Basic Quantum Field Theory

Abstract

The original objective of quantum field theory was to develop a quantum version of Maxwell’s electrodynamics. This was to some extent achieved between 1927 and 1931 by applying to the Maxwell theory the same formal rules of “quantization” which had proved so successful in the transition from classical to quantum mechanics. For the free Maxwell equations (absence of charged matter) this approach was a full success. It led to a correct description of the properties of light quanta. The two free Maxwell equations taken together imply that each component of the field strength tensor F_μν obeys the wave equation

$$\square F_{\mu \nu } = 0;\quad \square = g^{\mu \nu } \partial _\mu \partial _\nu .$$

(I.5.1)

In addition, the first order equations give constraints which couple the different components and yield the transversality of electromagnetic waves. Before discussing the complications due to the constraints let us treat as a preliminary exercise the case of a scalar field Φ obeying the Klein-Gordon equation, considered as a classical field equation

$$\square\Phi + m^2\Phi = 0.$$

(I.5.2)

This example will already provide the essential lesson to be learnt from the canonical quantization of a linear field theory. The Lagrange density is

$$L = \frac{1} {2}g^{\mu \nu } \partial _\mu \Phi \partial _\nu \Phi - \frac{1} {2}m^2 \Phi ^2 .$$

(I.5.3)

To copy the quantization rules of section 1 one has to treat space and time in an asymmetric fashion, considering at some fixed time the spatial argument x of Φ as a generalization of the index i so that the field is regarded as a mechanical system with continuously many degrees of freedom. The partial derivatives ∂/∂q _i are replaced by the variational derivatives ∂/∂Φ(x) and the Kronecker symbol δ _ik is replaced by the Dirac δ -function δ³(x − x′). The Lagrange function results from the Lagrange density (1.5.3) by integration over 3-dimensional space at a fixed time, say t = 0. The canonically conjugate momentum to Φ (at this time), denoted by π(x), turns out to be equal to the time derivative of Φ and one imposes the canonical commutation relations for the initial values at t = 0

$$\lbrack\pi({\bf x}), \Phi({\bf x}^\prime )\rbrack = -i\delta^3({\bf x - x}^\prime);\quad \lbrack\Phi({\bf x}), \Phi({\bf x}^\prime)\rbrack = \lbrack \pi ({\bf x}), \pi({\bf x}^\prime)\rbrack = 0.$$

(I.5.4)

The Hamiltonian is

$$H={1\over 2}\int (\pi^2+({\rm grad}\ \Phi)^2 + m^2\Phi^2)d^3x.$$

(I.5.4)

Since this procedure is described in all early presentations of quantum field theory, see e.g. [Wentzel 1949] there is no need to elaborate on it.