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. It comes from gauge theory on principal bundles whose vertical automorphisms, called the gauge transformations, are gauge symmetries of the Yang – Mills Lagrangian of gauge fields. Gauge symmetries of gravitation theory are general covariant transformations.

A gauge symmetry of a Lagrangian

*L*is defined as a differential operator on some vector bundle*E*taking its values in the linear space of (variational or exact) symmetries of*L*. Therefore, a gauge symmetry of*L*depends on sections of*E*and their partial derivatives. For instance, this is the case of gauge symmetries in classical field theory.Gauge symmetries possess the following two peculiarities.

(i) Being Lagrangian symmetries, gauge symmetries of a Lagrangian satisfy first Noether’s theorem, but the corresponding conserved current

*J*^{μ}takes a particular superpotential form*J*^{μ}=*W*^{μ}+*d*_{ν}*U*^{νμ}where the first term*W*^{μ}vanishes on solutions of the Euler – Lagrange equations and the second one is a boundary term, where*U*^{νμ}is called a superpotential.^{}(ii) In accordance with second Noether’s 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.

Note that, in quantum field theory, a generating functional 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.

References:

G.Gaichetta, L.Mangiarotti, G.Sardanashvily, Advanced Classical Field Theory (WS, 2009)

G.Gaichetta, L.Mangiarotti, G.Sardanashvily, On the notion of gauge symmetries of generic Lagrangian field theory, arXiv: 0807.3003

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