Transverse equal-time commutator ($[A_i^T(t,\mathbf{x}),{\Pi }^{Tj}(t,\mathbf{y})]=i{\delta }_{ij}^T(\mathbf{x}-\mathbf{y})$)

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.