Massive spin 1 propagator in imaginary time formalism

[Judas503]

Homework Statement

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.

Homework Equations

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.

 

[nrqed]

Homework Statement

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.

Homework Equations

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:
$$ \omega_{k}=\mathbf{k}^2 + m^2 $$

You mean $\omega_k^2$ on the left side, I guess.

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.

You cannot simplify the numerator. What you can do is to simply provide separately $D^{00}$, $D^{0i} = D^{i0}$ and $D^{ij}$.