Quantum Field Theory

so the requirement (4.44) becomes−[S ρσ , γ µ ] = (M ρσ ) µ νγ ν (4.47)where we’ve suppressed the α, β indices on γ µ and S µν , but otherwise left all otherindices explicit. In fact equation (4.47) follows from Claim 4.1 where we showed that[S ρσ , γ µ ] = γ ρ η σµ − γ σ η µρ . To see this, we write the right-hand side of (4.47) byexpanding out M,(M ρσ ) µ ν γν = (η ρµ δν σ − ησµ δν ρ )γν= η ρµ γ σ − η σµ γ ρ (4.48)which means that the proof follows if we can show−[S ρσ , γ µ ] = η ρµ γ σ − η σµ γ ρ (4.49)which is exactly what we proved in Claim 4.1.□Claim 4.5: ¯ψγ µ γ ν ψ transforms as a Lorentz tensor. More precisely, the symmetricpart is a Lorentz scalar, proportional to η µν ¯ψψ, while the antisymmetric part is aLorentz tensor, proportional to ¯ψS µν ψ.Proof: As above.□We are now armed with three bilinears of the Dirac field, ¯ψψ, ¯ψγ µ ψ and ¯ψγ µ γ ν ψ,each of which transforms covariantly under the Lorentz group. We can try to build aLorentz invariant action from these. In fact, we need only the first two. We choose∫S =d 4 x ¯ψ(x) (iγ µ ∂ µ − m) ψ(x) (4.50)This is the Dirac action. The factor of “i” is there to make the action real; upon complexconjugation, it cancels a minus sign that comes from integration by parts. (Said anotherway, it’s there for the same reason that the Hermitian momentum operator −i∇ inquantum mechanics has a factor i). As we will see in the next section, after quantizationthis theory describes particles and anti-particles of mass |m| and spin 1/2. Notice thatthe Lagrangian is first order, rather than the second order Lagrangians we were workingwith for scalar fields. Also, the mass appears in the Lagrangian as m, which can bepositive or negative.– 89 –