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 a1606 [and conserved quantities].

@ __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].

@ __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].

@ __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]; > 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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