Canonical quantization of the electromagnetic field

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.

For a spatial potential $\mathbf{A}$ with $\mathbf{E}=-\dot{\mathbf{A}}-\nabla A_0$, the Maxwell Lagrangian gives $\mathbf{\Pi }=-\mathbf{E}$ and ${\Pi }^0=0$. Its Hamiltonian is $\int [({\mathbf{\Pi }}^2+{\mathbf{B}}^2)/2+A_0\nabla \cdot \mathbf{\Pi }],d^{3}x$. Preserving the temporal momentum first-class constraint gives the Gauss law constraint in gauge theory. In free space, Coulomb gauge and decay at spatial infinity set $A_0=0$ and leave only transverse modes. The reduced Dirac bracket uses the projector ${\delta }_{ij}-k_i{k}_j/|\mathbf{k}{|}^2$, so quantization gives the transverse equal-time commutator and the positive Hamiltonian displayed above.

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 $A_0$, so its canonical momentum vanishes. The condition is a first-class constraint. Its time preservation yields the Gauss law constraint in gauge theory; $A_0$ acts as the corresponding multiplier in the Hamiltonian.