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 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 J^\mu takes a particular superpotential form J^\mu=W^\mu + d_\nu
U^{\nu\mu} where the first term  W^\mu vanishes on solutions of the Euler–Lagrange equations and the second one is a boundary term, where  U^{\nu\mu} 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[edit]

Notes[edit]

  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[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.