omega\left(t_{b}-t_{a}\right)}{2} $$

$$ \operatorname{Det}\left(\tilde{\mathcal{O}}^{-1} \mathcal{O}_{1}\right)=4 \cos ^{2} \frac{\phi\left(t_{b}\right)}{2} / 4 \cos ^{2} \frac{\omega\left(t_{b}-t_{a}\right)}{2} $$

$$ q(t) \equiv \sqrt{\frac{W}{\omega}} $$

$$ \phi(t)=\omega\left(t-t_{a}\right) $$

$$ \mathcal{A}^{x_{0}}=-\frac{M \omega}{2\left[\omega\left(t_{b}-t_{a}\right)-2 \tan \frac{\omega\left(t_{b}-t_{a}\right)}{2}\right]}\left[\omega\left(t_{b}-t_{a}\right) x_{0}-\tan \frac{\omega\left(t_{b}-t_{a}\right)}{2}\left(x_{a}+x_{b}\right)\right]^{2} .1 $$