Related papers: Reduction of Lie-Jordan algebras: Classical
We discuss the classical and quantum reduction to the space of physical degrees of freedom of Yang--Mills theory on a circle (so that space-time is a cylinder). Although the classical reduced phase space is finite-dimensional, the quantum…
I distinguish two types of reduction within the context of quantum-classical relations, which I designate "formal" and "empirical". Formal reduction holds or fails to hold solely by virtue of the mathematical relationship between two…
We first compare the mathematical structure of quantum and classical mechanics when both are formulated in a C*-algebraic framework. By using finite von Neumann algebras, a quantum mechanical analogue of Liouville's theorem is then…
Following the analysis we have presented in a previous paper (that we refer to as [I]), we describe a Noether theorem related to symmetries, with the associated reduction procedures, for classical dynamics within the Lagrangian and the…
The Lie algebra of the classical group SU(2) is constructed from two quon algebras for which the deformation parameter is a common root of unity. This construction leads to (i) a not very well-known polar decomposition of the ladder…
Exact procedures that follow Dirac's constraint quantization of gauge theories are usually technically involved and often difficult to implement in practice. We overview an "effective" scheme for obtaining the leading order semiclassical…
We present a framework for the reduction of various geometric structures extending the classical coisotropic Poisson reduction. For this we introduce constraint manifolds and constraint vector bundles. A constraint Serre-Swan theorem is…
We describe an algebra for composing automata which includes both classical and quantum entities and their communications. We illustrate by describing in detail a quantum protocol.
This work is intended as an attempt to extend the notion of bialgebra for Lie algebras to Leibniz algebras and also, the correspondence between the Leibniz bialgebras and its dual is investigated. Moreover, the coboundary Leibniz…
In this paper we propose a ``quantum reduction procedure'' based on the reduction of algebras of differential operators on a manifold. We use these techniques to show, in a systematic way, how to relate the hydrogen atom to a family of…
In this paper we study the variety of one dimensional representations of a finite $W$-algebra attached to a classical Lie algebra, giving a precise description of the dimensions of the irreducible components. We apply this to prove a…
The quantal algebra combines classical and quantum mechanics into an abstract structurally unified structure. The structure uses two products: one symmetric and one anti-symmetric. The local structure of spacetime is contained in the…
The Dirac method is used to analyze the classical and quantum dynamics of a particle constrained on a circle. The method of Lagrange multipliers is scrutinized, in particular in relation to the quantization procedure. Ordering problems are…
Several general properties, concerning reduction algebras - rings of definition and algorithmic efficiency of the set of ordering relations - are discussed. For the reduction algebras, related to the diagonal embedding of the Lie algebra…
We use the mathematical framework of loop quantum gravity (LQG) to study the quantization of three dimensional (Riemannian) gravity with positive cosmological constant (Lambda>0). We show that the usual regularization techniques (successful…
We propose a method for integrating the right-invariant geodesic flows on Lie groups based on the use of a special canonical transformation in the cotangent bundle of the group. We also describe an original method of constructing exact…
Supmech, an algebraic scheme of mechanics integrating noncommutative symplectic geometry and noncommutative probability, subsumes quantum and classical mechanics and permits consistent treatment of interaction of quantum and classical…
On the basis of a suggestive definition of a classical extension of quantum mechanics in terms of statistical models, we prove that every such classical extension is essentially given by the so-called Misra-Bugajski reduction map. We…
We create classical (non-quantum) dynamic data structures supporting queries for recommender systems and least-squares regression that are comparable to their quantum analogues. De-quantizing such algorithms has received a flurry of…
The purpose of this contribution is to provide an introduction for a general physics audience to the recent results of Emile Grgin that unifies quantum mechanics and relativity into the same mathematical structure. This structure is the…