On the single particle Green’s function

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

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

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

\[G_{a\bar{b}}(\tau_a, \tau_{\bar{b}}) = G_{a\bar{b}}(\tau_a - \tau_{\bar{b}}) \equiv G_{a\bar{b}}(\tau) \, .\]

Using the cyclicity of the trace (see the section on (anti-)periodicity), we can show that for \(0 < \tau < \beta\), the bosonic (fermionic) Green’s function is \(\beta\) (anti-)periodic, that is

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

with \(\xi = \pm 1\) for bosons (fermions). 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

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

with Fourier coefficients

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

where \(\nu_n\) are Matsubara frequencies

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

with \(\vartheta = (1-\xi)/2\). From now on, we employ the \(\nu \ (\omega)\) symbol to denote fermionic (bosonic) Matsubara frequencies.

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

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

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

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

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

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

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