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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send feedback and suggestions to bombelli at olemiss.edu – modified 22
oct
2009