Deformation Quantization
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Infinitely Many Star Products to Play With
DSF{40{01 BiBoS 01-12-068 ESI 1109 hep{th/0112092 December 2001 Infinitely many star products to play with J.M. Gracia-Bond´ıa a;b, F. Lizzi b, G. Marmo b and P. Vitale c a BiBoS, Fakult¨atder Physik, Universit¨atBielefeld, 33615 Bielefeld, Germany b Dipartimento di Scienze Fisiche, Universit`adi Napoli Federico II and INFN, Sezione di Napoli, Monte S. Angelo Via Cintia, 80126 Napoli, Italy [email protected], [email protected] c Dipartimento di Fisica, Universit`adi Salerno and INFN Gruppo Collegato di Salerno, Via S. Allende 84081 Baronissi (SA), Italy [email protected] Abstract While there has been growing interest for noncommutative spaces in recent times, most examples have been based on the simplest noncommutative algebra: [xi; xj] = iθij. Here we present new classes of (non-formal) deformed products k associated to linear Lie algebras of the kind [xi; xj] = icijxk. For all possible three- dimensional cases, we define a new star product and discuss its properties. To complete the analysis of these novel noncommutative spaces, we introduce noncom- pact spectral triples, and the concept of star triple, a specialization of the spectral triple to deformations of the algebra of functions on a noncompact manifold. We examine the generalization to the noncompact case of Connes' conditions for non- commutative spin geometries, and, in the framework of the new star products, we exhibit some candidates for a Dirac operator. On the technical level, properties of 2n the Moyal multiplier algebra M(Rθ ) are elucidated. 1 Introduction Over five years ago, Connes gave the first axiomatics for first-quantized fermion fields on (compact) noncommutative varieties, the so-called spectral triples [1]. -
Research Archive
http://dx.doi.org/10.1108/IJBM-05-2016-0075 View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Hertfordshire Research Archive Research Archive Citation for published version: Dimitris Kakofengitis, Maxime Oliva, and Ole Steuernagel, ‘Wigner's representation of quantum mechanics in integral form and its applications’, Physical Review A, Vol. 95, 022127, published 27 February 2017. DOI: https://doi.org/10.1103/PhysRevA.95.022127 Document Version: This is the Accepted Manuscript version. The version in the University of Hertfordshire Research Archive may differ from the final published version. Users should always cite the published version of record. Copyright and Reuse: ©2017 American Physical Society. This manuscript version is made available under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited. Enquiries If you believe this document infringes copyright, please contact the Research & Scholarly Communications Team at [email protected] Wigner's representation of quantum mechanics in integral form and its applications Dimitris Kakofengitis, Maxime Oliva, and Ole Steuernagel School of Physics, Astronomy and Mathematics, University of Hertfordshire, Hatfield, AL10 9AB, UK (Dated: February 28, 2017) We consider quantum phase-space dynamics using Wigner's representation of quantum mechanics. We stress the usefulness of the integral form for the description of Wigner's phase-space current J as an alternative to the popular Moyal bracket. The integral form brings out the symmetries between momentum and position representations of quantum mechanics, is numerically stable, and allows us to perform some calculations using elementary integrals instead of Groenewold star products. -
Phase Space Quantum Mechanics Is Such a Natural Formulation of Quantum Theory
Phase space quantum mechanics Maciej B laszak, Ziemowit Doma´nski Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Pozna´n, Poland Abstract The paper develope the alternative formulation of quantum mechanics known as the phase space quantum mechanics or deformation quantization. It is shown that the quantization naturally arises as an appropriate deformation of the classical Hamiltonian mechanics. More precisely, the deformation of the point-wise prod- uct of observables to an appropriate noncommutative ⋆-product and the deformation of the Poisson bracket to an appropriate Lie bracket is the key element in introducing the quantization of classical Hamiltonian systems. The formalism of the phase space quantum mechanics is presented in a very systematic way for the case of any smooth Hamiltonian function and for a very wide class of deformations. The considered class of deformations and the corresponding ⋆-products contains as a special cases all deformations which can be found in the literature devoted to the subject of the phase space quantum mechanics. Fundamental properties of ⋆-products of observables, associated with the considered deformations are presented as well. Moreover, a space of states containing all admissible states is introduced, where the admissible states are appropriate pseudo-probability distributions defined on the phase space. It is proved that the space of states is endowed with a structure of a Hilbert algebra with respect to the ⋆-multiplication. The most important result of the paper shows that developed formalism is more fundamental then the axiomatic ordinary quantum mechanics which appears in the presented approach as the intrinsic element of the general formalism. -
Weyl Quantization and Wigner Distributions on Phase Space
faculteit Wiskunde en Natuurwetenschappen Weyl quantization and Wigner distributions on phase space Bachelor thesis in Physics and Mathematics June 2014 Student: R.S. Wezeman Supervisor in Physics: Prof. dr. D. Boer Supervisor in Mathematics: Prof. dr H. Waalkens Abstract This thesis describes quantum mechanics in the phase space formulation. We introduce quantization and in particular the Weyl quantization. We study a general class of phase space distribution functions on phase space. The Wigner distribution function is one such distribution function. The Wigner distribution function in general attains negative values and thus can not be interpreted as a real probability density, as opposed to for example the Husimi distribution function. The Husimi distribution however does not yield the correct marginal distribution functions known from quantum mechanics. Properties of the Wigner and Husimi distribution function are studied to more extent. We calculate the Wigner and Husimi distribution function for the energy eigenstates of a particle trapped in a box. We then look at the semi classical limit for this example. The time evolution of Wigner functions are studied by making use of the Moyal bracket. The Moyal bracket and the Poisson bracket are compared in the classical limit. The phase space formulation of quantum mechanics has as advantage that classical concepts can be studied and compared to quantum mechanics. For certain quantum mechanical systems the time evolution of Wigner distribution functions becomes equivalent to the classical time evolution stated in the exact Egerov theorem. Another advantage of using Wigner functions is when one is interested in systems involving mixed states. A disadvantage of the phase space formulation is that for most problems it quickly loses its simplicity and becomes hard to calculate. -
Phase-Space Description of the Coherent State Dynamics in a Small One-Dimensional System
Open Phys. 2016; 14:354–359 Research Article Open Access Urszula Kaczor, Bogusław Klimas, Dominik Szydłowski, Maciej Wołoszyn, and Bartłomiej J. Spisak* Phase-space description of the coherent state dynamics in a small one-dimensional system DOI 10.1515/phys-2016-0036 space variables. In this case, the product of any functions Received Jun 21, 2016; accepted Jul 28, 2016 on the phase space is noncommutative. The classical alge- bra of observables (commutative) is recovered in the limit Abstract: The Wigner-Moyal approach is applied to investi- of the reduced Planck constant approaching zero ( ! 0). gate the dynamics of the Gaussian wave packet moving in a ~ Hence the presented approach is called the phase space double-well potential in the ‘Mexican hat’ form. Quantum quantum mechanics, or sometimes is referred to as the de- trajectories in the phase space are computed for different formation quantization [6–8]. Currently, the deformation kinetic energies of the initial wave packet in the Wigner theory has found applications in many fields of modern form. The results are compared with the classical trajecto- physics, such as quantum gravity, string and M-theory, nu- ries. Some additional information on the dynamics of the clear physics, quantum optics, condensed matter physics, wave packet in the phase space is extracted from the analy- and quantum field theory. sis of the cross-correlation of the Wigner distribution func- In the present contribution, we examine the dynam- tion with itself at different points in time. ics of the initially coherent state in the double-well poten- Keywords: Wigner distribution, wave packet, quantum tial using the phase space formulation of the quantum me- trajectory chanics. -
P. A. M. Dirac and the Maverick Mathematician
Journal & Proceedings of the Royal Society of New South Wales, vol. 150, part 2, 2017, pp. 188–194. ISSN 0035-9173/17/020188-07 P. A. M. Dirac and the Maverick Mathematician Ann Moyal Emeritus Fellow, ANU, Canberra, Australia Email: [email protected] Abstract Historian of science Ann Moyal recounts the story of a singular correspondence between the great British physicist, P. A. M. Dirac, at Cambridge, and J. E. Moyal, then a scientist from outside academia working at the de Havilland Aircraft Company in Britain (later an academic in Australia), on the ques- tion of a statistical basis for quantum mechanics. A David and Goliath saga, it marks a paradigmatic study in the history of quantum physics. A. M. Dirac (1902–1984) is a pre- neering at the Institut d’Electrotechnique in Peminent name in scientific history. In Grenoble, enrolling subsequently at the Ecole 1962 it was my privilege to acquire a set Supérieure d’Electricité in Paris. Trained as of the letters he exchanged with the then a civil engineer, Moyal worked for a period young mathematician, José Enrique Moyal in Tel Aviv but returned to Paris in 1937, (1910–1998), for the Basser Library of the where his exposure to such foundation works Australian Academy of Science, inaugurated as Georges Darmois’s Statistique Mathéma- as a centre for the archives of the history of tique and A. N. Kolmogorov’s Foundations Australian science. This is the only manu- of the Theory of Probability introduced him script correspondence of Dirac (known to to a knowledge of pioneering European stud- colleagues as a very reluctant correspondent) ies of stochastic processes. -
Matrix Bases for Star Products: a Review?
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 10 (2014), 086, 36 pages Matrix Bases for Star Products: a Review? Fedele LIZZI yzx and Patrizia VITALE yz y Dipartimento di Fisica, Universit`adi Napoli Federico II, Napoli, Italy E-mail: [email protected], [email protected] z INFN, Sezione di Napoli, Italy x Institut de Ci´enciesdel Cosmos, Universitat de Barcelona, Catalonia, Spain Received March 04, 2014, in final form August 11, 2014; Published online August 15, 2014 http://dx.doi.org/10.3842/SIGMA.2014.086 Abstract. 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. Key words: noncommutative geometry; star products; matrix models 2010 Mathematics Subject Classification: 58Bxx; 40C05; 46L65 1 Introduction Star products were originally introduced [34, 62] in the context of point particle quantization, they were generalized in [9, 10] to comprise general cases, the physical motivations behind this was the quantization of phase space. Later star products became a tool to describe possible noncommutative geometries of spacetime itself (for a review see [4]). In this sense they have become a tool for the study of field theories on noncommutative spaces [77]. In this short review we want to concentrate on one particular aspect of the star product, the fact that any action involving fields which are multiplied with a star product can be seen as a matrix model. -
The Moyal Equation in Open Quantum Systems
The Moyal Equation in Open Quantum Systems Karl-Peter Marzlin and Stephen Deering CAP Congress Ottawa, 13 June 2016 Outline l Quantum and classical dynamics l Phase space descriptions of quantum systems l Open quantum systems l Moyal equation for open quantum systems l Conclusion Quantum vs classical Differences between classical mechanics and quantum mechanics have fascinated researchers for a long time Two aspects: • Measurement process: deterministic vs contextual • Dynamics: will be the topic of this talk Quantum vs classical Differences in standard formulations: Quantum Mechanics Classical Mechanics Class. statistical mech. complex wavefunction deterministic position probability distribution observables = operators deterministic observables observables = random variables dynamics: Schrödinger Newton 2 Boltzmann equation for equation for state probability distribution However, some differences “disappear” when different formulations are used. Quantum vs classical First way to make QM look more like CM: Heisenberg picture Heisenberg equation of motion dAˆH 1 = [AˆH , Hˆ ] dt i~ is then similar to Hamiltonian mechanics dA(q, p, t) @A @H @A @H = A, H , A, H = dt { } { } @q @p − @p @q Still have operators as observables Second way to compare QM and CM: phase space representation of QM Most popular phase space representation: the Wigner function W(q,p) 1 W (q, p)= S[ˆ⇢](q, p) 2⇡~ iq0p/ Phase space S[ˆ⇢](q, p)= dq0 q + 1 q0 ⇢ˆ q 1 q0 e− ~ h 2 | | − 2 i Z The Weyl symbol S [ˆ ⇢ ]( q, p ) maps state ⇢ ˆ to a function of position and momentum (= phase space) Phase space The Wigner function is similar to the distribution function of classical statistical mechanics Dynamical equation = Liouville equation. -
A Functorial Construction of Quantum Subtheories
entropy Article A Functorial Construction of Quantum Subtheories Ivan Contreras 1 and Ali Nabi Duman 2,* 1 Department of Mathematics, University of Illinois at Urbana-Champain, Urbana, IL 61801, USA; [email protected] 2 Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia * Correspondence: [email protected]; Tel.: +966-138-602-632 Academic Editors: Giacomo Mauro D’Ariano, Andrei Khrennikov and Massimo Melucci Received: 2 March 2017; Accepted: 4 May 2017; Published: 11 May 2017 Abstract: We apply the geometric quantization procedure via symplectic groupoids to the setting of epistemically-restricted toy theories formalized by Spekkens (Spekkens, 2016). In the continuous degrees of freedom, this produces the algebraic structure of quadrature quantum subtheories. In the odd-prime finite degrees of freedom, we obtain a functor from the Frobenius algebra of the toy theories to the Frobenius algebra of stabilizer quantum mechanics. Keywords: categorical quantum mechanics; geometric quantization; epistemically-restricted theories; Frobenius algebras 1. Introduction The aim of geometric quantization is to construct, using the geometry of the classical system, a Hilbert space and a set of operators on that Hilbert space that give the quantum mechanical analogue of the classical mechanical system modeled by a symplectic manifold [1–3]. Starting with a symplectic space M corresponding to the classical phase space, the square integrable functions over M is the first Hilbert space in the construction, called the prequantum Hilbert space. In this case, the classical observables are mapped to the operators on this Hilbert space, and the Poisson bracket is mapped to the commutator. -
About the Concept of Quantum Chaos
entropy Concept Paper About the Concept of Quantum Chaos Ignacio S. Gomez 1,*, Marcelo Losada 2 and Olimpia Lombardi 3 1 National Scientific and Technical Research Council (CONICET), Facultad de Ciencias Exactas, Instituto de Física La Plata (IFLP), Universidad Nacional de La Plata (UNLP), Calle 115 y 49, 1900 La Plata, Argentina 2 National Scientific and Technical Research Council (CONICET), University of Buenos Aires, 1420 Buenos Aires, Argentina; [email protected] 3 National Scientific and Technical Research Council (CONICET), University of Buenos Aires, Larralde 3440, 1430 Ciudad Autónoma de Buenos Aires, Argentina; olimpiafi[email protected] * Correspondence: nachosky@fisica.unlp.edu.ar; Tel.: +54-11-3966-8769 Academic Editors: Mariela Portesi, Alejandro Hnilo and Federico Holik Received: 5 February 2017; Accepted: 23 April 2017; Published: 3 May 2017 Abstract: The research on quantum chaos finds its roots in the study of the spectrum of complex nuclei in the 1950s and the pioneering experiments in microwave billiards in the 1970s. Since then, a large number of new results was produced. Nevertheless, the work on the subject is, even at present, a superposition of several approaches expressed in different mathematical formalisms and weakly linked to each other. The purpose of this paper is to supply a unified framework for describing quantum chaos using the quantum ergodic hierarchy. Using the factorization property of this framework, we characterize the dynamical aspects of quantum chaos by obtaining the Ehrenfest time. We also outline a generalization of the quantum mixing level of the kicked rotator in the context of the impulsive differential equations. Keywords: quantum chaos; ergodic hierarchy; quantum ergodic hierarchy; classical limit 1. -
Phase-Space Formulation of Quantum Mechanics and Quantum State
formulation of quantum mechanics and quantum state reconstruction for View metadata,Phase-space citation and similar papers at core.ac.uk brought to you by CORE ysical systems with Lie-group symmetries ph provided by CERN Document Server ? ? C. Brif and A. Mann Department of Physics, Technion { Israel Institute of Technology, Haifa 32000, Israel We present a detailed discussion of a general theory of phase-space distributions, intro duced recently by the authors [J. Phys. A 31, L9 1998]. This theory provides a uni ed phase-space for- mulation of quantum mechanics for physical systems p ossessing Lie-group symmetries. The concept of generalized coherent states and the metho d of harmonic analysis are used to construct explicitly a family of phase-space functions which are p ostulated to satisfy the Stratonovich-Weyl corresp on- dence with a generalized traciality condition. The symb ol calculus for the phase-space functions is given by means of the generalized twisted pro duct. The phase-space formalism is used to study the problem of the reconstruction of quantum states. In particular, we consider the reconstruction metho d based on measurements of displaced pro jectors, which comprises a numb er of recently pro- p osed quantum-optical schemes and is also related to the standard metho ds of signal pro cessing. A general group-theoretic description of this metho d is develop ed using the technique of harmonic expansions on the phase space. 03.65.Bz, 03.65.Fd I. INTRODUCTION P function [?,?] is asso ciated with the normal order- ing and the Husimi Q function [?] with the antinormal y ordering of a and a . -
Abstracts for 18 Category Theory; Homological 62 Statistics Algebra 65 Numerical Analysis Syracuse, October 2–3, 2010
A bstr bstrActs A A cts mailing offices MATHEMATICS and additional of Papers Presented to the Periodicals postage 2010 paid at Providence, RI American Mathematical Society SUBJECT CLASSIFICATION Volume 31, Number 4, Issue 162 Fall 2010 Compiled in the Editorial Offices of MATHEMATICAL REVIEWS and ZENTRALBLATT MATH 00 General 44 Integral transforms, operational calculus 01 History and biography 45 Integral equations Editorial Committee 03 Mathematical logic and foundations 46 Functional analysis 05 Combinatorics 47 Operator theory Robert J. Daverman, Chair Volume 31 • Number 4 Fall 2010 Volume 06 Order, lattices, ordered 49 Calculus of variations and optimal Georgia Benkart algebraic structures control; optimization 08 General algebraic systems 51 Geometry Michel L. Lapidus 11 Number theory 52 Convex and discrete geometry Matthew Miller 12 Field theory and polynomials 53 Differential geometry 13 Commutative rings and algebras 54 General topology Steven H. Weintraub 14 Algebraic geometry 55 Algebraic topology 15 Linear and multilinear algebra; 57 Manifolds and cell complexes matrix theory 58 Global analysis, analysis on manifolds 16 Associative rings and algebras 60 Probability theory and stochastic 17 Nonassociative rings and algebras processes Abstracts for 18 Category theory; homological 62 Statistics algebra 65 Numerical analysis Syracuse, October 2–3, 2010 ....................... 655 19 K-theory 68 Computer science 20 Group theory and generalizations 70 Mechanics of particles and systems Los Angeles, October 9–10, 2010 .................. 708 22 Topological groups, Lie groups 74 Mechanics of deformable solids 26 Real functions 76 Fluid mechanics 28 Measure and integration 78 Optics, electromagnetic theory Notre Dame, November 5–7, 2010 ................ 758 30 Functions of a complex variable 80 Classical thermodynamics, heat transfer 31 Potential theory 81 Quantum theory Richmond, November 6–7, 2010 ..................