Free Fermion Fields

A free relativistic spin-$\tfrac12$ field is built from one tightly checked chain: Lorentz-covariant spinor dynamics, positive- and negative-frequency solutions, canonical anticommutation relations (CAR), particle and antiparticle operators, the Dirac propagator, and a regulated Grassmann integral. This chapter develops that chain in four-dimensional Minkowski spacetime while keeping representation-theory prerequisites, convention choices, and later interacting physics in their proper places.

The recurring lesson is that three statements are different. The conserved Dirac inner product is positive on ordinary complex solutions; the one-particle Dirac Hamiltonian nevertheless has both positive- and negative-energy branches; and a positive-energy particle–antiparticle Fock interpretation additionally requires a frequency split, the antiparticle creation-operator assignment, CAR, a vacuum representation, and a vacuum-energy prescription. Weyl chirality and a Majorana reality condition are also different structures: neither can be inferred from how a spinor happens to look in one matrix basis.

Choose an entry route · See every page · Check your entry point

Shared declarations, not shared derivations

This overview fixes the translation contract used by all seven leaves. It does not replace their derivations. Unless a page says otherwise, the chapter uses natural units, the site metric

$$\eta_{\mu\nu}=\operatorname{diag}(+,-,-,-),$$

and gamma matrices satisfying

$$\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}\mathbf 1, \qquad \overline\psi=\psi^\dagger\gamma^0.$$

No explicit gamma-matrix representation is selected for an invariant argument. The notation

$$p\!\!\!/\equiv \gamma^\mu p_\mu$$

is used throughout, and the positive-frequency Fourier phase is $e^{-ip\cdot x}$ with $p^0>0$. In four dimensions,

$$\gamma_5=i\gamma^0\gamma^1\gamma^2\gamma^3, \qquad P_L=\frac{1-\gamma_5}{2}, \qquad P_R=\frac{1+\gamma_5}{2}.$$

These declarations do not remove every convention choice. Spinor normalizations, charge conjugation, the order of Grassmann variables and sources, left or right differentiation, and pole prescriptions are stated again where they first affect a calculation. Dimension- or signature-dependent conclusions are never silently exported from the four-dimensional setup.

The Dirac Field develops the action, first-order equation, Klein–Gordon consequence, conserved current, boundary form, and one-particle Hamiltonian. The relevant textbook spine is Schwartz 2014, §§ 10.2–10.4, pp. 168–174; an independent Hamiltonian-spectrum check appears in Srednicki 2007, § 1, pp. 23–24, whose opposite metric signature must be translated. Canonical Quantization of the Free Dirac Field develops the mode-to-operator and positive-Fock-energy argument, with Schwartz 2014, §§ 12.3 and 12.5.2, pp. 211–212 and 217–218 as its teaching spine.

One symbol, four mathematical roles

The letter $\psi$ appears in several settings. Its type must be known before a positivity, ordering, or differentiation statement makes sense.

Setting	What $\psi$ means	Valid operation	Statement that does not follow yet
One-particle equation	An ordinary complex spinor-valued solution	Form the conserved solution inner product and spectral projectors of the Dirac Hamiltonian	A local quantum field, CAR, or a Hamiltonian bounded below
Classical action	A spinor field varied together with $\overline\psi$; its Grassmann parity and boundary data must be declared	Derive the Dirac and adjoint equations, including the boundary term	Quantum statistics or a Fock-space interpretation
Canonically quantized field	An operator-valued distribution expanded in particle annihilation and antiparticle creation modes	Impose CAR, smear fields, and construct Fock-space observables	Pointwise bounded operators or an interacting particle theory
Regulated functional integral	A finite collection of independent Grassmann generators $\psi$ and $\overline\psi$	Perform Berezin integration in a declared order and differentiate with respect to ordered sources	A continuum measure or an anomalous-Jacobian theorem

Choose a route

The arrows below are recommended dependency paths, not alternative definitions of the same object.

Reader goal	Suggested route	What you should be able to check at the exit
Build the free Dirac field canonically	Action principle plus one-particle states → the Dirac field → plane waves and bilinears → canonical quantization	Separate solution norm, frequency sign, particle or antiparticle label, Fock norm, charge, and energy
Work in two-component chiral language	Lorentz field and Poincaré representations plus spinors and bilinears → the Dirac field → Weyl fields	State when chirality is dynamically preserved and distinguish chirality from helicity
Compare Dirac, Weyl, and Majorana descriptions	Weyl fields → Majorana fields	Count independent degrees of freedom and state the dimension, signature, and charge-conjugation assumptions behind a reality condition
Derive the propagator from modes	Plane waves and bilinears → canonical quantization → the fermion propagator	Reproduce the numerator, time-ordering sign, contact equation, and $i0$ prescription from one convention set
Pass from canonical to functional quantization	Canonical quantization plus Gaussian fields and sources and Grassmann–Berezin algebra → Grassmann functional integrals	Recover the regulated determinant and inverse Dirac kernel with source order and every exchange sign explicit

Check your entry point

This overview has no prerequisite. Individual leaves do. Use these diagnostics to find the shortest repair path.

Try this	Ready when	Repair if unsure
Expand $(p\!\!\!/-m)(p\!\!\!/+m)$	You obtain $(p^2-m^2)\mathbf1$ from the Clifford relation without choosing a gamma basis	Review Clifford algebras and Pin and Spin groups
Integrate a conserved current over a spacetime slab	You can identify the lateral flux condition required for hypersurface independence	Review Classical Symmetries, Currents, and Stress Tensors and the action principle
Distinguish $e^{+ip\cdot x}$ from a negative-energy excitation	You say “negative frequency in the field expansion” and then identify $b^\dagger$ as creating a positive-energy antiparticle	Review one-particle states before the Dirac-to-CAR route
Exchange two Grassmann generators	You track the minus sign and know that the order of Berezin measures and source derivatives must be declared	Review Grassmann–Berezin algebra

The seven pages in order

The sidebar order supplies a coherent traversal, but the dependency graph branches after the Dirac page.

1. The Dirac Field. Establish the covariant action, equation, adjoint equation, conserved solution form, Hamiltonian, and its $\pm E_{\mathbf p}$ spectrum. Lorentz and Clifford machinery are imported from Mathematical Methods. Plane-wave normalization, CAR, propagators, chirality, and reality conditions remain downstream.

2. Plane Waves, Spin Sums, and Bilinears. Normalize $u$ and $v$ spinors, derive completeness and spin sums, and organize covariant bilinears from one declared convention set. This page is the common computational input to canonical quantization and the propagator.

3. Weyl Fields and Chirality. Decompose the massless four-dimensional Dirac equation with $P_L$ and $P_R$, compare chirality with helicity in its valid regime, and mark which claims change with dimension or signature. Chiral gauge theories and anomalies remain outside the chapter.

4. Majorana Fields and Reality Conditions. State a Lorentz-compatible charge-conjugation condition, count the resulting independent modes, and compare allowed free mass terms with the Dirac and Weyl cases. A basis with visually real components is neither the definition nor an existence proof.

5. Canonical Quantization of the Free Dirac Field. Impose equal-time CAR on the mode expansion, construct particle and antiparticle operators, and derive the free Hamiltonian, charge, and vacuum structure. The page does not prove the spin–statistics theorem or treat interacting fermion loops.

6. The Fermion Propagator. Derive the time-ordered two-point function from operator ordering and spin sums, then verify that it is the Feynman inverse of the first-order Dirac operator with the declared contact and pole conventions. Loop analyticity and finite-density propagators hand off elsewhere.

7. Grassmann Functional Integrals for Free Fermions. Build the finite-dimensional Berezin integral first, then apply it to a regulated free Dirac kernel with independent sources. The determinant, inverse kernel, boundary data, and sign of each contraction are checked before continuum notation is used.

Convention checks that travel with every calculation

Datum	Chapter convention or declaration	Quick invariant check
Signature and slash	$(+---)$ and $p\!\!\!/ = \gamma^\mu p_\mu$	$(p\!\!\!/)^2=p^2\mathbf1$
Gamma adjoints	$(\gamma^0)^\dagger=\gamma^0$, $(\gamma^i)^\dagger=-\gamma^i$	$\alpha^i$ and $\beta$ are Hermitian
Fourier phase	$e^{-ip\cdot x}$ for positive frequency, $p^0>0$	$i\partial_\mu\mapsto p_\mu$
Dirac adjoint	$\overline\psi=\psi^\dagger\gamma^0$	$\overline\psi\psi$ is a Lorentz scalar and $\overline\psi\gamma^\mu\psi$ a vector
Chirality	$\gamma_5=i\gamma^0\gamma^1\gamma^2\gamma^3$ and $P_{L,R}=(1\mp\gamma_5)/2$ in four dimensions	$P_LP_R=0$; for $m=0$, the Dirac equation decouples into chiral sectors, equivalently $[h_D,\gamma_5]=0$
Spinor normalization	Declared on the plane-wave page together with the invariant momentum measure	Spin sums reproduce the numerator used in the propagator
CAR normalization	Declared together with the field expansion and continuum delta function	Equal-time field CAR and positive number-operator coefficients agree
Charge conjugation	Matrix convention and action on fields declared before a Majorana condition	Final degree count is invariant under a change of gamma basis
Propagator	Ordering, Fourier transform, and $i0$ fixed together	Acting with $i\gamma^\mu\partial_\mu-m$ gives the declared delta-function contact term
Grassmann sources	Order of $\overline\eta,\eta,\overline\psi,\psi$, measure factors, and left or right derivatives declared	Source differentiation reproduces the same regulated inverse kernel and exchange signs as the operator calculation

Representation changes act by similarity transformations on the gamma matrices and by the corresponding transformations on spinors. A physical identity should survive that simultaneous change. A formula that depends on individual matrix entries must be identified as a basis computation and reduced to a representation-independent result before it is reused.

Four distinctions that prevent common errors

Positive norm is not positive one-particle energy. The density $j^0=\psi^\dagger\psi$ makes the conserved solution norm positive, but $h_D$ on the full four-component space still has both $+E_{\mathbf p}$ and $-E_{\mathbf p}$ branches. Restricting to the positive spectral subspace is a one-particle construction; it does not by itself produce a local quantum field.

Negative frequency is not negative physical energy. In the operator field, the $v e^{+ip\cdot x}$ term carries $d^\dagger$. With that mode assignment, CAR, and a chosen vacuum Fock representation, subtracting the state-independent vacuum energy leaves a nonnegative sum of particle and antiparticle number operators. This is an operator construction, not a consequence of relabeling a classical mode.

Chirality is not a reality condition. Chirality is an eigenspace decomposition of $\gamma_5$ in even dimension. A Majorana condition relates a spinor to an antilinear charge conjugate and exists only when the dimension, signature, and representation admit the required real structure. The two restrictions may be compatible in some settings and incompatible in others.

A fermion determinant is not a bosonic Gaussian normalization. A finite complex Grassmann Gaussian gives a determinant rather than an inverse square root. The sign and order of its inverse-kernel contractions depend on declared Berezin and source conventions. Continuum determinants, Pfaffians, phases, zero modes, and anomalous measures require further hypotheses.

What the chapter establishes

By the end of the chapter, the reader can carry out the following conditional chain.

• Start from the four-dimensional free Dirac action and derive its first-order equation, conserved solution form, Hamiltonian, and mass shell with the boundary domain visible.
• Construct and normalize positive- and negative-frequency spinors, then verify completeness, spin sums, and selected bilinears without relying on a preferred gamma basis.
• Impose CAR and obtain a positive-energy particle–antiparticle Fock representation, including the free Hamiltonian, charge, and vacuum prescription.
• Separate Dirac, Weyl, and Majorana field content while stating every dimension, signature, chirality, and charge-conjugation assumption.
• Derive the same regulated free propagator from operator time ordering and from a Grassmann source integral, with numerator, pole prescription, contact term, determinant, and exchange signs mutually consistent.

The chapter stops before interacting gauge or Yukawa couplings, chiral anomalies, the spin–statistics theorem, Standard Model representation assignments, fermion loops, scattering amplitudes, nonperturbative determinant phases, and continuum construction. Its canonical and functional agreement is a regulated free-field check, not a theorem equating all fermionic formulations.

Review the chapter

Use the stated success criteria to diagnose which step needs revision, then follow the corresponding repair route.

Review mode	Prompt	A successful response	Repair route
Explanation	Explain why a positive Dirac solution norm does not by itself resolve the negative-energy branch or determine quantum statistics	Separates slice positivity from flux-dependent conservation, the $\pm E_{\mathbf p}$ one-particle spectrum, and the mode assignment, CAR, vacuum representation, and energy prescription used in Fock space	The Dirac Field → Canonical Quantization
Convention translation	Translate a calculation from the opposite metric signature or a different gamma basis	Transforms all dependent definitions together and preserves the Clifford relation, mass shell, Hamiltonian spectrum, and final bilinears or propagator	The Dirac Field → Plane Waves, Spin Sums, and Bilinears
Comparison	Compare Dirac, Weyl, and Majorana field content without using basis-dependent component language	States the dimension and signature, distinguishes a chiral projection from an antilinear reality condition, counts independent modes, and identifies which mass terms are allowed	Weyl Fields and Chirality → Majorana Fields and Reality Conditions
Synthesis	Trace one regulated free two-point function from the operator field to the Grassmann source integral	Aligns spinor normalization, time-ordering signs, the numerator and $i0$, the inverse-operator contact equation, source order, and Berezin differentiation	The Fermion Propagator → Grassmann Functional Integrals
Failure diagnosis	Diagnose the claim “a fermionic Gaussian has the same determinant power as a bosonic Gaussian”	At finite regulator, distinguishes $\det D$ for a complex Grassmann pair from $(\det K)^{-1/2}$ for a real bosonic Gaussian, declares measure and source order, and does not infer a continuum determinant without further hypotheses	Grassmann Functional Integrals

Where to continue

• Develop interacting matter: Symmetry and Gauge Structure develops gauge representations, chiral symmetries, anomalies, and BRST-aware matter sectors; Perturbative QFT and Scattering develops fermion vertices, loops, and amplitudes; Renormalization and Effective Field Theory develops counterterms and renormalized perturbation theory.
• Use fermions in scattering: Scattering Amplitudes develops external-spinor conventions in amplitudes, crossing, polarization sums, and interacting analytic structure.
• Strengthen the mathematics: Spinors, Conjugations, Bilinears, Chirality, and Fierz Identities develops the reusable spinor algebra; Mathematical QFT develops theorem-level CAR algebras, domains, and inequivalent representations.
• Study determinants and anomalies: Symmetry and Gauge Structure develops anomalous fermion measures, while Nonperturbative QFT develops zero-mode, determinant-phase, and Pfaffian questions.
• Compare other free spins: Free Spin-One Fields and Constraints applies the same discipline—field content, constraints, positivity, propagators, and physical state space—to vector fields.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI.
• Srednicki, Mark. Quantum Field Theory. Cambridge: Cambridge University Press, 2007. DOI.