# Gauge symmetry (mathematics)

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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 ${\displaystyle L}$ is defined as a differential operator on some vector bundle ${\displaystyle E}$ taking its values in the linear space of (variational or exact) symmetries of ${\displaystyle L}$. Therefore, a gauge symmetry of ${\displaystyle L}$ depends on sections of ${\displaystyle E}$ 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 ${\displaystyle J^{\mu }}$ takes a particular superpotential form ${\displaystyle J^{\mu }=W^{\mu }+d_{\nu }U^{\nu \mu }}$ where the first term ${\displaystyle W^{\mu }}$ vanishes on solutions of the Euler–Lagrange equations and the second one is a boundary term, where ${\displaystyle U^{\nu \mu }}$ 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]

## Notes

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)

## 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.
• 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.