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].

@ __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].

**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].

@ __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].

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

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2016