Maxwell Theory

Maxwell theory is the free relativistic theory of an electromagnetic gauge field. In empty $3+1$-dimensional spacetime it has two propagating photon polarizations, although a vector potential has four components. The difference is the central lesson: gauge potentials provide a redundant description constrained by Gauss’s law.

Maxwell action and gauge redundancy

For a real potential $A_\mu$, let

$$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu, \qquad \mathcal L=-\frac14F_{\mu\nu}F^{\mu\nu}.$$

The transformation $A_\mu\mapsto A_\mu+\partial_\mu\alpha$ leaves $F_{\mu\nu}$ invariant. Variation gives $\partial_\mu F^{\mu\nu}=0$, while the definition of $F$ implies $\partial_{[\lambda}F_{\mu\nu]}=0$. These are the source-free Maxwell equations.

The canonical momentum of $A_0$ vanishes; $A_0$ enforces a constraint rather than adding an oscillator. For a nonzero-momentum plane wave, a gauge condition and the remaining gauge freedom leave two transverse polarizations. They carry helicities $+1$ and $-1$. This counting is developed in Srednicki 2007, §§54–55.

A positive-energy check

Write the electric and magnetic fields in the usual spatial notation. The physical Hamiltonian density is

$$\mathcal H=\frac12(\mathbf E^2+\mathbf B^2).$$

Thus the minus sign in $-\tfrac14F^2$ produces positive energy with the inherited metric convention. The longitudinal electric field is fixed by Gauss’s law when charges are present; it is not an additional radiation polarization.

Local observables include $F_{\mu\nu}$ and the stress tensor. On a closed curve $C$, the phase $\exp(iq\oint_C A)$ probes holonomy. Such nonlocal observables can retain information that a local field strength does not fully specify on a space with nontrivial topology.

What changes when matter or boundaries are added

Pure Maxwell theory has no photon self-interaction in its quadratic action. Nonlinear optical effects in vacuum arise after charged matter is included and, at sufficiently low energy, integrated out. A Proca mass changes the constraint structure and gives a massive vector three polarizations; it cannot be appended while keeping the same gauge interpretation without additional structure.

Boundaries, compact spatial directions, charged sectors, and zero modes require further data. The two-polarization count describes local propagating modes, not every possible global degree of freedom or boundary observable.

Develop the calculation

The Free Maxwell Field and Gauge Redundancy and Physical-Mode Quantization of the Free Electromagnetic Field develop the constraints and photon Hilbert space. Continue to QED for charged matter or Yang–Mills Theory for non-Abelian self-interaction.

References

• Srednicki, Mark. Quantum Field Theory. Cambridge University Press, 2007, §§54–58. DOI.