Gauge Groups, Transformations, Symmetry

In General > s.a. constraints; gauge choice; symmetry.
* Gauge transformation: A map between fields (or solutions of the dynamical equations) in a field theory under which the values of all physical observables are invariant; The concept applies to a field but is defined by a property of the theory.
* And physical theories: Gauge freedom shows up as canonical transformations generated by constraints; Many approaches to the quantization of a field theory require gauge fixing.
@ References: Earman PhSc(02)sep [constrained Hamiltonian formalism]; Martin PhSc(02)sep [meaning is heuristic]; Brading & Brown BJPS(04) [observability]; Suzuki & Sales ht/05 [and canonical transformations]; Leclerc CQG(07)gq [and types of theories]; Belot GRG(08); Guay SHPMP(08); Giachetta et al JMP(09)-a0807 [in Lagrangian field theories]; Jizba & Pons a0905 [constants of motion as constraints].
> Related topics: see symmetries [convention and objectivity].

For Gauge Theories
* On Lie-valued 1-forms: Local gauge transformations ("of the second kind") are fiber-preserving diffeomorphisms in the principal fiber bundle of a gauge theory, which can be written as G-valued functions g(x) on M; Under these, the fields transform as (D_a = _a– i eA_aⁱ T_i, with T_i a basis for )

A Ad(g^–1) A + g^–1dg ,   or   A_aⁱ D(g(x)) A_aⁱ D(g(x))^–1 + (i/e) D(g(x)) _a D(g(x))^–1,

D_a g(x) D_a g^–1(x) ,   F Ad(g^–1) F ,     g(x) .

* Dirac's approach: Gauge transformations are applied to fields at a given t, as opposed to spacetime fields; The dynamics is modified by substituting the extended Hamiltonian (including all first-class constraints) for the total Hamiltonian (including only the primary first-class constraints).
@ General references: Cirelli & Manià JMP(86); Abbati et al JMP(86) [action on connections]; Giulini MPLA(95)gq/94 [large transformations]; Wockel mp/05 [on manifolds with corners]; Salisbury a0904-in [Leon Rosenfeld as precursor].
@ Conceptual: Belot SHPMP(03); Pons SHPMP(05) [Dirac's analysis and dynamics]; Solomon PS(07)-a0706, a0708 [quantum field theory, non-gauge-invariance].
@ Generalized: Gastmans & Wu PRD(98) [point splitting]; Lahiri MPLA(02) [non-Abelian 2-forms]; Rossi m.DG/04 [groupoid structure]; Stoilov a0710-in [with higher-order time derivatives of the gauge parameters].
@ Maxwell theory: Dirac PRS(51), PRS(52), PRS(54) [and electrons]; Hojman AP(77), Gambini & Hojman AP(77) [true degrees of freedom, and quantization]; Potter a0903; > s.a. electromagnetism.
> Related topics: see conservation laws [currents, variational principles].
> Specific theories: see dirac fields; gauge theory; Gauge Theory of Gravity; yang-mills gauge theory.

For Gravity
* Classical: Issues are the exact relationship with diffeomorphisms and how to implement them in a canonical theory.
* And perturbations: In perturbative gravity there are two types of gauge transformations, which can be thought of as corresponding, respectively, to the coordinate system used (or a diffeomorphism), and the choice of background that the perturbed metric is a perturbation of.
@ Classical gauge and symmetries: Bergmann & Komar IJTP(72) [coordinate group symmetries]; Pons et al PRD(97)gq/96; Hall G&C(96) [survey]; Lusanna & Pauri GRG(06)gq/04, GRG(06)gq/04 [and observables]; Garfinkle AJP(06)mar-gq/05, Corda a0706 [and gravitational radiation]; Nakamura a0711-in [and perturbations]; > s.a. einstein equation [symmetries].
@ Quantum theory: Mercuri & Montani gq/04-in [need to fix before quantizing]; Leclerc gq/07 [need mixed momentum-coordinate representation for gauge invariance].
> Related topics: see embedding; Event; observables; perturbations in general relativity.
> Specific choices: see coordinate systems; gauge choices [including linearized and quantum gravity].
> Specific types of theories and aproaches: see canonical general relativity; finsler geometry; initial-value form; numerical general relativity.

Other Settings > s.a. lagrangian dynamics; quantum states; types of field theories.
@ References: Cutler & Wald CQG(87), Wald CQG(87) [collection of spin-2 fields]; Herrmann PLA(08)-a0708 [fractional wave equations].

main page – abbreviations – journals – comments – other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 3 jun 2009