Discreteness of space from GUP II: Relativistic wave equations

Various theories of Quantum Gravity predict modifications of the Heisenberg Uncertainty Principle near the Planck scale to a so-called Generalized Uncertainty Principle (GUP). In some recent papers, we showed that the GUP gives rise to corrections to the Schrödinger equation, which in turn affect all quantum mechanical Hamiltonians. In particular, by applying it to a particle in a one-dimensional box, we showed that the box length must be quantized in terms of a fundamental length (which could be the Planck length), which we interpreted as a signal of fundamental discreteness of space itself. In this Letter, we extend the above results to a relativistic particle in a rectangular as well as a spherical box, by solving the GUP-corrected Klein–Gordon and Dirac equations, and for the latter, to two and three dimensions. We again arrive at quantization of box length, area and volume and an indication of the fundamentally grainy nature of space. We discuss possible implications.     

Confined systems and the modified virial theorem from semiclassical considerations

We show that two simple semiclassical strategies, one based on the Wilson–Sommerfeld rule and the other on the uncertainty principle, yield the exact modified form of the virial theorem for confined systems. An alternative, easier quantum mechanical route to arrive at this result is also sketched. Pilot calculations on confined oscillators reveal decisive trends. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Partially harmonic forms and models of H-series

Let G   be a real reductive Lie group, let H=TA$H=TA$ be the identity component of a Cartan subgroup, and let h$\mathfrak{h}$ be the corresponding Cartan subalgebra. This leads to a parabolic subgroup of G whose identity component is MAN. The unitary G-representations induced by MAN are known as the H  -series. We study symplectic geometry of G×h$G\times \mathfrak{h}$ and apply geometric quantization to construct unitary G-representations by partially harmonic forms. They are direct integrals of the H-series, indexed by the image of the moment map. We also perform symplectic reduction and symplectic induction, and consider their analogues in representation theory via geometric quantization.     

Quantization of Boltzmann Entropy: Pairs and Correlation Function

We consider the Gibbs statistical ensemble. We introduce the statistical-weight operator , construct the entropy operator of the ensemble using the Boltzmann formula , and define the free-energy operator. We find asymptotic expressions for free-energy eigenvalues with pair correlations taken into account in the limit as the number of systems in the ensemble tends to infinity.     

Controller design for network‐based Markovian jump systems with unreliable communication links

This article is concerned with observer and controller design for networked control systems, where the considered plant refers to a class of discrete‐time communication delay Markovian jump systems. In the study, random packet losses and output quantization are considered simultaneously. The packet losses considered here includes sensor to controller and controller to actuator sides, which are modeled as two Bernoulli distributed white sequences, respectively. An observer‐based control scheme is developed to stabilize the closed‐loop systems. Finally, an illustrative example is provided to show the applicability of the proposed control method. © 2016 Wiley Periodicals, Inc. Complexity 21: 623–634, 2016     

About Nonassociativity in Mathematics and Physics

A short review about nonassociative algebraic systems (mainly nonassociative algebras) and their physical applications is presented. We begin with some motivations, then we give a brief historical overview about the formation and development of the concept of hypercomplex number system and about some earlier applications. The main directions discussed are the octonionic, Lie-admissible, and quasigroup approaches. Also, some problems investigated in Tartu, the octonionic approach, Moufang–Mal'tsev symmetry, and associator quantization are discussed. This review does not pretend to be complete as the accent is placed on ideas and not on the techniques, also the references are quite sporadic (there are many authors and results mentioned in the text without references).     

A Note on Higher Cohomology Groups of Kähler Quotients

Consider a holomorphic $#x2102;^×-action on a possibly noncompact Kähler manifold. We show that the highercohomology groups appearing in the geometric quantization of thesymplectic quotient are isomorphic to the invariant parts of thecorresponding cohomology groups of the original manifold. Fornon-Abelian group actions on compact Kähler manifolds, this resultwas proved recently by Teleman. Our approach is applying the holomorphicinstanton complex to the prequantum line bundles over the symplecticcuts. We also settle a conjecture of Zhang and the present author on theexact sequence of higher cohomology groups in the context of symplecticcutting.     

Bimodule Deformations, Picard Groups and Contravariant Connections

We study deformations of invertible bimodules and the behavior of Picard groups under deformation quantization. While K ₀-groups are known to be stable under formal deformations of algebras, Picard groups may change drastically. We identify the semiclassical limit of bimodule deformations as contravariant connections and study the associated deformation quantization problem. Our main focus is on formal deformation quantization of Poisson manifolds by star products.     

Deformation Quantizations of the Poisson Algebra of Laurent Polynomials

It is well known that the Moyal bracket gives a unique deformation quantization of the canonical phase space R2n up to equivalence. In his presentation of an interesting deformation quantization of the Poisson algebra of Laurent polynomials, Ovsienko discusses the equivalences of deformation quantizations of these algebras. We show that under suitable conditions, deformation quantizations of this algebra are equivalent. Though Ovsienko showed that there exists a deformation quantization of the Poisson algebra of Laurent polynomials which is not equivalent to the Moyal product, this is not correct. We show this equivalence by two methods: a direct construction of the intertwiner via the star exponential and a more standard approach using Hochschild 2-cocycles.     

A new energy source originating from extra dimensions

In this work the Einstein gravitational field equations and the Lichnerowicz boundary formalism in the extra dimensions are used to build up our black hole model from 6-dimensional space－time. From the internal stress－energy tensor the solutions with energy levels and semiclassical space-quantization are obtained, which combines with only one metric condition outside the defect. We show a new type of energy source, which originates from extra dimensions. A part of the energy source of quasi-stellar object (QSO) maybe come from extra dimensions in that way. The theoretical arithmetic upper limit is identical to that of the output energy of QSO.