From Gauge Covariance to Gauge Theory

Gauge covariance of a Dirac equation in a prescribed potential does not supply equations of motion or quantum states for that potential. Electromagnetic gauge theory adds a dynamical field, constraints, and a quantization prescription. Its free radiation sector has two physical polarizations, even though a four-potential has four components. The conceptual phase-to-gauge distinction belongs to From Phase Symmetry to Gauge Theory; this page follows the additional dynamics and the constraint through a Maxwell–Dirac example.

Required background. The phase-to-gauge bridge separates the symmetry notions; Minimal Coupling fixes the signed-charge convention; Why Fields Replace Wavefunctions explains what quantization adds.

Helpful background. Gauge Covariance handles time-dependent and global gauge issues; From Spinors to Fermion Fields introduces the charged operator field.

Maxwell dynamics is an additional physical input

Use rationalized natural units $\hbar=c=1$, metric $(+---)$, and a signed charge $q$. Retain the established package

$$\begin{aligned} D_\mu&=\partial_\mu+iqA_\mu,\\ A'_\mu&=A_\mu-\partial_\mu\chi,\qquad \psi'=e^{iq\chi}\psi . \end{aligned}$$

For a prescribed $A_\mu$, these transformations relate equivalent descriptions of the same background problem. They do not tell us how $A_\mu$ responds to matter.

Choose a local Maxwell kinetic term and minimally coupled Dirac matter:

$$\mathcal L =-\frac14F_{\mu\nu}F^{\mu\nu} +\bar\psi(i\gamma^\mu D_\mu-m)\psi,$$

where $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$. The interaction is

$$\mathcal L_{\rm int}=-j^\mu A_\mu, \qquad j^\mu=q\bar\psi\gamma^\mu\psi .$$

The sign of $q$ is already in this expression. For an electron, set $q=-e$ with $e>0$.

Varying the action with respect to $A_\nu$ and integrating by parts gives

$$\delta S=\int d^4x\, \bigl(\partial_\mu F^{\mu\nu}-j^\nu\bigr)\delta A_\nu,$$

and hence

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

This is a variational field equation. Turning it into quantum electrodynamics also requires the field algebra, physical-state construction, and a definition of operator products such as the current. A classical Maxwell–Dirac system or a semiclassical mean-current model is not automatically the fully quantized theory.

Gauge invariance alone does not select this action uniquely. With a fixed massless vector field content, the usual locality, Lorentz symmetry, and leading derivative assumptions motivate the Maxwell term. An effective theory can also contain gauge-invariant terms such as

$$\frac{c_8}{\Lambda^4} \bigl(F_{\mu\nu}F^{\mu\nu}\bigr)^2,$$

with a coefficient to be determined by matching or experiment. Gauge covariance does not fix that coefficient. Higher-order operators are organized by their regime of validity, not forbidden merely because they are absent from minimal coupling.

Gauss’s law constrains the initial data

Write $A^\mu=(\Phi,\mathbf A)$ and

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

The electromagnetic and current part of the Lagrangian is

$$\mathcal L_{\rm EM+int} =\frac12(\mathbf E^2-\mathbf B^2) -\rho\Phi+\mathbf j\cdot\mathbf A .$$

There is no $\partial_t\Phi$ in this expression, so its conjugate momentum is

$$\Pi_\Phi=\frac{\partial\mathcal L} {\partial(\partial_t\Phi)}=0.$$

Variation of $\Phi$ gives $\nabla\cdot\mathbf E=\rho$, the $\nu=0$ Maxwell equation. It constrains the fields on an initial slice; it is not an independent wave equation for a fourth radiation oscillator. The remaining Maxwell equations preserve this constraint when the current obeys $\partial_t\rho+\nabla\cdot\mathbf j=0$.

For example,

$$\partial_t(\nabla\cdot\mathbf E-\rho) =-\nabla\cdot\mathbf j-\partial_t\rho=0,$$

using $\partial_t\mathbf E=\nabla\times\mathbf B-\mathbf j$. An inconsistent initial Gauss constraint is therefore not repaired just by evolving the transverse radiation equations.

The singular canonical structure is why four copies of an unconstrained positive-norm scalar oscillator do not quantize electromagnetism. One can solve constraints and work with physical coordinates, or retain a covariant enlarged description with an appropriate physical-state condition. These are different ways to handle the same constraint (Tong, 2006, sections 6.1–6.2; Beisert, 2017, section 4.2).

Two transverse polarizations in the free radiation sector

To count local propagating modes, consider source-free Maxwell theory on $\mathbb R^3$ with suitable decay at infinity, and a nonzero spatial Fourier wave vector $\mathbf k$. This excludes boundary, global, and zero-mode questions from the counting argument.

In Coulomb gauge, $\nabla\cdot\mathbf A=0$, Gauss’s law reads $-\nabla^2\Phi=0$. With the stated boundary condition, $\Phi=0$ in the free radiation sector. The remaining Fourier amplitude satisfies

$$\mathbf k\cdot\widetilde{\mathbf A}(t,\mathbf k)=0.$$

Choose two orthonormal vectors perpendicular to $\mathbf k$. Then

$$\widetilde{\mathbf A}(t,\mathbf k) =\sum_{\lambda=1}^2 Q_\lambda(t,\mathbf k)\boldsymbol\epsilon_\lambda(\mathbf k),$$

and each physical mode obeys

$$\ddot Q_\lambda+|\mathbf k|^2Q_\lambda=0.$$

Quantizing these independent transverse oscillators produces the two photon polarizations. A circular polarization basis gives helicities $+1$ and $-1$. The choice of a spatial gauge makes this count transparent without changing the Lorentz-covariant content of physical predictions.

In the presence of charge, Coulomb gauge instead gives

$$-\nabla^2\Phi=\rho.$$

The longitudinal electric field is then fixed by the charge and boundary data. It cannot be set to zero merely because the free photon has two transverse polarizations. An instantaneous scalar potential in this gauge is not by itself a statement about the propagation speed of physical disturbances; the full electromagnetic solution and conserved source must be considered together.

Charged fields and physical observables

The field $\psi$ transforms under a local gauge change, so it is not itself a local gauge-invariant observable. It remains useful as a gauge-covariant field in a gauge-fixed calculation. Local quantities such as $F_{\mu\nu}$ and suitably defined neutral current operators have a different status. Their composite products need their own ultraviolet definition.

Gauss’s law also gives charged states a long-range electromagnetic structure. A physical charged excitation cannot generally be represented as a bare local matter insertion that leaves all distant electric flux unchanged. This is relevant to the infrared qualifications of the LSZ bridge (Buchholz, 1986). The present mode count is a free-radiation statement, not a construction of the full interacting charged Hilbert space.

The next QED application uses the dynamical photon exchange amplitude to recover a familiar static interaction. That calculation provides a direct check of how an external-potential approximation emerges from a theory with a quantum target.

Exercises

A forbidden initial condition. In a source-free Fourier mode with $\mathbf k\ne0$, suppose $\widetilde{\mathbf E}=E_\parallel\widehat{\mathbf k}$. Can this describe an additional free photon polarization?

Solution

Gauss’s law requires $i\mathbf k\cdot\widetilde{\mathbf E}=0$, so $E_\parallel=0$. A nonzero longitudinal electric field would require charge or different boundary/global data. It is not a third free-radiation polarization.

Constraint preservation. Derive the time derivative of $\nabla\cdot\mathbf E-\rho$ and identify precisely which matter equation is needed.

Solution

Taking the divergence of the Ampère–Maxwell equation gives $\partial_t\nabla\cdot\mathbf E=-\nabla\cdot\mathbf j$. Subtracting $\partial_t\rho$ yields zero when the current is conserved. The constraint must first hold in the initial data.

Gauge invariance versus a unique action. In four spacetime dimensions, check the power of $\Lambda$ in the displayed quartic field-strength term.

Solution

In natural mass dimensions, $[\mathcal L]=4$, $[A_\mu]=1$, and $[F_{\mu\nu}]=2$. Thus $(F_{\mu\nu}F^{\mu\nu})^2$ has dimension $8$, and $\Lambda^{-4}$ produces a dimension-four Lagrangian. The dimensionless $c_8$ is not fixed by gauge covariance.

References

• Beisert, Niklas. Quantum Field Theory II. ETH Zurich lecture notes, spring semester 2017, section 4.2. Lecture notes. Abelian gauge-field quantization and constraints.
• Buchholz, Detlev. “Gauss’ Law and the Infraparticle Problem.” Physics Letters B 174, 331–334 (1986). doi:10.1016/0370-2693(86)91110-X. Gauss-law and charged-sector infrared qualifications.
• Tong, David. Quantum Field Theory. University of Cambridge lecture notes (2006), sections 6.1–6.3. Quantum electrodynamics. Maxwell constraints, photon polarizations, and coupling to matter.