Kleinert · 제19장 보충

$$ t=-i \tau=-i x^{4} / c $$

$$ \mathcal{A}_{\mathrm{cl}, \mathrm{e}}=M c \sum_{n=1}^{N+1}\left|x_{n}-x_{n-1}\right| $$

$$ x^{2}=\mathbf{x}^{2}+\left(x^{4}\right)^{2}=\mathbf{x}^{2}+c^{2} \tau^{2} $$

$$ \left(x_{b} \mid x_{a}\right)=\mathcal{N} \prod_{n=1}^{N}\left[\int d^{D} x_{n}\right] e^{-M c \sum_{n=1}^{N+1}\left|x_{n}-x_{n-1}\right| / \hbar} $$

$$ S=\int_{\lambda_{a}}^{\lambda_{b}} d \lambda \sqrt{x^{\prime 2}(\lambda)} $$

$$ e^{-M c\left|x_{n}-x_{n-1}\right| / \hbar}=\sqrt{\frac{\epsilon M c}{2 \pi \hbar}} \int_{0}^{\infty} d h_{n} h_{n}^{-1 / 2} e^{-\epsilon h_{n} M c / 2 \hbar-M c\left(x_{n}-x_{n-1}\right)^{2} / 2 h_{n} \epsilon \hbar} $$

$$ \mathcal{A}_{\mathrm{cl}, \mathrm{e}}=M c S $$

$$ \begin{align*} \left(x_{b} \mid x_{a}\right) & =\mathcal{N} \prod_{n=1}^{N+1}\left[\int_{0}^{\infty} d h_{n} h_{n}^{(D-1) / 2}\right] e^{-M c \epsilon \Sigma_{n=1}^{N+1} h_{n} / 2} \\ & \times \frac{1}{{\sqrt{2 \pi \epsilon h_{N+1} / M c}}^{D}} \prod_{n=1}^{N}\left[\int \frac{d^{D} x_{n}}{{\sqrt{2 \pi \epsilon h_{n} / M c}}^{D}}\right] e^{-M c \Sigma_{n=1}^{N+1}\left(x_{n}-x_{n-1}\right)^{2} / 2 h_{n} \epsilon} \end{align*} $$

$$ \mathcal{A}_{\mathrm{cl}, \mathrm{e}}=M c \int_{\lambda_{a}}^{\lambda_{b}} d s(\lambda) $$

$$ \left(x_{b} S \mid x_{a} 0\right)=\frac{1}{\sqrt{2 \pi S \hbar / M c}^{D}} e^{-M c\left(x_{b}-x_{a}\right)^{2} / 2 S \hbar} $$

$$ d s(\lambda) \equiv d \lambda \sqrt{x^{\prime 2}(\lambda)}=d \lambda \sqrt{\mathbf{x}^{\prime 2}(\lambda)+c^{2} \tau^{\prime 2}(\lambda)} $$

$$ S \equiv \sum_{n=1}^{N+1} \epsilon h_{n} $$

$$ \lambda \rightarrow \bar{\lambda}=f(\lambda) $$

$$ \left(x_{b} \mid x_{a}\right)=\mathcal{N} \prod_{n=1}^{N+1}\left[\int_{0}^{\infty} d h_{n} h_{n}^{(D-1) / 2}\right] e^{-M c S / 2 \hbar} \int \frac{d^{D} p}{(2 \pi \hbar)^{D}} e^{-S p^{2} / 2 M c+i p\left(x_{b}-x_{a}\right) / \hbar} $$

$$ \begin{align*} x^{\prime 2} & \rightarrow \frac{1}{f^{\prime 2}} x^{\prime 2} \\ d \lambda & \rightarrow d \lambda f^{\prime} \end{align*} $$

$$ 1=\int_{0}^{\infty} d S \delta\left(\Sigma_{n=1}^{N+1} \epsilon h_{n}-S\right)=\int_{0}^{\infty} \frac{d S}{2 \lambda_{C}} \int_{-i \infty}^{\infty} \frac{d \sigma}{2 \pi i} e^{-\sigma\left(\Sigma_{n=1}^{N+1} \epsilon h_{n}-S\right) / 2 \lambda_{C}} $$

$$ \int_{0}^{\infty} \frac{d S}{2 \lambda_{C}}\left\{\int_{-i \infty}^{i \infty} \frac{d \sigma}{2 \pi i} e^{\sigma S / 2 \lambda_{C}} \prod_{n=1}^{N+1}\left[\int_{0}^{\infty} d h_{n} h_{n}^{(D-1) / 2} e^{-\sigma \epsilon h_{n} / 2 \lambda_{C}}\right]\right\} e^{-S / 2 \lambda_{C}} $$

$$ \overline{\mathcal{A}}_{\mathrm{e}}=\int_{\lambda_{a}}^{\lambda_{b}} d \lambda\left[\frac{M c}{2 h(\lambda)} x^{\prime 2}(\lambda)+h(\lambda) \frac{M c}{2}\right] $$

$$ \begin{align*} \int_{-i \infty}^{i \infty} \frac{d \sigma}{2 \pi i} & e^{\sigma S / 2 \lambda_{C}} \prod_{n=1}^{N+1}\left[\int_{-\infty}^{\infty} d r_{n} r_{n}^{D} e^{-\sigma \epsilon r_{n}^{2} / 2 \lambda_{C}}\right] \\ & =\left[\frac{2 \pi \lambda_{C}}{\Gamma(D+1) \epsilon}\right]^{N+1} \int_{-i \infty}^{i \infty} \frac{d \sigma}{2 \pi i} e^{\sigma S / 2 \lambda_{C}} \sigma^{-(N+1)(D+1) / 2} \end{align*} $$

$$ h(\lambda)=\sqrt{x^{\prime 2}(\lambda)} $$

$$ \int_{-i \infty}^{i \infty} \frac{d \sigma}{2 \pi i} e^{\sigma S / 2 \lambda_{C}} \sigma^{-(N+1)(D+1) / 2} \underset{\text { large } N}{\approx} \frac{1}{\sqrt{2 \pi}}\left[\frac{S}{(N+1)(D+1) \lambda_{C}}\right]^{(N+1)(D+1) / 2} $$

$$ \mathcal{A}_{\mathrm{cl}, \mathrm{e}}=M c \int_{\lambda_{a}}^{\lambda_{b}} d \lambda \sqrt{x^{\prime 2}(\lambda)} $$

$$ \frac{1}{\sqrt{2 \pi}}\left[\frac{(D+1) \lambda_{C}}{\bar{\epsilon}}\right]^{-S(D+1) / 2 \bar{\epsilon}}=\frac{1}{\sqrt{2 \pi}} e^{-z S / 2 \lambda_{C}} $$

$$ h \rightarrow h / f^{\prime} $$

$$ z \equiv \nu \log \nu \quad \text { with } \quad \nu \equiv(D+1) \lambda_{C} / \bar{\epsilon} $$

$$ \overline{\mathcal{A}}_{\mathrm{e}}[p, x]=\int_{\lambda_{a}}^{\lambda_{b}} d \lambda\left[-i p x^{\prime}+\frac{h(\lambda)}{2 M c}\left(p^{2}+M^{2} c^{2}\right)\right] $$

$$ \int_{0}^{\infty} \frac{d S}{2 \lambda_{C}} e^{-S M_{1} c / 2 \hbar} $$

$$ \overline{\mathcal{A}}_{\mathrm{e}}^{N}[p, x]=\sum_{n=1}^{N+1}\left[-i p_{n}\left(x_{n}-x_{n-1}\right)+h_{n} \epsilon_{n} \frac{p_{n}^{2}}{2 M c}+\epsilon_{n} h_{n} \frac{M c}{2}\right] $$

$$ \left(x_{b} \mid x_{a}\right)=\mathcal{N}^{\prime \prime} \int_{0}^{\infty} \frac{d S}{2 \lambda_{C}} e^{-M_{1} c S / 2 \hbar} \int \frac{d^{D} p}{(2 \pi \hbar)^{D}} e^{-S p^{2} / 2 M c+i p\left(x_{b}-x_{a}\right) / \hbar} $$

$$ \int \mathcal{D}^{D} x \int \frac{\mathcal{D}^{D} p}{(2 \pi \hbar)^{D}} e^{-\overline{\mathcal{A}}_{\mathrm{e}}[p, x] / \hbar} \approx \prod_{n=1}^{N}\left[\int d^{D} x_{n}\right] \prod_{n=1}^{N+1}\left[\int \frac{d^{D} p_{n}}{(2 \pi \hbar)^{D}} e^{-\overline{\mathcal{A}}_{\mathrm{e}}^{N}[p, x] / \hbar}\right] $$

$$ M_{R}=M(1+z)^{1 / 2} $$

$$ \frac{1}{\sqrt{2 \pi \hbar \epsilon_{b} h_{b} / M c}} \prod_{n=1}^{N}\left[\int \frac{d^{D} x_{n}}{{\sqrt{2 \pi \hbar \epsilon_{n} h_{n} / M c}}^{D}}\right] \exp \left(-\frac{1}{\hbar} \overline{\mathcal{A}}_{\mathrm{e}}^{N}[x]\right) $$

$$ \overline{\mathcal{A}}_{\mathrm{e}}^{N}[x]=\sum_{n=1}^{N+1}\left[\frac{M c}{2 h_{n} \epsilon_{n}}\left(\Delta x_{n}\right)^{2}+\epsilon_{n} h_{n} \frac{M c}{2}\right] $$

$$ \frac{1}{\sqrt{2 \pi \hbar S / M c}^{D}} \exp \left[-\frac{M c}{2 \hbar} \frac{\left(x_{b}-x_{a}\right)^{2}}{S}-\frac{M c}{2 \hbar} S\right] $$

$$ S \equiv \sum_{n=1}^{N+1} \epsilon_{n} h_{n} $$

$$ S=\int_{\lambda_{a}}^{\lambda_{b}} d \lambda h(\lambda) $$

$$ \left(x_{b} \mid x_{a}\right)=\mathcal{N} \int_{0}^{\infty} d S \int \mathcal{D} h \Phi[h] \int \mathcal{D}^{D} x e^{-\overline{\mathcal{A}}_{\mathrm{e}} / \hbar} $$

$$ \Phi[h]=\delta[h-1] $$

$$ S=\lambda_{b}-\lambda_{a} $$

$$ S=\lambda_{b}-\lambda_{a} \equiv c \hbar \beta $$

$$ \left(x_{b} \mid x_{a}\right)=\mathcal{N} c \hbar \int_{0}^{\infty} d \beta e^{-\beta M c^{2} / 2} \int \mathcal{D}^{D} x e^{-\mathcal{A}_{0, \mathrm{e}} / \hbar} $$

$$ \mathcal{A}_{0, \mathrm{e}}=\int_{0}^{\hbar \beta} d \lambda \frac{M}{2} \dot{x}^{2} $$

$$ \left(x_{b} \mid x_{a}\right)=\mathcal{N} c \hbar \int_{0}^{\infty} d \beta \frac{1}{{\sqrt{2 \pi \hbar^{2} \beta / M}}^{D}} \exp \left[-\frac{M}{2 \hbar} \frac{\left(x_{b}-x_{a}\right)^{2}}{\hbar \beta}-\beta \frac{M c^{2}}{2}\right] $$

$$ \left(x_{b} \mid x_{a}\right)=\mathcal{N} c \hbar \int_{0}^{\infty} d \beta e^{-\beta M c^{2} / 2} \int \frac{d^{D} k}{(2 \pi)^{D}} \exp \left[i k\left(x_{b}-x_{a}\right)-\beta \frac{\hbar^{2} k^{2}}{2 M}\right] $$

$$ \left(x_{b} \mid x_{a}\right)=\mathcal{N} \frac{2 M c}{\hbar} \int \frac{d^{D} k}{(2 \pi)^{D}} \frac{1}{k^{2}+M^{2} c^{2} / \hbar^{2}} e^{i k\left(x_{b}-x_{a}\right)} $$

$$ \lambda_{M}^{\mathrm{C}} \equiv \hbar / M c $$

$$ \left(-\partial_{b}^{2}+M^{2} c^{2} / \hbar^{2}\right)\left(x_{b} \mid x_{a}\right)=\delta^{(D)}\left(x_{b}-x_{a}\right) $$

$$ \left(x_{b} \mid x_{a}\right)=\frac{1}{(2 \pi)^{D / 2}}\left(\frac{M c}{\hbar \sqrt{x^{2}}}\right)^{D / 2-1} K_{D / 2-1}\left(M c \sqrt{x^{2}} / \hbar\right) $$

$$ \left(x_{b} \mid x_{a}\right)=\left(\mathbf{x}_{b} \tau_{a} \mid \mathbf{x}_{a} \tau_{a}\right) \xrightarrow{c \rightarrow \infty} \frac{\hbar}{2 M c} e^{-M c^{2}\left(\tau_{b}-\tau_{a}\right) / \hbar}\left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)_{\mathrm{Schr}} $$

$$ \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)_{\mathrm{Schr}}=\frac{1}{{\sqrt{2 \pi \hbar\left(\tau_{b}-\tau_{a}\right) / M}}^{D-1}} \exp \left\{-\frac{M}{2 \hbar} \frac{\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right)^{2}}{\tau_{b}-\tau_{a}}\right\} $$

$$ \beta=\frac{\sqrt{\left(x_{b}-x_{a}\right)^{2}}}{c \hbar}=\frac{\sqrt{\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right)^{2}+c^{2}\left(\tau_{b}-\tau_{a}\right)^{2}}}{c \hbar} \rightarrow \frac{\tau_{b}-\tau_{a}}{\hbar}+\frac{\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right)^{2}}{2 c^{2} \hbar\left(\tau_{b}-\tau_{a}\right)}+\ldots, $$

$$ Z_{1}=\int_{0}^{\infty} \frac{d \beta}{\beta} e^{-\beta M c^{2} / 2} \int \mathcal{D}^{D} x e^{-\mathcal{A}_{0, \mathrm{e}} / \hbar} $$

$$ Z_{1}=V_{D} \int_{0}^{\infty} \frac{d \beta}{\beta} e^{-\beta M c^{2} / 2} \int \frac{d^{D} k}{(2 \pi)^{D}} \exp \left(-\beta \frac{\hbar^{2} k^{2}}{2 M}\right) $$

$$ Z_{1}=V_{D} \int_{0}^{\infty} \frac{d \beta}{\beta} e^{-\beta M c^{2} / 2} \frac{1}{{\sqrt{2 \pi \hbar^{2} \beta / M}}^{D}}=\frac{V_{D}}{\lambda_{M}^{\mathrm{C}} D} \frac{\Gamma(1-D / 2)}{(4 \pi)^{D / 2}} $$

$$ Z_{1}=-V_{D} \int \frac{d^{D} k}{(2 \pi)^{D}} \log \left(k^{2}+M^{2} c^{2} / \hbar^{2}\right) $$

$$ Z_{1}=-\operatorname{Tr} \log \left(-\partial^{2}+M^{2} c^{2} / \hbar^{2}\right)=-\operatorname{Tr} \log \left(-\hbar^{2} \partial^{2}+M^{2} c^{2}\right) $$

$$ Z=e^{Z_{1}}=e^{-\operatorname{Tr} \log \left(-\hbar^{2} \partial^{2}+M^{2} c^{2}\right)} $$

$$ k^{2}+M^{2} c^{2} / \hbar^{2}=\left(k^{D}\right)^{2}+\omega_{\mathbf{k}}^{2} / c^{2} $$

$$ \omega_{\mathbf{k}} \equiv c \sqrt{\mathbf{k}^{2}+M^{2} c^{2} / \hbar^{2}} $$

$$ Z_{1}=-2 V_{D} \int \frac{d^{D-1} k}{(2 \pi)^{D-1}} \frac{\hbar \omega_{\mathbf{k}}}{2 c} $$

$$ -Z_{1}=-W[0] / \hbar=\Gamma_{\mathrm{e}} / \hbar $$

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E} \equiv-i \int_{\tau_{a}}^{\infty} d \tau_{b} e^{E\left(\tau_{b}-\tau_{a}\right) / \hbar}\left(x_{b} \mid x_{a}\right) $$

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=\frac{\hbar}{2 M c} \int_{0}^{\infty} d L \int \mathcal{D} h \Phi[h] \int \mathcal{D}^{D} x e^{-\overline{\mathcal{A}}_{\mathrm{e}, E} / \hbar} $$

$$ \overline{\mathcal{A}}_{\mathrm{e}, E}=\int_{\lambda_{a}}^{\lambda_{b}} d \lambda\left[\frac{M c}{2 h(\lambda)} \mathbf{x}^{\prime 2}(\lambda)-h(\lambda) \frac{E^{2}}{2 M c^{3}}+h(\lambda) \frac{M c}{2}\right] $$

$$ E_{\text {pair }}=2 M c^{2} $$

$$ |\mathbf{E}|>E_{c}=\frac{M^{2} c^{3}}{e \hbar} $$

$$ \mathcal{A}_{\mathrm{cl}, \mathrm{e}}=\int_{\tau_{a}}^{\tau_{b}} d \tau\left[M c^{2} \sqrt{1+\dot{\mathbf{x}}^{2}(\tau) / c^{2}}-e \mathbf{E} \cdot \mathbf{x}(\tau)\right] $$

$$ M \frac{d}{d \tau} \frac{\dot{\mathbf{x}}(\tau)}{\sqrt{1+\dot{\mathbf{x}}^{2}(\tau) / c^{2}}}=-e \mathbf{E} $$

$$ \left(\mathbf{x}-\mathbf{x}_{0}\right)^{2}+c^{2}\left(\tau-\tau_{0}\right)^{2}=l_{E}^{2} \equiv\left(\frac{M c^{2}}{e E}\right)^{2}, \quad E \equiv|\mathbf{E}| . $$

$$ \mathbf{x}(\theta)=l_{E} \hat{\mathbf{E}} \cos \theta+\mathbf{x}_{0}, \quad \tau(\theta)=\frac{l_{E}}{c} \sin \theta+\tau_{0} $$

$$ \mathcal{A}_{\mathrm{cl}, \mathrm{e}}=M c^{2} \frac{l_{E}}{c} \int_{0}^{2 \pi} d \theta \cos \theta\left[\frac{1}{\cos \theta}-\cos \theta\right]=M c l_{E} \pi=\hbar \frac{E_{c}}{E} \pi $$

$$ \Gamma \propto e^{-\pi E_{c} / E} $$

$$ \Gamma \propto \sum_{n=1}^{\infty} F_{n} e^{-n \pi E_{c} / E} $$

$$ \overline{\mathcal{A}}_{\mathrm{e}}=\int_{\lambda_{a}}^{\lambda_{b}} d \lambda\left[\frac{M}{2 h(\lambda)} x^{\prime 2}(\lambda)+h(\lambda) \frac{M c}{2}-i \frac{e}{c} A(x(\lambda)) x^{\prime}(\lambda)\right] $$

$$ Z_{1}=\int_{0}^{\infty} \frac{d \beta}{\beta} e^{-\beta M c^{2} / 2} \int \mathcal{D}^{4} x e^{-\overline{\mathcal{A}}_{\mathrm{e}} / \hbar} $$

$$ \overline{\mathcal{A}}_{\mathrm{e}}=\int_{0}^{\hbar \beta} d \tau\left[\frac{M}{2} x^{\prime 2}(\tau)-i \frac{e}{c} A(x(\tau)) x^{\prime}(\tau)\right] $$

$$ \mathbf{x}^{\prime \prime}=\frac{e}{M c}\left(x_{4}^{\prime} \mathbf{E}-i \mathbf{x}^{\prime} \times \mathbf{B}\right), \quad x_{4}^{\prime \prime}=-\frac{e}{M c} \mathbf{x}^{\prime} \cdot \mathbf{E}, $$

$$ \overline{\mathcal{A}}_{\mathrm{e}}=\int_{0}^{\hbar \beta} d \tau\left[\frac{M}{2} x^{\prime 2}-i \frac{e}{2 c} F_{\mu \nu} x^{\mu} x^{\prime \nu}\right] $$

$$ x^{\prime \prime \mu}=-i \frac{e}{c} F^{\mu \nu} x_{\nu}^{\prime}, \quad \text { with } \quad F_{i j}=-\epsilon_{i j k} B^{k}, \quad F^{i 4}=i F^{i 0}=i E^{i} $$

$$ \mathbf{x}(\tau)=\hat{\mathbf{E}} A \cos \omega_{L}^{E}\left(\tau-\tau_{0}\right)+c_{2}, \quad x_{4}(\tau)=A \sin \omega_{L}^{E}\left(\tau-\tau_{0}\right)+c_{4}, $$

$$ \omega_{L}^{E} \equiv \frac{e E}{M c} $$

$$ \begin{align*} Z_{1} & =\int_{0}^{\infty} \frac{d \beta}{\beta} e^{-\beta M c^{2} / 2} \int \mathcal{D}^{2} x^{(12)} e^{-\overline{\mathcal{A}}_{\mathrm{e}}^{(12)} / \hbar} \int \mathcal{D}^{2} x^{(34)} e^{-\overline{\mathcal{A}}_{\mathrm{e}}^{(34)} / \hbar} \\ & \equiv \int_{0}^{\infty} \frac{d \beta}{\beta} e^{-\beta M c^{2} / 2} Z^{(12)}(0) Z^{(34)}(E) \end{align*} $$

$$ \overline{\mathcal{A}}_{\mathrm{e}}^{(12)}=\int_{0}^{\hbar \beta} d \tau \frac{M}{2}\left(x_{1}^{\prime 2}+x_{2}^{\prime 2}\right) $$

$$ Z^{(12)}(0)=\Delta x_{1} \Delta x_{2}{\sqrt{\frac{M}{2 \pi \hbar^{2} \beta}}}_{2}^{2} $$

$$ \operatorname{Det}\left(\begin{array}{cc} -\partial_{\tau}^{2} & -\omega_{L}^{E} \partial_{\tau} \\ \omega_{L}^{E} \partial_{\tau} & -\partial_{\tau}^{2} \end{array}\right)=\operatorname{Det}\left(-\partial_{\tau}^{2}\right) \times \operatorname{Det}\left(-\partial_{\tau}^{2}-\omega_{L}^{E^{2}}\right)=1 \times \frac{\sin \hbar \omega_{L}^{E} \beta / 2}{\hbar \omega_{L}^{E} \beta / 2} $$

$$ Z^{(34)}(E)=\Delta x_{3} \Delta x_{4}{\sqrt{\frac{M}{2 \pi \hbar^{2} \beta}}}^{2} \frac{\sin \hbar \omega_{L}^{E} \beta / 2}{\hbar \omega_{L}^{E} \beta / 2}, $$

$$ \overline{\mathcal{A}}_{\mathrm{e}}^{(34)}=\int_{0}^{\hbar \beta} d \tau \frac{M}{2}\left[x_{3}^{\prime 2}+x_{4}^{\prime 2}+\frac{e}{c} E\left(x_{3} x_{4}^{\prime}-x_{4} x_{3}^{\prime}\right)\right] $$

$$ x_{3}^{\prime \prime}=\omega_{L}^{E} x_{4}^{\prime}, \quad x_{4}^{\prime \prime}=-\omega_{L}^{E} x_{3}^{\prime} $$

$$ Z^{(34)}(E)=\Delta x_{3} \Delta x_{4}{\sqrt{\frac{M}{2 \pi \hbar^{2} \beta}}}^{2} \hbar \omega_{L}^{E} \beta \sum_{m=0}^{\infty} e^{-i\left(m+\frac{1}{2}\right) \hbar \omega_{L}^{E}} $$

$$ Z_{1}=\Delta x_{4} V \int_{0}^{\infty} \frac{d \beta}{\beta} \sqrt{\frac{M}{2 \pi \hbar^{2} \beta}} \frac{\omega_{L}^{E} \hbar \beta / 2}{\sin \omega_{L}^{E} \hbar \beta / 2} e^{-\beta M c^{2} / 2} $$

$$ Z_{1}=i \Delta \mathcal{A}^{\mathrm{eff}} / \hbar=i \Delta t V \Delta \mathcal{L}^{\mathrm{eff}} / \hbar $$

$$ \Delta \mathcal{L}^{\mathrm{eff}}=\hbar c\left(\frac{M c}{\hbar}\right)^{4} \frac{1}{4(2 \pi)^{2}} \int_{0}^{\infty} \frac{d \zeta}{\zeta^{3}}\left[\frac{E \zeta / E_{c}}{\sin E \zeta / E_{c}}-1-\frac{1}{6}\left(\frac{E}{E_{c}}\right)^{2}\right] e^{-\zeta} $$

$$ \Delta \mathcal{L}_{\mathrm{div}}^{\mathrm{eff}}=\hbar c\left(\frac{M c}{\hbar}\right)^{4} \frac{1}{24(2 \pi)^{2}}\left(\frac{E}{E_{c}}\right)^{2} \int_{0}^{\infty} \frac{d \zeta}{\zeta} e^{-\zeta}=\frac{\alpha}{24 \pi} \int_{0}^{\infty} \frac{d \zeta}{\zeta} e^{-\zeta} $$

$$ Z_{A}=1+\frac{\alpha}{12 \pi} \int_{0}^{\infty} \frac{d \zeta}{\zeta} e^{-\zeta} $$

$$ \frac{\Gamma}{V}=\frac{2}{\hbar} \operatorname{Im} \Delta \mathcal{L}^{\mathrm{eff}} $$

$$ \frac{z}{\sin z}=1+2 \sum_{n=1}^{\infty}(-1)^{n} \frac{z^{2}}{z^{2}-n^{2} \pi^{2}}=2 \sum_{n=1}^{\infty}(-1)^{n} \frac{\zeta^{2}}{\zeta^{2}-\zeta_{n}^{2}}, \quad \zeta_{n} \equiv n \pi \frac{E_{c}}{E} $$

$$ \frac{\zeta}{\zeta^{2}-\zeta_{n}^{2}} \rightarrow \frac{\zeta}{\zeta^{2}-\zeta_{n}^{2}+i \eta}=i \frac{\pi}{2} \delta\left(\zeta+\zeta_{n}\right)-i \frac{\pi}{2} \delta\left(\zeta-\zeta_{n}\right)+\zeta \frac{\mathcal{P}}{\zeta^{2}-\zeta_{n}^{2}} $$

$$ \begin{align*} \frac{\Gamma}{V} & =\frac{2}{\hbar} \operatorname{Im} \mathcal{L}^{\mathrm{eff}}=c\left(\frac{M c}{\hbar}\right)^{4}\left(\frac{E}{E_{c}}\right)^{2} \frac{1}{4 \pi^{3}} \frac{1}{2} \sum_{n=1}^{\infty}(-1)^{n-1} \frac{(-1)^{n-1}}{n^{2}} e^{-n \pi E_{c} / E} \\ & =\frac{1}{8 \pi^{3}} \frac{e^{2}}{\hbar c} E^{2} \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{2}} e^{-n \pi E_{c} / E} \end{align*} $$

$$ \Delta \mathcal{L}_{\mathcal{P}}^{\mathrm{eff}}=\hbar c\left(\frac{M c}{\hbar}\right)^{4} \frac{1}{4(2 \pi)^{2}} \mathcal{P} \int_{0}^{\infty} \frac{d \zeta}{\zeta^{3}}\left(\frac{E \zeta / E_{c}}{\sin E \zeta / E_{c}}-1-\frac{E^{2} \zeta^{2}}{6 E_{c}^{2}}\right) e^{-\zeta} $$

$$ \frac{z}{\sin z}=1+\frac{z^{2}}{6}+\frac{7}{360} z^{4}+\frac{31}{15120} z^{6}+\ldots $$

$$ \Delta \mathcal{L}^{\mathrm{eff}}=\hbar c\left(\frac{M c}{\hbar}\right)^{4} \frac{1}{4(2 \pi)^{2}}\left[\frac{7}{360}\left(\frac{E}{E_{c}}\right)^{4}+\frac{31}{2520}\left(\frac{E}{E_{c}}\right)^{6}+\ldots\right] $$

$$ \mathcal{L}^{\mathrm{eff}}=\frac{1}{2}\left\{E^{2}+\frac{7 \alpha^{2}}{180} \frac{(\hbar c)^{3}}{\left(M c^{2}\right)^{4}} E^{4}+\frac{31 \pi \alpha^{3}}{315}\left[\frac{(\hbar c)^{3}}{\left(M c^{2}\right)^{4}}\right]^{2} E^{6}+\ldots\right\} $$

$$ \epsilon(E)=1+\frac{7 \alpha^{2}}{90} \frac{(\hbar c)^{3}}{\left(M c^{2}\right)^{4}} E^{2}+\frac{31 \pi \alpha^{3}}{105}\left[\frac{(\hbar c)^{3}}{\left(M c^{2}\right)^{4}}\right]^{2} E^{4}+\ldots $$

$$ Z_{1}=\Delta x_{4} V \int_{0}^{\infty} d \beta{\sqrt{\frac{M}{2 \pi \hbar^{2} \beta}}}^{4} \omega_{L}^{E} \hbar \sum_{n=0}^{\infty} e^{-i\left(n+\frac{1}{2}\right) \hbar \omega_{L}^{E} \beta} $$

$$ \Delta \mathcal{L}^{\mathrm{eff}}=\hbar c\left(\frac{M c}{\hbar}\right)^{4} \frac{1}{2(2 \pi)^{2}} \int_{0}^{\infty} \frac{d \zeta}{\zeta^{2}} \frac{E}{E_{c}}\left[\sum_{m=0}^{\infty} e^{i\left(m+\frac{1}{2}\right) 2 E \zeta / E_{c}}\right] e^{-\zeta} $$

$$ \Delta \mathcal{L}_{\mathrm{reg}}^{\mathrm{eff}}=\hbar c\left(\frac{M c}{\hbar}\right)^{4} \frac{i}{2(2 \pi)^{2}}\left[\sum_{k=4,6, \ldots}^{\infty} \frac{(-1)^{k / 2-1}}{(k-1)(k-2)}\left(2^{1-k}-1\right) \zeta(1-k) 2^{k-1}\left(\frac{E}{E_{c}}\right)^{k}\right]_{\alpha=1} $$

$$ \begin{align*} \Delta \mathcal{L}_{\mathrm{reg}}^{\mathrm{eff}} & =\hbar c\left(\frac{M c}{\hbar}\right)^{4} \frac{i}{2(2 \pi)^{2}}\left[\int d \alpha \int d \alpha \sum_{k=4,6, \ldots}^{\infty}\right. \\ & \left.\times \sum_{m=0}^{\infty} \frac{(-1)^{k / 2-1}}{\alpha^{k}}(-1)^{k / 2-1}\left[\left(m+\frac{1}{2}\right)\right]^{k-1} 2^{k-1}\left(\frac{E}{E_{c}}\right)^{k}\right]_{\alpha=1} \end{align*} $$

$$ \overline{\mathcal{A}}_{\mathrm{e}}^{(12)}=\int_{0}^{\hbar \beta} d \tau \frac{M}{2}\left[x_{1}^{\prime 2}+x_{2}^{\prime 2}+\frac{e}{c} i B\left(x_{1} x_{2}^{\prime}-x_{2} x_{1}^{\prime}\right)\right] $$

$$ Z^{(12)}(B)=Z^{(34)}(i B) $$

$$ Z_{1}=\Delta x_{4} V \int_{0}^{\infty} \frac{d \beta}{\beta}{\sqrt{\frac{M}{2 \pi \hbar^{2} \beta}}}^{4} \frac{\omega_{L}^{E} \hbar \beta / 2}{\sin \omega_{L}^{E} \hbar \beta / 2} \frac{\omega_{L}^{B} \hbar \beta / 2}{\sinh \omega_{L}^{B} \hbar \beta / 2} e^{-\beta M c^{2} / 2} $$

$$ \Delta \mathcal{L}^{\mathrm{eff}}=\hbar c\left(\frac{M c}{\hbar}\right)^{4} \frac{1}{4(2 \pi)^{2}} \int_{0}^{\infty} \frac{d \zeta}{\zeta^{3}}\left[\frac{E \zeta / E_{c}}{\sin E \zeta / E_{c}} \frac{B \zeta / E_{c}}{\sinh B \zeta / E_{c}}-1-\frac{\left(E^{2}-B^{2}\right) \zeta^{2}}{6 E_{c}^{3}}\right] e^{-\zeta} $$

$$ \begin{align*} \frac{1}{\tau^{3}} \frac{e \beta \tau}{\sin e \beta \tau} \frac{e \varepsilon \tau}{\sinh e \varepsilon \tau} & =\frac{1}{\tau^{3}}-\frac{e^{2}}{6 \tau}\left(\varepsilon^{2}-\beta^{2}\right)+e^{4} \frac{\tau}{360}\left(7 \varepsilon^{4}-10 \varepsilon^{2} \beta^{2}+7 \beta^{4}\right) \\ & -e^{6} \frac{\tau^{3}}{1520}\left(31 \varepsilon^{6}-49 \varepsilon^{4} \beta^{2}+49 \varepsilon^{2} \beta^{4}-31 \beta^{6}\right)+\ldots \end{align*} $$

$$ \begin{align*} \mathcal{L}^{\mathrm{eff}}= & \frac{1}{2}\left\{\left(\mathbf{E}^{2}-\mathbf{B}^{2}\right)+\frac{7 \alpha^{2}}{180} \frac{(\hbar c)^{3}}{\left(M c^{2}\right)^{4}}\left(\mathbf{E}^{2}-\mathbf{B}^{2}\right)^{2}\right. \\ & \left.+\frac{31 \pi \alpha^{3}}{315}\left[\frac{(\hbar c)^{3}}{\left(M c^{2}\right)^{4}}\right]^{2}\left(\mathbf{E}^{2}-\mathbf{B}^{2}\right)\left[2\left(\mathbf{E}^{2}-\mathbf{B}^{2}\right)^{2}-4(\mathbf{E B})^{2}\right]+\ldots\right\},(1 \end{align*} $$

$$ \mathcal{L}^{\mathrm{eff}} \equiv-\frac{e^{2}}{192 \pi^{2}}\left(\mathbf{E}^{2}-\mathbf{B}^{2}\right) \log \left[-4 e^{2} \frac{(\hbar c)^{3}}{\left(M c^{2}\right)^{4}}\left(\mathbf{E}^{2}-\mathbf{B}^{2}\right)\right]+\ldots $$

$$ 4 \operatorname{Det}^{1 / 2}\left(-g_{\mu \nu} \partial_{\lambda}+i \frac{e}{M c^{2}} F_{\mu \nu}\right)=4 \operatorname{det}^{1 / 2}\left(\cosh \frac{e}{M c} F_{\mu \nu} \frac{\hbar \beta}{2}\right) $$

$$ 4 \operatorname{Det}^{1 / 2}\left(-g_{\mu \nu} \partial_{\lambda}+i \frac{e}{M c^{2}} F_{\mu \nu}\right)=4 \cosh \left(\mu_{B} \mathcal{B} \beta / 2\right) \cos \left(\mu_{B} \mathcal{E} \beta / 2\right) $$

$$ z \frac{\cos z}{\sin z}=1+2 \sum_{n=1}^{\infty} \frac{z^{2}}{z^{2}-n^{2} \pi^{2}}=2 \sum_{n=1}^{\infty} \frac{\zeta^{2}}{\zeta^{2}-\zeta_{n}^{2}}, \quad \zeta_{n} \equiv n \pi \frac{E_{c}}{E} . $$

$$ \Delta \mathcal{L}_{\mathrm{spin} \frac{1}{2}}^{\mathrm{eff}}=-\hbar c\left(\frac{M c}{\hbar}\right)^{4} \frac{1}{2(2 \pi)^{2}} \int_{0}^{\infty} \frac{d \zeta}{\zeta^{3}}\left(\frac{E \zeta / E_{c}}{\tan E \zeta / E_{c}}-1+\frac{E^{2} \zeta^{2}}{3 E_{c}^{3}}\right) e^{-\zeta} $$

$$ \begin{align*} \frac{\Gamma_{\operatorname{spin} \frac{1}{2}}}{V} & =\frac{2}{\hbar} \operatorname{Im} \Delta \mathcal{L}_{\mathrm{spin} \frac{1}{2}}^{\mathrm{eff}}=c\left(\frac{M c}{\hbar}\right)^{4}\left(\frac{E}{E_{c}}\right)^{2} \frac{1}{4 \pi^{3}} \sum_{n=1}^{\infty} \frac{1}{n^{2}} e^{-n \pi E_{c} / E} \\ & =\frac{1}{4 \pi^{3}} \frac{e^{2}}{\hbar c} E^{2} \sum_{n=1}^{\infty} \frac{1}{n^{2}} e^{-n \pi E_{c} / E} \end{align*} $$

$$ \frac{z}{\tan z}=1-\frac{z^{2}}{3}-\frac{1}{45} z^{4}-\frac{2}{945} z^{6}-\ldots $$

$$ \Delta \mathcal{L}^{\mathrm{eff}}=\hbar c\left(\frac{M c}{\hbar}\right)^{4} \frac{2}{16 \pi^{2}}\left[\frac{1}{45}\left(\frac{E}{E_{c}}\right)^{4}+\frac{4}{315}\left(\frac{E}{E_{c}}\right)^{6}+\ldots\right] $$

$$ \mathcal{L}^{\mathrm{eff}}=\frac{1}{2}\left\{E^{2}+\frac{4 \alpha^{2}}{45} \frac{(\hbar c)^{3}}{\left(M c^{2}\right)^{4}} E^{4}+\frac{64 \pi \alpha^{3}}{315}\left[\frac{(\hbar c)^{3}}{\left(M c^{2}\right)^{4}}\right]^{2} E^{6}+\ldots\right\} $$

$$ \epsilon(E)=\frac{1}{E} \frac{\partial \mathcal{L}^{\mathrm{eff}}}{\partial E}=1+\frac{8 \alpha^{2}}{45} \frac{(\hbar c)^{3}}{\left(M c^{2}\right)^{4}} E^{2}+\frac{64 \pi \alpha^{3}}{105}\left[\frac{(\hbar c)^{3}}{\left(M c^{2}\right)^{4}}\right]^{2} E^{4}+\ldots $$

$$ \Delta \mathcal{L}_{\mathrm{spin} \frac{1}{2}}^{\mathrm{eff}}=-\hbar c\left(\frac{M c}{\hbar}\right)^{4} \frac{1}{2(2 \pi)^{2}} \int_{0}^{\infty} \frac{d \zeta}{\zeta^{3}}\left[\frac{E \zeta / E_{c}}{\tan E \zeta / E_{c}} \frac{B \zeta / E_{c}}{\tanh B \zeta / E_{c}}-1+\frac{\left(E^{2}-B^{2}\right) \zeta^{2}}{3 E_{c}^{3}}\right] e^{-\zeta} $$

$$ \frac{\Gamma_{\operatorname{spin} \frac{1}{2}}}{V}=\frac{2}{\hbar} \operatorname{Im} \Delta \mathcal{L}_{\mathrm{spin} \frac{1}{2}}^{\mathrm{eff}}=c\left(\frac{M c}{\hbar}\right)^{4}\left(\frac{\varepsilon}{E_{c}}\right)^{2} \frac{1}{4 \pi^{3}} \sum_{n=1}^{\infty} \frac{1}{n^{2}} \frac{n \pi \beta / \varepsilon}{\tanh n \pi \beta / \varepsilon} e^{-n \pi E_{c} / \varepsilon} \cdot( $$

$$ \mathcal{L}^{\mathrm{eff}} \equiv-\frac{e^{2}}{48 \pi^{2}}\left(\mathbf{E}^{2}-\mathbf{B}^{2}\right) \log \left[-4 e^{2} \frac{(\hbar c)^{3}}{\left(M c^{2}\right)^{4}}\left(\mathbf{E}^{2}-\mathbf{B}^{2}\right)\right]+\ldots $$

$$ g_{00}(x) \equiv 1 $$

$$ g_{0 i}(x) \equiv 0 $$

$$ \bar{\Gamma}_{00}^{\mu} \equiv \frac{1}{2} g^{\mu \nu}\left(\partial_{0} g_{0 \nu}+\partial_{0} g_{0 \nu}-g_{00}\right) \equiv 0 $$

$$ \frac{d u^{\mu}}{d \tau}=-\bar{\Gamma}_{00}^{\mu} c^{2}=0 $$

$$ d s^{2}=c^{2} d t^{2}-{ }^{(3)} g_{i j}(x) d x^{i} d x^{j} $$

$$ d l^{2}={ }^{(3)} g_{i j}(x) d x^{i} d x^{j} . $$

$$ { }^{(3)} R_{i j k l}(x)=\frac{1}{a^{2}}\left[{ }^{(3)} g_{i l}(x)^{(3)} g_{j k}(x)-{ }^{(3)} g_{i k}(x)^{(3)} g_{j l}(x)\right] \text {. } $$

$$ k=\left\{\begin{array}{cc} 1 & \text { spherical } \\ 0 & \text { parabolic } \\ -1 & \text { hyperbolic } \end{array}\right\} \quad \text { universe, } $$

$$ { }^{(3)} R_{i l}=k \frac{2}{a^{2}} g_{i l}(x), \quad{ }^{(3)} R=k \frac{6}{a^{2}} \text {. } $$

$$ { }^{4} S^{a}=2 \pi^{2} a^{3} $$

$$ \begin{align*} V_{r_{0}}^{a} & =\int_{0}^{2 \pi} d \varphi \int_{0}^{\pi} d \theta \sin \theta \int_{0}^{r} d r \frac{r^{2}}{\sqrt{1-r^{2} / a^{2}}} \\ & =4 \pi\left(\frac{a^{3}}{2} \arcsin \frac{r_{0}}{a}-\frac{a^{2} r_{0}}{2} \sqrt{1-\frac{r_{0}^{2}}{a^{2}}}\right) \end{align*} $$

$$ { }^{(3)} V_{r_{0}}^{a} \approx V_{r_{0}}=\frac{4 \pi}{3} r_{0}^{2} $$

$$ \begin{align*} d s^{2} & =c^{2} d t^{2}-d l^{2} \\ d l^{2} & =\frac{d r^{2}}{1-k r^{2} / a^{2}}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \varphi^{2}\right) \end{align*} $$

$$ r=a \sin \alpha $$

$$ d s^{2}=c^{2} d t^{2}-a^{2}(t)\left[d \alpha^{2}+f^{2}(\alpha)\left(d \theta^{2}+\sin ^{2} \theta d \varphi^{2}\right)\right] $$

$$ f(\alpha)= \begin{cases}\sin \alpha & k=1 \\ \alpha & k=0 \\ \sinh \alpha & k=-1\end{cases} $$

$$ c d t=a(\eta) d \eta $$

$$ d s^{2}=a^{2}(\eta)\left[d \eta^{2}-d \alpha^{2}-f^{2}(\alpha)\left(d \theta^{2}+\sin ^{2} \theta d \varphi^{2}\right)\right] . $$

$$ g_{\mu \nu}=a^{2}(\eta)\left(\begin{array}{llll} 1 & & & \\ & -1 & & \\ & & -f^{2}(\alpha) & \\ & & & -f^{2}(\alpha) \sin ^{2} \theta \end{array}\right) $$

$$ \Gamma_{00}^{0}=\frac{a_{\eta}}{a}, \quad \Gamma_{00}^{i}=0, \quad \Gamma_{0 i}^{0}=0, \quad \Gamma_{0 i}^{j}=\frac{a_{\eta}}{a} \delta_{i}^{j}, \quad \Gamma_{i j}^{0}=-\frac{a_{\eta}}{a^{3}} g_{i j}, \quad \Gamma_{i j}^{k}=0, $$

$$ a_{\eta} \equiv \frac{d a}{d \eta}=\frac{a}{c} \frac{d a}{d t} \equiv \frac{a}{c} a_{t} $$

$$ R_{00}=\partial_{\mu} \Gamma_{00}{ }^{\mu}-\partial_{0} \Gamma_{\mu 0}{ }^{\mu}-\Gamma_{\mu 0}{ }^{\nu} \Gamma_{0 \nu}{ }^{\mu}+\Gamma_{00}{ }^{\mu} \Gamma_{\nu \mu}{ }^{\nu} . $$

$$ \begin{align*} & \partial_{\mu} \Gamma_{00}{ }^{\mu}-\partial_{0} \Gamma_{\mu 0}{ }^{\mu}=-\partial_{0} \Gamma_{i 0}{ }^{i}=-3 \frac{d}{d \eta} \frac{a_{\eta}}{a}=-3 \frac{1}{a^{2}}\left(a_{\eta \eta} a-a_{\eta}^{2}\right) \\ & \Gamma_{\mu 0}{ }^{\nu} \Gamma_{0 \nu}{ }^{\mu}=\Gamma_{00}{ }^{0} \Gamma_{00}{ }^{0}+\Gamma_{00}{ }^{i} \Gamma_{0 i}{ }^{0}+\Gamma_{i 0}{ }^{0} \Gamma_{00}{ }^{i}+\Gamma_{i 0}{ }^{j} \Gamma_{0 j}{ }^{i}=\left(\frac{a_{\eta}}{a}\right)^{2}+3\left(\frac{a_{\eta}}{a}\right)^{2}, \\ & \Gamma_{00}{ }^{\mu} \Gamma_{\nu \mu}{ }^{\nu}=\Gamma_{00}{ }^{0} \Gamma_{00}{ }^{0}+\Gamma_{00}{ }^{0} \Gamma_{i 0}{ }^{i}+\Gamma_{00}{ }^{i} \Gamma_{0 i}{ }^{0}+\Gamma_{00}{ }^{i} \Gamma_{k i}{ }^{k}=\left(\frac{a_{\eta}}{a}\right)^{2}+3\left(\frac{a_{\eta}}{a}\right)^{2}, \end{align*} $$

$$ R_{00}=-\frac{3}{a^{2}}\left(a a_{\eta \eta}-a_{\eta}^{2}\right), \quad R_{0}^{0}=g^{00} R_{00}=-\frac{3}{a^{4}}\left(a a_{\eta \eta}-a_{\eta}^{2}\right) . $$

$$ \begin{align*} R_{i j}=R_{\mu i j}{ }^{\mu} & =R_{k i j}{ }^{k}+R_{0 i j}{ }^{0} \\ & ={ }^{(3)} R_{i j}-\Gamma_{k j}{ }^{0} \Gamma_{i 0}{ }^{k}+\Gamma_{i j}{ }^{0} \Gamma_{k 0}{ }^{k}+R_{0 i j}{ }^{0} \end{align*} $$

$$ \begin{gather*} R_{0 i j}{ }^{0}=\partial_{0} \Gamma_{i j}{ }^{0}-\partial_{i} \Gamma_{0 j}{ }^{0}-\Gamma_{0 j}{ }^{l} \Gamma_{i l}{ }^{0}-\Gamma_{0 j}{ }^{0} \Gamma_{i 0}{ }^{0}+\Gamma_{i j}{ }^{l} \Gamma_{0 l}{ }^{0}+\Gamma_{i j}{ }^{0} \Gamma_{00}{ }^{0}, \\ { }^{(3)} R_{i j}=k \frac{2}{a^{2}} g_{i j} \end{gather*} $$

$$ R_{i j}=-\frac{1}{a^{4}}\left(2 k a^{2}+a_{\eta}^{2}+a a_{\eta \eta}\right) g_{i j} $$

$$ \begin{align*} R & =g^{00} R_{00}+g^{i j} R_{i j}=-\frac{1}{a^{2}}\left[\frac{3}{a^{2}}\left(a a_{\eta \eta}-a_{\eta}^{2}\right)\right]-\frac{3}{a^{4}}\left(2 k a^{2}+a_{\eta}^{2}+a a_{\eta \eta}\right) \\ & =-\frac{6}{a^{3}}\left(a_{\eta \eta}+k a\right) \end{align*} $$

$$ \mathcal{A}=\int d^{4} x \sqrt{-g}{ }_{\mathcal{L}}^{f}=-\frac{1}{2 \kappa} \int d^{4} x \sqrt{-g}(R+2 \lambda) $$

$$ G_{N} \approx 6.673 \cdot 10^{-8} \mathrm{~cm}^{3} \mathrm{~g}^{-1} \mathrm{~s}^{-2} $$

$$ \frac{1}{\kappa}=\frac{c^{3}}{8 \pi G_{N}} $$

$$ l_{\mathrm{P}}=\left(\frac{c^{3}}{G_{N} \hbar}\right)^{-1 / 2} \approx 1.615 \times 10^{-33} \mathrm{~cm} $$

$$ m_{\mathrm{P}}=\left(\frac{c \hbar}{G_{N}}\right)^{1 / 2} \approx 2.177 \times 10^{-5} \mathrm{~g}=1.22 \times 10^{22} \mathrm{MeV} / \mathrm{c}^{2} $$

$$ \frac{1}{\kappa}=\frac{\hbar}{8 \pi l_{P}^{2}} $$

$$ \frac{1}{\kappa}\left(R_{\mu \nu}-\frac{1}{2} g_{\mu \nu} R-\lambda g_{\mu \nu}\right)=T_{\mu \nu} $$

$$ \Omega_{\lambda 0} \equiv \frac{\lambda c^{2}}{3 H_{0}^{2}} $$

$$ H_{0}^{-1} \approx 14 \times 10^{9} \text { years. } $$

$$ \Omega_{\lambda 0} \approx 0.68 \pm 0.10 $$

$$ \lambda=\Omega_{\lambda 0} \frac{3 H_{0}^{2}}{c^{2}} \approx \frac{\Omega_{\lambda 0}}{\left(6.55 \times 10^{27} \mathrm{~cm}\right)^{2}} \approx \frac{\Omega_{\lambda 0}}{\left(6.93 \times 10^{9} \mathrm{ly}\right)^{2}} \approx \frac{\Omega_{\lambda 0}}{\left(2.14 R_{\text {universe }}\right)^{2}} $$

$$ d s^{2}=B(r) c^{2} d t^{2}-B^{-1} d r^{2}-r^{2} d \theta^{2}-r^{2} \sin ^{2} \theta d \phi^{2} $$

$$ B(r)=1-\frac{2 M G_{N}}{c^{2} r}-\frac{2}{3} \lambda r^{2}=1-\frac{M}{m_{\mathrm{P}}} \frac{l_{\mathrm{P}}}{r}-\frac{2}{3} \Omega_{\lambda 0} \frac{r^{2}}{\left(2.14 R_{\text {universe }}\right)^{2}} $$

$$ \Lambda=\frac{\lambda}{\kappa}=\Omega_{\lambda 0} \frac{3 H_{0}^{2}}{c^{2}} \frac{l_{\mathrm{P}}^{2}}{8 \pi} \approx 10^{-122} \frac{\hbar}{l_{\mathrm{P}}^{4}} $$

$$ \frac{3}{a^{4}}\left(a_{\eta}^{2}+k a^{2}\right)-\lambda=\kappa T_{0}^{0} $$

$$ 3\left[\left(\frac{a_{t}}{a}\right)^{2}+k \frac{c^{2}}{a^{2}}\right]-\lambda c^{2}=c^{2} \kappa T_{0}^{0} $$

$$ T_{\mu}{ }^{\nu}=c \rho u_{\mu} u^{\nu}, $$

$$ T_{0}{ }^{0}=c \rho . $$

$$ \frac{3}{a^{4}}\left(a_{\eta}^{2}+k a^{2}\right)-\lambda=c \kappa \rho . $$

$$ \rho=\frac{M}{2 \pi^{2} a^{3}} $$

$$ \frac{3}{a^{4}}\left(a_{\eta}^{2}+k a^{2}\right)-\lambda=\frac{\kappa M c}{2 \pi^{2} a^{3}}=\frac{4 G_{\mathrm{N}} M}{\pi c^{2} a^{3}} $$

$$ \int d^{4} x \sqrt{-g}=\int d t^{(4)} S^{a}=2 \pi^{2} \int d \eta a^{4}(\eta) $$

$$ \stackrel{f}{\mathcal{A}}=\frac{2 \pi^{2}}{2 \kappa} \int d \eta\left[6 a\left(a_{\eta \eta}+k a\right)-2 \lambda a^{4}\right] \stackrel{\wedge}{=} \frac{2 \pi^{2}}{\kappa} \int d \eta\left[-3 a_{\eta}^{2}+3 k a^{2}-\lambda a^{4}\right] $$

$$ \stackrel{m}{\mathcal{A}}=-\int d^{4} x \sqrt{-g} c \rho=-2 \pi^{2} \int d \eta \frac{M c}{2 \pi^{2}} a(\eta) $$

$$ 6\left(a_{\eta \eta}+k a\right)-4 \lambda a^{3}-\frac{\kappa M c}{2 \pi^{2}}=0 $$

$$ \ddot{a}=\frac{\lambda}{3} a-\frac{1}{6} \frac{\kappa M c}{2 \pi^{2} a^{2}} $$

$$ \lambda=\lambda_{\text {Einstein }} \equiv \frac{\kappa M c}{2 \pi^{2} a^{3}}=\frac{4 G_{\mathrm{N}} M}{\pi c^{2} a^{3}}=\frac{4 \pi G_{\mathrm{N}} \rho}{c}, $$

$$ 3\left(a_{\eta}^{2}+k a^{2}\right)-\lambda a^{4}-\frac{\kappa M c}{2 \pi^{2}} a=\mathrm{const} $$

$$ a_{\eta}^{2}+k a^{2}-a_{\max } a-\frac{\lambda}{3} a^{4}=0 $$

$$ a_{\max } \equiv \frac{\kappa M c}{6 \pi^{2}}=\frac{4 G_{\mathrm{N}} M}{3 \pi c^{2}} $$

$$ V^{\text {univ }}(a)=k a^{2}-a_{\max } a-\frac{\lambda}{3} a^{4} $$

$$ a_{\eta}^{2}+k a^{2}-a_{\max } a=0 $$

$$ \eta=\int \frac{d a}{\sqrt{-V^{\text {univ }}(a)}}=\int \frac{d a}{\sqrt{-\left(a-a_{\max } / 2\right)^{2}+a_{\max }^{2} / 4}}=-\arccos \frac{2 a}{a_{\max }} $$

$$ a(\eta)=\frac{a_{\max }}{2}(1-\cos \eta) $$

$$ t=\frac{1}{c} \int d \eta a(\eta)=\frac{a_{\mathrm{max}}}{2 c} \int d \eta(1-\cos \eta)=\frac{a_{\mathrm{max}}}{2 c}(\eta-\sin \eta) $$

$$ a_{\eta}^{2}-a^{2}-a_{\max } a-\frac{\lambda}{3} a^{4}=0 $$

$$ \begin{align*} a(\eta) & =\frac{a_{\max }}{2}(\cosh \eta-1) \\ t & =\frac{a_{\max }}{2 c}(\sinh \eta-\eta) \end{align*} $$

$$ a_{\eta}^{2}-a_{\max } a-\frac{\lambda}{3} a^{4}=0 $$

$$ \eta=2 \sqrt{\frac{a}{a_{\max }}}, $$

$$ a(\eta)=a_{\max } \frac{\eta^{2}}{4} $$

$$ t=\frac{a_{\mathrm{max}}}{12 c} \eta^{3} $$

$$ a(t)=\left(\frac{9}{4} a_{\max }\right)^{1 / 3}(c t)^{2 / 3} $$

$$ \mathcal{A}^{\mathrm{int}}=-\int_{\lambda_{a}}^{\lambda_{b}} d \lambda h(\lambda) \frac{\left(E+e^{2} / r\right)^{2}}{2 M c^{3}} $$

$$ \tilde{\mathcal{A}}_{\mathrm{e}, E}=\int_{\lambda_{a}}^{\lambda_{b}} d \lambda\left\{\frac{4 M c \vec{u}^{2}}{2 h(\lambda)} \vec{u}^{\prime 2}(\lambda)+\frac{h(\lambda)}{2 M c^{3} \vec{u}^{2}}\left[\left(M^{2} c^{4}-E^{2}\right) \vec{u}^{2}-2 E e^{2}-\frac{e^{4}}{\vec{u}^{2}}\right]\right\} $$

$$ \overline{\mathcal{A}}_{\mathrm{e}, E}^{\mathrm{DK}}=\int_{s_{a}}^{s_{b}} \frac{d s}{c}\left\{\frac{4 M c^{2}}{2} \vec{u}^{\prime 2}(s)+\frac{1}{2 M c^{2}}\left[\left(M^{2} c^{4}-E^{2}\right) \vec{u}^{2}-2 E e^{2}-\frac{e^{4}}{\vec{u}^{2}}\right]\right\} . $$

$$ \omega=\frac{1}{2 M c^{2}} \sqrt{M^{2} c^{4}-E^{2}} $$

$$ V_{\text {extra }}=\hbar^{2} \frac{l_{\text {extra }}^{2}}{2 \mu \vec{u}^{2}} $$

$$ l_{\mathrm{extra}}^{2} \equiv-4 \alpha^{2} $$

$$ V_{\mathrm{const}}=-\frac{E}{M c^{2}} e^{2} $$

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

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

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

$$ h \equiv e^{-2 \omega S} $$

$$ \begin{align*} \nu & =\frac{e^{2}}{2 \omega \hbar} \frac{E}{M c^{2}}=\frac{\alpha}{\sqrt{M^{2} c^{4} / E^{2}-1}} \\ \kappa & =\frac{\mu \omega}{2 \hbar}=\frac{1}{\hbar c} \sqrt{M^{2} c^{4}-E^{2}}=\frac{E}{\hbar c} \frac{\alpha}{\nu} \end{align*} $$

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

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

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

$$ 2 l+1 \rightarrow 2 \tilde{l}+1 \equiv \sqrt{(2 l+1)^{2}+l_{\mathrm{extra}}^{2}} $$

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

$$ \delta_{l} \equiv l-\tilde{l}=l+1 / 2-\sqrt{(l+1 / 2)^{2}-\alpha^{2}} \approx \frac{\alpha^{2}}{2 l+1}+\mathcal{O}\left(\alpha^{4}\right) . $$

$$ \begin{align*} E_{n l} & = \pm M c^{2}\left[1+\frac{\alpha^{2}}{\left(n-\delta_{l}\right)^{2}}\right]^{-1 / 2} \\ & \approx \pm M c^{2}\left[1-\frac{\alpha^{2}}{2 n^{2}}-\frac{\alpha^{4}}{n^{3}}\left(\frac{1}{2 l+1}-\frac{3}{8 n}\right)+\mathcal{O}\left(\alpha^{6}\right)\right] \end{align*} $$

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

$$ \kappa=\frac{1}{\tilde{a}_{H}} \frac{1}{\nu} $$

$$ \tilde{a}_{H} \equiv a_{H} \frac{M c^{2}}{E} $$

$$ -i \Gamma(-\nu+\tilde{l}+1) \frac{M}{\hbar \kappa} \approx \frac{(-)^{n_{r}}}{\tilde{n}_{l}^{2} n_{r}!} \frac{1}{\tilde{a}_{H}}\left(\frac{E}{M c^{2}}\right)^{2} \frac{2 M c^{2} i \hbar}{E^{2}-E_{n l}^{2}} $$

$$ \left(r_{b} \mid r_{a}\right)_{E, l}=\frac{\hbar}{M c} \sum_{n=l+1}^{\infty}\left(\frac{E}{M c^{2}}\right)^{2} \frac{2 M c^{2} i \hbar}{E^{2}-E_{n l}^{2}} R_{n l}\left(r_{b}\right) R_{n l}\left(r_{a}\right)+\ldots $$

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

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

$$ \overline{\mathcal{A}}_{\mathrm{e}}[p, x]=\int_{\lambda_{a}}^{\lambda_{b}} d \lambda\left\{-i p \dot{x}+\frac{h(\lambda)}{2 M c}\left[\left(p-\frac{e}{c} A\right)^{2}+M^{2} c^{2}\right]\right\}, $$

$$ \left(x_{b} \mid x_{a}\right)=\frac{\hbar}{2 M c} \int_{0}^{\infty} d S \int \mathcal{D} h \Phi[h] \int \mathcal{D}^{D} x e^{-\overline{\mathcal{A}}_{\mathrm{e}} / \hbar} $$

$$ \overline{\mathcal{A}}_{\mathrm{e}}=\int_{\lambda_{a}}^{\lambda_{b}} d \lambda\left[\frac{M c}{2 h(\lambda)} \dot{x}^{2}(\lambda)+i \frac{e}{c} \dot{x}(\lambda) A(x(\lambda))+h(\lambda) \frac{M c}{2}\right] $$

$$ \left(x_{b} \mid x_{a}\right)=\frac{\hbar^{2}}{2 M} \int_{0}^{\infty} d \beta e^{-\beta M c^{2} / 2} \int \mathcal{D}^{4} x e^{-\mathcal{A}_{\mathrm{e}}} $$

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

$$ Z_{1}=\int_{0}^{\infty} \frac{d \beta}{\beta} e^{-\beta M c^{2} / 2} \int \mathcal{D}^{D} x e^{-\mathcal{A}_{\mathrm{e}} / \hbar} $$

$$ e^{-\mathcal{A}_{\mathrm{int}} / \hbar}=\sum_{n=0}^{\infty} \frac{(-i e / \hbar c)^{n}}{n!} \prod_{i=1}^{n}\left[\int_{0}^{\hbar \beta} d \tau_{i} \dot{x}\left(\tau_{i}\right) A\left(x\left(\tau_{i}\right)\right)\right] $$

$$ \frac{\Gamma_{\mathrm{e}, 0}}{\hbar} \equiv-Z_{1}=-\int_{0}^{\infty} \frac{d \beta}{\beta} e^{-\beta M c^{2} / 2} \frac{V_{D}}{{\sqrt{2 \pi \hbar^{2} \beta / M}}^{D}}=-\frac{V_{D}}{\lambda_{M}^{\mathrm{C}}} \frac{1}{(4 \pi)^{D / 2}} \Gamma(1-D / 2) $$

$$ \frac{\Gamma_{\mathrm{e}}}{\hbar}=\frac{\Gamma_{\mathrm{e}, 0}}{\hbar}-\int_{0}^{\infty} \frac{d \beta}{\beta} e^{-\beta M c^{2} / 2} \frac{V_{D}}{\sqrt{2 \pi \hbar^{2} \beta / M}} \sum_{n=1}^{\infty} \frac{(-i e / c)^{n}}{n!}\left\langle\prod_{i=1}^{n}\left[\int_{0}^{\hbar \beta} d \tau_{i} \dot{x}\left(\tau_{i}\right) A\left(x\left(\tau_{i}\right)\right)\right]\right\rangle_{0} $$

$$ \langle\mathcal{O}[x]\rangle_{0} \equiv \frac{\int \mathcal{D}^{D} x \mathcal{O}[x] e^{-\mathcal{A}_{\mathrm{e}, 0} / \hbar}}{\int \mathcal{D}^{D} x e^{-\mathcal{A}_{\mathrm{e}, 0} / \hbar}} $$

$$ A(x)=\int \frac{d^{D} k}{(2 \pi)^{D}} e^{i k x} A(k) $$

$$ \begin{align*} \frac{\Gamma_{\mathrm{e}}}{\hbar}= & \frac{\Gamma_{\mathrm{e}, 0}}{\hbar}-\int_{0}^{\infty} \frac{d \beta}{\beta} e^{-\beta M c^{2} / 2} \frac{V_{D}}{{\sqrt{2 \pi \hbar^{2} \beta / M}}^{D}} \\ & \times \sum_{n=1}^{\infty} \frac{(-i e / \hbar c)^{n}}{n!} \prod_{i=1}^{n}\left[\int \frac{d^{D} k_{i}}{(2 \pi)^{D}} A^{\mu_{i}}\left(k_{i}\right)\right]\left\langle\prod_{i=1}^{n}\left[\int_{0}^{\hbar \beta} d \tau_{i} \dot{x}^{\mu_{i}}\left(\tau_{i}\right) e^{i k_{i} x\left(\tau_{i}\right)}\right]\right\rangle_{0}^{.} \end{align*} $$

$$ \left\langle\dot{x}\left(\tau_{1}\right) e^{i k_{1} x\left(\tau_{1}\right)} \cdots \dot{x}\left(\tau_{n}\right) e^{i k_{n} x\left(\tau_{n}\right)}\right\rangle_{0} $$

$$ x(\tau)=x_{0}+\delta x(\tau) $$

$$ \left\langle e^{i\left(k_{1}+\ldots+k_{n}\right) x_{0}}\right\rangle_{0}\left\langle\delta \dot{x}\left(\tau_{1}\right) e^{i k_{1} \delta x\left(\tau_{1}\right)} \cdots \delta \dot{x}\left(\tau_{n}\right) e^{i k_{n} \delta x\left(\tau_{n}\right)}\right\rangle_{0} . $$

$$ \left\langle e^{i\left(k_{1}+\ldots+k_{n}\right) x_{0}}\right\rangle_{0}=\frac{1}{V_{D}}(2 \pi)^{D} \delta^{(D)}\left(k_{1}+\ldots+k_{n}\right) . $$

$$ \left\langle\delta x^{\mu}\left(\tau_{1}\right) \delta x^{\nu}\left(\tau_{2}\right)\right\rangle_{0}=\delta^{\mu \nu} G\left(\tau_{1}, \tau_{2}\right)=\delta^{\mu \nu} \frac{\hbar}{M} \bar{\Delta}\left(\tau_{1}, \tau_{2}\right) $$

$$ \bar{\Delta}\left(\tau, \tau^{\prime}\right) \equiv \bar{\Delta}\left(\tau-\tau^{\prime}\right)=\frac{\left(\tau-\tau^{\prime}\right)^{2}}{2 \hbar \beta}-\frac{\tau-\tau^{\prime}}{2}+\frac{\hbar \beta}{12}, \quad \tau \in[0, \hbar \beta] $$

$$ \bar{\Delta}\left(\tau, \tau^{\prime}\right)=-\bar{\Delta} \cdot\left(\tau, \tau^{\prime}\right) \equiv \frac{\tau-\tau^{\prime}}{\hbar \beta}-\frac{\epsilon\left(\tau-\tau^{\prime}\right)}{2}, \quad \tau, \tau^{\prime} \in[0, \hbar \beta] $$

$$ \left\langle e^{i k_{1} \delta x\left(\tau_{1}\right)} \cdots e^{i k_{n} \delta x\left(\tau_{n}\right)}\right\rangle_{0}=e^{-\frac{1}{2} \sum_{i, j=1}^{n} k_{i} k_{j} G\left(\tau_{i}, \tau_{j}\right)} . $$

$$ e^{-\frac{1}{2} \sum_{i, j=1}^{n} k_{i} k_{j} G\left(\tau_{i}, \tau_{j}\right)}=e^{-\frac{1}{2} \sum_{i, j=1}^{n} k_{i} k_{j}\left[G\left(\tau_{i}, \tau_{j}\right)-G\left(\tau_{i}, \tau_{i}\right)\right]-\frac{1}{2}\left(\sum_{i=1}^{n} k_{i}\right)^{2} G\left(\tau_{i}, \tau_{i}\right)}, $$

$$ \left\langle e^{i k_{1} \delta x\left(\tau_{1}\right)} \cdots e^{i k_{n} \delta x\left(\tau_{n}\right)}\right\rangle_{0}=\exp \left\{-\sum_{i