Quantum Field Theory

One can construct many other representations of the Clifford algebra by takingV γ µ V −1 for any invertible matrix V . However, up to this equivalence, it turns outthat there is a unique irreducible representation of the Clifford algebra. The matrices(4.16) provide one example, known as the Weyl or chiral representation (for reasonsthat will soon become clear). We will soon restrict ourselves further, and consider onlyrepresentations of the Clifford algebra that are related to the chiral representation bya unitary transformation V .So what does the Clifford algebra have to do with the Lorentz group? Consider thecommutator of two γ µ ,{ }S ρσ = 1 0 ρ = σ4 [γρ , γ σ ] == 1 12 γρ γ σ 2ρ ≠ σγρ γ σ − 1 2 ηρσ (4.18)Let’s see what properties these matrices have:Claim 4.1: [S µν , γ ρ ] = γ µ η νρ − γ ν η ρµProof: When µ ≠ ν we have[S µν , γ ρ ] = 1 2 [γµ γ ν , γ ρ ]= 1 2 γµ γ ν γ ρ − 1 2 γρ γ µ γ ν= 1 2 γµ {γ ν , γ ρ } − 1 2 γµ γ ρ γ ν − 1 2 {γρ , γ µ }γ ν + 1 2 γµ γ ρ γ ν= γ µ η νρ − γ ν η ρµ □Claim 4.2: The matrices S µν form a representation of the Lorentz algebra (4.11),meaning[S µν , S ρσ ] = η νρ S µσ − η µρ S νσ + η µσ S νρ − η νσ S µρ (4.19)Proof: Taking ρ ≠ σ, and using Claim 4.1 above, we have[S µν , S ρσ ] = 1 2 [Sµν , γ ρ γ σ ]= 1 2 [Sµν , γ ρ ]γ σ + 1 2 γρ [S µν , γ σ ]= 1 2 γµ γ σ η νρ − 1 2 γν γ σ η ρµ + 1 2 γρ γ µ η νσ − 1 2 γρ γ ν η σµ (4.20)Now using the expression (4.18) to write γ µ γ σ = 2S µσ + η µσ , we have[S µν , S ρσ ] = S µσ η νρ − S νσ η ρµ + S ρµ η νσ − S ρν η σµ (4.21)which is our desired expression.□– 84 –