Chapter 46
Quantization of the
Electromagnetic Field: the
Photon
Understanding the classical field theory of coupled dynamical scalar fields and
vector potentials is rather difficult, with the quantized theory even more so, due
to the fact that the Hamiltonian is no longer quadratic in the field variables
and the field equations are non-linear. Simplifying the problem by ignoring
the scalar fields and only considering the vector potentials gives a theory with
quadratic Hamiltonian that can be readily understood and quantized. The
classical equations of motion are the linear Maxwell equations in a vacuum,
with solutions electromagnetic waves. The corresponding quantum field theory
will be a relativistic theory of free, massless particles of helicity±1, the photons.
To get a physically sensible theory of photons, the infinite dimensional group
Gof gauge transformations that acts on the classical phase space of solutions
to the Maxwell equations must be taken into account. We will describe several
methods for doing this, carrying out the quantization in detail using one of them.
All of these methods have various drawbacks, with unitarity and explicit Lorentz
invariance seemingly impossible to achieve simultaneously. The reader should
be warned that due to the much greater complexities involved, this chapter and
succeeding ones will be significantly sketchier than most earlier ones.
46.1 Maxwell’s equations
We saw in chapter 45 that our quantum theories of free particles could be
coupled to a background electromagnetic field by introducing vector potential
fieldsAμ= (A 0 ,A). Electric and magnetic fields are defined in terms ofAμby
the equations
E=−∂A
∂t