Читать книгу Foundations of Quantum Field Theory - Klaus D Rothe - Страница 31

The Pauli matrices

together with the identity matrix 1 represent a complete basis for 2 × 2 hermitian matrices. Of these, the Pauli matrices satisfy the first of the conditions (4.3); however, the identity matrix cannot be identified with β, since trβ = 0. Since the dimension of the matrices must be even, we conclude that the dimension of these matrices must be at least four.

The following 4 × 4 matrices satisfy all the requirements (4.3):

The same applies of course to matrices obtained from the above ones via a unitary transformation (unitary, in order to preserve the hermiticity of the matrices). For the choice of basis (4.5), the equation reads

We can compactify the notation by introducing the definitions

where the subscript D stands for “Dirac representation”. Explicitly we have

We may collect these matrices into a 4-tuplet . This notation is justified since we shall show later that these matrices “transform” (in a sense to be made precise later) under Lorentz transformations as a “4-vector”. In terms of the matrices (4.6) the Dirac equation takes the compact form²

This equation implies that ψ(r, t) is also a solution of the Klein–Gordon equation (3.8). We thus have the following Fourier decomposition into positive and negative energy solutions,

where , and the sum in σ extends over the two spin orientations in the rest-frame of the particle, as we shall see. The reason for displaying explicitly the factor will become clear from the transformation (7.15) and canonical normalization (7.40) in Chapter 7.

For ψ(x) to be a solution of the Dirac equation (4.8), the (positive and negative energy) Dirac spinors U(p, σ) and V(p, σ) must satisfy the equations

Recalling the explicit form (4.7) of the -matrices, we obtain for the independent solutions in the Dirac representation,

where , and N_± are normalization constants to be determined below. χ(σ) and χc(σ) denote the spinor and its conjugate in the rest frame of the particle,

with c the “charge conjugation” matrix defined by

Notice that

The matrix c has the fundamental property³

The following algebraic relations will turn out to be useful:

where

with

and

We further have

In the Dirac representation, γ⁵ is the off-diagonal 4 × 4 matrix

In order to fix the normalization constants in (4.11), we need to choose a scalar product. To this end we observe that

Hence the Dirac operator iγμ∂μ − m is hermitian with respect to the “Dirac” scalar product

Correspondingly we normalize the Dirac spinors by requiring⁴

which finally leads in the Dirac representation to the normalized Dirac spinors