1,285 research outputs found
Symmetrization of Berezin Star Product and Path-Integral Quantization
We propose a new star pruduct which interpolates the Berezin and Moyal
quantization. A multiple of this product is shown to reduce to a path-integral
quantization in the continuous time limit. In flat space the action becomes the
one of free bosonic strings. Relation to Kontsevich prescription is also
discussed.Comment: 11 pages, Latex2
Coordinate-Free Quantization of Second-Class Constraints
The conversion of second-class constraints into first-class constraints is
used to extend the coordinate-free path integral quantization, achieved by a
flat-space Brownian motion regularization of the coherent-state path integral
measure, to systems with second-class constraints.Comment: 21 pages, plain LaTeX, no figure
Formal Groups and -Entropies
We shall prove that the celebrated R\'enyi entropy is the first example of a
new family of infinitely many multi-parametric entropies. We shall call them
the -entropies. Each of them, under suitable hypotheses, generalizes the
celebrated entropies of Boltzmann and R\'enyi.
A crucial aspect is that every -entropy is composable [1]. This property
means that the entropy of a system which is composed of two or more independent
systems depends, in all the associated probability space, on the choice of the
two systems only. Further properties are also required, to describe the
composition process in terms of a group law.
The composability axiom, introduced as a generalization of the fourth
Shannon-Khinchin axiom (postulating additivity), is a highly non-trivial
requirement. Indeed, in the trace-form class, the Boltzmann entropy and Tsallis
entropy are the only known composable cases. However, in the non-trace form
class, the -entropies arise as new entropic functions possessing the
mathematical properties necessary for information-theoretical applications, in
both classical and quantum contexts.
From a mathematical point of view, composability is intimately related to
formal group theory of algebraic topology. The underlying group-theoretical
structure determines crucially the statistical properties of the corresponding
entropies.Comment: 20 pages, no figure
Inductive Definition and Domain Theoretic Properties of Fully Abstract
A construction of fully abstract typed models for PCF and PCF^+ (i.e., PCF +
"parallel conditional function"), respectively, is presented. It is based on
general notions of sequential computational strategies and wittingly consistent
non-deterministic strategies introduced by the author in the seventies.
Although these notions of strategies are old, the definition of the fully
abstract models is new, in that it is given level-by-level in the finite type
hierarchy. To prove full abstraction and non-dcpo domain theoretic properties
of these models, a theory of computational strategies is developed. This is
also an alternative and, in a sense, an analogue to the later game strategy
semantics approaches of Abramsky, Jagadeesan, and Malacaria; Hyland and Ong;
and Nickau. In both cases of PCF and PCF^+ there are definable universal
(surjective) functionals from numerical functions to any given type,
respectively, which also makes each of these models unique up to isomorphism.
Although such models are non-omega-complete and therefore not continuous in the
traditional terminology, they are also proved to be sequentially complete (a
weakened form of omega-completeness), "naturally" continuous (with respect to
existing directed "pointwise", or "natural" lubs) and also "naturally"
omega-algebraic and "naturally" bounded complete -- appropriate generalisation
of the ordinary notions of domain theory to the case of non-dcpos.Comment: 50 page
Matrix Bases for Star Products: a Review
We review the matrix bases for a family of noncommutative products
based on a Weyl map. These products include the Moyal product, as well as the
Wick-Voros products and other translation invariant ones. We also review the
derivation of Lie algebra type star products, with adapted matrix bases. We
discuss the uses of these matrix bases for field theory, fuzzy spaces and
emergent gravity
Quantum theta functions and Gabor frames for modulation spaces
Representations of the celebrated Heisenberg commutation relations in quantum
mechanics and their exponentiated versions form the starting point for a number
of basic constructions, both in mathematics and mathematical physics (geometric
quantization, quantum tori, classical and quantum theta functions) and signal
analysis (Gabor analysis).
In this paper we try to bridge the two communities, represented by the two
co--authors: that of noncommutative geometry and that of signal analysis. After
providing a brief comparative dictionary of the two languages, we will show
e.g. that the Janssen representation of Gabor frames with generalized Gaussians
as Gabor atoms yields in a natural way quantum theta functions, and that the
Rieffel scalar product and associativity relations underlie both the functional
equations for quantum thetas and the Fundamental Identity of Gabor analysis.Comment: 38 pages, typos corrected, MSC class change
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