Observables in Quantum Theory

In General > s.a. observable algebras; operators [hermitian and self-adjoint]; quantum measurements; wave-function collapse.
* Idea: 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]; Contrary to the situation in classical theory, observables are not exactly the same as generators of transformations on the set of states, since those generators are skew-adjoint (anti-self-adjoint); Observables for a Jordan algebra, generators a Lie algebra.
* For theories with constraints: In addition, the commutators of the observable operators 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 $$|i\rangle$$, and construct the operator $$A:= \sum_i |i\rangle\,i\,\langle i|$$.
* Remark: Projection operators onto states of infinite energy or, with superselection rules, projection operators onto states which mix sectors, are not observable.
@ General references: de Oliveira JMP(90) [complete set]; Busch & Jaeger FP(10)-a1005 [observables as positive operator-valued measures, and unsharp reality]; Hu et al QS:MF(17)-a1601 [observables as normal operators]; Jurić a2103 [symmetric, hermitian, and self-adjoint operators].
@ Classical-quantum relation: Todorov IJTP(77) [inequivalent procedures]; Ashtekar CMP(80); Peres FP(03)qp/02 [measurements and values]; Luis PRA(03); de Groote mp/05, mp/05; Paugam JGP(11)-a1010 [histories and non-local observables]; Wouters a1404 ["classical observables"].
@ Weak observables: Parks JPA(00), JPA(03), JPA(06) [weak energy]; > s.a. contextuality.
@ 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]; Białynicki-Birula NJP(14)-a1412 [local-nonlocal, in quantum optics]; Gudder a2010 [combinations of observables and instruments]; > s.a. uncertainty relations.
@ Non-selfadjoint operators as observables: Recami et al IJMPA(10)-a0903; Roberts a1610.
@ Related topics: Ni PRA(86) [limit on measurement]; Gudder IJTP(00) [combinations]; Lanz & Vacchini IJMPA(02) [subdynamics, relevant observables]; Dubin et al JPA(02)qp [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]; Campos Venuti & Zanardi PLA(13)-a1202 [probability density for the expectation value in a random state]; Loveridge & Miyadera FP(19)-a1905 [relative time observables].
> Related topics: see pilot-wave interpretation; quantum chaos.

In Quantum Field Theory > s.a. measurement in quantum theory; discrete spacetime.
@ General references: Kuckert CMP(00) [smallest localization region]; Srikanth qp/01; Ojima & Takeori mp/06 [macroscopic manifestations]; Gambini & Porto NJP(03) [causality restrictions and covariance]; Oeckl in(12)-a1101-proc [in the general-boundary formulation].
@ Quantum gravity: Pérez & Rovelli in(11)gq/01 [n-net transition amplitudes]; Giddings et al PRD(06)ht/05 [low-energy effective theory]; Donnelly & Giddings PRD(16)-a1607 [implications of diffeomorphism invariance, non-locality].
@ Weyl algebra of quantum geometry: Fleischhack CMP(09)mp/04.
> Quantum gravity: see quantum gravity, 3D quantum gravity, canonical quantum gravity [reference matter]; quantum-gravity phenomenology.
> Other related topics: see approaches to quantum field theory; Coarse-Graining; Covariance.

Related Topics > see conservation laws; fock space; observers; Phase of a quantum state.