The electromagnetic four-potential has a vanishing temporal momentum and a Gauss law constraint in gauge theory. Gauge reduction leaves two transverse canonical pairs. Their canonical commutation relations give the photon creation and annihilation operators. Covariant Gupta-Bleuler quantization instead retains auxiliary polarization modes and selects the same physical state space by a subsidiary condition and null-state quotient.
The free transverse photon field is a Hermitian sum of two transverse-polarization plane-wave modes with creation and annihilation operators. Their standard bosonic commutator gives the transverse equal-time commutator. The normal-ordered Hamiltonian is a sum of positive oscillator energies.
The reduced canonical commutation relation for a transverse field uses a transverse delta function instead of an unconstrained componentwise delta. Its Fourier transform is the spatial transverse projector. It can be derived from the two-polarization oscillator expansion or from the Dirac bracket after gauge fixing.
The Maxwell Lagrangian density contains no time derivative of , so its canonical momentum vanishes. The condition is a first-class constraint. Its time preservation yields the Gauss law constraint in gauge theory; acts as the corresponding multiplier in the Hamiltonian.
Articles by others on the same topic
There are currently no matching articles.