- #1
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Hi all,
I'm stuck with this following problem:
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.
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).
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}.
I'm stuck with this following problem:
Homework Statement
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.
Homework 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).
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}.
