Particle physics

The Dirac field and fermions

The quantized Dirac field describes spin-1/2 particles and antiparticles; anticommutators enforce Fermi–Dirac statistics and the Pauli exclusion principle.

In 3+1 dimensions the Dirac spinor ψ has four components, describing two spin states each for particles and antiparticles. The Dirac equation combines quantum dynamics with Lorentz covariance; its positive- and negative-frequency modes are quantized as particle and antiparticle fields.

(i\gamma^\mu\partial_\mu-m)\psi=0,\qquad \{b_r(\mathbf p),b_s^\dagger(\mathbf q)\}=(2\pi)^3\delta_{rs}\delta^3(\mathbf p-\mathbf q)

Definition: Definition and physical meaning

The field expands in particle annihilation b and antiparticle creation d† operators, with anticommutators {b,b†}=δ. Anticommutation replaces bosonic commutation and prevents two fermions from occupying the same mode.

Adjust the parameters and drag the figure to explore the model; readouts are quantitative illustrations under the stated assumptions.

Structure and interpretation

The free Lagrangian L=ψ̄(iγμ∂μ−m)ψ yields the Dirac equation. Electromagnetic interactions follow from ∂μ→Dμ=∂μ+ieAμ, producing the photon–fermion vertex; gauge invariance conserves electric current.

Example: Quantitative example

For a rest-frame momentum p=(m,0,0,0), the Dirac equation separates u and v spinors. Upon quantization, b† creates a particle and d† its antiparticle; the antiparticle carries opposite charge.

Solution

The v modes are not exploitable classical negative-energy particles; in quantum field theory they correspond to antiparticle creation operators and a positive-energy spectrum.

Lorentz covariance is more than formal bookkeeping: spinor components transform together so the equation keeps its form in every inertial frame. The one-particle probability current generalizes to the four-current $j^\mu=\bar\psi\gamma^\mu\psi$ , with $\partial_\mu j^\mu=0$ when the field equation holds. In field theory it becomes a conserved charge current and provides the basis for coupling the Dirac field to gauge fields.

Anticommutation also resolves the instability suggested by negative one-particle energies: the field expansion uses fermion annihilation and antiparticle creation operators, giving the Hamiltonian a positive spectrum. Exchanging two fermion operators introduces a minus sign; acting on many-particle states yields exclusion and underlies the sign structure of Slater determinants in atomic physics.

Quick check

Which statistics do fermions obey?

What charge does the electron antiparticle carry?

References

Michael E. Peskin and Daniel V. Schroeder (1995). An Introduction to Quantum Field Theory
Steven Weinberg (1995). The Quantum Theory of Fields, Volume I