Gauge fields and quantum electrodynamics

Use natural units and metric $(+---)$. Let $q$ denote the signed matter charge in the convention $D_\mu=\partial_\mu+iqA_\mu$. Choose $\psi\mapsto e^{-iq\alpha(x)}\psi$ and $A_\mu\mapsto A_\mu+\partial_\mu\alpha$. Differentiating the local phase produces a term proportional to $\partial_\mu\alpha$, which the transformed connection cancels. The covariant derivative therefore transforms with the same phase as the matter field.

The field strength $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$ stays unchanged because mixed ordinary derivatives commute. The QED action combines its quadratic term with the Dirac kinetic term containing the covariant derivative. Expanding gives an interaction $-q\bar\psi\gamma^\mu\psi A_\mu$. A book that reverses the derivative or phase convention can write a different-looking vertex sign; translating all definitions together resolves the discrepancy.

A four-component potential does not describe four independent propagating photon polarizations. Gauge freedom and constraints leave two transverse physical polarizations for a freely propagating massless photon in four-dimensional vacuum spacetime. The time component enforces a constraint in a canonical treatment; the division among components can depend on the gauge. Counting symbols before accounting for constraints gives the wrong physical state count.

Gauge fixing selects representatives so that propagators and perturbative calculations can be defined. It does not remove a physical polarization or allow a gauge-dependent intermediate field component to become a detector reading. In a covariant calculation, unphysical contributions cancel from suitable physical observables when the theory and calculation respect the gauge structure. Residual gauge freedom, boundaries, and zero modes require additional attention in a full treatment.

A practical gauge-invariance check replaces an external photon polarization by its momentum in an appropriately assembled on-shell amplitude; the result should vanish under the applicable Ward identity. Individual diagrams may fail this check until all required terms are added. This tests the full coherent amplitude and conservation structure, rather than the artistic appearance of a diagram. Loop-level identities also demand a compatible regulator and renormalization procedure.

Gauge transformations that are trivial at the boundary are normally treated as redundancy. Transformations with nontrivial boundary behavior can instead be associated with physical charges; “every gauge transformation is unphysical” is too broad without boundary qualifications. QED also has soft-photon infrared subtleties: experimentally meaningful inclusive rates sum over unresolved radiation. Our tree-level construction introduces the theory’s grammar, while precise comparison with charged-particle scattering requires those additional ingredients.