in the second relation we have used the fact that γ5 anti-commutes with γµ 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 γµ 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
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 γµ matrices, namely,
where
This is known as the Weyl representation for the Dirac matrices and is quite useful in the study of massless fermions. It can be checked that the Weyl representation is related to the standard Pauli-Dirac representation through the similarity (unitary) transformation
2.6.1 Fierz rearrangement. As we have pointed out in (2.101), the sixteen Dirac matrices Γ(a), a = S, V, T, A, P define a complete basis for 4 × 4 matrices. This is easily demonstrated by showing that they are linearly independent which is seen as follows.
We have explicitly constructed the sixteen matrices to correspond to the set
From the properties of the γµ matrices, it can be easily checked that
where “Tr” denotes trace over the matrix indices. As a result, given this set of matrices, we can construct the inverse set of matrices as
such that
Explicitly, we can write the inverse set of matrices as
With this, the linear independence of the set of matrices in (2.123) is straightforward. For example, it follows now that if
then, multiplying (2.128) with Γ(b), where b is arbitrary, and taking trace over the matrix indices and using (2.126) we obtain
for any b = S, V, T, A, P . Therefore, (2.128) implies that all the coefficients of expansion must vanish which shows that the set of sixteen matrices Γ(a) in (2.123) are linearly independent. As a result they constitute a basis for 4 × 4 matrices.
Since the set of matrices in (2.123) provide a basis for the 4 × 4 matrix space, any arbitrary 4 × 4 matrix M can be expanded as a linear superposition of these matrices, namely,
Multiplying this expression with Γ(b) and taking trace over the matrix indices, we obtain
Substituting (2.131) into the expansion (2.130), we obtain
Introducing the matrix indices explicitly, this leads to
Here α, β, γ, η = 1, 2, 3, 4 and correspond to the matrix indices of the 4 × 4 matrices and we are assuming that the repeated indices are being summed.
Equation (2.133) describes a fundamental relation which expresses the completeness relation for the sixteen basis matrices. Just like any other completeness relation, it can be used effectively in many ways. For example, we note that if M and N denote two arbitrary 4 × 4 matrices, then using (2.133) we can derive (for simplicity, we will use the standard convention that the repeated index (a) as well as the matrix indices are being summed)
Using
