Turning now from classical mechanics to quantum mechanics, a free photon can be represented by a wave function , which satisfies the equation
A plane-wave solution to this equation is
where is the four-momentum and is the unit polarization four-vector of the photon. We will discuss the polarization and normalization presently.
On substituting into the equation, we fine satisfies
The polarization vector has four components and yet it describes a spin-1 particle. The Lorentz condition , gives
and this reduces the number of independent components of to three. Moreover, we have to explore the consequences of the additional gauge freedom. Choose a gauge parameter
with constant so that is satisfied. This, together with the solution for shows that the physics is unchanged by the replacement
In other words, two polarization vectors which differ by a multiple of describe the same photon. We may use this freedom to ensure that the time component of vanishes, ; and then the Lorentz condition reduces to
This (noncovariant) choice of gauge is known as the Coulomb gauge.
We see that there are only two independent polarization vectors and that they are both transverse to the three-momentum of the photon. For example, for a photon traveling along the -axis, we may take
These are referred to as linear polarization vectors. The linear combinations
are called circular polarization vectors. A free photon is thus described by its momentum and a polarization vector . Since transforms as a vector, we anticipate that it is associated with a particle of spin-1. If were along , it would be associated with a helicity-zero photon. This state is missing because of the transversability condition . It can only be absent because the photon is massless.
In a special Lorentz frame is pure space-like, , with . In an arbitrary Lorentz frame is space-like and normalized to . The normalization constant of the plane wave for a photon is chosen such that the energy in the wave is just (ie. for a single photon). To verify this, we compute
This if for the Heaviside-Lorentz system of units. In the Gaussian system of units the constant in front of the integral would be . Since ( in the Lorentz gauge)
where we have taken the time average.
The completeness relationship for polarization vectors is