4 - Quantum fields

Abstract

The formalism of second quantization opens this chapter; for the boson and fermion fields, we introduce the corresponding operators of creation and destruction. Our main purpose is to construct the general equations that describe the interaction of electron–positron and electromagnetic fields. Being the simplest example, a free scalar field with zero mass is initially discussed. We proceed to a canonical quantization of an electromagnetic field for which four components of a potential are considered as independent scalar fields; we substantially follow the treatment by Fermi. The canonical formalism allows one to also correctly quantize the Dirac field of electrons and positrons. We take into account the interaction, which causes the notable effects of quantum electrodynamics, in a framework of Heisenberg’s picture according to the perturbation theory. As a result, performing a normal ordering of the operators of creation and destruction, we generally obtain the working formulae that represent a basis of the quantum field theory by Dirac. The discussion of the problem of ultraviolet divergences closes this chapter. Having estimated the limits of the application of the developed quantum electrodynamics, we test and compare some more effective methods of regularization of divergent expressions that arise in a calculation of the observable quantities.