Griffiths Introduction to Elementary Particles · Problem 7.21

Electrodynamics of Quarks and Hadrons — David J. Griffiths

Physics — QFT · gauge-transformation-photon

Problem

Suppose we apply a gauge transformation (7.81) to the plane-wave potential (7.88), using as the gauge function λ=iℏκa e−(i/ℏ)p⋅x\lambda = i\hbar\kappa a\, e^{-(i/\hbar)p\cdot x} where κ\kappa is an arbitrary constant and pp is the photon four-momentum. (a) Show that this λ\lambda satisfies equation (7.84). (b) Show that this gauge transformation has the effect of modifying ϵμ\epsilon^\mu: ϵμ→ϵμ+κpμ\epsilon^\mu \to \epsilon^\mu + \kappa p^\mu. [In particular, if we choose κ=−ϵ0/p0\kappa = -\epsilon^0/p^0 we obtain the Coulomb gauge polarization vector (7.91).] This observation leads to a beautifully simple test for the gauge invariance of QED results: the answer must be unchanged if you replace ϵμ\epsilon^\mu by ϵμ+κpμ\epsilon^\mu + \kappa p^\mu.

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