Related papers: Positive Kernels and Quantization
The paper is devoted to integral quantization, a procedure based on operator-valued measure and resolution of the identity. We insist on covariance properties in the important case where group representation theory is involved. We also…
We consider questions related to quantizing complex valued functions defined on a locally compact topological group. In the case of bounded functions, we generalize R. Werner's approach to prove the characterization of the associated normal…
For the real, continuous, isotropic and positive definite kernels on a product of spheres, one may consider not only its usual strict positive definiteness but also strict positive definiteness restrict to the points of the product that…
This paper describes the concepts of Universal/ Integrally Strictly Positive Definite/ $C_{0}$-Universal for the Gaussian kernel on a Hilbert space. As a consequence we obtain a similar characterization for an important family of kernels…
We propose a new formalism of quantum subsystems which allows to unify the existing and new methods of reduced description of quantum systems. The main mathematical ingredients are completely positive maps and correlation functions. In this…
We present an explicit characterization for the real, continuous, isotropic and strictly positive definite kernels on a product of compact two-point homogeneous spaces, in the cases in which at least one of the spaces is a sphere of…
For two continuous and isotropic positive definite kernels on the same compact two-point homogeneous space, we determine necessary and sufficient conditions in order that their product be strictly positive definite. We also provide a…
In this review the foundations of Geometric Quantization are explained and discussed. In particular, we want to clarify the mathematical aspects related to the geometrical structures involved in this theory: complex line bundles, hermitian…
I present a method of quantization using cohomology groups extended via coefficient groups of different types. This is possible according to the Universal Coefficient Theorem (UCT). I also show that by using this method new features of…
A fully geometric procedure of quantization that utilizes a natural and necessary metric on phase space is reviewed and briefly related to the goals of the program of geometric quantization.
Quantization with coherent states allows to " quantize " any space X of parameters. In the case where X is a phase space, this leads to the usual quantum mechanics. But the procedure is much more general, and does not require a symplectic,…
We study a positive reproducing kernel for holomorphic functions on a domain in a complex space. The technique is based on an idea of L. Hua. Applications are provided. These ideas were developed in another context (quantization of…
We supply a Fourier characterization for the real, continuous, isotropic and strictly positive definite kernels on a product of circles.
A discussion of the meaning of a physical concept cannot be separated from discussion of the conditions for its ideal measurement. We assert that quantization is no more than the invocation of the quantum of action in the explanation of…
Quantum kernel methods offer significant theoretical benefits by rendering classically inseparable features separable in quantum space. Yet, the practical application of Quantum Machine Learning (QML), currently constrained by the…
This work aims to prove that the classical Gaussian kernel, when defined on a non-Euclidean symmetric space, is never positive-definite for any choice of parameter. To achieve this goal, the paper develops new geometric and analytical…
An affine quantization approach leads to a genuine quantum theory of general relativity by extracting insights from a short list of increasingly more complex, soluble, perturbably nonrenormalizable models.
A class of negative definite kernels is defined in terms of measure spaces. Using this concept, property (T) for a countable group $\G$ is characterized in terms of measure preserving actions of $\G$, as follows. If a set $S$ is translated…
The affine coherent states quantization is a promising integral quantization of Hamiltonian systems when the phase space includes at least one conjugate pair of variables which takes values from a half-plane. Such a situation is common for…
Let $G$ be the complex general linear group and $g$ its Lie algebra equipped with a factorizable Lie bialgebra structure; let $U_h$ be the corresponding quantum group. We construct explicit $U_h$-equivariant quantization of Poisson orbit…