Kleinert · 제7장 구속 조건

$$ \left(\frac{\partial C_{>}}{\partial T}\right)=\left(\frac{\partial C_{<}}{\partial T}\right)-\Delta\left(\frac{\partial C}{\partial T}\right) \approx-0.7715 \frac{N k_{B}}{T_{c}}, \quad T>T_{c} $$

$$ N=\frac{V_{D}}{l_{\mathrm{e}}^{D}(\hbar \beta)} \zeta_{D / 2}^{\mathrm{box}}(z)=\sum_{\mathbf{p}_{\mathbf{n}}} \frac{1}{e^{\beta \mathbf{p}_{\mathbf{n}}^{2} / 2 M-\beta \mu}-1} $$

$$ Z_{1}(b) \equiv \sum_{n=1}^{\infty} e^{-b n^{2} / 2}, \quad b \equiv \beta \hbar^{2} \pi^{2} / M L^{2}=\pi l_{\mathrm{e}}^{2}(\hbar \beta) / 2 L^{2} $$

$$ N=\frac{V_{D}}{l_{\mathrm{e}}^{D}(\hbar \beta)} \zeta_{D / 2}^{\mathrm{box}}(z)=\sum_{w} Z_{1}^{D}(w b) z^{w} $$

$$ \vartheta_{3}(u, z) \equiv 1+2 \sum_{n=1}^{\infty} z^{n^{2}} \cos 2 n u $$

$$ \begin{align*} \vartheta_{3}\left(0, e^{-b / 2}\right) & =\sum_{k=-\infty}^{\infty} e^{-k^{2} b / 2}=\sum_{m=-\infty}^{\infty} \int_{-\infty}^{\infty} d k e^{-k^{2} b / 2+2 \pi i k m} \\ & =\sqrt{\frac{2 \pi}{b}}\left(1+2 \sum_{m=1}^{\infty} e^{-2 \pi^{2} m^{2} / b}\right) \end{align*} $$

$$ Z_{1}(b)=\sqrt{\frac{\pi}{2 b}}-\frac{1}{2}+\mathcal{O}\left(e^{-2 \pi^{2} / b}\right) $$

$$ N_{\mathrm{cond}}(T)=\frac{1}{e^{D b / 2-\beta \mu}-1}=\frac{z_{D}}{1-z_{D}}, \quad z_{D} \equiv e^{\beta \mu-D b / 2} $$

$$ N_{\mathrm{n}}(T)=\frac{V_{D}}{l_{\mathrm{e}}^{D}(\hbar \beta)} \bar{\zeta}_{D / 2}^{\mathrm{box}}(z(\beta)) $$

$$ N_{\mathrm{n}}(T)=S_{D}\left(z_{D}\right) \equiv \sum_{w=1}^{\infty}\left[Z_{1}^{D}(w b) e^{w D b / 2}-1\right] z_{D}^{w} $$

$$ N={\sqrt{\frac{\pi}{2 b_{c}}}}^{3} \zeta(3 / 2)+\frac{3 \pi}{4 b_{c}^{(1)}} \log C_{3} b_{c}+\ldots $$

$$ N={\sqrt{\frac{\pi}{2 b_{c}^{(0)}}}}^{3} \zeta(3 / 2) $$

$$ 1 \equiv\left(\frac{T_{c}^{(1)}}{T_{c}^{(0)}}\right)^{3 / 2}+\frac{3}{2 N} \frac{\pi}{2 b_{c}^{(0)}} \log C_{3} b_{c}^{(0)} $$

$$ \frac{\delta T_{c}^{(1)}}{T_{c}^{(0)}} \approx \frac{1}{\zeta^{2 / 3}(3 / 2) N^{1 / 3}} \log \frac{2}{\pi C_{3}} \frac{N^{2 / 3}}{\zeta^{2 / 3}(3 / 2)} $$

$$ \tilde{V}(\mathbf{k})=g\left[1-r_{\mathrm{eff}}^{2} \mathbf{k}^{2} / 6+\ldots\right] $$

$$ \frac{\Delta T_{c}}{T_{c}^{(0)}}=-\frac{1}{3 \hbar^{2}} M r_{\mathrm{eff}}^{2} g \frac{N}{V} $$

$$ g=\frac{2 \pi \hbar^{2}}{M / 2} a $$

$$ g=\frac{\pi^{D / 2-1}}{2^{2-D} \Gamma(1-D / 2)\left(n a^{D}\right)^{D-2}+\Gamma(D / 2-1)} \frac{2 \pi \hbar^{2}}{M / 2} a^{D-2} \equiv \gamma\left(D, n a^{D}\right) \frac{2 \pi \hbar^{2}}{M / 2} a^{D-2} $$

$$ g=-\frac{2 \pi \hbar^{2} / M}{\ln \left(e^{\gamma} n a^{2} / 2\right)} $$

$$ V\left(\mathbf{x}-\mathbf{x}^{\prime}\right)=g \delta^{(3)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right) $$

$$ \Delta E=g \int d^{D} x n^{2}(\mathbf{x}) $$

$$ F_{G}=-\frac{1}{\beta} \frac{V_{D}}{l_{\mathrm{e}}^{D}(\hbar \beta)} \zeta_{D / 2+1}(z)+g V_{D}\left[\frac{\zeta_{D / 2}(z)}{l_{\mathrm{e}}^{D}(\hbar \beta)}\right]^{2}+\mathcal{O}\left(g^{2}\right) $$

$$ g \equiv \frac{2}{\beta} \frac{\alpha}{l_{\mathrm{e}}(\hbar \beta)} l_{\mathrm{e}}^{D}(\hbar \beta) $$

$$ \alpha=\gamma\left(D, n a^{D}\right)\left[\frac{a}{l_{\mathrm{e}}(\hbar \beta)}\right]^{D-3} a $$

$$ \alpha=-\frac{l_{\mathrm{e}}(\hbar \beta)}{\ln \left(e^{\gamma} n a^{2} / 2\right)} $$

$$ F_{G}=-\frac{1}{\beta} \frac{V_{D}}{l_{\mathrm{e}}^{D}(\hbar \beta)}\left\{\zeta_{D / 2+1}(z)-2 \hat{\alpha}\left[\zeta_{D / 2}(z)\right]^{2}\right\}+\mathcal{O}\left(\alpha^{2}\right) $$

$$ \Delta F_{G}=-\frac{1}{\beta} \frac{V_{D}}{l_{\mathrm{e}}^{D}(\hbar \beta)}\left\{8 \hat{\alpha}^{2} h_{D}(z)\right\} $$

$$ h_{D}^{(1)}(z) \equiv\left[\zeta_{D / 2}(z)\right]^{2} \zeta_{D / 2-1}(z) $$

$$ h_{3}^{(2)}(z) \equiv \sum_{n_{1}, n_{2}, n_{3}=1}^{\infty} \frac{z^{n_{1}+n_{2}+n_{3}}}{\sqrt{n_{1} n_{2} n_{3}}\left(n_{1}+n_{2}\right)\left(n_{1}+n_{3}\right)} $$

$$ N=\frac{V_{D}}{l_{\mathrm{e}}^{D}(\hbar \beta)}\left\{\zeta_{D / 2}(z)-4 \hat{\alpha} \zeta_{D / 2-1}(z) \zeta_{D / 2}(z)+8 \hat{\alpha}^{2} z \frac{d h_{D}(z)}{d z}\right\}+\mathcal{O}\left(\alpha^{3}\right) $$

$$ \frac{\Delta T_{c}}{T_{c}^{(0)}} \approx-\frac{2}{D} \frac{\Delta n}{n} $$

$$ \frac{\Delta T_{c}}{T_{c}^{(0)}}=c[\zeta(3 / 2)]^{\kappa / 3}\left[\frac{a}{\ell_{c}}\right]^{\kappa}=c a^{\kappa}\left(\frac{N}{V}\right)^{\kappa / 3} $$

$$ F_{G}^{\sigma}=-\frac{1}{\beta} \frac{V_{D}}{l_{\mathrm{e}}^{D}(\hbar \beta)}\left[\zeta_{D / 2+1}\left(z e^{\sigma}\right)-\frac{\sigma^{2}}{8 \hat{\alpha}}\right] $$

$$ \Sigma \equiv-4 \hat{\alpha} \zeta_{D / 2}\left(z e^{\Sigma}\right) $$

$$ F_{G}^{\Sigma}=-\frac{1}{\beta} \frac{V_{D}}{l_{\mathrm{e}}^{D}(\hbar \beta)}\left\{\zeta_{D / 2+1}(Z)-2 \hat{\alpha}\left[\zeta_{D / 2}(Z)\right]^{2}\right\}, \quad Z \equiv z e^{\Sigma} $$

$$ N=\frac{V_{D}}{l_{\mathrm{e}}^{D}(\hbar \beta)} \zeta_{D / 2}(Z) $$

$$ \epsilon(\mathbf{p})=\sqrt{\epsilon^{2}(\mathbf{p})+2 \operatorname{gn} \epsilon(\mathbf{p})}, $$

$$ f \equiv \epsilon(\mathbf{p})-\mathbf{v} \cdot \mathbf{p} $$

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

$$ \begin{align*} \left(\mathbf{x}_{b}^{(p(\nu))} \hbar \beta \mid \mathbf{x}_{a}^{(\nu)} 0\right)_{\omega} & =\frac{1}{\sqrt{2 \pi \hbar / M}} \sqrt{\frac{\omega}{\sinh \beta \hbar \omega}}^{D} \\ & \left.\times \exp \left\{-\frac{1}{2 \hbar} \frac{M \omega}{\sinh \beta \hbar \omega}\left[\mathbf{x}_{b}^{(p(\nu)) 2}+\mathbf{x}_{a}^{(\nu) 2}\right) \cosh \beta \hbar \omega-2 \mathbf{x}_{b}^{(p(\nu))} \mathbf{x}_{a}^{(\nu)}\right]\right\} \end{align*} $$

$$ \begin{align*} Z_{\omega}^{(N)} & =\sqrt{\frac{M \omega}{2 \pi \hbar \sinh \beta \hbar \omega}}^{N D} \frac{1}{N!} \int d^{D} x^{(1)} \cdots d^{D} x^{(N)} \\ & \times \sum_{p(\nu)} \exp \left\{-\frac{1}{2 \hbar} \frac{M \omega}{\sinh \beta \hbar \omega}\left[\left(\mathbf{x}^{(p(\nu)) 2}+\mathbf{x}^{(\nu) 2}\right) \cosh \beta \hbar \omega-2 \mathbf{x}^{(p(\nu))} \mathbf{x}^{(\nu)}\right]\right\} \end{align*} $$

$$ \Delta Z_{\omega}^{(N) w}=Z_{\omega}(w \beta) \frac{1}{w}=\int d^{D} x z_{\omega}(w \beta ; \mathbf{x}) \frac{1}{w} $$

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

$$ z_{\omega}(\beta ; \mathbf{x})=\sqrt{\frac{\omega M}{2 \pi \hbar \sinh \beta \hbar \omega}}^{D} \exp \left[-\frac{M \omega}{\hbar} \tanh \frac{\beta \hbar \omega}{2} \mathbf{x}^{2}\right] . $$

$$ F_{G}=-\frac{1}{\beta} \sum_{w=1}^{\infty} Z_{\omega}(w \beta) \frac{e^{w \beta \mu}}{w}=-\frac{1}{\beta} \sum_{w=1}^{\infty} \int d^{D} x z_{\omega}(w \beta ; \mathbf{x}) \frac{e^{w \beta \mu}}{w} $$

$$ N=-\frac{\partial}{\partial \mu} F_{G}=\sum_{w=1}^{\infty} Z_{\omega}(w \beta) e^{w \beta \mu}=\int d^{D} x \sum_{w=1}^{\infty} z_{\omega}(w \beta ; \mathbf{x}) e^{w \beta \mu} $$

$$ z_{D}(\beta)=e^{-(D \hbar \omega / 2-\mu) \beta}=e^{-D \beta \hbar \omega / 2} z $$

$$ N=-\frac{\partial}{\partial \mu} F_{G}=Z_{\omega}(\beta) \zeta_{D}\left(\beta \hbar \omega ; z_{D}\right) $$

$$ \zeta_{D}\left(\beta \hbar \omega ; z_{D}\right) \equiv \sum_{w=1}^{\infty}\left[\frac{\sinh (\omega \hbar \beta / 2)}{\sinh (w \omega \hbar \beta / 2)}\right]^{D} e^{w \beta \mu}=Z_{\omega}^{-1}(\beta) \sum_{w=1}^{\infty} \frac{1}{\left(1-e^{-2 w \beta \hbar \omega}\right)^{D / 2}} z_{D}^{w} $$

$$ \zeta_{D}\left(\beta \hbar \omega ; z_{D} ; \mathbf{x}\right) \equiv Z_{\omega}^{-1}(\beta) \sum_{w=1}^{\infty} \frac{1}{\left(1-e^{-2 w \beta \hbar \omega}\right)^{D / 2}}\left(\frac{\omega M}{\pi \hbar}\right)^{D / 2} e^{-M \omega \tanh (w \beta \hbar \omega / 2) \mathbf{x}^{2} / \hbar} z_{D}^{w},( $$

$$ N=-\frac{\partial}{\partial \mu} F_{G}=\int d^{D} x n_{\omega}(\mathbf{x}) \equiv Z_{\omega}(\beta) \int d^{D} x \zeta_{D}\left(\beta \hbar \omega ; z_{D} ; \mathbf{x}\right) $$

$$ F_{G}=\int d^{D} x f(\mathbf{x}) \equiv-\frac{1}{\beta} Z_{\omega}(\beta) \int d^{D} x \int_{0}^{z_{D}} \frac{d z}{z} \zeta_{D}(\beta \hbar \omega ; z ; \mathbf{x}) $$

$$ \zeta_{D}\left(\beta \hbar \omega ; z_{D} ; \mathbf{x}\right) \stackrel{\omega \approx 0}{\approx}\left(\frac{\beta \hbar \omega}{2 \pi}\right)^{D / 2} \frac{1}{\lambda_{\omega}^{D}} \sum_{w=1}^{\infty} \frac{1}{w^{D / 2}} e^{-w \beta\left(M \omega^{2} \mathbf{x}^{2} / 2-\mu\right)} $$

$$ n_{0}(\beta ; z ; \mathbf{x})=\frac{1}{l_{\mathrm{e}}^{D}(\hbar \beta)} \sum_{w=1}^{\infty} \frac{1}{w^{D / 2}} e^{-w \beta[V(\mathbf{x})-\mu]}=\frac{1}{l_{\mathrm{e}}^{D}(\hbar \beta)} \zeta_{D / 2}\left(e^{-\beta[V(\mathbf{x})-\mu]}\right) $$

$$ z(\mathbf{x}) \equiv e^{-\beta[V(\mathbf{x})-\mu]} $$

$$ N_{\mathrm{cond}}=\frac{1}{e^{D \beta \hbar \omega / 2-\beta \mu}-1} $$

$$ \bar{\zeta}_{D}\left(\beta \hbar \omega ; z_{D}\right) \equiv \zeta_{D}\left(\beta \hbar \omega ; z_{D}\right)-Z_{\omega}^{-1}(\beta) \frac{z_{D}}{1-z_{D}}=Z_{\omega}^{-1}(\beta) S_{D}\left(\beta \hbar \omega ; z_{D}\right) $$

$$ S\left(\beta \hbar \omega, z_{D}\right) \equiv \sum_{w=1}^{\infty}\left[\frac{1}{\left(1-e^{-w \beta \hbar \omega}\right)^{D}}-1\right] z_{D}^{w} $$

$$ N_{\mathrm{n}}(T)=Z_{\omega}(\beta) \bar{\zeta}_{D}\left(\beta \hbar \omega ; z_{D}\right) \equiv S_{D}\left(\beta \hbar \omega, z_{D}\right) $$

$$ N_{\mathrm{n}}=\frac{1}{(\beta \hbar \omega)^{D}} \zeta_{D}\left(z_{D}\right) $$

$$ N_{\mathrm{n}}=\frac{1}{(\beta \hbar \omega)^{D}} \zeta(D) $$

$$ k_{B} T_{c}^{(0)}=\hbar \omega\left[\frac{N}{\zeta(D)}\right]^{1 / D} $$

$$ \frac{N_{\mathrm{n}}^{(0)}}{N} \approx\left(\frac{T}{T_{c}^{(0)}}\right)^{D} $$

$$ \frac{N_{\mathrm{cond}}^{(0)}}{N} \approx 1-\left(\frac{T}{T_{c}^{(0)}}\right)^{D} $$

$$ \frac{N_{\mathrm{cond}}^{(0)}}{N} \approx 1-\left(\frac{T}{T_{c}^{(0)}+\delta T_{c}}\right)^{D} . $$

$$ n_{0}(\mathbf{0})=\frac{1}{l_{\mathrm{e}}^{D}\left(\hbar \beta_{c}\right)} \zeta(D / 2) $$

$$ k_{B} T_{c}^{(0)}=\frac{2 \pi \hbar^{2}}{M}\left[\frac{n_{0}(\mathbf{0})}{\zeta(D / 2)}\right]^{2 / D} $$

$$ \rho_{\mathrm{cl}}(E)=\left(\frac{M}{2 \pi \hbar^{2}}\right)^{D / 2} \frac{1}{\Gamma(D / 2)} \int d^{D} x[E-V(\mathbf{x})]^{D / 2-1} . $$

$$ N_{\mathrm{n}}=\int_{E_{\min }}^{\infty} d E \frac{\rho_{\mathrm{cl}}(E)}{e^{E / k_{B} T}-1} $$

$$ \begin{align*} N_{\mathrm{n}} & =\int d^{D} x \frac{d^{D} p}{(2 \pi \hbar)^{D}} \frac{1}{e^{\beta\left[p^{2} / 2 M+V(\mathbf{x})\right]}-1}=\sum_{n=1}^{\infty} \int d^{D} x \frac{d^{D} p}{(2 \pi \hbar)^{D}} e^{-n \beta\left[p^{2} / 2 M+V(\mathbf{x})\right]} \\ & =\sum_{n=1}^{\infty} \frac{1}{{\sqrt{2 \pi \hbar^{2} n \beta / M}}^{D}} \int d^{D} x e^{-n \beta V(\mathbf{x})} \end{align*} $$

$$ V(\mathbf{x})=\frac{M}{2} \tilde{\omega}^{2} \tilde{a}^{2} \sum_{i=1}^{D}\left(\frac{\left|x_{i}\right|}{a_{i}}\right)^{p_{i}} $$

$$ \prod_{i=1}^{D} \int_{-\infty}^{\infty} d x e^{-n \beta M \tilde{\omega}^{2} \tilde{a}^{2}\left(\left|x_{i}\right| / a_{i}\right)^{p_{i}} / 2}=\prod_{i=1}^{D} \frac{a_{i}}{\left(\beta M \tilde{\omega}^{2} \tilde{a}^{2} / 2\right)^{1 / p_{i}}} \Gamma\left(1+1 / p_{i}\right) $$

$$ k_{B} T_{c}^{(0)}=\frac{M \tilde{a}^{2} \tilde{\omega}^{2}}{2}\left(\frac{\hbar \tilde{\omega}}{M \tilde{a}^{2} \tilde{\omega}^{2}}\right)^{D / \tilde{D}}\left[\frac{N \pi^{D / 2}}{\zeta(\tilde{D}) \prod_{i=1}^{D} \Gamma\left(1+1 / p_{i}\right)}\right]^{1 / \tilde{D}} $$

$$ \tilde{D} \equiv \frac{D}{2}+\sum_{i=1}^{D} \frac{1}{p_{i}} $$

$$ k_{B} T_{c}^{(0)}=\frac{\pi \hbar^{2}}{2 M \tilde{a}^{2}}\left[\frac{N}{\zeta(D / 2)}\right]^{2 / D}=\frac{2 \pi \hbar^{2}}{M}\left[\frac{N}{V_{D} \zeta(D / 2)}\right]^{2 / D} $$

$$ k_{B} T_{c}^{(0)}=\hbar \tilde{\omega}\left(\frac{\pi \hbar}{2 M \tilde{\omega}}\right)^{1 / 5}\left[\frac{N}{a_{1} \zeta(5 / 2)}\right]^{2 / 5}=\hbar \tilde{\omega}\left(\frac{2 \pi \lambda_{\omega_{1}} \lambda_{\omega_{2}}}{L^{2}}\right)^{1 / 5}\left[\frac{N}{\zeta(5 / 2)}\right]^{2 / 5} . $$

$$ V(\mathbf{x})=\frac{M \omega_{z}^{2}}{2}\left(z^{2}+36 \eta r_{\perp}^{2}+\frac{\kappa}{2} \frac{r_{\perp}^{4}}{\lambda_{\omega_{z}}^{2}}\right)=\frac{\hbar \omega_{z}}{2}\left(\frac{z^{2}}{\lambda_{\omega_{z}}^{2}}+36 \eta \frac{r_{\perp}^{2}}{\lambda_{\omega_{z}}^{2}}+\frac{\kappa}{2} \frac{r_{\perp}^{4}}{\lambda_{\omega_{z}}^{4}}\right) $$

$$ k_{B} T_{c}^{(0)} \approx \hbar \omega_{z}\left(\frac{\kappa}{\pi}\right)^{1 / 5}\left[\frac{N}{\zeta(5 / 2)}\right]^{2 / 5} $$

$$ k_{B} T_{c}^{(0)}=\hbar \omega_{z}\left[\frac{\pi^{2} \kappa}{16 \Gamma^{4}(5 / 4)}\right]^{1 / 5}\left[\frac{N}{\zeta(5 / 2)}\right]^{2 / 5} $$

$$ \frac{2 \pi \int r d r d x d y e^{-r^{4}}}{\int d x d y e^{-x^{4}-y^{4}}}=\frac{\pi^{3 / 2}}{\Gamma[5 / 4]^{2}} $$

$$ k_{B} T_{c}^{(0)}=\hbar \omega_{z}\left(\frac{4 \kappa}{\pi}\right)^{1 / 5}\left[\frac{N}{\zeta(5 / 2)}\right]^{2 / 5} $$

$$ \begin{align*} N_{\mathrm{n}}\left(T_{c}\right) & =\frac{1}{(\beta \hbar \omega)^{D}}\left[\zeta(D)+\frac{\beta \hbar \omega}{2} D \zeta(D-1)+\frac{(\omega \hbar \beta)^{2}}{24} D(3 D-1) \zeta(D-2)+\ldots\right] \\ & -\zeta(0) \end{align*} $$

$$ \frac{N_{\mathrm{n}}}{N}=\left(\frac{T}{T_{c}^{(0)}}\right)^{D}+\frac{D}{2} \frac{\zeta(D-1)}{\zeta^{1-1 / D}(D)} \frac{1}{N^{1 / D}} $$

$$ \frac{\delta T_{c}}{T_{c}^{(0)}}=-\frac{1}{2} \frac{\zeta(D-1)}{\zeta^{(D-1) / D}(D)} \frac{1}{N^{1 / D}}+\ldots $$

$$ \begin{align*} & N_{\mathrm{n}}=\frac{1}{\beta \hbar \omega}\left[-(\log \beta \hbar \omega-\gamma)-\frac{\beta \hbar \omega}{2} \zeta(0)+\frac{(\beta \hbar \omega)^{2}}{12} \zeta(-1)+\ldots\right] \\ & N_{\mathrm{n}}=\frac{1}{(\beta \hbar \omega)^{2}}\left[\zeta(2)-\beta \hbar \omega\left(\log \beta \hbar \omega-\gamma+\frac{1}{2}\right)-\frac{7(\beta \hbar \omega)^{2}}{12} \zeta(0)+\ldots\right] \\ & N_{\mathrm{n}}=\frac{1}{(\beta \hbar \omega)^{3}}\left[\zeta(3)+\frac{3 \beta \hbar \omega}{2} \zeta(2)-(\beta \hbar \omega)^{2}\left(\log \beta \hbar \omega-\gamma+\frac{19}{24}\right)+\ldots\right] \end{align*} $$

$$ k_{B} T_{c}^{(0)}=\hbar \omega N \frac{1}{(-\log \beta \hbar \omega+\gamma)} \approx \hbar \omega N \frac{1}{\log N}, \quad D=1 $$

$$ F_{G}\left(\beta, \mu_{c}\right)=-\frac{1}{\beta(\beta \hbar \omega)^{D}}\left[\zeta(D+1)+\beta \hbar \omega \frac{D}{2} \zeta(D)+\ldots\right] $$

$$ S=-k_{B}(D+1) \beta F_{G}=k_{B}(D+1) \frac{1}{(\beta \hbar \omega)^{D}}\left[\zeta(D+1)+\beta \hbar \omega \frac{D}{2} \zeta(D)+\ldots\right] $$

$$ S=k_{B} N(D+1)\left\{\left(\frac{T}{T_{c}^{(0)}}\right)^{D} \frac{\zeta(D+1)}{\zeta(D)}+\frac{D}{2}\left[\frac{\zeta(D)}{N}\right]^{1 / D}\left(\frac{T}{T_{c}^{(0)}}\right)^{D-1}+\ldots\right\} . $$

$$ C=k_{B} N(D+1)\left\{D\left(\frac{T}{T_{c}^{(0)}}\right)^{D} \frac{\zeta(D+1)}{\zeta(D)}+\frac{D(D-1)}{2}\left[\frac{\zeta(D)}{N}\right]^{1 / D}\left(\frac{T}{T_{c}^{(0)}}\right)^{D-1}+\ldots\right\} . $$

$$ C_{\max } \approx k_{B} N(D+1) D \frac{\zeta(D+1)}{\zeta(D)}\left\{1+\left[\frac{D-1}{2} \frac{\zeta^{1+1 / D}(D)}{\zeta(D+1)}-\frac{D}{2} \frac{\zeta(D-1)}{\zeta^{1-1 / D}(D)}\right] \frac{1}{N^{1 / D}}\right\} . $$

$$ C_{\max }^{(0)} \approx k_{B} N 10.805, \quad C_{\max }^{(1)} \approx k_{B} N 9.556 $$

$$ N(\beta, \mu)=\frac{1}{(\beta \hbar \omega)^{D}}\left[\zeta_{D}\left(z_{D}\right)+\beta \hbar \omega \frac{D}{2} \zeta_{D-1}\left(z_{D}\right)+\ldots\right] . $$

$$ F_{G}(\beta, \mu)=-\frac{1}{\beta(\beta \hbar \omega)^{D}}\left[\zeta_{D+1}\left(z_{D}\right)+\beta \hbar \omega \frac{D}{2} \zeta_{D}\left(z_{D}\right)+\ldots\right] $$

$$ S(\beta, \mu)=k_{B} \frac{1}{(\beta \hbar \omega)^{D}}\left\{(D+1) \zeta_{D+1}\left(z_{D}\right)+\frac{1}{2}\left(\beta \hbar \omega D^{2}-2 \beta \mu\right) \zeta_{D}\left(z_{D}\right)+\ldots\right\} $$

$$ \begin{align*} C(\beta, \mu) & =k_{B} \frac{1}{(\beta \hbar \omega)^{D}}\left\{(D+1) D \zeta_{D+1}\left(z_{D}\right)\right. \\ & \left.+\frac{1}{2}\left[D\left(D^{2}+1\right) \beta \hbar \omega-D \beta \mu-\beta \partial_{\beta}(\beta \mu)\right] \zeta_{D}\left(z_{D}\right)+\ldots\right\} \end{align*} $$

$$ \begin{align*} 0= & \frac{1}{(\beta \hbar \omega)^{D}}\left\{-\left[D \zeta_{D}\left(z_{D}\right)+\beta \hbar \omega \frac{D}{2}(D-1) \zeta_{D-1}\left(z_{D}\right)\right]\right. \\ & \left.+\left[\zeta_{D-1}\left(z_{D}\right)+\beta \hbar \omega \frac{D}{2} \zeta_{D-2}\left(z_{D}\right)\right]\left[\beta \partial_{\beta}(\beta \mu)-\beta \hbar \omega \frac{D}{2}\right]+\ldots\right\}, \end{align*} $$

$$ \beta \partial_{\beta}(\beta \mu)=D \frac{\zeta_{D}\left(z_{D}\right)+\beta \hbar \omega \frac{D}{2} \zeta_{D-2}\left(z_{D}\right)+\ldots}{\zeta_{D-1}\left(z_{D}\right)+\beta \hbar \frac{D}{2} \omega \zeta_{D-2}\left(z_{D}\right)+\ldots} $$

$$ \begin{align*} N(\beta, \mu) & \approx \frac{1}{(\beta \hbar \omega)^{D}} \zeta_{D}(z) \\ F_{G}(\beta, \mu) & \approx-\frac{1}{\beta} \frac{1}{(\beta \hbar \omega)^{D}} \zeta_{D+1}(z)=-N \frac{1}{\beta} \frac{\zeta_{D+1}(z)}{\zeta_{D}(z)} \\ S(\beta, \mu) & \approx \frac{1}{T}\left[\frac{D+1}{\beta} \frac{1}{(\beta \hbar \omega)^{D}} \zeta_{D+1}(z)-\mu N\right]=N k_{B}\left[\frac{D+1}{(\beta \hbar \omega)^{D}} \frac{\zeta_{D+1}(z)}{\zeta_{D}(z)}-\beta \mu\right] \end{align*} $$

$$ E(\beta, \mu) \approx \frac{D}{\beta} \frac{1}{(\beta \hbar \omega)^{D}} \zeta_{D+1}(z)=N \frac{D}{\beta} \frac{\zeta_{D+1}(z)}{\zeta_{D}(z)} . $$

$$ \beta \partial_{\beta}(\beta \mu)=D \frac{\zeta_{D}(z)}{\zeta_{D-1}(z)} $$

$$ C=k_{B} N\left[(D+1) D \frac{\zeta_{D+1}(z)}{\zeta_{D}(z)}-D^{2} \frac{\zeta_{D}(z)}{\zeta_{D-1}(z)}\right] $$

$$ C_{\max }^{(0)}=k_{B} N\left[12 \frac{\zeta_{4}(1)}{\zeta_{3}(1)}-9 \frac{\zeta_{3}(1)}{\zeta_{2}(1)}\right] \approx k_{B} N 4.22785 . $$

$$ F_{G}=\int d^{D} x\left\{-\frac{1}{\beta} Z_{\omega}(\beta) \int_{0}^{z_{D}} \frac{d z}{z} \zeta_{D}(\beta \hbar \omega ; z ; \mathbf{x})+g\left[Z_{\omega}(\beta) \zeta_{D}\left(\beta \hbar \omega ; z_{D} ; \mathbf{x}\right)\right]^{2}\right\} . $$

$$ F_{G}=-\frac{1}{\beta} Z_{\omega}(\beta) \int d^{D} x\left\{\int_{0}^{z_{D}} \frac{d z}{z} \zeta_{D}(\beta \hbar \omega ; z ; \mathbf{x})-2 \hat{a} l_{\mathrm{e}}^{D}(\hbar \beta) Z_{\omega}(\beta)\left[\zeta_{D}\left(\beta \hbar \omega ; z_{D} ; \mathbf{x}\right)\right]^{2}\right\} . $$

$$ F_{G}^{\sigma}=-\frac{1}{\beta} Z_{\omega}(\beta) \int d^{3} x\left[\int_{0}^{z_{D} e^{\sigma(\mathbf{x})}} \frac{d z}{z} \zeta_{D}(\beta \hbar \omega ; z ; \mathbf{x})-\frac{\sigma^{2}(\mathbf{x})}{8 \tilde{a}}\right], $$

$$ \tilde{a} \equiv \hat{a} l_{\mathrm{e}}^{D}(\hbar \beta) Z_{\omega}(\beta)=\frac{a}{l_{\mathrm{e}}(\hbar \beta)} l_{\mathrm{e}}^{D}(\hbar \beta) Z_{\omega}(\beta) . $$

$$ \Sigma(\mathbf{x}) \equiv-4 \tilde{a} \zeta_{D}\left(\beta \hbar \omega ; z_{D} e^{\Sigma(\mathbf{x})} ; \mathbf{x}\right) . $$

$$ \zeta_{D}\left(\beta \hbar \omega ; z_{D} ; \mathbf{x}\right) \equiv \sum_{w=1}^{\infty}{\sqrt{\frac{\omega^{2} M \beta}{2 \pi w}}}^{D} z_{D}^{w} e^{-M w \beta \omega^{2} \mathbf{x}^{2} / 2} $$

$$ \Sigma(\mathbf{x}) \stackrel{\omega \approx 0}{\approx}-4 \tilde{a} \sum_{w=1}^{\infty}{\sqrt{\frac{\omega^{2} M \beta}{2 \pi w}}}^{D} z_{D}^{w} e^{w \Sigma(\mathbf{x})} e^{-M w \beta \omega^{2} \mathbf{x}^{2} / 2} $$

$$ N=\int d^{D} x n(\mathbf{x})=Z_{\omega}(\beta) \int d^{D} x \bar{\zeta}_{D}\left(\beta \hbar \omega ; z_{D} e^{\Sigma(\mathbf{x})} ; \mathbf{x}\right) $$

$$ 1=\frac{Z_{\omega}(\beta)}{Z_{\omega}\left(\beta_{c}^{(0)}\right)} \int d^{D} x \frac{\bar{\zeta}_{D}\left(\beta \hbar \omega ; z_{D} e^{\Sigma(\mathbf{x})} ; \mathbf{x}\right)}{\bar{\zeta}_{D}\left(\beta_{c}^{(0)} \hbar \omega ; 1\right)} $$

$$ 1=\frac{Z_{\omega}\left(\beta_{c}\right)}{Z_{\omega}\left(\beta_{c}^{(0)}\right)} \int d^{D} x \frac{\bar{\zeta}_{D}\left(\beta_{c} \hbar \omega ; e^{\Sigma_{c}(\mathbf{x})-\Sigma_{c}(\mathbf{0})} ; \mathbf{x}\right)}{\bar{\zeta}_{D}\left(\beta_{c}^{(0)} \hbar \omega ; 1\right)} $$

$$ 1 \approx \frac{Z_{\omega}\left(\beta_{c}\right)}{Z_{\omega}\left(\beta_{c}^{(0)}\right)}\left\{1+\frac{1}{\bar{\zeta}_{D}\left(\beta_{c}^{(0)} \hbar \omega ; 1\right)} \int d^{D} x \Delta \bar{\zeta}_{D}\left(\beta_{c}^{(0)} \hbar \omega ; 1 ; \mathbf{x}\right)\right\} $$

$$ \int d^{D} x \sum_{w=1}^{\infty}{\sqrt{\frac{\omega^{2} M \beta_{c}^{(0)}}{2 \pi w}}}^{D} e^{-M w \beta_{c}^{(0)} \omega^{2} \mathbf{x}^{2} / 2}\left(e^{w\left[\Sigma_{c}(\mathbf{x})-\Sigma_{c}(\mathbf{0})\right]}-1\right) $$

$$ \frac{Z_{\omega}\left(\beta_{c}\right)}{Z_{\omega}\left(\beta_{c}^{(0)}\right)} \equiv 1+D{\frac{\Delta T_{c}}{T_{c}^{(0)}}}_{\tanh \left(\beta_{c}^{(0)} \hbar \omega / 2\right)}^{\beta_{c}^{(0)} \hbar \omega / 2}{ }^{\omega} \approx 01+D \frac{\Delta T_{c}}{T_{c}^{(0)}}, $$

$$ \frac{\Delta T_{c}}{T_{c}^{(0)}} \approx-\frac{1}{D} \frac{1}{\zeta(D)} \int d^{D} x \Delta \bar{\zeta}_{D}\left(\beta_{c}^{(0)} \hbar \omega ; 1 ; \mathbf{x}\right) $$

$$ -4 \tilde{a}_{c}{\sqrt{\frac{\omega^{2} M \beta_{c}^{(0)}}{2 \pi}}}^{D} \sum_{w, w^{\prime}=1}^{\infty} \frac{1}{w^{D / 2-1} w^{D / 2}}\left[\frac{1}{\left(w+w^{\prime}\right)^{D / 2}}-\frac{1}{w^{D / 2}}\right] \equiv-4 \hat{a}_{c} S(D) $$

$$ \frac{\Delta T_{c}}{T_{c}^{(0)}} \approx \frac{4 \hat{a}_{c}}{D} \frac{S(D)}{\zeta(D)} \underset{D=3}{\approx}-3.427 \frac{a}{\ell_{c}} $$

$$ \frac{\Delta T_{c}}{T_{c}^{(0)}} \approx-3.427 \frac{1}{\sqrt{2 \pi}[\zeta(3)]^{1 / 6}} \frac{a}{\lambda_{\omega}} N^{1 / 6} \approx-1.326 \frac{a}{\lambda_{\omega}} N^{1 / 6} $$

$$ \Delta T_{c} \stackrel{\omega, \tilde{a} \approx 0}{\approx} 2 g \frac{\int d^{3} x\left[\partial_{\mu} n_{0}(\mathbf{x})\right]\left[n_{0}(\mathbf{x})-n_{0}(\mathbf{0})\right]}{\int d^{3} x \partial_{T} n_{0}(\mathbf{x})} $$

$$ F_{G}=-\frac{1}{\beta} \int d^{3} x\left[\int_{0}^{z e^{\beta \nu(\mathbf{x})}} \frac{d z^{\prime}}{z^{\prime}} n_{0}\left(\beta ; z^{\prime} ; \mathbf{x}\right)-\beta \frac{\nu^{2}(\mathbf{x})}{4 g}\right] $$

$$ N=\int d^{3} x n_{0}\left(\beta ; z e^{\beta \nu(\mathbf{x})} ; \mathbf{x}\right) $$

$$ \nu(\mathbf{x})=-2 g n_{0}\left(\beta ; z e^{\beta \nu(\mathbf{x})} ; \mathbf{x}\right) $$

$$ N=\int d^{3} x n_{0}\left(\beta_{c} ; z e^{-2 \beta g\left[n_{0}(\mathbf{x})-n_{0}(\mathbf{0})\right]} ; \mathbf{x}\right) $$

$$ \frac{\Delta T_{c}}{T_{c}^{(0)}} \approx \frac{4 \hat{a}_{c}}{\tilde{D}} \frac{1}{\zeta(\tilde{D})} \sum_{w, w^{\prime}=1}^{\infty} \frac{1}{w^{D / 2-1} w^{\prime D / 2}}\left[\frac{1}{\left(w+w^{\prime}\right)^{\tilde{D}-D / 2}}-\frac{1}{w^{\tilde{D}-D / 2}}\right] $$

$$ \frac{a}{\ell_{c}}=\frac{2 \pi \hbar^{2}}{M k_{B} T_{c}}=\sqrt{2 \pi} \sqrt{\frac{E_{H}}{k_{B} T_{c}}} \sqrt{\frac{M_{e}}{87 M_{p}}} a_{H} $$

$$ \frac{\Delta T_{c}}{T_{c}^{(0)}} \approx-5.4 \% $$

$$ \zeta_{\nu}^{\mathrm{f}}(z) \equiv \sum_{w=1}^{\infty}(-1)^{w-1} \frac{z^{w}}{w^{\nu}} $$

$$ \zeta_{\nu}^{\mathrm{f}}(1)=\frac{1}{1}-\frac{1}{2^{\nu}}+\frac{1}{3^{\nu}}-\frac{1}{4^{\nu}}+\ldots=\sum_{k=0}^{\infty} \frac{1}{k^{\nu}}-2^{1-\nu} \sum_{k=0}^{\infty} \frac{1}{k^{\nu}}=\left(1-2^{1-\nu}\right) \zeta(\nu) $$

$$ \zeta_{n}^{\mathrm{f}}(z) \equiv \frac{1}{\Gamma(n)} i_{n}^{\mathrm{f}}(\alpha) $$

$$ i_{n}^{\mathrm{f}}(\alpha) \equiv \int_{0}^{\infty} d \varepsilon \frac{\varepsilon^{n-1}}{e^{\varepsilon-\alpha}+1} $$

$$ n_{\varepsilon}^{\mathrm{f}}=\frac{1}{e^{\varepsilon-\alpha}+1} $$

$$ F_{G}=-\frac{1}{\beta \Gamma(D / 2)} \frac{g_{S} V_{D}}{{\sqrt{2 \pi \hbar^{2} \beta / M}}^{D}} \int_{0}^{\infty} d \varepsilon \varepsilon^{D / 2-1} \log \left(1+e^{-\varepsilon+\beta \mu}\right) $$

$$ F_{G}=-\frac{g_{S}}{\beta} \sum_{\mathbf{p}} \log \left(1+e^{-\beta \hbar \omega_{\mathbf{p}}+\beta \mu}\right) $$

$$ N=-\frac{\partial}{\partial \mu} F_{G}=\frac{1}{\Gamma(D / 2)} \frac{g_{S} V_{D}}{\sqrt{2 \pi \hbar^{2} \beta / M}} \int_{0}^{\infty} d \varepsilon \frac{\varepsilon^{D / 2-1}}{e^{\varepsilon+\beta \mu}-1}=g_{S} \sum_{\mathbf{p}} \frac{1}{e^{\beta \hbar \omega_{\mathbf{p}}+\beta \mu}-1} $$

$$ N=g_{S} V_{D} \int_{0}^{\infty} d \varepsilon N_{\varepsilon} n_{\varepsilon}^{\mathrm{f}} $$

$$ n_{\varepsilon}^{\mathrm{f}}=\left\{\begin{array}{lll} 1 & \text { for } & \varepsilon<\alpha \\ 0 & & \varepsilon>\alpha \end{array}\right\}=\Theta(\varepsilon-\alpha) $$

$$ \left.\mu\right|_{T=0} \equiv E_{\mathrm{F}} $$

$$ N=g_{S} V_{D} \int_{0}^{E_{\mathrm{F}}} d \varepsilon N_{\varepsilon}=\frac{1}{\Gamma(D / 2+1)} \frac{g_{S} V_{D} E_{F}^{D / 2}}{{\sqrt{2 \pi \hbar^{2} / M}}^{D}}=\frac{1}{\Gamma(D / 2+1)} \frac{g_{S} V_{D}}{\sqrt{4 \pi}^{D}}\left(\frac{p_{F}}{\hbar}\right)^{D} $$

$$ p_{\mathrm{F}} \equiv \sqrt{2 M E_{\mathrm{F}}} $$

$$ E_{\mathrm{F}}=\frac{2 \pi \hbar^{2}}{M}\left[\frac{\Gamma(D / 2+1)}{g_{S}}\right]^{2 / D}\left(\frac{N}{V_{D}}\right)^{2 / D} $$

$$ p_{\mathrm{F}}=2 \sqrt{\pi} \hbar\left[\frac{\Gamma(D / 2+1)}{g_{S}}\right]^{1 / D}\left(\frac{N}{V_{D}}\right)^{1 / D} \hbar $$

$$ N_{\varepsilon} \equiv \frac{d}{2} \frac{N}{V_{D}}{\sqrt{\frac{k_{B} T}{E_{\mathrm{F}}}}}^{D / 2} \varepsilon^{D / 2-1} $$

$$ T_{\mathrm{F}} \equiv \frac{E_{\mathrm{F}}}{k_{B}}=\frac{1}{k_{B}} \frac{p_{\mathrm{F}}^{2}}{2 M} $$

$$ T_{\mathrm{F}} \approx 44000 \mathrm{~K} . $$

$$ N=N(T, \mu) \equiv \frac{g_{S} V}{l_{\mathrm{e}}^{3}(\hbar \beta)} \frac{2}{\sqrt{\pi}} i_{3 / 2}^{\mathrm{f}}\left(\frac{\mu}{k_{B} T}\right) $$

$$ 1=\left(\frac{k_{B} T}{E_{\mathrm{F}}}\right)^{3 / 2} \frac{3}{2} i_{3 / 2}^{\mathrm{f}}\left(\frac{\mu}{k_{B} T}\right) . $$

$$ \begin{align*} i_{n}^{\mathrm{f}}(\alpha) & =\int_{-\alpha}^{\infty} d x \frac{(\alpha+x)^{n-1}}{e^{x}+1} \\ & =\int_{0}^{\alpha} d x \frac{(\alpha-x)^{n-1}}{e^{-x}+1}+\int_{0}^{\infty} d x \frac{(\alpha+x)^{n-1}}{e^{x}+1} \end{align*} $$

$$ i_{n}^{\mathrm{f}}(\alpha)=\int_{0}^{\alpha} d x x^{n-1}+\int_{0}^{\infty} d x \frac{(\alpha+x)^{n-1}-(\alpha-x)^{n-1}}{e^{x}+1}+\int_{\alpha}^{\infty} d x \frac{(\alpha-x)^{n-1}}{e^{x}+1} $$

$$ 2 \sum_{k=\text { odd }}^{\infty}\binom{n-1}{k} \alpha^{n-1-k} \int_{0}^{\infty} d x \frac{x^{k}}{e^{x}+1}=2 \sum_{k=\text { odd }} \frac{(n-1)!}{(n-1-k)!} \alpha^{n-1-k}\left(1-2^{-k}\right) \zeta(k+1) . $$

$$ \int_{0}^{\infty} d x \frac{x^{\nu-1}}{e^{\mu x}+1}=\mu^{-\nu}\left(1-2^{1-\nu}\right) \zeta(\nu) $$

$$ i_{n}^{\mathrm{f}}(\alpha)=\frac{1}{n} \alpha^{n}+2(n-1) \frac{1}{2} \zeta(2) \alpha^{n-2}+2(n-1)(n-2)(n-3) \frac{7}{8} \zeta(4) \alpha^{n-4}+\ldots . $$

$$ 1=\left(\frac{k_{B} T}{E_{\mathrm{F}}}\right)^{3 / 2} \frac{3}{2}\left[\frac{2}{3}\left(\frac{\mu}{k_{B} T}\right)^{3 / 2}+\frac{\pi^{2}}{12}\left(\frac{\mu}{k_{B} T}\right)^{-1 / 2}+\frac{7 \pi^{4}}{3 \cdot 320}\left(\frac{\mu}{k_{B} T}\right)^{-5 / 2} \ldots\right] $$

$$ \mu=E_{\mathrm{F}}\left[1-\frac{\pi^{2}}{12}\left(\frac{k_{B} T}{E_{\mathrm{F}}}\right)^{2}+\frac{7 \pi^{4}}{720}\left(\frac{k_{B} T}{E_{\mathrm{F}}}\right)^{4}+\ldots\right] $$

$$ F_{G}=-\frac{1}{\beta} \frac{g_{S} V}{l_{\mathrm{e}}^{3}(\hbar \beta)} \frac{1}{\Gamma(5 / 2)} i_{5 / 2}^{\mathrm{f}}(\alpha) $$

$$ F_{G}(T, \mu, V)=F_{G}(0, \mu, V)\left[1+\frac{5 \pi^{2}}{8}\left(\frac{k_{B} T}{\mu}\right)^{2}-\frac{7 \pi^{4}}{384}\left(\frac{k_{B} T}{\mu}\right)^{4}+\ldots\right] $$

$$ S=k_{B} \frac{\pi^{2}}{2} \frac{k_{B} T}{E_{\mathrm{F}}} N+\ldots $$

$$ C=\left.T \frac{\partial S}{\partial T}\right|_{V, N} \stackrel{T \approx 0}{\approx} S=k_{B} \frac{\pi^{2}}{2} \frac{k_{B} T}{E_{\mathrm{F}}} N+\ldots . $$

$$ F_{G}(T, \mu, V)=F_{G}\left(0, E_{\mathrm{F}}, V\right)\left[1+\frac{5 \pi^{2}}{12}\left(\frac{k_{B} T}{E_{\mathrm{F}}}\right)^{2}-\frac{\pi^{4}}{16}\left(\frac{k_{B} T}{E_{F}}\right)^{4}+\ldots\right], $$

$$ F_{G}\left(0, E_{\mathrm{F}}, V\right)=-\frac{2}{5} N E_{\mathrm{F}} $$

$$ E=-\frac{3}{2} F_{G} $$

$$ E=\frac{3}{5} N E_{\mathrm{F}}\left[1+\frac{5 \pi^{2}}{12}\left(\frac{k_{B} T}{E_{F}}\right)^{2}-\frac{\pi^{4}}{16}\left(\frac{k_{B} T}{E_{F}}\right)^{4}+\ldots\right] $$

$$ \mathcal{A}=\int_{t_{a}}^{t_{b}} d t\left[\frac{M^{(1)}}{2} \dot{\mathbf{x}}^{(1) 2}+\frac{M^{(2)}}{2} \dot{\mathbf{x}}^{(2) 2}-V_{\mathrm{int}}\left(\mathbf{x}^{(1)}-\mathbf{x}^{(2)}\right)\right] $$

$$ \mathbf{X}=\left(M^{(1)} \mathbf{x}^{(1)}+M^{(2)} \mathbf{x}^{(2)}\right) /\left(M^{(1)}+M^{(2)}\right), \quad \mathbf{x}=\left(\mathbf{x}^{(1)}-\mathbf{x}^{(2)}\right), $$

$$ \mathcal{A}=\mathcal{A}_{\mathrm{CM}}+\mathcal{A}_{\mathrm{rel}}=\int_{t_{a}}^{t_{b}} d t \frac{M}{2} \dot{\mathbf{X}}^{2}+\int_{t_{a}}^{t_{b}} d t\left[\frac{\mu}{2} \dot{\mathbf{x}}^{2}-V_{\mathrm{int}}(\mathbf{x})\right] $$

$$ \int_{0}^{\infty} d r|r\rangle\langle r|=1 $$

$$ \left\langle r_{b} \mid r_{a}\right\rangle=2 \int_{0}^{\infty} \frac{d p}{\pi \hbar}\left\{\begin{array}{cc} \cos p r_{b} / \hbar & \cos p r_{a} / \hbar \\ \sin p r_{b} / \hbar & \sin p r_{a} / \hbar \end{array}\right\} $$

$$ \left\langle r_{b} \mid r_{a}\right\rangle=\int_{-\infty}^{\infty} \frac{d p}{2 \pi \hbar}\left(e^{i p\left(r_{b}-r_{a}\right) / \hbar} \pm e^{i p\left(r_{b}+r_{a}\right) / \hbar}\right)=\delta\left(r_{b}-r_{a}\right) \pm \delta\left(r_{b}+r_{a}\right) $$

$$ \begin{align*} \left(r_{n} \epsilon \mid r_{n-1} 0\right) & =\left\langle r_{n}\right| e^{-i \epsilon \hat{H}_{\mathrm{rel}} / \hbar}\left|r_{n-1}\right\rangle \\ & =\int_{-\infty}^{\infty} \frac{d p}{2 \pi \hbar}\left(e^{i p\left(r_{n}-r_{n-1}\right) / \hbar} \pm e^{i p\left(r_{n}+r_{n-1}\right) / \hbar}\right) e^{-i \epsilon H_{\mathrm{rel}}\left(p, r_{n}\right) / \hbar} \end{align*} $$

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

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

$$ \sigma(x)=\Theta(-x) $$

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

$$ e^{i \pi\left(\sigma\left(x_{b}\right)-\sigma\left(x_{a}\right)\right)} $$

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

$$ \mathcal{A}=\mathcal{A}_{\mathrm{rel}}+\mathcal{A}_{\mathrm{f}}=\int_{t_{a}}^{t_{b}} d t\left[p \dot{x}-H_{\mathrm{rel}}(p, x)+\hbar \pi \dot{x}(t) \partial_{x} \sigma(x(t))\right] $$

$$ \mathcal{A}_{\mathrm{f}}=-\hbar \pi \int_{t_{a}}^{t_{b}} d t \dot{x}(t) \delta(x(t))=\hbar \pi \int_{t_{a}}^{t_{b}} d t \partial_{t} \Theta(-x(t)) $$

$$ \begin{align*} &\left(x_{b}^{\left(\nu_{b}\right)} ; t_{b} \mid x_{a}^{\left(\nu_{a}\right)} ; t_{a}\right)=\sum_{p\left(\nu_{b}\right)} \prod_{\nu=1}^{n}\left\{\prod_{n=1}^{N}\left[\int_{-\infty}^{\infty} d x_{n}^{(\nu)}\right] \prod_{n=1}^{N+1}\left[\int_{-\infty}^{\infty} \frac{d p_{n}^{(\nu)}}{2 \pi \hbar}\right]\right\} \\ & \times \exp \left(\frac { i } { \hbar } \sum _ { n = 1 } ^ { N + 1 } \left\{\sum_{\nu=1}^{n}\left[p_{n}^{(\nu)}\left(x_{n}^{(\nu)}-x_{n-1}^{(\nu)}\right)-\epsilon H_{\mathrm{rel}}\left(p_{n}^{(\nu)}, x_{n}^{(\nu)}\right)\right]\right.\right. \\ &\left.\left.\quad+\hbar \pi \sum_{\nu<\nu^{\prime}}\left[\sigma\left(x_{n}^{\left(\nu, \nu^{\prime}\right)}\right)-\sigma\left(x_{n-1}^{\left(\nu, \nu^{\prime}\right)}\right)\right]\right\}\right) \end{align*} $$

$$ \begin{align*} \left\langle r_{b} \varphi_{b} \mid r_{a} \varphi_{a}\right\rangle & =\int_{0}^{\infty} d k k \sum_{m=-\infty}^{\infty} i_{m}\left(k r_{b}\right) i_{m}\left(k r_{a}\right) \frac{1}{2 \pi} e^{i m\left(\varphi_{b}-\varphi_{a}\right)} \\ & =\frac{1}{\sqrt{r_{b} r_{a}}} \delta\left(r_{b}-r_{a}\right) \delta\left(\varphi_{b}-\varphi_{a}\right) \end{align*} $$

$$ \left\langle\mathbf{x}_{b} \mid \mathbf{x}_{a}\right\rangle=\int \frac{d^{2} k}{(2 \pi)^{2}} e^{i \mathbf{k} \mathbf{x}_{b}} e^{-i \mathbf{k} \mathbf{x}_{a}}=\delta^{(2)}\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right) $$

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

$$ \left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t\right)=\int \mathcal{D}^{2} x \int \frac{\mathcal{D}^{2} p}{2 \pi}\left\{\exp \left[\frac{i}{\hbar} \mathcal{A}_{\mathrm{rel}}+\frac{i}{\hbar} \mathcal{A}_{\mathrm{f}}\right]+\left(\mathbf{x}_{b} \rightarrow-\mathbf{x}_{b}\right)\right\} $$

$$ \mathcal{A}_{\mathrm{f}}=\hbar \pi \int_{t_{a}}^{t_{b}} d t \dot{\varphi}(t) \partial_{\varphi} \sigma(\varphi(t)) $$

$$ \mathcal{A}_{\mathrm{f}}=\hbar \int_{t_{a}}^{t_{b}} d t \dot{\mathbf{x}}(t) \cdot \mathbf{a}(\mathbf{x}(t)) $$

$$ \mathbf{a}(\mathbf{x}) \equiv \pi \boldsymbol{\nabla} \sigma_{\mathrm{p}}(\varphi) $$

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

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

$$ \sigma_{\mathrm{p}}(\mathbf{x})=\frac{1}{\pi} \varphi(\mathbf{x}) \equiv \frac{1}{\pi} \arctan \frac{x_{2}}{x_{1}} $$

$$ \mathcal{A}_{\mathrm{f}}=\hbar \int_{t_{a}}^{t_{b}} d t \dot{\mathbf{x}} \partial_{\mathbf{x}} \varphi(\mathbf{x})=\hbar \int_{t_{a}}^{t_{b}} d t \epsilon_{i j} \frac{x_{i} \dot{x}_{j}}{\mathbf{x}^{2}} $$

$$ a_{i}(\mathbf{x})=\partial_{i} \varphi=-\epsilon_{i j} \frac{x_{j}}{\mathbf{x}^{2}} $$

$$ a_{i}(\mathbf{x})=\partial_{i} \varphi=-(2 n+1) \epsilon_{i j} \frac{x_{j}}{\mathbf{x}^{2}}, \quad n=0, \pm 1, \pm 2, \ldots $$

$$ a_{i}(\mathbf{x})=\partial_{i} \varphi=-2 n \epsilon_{i j} \frac{x_{j}}{\mathbf{x}^{2}}, \quad n=0, \pm 1, \pm 2, \ldots $$

$$ \mathcal{A}_{\mathrm{f}}=\hbar \mu_{0} \int_{t_{a}}^{t_{b}} d t \dot{\mathbf{x}} \cdot \nabla \varphi(\mathbf{x})=\hbar \mu_{0} \int_{t_{a}}^{t_{b}} d t \epsilon_{i j} \frac{x_{i} \dot{x}_{j}}{\mathbf{x}^{2}} $$

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

$$ A_{i}(\mathbf{x})=\frac{\Phi}{2 \pi} \partial_{i} \varphi=-\Phi \epsilon_{i j} \frac{x_{j}}{\mathbf{x}^{2}} $$

$$ \begin{align*} {\left[\hat{\psi}(\mathbf{x}, t), \hat{\psi}^{\dagger}\left(\mathbf{x}^{\prime}, t\right)\right] } & =\delta_{\mathbf{x x}^{\prime}} \\ {\left[\hat{\psi}^{\dagger}(\mathbf{x}, t), \hat{\psi}^{\dagger}\left(\mathbf{x}^{\prime}, t\right)\right] } & =0 \\ {\left[\hat{\psi}(\mathbf{x}, t), \hat{\psi}\left(\mathbf{x}^{\prime}, t\right)\right] } & =0 \end{align*} $$

$$ |n, \mathbf{x}\rangle=\frac{1}{\sqrt{n!}}\left[\hat{\psi}^{\dagger}(\mathbf{x}, 0)\right]^{n}|0\rangle $$

$$ |0\rangle \equiv \prod_{\mathbf{x}}|0\rangle_{\mathbf{x}} $$

$$ \hat{N}(t)=\sum_{\mathbf{x}} \hat{\psi}^{\dagger}(\mathbf{x}, t) \hat{\psi}(\mathbf{x}, t) $$

$$ \mathcal{A}\left[\psi^{*}, \psi\right]=\sum_{\mathbf{x}} \int_{t_{a}}^{t_{b}} d t\left[\psi^{*}\left(i \hbar \partial_{t}+\mu\right) \psi(\mathbf{x}, t)-\frac{\hbar^{2}}{2 M} \psi^{*} \nabla_{\mathbf{x}} \bar{\nabla}_{\mathbf{x}} \psi(\mathbf{x}, t)\right] $$

$$ -i \hbar \nabla_{i} e^{i \mathbf{p} \mathbf{x} / \hbar}=P_{i} e^{i \mathbf{p} \mathbf{x} / \hbar}, \quad-i \hbar \bar{\nabla}_{i} e^{i \mathbf{p} \mathbf{x} / \hbar}=\bar{P}_{i} e^{i \mathbf{p} \mathbf{x} / \hbar} $$

$$ P_{i}=-i \frac{\hbar}{\epsilon}\left[e^{i \epsilon p_{i} / \hbar}-1\right], \quad \bar{P}_{i}=P_{i}^{*} $$

$$ \psi(\mathbf{x}, t)=\sqrt{\frac{\epsilon^{3}}{V}} \sum_{\mathbf{p}} e^{i \mathbf{p x} / \hbar} a_{\mathbf{p}}(t) $$

$$ \mathcal{A}\left[a^{*}, a\right]=\hbar \sum_{\mathbf{p}} \int_{t_{a}}^{t_{b}} d t\left[a_{\mathbf{p}}^{*}(t) i \partial_{t} a_{\mathbf{p}}(t)-\omega(\mathbf{p}) a_{\mathbf{p}}^{*}(t) a_{\mathbf{p}}(t)\right] $$

$$ \omega(\mathbf{p}) \equiv \frac{1}{\hbar}\left[\frac{|\mathbf{P}|^{2}}{2 M}-\mu\right] $$

$$ |\mathbf{P}|^{2} \equiv \frac{\hbar^{2}}{\epsilon^{2}} \sum_{i} 2\left[1-\cos \left(\frac{\epsilon p_{i}}{\hbar}\right)\right] $$

$$ \left(i \hbar \partial_{t}+\mu+\frac{\hbar^{2}}{2 M} \nabla_{\mathbf{x}} \bar{\nabla}_{\mathbf{x}}\right) \psi(\mathbf{x}, t)=0 $$

$$ \begin{align*} {\left[\hat{a}_{\mathbf{p}}(t), \hat{a}_{\mathbf{p}^{\prime}}^{\dagger}(t)\right] } & =\delta_{\mathbf{p p}^{\prime}} \\ {\left[\hat{a}_{\mathbf{p}}^{\dagger}(t), \hat{a}_{\mathbf{p}^{\prime}}^{\dagger}(t)\right] } & =0 \\ {\left[\hat{a}_{\mathbf{p}}(t), \hat{a}_{\mathbf{p}^{\prime}}(t)\right] } & =0 \end{align*} $$

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

$$ \hat{a}^{\dagger}=\sqrt{M / 2 \hbar \omega}(\omega \hat{x}-i \hat{p} / M), \quad \hat{a}=\sqrt{M / 2 \hbar \omega}(\omega \hat{x}+i \hat{p} / M) $$

$$ \hat{H}_{\omega}=\frac{1}{2 M} \hat{p}^{2}+\frac{M \omega^{2}}{2} \hat{x}^{2} $$

$$ \hat{H}_{\omega}=\frac{\hbar \omega}{2}\left(\hat{a}^{\dagger} \hat{a}+\hat{a} \hat{a}^{\dagger}\right) $$

$$ \mathcal{A}[p, q]=\int_{t_{a}}^{t_{b}} d t\left[p \dot{q}-H_{\omega}(p, q)\right] $$

$$ \mathcal{A}\left[a^{*}, a\right]=\hbar \int_{t_{a}}^{t_{b}} d t\left(a^{*} i \partial_{t} a-\omega a^{*} a\right) $$

$$ \mathcal{A}_{\mathrm{e}}\left[a^{*}, a\right]=\hbar \int_{0}^{\hbar \beta} d \tau\left(a^{*} \partial_{\tau} a+\omega a^{*} a\right) $$

$$ Z_{\omega}=\oint \mathcal{D} x(\tau) \int \frac{\mathcal{D} p(\tau)}{2 \pi \hbar} \exp \left[-\int_{0}^{\hbar \beta} d \tau\left(a^{*} \partial_{\tau} a+\omega a^{*} a\right)\right] $$

$$ x(\tau)=\frac{1}{\sqrt{\hbar \beta}} \sum_{m=-\infty}^{\infty} x_{m} e^{-i \omega_{m} \tau}, \quad \omega_{m}=2 \pi m / \hbar \beta $$

$$ Z_{\omega} \equiv \oint \frac{\mathcal{D} a^{*}(\tau) \mathcal{D} a(\tau)}{\pi} \exp \left\{-\int_{0}^{\hbar \beta} d \tau\left(a^{*} \partial_{\tau} a+\omega a^{*} a\right)\right\} $$

$$ \oint \mathcal{D} a^{*} \mathcal{D} a=\oint_{-\infty}^{\infty} \mathcal{D} \operatorname{Re} a \int_{-\infty}^{\infty} \mathcal{D} \operatorname{Im} a $$

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

$$ Z_{\omega} \rightarrow e^{-\hbar \omega \beta / 2} $$

$$ \hat{H}=\hbar \omega \hat{a}^{\dagger} \hat{a}, $$

$$ Z_{\omega}^{N}=\prod_{n=0}^{N}\left[\int \frac{d a_{n}^{*} d a_{n}}{\pi}\right] \exp \left\{-\frac{1}{\hbar} \mathcal{A}_{\omega}^{N}\right\} $$

$$ \mathcal{A}_{\omega}^{N}=\hbar \sum_{n=1}^{N}\left[a_{n}^{*}\left(a_{n}-a_{n-1}\right)+\epsilon \omega a_{n}^{*} a_{n-1}\right] $$

$$ \mathcal{A}_{\omega}^{N}=\hbar \epsilon \sum_{n=1}^{N} a_{n}^{*}[(1-\epsilon \omega) \bar{\nabla}+\omega] a_{n} $$

$$ a(\tau)=\frac{1}{\sqrt{\hbar \beta}} \sum_{m=-\infty}^{\infty} a_{m} e^{-i \omega_{m} \tau}, \quad \omega_{m}=2 \pi m / \hbar \beta $$

$$ \int \frac{d a_{n}^{*} d a_{n}}{\pi} e^{-a_{n}^{*} A_{n} a_{n}}=\frac{1}{A_{n}}, \quad \operatorname{Re} A_{n}>0 $$

$$ \prod_{n=0}^{N}\left[\int \frac{d a_{n}^{*} d a_{n}}{\pi}\right] e^{-\sum_{n} a_{n}^{*} A_{n} a_{n}}=\prod_{n=1}^{N+1} \frac{1}{A_{n}}, \quad \operatorname{Re} A_{n}>0 $$

$$ Z=\prod_{n=0}^{N}\left[\int \frac{d a_{n}^{*} d a_{n}}{\pi}\right] e^{-\sum_{n, m} a_{n}^{*} A_{n m} a_{m}}=\frac{1}{\operatorname{det} A} $$

$$ a_{n} \rightarrow \sum_{n^{\prime}} U_{n, n^{\prime}} a_{n^{\prime}} $$

$$ \operatorname{det} A \rightarrow \operatorname{det}\left(U A^{d} U^{\dagger}\right)=\operatorname{det} A^{d} $$

$$ A=\epsilon(1-\epsilon \omega) \bar{\nabla}+\epsilon \omega=\left(\begin{array}{cccccc} 1 & 0 & 0 & \cdots & 0 & -1+\epsilon \omega \\ -1+\epsilon \omega & 1 & 0 & \cdots & 0 & 0 \\ 0 & -1+\epsilon \omega & 1 & \cdots & 0 & 0 \\ 0 & 0 & -1+\epsilon \omega & \cdots & 0 & 0 \\ \vdots & & & & & \vdots \\ 0 & 0 & 0 & \cdots & -1+\epsilon \omega & 1 \end{array}\right) . $$

$$ \operatorname{det}_{N+1} A=1-(1-\epsilon \omega)^{N+1} $$

$$ Z_{\omega}^{N}=\frac{1}{\operatorname{det}_{N+1}[\epsilon(1-\epsilon \omega) \bar{\nabla}+\epsilon \omega]}=\frac{1}{1-(1-\epsilon \omega)^{N+1}} $$

$$ \bar{\omega}_{\mathrm{e}} \equiv-\frac{1}{\epsilon} \log (1-\epsilon \omega) $$

$$ Z_{\omega}^{N}=\frac{1}{1-e^{-\beta \hbar \bar{\omega}_{\mathrm{e}}}} $$

$$ Z_{\omega}=1+e^{-\beta \hbar \bar{\omega}_{\mathrm{e}}}+e^{-2 \beta \hbar \bar{\omega}_{\mathrm{e}}}+\ldots $$

$$ \hat{H}_{\omega}=\hbar \bar{\omega}_{\mathrm{e}} \hat{N}=\hbar \bar{\omega}_{\mathrm{e}} a^{\dagger} a $$

$$ \bar{\omega}_{\mathrm{e}} \xrightarrow{\epsilon \rightarrow 0} \omega, $$

$$ Z_{\omega}=\frac{1}{1-e^{-\beta \hbar \omega}} $$

$$ Z_{\omega}^{N}=\prod_{n=0}^{N}\left[\int \frac{d a_{n}^{\dagger} d a_{n}}{\pi}\right] \exp \left\{-\frac{1}{\hbar} \mathcal{A}^{N}\right\} $$

$$ \mathcal{A}_{\omega}^{N}=\hbar \sum_{n=1}^{N}\left[a_{n}^{*}\left(a_{n}-a_{n-1}\right)+\epsilon \Omega_{n} a_{n}^{*} a_{n-1}\right] $$

$$ \mathcal{A}_{\omega}^{N}=\hbar \epsilon \sum_{n=1}^{N} a_{n}^{*}\left[\left(1-\epsilon \Omega_{n}\right) \bar{\nabla}+\Omega_{n}\right] a_{n} . $$

$$ Z_{\omega}^{N}=\frac{1}{\operatorname{det}_{N+1}[\epsilon(1-\epsilon \Omega) \bar{\nabla}+\epsilon \Omega]}=\frac{1}{1-\prod_{n=0}^{N}\left(1-\epsilon \Omega_{n}\right)} $$

$$ \bar{\Omega}_{\mathrm{e}} \equiv-\frac{1}{(N+1) \epsilon} \sum_{n=0}^{N} \log \left(1-\epsilon \Omega_{n}\right), $$

$$ Z_{\omega}^{N}=\frac{1}{1-e^{-\beta \hbar \bar{\Omega}_{\mathrm{e}}}} $$

$$ (1-\epsilon \omega) \bar{\nabla}+\omega \rightarrow \partial_{\tau}+\omega $$

$$ \begin{align*} Z_{\omega} & =\oint \frac{\mathcal{D} a^{*} \mathcal{D} a}{\pi} \exp \left[-\int_{0}^{\hbar \beta} d \tau\left(a^{*} \partial_{\tau} a+\omega a^{*} a\right)\right] \\ & =\mathcal{N}_{\omega} \frac{1}{\operatorname{det}\left(\partial_{\tau}+\omega\right)} \end{align*} $$

$$ \prod_{m=-\infty, \neq 0}^{\infty} \frac{-i \omega_{m}+\omega}{-i \omega_{m}}=\frac{\sinh (\hbar \omega \beta / 2)}{\hbar \omega \beta / 2} $$

$$ \frac{\operatorname{det}\left(\partial_{\tau}+\omega\right)}{\operatorname{det}^{\prime}\left(\partial_{\tau}\right)}=\omega \frac{\sinh (\hbar \omega \beta / 2)}{\hbar \omega \beta / 2} $$

$$ \begin{align*} Z_{\omega} & =\oint \frac{\mathcal{D} a^{*} \mathcal{D} a}{\pi} \exp \left[-\int_{0}^{\hbar \beta} d \tau\left(a^{*} \partial_{\tau} a+\omega a^{*} a\right)\right] \\ & =\frac{k_{B} T}{\hbar} \frac{\operatorname{det}^{\prime}\left(\partial_{\tau}\right)}{\operatorname{det}\left(\partial_{\tau}+\omega\right)}=\frac{1}{2 \sinh (\hbar \omega \beta / 2)} \end{align*} $$

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

$$ \begin{align*} Z_{\omega} & =\frac{k_{B} T}{\hbar} \frac{\operatorname{det}^{\prime}\left(\partial_{\tau}\right)}{\operatorname{det}\left(\partial_{\tau}+\omega\right)} \\ & =\frac{k_{B} T}{\hbar}\left[\frac{\operatorname{det}^{\prime}\left(-\partial_{\tau}^{2}\right)}{\operatorname{det}\left(-\partial_{\tau}^{2}+\omega^{2}\right)}\right]^{1 / 2}=\frac{1}{2 \sinh (\hbar \omega \beta / 2)} \end{align*} $$

$$ \begin{align*} Z_{\Omega(\tau)} & =\oint \frac{\mathcal{D} a^{*}(\tau) \mathcal{D} a(\tau)}{\pi} \exp \left\{-\int_{0}^{\hbar \beta} d \tau\left[a^{\dagger} \partial_{\tau} a+\Omega(\tau) a^{\dagger} a\right]\right\} \\ & =\frac{k_{B} T}{\hbar}\left[\frac{\operatorname{det}^{\prime}\left(-\partial_{\tau}^{2}\right)}{\operatorname{det}\left(-\partial_{\tau}^{2}+\Omega^{2}(\tau)\right)}\right]^{1 / 2}=\frac{1}{2 \sinh (\hbar \omega \beta / 2)}\left[\frac{\operatorname{det}\left(-\partial_{\tau}^{2}+\omega^{2}\right)}{\operatorname{det}\left(-\partial_{\tau}^{2}+\Omega^{2}(\tau)\right)}\right]^{1 / 2} \end{align*} $$

$$ \int d x|x\rangle\langle x|=1, \quad \int \frac{d p}{2 \pi}|p\rangle\langle p|=1 $$

$$ \int d x \int \frac{d p}{2 \pi}|x p\rangle\langle x p|=1 $$

$$ |z\rangle \equiv e^{z \hat{a}^{\dagger}-z^{*} \hat{a}}|0\rangle, \quad\langle z| \equiv\langle 0| e^{-z \hat{a}^{\dagger}+z^{*} \hat{a}} $$

$$ e^{z \hat{a}^{\dagger}-z^{*} \hat{a}}=e^{z^{*} z\left[\hat{a}^{\dagger}, \hat{a}\right] / 2} e^{z \hat{a}^{\dagger}} e^{-z^{*} \hat{a}}=e^{-z^{*} z / 2} e^{z \hat{a}^{\dagger}} e^{-z^{*} \hat{a}} $$

$$ |z\rangle=e^{-z^{*} z / 2} e^{z \hat{a}^{\dagger}}|0\rangle=e^{-z^{*} z / 2} \sum_{n=0}^{\infty} \frac{z^{n}}{\sqrt{n!}}|n\rangle . $$

$$ |n\rangle=\left[|z\rangle e^{z^{*} z / 2} \overleftarrow{\partial}_{z}^{n}\right]_{z=0} \frac{1}{\sqrt{n!}}, \quad\langle n|=\frac{1}{\sqrt{n!}}\left[\partial_{z^{*}}^{n} e^{z^{*} z / 2}\langle z|\right]_{z=0} $$

$$ \operatorname{tr} \hat{\mathcal{O}}=\int \frac{d z^{*} d z}{\pi}\langle z| \hat{\mathcal{O}}|z\rangle=\int \frac{d z^{*} d z}{\pi} e^{-z^{*} z} \sum_{m, n=0}^{\infty} \frac{z^{* m}}{\sqrt{m!}} \frac{z^{n}}{\sqrt{n!}}\langle m| \hat{\mathcal{O}}|n\rangle $$

$$ \operatorname{tr} \hat{\mathcal{O}}=\int d r^{2}\left[\frac{d \phi}{2 \pi} e^{-i(m-n) \phi}\right] e^{-r^{2}} \sum_{m, n=0}^{\infty}\left(r^{2}\right)^{(m+n) / 2} \frac{1}{\sqrt{m!}} \frac{1}{\sqrt{n!}}\langle m| \hat{\mathcal{O}}|n\rangle $$

$$ \operatorname{tr} \hat{\mathcal{O}}=\int d r^{2} e^{-r^{2}} \sum_{n=0}^{\infty}\left(r^{2}\right)^{n} \frac{1}{n!}\langle n| \hat{\mathcal{O}}|n\rangle=\sum_{n=0}^{\infty}\langle n| \hat{\mathcal{O}}|n\rangle $$

$$ \left\langle z_{1} \mid z_{2}\right\rangle=e^{-z_{1}^{*} z_{1} / 2-z_{2}^{*} z_{2} / 2+z_{1}^{*} z_{2}} $$

$$ |x p\rangle \equiv|z\rangle, \quad \text { where } z \equiv(x+i p) / \sqrt{2} $$

$$ |x p\rangle=e^{-\left(x^{2}+p^{2}\right) / 4} \sum_{n=0}^{\infty} \frac{(x+i p)^{n}}{\sqrt{2^{n} n!}}|0\rangle $$

$$ \int d x \frac{d p}{2 \pi}|x p\rangle\langle x p|=\int d x \frac{d p}{2 \pi}|x p\rangle\langle x p| e^{-\left(x^{2}+p^{2}\right) / 2} \sum_{m, n=0}^{\infty} \frac{(x-i p)^{m}}{\sqrt{2^{m} m!}} \frac{(x+i p)^{n}}{\sqrt{2^{n} n!}}|m\rangle\langle n| $$

$$ \int \frac{d z^{*} d z}{\pi}|z\rangle\langle z|=1 $$

$$ \left\langle z_{b}\right| e^{-\beta \hat{H}_{\omega}}\left|z_{a}\right\rangle=\left\langle z_{b}\right| e^{-\beta \hat{H}_{\omega} /(N+1)} e^{-\beta \hat{H}_{\omega} /(N+1)} \cdots e^{-\beta \hat{H}_{\omega} /(N+1)}\left|z_{a}\right\rangle $$

$$ \left\langle z_{b}\right| e^{-\beta \hat{H}_{\omega}}\left|z_{a}\right\rangle=\prod_{n=1}^{N}\left[\int \frac{d z_{n}^{*} d z_{n}}{\pi}\right] \prod_{n=1}^{N+1}\left\langle z_{n}\right| e^{-\epsilon \hat{H}_{\omega}}\left|z_{n-1}\right\rangle, z_{0}=z_{a}, z_{N+1}=z_{b}, \epsilon \equiv \beta /(N+1) $$

$$ \left\langle z_{n}\right| e^{-\epsilon \hat{H}_{\omega}}\left|z_{n-1}\right\rangle \approx\left\langle z_{n}\right| 1-\epsilon \hat{H}_{\omega}\left|z_{n-1}\right\rangle=\left\langle z_{n} \mid z_{n-1}\right\rangle-\epsilon\left\langle z_{n}\right| \hat{H}_{\omega}\left|z_{n-1}\right\rangle $$

$$ \left\langle z_{n} \mid z_{n-1}\right\rangle=e^{-z_{n}^{*} z_{n} / 2-z_{n-1}^{*} z_{n-1} / 2+z_{n}^{*} z_{n-1}}=e^{-(1 / 2)\left[z_{n}^{*}\left(z_{n}-z_{n-1}\right)-\left(z_{n}^{*}-z_{n-1}^{*}\right) z_{n-1}\right]} $$

$$ \hat{a}|z\rangle=e^{-z^{*} z / 2} \sum_{n=0}^{\infty} \frac{z^{n}}{\sqrt{n!}} \hat{a}|n\rangle=e^{-z^{*} z / 2} \sum_{n=1}^{\infty} \frac{z^{n}}{\sqrt{(n-1)!}}|n-1\rangle=z|z\rangle $$

$$ \left\langle z_{n}\right| \hat{H}_{\omega}\left|z_{n-1}\right\rangle=\hbar \omega\left\langle z_{n}\right|\left(\hat{a}^{\dagger} \hat{a}+\hat{a} \hat{a}^{\dagger}\right)\left|z_{n-1}\right\rangle=\hbar \omega\left(z_{n}^{\dagger} z_{n-1}+\frac{1}{2}\right) $$

$$ \left\langle z_{b}\right| e^{-\beta \hat{H}_{\omega}}\left|z_{a}\right\rangle=\prod_{n=1}^{N}\left[\int \frac{d z_{n}^{*} d z_{n}}{\pi}\right] e^{-\mathcal{A}_{\omega}^{N}\left[z^{*}, z\right] / \hbar} $$

$$ \mathcal{A}_{\omega}^{N}\left[z^{*}, z\right]=\hbar \epsilon \sum_{n=1}^{N+1}\left\{\frac{1}{2}\left[z_{n}^{*} \bar{\nabla} z_{n}-\left(\bar{\nabla} z_{n}^{*}\right) z_{n-1}\right]+\omega\left(z_{n}^{*} z_{n-1}+\frac{1}{2}\right)\right\} . $$

$$ \mathcal{A}_{\omega}^{N}\left[z^{*}, z\right]=\frac{\hbar}{2}\left(-z_{b}^{*} z_{b}+z_{a}^{*} z_{a}\right)+\hbar \epsilon \sum_{n=1}^{N+1}\left\{z_{n}^{*} \bar{\nabla} z_{n}+\omega\left(z_{n}^{*} z_{n-1}+\frac{1}{2}\right)\right\} . $$

$$ \mathcal{A}_{\omega}^{0}\left[z^{*}, z\right]=\frac{\hbar}{2}\left(-z_{b}^{*} z_{b}+z_{a}^{*} z_{a}\right)+\hbar z_{b}^{*}\left(z_{b}-z_{a}\right)+\omega\left(z_{b}^{*} z_{a}+\frac{1}{2}\right) $$

$$ \begin{align*} \langle 0| e^{-\epsilon \hat{H}_{\omega}}|0\rangle & =\left[e^{\left(z_{b}^{*} z_{b}+z_{a}^{*} z_{a}\right) / 2}\left\langle z_{b}\right| e^{-\epsilon \hat{H}_{\omega}}\left|z_{a}\right\rangle\right]_{z^{*}=0, z=0}=e^{-\epsilon \hbar \omega / 2} \\ \langle 1| e^{-\epsilon \hat{H}_{\omega}}|1\rangle & =\left\{\partial_{z}\left[e^{\left(z_{b}^{*} z_{b}+z_{a}^{*} z_{a}\right) / 2}\left\langle z_{b}\right| e^{-\epsilon \hat{H}_{\omega}}\left|z_{a}\right\rangle\right] \overleftarrow{\partial}_{z}^{*}\right\}_{z^{*}=0, z=0}=e^{-\epsilon 3 \hbar \omega / 2} \\ \langle 0| e^{-\epsilon \hat{H}_{\omega}}|1\rangle & =\langle 1| e^{-\epsilon \hat{H}_{\omega}}|0\rangle=0 \end{align*} $$

$$ Z_{\omega}=\int \frac{d z^{*} d z}{\pi}\langle z| e^{-\beta \hat{H}_{\omega}}|z\rangle $$

$$ \begin{align*} {\left[\hat{\psi}(\mathbf{x}, t), \hat{\psi}^{\dagger}\left(\mathbf{x}^{\prime}, t\right)\right]_{+} } & =\delta_{\mathbf{x}^{\prime}} \\ {\left[\hat{\psi}^{\dagger}(\mathbf{x}, t), \hat{\psi}^{\dagger}\left(\mathbf{x}^{\prime}, t\right)\right]_{+} } & =0 \\ {\left[\hat{\psi}(\mathbf{x}, t), \hat{\psi}\left(\mathbf{x}^{\prime}, t\right)\right]_{+} } & =0 \end{align*} $$

$$ \begin{align*} {\left[\hat{a}_{\mathbf{p}}(t), \hat{a}_{\mathbf{p}^{\prime}}^{\dagger}(t)\right]_{+} } & =\delta_{\mathbf{p p}^{\prime}} \\ {\left[\hat{a}_{\mathbf{p}}^{\dagger}(t), \hat{a}_{\mathbf{p}^{\prime}}^{\dagger}(t)\right]_{+} } & =0 \\ {\left[\hat{a}_{\mathbf{p}}(t), \hat{a}_{\mathbf{p}^{\prime}}(t)\right]_{+} } & =0 \end{align*} $$

$$ [\hat{A}, \hat{B}]_{+} \equiv \hat{A} \hat{B}+\hat{B} \hat{A} $$

$$ \theta_{1} \theta_{2}=-\theta_{2} \theta_{1}, $$

$$ \theta^{2}=0 . $$

$$ F(\theta)=F_{0}+F_{1} \theta $$

$$ \begin{align*} F_{0} & =F(0) \\ F_{1} & =F^{\prime} \equiv \frac{\partial}{\partial \theta} F \end{align*} $$

$$ \begin{align*} \int \frac{d \theta}{\sqrt{2 \pi}} & =0 \\ \int \frac{d \theta}{\sqrt{2 \pi}} \theta & =1 \end{align*} $$

$$ \int \frac{d \theta}{\sqrt{2 \pi}} F(\theta)=F_{1}=F^{\prime} $$

$$ \int \frac{d \theta}{\sqrt{2 \pi}} F(c \cdot \theta)=c \cdot F^{\prime}=c \cdot \int \frac{d \theta^{\prime}}{\sqrt{2 \pi}} F\left(\theta^{\prime}\right) $$

$$ \int \frac{d \theta}{\sqrt{2 \pi}} F\left(\theta^{\prime}(\theta)\right)=\int \frac{d \theta^{\prime}}{\sqrt{2 \pi}}\left[\frac{d \theta}{d \theta^{\prime}}\right]^{-1} F\left(\theta^{\prime}\right) $$

$$ \int \frac{d \theta}{\sqrt{2 \pi}} G(\theta) \frac{\partial}{\partial \theta} F(\theta)=\int \frac{d \theta}{\sqrt{2 \pi}}\left[\frac{\partial}{\partial \theta} G(\theta)\right] F(\theta) $$

$$ \int \frac{d \theta^{\prime}}{\sqrt{2 \pi}} \delta\left(\theta-\theta^{\prime}\right) F\left(\theta^{\prime}\right) \equiv F(\theta) $$

$$ \delta\left(\theta-\theta^{\prime}\right)=\theta^{\prime}-\theta $$

$$ \delta^{\prime}\left(\theta-\theta^{\prime}\right) \equiv \partial_{\theta} \delta\left(\theta-\theta^{\prime}\right)=-1 $$

$$ \int \frac{d \theta^{\prime}}{\sqrt{2 \pi}} \delta^{\prime}\left(\theta-\theta^{\prime}\right) F\left(\theta^{\prime}\right)=-F^{\prime}(\theta) $$

$$ a^{*}=\frac{1}{\sqrt{2}}\left(\theta_{1}-i \theta_{2}\right), \quad a=\frac{1}{\sqrt{2}}\left(\theta_{1}+i \theta_{2}\right) $$

$$ \int \frac{d a^{*} d a}{\pi} \equiv \int \frac{d \theta_{2} d \theta_{1}}{2 \pi i} \equiv-\int \frac{d a d a^{*}}{\pi} $$

$$ \begin{align*} \int \frac{d a^{*} d a}{\pi} & =0, \quad \int \frac{d a^{*} d a}{\pi} a=0, \quad \int \frac{d a^{*} d a}{\pi} a^{*}=0 \\ \int \frac{d a^{*} d a}{\pi} a^{*} a & =\int \frac{d \theta_{2} d \theta_{1}}{2 \pi i} i \theta_{1} \theta_{2}=1 \end{align*} $$

$$ F\left(a^{*} a\right)=F_{0}+F_{1} a^{*} a $$

$$ e^{-a^{*} A a}=1-a^{*} A a $$

$$ \int \frac{d a^{*} d a}{\pi} e^{-a^{*} A a}=A $$

$$ Z^{\mathrm{f}}=\prod_{n}\left[\int \frac{d a_{n}^{*} d a_{n}}{\pi}\right] e^{i \Sigma_{n, n^{\prime}} a_{n}^{*} A_{n, n^{\prime}} a_{n^{\prime}}}=\operatorname{det} A $$

$$ a(\hbar \beta)=-a(0) $$

$$ a_{N+1}=-a_{0} $$

$$ A^{\mathrm{f}}=\epsilon(1-\epsilon \omega) \bar{\nabla}_{\tau}+\epsilon \omega=\left(\begin{array}{cccccc} 1 & 0 & 0 & \cdots & 0 & 1-\epsilon \omega \\ -1+\epsilon \omega & 1 & 0 & \cdots & 0 & 0 \\ 0 & -1+\epsilon \omega & 1 & \cdots & 0 & 0 \\ 0 & 0 & -1+\epsilon \omega & \cdots & 0 & 0 \\ \vdots & & & & & \vdots \\ 0 & 0 & 0 & \cdots & -1+\epsilon \omega & 1 \end{array}\right), $$

$$ \operatorname{det}(-\epsilon \bar{\nabla})_{\omega=0}=0 $$

$$ \operatorname{det}(-\epsilon \bar{\nabla})_{\omega=0}=2 $$

$$ \operatorname{det}_{N+1} A=1+(1-\epsilon \omega)^{N+1} $$

$$ Z_{\omega}^{\mathrm{f}, N}=\operatorname{det}_{N+1}[\epsilon(1-\epsilon \omega) \bar{\nabla}+\epsilon \omega]=1+(1-\epsilon \omega)^{N+1} . $$

$$ \bar{\omega}_{\mathrm{e}} \equiv-\frac{1}{\epsilon} \log (1-\epsilon \omega) $$

$$ Z_{\omega}^{N}=1+e^{-\beta \hbar \bar{\omega}_{\mathrm{e}}} . $$

$$ \hat{H}_{\omega}=\hbar \bar{\omega}_{e} \hat{N}=\hbar \bar{\omega}_{e} a^{\dagger} a $$

$$ Z_{\omega}=1+e^{-\beta \hbar \omega} $$

$$ Z_{\omega}^{\mathrm{f}, N}=\prod_{n=0}^{N}\left[\int \frac{d a_{n}^{*} d a_{n}}{\pi}\right] \exp \left(-\frac{1}{\hbar} \mathcal{A}_{\omega}^{N}\right), $$

$$ \mathcal{A}_{\omega}^{N}=\hbar \sum_{n=1}^{N}\left[a_{n}^{*}\left(a_{n}-a_{n-1}\right)+\epsilon \Omega_{n} a_{n}^{*} a_{n-1}\right] $$

$$ \mathcal{A}_{\omega}^{N}=\hbar \epsilon \sum_{n=1}^{N} a_{n}^{*}\left[\left(1-\epsilon \Omega_{n}\right) \bar{\nabla}+\Omega_{n}\right] a_{n} . $$

$$ Z_{\omega}^{\mathrm{f}, N}=\operatorname{det}_{N+1}[\epsilon(1-\epsilon \Omega) \bar{\nabla}+\epsilon \omega]=1-\prod_{n=0}^{N}\left(1-\epsilon \Omega_{n}\right) $$

$$ \bar{\Omega}_{\mathrm{e}} \equiv-\frac{1}{(N+1) \epsilon} \sum_{n=0}^{N} \log \left(1-\epsilon \Omega_{n}\right) $$

$$ Z_{\omega}^{\mathrm{f}, N}=1+e^{-\beta \hbar \bar{\Omega}_{\mathrm{e}}} $$

$$ (1-\epsilon \omega) \bar{\nabla}+\omega \rightarrow \partial_{\tau}+\omega $$

$$ \omega_{m}^{\mathrm{f}}=\pi(2 m+1) k_{B} T / \hbar, \quad m=0, \pm 1, \pm 2, \ldots $$

$$ \begin{align*} Z_{\omega}^{\mathrm{f}} & =\oint \frac{\mathcal{D} a^{*} \mathcal{D} a}{\pi} \exp \left[-\int_{0}^{\hbar \beta} d \tau\left(a^{*} \partial_{\tau} a+\omega a^{*} a\right)\right] \\ & =\mathcal{N}_{\omega} \operatorname{det}\left(\partial_{\tau}+\omega\right) \end{align*} $$

$$ \prod_{m=-\infty}^{\infty} \frac{-i \omega_{m}^{\mathrm{f}}+\omega}{-i \omega_{m}^{\mathrm{f}}}=\cosh (\hbar \omega \beta / 2) $$

$$ \frac{\operatorname{det}\left(\partial_{\tau}+\omega\right)}{\operatorname{det}\left(\partial_{\tau}\right)}=\cosh (\hbar \omega \beta / 2) $$

$$ Z_{\omega}^{f}=2 \cosh (\hbar \omega \beta / 2) $$

$$ \begin{align*} Z_{\omega}^{\mathrm{f}} & =\oint \frac{\mathcal{D} a^{*} \mathcal{D} a}{\pi} \exp \left[-\int_{0}^{\hbar \beta} d \tau\left(a^{*} \partial_{\tau} a+\omega a^{*} a\right)\right]=2 \frac{\operatorname{det}\left(\partial_{\tau}+\omega\right)}{\operatorname{det}\left(\partial_{\tau}\right)} \\ & =2 \cosh (\hbar \omega \beta / 2) \end{align*} $$

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

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

$$ Z_{\omega}^{\mathrm{f}, N}=2 \cosh \left(\hbar \tilde{\omega}_{\mathrm{e}} \beta\right) $$

$$ \sinh \left(\tilde{\omega}_{\mathrm{e}} / 2\right)=\epsilon \omega / 2 $$

$$ \begin{gather*} \prod_{m=0}^{N / 2-1}\left(1-\frac{\sin ^{2} x}{\sin ^{2} \frac{(2 m+1) \pi}{2(N+1)}}\right)=\frac{\cos (N+1) x}{\cos x}, \quad N=\text { even } \\ \prod_{m=0}^{(N-1) / 2}\left(1-\frac{\sin ^{2} x}{\sin ^{2} \frac{(2 m+1) \pi}{2(N+1)}}\right)=\cos (N+1) x, \quad N=\operatorname{odd} \end{gather*} $$

$$ \prod_{m=0}^{N}\left(1-\frac{\sin ^{2} x}{\sin ^{2} \frac{(2 m+1) \pi}{2(N+1)}}\right)^{1 / 2}=\cos (N+1) x $$

$$ E^{(0)}=-\frac{\hbar \omega}{2} $$

$$ |\zeta\rangle \equiv e^{-\zeta^{*} \zeta / 2} e^{a^{\dagger} \zeta}|0\rangle=e^{-\zeta^{*} \zeta / 2}(|0\rangle-\zeta|1\rangle) . $$

$$ \langle\zeta| \equiv e^{-\zeta^{*} \zeta / 2}\langle 0| e^{\zeta^{*} a}=e^{-\zeta^{*} \zeta / 2}\left(\langle 0|+\zeta^{*}\langle 1|\right) . $$

$$ |n\rangle=\left[|\zeta\rangle e^{\zeta^{*} \zeta / 2} \overleftarrow{\partial}_{\zeta}^{n}\right]_{\zeta=0} \frac{1}{\sqrt{n!}}, \quad\langle n|=\frac{1}{\sqrt{n!}}\left[\partial_{\zeta^{*}}^{n} e^{\zeta^{*} \zeta / 2}\langle\zeta|\right]_{\zeta=0} . $$

$$ \begin{array}{ll} |0\rangle=\left[|\zeta\rangle e^{c^{*} \zeta / 2}\right]_{\zeta=0}, & \langle 0|=\left[e^{\zeta^{*} \zeta / 2}\langle\zeta|\right]_{\zeta=0} \\ |1\rangle=\left[|\zeta\rangle e^{c^{*} \zeta / 2} \overleftarrow{\partial}_{\zeta}\right]_{\zeta=0}, & \langle 1|=\left[\partial_{\zeta^{*}} e^{\zeta^{*} \zeta / 2}\langle\zeta|\right]_{\zeta=0} \end{array} $$

$$ \operatorname{tr} \hat{\mathcal{O}}=\int \frac{d \zeta^{*} d \zeta}{\pi}\langle-\zeta| \hat{\mathcal{O}}|\zeta\rangle=\int \frac{d \zeta^{*} d \zeta}{\pi} e^{-\zeta^{*} \zeta}\left(\langle 0|-\zeta^{*}\langle 1|\right) \hat{\mathcal{O}}(|0\rangle-\zeta|1\rangle) $$

$$ \operatorname{tr} \hat{\mathcal{O}}=\langle 0| \hat{\mathcal{O}}|0\rangle+\langle 1| \hat{\mathcal{O}}|1\rangle $$

$$ \begin{align*} \left\langle\zeta_{1} \mid \zeta_{2}\right\rangle & =e^{-\zeta_{1}^{*} \zeta_{1} / 2-\zeta_{2}^{*} \zeta_{2} / 2+\zeta_{1}^{*} \zeta_{2}} \\ & =e^{-\zeta_{1}^{*}\left(\zeta_{1}-\zeta_{2}\right) / 2+\left(\zeta_{1}^{*}-\zeta_{2}^{*}\right) \zeta_{2} / 2} \end{align*} $$

$$ \begin{align*} & \int \frac{d \zeta^{*} d \zeta}{\pi}|\zeta\rangle\langle\zeta|=\int \frac{d \zeta^{*} d \zeta}{\pi} e^{-\zeta^{*} \zeta}\left[|0\rangle\langle 0|-\zeta|1\rangle\langle 0|+\zeta^{*}|0\rangle\langle 1|\right] \\ & \quad=\int \frac{d \zeta^{*} d \zeta}{\pi}\left[|0\rangle\langle 0|+|1\rangle\langle 0| \zeta+\zeta^{*}|0\rangle\langle 1|+\zeta \zeta^{*}(|0\rangle\langle 0|+|1\rangle\langle 0|)\right]=1 \end{align*} $$

$$ \left\langle\zeta_{b}\right| e^{-\beta \hat{H}_{\omega}}\left|\zeta_{a}\right\rangle=\left\langle\zeta_{b}\right| e^{-\epsilon \hat{H}_{\omega}} e^{-\epsilon \hat{H}_{\omega}} \cdots e^{-\epsilon \hat{H}_{\omega}}\left|\zeta_{a}\right\rangle $$

$$ \left\langle z_{b}\right| e^{-\beta \hat{H}_{\omega}}\left|z_{a}\right\rangle=\prod_{n=1}^{N}\left[\int \frac{d z_{n}^{*} d z_{n}}{\pi}\right] e^{-\mathcal{A}_{\mathrm{e}}\left[z^{*}, z\right] / \hbar} $$

$$ \mathcal{A}_{\omega}^{N}\left[\zeta^{*}, \zeta\right]=\frac{\hbar}{2}\left(-\zeta_{b}^{*} \zeta_{b}+\zeta_{a}^{*} \zeta_{a}\right)+\hbar \epsilon \sum_{n=1}^{N+1}\left\{\zeta_{n}^{*} \bar{\nabla} \zeta_{n}+\omega\left(\zeta_{n}^{*} \zeta_{n-1}+\frac{1}{2}\right)\right\} $$

$$ Z_{\omega}^{\mathrm{f}}=\int \frac{d \zeta^{*} d \zeta}{\pi}\langle-\zeta| e^{-\beta \hat{H}_{\omega}}|\zeta\rangle $$

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

$$ \int \frac{\mathcal{D} \theta}{2 \pi} \exp \left[\frac{i}{\hbar} \int d t \frac{i \hbar}{2} \theta(t) \dot{\theta}(t)\right] $$

$$ \mathcal{L}(t)=\frac{i}{2} \hbar \theta(t) \dot{\theta}(t) $$

$$ p_{\theta}=\frac{\partial \mathcal{L}}{\partial \dot{\theta}}=-\frac{i \hbar}{2} \theta . $$

$$ \chi=p_{\theta}+\frac{i \hbar}{2} \theta=0 . $$

$$ \{A, B\}_{D}=\{A, B\}-\left\{A, \chi_{p}\right\} C^{p q}\left\{\chi_{q}, B\right\} $$

$$ C^{p q}=\left\{\chi_{p}, \chi_{q}\right\} $$

$$ \{\chi, \chi\}=\left\{p_{\theta}+\frac{i \hbar}{2} \theta, p_{\theta}+\frac{i \hbar}{2} \theta\right\}=-i \hbar\left\{p_{\theta}, \theta\right\}=-i \hbar $$

$$ \left\{p_{\theta}, \theta\right\}_{D}=\left\{p_{\theta}, \theta\right\}-\frac{i}{\hbar}\left\{p_{\theta}, \chi\right\}\{\chi, \theta\}=0-\frac{-i}{\hbar}\left(\frac{i \hbar}{2}\right)(\hbar)=-\frac{\hbar}{2} $$

$$ [\hat{p}(t), \hat{\theta}(t)]_{+}=-\frac{i \hbar}{2} $$

$$ [\hat{\theta}(t), \hat{\theta}(t)]_{+}=1 $$

$$ \psi(\theta)=\psi_{0}+\psi_{1} \theta . $$

$$ \left\langle\psi^{\prime} \mid \psi\right\rangle \equiv \int \frac{d \theta}{2 \pi} \psi^{\prime *}(\theta) \psi(\theta)=\psi_{0}^{\prime *} \psi_{1}+\psi_{1}^{\prime *} \psi_{0} $$

$$ \hat{p}=i \hbar \partial_{\theta}, $$

$$ \left\langle\psi^{\prime}\right| \hat{p}|\psi\rangle \equiv \int \frac{d \theta}{2 \pi} \psi^{\prime *}(\theta) i \hbar \frac{\partial}{\partial \theta} \psi(\theta)=i \hbar \psi_{1}^{\prime *} \psi_{1} $$

$$ \left\langle\hat{p} \psi^{\prime} \mid \psi\right\rangle \equiv \int \frac{d \theta}{2 \pi}\left[i \hbar \frac{\partial}{\partial \theta} \psi^{\prime}(\theta)\right]^{*} \psi(\theta)=-i \hbar \psi_{1}^{\prime *} \psi_{1} $$

$$ \hat{\theta}|\theta\rangle=\theta|\theta\rangle . $$

$$ \langle\theta| \hat{\theta}=\langle\theta| \theta=\theta\langle\theta| . $$

$$ \left(\theta^{\prime}-\theta\right)\left\langle\theta^{\prime} \mid \theta\right\rangle=0 $$

$$ \left\langle\theta^{\prime} \mid \theta\right\rangle=-\theta^{\prime}+\theta+S_{2} \theta \theta^{\prime} $$

$$ \left\langle\theta^{\prime} \mid \theta\right\rangle=\int \frac{d \theta^{\prime \prime}}{2 \pi}\left\langle\theta^{\prime} \mid \theta^{\prime \prime}\right\rangle\left\langle\theta^{\prime \prime} \mid \theta\right\rangle $$

$$ \left\langle\theta^{\prime} \mid \theta\right\rangle=\delta\left(\theta-\theta^{\prime}\right) $$

$$ \left\langle\theta^{\prime} \mid \theta\right\rangle^{*}=-\left\langle\theta \mid \theta^{\prime}\right\rangle=\left\langle-\theta \mid-\theta^{\prime}\right\rangle, $$

$$ \hat{\theta}|\theta\rangle=\theta|\theta\rangle=-|\theta\rangle \theta $$

$$ \left\langle\theta^{\prime}\right| \hat{p}|\theta\rangle=-i \hbar \partial_{\theta^{\prime}}\left\langle\theta^{\prime} \mid \theta\right\rangle=i \hbar $$

$$ \langle\theta| \hat{p}|p\rangle=\int \frac{d \theta^{\prime}}{2 \pi}\langle\theta| \hat{p}\left|\theta^{\prime}\right\rangle\left\langle\theta^{\prime} \mid p\right\rangle=i \hbar \int \frac{d \theta^{\prime}}{2 \pi}\left\langle\theta^{\prime} \mid p\right\rangle=i \hbar \partial_{\theta^{\prime}}\left\langle\theta^{\prime} \mid p\right\rangle $$

$$ \langle\theta \mid p\rangle=e^{i \theta p / \hbar} $$

$$ \psi_{+}(\theta) \equiv \frac{1}{\sqrt{2}}(1+\theta), \quad \psi_{-}(\theta) \equiv \frac{1}{\sqrt{2}}(1-\theta) . $$

$$ \int \frac{d \theta}{2 \pi} \psi_{ \pm}^{*}(\theta) \psi_{ \pm}(\theta)= \pm 1 $$

$$ \mathcal{L}(t)=\hbar\left[a^{*}(t) i \partial_{t} a(t)-\omega a^{*}(t) a(t)\right] $$

$$ i \dot{a}(t)=\omega a(t) $$

$$ a(t)=e^{-i \omega t} a(0), \quad a^{\dagger}(t)=e^{-i \omega t} a^{\dagger}(0) $$

$$ p_{a}(t)=\frac{\partial \mathcal{L}(t)}{\partial \dot{a}(t)}=-i \hbar a(t) $$

$$ \left[\hat{p}_{a}(t), \hat{a}(t)\right]_{+}=-i \hbar $$

$$ \left[\hat{a}^{\dagger}(t), \hat{a}(t)\right]_{+}=1 $$

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

$$ \hat{N} \equiv a^{\dagger}(t) a(t) $$

$$ \left[\hat{N}, a^{\dagger}(t)\right]=a^{\dagger}(t), \quad[\hat{N}, a(t)]=-a(t) $$

$$ a|0\rangle=0 $$

$$ |1\rangle \equiv a^{\dagger}|0\rangle $$

$$ \hat{p}_{a} \equiv-i \hbar \partial_{a} $$

$$ \prod_{i=1}^{3}\left[\int \mathcal{D} \theta^{i}\right] \exp \left[\frac{i}{\hbar} \int d t \frac{i \hbar}{2} \theta^{i}(t) \dot{\theta}^{i}(t)\right] $$

$$ \dot{\theta}^{i}(t)=0 $$

$$ \left[\hat{\theta}^{i}(t), \hat{\theta}^{j}(t)\right]_{+}=2 \delta^{i j} $$

$$ \langle B| \hat{\theta}^{i}|A\rangle=\sigma_{B A}^{i}, \quad A, B=1,2 $$

$$ H_{B}=-\mathbf{S} \cdot \mathbf{B}(t) $$

$$ S^{i} \equiv-\frac{i}{4} \epsilon^{i j k} \theta^{j} \theta^{k} $$

$$ \left[\hat{S}^{i}, \hat{S}^{j}\right]=i \epsilon_{i j k} \hat{S}^{k} $$

$$ \left[\hat{S}^{i}, \hat{\theta}^{j}\right]=i \epsilon_{i j k} \hat{\theta}^{k} $$

$$ \dot{\boldsymbol{\theta}}=\mathbf{B} \times \boldsymbol{\theta}, $$

$$ \dot{\mathbf{S}}=\mathbf{B} \times \mathbf{S} . $$

$$ \int_{\theta_{a}^{i}=\theta^{i}\left(\tau_{a}\right)}^{\theta_{b}^{i}=\theta^{i}\left(\tau_{b}\right)} \mathcal{D}^{3} \theta \exp \left[\frac{i}{\hbar} \int d t\left(\frac{i \hbar}{4} \theta^{i} \dot{\theta}^{i}+B^{i} \frac{i}{4} \epsilon^{i j k} \theta^{j} \theta^{k}\right)\right] $$

$$ \hat{T} \exp \left(-\frac{i}{\hbar} \int d t \mathbf{B}(t) \cdot \frac{\boldsymbol{\sigma}}{2}\right) $$

$$ S^{i j} \equiv \epsilon^{i j k} S^{k}=\frac{1}{2 i} \theta^{i} \theta^{j} $$

$$ \left[\hat{S}^{i j}, \hat{S}^{k l}\right]=i\left(\delta^{i k} \hat{S}^{j l}-\delta^{i l} \hat{S}^{j k}+\delta^{j l} \hat{S}^{i k}-\delta^{j k} \hat{S}^{i l}\right) $$

$$ \langle B| \hat{S}^{i j}|A\rangle=\frac{1}{2} \sigma_{B A}^{i j} \equiv \frac{1}{4 i}\left[\sigma^{i}, \sigma^{j}\right]_{B A}=\epsilon^{i j k} \frac{\sigma^{k}}{2} $$

$$ \sigma^{12}=\sigma^{3} $$

$$ F_{i j} \equiv \epsilon^{i j k} B^{k} $$

$$ \hat{T} \exp \left(-\frac{i}{4 \hbar} \int d t F_{i j}(t) \sigma^{i j}\right)=\int_{\theta_{a}^{i}=\theta^{i}\left(\tau_{a}\right)}^{\theta_{b}^{i}=\theta^{i}\left(\tau_{b}\right)} \mathcal{D}^{3} \theta \exp \left[\frac{i}{\hbar} \int d t\left(\frac{i \hbar}{4} \theta^{i} \dot{\theta}^{i}+\frac{i}{4} F_{i j} \theta^{i} \theta^{j}\right)\right] $$

$$ \oint \mathcal{D}^{3} \theta \exp \left[\frac{i}{\hbar} \int d t\left(\frac{i \hbar}{4} \theta^{i} \dot{\theta}^{i}+B^{i} \frac{i}{4} \epsilon^{i j k} \theta^{j} \theta^{k}\right)\right]=2 \cos \left[|\mathbf{B}|\left(t_{b}-t_{a}\right) / 2 \hbar\right] $$

$$ 2 \operatorname{Det}^{1 / 2}\left[\delta_{i j} i \partial_{t}+\frac{i}{\hbar} F_{i j}(x(t))\right] $$

$$ \int \mathcal{D}^{3} \theta \exp \left\{\frac{i}{\hbar} \int d t\left[\frac{i \hbar}{4} \theta^{i} \dot{\theta}^{i}\right]\right\}=2 $$

$$ \left(\begin{array}{ccc} \omega_{m}^{\mathrm{f}} & i B & 0 \\ -i B & \omega_{m}^{\mathrm{f}} & 0 \\ 0 & 0 & \omega_{m}^{\mathrm{f}} \end{array}\right), \quad \omega_{m}^{\mathrm{f}}=\frac{\pi(2 m+1)}{t_{b}-t_{a}} $$

$$ \operatorname{Det}^{1 / 2}\left[\delta_{i j} i \partial_{t}+\frac{i}{\hbar} F_{i j}\right]=\left[\prod_{m=-\infty}^{\infty}\left|\omega_{m}\right|\left(\omega_{m}^{2}-B^{2} / \hbar^{2}\right)\right]^{1 / 2}=\cos \left[B\left(t_{b}-t_{a}\right) / 2 \hbar\right] $$

$$ \prod_{\mu=0}^{3}\left[\int \mathcal{D} \theta^{\mu}\right] \exp \left\{\frac{i}{\hbar} \int d t\left[-\frac{i \hbar}{4} \theta_{\mu}(t) \dot{\theta}^{\mu}(t)\right]\right\} $$

$$ \dot{\theta}^{\mu}(t)=0 $$

$$ \left\{\hat{\theta}^{\mu}(t), \hat{\theta}^{\nu}(t)\right\}=2 g^{\mu \nu} $$

$$ g^{\mu \nu}=\left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{array}\right) . $$

$$ \langle\beta| \hat{\theta}^{\mu}(t)|\alpha\rangle=\left(\gamma^{\mu}\right)_{\beta \alpha}, \quad \beta, \alpha=1,2,3,4 $$

$$ \gamma^{0} \equiv\left(\begin{array}{rr} 0 & 1 \\ -1 & 0 \end{array}\right), \gamma^{i} \equiv\left(\begin{array}{cc} 0 & \sigma^{i} \\ -\sigma^{i} & 0 \end{array}\right), \gamma_{5} \equiv i \gamma^{0} \gamma^{1} \gamma^{2} \gamma^{3}=\left(\begin{array}{rr} -1 & 0 \\ 0 & 1 \end{array}\right) \equiv \gamma^{5} . $$

$$ \prod_{\mu=0}^{3}\left[\int \mathcal{D} \theta_{5}\right] \exp \left[\frac{i}{\hbar} \int d t \frac{i \hbar}{4} \theta_{5}(t) \dot{\theta}_{5}(t)\right] $$

$$ \langle\beta| \hat{\theta}_{5}(t)|\alpha\rangle=\left(\gamma_{5}\right)_{\beta \alpha}, \quad \beta, \alpha=1,2,3,4 $$

$$ \hat{T} \exp \left(-\frac{i}{2 \hbar} \int d t F_{\mu \nu} \Sigma^{\mu \nu}\right)=\int \mathcal{D}^{4} \theta \exp \left\{\frac{i}{\hbar} \int d t\left[-\frac{i \hbar}{2} \theta_{\mu} \dot{\theta}^{\mu}+\frac{i}{4} F_{\mu \nu} \theta^{\mu} \theta^{\nu}\right]\right\} $$

$$ \Sigma^{\mu \nu} \equiv \frac{i}{4}\left[\gamma^{\mu}, \gamma^{\nu}\right]=-\Sigma^{\nu \mu} $$

$$ \Sigma^{12}=\frac{1}{2}\left(\begin{array}{cc} \sigma^{12} & 0 \\ 0 & \sigma^{12} \end{array}\right)=\frac{1}{2}\left(\begin{array}{cc} \sigma^{3} & 0 \\ 0 & \sigma^{3} \end{array}\right) $$

$$ \int \mathcal{D}^{4} \theta \exp \left\{\frac{i}{\hbar} \int d t\left[-\frac{i \hbar}{4} \theta_{\mu} \dot{\theta}^{\mu}\right]\right\}=4 $$

$$ \int \mathcal{D}^{4} \theta e^{(i / 4 \hbar) \int d t\left\{\left[-i \hbar \theta_{\mu} \dot{\theta}^{\mu}+i F_{\mu \nu} \theta^{\mu} \theta^{\nu}\right]\right\}}=4 \operatorname{Det}^{1 / 2}\left[-g_{\mu \nu} i \partial_{t}+\frac{i}{\hbar} F_{\mu \nu}(x(t))\right] $$

$$ F_{\mu \nu}=\left(\begin{array}{cccc} 0 & 0 & 0 & E \\ 0 & 0 & -B & 0 \\ 0 & B & 0 & 0 \\ E & 0 & 0 & 0 \end{array}\right) $$

$$ e^{F_{\mu \nu} t}=\left(\begin{array}{cccc} \cosh E t & 0 & 0 & \sinh E t \\ 0 & \cos B t & -\sin B t & 0 \\ 0 & \sin B t & \cos B t & 0 \\ \sinh E t & 0 & 0 & \cosh E t \end{array}\right), $$

$$ \operatorname{Det}\left(-g_{\mu \nu} i \partial_{t}+\frac{i}{\hbar} F_{\mu \nu}\right)=\operatorname{det} \cos \left(F_{\mu \nu} \frac{t_{b}-t_{a}}{2 \hbar}\right)=\cos ^{2}\left(B \frac{t_{b}-t_{a}}{2 \hbar}\right) \cosh ^{2}\left(E \frac{t_{b}-t_{a}}{2 \hbar}\right) . $$

$$ \begin{align*} \mathbf{E}_{\mathrm{CF}} & =\gamma\left(\mathbf{E}+\frac{\mathbf{v}}{c} \times \mathbf{B}\right)-\frac{\gamma^{2}}{\gamma+1} \frac{\mathbf{v}}{c}\left(\frac{\mathbf{v}}{c} \cdot \mathbf{E}\right) \\ \mathbf{B}_{\mathrm{CF}} & =\gamma\left(\mathbf{B}-\frac{\mathbf{v}}{c} \times \mathbf{E}\right)-\frac{\gamma^{2}}{\gamma+1} \frac{\mathbf{v}}{c}\left(\frac{\mathbf{v}}{c} \cdot \mathbf{B}\right) \end{align*} $$

$$ \frac{\mathbf{v} / c}{1+(|\mathbf{v}| / c)^{2}}=\frac{\mathbf{E} \times \mathbf{B}}{|\mathbf{E}|^{2}+|\mathbf{B}|^{2}}, $$

$$ S \equiv-\frac{1}{4} F_{\mu \nu} F^{\mu \nu}=\frac{1}{2}\left(\mathbf{E}^{2}-\mathbf{B}^{2}\right)=\frac{1}{2}\left(\mathcal{E}^{2}-\mathcal{B}^{2}\right), \quad P \equiv-\frac{1}{4} F_{\mu \nu} \tilde{F}^{\mu \nu}=\mathbf{E} \mathbf{B}=\mathcal{E} \mathcal{B} $$

$$ \left\{\begin{array}{l} \mathcal{E} \\ \mathcal{B} \end{array}\right\} \equiv \sqrt{\sqrt{S^{2}+P^{2}} \pm S}=\frac{1}{\sqrt{2}} \sqrt{\sqrt{\left(\mathbf{E}^{2}-\mathbf{B}^{2}\right)^{2}+4(\mathbf{E} \mathbf{B})^{2}} \pm\left(\mathbf{E}^{2}-\mathbf{B}^{2}\right)} . $$

$$ \hat{a}^{\dagger}=\frac{1}{2}\left(\sigma^{1}+i \sigma^{2}\right)=\frac{1}{2} \sigma^{+}=\left(\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right), \hat{a}=\frac{1}{2}\left(\sigma^{1}-i \sigma^{2}\right)=\frac{1}{2} \sigma^{-}=\left(\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right), $$

$$ \left[\hat{a}, \hat{a}^{\dagger}\right]_{+}=1, \quad\left[\hat{a}^{\dagger}, \hat{a}^{\dagger}\right]_{+}=1, \quad[\hat{a}, \hat{a}]_{+}=0 $$

$$ |0\rangle=\binom{0}{1}, \quad|1\rangle=\binom{1}{0} $$

$$ \hat{a}^{\dagger}=\frac{1}{2}\left(\gamma^{1}+i \gamma^{2}\right)=\frac{1}{2}\left(\begin{array}{cc} 0 & \sigma^{+} \\ -\sigma^{+} & 0 \end{array}\right), \hat{a}=\frac{1}{2}\left(-\gamma^{1}+i \gamma^{2}\right)=\frac{1}{2}\left(\begin{array}{cc} 0 & -\sigma^{-} \\ \sigma^{-} & 0 \end{array}\right), $$

$$ \hat{b}^{\dagger}=\frac{1}{2}\left(\gamma^{0}+\gamma^{3}\right)=\frac{1}{2}\left(\begin{array}{cc} 0 & i \sigma^{2} \sigma^{+} \\ -i \sigma^{2} \sigma^{+} & 0 \end{array}\right), \hat{b}=\frac{1}{2}\left(\gamma^{1}-\gamma^{3}\right)=\frac{1}{2}\left(\begin{array}{cc} 0 & i \sigma^{2} \sigma^{-} \\ -i \sigma^{2} \sigma^{-} & 0 \end{array}\right) . $$

$$ \begin{array}{ll} |0,0\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}{l} 0 \\ 1 \\ 0 \\ 1 \end{array}\right), & |1,0\rangle=a^{\dagger}|0,0\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}{r} 1 \\ 0 \\ -1 \\ 0 \end{array}\right), \\ |0,1\rangle=b^{\dagger}|0,0\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}{r} 0 \\ -1 \\ 0 \\ 1 \end{array}\right), & |1,1\rangle=a^{\dagger} b^{\dagger}|0,0\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}{l} 1 \\ 0 \\ 1 \\ 0 \end{array}\right) . \end{array} $$

$$ \mathcal{A}\left[a^{*}, a\right]+\mathcal{A}^{\text {source }}=\hbar \int_{0}^{\hbar \beta} d \tau\left[a^{*}(\tau) \partial_{\tau} a(\tau)+\omega a^{*} a(\tau)-\left(\eta^{*}(\tau) a(\tau)+\text { c.c. }\right)\right] $$

$$ Z_{\omega}\left[\eta^{*}, \eta\right]=\oint \frac{\mathcal{D} a^{*} \mathcal{D} a}{\pi} \exp \left\{-\frac{1}{\hbar}\left(\mathcal{A}\left[a^{*}, a\right]+\mathcal{A}^{\text {source }}\right)\right\} $$

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

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

$$ a^{\prime}(\tau)=a(\tau)-\int_{0}^{\hbar \beta} d \tau^{\prime} G_{\omega, \mathrm{e}}^{\mathrm{p}}\left(\tau, \tau^{\prime}\right) \eta\left(\tau^{\prime}\right) $$

$$ \mathcal{A}_{\mathrm{e}}=\hbar \int_{0}^{\hbar \beta} d \tau \int_{0}^{\hbar \beta} d \tau^{\prime}\left[a^{\prime *}(\tau) D_{\omega, \mathrm{e}}\left(\tau, \tau^{\prime}\right) a^{\prime}\left(\tau^{\prime}\right)-\eta^{*}(\tau) G_{\omega, \mathrm{e}}^{\mathrm{p}}\left(\tau, \tau^{\prime}\right) \eta\left(\tau^{\prime}\right)\right] $$

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

$$ G_{\omega, \mathrm{e}}^{\mathrm{p}}\left(\tau, \tau^{\prime}\right)=G_{\omega, \mathrm{e}}^{\mathrm{p}}\left(\tau-\tau^{\prime}\right)=\sum_{n=-\infty}^{\infty} e^{-\omega\left(\tau-\tau^{\prime}-n \hbar \beta\right)} \Theta\left(\tau-\tau^{\prime}-n \hbar \beta\right) $$

$$ G_{\omega, \mathrm{e}}^{\mathrm{p}}(\tau)=\frac{e^{-\omega(\tau-\hbar \beta / 2)}}{2 \sinh (\omega \hbar \beta / 2)}=\left(1+n_{\omega}^{\mathrm{b}}\right) e^{-\omega \tau} $$

$$ n_{\omega}=\frac{1}{e^{\beta \hbar \omega}-1} $$

$$ \begin{align*} \eta & =\sqrt{\omega / 2 M \hbar}(j-i \omega M k) \\ \eta^{*} & =\sqrt{\omega / 2 M \hbar}(j+i \omega M k) \end{align*} $$

$$ \mathcal{A}^{\text {source }}=\hbar \int_{0}^{\hbar \beta} d \tau\left(a^{*} \eta+\eta^{*} a\right)=\int_{0}^{\hbar \beta} d \tau(j x+k p) $$

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

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

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

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

$$ \hbar \int d \tau\left[\Omega^{2}(\tau) a^{*}(\tau) a(\tau)+\frac{1}{2} \Delta^{*}(\tau) a^{2}(\tau)+\frac{1}{2} \Delta(\tau) a^{* 2}(\tau)\right] $$

$$ Z=\oint \frac{\mathcal{D} a^{*} \mathcal{D} a}{\pi} \exp \left[-\int_{0}^{\hbar \beta} d \tau\left(a^{*} \partial_{\tau} a+\Omega^{2} a^{*} a+\frac{1}{2} \Delta^{*} a^{2}+\frac{1}{2} \Delta a^{* 2}\right)\right] $$

$$ f(\tau) \equiv\binom{a(\tau)}{a^{*}(\tau)} $$

$$ \mathcal{A}_{\mathrm{e}}=\frac{\hbar}{2} \int_{0}^{\hbar \beta} d \tau f^{* T}(\tau)\left(\begin{array}{cc} \partial_{\tau}+\Omega(\tau) & \Delta(\tau) \\ \Delta^{*}(\tau) & \mp\left(\partial_{\tau} \pm \Omega(\tau)\right) \end{array}\right) f(\tau) $$

$$ Z=\int \frac{\mathcal{D} f^{*} \mathcal{D} f}{\pi} e^{-\frac{1}{2} f^{*} M f} $$

$$ M=\left(\begin{array}{cc} \partial_{\tau}+\Omega(\tau) & \Delta(\tau) \\ \Delta^{*}(\tau) & \mp\left(\partial_{\tau} \pm \Omega(\tau)\right) \end{array}\right) $$

$$ f^{*}=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right) f $$

$$ Z=\mathcal{N}^{b, f} \operatorname{det}\left(\begin{array}{cc} \partial_{\tau}+\Omega(\tau) & \Delta(\tau) \\ \Delta^{*}(\tau) & \mp\left(\partial_{\tau}+\Omega(\tau)\right) \end{array}\right)^{\mp 1 / 2} $$

$$ M=\left(\begin{array}{cc} -i \omega_{m}^{\mathrm{f}}+\omega & \Delta \\ \Delta^{*} & \mp\left(-i \omega_{m}^{\mathrm{f}} \pm \omega\right) \end{array}\right) $$

$$ M \rightarrow M^{d}=\left(\begin{array}{cc} -i \omega_{m}^{\mathrm{f}}+\omega_{\Delta} & 0 \\ 0 & \mp\left(-i \omega_{m}^{\mathrm{f}} \pm \omega_{\Delta}\right) \end{array}\right) $$

$$ \omega_{\Delta}=\sqrt{\omega^{2}+\Delta^{2}} $$

$$ \begin{align*} Z_{\omega, \Delta} & =\mathcal{N}^{b, f} \operatorname{det}\left(\begin{array}{cc} \partial_{\tau}+\omega & \Delta \\ \Delta^{*} & \mp \partial_{\tau} \pm \omega \end{array}\right)^{\mp 1 / 2} \\ & =N^{b, f}\left[\operatorname{det}\left(-\partial_{\tau}^{2}+\omega_{\Delta}^{2}\right)\right]^{\mp 1 / 2}=\left\{\begin{array}{c} {\left[2 \sinh \left(\hbar \beta \omega_{\Delta} / 2\right)\right]^{-1}} \\ 2 \cosh \left(\hbar \beta \omega_{\Delta} / 2\right) \end{array}\right. \end{align*} $$

$$ \hat{H}=\frac{\omega_{\Delta}}{2} \Delta\left(\hat{a}^{\dagger} \hat{a}+\hat{a} \hat{a}^{\dagger}\right)=\omega_{\Delta}\left(\hat{a}^{\dagger} \hat{a} \pm \frac{1}{2}\right) $$

$$ Z_{\omega, \Delta}=\sum_{n}^{\infty, 1} e^{-\hbar \omega_{\Delta}(n \pm 1 / 2) \beta} $$

$$ Z_{\omega, \Delta}=\left\{\begin{array}{l} \left(1-e^{-\hbar \beta \omega_{\Delta}}\right)^{-1} \\ \left(1+e^{-\hbar \beta \omega_{\Delta}}\right) \end{array}\right. $$

$$ \mathcal{A}_{\mathrm{e}}\left[a^{*}, a\right]=\hbar \int_{0}^{\hbar \beta} d \tau \sum_{\mathbf{p}}\left[a_{\mathbf{p}}^{*} \partial_{\tau} a_{\mathbf{p}}+\omega(\mathbf{p}) a_{\mathbf{p}}^{*} a_{\mathbf{p}}\right] $$

$$ \begin{align*} Z & =\prod_{\mathbf{p}}\{\operatorname{det}[\epsilon(1-\epsilon \omega) \bar{\nabla}+\epsilon \omega(\mathbf{p})]\}^{\mp 1} \\ & = \begin{cases}\exp \left[-\sum_{\mathbf{p}} \log \left(1-e^{-\hbar \beta \bar{\omega}_{\mathrm{e}}(\mathbf{p})}\right)\right] & \text { for bosons, } \\ \exp \left[\sum_{\mathbf{p}} \log \left(1+e^{-\hbar \beta \bar{\omega}_{\mathrm{e}}(\mathbf{p})}\right)\right] & \text { for fermions. }\end{cases} \end{align*} $$

$$ F=-k_{B} T \log Z $$

$$ F=k_{B} T \sum_{\mathbf{p}} \log \left(1-e^{-\hbar \beta \bar{\omega}_{\mathrm{e}}(\mathbf{p})}\right), $$

$$ F=-k_{B} T \sum_{\mathbf{p}} \log \left(1+e^{-\hbar \beta \bar{\omega}_{\mathrm{e}}(\mathbf{p})}\right) . $$

$$ \sum_{\mathbf{p}} \rightarrow \int \frac{d^{D} p V}{(2 \pi \hbar)^{D}} $$

$$ \psi(\mathbf{x}, \tau)=\frac{1}{\sqrt{V}} \sum_{\mathbf{p}} e^{i \mathbf{p} \mathbf{x}} a_{\mathbf{p}}(\tau) $$

$$ \mathcal{A}_{\mathrm{e}}\left[\psi^{*}, \psi\right]=\int_{0}^{\hbar \beta} d \tau \int d^{D} x\left[\psi^{*}(\mathbf{x}, \tau) \hbar \partial_{\tau} \psi(\mathbf{x}, \tau)+\frac{\hbar^{2}}{2 M} \boldsymbol{\nabla} \psi^{*}(\mathbf{x}, \tau) \boldsymbol{\nabla} \psi(\mathbf{x}, \tau)\right] $$

$$ Z=\oint \mathcal{D} \psi \mathcal{D} \psi^{*} e^{-\mathcal{A}_{\mathrm{e}}\left[\psi^{*}, \psi\right] / \hbar} $$

$$ \mathcal{A}_{\mathrm{e}}\left[\psi^{*}, \psi\right]=\int_{0}^{\hbar \beta} d \tau \int d^{D} x\left\{\psi^{*}(\mathbf{x}, \tau) \hbar \partial_{\tau} \psi(\mathbf{x}, \tau)+\psi^{*}(\mathbf{x}, \tau)[\hat{H}(\tau)-\mu] \psi(\mathbf{x}, \tau)\right\} $$

$$ F=\frac{1}{\beta} \operatorname{Tr} \log \left[\hbar \partial_{\tau}+\hat{H}(\tau)-\mu\right] $$

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

$$ F=\frac{1}{2} \operatorname{Tr} \hat{H}-\frac{1}{\beta} \frac{1}{{\sqrt{2 \pi \hbar^{2} \beta / M}}^{D}} \sum_{n=1}^{\infty} \int d^{D} x \frac{[z(\mathbf{x})]^{n}}{n^{D / 2+1}} $$

$$ \delta^{(D)}\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right)=\int_{\mathbf{x}\left(t_{a}\right)=\mathbf{x}_{a}}^{\mathbf{x}\left(t_{b}\right)=\mathbf{x}_{b}} \mathcal{D}^{D} x \oint \frac{\mathcal{D}^{D} p}{(2 \pi \hbar)^{D}} e^{(i / \hbar) \int_{t_{a}}^{t_{b}} d t \mathbf{p} \dot{\mathbf{x}}} $$

$$ i \hbar \delta^{(D)}\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right) \delta\left(t_{b}-t_{a}\right)=\oint \mathcal{D} \psi \mathcal{D} \psi^{*} \psi\left(\mathbf{x}_{b}, t_{b}\right) \psi^{*}\left(\mathbf{x}_{a}, t_{a}\right) e^{(i / \hbar) \int d^{D} x \int_{-\infty}^{\infty} d t \psi^{*}(\mathbf{x}, t) \psi(\mathbf{x}, t)} $$

$$ \left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)=e^{i \hat{H}\left(t_{b}-t_{a}\right)} \delta^{(D)}\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right)=\int \mathcal{D}^{D} x \oint \frac{\mathcal{D}^{D} p}{(2 \pi \hbar)^{D}} e^{(i / \hbar) \int_{t_{a}}^{t_{b}} d t(\mathbf{p} \dot{\mathbf{x}}-H)} $$

$$ \left(i \hbar \partial_{t}-\hat{H}\right) \Theta\left(t_{b}-t_{a}\right)\left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)=i \hbar \delta^{(D)}\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right) \delta\left(t_{b}-t_{a}\right) $$

$$ \left\langle\mathbf{x}_{b} t_{b}\right| \frac{i \hbar}{i \hbar \partial_{t}-\hat{H}}\left|\mathbf{x}_{a} t_{a}\right\rangle=\Theta\left(t_{b}-t_{a}\right) \int_{\mathbf{x}\left(t_{a}\right)=\mathbf{x}_{a}}^{\mathbf{x}\left(t_{b}\right)=\mathbf{x}_{b}} \mathcal{D}^{D} x \oint \frac{\mathcal{D}^{D} p}{(2 \pi \hbar)^{D}} e^{(i / \hbar) \int_{t_{a}}^{t_{b}} d t(\mathbf{p} \dot{\mathbf{x}}-H)} $$

$$ \begin{align*} & \left\langle\mathbf{x}_{b} t_{b}\right| \frac{i \hbar}{i \hbar \partial_{t}-\hat{H}}\left|\mathbf{x}_{a} t_{a}\right\rangle \\ & \quad=\oint \mathcal{D} \psi \mathcal{D} \psi^{*} \psi\left(\mathbf{x}_{b}, t_{b}\right) \psi^{*}\left(\mathbf{x}_{a}, t_{a}\right) e^{(i / \hbar) \int d^{D} x \int_{-\infty}^{\infty} d t \psi^{*}(\mathbf{x}, t)\left(i \hbar \partial_{t}-\hat{H}\right) \psi(\mathbf{x}, t)} \end{align*} $$

$$ \mathcal{A}_{\mathrm{e}}^{\mathrm{int}}\left[\psi^{*}, \psi\right]=-\int_{0}^{\hbar \beta} d \tau \Delta E=-\frac{g}{2} \int_{0}^{\hbar \beta} d \tau \int d^{3} x \psi^{*}(\mathbf{x}, \tau+\eta) \psi^{*}(\mathbf{x}, \tau+\eta) \psi(\mathbf{x}, \tau) \psi(\mathbf{x}, \tau) $$

$$ \left\langle\psi(\mathbf{x}, \tau) \psi^{*}\left(\mathbf{x}^{\prime}, \tau^{\prime}\right)\right\rangle=\sum_{\mathbf{p}, \mathbf{p}^{\prime}}\left\langle a_{\mathbf{p}}(\tau) a_{\mathbf{p}^{\prime}}^{*}\left(\tau^{\prime}\right)\right\rangle e^{i\left(\mathbf{p x}-\mathbf{p}^{\prime} \mathbf{x}^{\prime}\right)}=\sum_{\mathbf{p}}\left\langle a_{\mathbf{p}}(\tau) a_{\mathbf{p}}^{*}\left(\tau^{\prime}\right)\right\rangle e^{i \mathbf{p}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)} . $$

$$ \left\langle\psi(\mathbf{x}, \tau) \psi^{*}\left(\mathbf{x}^{\prime}, \tau^{\prime}\right)\right\rangle=\sum_{\mathbf{p}}\left(1+n_{\omega_{\mathbf{p}}}\right) e^{-\omega_{\mathbf{p}}\left(\tau-\tau^{\prime}\right)+i \mathbf{p}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)} $$

$$ \left\langle\psi(\mathbf{x}, \tau) \psi^{*}\left(\mathbf{x}^{\prime}, \tau^{\prime}\right)\right\rangle=\frac{1}{\hbar \beta} \sum_{\omega_{m}, \mathbf{p}} \frac{-1}{i \omega_{m}-\omega_{\mathbf{p}}} e^{-i \omega_{m}\left(\tau-\tau^{\prime}\right)+i \mathbf{p}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)} $$

$$ n_{\omega_{\mathbf{p}}-\mu / \hbar}=\frac{1}{z^{-1} e^{\beta \hbar \omega_{\mathbf{p}}}-1} $$

$$ Z=\oint \mathcal{D} \psi \mathcal{D} \psi^{*} e^{-\mathcal{A}_{\mathrm{e}}\left[\psi^{*}, \psi\right]}=\oint \mathcal{D} \psi_{0} \mathcal{D} \psi_{0}^{*} \oint \mathcal{D}^{\prime} \psi \mathcal{D}^{\prime} \psi^{*} e^{-\mathcal{A}_{\mathrm{e}}\left[\psi^{*}, \psi\right]} $$

$$ B\left[\psi_{0}^{*}, \psi_{0}\right] \equiv e^{-\mathcal{A}^{\mathrm{eff} \mathrm{cl}}\left[\psi_{0}^{*}, \psi_{0}\right]} \equiv \oint \mathcal{D}^{\prime} \psi \mathcal{D}^{\prime} \psi^{*} e^{-\mathcal{A}_{\mathrm{e}}\left[\psi^{*}, \psi\right]} $$

$$ Z=\oint \mathcal{D} \psi_{0} \mathcal{D} \psi_{0}^{*} e^{-\mathcal{A}^{\mathrm{eff} \mathrm{cl}}\left[\psi_{0}^{*}, \psi_{0}\right]} $$

$$ \mathcal{A}_{\mathrm{b}}^{\mathrm{eff} \mathrm{cl}}\left[\psi_{0}^{*}, \psi_{0}\right]=\beta \int d^{3} x\left\{\psi_{0}^{*}(\mathbf{x})\left(-\frac{1}{2 m} \nabla^{2}-\mu\right) \psi_{0}(\mathbf{x})+\frac{2 \pi a}{m}\left[\psi_{0}^{*}(\mathbf{x}) \psi_{0}(\mathbf{x})\right]^{2}\right\} . $$

$$ \mathcal{A}_{\mathrm{b}}^{\mathrm{eff} \mathrm{cl}}\left[\psi_{0}^{*}, \psi_{0}\right]=\mathcal{A}[\phi]=\int d^{3} x\left[\frac{1}{2}|\nabla \phi|^{2}+\frac{1}{2} m^{2} \phi^{2}+\frac{u}{4!}\left(\phi^{2}\right)^{2}\right] . $$

$$ \begin{align*} \frac{\Delta T_{c}}{T_{c}^{(0)}} & \approx-\frac{2}{3} \frac{m T_{c}^{(0)}}{n}\left\langle\Delta \phi^{2}\right\rangle=-\frac{4 \pi}{3} \frac{\left(m T_{c}^{(0)}\right)^{2}}{n} 4!\left\langle\frac{\Delta \phi^{2}}{u}\right\rangle a \\ & =-\frac{4 \pi}{3} \frac{(2 \pi)^{2}}{[\zeta(3 / 2)]^{4 / 3}} 4!\left\langle\frac{\Delta \phi^{2}}{u}\right\rangle a n^{1 / 3} \end{align*} $$

$$ L(t)=a^{*}(t) i \partial_{t} a(t)-\frac{\varepsilon}{2}\left[a^{*}(t) a(t)\right]^{2} $$

$$ Z\left[\eta^{*}, \eta\right]=N \int \mathcal{D} a^{*} D a \exp \left[i \int d t\left(L+\eta^{*} a+a^{*} \eta\right)\right] $$

$$ \hat{H}=\varepsilon\left(\hat{a}^{\dagger} \hat{a}\right)^{2} / 2 $$

$$ |n\rangle=\frac{1}{\sqrt{n!}}\left(\hat{a}^{\dagger}\right)^{n}|0\rangle, \quad n=0,1,2, \ldots $$

$$ E_{n}=\varepsilon \frac{n^{2}}{2} $$

$$ \begin{align*} & |0\rangle \quad \text { with } \quad E_{0}=0 \\ & |1\rangle=a^{\dagger}|0\rangle \quad \text { with } \quad E_{1}=\frac{\varepsilon}{2} \end{align*} $$

$$ Z\left[\eta^{\dagger}, \eta\right]=\langle 0| \hat{T} \exp \left[i \int d t\left(\eta^{*} \hat{a}+\hat{a}^{\dagger} \eta\right)\right]|0\rangle $$

$$ \exp \left\{-i \frac{\varepsilon}{2} \int d t\left[a^{*}(t) a(t)\right]^{2}\right\}=N \int \mathcal{D} \rho(t) \exp \left\{i \int d t\left[\frac{\rho^{2}(t)}{2 \varepsilon}-\rho(t) a^{*}(t) a(t)\right]\right\} $$

$$ 1=N \int \mathcal{D} \rho(t) \exp \left\{\int d t \frac{1}{2 \varepsilon}\left[\rho(t)-\varepsilon a^{\dagger}(t) a(t)\right]^{2}\right\} $$

$$ \begin{align*} & Z\left[\eta^{\dagger}, \eta\right]=N \int \mathcal{D} a^{*} \mathcal{D} a \mathcal{D} \rho \\ & \quad \times \exp \left\{\int d t\left[a^{*}(t) i \partial_{t} a(t)-\varepsilon \rho(t) a^{*}(t) a(t)+\frac{\rho^{2}(t)}{2 \varepsilon}+\eta^{*}(t) a(t)+a^{*}(t) \eta(t)\right]\right\} \end{align*} $$

$$ \rho(t)=\varepsilon a^{*}(t) a(t) $$

$$ Z\left[\eta^{*}, \eta\right]=N \int \mathcal{D} \rho \exp \left\{i \mathcal{A}[\rho]-\int d t d t^{\prime} \eta^{*}(t) G_{\rho}\left(t, t^{\prime}\right) \eta\left(t^{\prime}\right)\right\}, $$

$$ \mathcal{A}[\rho]= \pm i \operatorname{Tr} \log \left(i G_{\rho}^{-1}\right)+\int d t \frac{\rho^{2}(t)}{2} $$

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

$$ \varphi(t)=\int^{t} \rho\left(t^{\prime}\right) d t^{\prime} $$

$$ G_{\rho}\left(t, t^{\prime}\right)=e^{-i \varphi(t)} e^{i \varphi\left(t^{\prime}\right)} G_{0}\left(t-t^{\prime}\right) $$

$$ G_{\rho}\left(t, t^{\prime}\right)=e^{-i \varphi(t)} e^{i \varphi\left(t^{\prime}\right)} \bar{\Theta}\left(t-t^{\prime}\right) . $$

$$ \frac{\delta}{\delta \rho(t)}\left\{ \pm i \operatorname{Tr} \log \left(i G_{\rho}^{-1}\right)\right\}=\frac{\delta}{\delta \rho(t)}\left\{ \pm i \operatorname{Tr} \log \left[i \partial_{t}-\rho(t)\right]\right\}=\left.\mp G_{\rho}\left(t, t^{\prime}\right)\right|_{t^{\prime}=t}=0 $$

$$ Z\left[\eta^{\dagger}, \eta\right]=N \int \mathcal{D} \varphi(t) \exp \left\{\frac{i}{2 \varepsilon} \int d t \dot{\varphi}(t)^{2}-\int d t d t^{\prime} \eta^{\dagger}(t) \eta\left(t^{\prime}\right) e^{-i \varphi(t)} e^{-i \varphi\left(t^{\prime}\right)} \Theta\left(t-t^{\prime}\right)\right\} $$

$$ \mathcal{D} \rho=\mathcal{D} \varphi \operatorname{Det}\left(\partial_{t}\right)=\mathcal{D} \varphi, $$

$$ L_{0}(t)=\frac{1}{2 \varepsilon} \dot{\varphi}(t)^{2} $$

$$ \begin{align*} \left\langle\varphi(t) \varphi\left(t^{\prime}\right)\right\rangle & =\varepsilon \int \frac{d \omega}{2 \pi} \frac{i}{\omega^{2}-\kappa^{2}+i \epsilon} e^{-i \omega\left(t-t^{\prime}\right)} \\ & =\frac{\varepsilon}{2 \kappa} e^{-\kappa\left|t-t^{\prime}\right|}=\frac{\varepsilon}{2 \kappa}-\varepsilon \frac{i}{2}\left|t-t^{\prime}\right|+\mathcal{O}(\kappa) \end{align*} $$

$$ \begin{align*} G^{(n)}\left(t_{n}, \ldots, t_{1} ; t_{n}^{\prime}, \ldots, t_{1}^{\prime}\right) & =(-1)^{n}\left\langle e^{-i \varphi\left(t_{n}\right)} \cdots e^{-i \varphi\left(t_{1}\right)} e^{i \varphi\left(t_{n}^{\prime}\right)} \cdots e^{i \varphi\left(t_{1}^{\prime}\right)}\right\rangle \\ & \times \sum_{p} \epsilon_{p} \Theta\left(t_{p_{1}}-t_{1}\right) \cdots \Theta\left(t_{p_{n}}-t_{1}\right) \end{align*} $$

$$ \langle\ldots\rangle \equiv \frac{1}{Z[0,0]} \int \mathcal{D} \varphi(t) \ldots e^{(i / 2 \varepsilon) \int d t \dot{\varphi}(t)^{2}} $$

$$ \left\langle\exp \left[i \sum_{i=1}^{2 n} q_{i} \varphi\left(t_{i}\right)\right]\right\rangle=\left\langle\exp \left[\sum_{i} \int d t \delta\left(t-t_{i}\right) q_{i} \varphi(t)\right]\right\rangle $$

$$ \begin{align*} \left\langle e^{i \sum_{i=1}^{2 n} q_{i} \varphi\left(t_{i}\right)}\right\rangle & =\exp \left[-\frac{1}{2} \int d t d t^{\prime} \sum_{i=1}^{2 n} q_{i} \delta\left(t-t_{i}\right)\left\langle\varphi(t) \varphi\left(t^{\prime}\right)\right\rangle\left(t^{\prime}\right) \sum_{j} q_{j} \delta\left(t-t_{j}\right)\right] \\ & =\exp \left[-\frac{1}{2} \sum_{i, j=1}^{2 n} q_{i} q_{j}\left\langle\varphi\left(t_{i}\right) \varphi\left(t_{j}\right)\right\rangle\right] \end{align*} $$

$$ \exp \left[-\frac{\varepsilon}{4 \kappa}\left(\sum_{i} q_{i}\right)^{2}\right] $$

$$ \left\langle\exp \left[i \sum_{i=1}^{2 n} q_{i} \varphi\left(t_{i}\right)\right]\right\rangle=\exp \left[\varepsilon \frac{i}{2} \sum_{i>j} q_{i} q_{j}\left|t_{i}-t_{j}\right|\right] $$

$$ G^{(2)}\left(t, t^{\prime}\right)=e^{-i \varepsilon\left(t-t^{\prime}\right) / 2} \Theta\left(t-t^{\prime}\right) $$

$$ G^{(2)}\left(t, t^{\prime}\right)=\langle 0| \hat{T} \hat{a}(t) \hat{a}^{\dagger}\left(t^{\prime}\right)|0\rangle=\Theta\left(t-t^{\prime}\right)\langle 0| \hat{a}(t) \hat{a}^{\dagger}\left(t^{\prime}\right)|0\rangle e^{-i \varepsilon\left(t-t^{\prime}\right) / 2} \Theta\left(t-t^{\prime}\right) $$

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

$$ \begin{align*} \langle 0| \hat{T} \hat{a}(t) \hat{a}^{\dagger}\left(t^{\prime}\right)|0\rangle & =\Theta\left(t-t^{\prime}\right)\langle 0| e^{i \varepsilon\left(\hat{a}^{\dagger} \hat{a}\right)^{2} t / 2} a e^{-\frac{i}{2}\left(\hat{a}^{\dagger} \hat{a}\right)^{2}\left(t-t^{\prime}\right)} \hat{a}^{\dagger} e^{-i\left(\hat{a}^{\dagger} \hat{a}\right)^{2} t^{\prime} / 2}|0\rangle \\ & =\Theta\left(t-t^{\prime}\right)\langle 1| e^{-\frac{i}{2}\left(\hat{a}^{\dagger} \hat{a}\right)^{2}\left(t-t^{\prime}\right)}|1\rangle=\Theta\left(t-t^{\prime}\right) e^{-i \varepsilon\left(t-t^{\prime}\right) / 2} \end{align*} $$

$$ \{\varphi \mid p\}=\frac{1}{\sqrt{2 \pi}} e^{i p \varphi} $$

$$ Z\left[\eta^{\dagger}, \eta\right]=\frac{1}{\{0 \mid 0\}}\left\{0\left|\hat{T} \exp \left[-\int d t d t^{\prime} \eta^{\dagger}(t) \eta\left(t^{\prime}\right) e^{-i \hat{\varphi}(t)} e^{i \hat{\varphi}\left(t^{\prime}\right)} \Theta\left(t-t^{\prime}\right)\right]\right| 0\right\} $$

$$ \langle 0| \hat{T} \hat{a}(t) \hat{a}^{\dagger}\left(t^{\prime}\right)|0\rangle=-\left.\frac{\delta^{(2)} Z}{\delta \eta^{\dagger}(t) \delta \eta\left(t^{\prime}\right)}\right|_{\eta^{\dagger}, \eta=0}=\frac{1}{\{0 \mid 0\}}\left\{0\left|e^{-i \hat{\varphi}(t)} e^{i \hat{\varphi}\left(t^{\prime}\right)}\right| 0\right\} \Theta\left(t-t^{\prime}\right) $$

$$ \hat{\varphi}(t)=e^{i \varepsilon \frac{\hat{p}^{2}}{2} t} \hat{\varphi}(0) e^{-i \varepsilon \frac{\hat{p}^{2}}{2} t} $$

$$ \left\{0\left|e^{i \varepsilon \frac{\hat{p}^{2}}{2} t} 2 e^{-i \hat{\varphi}(0)} e^{-i \varepsilon \frac{\hat{p}^{2}}{2}\left(t-t^{\prime}\right)} e^{i \hat{\varphi}(0)} e^{-i \varepsilon \frac{\hat{p}^{2}}{2} t^{\prime}}\right| 0\right\}=\left\{0\left|e^{-i \hat{\varphi}(0)} e^{-i \varepsilon \frac{\hat{p}^{2}}{2}\left(t-t^{\prime}\right)} e^{i \hat{\varphi}(0)}\right| 0\right\} $$

$$ \frac{1}{\{0 \mid 0\}}\{1 \mid 1\} e^{-i \varepsilon\left(t-t^{\prime}\right) / 2}=e^{-i \varepsilon\left(t-t^{\prime}\right) / 2} $$

$$ \langle 0| \hat{T} \hat{a}(t) \hat{a}^{\dagger}\left(t^{\prime}\right)|0\rangle=e^{-i \varepsilon\left(t-t^{\prime}\right) / 2} \Theta\left(t-t^{\prime}\right) $$