Dirac and Maxwell Fields

Scalar fields conceal two structures that become unavoidable for fields with spin or gauge freedom. A Dirac field needs a spin bundle and a first-order evolution operator; a Maxwell potential has redundant representatives and a constraint that survives on every Cauchy surface. Both theories still use causal Green operators, but they reach the physical solution space by different routes.

For the local PDE bridge, a differential operator is tested on compactly supported smooth sections, and its advanced or retarded inverse must place the resulting support in the causal past or future of the source. A solution is spacelike compact when its intersection with every Cauchy surface is compact. These facts justify Green identities and the surface integrals below without presuming a developed PDE course.

Required background. Partial Differential Equations and Green Functions provides distributional Green identities, causal support, and Green operators; Classical Field Theory supplies Euler–Lagrange equations, Noether currents, and gauge equivalence. Helpful background. Flat-Spacetime Quantum Field Theory provides spinor normalization and canonical electromagnetic constraints; Causality and Global Structure establishes Cauchy hypersurfaces and causal domains.

Dirac geometry and its hyperbolic square

Let $(M,g)$ be an oriented, time-oriented, four-dimensional globally hyperbolic spacetime with a chosen spin structure. The associated complex spinor bundle $S\to M$ carries a spin connection $\nabla$ and Clifford multiplication $\gamma^\mu$ satisfying

$$\{\gamma^\mu,\gamma^\nu\}=2g^{\mu\nu}\mathbf 1, \qquad \nabla_\mu\gamma^\nu=0.$$

To keep the sign calculation visible, set

$$D_-=-\gamma^\mu\nabla_\mu+m, \qquad D_+=\gamma^\mu\nabla_\mu+m.$$

The curvature of the spin connection gives the Lorentzian Lichnerowicz identity in these conventions,

$$(\gamma^\mu\nabla_\mu)^2 =\nabla_\mu\nabla^\mu-\frac14R.$$

Consequently,

$$D_+D_-=D_-D_+ =-\nabla_\mu\nabla^\mu+\frac14R+m^2.$$

The right-hand side is normally hyperbolic up to an overall sign; its Green operators must invert that signed operator. If $Q=D_+D_-$ and $G_Q^{\mathrm{ret/adv}}$ obey $QG_Qf=f$, then $S^{\mathrm{ret/adv}}=D_+G_Q^{\mathrm{ret/adv}}$ solves $D_-S f=f$ with the required causal support. Uniqueness gives the other inverse identity. The curvature sign and this construction are checked in Dappiaggi, Hack, and Pinamonti 2009, Lemma 2.1, pp. 14–16, and Theorem 2.3, pp. 17–18, PDF.

The absence of an $i$ in $D_-$ is deliberate: the Clifford matrices here satisfy $\{\gamma^\mu,\gamma^\nu\}=2g^{\mu\nu}$. With $\Gamma^\mu=-i\gamma^\mu$, the equivalent equation is $(i\Gamma^\mu\nabla_\mu-m)\psi=0$, and $\{\Gamma^\mu,\Gamma^\nu\}=-2g^{\mu\nu}$. One must translate both formulas together. In flat spacetime either version implies $(\Box-m^2)\psi=0$ and hence $\omega^2=\lvert\mathbf k\rvert^2+m^2$.

The conserved Dirac current is

$$j^\mu(\psi)=\bar\psi\gamma^\mu\psi, \qquad \nabla_\mu j^\mu=0$$

for a solution and its Dirac adjoint. More explicitly, in an orthonormal frame take $(\gamma^{\hat0})^\dagger=-\gamma^{\hat0}$, $(\gamma^{\hat i})^\dagger=\gamma^{\hat i}$ and $\bar\psi=-\psi^\dagger\gamma^{\hat0}$. Then $j^{\hat0}=\psi^\dagger\psi$, and the spatial current satisfies $\lvert\mathbf j\rvert\leq j^{\hat0}$. Thus $j^\mu$ is future causal. Since our future unit normal obeys $n_\mu n^\mu=-1$, the positive Cauchy-surface density is $-n_\mu j^\mu$. Thus the Hermitian form

$$(\psi_1,\psi_2)_\Sigma =-\int_\Sigma d\Sigma\, n_\mu\bar\psi_1\gamma^\mu\psi_2$$

is independent of the Cauchy surface for spacelike-compact solutions. Unlike the scalar symplectic form, it is already adapted to fermionic CAR quantization.

Maxwell fields: equations, gauge freedom, and constraints

On a topologically trivial $U(1)$ bundle, a connection may be represented by a one-form $A$. Define the codifferential on a $p$-form by $(\delta\alpha)_{\mu_2\ldots\mu_p}=-\nabla^{\mu_1}\alpha_{\mu_1\mu_2\ldots\mu_p}$, so $\delta d\chi=-\Box\chi$ on scalars. Its curvature and vacuum equation are

$$F=dA, \qquad dF=0, \qquad \delta F=0,$$

and $A$ and $A+d\chi$ describe the same local electromagnetic field. The operator $\delta d$ acting on potentials is not normally hyperbolic: every exact one-form is a gauge direction in its principal symbol. Imposing Lorenz gauge, $\delta A=0$, replaces the equation by

$$(\delta d+d\delta)A=0,$$

whose Hodge–d’Alembert operator on one-forms is $((\delta d+d\delta)A)_\nu=-\Box A_\nu+R_\nu{}^\mu A_\mu$. It is normally hyperbolic up to the displayed overall minus sign. One may solve this hyperbolic equation, but one must impose initial data that preserve $\delta A=0$ and still quotient residual transformations with $\Box\chi=0$. Gauge fixing is therefore a device for finding representatives, not a creation of new physical degrees of freedom.

The same distinction appears on a Cauchy surface. The tangential vector potential and electric field cannot be chosen freely: Gauss’ law requires the electric field to be divergence-free in vacuum. Evolution preserves this constraint because applying $\delta$ to the field equation gives an identity. A set of unconstrained wave-equation data would include solutions that do not solve Maxwell’s equations.

Globally, connections form an affine space rather than a vector space, and large gauge transformations need not be of the form $d\chi$. Benini, Dappiaggi, and Schenkel construct the gauge-invariant phase space from compactly supported affine observables, obtain its Green-operator presymplectic form, and characterize its radical cohomologically in Benini, Dappiaggi, and Schenkel 2013, §§2–4, pp. 3–16. Their analysis is a warning against identifying “Lorenz-gauge solutions modulo exact forms” with the full global theory without checking the bundle and the cohomology.

Two routes to the physical phase space

Question	Dirac field	Maxwell field
Geometric input	A spin structure and spinor bundle	A principal bundle and its affine space of connections
Evolution operator	First order and Green-hyperbolic through its square	Gauge-degenerate until a hyperbolic gauge is chosen
Initial data	One spinor on a Cauchy surface	Potential and electric data subject to Gauss’ law
Redundancy	No classical gauge quotient for the free neutral spinor considered here	Quotient by small and, when present, large gauge transformations
Bilinear structure	Positive Hermitian surface form	Antisymmetric presymplectic form, possibly with a radical
Global sensitivity	Choice of spin structure can matter	Cohomology and bundle topology can carry flux sectors

The common causal mechanism is now precise. For Dirac fields, factorization transfers Green operators from a second-order normally hyperbolic operator to a first-order one. For Maxwell fields, gauge fixing exposes a normally hyperbolic operator on representatives, while constraint reduction and quotienting recover the physical space. Treating these as the same construction loses either the fermionic Hermitian structure or the gauge radical.

Topology is not merely another gauge choice

Suppose $M$ has a closed one-form $a$ that is not exact. Then $A$ and $A+a$ have identical curvature, yet no globally defined real function $\chi$ satisfies $a=d\chi$. For a compact gauge group, some such shifts are large gauge transformations and others label distinct flat connections. Local observables built only from $F$ cannot distinguish them.

This gives a useful adverse test. On a spacetime with a spatial circle, $a=c\,d\theta$ is closed. Its loop integral is $\oint a=2\pi c$, whereas every exact form has zero loop integral. Thus a claim that all flat potentials are pure gauge is false unless the allowed gauge group and its periods have been specified. In the affine-observable construction, gauge-invariant linear functionals can also fail to separate gauge-inequivalent flat connections when the Abelian group has a compact factor; the precise obstruction and the resulting radical are identified in Benini, Dappiaggi, and Schenkel 2013, §4, especially Remark 4.5 and Theorem 4.6, pp. 13–16.

Diagnostic: why the two reductions cannot be interchanged

Consider a Maxwell potential $A$ satisfying Lorenz gauge and the wave equation. Show that $A'=A+d\chi$ has the same properties precisely when $\Box\chi=0$, and explain why this residual quotient has no Dirac analogue in the free theory above.

Solution

Because $d^2=0$, both potentials give the same curvature. Moreover $\delta A'=\delta A-\Box\chi$, so Lorenz gauge is preserved exactly when $\Box\chi=0$. The Hodge wave operator commutes with $d$, so the wave equation is then preserved as well. This checks that gauge fixing leaves a residual orbit that must still be quotiented. A free Dirac spinor is instead a section of a fixed spin bundle with a nondegenerate conserved Hermitian form; the factorization of its equation introduces no equivalence $\psi\sim\psi+D f$. Spin-frame covariance changes local components but not the geometric section and is not a Maxwell-like gauge degeneracy of the solution space.

References

• Benini, Marco, Claudio Dappiaggi, and Alexander Schenkel. “Quantized Abelian Principal Connections on Lorentzian Manifolds.” arXiv:1303.2515 [math-ph], 2013. arXiv.
• Dappiaggi, Claudio, Thomas-Paul Hack, and Nicola Pinamonti. “The Extended Algebra of Observables for Dirac Fields and the Trace Anomaly of Their Stress-Energy Tensor.” arXiv:0904.0612 [math-ph], 2009. arXiv.