Quantization of the Radiation Field
This chapter describes the quantization of the free radiation field. Since, for certain aspects, it is necessary to include the coupling to external current densities, a separate chapter is devoted to this subject. Starting from the classical Maxwell equations and a discussion of gauge transformations, the quantization will be carried out in the Coulomb gauge. The principal aim in this chapter is to calculate the propagator for the radiation field. In the Coulomb gauge, one initially obtains a propagator that is not Lorentz invariant. However, when one includes the effect of the instantaneous Coulomb interaction in the propagator and notes that the terms in the propagator that are proportional to the wave vector yield no contribution in perturbation theory, one then finds that the propagator is equivalent to using a covariant one. The difficulty in quantizing the radiation field arises from the massless nature of the photons and from gauge invariance. Therefore, the vector potential A μ(χ) has, in effect, only two dynamical degrees of freedom and the instantaneous Coulomb interaction.
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