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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]]

Latest revision as of 07:19, 20 August 2013

Template:More footnotes 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 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.[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.

(i) Being Lagrangian symmetries, gauge symmetries of a Lagrangian satisfy first Noether's theorem, but the corresponding conserved current Jμ takes a particular superpotential form Jμ=Wμ+dνUνμ where the first term Wμ vanishes on solutions of the Euler–Lagrange equations and the second one is a boundary term, where Uνμ is called a superpotential.[4]

(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.[5]

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.[6]

See also

Notes

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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; arXiv: hep-th/9412228.
  • Giachetta, G. (2008), Mangiarotti, L., Sardanashvily, G., On the notion of gauge symmetries of generic Lagrangian field theory, J. Math. Phys. 50 (2009) 012903; arXiv: 0807.3003.
  • Giachetta, G. (2009), Mangiarotti, L., Sardanashvily, G., Advanced Classical Field Theory (World Scientific, 2009) ISBN 978-981-2838-95-7.
  1. ↑ Giachetta (2008)
  2. ↑ Giachetta (2009)
  3. ↑ Daniel (1980), Eguchi (1980), Marathe (1992), Giachetta (2009)
  4. ↑ Gotay (1992), Fatibene (1994)
  5. ↑ Gomis (1995), Giachetta (2009)
  6. ↑ Gomis (1995)
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