Reduction of the electromagnetic vector potential to the irreducible representations of the inhomogeneous Lorentz group

In a previous paper we showed how the electromagnetic vector potential could be reduced to the irreducible representation of the inhomogeneous Lorentz group corresponding to particles with zero mass and spin 1, which are identified with photons. When the photon amplitudes were replaced in a suitable fashion by annihilation and creation operators, a covariant second-quantized theory was obtained in which the Lorentz condition is satisfied identically and in which the norm of the Hilbert space is positive definite. In the present paper we use the results of another of our papers to transform the photon wave functions from the linear momentum representation to the angular momentum representation. The vector potential, when second quantized, then takes the form of an expansion in terms of vector spherical harmonics in which the amplitudes are an nihilation and creation operators which create and destroy photons with a specific angular momentum. This expansion is a covariant quantum analogue of the expansion in multipoles of classical electromagnetic theory due to Blatt and Weisskopf and leads to the usual selection rules for atomic transitions.