Potentials, Gauge Freedom, and Constraints

An electromagnetic potential contains both indispensable structure and redundant description. Locally, gauge-related potentials produce the same electric and magnetic fields. Globally, topology, boundary conditions, and charged probes determine which transformations are true redundancies and which data survive. A good gauge choice simplifies the equations without changing any field measurement.

Required background. Special Relativity supplies Lorentz covariance, four-vectors, contractions, and index raising and lowering; Calculus and Linear Algebra supplies partial derivatives, vector identities, and linear constraints. Helpful background. Ordinary Differential Equations supplies initial-value reasoning; Differential Forms and Integration supplies exterior derivatives and global integration.

Gauge-equivalent potentials

A one-form is a covector field written here as $A=A_\mu dx^\mu$. Its exterior derivative antisymmetrizes one derivative, so

$$A=-\Phi\,dt+A_i\,dx^i, \qquad F=dA$$

means

$$\mathbf E=-\boldsymbol\nabla\Phi-\partial_t\mathbf A, \qquad \mathbf B=\boldsymbol\nabla\times\mathbf A.$$

For any smooth scalar function $\chi$,

$$\boxed{A\longmapsto A+d\chi}$$

leaves $F$ unchanged because $d^2\chi=0$. In components,

$$\Phi\longmapsto\Phi-\partial_t\chi, \qquad \mathbf A\longmapsto\mathbf A+\boldsymbol\nabla\chi.$$

This local construction is derived in Tong 2015, §5.3.1, printed pp. 105–106, PDF.

An ordinary global symmetry maps one physical state to another. A gauge transformation that preserves the stipulated boundary data changes the representative of the same electromagnetic state: $A$ changes, but $F$, forces, and local energy densities do not. If a transformation changes fixed boundary data, then it is not a redundancy of that boundary-value problem. Gauge fixing must therefore be stated together with its boundary or falloff conditions.

The source coupling reveals the same consistency condition. Up to a boundary term,

$$\Delta_\chi S_{\mathrm{source}} =-\int_\Omega d^4x\,\chi\,\partial_\mu J^\mu.$$

Thus a conserved current and compatible boundary behavior are required for gauge invariance. This is the potential-language counterpart of the current conservation derived from the antisymmetric Maxwell tensor.

Loops, holonomy, and the gauge group

On a multiply connected region, a flat field $F=0$ need not make every potential globally removable. For a real gauge group $\mathbb R$, the allowed $\chi$ is a globally single-valued real function, and

$$\oint_\gamma(A+d\chi)=\oint_\gamma A.$$

For the compact gauge group $U(1)$, the gauge transformation $e^{iq\chi}$ must be single-valued, but the real parameter $\chi$ can change by $2\pi n/q$ around a noncontractible loop. The raw integral may then shift by $2\pi n/q$. The gauge-invariant charged-probe datum is the holonomy

$$\boxed{ \operatorname{Hol}_\gamma(A) =\exp\left(iq\oint_\gamma A\right).}$$

This phase can distinguish flat $U(1)$ connections even when every local measurement of $F$ vanishes. Tong develops the loop phase and the distinction between $U(1)$ and $\mathbb R$ gauge groups in Tong 2018, §§1.1.1–1.1.2, printed pp. 4–8, PDF. This page uses that global distinction without entering quantum electrodynamics.

Lorenz and Coulomb gauge

From the covariant Maxwell equation,

$$\Box A^\nu-\partial^\nu(\partial_\mu A^\mu)=-J^\nu, \qquad \Box=-\partial_t^2+\nabla^2.$$

Two useful gauge choices expose different features:

Gauge	Condition	Equation that reaches it	Residual freedom	Main advantage
Lorenz	$\partial_\mu A^\mu=0$	$\Box\chi=-\partial_\mu A^\mu$	$\Box\chi=0$	Lorentz-covariant wave equations
Coulomb	$\boldsymbol\nabla\mathbin{\cdot}\mathbf A=0$	$\nabla^2\chi=-\boldsymbol\nabla\mathbin{\cdot}\mathbf A$	$\nabla^2\chi=0$	Direct separation of the scalar constraint

Boundary or falloff data decide whether the equation for $\chi$ has a solution and whether a residual solution remains. A differential condition alone need not select exactly one representative.

In Lorenz gauge the potential satisfies

$$\boxed{\Box A^\nu=-J^\nu.}$$

Conversely, solving these four reduced wave equations is not enough unless the gauge condition is also enforced. If $C=\partial_\mu A^\mu$ and the current is conserved, then

$$\Box C=-\partial_\mu J^\mu=0.$$

Compatible Cauchy data set both $C$ and its normal derivative to zero; the homogeneous wave equation then propagates $C=0$. The reachability equation, residual freedom, Coulomb gauge, and two-polarization count are treated in Tong 2006–2007, §6.1.1, printed pp. 125–127, PDF.

In Coulomb gauge, Gauss’s law gives

$$\boldsymbol\nabla\mathbin{\cdot}\mathbf E =-\nabla^2\Phi=\rho, \qquad \boxed{\nabla^2\Phi=-\rho.}$$

The scalar potential is therefore determined on each spatial slice by the charge density and spatial boundary data. Its instantaneous Poisson form does not permit instantaneous signaling. In Coulomb gauge, $\mathbf A=\mathbf A_{\rm T}$ is transverse, meaning $\boldsymbol\nabla\mathbin{\cdot}\mathbf A_{\rm T}=0$; Ampère–Maxwell evolution gives

$$(\partial_t^2-\nabla^2)\mathbf A_{\rm T} =\mathbf J-\boldsymbol\nabla\partial_t\Phi \equiv\mathbf J_{\rm T}.$$

The source on the right is transverse because charge conservation and $\nabla^2\Phi=-\rho$ imply $\boldsymbol\nabla\mathbin{\cdot}\mathbf J_{\rm T}=0$. It depends nonlocally on the same instantaneous Poisson solution. Therefore neither the scalar potential nor the transverse-potential contribution can be interpreted as an independent local signal. In the measurable field $\mathbf E=-\boldsymbol\nabla\Phi-\partial_t\mathbf A_{\rm T}$, the apparently instantaneous changes cancel outside the causal domain for compatible retarded data. Maxwell evolution, or the equivalent Lorenz-gauge retarded solution, fixes that causal domain. Jackson 2002, §II, Eq. (2.7), and §V.A, Eqs. (5.1)–(5.3), preprint pp. 4 and 11–12, PDF derives this transverse source and explicitly checks the cancellation; his Gaussian factors are translated here to rationalized units with $c=1$.

Gauss’s law and the physical degrees of freedom

The action

$$S[A]=\int d^4x\left( -\frac14F_{\mu\nu}F^{\mu\nu}+J^\mu A_\mu \right)$$

contains no independent time derivative of $A_0$. Varying $A_0$ imposes Gauss’s law rather than evolving a fourth radiative degree of freedom. The spatial potential evolves subject to that constraint; Tong 2015, §5.5.3, printed pp. 125–126, PDF gives the action variation, while Tong 2006–2007, §6.2, printed pp. 127–128, PDF makes the nondynamical role of $A_0$ explicit.

Define the Gauss residual

$$G=\boldsymbol\nabla\mathbin{\cdot}\mathbf E-\rho.$$

Ampère–Maxwell evolution and current conservation give

$$\partial_tG =-\boldsymbol\nabla\mathbin{\cdot}\mathbf J-\partial_t\rho=0.$$

Gauss’s law is therefore a constraint on initial data that exact evolution preserves. It is not an equation that can be discarded after a gauge choice.

The degree count is now transparent. The potential begins with four components. The $A_0$ equation is a constraint rather than an independent evolution equation, and gauge freedom removes one longitudinal description from the remaining spatial potential. A source-free electromagnetic wave consequently has two transverse polarization amplitudes.

Common pitfalls

Calling every gauge transformation physically empty. The allowed transformations depend on topology and boundary conditions. In compact $U(1)$ theory, holonomy—not the unexponentiated loop integral—is the globally invariant charged-probe quantity.

Confusing gauge fixing with symmetry breaking. A gauge condition selects a representative. It does not change $F$, create a mass, or alter a force measurement.

Using the reduced wave equation without its constraint. The equation $\Box A^\mu=-J^\mu$ is equivalent to Maxwell’s equation only with Lorenz gauge and compatible data.

Exercises

Two potentials for one magnetic field

For a uniform field $\mathbf B=B\widehat{\mathbf z}$, compare

$$\mathbf A_1=\frac{B}{2}(-y,x,0), \qquad \mathbf A_2=(0,Bx,0).$$

Show that they are gauge-related.

Solution

Both curls equal $B\widehat{\mathbf z}$. Their difference is

$$\mathbf A_2-\mathbf A_1 =\left(\frac{By}{2},\frac{Bx}{2},0\right) =\boldsymbol\nabla\left(\frac{Bxy}{2}\right).$$

Thus $\mathbf A_2=\mathbf A_1+\boldsymbol\nabla\chi$ with $\chi=Bxy/2$. Because $\chi$ is time independent, the scalar potential is unchanged. This $\chi$ is globally single-valued on ordinary Cartesian space; on a region with nontrivial loops, that global check cannot be skipped.

Counting plane-wave polarizations

Let a vacuum plane wave have $A^\mu=\varepsilon^\mu e^{ik_\alpha x^\alpha}$ with $k^\mu=(\omega,0,0,\omega)$ and $k^2=0$. Use Lorenz gauge and residual gauge freedom to identify its independent amplitudes.

Solution

Lorenz gauge gives

$$k_\mu\varepsilon^\mu =-\omega\varepsilon^0+\omega\varepsilon^3=0,$$

so $\varepsilon^3=\varepsilon^0$. A residual plane-wave gauge transformation shifts $\varepsilon^\mu\mapsto\varepsilon^\mu+\alpha k^\mu$. Choosing $\alpha=-\varepsilon^0/\omega$ sets both $\varepsilon^0$ and $\varepsilon^3$ to zero. The independent components are therefore

$$\varepsilon^1, \qquad \varepsilon^2,$$

the two transverse polarization amplitudes. The count used both the Lorenz condition and residual gauge freedom.

Electromagnetism with Differential Forms now expresses the local equations, conserved current, and global obstruction in a coordinate-free language.

References

• Jackson, John D. “From Lorenz to Coulomb and Other Explicit Gauge Transformations.” American Journal of Physics 70, 917–928 (2002). DOI 10.1119/1.1491265. Open PDF, arXiv:physics/0204034.
• David Tong, Electromagnetism (PDF), §§5.3.1 and 5.5.3, printed pp. 105–106 and 125–126, University of Cambridge Mathematical Tripos lecture notes, Lent Term 2015.
• David Tong, Gauge Theory (PDF), §§1.1.1–1.1.2, printed pp. 4–8, University of Cambridge lecture notes, 2018.
• David Tong, Quantum Field Theory (PDF), §§6.1.1–6.2, printed pp. 125–128, University of Cambridge Part III Mathematical Tripos lecture notes, Michaelmas Terms 2006 and 2007.