Second quantization and field theory

Luca Salasnich

Mario Schönberg

Robert Golub, S. K. Lamoreaux

Abstract We continue the discussion of the abstract transformation theory which led to the development of second quantization. Jordan had already suggested (in the foundational papers on matrix mechanics by Heisenberg, Born, and himself) that the electromagnetic field should be represented by a matrix, i.e., quantized. Dirac eventually took up this idea and proposed quantizing the field by treating the field as an operator. Jordan (with others) showed how the same technique could be applied to Bosons and then Fermions. This established the roots of quantum field theory. It also cleared up some of the interpretational issues of the time.

Radi A. Jishi

Nothing can be made out of nothing. William Shakespeare , King Lear Historically, quantization of the motion of particles was developed first. The state was described by a wave function and observables by operators. When dealing with interactions between particles and fields, such as the electromagnetic field, the fields were treated classically. Classical field equations look like the quantum mechanical equations for the wave function of the field quanta. For example, the Klein–Gordon classical field equation is similar to the quantum mechanical wave equation for a relativistic spinless particle. Quantizing the fields, leading to quantum field theory, appears to be quantizing a theory that has already been quantized; hence the name “second quantization.” In reality, there is only one quantization and one quantum theory. The method of second quantization is important in the study of many-particle systems. It enables us to express many-body operators in terms of creation and annihilation operators, thus rendering calculations less cumbersome. Moreover, the method makes it possible to treat systems with a variable number of particles; that is why the method initially emerged in the context of quantum field theory. In Chapter 1 we indicated that any one-particle wave function may be expanded in a complete set of states. In this chapter, we show that products of single-particle states, when properly symmetrized, form an orthonormal basis for the expansion of the wave function of an N -particle system.