In mathematics, any Lagrangian system generally admits gauge symmetries, though it may happen that they are trivial. In theoretical physics, the notion of gauge symmetries depending on parameter functions is a cornerstone of contemporary field theory.
A gauge symmetry of a Lagrangian L {\displaystyle L} is defined as a differential operator on some vector bundle E {\displaystyle E} taking its values in the linear space of (variational or exact) symmetries of L {\displaystyle L} . Therefore, a gauge symmetry of L {\displaystyle L} depends on sections of E {\displaystyle E} and their partial derivatives.[1] For instance, this is the case of gauge symmetries in classical field theory.[2] Yang–Mills gauge theory and gauge gravitation theory exemplify classical field theories with gauge symmetries.[3]
Gauge symmetries possess the following two peculiarities.
Note that, in quantum field theory, a generating functional may fail to be invariant under gauge transformations, and gauge symmetries are replaced with the BRST symmetries, depending on ghosts and acting both on fields and ghosts.[6]
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