Foundations of Modern Physics 11: Gauge symmetries

In mathematics, any Lagrangian system 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. It comes from gauge theory on principal bundles whose vertical automorphisms, called the gauge transformations, are gauge symmetries of the Yang – Mills Lagrangian of gauge fields. Gauge symmetries of gravitation theory are general covariant transformations.

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. For instance, this is the case of gauge symmetries in classical field theory.

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^μ takes a particular superpotential form J^μ = W^μ + d_νU^νμ where the first term W^μ vanishes on solutions of the Euler–Lagrange equations and the second one is a boundary term, where U^νμ is called a superpotential.

(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.

References:

WikipediA Gauge symmetry (mathematics)

G.Gaichetta, L.Mangiarotti, G.Sardanashvily, Advanced Classical Field Theory (WS, 2009)

G.Gaichetta, L.Mangiarotti, G.Sardanashvily, On the notion of gauge symmetries of generic Lagrangian field theory, J. Math. Phys. 50 (2009) 012903 (#)
G.Sardanashvily, Gauge conservation law in a general setting: Superpotential, Int. J. Geom. Methods Mod. Phys. 6 (2009) 1047 (#)

FMP 10: Relativity Principle

FMP 9: Gauge gravitation theory

FMP 8: Relativistic mechanics

FMP 7: Non-relativistic time-dependent mechanics 

FMP 6: Lagrangian formalism

FMP 5: Supergeometry

FMP 4: Equivalence principle

FMP 3: Classical field theory

FMP 2: The notion of an inertial frame

FMP 1: The differential calculus on modules

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My recent article: Classical Higgs fields

My recent article: G.Sardanashvily, “Classical Higgs fields”  in arXiv: 1602.03818

We consider classical gauge theory on a principal bundle P->X in a case of spontaneous symmetry breaking characterized by the reduction of a structure group G of P->X to its closed subgroup H. This reduction is ensured by the existence of global sections of the quotient bundle P/H->X treated as classical Higgs fields. Matter fields with an exact symmetry group H in such gauge theory are considered in the pairs with Higgs fields, and they are represented by sections of a composite bundle Y->P/H->X, where Y->P/H is a fiber bundle associated to a principal bundle P->P/H with a structure group H. A key point is that a composite bundle Y->X is proved to be associated to a principal G-bundle P->X. Therefore, though matter fields possess an exact symmetry group H, their gauge G-invariant theory in the presence of Higgs fields can be developed. Its gauge invariant Lagrangian factorizes through the vertical covariant differential determined by a connection on a principal H-bundle P->P/H. In a case of the Cartan decomposition of a Lie algebra of G, this connection can be expressed in terms of a connection on a principal bundle P->X, i.e., gauge potentials for a group of broken symmetries G.

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