Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 301 / 4 / Solution

In Feynman gauge, the Lagrangian differs by a total divergence fromIts Euler-Lagrange equation is $\square A^{\mu }=0$. Directly from the original gauge-fixed Lagrangian, the canonical momenta areequivalently ${\pi }^{\mu }=-{\partial }^0{A}^{\mu }$ after using the total-divergence form.

Using the mode expansions and oscillator commutator, the two nonzero cross terms combine towhere polarization completeness was used. The other equal-time field commutators vanish.

Substitution into the Hamiltonian and normal ordering giveEvery oscillator excitation carries positive energy $|\mathbf{p}|$, but the covariant state space has an indefinite inner product: time-like excitations have negative norm, and time-like and longitudinal polarizations are unphysical. Gupta-Bleuler quantization imposesand quotients by null states. The physical state space contains only the two transverse photon polarizations with positive norm, as required by gauge invariance.