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Massive spin 1 propagator in imaginary time formalism

  1. May 25, 2015 #1
    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. jcsd
  3. May 26, 2015 #2

    nrqed

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