en.wikipedia.org/wiki/Dirac_matter
1 correction found
{ γ μ , γ ν } = γ μ γ ν + γ ν γ μ = η μ ν I d .
The gamma-matrix anticommutation relation is missing a factor of 2. The standard Clifford algebra is {γ^μ,γ^ν} = 2η^μν I, not η^μν I.
Full reasoning
This equation is the defining Clifford-algebra relation for Dirac/gamma matrices, and the standard relation includes a factor of 2:
[
{\gamma^\mu,\gamma^\nu}=\gamma^\mu\gamma^\nu+\gamma^\nu\gamma^\mu = 2\eta^{\mu\nu} I.
]
Leaving out that factor changes the algebra itself. For example, with the usual Minkowski signature, the standard definition gives ((\gamma^0)^2=+I) and ((\gamma^i)^2=-I). The article's version would instead imply half those values, so it is not the standard Dirac/gamma-matrix relation used in relativistic quantum mechanics or condensed-matter Dirac Hamiltonians.
Multiple physics references state the defining relation with the factor of 2 explicitly.
2 sources
- Optimally scrambling chiral spin-chain with effective black hole geometry | Scientific Reports
the gamma matrices { γ^μ ≡ e_a^μ γ^a } are the curved space gamma matrices which obey the Clifford algebra { γ^μ , γ^ν } = 2 g^{μν } and are related to the local flat Minkowski space gamma matrices { γ^a } which obey the flat space Clifford algebra { γ^a, γ^b } = 2 η^{ab }.
- Reviews of Modern Physics 90, 035007 (2018)
The gamma matrices obey the anticommutation relations {Γ_A; Γ_B} = 2η_AB for the metric η = diag(−1, 1, 1, …, 1).