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.

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