1,585 research outputs found
Three superposition principles: currents, continuity equations and curves of measures
We establish a general superposition principle for curves of measures solving
a continuity equation on metric spaces without any smooth structure nor
underlying measure, representing them as marginals of measures concentrated on
the solutions of the associated ODE defined by some algebra of observables. We
relate this result with decomposition of acyclic normal currents in metric
spaces.
As an application, a slightly extended version of a probabilistic
representation for absolutely continuous curves in Kantorovich-Wasserstein
spaces, originally due to S. Lisini, is provided in the metric framework. This
gives a hierarchy of implications between superposition principles for curves
of measures and for metric currents
Generalized compactness in linear spaces and its applications
The class of subsets of locally convex spaces called -compact sets is
considered. This class contains all compact sets as well as several noncompact
sets widely used in applications. It is shown that many results well known for
compact sets can be generalized to -compact sets. Several examples are
considered.
The main result of the paper is a generalization to -compact convex sets
of the Vesterstrom-O'Brien theorem showing equivalence of the particular
properties of a compact convex set (s.t. openness of the mixture map, openness
of the barycenter map and of its restriction to maximal measures, continuity of
a convex hull of any continuous function, continuity of a convex hull of any
concave continuous function). It is shown that the Vesterstrom-O'Brien theorem
does not hold for pointwise -compact convex sets defined by the slight
relaxing of the -compactness condition. Applications of the obtained
results to quantum information theory are considered.Comment: 27 pages, the minor corrections have been mad
Field-Theoretic Weyl Deformation Quantization of Enlarged Poisson Algebras
-algebraic Weyl quantization is extended by allowing also degenerate
pre-symplectic forms for the Weyl relations with infinitely many degrees of
freedom, and by starting out from enlarged classical Poisson algebras. A
powerful tool is found in the construction of Poisson algebras and
non-commutative twisted Banach--algebras on the stage of measures on the not
locally compact test function space. Already within this frame strict
deformation quantization is obtained, but in terms of Banach--algebras
instead of -algebras. Fourier transformation and representation theory of
the measure Banach--algebras are combined with the theory of continuous
projective group representations to arrive at the genuine -algebraic
strict deformation quantization in the sense of Rieffel and Landsman. Weyl
quantization is recognized to depend in the first step functorially on the (in
general) infinite dimensional, pre-symplectic test function space; but in the
second step one has to select a family of representations, indexed by the
deformation parameter . The latter ambiguity is in the present
investigation connected with the choice of a folium of states, a structure,
which does not necessarily require a Hilbert space representation.Comment: This is a contribution to the Special Issue on Deformation
Quantization, published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
Function spaces from coherent continuous domains to RB-domains
In this paper, continuing the work of the first and third authors, we study the function spaces from coherent continuous domains to RB-domains. Firstly, we prove that the function spaces from coherent core compact spaces with compact open sets as a basis to bifinite domains (algebraic RB-domains) are algebraic. As an application, we give an example which shows that a function space from a coherent quasi-algebraic domain to a finite domain might not be coherent. Finally, we show that the function space from a coherent continuous domain to an RB-domain is an RB-domain. Particularly, a function space from an FS-domain (introduced by Achim Jung) to an RB-domain is an RB-domain. This addresses an old open problem of whether FS-domains are RB-domains
Geometric Modular Action and Spacetime Symmetry Groups
A condition of geometric modular action is proposed as a selection principle
for physically interesting states on general space-times. This condition is
naturally associated with transformation groups of partially ordered sets and
provides these groups with projective representations. Under suitable
additional conditions, these groups induce groups of point transformations on
these space-times, which may be interpreted as symmetry groups. The
consequences of this condition are studied in detail in application to two
concrete space-times -- four-dimensional Minkowski and three-dimensional de
Sitter spaces -- for which it is shown how this condition characterizes the
states invariant under the respective isometry group. An intriguing new
algebraic characterization of vacuum states is given. In addition, the logical
relations between the condition proposed in this paper and the condition of
modular covariance, widely used in the literature, are completely illuminated.Comment: 83 pages, AMS-TEX (format changed to US letter size
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