Matsubara sum of thermal Green's function

Question

I need to retrieve a Matsubara sum representation of the thermal Green's function

$$G_{ij}(\tau)=-\frac{1}{Z}\int \mathcal{D}(\overline{\psi},\psi)\psi_i(\tau)\overline{\psi}_j(0)\exp(-\sum_k\int_0^\beta\mathrm{d}\tau\,\overline{\psi_k}(\partial_\tau+\epsilon_k-\mu)\psi_k))$$

I want to use the fact that the Gaussian functional integral reads

$$\int\mathrm{d}(\overline{\psi},\psi)\exp(-\overline{\psi}A\psi)=\mathrm{det}A$$

I do not understand how to apply the Gaussian integral, since the above expression does not seem to be in the correct form for that: There are $\psi_i(\tau)\overline{\psi}_j(0)$ outside the exponential and there is an integral over imaginary time within the exponential. How is it possible to rewrite the integral in order to use the functional Gaussian integral? Will the result be somehow ~$\mathrm{det}(\partial_\tau+\epsilon_k-\mu)$?

Answer 1

Take a look into the book from Alexander Altland in Chapter 4. This is a one-body-Green-function s.t. the result will not be proportional to the (ill-divergent) determinant, since this factor will cancel out. (The one-body-Green-function is a special case of the Wick-theorem, which is being introduced in the very same chapter for fermions, assuming your indices refer to electronic wave-vectors.) Notice however that you will need to introduce an convergence-generating-factor in your functional integral representation, as this is a problem that is "a little bit too simple" for the functional integral formalism to be really fruitful.