On the single particle Green’s function

The imaginary time single particle Green’s function is defined as

(1)\[G_{a\bar{b}}(\tau_1, \tau_2) \equiv - \langle \mathcal{T} c_{a}(\tau_1) c^\dagger_{\bar{b}}(\tau_2) \rangle \, .\]

It is time translational invariant and hence only depends on the time difference

(2)\[G_{a\bar{b}}(\tau_1, \tau_2) = G_{a\bar{b}}(\tau_1 - \tau_2) \, .\]

Using the cyclicity of the trace, see the section on (anti-)periodicity, we can show that it is \(\beta\) (anti-)periodic.

(3)\[G_{a\bar{b}}(- \tau) = \xi G_{a\bar{b}}(\beta - \tau) \, .\]

Hence, extending the function as an (anti-)periodic function to all real valued imaginary times \(\tau \in (-\infty, \infty)\) the Green’s function can be expanded in the Matsubara Fourier series

(4)\[G_{a\bar{b}}(\tau) = \frac{1}{\beta} \sum_{n=-\infty}^\infty e^{- i\nu_n \tau} G_{a\bar{b}}(i\nu_n) \, ,\]

with Fourier coefficients

(5)\[G(i\nu_n) = \int_0^\beta d\tau e^{i\nu_n \tau} G_{a\bar{b}}(\tau)\]

where \(\nu_n\) are Matsubara frequencies

(6)\[\nu_n = \frac{\pi}{\beta}(2n + \vartheta)\]

with \(\vartheta = (1-\xi)/2\) and \(\xi = \pm 1\) for bosons/fermions) exploiting \(\beta\) (anti)periodicity.

Field operator Matsubara transforms

The notion of the Fourier series can be generalized to the second quantized (field) operators \(c(\tau)\) and \(c^\dagger(\tau)\) by introducing the transform relations

(7)\[c(i\nu_n) \equiv \frac{1}{\sqrt{\beta}} \int_0^\beta d\tau \, e^{i\nu_n \tau} c(\tau) \, , \quad c^\dagger(i\nu_n) \equiv \frac{1}{\sqrt{\beta}} \int_0^\beta d\tau \, e^{-i\nu_n \tau} c^\dagger(\tau)\]

(8)\[c(\tau) = \frac{1}{\sqrt{\beta}} \sum_{n=-\infty}^{\infty} e^{-i\nu_n \tau} c(i\nu_n) \, , \quad c^\dagger(\tau) = \frac{1}{\sqrt{\beta}} \sum_{n=-\infty}^{\infty} e^{i\nu_n \tau} c^\dagger(i\nu_n)\]

The symmetic definition of the field operator transforms results in trivial relations for the two frequency single particle Green’s function

(9)\[\begin{split}G(i\nu, i\nu') = \\ = \frac{1}{\sqrt{\beta}} \int_0^\beta d\tau e^{i\nu\tau} \frac{1}{\sqrt{\beta}} \int_0^\beta d\tau' e^{-i\nu'\tau'} G(\tau, \tau') \\ = \frac{1}{\sqrt{\beta}} \int_0^\beta d\tau e^{i\nu\tau} \frac{1}{\sqrt{\beta}} \int_0^\beta d\tau' e^{-i\nu'\tau'} \frac{1}{\beta} \sum_{n=-\infty}^\infty e^{-i \omega (\tau - \tau')} G(i\omega) \\ = \frac{1}{\beta^2} \sum_{n=-\infty}^\infty G(i\omega) \int_0^\beta d\tau e^{(i\nu - i\omega)\tau} \int_0^\beta d\tau' e^{(-i\nu' + i\omega)\tau'} \\ = \frac{1}{\beta^2} \sum_{n=-\infty}^\infty G(i\omega) \cdot \beta \delta_{\nu, \omega} \cdot \beta \delta_{\nu', \omega} \\ = \delta_{\nu, \nu'} G(i\nu)\end{split}\]

Thus there is no scale factor relating the one and two frequency single particle Green’s function

(10)\[G(i\nu, i\nu') = \delta_{\nu, \nu'} G(i\nu)\]