Covariance of a Physical Theory |

**In General ** > s.a. coordinates; Event; Hole Argument; reference frame; regularization; Relativity Principle.

* __Idea__: A physical theory is said to be covariant
with respect to a certain class of transformations if its basic equations retain their form under those
transformations; If the transformations are changes of reference frame, then covariance amounts to the
theory satifying the principle of relativity with respect to those transformations; The main examples
are Lorentz covariance and general covariance.

* __Origin__: The term comes from the covariance (and contravariance) of tensors.

@ __References__: Frewer a1611 [and objectivity].

**General Covariance**

* __Idea__:
A theory is generally covariant iff it is (a) Invariant under all changes of
coordinate system, which is similar to saying that it is diffeomorphism-invariant, or
(b) Expressed in terms of only the spacetime metric and other dynamical fields, with no background geometry.
To implement it, one usually requires that all fundamental theories be expressed in terms of spacetime tensors,
or other objects with well-defined transformation properties under spacetime coordinate trasnformations.

* __Remark__: This is not always the same as
saying that no preferred observer is selected (e.g., such a selection may be possible for cobordisms).

* __Remark__: Any theory can be reformulated
(by putting enough structure among
the "variables") so as to satisfy the definition.

@ __Background independence__:
Gryb CQG(10)-a1003 [definition]; Belot GRG(11)-a1106 [explication];
Bärenz a1207;
Anderson a1310;
Vassallo a1410-in [5D];
Pooley a1506, Cartwright & Flournoy a1512 [vs diffeomorphism invariance].

@ __Related topics__: 't Hooft pr(89) [2D, discrete
model]; Mack gq/97;
Bing gq/98 [??];
Francis gq/02 [quantum
proposal]; Lusanna & Pauri gq/03 [and
gauge]; Mekhitarian & Mkrtchian mp/04 [applications];
Colosi et al CQG(05)gq/04 [model,
info and evolution]; Treder & von Borzeszkowski FP(06)
[and spacetime structure]; Klajn & Smolić EJP(13) [invariance, covariance and observer independence]; Fatibene et al a1605 [freedom in defining physical states].

> __Online resources__: see Wikipedia page.

**In Different Theories**

@ __In general relativity__: Norton FP(89)
[Einstein's view and modern view]; Ellis
and Matravers GRG(95)
[questioning]; Zalaletdinov et al GRG(96);
Guo et al PRD(03)
[and Noether charges]; Wu & Ruan ht/03 [and
general relativity, ??]; Earman in(07) [implications for the ontology and ideology of spacetime]; Lusanna JPCS(06)gq/05 [rev];
Dieks SHPMP(06)
[vs equivalence of reference frames]; Giulini LNP(07)gq/06 [issues
+ historical]; Mashkevich gq/06 ["geometricity"];
Gao & Zhang PRD(07)gq,
Sotiriou & Liberati PRD(07)gq [relationship
with gravitational dynamics]; Pitts a0911 [artificial gauge freedom and Kretschmann
objection]; Chamorro IJTP(13)-a1106; Pitts SHPMP-a1111 [and Ogievetsky-Polubarinov spinors]; Herrera IJMPD(11)-a1111 [and the relevance of observers]; Khoury et al CQG(14)-a1305 [as an accidental or emergent symmetry].

@ __Classical field theory__: Castrillón-López & Gotay
a1008 [covariantizing theories];
Pitts SHPMP(12) [and spinors];
> s.a. types of field theories.

@ __Quantum field theory__: Brunetti et al CMP(03)mp/01 [algebraic],
mp/05 [rev];
Noldus a1102 [and causality];
Fredenhagen & Rejzner a1102-proc [and background independence];
Fewster a1105-proc [vs dynamical locality];
Verch a1105-proc [renormalization ambiguity,
and local thermal equilibrium in cosmology]; > s.a. types
of quantum field theories [diffeomorphism-invariant].

@ __Quantum gravity__: Padmanabhan MPLA(88);
Kazakov CQG(02);
Christodoulakis & Papadopoulos gq/04 [and observables];
Bärenz a1207;
Bojowald & Brahma a1507 [obstacles in lqg, example of Gowdy systems].

**Generalizations and Violations**

@ __Generalied forms__: Dąbrowski et al PRD(10)-a0912 [*k*-deformed covariance].

@ __Violations of general covariance__: Pirogov gq/06-conf
[and extra particles]; Anber et al PRD(10)-a0911 [phenomenology].

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