Gauge symmetry (mathematics)

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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 ${\displaystyle L}$ is defined as a differential operator on some vector bundle ${\displaystyle E}$ taking its values in the linear space of (variational or exact) symmetries of ${\displaystyle L}$. Therefore, a gauge symmetry of ${\displaystyle L}$ depends on sections of ${\displaystyle E}$ 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.

1. Being Lagrangian symmetries, gauge symmetries of a Lagrangian satisfy Noether's first theorem, but the corresponding conserved current ${\displaystyle J^{\mu }}$ takes a particular superpotential form ${\displaystyle J^{\mu }=W^{\mu }+d_{\nu }U^{\nu \mu }}$ where the first term ${\displaystyle W^{\mu }}$ vanishes on solutions of the Euler–Lagrange equations and the second one is a boundary term, where ${\displaystyle U^{\nu \mu }}$ is called a superpotential.^[4]
2. In accordance with Noether's second theorem, there is one-to-one correspondence between the gauge symmetries of a Lagrangian and the Noether identities which the Euler–Lagrange operator satisfies. Consequently, gauge symmetries characterize the degeneracy of a Lagrangian system.^[5]

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]