Gauge theory

A field theory whose descriptions are invariant under local transformations of internal degrees of freedom. Gauge connections define covariant derivatives and field strengths on bundles.

Gauge symmetry: A redundancy in a field theory’s description associated with transformations that can vary across spacetime. Its local form is the organizing principle behind gauge theories.
Principal bundle: A space assembled from fibers carrying a group action, locally resembling a product of base space and group. Gauge fields are geometrically formulated as connections on principal bundles.
Quantum electrodynamics: The quantum field theory of electromagnetic interactions between charged particles and photons. It is the abelian gauge theory associated with electromagnetic phase symmetry.
Global symmetry: A symmetry transformation whose parameters are constant across spacetime. Unlike gauge symmetry, it relates physical configurations rather than merely redundant descriptions.
Gauge connection: A geometric object that compares internal states at neighboring points and defines parallel transport. It lets derivatives respect local changes of field description.
Lie group: A group that is also a smooth manifold, with smoothly varying multiplication and inversion. Its elements specify the continuous transformations allowed by a gauge theory.
Quantum chromodynamics: The quantum field theory of quarks and gluons, based on the color gauge group SU(3). It shows how a non-abelian gauge theory produces the strong interaction.
General relativity: A theory of gravity in which spacetime geometry is dynamical and matter follows its curvature. Its spacetime diffeomorphism symmetry differs from internal gauge symmetry.
Covariant derivative: A derivative modified by a connection so that it transforms appropriately under local changes of basis. It couples matter fields to the gauge connection without breaking local symmetry.
Lie algebra: The tangent space at a Lie group’s identity, equipped with a bracket encoding infinitesimal group structure. Gauge potentials and infinitesimal gauge transformations take values in this algebra.

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