# Massive vector (Proca) propagator

 Sci Advisor HW Helper P: 4,300 Hi all, I'm stuck with this following problem: 1. The problem statement, all variables and given/known data Consider the Proca action, $$S[A_\mu] = \int \, \mathrm d^4x \left[ - \frac14 F_{\mu\nu} F^{\mu\nu} + \frac12 m^2 A_\mu A^\mu \right]$$ where $F_{\mu\nu} = 2 \partial_{[\mu} A_{\nu]}$ is the anti-symmetric electromagnetic field tensor. Derive the propagator for the vector field $A_\mu$. 2. Relevant equations I did a Fourier transform to get $$\left[ (- k^2 + m^2) g^{\mu\nu} + k^\mu k^\nu \right] \tilde D_{\nu\lambda}(k) = \delta^\mu_\lambda.$$ (*) 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 $k^\nu \tilde D_{\nu\mu}$". I contracted (*) with $k_\mu$ which got me $$k^\nu \tilde D_{\nu\lambda} = k_\lambda$$ or (contracting with $k^\lambda$) $$k^\lambda k^\nu D_{\nu\lambda} = k^2$$ but I still didn't really see how to solve for $\tilde D_{\nu\lambda}$.
 Mentor P: 6,248 I get $$k^\nu \tilde D_{\nu\lambda} = \frac{k_\lambda}{m^2},$$ and then I think everything works out okay.
 Sci Advisor HW Helper P: 4,300 Thanks, I'll check that calculation. My problem was how to extract the propagator from that contraction, though. Anyway, let me get some sleep now, as it's 1:30
Mentor
P: 6,248
Massive vector (Proca) propagator

 Quote by CompuChip Thanks, I'll check that calculation. My problem was how to extract the propagator from that contraction, though. Anyway, let me get some sleep now, as it's 1:30
Substitute the identity and then contract with the metric.
 Sci Advisor HW Helper P: 4,300 I checked my earlier calculation and the 1/m^2 missing was just a typo. Also, I see what you mean now and it turns out to be quite easy indeed. Thank you very much George!
 P: 2 I am having the same problem. Could you elaborate on what you mean by substituting the identity? Edit: scratch that. I figured it out.

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