Observers and Observables in Physical Theory |

**Observers** > s.a. Covariance;
reference frames [including accelerated].

* __Types of observers__: Lagrangian (non surface-forming
observers) or Eulerian (surface-forming ones, or hypersurface-orthogonal), inertial or non-inertial.

* __Observer space__: The 'observer space' of a Lorentzian
spacetime is the space of future-timelike unit tangent vectors.

@ __Role of observers__: Klajn & Smolić EJP(13)-a1302 [comments];
Knuth CP(14)-a1310 [observer-centric physics];
Clough a1801 [3+1 description vs 4D spacetime description].

@ __Accelerated observers__: Mashhoon AdP(13)-a1211
[non-local connection between non-inertial and inertial observers];
Kolekar PRD(14) [in a thermal bath];
> s.a. observables in classical gravity [non-inertial].

@ __In general spacetimes__: Page CQG(98)gq/97 [stationary axisymmetric, maximal acceleration];
Garat JMP(05)-a1306 [Euler observers in geometrodynamics];
Dahia & Felix da Silva GRG(11)-a1004 [static];
Dupré a1403 [two symmetric tensors
observed to be equal by all observers at a specific event are necessarily equal at that event];
Chęcińska & Dragan PRA(15)-a1509
[communication between observers without a shared reference frame].

@ __Physics in observer space__: Gielen & Wise JMP(13)-a1210 [observer-space formulation of general relativity];
Gielen PRD(13)-a1301 [observer-space geometry].

@ __In quantum theory__: Konishi IJMPB(12)-a1212 [and time-reparametrization symmetry];
Ahluwalia IJMPD(17)-a1706-GRF [in quantum gravity];
Vedral a1803
[can an observer know he/she is in a superposition?].

**Observables**
> s.a. information; Observers.

$ __Idea__: When there are
no constraints, an observable for a classical theory is any measurable function
on the phase space *Γ* for a theory; In quantum theory, this leads
to operators on the Hilbert space of the theory; Since classical observables are
usually real, quantum operators are usually self-adjoint.

* __Remark__: There are observables
for which actually constructing a measuring apparatus is difficult or impossible.

* __Linear theory__: Linear
observables are labelled by vectors *X* ∈ *Γ*, and
given by *w*_{X}(*V*)
= Ω(*X*,* V*), for all *V* ∈ *Γ*.

* __Theory with constraints__:
In addition, (for Dirac observables) the Poisson brackets with the constraints must weakly vanish.

@ __General references__:
Fernández PLA(03) [perturbative];
de Groote mp/06;
Hartmann FP-a1504 [and foundations of physics];
Anderson a1505 [differential equations];
Zalamea a1711 [two-fold role of observables].

@ __Theories with constraints__:
Lusanna ht/95-conf [presymplectic approach];
Hájíček CQG(96)gq/95 [and time evolution];
Lucenti et al JPA(98) [*N* relativistic particles];
Dütsch & Fredenhagen CMP(99)ht/98 [gauge theories];
Bratchikov IJGMP(07)ht/04 [space of orbits vs gauge fixing],
JGP(06) [second-class];
Hellmann a0812
[kinematic observables, physical interpretation];
Pons et al PRD(09)-a0905 [generally covariant theories and gauge];
Quadri EPJC(10)-a1007 [non-linearly realized gauge theories]; Pitts FP-a1907 [Hamiltonian Einstein-Maxwell theory].

**Related Topics**
> see Coarse-Graining; conservation laws;
lattice theories [observable currents].

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