I am asked to show that the matrix S = 1 &#x2212;<!-- − --> i 4 </mfr

lurtzslikgtgjd 2022-05-18 Answered
I am asked to show that the matrix
S = 1 i 4 σ μ ν ε μ ν
represents the infinitesimal Lorentz transformation
Λ μ ν = δ μ ν + ε μ ν ,
in the sense that
S 1 γ μ S = Λ μ ν γ ν .
I have already proven that S 1 = γ 0 S γ 0 , so I can begin with the left-hand side of this third equation:
S 1 γ μ S = γ 0 S γ 0 γ μ S = γ 0 ( 1 + i 4 σ μ ν ε μ ν ) γ 0 γ μ ( 1 i 4 σ μ ν ε μ ν ) = γ 0 γ 0 γ μ γ 0 γ 0 γ μ i 4 σ μ ν ε μ ν + γ 0 i 4 σ μ ν ε μ ν γ 0 γ μ γ 0 i 4 σ ρ τ ε ρ τ γ 0 γ μ i 4 σ μ ν ε μ ν = γ μ + 1 16 γ 0 σ ρ τ ε ρ τ γ 0 γ μ σ μ ν ε μ ν = γ μ + 1 16 ( σ ρ τ ε ρ τ σ μ ν ε μ ν ) γ μ .
Remember, we want this to be equal to
Λ μ ν γ ν = ( δ μ ν + ε μ ν ) γ ν = γ μ + ε μ ν γ ν .
The first term is there already, but I have no idea how that second term is going to work out to ε μ ν γ ν . Can anyone give me a hint, or tell me what I’ve done wrong?
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Answers (1)

Raiden Williamson
Answered 2022-05-19 Author has 18 answers
To order S 1 , isn't S 1 just 1 + i 4 σ μ ν ϵ μ ν ? Then try expanding out S 1 γ μ S keeping terms of order ϵ and using the Clifford algebra relations when you have to commute γs through σ μ ν terms.
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