Free Spin-One Fields and Constraints

A four-component vector potential need not describe four physical polarizations. Proca and Maxwell theory expose the constraints, gauge redundancy, and auxiliary states hidden by simpler free-field examples, and show why the massless limit needs a specified physical question.

Enter this chapter

Begin with The Proca Field for a massive vector with a direct polarization count. Then compare it with Maxwell theory before choosing physical-mode or covariant quantization.

Topics

1. The Proca Field — Derives the massive vector equation and its three physical polarizations.
2. The Free Maxwell Field and Gauge Redundancy — Separates the electromagnetic potential from gauge-invariant field strength.
3. Massive and Massless Spin-One Polarizations — Compares massive polarization spaces with massless gauge equivalence classes.
4. Maxwell Constraints as a Worked Application — Follows Gauss’s law and the constrained Hamiltonian construction explicitly.
5. Physical-Mode Quantization of the Free Electromagnetic Field — Quantizes the transverse modes after reduction.
6. Covariant Free-Photon Quantization and Propagator — Uses an auxiliary covariant description and identifies its physical quotient.

How the ideas fit together

For nonzero mass in four dimensions, the Proca equation imposes a subsidiary condition leaving three polarizations. Maxwell theory instead has a constraint and a gauge identification, leaving two radiative polarizations. The count therefore changes through a change of structure, not by simply discarding an oscillator. See Schwartz 2014, §§ 8.1–8.4.

Covariant photon quantization retains extra components to preserve a convenient local description. Positivity applies to physical states after the constraint and quotient, not to each auxiliary component. Gauge-dependent propagators can differ while giving the same result for an appropriately conserved source.

Review the chapter

Why does a longitudinal propagator term proportional to $k_\mu k_\nu$ not contribute between conserved external currents?

Solution

The contraction is proportional to $(k\cdot J)(k\cdot J')$, which vanishes when both currents are conserved. This checks that particular observable. It does not prove that every component of the propagator has a smooth massless limit or is gauge invariant.

Where to go next

Continue to Gauge Structure, Global Form, and Observables for the wider distinction between redundant descriptions and physical operators.

References

• Matthew D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014. DOI.