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];
Anderson a1907 [higher Lie theory and problem of time];
Anderson a1911 [theory].

@ __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 AP(17)-a1605 [freedom in defining physical states].

> __Online resources__:
see Wikipedia page.

**In Different Theories** > s.a. reference
frames [with quantum reference frames]; relativistic quantum mechanics.

@ __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(12)-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];
Deser GRG(19) [covariantizing matter fields].

@ __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 PRD(15)-a1507 [obstacles in lqg, example of Gowdy systems];
Finn et al PRD(20)-a1910 [frame covariance and curved field space];
Chishtie a2102 [loss of general covariance].

**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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