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 (*D*_{a}
= ∂_{a} −
i *eA*_{a}^{i}
*T*_{i}, with
*T*_{i}a basis for \(\cal G\))

*A* \(\mapsto\) Ad(*g*^{−1}) *A*
+ *g*^{−1}d*g* , or
*A*_{a}^{i}
\(\mapsto\) *D*(*g*(*x*))
*A*_{a}^{i}
*D*(*g*(*x*))^{−1}
+ (i/*e*) *D*(*g*(*x*)) ∂_{a}
*D*(*g*(*x*))^{−1},

*D*_{a} \(\mapsto\) *g*(*x*)
*D*_{a} *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 *A*^{2}];
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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