Quantum Field Theory

5. Quantizing the Dirac FieldWe would now like to quantize the Dirac Lagrangian,L = ¯ψ(x) ( i /∂ − m ) ψ(x) (5.1)We will proceed naively and treat ψ as we did the scalar field. But we’ll see that thingsgo wrong and we will have to reconsider how to quantize this theory.5.1 A Glimpse at the Spin-Statistics TheoremWe start in the usual way and define the momentum,π = ∂L∂ ˙ ψ = i ¯ψγ 0 = iψ † (5.2)For the Dirac Lagrangian, the momentum conjugate to ψ is iψ † . It does not involvethe time derivative of ψ. This is as it should be for an equation of motion that is firstorder in time, rather than second order. This is because we need only specify ψ andψ † on an initial time slice to determine the full evolution.To quantize the theory, we promote the field ψ and its momentum ψ † to operators,satisfying the canonical commutation relations, which read[ψ α (⃗x), ψ β (⃗y)] = [ψ † α (⃗x), ψ† β (⃗y)] = 0It’s this step that we’ll soon have to reconsider.[ψ α (⃗x), ψ † β (⃗y)] = δ αβ δ (3) (⃗x − ⃗y) (5.3)Since we’re dealing with a free theory, where any classical solution is a sum of planewaves, we may write the quantum operators asψ(⃗x) =ψ † (⃗x) =2∑∫s=12∑∫s=1d 3 p 1[]√ b s(2π) 3 ⃗p us (⃗p)e +i⃗p·⃗x + c s †⃗p vs (⃗p)e −i⃗p·⃗x2E⃗pd 3 p 1[]√ b s †(2π) 3 ⃗p us (⃗p) † e −i⃗p·⃗x + c s ⃗p vs (⃗p) † e +i⃗p·⃗x2E⃗p(5.4)where the operators b s †⃗pcreate particles associated to the spinors u s (⃗p), while c s †⃗pcreateparticles associated to v s (⃗p). As with the scalars, the commutation relations of thefields imply commutation relations for the annihilation and creation operators– 106 –