Srednicki — Quantum Field Theory · Problem 47.3

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · gamma-matrix-technology-basis

Problem

Show that the most general 4×44 \times 4 matrix can be written as a linear combination (with complex coefficients) of 11, γμ\gamma^\mu, SμνS^{\mu\nu}, γμγ5\gamma^\mu\gamma_5, and γ5\gamma_5, where 11 is the identity matrix and Sμν=i4[γμ,γν]S^{\mu\nu} = \frac{i}{4}[\gamma^\mu,\gamma^\nu]. Hint: if AA and BB are two different members of this set, prove linear independence by showing that Tr A†B=0\mathrm{Tr}\,A^\dagger B = 0 vanishes. Then count.
Intuitive explanation

The key insight is that these sixteen objects—identity, four vectors, six antisymmetric pairs, four axial vectors, and pseudoscalar—form a complete orthogonal basis for the space of all 4×44 \times 4 matrices. The hint tells you to verify orthogonality using the trace inner product: if distinct basis elements have vanishing Tr A†B\mathrm{Tr}\,A^\dagger B, they're linearly independent. Since you have exactly 16 linearly independent objects and 4×44 \times 4 matrices form a 16-dimensional vector space, you're done—any matrix must be expressible as their linear combination.

Intuitive explanation

The key insight is that these sixteen objects—identity, four vectors, six antisymmetric pairs, four axial vectors, and pseudoscalar—form a complete orthogonal basis for the space of all 4×44 \times 4 matrices. The hint tells you to verify orthogonality using the trace inner product: if distinct basis elements have vanishing Tr A†B\mathrm{Tr}\,A^\dagger B, they're linearly independent. Since you have exactly 16 linearly independent objects and 4×44 \times 4 matrices form a 16-dimensional vector space, you're done—any matrix must be expressible as their linear combination.

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