Kleinert · 제12장 양자장

$$ Z_{\mathrm{cl}}^{\prime} \equiv Z_{\mathrm{cl}}-\left.Z_{\mathrm{cl}}\right|_{e=0}=\int \frac{d^{3} x}{{\sqrt{2 \pi \hbar^{2} \beta / M}}^{3}}\left[\exp \left(\beta \frac{e^{2}}{r}\right)-1\right] $$

$$ \mathcal{A}=\int_{\tau_{a}}^{\tau_{b}} d \tau\left[\frac{M}{2} \mathbf{x}^{\prime 2}(\tau)-\frac{e^{2}}{r(\tau)}\right] $$

$$ S=n \log (2 D) $$

$$ \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right) \equiv \int \mathcal{D}^{D} x(\tau) \int \frac{\mathcal{D}^{D} p(\tau)}{(2 \pi \hbar)^{D}} \exp \left\{\frac{1}{\hbar} \int_{\tau_{a}}^{\tau_{b}} d \tau[i \mathbf{p} \dot{\mathbf{x}}-H(\mathbf{p}, \mathbf{x})]\right\} $$

$$ \left(\hbar \partial_{\tau}+\hat{H}\right)\left(\mathbf{x} \tau \mid \mathbf{x}_{a} \tau_{a}\right)=\hbar \delta\left(\tau-\tau_{a}\right) \delta^{(D)}\left(\mathbf{x}-\mathbf{x}_{a}\right) $$

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=\left\langle\mathbf{x}_{b}\right| \hat{R}\left|\mathbf{x}_{a}\right\rangle $$

$$ \hat{R}=\frac{i \hbar}{E-\hat{H}+i \eta} $$

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=\int_{t_{a}}^{\infty} d t_{b}\left\langle\mathbf{x}_{b}\right| \hat{U}_{E}\left(t_{b}-t_{a}\right)\left|\mathbf{x}_{a}\right\rangle $$

$$ \hat{U}_{E}(t) \equiv e^{-i t(\hat{H}-E) / \hbar} $$

$$ \hat{H}_{E} \equiv \hat{H}-E . $$

$$ e^{-i t \hat{H}_{E} / \hbar}=e^{-i \epsilon \hat{H}_{E} / \hbar} \cdots e^{-i \epsilon \hat{H}_{E} / \hbar} $$

$$ \prod_{n=1}^{N} \int d^{D} x_{n}\left|\mathbf{x}_{n}\right\rangle\left\langle\mathbf{x}_{n}\right|=1 $$

$$ \left\langle\mathbf{x}_{b}\right| \hat{U}_{E}^{N}\left(t_{b}-t_{a}\right)\left|\mathbf{x}_{a}\right\rangle=\prod_{n=1}^{N}\left[\int d^{D} x_{n}\right] \prod_{n=1}^{N+1}\left[\int \frac{d^{D} p_{n}}{(2 \pi \hbar)^{D}}\right] \exp \left(\frac{i}{\hbar} A_{E}^{N}\right), $$

$$ A_{E}^{N}=\sum_{n=1}^{N+1}\left\{\mathbf{p}_{n}\left(\mathbf{x}_{n}-\mathbf{x}_{n-1}\right)-\epsilon\left[H\left(\mathbf{p}_{n}, \mathbf{x}_{n}\right)-E\right]\right\} $$

$$ \left\langle\mathbf{x}_{b}\right| \hat{U}_{E}(t)\left|\mathbf{x}_{a}\right\rangle=\int \mathcal{D}^{D} x\left(t^{\prime}\right) \int \frac{\mathcal{D}^{D} p\left(t^{\prime}\right)}{(2 \pi \hbar)^{D}} \exp \left\{\frac{i}{\hbar} \int_{0}^{t} d t^{\prime}\left[\mathbf{p} \dot{\mathbf{x}}\left(t^{\prime}\right)-H_{E}\left(\mathbf{p}\left(t^{\prime}\right), \mathbf{x}\left(t^{\prime}\right)\right)\right]\right\} $$

$$ \int_{t_{a}}^{\infty} d t_{b}=(N+1) \int_{0}^{\infty} d \epsilon $$

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}^{N} \equiv(N+1) \int_{0}^{\infty} d \epsilon\left\langle\mathbf{x}_{b}\right| \hat{U}_{E}^{N}(\epsilon(N+1))\left|\mathbf{x}_{a}\right\rangle=\int_{t_{a}}^{\infty} d t_{b}\left\langle\mathbf{x}_{b}\right| \hat{U}_{E}^{N}\left(t_{b}-t_{a}\right)\left|\mathbf{x}_{a}\right\rangle $$

$$ \begin{align*} \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}^{N} & =(N+1) \int_{0}^{\infty} d \epsilon \frac{1}{\sqrt{2 \pi i(N+1) \epsilon \hbar / M}}{ }^{D} \\ & \times \exp \left[i \frac{M}{2(N+1) \epsilon}\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right)^{2}+i E(N+1) \epsilon\right] \end{align*} $$

$$ \hat{R}=\frac{i \hbar}{\hat{f}_{l}(E-\hat{H}+i \eta)} \hat{f}_{l} $$

$$ \hat{R}=\hat{f}_{r} \frac{i \hbar}{(E-\hat{H}+i \eta) \hat{f}_{r}} $$

$$ \hat{R}=\hat{f}_{r} \frac{i \hbar}{\hat{f}_{l}(E-\hat{H}+i \eta) \hat{f}_{r}} \hat{f}_{l} $$

$$ \hat{f}_{l}=\hat{f}^{1-\lambda}, \quad \hat{f}_{r}=\hat{f}^{\lambda} $$

$$ \hat{f}_{l} \hat{f}_{r}=\hat{f} $$

$$ \left\langle\mathbf{x}_{b}\right| \hat{R}\left|\mathbf{x}_{a}\right\rangle=\left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=\int_{s_{a}}^{\infty} d s_{b}\left\langle\mathbf{x}_{b}\right| \hat{\mathcal{U}}_{E}\left(s_{b}-s_{a}\right)\left|\mathbf{x}_{a}\right\rangle $$

$$ \hat{\mathcal{U}}_{E}(s) \equiv f_{r}(\mathbf{x}) e^{-i s f_{l}(\mathbf{x})(\hat{H}-E) f_{r}(\mathbf{x})} f_{l}(\mathbf{x}) $$

$$ \hat{\mathcal{H}}_{E} \equiv f_{l}(\mathbf{x})(\hat{H}-E) f_{r}(\mathbf{x}) $$

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E} \approx(N+1) \int_{0}^{\infty} d \epsilon_{s}\left\langle\mathbf{x}_{b}\right| \hat{\mathcal{U}}_{E}^{N}\left(\epsilon_{s}(N+1)\right)\left|\mathbf{x}_{a}\right\rangle $$

$$ \left\langle\mathbf{x}_{b}\right| \hat{\mathcal{U}}_{E}^{N}\left(\epsilon_{s}(N+1)\right)\left|\mathbf{x}_{a}\right\rangle=f_{r}\left(\mathbf{x}_{b}\right) f_{l}\left(\mathbf{x}_{a}\right) \prod_{n=1}^{N}\left[\int d^{D} x_{n}\right] \prod_{n=1}^{N+1}\left[\int \frac{d^{D} p_{n}}{(2 \pi \hbar)^{D}}\right] e^{i \mathcal{A}_{E}^{N} / \hbar} $$

$$ \mathcal{A}_{E}^{N}=\sum_{n=1}^{N+1}\left\{\mathbf{p}_{n}\left(\mathbf{x}_{n}-\mathbf{x}_{n-1}\right)-\epsilon_{s} f_{l}\left(\mathbf{x}_{n}\right)\left[H\left(\mathbf{p}_{n}, \mathbf{x}_{n}\right)-E\right] f_{r}\left(\mathbf{x}_{n}\right)\right\} $$

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=\int_{0}^{\infty} d S\left\langle\mathbf{x}_{b}\right| \hat{\mathcal{U}}_{E}(S)\left|\mathbf{x}_{a}\right\rangle $$

$$ \left\langle\mathbf{x}_{b}\right| \hat{\mathcal{U}}_{E}(S)\left|\mathbf{x}_{a}\right\rangle=f_{r}\left(\mathbf{x}_{b}\right) f_{l}\left(\mathbf{x}_{a}\right) \int \mathcal{D} x(s) \int \frac{\mathcal{D} p(s)}{2 \pi \hbar} \exp \left\{\frac{i}{\hbar} \int_{0}^{S} d s\left[\mathbf{p x}^{\prime}-\mathcal{H}_{E}(\mathbf{p}, \mathbf{x})\right]\right\} $$

$$ H=T(\mathbf{p})+V(\mathbf{x}) $$

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

$$ \begin{align*} \left\langle\mathbf{x}_{b}\right| \hat{\mathcal{U}}_{E}^{N}\left(\epsilon_{s}(N+1)\right)\left|\mathbf{x}_{a}\right\rangle= & \frac{f_{r}\left(\mathbf{x}_{b}\right) f_{l}\left(\mathbf{x}_{a}\right)}{{\sqrt{2 \pi i \epsilon_{s} f_{l}\left(\mathbf{x}_{b}\right) f_{r}\left(\mathbf{x}_{a}\right) \hbar / M}}^{D}} \\ & \times \prod_{n=1}^{N}\left[\int \frac{d^{D} x_{n}}{{\sqrt{2 \pi i \epsilon_{s} f\left(\mathbf{x}_{n}\right) \hbar / M}}^{D}}\right] e^{i \mathcal{A}_{E}^{N} / \hbar} \end{align*} $$

$$ \mathcal{A}_{E}^{N}=\sum_{n=1}^{N+1}\left\{\frac{M}{2 \epsilon_{s} f_{l}\left(\mathbf{x}_{n}\right) f_{r}\left(\mathbf{x}_{n-1}\right)}\left(\mathbf{x}_{n}-\mathbf{x}_{n-1}\right)^{2}-\epsilon_{s} f_{l}\left(\mathbf{x}_{n}\right)\left[V\left(\mathbf{x}_{n}\right)-E\right] f_{r}\left(\mathbf{x}_{n-1}\right)\right\} $$

$$ \left\langle\mathbf{x}_{b}\right| \hat{\mathcal{U}}_{E}(S)\left|\mathbf{x}_{a}\right\rangle=f_{r}\left(\mathbf{x}_{b}\right) f_{l}\left(\mathbf{x}_{a}\right) \int \mathcal{D} x(s) \exp \left\{\frac{i}{\hbar} \int_{0}^{S} d s\left[\frac{M}{2 f_{l} f_{r}} \mathbf{x}^{\prime 2}-f_{l}(V-E) f_{r}\right]\right\} $$

$$ d t=d s f_{l}\left(\mathbf{x}_{n}\right) f_{r}\left(\mathbf{x}_{n-1}\right) $$

$$ \hat{\mathcal{H}}=f_{l}(\mathbf{x}, t)[H(\hat{\mathbf{p}}, \mathbf{x}, t)-\hat{E}] f_{r}(\mathbf{x}, t), $$

$$ \hat{E} \equiv i \hbar \partial_{t} . $$

$$ \left\{\mathbf{x} t \mid \mathbf{x}^{\prime} t^{\prime}\right\}=\delta^{(D)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right) \delta\left(t-t^{\prime}\right) $$

$$ \left.\int d^{D} x \int d t \mid \mathbf{x} t\right\}\{\mathbf{x} t \mid=1 $$

$$ \hat{\mathcal{U}}(s) \equiv f_{r}(\mathbf{x}, t) e^{-i s f_{l}(\mathbf{x}, t)(\hat{H}-\hat{E}) f_{r}(\mathbf{x}, t)} f_{l}(\mathbf{x}, t) . $$

$$ \begin{align*} \left\{\mathbf{x}_{b} t_{b}\left|\hat{\mathcal{U}}^{N}(s)\right| \mathbf{x}_{a} t_{a}\right\}= & f_{r}\left(\mathbf{x}_{b}, t_{b}\right) f_{l}\left(\mathbf{x}_{a}, t_{a}\right) \\ & \times \prod_{n=1}^{N}\left[\int d^{D} x_{n} d t_{n}\right] \prod_{n=1}^{N+1}\left[\int \frac{d^{D} p_{n}}{(2 \pi \hbar)^{D}} \frac{d E_{n}}{2 \pi \hbar}\right] e^{i \mathcal{A}^{N} / \hbar} \end{align*} $$

$$ \begin{align*} \mathcal{A}^{N}=\sum_{n=1}^{N+1} & \left\{\mathbf{p}_{n}\left(\mathbf{x}_{n}-\mathbf{x}_{n-1}\right)-E_{n}\left(t_{n}-t_{n-1}\right)\right. \\ & \left.-f_{l}\left(\mathbf{x}_{n}, t_{n}\right)\left[H\left(\mathbf{p}_{n}, \mathbf{x}_{n}, t_{n}\right)-E_{n}\right] f_{r}\left(\mathbf{x}_{n-1}, t_{n-1}\right)\right\} \end{align*} $$

$$ \begin{align*} \left\{\mathbf{x}_{b} t_{b}|\hat{\mathcal{U}}(S)| \mathbf{x}_{a} t_{a}\right\} & =f_{r}\left(\mathbf{p}_{b}, \mathbf{x}_{b}, t_{b}\right) f_{l}\left(\mathbf{p}_{a}, \mathbf{x}_{a}, t_{a}\right) \\ & \times \int \mathcal{D}^{D} x(s) \mathcal{D} t(s) \int \frac{\mathcal{D}^{D} p(s)}{(2 \pi \hbar)^{D}} \frac{\mathcal{D} E(s)}{2 \pi \hbar} e^{i \mathcal{A} / \hbar} \end{align*} $$

$$ \begin{align*} & \mathcal{A}[\mathbf{p}, \mathbf{x}, E, t]=\int_{0}^{S} d s\left\{\mathbf{p}(s) \mathbf{x}^{\prime}(s)-E(s) t^{\prime}(s)\right. \\ & \left.\quad-f_{l}(\mathbf{p}(s), \mathbf{x}(s), t(s))[H(\mathbf{p}(s), \mathbf{x}(s), t(s))-E(s)] f_{r}(\mathbf{p}(s), \mathbf{x}(s), t(s))\right\} \end{align*} $$

$$ \begin{align*} &\left\{\mathbf{x}_{b} t_{b}|\hat{\mathcal{U}}(S)| \mathbf{x}_{a} t_{a}\right\}=\prod_{n=1}^{N}\left[\int d^{D} x_{n}\right] \prod_{n=1}^{N+1}\left[\int \frac{d^{D} p_{n}}{(2 \pi \hbar)^{D}}\right] \\ & \times \delta\left(t_{b}-t_{a}-\epsilon_{s} \sum_{n=1}^{N+1} f_{l}\left(\mathbf{p}_{n}, \mathbf{x}_{n}, t_{n}\right) f_{r}\left(\mathbf{p}_{n-1}, \mathbf{x}_{n-1}, t_{n-1}\right)\right) e^{i \tilde{\mathcal{A}}^{N} / \hbar} \end{align*} $$

$$ \tilde{\mathcal{A}}^{N}=\sum_{n=1}^{N+1}\left[\mathbf{p}_{n}\left(\mathbf{x}_{n}-\mathbf{x}_{n-1}\right)-\epsilon_{s} f_{l}\left(\mathbf{p}_{n} \mathbf{x}_{n}, t_{n}\right) H\left(\mathbf{p}_{n}, \mathbf{x}_{n}, t_{n}\right) f_{r}\left(\mathbf{p}_{n-1}, \mathbf{x}_{n-1}, t_{n-1}\right)\right] $$

$$ \epsilon \rightarrow \epsilon_{s} f_{l}\left(\mathbf{p}_{n} \mathbf{x}_{n}, t_{n}\right) f_{r}\left(\mathbf{p}_{n-1}, \mathbf{x}_{n-1}, t_{n-1}\right) $$

$$ \left\{\mathbf{x}_{b} t_{b}|\hat{\mathcal{U}}(S)| \mathbf{x}_{a} t_{a}\right\}=\int \mathcal{D}^{D} x(s) \int \frac{\mathcal{D}^{D} p(s)}{(2 \pi \hbar)^{D}} \delta\left(t_{b}-t_{a}-\int_{0}^{S} d s f(\mathbf{x}, t)\right) e^{i \tilde{\mathcal{A}} / \hbar} $$

$$ \tilde{\mathcal{A}}[\mathbf{p}, \mathbf{x}, t]=\int_{0}^{S} d s\left[\mathbf{p} \mathbf{x}^{\prime}-f_{l}(\mathbf{x}, t) H(\mathbf{p}, \mathbf{x}, t) f_{r}(\mathbf{x}, t)\right] $$

$$ \left(\mathbf{x}_{b} t_{a} \mid \mathbf{x}_{a} t_{a}\right)=\int_{0}^{\infty} d S\left\{\mathbf{x}_{b} t_{b}|\hat{\mathcal{U}}(S)| \mathbf{x}_{a} t_{a}\right\}=\left\{\mathbf{x}_{b} t_{b}\left|\frac{i \hbar}{\hat{H}-\hat{E}}\right| \mathbf{x}_{a} t_{a}\right\} $$

$$ \left\{\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right\}=\int \frac{d^{D} p}{(2 \pi \hbar)^{D}} \int \frac{d E}{2 \pi \hbar} e^{i \mathbf{p}\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right) / \hbar-i E\left(t_{b}-t_{a}\right) / \hbar} $$

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

$$ \left(\mathbf{x}_{b} t_{a} \mid \mathbf{x}_{a} t_{a}\right)=\int_{-\infty}^{\infty} d E e^{-i E\left(t_{b}-t_{a}\right) / \hbar} \int_{0}^{\infty} d S\left\{\mathbf{x}_{b} t_{b}\left|\hat{\mathcal{U}}_{E}(S)\right| \mathbf{x}_{a} t_{a}\right\} $$

$$ \hat{\mathcal{U}}_{E}(s) \equiv f_{r}(\hat{\mathbf{p}}, \mathbf{x}, t) e^{-i s f_{l}(\hat{\mathbf{p}}, \mathbf{x}, t)(\hat{H}-E) f_{r}(\hat{\mathbf{p}}, \mathbf{x}, t)} f_{l}(\hat{\mathbf{p}}, \mathbf{x}, t) . $$

$$ \mathcal{H}(\hat{\mathbf{p}}, \mathbf{x}, \hat{E}, t) \phi(\mathbf{x}, t, s)=i \hbar \partial_{s} \phi(\mathbf{x}, t, s) $$

$$ f_{l}(\mathbf{x}, t)\left[H(\hat{\mathbf{p}}, \mathbf{x}, t)-i \hbar \partial_{t}\right] f_{r}(\mathbf{x}, t) \phi(\mathbf{x}, t, s)=i \hbar \partial_{s} \phi(\mathbf{x}, t, s) $$

$$ \phi(\mathbf{x}, t, s)=\phi_{\mathcal{E}}(\mathbf{x}, t) e^{-i \mathcal{E} s / \hbar} $$

$$ \phi_{\mathcal{E}}(\mathbf{x}, t)=\phi_{\mathcal{E}, E}(\mathbf{x}) e^{-i E t / \hbar} $$

$$ \begin{align*} \mathcal{H}(\hat{\mathbf{p}}, \mathbf{x}, E) \phi_{\mathcal{E}, E}(\mathbf{x}) & =f_{l}(\mathbf{x})[H(\hat{\mathbf{p}}, \mathbf{x})-E] f_{r}(\mathbf{x}) \phi_{\mathcal{E}, E}(\mathbf{x}) \\ & =\mathcal{E} \phi_{\mathcal{E}, E}(\mathbf{x}) \end{align*} $$

$$ \left\langle\mathbf{x}_{b}\right| \hat{\mathcal{U}}_{E}(S)\left|\mathbf{x}_{a}\right\rangle=f_{r}\left(\mathbf{x}_{b}\right) f_{l}\left(\mathbf{x}_{a}\right) \sum_{n} \phi_{\mathcal{E}_{n}(E)}\left(\mathbf{x}_{b}\right) \phi_{\mathcal{E}_{n}(E)}^{*}\left(\mathbf{x}_{a}\right) e^{-i S \mathcal{E}_{n}(E) / \hbar} $$

$$ \left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)=f_{r}\left(\mathbf{x}_{b}\right) f_{l}\left(\mathbf{x}_{a}\right) \int_{-\infty}^{\infty} \frac{d E}{2 \pi \hbar} e^{i E\left(t_{b}-t_{a}\right) / \hbar} \sum_{n} \phi_{\mathcal{E}_{n}(E)}\left(\mathbf{x}_{b}\right) \phi_{\mathcal{E}_{n}(E)}^{*}\left(\mathbf{x}_{a}\right) \frac{i \hbar}{\mathcal{E}_{n}(E)} $$

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

$$ H(\hat{\mathbf{p}}, \mathbf{x}) \psi_{n}(\mathbf{x})=E_{n} \psi_{n}(\mathbf{x}) $$

$$ \frac{i \hbar}{\mathcal{E}_{n}(E)} \approx \frac{1}{\mathcal{E}_{n}^{\prime}\left(E_{n}\right)} \frac{i \hbar}{E-E_{n}+i \eta} $$

$$ \left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right) \sim f_{r}\left(\mathbf{x}_{b}\right) f_{l}\left(\mathbf{x}_{a}\right) \sum_{n} \phi_{\mathcal{E}_{n}\left(E_{n}\right)}\left(\mathbf{x}_{b}\right) \phi_{\mathcal{E}_{n}\left(E_{n}\right)}^{*}\left(\mathbf{x}_{a}\right) e^{-i E_{n}\left(t_{b}-t_{a}\right) / \hbar} $$