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Quantum Electrodynamics — Robert Mann
Physics — Particle Physics · gamma-matrix-trace-identities
Problem Prove the following:
T r [ I ] = 4 and T r [ γ μ γ ν ] = 4 g μ ν \mathrm{Tr}[I] = 4 \quad \text{and} \quad \mathrm{Tr}[\gamma^\mu\gamma^\nu] = 4g^{\mu\nu} Tr [ I ] = 4 and Tr [ γ μ γ ν ] = 4 g μν T r [ odd # of γ -matrices ] = 0 \mathrm{Tr}[\text{odd }\# \text{ of }\gamma\text{-matrices}] = 0 Tr [ odd # of γ -matrices ] = 0 T r [ γ μ γ ν γ α γ β ] = 4 ( g μ ν g α β − g μ α g ν β + g μ β g ν α ) \mathrm{Tr}[\gamma^\mu\gamma^\nu\gamma^\alpha\gamma^\beta] = 4\left(g^{\mu\nu}g^{\alpha\beta} - g^{\mu\alpha}g^{\nu\beta} + g^{\mu\beta}g^{\nu\alpha}\right) Tr [ γ μ γ ν γ α γ β ] = 4 ( g μν g α β − g μα g ν β + g μ β g ν α ) T r [ γ 5 γ μ γ ν γ α γ β ] = − 4 i ε μ ν α β with ε 0123 = − 1 \mathrm{Tr}[\gamma^5\gamma^\mu\gamma^\nu\gamma^\alpha\gamma^\beta] = -4i\varepsilon^{\mu\nu\alpha\beta} \quad \text{with} \quad \varepsilon^{0123} = -1 Tr [ γ 5 γ μ γ ν γ α γ β ] = − 4 i ε μν α β with ε 0123 = − 1 γ α γ ν γ α = − 2 γ ν \gamma_\alpha\gamma^\nu\gamma^\alpha = -2\gamma^\nu γ α γ ν γ α = − 2 γ ν γ α γ μ γ ν γ α = 4 g μ ν \gamma_\alpha\gamma^\mu\gamma^\nu\gamma^\alpha = 4g^{\mu\nu} γ α γ μ γ ν γ α = 4 g μν T r [ γ 5 ] = 0 \mathrm{Tr}[\gamma^5] = 0 Tr [ γ 5 ] = 0 ε μ ν α β \varepsilon^{\mu\nu\alpha\beta} ε μν α β ε 0123 = − 1 ε 0123 = + 1 \varepsilon^{0123} = -1 \qquad \varepsilon_{0123} = +1 ε 0123 = − 1 ε 0123 = + 1 ε μ ν α β = { − 1 ( if μ ν α β is an even permutation of 0123 ) + 1 ( if μ ν α β is an odd permutation of 0123 ) \varepsilon^{\mu\nu\alpha\beta} = \begin{cases} -1 & (\text{if }\mu\nu\alpha\beta\text{ is an even permutation of }0123) \\ +1 & (\text{if }\mu\nu\alpha\beta\text{ is an odd permutation of }0123) \end{cases} ε μν α β = { − 1 + 1 ( if μν α β is an even permutation of 0123 ) ( if μν α β is an odd permutation of 0123 ) ε μ ν α β = { + 1 ( if μ ν α β is an even permutation of 0123 ) − 1 ( if μ ν α β is an odd permutation of 0123 ) \varepsilon_{\mu\nu\alpha\beta} = \begin{cases} +1 & (\text{if }\mu\nu\alpha\beta\text{ is an even permutation of }0123) \\ -1 & (\text{if }\mu\nu\alpha\beta\text{ is an odd permutation of }0123) \end{cases} ε μν α β = { + 1 − 1 ( if μν α β is an even permutation of 0123 ) ( if μν α β is an odd permutation of 0123 ) Browse the problem library