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I'm stuck with this following problem:

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

Consider the Proca action,

[tex] S[A_\mu] = \int \, \mathrm d^4x \left[ - \frac14 F_{\mu\nu} F^{\mu\nu} + \frac12 m^2 A_\mu A^\mu \right] [/tex]

where [itex]F_{\mu\nu} = 2 \partial_{[\mu} A_{\nu]}[/itex] is the anti-symmetric electromagnetic

field tensor.

Derive the propagator for the vector field [itex]A_\mu[/itex].

2. Relevant equations

I did a Fourier transform to get

[tex] \left[ (- k^2 + m^2) g^{\mu\nu} + k^\mu k^\nu \right] \tilde D_{\nu\lambda}(k) = \delta^\mu_\lambda. [/tex] (*)

Zee's book on QFT gives the result on page 13, as if it were trivial, but I can't do the calculation (satisfactorily).

3. The attempt at a solution

I tried to follow the hint in the question: "the calculation involves deriving an identity for [itex]k^\nu \tilde D_{\nu\mu}[/itex]".

I contracted (*) with [itex]k_\mu[/itex] which got me

[tex]k^\nu \tilde D_{\nu\lambda} = k_\lambda[/tex]

or (contracting with [itex]k^\lambda[/itex])

[tex]k^\lambda k^\nu D_{\nu\lambda} = k^2[/tex]

but I still didn't really see how to solve for [itex]\tilde D_{\nu\lambda}[/itex].

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# Massive vector (Proca) propagator

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