Lectures on Quantum Field Theory. Ashok Das

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 γ^µ 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 the relations in (2.134), it is now straightforward to obtain

      The two relations in (2.135) are known as the Fierz rearrangement identities which are very useful in calculating cross sections. In deriving these identities, we have assumed that the spinors are ordinary functions. On the other hand, if they correspond to anti-commuting fermion operators, the right-hand sides of the identities in (2.135) pick up a negative sign which arises from commuting the fermionic fields past one another.

      Let us note that using the explicit forms for Γ^(a) and Γ_(a) in (2.123) and (2.125) respectively, we can write the first of the Fierz rearrangement identities in (2.135) as (assuming the spinors to be ordinary functions and not anti-commuting fermion fields which will introduce an overall negative sign, for example, in commuting ψ₂ past )

      Since this is true for any matrices M, N and any spinors, we can define a new spinor = Nψ₄ to write the identity in (2.136) equivalently as

      which is often calculationally simpler. Thus, for example, if we choose

      then using various properties of the gamma matrices derived earlier as well as (2.111) and (