Kleinert · 제18장 고급 주제

$$ \left|\Psi_{S}(t)\right\rangle=e^{-i \hat{H} t}\left|\Psi_{S}(0)\right\rangle $$

$$ \langle\ldots\rangle_{T}=\operatorname{Tr}[\exp (-\hat{H} / T) \ldots] / \operatorname{Tr}[\exp (-\hat{H} / T)]=\langle\ldots\rangle_{T} $$

$$ \begin{align*} G_{\Omega}^{R}\left(t, t^{\prime}\right) & =\Theta\left(t-t^{\prime}\right) e^{-i \Omega\left(t-t^{\prime}\right)} \\ G_{\Omega}^{A}\left(t, t^{\prime}\right) & =-\Theta\left(t^{\prime}-t\right) e^{-i \Omega\left(t-t^{\prime}\right)} \\ C_{\Omega}\left(t, t^{\prime}\right) & =e^{-i \Omega\left(t-t^{\prime}\right)} \\ A_{\Omega}\left(t, t^{\prime}\right) & =\left(\tanh \frac{\Omega}{2 T}\right)^{\mp 1} e^{-i \Omega\left(t-t^{\prime}\right)} \end{align*} $$

$$ \begin{align*} \hat{A}(t) & =\alpha_{1} \hat{a}(t)+\alpha_{2} \hat{a}^{\dagger}(t), \\ \hat{B}(t) & =\beta_{1} \hat{a}(t)+\beta_{2} \hat{a}^{\dagger}(t) . \end{align*} $$

$$ \hat{H} \rightarrow \hat{H}+\hat{H}^{\mathrm{ext}}(t) $$

$$ \begin{align*} c & \equiv\left\langle\left[\hat{\psi}, \hat{\psi}^{\dagger}\right]_{\mp}\right\rangle_{T}=\int_{-\infty}^{\infty} \frac{d \omega}{2 \pi} \rho_{12}(\omega) \\ a & \equiv\left\langle\left[\hat{\psi}, \hat{\psi}^{\dagger}\right]_{ \pm}\right\rangle_{T}=\int_{-\infty}^{\infty} \frac{d \omega}{2 \pi} \rho_{12}(\omega) \tanh ^{\mp 1} \frac{\omega}{2 T} \end{align*} $$

$$ Z_{0}\left[\eta_{\mathrm{P}}, \eta_{\mathrm{P}}^{\dagger}\right]=\operatorname{Tr}\left\{\hat{\rho} \hat{T}_{\mathrm{P}} \exp \left[-i \int_{t_{a}}^{t_{b}} d t\left(\hat{a}^{\dagger} \eta+\eta^{\dagger} \hat{a}\right)\right]\right\} $$

$$ \hat{T} \hat{A}(t) \hat{B}(t)=\langle\hat{T} \hat{A}(t) \hat{B}(t)\rangle_{0}+\hat{N} \hat{A}(t) \hat{B}(t) $$

$$ \left|\Psi_{S}^{\text {dist }}(t)\right\rangle=e^{-i \hat{H} t} \hat{U}_{H}(t)\left|\Psi_{S}(0)\right\rangle, $$

$$ \begin{align*} g & \equiv G\left(\omega_{m}=0\right)=\int_{0}^{1 / T} d \tau\left\langle\hat{\psi}(\tau) \hat{\psi}^{\dagger}(0)\right\rangle_{T} \\ & =\int_{-\infty}^{\infty} \frac{d \omega}{2 \pi} \rho_{12}(\omega) \frac{1}{\omega}\left\{\begin{array}{c} 1 \\ \tanh (\omega / 2 T) \end{array}\right\} \geq 0 \end{align*} $$

$$ Z_{0}\left[\eta_{\mathrm{P}}, \eta_{\mathrm{P}}^{\dagger}\right]=\exp \left\{-\int d t \int d t^{\prime} \eta_{\mathrm{P}}^{\dagger}(t) G_{\mathrm{P}}\left(t, t^{\prime}\right) \eta_{\mathrm{P}}\left(t^{\prime}\right)\right\} $$

$$ \begin{align*} \hat{T} \hat{a}(t) \hat{a}^{\dagger}\left(t^{\prime}\right) & =\Theta\left(t-t^{\prime}\right) \hat{a}(t) \hat{a}^{\dagger}\left(t^{\prime}\right) \pm \Theta\left(t^{\prime}-t\right) \hat{a}^{\dagger}\left(t^{\prime}\right) \hat{a}(t) \\ & =\Theta\left(t-t^{\prime}\right)\left[\hat{a}(t) \hat{a}^{\dagger}\left(t^{\prime}\right)\right]_{\mp} \pm \hat{a}^{\dagger}\left(t^{\prime}\right) \hat{a}(t) . \end{align*} $$

$$ i \hat{U}_{H}(t)=\hat{H}_{H}^{\mathrm{ext}}(t) \hat{U}_{H}(t) $$

$$ \begin{align*} i \int_{-\infty}^{\infty} d t \Theta(t)\left\langle\left[\hat{\psi}(t), \hat{\psi}^{\dagger}(0)\right]_{\mp}\right\rangle_{T} & =\int \frac{d \omega}{2 \pi} \rho_{12}(\omega) \frac{1}{\omega} \\ i \int_{-\infty}^{\infty} d t \Theta(t)\left\langle\left[\hat{\psi}(t), \hat{\psi}^{\dagger}(0)\right]_{ \pm}\right\rangle_{T} & =\int \frac{d \omega}{2 \pi} \rho_{12}(\omega) \frac{1}{\omega} \tanh ^{\mp 1} \frac{\omega}{2 T} \end{align*} $$

$$ G_{\mathrm{p}}=\frac{1}{2}\left(\begin{array}{cc} A_{\Omega}+G_{\Omega}^{R}+G_{\Omega}^{A} & A_{\Omega}-G_{\Omega}^{R}+G_{\Omega}^{A} \\ A_{\Omega}+G_{\Omega}^{R}-G_{\Omega}^{A} & A_{\Omega}-G_{\Omega}^{R}-G_{\Omega}^{A} \end{array}\right) $$

$$ \hat{T} \hat{a}(t) \hat{a}^{\dagger}\left(t^{\prime}\right)=\left\langle\hat{T} \hat{a}(t) \hat{a}^{\dagger}\left(t^{\prime}\right)\right\rangle_{0}+\hat{N} \hat{a}(t) \hat{a}^{\dagger}\left(t^{\prime}\right) . $$

$$ \hat{H}_{H}^{\mathrm{ext}}(t) \equiv e^{i \hat{H} t} \hat{H}^{\mathrm{ext}}(t) e^{-i \hat{H} t} . $$

$$ -\left\langle\dot{\hat{\psi}}(0), \hat{\psi}^{\dagger}(0)\right\rangle_{T}=\int_{-\infty}^{\infty} \frac{d \omega}{2 \pi} \rho_{12}(\omega) \omega\left(1 \pm n_{\omega}\right) $$

$$ \begin{align*} G_{\mathrm{p}}\left(t, t^{\prime}\right) & =\left(\begin{array}{cc} \left\langle\hat{T}_{\mathrm{P}} \hat{a}_{H}\left(t_{+}\right) \hat{a}_{H}^{\dagger}\left(t_{+}^{\prime}\right)\right\rangle_{T} & \left\langle\hat{T}_{\mathrm{P}} \hat{a}_{H}\left(t_{+}\right) \hat{a}_{H}^{\dagger}\left(t_{-}^{\prime}\right)\right\rangle_{T} \\ \left\langle\hat{T}_{\mathrm{P}} \hat{a}_{H}\left(t_{-}\right) \hat{a}_{H}^{\dagger}\left(t_{+}^{\prime}\right)\right\rangle_{T} & \left\langle\hat{T}_{\mathrm{P}} \hat{a}_{H}\left(t_{-}\right) \hat{a}_{H}^{\dagger}\left(t_{-}^{\prime}\right)\right\rangle_{T} \end{array}\right) \\ & =\left(\begin{array}{cc} \left\langle\hat{T} \hat{a}_{H}\left(t_{+}\right) \hat{a}_{H}^{\dagger}\left(t_{+}^{\prime}\right)\right\rangle_{T} & \pm\left\langle\hat{a}_{H}^{\dagger}\left(t_{-}^{\prime}\right) \hat{a}_{H}\left(t_{+}\right)\right\rangle_{T} \\ \left\langle\hat{a}_{H}\left(t_{-}\right) \hat{a}_{H}^{\dagger}\left(t_{+}^{\prime}\right)\right\rangle_{T} & \left\langle\overline{\bar{T}} \hat{a}_{H}\left(t_{-}\right) \hat{a}_{H}^{\dagger}\left(t_{-}^{\prime}\right)\right\rangle_{T} \end{array}\right) . \end{align*} $$

$$ \hat{T} \hat{A}\left(t_{1}\right) \ldots \hat{A}\left(t_{n}\right)=\sum_{i=2}^{n} \hat{N} \dot{\hat{A}}\left(t_{1}\right) \ldots \dot{\hat{A}}\left(t_{i}\right) \ldots \hat{A}\left(t_{n}\right) $$

$$ \hat{U}_{H}(t)=1-i \int_{t_{0}}^{t} d t^{\prime} \hat{H}_{H}^{\mathrm{ext}}\left(t^{\prime}\right)+\cdots $$

$$ \begin{align*} d & \equiv i\left\langle\left[\dot{\hat{\psi}}(0), \hat{\psi}^{\dagger}(0)\right]_{\mp}\right\rangle_{T}=\int_{-\infty}^{\infty} \frac{d \omega}{2 \pi} \rho_{12}(\omega) \omega \\ e & \equiv i\left\langle\left[\dot{\hat{\psi}}(0), \hat{\psi}^{\dagger}(0)\right]_{ \pm}\right\rangle_{T}=\int_{-\infty}^{\infty} \frac{d \omega}{2 \pi} \rho_{12}(\omega) \omega \tanh ^{\mp 1} \frac{\omega}{2 T} \end{align*} $$

$$ \begin{align*} \hat{H}_{+} & \equiv \frac{\Omega}{2}\left[\hat{a}_{H}^{\dagger}\left(t_{+}\right) \hat{a}_{H}\left(t_{+}\right) \pm \hat{a}_{H}\left(t_{+}\right) \hat{a}_{H}^{\dagger}\left(t_{+}\right)\right] \\ \hat{H}_{-} & \equiv-\frac{\Omega}{2}\left[\hat{a}_{H}^{\dagger}\left(t_{-}\right) \hat{a}_{H}\left(t_{-}\right) \pm \hat{a}_{H}\left(t_{-}\right) \hat{a}_{H}^{\dagger}\left(t_{-}\right)\right] \end{align*} $$

$$ \hat{T} \hat{A}\left(t_{1}\right) \cdots \hat{A}\left(t_{n}\right) $$

$$ \hat{O}_{H}(t)=e^{i \hat{H} t} \hat{O} e^{-i \hat{H} t} . $$

$$ \mu(\omega)=\frac{1}{g} \frac{1}{2 \pi} \rho_{12}(\omega) \frac{1}{\omega}\left\{\begin{array}{c} 1 \\ \tanh (\omega / 2 T) \end{array}\right\} $$

$$ G_{\mathrm{P}}\left(t, t^{\prime}\right)=e^{-i \Omega\left(t-t^{\prime}\right)}\left(\begin{array}{cc} \Theta\left(t-t^{\prime}\right) \pm n_{\Omega} & \pm n_{\Omega} \\ 1 \pm n_{\Omega} & \Theta\left(t^{\prime}-t\right) \pm n_{\Omega} \end{array}\right) $$

$$ \hat{T} e^{i \int_{-\infty}^{\infty} d t \hat{A}(t) j(t)}=e^{-\frac{1}{2} \int_{-\infty}^{\infty} d t d t^{\prime} j(t)\left\langle\hat{T} \hat{A}(t) \hat{A}\left(t^{\prime}\right)\right\rangle_{0} j\left(t^{\prime}\right)} \hat{N}\left(e^{i \int_{-\infty}^{\infty} d t \hat{A}(t) j(t)}\right) $$

$$ \begin{align*} & \left\langle\Psi_{S}^{\mathrm{dist}}(t)\right| \hat{O}\left|\Psi_{S}^{\mathrm{dist}}(t)\right\rangle=\left\langle\Psi_{S}(0)\right| \hat{U}_{H}^{\dagger}(t) e^{i \hat{H} t} \hat{O} e^{-i \hat{H} t} \hat{U}_{H}(t)\left|\Psi_{S}(0)\right\rangle \\ & \quad \approx\left\langle\Psi_{S}(0)\right|\left(1+i \int_{-\infty}^{t} d t^{\prime} \hat{H}_{H}^{\mathrm{ext}}\left(t^{\prime}\right)+\ldots\right) \hat{O}_{H}(t) \\ & \quad \times\left(1-i \int_{-\infty}^{t} d t^{\prime} \hat{H}_{H}^{\mathrm{ext}}\left(t^{\prime}\right)+\ldots\right)\left|\Psi_{S}(0)\right\rangle \\ & \quad=\left\langle\Psi_{H}\right| \hat{O}_{H}(t)\left|\Psi_{H}\right\rangle-i\left\langle\Psi_{H}\right| \int_{-\infty}^{t} d t^{\prime}\left[\hat{O}_{H}(t), \hat{H}_{H}^{\mathrm{ext}}\left(t^{\prime}\right)\right]\left|\Psi_{H}\right\rangle+\ldots . \end{align*} $$

$$ \int_{-\infty}^{\infty} d \omega \mu(\omega)=1 $$

$$ G_{\mathrm{P}}^{0}\left(t, t^{\prime}\right)+G_{\mathrm{P}}^{N}\left(t, t^{\prime}\right) $$

$$ \left\langle\hat{T} e^{i \int_{-\infty}^{\infty} d t \hat{A}(t) j(t)}\right\rangle_{T}=e^{-\frac{i}{2} \int_{-\infty}^{\infty} d t d t^{\prime} j(t) G\left(t, t^{\prime}\right) j\left(t^{\prime}\right)} $$

$$ \begin{align*} \delta\left\langle\Psi_{S}(t)\right| \hat{O}\left|\Psi_{S}(t)\right\rangle & \equiv\left\langle\Psi_{S}^{\text {dist }}(t)\right| \hat{O}(t)\left|\Psi_{S}^{\text {dist }}(t)\right\rangle-\left\langle\Psi_{S}(t)\right| \hat{O}(t)\left|\Psi_{S}(t)\right\rangle \\ & =-i \int_{-\infty}^{t} d t^{\prime}\left\langle\Psi_{H}\right|\left[\hat{O}_{H}(t), \hat{H}_{H}^{\mathrm{ext}}\left(t^{\prime}\right)\right]\left|\Psi_{H}\right\rangle \end{align*} $$

$$ \begin{align*} & \frac{c}{g}=\int_{-\infty}^{\infty} d \omega \mu(\omega) \omega\left\{\begin{array}{c} 1 \\ \operatorname{coth}(\omega / 2 T) \end{array}\right\} \\ & \frac{a}{g}=\int_{-\infty}^{\infty} d \omega \mu(\omega) \omega\left\{\begin{array}{c} \operatorname{coth}(\omega / 2 T) \\ 1 \end{array}\right\} \\ & \frac{d}{g}=\int_{-\infty}^{\infty} d \omega \mu(\omega) \omega^{2}\left\{\begin{array}{c} 1 \\ \operatorname{coth}(\omega / 2 T) \end{array}\right\} \\ & \frac{e}{g}=\int_{-\infty}^{\infty} d \omega \mu(\omega) \omega^{2}\left\{\begin{array}{c} \operatorname{coth}(\omega / 2 T) \\ 1 \end{array}\right\} \end{align*} $$

$$ \begin{align*} G_{\mathrm{P}}^{N}\left(t, t^{\prime}\right) & \equiv\left(\begin{array}{ll} \left\langle\hat{N} \hat{a}_{H}\left(t_{+}\right) \hat{a}_{H}^{\dagger}\left(t_{+}^{\prime}\right)\right\rangle_{T} & \left\langle\hat{N} \hat{a}_{H}\left(t_{+}\right) \hat{a}_{H}^{\dagger}\left(t_{-}^{\prime}\right)\right\rangle_{T} \\ \left\langle\hat{N} \hat{a}_{H}\left(t_{-}\right) \hat{a}_{H}^{\dagger}\left(t_{+}^{\prime}\right)\right\rangle_{T} & \left\langle\hat{N} \hat{a}_{H}\left(t_{-}\right) \hat{a}_{H}^{\dagger}\left(t_{-}^{\prime}\right)\right\rangle_{T} \end{array}\right) \\ & \equiv \pm\left(\begin{array}{ll} \left\langle\hat{a}_{H}^{\dagger}\left(t_{+}^{\prime}\right) \hat{a}_{H}\left(t_{+}\right)\right\rangle_{T} & \left\langle\hat{a}_{H}^{\dagger}\left(t_{-}^{\prime}\right) \hat{a}_{H}\left(t_{+}\right)\right\rangle_{T} \\ \left\langle\hat{a}_{H}^{\dagger}\left(t_{+}^{\prime}\right) \hat{a}_{H}\left(t_{-}\right)\right\rangle_{T} & \left\langle\hat{a}_{H}^{\dagger}\left(t_{-}^{\prime}\right) \hat{a}_{H}\left(t_{-}\right)\right\rangle_{T} \end{array}\right) . \end{align*} $$

$$ G\left(t, t^{\prime}\right)=\left\langle\hat{T} \hat{A}(t) \hat{A}\left(t^{\prime}\right)\right\rangle_{0}+\left\langle\hat{N} \hat{A}(t) \hat{A}\left(t^{\prime}\right)\right\rangle_{T} $$

$$ \left\langle\Psi_{H}\right| \delta \hat{O}_{H}(t)\left|\Psi_{H}\right\rangle=-i \int_{-\infty}^{t} d t^{\prime}\left\langle\Psi_{H}\right|\left[\hat{O}_{H}(t), \hat{H}_{H}^{\mathrm{ext}}\left(t^{\prime}\right)\right]\left|\Psi_{H}\right\rangle $$

$$ \hat{T} \hat{A}(t) \hat{B}\left(t^{\prime}\right)=\left\langle\hat{T} \hat{A}(t) \hat{B}\left(t^{\prime}\right)\right\rangle_{0}+\hat{N} \hat{A}(t) \hat{B}\left(t^{\prime}\right) $$

$$ G_{O H}^{R}\left(t, t^{\prime}\right) \equiv \Theta\left(t-t^{\prime}\right)\left\langle\Psi_{H}\right|\left[\hat{O}_{H}(t), \hat{H}_{H}\left(t^{\prime}\right)\right]\left|\Psi_{H}\right\rangle $$

$$ \tilde{\eta}_{\mathrm{P}}=Q \eta_{\mathrm{P}}=\frac{1}{\sqrt{2}}\left(\begin{array}{rr} 1 & -1 \\ 1 & 1 \end{array}\right) \hat{\eta}_{\mathrm{P}} $$

$$ \left\langle\Psi_{H}\right| \delta \hat{O}_{H}(t)\left|\Psi_{H}\right\rangle=-i \int_{-\infty}^{\infty} d t^{\prime} G_{O H}^{R}\left(t, t^{\prime}\right) $$

$$ \begin{array}{r} Z_{0}\left[\eta_{\mathrm{P}}^{*}, \eta_{\mathrm{P}}\right]=\exp \left[-\int d t \int d t^{\prime}\left(\eta_{+}^{*},-\eta_{-}^{*}\right) Q^{-1}\left(\begin{array}{cc} 0 & G_{\Omega}^{A} \\ G_{\Omega}^{R} & A \end{array}\right) Q\binom{\eta_{+}}{-\eta_{-}}\right] \\ =\exp \left\{-\frac{1}{2} \int d t \int d t^{\prime}\left[\left(\eta_{+}^{*}-\eta_{-}^{*}\right)(t) G_{\Omega}^{R}\left(t-t^{\prime}\right)\left(\eta_{+}+\eta_{-}\right)\left(t^{\prime}\right)\right.\right. \\ +\left(\eta_{+}^{*}+\eta_{-}^{*}\right)(t) G_{\Omega}^{A}\left(t-t^{\prime}\right)\left(\eta_{+}-\eta_{-}\right)\left(t^{\prime}\right) \\ \left.\left.+\left(\eta_{-}^{*}-\eta_{-}^{*}\right)(t) A_{\Omega}\left(t-t^{\prime}\right)\left(\eta_{+}-\eta_{-}\right)\left(t^{\prime}\right)\right]\right\} \end{array} $$

$$ \hat{H}^{\mathrm{ext}}(t)=-\hat{O}_{H}(t) \delta j(t) $$

$$ f\left(\frac{\omega_{1}+\omega_{2}}{2}\right) \leq \frac{f\left(\omega_{1}\right)+f\left(\omega_{2}\right)}{2} $$

$$ \eta_{+}(t) \equiv \eta\left(t_{+}\right), \quad \eta_{-}(t) \equiv \eta\left(t_{-}\right) $$

$$ \left\langle\Psi_{H}\right| \delta \hat{O}_{H}(t)\left|\Psi_{H}\right\rangle=i \int_{-\infty}^{\infty} d t^{\prime} G_{O O}^{R}\left(t, t^{\prime}\right) \delta j\left(t^{\prime}\right) $$

$$ f\left(\sum_{i} \mu_{i} \omega_{i}\right) \leq \sum_{i} \mu_{i} f\left(\omega_{i}\right) $$

$$ A_{\Omega}\left(t, t^{\prime}\right)=\Theta\left(t, t^{\prime}\right) A_{\Omega}\left(t, t^{\prime}\right) \pm \Theta\left(t^{\prime}-t\right) A_{\Omega}\left(t^{\prime}, t\right) $$

$$ G_{O O}^{R}\left(t, t^{\prime}\right)=\Theta\left(t-t^{\prime}\right)\left\langle\Psi_{H}\right|\left[\hat{O}_{H}(t), \hat{O}_{H}\left(t^{\prime}\right)\right]\left|\Psi_{H}\right\rangle $$

$$ f\left(\int_{-\infty}^{\infty} d \omega \mu(\omega) \omega\right) \leq \int_{-\infty}^{\infty} d \omega \mu(\omega) f(\omega) $$

$$ \begin{align*} Z_{0}\left[\eta_{\mathrm{P}}^{*}, \eta_{\mathrm{P}}\right]=\exp \{ & -\frac{1}{2} \int_{-\infty}^{\infty} d t \int_{-\infty}^{t} d t^{\prime} \\ & \times\left[\left(\eta_{+}-\eta_{-}\right)^{*}(t) G_{\Omega}^{R}\left(t, t^{\prime}\right)\left(\eta_{+}+\eta_{-}\right)\left(t^{\prime}\right)\right. \\ & -\left(\eta_{+}-\eta_{-}\right)(t) G_{\Omega}^{R}\left(t, t^{\prime}\right)^{*}\left(\eta_{+}+\eta_{-}\right)^{*}\left(t^{\prime}\right) \\ & +\left(\eta_{+}-\eta_{-}\right)^{*}(t) A_{\Omega}\left(t, t^{\prime}\right)\left(\eta_{+}-\eta_{-}\right)\left(t^{\prime}\right) \\ & \left.\left.+\left(\eta_{+}-\eta_{-}\right)(t) A_{\Omega}\left(t, t^{\prime}\right)^{*}\left(\eta_{+}-\eta_{-}\right)^{*}\left(t^{\prime}\right)\right]\right\} \end{align*} $$

$$ \langle\hat{O}\rangle_{T}=e^{F / T} \operatorname{Tr}\left(e^{-\hat{H} / T} \hat{O}\right) $$

$$ f(\omega)=\omega \operatorname{coth} \frac{\omega}{2 T} $$

$$ Z\left[j_{\mathrm{P}}, k_{\mathrm{P}}\right]=\operatorname{Tr}\left(\hat{\rho} \hat{T}_{\mathrm{P}} \exp \left\{i \int_{\mathrm{P}} d x\left[j_{\mathrm{P}}(t) x_{\mathrm{P}}(t)+k_{\mathrm{P}}(t) p_{\mathrm{P}}(t)\right]\right\}\right) $$

$$ \delta\langle\hat{O}(t)\rangle_{T}=i \int_{-\infty}^{\infty} d t^{\prime} G_{O O}^{R}\left(t, t^{\prime}\right) \delta j\left(t^{\prime}\right) $$

$$ c \operatorname{coth} \frac{c}{2 T g} \leq a $$

$$ \hat{x}(t)=\sqrt{\frac{\hbar}{2 M \Omega}}\left[\hat{a} e^{-i \Omega t}+\hat{a}^{\dagger} e^{i \Omega t}\right] $$

$$ G_{O O}^{R}\left(t, t^{\prime}\right) \equiv G_{O O}^{R}\left(t-t^{\prime}\right) \equiv \Theta\left(t-t^{\prime}\right) e^{F / T} \operatorname{Tr}\left\{e^{-\hat{H} / T}\left[\hat{O}_{H}(t), \hat{O}_{H}\left(t^{\prime}\right)\right]\right\} $$

$$ \left\langle\left[\hat{\psi}, \hat{\psi}^{\dagger}\right]_{+}\right\rangle_{T} \geq\left\langle\left[\hat{\psi}, \hat{\psi}^{\dagger}\right]_{-}\right\rangle_{T} \operatorname{coth} \frac{\left\langle\left[\hat{\psi}, \hat{\psi}^{\dagger}\right]_{-}\right\rangle_{T}}{2 T \int_{0}^{1 / T} d \tau\left\langle\hat{\psi}(\tau) \hat{\psi}^{\dagger}(0)\right\rangle_{T}} $$

$$ \left\{\begin{array}{l} \hat{a} \\ \hat{a}^{\dagger} \end{array}\right\}=(M \Omega \hat{\varphi} \pm i \hat{p}) / \sqrt{2 M \Omega \hbar}, $$

$$ G_{i j}^{R}\left(t, t^{\prime}\right) \equiv G_{i j}^{R}\left(t-t^{\prime}\right) \equiv \Theta\left(t-t^{\prime}\right) e^{F / T} \operatorname{Tr}\left\{e^{-\hat{H} / T}\left[\hat{O}_{H}^{i}(t), \hat{O}_{H}^{j}\left(t^{\prime}\right)\right]\right\} $$

$$ 1+2\left\langle\hat{\psi}_{\mathbf{p}}^{\dagger} \hat{\psi}_{\mathbf{p}}\right\rangle_{T} \geq \operatorname{coth}(1 / 2 T g)=1+\frac{2}{e^{1 / g T}-1}=1+2 n_{g^{-1}} $$

$$ \left\{\begin{array}{l} \eta \\ \eta^{\dagger} \end{array}\right\}=(j \pm i M \Omega k) / \sqrt{2 M \Omega \hbar} $$

$$ G_{12}(\tau, 0) \equiv G_{12}(\tau) \equiv e^{F / T} \operatorname{Tr}\left[e^{-\hat{H} / T} \hat{T}_{\tau} \hat{O}_{H}^{1}(\tau) \hat{O}_{H}^{2}(0)\right] $$

$$ \left\langle\hat{\psi}_{\mathbf{p}}^{\dagger} \hat{\psi}_{\mathbf{p}}\right\rangle_{T} \geq \frac{1}{e^{1 / g T}-1} \equiv n_{g^{-1}} $$

$$ \begin{gather*} Z_{0}\left[j_{\mathrm{P}}, k_{\mathrm{P}}\right]=\exp \left\{-\frac{1}{2 M \Omega} \int_{-\infty}^{\infty} d t \int_{-\infty}^{t} d t^{\prime}\left(j_{+}-j_{-}\right)(t)\right. \\ \times\left\{\left[\operatorname{Re} A_{\Omega}\left(t, t^{\prime}\right)+i \operatorname{Im} G_{\Omega}^{R}\left(t, t^{\prime}\right)\right] j_{+}\left(t^{\prime}\right)\right. \\ \left.-\left[\operatorname{Re} A_{\Omega}\left(t, t^{\prime}\right)-i \operatorname{Im} G_{\Omega}^{R}\left(t, t^{\prime}\right)\right] j_{-}\left(t^{\prime}\right)\right\} \\ -\frac{1}{2} \int_{-\infty}^{\infty} d t \int_{-\infty}^{t} d t^{\prime}\left(k_{+}-k_{-}\right)(t)\left\{\left[\operatorname{Im} A_{\Omega}\left(t, t^{\prime}\right)-i \operatorname{Re} G_{\Omega}^{R}\left(t, t^{\prime}\right)\right] j_{+}\left(t^{\prime}\right)\right. \\ \left.\left.-\left[\operatorname{Im} A_{\Omega}\left(t, t^{\prime}\right)+i \operatorname{Re} G_{\Omega}^{R}\left(t, t^{\prime}\right)\right] j_{-}\left(t^{\prime}\right)\right\}+(j \leftrightarrow k M \Omega)\right\} \end{gather*} $$

$$ \hat{O}_{H}(\tau) \equiv e^{\hat{H} \tau} \hat{O} e^{-\hat{H} \tau} $$

$$ g^{-1}=G(0, \mathbf{p})^{-1}=\frac{\mathbf{p}^{2}}{2 M}-\mu \equiv \xi(\mathbf{p}) $$

$$ \begin{align*} \alpha\left(t, t^{\prime}\right) & =\frac{1}{2 M \Omega}\left[\operatorname{Re} A_{\Omega}\left(t, t^{\prime}\right)+i \operatorname{Im} G_{\Omega}^{R}\left(t, t^{\prime}\right)\right], \\ \beta\left(t, t^{\prime}\right) & =\frac{1}{2 M \Omega}\left[\operatorname{Im} A_{\Omega}\left(t, t^{\prime}\right)-i \operatorname{Re} G_{\Omega}^{R}\left(t, t^{\prime}\right)\right] . \end{align*} $$

$$ G_{12}(\tau)=e^{F / T} \sum_{n, n^{\prime}} e^{-E_{n} / T} e^{\left(E_{n}-E_{n^{\prime}}\right) \tau}\langle n| \hat{O}^{1}\left|n^{\prime}\right\rangle\left\langle n^{\prime}\right| \hat{O}^{2}|n\rangle $$

$$ \left\langle\hat{\psi}_{\mathbf{p}}^{\dagger} \hat{\psi}_{\mathbf{p}}\right\rangle_{T}=n_{\xi(\mathbf{p})} $$

$$ \begin{align*} Z_{0}\left[j_{+}, j_{-}, k_{+}, k_{-}\right]= & \exp \left\{-\int_{-\infty}^{\infty} d t \int_{-\infty}^{t} d t^{\prime}\left(j_{+}-j_{-}\right)(t)\left[\alpha\left(t, t^{\prime}\right) j_{+}\left(t^{\prime}\right)-\alpha^{*}\left(t, t^{\prime}\right) j_{-}\left(t^{\prime}\right)\right]+(j \leftrightarrow k M \Omega)\right. \\ & \left.-M \Omega \int_{-\infty}^{\infty} d t \int_{-\infty}^{t} d t^{\prime}\left(k_{+}-k_{-}\right)(t)\left[\beta\left(t, t^{\prime}\right) j_{+}\left(t^{\prime}\right)-\beta^{*}\left(t, t^{\prime}\right) j_{-}\left(t^{\prime}\right)\right]+(j \leftrightarrow k M \Omega)\right\} \end{align*} $$

$$ \begin{align*} G_{12}\left(\omega_{m}\right)= & \int_{0}^{1 / T} d \tau e^{i \omega_{m} \tau} G_{12}(\tau) \\ = & e^{F / T} \sum_{n, n^{\prime}} e^{-E_{n} / T}\left(1-e^{\left(E_{n}-E_{n^{\prime}}\right) / T}\right)\langle n| \hat{O}^{1}\left|n^{\prime}\right\rangle\left\langle n^{\prime}\right| \hat{O}^{2}|n\rangle \\ & \times \frac{-1}{i \omega_{m}-E_{n^{\prime}}+E_{n}} \end{align*} $$

$$ \bar{f}(y)=\sqrt{y} \operatorname{coth} \frac{\sqrt{y}}{2 T} $$

$$ Z\left[j_{\mathrm{P}}\right]=\operatorname{Tr}\left\{\hat{\rho} \hat{T}_{\mathrm{P}} \exp \left[i \int_{\mathrm{P}} d x j_{\mathrm{P}}(t) \hat{x}_{\mathrm{P}}(t)\right]\right\} $$

$$ \begin{align*} G_{12}^{R}(\omega)= & \int_{-\infty}^{\infty} d t e^{i \omega t} \Theta(t) e^{F / T} \\ = & \operatorname{Tr}\left\{e^{-\hat{H} / T}\left[\hat{O}_{H}^{1}(t), \hat{O}_{H}^{2}(0)\right]_{\mp}\right\} \\ = & e^{F / T} \int_{0}^{\infty} d t e^{i \omega t} \sum_{n, n^{\prime}}[ \\ & \quad e^{-E_{n} / T} e^{i\left(E_{n}-E_{n^{\prime}}\right) t}\langle n| \hat{O}^{1}\left|n^{\prime}\right\rangle\left\langle n^{\prime}\right| \hat{O}^{2}|n\rangle \\ & \left.\quad e^{-E_{n} / T} e^{-i\left(E_{n}-E_{n^{\prime}}\right) t}\langle n| \hat{O}^{2}\left|n^{\prime}\right\rangle\left\langle n^{\prime}\right| \hat{O}^{1}|n\rangle\right] \end{align*} $$

$$ \int_{-\infty}^{\infty} d \omega \mu(\omega)=\int_{0}^{\infty} \frac{d y}{\sqrt{y}} \mu(\sqrt{y})=1 $$

$$ Z_{0}\left[j_{+}, j_{-}\right]=\exp \left\{-\int_{-\infty}^{\infty} d t \int_{-\infty}^{t} d t^{\prime}\left(j_{+}-j_{-}\right)(t)\left[\alpha\left(t, t^{\prime}\right) j_{+}\left(t^{\prime}\right)-\alpha^{*}\left(t, t^{\prime}\right) j_{-}\left(t^{\prime}\right)\right]\right\} $$

$$ \begin{align*} G_{12}^{R}(\omega)=e^{F / T} \sum_{n, n^{\prime}} e^{-E_{n} / T} & {\left[1-e^{\left(E_{n}-E_{n^{\prime}}\right) / T}\right]\langle n| \hat{O}^{1}\left|n^{\prime}\right\rangle\left\langle n^{\prime}\right| \hat{O}^{2}|n\rangle } \\ & \times \frac{i}{\omega-E_{n^{\prime}}+E_{n}+i \eta} \end{align*} $$

$$ \bar{f}\left(\int_{0}^{\infty} \frac{d y}{\sqrt{y}} \mu(\sqrt{y}) y\right) \geq \int_{0}^{\infty} \frac{d y}{\sqrt{y}} \mu(\sqrt{y}) \bar{f}(y) $$

$$ \begin{align*} \alpha\left(t, t^{\prime}\right) & =\frac{1}{2 M \Omega}\left[\operatorname{Re} A_{\Omega}\left(t, t^{\prime}\right)+i \operatorname{Im} C_{\Omega}\left(t, t^{\prime}\right)\right], \quad t>t^{\prime}, \\ \beta\left(t, t^{\prime}\right) & =\frac{1}{2 M \Omega}\left[\operatorname{Im} A_{\Omega}\left(t, t^{\prime}\right)-i \operatorname{Re} C_{\Omega}\left(t, t^{\prime}\right)\right], \quad t>t^{\prime} . \end{align*} $$

$$ \frac{i}{\omega-E_{n^{\prime}}+E_{n}+i \eta} \rightarrow \frac{-1}{i \omega_{m}-E_{n^{\prime}}+E_{n}} $$

$$ \bar{f}\left(\int_{-\infty}^{\infty} d \omega \mu(\omega) \omega^{2}\right) \geq \int_{-\infty}^{\infty} d \omega \mu(\omega) \bar{f}\left(\omega^{2}\right) $$

$$ \begin{align*} \alpha\left(t, t^{\prime}\right) & =\frac{1}{2 M \Omega}\left[\operatorname{Re} e^{-i \Omega\left(t-t^{\prime}\right)}\left\{\begin{array}{c} \operatorname{coth} \frac{\Omega}{2 T} \\ \tanh \frac{\Omega}{2 T} \end{array}\right\}+i \operatorname{Im} e^{-i \Omega\left(t-t^{\prime}\right)}\right] \\ & =\frac{1}{2 M \Omega}\left[\cos \Omega\left(t-t^{\prime}\right)\left\{\begin{array}{c} \operatorname{coth} \frac{\Omega}{2 T} \\ \tanh \frac{\Omega}{2 T} \end{array}\right\}-i \sin \Omega\left(t-t^{\prime}\right)\right], \\ \beta\left(t, t^{\prime}\right) & =\frac{1}{2 M \Omega}\left[\operatorname{Im} e^{-i \Omega\left(t-t^{\prime}\right)}\left\{\begin{array}{c} \operatorname{coth} \frac{\Omega}{2 T} \\ \tanh \frac{\Omega}{2 T} \end{array}\right\}-i \operatorname{Re} e^{-i \Omega\left(t-t^{\prime}\right)}\right] \\ & =-\frac{1}{2 M \Omega}\left[\sin \Omega\left(t-t^{\prime}\right)\left\{\begin{array}{c} \operatorname{coth} \frac{\Omega}{2 T} \\ \tanh \frac{\Omega}{2 T} \end{array}\right\}+i \cos \Omega\left(t-t^{\prime}\right)\right] . \end{align*} $$

$$ G_{i j}^{R}\left(t, t^{\prime}\right) \equiv G_{i j}^{R}\left(t-t^{\prime}\right) \equiv \Theta\left(t-t^{\prime}\right) e^{F / T} \operatorname{Tr}\left\{e^{-\hat{H} / T}\left[\hat{O}_{H}^{i}(t), \hat{O}_{H}^{j}\left(t^{\prime}\right)\right]_{+}\right\} $$

$$ \frac{a}{g} \leq \sqrt{\frac{d}{g}} \operatorname{coth}\left(\frac{1}{2 T} \sqrt{\frac{d}{g}}\right) $$

$$ \begin{align*} \rho_{12}\left(\omega^{\prime}\right)= & \left(1 \mp e^{-\omega^{\prime} / T}\right) e^{F / T} \\ & \times \sum_{n, n^{\prime}} e^{-E_{n} / T} 2 \pi \delta\left(\omega-E_{n^{\prime}}+E_{n}\right)\langle n| \hat{O}^{1}\left|n^{\prime}\right\rangle\left\langle n^{\prime}\right| \hat{O}^{2}|n\rangle, \end{align*} $$

$$ c \operatorname{coth} \frac{c}{2 T g} \leq a \leq \sqrt{d g} \operatorname{coth}\left(\frac{1}{2 T} \sqrt{\frac{d}{g}}\right) $$

$$ \bar{\alpha}\left(t-t^{\prime}\right)=\frac{1}{2 M \Omega}\left\{\begin{array}{lll} \frac{\cosh \left[\Omega\left(\beta / 2-i\left(t-t^{\prime}\right)\right]\right.}{\sinh (\Omega \beta / 2)} & \text { for bosons, } \\ \frac{\sinh \left[\Omega\left(\beta / 2-i\left(t-t^{\prime}\right)\right]\right.}{\cosh (\Omega \beta / 2)} & \text { for } & \text { fermions. } \end{array}\right. $$

$$ \rho_{12}\left(\omega^{\prime}\right)=\mp \rho_{12}\left(-\omega^{\prime}\right) . $$

$$ \begin{align*} c^{2} & \leq d g, \\ c \operatorname{coth}(d / 2 T c) & \leq a, \\ g & \leq \operatorname{coth}(c / 2 T a), \\ c & \leq d \tanh (c / 2 T a), \\ c & \leq a \tanh (d / 2 T c) . \end{align*} $$

$$ \begin{align*} G_{12}^{R}(\omega) & =\int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \rho_{12}\left(\omega^{\prime}\right) \frac{i}{\omega-\omega^{\prime}+i \eta} \\ G_{12}\left(\omega_{m}\right) & =\int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \rho_{12}\left(\omega^{\prime}\right) \frac{-1}{i \omega_{m}-\omega^{\prime}} \end{align*} $$

$$ a \operatorname{coth} \frac{a}{2 T g} \leq c $$

$$ \left\langle\left[\hat{\psi}, \hat{\psi}^{\dagger}\right]_{-}\right\rangle_{T} \leq\left\langle\left[\hat{\psi}, \hat{\psi}^{\dagger}\right]_{+}\right\rangle_{T} \tanh \frac{\left\langle\left[\hat{\psi}, \hat{\psi}^{\dagger}\right]_{+}\right\rangle_{T}}{2 T \int_{0}^{1 / T} d \tau\left\langle\hat{\psi}(\tau) \hat{\psi}^{\dagger}(0)\right\rangle_{T}} $$

$$ \begin{align*} G_{12}^{R}(t) & =\Theta(t) \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \rho_{12}\left(\omega^{\prime}\right) e^{-i \omega^{\prime} t} \\ G_{12}(\tau) & =\int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \rho_{12}\left(\omega^{\prime}\right) T \sum_{\omega_{m}} e^{-i \omega_{m} \tau} \frac{-1}{i \omega_{m}-\omega^{\prime}} \end{align*} $$

$$ 1-2\left\langle\hat{\psi}^{\dagger} \hat{\psi}\right\rangle_{T} \leq \quad \tanh (1 / 2 g T)=1-\frac{2}{e^{1 / g T}+1} $$

$$ \left\langle\hat{\psi}_{\mathbf{p}}^{\dagger} \hat{\psi}_{\mathbf{p}}\right\rangle_{T} \leq \frac{1}{e^{1 / g T}+1}=n_{g^{-1}} $$

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

$$ \left\langle\hat{\psi}_{\mathbf{p}}^{\dagger} \hat{\psi}_{\mathbf{p}}\right\rangle_{T}=n_{\xi(\mathbf{p})} $$

$$ \begin{align*} T \sum_{n} e^{-i \omega_{m} \tau} \frac{-1}{i \omega_{m}-\omega} & =G_{\omega, e}^{a}(\tau)=e^{-\omega(\tau-1 / 2 T)} \frac{1}{2 \cos (\omega / 2 T)} \\ & =e^{-\omega \tau}\left(1-n_{\omega}\right) \end{align*} $$

$$ n_{\omega}=\frac{1}{e^{\omega / T} \mp 1} $$

$$ G_{12}^{A}\left(t, t^{\prime}\right) \equiv G_{12}^{A}\left(t-t^{\prime}\right)=-\Theta\left(t^{\prime}-t\right) e^{F / T} \operatorname{Tr}\left\{e^{-\hat{H} / T}\left[\hat{O}_{H}^{1}(t), \hat{O}_{H}^{2}\left(t^{\prime}\right)\right]_{\mp}\right\} $$

$$ G_{12}^{A}(\omega)=\int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \rho_{12}\left(\omega^{\prime}\right) \frac{i}{\omega-\omega^{\prime}-i \eta} $$

$$ G_{12}^{A}(t)=-\Theta(-t) \int_{-\infty}^{\infty} \frac{d \omega}{2 \pi} \rho_{12}(\omega) e^{-i \omega t} $$

$$ C_{12}\left(t, t^{\prime}\right)=e^{F / T} \operatorname{Tr}\left\{e^{-\hat{H} / T}\left[\hat{O}_{H}^{1}(t), \hat{O}_{H}^{2}\left(t^{\prime}\right)\right]_{\mp}\right\}=G_{12}^{R}\left(t, t^{\prime}\right)-G_{12}^{A}\left(t, t^{\prime}\right) $$

$$ \begin{align*} G_{12}^{R}\left(t, t^{\prime}\right) & =\Theta\left(t-t^{\prime}\right) C_{12}\left(t, t^{\prime}\right) \\ G_{12}^{A}\left(t, t^{\prime}\right) & =-\Theta\left(t^{\prime}-t\right) C_{12}\left(t, t^{\prime}\right) \end{align*} $$

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

$$ C_{12}(t)=\int_{-\infty}^{\infty} \frac{d \omega}{2 \pi} \rho_{12}(\omega) e^{-i \omega t} $$

$$ C_{12}(\omega)=\rho_{12}(\omega) $$

$$ G_{12}\left(t, t^{\prime}\right) \equiv G_{12}\left(t-t^{\prime}\right)=e^{F / T} \operatorname{Tr}\left[e^{-\hat{H} / T} \hat{T} \hat{O}_{H}^{1}(t) \hat{O}_{H}^{2}\left(t^{\prime}\right)\right] $$

$$ \begin{align*} G_{12}(\omega) & =\int_{-\infty}^{\infty} d t e^{i \omega t} \Theta(t) e^{F / T} \operatorname{Tr}\left\{e^{-\hat{H} / T} \hat{O}_{H}^{1}(t) \hat{O}_{H}^{2}(0)\right\} \\ & +\int_{-\infty}^{\infty} d t e^{i \omega t} \Theta(-t) e^{F / T} \operatorname{Tr}\left\{e^{-\hat{H} / T} \hat{O}_{H}^{2}(t) \hat{O}_{H}^{1}(0)\right\} \\ & =e^{F / T} \int_{0}^{\infty} d t e^{i \omega t} \sum_{n, n^{\prime}} e^{-E_{n} / T} e^{i\left(E_{n}-E_{n^{\prime}}\right) t}\langle n| \hat{O}^{1}\left|n^{\prime}\right\rangle\left\langle n^{\prime}\right| \hat{O}^{2}|n\rangle \\ & \pm e^{F / T} \int_{-\infty}^{0} d t e^{i \omega t} \sum_{n, n^{\prime}} e^{-E_{n} / T} e^{-i\left(E_{n}-E_{n^{\prime}}\right) t}\langle n| \hat{O}^{2}\left|n^{\prime}\right\rangle\left\langle n^{\prime}\right| \hat{O}^{1}|n\rangle \end{align*} $$

$$ G_{12}(\omega)=\int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \rho_{12}\left(\omega^{\prime}\right)\left[\frac{1}{1 \mp e^{-\omega^{\prime} / T}} \frac{i}{\omega-\omega^{\prime}+i \eta}+\frac{1}{1 \mp e^{\omega^{\prime} / T}} \frac{i}{\omega-\omega^{\prime}-i \eta}\right] $$

$$ A_{12}\left(t-t^{\prime}\right) \equiv e^{F / T} \operatorname{Tr}\left\{e^{-\hat{H} / T}\left[\hat{O}_{H}^{1}(t), \hat{O}_{H}^{2}\left(t^{\prime}\right)\right]_{ \pm}\right\} $$

$$ \begin{align*} & A_{12}(\omega)= \int_{-\infty}^{\infty} d t e^{i \omega t} e^{F / T} \operatorname{Tr}\left\{e^{-\hat{H} / T}\left[\hat{O}_{H}^{1}(t), \hat{O}_{H}^{2}(0)\right]_{ \pm}\right\} \\ &= e^{F / T} \int_{-\infty}^{\infty} d t e^{i \omega t} \sum_{n, n^{\prime}}[ \\ & \quad e^{-E_{n} / T} e^{i\left(E_{n}-E_{n^{\prime}}\right) t}\langle n| \hat{O}^{1}\left|n^{\prime}\right\rangle\left\langle n^{\prime}\right| \hat{O}^{2}|n\rangle \\ & \quad e_{n} / T \\ &\left.e^{-i\left(E_{n}-E_{n^{\prime}}\right) t}\langle n| \hat{O}^{2}\left|n^{\prime}\right\rangle\left\langle n^{\prime}\right| \hat{O}^{1}|n\rangle\right] \end{align*} $$

$$ \begin{gather*} A_{12}(\omega)=e^{F / T} \sum_{n, n^{\prime}} e^{-E_{n} / T}\left[1 \pm e^{\left(E_{n}-E_{n^{\prime}}\right) / T}\right]\langle n| \hat{O}^{1}\left|n^{\prime}\right\rangle\left\langle n^{\prime}\right| \hat{O}^{2}|n\rangle \\ \times 2 \pi \delta\left(\omega-E_{n^{\prime}}+E_{n}\right) \end{gather*} $$

$$ A_{12}(\omega)=\int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \tanh ^{\mp 1} \frac{\omega^{\prime}}{2 T} \rho_{12}\left(\omega^{\prime}\right) 2 \pi \delta\left(\omega-\omega^{\prime}\right)=\tanh ^{\mp 1} \frac{\omega}{2 T} \rho_{12}(\omega) $$

$$ A_{12}\left(t, t^{\prime}\right) \equiv A_{12}\left(t-t^{\prime}\right)=\int_{-\infty}^{\infty} \frac{d \omega}{2 \pi} \rho_{12}(\omega) \tanh ^{\mp 1} \frac{\omega}{2 T} e^{-i \omega\left(t-t^{\prime}\right)} $$

$$ \frac{i}{\omega-\omega^{\prime} \pm i \eta}=i\left[\frac{\mathcal{P}}{\omega-\omega^{\prime}} \mp i \pi \delta\left(\omega-\omega^{\prime}\right)\right] $$

$$ G_{12}^{R, A}(\omega)=i \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \rho_{12}\left(\omega^{\prime}\right)\left[\frac{\mathcal{P}}{\omega-\omega^{\prime}} \mp i \pi \delta\left(\omega-\omega^{\prime}\right)\right] $$

$$ G_{12}(\omega)=i \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \rho_{12}\left(\omega^{\prime}\right)\left[\frac{\mathcal{P}}{\omega-\omega^{\prime}}-i \pi \tanh ^{\mp 1} \frac{\omega}{2 T} \delta\left(\omega-\omega^{\prime}\right)\right] $$

$$ G_{12}(\omega)=\int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \rho_{12}\left(\omega^{\prime}\right)\left[\frac{i}{\omega-\omega^{\prime}+i \eta} \pm 2 \pi n_{\omega} \delta\left(\omega-\omega^{\prime}\right)\right] . $$

$$ \hat{O}_{H}^{2}(t)=\left[\hat{O}_{H}^{1}(t)\right]^{\dagger} $$

$$ \begin{align*} \rho_{12}\left(\omega^{\prime}\right)= & \left(1 \mp e^{-\omega^{\prime} / T}\right) e^{F / T} \\ & \times \sum_{n, n^{\prime}} e^{-E_{n} / T} 2 \pi \delta\left(\omega^{\prime}-E_{n^{\prime}}+E_{n}\right) \mid\langle n| \hat{O}_{H}^{1}(t)\left|n^{\prime}\right\rangle \|^{2} \end{align*} $$

$$ \begin{align*} \rho_{12}\left(\omega^{\prime}\right) \omega^{\prime} \geq 0 \quad & \text { for bosons } \\ \rho_{12}\left(\omega^{\prime}\right) \geq 0 \quad & \text { for fermions. } \end{align*} $$

$$ \begin{align*} G_{12}^{A}\left(t, t^{\prime}\right) & =\mp G_{21}^{R}\left(t^{\prime}, t\right)^{*} \\ A_{12}\left(t, t^{\prime}\right) & = \pm A_{21}\left(t^{\prime}, t\right)^{*} \\ C_{12}\left(t, t^{\prime}\right) & =\mp C_{21}\left(t^{\prime}, t\right)^{*} \\ G_{12}\left(t, t^{\prime}\right) & = \pm G_{21}\left(t^{\prime}, t\right)^{*} \end{align*} $$

$$ \left[\hat{\psi}_{\mathbf{p}}(t), \hat{\psi}_{\mathbf{p}}^{\dagger}(t)\right]=1 $$

$$ \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \rho_{12}\left(\omega^{\prime}\right)=1 $$

$$ G_{12}\left(\omega_{m}\right) \underset{\omega_{m} \rightarrow \infty}{\longrightarrow} \frac{i}{\omega_{m}}, \quad G_{12}^{A, R}(\omega) \underset{\omega \rightarrow \infty}{\longrightarrow} \frac{1}{\omega} . $$

$$ \hat{a}_{H}^{\dagger}(t)=\hat{a}^{\dagger} e^{i \Omega t}, \quad \hat{a}_{H}(t)=\hat{a} e^{-i \Omega t} $$

$$ \hat{H}=\frac{1}{2}\left(\hat{p}^{2}+\Omega^{2} \hat{x}^{2}\right)=\frac{\omega}{2}\left(\hat{a}^{\dagger} \hat{a}+\hat{a} \hat{a}^{\dagger}\right)=\omega\left(\hat{a}^{\dagger} \hat{a} \pm \frac{1}{2}\right) $$

$$ |n\rangle=\frac{1}{\sqrt{n!}}\left(\hat{a}^{\dagger}\right)^{n}|0\rangle $$

$$ \rho_{12}\left(\omega^{\prime}\right)=2 \pi \delta\left(\omega^{\prime}-\Omega\right)\left(1 \mp e^{-\Omega / T}\right) e^{F / T} \sum_{n=0}^{\infty, 0} e^{-(n \pm 1 / 2) \Omega / T}(n+1) $$

$$ Z_{\Omega} \equiv e^{-F / T}=\sum_{n=0}^{\infty, 1} e^{-(n \pm 1 / 2) \Omega / T}=\left\{\begin{array}{ccc} {[2 \sinh (\Omega / 2 T)]^{-1}} & \text { for } & \text { bosons } \\ 2 \cosh (\Omega / 2 T) & & \text { fermions } \end{array}\right\} $$

$$ \begin{align*} \sum_{n=0}^{\infty} e^{-(n+1 / 2) \Omega / T}(n+1) & =\left(-T \frac{\partial}{\partial \Omega}+\frac{1}{2}\right) e^{-F / T}=\left(1 \mp e^{-\Omega / T}\right)^{-1} e^{-F / T} \\ \sum_{n=0}^{0} e^{-(n-1 / 2) \Omega / T}(n+1) & =e^{\Omega / 2 T}=\left(1+e^{-\Omega / T}\right)^{-1} e^{-F / T} \end{align*} $$

$$ \rho_{12}\left(\omega^{\prime}\right)=2 \pi \delta\left(\omega^{\prime}-\Omega\right) $$

$$ \begin{align*} G_{\Omega}^{R}\left(t, t^{\prime}\right) & =\Theta\left(t-t^{\prime}\right) e^{-\Omega\left(t-t^{\prime}\right)} \\ G_{\Omega}\left(\tau, \tau^{\prime}\right) & =-T \sum_{m=-\infty}^{\infty} e^{-i \omega_{m}\left(\tau-\tau^{\prime}\right)} \frac{1}{i \Omega_{m}-\Omega} \\ & =e^{-\Omega\left(\tau-\tau^{\prime}\right)}\left\{\begin{array}{c} 1 \pm n_{\Omega} \\ \pm n_{\Omega} \end{array} \text { for } \tau \geq \tau^{\prime}\right. \end{align*} $$

$$ C_{12}\left(t, t^{\prime}\right)=e^{-i \Omega\left(t-t^{\prime}\right)} $$

$$ A_{\Omega}\left(t, t^{\prime}\right)=\tanh ^{\mp 1} \frac{\Omega}{2 T} e^{-i \Omega\left(t-t^{\prime}\right)} $$

$$ C_{\Omega}\left(t, t^{\prime}\right)=e^{F / T} \operatorname{Tr}\left\{e^{-\hat{H} / T}\left[\hat{a}_{H}(t), \hat{a}_{H}^{\dagger}\left(t^{\prime}\right)\right]_{\mp}\right\} $$

$$ \left[\hat{a}_{H}(t), \hat{a}_{H}^{\dagger}\left(t^{\prime}\right)\right]=e^{-i \Omega\left(t-t^{\prime}\right)} $$

$$ e^{F / T} \operatorname{Tr}\left(e^{-\hat{H} / T}\right)=1 $$

$$ G_{\Omega}^{R}\left(t-t^{\prime}\right)=\Theta\left(t-t^{\prime}\right) e^{-i \Omega\left(t-t^{\prime}\right)}, \quad G_{\Omega}^{A}\left(t-t^{\prime}\right)=-\Theta\left(t^{\prime}-t\right) e^{-i \Omega\left(t-t^{\prime}\right)} $$

$$ G_{\Omega}\left(\tau, \tau^{\prime}\right) \equiv e^{F / T} \operatorname{Tr}\left[e^{-\hat{H} / T} \hat{T}_{\tau} \hat{a}_{H}(\tau) \hat{a}_{H}^{\dagger}\left(\tau^{\prime}\right)\right] $$

$$ \begin{align*} & \hat{a}_{H}^{\dagger}(\tau) \equiv e^{\hat{H} \tau} \hat{a}^{\dagger} e^{-\hat{H} \tau}=\hat{a}^{\dagger} e^{\Omega \tau} \\ & \hat{a}_{H}(\tau) \equiv e^{\hat{H} \tau} \hat{a} e^{-\hat{H} \tau}=\hat{a} e^{-\Omega \tau} \end{align*} $$

$$ A_{12}\left(t-t^{\prime}\right) \equiv e^{F / T} \operatorname{Tr}\left\{e^{-\hat{H} / T}\left[\hat{a}_{H}(t), \hat{a}_{H}^{\dagger}\left(t^{\prime}\right)\right]_{ \pm}\right\} $$

$$ G_{\Omega}(\omega)=\left(1 \mp e^{-\Omega / T}\right)^{-1} G_{\Omega}^{R}(\omega)+\left(1 \mp e^{\Omega / T}\right)^{-1} G_{\Omega}^{A}(\omega) $$

$$ \begin{align*} G_{\Omega}\left(t, t^{\prime}\right) & =\left(1 \mp e^{-\Omega / T}\right)^{-1} \Theta\left(t-t^{\prime}\right) e^{-i \Omega\left(t-t^{\prime}\right)}-\left(1 \mp e^{\Omega / T}\right)^{-1} \Theta\left(t^{\prime}-t\right) e^{-i \Omega\left(t-t^{\prime}\right)} \\ & =\left[\Theta\left(t-t^{\prime}\right) \pm\left(e^{\Omega / T} \mp 1\right)^{-1}\right] e^{-i \Omega\left(t-t^{\prime}\right)}=\left[\Theta\left(t-t^{\prime}\right) \pm n_{\Omega}\right] e^{-i \Omega\left(t-t^{\prime}\right)} \end{align*} $$

$$ \begin{align*} G_{\Omega}\left(t, t^{\prime}\right) & \equiv G_{\Omega}\left(t-t^{\prime}\right)=e^{F / T} \operatorname{Tr}\left[e^{-\hat{H} / T} \hat{T} \hat{a}_{H}(t) \hat{a}_{H}^{\dagger}\left(t^{\prime}\right)\right] \\ & =\Theta\left(t-t^{\prime}\right)\left\langle\hat{a} \hat{a}^{\dagger}\right\rangle e^{-i \Omega\left(t-t^{\prime}\right)} \pm \Theta\left(t^{\prime}-t\right)\left\langle\hat{a}^{\dagger} \hat{a}\right\rangle e^{-i \Omega\left(t-t^{\prime}\right)} \\ & =\Theta\left(t-t^{\prime}\right)\left(1 \pm n_{\Omega}\right) e^{-i \Omega\left(t-t^{\prime}\right)} \pm \Theta\left(t^{\prime}-t\right) n_{\Omega} e^{-i \Omega\left(t-t^{\prime}\right)} \end{align*} $$

$$ \bar{G}_{\Omega}\left(t, t^{\prime}\right) \equiv G_{\Omega}\left(t-t^{\prime}\right)=e^{F / T} \operatorname{Tr}\left[e^{-\hat{H} / T} \hat{T} \hat{a}_{H}^{\dagger}(t) \hat{a}_{H}\left(t^{\prime}\right)\right] $$

$$ \begin{align*} \bar{G}_{\Omega}\left(t, t^{\prime}\right) & =\Theta\left(t-t^{\prime}\right)\left\langle\hat{a}^{\dagger} \hat{a}\right\rangle e^{-i \Omega\left(t-t^{\prime}\right)} \pm \Theta\left(t^{\prime}-t\right)\left\langle\hat{a} \hat{a}^{\dagger}\right\rangle e^{-i \Omega\left(t-t^{\prime}\right)} \\ & =\Theta\left(t-t^{\prime}\right) n_{\Omega} e^{-i \Omega\left(t-t^{\prime}\right)} \pm \Theta\left(t^{\prime}-t\right)\left(1 \pm n_{\Omega}\right) e^{-i \Omega\left(t-t^{\prime}\right)} \end{align*} $$

$$ \hat{x}(t)=\sqrt{\frac{\hbar}{2 M \Omega}}\left[\hat{a} e^{-i \Omega t}+\hat{a}^{\dagger} e^{i \Omega t}\right] $$

$$ C\left(t, t^{\prime}\right) \equiv\left\langle\left[\hat{\varphi}(t), \hat{\varphi}\left(t^{\prime}\right)\right]_{\mp}\right\rangle_{\rho}=-\frac{\hbar}{2 M \Omega} 2 i \sin \Omega\left(t-t^{\prime}\right) $$

$$ \rho\left(\omega^{\prime}\right)=\frac{1}{2 M \Omega} 2 \pi\left[\delta\left(\omega^{\prime}-\Omega\right)-\delta\left(\Omega^{\prime}+\Omega\right)\right] $$

$$ \begin{align*} G^{R}\left(t, t^{\prime}\right) & =\frac{\hbar}{2 M \Omega}\left[G_{\Omega}^{R}\left(t, t^{\prime}\right)-G_{-\Omega}^{R}\left(t, t^{\prime}\right)\right] \\ G^{A}\left(t, t^{\prime}\right) & =-\frac{\hbar}{2 M \Omega} \Theta\left(t-t^{\prime}\right) 2 i \sin \Omega\left(t-t^{\prime}\right) \\ 2 M \Omega & \left.G_{\Omega}^{A}\left(t, t^{\prime}\right)-G_{-\Omega}^{A}\left(t, t^{\prime}\right)\right]=\frac{\hbar}{2 M \Omega} \Theta\left(t-t^{\prime}\right) 2 i \sin \Omega\left(t^{\prime}-t\right) \end{align*} $$

$$ A\left(t, t^{\prime}\right)=\left\langle\left[\hat{\varphi}(t), \hat{\varphi}\left(t^{\prime}\right)\right]_{\mp}\right\rangle=\frac{\hbar}{2 M \Omega} \operatorname{coth}^{ \pm 1} \frac{\Omega}{2 k_{B} T} 2 \cos \Omega\left(t-t^{\prime}\right) $$

$$ G^{P}\left(t, t^{\prime}\right) \equiv\left\langle\hat{\varphi}(t) \hat{\varphi}\left(t^{\prime}\right)\right\rangle=\frac{\hbar}{2 M \Omega}\left[\left(1 \pm 2 n_{\Omega}\right) \cos \Omega\left(t-t^{\prime}\right)-i \sin \Omega\left(t-t^{\prime}\right)\right] $$

$$ G^{P}\left(t, t^{\prime}\right)=\left\langle\hat{\varphi}(t) \hat{\varphi}\left(t^{\prime}\right)\right\rangle=\frac{\hbar}{2 M \Omega} e^{-i \Omega\left(t-t^{\prime}\right)} $$

$$ G\left(t, t^{\prime}\right)=\Theta\left(t-t^{\prime}\right) G^{P}\left(t, t^{\prime}\right) \pm \Theta\left(t^{\prime}-t\right) G^{P}\left(t^{\prime}, t\right)=\frac{1}{2}\left[A\left(t, t^{\prime}\right)+\epsilon\left(t-t^{\prime}\right) C\left(t, t^{\prime}\right)\right] $$

$$ G\left(t, t^{\prime}\right) \equiv\left\langle\hat{T} \hat{\varphi}(t) \hat{\varphi}\left(t^{\prime}\right)\right\rangle=\frac{\hbar}{2 M \Omega}\left[\left(1 \pm 2 n_{\Omega}\right) \cos \Omega\left|t-t^{\prime}\right|-i \sin \Omega\left|t-t^{\prime}\right|\right] $$

$$ G\left(t, t^{\prime}\right)=\left\langle\hat{T} \hat{\varphi}(t) \hat{\varphi}\left(t^{\prime}\right)\right\rangle=\frac{\hbar}{2 M \Omega} e^{-i \Omega\left|t-t^{\prime}\right|} $$

$$ G\left(t, t^{\prime}\right) \equiv\left\langle\hat{T} \hat{\varphi}(t) \hat{\varphi}\left(t^{\prime}\right)\right\rangle=\frac{\hbar}{2 M \Omega} \frac{\cosh \left[\frac{\Omega}{2}\left(\hbar \beta-i\left|t-t^{\prime}\right|\right)\right]}{\sinh \frac{\hbar \Omega \beta}{2}} $$

$$ A\left(t, t^{\prime}\right)=2 \operatorname{Re} G\left(t, t^{\prime}\right), \quad C\left(t, t^{\prime}\right)=2 i \operatorname{Im} G\left(t, t^{\prime}\right) . $$

$$ \begin{align*} G^{A}\left(t, t^{\prime}\right) & =\mp G^{R}\left(t^{\prime}, t\right) \\ A\left(t, t^{\prime}\right) & = \pm A\left(t^{\prime}, t\right) \\ C\left(t, t^{\prime}\right) & =\mp C\left(t^{\prime}, t\right) \\ G\left(t, t^{\prime}\right) & = \pm G\left(t^{\prime}, t\right) \end{align*} $$

$$ \hat{\rho}=\sum_{n} \rho_{n}|n\rangle\langle n|, $$

$$ \rho_{n}=e^{-E_{n} / T} $$

$$ \mathcal{A}_{0}=\int d t d^{D} x \mathcal{L}_{0}(\mathbf{x}, t) \equiv \frac{1}{2} \int d t d^{D} x\left\{[\dot{\varphi}(\mathbf{x}, t)]^{2}-[\nabla \varphi(\mathbf{x}, t)]^{2}-m^{2} \varphi^{2}(\mathbf{x}, t)\right\} $$

$$ \ddot{\varphi}(\mathbf{x}, t)+\left(-\partial_{\mathbf{x}}^{2}+m^{2}\right) \varphi(\mathbf{x}, t)=0 $$

$$ f_{\mathbf{p}}(\mathbf{x}, t)=\frac{1}{\sqrt{2 \omega_{\mathbf{p}} V}} e^{-i \omega_{\mathbf{p}} t+i \mathbf{p x}}, \quad \bar{f}_{\mathbf{p}}(\mathbf{x}, t)=\frac{1}{\sqrt{\omega_{\mathbf{p}} V}} e^{i \omega_{\mathbf{p}} t+i \mathbf{p} \mathbf{x}} $$

$$ \omega_{\mathbf{p}} \equiv \sqrt{\mathbf{p}^{2}+m^{2}} $$

$$ \varphi(\mathbf{x}, t)=\sum_{\mathbf{p}} \frac{1}{2 \omega_{\mathbf{p}} V}\left(a_{\mathbf{p}} e^{-i \omega_{\mathbf{p}}+i \mathbf{p x}}+a_{\mathbf{p}}^{*} e^{i \omega_{\mathbf{p}} t+i \mathbf{p x}}\right) . $$

$$ \pi(\mathbf{x}, t) \equiv p_{\mathbf{x}}(t) \equiv \dot{\varphi}(\mathbf{x}, t) $$

$$ [\hat{\pi}(\mathbf{x}, t), \varphi(\mathbf{x}, t)]=-i \delta_{\mathbf{x z}^{\prime}} $$

$$ \left[\hat{a}_{\mathbf{p}}(t), \hat{a}_{\mathbf{p}^{\prime}}^{\dagger}(t)\right]=\delta_{\mathbf{p p}^{\prime}}, \quad\left[\hat{a}_{\mathbf{p}}^{\dagger}(t), \hat{a}_{\mathbf{p}^{\prime}}^{\dagger}(t)\right]=0, \quad\left[\hat{a}_{\mathbf{p}}(t), \hat{a}_{\mathbf{p}^{\prime}}(t)\right]=0 $$

$$ \begin{align*} \left\langle\hat{\varphi}_{H}(x)\right\rangle_{\rho} & =\operatorname{Tr}\left[\left(\hat{\rho} \hat{\varphi}_{H}(x)\right]\right. \\ \left\langle\hat{\varphi}_{H}(x) \hat{\varphi}_{H}(y)\right\rangle_{\rho} & =\operatorname{Tr}\left[\hat{\rho} \hat{\varphi}_{H}(x) \hat{\varphi}_{H}(y)\right] \end{align*} $$

$$ \hat{H}_{0}=\int d^{D} x \hat{\mathcal{H}}_{0}(\mathbf{x}, t) \equiv \frac{1}{2} \int d^{D} x\left\{[\dot{\hat{\varphi}}(\mathbf{x}, t)]^{2}+[\nabla \hat{\varphi}(\mathbf{x}, t)]^{2}+m^{2} \hat{\varphi}^{2}(\mathbf{x}, t)\right\} . $$

$$ \hat{\varphi}(x) \equiv e^{i \hat{H}_{0}\left(t-t_{0}\right)} \hat{\varphi}_{H}\left(x, t_{0}\right) e^{-i \hat{H}_{0}\left(t-t_{0}\right)} $$

$$ \hat{H}_{I}^{\mathrm{int}}(t) \equiv e^{i \hat{H} t} \hat{H}^{\mathrm{int}}(t) e^{-i \hat{H} t} $$

$$ \hat{U}\left(t, t_{0}\right) \equiv \hat{T} \exp \left[i \int_{t_{0}}^{t} d t^{\prime} \hat{H}_{I}^{\mathrm{int}}\left(t^{\prime}\right)\right] $$

$$ \hat{\varphi}_{H}(x)=\hat{U}\left(t_{0}, t\right) \hat{\varphi}(x) \hat{U}\left(t, t_{0}\right) $$

$$ \begin{align*} \left\langle\hat{\varphi}_{H}(x)\right\rangle_{\rho} & =\operatorname{Tr}\left[\hat{\rho} \hat{U}\left(t_{0}, t\right) \hat{\varphi}(x) \hat{U}\left(t, t_{0}\right)\right] \\ \left\langle\hat{\varphi}_{H}(x) \hat{\varphi}_{H}\left(x^{\prime}\right)\right\rangle_{\rho} & = \begin{cases}\operatorname{Tr}\left[\hat{\rho} \hat{U}\left(t_{0}, t\right) \hat{\varphi}(x) \hat{U}\left(t, t^{\prime}\right) \hat{\varphi}\left(x^{\prime}\right) \hat{U}\left(t^{\prime}, t_{0}\right)\right], & t>t^{\prime} \\ \operatorname{Tr}\left[\hat{\rho} \hat{U}\left(t_{0}, t^{\prime}\right) \hat{\varphi}\left(x^{\prime}\right) \hat{U}\left(t^{\prime}, t\right) \hat{\varphi}(x) \hat{U}\left(t, t_{0}\right)\right], & t^{\prime}>t\end{cases} \end{align*} $$

$$ \begin{align*} \left\langle\hat{\varphi}_{H}(x)\right\rangle_{\rho} & =\operatorname{Tr}\left[\hat{\rho} \hat{S}^{\dagger} \hat{T} \hat{S} \hat{\varphi}(x)\right] \\ \left\langle\hat{\varphi}_{H}(x) \hat{\varphi}_{H}(y)\right\rangle_{\rho} & =\operatorname{Tr}\left[\hat{\rho} \hat{S}^{\dagger} \hat{T} \hat{S} \hat{\varphi}(x) \hat{\varphi}(y)\right] \end{align*} $$

$$ \begin{align*} \hat{S}^{\dagger} \hat{T}(\hat{S} \hat{\varphi}(x)) & =\hat{U}(-\infty, t) \hat{U}(t, \infty) \hat{T}(\hat{U}(\infty, t) \hat{\varphi}(x) \hat{U}(t,-\infty)) \\ & =\hat{U}(-\infty, t) \hat{\varphi}(x) \hat{U}(t,-\infty) \end{align*} $$

$$ \hat{S}^{\dagger} \hat{T}(\hat{S} \hat{\varphi}(x)) $$

$$ \hat{T}_{\mathrm{P}}\left(\hat{S}^{\dagger} \hat{S} \hat{\varphi}\left(x_{+}\right)\right) $$

$$ \hat{T}_{\mathrm{P}}\left\{\hat{S}^{\dagger} \hat{S} \exp \left[i \int d x j\left(x_{+}\right) \hat{\varphi}\left(x_{+}\right)\right]\right\} $$

$$ Z\left[j_{\mathrm{P}}\right]=\operatorname{Tr}\left\{\hat{\rho} \hat{T}_{\mathrm{P}} \hat{S}^{\dagger} \hat{S} \exp \left[i \int_{\mathrm{P}} d x j_{\mathrm{P}}(x) \hat{\varphi}_{\mathrm{P}}(x)\right]\right\} $$

$$ \hat{S}^{\dagger} \hat{S}=\hat{T}_{\mathrm{P}} \exp \left[-i \int_{\mathrm{P}} d t \hat{H}_{I}^{\mathrm{int}}(t)\right] $$

$$ Z\left[j_{\mathrm{P}}\right]=\operatorname{Tr}\left\{\hat{\rho} \hat{T}_{\mathrm{P}} \exp \left[-i \int_{\mathrm{P}} d t \hat{H}_{I}^{\mathrm{int}}(t)+i \int_{\mathrm{P}} d x j_{\mathrm{P}}(x) \hat{\varphi}_{\mathrm{P}}(x)\right]\right\} $$

$$ \int_{\mathrm{P}} d x j_{\mathrm{P}}(x) \hat{\varphi}_{\mathrm{P}}(x)=\int d^{3} x\left[\int_{-\infty}^{\infty} d t j\left(\mathbf{x}, t_{+}\right) \hat{\varphi}\left(x_{+}\right)+\int_{\infty}^{-\infty} d t j\left(\mathbf{x}, t_{-}\right) \hat{\varphi}\left(x_{-}\right)\right] $$

$$ \int_{\mathrm{P}} d x j_{\mathrm{P}}(x) \hat{\varphi}_{\mathrm{P}}(x)=\int d^{3} x \int_{-\infty}^{\infty} d t\left[j\left(\mathbf{x}, t_{+}\right) \hat{\varphi}\left(x_{+}\right)-j\left(\mathbf{x}, t_{-}\right) \hat{\varphi}\left(x_{-}\right)\right] $$

$$ \hat{\vec{\varphi}}(x)=\binom{\hat{\varphi}\left(x_{+}\right)}{\hat{\varphi}\left(x_{-}\right)} $$

$$ \vec{\jmath}(x)=\binom{j\left(x_{+}\right)}{-j\left(x_{-}\right)} $$

$$ \int d x \vec{\jmath}(x) \hat{\vec{\varphi}}(x) $$

$$ \int_{\mathrm{P}} d x j_{\mathrm{P}}(x) G_{\mathrm{P}}\left(x, x^{\prime}\right) j_{\mathrm{P}}\left(x^{\prime}\right)=\int d x \vec{\jmath}(x) G\left(x, x^{\prime}\right) \vec{\jmath}\left(x^{\prime}\right) $$

$$ G(x, y)=\left(\begin{array}{cc} G_{++}(x, y) & G_{+-}(x, y) \\ G_{-+}(x, y) & G_{--}(x, y) \end{array}\right) \equiv\left(\begin{array}{cc} G\left(x_{+}, y_{+}\right) & G\left(x_{+}, y_{-}\right) \\ G\left(x_{-}, y_{+}\right) & G\left(x_{-}, y_{-}\right) \end{array}\right) $$

$$ G_{\mathrm{P}}(x, y)=\left.\frac{\delta}{i \delta j_{\mathrm{P}}(x)} \frac{\delta}{i \delta j_{\mathrm{P}}(y)} Z\left[j_{\mathrm{P}}\right]\right|_{j_{\mathrm{P}}=0}=\operatorname{Tr}\left[\hat{\rho} \hat{T}_{\mathrm{P}} \hat{S}^{\dagger} \hat{S} \hat{\varphi}_{\mathrm{P}}(x) \hat{\varphi}_{\mathrm{P}}(y)\right] $$

$$ G_{\mathrm{P}}(x, y)=\left(\begin{array}{cc} G_{++}(x, y) & G_{+-}(x, y) \\ G_{-+}(x, y) & G_{--}(x, y) \end{array}\right) $$

$$ G_{\mathrm{P}}(x, y)=\left\langle\hat{T}_{\mathrm{P}} \hat{\varphi}_{H}\left(x_{\mathrm{P}}\right) \hat{\varphi}_{H}\left(y_{\mathrm{P}}\right)\right\rangle_{\rho}, $$

$$ G_{-+}(x, y)=\left\langle\hat{\varphi}_{H}(x) \hat{\varphi}_{H}(y)\right\rangle_{\rho} . $$

$$ G_{+-}(x, y)=\left\langle\hat{\varphi}_{H}(y) \hat{\varphi}_{H}(x)\right\rangle_{\rho}= \pm\left\langle\hat{\varphi}_{H}(x) \hat{\varphi}_{H}(y)\right\rangle_{\rho} . $$

$$ G_{++}(x, y)=\left\langle\hat{T} \hat{\varphi}_{H}(x) \hat{\varphi}_{H}(y)\right\rangle_{\rho} \equiv G(x, y), $$

$$ G_{--}(x, y)=\left\langle\hat{\bar{T}} \hat{\varphi}_{H}(x) \hat{\varphi}_{H}(y)\right\rangle_{\rho} \equiv \bar{G}(x, y) $$

$$ G_{++}+G_{--}=G_{+-}+G_{-+} . $$

$$ \begin{align*} G^{R}(x, y) & =\Theta(x-y)\left\langle\left[\hat{\varphi}_{H}(x), \hat{\varphi}_{H}(y)\right]_{\mp}\right\rangle_{\rho}, \\ G^{A}(x, y) & =-\Theta(y-x)\left\langle\left[\hat{\varphi}_{H}(x), \hat{\varphi}_{H}(y)\right]_{\mp}\right\rangle_{\rho}, \\ A(x, y) & =\left\langle\left[\hat{\varphi}_{H}(x), \hat{\varphi}_{H}(y)\right]_{ \pm}\right\rangle_{\rho} . \end{align*} $$

$$ C(x, y)=\left\langle\left[\hat{\varphi}_{H}(x), \hat{\varphi}_{H}(y)\right]_{\mp}\right\rangle_{\rho}, $$

$$ C(x, y)=G^{R}(x, y)-G^{A}(x, y) . $$

$$ G^{R}\left(x, x^{\prime}\right)=\left.\sum_{\mathbf{p}} \frac{M}{\hbar V} e^{i \mathbf{p}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)} G^{R}\left(t, t^{\prime}\right)\right|_{\Omega=\omega_{\mathbf{p}}} . $$

$$ G^{R}\left(x, x^{\prime}\right)=-\Theta\left(x-x^{\prime}\right) \int \frac{d^{D} p}{2 \omega_{\mathbf{p}}(2 \pi)^{D}} e^{i \mathbf{k}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)} 2 i \sin \omega_{\mathbf{p}}\left(t-t^{\prime}\right) $$

$$ A\left(x, x^{\prime}\right)=\int \frac{d^{D} p}{2 \omega_{\mathbf{p}}(2 \pi)^{D}} e^{i \mathbf{k}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)} 2 \cos \omega_{\mathbf{p}}\left(t-t^{\prime}\right) $$

$$ \begin{align*} G^{A}\left(x, x^{\prime}\right) & =\mp G^{R}\left(x^{\prime}, x\right) \\ A\left(x, x^{\prime}\right) & = \pm A\left(x^{\prime}, x\right) \\ C\left(x, x^{\prime}\right) & =\mp C\left(x^{\prime}, x\right) \\ G\left(x, x^{\prime}\right) & = \pm G\left(x^{\prime}, x\right)^{*} \end{align*} $$

$$ \begin{align*} G^{R} & =G_{-+}-G_{--}=G_{++}-G_{+-}, \\ G^{A} & =G_{+-}-G_{--}=G_{++}-G_{-+}, \\ A & =G_{-+}+G_{+-}=G_{++}+G_{--}, \\ C & =G_{-+}-G_{+-}=G^{R}-G^{A}, \end{align*} $$

$$ \begin{align*} & G_{-+}=\frac{1}{2}(A+C)=\frac{1}{2}\left(A+G^{R}-G^{A}\right) \\ & G_{+-}=\frac{1}{2}(A-C)=\frac{1}{2}\left(A-G^{R}+G^{A}\right) \end{align*} $$

$$ \begin{align*} G_{++} & =G^{R}+G_{+-}=\frac{1}{2}\left(A+G^{R}+G^{A}\right) \\ G_{--} & =G_{+-}+G_{-+}-G_{++} \\ & =A-G_{++}=\frac{1}{2}\left(A-G^{R}-G^{A}\right) \end{align*} $$

$$ G_{\mathrm{P}}=\frac{1}{2}\left(\begin{array}{cc} A+G^{R}+G^{A} & A-G^{R}+G^{A} \\ A+G^{R}-G^{A} & A-G^{R}-G^{A} \end{array}\right) . $$

$$ \tilde{G}=Q G_{\mathrm{P}} Q^{-1}, \quad \text { with } \quad Q=\frac{1}{\sqrt{2}}\left(\begin{array}{rr} 1 & -1 \\ 1 & 1 \end{array}\right)=\left(Q^{T}\right)^{-1} $$

$$ \begin{gather*} \tilde{G}(x, y)=\frac{1}{\sqrt{2}}\left(\begin{array}{rr} 1 & -1 \\ 1 & 1 \end{array}\right) \frac{1}{2}\left(\begin{array}{cc} A+G^{R}+G^{A} & A-G^{R}+G^{A} \\ A+G^{R}-G^{A} & A-G^{R}-G^{A} \end{array}\right) \\ \times \frac{1}{\sqrt{2}}\left(\begin{array}{rr} 1 & 1 \\ -1 & 1 \end{array}\right)=\left(\begin{array}{rc} 0 & G^{A} \\ G^{R} & A \end{array}\right) \end{gather*} $$

$$ \begin{align*} \int_{\mathrm{P}} d x j_{\mathrm{P}}(x) \hat{\varphi}_{\mathrm{P}}(x) & =\int d x\left(j\left(x_{+}\right),-j\left(x_{-}\right)\right)\binom{\hat{\varphi}\left(x_{+}\right)}{\hat{\varphi}\left(x_{-}\right)} \\ & =\int d x \tilde{\jmath}(x) \hat{\tilde{\varphi}}(x) \end{align*} $$

$$ \tilde{\hat{\varphi}}(x) \equiv Q\binom{\hat{\varphi}\left(x_{+}\right)}{\hat{\varphi}\left(x_{-}\right)}=\frac{1}{\sqrt{2}}\binom{\hat{\varphi}\left(x_{+}\right)-\hat{\varphi}\left(x_{-}\right)}{\hat{\varphi}\left(x_{+}\right)+\hat{\varphi}\left(x_{-}\right)} $$

$$ \tilde{\jmath}(x) \equiv\binom{j_{1}(x)}{j_{2}(x)}=Q\binom{j\left(x_{+}\right)}{-j\left(x_{-}\right)}=\frac{1}{\sqrt{2}}\binom{j\left(x_{+}\right)+j\left(x_{-}\right)}{j\left(x_{+}\right)-j\left(x_{-}\right)} . $$

$$ \begin{align*} & \int d x d x^{\prime} j_{\mathrm{P}}(x) G_{\mathrm{P}}\left(x, x^{\prime}\right) j_{\mathrm{P}}\left(x^{\prime}\right) \\ & \quad=\int d x d x^{\prime}\left(j\left(x_{+}\right),-j\left(x_{-}\right)\right)\left(\begin{array}{ll} G_{++} & G_{+-} \\ G_{-+} & G_{--} \end{array}\right)\left(x, x^{\prime}\right)\binom{j\left(x_{+}^{\prime}\right)}{-j\left(x_{-}^{\prime}\right)} \end{align*} $$

$$ \int d x d x^{\prime} \tilde{\jmath}^{T}(x) \tilde{G}\left(x, x^{\prime}\right) \tilde{\jmath}\left(x^{\prime}\right) $$

$$ \begin{align*} \tilde{G}^{12} & =\tilde{G}^{(1)} \tilde{G}^{(2)}=Q G_{\mathrm{P}}^{(1)} Q^{-1} Q G_{\mathrm{P}}^{(2)} Q^{-1} \\ & =\left(\begin{array}{cc} 0 & G_{1}^{A} G_{2}^{A} \\ G_{1}^{R} G_{2}^{R} & G_{1}^{R} A_{2}+A_{1} G_{2}^{A} \end{array}\right) \end{align*} $$

$$ \exp \left[-i \int_{\mathrm{P}} d t \hat{H}_{I}^{\mathrm{int}}(t)\right]=\exp \left[i \int_{\mathrm{P}} d t \int d^{3} x L^{\mathrm{int}}\left(\hat{\varphi}_{\mathrm{P}}(\mathbf{x}, t)\right)\right] $$

$$ \exp \left\{i \mathcal{A}_{\mathrm{P}}^{\mathrm{int}}\left[\hat{\varphi}_{\mathrm{P}}\right]\right\} $$

$$ Z\left[j_{\mathrm{P}}\right]=\exp \left\{i \mathcal{A}_{\mathrm{P}}^{\mathrm{int}}\left[\delta / i \delta j_{\mathrm{P}}\right]\right\} Z_{0}\left[j_{\mathrm{P}}\right] $$

$$ Z_{0}\left[j_{\mathrm{P}}\right]=\operatorname{Tr}\left\{\hat{\rho} \hat{T}_{\mathrm{P}} \exp \left[i \int_{\mathrm{P}} d x \hat{\varphi}_{\mathrm{P}}(x) j_{\mathrm{P}}(x)\right]\right\} $$

$$ Z_{0}\left[j_{\mathrm{P}}\right]=\exp \left[-\frac{1}{2} \int d x d y j_{\mathrm{P}}(x) G_{\mathrm{P}}(x, y) j_{\mathrm{P}}(y)\right] $$

$$ \begin{array}{r} Z_{0}\left[j_{+}, j_{-}\right]=\exp \left[-\frac{1}{2} \int d x \int d x^{\prime}\left(j_{+},-j_{-}\right) Q^{-1}\left(\begin{array}{cc} 0 & G^{A} \\ G^{R} & A \end{array}\right) Q\binom{j_{+}}{-j_{-}}\right] \\ =\exp \left\{-\frac{1}{4} \int d x \int d x^{\prime}\left[\left(j_{+}+j_{-}\right)(x) G^{A}\left(x, x^{\prime}\right)\left(j_{+}-j_{-}\right)\left(x^{\prime}\right)\right.\right. \\ +\left(j_{+}-j_{-}\right)(x) G^{R}\left(x, x^{\prime}\right)\left(j_{+}+j_{-}\right)\left(x^{\prime}\right) \\ \left.\left.+\left(j_{+}-j_{-}\right)(x) A\left(x, x^{\prime}\right)\left(j_{+}-j_{-}\right)\left(x^{\prime}\right)\right]\right\} \end{array} $$

$$ j_{+}(x) \equiv j\left(x_{+}\right), \quad j_{-}(x) \equiv j\left(x_{-}\right) $$

$$ \begin{align*} & Z_{0}\left[j_{+}, j_{-}\right]=\exp \left\{-\frac{1}{2} \int d x \int d x^{\prime} \Theta\left(x^{\prime}-x\right)\right. \\ & \left.\quad \times\left[\left(j_{+}-j_{-}\right)(x) G^{R}\left(x, x^{\prime}\right)\left(j_{+}+j_{-}\right)\left(x^{\prime}\right)+\left(j_{+}-j_{-}\right)(x) A\left(x, x^{\prime}\right)\left(j_{+}-j_{-}\right)\left(x^{\prime}\right)\right]\right\} \end{align*} $$

$$ \begin{align*} & Z_{0}\left[j_{+}, j_{-}\right]=\exp \left\{-\frac{1}{2 \hbar^{2}} \int d t \int d t^{\prime} \Theta\left(t-t^{\prime}\right)\right. \\ & \left.\quad \times\left[\left(j_{+}-j_{-}\right)(t) C\left(t, t^{\prime}\right)\left(j_{+}+j_{-}\right)\left(t^{\prime}\right)+\left(j_{+}-j_{-}\right)(t) A\left(t, t^{\prime}\right)\left(j_{+}-j_{-}\right)\left(t^{\prime}\right)\right]\right\} \end{align*} $$

$$ \begin{align*} Z_{0}\left[j_{+}, j_{-}\right]=\exp & \left\{-\frac{1}{2 M \Omega \hbar} \int d t \int d t^{\prime} \Theta\left(t^{\prime}-t\right)\right. \\ \times & {\left[-\left(j_{+}-j_{-}\right)(t) \quad i \sin \Omega\left(t-t^{\prime}\right) \quad\left(j_{+}+j_{-}\right)\left(t^{\prime}\right)\right.} \\ & \left.\left.\quad+\left(j_{+}-j_{-}\right)(t) \operatorname{coth} \frac{\hbar \Omega}{2 k_{B} T} \cos \Omega\left(t-t^{\prime}\right)\left(j_{+}-j_{-}\right)\left(t^{\prime}\right)\right]\right\} \end{align*} $$

$$ G\left(t, t^{\prime}\right)=\frac{1}{2}\left[A\left(t, t^{\prime}\right)+C\left(t, t^{\prime}\right)\right]=\frac{\hbar}{2 M \Omega} \frac{\cosh \frac{\Omega}{2}\left[\hbar \beta-i\left(t-t^{\prime}\right)\right]}{\sinh \frac{\hbar \Omega \beta}{2}} $$

$$ \begin{align*} \left(X_{b} t_{b} \mid X_{a} t_{a}\right)_{\Omega}^{j} & =\int \mathcal{D} X(t) \exp \left\{\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t\left[\frac{M}{2}\left(\dot{X}^{2}-\Omega^{2} X^{2}\right)+j X\right]\right\} \\ & =e^{(i / \hbar) \mathcal{A}_{\mathrm{cl}, j}} F_{\Omega, j}\left(t_{b}, t_{a}\right) \end{align*} $$

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

$$ +\frac{1}{D_{a}\left(t_{b}\right)} \int_{t_{a}}^{t_{b}} d t\left[X_{b} D_{a}(t)+X_{a} D_{b}(t)\right] j(t) $$

$$ Z_{0}\left[j_{+}, j_{-}\right]=\int d X_{b} d X_{a}\left(X_{b} \hbar \beta \mid X_{a} 0\right)_{\Omega}\left(X_{b} t_{b} \mid X_{a} t_{a}\right)_{\Omega}^{j+}\left(X_{b} t_{b} \mid X_{a} t_{a}\right)_{\Omega}^{j_{--} *} $$

$$ \begin{align*} \left(X_{b} \hbar \beta \mid X_{a} 0\right) & =\frac{1}{\sqrt{2 \pi \hbar / M}} \sqrt{\frac{\Omega}{\sinh \hbar \beta}} \\ & \times \exp \left\{-\frac{1}{2 \hbar} \frac{M \Omega}{\sinh \hbar \beta \Omega}\left[\left(X_{b}^{2}+X_{a}^{2}\right) \cosh \hbar \beta \Omega-2 X_{b} X_{a}\right]\right\} \end{align*} $$

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

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

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

$$ \begin{align*} \left|\left(x_{b} t_{b} \mid x_{a} t_{a}\right)\right|^{2}=\int \mathcal{D} x_{\mathrm{P}} & \exp \left\{\frac{i}{\hbar} \int_{\mathrm{P}} d t\left[\frac{M}{2} \dot{x}_{\mathrm{P}}^{2}-V\left(x_{\mathrm{P}}\right)\right]\right\} \\ & \times \operatorname{Tr}\left\{\hat{\rho} \hat{T}_{\mathrm{P}} \exp \left[\frac{i}{\hbar} \sum_{i} c_{i} \int_{\mathrm{P}} d t \hat{\varphi}_{\mathrm{P}}^{i}(t) x_{\mathrm{P}}(t)\right]\right\} \end{align*} $$

$$ \operatorname{Tr}\left\{\hat{\rho} \hat{T}_{\mathrm{P}} \exp \left[\frac{i}{\hbar} \sum_{i} c_{i} \int_{\mathrm{P}} d t \hat{\varphi}_{\mathrm{P}}^{i}(t) x_{\mathrm{P}}(t)\right]\right\}=\prod_{i} \operatorname{Tr}\left\{\hat{\rho} \hat{T}_{\mathrm{P}} \exp \left[\frac{i}{\hbar} c_{i} \int_{\mathrm{P}} d t \hat{\varphi}_{\mathrm{P}}^{i}(t) x_{\mathrm{P}}(t)\right]\right\} $$

$$ \begin{align*} & Z_{0}^{\mathrm{b}}\left[x_{+}, x_{-}\right]=\exp \left\{-\frac{1}{2 \hbar^{2}} \int d t \int d t^{\prime} \Theta\left(t-t^{\prime}\right)\right. \\ & \left.\quad \times\left[\left(x_{+}-x_{-}\right)(t) C_{\mathrm{b}}\left(t, t^{\prime}\right)\left(x_{+}+x_{-}\right)\left(t^{\prime}\right)+\left(x_{+}-x_{-}\right)(t) A_{\mathrm{b}}\left(t, t^{\prime}\right)\left(x_{+}-x_{-}\right)\left(t^{\prime}\right)\right]\right\} \end{align*} $$

$$ \begin{align*} C_{\mathrm{b}}\left(t, t^{\prime}\right) & =\sum_{i} c_{i}^{2}\left\langle\left[\hat{\varphi}_{i}(t), \hat{\varphi}_{i}\left(t^{\prime}\right)\right]\right\rangle_{T}=-\hbar \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \rho_{\mathrm{b}}\left(\omega^{\prime}\right) i \sin \omega^{\prime}\left(t-t^{\prime}\right) \\ A_{\mathrm{b}}\left(t, t^{\prime}\right) & =\sum_{i} c_{i}^{2}\left\langle\left\{\hat{\varphi}_{i}(t), \hat{\varphi}_{i}\left(t^{\prime}\right)\right\}\right\rangle_{T}=\hbar \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \rho_{\mathrm{b}}\left(\omega^{\prime}\right) \operatorname{coth} \frac{\hbar \omega^{\prime}}{2 k_{B} T} \cos \omega^{\prime}\left(t-t^{\prime}\right) \end{align*} $$

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

$$ Z_{0}\left[x_{+}, x_{-}\right]=\exp \left\{\frac{i}{\hbar} \mathcal{A}^{\mathrm{FV}}\left[x_{+}, x_{-}\right]\right\}=\exp \left\{\frac{i}{\hbar}\left(\mathcal{A}_{D}^{\mathrm{FV}}\left[x_{+}, x_{-}\right]+\mathcal{A}_{F}^{\mathrm{FV}}\left[x_{+}, x_{-}\right]\right)\right\} $$

$$ \begin{align*} & \left|\left(x_{b} t_{b} \mid x_{a} t_{a}\right)\right|^{2}=\int \mathcal{D} x_{+}(t) \int \mathcal{D} x_{-}(t) \times \\ & \quad \times \exp \left\{\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t\left[\frac{M}{2}\left(\dot{x}_{+}^{2}-\dot{x}_{-}^{2}\right)-\left(V\left(x_{+}\right)-V\left(x_{-}\right)\right)\right]+\frac{i}{\hbar} \mathcal{A}^{\mathrm{FV}}\left[x_{+}, x_{-}\right]\right\} \end{align*} $$

$$ \gamma\left(t-t^{\prime}\right) \equiv \Theta\left(t-t^{\prime}\right) \frac{1}{M} \int_{-\infty}^{\infty} \frac{d \omega}{2 \pi} \frac{\sigma_{\mathrm{b}}(\omega)}{\omega} e^{-i \omega\left(t-t^{\prime}\right)} $$

$$ \Theta\left(t-t^{\prime}\right) C_{\mathrm{b}}\left(t, t^{\prime}\right)=i \hbar M \dot{\gamma}\left(t-t^{\prime}\right)+i \hbar M \Delta \omega^{2} \delta\left(t-t^{\prime}\right) $$

$$ \Delta \omega^{2} \equiv-\frac{1}{M} \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \frac{\sigma_{\mathrm{b}}\left(\omega^{\prime}\right)}{\omega^{\prime}}=-\frac{1}{M} \sum_{i} \frac{c_{i}^{2}}{M_{i} \Omega_{i}^{2}} $$

$$ \begin{align*} \mathcal{A}_{D}^{\mathrm{FV}}\left[x_{+}, x_{-}\right]= & -\frac{M}{2} \int_{t_{a}}^{t_{b}} d t \int_{t_{a}}^{t_{b}} d t^{\prime}\left(x_{+}-x_{-}\right)(t) \gamma\left(t-t^{\prime}\right)\left(\dot{x}_{+}+\dot{x}_{-}\right)\left(t^{\prime}\right) \\ & +\frac{M}{2} \int_{t_{a}}^{t_{b}} d t\left(x_{+}-x_{-}\right)(t) \gamma\left(t-t_{b}\right)\left(x_{+}+x_{-}\right)\left(t_{a}\right) \end{align*} $$

$$ \Delta \mathcal{A}_{\mathrm{loc}}\left[x_{+}, x_{-}\right]=\frac{M}{2} \int_{t_{a}}^{t_{b}} d t \Delta \omega^{2}\left(x_{+}^{2}-x_{-}^{2}\right)(t) $$

$$ -\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t\left[V_{\mathrm{ren}}\left(x_{+}\right)-V_{\mathrm{ren}}\left(x_{-}\right)\right] $$

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

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

$$ \gamma_{D}^{R}(t) \equiv \Theta(t) \gamma \omega_{D} e^{-\omega_{D} t} $$

$$ \gamma_{D}^{R}(t)=\int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \gamma_{D}^{R}\left(\omega^{\prime}\right) e^{-i \omega^{\prime} t} $$

$$ \gamma_{D}^{R}\left(\omega^{\prime}\right)=\gamma \frac{i \omega_{D}}{\omega^{\prime}+i \omega_{D}} $$

$$ \gamma_{m}=\left.\gamma\left(\omega^{\prime}\right)\right|_{\omega^{\prime}=i\left|\omega_{m}\right|} $$

$$ \gamma_{D}^{R}(t) \rightarrow \gamma \delta^{R}(t) $$

$$ \mathcal{A}_{D}^{\mathrm{FV}}\left[x_{+}, x_{-}\right]=-\frac{M}{2} \gamma \int_{t_{a}}^{t_{b}} d t\left(x_{+}-x_{-}\right)\left(\dot{x}_{+}+\dot{x}_{-}\right)^{R}-\frac{M}{2} \gamma\left(x_{+}^{2}-x_{-}^{2}\right)\left(t_{a}\right) $$

$$ A_{\mathrm{b}}\left(t, t^{\prime}\right)=2 M \gamma k_{B} T K\left(t, t^{\prime}\right) $$

$$ K\left(t, t^{\prime}\right)=K\left(t-t^{\prime}\right) \equiv \frac{1}{2 M \gamma k_{B} T} \sum_{i} c_{i}^{2}\left\langle\left\{\hat{\varphi}_{i}(t), \hat{\varphi}_{i}\left(t^{\prime}\right)\right\}\right\rangle_{T} $$

$$ w \equiv 2 M \gamma k_{B} T $$

$$ D \equiv k_{B} T / M \gamma $$

$$ w=2 \gamma^{2} M^{2} D $$

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

$$ K\left(\omega^{\prime}\right) \equiv \frac{1}{2 M \gamma} \frac{\rho_{\mathrm{b}}\left(\omega^{\prime}\right)}{\omega^{\prime}} \frac{\hbar \omega^{\prime}}{2 k_{B} T} \operatorname{coth} \frac{\hbar \omega^{\prime}}{2 k_{B} T} $$

$$ K\left(\omega^{\prime}\right) \rightarrow K^{\mathrm{Ohm}}\left(\omega^{\prime}\right) \equiv \frac{\hbar \omega^{\prime}}{2 k_{B} T} \operatorname{coth} \frac{\hbar \omega^{\prime}}{2 k_{B} T} $$

$$ \int_{-\infty}^{\infty} d t K\left(t-t^{\prime}\right)=1 $$

$$ K_{D}^{\mathrm{cl}}\left(\omega^{\prime}\right)=\frac{\omega_{D}^{2}}{\omega^{\prime 2}+\omega_{D}^{2}} $$

$$ K_{D}^{\mathrm{cl}}\left(t-t^{\prime}\right)=\frac{1}{2 \omega_{D}} e^{-\omega_{D}\left(t-t^{\prime}\right)} $$

$$ \mathcal{A}_{F}^{\mathrm{FV}}\left[x_{+}, x_{-}\right]=i \frac{w}{2 \hbar} \int_{t_{a}}^{t_{b}} d t \int_{t_{a}}^{t_{b}} d t^{\prime}\left(x_{+}-x_{-}\right)(t) K\left(t, t^{\prime}\right)\left(x_{+}-x_{-}\right)\left(t^{\prime}\right) $$

$$ \begin{align*} & \left|\left(x_{b} t_{b} \mid x_{a} t_{a}\right)\right|^{2}=\int \mathcal{D} x_{+}(t) \int \mathcal{D} x_{-}(t) \\ & \quad \quad \times \exp \left\{\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t\left[\frac{M}{2}\left(\dot{x}_{+}^{2}-\dot{x}_{-}^{2}\right)-\left(V\left(x_{+}\right)-V\left(x_{-}\right)\right)\right]\right\} \\ & \quad \times \exp \left\{-i \int_{t_{a}}^{t_{b}} d t \frac{M \gamma}{2 \hbar}\left(x_{+}-x_{-}\right)(t)\left(\dot{x}_{+}+\dot{x}_{-}\right)^{R}(t)\right. \\ & \left.\quad-\frac{w}{2 \hbar^{2}} \int_{t_{a}}^{t_{b}} d t \int_{t_{a}}^{t_{b}} d t^{\prime}\left(x_{+}-x_{-}\right)(t) K^{\mathrm{Ohm}}\left(t, t^{\prime}\right)\left(x_{+}-x_{-}\right)\left(t^{\prime}\right)\right\} \end{align*} $$

$$ \begin{align*} x & \equiv\left(x_{+}+x_{-}\right) / 2 \\ y & \equiv x_{+}-x_{-} \end{align*} $$

$$ \begin{align*} \left|\left(x_{b} t_{b} \mid x_{a} t_{a}\right)\right|^{2}= & \int \mathcal{D} x(t) \int \mathcal{D} y(t) \\ \times & \exp \left\{-\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t\left[M\left(-\dot{y} \dot{x}+\gamma y \dot{x}^{R}\right)+V\left(x+\frac{y}{2}\right)-V\left(x-\frac{y}{2}\right)\right]\right. \\ & \left.-\frac{w}{2 \hbar^{2}} \int_{t_{a}}^{t_{b}} d t \int_{t_{a}}^{t_{b}} d t^{\prime} y(t) K^{\mathrm{Ohm}}\left(t, t^{\prime}\right) y\left(t^{\prime}\right)\right\} \end{align*} $$

$$ \begin{align*} & P\left(x_{b} t_{b} \mid x_{a} t_{a}\right) \equiv\left|\left(x_{b} t_{b} \mid x_{a} t_{a}\right)\right|^{2}=\int \mathcal{D} x(t) \int \mathcal{D} y(t) \\ & \quad \times \exp \left\{-\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t y\left[M \ddot{x}+M \gamma \dot{x}^{R}+V^{\prime}(x)\right]-\frac{w}{2 \hbar^{2}} \int_{t_{a}}^{t_{b}} d t y^{2}\right\} \end{align*} $$

$$ P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\mathcal{N} \int \mathcal{D} x(t) \exp \left\{-\frac{1}{2 w} \int_{t_{a}}^{t_{b}} d t\left[M \ddot{x}+M \gamma \dot{x}^{R}+V^{\prime}(x)\right]^{2}\right\} $$

$$ \int d x_{b} P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=1 $$

$$ \lim _{t_{b} \rightarrow t_{a}} P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\delta\left(x_{b}-x_{a}\right) $$

$$ L_{\mathrm{e}}=\frac{1}{2 w}\left[M \ddot{x}+M \gamma \dot{x}+V^{\prime}(x)\right]^{2} $$

$$ \frac{\partial L}{\partial x}-\frac{d}{d t} \frac{\partial L}{\partial \dot{x}}+\frac{d^{2}}{d t^{2}} \frac{\partial L}{\partial \ddot{x}}=0 $$

$$ \begin{align*} p & =i \frac{\partial L}{\partial \dot{x}}=i \frac{M \gamma}{w}\left[M \ddot{x}+M \gamma \dot{x}+V^{\prime}(x)\right]=i \frac{M \gamma}{w}\left[M \dot{v}+M \gamma v+V^{\prime}(x)\right] \\ p_{v} & =i \frac{\partial L}{\partial \ddot{x}}=\frac{1}{\gamma} p \end{align*} $$

$$ H\left(p, p_{v}, x, v\right)=L_{\mathrm{e}}(\dot{x}, \ddot{x})-\sum_{i=1}^{2} \frac{\partial L_{\mathrm{e}}}{\partial \dot{x}_{i}} \dot{x}_{i}=L_{\mathrm{e}}(v, \dot{v})+i p v+i p_{v} \dot{v} $$

$$ H\left(p, p_{v}, x, v\right)=\frac{w}{2 M^{2}} p_{v}^{2}-i p_{v}\left[\gamma v+\frac{1}{M} V^{\prime}(x)\right]+i p v $$

$$ \begin{align*} P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)= & \int \mathcal{D} x \int \frac{\mathcal{D} p}{2 \pi} \int \mathcal{D} v \int \frac{\mathcal{D} p_{v}}{2 \pi} \\ & \times \exp \left\{\int_{t_{a}}^{t_{b}} d t\left[i\left(p \dot{x}+p_{v} \dot{v}\right)-H\left(p, p_{v}, x, v\right)\right]\right\} \end{align*} $$

$$ P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int_{-\infty}^{\infty} d v_{b} \int_{-\infty}^{\infty} d v_{a} P\left(x_{b} v_{b} t_{b} \mid x_{a} v_{a} t_{a}\right) $$

$$ \begin{align*} P\left(x_{b} v_{b} t_{b} \mid x_{a} v_{a} t_{a}\right)= & \left|\left(x_{b} v_{b} t_{b} \mid x_{a} v_{a} t_{a}\right)\right|^{2}=\int \mathcal{D} x \int \frac{\mathcal{D} p}{2 \pi} \int \mathcal{D} v \int \frac{\mathcal{D} p_{v}}{2 \pi} \\ & \times \exp \left\{\int_{t_{a}}^{t_{b}} d t\left[i\left(p \dot{x}+p_{v} \dot{v}\right)-H\left(p, p_{v}, x, v\right)\right]\right\} \end{align*} $$

$$ H\left(\hat{p}, \hat{p}_{v}, x, v\right) P\left(x v t_{b} \mid x_{a} v_{a} t_{a}\right)=-\partial_{t} P\left(x v t \mid x_{a} v_{a} t_{a}\right) . $$

$$ \partial_{t} P\left(\mathbf{x} t \mid \mathbf{x}_{a} v_{a} t_{a}\right)=\left[-\partial_{i} D_{i}(\mathbf{x})+\partial_{i} \partial_{j} D_{i j}^{(2)}(\mathbf{x})\right] P\left(\mathbf{x} t \mid \mathbf{x}_{a} v_{a} t_{a}\right) $$

$$ \partial_{t} P\left(\mathbf{x} t \mid t_{a} \mathbf{x}_{a}\right)=\left(-\kappa_{i j} \partial_{i} x_{j}+D_{i j} \partial_{i} \partial_{j}\right) P\left(\mathbf{x} t \mid t_{a} \mathbf{x}_{a}\right) $$

$$ \mathbf{D}=\left(\begin{array}{cc} 0 & 0 \\ 0 & w / 2 M^{2} \end{array}\right)=\left(\begin{array}{cc} 0 & 0 \\ 0 & \gamma k_{B} T / M \end{array}\right)=\left(\begin{array}{cc} 0 & 0 \\ 0 & \gamma^{2} D \end{array}\right), $$

$$ \boldsymbol{\kappa}=\left(\begin{array}{cc} 0 & -1 \\ V^{\prime}(x) / M & \gamma \end{array}\right) . $$

$$ \int d x d v P\left(x v t_{b} \mid x_{a} v_{a} t_{a}\right)=1 $$

$$ \tilde{H}_{0}\left(p, p_{v}, x, v\right)=\frac{w}{2 M^{2}} p_{v}^{2}-i \gamma p_{v} v+i p v, $$

$$ P_{0}\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\mathcal{N} \int \mathcal{D} x(t) \exp \left\{-\frac{1}{2 w} \int_{t_{a}}^{t_{b}} d t\left[M \ddot{x}+M \gamma \dot{x}^{R}\right]^{2}\right\} $$

$$ \begin{align*} P_{0}\left(0 t_{b} \mid 0 t_{a}\right) & \propto \operatorname{Det}^{-1}\left(-\partial_{t}^{2}-\gamma \partial_{t}\right) \\ & \propto \exp \left[-\left(t_{b}-t_{a}\right) \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \log \left(\omega^{\prime 2}-i \gamma \omega^{\prime}\right)\right] \end{align*} $$

$$ \frac{1}{2} \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \log \left[\tilde{\omega}^{\prime 4}+\gamma^{2} \tilde{\omega}^{\prime 2}\right]=\frac{1}{2} \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \log \tilde{\omega}^{\prime 2}+\frac{1}{2} \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \log \left[\tilde{\omega}^{\prime 2}+\gamma^{2}\right]=0+\frac{\gamma}{2} $$

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

$$ \begin{align*} \operatorname{Det}\left(-\partial_{t}^{2}-\gamma \partial_{t}\right) & =\operatorname{Det}\left(i \partial_{t}\right) \operatorname{Det}\left(i \partial_{t}+i \gamma\right)=\exp \left[\operatorname{Tr} \log \left(i \partial_{t}\right)+\operatorname{Tr} \log \left(i \partial_{t}+i \gamma\right)\right] \\ & =\exp \left[\left(t_{b}-t_{a}\right) \frac{\gamma}{2}\right] \end{align*} $$

$$ P_{0}\left(0 t_{b} \mid 0 t_{a}\right) \propto \exp \left[-\left(t_{b}-t_{a}\right) \frac{\gamma}{2}\right] . $$

$$ \gamma y \dot{x}^{R}(t) \rightarrow \int d t^{\prime} y(t) \gamma_{D}^{R}\left(t-t^{\prime}\right) x\left(t^{\prime}\right) $$

$$ \log \left(\omega^{\prime 2}+i \omega^{\prime} \omega_{D}-\gamma \omega_{D}\right)=\log \left(\omega^{\prime}+i \omega_{1}\right)+\log \left(\omega^{\prime}+i \omega_{2}\right) $$

$$ \omega_{1,2}=\frac{\omega_{D}}{2}\left(1 \pm \sqrt{1-\frac{4 \gamma}{\omega_{D}}}\right), $$

$$ \int_{-\infty}^{\infty} \frac{d \omega}{2 \pi}\left[-\log \left(\omega^{\prime}+i \omega_{D}\right)+\log \left(\omega^{\prime 2}+i \omega^{\prime} \omega_{D}-\gamma \omega_{D}\right)\right]=-\frac{\omega_{D}}{2}+\frac{\omega_{1}}{2}+\frac{\omega_{2}}{2}=0 $$

$$ \operatorname{Det}\left(-\partial_{t}^{2}-\gamma \partial_{t}^{R}\right)=\exp \left[\operatorname{Tr} \log \left(-\partial_{t}^{2}-\gamma \partial_{t}^{R}\right)\right]=1 $$

$$ P_{0}\left(0 t_{b} \mid 0 t_{a}\right)=\text { const. } $$

$$ \operatorname{Det}\left(-\partial_{t}^{2}-\gamma \partial_{t}^{A}\right)=\exp \left[\operatorname{Tr} \log \left(-\partial_{t}^{2}-\gamma \partial_{t}^{A}\right)\right]=\exp \left[\left(t_{b}-t_{a}\right) \gamma\right] $$

$$ P_{0}\left(0 t_{b} \mid 0 t_{a}\right) \propto \exp \left[-\left(t_{b}-t_{a}\right) \gamma\right] $$

$$ P_{0}\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\mathcal{N} \int \mathcal{D} x(t) \exp \left\{-\frac{1}{2 w} \int_{t_{a}}^{t_{b}} d t\left[M \ddot{x}+M \gamma \dot{x}^{R}+\omega_{0}^{2} x\right]^{2}\right\} $$

$$ \begin{align*} P_{0}\left(0 t_{b} \mid 0 t_{a}\right) & \propto \operatorname{Det}^{-1}\left(-\partial_{t}^{2}-\gamma \partial_{t}+\omega_{0}^{2}\right) \\ & \propto \exp \left[-\left(t_{b}-t_{a}\right) \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \log \left(\omega^{\prime 2}-i \gamma \omega^{\prime}-\omega_{0}^{2}\right)\right] \end{align*} $$

$$ \omega_{1,2}=\frac{\gamma}{2}\left(1 \pm \sqrt{1-\frac{4 \omega_{0}^{2}}{\gamma^{2}}}\right) . $$

$$ \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi}\left[\log \left(\omega^{\prime}+i \omega_{1}\right)+\log \left(\omega^{\prime}+i \omega_{2}\right)\right]=\frac{\omega_{1}}{2}+\frac{\omega_{2}}{2}=\gamma $$

$$ \operatorname{Det}\left(-\partial_{t}^{2}-\gamma \partial_{t}-\omega_{0}^{2}\right)=\exp \left[\operatorname{Tr} \log \left(-\partial_{t}^{2}-\gamma \partial_{t}-\omega_{0}^{2}\right)\right]=\exp \left[\left(t_{b}-t_{a}\right) \frac{\gamma}{2}\right] $$

$$ \frac{\partial}{\partial \omega_{0}^{2}} \operatorname{Tr} \log \left(-\partial_{t}^{2}-\gamma \partial_{t}-\omega_{0}^{2}\right)=-\int d t\left(-\partial_{t}^{2}-\gamma \partial_{t}-\omega_{0}^{2}\right)^{-1}(t, t) $$

$$ -\int \frac{d \omega^{\prime}}{2 \pi} \frac{1}{\left(\omega^{\prime}+i \omega_{1}\right)\left(\omega^{\prime}+i \omega_{1}\right)} $$

$$ \frac{\partial}{\partial \gamma} \log \operatorname{Det} \partial_{t}\left(-\partial_{t}^{2}-\gamma \partial_{t}-\omega_{0}^{2}\right)=-\int d t\left[\partial_{t}\left(-\partial_{t}^{2}-\gamma \partial_{t}-\omega_{0}^{2}\right)^{-1}\right](t, t) $$

$$ i \int \frac{d \omega^{\prime}}{2 \pi} \frac{\omega^{\prime}}{\left(\omega^{\prime}+i \omega_{1}\right)\left(\omega^{\prime}+i \omega_{1}\right)} $$

$$ \operatorname{Det}\left[-\partial_{t}^{2}-\gamma(t) \partial_{t}-\Omega^{2}(t)\right]=\operatorname{Det}\left[\partial_{t}+\Omega_{1}(t)\right] \operatorname{Det}\left[\partial_{t}+\Omega_{2}(t)\right] $$

$$ \Omega_{1}(t)+\Omega_{2}(t)=\gamma(t), \quad \partial_{t} \Omega_{2}(t)+\Omega_{1}(t) \Omega_{2}(t)=\Omega^{2}(t) $$

$$ P_{0}\left(0 t_{b} \mid 0 t_{a}\right) \propto \exp \left[-\left(t_{b}-t_{a}\right) \frac{\gamma}{2}\right] $$

$$ P_{0}\left(0 t_{b} \mid 0 t_{a}\right) \propto \exp \left\{-\left(t_{b}-t_{a}\right) \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi} \log \left[\omega^{\prime 2}-i \gamma_{D}^{R}\left(\omega^{\prime}\right) \omega^{\prime}-\omega_{0}^{2}\right]\right\} $$

$$ \omega_{1,2}=\frac{\gamma}{2}\left(1 \pm \sqrt{1-\frac{4 \omega_{0}^{2}}{\gamma^{2}}}\right), \quad \omega_{3}=\omega_{D}-\gamma $$

$$ \int_{-\infty}^{\infty} \frac{d \omega^{\prime}}{2 \pi}\left[-\log \left(\omega^{\prime}+i \omega_{D}\right)+\sum_{k=1}^{3} \log \left(\omega^{\prime}+i \omega_{k}\right)\right]=-\omega_{D}+\sum_{k=1}^{3} \frac{\omega_{k}}{2}=0 . $$

$$ P_{0}\left(0 t_{b} \mid 0 t_{a}\right)=\text { const. } $$

$$ \operatorname{Det}\left(\partial_{t}^{2}+i \gamma \partial_{t}^{R}+\omega_{0}^{2}\right)=1 $$

$$ \frac{\partial}{\partial \gamma} \operatorname{Det}\left(-\partial_{t}^{2}-\gamma \partial_{t}^{R}-\omega_{0}^{2}\right)=-\int d t\left[\partial_{t}^{R}\left(\partial_{t}^{2}-\gamma \partial_{t}-\omega_{0}^{2}\right)^{-1}(t, t)\right. $$

$$ \operatorname{Det}\left(\partial_{t}^{2}+i \gamma \partial_{t}^{A}+\omega_{0}^{2}\right)=\gamma $$

$$ \operatorname{Det}\left[-\partial_{t}^{2}-\gamma(t) \partial_{t}^{R}-\Omega^{2}(t)\right]=1 . $$

$$ \operatorname{Det}\left[-\partial_{t}^{2}-\gamma(t) \partial_{t}^{A}-\Omega^{2}(t)\right]=\exp \left[\int d t \gamma(t)\right] $$

$$ L_{\mathrm{e}}(x, \dot{x})=\frac{1}{2 w}\left[\ddot{x}+M \gamma \dot{x}+V^{\prime}(x)\right]^{2}-\frac{\gamma}{2} . $$

$$ H_{0}\left(p, p_{v}, x, v\right)=\frac{w}{2 M^{2}} p_{v}^{2}-i \gamma p_{v} v+i p v-\frac{\gamma}{2} . $$

$$ L_{\mathrm{e}}(x, \dot{x})=\frac{1}{2 w}\left[M \gamma \dot{x}^{R}+V^{\prime}(x)\right]^{2}=\frac{1}{4 D}\left[\dot{x}^{R}+\frac{1}{M \gamma} V^{\prime}(x)\right]^{2}, $$

$$ P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\mathcal{N} \int \mathcal{D} x \exp \left[-\int_{t_{a}}^{t_{b}} d t L_{\mathrm{e}}\left(x, \dot{x}^{R}\right)\right] $$

$$ P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int \mathcal{D} x \int \frac{\mathcal{D} p}{2 \pi} \exp \left\{\int_{t_{a}}^{t_{b}} d t\left[i p \dot{x}-2 D \frac{p^{2}}{2}+i p \frac{1}{M \gamma} V^{\prime}(x)\right]\right\} $$

$$ H\left(\hat{p}_{b}, x_{b}\right) P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=-\partial_{t_{b}} P\left(x_{b} t_{b} \mid x_{a} t_{a}\right) $$

$$ H(\hat{p}, x) \equiv 2 D \frac{\hat{p}^{2}}{2}-i \hat{p} \frac{1}{M \gamma} V^{\prime}(x)=-D \partial_{x}\left[\partial_{x}+\frac{1}{D M \gamma} V^{\prime}(x)\right] $$

$$ P_{0}\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\mathcal{N} \int \mathcal{D} x \exp \left\{-\frac{1}{2 w} \int_{t_{a}}^{t_{b}} d t\left[M \gamma \dot{x}+V^{\prime}(x)\right]^{2}\right\} $$

$$ P_{0}\left(0 t_{b} \mid 0 t_{a}\right)=\operatorname{Det}^{-1}\left[\partial_{t}+V^{\prime \prime}(x) / M \gamma\right] $$

$$ \operatorname{Det}\left[\partial_{t}+V^{\prime \prime}(x) / M \gamma\right]=\exp \left[\int d t V^{\prime \prime}(x) / 2 M \gamma\right] $$

$$ \operatorname{Det}\left[\partial_{t}^{R}+V^{\prime \prime}(x) / M \gamma\right]=1 $$

$$ \operatorname{Det}\left[\partial_{t}^{A}+V^{\prime \prime}(x) / M \gamma\right]=\exp \left[\int d t V^{\prime \prime}(x) / M \gamma\right] $$

$$ L_{\mathrm{e}}(x, \dot{x})=\frac{1}{4 D}\left[\dot{x}+\frac{1}{M \gamma} V^{\prime}(x)\right]^{2}-\frac{1}{2 M \gamma} V^{\prime \prime}(x) $$

$$ P_{0}\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\mathcal{N} \int \mathcal{D} x(t) \exp \left[-\int_{t_{a}}^{t_{b}} d t\left\{\frac{1}{4 D}\left[\dot{x}+\frac{V^{\prime}(x)}{M \gamma}\right]^{2}-\frac{V^{\prime \prime}(x)}{2 M \gamma}\right\}\right] $$

$$ L_{\mathrm{e}}(x, \dot{x})=\frac{1}{4 D}(\dot{x}+\kappa x)^{2}-\frac{\kappa}{2}, $$

$$ -\ddot{x}+\kappa^{2} x=0, $$

$$ x(t)=\frac{1}{e^{2 \kappa t_{a}}-e^{2 \kappa t_{b}}}\left[e^{\kappa\left(t+t_{a}\right)} x_{a}-e^{\kappa\left(-t+t_{a}+2 \kappa t_{b}\right)} x_{a}-e^{\kappa\left(t+t_{b}\right)} x_{b}+e^{\kappa\left(-t+2 t_{a}+t_{b}\right)} x_{b}\right] . $$

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

$$ F_{\kappa}\left(t_{b}-t_{a}\right)=\frac{1}{\sqrt{2 \pi \sinh \kappa\left(t_{b}-t_{a}\right)}} $$

$$ P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=F_{\kappa}\left(t_{b}-t_{a}\right) e^{-\mathcal{A}_{\mathrm{e}}}=\frac{1}{\sqrt{2 \pi \sigma^{2}\left(t_{b}-t_{a}\right)}} \exp \left\{-\frac{\left[x_{b}-\bar{x}\left(t_{b}-t_{a}\right)\right]^{2}}{2 \sigma^{2}\left(t_{b}-t_{a}\right)}\right\} $$

$$ \bar{x}(t) \equiv\langle x(t)\rangle=x_{a} e^{-\kappa t}, \quad \sigma^{2}(t) \equiv\left\langle[x(t)-\bar{x}(t)]^{2}\right\rangle=\frac{D}{\kappa}\left(1-e^{-2 \kappa t}\right), $$

$$ \begin{align*} \bar{x}\left(t_{b}-t_{a}\right) \equiv\left\langle x\left(t_{b}-t_{a}\right)\right\rangle & \equiv \int_{-\infty}^{\infty} x_{b} P\left(x_{b} t_{b} \mid x_{a} t_{a}\right) \\ \left\langle\left[x\left(t_{b}-t_{a}\right)-\bar{x}\left(t_{b}-t_{a}\right)\right]^{2}\right\rangle & \equiv \int_{-\infty}^{\infty}\left[x_{b}-\bar{x}\left(t_{b}-t_{a}\right)\right]^{2} P\left(x_{b} t_{b} \mid x_{a} t_{a}\right) \end{align*} $$

$$ \hat{H}(p, x)=D \hat{p}^{2}+i \kappa \hat{p} x, \quad p \equiv-i \partial_{x} $$

$$ \left(-D \partial_{x_{b}}^{2}+\kappa \partial_{x_{b}} x_{b}\right) P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=-\partial_{t_{b}} P\left(x_{b} t_{b} \mid x_{a} t_{a}\right) $$

$$ \lim _{t_{b} \rightarrow \infty} P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\sqrt{\frac{\kappa}{2 \pi D}} \exp \left\{-\kappa \frac{x_{b}^{2}}{2 D}\right\} $$

$$ \lim _{t_{b} \rightarrow \infty} P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\sqrt{\frac{V^{\prime \prime}(0)}{2 \pi k_{B} T}} \exp \left\{-\frac{1}{k_{B} T} V\left(x_{b}\right)\right\} $$

$$ \left\langle x^{n}\right\rangle=\lim _{t_{b} \rightarrow \infty}\left\langle x^{n}\left(t_{b}\right)\right\rangle=\lim _{t_{b} \rightarrow \infty} \frac{\int \mathcal{D} x x^{n}\left(t_{b}\right) e^{-\int_{t_{a}}^{t_{b}} d t L_{\mathrm{e}}\left(x, \dot{x}^{R}\right)}}{\int \mathcal{D} x e^{-\int_{t_{a}}^{t_{b}} d t L_{\mathrm{e}}\left(x, \dot{x}^{R}\right)}} $$

$$ \lim _{t_{b} \rightarrow \infty}\left\langle x^{n}\left(t_{b}\right)\right\rangle=\left\langle x^{n}\right\rangle=\frac{\int d x x^{n} e^{-V(x) / k_{B} T}}{\int d x e^{-V(x) / k_{B} T}} $$

$$ V\left(x+\frac{y}{2}\right)-V\left(x-\frac{y}{2}\right) \sim y V^{\prime}(x)+\frac{y^{3}}{24} V^{\prime \prime \prime}(x)+\ldots, $$

$$ \eta(t) \equiv M \ddot{x}(t)+M \gamma \dot{x}^{R}(t)+V^{\prime}(x(t)) . $$

$$ \exp \left\{-\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t y \eta-\frac{w}{2 \hbar^{2}} \int_{t_{a}}^{t_{b}} d t \int_{t_{a}}^{t_{b}} d t^{\prime} y(t) K\left(t, t^{\prime}\right) y\left(t^{\prime}\right)\right\} $$

$$ P[\eta] \propto \exp \left\{-\frac{1}{2 w} \int_{t_{a}}^{t_{b}} d t \int_{t_{a}}^{t_{b}} d t^{\prime} \eta(t) K^{-1}\left(t, t^{\prime}\right) \eta\left(t^{\prime}\right)\right\} $$

$$ \langle\eta(t)\rangle_{\eta}=0, \quad\left\langle\eta(t) \eta\left(t^{\prime}\right)\right\rangle_{\eta}=w K\left(t-t^{\prime}\right) $$

$$ \langle F[x]\rangle_{\eta} \equiv \mathcal{N} \int_{x\left(t_{a}\right)=x_{a}} \mathcal{D} x P[\eta] F[x] $$

$$ \begin{align*} M \dot{v}(t)+M \gamma v^{R}(t)+V^{\prime}(x(t)) & =\eta(t), \\ \dot{x}(t) & =v(t), \end{align*} $$

$$ P_{\eta}\left(x_{b} v_{b} t_{b} \mid x_{a} v_{a} t_{a}\right)=\delta\left(x_{\eta}(t)-x_{b}\right) \delta\left(\dot{x}_{\eta}(t)-v_{b}\right) . $$

$$ P\left(x_{b} v_{b} t_{b} \mid x_{a} v_{a} t_{a}\right)=\left\langle P_{\eta}\left(x_{b} v_{b} t_{b} \mid x_{a} v_{a} t_{a}\right)\right\rangle_{\eta} . $$

$$ J[x] \equiv \operatorname{Det}\left[\delta \eta(t) / \delta x\left(t^{\prime}\right)\right]=\operatorname{det}\left[M \partial_{t}^{2}+M \gamma \partial_{t}^{R}+V^{\prime \prime}(x(t))\right] . $$

$$ \left.\langle F[x]\rangle_{\eta} \equiv \int \mathcal{D} \eta P[\eta] F[x]\right|_{x\left(t_{a}\right)=x_{a}} $$

$$ \lim _{T \rightarrow \infty} K\left(\omega^{\prime}\right) \equiv 1 $$

$$ \langle\eta(t)\rangle_{\eta}=0, \quad\left\langle\eta(t) \eta\left(t^{\prime}\right)\right\rangle_{\eta}=w \delta\left(t-t^{\prime}\right) $$

$$ K\left(\omega^{\prime}\right) \underset{T \rightarrow 0}{\longrightarrow} \frac{\hbar\left|\omega^{\prime}\right|}{2 k_{B} T}, $$

$$ \lim _{T \rightarrow 0} w K\left(\omega^{\prime}\right)=M \gamma \hbar\left|\omega^{\prime}\right| $$

$$ \Theta\left(\omega^{\prime}\right)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} d t e^{-i \omega^{\prime} t} \frac{i}{t+i \eta} $$

$$ \begin{align*} \Theta\left(\omega^{\prime}\right)-\Theta\left(-\omega^{\prime}\right) & =\frac{1}{2 \pi} \int_{-\infty}^{\infty} d t e^{-i \omega^{\prime} t}\left(\frac{i}{t+i \eta}+\frac{i}{t-i \eta}\right) \\ & \equiv \frac{i}{\pi} \int_{-\infty}^{\infty} d t e^{-i \omega^{\prime} t} \frac{\mathcal{P}}{t} \end{align*} $$

$$ \begin{align*} \left|\omega^{\prime}\right|=\omega^{\prime}\left[\Theta\left(\omega^{\prime}\right)-\Theta\left(-\omega^{\prime}\right)\right] & =-\frac{1}{\pi} \int_{-\infty}^{\infty} d t \partial_{t} e^{-i \omega^{\prime} t} \frac{\mathcal{P}}{t}=\frac{1}{\pi} \int_{-\infty}^{\infty} d t e^{-i \omega^{\prime} t} \partial_{t} \frac{\mathcal{P}}{t} \\ & =-\frac{1}{\pi} \int_{-\infty}^{\infty} d t e^{-i \omega^{\prime} t} \frac{\mathcal{P}}{t^{2}} \end{align*} $$

$$ w K\left(t-t^{\prime}\right) \underset{T=0}{=}-\frac{M \gamma \hbar}{\pi} \frac{\mathcal{P}}{\left(t-t^{\prime}\right)^{2}} $$

$$ \left\langle\eta(t) \eta\left(t^{\prime}\right)\right\rangle_{\eta}=-\frac{M \gamma \hbar}{\pi} \frac{\mathcal{P}}{\left(t-t^{\prime}\right)^{2}} $$

$$ \left\langle(\Delta t)^{2}\right\rangle_{t} \equiv \int_{-\infty}^{\infty} d \Delta t(\Delta t)^{2} K(\Delta t)=-\left.\frac{\partial^{2}}{\partial \omega^{\prime 2}} K\left(\omega^{\prime}\right)\right|_{\omega^{\prime}=0}=-\frac{1}{6}\left(\frac{\hbar}{k_{B} T}\right)^{2} $$

$$ \dot{x}(t)=-V^{\prime}(x(t)) / M \gamma+\eta(t) / M \gamma $$

$$ P\left(x_{b} t_{a} \mid x_{a} t_{a}\right)=\int \mathcal{D} \eta P[\eta] \delta\left(x_{\eta}(t)-x_{b}\right) $$

$$ \int_{x\left(t_{a}\right)=x_{a}}^{x\left(t_{b}\right)=x_{b}} \mathcal{D} x \delta[\dot{x}-\eta]=\delta\left(x_{\eta}\left(t_{b}\right)-x_{b}\right) $$

$$ \delta[\dot{x}-\eta]=\int \mathcal{D} p e^{i \int d t p(\dot{x}-\eta)} $$

$$ P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int_{x\left(t_{a}\right)=x_{a}}^{x\left(t_{b}\right)=x_{b}} \mathcal{D} x \int \mathcal{D} \eta P[\eta] \delta[\dot{x}-\eta] $$

$$ \dot{x}(t)=-\omega_{0}^{2} x(t) / M \gamma+\eta(t) / M \gamma=-\kappa x(t) / M \gamma+\bar{\eta}(t), $$

$$ \langle\bar{\eta}(t)\rangle=0, \quad\left\langle\bar{\eta}(t) \bar{\eta}\left(t^{\prime}\right)\right\rangle_{\eta}=\frac{w}{M^{2} \gamma^{2}} \delta\left(t-t^{\prime}\right)=2 D \delta\left(t-t^{\prime}\right) $$

$$ \mathcal{A}_{\mathrm{e}}=\int_{t_{a}}^{t_{b}} d t\left[-i\left(p \dot{x}+p_{v} \dot{v}\right)+H\left(p, p_{v}, v, x\right)\right] $$

$$ H\left(p, p_{v}, v, x\right)=\gamma v_{T}^{2}\left(p_{v}-i \frac{v}{2 v_{T}^{2}}\right)^{2}+\frac{\gamma}{4 v_{T}^{2}}\left(v+2 i \frac{v_{T}^{2}}{\gamma} p\right)^{2}+\frac{v_{T}^{2}}{\gamma} p^{2}+\frac{\gamma}{2} . $$

$$ P\left(x_{b} v_{b} t_{b} \mid x_{a} v_{a} t_{a}\right)=\int \frac{d p}{2 \pi} P\left(v_{b} t_{b} \mid v_{a} p_{a}\right)_{p} e^{i p\left(x_{b}-x_{a}\right)-v_{T}^{2} p^{2}\left(t_{b}-t_{a}\right) / \gamma} $$

$$ P_{p}\left(v_{b} t_{b} \mid v_{a} t_{a}\right)=\int \mathcal{D} v \int \frac{\mathcal{D} p_{v}}{2 \pi} \exp \left\{\int_{t_{a}}^{t_{b}} d t\left[i p_{v} \dot{v}-H_{p}\left(p_{v}, v\right)\right]\right\} $$

$$ H_{p}\left(p_{v}, v\right) \equiv \gamma v_{T}^{2}\left(p_{v}-i \frac{v}{2 v_{T}^{2}}\right)^{2}+\frac{\gamma}{4 v_{T}^{2}}\left(v+2 i \frac{v_{T}^{2}}{\gamma} p\right)^{2}-\frac{\gamma}{2} $$

$$ \tilde{H}_{p}\left(p_{v}, v\right) \equiv e^{v^{2} / 4 v_{T}^{2}} \hat{H}_{p} e^{-v^{2} / 4 v_{T}^{2}}=\gamma v_{T}^{2} p_{v}^{2}+\frac{\gamma}{4 v_{T}^{2}}\left(v+2 i \frac{v_{T}^{2}}{\gamma} p\right)^{2}-\frac{\gamma}{2} . $$

$$ P_{p}\left(v_{b} t_{b} \mid v_{a} t_{a}\right)=e^{-v_{b}^{2} / 4 v_{T}^{2}} \tilde{P}_{p}\left(v_{b} t_{b} \mid v_{a} t_{a}\right) e^{v_{a}^{2} / 4 v_{T}^{2}} $$

$$ \begin{align*} P\left(x_{b} v_{b} t_{b} \mid x_{a} v_{a} t_{a}\right) & =e^{-\left(v_{b}^{2}-v_{a}^{2}\right) / 4 v_{T}^{2}} \int \frac{d p}{2 \pi} \sum_{n=0}^{\infty} \psi_{n}\left(v_{b}-v_{p}\right) \psi_{n}\left(v_{a}-v_{p}\right) e^{-n \gamma\left(t_{b}-t_{a}\right)} \\ & \times e^{i p\left(x_{b}-x_{a}\right)-v_{T}^{2} p^{2}\left(t_{b}-t_{a}\right) / \gamma} \end{align*} $$

$$ \psi_{n}(v)=\frac{1}{\left(2^{n} n!\sqrt{\pi}\right)^{1 / 2}\left(\sqrt{2} v_{T}\right)^{1 / 2}} e^{-v^{2} / 4 v_{T}^{2}} H_{n}\left(v / \sqrt{2} v_{T}\right) $$

$$ P\left(x_{b} t_{b} \mid x_{a} v_{a} t_{a}\right)=\int \frac{d p}{2 \pi} e^{i p\left(x_{b}-x_{a}-v_{a} / \gamma\right)-v_{T}^{2} p^{2}\left(t_{b}-t_{a}\right) / \gamma}=\frac{1}{\sqrt{4 \pi v_{T}^{2} / \gamma}} e^{-\frac{\gamma\left(x_{b}-x_{a}-v_{a} / \gamma\right)^{2}}{4 v_{T}^{2}\left(t_{b}-t_{a}\right)}} $$

$$ \left\langle x^{n}\right\rangle=\lim _{s \rightarrow \infty}\left\langle x^{n}(s)\right\rangle=\lim _{s \rightarrow \infty} \int \mathcal{D} \eta x_{\eta}^{n}(s) P[\eta] $$

$$ P[\eta] \equiv \int \mathcal{D} \eta e^{-\left(1 / 4 k_{B} T\right) \int_{s_{a}}^{s} d s^{\prime} \eta^{2}\left(s^{\prime}\right)} $$

$$ x^{\prime}(s)=-V^{\prime}(x)+\eta(s) $$

$$ \langle\eta(s)\rangle_{T}=0, \quad\left\langle\eta(s) \eta\left(s^{\prime}\right)\right\rangle_{T}=2 k_{B} T \delta\left(s-s^{\prime}\right), $$

$$ \left\langle x\left(\tau_{1}\right) x\left(\tau_{2}\right) \cdots x\left(\tau_{n}\right)\right\rangle \equiv Z^{-1} \int \mathcal{D} x x\left(\tau_{1}\right) x\left(\tau_{2}\right) \cdots x\left(\tau_{n}\right) \exp \left(-\frac{1}{\hbar} \mathcal{A}_{\mathrm{e}}\right) $$

$$ \partial_{s} x(\tau ; s)=-\frac{\delta \mathcal{A}_{\mathrm{e}}}{\delta x(\tau ; s)}+\eta(\tau ; s) $$

$$ \langle\eta(\tau ; s)\rangle=0, \quad\left\langle\eta(\tau ; s) \eta\left(\tau^{\prime} ; s^{\prime}\right)\right\rangle=2 \hbar \delta\left(\tau-\tau^{\prime}\right) \delta\left(s-s^{\prime}\right) $$

$$ \left\langle x\left(\tau_{1}\right) x\left(\tau_{2}\right) \cdots x\left(\tau_{n}\right)\right\rangle=\lim _{s \rightarrow \infty}\left\langle x\left(\tau_{1}, s\right) x\left(\tau_{2}, s\right) \cdots x\left(\tau_{n}, s\right)\right\rangle $$

$$ \begin{align*} P\left[x_{b}\left(\tau_{b}\right), s ; x_{a}(\tau), s_{a}\right] & =\mathcal{N} \int \mathcal{D} x(\tau ; s) \\ & \times e^{-\int_{s_{a}}^{s_{b}} d s\left\{\frac{1}{4 \hbar} \int_{-\infty}^{\infty} d \tau\left[\partial_{s} x(\tau ; s)+\frac{\delta}{\delta x(\tau ; s)} \mathcal{A}_{\mathrm{e}}\right]-\frac{1}{2 \hbar} \frac{\delta^{2}}{\delta x(\tau ; s)^{2}} \mathcal{A}_{\mathrm{e}}\right\}} \end{align*} $$

$$ H[\hat{p}(\tau), x(\tau)] P\left(x(\tau) s \mid x_{a}(\tau) ; s_{a}\right)=-\partial_{s} P\left(x(\tau) s \mid x_{a}(\tau) ; s_{a}\right) $$

$$ H[\hat{p}(\tau), x(\tau)]=\int_{-\infty}^{\infty} d \tau\left[\hbar \hat{p}^{2}(\tau)-i \hat{p}(\tau) \frac{\delta}{\delta x(\tau)} \mathcal{A}_{\mathrm{e}}\right] $$

$$ -\int_{-\infty}^{\infty} d \tau \frac{\hbar \delta}{\delta x(\tau)}\left[\frac{\hbar \delta}{\delta x(\tau)}+\frac{\delta \mathcal{A}_{\mathrm{e}}}{\delta x(\tau)}\right] P\left[x(\tau), s ; x_{a}(\tau), s_{a}\right]=-\hbar \partial_{s} P\left[x(\tau), s ; x_{a}(\tau), s_{a}\right] $$

$$ \lim _{s \rightarrow \infty} P\left[x(\tau), s ; x_{a}(\tau), s_{a}\right]=\frac{e^{-\mathcal{A}_{\mathrm{e}}[x] / \hbar}}{\int \mathcal{D} x(\tau) e^{-\mathcal{A}_{\mathrm{e}}[x] / \hbar}} $$

$$ \partial_{s} x(\tau ; s)=-M\left(-\partial_{\tau}^{2}+\omega^{2}\right) x(\tau ; s)+\eta(\tau ; s) $$

$$ x(\tau ; s)=\int_{0}^{s} d s^{\prime} e^{-M\left(-\partial_{\tau}^{2}+\omega^{2}\right)\left(s^{\prime}-s\right)} \eta\left(\tau ; s^{\prime}\right) $$

$$ \left\langle x\left(\tau_{1} ; s_{1}\right) x\left(\tau_{2} ; s_{2}\right)\right\rangle=\int_{0}^{s_{1}} d s_{1}^{\prime} \int_{0}^{s_{2}} d s_{2}^{\prime} e^{M\left(-\partial_{\tau}^{2}+\omega^{2}\right)\left(s_{1}^{\prime}+s_{2}^{\prime}-s_{1}-s_{2}\right)}\left\langle\eta\left(\tau_{1} ; s_{1}^{\prime}\right) \eta\left(\tau_{2} ; s_{2}^{\prime}\right)\right\rangle . $$

$$ \left\langle x\left(\tau_{1} ; s_{1}\right) x\left(\tau_{2} ; s_{2}\right)\right\rangle=\hbar \int_{0}^{\infty} d s\left[e^{-M\left(-\partial_{\tau}^{2}+\omega^{2}\right)\left(s+\left|s_{1}-s_{2}\right|\right)}-e^{-M\left(-\partial_{\tau}^{2}+\omega^{2}\right)\left(s+s_{1}+s_{2}\right)}\right] $$

$$ \left\langle x\left(\tau_{1} ; s_{1}\right) x\left(\tau_{2} ; s_{2}\right)\right\rangle=\frac{\hbar}{M} \frac{1}{-\partial_{\tau}^{2}+\omega^{2}}\left[e^{-M\left(-\partial_{\tau}^{2}+\omega^{2}\right)\left|s_{1}-s_{2}\right|}-e^{-M\left(-\partial_{\tau}^{2}+\omega^{2}\right)\left(s_{1}+s_{2}\right)}\right] $$

$$ \begin{align*} \left\langle x\left(\tau_{1} ; s_{1}\right) x\left(\tau_{2} ; s_{2}\right)\right\rangle & =\frac{\hbar}{M} \frac{2}{t_{b}-t_{a}} \sum_{n=1}^{\infty} \frac{1}{\nu_{n}^{2}+\omega^{2}} \sin \nu_{n}\left(\tau_{1}-\tau_{a}\right) \sin \nu_{n}\left(\tau_{2}-\tau_{a}\right) \\ & \times\left[e^{-M\left(\nu_{n}^{2}+\omega^{2}\right)\left|s_{1}-s_{2}\right|}-e^{-M\left(-\nu_{n}^{2}+\omega^{2}\right)\left(s_{1}+s_{2}\right)}\right] \end{align*} $$

$$ \begin{align*} \lim _{s_{1}=s_{2} \rightarrow \infty}\left\langle x\left(\tau_{1} ; s\right) x\left(\tau_{2} ; s\right)\right\rangle & =\left\langle x\left(\tau_{1}\right) x\left(\tau_{2}\right)\right\rangle=\frac{\hbar}{M} \frac{1}{-\partial_{\tau}^{2}+\omega^{2}}\left(\tau_{1}, \tau_{2}\right) \\ & =\frac{\hbar}{M} \frac{\sinh \omega\left(\tau_{b}-\tau_{>}\right) \sinh \omega\left(\tau_{<}-\tau_{a}\right)}{\omega \sinh \omega\left(\tau_{b}-\tau_{a}\right)} \end{align*} $$

$$ \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)=C\left(\mathbf{x}_{b}, \mathbf{x}_{a}\right) e^{-A_{\mathrm{e}}\left(\mathbf{x}_{b}, \mathbf{x}_{a} ; \tau_{b}-\tau_{a}\right) / \hbar} e^{-\int_{\tau_{a}}^{\tau_{b}} \frac{M}{2} d \tau_{b}^{\prime}\left\langle L_{\mathrm{e}, \mathrm{fi}}\left(\mathbf{x}_{b}, \dot{\mathbf{x}}_{b}\right)\right\rangle / \hbar} $$

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

$$ \frac{M}{2}\left\langle\delta \dot{\mathbf{x}}_{b}^{2}\right\rangle=\frac{\hbar \omega}{2} D \operatorname{coth} \omega\left(\tau_{b}-\tau_{a}\right) $$

$$ \left\langle x\left(t_{1}\right) x\left(t_{2}\right) \cdots x\left(\tau_{n}\right)\right\rangle=\lim _{s \rightarrow \infty}\left\langle x\left(t_{1}, s\right) x\left(t_{2}, s\right) \cdots x\left(t_{n}, s\right)\right\rangle $$

$$ \hbar \partial_{s} x(t ; s)=i \frac{\delta \mathcal{A}}{\partial x(t ; s)}+\eta(t ; s) $$

$$ \partial_{t} P_{\eta}\left(x v t \mid x_{a} v_{a} t_{a}\right)=\dot{x}_{\eta}(t) \delta^{\prime}\left(x_{\eta}(t)-x\right) \delta\left(\dot{x}_{\eta}(t)-v\right)+\ddot{x}_{\eta}(t) \delta\left(x_{\eta}(t)-x\right) \delta^{\prime}\left(\dot{x}_{\eta}(t)-v\right) . $$

$$ \frac{d}{d t} \delta(z(t)-z)=\dot{z}(t) \frac{\partial}{\partial z(t)} \delta(z(t)-z)=-\frac{\partial}{\partial z}[\dot{z}(t) \delta(z(t)-z)]=-\frac{\partial}{\partial z}[F(z) \delta(z(t)-z)] $$

$$ \partial_{t} P_{\eta}\left(x v t \mid x_{a} v_{a} t_{a}\right)=-\left[\partial_{x} \dot{x}_{\eta}(t)+\partial_{v} \ddot{x}_{\eta}(t)\right] P_{\eta}\left(x v t \mid x_{a} v_{a} t_{a}\right) . $$

$$ \partial_{t} P_{\eta}\left(x v t \mid x_{a} v_{a} t_{a}\right)=-\left\{v \partial_{x}+\frac{1}{M}[\eta(t)+f(x, v)]\right\} P_{\eta}\left(x v t \mid x_{a} v_{a} t_{a}\right) $$

$$ f(x, v) \equiv-M \gamma v-V^{\prime}(x) $$

$$ \langle\eta(t) F[\eta]\rangle_{\eta}=\int d t^{\prime}\left\langle\eta(t) \eta\left(t^{\prime}\right)\right\rangle_{\eta}\left\langle\frac{\delta \eta(t)}{\delta \eta\left(t^{\prime}\right)} F[\eta]\right\rangle_{\eta} $$

$$ \eta(t) e^{-\frac{1}{2 w} \int d t d t^{\prime} \eta(t) K^{-1}\left(t, t^{\prime}\right) \eta\left(t^{\prime}\right)}=-w \int d t^{\prime} K\left(t, t^{\prime}\right) \frac{\delta}{\delta \eta\left(t^{\prime}\right)} e^{-\frac{1}{2 w} \int d t d t^{\prime} \eta(t) K^{-1}\left(t, t^{\prime}\right) \eta\left(t^{\prime}\right)} $$

$$ \partial_{t} P_{\eta}\left(x v t \mid x_{a} v_{a} t_{a}\right)=-\left\{v \partial_{x}+\frac{1}{M} \partial_{v}\left[w \int d t^{\prime} K\left(t, t^{\prime}\right) \frac{\delta}{\delta \eta\left(t^{\prime}\right)}+f(x, v)\right]\right\} P_{\eta}\left(x v t \mid x_{a} v_{a} t_{a}\right) $$

$$ \frac{\delta}{\delta \eta\left(t^{\prime}\right)} \delta\left(x_{\eta}(t)-x\right) \delta\left(\dot{x}_{\eta}(t)-v\right)=-\left[\frac{\delta x_{\eta}(t)}{\delta \eta\left(t^{\prime}\right)} \partial_{x}+\frac{\delta \dot{x}_{\eta}(t)}{\delta \eta\left(t^{\prime}\right)} \partial_{v}\right] \delta\left(x_{\eta}(t)-x\right) \delta\left(\dot{x}_{\eta}(t)-v\right) $$

$$ \begin{align*} \frac{\delta \ddot{x}_{\eta}(t)}{\delta \eta\left(t^{\prime}\right)} & =\frac{1}{M} \delta\left(t-t^{\prime}\right)-\gamma \Theta\left(t-t^{\prime}\right)+\text { smooth function of } t-t^{\prime} \\ \frac{\delta \dot{x}_{\eta}(t)}{\delta \eta\left(t^{\prime}\right)} & =\frac{1}{M} \Theta\left(t-t^{\prime}\right)+\mathcal{O}\left(t-t^{\prime}\right) \\ \frac{\delta x_{\eta}(t)}{\delta \eta\left(t^{\prime}\right)} & =\mathcal{O}\left(\left(t-t^{\prime}\right)^{2}\right) \end{align*} $$

$$ \begin{align*} & \int d t^{\prime} K\left(t, t^{\prime}\right) \frac{\delta}{\delta \eta\left(t^{\prime}\right)} \delta\left(x_{\eta}(t)-x\right) \delta\left(\dot{x}_{\eta}(t)-v\right) \\ & =-\int d t^{\prime} \delta_{\epsilon}\left(t-t^{\prime}\right) \frac{\delta \dot{x}_{\eta}(t)}{\delta \eta\left(t^{\prime}\right)} \partial_{v} \delta\left(x_{\eta}(t)-x\right) \delta\left(\dot{x}_{\eta}(t)-v\right)=-\frac{1}{2 M} \partial_{v} \delta\left(x_{\eta}(t)-x\right) \delta\left(\dot{x}_{\eta}(t)-v\right) \end{align*} $$

$$ \partial_{t} P\left(x v t \mid x_{a} v_{a} t_{a}\right)=\left\{-v \partial_{x}+\frac{1}{M} \partial_{v}\left[\frac{w}{2 M} \partial_{v}-f(x, v)\right]\right\} P\left(x v t \mid x_{a} v_{a} t_{a}\right) $$

$$ P_{\eta}\left(x t \mid x_{a} t_{a}\right)=\int d v P_{\eta}\left(x v t \mid x_{a} v_{a} t_{a}\right)=\delta\left(x_{\eta}(t)-x\right) $$

$$ \begin{align*} \partial_{t} P_{\eta}\left(x t \mid x_{a} v_{a} t_{a}\right) & =-\partial_{x} \dot{x}_{\eta}(t) P_{\eta}\left(x t \mid x_{a} v_{a} t_{a}\right) \\ & =-\frac{1}{M \gamma} \partial_{x}\left[\eta(t)-V^{\prime}(x)\right] P_{\eta}\left(x t \mid x_{a} v_{a} t_{a}\right) \end{align*} $$

$$ \eta(t) \rightarrow w \int d t^{\prime} \delta_{\epsilon}\left(t-t^{\prime}\right) \frac{\delta}{\delta \eta\left(t^{\prime}\right)} $$

$$ \frac{\delta}{\delta \eta\left(t^{\prime}\right)} \delta\left(x_{\eta}(t)-x\right)=-\frac{\delta x_{\eta}(t)}{\delta \eta\left(t^{\prime}\right)} \delta\left(x_{\eta}(t)-x\right) $$

$$ \begin{align*} \frac{\delta \dot{x}_{\eta}(t)}{\delta \eta\left(t^{\prime}\right)} & =\frac{1}{M \gamma} \delta\left(t-t^{\prime}\right)+\text { smooth function of } t-t^{\prime} \\ \frac{\delta x_{\eta}(t)}{\delta \eta\left(t^{\prime}\right)} & =\frac{1}{M \gamma} \Theta\left(t-t^{\prime}\right)+\mathcal{O}\left(t-t^{\prime}\right) \end{align*} $$

$$ \partial_{t} P\left(x t \mid x_{a} t_{a}\right)=\left[D \partial^{2}+\frac{1}{M \gamma} V^{\prime}(x)\right] P\left(x t \mid x_{a} t_{a}\right) $$

$$ \begin{align*} P(x, t) & =\int \mathcal{D} \eta e^{-(1 / 2 w) \int d t d t^{\prime} \eta(t) K^{-1}\left(t, t^{\prime}\right) \eta\left(t^{\prime}\right)} P\left(x_{a \eta}(t), t_{a}\right) \\ P(x v, t) & =\int \mathcal{D} \eta e^{-(1 / 2 w) \int d t d t^{\prime} \eta(t) K^{-1}\left(t, t^{\prime}\right) \eta\left(t^{\prime}\right)} P\left(x_{a \eta}(t), v_{a \eta}, t\right) \end{align*} $$

$$ x_{a \eta}(t)=x-\int_{t_{a}}^{t} d t^{\prime} \dot{x}\left(t^{\prime}\right), \quad v_{a \eta}(t)=x-\int_{t_{a}}^{t} d t^{\prime} \dot{v}\left(t^{\prime}\right) $$

$$ P(x, t)=\int \mathcal{D} \eta e^{-(1 / 2 w) \int d t \eta^{2}(t)} P\left(x-\frac{1}{M \gamma} \int_{t_{a}}^{t} d t^{\prime}\left[\eta\left(t^{\prime}\right)-V^{\prime}\left(x\left(t^{\prime}\right)\right)\right], t\right) $$

$$ \begin{gather*} P(x, t+\epsilon)=\int \mathcal{D} \eta e^{-(1 / 2 w) \int d t \eta^{2}(t)}\left\{-\frac{\epsilon}{M \gamma} \int_{t}^{t+\epsilon} d t^{\prime}\left[\eta\left(t^{\prime}\right)-V^{\prime}\left(x\left(t^{\prime}\right)\right)\right] \partial_{x}\right. \\ \left.+\frac{1}{2 M^{2} \gamma^{2}} \int_{t}^{t+\epsilon} d t^{\prime} \int_{t}^{t+\epsilon} d t^{\prime \prime}\left[\eta\left(t^{\prime}\right)-V^{\prime}\left(x\left(t^{\prime}\right)\right)\right]\left[\eta\left(t^{\prime \prime}\right)-V^{\prime}\left(x\left(t^{\prime \prime}\right)\right)\right] \partial_{x}^{2}+\ldots\right\} \\ \quad \times P\left(x-\frac{1}{M \gamma} \int_{t_{a}}^{t} d t^{\prime}\left[\eta\left(t^{\prime}\right)-V^{\prime}\left(x\left(t^{\prime}\right)\right)\right], t\right) \end{gather*} $$

$$ \dot{x}(t)=\langle\dot{x}(t)\rangle+\eta(t)=r_{x}+\eta(t) $$

$$ \left\langle\eta(t) \eta\left(t^{\prime}\right)\right\rangle=\sigma^{2} \delta\left(t-t^{\prime}\right) $$

$$ \Delta x(t) \equiv \int_{t}^{t+\epsilon} d t^{\prime} \dot{x}\left(t^{\prime}\right)=\epsilon r_{x}+\int_{t}^{t+\epsilon} d t^{\prime} \eta\left(t^{\prime}\right) $$

$$ \begin{align*} f(x(t+\epsilon))=f(x(t)) & +f^{\prime}(x(t)) \Delta x(t) \\ & +\frac{1}{2} f^{\prime \prime}(x(t))[\Delta x(t)]^{2}+\frac{1}{3!} f^{(3)}[\Delta x(t)]^{3}+\ldots \end{align*} $$

$$ \langle\Delta x(t)\rangle=\int_{t}^{t+\epsilon} d t^{\prime}\left\langle\dot{x}\left(t^{\prime}\right)+\eta\left(t^{\prime}\right)\right\rangle=\int_{t}^{t+\epsilon} d t^{\prime}\left\langle\dot{x}\left(t^{\prime}\right)\right\rangle \approx \epsilon r_{x} $$

$$ \left\langle[\Delta x(t)]^{2}\right\rangle=\epsilon \sigma^{2}+\mathcal{O}\left(\epsilon^{2}\right) $$

$$ \begin{align*} & \int_{t}^{t+\epsilon} d t_{1} \int_{t}^{t+\epsilon} d t_{2} \int_{t}^{t+\epsilon} d t_{3}\left\langle\left[\left\langle\dot{x}\left(t_{1}\right)\right\rangle+\eta\left(t_{1}\right)\right]\left[\left\langle\dot{x}\left(t_{2}\right)\right\rangle+\eta\left(t_{2}\right)\right]\left[\left\langle\dot{x}\left(t_{3}\right)\right\rangle+\eta\left(t_{2}\right)\right]\right\rangle \\ & =\int_{t}^{t+\epsilon} d t_{1} \int_{t}^{t+\epsilon} d t_{2} \int_{t}^{t+\epsilon} d t_{3}\left[\left\langle\dot{x}\left(t_{1}\right)\right\rangle\left\langle\dot{x}\left(t_{2}\right)\right\rangle\left\langle\dot{x}\left(t_{3}\right)\right\rangle+\left\langle\dot{x}\left(t_{1}\right)\right\rangle\left\langle\eta\left(t_{2}\right) \eta\left(t_{3}\right)\right\rangle\right. \\ & \left.\quad+\left\langle\dot{x}\left(t_{2}\right)\right\rangle\left\langle\eta\left(t_{1}\right) \eta\left(t_{3}\right)\right\rangle+\left\langle\dot{x}\left(t_{3}\right)\right\rangle\left\langle\eta\left(t_{1}\right) \eta\left(t_{2}\right)\right\rangle\right] \\ & =\epsilon^{3} r_{x}^{3}+3 \epsilon^{2} r_{x} \sigma^{2}=\mathcal{O}\left(\epsilon^{2}\right) \end{align*} $$

$$ \langle\dot{f}(x(t))\rangle=\left\langle f^{\prime}(x(t))\right\rangle\langle\dot{x}(t)\rangle+\frac{\sigma^{2}}{2}\left\langle f^{\prime \prime}(x(t))\right\rangle . $$

$$ \dot{f}(x(t))=f^{\prime}(x(t)) \dot{x}(t)+\frac{\sigma^{2}}{2} f^{\prime \prime}(x(t)), $$

$$ z_{1}(t)=\int_{t}^{t+\epsilon} d t \eta(t), \quad z_{2}(t) \equiv\left[z_{2,1}(t)+z_{2,2}(t)\right] $$

$$ z_{2,1}(t)=2 \int_{t}^{t+\epsilon} d t_{1}\left\langle\dot{x}\left(t_{1}\right)\right\rangle z_{1}(t) \approx 2 \epsilon r_{x} z_{1}(t), \quad z_{2,2}(t)=\left[z_{1}(t)\right]^{2} $$

$$ \left\langle\left[z_{2,2}(t)\right]^{2}\right\rangle=\int_{t}^{t+\epsilon} d t_{1} \int_{t}^{t+\epsilon} d t_{2} \int_{t}^{t+\epsilon} d t_{3} \int_{t}^{t+\epsilon} d t_{4}\left\langle\eta\left(t_{1}\right) \eta\left(t_{2}\right) \eta\left(t_{3}\right) \eta\left(t_{4}\right)\right\rangle $$

$$ \left\langle\eta\left(t_{1}\right) \eta\left(t_{2}\right)\right\rangle\left\langle\eta\left(t_{3}\right) \eta\left(t_{4}\right)\right\rangle+\left\langle\eta\left(t_{1}\right) \eta\left(t_{3}\right)\right\rangle\left\langle\eta\left(t_{2}\right) \eta\left(t_{4}\right)\right\rangle\left\langle\eta\left(t_{1}\right) \eta\left(t_{4}\right)\right\rangle\left\langle\eta\left(t_{2}\right) \eta\left(t_{3}\right)\right\rangle . $$

$$ \left\langle\left[z_{2,2}(t)\right]^{2}\right\rangle=3 \epsilon^{2} \sigma^{4} $$

$$ \left\langle z_{2,2}(t)\right\rangle^{2}=\left\langle z_{1}^{2}(t)\right\rangle^{2}=\left[\int_{t}^{t+\epsilon} d t_{1} \int_{t}^{t+\epsilon} d t_{2}\left\langle\eta\left(t_{1}\right) \eta\left(t_{2}\right)\right\rangle\right]^{2}=\epsilon^{2} \sigma^{4} $$

$$ \left\langle\left[z_{2,2}(t)\right]^{2}\right\rangle-\left\langle z_{2,2}(t)\right\rangle^{2}=2 \sigma^{4} \epsilon^{2} $$

$$ \left\langle\left[z_{1}(t)\right]^{2}\right\rangle-\left\langle z_{1}(t)\right\rangle^{2}=\int_{t}^{t+\epsilon} d t_{1} \int_{t}^{t+\epsilon} d t_{2}\left\langle\eta\left(t_{1}\right) \eta\left(t_{2}\right)\right\rangle=\epsilon \sigma^{2} $$

$$ [\Delta x(t)]^{2}=\epsilon \sigma^{2}+\mathcal{O}\left(\epsilon^{2}\right) $$

$$ \frac{d}{d t} e^{P x}=\left(P \dot{x}+\frac{\sigma^{2} P^{2}}{2}\right) e^{P x} $$

$$ e^{P x}=e^{\int_{0}^{t} d t^{\prime} P \dot{x}} e^{P^{2} \sigma^{2} t / 2} $$

$$ \left\langle e^{P \int_{0}^{t} d t^{\prime} \eta\left(t^{\prime}\right)}\right\rangle=e^{P^{2} \int_{0}^{t} d t^{\prime} \int_{0}^{t} d t^{\prime \prime}\left\langle\eta\left(t^{\prime}\right) \eta\left(t^{\prime \prime}\right)\right\rangle}=e^{P^{2} \sigma^{2} t / 2} $$

$$ f(x(t+d t))=f(x(t)+\dot{x} d t)=f(x(t))+f^{\prime}(x(t)) \dot{x}(t) d t+\frac{1}{2} f^{\prime \prime}(x(t)) \dot{x}^{2}(t) d t^{2}+\ldots, $$

$$ \left\langle\eta^{2}(t)\right\rangle d t=\sigma^{2} $$

$$ \int_{t}^{t+\epsilon} d t^{\prime}\left\langle\eta\left(t^{\prime}\right) \eta(t)\right\rangle=\int_{t}^{t+\epsilon} d t^{\prime} \sigma^{2} \delta\left(t^{\prime}-t\right)=\sigma^{2} $$

$$ \dot{x}^{2}(t) d t^{2} \rightarrow \sigma^{2} d t / 2 $$

$$ z_{n} \approx \mathcal{O}\left((\sigma \sqrt{\epsilon})^{n}\right) $$

$$ \dot{x}^{n}(t) d t^{n} \approx \mathcal{O}\left((\sigma \sqrt{d t})^{n}\right) $$

$$ \frac{\Delta f\left(x\left(t_{n}\right)\right)}{\Delta t}=f^{\prime}\left(x\left(t_{n}\right)\right) \frac{\Delta x\left(t_{n}\right)}{\Delta t}+\frac{\sigma^{2}}{2} f^{\prime \prime}\left(x\left(t_{n}\right)\right)+\mathcal{O}(\sigma \sqrt{\Delta t}) . $$

$$ \dot{x}(t)=-\kappa x(t)+\bar{\eta}(t) $$

$$ \langle\bar{\eta}(t)\rangle_{\eta}, \quad\left\langle\bar{\eta}(t) \bar{\eta}\left(t^{\prime}\right)\right\rangle_{\eta}=2 D \delta\left(t-t^{\prime}\right) $$

$$ x(t)=x_{0} e^{-\kappa t}+\int_{0}^{t} d t_{1} e^{-\kappa\left(t-t_{1}\right)} \bar{\eta}\left(t_{1}\right) $$

$$ \begin{align*} \left\langle x(t) x\left(t^{\prime}\right)\right\rangle_{\eta} & =x_{0}^{2} e^{-\kappa\left(t+t^{\prime}\right)}+2 D \int_{0}^{t} d t_{1} e^{-\kappa\left(t-t_{1}\right)} \int_{0}^{t^{\prime}} d t_{2} e^{-\left(t^{\prime}-t_{2}\right)} \delta\left(t_{1}-t_{2}\right) \\ & =x_{0}^{2} e^{-\kappa\left(t+t^{\prime}\right)}+\kappa^{-1} D\left(e^{-\kappa\left|t-t^{\prime}\right|}-e^{-\kappa\left(t+t^{\prime}\right)}\right) \end{align*} $$

$$ \left\langle[x(t)-\langle x(t)\rangle]^{2}\right\rangle_{\eta}=\kappa^{-1} D\left(1-e^{-2 \kappa t}\right) $$

$$ \dot{\mathbf{x}}(t)=-\boldsymbol{\kappa} \mathbf{x}(t)+\overline{\boldsymbol{\eta}}(t) $$

$$ \langle\overline{\boldsymbol{\eta}}(t)\rangle=0, \quad\left\langle\overline{\boldsymbol{\eta}}(t) \overline{\boldsymbol{\eta}}^{T}\left(t^{\prime}\right)\right\rangle_{\boldsymbol{\eta}}=2 \mathbf{D} \delta\left(t-t^{\prime}\right) $$

$$ \begin{align*} P\left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right) & =\frac{1}{\sqrt{2 \pi}^{D}} \frac{1}{{\sqrt{\operatorname{det}\left[\sigma^{2}\left(t_{b}-t_{a}\right)\right]}}^{D}} \\ & \times \exp \left\{-\frac{1}{2}\left[x_{b}-\bar{x}\left(t_{b}-t_{a}\right)\right]^{i}\left[\sigma_{i j}^{2}\left(t_{b}-t_{a}\right)\right]^{-1}\left[x_{b}-\bar{x}\left(t_{b}-t_{a}\right)\right]^{j}\right\} \end{align*} $$

$$ \overline{\mathbf{x}}(t)=e^{-\boldsymbol{\kappa} t} \mathbf{x}_{a} $$

$$ \sigma_{i j}^{2}(t) \equiv\left\langle[x(t)-\bar{x}(t)]^{i}[x(t)-\bar{x}(t)]^{j}\right\rangle_{\boldsymbol{\eta}} $$

$$ \partial_{t} P\left(\mathbf{x} t \mid t_{a} \mathbf{x}_{a}\right)=\left(-\kappa_{i j} \partial_{i} x_{j}+D_{i j} \partial_{i} \partial_{j}\right) P\left(\mathbf{x} t \mid t_{a} \mathbf{x}_{a}\right) $$

$$ \boldsymbol{\kappa}=\left(\begin{array}{cc} 0 & -1 \\ \omega_{0}^{2} & \gamma \end{array}\right), \quad \bar{\eta}(t)=\frac{1}{M}\binom{0}{\eta(t)}, $$

$$ e^{-\kappa t}=\frac{1}{\kappa_{1}-\kappa_{2}}\left(\begin{array}{cc} \kappa_{1} e^{-\kappa_{2} t}-\kappa_{2} e^{-\kappa_{1} t} & e^{-\kappa_{2} t}-e^{-\kappa_{1} t} \\ \omega_{0}^{2}\left(e^{-\kappa_{1} t}-e^{-\kappa_{2} t}\right) & \kappa_{1} e^{-\kappa_{1} t}-\kappa_{2} e^{-\kappa_{2} t} \end{array}\right) $$

$$ \left[\sigma_{i j}^{2}(t)\right]^{-1}=\left[\operatorname{det} \sigma_{i j}^{2}(t)\right]^{-1}\left(\begin{array}{cc} \sigma_{v v}^{2}(t) & -\sigma_{x v}^{2}(t) \\ -\sigma_{x v}^{2}(t) & \sigma_{x x}^{2}(t) \end{array}\right) $$

$$ \mathbf{x}(t)=e^{-\mathbf{\kappa} t} \mathbf{x}_{a}+\int_{t_{a}}^{t} d t \overline{\boldsymbol{\eta}}(t) $$

$$ \begin{align*} \sigma_{x x}^{2}(t) & =\frac{\gamma^{2} D}{\left(\kappa_{1}-\kappa_{2}\right)^{2}}\left[\frac{1}{\kappa_{1}}\left(1-e^{-2 \kappa_{1} t}\right)+\frac{1}{\kappa_{2}}\left(1-e^{-2 \kappa_{2} t}\right)-\frac{4}{\kappa_{1}+\kappa_{2}}\left(1-e^{\left.-\kappa_{1}+\kappa_{2}\right) t}\right)\right] \\ \sigma_{x v}^{2}(t) & =\frac{\gamma^{2} D}{\left(\kappa_{1}-\kappa_{2}\right)^{2}}\left(e^{-\kappa_{1} t}-e^{-\kappa_{2} t}\right)^{2} \\ \sigma_{v v}^{2}(t) & =\frac{\gamma^{2} D}{\left(\kappa_{1}-\kappa_{2}\right)^{2}}\left[\kappa_{1}\left(1-e^{-2 \kappa_{1} t}\right)+\kappa_{2}\left(1-e^{-2 \kappa_{2} t}\right)-\frac{4}{\kappa_{1}^{-1}+\kappa_{2}^{-1}}\left(1-e^{\left.-\kappa_{1}+\kappa_{2}\right) t}\right)\right] \end{align*} $$

$$ \sigma_{x x}^{2}(t) \rightarrow \frac{\gamma D}{\kappa_{1} \kappa_{2}}=\frac{\gamma D}{\omega_{0}^{2}}, \quad \sigma_{x v}^{2}(t) \rightarrow 0, \quad \sigma_{v v}^{2}(t) \rightarrow \gamma D $$

$$ \lim _{t_{b} \rightarrow \infty} P\left(x_{b} v_{b} t_{b} \mid x_{a} v_{a} t_{a}\right)=\frac{\omega_{0}}{2 \pi \gamma D} e^{-\left(v_{b}^{2}+\omega_{0}^{2} x_{b}^{2}\right) / 2 \gamma D}=\frac{M \omega_{0}}{2 \pi k_{B} T} e^{-M\left(v_{b}^{2}+\omega_{0}^{2} x_{b}^{2}\right) / 2 k_{B} T} $$

$$ P\left(v_{b}\right)=\frac{1}{\sqrt{2 \pi \gamma D}} e^{-v_{b}^{2} / 2 \gamma D}=\frac{1}{\sqrt{2 \pi k_{B} T / M}} e^{-M v_{b}^{2} / 2 k_{B} T}=\frac{1}{\sqrt{2 \pi} v_{T}} e^{-v_{b}^{2} / 2 v_{T}^{2}},( $$

$$ v_{T} \equiv \sqrt{k_{B} T / M} $$

$$ P\left(x_{b} t_{b} \mid x_{a} v_{a} t_{a}\right)=\int d v_{b} P\left(x_{b} v_{b} t_{b} \mid x_{a} v_{a} t_{a}\right)=\frac{1}{\sqrt{2 \pi \sigma_{x x}^{2}\left(t_{b}-t_{a}\right)}} \exp \left\{-\frac{1}{2} \frac{\left[x_{b}-\bar{x}\left(t_{b}-t_{a}\right)\right]^{2}}{\sigma_{x x}^{2}\left(t_{b}-t_{a}\right)}\right\} $$

$$ e^{-\kappa t}=\left(\begin{array}{cc} 1 & \gamma^{-1}\left(1-e^{-\gamma t}\right) \\ 0 & e^{-\gamma t} \end{array}\right) $$

$$ \sigma_{x x}^{2}(t)=\gamma^{-1} D\left(2 \gamma t-3+4 e^{-\gamma t}-e^{-2 \gamma t}\right), \sigma_{x v}^{2}(t)=D\left(1-e^{-\gamma t}\right)^{2}, \sigma_{v v}^{2}(t)=\gamma D\left(1-e^{-2 \gamma t}\right) $$

$$ \operatorname{det} \sigma_{i j}^{2}(t)=D^{2}\left[2 \gamma t\left(1-e^{-2 \gamma t}\right)+\left(1-e^{-\gamma t}\right)^{2}\left(-4-2 e^{-\gamma t}+e^{-3 \gamma t}\right)\right] $$

$$ \sigma_{x x}^{2}(t) \rightarrow 2 D t, \quad \sigma_{x v}^{2}(t) \rightarrow D, \quad \sigma_{v v}^{2}(t) \rightarrow \gamma D, \quad \operatorname{det} \sigma_{i j}^{2}(t) \rightarrow 2 \gamma t D^{2} $$

$$ \begin{align*} \langle x\rangle & \equiv \int_{-\infty}^{\infty} d x_{b} x_{b} P\left(x_{b} t_{b} \mid x_{a} t_{a}\right) \\ \left\langle x^{2}\right\rangle & \equiv \int_{-\infty}^{\infty} d x_{b} x_{b}^{2} P\left(x_{b} t_{b} \mid x_{a} t_{a}\right) \end{align*} $$

$$ \left\langle x_{b} t_{b} \mid x_{a} t_{a}\right\rangle \equiv\left|\left(x_{b} t_{b} \mid x_{a} t_{a}\right)\right|^{2} $$

$$ \left\langle x_{b} t_{b} \mid x_{a} t_{a}\right\rangle=e^{-\left(t_{b}-t_{a}\right) H\left(\hat{p}_{b}, x_{b}\right)} \delta\left(x_{b}-x_{a}\right) $$

$$ \left\langle x_{b} \mid x_{a}\right\rangle=\delta\left(x_{b}-x_{a}\right) $$

$$ \left\langle x_{b}\right| \hat{x}=x_{b}\left\langle x_{b}\right|, \quad\left\langle x_{b}\right| \hat{p}=-i \frac{\partial}{\partial x_{b}}\left\langle x_{b}\right| $$

$$ \left\langle x_{b} t_{b} \mid x_{a} t_{a}\right\rangle=\left\langle x_{b}\right| e^{-H(\hat{p}, \hat{x})\left(t_{b}-t_{a}\right)}\left|x_{a}\right\rangle . $$

$$ \begin{align*} \langle f(x)\rangle & =\int_{-\infty}^{\infty} d x_{b} f\left(x_{b}\right)\left\langle x_{b}\right| e^{-\left(t_{b}-t_{a}\right) H(\hat{p}, \hat{x})}\left|x_{a}\right\rangle \\ & =\int_{-\infty}^{\infty} d x_{b}\left\langle x_{b}\right| f(\hat{x}) e^{-\left(t_{b}-t_{a}\right) H(\hat{p}, \hat{x})}\left|x_{a}\right\rangle \\ & =\int_{-\infty}^{\infty} d x_{b} \int_{-\infty}^{\infty} d x\left\langle x_{b}\right| e^{-\left(t_{b}-t_{a}\right) H(\hat{p}, \hat{x})}|x\rangle\langle x| f\left(\hat{x}\left(t_{b}-t_{a}\right)\right)\left|x_{a}\right\rangle \end{align*} $$

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

$$ \begin{align*} \int_{-\infty}^{\infty} d x_{b} P\left(x_{b} t_{b} \mid x_{a} t_{a}\right) & =\int_{-\infty}^{\infty} d x_{b}\left\langle x_{b} t_{b} \mid x_{a} t_{a}\right\rangle \\ & =\int_{-\infty}^{\infty} d x_{b}\left\langle x_{b}\right| e^{-\left(t_{b}-t_{a}\right) H(\hat{p}, \hat{x})}\left|x_{a}\right\rangle=1 \end{align*} $$

$$ \langle f(x)\rangle=\int_{-\infty}^{\infty} d x_{b}\left\langle x_{b}\right| f\left(\hat{x}\left(t_{b}-t_{a}\right)\right)\left|x_{a}\right\rangle $$

$$ L_{\mathrm{e}}=\frac{\dot{x}^{2}}{4 D}, \quad H=D p^{2} $$

$$ \hat{p}(t)=\hat{p}, \quad \hat{x}(t)=e^{\hat{H} t} \hat{x} e^{-\hat{H} t}=\hat{x}-i 2 D \hat{p} t, $$

$$ \begin{align*} \hat{x}^{2}(t) & =\hat{x}^{2}-i 2 D \cdot(\hat{p} \hat{x}+\hat{x} \hat{p}) t-4 D^{2} \hat{p}^{2} t^{2} \\ & =\hat{x}^{2}+2 D t-i 2 D \cdot 2 \hat{x} \hat{p}-4 D^{2} \hat{p}^{2} t \end{align*} $$

$$ \begin{align*} \int_{-\infty}^{\infty} d x_{b}\left\langle x_{b}\right| \hat{x}\left|x_{a}\right\rangle & =x_{a} \\ \int_{-\infty}^{\infty} d x_{b}\left\langle x_{b}\right| \hat{p}\left|x_{a}\right\rangle & =-i \int_{-\infty}^{\infty} d x_{b} \frac{\partial}{\partial x_{b}} \delta\left(x_{b}-x_{a}\right)=0 \\ \int_{-\infty}^{\infty} d x_{b}\left\langle x_{b}\right| \hat{x}^{2}\left|x_{a}\right\rangle & =\int_{-\infty}^{\infty} d x_{b} x_{b}^{2} \delta\left(x_{b}-x_{a}\right)=x_{a}^{2} \\ \int_{-\infty}^{\infty} d x_{b}\left\langle x_{b}\right| \hat{p}^{2}\left|x_{a}\right\rangle & =-\int_{-\infty}^{\infty} d x_{b} \frac{\partial^{2}}{\partial x_{b}^{2}} \delta\left(x_{b}-x_{a}\right)=0 \\ \int_{-\infty}^{\infty} d x_{b}\left\langle x_{b}\right| \hat{p} \hat{x}\left|x_{a}\right\rangle & =-i \int_{-\infty}^{\infty} d x_{b} \frac{\partial}{\partial x_{b}} \delta\left(x_{b}-x_{a}\right) x_{a}=0 \end{align*} $$

$$ \int_{-\infty}^{\infty} d x_{b}\left\langle x_{b}\right| \hat{p}=0 $$

$$ \langle x\rangle=x_{a}, \quad\left\langle x^{2}\right\rangle=x_{a}^{2}+2 D\left(t_{b}-t_{a}\right), $$

$$ \left\langle\left(x-x_{a}\right)^{2}\right\rangle=2 D\left(t_{b}-t_{a}\right) . $$

$$ H=\frac{w}{2 M^{2}} p_{v}^{2}-i \gamma p_{v} v+i p v $$

$$ \begin{align*} \langle f(x, v)\rangle & =\int_{-\infty}^{\infty} d x_{b} \int_{-\infty}^{\infty} d v_{b} f\left(x_{b}, v_{b}\right) P\left(x_{b} v_{b} t_{b} \mid x_{a} v_{a} t_{a}\right) \\ & =\int_{-\infty}^{\infty} d x_{b} \int_{-\infty}^{\infty} d v_{b}\left\langle x_{b} v_{b}\right| f\left(\hat{x}\left(t_{b}-t_{a}\right), \hat{v}\left(t_{b}-t_{a}\right)\right)\left|x_{a} v_{a}\right\rangle \end{align*} $$

$$ \langle x v| \hat{p}=-\frac{\partial}{\partial x}\langle x v|, \quad\langle x v| \hat{p}_{v}=-\frac{\partial}{\partial v}\langle x v| . $$

$$ \left\langle x^{2}\right\rangle=\int_{-\infty}^{\infty} d x_{b} \int_{-\infty}^{\infty} d v_{b}\left\langle x_{b} v_{b}\right| \hat{x}^{2}\left(t_{b}-t_{a}\right)\left|x_{a} v_{a}\right\rangle $$

$$ \hat{x}(t)=e^{t H\left(\hat{p}, \hat{p}_{v}, \hat{x}, \hat{v}\right)} \hat{x} e^{-t H\left(\hat{p}, \hat{p}_{v}, \hat{x}, \hat{v}\right)} $$

$$ \begin{align*} \dot{\hat{p}}(t) & =[\hat{H}, \hat{p}(t)]=0 \\ \dot{\hat{p}}(t) & =\left[\hat{H}, \hat{p}_{v}(t)\right]=\gamma \hat{p}_{v}(t)-\hat{p}(t) \\ \dot{\hat{x}}(t) & =[\hat{H}, \hat{x}(t)]=\hat{v}(t) \\ \dot{\hat{v}}(t) & =[\hat{H}, \hat{v}(t)]=-i \frac{w}{M^{2}} \hat{p}_{v}(t)-\gamma \hat{v}(t) \end{align*} $$

$$ \hat{p}_{v}(t)=\hat{p}_{v} e^{\gamma t}-\frac{1}{\gamma} \hat{p}\left(e^{\gamma t}-1\right) $$

$$ \begin{align*} \hat{v}(t) & =\hat{v} e^{-\gamma t}-i \frac{w}{M^{2}} \int_{0}^{t} d t^{\prime} e^{-\gamma\left(t-t^{\prime}\right)} \hat{p}_{v}\left(t^{\prime}\right) \\ & =\hat{v} e^{-\gamma t}-i \frac{w}{\gamma M^{2}}\left[\hat{p}_{v} \sinh \gamma t-\frac{1}{\gamma} \hat{p}(\cosh \gamma t-1)\right] \end{align*} $$

$$ \hat{x}(t)=\hat{x}+\hat{v} \frac{1}{\gamma}\left(1-e^{-\gamma t}\right)-i \frac{w}{\gamma M^{2}}\left[p_{v} \cosh \gamma t-\frac{1}{\gamma} p(\sinh \gamma t-\gamma t)\right] . $$

$$ \int_{-\infty}^{\infty} d x_{b} \int_{-\infty}^{\infty} d x_{2 b}\left\langle x_{b} x_{2 b}\right|\left\{\begin{array}{c} \hat{p} \\ \hat{p}_{v} \end{array}\right\}=0 $$

$$ \langle x\rangle=x_{a}+\dot{x}_{a} \frac{1}{\gamma}\left(1-e^{-\gamma\left(t_{b}-t_{a}\right)}\right), \quad\langle v\rangle=v_{a} e^{-\gamma\left(t_{b}-t_{a}\right)}, $$

$$ P_{0}\left(x_{b} t_{b} \mid x_{a} t_{a}\right) \propto \int \mathcal{D} x(t) \operatorname{Det}\left[\partial_{t}+\frac{V^{\prime \prime}(x)}{M \gamma}\right] \exp \left\{-\int_{t_{a}}^{t_{b}} d t \frac{1}{4 D}\left[\dot{x}+\frac{V^{\prime}(x)}{M \gamma}\right]^{2}\right\} $$

$$ \operatorname{det}\left[\partial_{t}+V^{\prime \prime}(x(t)) / M \gamma\right] \propto \int \mathcal{D} c \mathcal{D} \bar{c} e^{-\int d t \bar{c}(t)\left[M \gamma \partial_{t}+V^{\prime \prime}(x(t))\right] c(t)} $$

$$ P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int \mathcal{D} x \int \mathcal{D} c \mathcal{D} \bar{c} \exp \left\{-\mathcal{A}_{\mathrm{PS}}[x, c, \bar{c}]\right\} $$

$$ \mathcal{A}_{\mathrm{PS}}=\frac{1}{2 D M^{2} \gamma^{2}} \int_{t_{a}}^{t_{b}} d t\left\{\frac{1}{2}\left[M \gamma \dot{x}+V^{\prime}(x)\right]^{2}+\bar{c}(t)\left[M \gamma \partial_{t}+V^{\prime \prime}(x(t))\right] c(t)\right\} $$

$$ U_{x} \equiv M \gamma \partial_{t} x+V^{\prime}(x) $$

$$ U_{x y} \equiv \frac{\delta U_{x}}{\delta y}=M \gamma \partial_{t}+V^{\prime \prime}(x) $$

$$ \mathcal{A}_{\mathrm{PS}}=\frac{1}{2 D} \int_{t_{a}}^{t_{b}} d t\left[\frac{1}{2} U_{x}^{2}+\bar{c}(t) U_{x y} c(t)\right] $$

$$ \begin{align*} \delta x(t) & =\bar{\varepsilon} c(t)+\bar{c}(t) \varepsilon, \\ \delta \bar{c}(t) & =-\bar{\varepsilon} U_{x}, \\ \delta c(t) & =U_{x} \varepsilon . \end{align*} $$

$$ \delta U_{x}=\bar{\varepsilon} U_{x y} c(t)+\bar{c}(t) U_{x y} \varepsilon $$

$$ P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\mathcal{N} \int \mathcal{D} x J[x] \exp \left\{-\frac{1}{2 w} \int_{t_{a}}^{t_{b}} d t\left[M \ddot{x}+M \gamma \dot{x}+V^{\prime}(x)\right]^{2}\right\} $$

$$ J[x]=\operatorname{det}\left[M \partial_{t}^{2}+M \gamma \partial_{t}+V^{\prime \prime}(x(t))\right], $$

$$ J[x]=\operatorname{det}\left[M \partial_{t}^{2}+M \gamma \partial_{t}+V^{\prime \prime}(x(t))\right] \propto \int \mathcal{D} c \mathcal{D} \bar{c} e^{-\int d t \bar{c}(t)\left[M \partial_{t}^{2}+M \gamma \partial_{t}+V^{\prime \prime}(x(t))\right] c(t)} $$

$$ P\left(x_{b} t_{b} \mid x_{a} t_{a}\right) \propto \int \mathcal{D} x \int \mathcal{D} c \mathcal{D} \bar{c} \exp \left\{-\mathcal{A}^{\mathrm{KS}}\left[x,{ }_{s} \bar{c}\right]\right\} $$

$$ \mathcal{A}^{\mathrm{KS}}\left[x,{ }_{s} \bar{c}\right] \equiv \int_{t_{a}}^{t_{b}} d t\left\{\frac{1}{2 w}\left[M \ddot{x}+M \gamma \dot{x}+V^{\prime}(x)\right]^{2}+\bar{c}(t)\left[M \partial_{t}^{2}+M \gamma \partial_{t}+V^{\prime \prime}(x(t))\right] c(t)\right\} $$

$$ \begin{align*} P\left(x_{b} t_{b} \mid x_{a} t_{a}\right) & \underset{\gamma \rightarrow 0}{\propto} \\ & \times \int \mathcal{D} x \delta[\delta \mathcal{A} / \delta x] \\ & \times \mathcal{D} c \mathcal{D} \bar{c} \exp \left\{-\int_{t_{a}}^{t_{b}} d t \int_{t_{a}}^{t_{b}} d t^{\prime} \bar{c}(t) \delta^{2} \mathcal{A} / \partial x(t) \partial x\left(t^{\prime}\right) c\left(t^{\prime}\right)\right\} \end{align*} $$

$$ P\left(x_{b} t_{b} \mid x_{a} t_{a}\right) \underset{\gamma \rightarrow 0}{\propto} \int \mathcal{D} x \delta[\delta \mathcal{A} / \delta x] \operatorname{Det}\left[\delta^{2} \mathcal{A} / \partial x(t) \partial x\left(t^{\prime}\right)\right] $$

$$ \delta\left[M \ddot{x}+V^{\prime}(x)\right]=\delta\left[x-x_{\mathrm{cl}}\right] \times \operatorname{Det}^{-1}\left[M \ddot{x}+V^{\prime \prime}(x)\right], $$

$$ P\left(x_{b} t_{b} \mid x_{a} t_{a}\right) \underset{\gamma \rightarrow 0}{\propto} \int \mathcal{D} x \delta\left[x-x_{\mathrm{cl}}\right] $$

$$ P\left(x_{b} t_{b} \mid x_{a} t_{a}\right) \underset{\gamma \rightarrow 0}{\propto} \int \mathcal{D} x \mathcal{D} \lambda \mathcal{D} c \mathcal{D} \bar{c} e^{-\int_{t_{a}}^{t_{b}} d t \delta \mathcal{A} / \delta x(t) \lambda(t)-\int_{t_{a}}^{t_{b}} d t \int_{t_{a}}^{t_{b}} d t^{\prime} \bar{c}(t) \delta^{2} \mathcal{A} / \delta x(t) \delta x\left(t^{\prime}\right) c\left(t^{\prime}\right)} $$

$$ \delta x=\bar{\varepsilon} c, \quad \delta c=0, \quad \delta \bar{c}=-\bar{\varepsilon} \lambda, \quad \delta \lambda=0, $$

$$ X(t) \equiv x(t)+i \bar{\theta} c(t)-i \bar{\theta} c(t)-\bar{\theta} \lambda(t) $$

$$ \mathcal{A}^{\text {super }} \equiv \int d \bar{\theta} d \theta \mathcal{A}[X] \equiv \int d \bar{\theta} d \theta \mathcal{A}[x+i \bar{\theta} c-i \theta \bar{c}-\bar{\theta} \theta \lambda] $$

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

$$ H_{T}=\frac{1}{M} p_{y} p_{x}+\gamma p_{y} y+V(x+y / 2)-V(x-y / 2)-i \frac{w}{2 \hbar} y \hat{K}^{\mathrm{Ohm}} y $$

$$ U\left(x_{b} y_{b} t_{b} \mid x_{a} y_{a} t_{a}\right) \equiv\left(x_{b}+y_{b} / 2 t_{b} \mid x_{a}+y_{a} / 2 t_{a}\right)\left(x_{b}-y_{b} / 2 t_{b} \mid x_{a}-y_{a} / 2 t_{a}\right)^{*} . $$

$$ \rho(x+y / 2, x-y / 2 ; t)=\int d x_{a} d y_{a} U\left(x y t \mid x_{a} y_{a} t_{a}\right) \rho\left(x_{a}+y_{a} / 2, x_{a}-y_{a} / 2 ; t_{a}\right) . $$

$$ \left\langle\eta(t) \eta\left(t^{\prime}\right)\right\rangle_{T}=\frac{w}{2}\left[K^{\mathrm{Ohm}}\right]^{-1}\left(t, t^{\prime}\right) . $$

$$ \hat{H}_{\eta} \equiv \frac{1}{M}\left(\hat{p}_{x}+\gamma y\right) \hat{p}_{y}+V(x+y / 2)-V(x-y / 2)-y \eta, $$

$$ i \hbar \partial_{t} U_{\eta}\left(x y t \mid x_{a} y_{a} t_{a}\right)=\hat{H}_{\eta} U_{\eta}\left(x y t \mid x_{a} y_{a} t_{a}\right) . $$

$$ \left|\left(x_{b} t_{b} \mid x_{a} t_{a}\right)\right|^{2}=U\left(x_{b} 0 t_{b} \mid x_{a} 0 t_{a}\right) \equiv\left\langle U\left(x_{b} 0 t_{b} \mid x_{a} y_{a} t_{a}\right)\right\rangle_{\eta} . $$

$$ i \hbar \partial_{t} U\left(x y t \mid x_{a} y_{a} t_{a}\right)=\hat{\bar{H}}_{T} U\left(x y t \mid x_{a} y_{a} t_{a}\right), $$

$$ \hat{\bar{H}}_{T} \equiv \frac{1}{M} \hat{p}_{y} \hat{p}_{x}+\gamma y \hat{p}_{y}+V(x+y / 2)-V(x-y / 2)-i \frac{w}{2 \hbar} y^{2}, $$

$$ \hat{\bar{H}}_{T} \equiv \frac{1}{2 M}\left(\hat{p}_{+}^{2}-\hat{p}_{-}^{2}\right)+V\left(x_{+}\right)-V\left(x_{-}\right)+\frac{\gamma}{2}\left(x_{+}-x_{-}\right)\left(\hat{p}_{+}-\hat{p}_{-}\right)-i \frac{w}{2 \hbar}\left(x_{+}-x_{-}\right)^{2} . $$

$$ \Lambda \equiv \frac{w}{2 \hbar^{2}}=\frac{M \gamma k_{B} T}{\hbar^{2}} $$

$$ \Lambda=\frac{2 \pi \gamma}{l_{\mathrm{e}}^{2}(\hbar \beta)} $$

$$ i \hbar \partial_{t} \rho\left(x_{+}, x_{-} ; t_{a}\right)=\hat{\bar{H}}_{T} \rho\left(x_{+}, x_{-} ; t_{a}\right) $$

$$ \begin{align*} \hat{\bar{H}}_{T} \equiv \frac{1}{2 M}\left(\hat{\mathbf{p}}_{+}^{2}\right. & \left.-\hat{\mathbf{p}}_{-}^{2}\right)+V\left(\mathbf{x}_{+}\right)-V\left(\mathbf{x}_{-}\right)+\frac{M \gamma}{2}\left(\hat{\mathbf{x}}_{+}-\hat{\mathbf{x}}_{-}\right)\left(\hat{\dot{\mathbf{x}}}_{+}+\hat{\dot{\mathbf{x}}}_{-}\right)^{R} \\ & -i \frac{w}{2 \hbar}\left(\hat{\mathbf{x}}_{+}-\hat{\mathbf{x}}_{-}\right) \hat{K}^{\mathrm{Ohm}}\left(\hat{\mathbf{x}}_{+}-\hat{\mathbf{x}}_{-}\right) \end{align*} $$

$$ K^{\mathrm{Ohm}}\left(\omega^{\prime}\right)=1+\frac{1}{3}\left(\frac{\hbar \omega^{\prime}}{2 k_{B} T}\right)^{2}+\ldots $$

$$ -i\left(\hat{\mathbf{x}}_{+}-\hat{\mathbf{x}}_{-}\right) \hat{K}^{\mathrm{Ohm}}\left(\hat{\mathbf{x}}_{+}-\hat{\mathbf{x}}_{-}\right)=-i\left(\hat{\mathbf{x}}_{+}-\hat{\mathbf{x}}_{-}\right)^{2}+i \frac{w \hbar}{24\left(k_{B} T\right)^{2}}\left(\hat{\dot{\mathbf{x}}}_{+}-\hat{\dot{\mathbf{x}}}_{-}\right)^{2}+\ldots $$

$$ \hat{\dot{\mathbf{x}}} \equiv \frac{i}{\hbar}\left[\hat{\bar{H}}_{T}, \hat{\mathbf{x}}\right] $$

$$ \hat{\dot{\mathbf{x}}} \equiv \frac{i}{\hbar}\left[\hat{\bar{H}}_{T}, \hat{\dot{\mathbf{x}}}\right], \quad \hat{\ddot{\mathbf{x}}} \equiv \frac{i}{\hbar}\left[\hat{\bar{H}}_{T}, \hat{\dot{\mathbf{x}}}\right], \ldots . $$

$$ \begin{align*} i \hbar \partial_{t} \hat{\rho}=\hat{\bar{H}}_{T} \hat{\rho} & \equiv[\hat{H}, \hat{\rho}]+\frac{M \gamma}{2}(\hat{\mathbf{x}} \hat{\dot{\mathbf{x}}} \hat{\rho}-\hat{\rho} \hat{\dot{\mathbf{x}}} \hat{\mathbf{x}}+\hat{\mathbf{x}} \hat{\rho} \hat{\dot{\mathbf{x}}}-\hat{\dot{\mathbf{x}}} \hat{\rho} \hat{\mathbf{x}}) \\ & -\frac{i w}{2 \hbar}[\hat{\mathbf{x}},[\hat{\mathbf{x}}, \hat{\rho}]]-\frac{i w \hbar^{2}}{24\left(k_{B} T\right)^{2}}[\hat{\dot{\mathbf{x}}},[\hat{\dot{\mathbf{x}}}, \hat{\rho}]]+\ldots \end{align*} $$

$$ \begin{align*} & U\left(\mathbf{x}_{+b}, \mathbf{x}_{-b}, t_{b} \mid \mathbf{x}_{+a}, \mathbf{x}_{-a}, t_{b}-\epsilon\right) \\ & \quad=\int \frac{d \mathbf{p}_{+}\left(t_{b}\right)}{(2 \pi)^{3}} \int \frac{d \mathbf{p}_{-}\left(t_{b}\right)}{(2 \pi)^{3}} e^{\frac{i}{\hbar}\left\{\mathbf{p}_{+}\left(t_{b}\right)\left[\mathbf{x}_{+}\left(t_{b}\right)-\mathbf{x}_{+}\left(t_{b}-\epsilon\right)\right]-\mathbf{p}_{-} \dot{\mathbf{x}}_{-}-\bar{H}_{T}\left(t_{b}\right)\right\}} \end{align*} $$

$$ \epsilon^{-1}\left[F_{+}\left(\mathbf{x}_{+}\left(t_{b}\right)\right) U-U F_{+}\left(\mathbf{x}_{+}\left(t_{b}-\epsilon\right)\right)\right] F_{-}\left(\mathbf{x}_{-}\left(t_{b}\right)\right) $$

$$ \frac{i}{\hbar}\left[\hat{\bar{H}}_{T}, \hat{F}_{+}\left(\mathbf{x}_{+}\right)\right] F_{-}\left(\mathbf{x}_{-}\right) $$

$$ \partial_{t} \hat{\rho}=-\frac{i}{\hbar}[\hat{H}, \hat{\rho}]-\sum_{n=1}^{2}\left(\frac{1}{2} \hat{L}_{n} \hat{L}_{n}^{\dagger} \hat{\rho}+\frac{1}{2} \hat{\rho} \hat{L}_{n} \hat{L}_{n}^{\dagger}-\hat{L}_{n}^{\dagger} \hat{\rho} \hat{L}_{n}\right), $$

$$ \hat{L}_{1} \equiv \frac{\sqrt{w}}{2 \hbar} \hat{\mathbf{x}}, \quad \hat{L}_{2} \equiv \frac{\sqrt{3 w}}{2 \hbar}\left(\hat{\mathbf{x}}-i \frac{\hbar}{3 k_{B} T} \hat{\dot{\mathbf{x}}}\right) $$

$$ \hat{H}_{\gamma}=\gamma M \frac{1}{4}[\hat{\mathbf{x}}, \hat{\dot{\mathbf{x}}}] $$

$$ M \ddot{\hat{x}}(t)+M \gamma \dot{\hat{x}}(t)+V^{\prime}(\hat{x}(t))=\hat{\eta}(t) $$

$$ \left[\hat{\eta}_{t}, \hat{\eta}_{t^{\prime}}\right]=w \frac{i \hbar}{k_{B} T} \partial_{t} \delta\left(t-t^{\prime}\right) $$

$$ \frac{1}{2}\left\langle\left[\hat{\eta}_{t}, \hat{\eta}_{t^{\prime}}\right]_{+}\right\rangle_{\hat{\eta}}=w K\left(t, t^{\prime}\right) $$

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

$$ \hat{\eta}(t)=-i \sqrt{\frac{M \hbar \gamma}{\pi}} \int_{0}^{\infty} d \Omega^{\prime} \sqrt{\Omega^{\prime}}\left[a_{\Omega^{\prime}} e^{-i \omega^{\prime} t}-a_{\omega^{\prime}}^{\dagger} e^{i \omega^{\prime} t}\right] $$

$$ \rho\left(\mathbf{x}_{+b}, \mathbf{x}_{-a} ; t_{b}\right)=\int d \mathbf{x}_{+a} d \mathbf{x}_{-a} U\left(\mathbf{x}_{+b}, \mathbf{x}_{-b}, t_{b} \mid \mathbf{x}_{+a}, \mathbf{x}_{-a}, t_{a}\right) \rho\left(\mathbf{x}_{+a}, \mathbf{x}_{-a} ; t_{a}\right) $$

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

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

$$ G_{\mathbf{k k}}^{i j}\left(t, t^{\prime}\right)=\left\langle\hat{T} \hat{X}_{\mathbf{k}}^{i}(t), \hat{X}_{-\mathbf{k}^{\prime}}^{j}\left(t^{\prime}\right)\right\rangle=\delta_{\mathbf{k k}^{\prime}}^{i j \operatorname{tr}} G_{\Omega_{\mathbf{k}}}\left(t, t^{\prime}\right) \equiv \delta_{\mathbf{k k}^{\prime}} P_{\mathbf{k}}^{\perp i j} G_{\Omega_{\mathbf{k}}}\left(t, t^{\prime}\right) $$

$$ P_{\mathbf{k}}^{\perp i j}=\sum_{h= \pm} \epsilon^{i}(\mathbf{k}, h) \epsilon^{j *}(\mathbf{k}, h)=\left(\delta^{i j}-k^{i} k^{j} / \mathbf{k}^{2}\right) $$

$$ \begin{align*} & U\left(\mathbf{x}_{+b}, \mathbf{x}_{-b}, t_{b} \mid \mathbf{x}_{+a}, \mathbf{x}_{-a}, t_{a}\right)=\int \mathcal{D} \mathbf{x}_{+}(t) \int \mathcal{D} \mathbf{x}_{-}(t) \\ & \quad \times \exp \left\{\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t\left[\frac{M}{2}\left(\dot{\mathbf{x}}_{+}^{2}-\dot{\mathbf{x}}_{-}^{2}\right)-\left(V\left(\mathbf{x}_{+}\right)-V\left(\mathbf{x}_{-}\right)\right)\right]+\frac{i}{\hbar} \mathcal{A}^{\mathrm{FV}}\left[\mathbf{x}_{+}, \mathbf{x}_{-}\right]\right\} \end{align*} $$

$$ \begin{align*} \mathcal{A}_{D}^{\mathrm{FV}}\left[\mathbf{x}_{+}, \mathbf{x}_{-}\right]= & \frac{i e^{2}}{2 \hbar c^{2}} \int d t \int d t^{\prime} \Theta\left(t-t^{\prime}\right) \\ \times & {\left[\dot{\mathbf{x}}_{+}(t) \mathbf{C}_{\mathrm{b}}\left(\mathbf{x}_{+} t, \mathbf{x}_{+}^{\prime} t^{\prime}\right) \dot{\mathbf{x}}_{+}\left(t^{\prime}\right)-\dot{\mathbf{x}}_{+}(t) \mathbf{C}_{\mathrm{b}}\left(\mathbf{x}_{+} t, \mathbf{x}_{-}^{\prime} t^{\prime}\right) \dot{\mathbf{x}}_{-}\left(t^{\prime}\right)\right.} \\ & \left.-\dot{\mathbf{x}}_{-}(t) \mathbf{C}_{\mathrm{b}}\left(\mathbf{x}_{-} t, \mathbf{x}_{+}^{\prime} t^{\prime}\right) \dot{\mathbf{x}}_{+}\left(t^{\prime}\right)+\dot{\mathbf{x}}_{-}(t) \mathbf{C}_{\mathrm{b}}\left(\mathbf{x}_{-} t, \mathbf{x}_{-}^{\prime} t^{\prime}\right) \dot{\mathbf{x}}_{-}\left(t^{\prime}\right)\right] \end{align*} $$

$$ \begin{align*} \mathbf{A}_{F}^{\mathrm{FV}}\left[\mathbf{x}_{+}, \mathbf{x}_{-}\right]= & \frac{i e^{2}}{2 \hbar c^{2}} \int d t \int d t^{\prime} \Theta\left(t-t^{\prime}\right) \\ \times & {\left[\dot{\mathbf{x}}_{+}(t) \mathbf{A}_{\mathrm{b}}\left(\mathbf{x}_{+} t, \mathbf{x}_{+}^{\prime} t^{\prime}\right) \dot{\mathbf{x}}_{+}\left(t^{\prime}\right)+\dot{\mathbf{x}}_{+}(t) \mathbf{A}_{\mathrm{b}}\left(\mathbf{x}_{+} t, \mathbf{x}_{-}^{\prime} t^{\prime}\right) \dot{\mathbf{x}}_{-}\left(t^{\prime}\right)\right.} \\ & \left.+\dot{\mathbf{x}}_{-}(t) \mathbf{A}_{\mathrm{b}}\left(\mathbf{x}_{-} t, \mathbf{x}_{+}^{\prime} t^{\prime}\right) \dot{\mathbf{x}}_{+}\left(t^{\prime}\right)+\dot{\mathbf{x}}_{-}(t) \mathbf{A}_{\mathrm{b}}\left(\mathbf{x}_{-} t, \mathbf{x}_{-}^{\prime} t^{\prime}\right) \dot{\mathbf{x}}_{-}\left(t^{\prime}\right)\right] \end{align*} $$

$$ \begin{align*} C_{\mathrm{b}}^{i j}\left(\mathbf{x} t, \mathbf{x}^{\prime} t^{\prime}\right) & =\sum_{\mathbf{k}} c_{-\mathbf{k}}(\mathbf{x}) c_{\mathbf{k}}\left(\mathbf{x}^{\prime}\right)\left\langle\left[\hat{X}_{-\mathbf{k}}^{i}(t), \hat{X}_{\mathbf{k}}^{j}\left(t^{\prime}\right)\right]\right\rangle_{T} \\ & =-i \hbar \int \frac{d \omega^{\prime} d^{3} k}{(2 \pi)^{4}} \rho_{\mathbf{k}}\left(\omega^{\prime}\right) P_{\mathbf{k}}^{\perp i j} e^{i \mathbf{k}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)} \sin \omega^{\prime}\left(t-t^{\prime}\right) \\ A_{\mathbf{b}}^{i j}\left(\mathbf{x} t, \mathbf{x}^{\prime} t^{\prime}\right) & =\sum_{\mathbf{k}} c_{-\mathbf{k}}(\mathbf{x}) c_{\mathbf{k}}\left(\mathbf{x}^{\prime}\right)\left\langle\left\{\hat{X}_{-\mathbf{k}}^{i}(t), \hat{X}_{\mathbf{k}}^{j}\left(t^{\prime}\right)\right\}\right\rangle_{T} \\ & =\hbar \int \frac{d \omega^{\prime} d^{3} k}{(2 \pi)^{4}} \rho_{\mathbf{k}}\left(\omega^{\prime}\right) P_{\mathbf{k}}^{\perp i j} \operatorname{coth} \frac{\hbar \omega^{\prime}}{2 k_{B} T} e^{i \mathbf{k}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)} \cos \omega^{\prime}\left(t-t^{\prime}\right) \end{align*} $$

$$ \rho_{\mathbf{k}}\left(\omega^{\prime}\right) \equiv \frac{2 \pi}{2 \Omega_{\mathbf{k}}}\left[\delta\left(\omega^{\prime}-\Omega_{\mathbf{k}}\right)-\delta\left(\omega^{\prime}+\Omega_{\mathbf{k}}\right)\right] $$

$$ \begin{align*} G\left(x, x^{\prime}\right) & =\frac{1}{2}\left[A\left(x, x^{\prime}\right)+C\left(x, x^{\prime}\right)\right]=\int \frac{d^{4} k}{(2 \pi)^{4}} e^{i k\left(x-x^{\prime}\right)} \frac{i \hbar}{k^{2}+i \eta} \\ & =\int \frac{d \omega d^{3} k}{(2 \pi)^{4}} \frac{i c \hbar}{\omega^{2}-\Omega_{\mathbf{k}}^{2}+i \eta} e^{-i\left[\omega\left(t-t^{\prime}\right)-\mathbf{k}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)\right]} \end{align*} $$

$$ C_{\mathrm{b}}^{i j}\left(\mathbf{x} t, \mathbf{x}^{\prime} t^{\prime}\right) \approx C_{\mathrm{b}}^{i j}\left(t, t^{\prime}\right)=i \frac{\hbar}{2 \pi c} \frac{2}{3} \delta^{i j} \partial_{t} \delta\left(t-t^{\prime}\right) $$

$$ \Delta \mathcal{A}_{\mathrm{loc}}\left[\mathbf{x}_{+}, \mathbf{x}_{-}\right]=\frac{\Delta M}{2} \int_{t_{a}}^{t_{b}} d t\left(\dot{\mathbf{x}}_{+}^{2}-\dot{\mathbf{x}}_{-}^{2}\right)(t) $$

$$ \Delta M \equiv-\frac{e^{2}}{c^{2}} \int \frac{d \omega^{\prime} d^{3} k}{(2 \pi)^{4}} \frac{\sigma_{\mathbf{k}}\left(\omega^{\prime}\right)}{\omega^{\prime}} \delta_{\mathbf{k k}}^{i j \operatorname{tr}}=-\frac{e^{2}}{3 \pi^{2} c^{3}} \int_{0}^{\infty} d k $$

$$ \frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t \frac{M_{\mathrm{ren}}}{2}\left(\dot{\mathrm{x}}_{+}^{2}-\dot{\mathrm{x}}_{-}^{2}\right) $$

$$ \mathcal{A}_{D}^{\mathrm{FV}}\left[\mathbf{x}_{+}, \mathbf{x}_{-}\right]=-\gamma \frac{M}{2} \int_{t_{a}}^{t_{b}} d t\left(\dot{\mathbf{x}}_{+}-\dot{\mathbf{x}}_{-}\right)(t)\left(\ddot{\mathbf{x}}_{+}+\ddot{\mathbf{x}}_{-}\right)^{R}(t) $$

$$ \gamma \equiv \frac{e^{2}}{6 \pi c^{3} M}=\frac{2}{3} \frac{\alpha}{\omega_{M}}, $$

$$ \frac{e^{2}}{c^{2}} A_{\mathrm{b}}\left(\mathbf{x} t, \mathbf{x}^{\prime} t^{\prime}\right) \approx 2 \gamma k_{B} T K^{\mathrm{Ohm}}\left(t, t^{\prime}\right) $$

$$ \mathcal{A}_{F}^{\mathrm{FV}}\left[\mathbf{x}_{+}, \mathbf{x}_{-}\right]=i \frac{w}{2 \hbar} \int_{t_{a}}^{t_{b}} d t \int_{t_{a}}^{t_{b}} d t^{\prime}\left(\dot{\mathbf{x}}_{+}-\dot{\mathbf{x}}_{-}\right)(t) K^{\mathrm{Ohm}}\left(t, t^{\prime}\right)\left(\dot{\mathbf{x}}_{+}-\dot{\mathbf{x}}_{-}\right)\left(t^{\prime}\right) $$

$$ w \equiv 2 M k_{B} T \gamma $$

$$ \begin{align*} & U\left(\mathbf{x}_{+b}, \mathbf{x}_{-b}, t_{b} \mid \mathbf{x}_{+a}, \mathbf{x}_{-a}, t_{a}\right)=\int \mathcal{D} \mathbf{x}_{+}(t) \int \mathcal{D} \mathbf{x}_{-}(t) \\ & \quad \times \exp \left\{\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t\left[\frac{M}{2}\left(\dot{\mathbf{x}}_{+}^{2}-\dot{\mathbf{x}}_{-}^{2}\right)-\left(V\left(\mathbf{x}_{+}\right)-V\left(\mathbf{x}_{-}\right)\right)\right]\right\} \\ & \quad \times \exp \left\{-\frac{i}{2 \hbar} M \gamma \int_{t_{a}}^{t_{b}} d t\left(\dot{\mathbf{x}}_{+}-\dot{\mathbf{x}}_{-}\right)\left(\ddot{\mathbf{x}}_{+}+\ddot{\mathbf{x}}_{-}\right)^{R}-\frac{w}{2 \hbar^{2}} \int_{t_{a}}^{t_{b}} d t\left(\dot{\mathbf{x}}_{+}-\dot{\mathbf{x}}_{-}\right)^{2}\right\} \end{align*} $$

$$ -\frac{w}{24\left(k_{B} T\right)^{2}} \int_{t_{a}}^{t_{b}} d t\left(\ddot{\mathbf{x}}_{+}-\ddot{\mathbf{x}}_{-}\right)^{2} $$

$$ \hat{\mathcal{H}} \equiv \frac{1}{2 M}\left(\hat{\mathbf{p}}_{+}^{2}-\hat{\mathbf{p}}_{-}^{2}\right)+V\left(\mathbf{x}_{+}\right)-V\left(\mathbf{x}_{-}\right)+\frac{M \gamma}{2}\left(\hat{\dot{\mathbf{x}}}_{+}-\hat{\dot{\mathbf{x}}}_{-}\right)\left(\hat{\dot{\mathbf{x}}}_{+}+\hat{\dot{\mathbf{x}}}_{-}\right)^{R}-i \frac{w}{2 \hbar}\left(\hat{\dot{\mathbf{x}}}_{+}-\hat{\dot{\mathbf{x}}}_{-}\right)^{2}, $$

$$ \Delta \hat{\bar{H}}_{T} \equiv i \frac{w \hbar}{24\left(k_{B} T\right)^{2}}\left(\hat{\overline{\mathbf{x}}}_{+}-\hat{\dot{\mathbf{x}}}_{-}\right)^{2} $$

$$ \begin{align*} i \hbar \partial_{t} \hat{\rho}=\hat{\bar{H}}_{T} \hat{\rho} & \equiv[\hat{H}, \hat{\rho}]+\frac{M \gamma}{2}(\hat{\dot{\mathbf{x}}} \hat{\dot{\mathbf{x}}} \hat{\rho}-\hat{\rho} \hat{\dot{\mathbf{x}}} \hat{\dot{\mathbf{x}}}+\hat{\dot{\mathbf{x}}} \hat{\rho} \hat{\ddot{\mathbf{x}}}-\hat{\dot{\mathbf{x}}} \hat{\rho} \hat{\dot{\mathbf{x}}}) \\ & -\frac{i w}{2 \hbar}[\hat{\dot{\mathbf{x}}},[\hat{\dot{\mathbf{x}}}, \hat{\rho}]]-\frac{i w \hbar^{2}}{24\left(k_{B} T\right)^{2}}[\hat{\ddot{\mathbf{x}}},[\hat{\ddot{\mathbf{x}}}, \hat{\rho}]] \end{align*} $$

$$ \partial_{t} \hat{\rho}=-\frac{i}{\hbar}[\hat{H}, \hat{\rho}]-\sum_{n=1}^{2}\left(\frac{1}{2} \hat{L}_{n} \hat{L}_{n}^{\dagger} \hat{\rho}+\frac{1}{2} \hat{\rho} \hat{L}_{n} \hat{L}_{n}^{\dagger}-\hat{L}_{n}^{\dagger} \hat{\rho} \hat{L}_{n}\right), $$

$$ \hat{L}_{1} \equiv \frac{\sqrt{w}}{2 \hbar} \hat{\dot{\mathbf{x}}}, \quad \hat{L}_{2} \equiv \frac{\sqrt{3 w}}{2 \hbar}\left(\hat{\dot{\mathbf{x}}}-i \frac{\hbar}{3 k_{B} T} \hat{\dot{\mathbf{x}}}\right) . $$

$$ \hat{H}_{\gamma}=\gamma M \frac{1}{4}[\hat{\dot{\mathbf{x}}}, \hat{\dot{\mathbf{x}}}] $$

$$ i \hbar \partial_{t} \hat{\rho}=[\hat{H}, \hat{\rho}]-\frac{i w}{2 M^{2} \hbar}[\hat{\mathbf{p}},[\hat{\mathbf{p}}, \hat{\rho}]] $$

$$ \begin{align*} \partial_{t}\langle i| \hat{\rho}(t)|i\rangle & =-\frac{\gamma}{\hbar M}\langle i|[\hat{H}, \hat{\mathbf{p}}] \hat{\mathbf{p}} \hat{\rho}(0)|i\rangle=\frac{\gamma}{M} \sum_{f \neq i} \omega_{i f}\langle i| \mathbf{p}|f\rangle\langle f| \mathbf{p}|i\rangle \\ & =-M \gamma \sum_{f} \omega_{i f}^{3}\left|\mathbf{x}_{f i}\right|^{2} \end{align*} $$

$$ \begin{align*} \partial_{t}\langle i| \hat{\rho}(t)|i\rangle & =-\frac{w}{M^{2} \hbar^{2}}\langle i| \mathbf{p}^{2}|i\rangle-\frac{w}{12 M^{2}\left(k_{B} T\right)^{2}}\langle i| \dot{\mathbf{p}}^{2}|i\rangle \\ & =-w \sum_{n} \omega_{i f}^{2}\left[1+\frac{\hbar^{2} \omega_{i f}^{2}}{12\left(k_{B} T\right)^{2}}\right]\left|\mathbf{x}_{f i}\right|^{2} \end{align*} $$

$$ \begin{align*} & C_{b}\left(t, t^{\prime}\right)+A_{b}\left(t, t^{\prime}\right) \\ & =\frac{4 \pi}{3} \hbar \int \frac{d \omega^{\prime} d^{3} k}{(2 \pi)^{4}} \frac{\pi}{2 M \Omega_{\mathbf{k}}}\left\{1+\operatorname{coth} \frac{\hbar \omega^{\prime}}{2 k_{B} T}\right\}\left[\delta\left(\omega^{\prime}-\Omega_{\mathbf{k}}\right)-\delta\left(\omega^{\prime}+\Omega_{\mathbf{k}}\right)\right] e^{-i \omega^{\prime}\left(t-t^{\prime}\right)} \end{align*} $$

$$ \begin{align*} & C_{b}\left(t, t^{\prime}\right)+A_{b}\left(t, t^{\prime}\right) \\ & =\frac{4 \pi}{3} \hbar \int \frac{d \omega^{\prime} d^{3} k}{(2 \pi)^{4}} \frac{\pi}{2 M \Omega_{\mathbf{k}}}\left\{2 \delta\left(\omega^{\prime}-\Omega_{\mathbf{k}}\right)+\frac{2}{e^{\hbar \Omega_{\mathbf{k}} / k_{B} T}-1}\left[\delta\left(\omega^{\prime}-\Omega_{\mathbf{k}}\right)+\delta\left(\omega^{\prime}+\Omega_{\mathbf{k}}\right)\right]\right\} e^{-i \omega^{\prime}\left(t-t^{\prime}\right)} \end{align*} $$

$$ \begin{align*} & \frac{4 \pi}{3} \hbar \int \frac{d \omega^{\prime} d^{3} k}{(2 \pi)^{4}} \frac{\pi}{2 M \Omega_{\mathbf{k}}}\left\{2 \delta\left(\omega^{\prime}-\Omega_{\mathbf{k}}\right)\right. \\ & \left.\quad+\left(\frac{2 k_{B} T}{\hbar \Omega_{\mathbf{k}}}-1+\frac{1}{6} \frac{\hbar \Omega_{\mathbf{k}}}{k_{B} T}\right)\left[\delta\left(\omega^{\prime}-\Omega_{\mathbf{k}}\right)+\delta\left(\omega^{\prime}+\Omega_{\mathbf{k}}\right)\right]\right\} e^{-i \omega^{\prime}\left(t-t^{\prime}\right)} \end{align*} $$

$$ \Gamma=2 M \gamma \sum_{f