Quantum Field Theory and the Standard Model by Schwartz M.D. z-lib.org · Problem 3.9

Classical field theory — Schwartz M.D. z-lib.org

Physics — QFT · photon-polarizations

Problem

Photon polarizations (this problem follows the approach in [Feynman et al., 1996]). (a) Starting with L=−14Fμν2+JμAμ\mathcal{L} = -\frac{1}{4}F^2_{\mu\nu} + J_\mu A_\mu, substitute in AμA_\mu's equations of motion. This is called integrating out AμA_\mu. In momentum space, you should get something like Jμ1k2JμJ_\mu \frac{1}{k^2} J_\mu. (b) Choose kμ=(ω,κ,0,0)k_\mu = (\omega, \kappa, 0, 0). Use current conservation (∂μJμ=0\partial_\mu J^\mu = 0) to formally solve for J1J_1 in terms of J0J_0, ω\omega and κ\kappa in this coordinate system. (c) Rewrite the interaction Jμ1k2JμJ_\mu \frac{1}{k^2} J_\mu in terms of J0,J2,J3,ωJ_0, J_2, J_3, \omega and κ\kappa. (d) In what way is a term without time derivatives instantaneous (non-causal)? How many causally propagating degrees of freedom are there? (e) How do we know that the instantaneous term(s) do not imply that you can communicate faster than the speed of light?

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