Related papers: Reduction of Lie-Jordan algebras: Classical
Quantum Reduced Loop Gravity is a promising framework for linking Loop Quantum Gravity and the effective semiclassical dynamics of Loop Quantum Cosmology. We review its basic achievements and its main perspectives, outlining how it provides…
The cosmological singularities of the Bianchi I universe are analyzed in the setting of loop geometry underlying the loop quantum cosmology. We solve the Hamiltonian constraint of the theory and find the Lie algebra of elementary…
We obtain a condensed reconstruction of algebraic quantum theory, emphasizing its foundational aspects and algebraic structure. We obtain the $W^*$-algebra structure from elementary assumptions about observers and how they can observe…
Our aim in this paper is to present a new type of the modular space. This space contains the classical modular space. There are some mappings that do not have contractive condition in the usual modular space but become contraction in this…
This is a self-contained introduction to quantum Riemannian geometry based on quantum groups as frame groups, and its proposed role in quantum gravity. Much of the article is about the generalisation of classical Riemannian geometry that…
We propose a method of constructing completely integrable systems based on reduction of bihamiltonian structures. More precisely, we give an easily checkable necessary and sufficient conditions for the micro-kroneckerity of the reduction…
Dirac's method of classical analogy is employed to incorporate quantum degrees of freedom into modern nonequilibrium thermodynamics. The proposed formulation of dissipative quantum mechanics builds entirely upon the geometric structures…
A novel reduction procedure for covariant classical field theories, reflecting the generalized symplectic reduction theory of Hamiltonian systems, is presented. The departure point of this reduction procedure consists in the choice of a…
An analogue of geometric quantization of Poisson algebras obtained by algebraic reduction of symmetries is developed. Interpretation of the obtained results and their application to the problem of commutativity of quantization and reduction…
Quadratic algebras are generalizations of Lie algebras; they include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical…
Lie algebra contractions on the classical Drinfel'd Double of a given Lie bialgebra are introduced and compared to the usual Lie bialgebra contraction theory. The connection between both approaches turns out to be intimately linked to…
Quantum advantage in computation refers to the existence of computational tasks that can be performed efficiently on a quantum computer but cannot be efficiently simulated on any classical computer. Identifying the precise boundary of…
We prove the existence of a linear isometric correspondence between the Banach space of all symmetric orthogonal forms on a JB$^*$-algebra $\mathcal{J}$ and the Banach space of all purely Jordan generalized derivations from $\mathcal{J}$…
This paper develops the notion of implicit Lagrangian systems on Lie algebroids and a Hamilton--Jacobi theory for this type of system. The Lie algebroid framework provides a natural generalization of classical tangent bundle geometry. We…
Classical algebraic structures require exact satisfaction of their defining axioms. We propose similarity algebra, a framework extending algebraic and Lie structures to settings where operations satisfy quantitative bounds up to a tolerance…
Quantum physics has revealed many interesting formal properties associated with the algebra of two operators, A and B, satisfying the partial commutation relation AB-BA=1. This study surveys the relationships between classical combinatorial…
A noncommutative-geometric generalization of the classical concept of spinor structure is presented. This is done in the framework of the formalism of quantum principal bundles. In particular, analogs of the Dirac operator and the Laplacian…
GUP is a phenomenological model aimed for a description of a minimal length in quantum and classical systems. However, the analysis of problems in classical physics is usually approached preferring a different formalism than the one used…
This survey article is concerned with the modeling of the kinematical structure of quantum systems in an algebraic framework which eliminates certain conceptual and computational difficulties of the conventional approaches. Relying on the…
This paper serves as a preparation of work that focuses on extracting cosmological sectors from Loop Quantum Gravity. We start with studying the extraction of subsystems from classical systems. A classical Hamiltonian system can be reduced…