The Dirac Field

In four-dimensional Minkowski spacetime, the free Dirac field is a complex four-component spinor governed by a Lorentz-covariant first-order action. The Clifford relation makes its equation imply the relativistic mass shell, a global phase symmetry supplies a conserved positive inner product on ordinary complex solutions, and the corresponding one-particle Hamiltonian is self-adjoint but has both $+E_{\mathbf p}$ and $-E_{\mathbf p}$ branches. Selecting the positive spectral branch gives positive-energy one-particle dynamics; constructing a local quantum field with particles and antiparticles requires both frequency branches and the canonical anticommutation relations developed later in the chapter.

This page builds that free model and makes the boundary and domain assumptions visible. It does not yet normalize explicit $u$ and $v$ spinors, impose quantum statistics, construct Fock space, or derive the propagator.

Required background. The Action Principle and Field Equations supplies independent variations, integration by parts, and the distinction between bulk equations and a complete boundary variational problem. One-Particle States: Mass, Spin, and Relativistic Normalization supplies the positive-energy mass shell, spin-$\tfrac12$ particle interpretation, and the distinction between a covariant field representation and a unitary particle representation.

Helpful background. Lorentz Field Representations and Poincaré Particle Representations and Clifford Algebras and Pin and Spin Groups supply the representation-theory and Clifford-algebra machinery used here.

The free four-dimensional model

The calculation uses the inherited metric, Fourier, Dirac-adjoint, gamma-matrix, and slash conventions summarized in the chapter overview. No explicit gamma-matrix basis is chosen. We work on four-dimensional flat spacetime with $m\ge0$ and no background or interaction. Proper orthochronous Lorentz covariance is part of the model; parity, time reversal, and charge conjugation require additional choices and are not assumed here.

There are two related mathematical uses of $\psi$.

• For the linear equation and its positive solution inner product, $\psi$ is an ordinary $\mathbb C^4$-valued spinor field and $\overline\psi$ is its Dirac adjoint.
• As a classical precursor of a fermionic quantum field, $\psi$ and the variable denoted $\overline\psi$ may instead be varied as independent Grassmann-odd fields. Their physical complex slice restores the Dirac-adjoint relation. The displayed ordering then matters, and no ordered-number inequality such as $\psi^\dagger\psi\ge0$ applies to the Grassmann-valued field itself.

The equations are the same in both uses. Positivity statements below always refer to ordinary complex solutions. The model’s defining data are:

Datum	Free Dirac model
Spacetime	Four-dimensional Minkowski space
Field	A complex four-component spinor $\psi$ and its Dirac adjoint $\overline\psi$
Parameter	A real mass $m\ge0$
Defining action	The symmetrized first-order action displayed below
Continuous spacetime symmetry	Proper orthochronous Poincaré covariance
Internal symmetry	Global vector $U(1)$ for every $m$; independent chiral phase rotations at $m=0$, developed on the Weyl page
On-shell data	Four complex initial components, with no independent initial $\partial_t\psi$; at nonzero on-shell momentum, two spin amplitudes in each energy branch
Reusable checks	Clifford factorization, current conservation, boundary form, $h_D^2=-\nabla^2+m^2$, and the $m\to0$ and $
Scope ceiling	No CAR, Fock representation, spin–statistics proof, Weyl or Majorana classification, propagator, interaction, or anomaly

In four dimensions the action is dimensionless, so the kinetic term gives the engineering dimension

$$[\psi]=[\overline\psi]=\frac32.$$

This is an immediate normalization check on every term in the free density.

Lorentz covariance of the first-order operator

Let $\Lambda\in SO^+(1,3)$ and let $S(\Lambda)$ be its spinor representative. For the active transformation

$$\psi_\Lambda(x) =S(\Lambda)\psi(\Lambda^{-1}x),$$

the required intertwining and adjoint relations are

$$\begin{aligned} S(\Lambda)^{-1}\gamma^\mu S(\Lambda) &=\Lambda^\mu{}_{\nu}\gamma^\nu, \\ S(\Lambda)^\dagger\gamma^0S(\Lambda) &=\gamma^0. \end{aligned}$$

The second identity gives

$$\overline{\psi_\Lambda}(x) = \overline\psi(\Lambda^{-1}x)S(\Lambda)^{-1}.$$

Writing $y=\Lambda^{-1}x$ and using the chain rule, the first identity yields the covariance check

$$\left(i\gamma^\mu\partial_\mu-m\right) \psi_\Lambda(x) = S(\Lambda) \left(i\gamma^\nu\partial_\nu-m\right) \psi(y).$$

Thus the first-order equation transforms into itself. The matrices $S(\Lambda)$ need not be unitary under the finite-dimensional Euclidean form $\psi^\dagger\psi$; their $\gamma^0$-pseudo-unitarity is exactly what makes the Dirac adjoint work. The covariant action and these gamma-matrix adjoint relations are developed in Schwartz 2014, §§ 10.2–10.3, pp. 168–172.

The action, its boundary term, and the field equation

Use the symmetrized action on a spacetime region $M$,

$$\begin{aligned} S_D[\overline\psi,\psi] = \int_M\mathrm d^4x\, \bigg[ &\frac{i}{2} \left( \overline\psi\gamma^\mu\partial_\mu\psi -(\partial_\mu\overline\psi)\gamma^\mu\psi \right) \\ &-m\overline\psi\psi \bigg]. \end{aligned}$$

Vary $\overline\psi$ and $\psi$ independently. Keeping $\delta\overline\psi$ on the left and $\delta\psi$ on the right gives an ordering that also works for Grassmann-odd variables:

$$\begin{aligned} \delta S_D ={}& \int_M\mathrm d^4x\, \delta\overline\psi \left(i\gamma^\mu\partial_\mu-m\right)\psi \\ &+ \int_M\mathrm d^4x\, \left[ -i(\partial_\mu\overline\psi)\gamma^\mu -m\overline\psi \right]\delta\psi \\ &+ \frac{i}{2} \int_{\partial M}\mathrm d\Sigma_\mu \left( \overline\psi\gamma^\mu\delta\psi -\delta\overline\psi\gamma^\mu\psi \right). \end{aligned}$$

For compactly supported variations, or for boundary data that make the last line vanish, stationarity gives the Dirac and adjoint equations

$$\begin{aligned} \left(i\gamma^\mu\partial_\mu-m\right)\psi&=0, \\ i(\partial_\mu\overline\psi)\gamma^\mu +m\overline\psi&=0. \end{aligned}$$

On the physical complex slice $\overline\psi=\psi^\dagger\gamma^0$, the second equation is the Hermitian adjoint of the first because

$$(\gamma^\mu)^\dagger =\gamma^0\gamma^\mu\gamma^0.$$

The frequently used compact density differs from the symmetrized density by a total derivative:

$$\overline\psi \left(i\gamma^\mu\partial_\mu-m\right)\psi = \mathcal L_{\mathrm{sym}} +\frac{i}{2} \partial_\mu \left(\overline\psi\gamma^\mu\psi\right).$$

They therefore produce the same bulk equations for compactly supported variations, but not automatically the same boundary variational problem. This is the fermionic instance of the bulk–boundary distinction established on the action-principle page. The standard compact action and equation appear in Schwartz 2014, § 10.2, p. 168.

Clifford factorization and the mass shell

Because partial derivatives commute, only the symmetric gamma product contributes when the first-order operators are multiplied:

$$\begin{aligned} &\left(i\gamma^\mu\partial_\mu+m\right) \left(i\gamma^\nu\partial_\nu-m\right) \\ &\qquad = -\frac12 \{\gamma^\mu,\gamma^\nu\} \partial_\mu\partial_\nu -m^2 \\ &\qquad =-(\Box+m^2). \end{aligned}$$

Every Dirac solution consequently satisfies

$$(\Box+m^2)\psi=0$$

component by component. For a plane wave $\psi(x)=u(p)e^{-ip\cdot x}$, the two conditions are

$$p^2=m^2, \qquad (p\!\!\!/-m)u(p)=0.$$

The first is necessary but not sufficient. For $m>0$ at rest, $p=(m,\mathbf0)$, every constant spinor amplitude lies on the Klein–Gordon mass shell, whereas the Dirac equation requires

$$(\gamma^0-\mathbf1)u=0.$$

Since $\gamma^0$ has two $+1$ and two $-1$ eigenvalues in the four-component irreducible complex Clifford representation, only a two-dimensional subspace survives. The first-order equation therefore removes half of the arbitrary component amplitudes on each chosen energy branch. The factorization and its sign are checked in Schwartz 2014, § 10.3, p. 172.

One can construct a Dirac solution from a Klein–Gordon spinor by applying the complementary first-order factor, for example $\psi=(i\gamma^\mu\partial_\mu+m)\chi$. This map is not an idempotent projector and does not make the unrestricted solution spaces equivalent. With gauge or spin covariant derivatives, their commutator also adds field-strength or curvature terms, so the free factorization cannot be exported unchanged.

The conserved solution inner product

The action is invariant under the global phase rotation

$$\psi\longmapsto e^{-i\alpha}\psi, \qquad \overline\psi\longmapsto \overline\psi e^{i\alpha}.$$

Its current is

$$j^\mu=\overline\psi\gamma^\mu\psi.$$

Using the Dirac and adjoint equations,

$$\begin{aligned} \partial_\mu j^\mu &= (\partial_\mu\overline\psi)\gamma^\mu\psi + \overline\psi\gamma^\mu\partial_\mu\psi \\ &= im\overline\psi\psi -im\overline\psi\psi =0. \end{aligned}$$

More generally, two ordinary complex solutions $\phi$ and $\psi$ of the same real mass define the polarized conserved current

$$J^\mu[\phi,\psi] =\overline\phi\gamma^\mu\psi.$$

On a future-oriented spacelike Cauchy surface $\Sigma$, define

$$(\phi,\psi)_\Sigma = \int_\Sigma \mathrm d\Sigma_\mu\, \overline\phi\gamma^\mu\psi.$$

In the local Lorentz frame where the future unit normal is $n^\mu=(1,\mathbf0)$,

$$n_\mu J^\mu[\psi,\psi] =\psi^\dagger\psi\ge0.$$

Thus this is a positive solution norm. For two Cauchy surfaces bounding a slab, the divergence theorem gives

$$(\phi,\psi)_{\Sigma_2} -(\phi,\psi)_{\Sigma_1} = -\int_B \mathrm d\Sigma_\mu\, J^\mu[\phi,\psi],$$

where $B$ is the lateral boundary. The norm is surface-independent when this flux vanishes—for example for adequate decay at spatial infinity, periodic spatial directions, or a boundary condition that kills the bilinear flux for every pair in the domain. On a constant-time slice,

$$(\phi,\psi)_t = \int\mathrm d^3\mathbf x\, \phi^\dagger(t,\mathbf x) \psi(t,\mathbf x).$$

The density $j^0=\psi^\dagger\psi$ is positive but is not a Lorentz scalar; the covariant object is the hypersurface contraction and integral. After second quantization, the normal-ordered $U(1)$ charge counts particles minus antiparticles and is not positive. These are different statements. The free current and its positive time component are derived in Schwartz 2014, § 10.4, p. 174.

Hamiltonian domain and the two energy branches

Define

$$\alpha^i=\gamma^0\gamma^i, \qquad \beta=\gamma^0.$$

The field equation becomes

$$i\partial_t\psi=h_D\psi, \qquad h_D=-i\boldsymbol\alpha\cdot\boldsymbol\nabla+\beta m.$$

The mostly-minus adjoint relations imply

$$\begin{gathered} (\alpha^i)^\dagger=\alpha^i, \qquad \beta^\dagger=\beta, \\ \{\alpha^i,\alpha^j\}=2\delta^{ij}\mathbf1, \qquad \{\alpha^i,\beta\}=0, \qquad \beta^2=\mathbf1. \end{gathered}$$

Formal integration by parts on a spatial region $\Omega$ exposes the boundary form

$$\begin{aligned} \langle\phi,h_D\psi\rangle_\Omega -\langle h_D\phi,\psi\rangle_\Omega =-i\int_{\partial\Omega}\mathrm dS\, \phi^\dagger (\boldsymbol\alpha\cdot\mathbf n)\psi. \end{aligned}$$

Calling $h_D$ self-adjoint therefore requires a domain, not only Hermitian coefficient matrices. On $\mathbb R^3$, Fourier transformation turns $h_D$ into multiplication by the Hermitian matrix

$$h_D(\mathbf p) =\boldsymbol\alpha\cdot\mathbf p+\beta m.$$

Its maximal square-integrable domain is

$$\begin{aligned} \mathcal D(h_D) &= \left\{ \psi\in L^2(\mathbb R^3,\mathbb C^4): E_{\mathbf p}\widetilde\psi(\mathbf p) \in L^2 \right\} \\ &=H^1(\mathbb R^3,\mathbb C^4), \end{aligned}$$

where $E_{\mathbf p}=\sqrt{\mathbf p^2+m^2}$. A Hermitian multiplication matrix on this maximal domain is self-adjoint, so the free evolution is unitary. Periodic $H^1$ data give the analogous result on a spatial torus. On a bounded region, one must instead choose a domain that makes the bilinear boundary form vanish and is maximal with that property; checking only the diagonal flux is not a complete self-adjointness argument.

The Clifford relations now give the decisive spectral check:

$$h_D(\mathbf p)^2 =(\mathbf p^2+m^2)\mathbf1 =E_{\mathbf p}^2\mathbf1.$$

For $E_{\mathbf p}>0$, the spectral projectors are

$$P_\pm(\mathbf p) = \frac12 \left( \mathbf1 \pm \frac{h_D(\mathbf p)}{E_{\mathbf p}} \right).$$

They obey

$$P_\pm^2=P_\pm, \qquad P_+P_-=0, \qquad P_++P_-=\mathbf1.$$

In that four-component complex Clifford representation $\operatorname{tr}h_D(\mathbf p)=0$, so $\operatorname{tr}P_+=\operatorname{tr}P_-=2$. There are two independent spin amplitudes with energy $+E_{\mathbf p}$ and two with energy $-E_{\mathbf p}$. The full one-particle Hamiltonian is self-adjoint but not bounded below. Its positive spectral subspace is invariant under free evolution and carries positive energy; the local field equation still contains both branches.

This conclusion is sometimes obscured by the positive solution norm. A positive inner product makes unitary evolution possible, but it does not force the generator’s spectrum to be positive. Srednicki derives the same $\pm E_{\mathbf p}$ spectrum in an opposite-signature convention; translating the Clifford relation leaves the invariant equation $E_{\mathbf p}^2=\mathbf p^2+m^2$ and the multiplicities unchanged Srednicki 2007, § 1, pp. 23–24.

Controlled limits and independent checks

Rest frame. For $m>0$ and $\mathbf p=0$,

$$P_\pm(\mathbf0)=\frac12(\mathbf1\pm\beta).$$

This reproduces the two-dimensional rest-frame constraint found from $p\!\!\!/-m$ and checks the relative signs of $\beta$ and the mass term.

Massless limit. When $m=0$,

$$[h_D,\gamma_5]=0.$$

Hamiltonian evolution therefore preserves the left- and right-chiral subspaces. Equivalently, although the covariant kinetic operator anticommutes with $\gamma_5$, each projected component of a massless solution is again a solution. At $m>0$ the mass term couples the two chiralities. The two-component formulation, helicity comparison, and the exceptional point $m=|\mathbf p|=0$ belong to Weyl Fields and Chirality.

Low-momentum positive branch. For $m>0$ and $|\mathbf p|\ll m$,

$$E_{\mathbf p} =m+\frac{\mathbf p^2}{2m} -\frac{\mathbf p^4}{8m^3} +O\!\left(\frac{|\mathbf p|^6}{m^5}\right).$$

After the rest energy is removed, the leading dispersion is the nonrelativistic kinetic energy. This is a check on the positive branch only; it does not eliminate the negative-frequency sector of the relativistic field.

Boundary round trip. The same matrix $\boldsymbol\alpha\cdot\mathbf n$ appears in both the Hamiltonian boundary form and the spatial flux $\mathbf j\cdot\mathbf n$. A domain that makes the sesquilinear Hamiltonian boundary form vanish also removes probability flux for every pair of allowed solutions. This agreement checks the current and Hamiltonian signs independently.

What this page establishes—and where it stops

Lorentz covariance selects a spinor transformation law compatible with the Clifford algebra. The symmetrized first-order action then gives the Dirac equation and an explicit surface variation. Clifford factorization fixes the relativistic mass shell but leaves only two spin amplitudes on each energy branch. The $U(1)$ current gives a positive conserved solution form under a no-flux hypothesis, while the Hamiltonian domain and projectors separate the $\pm E_{\mathbf p}$ spectra.

These results define the free Dirac dynamics, not the quantized fermion field by themselves. In particular:

• Plane Waves, Spin Sums, and Bilinears chooses and checks explicit $u$ and $v$ normalizations, completeness relations, helicity or spin labels, and covariant bilinears.
• Canonical Quantization of the Free Dirac Field assigns the negative-frequency coefficients to antiparticle creation operators, imposes CAR, and derives positive Fock energy and the particle-minus-antiparticle charge.
• Weyl Fields and Chirality develops the massless two-component theory; Majorana Fields and Reality Conditions develops Lorentz-compatible reality conditions.
• Gauge and Yukawa couplings, chiral anomalies, and interacting matter belong to Symmetry and Gauge Structure and the later interacting volumes.

The Dirac equation alone does not prove Fermi statistics, the spin–statistics theorem, vacuum stability, or locality of the quantized field.

Common pitfalls

“Dirac is just four Klein–Gordon equations.” Every Dirac solution is Klein–Gordon, but the first-order condition restricts the allowed component amplitudes and initial data. The rest-frame kernel gives a direct counterexample to the converse.

“A positive density means a positive Hamiltonian.” The solution norm is positive, while the self-adjoint one-particle Hamiltonian has both energy signs. Positive Fock energy is a later result using the frequency split, antiparticle assignment, CAR, a vacuum representation, and a vacuum-energy prescription.

“Hermitian gamma matrices make the Hamiltonian self-adjoint.” In the $(+---)$ convention the spatial gamma matrices are anti-Hermitian, while $\alpha^i$ and $\beta$ are Hermitian. More importantly, self-adjointness is a property of the differential operator together with its domain and boundary conditions.

“$j^0$ is a Lorentz scalar probability density.” It is the time component of a vector current. Positivity on an arbitrary spacelike Cauchy surface is expressed covariantly by $n_\mu j^\mu\ge0$, and conservation of the integrated norm also requires control of lateral flux.

“A classical Grassmann field has a positive pointwise density.” Positivity belongs to the ordinary complex solution space used for the one-particle equation. Grassmann-valued fields have no ordered-number inequality; their quantum interpretation is supplied by CAR and a state representation.

Check your understanding

1. At rest with $m>0$, compare the Klein–Gordon and Dirac conditions on a constant spinor amplitude.

Solution

The mass-shell equation $p^2=m^2$ is already satisfied at $p=(m,\mathbf0)$ and places no condition on a four-component amplitude $u$. The Dirac equation adds $(\gamma^0-\mathbf1)u=0$. Since $\gamma^0$ has two $+1$ eigenvectors and two $-1$ eigenvectors, the allowed amplitude space is two-dimensional. Klein–Gordon is necessary but not sufficient.

2. Show that the positive solution norm is conserved in a periodic spatial box.

Solution

Integrating $\partial_tj^0+\boldsymbol\nabla\cdot\mathbf j=0$ over the box gives

$$\frac{\mathrm d}{\mathrm dt} \int_\Omega\mathrm d^3\mathbf x\,j^0 =- \int_{\partial\Omega}\mathrm dS\, \mathbf j\cdot\mathbf n.$$

Opposite faces have equal current values and opposite outward normals, so their fluxes cancel pairwise. The integral is constant. Its integrand is $j^0=\psi^\dagger\psi\ge0$ for ordinary complex solutions.

3. Verify that $P_\pm(\mathbf p)$ are rank-two orthogonal projectors when $E_{\mathbf p}>0$.

Solution

Set $A=h_D(\mathbf p)/E_{\mathbf p}$. Since $A^2=\mathbf1$ and $A=A^\dagger$,

$$P_\pm^2 =\frac14(\mathbf1\pm A)^2 =\frac12(\mathbf1\pm A) =P_\pm,$$

and $P_+P_-=0$. The trace identity $\operatorname{tr}h_D=0$ gives $\operatorname{tr}P_\pm=2$, which equals the rank of an orthogonal projector.

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.