# Gauge symmetry (mathematics)

This article includes a list of general references, but it lacks sufficient corresponding inline citations. (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 symmetries depending on parameter functions is a cornerstone of contemporary field theory.

A gauge symmetry of a Lagrangian is defined as a differential operator on some vector bundle taking its values in the linear space of (variational or exact) symmetries of . Therefore, a gauge symmetry of
depends on sections of 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.

- Being Lagrangian symmetries, gauge symmetries of a Lagrangian satisfy Noether's first theorem, but the corresponding conserved current takes a particular superpotential form where the first term vanishes on solutions of the Euler–Lagrange equations and the second one is a boundary term, where is called a superpotential.
^{[4]} - 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]}

## See also[edit]

- Gauge theory (mathematics)
- Lagrangian system
- Noether identities
- Gauge theory
- Gauge symmetry
- Yang–Mills theory
- Gauge group (mathematics)
- Gauge gravitation theory

## Notes[edit]

## References[edit]

- 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.
- 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. S2CID 119268222. - 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. S2CID 53531742.