Observables in Physical Theory  

In Classical Theory
$ Idea: When there are no constraints, an observable is any measurable function on phase space .
* Remark: There are observables for which actually constructing a measuring apparatus is difficult or impossible.
* Linear theory: Linear observables are labelled by vectors X , and given by wX(V) = (X, V), for all V .
* 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.
@ Theory with constraints: Lusanna ht/95-in [presymplectic approach]; Hájícek 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-a0905 [generally covariant theories and gauge].

In Gravity > s.a. 3D gravity; canonical general relativity [diffeomorphisms]; parametrized theories; unimodular relativity.
* Idea: In an abstract sense, observables are as defined above for a theory with constraints, in this case diffeomorphism-invariant functions of the dynamical variables; But this may not be very useful physically, so (as emphasized by Ehlers) keep in mind that the real observables are those related to proper times and lengths, and things one can actually measure.
* How to find observables: One way to define observables is to deparametrize the theory and obtain a preferred time; But this cannot be done in the full theory, only in some sectors; Another approach is that of relational observables, proposed by Rovelli and generalized to field theory by Dittrich; Given a function f on phase space and a "clock" function T, the observable F gives the value of f when T has some given value .
@ General references: Bergmann RMP(61), in(65); Ohta & Kimura NCB(91); Rovelli CQG(91); Torre CQG(91), gq/94-in; Anderson gq/92; Efroimsky & Lazarian CQG(93); Husain in(94); Pons CQG(01)gq [and gauge]; Rovelli PRD(02)gq/01 [GPS-type]; Lusanna gq/02-in [and non-inertial observers], gq/03-in [and Einstein's hole argument]; Lusanna & Pauri GRG(06)gq/04 [Bergmann vs Dirac observables]; Giesel IJMPA(08)-in [rev]; Pons et al MPLA(09)-a0902.
@ Relational: Westman & Sonego FP(08)-a0708, AP(09)-a0711 [and symmetries, coordinates]; Girelli & Poulin PRD(08) [and deformed symmetries]; > s.a. FRW quantum cosmology.
@ Evolving constants, t-dependent invariants: Page & Wootters PRD(83); Anderson gq/05; Pons & Salisbury PRD(05)gq; Gambini et al PRD(09)-a0809 [and relational time].
@ No known for closed spacetimes: Torre PRD(93)gq.
@ Partial observables: Rovelli PRD(02)gq/01; Dittrich GRG(07)gq/04, CQG(06)gq/05.
@ Dirac eigenvalues: Landi & Rovelli PRL(97)gq/96; Abdalla et al PLB(02)gq [and symplectic form]; Dittrich & Tambornino CQG(07)gq/06 [approximation scheme].
@ Other observables: Smolin PRD(94)gq/93; De Pietri & Rovelli CQG(95)gq/94; Rovelli gq/01 [general relativity + particles]; Szabados CQG(06)gq/05 [2-surfaces].
@ Special systems: Neville CQG(93) [plane waves]; Husain PRD(94)gq, PRD(97)gq [2-Killing vector field reduction]; Damour et al PRD(00)gq/99 [2-body]; Salisbury et al GRG(08)gq/05 [Bianchi I + scalar, t-dependent invariants]; > s.a. minisuperspace; spin-foam models.
@ Related topics: Feng & Huang IJTP(97) ["Dirac condition does not apply"??]; > s.a. Event, gauge transformations; riemann tensor [invariants]; time in gravitation.

In Quantum Gravity > s.a. [quantum gravity]; 3D quantum gravity; approaches to quantum field theory; Covariance; discrete spacetime; measurement in quantum mechanics; quantum-gravity phenomenology.
@ Proposals: Pérez & Rovelli gq/01 [n-net transition amplitudes]; Giddings et al PRD(06)ht/05 [low-energy effective quantum gravity].
@ Weyl algebra of quantum geometry: Fleischhack CMP(09)mp/04.

In Quantum Theory > s.a. measurement in quantum mechanics; observable algebras; operators; pilot-wave interpretation; wave-function collapse.
$ Without constraints: An observable is any self-adjoint operator (not necessarily bounded) on the Hilbert space describing the states of a physical system, such that all of its eigenvectors are in the domain of the Hamiltonian; Otherwise, their measurement would yield unphysical states, with infinite energy or something like that [@ in Reed & Simon 75, v2; in Balachandran et al NPB(95)gq].
$ With constraints: In addition, its commutators with the constraint operators must weakly vanish.
* Complete commuting sets: There may exist such a set of observables, but the number of operators in it need not be fixed, for a given physical system; Think of a Hilbert space with a countable basis | i , and construct the operator A:= i |i i i|.
* Remark: Projection operators onto states of infinite energy or, with superselection rules, projection operators onto states which mix sectors, are not observable.
@ Complete set: de Oliveira JMP(90).
@ Classical-quantum relation: Ashtekar CMP(80); Peres FP(03)qp/02 [measurements and values]; Luis PRA(03); de Groote mp/05, mp/05.
@ In relativistic quantum theory / quantum field theory: Kuckert CMP(00) [smallest localization region]; Srikanth qp/01; Ojima & Takeori mp/06 [macroscopic manifestations]; Gambini & Porto NJP(03) [causality restrictions and covariance].
@ Weak observables: Parks JPA(00), JPA(03), JPA(06) [weak energy].
@ Multiple-time: Aharonov & Albert PRD(84), PRD(84) [and t-evolution]; Sokolovski PRA(98) [defined by histories].
@ Relationships: Correggi & Morchio AP(02) [correlations at different times]; García Díaz et al NJP(05) [local-nonlocal complementarity]; > s.a. uncertainty relations.
@ Related topics: Ni PRA(86) [limit on measurement]; Gudder IJTP(00) [combinations]; Lanz & Vacchini IJMPA(02) [subdynamics, relevant obs]; Dubin et al qp/02 [and measures, dilemma]; Zafiris IJTP(04) [categorical viewpoint]; Heinonen et al RPMP(04) [covariant, fuzzy]; de Groote mp/05 [Stone spectra]; Gary & Giddings PRD(07)ht/06 [2D, relational]; de Groote a0708 [presheaf perspective]; Recami et al a0903 [use of non-selfadjoint operators].

Related Topics > see Coarse-Graining; conservation laws; fock space; Phase; quantum chaos.

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