Lectures on Quantum Field Theory. Ashok Das

      can be identified with the spin operators for the Dirac particle. (This relation can be obtained from (2.99) using the identity for products of Levi-Civita tensors, namely, ϵ_ijkϵ_ℓjk = 2 δ_iℓ.)

      We have thus constructed a set of sixteen Dirac matrices, namely,

      where the numbers on the right denote the number of matrices in each category and these, in fact, provide a basis for all the 4 × 4 matrices. Here, the notation is suggestive and stands for the fact that transforms like a scalar under Lorentz and parity transformations. Similarly, and behave respectively like a vector, tensor, axial-vector and a pseudo-scalar under Lorentz and parity transformations as we will see in the next chapter.

      Let us note here that each of the matrices, even within a given class, has its own Hermiticity property. However, it can be checked that except for γ₅, which is defined to be Hermitian, all other matrices satisfy

      In fact, it follows easily that

      where we have used the fact that γ₅ is Hermitian and it anti-commutes with γ^µ. Finally, from

      it follows that

      The Dirac matrices satisfy nontrivial (anti) commutation relations. We already know that

      We can also calculate various other commutation relations in a straightforward and representation independent manner. For example,

      In this derivation, we have used the fact that

      We note here parenthetically that the commutator in (2.108) can also be expressed in terms of commutators (instead of anti-commutators) as

      However, since γ^µ matrices satisfy simple anti-commutation relations, the form in (2.108) is more useful for our purpose.

      Similarly, for the commutator of two σ^µν matrices, we obtain

      Thus, we see that the σ^µν matrices satisfy an algebra in the sense that the commutator of any two of them gives back a σ^µν matrix. We will see in the next chapter that they provide a representation for the Lorentz algebra.

      The various commutation and anti-commutation relations also lead to many algebraic simplifications in dealing with such matrices. This becomes particularly useful in calculating various amplitudes involving Dirac particles. Thus, for example, (these relations are true only in 4-dimensions)

      where we have used (γ_µ = η_µνγ^ν)

      and it follows now that,

      Similarly,

      and so on.

      The commutation and anti-commutation relations also come in handy when we are evaluating traces of products of such matrices. For example, we know from the cyclicity of traces that

      Therefore, it follows (in 4-dimensions) that

      Here in the second relation we have used the fact that γ₅ anti-commutes with γ^µ in addition to the cyclicity of trace. Even more complicated traces can be evaluated by using the basic relations we have developed so far. For example, we note that

      and so on. We would use all these properties in the next chapter to study the covariance of the Dirac equation under a Lorentz transformation.

      To conclude this section, let us note that we have constructed a particular representation for the Dirac matrices commonly known as the Pauli-Dirac representation. However, there are other equivalent representations possible which may be more useful for a particular system under study. For example, there exists a representation for the Dirac matrices where γ^µ are all purely imaginary. This is known as the Majorana representation and is quite useful in the study of Majorana fermions which are charge neutral fermions. Explicitly, the matrices have the forms

      It can be checked that the Dirac matrices in the Pauli-Dirac representation and the Majorana representation are related by the similarity (unitary) transformation (see (1.93))

      Similarly, there exists yet another representation for the γ^µ matrices, namely,

      where

      This is known as the Weyl representation for