Quantization of the Klein–Gordon field

Abstract

In ordinary non-relativistic quantum mechanics, quantizing a point particle, we interpret the generalized coordinates q n (normally the three Cartesian spatial coordinates x, y , and z ) and their corresponding 3-momenta p n as operators, which act on the wave function Ψ that is a representation of the vector of states, and they fulfil the commutation relations [ q n , p m ] = iħδ nm . However, in field theory, the position coordinates x i have a different meaning and are used to label the infinitely many ‘coordinates’ φ( x ) and their corresponding conjugate momenta π( x ). In order to quantize a field theory, we impose on the fields φ( x ) and π( x ) the commutation relations [φ( t , x ), π( t , y )] = iħδ( x - y ), which are the natural generalizations of the canonical commutation relations for ordinary quantum mechanics. Note that, in what follows, we will again set ħ = 1. In addition, the discussion in this chapter will be performed for a free neutral Klein–Gordon field. However, in the last two sections, we will present two natural extensions. First, in Section 6.5, we will extend the discussion with an example of interactions in the form of a classical external source, and second, in Section 6.6, we will describe how a free charged Klein–Gordon field can be treated. Canonical quantization Next, we want to develop the canonical quantum field theory for the neutral Klein–Gordon field. This theory was invented in the 1930s and it is indeed a very successful theory.