# Massive spin 1 propagator in imaginary time formalism

1. May 25, 2015

### Judas503

1. The problem statement, all variables and given/known data

I have the following massive spin-1 propagator-
$$D^{\mu\nu}(k)=\frac{\eta^{\mu\nu}-\frac{k^{\mu}k^{\nu}}{m^2}}{k^2 - m^2}$$
I want to write down the propagator in the imaginary time formalism commonly used in thermal field theories.

2. Relevant equations

3. The attempt at a solution

Factorizing the denominator is easy:
$$k^2 - m^2 = (k^0)^2 - \mathbf{k}^2 - m^2$$
Then, the following substitution can be used:
$$\omega_{k}=\mathbf{k}^2 + m^2$$
and, $$k^0 = i\omega_{n}=\frac{2n\pi i}{\beta}$$
However, the problem lies in simplifying the numerator. Any help would be really appreciated.

2. May 26, 2015

### nrqed

You mean $\omega_k^2$ on the left side, I guess.
You cannot simplify the numerator. What you can do is to simply provide separately $D^{00}$, $D^{0i} = D^{i0}$ and $D^{ij}$.