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. These are two first-class constraints; they generate the gauge freedom and remove two canonical pairs from the four potential components. The reduced phase space has four dimensions per spatial mode, hence two propagating photon degrees of freedom.

Impose Coulomb gauge,

\partial_{i} A_{i} = 0

. With no charges, Gauss's law then gives

\nabla^{2} A_{0} = 0

; vanishing boundary conditions set

A_{0} = 0

. This is radiation gauge. The remaining components are transverse and obey the massless wave equation. Let

ϵ_{i}^{(r)} (k)

r = 1, 2

, be orthonormal transverse polarization vectors. Their photon polarization completeness relation is

k_{i} ϵ_{i}^{(r)} = 0, \sum_{r = 1}^{2} ϵ_{i}^{(r)} ϵ_{j}^{(r) *} = P_{ij}^{T} (k) := δ_{ij} - \frac{k _{i} k _{j}}{∣ k ∣ ^{2}} .

(3)

The canonical transverse photon field is the Hermitian operator

A_{i}^{T} (x) = \sum_{r = 1}^{2} \int \frac{d ^{3} k}{( 2 π ) ^{3} 2 ω _{k}} [ϵ_{i}^{(r)} a_{r} (k) e^{- ik \cdot x} + ϵ_{i}^{(r) *} a_{r}^{†} (k) e^{ik \cdot x}], ω_{k} = ∣ k ∣,

(4)

where

[a_{r} (k), a_{s}^{†} (k^{'})] = (2 π)^{3} δ_{rs} δ^{3} (k - k^{'}), [a_{r}, a_{s}] = [a_{r}^{†}, a_{s}^{†}] = 0.

(5)

The field and its conjugate momentum have the transverse equal-time commutator

[A_{i}^{T} (t, x), Π^{T j} (t, y)] = i δ_{ij}^{T} (x - y), δ_{ij}^{T} (x) = \int \frac{d ^{3} k}{( 2 π ) ^{3}} P_{ij}^{T} (k) e^{i k \cdot x} .

(6)

This is the quantized reduced bracket, or equivalently the Dirac bracket after imposing the constraints and gauge conditions. The normal-ordered Hamiltonian is

\sum_{r} \int d^{3} k ω_{k} a_{r}^{†} a_{r} / (2 π)^{3}

. Its excitations are photons; circular combinations of the two transverse polarizations have helicity

+ 1

and

- 1

. The scalar and longitudinal potential components do not create additional physical photons.

The Feynman propagator is the vacuum expectation of a time-ordered product. The mode expansion directly gives the radiation-gauge photon propagator, with

z = x - y

D_{ij}^{F} (z) = \int \frac{d ^{3} k}{( 2 π ) ^{3}} \frac{P _{ij}^{T} ( k )}{2 ω _{k}} e^{i k \cdot z} [θ (z^{0}) e^{- i ω_{k} z^{0}} + θ (- z^{0}) e^{i ω_{k} z^{0}}] = \int \frac{d ^{4} k}{( 2 π ) ^{4}} \frac{i P _{ij}^{T} ( k )}{k ^{2} + i 0} e^{- ik \cdot z} .

(7)

The first expression comes from the creation-annihilation commutator; the second is its contour-integral representation. The positive-energy pole lies below the real axis and the negative-energy pole above it. In this reduced free-field description, the temporal operator is zero. A photon propagator must specify its gauge; the spatial transverse propagator is not the same tensor as the covariant four-potential propagator.

For the commonly used covariant form, add the gauge fixing term

- (\partial_{μ} A^{μ})^{2} / (2 ξ)

. The resulting Fourier-space kinetic operator is

K^{μν} (k) = - k^{2} η^{μν} + (1 - ξ^{- 1}) k^{μ} k^{ν} .

(8)

The inversion of the gauge-fixed Maxwell kinetic operator gives

K^{μν} D_{ν ρ}^{F} = i δ^{μ}_{ρ}

. In Feynman gauge,

ξ = 1

, the photon propagator is

D_{μν}^{F} (x - y) = \int \frac{d ^{4} k}{( 2 π ) ^{4}} \frac{- i η _{μν}}{k ^{2} + i 0} e^{- ik \cdot (x - y)} .

(9)

Equivalently,

□ D_{μν}^{F} = i η_{μν} δ^{4} (x - y)

with the vacuum pole prescription. For general

ξ

, the momentum-space numerator is

η_{μν} - (1 - ξ) k_{μ} k_{ν} / (k^{2} + i 0)

A covariant canonical realization uses four polarization oscillators with

[a_{r}, a_{s}^{†}] = - (2 π)^{3} η_{rs} δ^{3} (k - k^{'})

. The resulting indefinite inner product is auxiliary. In Gupta-Bleuler quantization, impose

(\partial_{μ} A^{μ})^{(+)} ∣ phys ⟩ = 0

and take the Gupta-Bleuler null-state quotient. The scalar-longitudinal combination is thereby removed from the physical state space, leaving the same two transverse photon states. Thus the four-component Feynman-gauge numerator does not imply four physical polarization states.

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