Quantum Field Theory

4.3 The Dirac EquationThe equation of motion follows from the action (4.50) by varying with respect to ψ and¯ψ independently. Varying with respect to ¯ψ, we have(iγ µ ∂ µ − m) ψ = 0 (4.51)This is the Dirac equation. It’s completely gorgeous. Varying with respect to ψ givesthe conjugate equationi∂ µ ¯ψ γ µ + m ¯ψ = 0 (4.52)The Dirac equation is first order in derivatives, yet miraculously Lorentz invariant. Ifwe tried to write down a first order equation of motion for a scalar field, it would looklike v µ ∂ µ φ = . . ., which necessarily includes a privileged vector in spacetime v µ and isnot Lorentz invariant. However, for spinor fields, the magic of the γ µ matrices meansthat the Dirac Lagrangian is Lorentz invariant.The Dirac equation mixes up different components of ψ through the matrices γ µ .However, each individual component itself solves the Klein-Gordon equation. To seethis, write(iγ ν ∂ ν + m)(iγ µ ∂ µ − m)ψ = − ( γ µ γ ν ∂ µ ∂ ν + m 2) ψ = 0 (4.53)But γ µ γ ν ∂ µ ∂ ν = 1 2 {γµ , γ ν }∂ µ ∂ ν = ∂ µ ∂ µ , so we get−(∂ µ ∂ µ + m 2 )ψ = 0 (4.54)where this last equation has no γ µ matrices, and so applies to each component ψ α , withα = 1, 2, 3, 4.The SlashLet’s introduce some useful notation. We will often come across 4-vectors contractedwith γ µ matrices. We writeso the Dirac equation readsA µ γ µ ≡ /A (4.55)(i /∂ − m)ψ = 0 (4.56)– 90 –