Gauge symmetry

A redundancy in describing physical fields, expressed through transformations that may vary across spacetime while leaving observable predictions unchanged. In gauge theories, this redundancy constrains interactions and relates different mathematical descriptions of the same physical state.

Gauge transformation: A change in field variables that preserves the observable content of a gauge theory. It is the transformation whose local form defines gauge symmetry.
Lagrangian field theory: A framework describing fields and their dynamics through a Lagrangian density and an action. Gauge symmetry is usually stated as invariance of a field-theory action.
Electromagnetism: The physical theory of electric and magnetic fields and their interactions with electric charge. Its potential admits local phase changes when the electromagnetic field transforms with it.
Global symmetry: A transformation symmetry whose parameters are constant across spacetime. Unlike gauge symmetry, it relates physical states rather than redundant descriptions.
Gauge field: A field that mediates interactions and compensates for local changes in matter-field variables. It makes spacetime-dependent transformations compatible with invariant dynamics.
Action (physics): A quantity obtained by integrating a system's Lagrangian over time or spacetime, whose stationary values determine its dynamics. Gauge transformations are tested by whether they preserve the action.
Quantum electrodynamics: The quantum field theory of electromagnetic interactions between charged particles and photons. Its local U(1) symmetry governs the coupling of charged matter to photons.
Spontaneous symmetry breaking: A situation in which the laws of a system have a symmetry that its chosen state does not display. Its usual physical interpretation differs from local gauge redundancy, which is not a symmetry between distinct observable states.
Covariant derivative: A derivative modified by a connection so that it transforms consistently with the fields it acts on. It lets local symmetry coexist with derivatives in the equations of motion.
Field (physics): A physical quantity represented by a value at each point in space and time. Gauge transformations act on fields rather than directly on measured outcomes.

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