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Free Fermion Fields

Fermionic fields combine Lorentz spinor structure with anticommuting quantum operators. This chapter connects the Dirac equation, spin sums, Weyl and Majorana conditions, canonical quantization, and Grassmann integration for free fields in four-dimensional Minkowski spacetime.

Enter this chapter

Section titled “Enter this chapter”

Begin with The Dirac Field and its plane-wave solutions. Review Spinors, Conjugations, Bilinears, Chirality, and Fierz Identities when the underlying spinor operations are unfamiliar.

Topics

Section titled “Topics”
  1. The Dirac Field — Derives the free first-order field equation and its conserved current.
  2. Plane Waves, Spin Sums, and Bilinears — Uses complete spin sums to remove basis choices from calculations.
  3. Weyl Fields and Chirality — Distinguishes chiral Lorentz representations from helicity labels.
  4. Majorana Fields and Reality Conditions — Defines a Lorentz-compatible reality condition and its restrictions.
  5. Canonical Quantization of the Free Dirac Field — Builds positive-energy particles and antiparticles with canonical anticommutators.
  6. The Fermion Propagator — Connects ordered contractions to the Dirac Green kernel.
  7. Grassmann Functional Integrals for Free Fermions — Reproduces free correlators using finite regulated Berezin integrals.

How the ideas fit together

Section titled “How the ideas fit together”

The spin sum converts normalized on-shell solutions into a covariant matrix numerator. Canonical anticommutation and fermionic time ordering then supply the relative signs needed for the propagator. A matrix inverse of the Dirac differential expression is incomplete until its boundary prescription is fixed. See Schwartz 2014, §§ 12.1–12.6.

Chirality, helicity, and Majorana reality answer different questions. Chirality labels a Lorentz representation, helicity projects spin along momentum, and a Majorana condition relates a field to its charge conjugate. Their relations depend on dimension, mass, and signature; a basis of apparently real components proves none of them by itself.

Review the chapter

Section titled “Review the chapter”

Why does a massive Dirac equation couple its left- and right-chiral parts?

Solution

The mass term pairs opposite chiral components, and the first-order equations express the derivative of each component in terms of the other. Setting one component identically to zero at nonzero mass forces the other to vanish as well. The massless limit removes that coupling and permits an independent Weyl equation.

Where to go next

Section titled “Where to go next”

Continue to Free Spin-One Fields and Constraints to see how gauge redundancy changes the free-field construction.

References

Section titled “References”
  • Matthew D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014. DOI.