Quantization of the electromagnetic field

The quantization of the electromagnetic field is a procedure in physics turning Maxwell's classical electromagnetic waves into particles called photons. Photons are massless particles of definite energy, definite momentum, and definite spin.

To explain the photoelectric effect, Albert Einstein assumed heuristically in 1905 that an electromagnetic field consists of particles of energy of amount hν, where h is the Planck constant and ν is the wave frequency. In 1927 Paul A. M. Dirac was able to weave the photon concept into the fabric of the new quantum mechanics and to describe the interaction of photons with matter. He applied a technique which is now generally called second quantization, although this term is somewhat of a misnomer for electromagnetic fields, because they are solutions of the classical Maxwell equations. In Dirac's theory the fields are quantized for the first time and it is also the first time that the Planck constant enters the expressions. In his original work, Dirac took the phases of the different electromagnetic modes (Fourier components of the field) and the mode energies as dynamic variables to quantize (i.e., he reinterpreted them as operators and postulated commutation relations between them). At present it is more common to quantize the Fourier components of the vector potential. This is what is done below.

A quantum mechanical photon state ${\displaystyle |\mathbf {k} ,\mu \rangle }$ belonging to mode ${\displaystyle (\mathbf {k} ,\mu )}$ is introduced below, and it is shown that it has the following properties:

These equations say respectively: a photon has zero rest mass; the photon energy is hν = hc|k| (k is the wave vector, c is speed of light); its electromagnetic momentum is ħk [ħ = h/(2π)]; the polarization μ = ±1 is the eigenvalue of the z-component of the photon spin.

Second quantization starts with an expansion of a scalar or vector field (or wave functions) in a basis consisting of a complete set of functions. These expansion functions depend on the coordinates of a single particle. The coefficients multiplying the basis functions are interpreted as operators and (anti)commutation relations between these new operators are imposed, commutation relations for bosons and anticommutation relations for fermions (nothing happens to the basis functions themselves). By doing this, the expanded field is converted into a fermion or boson operator field. The expansion coefficients have been promoted from ordinary numbers to operators, creation and annihilation operators. A creation operator creates a particle in the corresponding basis function and an annihilation operator annihilates a particle in this function.

In the case of EM fields the required expansion of the field is the Fourier expansion.

As the term suggests, an EM field consists of two vector fields, an electric field ${\displaystyle \mathbf {E} (\mathbf {r} ,t)}$ and a magnetic field ${\displaystyle \mathbf {B} (\mathbf {r} ,t)}$. Both are time-dependent vector fields that in vacuum depend on a third vector field ${\displaystyle \mathbf {A} (\mathbf {r} ,t)}$ (the vector potential), as well as a scalar field ${\displaystyle \phi (\mathbf {r} ,t)}$

where ∇ × A is the curl of A.