Propagator of the Proca Lagrangian

  • #1
5
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Homework Statement


I want to show that the propagator of Proca Lagrangian:

[tex] \mathcal{L}=-\frac{1}{4}F_{\mu \nu}F^{\mu \nu}+\frac{1}{2}M^2A_\mu A^\mu[/tex]

Is given by:

[tex]\widetilde{D}_{\mu \nu}(k)=\frac{i}{k^2-M^2+i\epsilon}[-g_{\mu\nu}+\frac{k_\mu k_\nu}{M^2}][/tex]


Homework Equations



Remember that: [tex]F_{\mu \nu}=\partial_\mu A_\nu - \partial_\nu A_\mu[/tex]

The Attempt at a Solution



I tried to use the Euler-Lagrange equation, and I obtained:

[tex]\partial_{\mu} (\partial^{\mu} A_{\nu} - \partial_{\nu} A^{\mu} ) + M^2 A^{\nu} = 0[/tex]

I suppose I have to do a Fourier Transform in order to express that equation in terms of [tex]k^\mu[/tex]
but I don't know how to do it. I don't even know if I have started the problem properly, or if there's another way.
Can anyone help me, please?
 

Answers and Replies

  • #2
MathematicalPhysicist
Gold Member
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Check it in Problem Book in QFT of Voja's the answers to problems 5.2b, 5.7,6.15.

This should answer your questions.
 
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  • #3
5
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Check it in Problem Book in QFT of Voja's the answers to problems 5.2b, 5.7,6.15.

This should answer your questions.
Sorry for the late answer, but I wanted to check all the steps with calm.
The problems you told me helped a lot, thanks!
 

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