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
freedom shows up as canonical transformations generated by constraints; Many
approaches to the quantization of 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];
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 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-a1801 [artificial vs substantial].

@ __And canonical transformations__: Suzuki & Sales ht/05; Silagadze a1409.

**For Gauge Theories** > s.a. gauge [emergence of symmetries].

* __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].

@ __Conceptual__: Belot SHPMP(03);
Pons SHPMP(05) [Dirac's analysis and dynamics];
Solomon PS(07)-a0706,
a0708 [quantum field theory, non-gauge-invariance];
Schwichtenberg a1901 [nature of gauge symmetries].

@ __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.

> __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**

* __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.

**Other Settings** > s.a. lagrangian dynamics;
quantum states; types of field theories.

* __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.

@ __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 [in locally covariant quantum field theory];
Rejzner a1301
[in perturbative algebraic quantum field theory].

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