Dirac Field Quantization

Core Idea

Quantizing the Dirac field requires anticommutation relations (not commutation relations) for the creation and annihilation operators. This produces fermions obeying the Pauli exclusion principle and naturally yields both particles (electrons) and antiparticles (positrons) with opposite charge.

Explainer

Quantizing the Dirac field follows the same canonical procedure as for the Klein-Gordon field, but with a critical difference: you must use anticommutation relations instead of commutation relations. The classical Dirac field psi(x) is a four-component spinor satisfying (i gamma^mu partial_mu - m)psi = 0. Its conjugate momentum is pi = i psi-dagger. The equal-time anticommutation relation is {psi_alpha(x, t), psi-dagger_beta(y, t)} = delta_{alpha beta} delta^3(x - y), where alpha and beta are spinor indices.

The field operator expands into positive- and negative-frequency parts: psi(x) = sum over spins s of integral [b_{p,s} u_s(p) e^-ipx + d-dagger_{p,s} v_s(p) e^+ipx] d^3p / ((2pi)³ 2E_p). Here u_s(p) and v_s(p) are the positive- and negative-frequency Dirac spinors, b_{p,s} destroys an electron with momentum p and spin s, and d-dagger_{p,s} creates a positron. The anticommutation relations are {b_{p,s}, b-dagger_{q,r}} = (2pi)³ delta^3(p-q) delta_{sr} and {d_{p,s}, d-dagger_{q,r}} = (2pi)³ delta^3(p-q) delta_{sr}, with all other anticommutators vanishing.

The reason anticommutation is mandatory (not a choice) is stability. The Dirac Hamiltonian has both positive and negative energy solutions. If you used bosonic commutation relations, each negative-energy mode would contribute -E_p per quantum, and since bosonic statistics allow unlimited occupation, you could drive the energy to negative infinity. With fermionic anticommutation relations, the reinterpretation of negative-frequency modes as antiparticle creation operators flips the energy sign: d-dagger creates a positron with positive energy +E_p. The Pauli exclusion principle then prevents unlimited occupation, and the vacuum is stable. This is a concrete manifestation of the spin-statistics theorem: half-integer spin fields must be quantized as fermions.

After quantization, the Dirac field naturally describes both particles and antiparticles. The electron field psi has two types of creation operators (b-dagger for electrons, d-dagger for positrons) and two types of annihilation operators (b for electrons, d for positrons). The conserved Noether current from the U(1) symmetry psi -> e^{i alpha} psi gives the electric charge operator Q = integral (b-dagger b - d-dagger d) d^3p, which counts electrons minus positrons. Every interaction vertex in QED involves psi and psi-bar, which is why every QED process conserves the number of electrons minus positrons (electric charge conservation).