Reduction of the electromagnetic vector potential to the irreducible representations of the inhomogeneous Lorentz group and manifestly covariant quantization with a positive-definite metric for the hilbert space

Abstract

In the present paper we expand the vector potential into modes, the amplitudes of which are wave functions for massless particles of spin 0 and 1 in Wigner’s classification of particles according to the irreducible representations of the inhomogeneous Lorentz group. It is shown that only the spin-1 wave functions contribute to the electromagnetic fields and that they represent circularly polarized electro magnetic waves. On replacing the wave functions by operators, one obtains a manifestly covariant quantization of the vector potential acting on a Hilbert space in which the norm of the Hilbert space and the energy are positive-definite. Unlike the treatments of Fermi and Bleuler and Gupta where the Lorentz condition is required to be satisfied in a subspace of the Hilbert space in order to obtain positive-definite energies and norms, in our treatment the Lorentz condition is satisfied identically over the entire Hilbert space. Indeed, our quantization procedure resembles that used for the Dirac electron field and has no more difficulties. It is also shown that the new method of quantization does not change in any way the Feynman-Dyson-Wick calculations for the matrix elements of the scattering operator, except that the photon wave functions are now explicit instead of implicit as in earlier treatments.