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{{More footnotes|date=October 2009}}
In mathematics, any [[Lagrangian system]] generally admits gauge
symmetries, though it may happen that they are trivial. In
[[theoretical physics]], the notion of [[gauge symmetry|gauge
symmetries]] depending on parameter functions is a cornerstone of
contemporary [[field theory (physics)|field theory]]. 

A gauge symmetry of a [[Lagrangian system|Lagrangian]] <math>L</math> is defined as a differential operator on some [[fiber bundle|vector bundle]] <math>E</math> taking its values in the linear space of (variational or exact) symmetries of <math>L</math>. Therefore, a gauge symmetry of <math>L</math>
depends on sections of <math>E</math> and their partial derivatives.<ref>Giachetta (2008)</ref> For instance, this is the case of gauge symmetries in [[classical field theory]].<ref>Giachetta (2009)</ref> [[Yang–Mills theory|Yang–Mills gauge theory]] and [[gauge gravitation theory]] exemplify classical field theories with gauge symmetries.<ref>Daniel (1980), Eguchi (1980), Marathe (1992), Giachetta (2009)</ref> Gauge symmetries possess the following two peculiarities.

(i) Being Lagrangian symmetries, gauge symmetries of a [[Lagrangian system|Lagrangian]]
satisfy [[Noether's theorem|first Noether's theorem]], but the
corresponding conserved current <math>J^\mu</math> takes a
particular superpotential form  <math>J^\mu=W^\mu + d_\nu
U^{\nu\mu}</math> where the first term <math> W^\mu</math>
vanishes on solutions of the [[Lagrangian system|Euler–Lagrange equations]] and the
second one is a boundary term, where <math> U^{\nu\mu}</math> is
called a superpotential.<ref>Gotay (1992), Fatibene (1994)</ref>

(ii) In accordance with [[Noether's second theorem|second
Noether's theorem]], there is one-to-one correspondence between
the gauge symmetries of a [[Lagrangian system|Lagrangian]] and the [[Noether identities]] which the [[Lagrangian system|Euler–Lagrange operator]] satisfies.
Consequently, gauge symmetries characterize the degeneracy of a
[[Lagrangian system]].<ref>Gomis (1995), Giachetta (2009)</ref>

Note that, in [[quantum field theory]], a generating functional
fail to be invariant under gauge transformations, and gauge
symmetries are replaced with the [[Faddeev–Popov ghost|BRST
symmetries]], depending on ghosts and acting both on fields and
ghosts.<ref>Gomis (1995)</ref>

==See also==
*[[Lagrangian system]]
*[[Noether identities]]
*[[Gauge theory]]
*[[Gauge symmetry]]
*[[Yang–Mills theory]]
*[[Gauge group (mathematics)]]
*[[Gauge gravitation theory]]

==Notes==
{{reflist}}

==References==
* Daniel, M., Viallet, C., The geometric setting of gauge symmetries of the Yang–Mills type, Rev. Mod. Phys. '''52''' (1980) 175.
* Eguchi, T., Gilkey, P., Hanson, A., Gravitation, gauge theories and differential geometry, Phys. Rep. '''66''' (1980) 213.
* Gotay, M., Marsden, J.,  Stress-energy-momentum tensors and the Belinfante–Rosenfeld formula,  Contemp. Math. '''132''' (1992) 367.
* Marathe, K., Martucci, G., The Mathematical Foundation of Gauge Theories (North Holland, 1992)  ISBN 0-444-89708-9.
* Fatibene, L., Ferraris, M., Francaviglia, M., Noether formalism for conserved quantities in classical gauge field theories, J. Math. Phys. '''35''' (1994) 1644.
* Gomis, J., Paris, J., Samuel, S., Antibracket, antifields and gauge theory quantization, Phys. Rep. '''295''' (1995) 1; [http://xxx.lanl.gov/abs/hep-th/9412228 arXiv: hep-th/9412228].
* Giachetta, G. (2008), Mangiarotti, L., [[Gennadi Sardanashvily|Sardanashvily, G.]], On the notion of gauge symmetries of generic Lagrangian field theory,  J. Math. Phys. '''50''' (2009) 012903; [http://xxx.lanl.gov/abs/0807.3003 arXiv: 0807.3003].
* Giachetta, G. (2009), Mangiarotti, L., [[Gennadi Sardanashvily|Sardanashvily, G.]], Advanced Classical Field Theory (World Scientific, 2009) ISBN 978-981-2838-95-7.

[[Category:Symmetry]]
[[Category:Gauge theories]]