Canonical transverse photon field ($A_i^T={\sum }_{r=1}^2\int \frac{d^{3}k}{(2\pi {)}^3\sqrt{2|\mathbf{k}|}}({ϵ}_i^{(r)}{a}_r{e}^{-ikx}+h.c.)$)

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.