← Quantum Field Theory
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Topic 5 of 22

The Quantized Dirac Field

The Dirac equation describes a single relativistic particle. Elevating it to a quantum field requires second quantization: the field becomes an operator built from fermionic creation and annihilation operators that anticommute rather than commute. This anticommutation is not a choice — it is forced by the requirement of a stable vacuum and is the algebraic statement of the Pauli exclusion principle.

Key Concepts

  • ψ(x) = ∫ d³p/(2π)³ 1/√(2Eₚ) Σₛ [aₚₛ uˢ e^{−ipx} + b†ₚₛ vˢ e^{ipx}]
  • Anticommutators: {aₚₛ, a†qᵣ} = (2π)³δ³(p−q)δₛᵣ — fermionic algebra encodes Pauli principle
  • Dirac propagator: SF(p) = i(p̸+m)/(p²−m²+iε) — a 4×4 matrix in spinor space
  • Normal ordering for fermions flips sign: :aa†: = −a†a (unlike bosons)
  • Spin-Statistics Theorem: half-integer spin must anticommute for the Hamiltonian to be bounded below

Key Equations

Dirac field expansion
ψ(x)=∫ ⁣d3p(2π)312Ep∑s[ap,s use−ipx+bp,s† vse+ipx]\psi(x)=\int\!\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_p}}\sum_s\bigl[a_{p,s}\,u^s e^{-ipx}+b^\dagger_{p,s}\,v^s e^{+ipx}\bigr]
Fermionic anticommutator
{ap,s, aq,r†}=(2π)3δ3(p−q) δsr\{a_{p,s},\,a^\dagger_{q,r}\}=(2\pi)^3\delta^3(\mathbf{p}-\mathbf{q})\,\delta_{sr}
Dirac propagator
SF(p)=i(̸ ⁣p+m)p2−m2+iεS_F(p)=\frac{i(\not\!p+m)}{p^2-m^2+i\varepsilon}
Free Hamiltonian
H=∫ ⁣d3p(2π)3Ep∑s[ap,s†ap,s+bp,s†bp,s]H=\int\!\frac{d^3p}{(2\pi)^3}E_p\sum_s\bigl[a^\dagger_{p,s}a_{p,s}+b^\dagger_{p,s}b_{p,s}\bigr]
Worked Example

Example Problem

Problem

Show that {a†, a†} = 0 implies (a†)² = 0. What does this mean physically?

Solution

{a†_{p,s}, a†_{p,s}} = 2(a†_{p,s})² = 0, so (a†)² = 0. Acting twice with the same creation operator annihilates the state — you cannot put two identical fermions in the same mode. The Pauli exclusion principle is a consequence of the anticommutation algebra.

Key Takeaways

  • The Dirac field requires anticommutators — not commutators — to ensure the Hamiltonian is bounded below and the vacuum is stable
  • Antiparticle (b†) and particle (a†) operators are independent; both create positive-energy states
  • The Spin-Statistics Theorem: Lorentz invariance + positive energy forces half-integer spin → anticommutators (fermions)
  • The Dirac propagator SF(p) = i(p̸+m)/(p²−m²+iε) is a 4×4 spinor matrix, carrying helicity information through loop diagrams
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