Question

In the Dirac matrices we defineγ5γ5matrix, which transforms like ψ¯′γ5ψ′=ψ¯S−1γ5Sψ=det(a)ψ¯γ5ψψ¯′γ5ψ′=ψ¯S−1γ5Sψ=det(a)ψ¯γ5ψ

You try the following: ψ¯′γ5ψ′=ψ¯iS−1γ0γ1γ2γ3Sψψ¯′γ5ψ′=ψ¯iS−1γ0γ1γ2γ3Sψ ψ¯′γ5ψ′=ψ¯iS−1γ0SS−1γ1SS−1γ2SS−1γ3Sψψ¯′γ5ψ′=ψ¯iS−1γ0SS−1γ1SS−1γ2SS−1γ3Sψ Now,S−1γμS=aμvγvS−1γμS=avμγv(aμvavμ- Lorentz $matrix element). Thus: S−1γ5S=a0μa1νa2λa3αiγμγνγλγαS−1γ5S=aμ0aν1aλ2aα3iγμγνγλγα

Just rearrange gammas on the rhs to put them in the order 0,1,2,3 - this involves the inversions (which change sign due to commutation rules), required in the permutation definition of determinant. So, everything is almost fine. Except for: this sum will display also repeated indices inγγ(and so lower indices of aaas well). E.g. terms like a00a10a20a30γ0γ0γ0γ0a00a01a02a03γ0γ0γ0γ0which should not appear in the determinant sum! How can make them vanish?

Answer 1

The student's question about the Dirac matrices and the γ5 matrix can be resolved by applying the anticommutation relations of the gamma matrices, ensuring that only terms without repeated indices contribute to the determinant in the Lorentz transformation.

Step-by-step explanation:

The question involves the properties of the Dirac matrices and a specific matrix, γ5, within the framework of quantum field theory. The γ5 matrix is important in various physical contexts, including the study of chirality and parity violation in particle interactions. The student is asking about the transformation properties of γ5 under Lorentz transformations and how to address problematic terms in a specific calculation that should not appear when considering the determinant related to the Lorentz transformation.

To correct the issue with non-vanishing terms such as a00a10a20a30γ0γ0γ0γ0, one must recall that the indices in the Lorentz transformation matrix aμν are summation indices that follow the Einstein summation convention, where repeated indices imply a sum over all possible values. Since the gamma matrices follow anticommutation relations, only terms where the matrices are pairwise different contribute. Terms with repeated indices like γ0γ0 will square to the identity and not contribute to the determinant in the desired manner.

To ensure the correct evaluation of the determinant, the calculation must only include terms where the Lorentz transformation matrix elements' indices are not repeated. In other words, only terms that correspond to permutations of the Lorentz indices without repetition should be taken into account. This restriction naturally arises from the properties of the determinant and the antisymmetry of the gamma matrices under exchange, which ensures that the correct Lorentz invariant behavior of γ5 is respected.