They provide an orthonormal basis of the tangent space atpto the sphere of
radius|p|.
p 2 (p) 1 (p)Figure 46.1: Polarization vectors at a pointpin momentum space.The space of solutions is thus two copies of the space of solutions of the
massless Klein-Gordon case. The quantum field for the theory of photons is
then
Â(t,x) =^1
(2π)^3 /^2∫
R^3∑
σ=1, 2(σ(p)aσ(p)e−iωpteip·x+σ(p)a†σ(p)eiωpte−ip·x)
d^3 p
√
2 ωp(46.17)
whereaσ,a†σare annihilation and creation operators satisfying
[aσ(p),a†σ′(p′)] =δσσ′δ^3 (p−p′)The state space of the theory will describe an arbitrary number of particles for
each value of the momentump(called photons), obeying the energy-momentum
relationωp=|p|, with a two dimensional degree of freedom describing their
polarization.
Note the appearance here of the following problem: unlike the scalar field
case (equation 43.8) where the Fourier coefficients were unconstrained functions,
here they satisfy a condition (equation 46.16), and theαj(p) cannot simply be
quantized as independent annihilation operators for eachj. Solving equation
46.16 and reducing the number of degrees of freedom by introducing the polar-
ization vectorsσinvolves an arbitrary choice and makes the properties of the
theory under the action of the Lorentz group much harder to understand. A
similar problem for solutions to the Dirac equation will appear in chapter 47.