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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 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.

1. Being Lagrangian symmetries, gauge symmetries of a Lagrangian satisfy first Noether's theorem, but the corresponding conserved current Jμ{displaystyle J^{mu }} takes a particular superpotential form Jμ=Wμ+dνUνμ{displaystyle J^{mu }=W^{mu }+d_{nu }U^{nu mu }} where the first term Wμ{displaystyle W^{mu }} vanishes on solutions of the Euler–Lagrange equations and the second one is a boundary term, where Uνμ{displaystyle U^{nu mu }} is called a superpotential.^[4]
   


2. 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

• Lagrangian system


• Noether identities


• Gauge theory


• Gauge symmetry


• Yang–Mills theory


• Gauge group (mathematics)


• Gauge gravitation theory

Notes

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) .mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
  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.


• 
  Montesinos, Merced; Gonzalez, Diego; Celada, Mariano; Diaz, Bogar (2017). "Reformulation of the symmetries of first-order general relativity". Classical and Quantum Gravity. 34 (20): 205002. arXiv:1704.04248. Bibcode:2017CQGra..34t5002M. doi:10.1088/1361-6382/aa89f3.
  


• 
  Montesinos, Merced; Gonzalez, Diego; Celada, Mariano (2018). "The gauge symmetries of first-order general relativity with matter fields". Classical and Quantum Gravity. 35 (20): 205005. arXiv:1809.10729. Bibcode:2018CQGra..35t5005M. doi:10.1088/1361-6382/aae10d.