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.

main page
– abbreviations
– journals – comments
– other sites – acknowledgements

send feedback and suggestions to bombelli at olemiss.edu – modified 11 dec 2016