# Trouble following the Saclay method (spectral representation of thermal Green functions)

Note: I just answered my own mathematical question by writing it up, but I thought I'd share it anyway in case someone else has a similar difficulty. :) I'm still left with my real physical question: why do the distribution functions appear in the thermal Green function the way they do? It is not intuitively obvious to me at all. Is there any way of understanding this on an intuitive or physical level?

Here follows the solved question: I'm trying to follow a derivation in Parwani, "Resummation in a Hot Scalar Field Theory" and having trouble following the "Saclay method" he introduces for simplyfing the Green functions for finite temperature scalar field theory. (The reference he gives for this is his own work here but I can't seem to get the full-text through my university account for some reason.)

If this is too much of a pure math problem I'm happy to move it over to the other stackexchange, but I'm really trying to teach myself thermal field theory. The spectral representation mentioned in this paper looked interesting to me because of the way the distribution functions come in and how it is supposed to simplify the Matsubara sums, but trying to reproduce the result is giving me trouble.

I'm getting stuck between equations 2.3 and 2.4. Equation 2.3 is a propagator:

$$\Delta(K^2) = \frac{1}{K^2 + M^2} \equiv \frac{1}{(k^0)^2 + k^2 + M^2} \;\;\;\; (2.3)$$

and in equation 2.4 he gives a spectral representation for it:

$$\Delta(K^2) = \int_0^{1/T} \mathrm{d}\tau\ \mathrm{e}^{i k^0 \tau} \Delta(\tau, k) \;\;\;\; (2.4)$$

where

$$\Delta(\tau, k) \equiv \frac{(1+n_k)\mathrm{e}^{- E_k \tau}+n_k\mathrm{e}^{E_k \tau}}{2 E_k}$$

and $E_k^2 = k^2 + M^2$ and $n_k = [\exp(E_k/T)-1]^{-1}$ is the Bose-Einstein distribution.

I'm trying to verify the formula by going from 2.4 to 2.3. The $\tau$ integral is trivial. I get:

$$\int{\cdots} = \frac{1}{ik^0 - E_k} \frac{1+n_k}{2 E_k} \left[ \mathrm{e}^{(i k^0 - E_k)/T} - 1 \right] + \frac{1}{ik^0 + E_k} \frac{n_k}{2 E_k} \left[ \mathrm{e}^{(i k^0 + E_k)/T} - 1 \right]$$

Combining denominators I need to show that:

$$f(k,k^0,T)\equiv (ik^0 + E_k)\frac{1+n_k}{2 E_k} \left[ \mathrm{e}^{(i k^0 - E_k)/T} - 1 \right] + (ik^0 - E_k) \frac{n_k}{2 E_k} \left[ \mathrm{e}^{(i k^0 + E_k)/T} - 1 \right] \overset{?}{=} -1$$

I don't see how this could possibly be a constant. I've tried plugging in $n_k$ and simplifying but it doesn't work, either by hand or in Mathematica. In fact, putting in simple values like $M=0,\ T=1$ and plotting (the real part, and I've called $k^0\rightarrow\omega$) I get:

There's also a non-trivial imaginary part, which looks like that but shifted in $\omega$.

So clearly I'm missing something basic or the method doesn't actually work. I've also tried changing signs in a few places in case there was a typo, to no avail. Anybody know where I'm going wrong? Or even better if you have a good technique that accomplishes the same thing this supposedly does. :-)

So of course $k^0$ is restricted to be an integer multiple of $2\pi T$ by the periodicity condition on the imaginary time axis. If you substitute this in and use the identity
$$\mathrm{e}^{-E_k/T} (n_k + 1) = n_k$$