Observables
in Quantum Theory |

**In General** > s.a. observable
algebras; operators; quantum
measurements; wave-function collapse.

$ __Theories 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].

$ __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*:= ∑_{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 a1601
[observables as normal 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]; > 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].

> __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; Phase of a quantum state.

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