The Free Maxwell Field and Gauge Redundancy

Free Maxwell theory uses a four-component potential $A_\mu$ to describe source-free electromagnetic radiation, but the potential is redundant: admissibly related fields $A_\mu$ and $A_\mu+\partial_\mu\alpha$ determine the same field strength $F_{\mu\nu}$. The action depends only on $F$, stationarity gives $\partial_\mu F^{\mu\nu}=0$, and the time component of that equation is Gauss law. The electric and magnetic fields are gauge invariant, while $A_0$ is nondynamical and does not represent an additional photon.

For a nonzero plane wave, the equation of motion and the gauge quotient leave two physical polarizations. This page derives that free Abelian result in a local-potential, topologically trivial setting with no dynamical charged matter. It does not define gauge theory in general or develop gauge fixing, global sectors, BRST/BV, or the quantum photon state space.

Required background. The Action Principle and Field Equations supplies boundary-aware first variation, integration by parts, and Euler–Lagrange equations.

Helpful background. Differential Forms, Integration, Orientation, and Stokes Theorem supplies alternating forms, the exterior derivative and $\mathrm d^2=0$, pullback, orientation, and Stokes’ theorem for the compact notation $F=\mathrm dA$ and the Bianchi identity.

The source-free Maxwell model

Let $A_\mu$ be a real Abelian potential on four-dimensional Minkowski space. Work either on a contractible region or in one fixed bundle trivialization, so a single potential describes the configurations under discussion. With the inherited site conventions,

$$\begin{aligned} F_{\mu\nu} &=\partial_\mu A_\nu-\partial_\nu A_\mu,\\ S_{\mathrm M}[A] &=-\frac14\int_\Omega\mathrm d^4x\, F_{\mu\nu}F^{\mu\nu}. \end{aligned}$$

There is no mass parameter or interaction coupling in this normalized free action. Its local redundancy follows directly from commuting partial derivatives:

$$\begin{aligned} A_\mu&\longmapsto A_\mu+\partial_\mu\alpha,\\ F_{\mu\nu}&\longmapsto F_{\mu\nu} +\partial_\mu\partial_\nu\alpha -\partial_\nu\partial_\mu\alpha =F_{\mu\nu}. \end{aligned}$$

Thus the action is invariant, and a pure-gradient potential $A_\mu=\partial_\mu\alpha$ has $F_{\mu\nu}=0$. This is an off-shell identity, not a consequence of the field equation.

The formula alone does not decide which transformations are quotiented out. The allowed fields, boundary or falloff data, and admissible functions $\alpha$ must first be fixed. On this page, gauge redundancy means transformations that preserve those data and are declared physically trivial; compactly supported $\alpha$ provide the clean local example. Transformations with nontrivial boundary or global action belong to the general gauge-theory treatment.

Schwartz 2014, §§ 8.2.3–8.2.4, pp. 118–120 develops the massless action, local transformation, and two-mode interpretation. Zinn-Justin 2021, § 21.4, pp. 514–515 independently separates the redundant potential from the physical field strength. Zinn-Justin uses Euclidean notation and $A_\mu\mapsto A_\mu-e^{-1}\partial_\mu\chi$; setting $\alpha=-\chi/e$ gives the same local orbit, while no Euclidean action sign is imported here.

Dynamics and the Bianchi identity have different origins

For the variational problem, take $\delta A_\mu$ to have compact support in $\Omega$ or to vanish on all of $\partial\Omega$. If the region extends to spatial infinity, suitable spatial falloff controls that part of the boundary, while variations still vanish on the initial and final time slices. The complete first variation is

$$\begin{aligned} \delta S_{\mathrm M} = {}&\int_\Omega\mathrm d^4x\, (\partial_\mu F^{\mu\nu})\delta A_\nu\\ &-\int_{\partial\Omega}\mathrm d\Sigma_\mu\, F^{\mu\nu}\delta A_\nu. \end{aligned}$$

Under the declared boundary conditions, stationarity for arbitrary interior variations gives the source-free Maxwell equation

$$\partial_\mu F^{\mu\nu}=0.$$

The homogeneous equation has a different status:

$$\partial_\rho F_{\mu\nu} +\partial_\mu F_{\nu\rho} +\partial_\nu F_{\rho\mu}=0.$$

It is the Bianchi identity, which follows from $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$ before any action is varied. In form notation these two facts begin with $F=\mathrm dA$ and $\mathrm dF=\mathrm d^2A=0$; the component derivation above requires no form language.

Expanding the dynamical equation in terms of the potential gives

$$\Box A^\nu-\partial^\nu(\partial\cdot A)=0.$$

Taking a divergence now produces $0=0$. Unlike the massive equation on The Proca Field, Maxwell dynamics does not derive $\partial\cdot A=0$. Lorenz gauge is an optional representative condition, with its own residual transformations and boundary requirements, not another field equation. The action, sourced equation specialized here to zero current, and homogeneous identity are given structurally in Weinberg 1995, § 8.1, pp. 339–342.

Gauss law constrains initial data

To identify the familiar electric and magnetic fields without hidden sign choices, define the page-local spatial symbol by $\epsilon_{123}=+1$ and set

$$E_i=F_{0i}, \qquad B_i=-\frac12\epsilon_{ijk}F_{jk}.$$

Then

$$\mathcal L_{\mathrm M} =\frac12\mathbf E^2-\frac12\mathbf B^2.$$

The Euler–Lagrange equations become

$$\boldsymbol\nabla\cdot\mathbf E=0, \qquad \boldsymbol\nabla\times\mathbf B -\partial_t\mathbf E=0,$$

while the Bianchi identity becomes

$$\boldsymbol\nabla\cdot\mathbf B=0, \qquad \boldsymbol\nabla\times\mathbf E +\partial_t\mathbf B=0.$$

The first dynamical equation is source-free Gauss law. It contains no second time derivative and restricts admissible initial electric fields. Its preservation is visible without a Hamiltonian formalism: taking the divergence of the Ampère equation gives

$$\partial_t(\boldsymbol\nabla\cdot\mathbf E) =\boldsymbol\nabla\cdot (\boldsymbol\nabla\times\mathbf B)=0.$$

The canonical preview reaches the same conclusion. Since no $\dot A_0$ occurs,

$$\pi^0=0, \qquad \pi^i=\frac{\partial\mathcal L_{\mathrm M}} {\partial\dot A_i}=E_i.$$

Consistency of the primary condition $\pi^0=0$ yields the secondary condition $\partial_i\pi^i=0$, exactly Gauss law. In Proca theory the corresponding equation contains $m^2A_0$ and solves for $A_0$; here it does not, so $A_0$ remains a multiplier and the constraint pair is first class. This is the bounded sense in which free Maxwell theory is the canonical constrained vector example. Weinberg 1995, § 8.2, pp. 344–346 gives the primary and Gauss constraints and their first-class character. Weinberg uses a mostly-plus metric, so only the translated structural statements are used here. The full bracket algebra, gauge generator, and reduced phase space belong to Maxwell Constraints as a Worked Application.

Two physical plane-wave modes survive the quotient

A short covariant count checks the constraint interpretation without choosing an explicit polarization basis. Let

$$A^\mu(x)=\varepsilon^\mu(p)e^{-ip\cdot x}.$$

The equation of motion is

$$-p^2\varepsilon^\nu +p^\nu(p\cdot\varepsilon)=0.$$

If $p^2\ne0$, this equation makes $\varepsilon^\mu$ proportional to $p^\mu$, so $F_{\mu\nu}=0$ and the mode is pure gauge. A mode with nonzero field strength therefore has $p^2=0$, after which the equation requires $p\cdot\varepsilon=0$.

The transverse subspace $p\cdot\varepsilon=0$ is three-dimensional, but it contains the null direction $\varepsilon^\mu\propto p^\mu$. A plane-wave gauge transformation identifies

$$\varepsilon^\mu\sim\varepsilon^\mu+c\,p^\mu.$$

Quotienting the three-dimensional transverse space by this one pure-gauge direction leaves two physical modes. This count concerns nonzero momentum in the local free theory; boundaries, topology, and zero modes require separate analysis. Schwartz 2014, §§ 8.2.3–8.2.4, pp. 119–120 identifies the pure-gauge polarization and the two transverse modes, while Zinn-Justin 2021, § 21.5.1, pp. 516–517 independently obtains $d-2$ physical components. Explicit bases, helicities, and completeness relations belong to Massive and Massless Spin-One Polarizations.

Field strengths carry the local observable content

Within this free classical model, $F_{\mu\nu}$ and therefore $\mathbf E$ and $\mathbf B$ are unchanged along every admitted gauge orbit. They supply the local gauge-invariant content, whereas an individual potential $A_\mu$ is a representative. After imposing source-free Gauss law and using boundary conditions that remove the spatial surface term, the reduced physical energy is

$$H=\frac12\int\mathrm d^3x\, (\mathbf E^2+\mathbf B^2),$$

which depends only on gauge-invariant fields and provides a sign check on the action. Zinn-Justin 2021, § 21.5.2, p. 518 gives the source-free Maxwell Hamiltonian in real-time notation; using $F_{ij}F_{ij}=2\mathbf B^2$ yields this form.

This statement is intentionally local and classical. It does not claim that point fields are ordinary quantum observables, that $F$ exhausts global electromagnetic data on every spacetime, or that every boundary transformation is redundant. The primary treatment of gauge connections, admissible transformation groups, global form, boundary qualifications, and observable content is Gauge Fields, Redundancy, and Observable Content. Gauge fixing, ghosts, BRST/BV, and generalized symmetries remain with Symmetry and Gauge Structure.

What this worked model establishes

Maxwell dynamics, redundancy, Gauss law, and field-strength observables are not competing descriptions. They are four views of the same degenerate free action:

• dependence on $F$ makes the local potential redundant;
• varying the action gives the dynamical Maxwell equation;
• its time component constrains initial data rather than evolving $A_0$;
• quotienting the on-shell potential by pure-gauge directions leaves two radiative modes; and
• $F$, $\mathbf E$, and $\mathbf B$ remain unchanged along the quotient.

The next steps divide by scientific task. Maxwell Constraints as a Worked Application performs the full canonical reduction, while Massive and Massless Spin-One Polarizations constructs explicit mode bases. Quantization of the reduced radiative modes begins on Physical-Mode Quantization of the Free Electromagnetic Field. Interacting gauge fields and matter belong to Dynamical Gauge Fields and Matter.

Common pitfalls

Treating the Bianchi identity as a second equation of motion. The equation $\partial_{[\rho}F_{\mu\nu]}=0$ follows from $F=\mathrm dA$. Only $\partial_\mu F^{\mu\nu}=0$ follows from varying the source-free Maxwell action.

Calling $\partial\cdot A=0$ a Maxwell equation. The divergence of the Maxwell equation is an identity. A Lorenz condition selects representatives and retains residual gauge freedom; it is not the Proca subsidiary condition.

Declaring every smooth $\alpha$ redundant. The local formula leaves $F$ invariant, but the physical quotient depends on allowed fields, boundary data, falloff, and global sector. This page quotients only the admitted transformations declared trivial.

Counting four potential components as four photons. Gauss law constrains the initial data, and the gauge quotient identifies a further direction. The plane-wave check leaves two physical modes.

Check your understanding

1. Show directly that a pure-gradient potential has vanishing field strength and solves the source-free Maxwell equation.

   Solution

   For $A_\mu=\partial_\mu\alpha$, commuting derivatives gives $F_{\mu\nu}=\partial_\mu\partial_\nu\alpha-\partial_\nu\partial_\mu\alpha=0$. Hence $\partial_\mu F^{\mu\nu}=0$ identically. This proves a local algebraic fact; treating the transformation as redundancy still requires the stated boundary and admissibility conditions.

2. Derive the four component Maxwell equations from the covariant equation and Bianchi identity, then verify preservation of Gauss law.

   Solution

   With $E_i=F_{0i}$ and $F_{ij}=-\epsilon_{ijk}B_k$, the $\nu=0$ Euler–Lagrange equation gives $\boldsymbol\nabla\cdot\mathbf E=0$, and its spatial components give $\boldsymbol\nabla\times\mathbf B-\partial_t\mathbf E=0$. The spatial Bianchi identity gives $\boldsymbol\nabla\cdot\mathbf B=0$, while its one-time-index components give $\boldsymbol\nabla\times\mathbf E+\partial_t\mathbf B=0$. Taking the divergence of the Ampère equation shows $\partial_t(\boldsymbol\nabla\cdot\mathbf E)=0$.

3. Explain the two-mode count without choosing Coulomb or Lorenz gauge.

   Solution

   A non-pure-gauge plane wave has $p^2=0$ and $p\cdot\varepsilon=0$. The latter condition leaves a three-dimensional subspace of four-vector polarizations. Because $p^2=0$, the vector $p^\mu$ lies in that subspace, and $\varepsilon^\mu\sim\varepsilon^\mu+c\,p^\mu$ identifies it as pure gauge. The quotient therefore has dimension $3-1=2$.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI.
• Weinberg, Steven. The Quantum Theory of Fields. Volume I: Foundations. Cambridge: Cambridge University Press, 1995. DOI.
• Zinn-Justin, Jean. Quantum Field Theory and Critical Phenomena. 5th ed. Oxford: Oxford University Press, 2021. DOI.