An Interaction with a Quantum Electromagnetic Field

Abstract

A quantization of an interaction with the electromagnetic field poses some difficulties. The problem is that in order to describe such interactions the field strength variables are not sufficient. We need the vector potentials. The problem arises with the covariant quantization of the four-potentials. We must choose the correct number of degrees of freedom for quantization. We can distinguish physical components imposing a gauge condition. In this chapter we work in various gauges and postpone the proof till Chap. 11 of the fact that the physics does not depend on the choice of gauge. We first quantize the electromagnetic field defining the functional integral by an introduction of a particular gauge fixing (the $$\xi $$ -gauge). The functional integral is Lorentz (or Euclidean) invariant. However, the corresponding Wightman functions (or Euclidean Schwinger functions) do not satisfy the positivity requirement. These positivity property is satisfied in the $$A_{0}=0$$ gauge. We define the Higgs model and represent the correlation functions of this model as a Gibbs ensemble of polymers. In the subsequent sections we discuss a particle in a quantum electromagnetic field quantizing the field in the radiation gauge. We calculate the transition amplitudes and the density matrix in an environment of photons. Then, we study the particle motion in quantum electromagnetic field working in the Heisenberg picture. In such a description the creation and annihilation operators of the electromagnetic field play the role of the quantum noise (quantum Brownian motion). We derive a stochastic equation describing the radiation damping of the electron motion in the photon thermal state. The photon noise leads to the decoherence. In the relation to noise we discuss the notion of entropy in quantum field theory. The functional integral approach leads to a definition of entropy similar to the one in classical random systems. We calculate the entropy of a Gaussian Wigner function.