Gauge Groups, Transformations, Symmetry  

In General > s.a. constraints; gauge choice; symmetries [convention and objectivity].
* 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 symmetries are typically interpreted as redundancies in our description of a physical system, needed in order to make Lorentz invariance explicit when working with fields of spin 1 or higher; Gauge freedom shows up as canonical transformations generated by constraints; Many approaches to quantizing a field theory require gauge fixing; In condensed matter physics, gauge symmetries other than the U(1) of electromagnetism are of an emergent nature.
@ General references: Earman PhSc(02)sep [constrained Hamiltonian formalism]; Martin PhSc(02)sep [meaning is heuristic]; Brading & Brown BJPS(04) [observability]; Belot GRG(08); Guay SHPMP(08); Giachetta et al JMP(09)-a0807 [in Lagrangian field theories]; Jizba & Pons JPA(10)-a0905 [constants of motion as constraints]; Zaanen & Beekman AP(12) [emergence of gauge invariance, condensed matter]; Barbero et al EJP(15)-a1506 [simple mechanical systems as examples]; Berche et al AJP(16)aug-a1606 [and conserved quantities]; François PhSc(19)-a1801 [artificial vs substantial].
@ Types of theories: Leclerc CQG(07)gq; Arrighi et al a2004 [discrete, in terms of cellular automata]; > s.a. conformal transformations in physics.
@ And canonical transformations: Suzuki & Sales ht/05; Silagadze a1409.

For Gauge Theories > s.a. conservation laws [currents, variational principles]; 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 (Da = ∂a − i eAai Ti, with Tia basis for \(\cal G\))

A \(\mapsto\) Ad(g−1) A + g−1dg ,   or   Aai \(\mapsto\) D(g(x)) Aai D(g(x))−1 + (i/e) D(g(x)) ∂a D(g(x))−1,

Da \(\mapsto\) g(x) Da g−1(x) ,   F \(\mapsto\) Ad(g−1) F ,   φ \(\mapsto\) g(x) φ .

* Dirac's approach: Gauge transformations are applied to fields at a given time, 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 SHPMP(09)-a0904-proc [Leon Rosenfeld as precursor]; Lorcé PRD(13)-a1302 [gauge-covariant canonical formalism]; Solomon a1306 [second quantization and gauge invariance]; Gomes & Riello a2007 [large transformations and QCD θ-sectors].
@ Origin, gauge symmetry as emergent: 't Hooft AIP(07)-a0707; Donoghue et al proc(10)-a1007 [and violations]; Bjorken a1008-conf [vacuum condensate and QED]; Freund a1008 [extension of Verlinde's entropic gravity proposal]; Zaanen & Beekman AP(12)-a1108; Kirillov et al PLB(12)-a1205; Chkareuli PLB(13) [from spontaneously broken supersymmetry]; Levin & Wen PRB(05), RMP(05)cm/04 [gauge bosons and fermions from "string-net condensation" in condensed-matter theory]; Canarutto IJGMP(14)-a1404-conf [from the geometry of Weyl spinors]; Arias et al PRB(15)-a1511 [elastic deformations in graphene]; Urrutia a1607-conf [from Nambu models]; Wetterich NPB(17)-a1608 [from decoupling]; Barceló et al JHEP(16)-a1608 [systematic study]; Sachdev RPP(18)-a1801; Galitski et al PT(19)jan [artificial gauge fields with ultracold atoms]; Balachandran et al JPA(20)-a1906 [entropy production and quantum operations]; Barceló et al a2101.
@ Meaning, conceptual: de Souza ht/98, ht/99, ht/99 [discrete]; Guttmann & Lyre phy/00 [physics vs math]; Gubarev et al PRL(01)hp/00 [of A2]; Lyre PhSc(01)qp-conf; Healey PhSc(01)dec [reality of A]; Belot SHPMP(03); Pons SHPMP(05) [Dirac's analysis and dynamics]; Solomon PS(07)-a0706, a0708 [quantum field theory, non-gauge-invariance]; Sánchez a0803; Rovelli FP(14)-a1308 [why gauge?]; Weatherall a1411, a1505-conf; Afriat a1706; Nguyen et al a1712 [need for surplus structure]; Schwichtenberg a1901 [nature of gauge symmetries]; Gomes a1910 [holism]; Rovelli a2009-proc [gauge invariance and relations between subsystems].
@ Generalized: Gastmans & Wu PRD(98) [point splitting]; Lahiri MPLA(02) [non-Abelian 2-forms]; Rossi m.DG/04 [groupoid structure]; Stoilov MPLA(08)-a0710-in [with higher-order time derivatives of the gauge parameters]; Costa et al a1806 [Lie groupoids as generalized symmetries].
@ 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.
> Specific theories: see dirac fields; gauge theory; Gauge Theory of Gravity; yang-mills gauge theory.

For Gravity
* Rem: Historically, the difficulty in an effective separation of the gauge and physical degrees of freedom has lead to various confusions about the physical significance of ideas as varied as the hole argument, coordinate singularities, gravitation waves, the problem of time and the relation between general covariance and quantization.
* 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-wd [and gravitational radiation]; Nakamura a0711-proc [and perturbations]; Pitts a0911 [artificial gauge freedom]; Gielen et al a1805 ["inessential gauge" and global properties]; Montesinos et al CQG(18) [first-order general relativity with matter fields, diffeomorphisms as a derived symmetry]; > s.a. einstein equation [symmetries].
@ Quantum theory: Mercuri & Montani gq/04-MGX [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; Relativity Principle.
> 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.

In Quantum Field Theories > s.a. lagrangian dynamics; quantum states; types of field theories.
* Gauge theories: The presence of gauge symmetries at the quantum level induces symmetries between renormalized Green's functions; These symmetries are known as Ward-Takahashi and Slavnov-Taylor identities; At the perturbative level, they can be implemented as Hopf ideals in the Connes-Kreimer renormalization Hopf algebra.
* In locally covariant quantum field theory: A theory is described as a functor from a category of spacetimes to a category of *-algebras, and the global gauge group of such a theory can be identified as the group of automorphisms of the defining functor.
@ General references: Cutler & Wald CQG(87), Wald CQG(87) [collection of spin-2 fields]; Herrmann PLA(08)-a0708 [fractional wave equations]; Banerjee et al JHEP(11)-a1012 [higher-derivative Lagrangian systems]; Fewster RVMP(13)-a1201 [locally covariant]; Rejzner a1301 [perturbative, algebraic].
@ In gauge theories: Prinz a2001 [and renormalization]; > s.a. Ward-Takahashi Identities.

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