Lectures on Quantum Field Theory. Ashok Das

brackets which we will not use to avoid confusion with Poisson brackets) and this algebra is known as the Clifford algebra. We see that if we can find a set of four linearly independent constant matrices satisfying the Clifford algebra, then, we can obtain a matrix square root of p² which would be linear in energy and momentum.

      Before going into an actual determination of such matrices, let us look at the consequences of such a possibility. In this case, the solutions of the equation (sign of the mass term is irrelevant and the wave function is a matrix in this case)

      would automatically satisfy the Einstein relation. Namely,

      Furthermore, since the new equation, (1.80), is linear in the energy and momentum variables, it will, consequently, be linear in the space and time derivatives. This is, of course, what we would like for a consistent definition of the probability density. The equation (1.80) (or its coordinate representation) is known as the Dirac equation.

      To determine the matrices, γ^µ, and their dimensionality, let us note that the Clifford algebra in (1.79)

      can be written out explicitly as

      We can choose any one of the matrices to be diagonal and without loss of generality, let us choose

      From the fact that we conclude that each of the diagonal elements in γ⁰ must be ±1, namely,

      Let us next note that using the relations from the Clifford algebra in (1.83), for a fixed i, we obtain

      where “Tr” denotes trace over the matrix indices. On the other hand, the cyclicity property of the trace, namely,

      leads to

      Thus, comparing Eqs. (1.86) and (1.88), we obtain

      For this to be true, we conclude that γ⁰ must have as many diagonal elements with value +1 as with −1. Consequently, the γ^µ matrices must be even dimensional.

      Let us assume that n = 2N. The simplest nontrivial matrix structure would arise for N = 1 when the matrices would be two dimensional (namely, 2 × 2 matrices). We know that the three Pauli matrices along with the identity matrix define a complete basis for 2 × 2 matrices. However, as we know, they do not satisfy the Clifford algebra. Namely, if we define then,

      In fact, we know that in two dimensions, there cannot exist four anti-commuting matrices.

      The next choice is N = 2 for which the matrices will be four dimensional (4 × 4 matrices). In this case, we can find a set of four linearly independent, constant matrices which satisfy the Clifford algebra. A particular choice of these matrices, for example, has the form

      where each element of the 4 × 4 matrices represents a 2 × 2 matrix and the σ_i correspond to the three Pauli matrices. This particular choice of the Dirac matrices is commonly known as the Pauli-Dirac representation.

      There are, of course, other representations for the γ^µ matrices. However, the physics of Dirac equation is independent of any particular representation for the γ^µ matrices. This can be easily seen by invoking Pauli’s fundamental theorem which says that if there are two sets of (constant) matrices γ^µ and γ′^µ satisfying the Clifford algebra, then, they must be related by a similarity transformation. Namely, if

      then, there exists a constant, nonsingular matrix S such that (in fact, the similarity transformation is really a unitary transformation if we take the Hermiticity properties of the γ-matrices into account)

      Therefore, given the equation

      we obtain

      with ψ = S⁻¹ψ′. (The matrix S⁻¹ can be moved past the momentum operator since it is assumed to be constant.) This shows that different representations of the γ^µ matrices are equivalent and merely correspond to a change in the basis of the wave function. As we know, a change of basis does not change physics.

      To obtain the Hamiltonian for the Dirac equation, let us go to the coordinate representation where the Dirac equation (1.80) takes the form (remember ħ = 1)

      Multiplying with γ⁰ from the left and using the fact that we obtain

      Conventionally, one denotes

      In terms of these matrices, then, we can write (1.97) as

      This is a first order equation (in time derivative) like the