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In mathematics, anyLagrangian system generally admits gauge symmetries, though it may happen that they are trivial. Intheoretical physics, the notion ofgauge symmetries depending on parameter functions is a cornerstone of contemporaryfield theory.

A gauge symmetry of aLagrangian${\displaystyle L}$ is defined as a differential operator on somevector 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 inclassical field theory.^[2]Yang–Mills gauge theory andgauge gravitation theory exemplify classical field theories with gauge symmetries.^[3]

Gauge symmetries possess the following two peculiarities.

1. Being Lagrangian symmetries, gauge symmetries of aLagrangian satisfyNoether's first 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 theEuler–Lagrange equations and the second one is a boundary term, where${\displaystyle U^{\nu \mu }}$ is called a superpotential.^[4]
2. In accordance withNoether's second theorem, there is one-to-one correspondence between the gauge symmetries of aLagrangian and theNoether identities which theEuler–Lagrange operator satisfies. Consequently, gauge symmetries characterize the degeneracy of aLagrangian system.^[5]

Note that, inquantum field theory, a generating functional may fail to be invariant under gauge transformations, and gauge symmetries are replaced with theBRST symmetries, depending on ghosts and acting both on fields and ghosts.^[6]

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

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• 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.

• Symmetry
• Gauge theories

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