46,763 research outputs found
Smearing of Observables and Spectral Measures on Quantum Structures
An observable on a quantum structure is any -homomorphism of quantum
structures from the Borel -algebra of the real line into the quantum
structure which is in our case a monotone -complete effect algebras
with the Riesz Decomposition Property. We show that every observable is a
smearing of a sharp observable which takes values from a Boolean
-subalgebra of the effect algebra, and we prove that for every element
of the effect algebra there is its spectral measure
Generalization of entanglement to convex operational theories: Entanglement relative to a subspace of observables
We define what it means for a state in a convex cone of states on a space of
observables to be generalized-entangled relative to a subspace of the
observables, in a general ordered linear spaces framework for operational
theories. This extends the notion of ordinary entanglement in quantum
information theory to a much more general framework. Some important special
cases are described, in which the distinguished observables are subspaces of
the observables of a quantum system, leading to results like the identification
of generalized unentangled states with Lie-group-theoretic coherent states when
the special observables form an irreducibly represented Lie algebra. Some open
problems, including that of generalizing the semigroup of local operations with
classical communication to the convex cones setting, are discussed.Comment: 19 pages, to appear in proceedings of Quantum Structures VII, Int. J.
Theor. Phy
The Two-fold Role of Observables in Classical and Quantum Kinematics
Observables have a dual nature in both classical and quantum kinematics: they
are at the same time \emph{quantities}, allowing to separate states by means of
their numerical values, and \emph{generators of transformations}, establishing
relations between different states. In this work, we show how this two-fold
role of observables constitutes a key feature in the conceptual analysis of
classical and quantum kinematics, shedding a new light on the distinguishing
feature of the quantum at the kinematical level. We first take a look at the
algebraic description of both classical and quantum observables in terms of
Jordan-Lie algebras and show how the two algebraic structures are the precise
mathematical manifestation of the two-fold role of observables. Then, we turn
to the geometric reformulation of quantum kinematics in terms of K\"ahler
manifolds. A key achievement of this reformulation is to show that the two-fold
role of observables is the constitutive ingredient defining what an observable
is. Moreover, it points to the fact that, from the restricted point of view of
the transformational role of observables, classical and quantum kinematics
behave in exactly the same way. Finally, we present Landsman's general
framework of Poisson spaces with transition probability, which highlights with
unmatched clarity that the crucial difference between the two kinematics lies
in the way the two roles of observables are related to each other.Comment: Corrected typos; revised final arguments of section 2.2 and added a
figure at the end of this sectio
Generalized Quantum Theory: Overview and Latest Developments
The main formal structures of Generalized Quantum Theory are summarized.
Recent progress has sharpened some of the concepts, in particular the notion of
an observable, the action of an observable on states (putting more emphasis on
the role of proposition observables), and the concept of generalized
entanglement. Furthermore, the active role of the observer in the structure of
observables and the partitioning of systems is emphasized.Comment: 14 pages, update in reference
Symplectic quantization, inequivalent quantum theories, and Heisenberg's principle of uncertainty
We analyze the quantum dynamics of the non-relativistic two-dimensional
isotropic harmonic oscillator in Heisenberg's picture. Such a system is taken
as toy model to analyze some of the various quantum theories that can be built
from the application of Dirac's quantization rule to the various symplectic
structures recently reported for this classical system. It is pointed out that
that these quantum theories are inequivalent in the sense that the mean values
for the operators (observables) associated with the same physical classical
observable do not agree with each other. The inequivalence does not arise from
ambiguities in the ordering of operators but from the fact of having several
symplectic structures defined with respect to the same set of coordinates. It
is also shown that the uncertainty relations between the fundamental
observables depend on the particular quantum theory chosen. It is important to
emphasize that these (somehow paradoxical) results emerge from the combination
of two paradigms: Dirac's quantization rule and the usual Copenhagen
interpretation of quantum mechanics.Comment: 8 pages, LaTex file, no figures. Accepted for publication in Phys.
Rev.
Can chaos be observed in quantum gravity?
Full general relativity is almost certainly 'chaotic'. We argue that this
entails a notion of nonintegrability: a generic general relativistic model, at
least when coupled to cosmologically interesting matter, likely possesses
neither differentiable Dirac observables nor a reduced phase space. It follows
that the standard notion of observable has to be extended to include
non-differentiable or even discontinuous generalized observables. These cannot
carry Poisson-algebraic structures and do not admit a standard quantization;
one thus faces a quantum representation problem of gravitational observables.
This has deep consequences for a quantum theory of gravity, which we
investigate in a simple model for a system with Hamiltonian constraint that
fails to be completely integrable. We show that basing the quantization on
standard topology precludes a semiclassical limit and can even prohibit any
solutions to the quantum constraints. Our proposed solution to this problem is
to refine topology such that a complete set of Dirac observables becomes
continuous. In the toy model, it turns out that a refinement to a polymer-type
topology, as e.g. used in loop gravity, is sufficient. Basing quantization of
the toy model on this finer topology, we find a complete set of quantum Dirac
observables and a suitable semiclassical limit. This strategy is applicable to
realistic candidate theories of quantum gravity and thereby suggests a solution
to a long-standing problem which implies ramifications for the very concept of
quantization. Our work reveals a qualitatively novel facet of chaos in physics
and opens up a new avenue of research on chaos in gravity which hints at deep
insights into the structure of quantum gravity.Comment: 6 pages + references -- matches published version (clarifications
added for why GR with cosmologically interesting matter likely fails our
notion of weak-integrability
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