松原函数的性质

统计物理 格林函数

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松原函数

定义松原函数：

$$- \mathscr{G}(1,2) = \left\langle \mathscr{T} \psi_1 \psi_2^\dagger \right\rangle$$

展开：

$$\mathscr{G}(1,2) = - \theta(\tau_1 - \tau_2) \left\langle e^{K(\tau_1 - \tau_2)}\psi(x_1) e^{-K(\tau_1 - \tau_2)} \psi^\dagger(x_2) \right\rangle \\ \pm \theta(\tau_2 - \tau_1) \left\langle e^{-K(\tau_1 - \tau_2)} \psi^\dagger(x_2) e^{K(\tau_1 - \tau_2)} \psi(x_1) \right\rangle$$

热力学平均意味着：

$$Tr e^{\beta \Omega} e^{- \beta K} e^{K(\tau_1 - \tau_2)} ... = e^{\beta \Omega} Tr e^{(\tau_1 - \tau_2 - \beta)K} ...$$

及

$$Tr e^{\beta \Omega} e^{- \beta K} e^{- K(\tau_1 - \tau_2)} ... = e^{\beta \Omega} Tr e^{-(\tau_1 - \tau_2 + \beta)K} ...$$

这里$K$是算符，求Trace收敛需要：

$\tau_1 - \tau_2 - \beta < 0$，及：$\tau_1 -\tau_2 + \beta > 0$

这意味着：

$\tau_1 - \tau_2 < \beta$，及：$\tau_1 - \tau_2 > - \beta$

松原函数的性质

考虑：$\mathscr{G}(x 0,x'\tau')$，$\tau'>0$

$$\mathscr{G}(x 0,x'\tau') = \pm \left\langle \psi^\dagger(x' \tau') \psi(x0) \right\rangle \\ = \pm Tr e^{\beta \Omega} e^{- \beta K} \psi^\dagger(x' \tau') \psi(x0) \\ = \pm e^{\beta \Omega} Tr \psi(x0) e^{- \beta K} \psi^\dagger(x' \tau') \\ = \pm e^{\beta \Omega} Tr e^{- \beta K } e^{\beta K} \psi(x0) e^{- \beta K} \psi^\dagger(x' \tau') \\ =\pm Tr e^{\beta \Omega} e^{- \beta K} \psi(x \beta)\psi^\dagger(x' \tau') \\ = \pm \left\langle \psi(x \beta) \psi^\dagger(x' \tau') \right\rangle \\ = \mp \mathscr{G}(x \beta, x' \tau')$$

改写为：

$$\mathscr{G}(x-x', 0 - \tau') = \mp \mathscr{G}(x-x',\beta - \tau')$$

利用：$0-\tau' = \tau - \tau'$进一步改写，

$$\mathscr{G}(x-x',\tau-\tau'<0) =\mp \mathscr{G}(x- x',\tau-\tau'+ \beta>0 )$$

可简写为：

$$\mathscr{G}(\tau < 0) = \mp \mathscr{G}(\tau+ \beta > 0)$$

这里“-”对应费米子，“+”对应玻色子。

傅里叶变换

因为$\mathscr{G}(\tau)$是定义在$- \beta < \tau < \beta$，定义傅里叶变换为：

$$\mathscr{G}(i \omega_n) = \frac{1}{2} \int_{- \beta}^{\beta} d \tau e^{i \omega_n \tau} \mathscr{G}(\tau)$$

$$=\frac{1}{2}\left( \int_{- \beta}^0 ... + \int_0^\beta ... \right)$$

其中：

$$\int_{-\beta}^0 d \tau e^{i \omega_n \tau} \mathscr{G}(\tau < 0) = \mp \int_{-\beta}^0 d \tau e^{i \omega_n \tau} \mathscr{G}(\tau + \beta)$$

变量变换：

$$\tau + \beta \to \tau$$

$$I = \mp \int_0^\beta d \tau e^{i \omega_n (\tau - \beta) } \mathscr{G}(\tau)$$

最后：

$$\mathscr{G}(i \omega_n) = \frac{1 \mp e^{-i \omega_n \beta}}{2} \int_0^\beta e^{i \omega_n \tau} \mathscr{G}(\tau)$$

讨论因子：

$$\frac{1 \mp e^{-i \omega_n \beta}}{2}, \omega_n = \frac{n \pi}{\beta}$$

费米子：“-”

$$e^{-i \omega_n \beta} = -1$$

对应n取奇数。

玻色子：“+”

$$e^{-i \omega_n \beta} = 1$$

对应n取偶数。

利用完松原函数的“奇偶性”，剩下：

$$\mathscr{G}(i \omega_n) = \int_0^\beta d \tau e^{i \omega_n \tau } \mathscr{G}(\tau)$$