“Quantization Is a Mystery”∗
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Strocchi's Quantum Mechanics
Strocchi’s Quantum Mechanics: An alternative formulation to the dominant one? Antonino Drago ‒ Formerly at Naples University “Federico II”, Italy ‒ drago@un ina.it Abstract: At first glance, Strocchi’s formulation presents several characteri- stic features of a theory whose two choices are the alternative ones to the choices of the paradigmatic formulation: i) Its organization starts from not axioms, but an operative basis and it is aimed to solve a problem (i.e. the indeterminacy); moreover, it argues through both doubly negated proposi- tions and an ad absurdum proof; ii) It put, before the geometry, a polyno- mial algebra of bounded operators; which may pertain to constructive Mathematics. Eventually, it obtains the symmetries. However one has to solve several problems in order to accurately re-construct this formulation according to the two alternative choices. I conclude that rather than an al- ternative to the paradigmatic formulation, Strocchi’s represents a very inter- esting divergence from it. Keywords: Quantum Mechanics, C*-algebra approach, Strocchi’s formula- tion, Two dichotomies, Constructive Mathematics, Non-classical Logic 1. Strocchi’s Axiomatic of the paradigmatic formulation and his criticisms to it Segal (1947) has suggested a foundation of Quantum Mechanics (QM) on an algebraic approach of functional analysis; it is independent from the space-time variables or any other geometrical representation, as instead a Hilbert space is. By defining an algebra of the observables, it exploits Gelfand-Naimark theorem in order to faithfully represent this algebra into Hilbert space and hence to obtain the Schrödinger representation of QM. In the 70’s Emch (1984) has reiterated this formulation and improved it. -
Geometric Quantization of Chern Simons Gauge Theory
J. DIFFERENTIAL GEOMETRY 33(1991) 787 902 GEOMETRIC QUANTIZATION OF CHERN SIMONS GAUGE THEORY SCOTT AXELROD, STEVE DELLA PIETRA & EDWARD WITTEN Abstract We present a new construction of the quantum Hubert space of Chern Simons gauge theory using methods which are natural from the three dimensional point of view. To show that the quantum Hubert space associated to a Riemann surface Σ is independent of the choice of com plex structure on Σ, we construct a natural projectively flat connection on the quantum Hubert bundle over Teichmuller space. This connec tion has been previously constructed in the context of two dimensional conformal field theory where it is interpreted as the stress energy tensor. Our construction thus gives a (2 + 1 ) dimensional derivation of the basic properties of (1 + 1) dimensional current algebra. To construct the con nection we show generally that for affine symplectic quotients the natural projectively flat connection on the quantum Hubert bundle may be ex pressed purely in terms of the intrinsic Kahler geometry of the quotient and the Quillen connection on a certain determinant line bundle. The proof of most of the properties of the connection we construct follows surprisingly simply from the index theorem identities for the curvature of the Quillen connection. As an example, we treat the case when Σ has genus one explicitly. We also make some preliminary comments con cern ing the Hubert space structure. Introduction Several years ago, in examining the proof of a rather surprising result about von Neumann algebras, V. F. R. Jones [20] was led to the discovery of some unusual representations of the braid group from which invariants of links in S3 can be constructed. -
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. -
Geometric Quantization
GEOMETRIC QUANTIZATION 1. The basic idea The setting of the Hamiltonian version of classical (Newtonian) mechanics is the phase space (position and momentum), which is a symplectic manifold. The typical example of this is the cotangent bundle of a manifold. The manifold is the configuration space (ie set of positions), and the tangent bundle fibers are the momentum vectors. The solutions to Hamilton's equations (this is where the symplectic structure comes in) are the equations of motion of the physical system. The input is the total energy (the Hamiltonian function on the phase space), and the output is the Hamiltonian vector field, whose flow gives the time evolution of the system, say the motion of a particle acted on by certain forces. The Hamiltonian formalism also allows one to easily compute the values of any physical quantities (observables, functions on the phase space such as the Hamiltonian or the formula for angular momentum) using the Hamiltonian function and the symplectic structure. It turns out that various experiments showed that the Hamiltonian formalism of mechanics is sometimes inadequate, and the highly counter-intuitive quantum mechanical model turned out to produce more correct answers. Quantum mechanics is totally different from classical mechanics. In this model of reality, the position (or momentum) of a particle is an element of a Hilbert space, and an observable is an operator acting on the Hilbert space. The inner product on the Hilbert space provides structure that allows computations to be made, similar to the way the symplectic structure is a computational tool in Hamiltonian mechanics. -
Some Background on Geometric Quantization
SOME BACKGROUND ON GEOMETRIC QUANTIZATION NILAY KUMAR Contents 1. Introduction1 2. Classical versus quantum2 3. Geometric quantization4 References6 1. Introduction Let M be a compact oriented 3-manifold. Chern-Simons theory with gauge group G (that we will take to be compact, connected, and simply-connected) on M is the data of a principal G-bundle π : P ! M together with a Lagrangian density L : A ! Ω3(P ) on the space of connections A on P given by 2 LCS(A) = hA ^ F i + hA ^ [A ^ A]i: 3 Let us detail the notation used here. Recall first that a connection A 2 A is 1 ∗ a G-invariant g-valued one-form, i.e. A 2 Ω (P ; g) such that RgA = Adg−1 A, satisfying the additional condition that if ξ 2 g then A(ξP ) = ξ if ξP is the vector field associated to ξ. Notice that A , though not a vector space, is an affine space 1 modelled on Ω (M; P ×G g). The curvature F of a connection A is is the g-valued two-form given by F (v; w) = dA(vh; wh), where •h denotes projection onto the 1 horizontal distribution ker π∗. Finally, by h−; −i we denote an ad-invariant inner product on g. The Chern-Simons action is now given Z SSC(A) = LSC(A) M and the quantities of interest are expectation values of observables O : A ! R Z hOi = O(A)eiSSC(A)=~: A =G Here G is the group of automorphisms of P ! Σ, which acts by pullback on A { the physical states are unaffected by these gauge transformations, so we integrate over the quotient A =G to eliminate the redundancy. -
Geometric Quantization
June 16, 2016 BSC THESIS IN PHYSICS, 15 HP Geometric quantization Author: Fredrik Gardell Supervisor: Luigi Tizzano Subject evaluator: Maxim Zabzine Uppsala University, Department of Physics and Astronomy E-mail: [email protected] Abstract In this project we introduce the general idea of geometric quantization and demonstrate how to apply the process on a few examples. We discuss how to construct a line bundle over the symplectic manifold with Dirac’s quantization conditions and how to determine if we are able to quantize a system with the help of Weil’s integrability condition. To reduce the prequantum line bundle we employ real polarization such that the system does not break Heisenberg’s uncertainty principle anymore. From the prequantum bundle and the polarization we construct the sought after Hilbert space. Sammanfattning I detta arbete introducerar vi geometrisk kvantisering och demonstrerar hur man utför denna metod på några exempel. Sen diskuterar vi hur man konstruerar ett linjeknippe med hjälp av Diracs kvantiseringskrav och hur man bedömer om ett system är kvantiserbart med hjälp av Weils integrarbarhetskrav. För att reducera linjeknippet så att Heisenbergs osäkerhetsrelation inte bryts, använder vi oss av reell polarisering. Med det polariserade linjeknippet och det ursprungliga linjeknippet kan vi konstruera det eftersökta Hilbert rummet. Innehåll 1 Introduction2 2 Symplectic Geometry3 2.1 Symplectic vector space3 2.2 Symplectic manifolds4 2.3 Cotangent bundles and Canonical coordinates5 2.4 Hamiltonian vector -
Turbulence, Entropy and Dynamics
TURBULENCE, ENTROPY AND DYNAMICS Lecture Notes, UPC 2014 Jose M. Redondo Contents 1 Turbulence 1 1.1 Features ................................................ 2 1.2 Examples of turbulence ........................................ 3 1.3 Heat and momentum transfer ..................................... 4 1.4 Kolmogorov’s theory of 1941 ..................................... 4 1.5 See also ................................................ 6 1.6 References and notes ......................................... 6 1.7 Further reading ............................................ 7 1.7.1 General ............................................ 7 1.7.2 Original scientific research papers and classic monographs .................. 7 1.8 External links ............................................. 7 2 Turbulence modeling 8 2.1 Closure problem ............................................ 8 2.2 Eddy viscosity ............................................. 8 2.3 Prandtl’s mixing-length concept .................................... 8 2.4 Smagorinsky model for the sub-grid scale eddy viscosity ....................... 8 2.5 Spalart–Allmaras, k–ε and k–ω models ................................ 9 2.6 Common models ........................................... 9 2.7 References ............................................... 9 2.7.1 Notes ............................................. 9 2.7.2 Other ............................................. 9 3 Reynolds stress equation model 10 3.1 Production term ............................................ 10 3.2 Pressure-strain interactions -
Supplementvolum-E18 Nu-Mber-41989 .__L,___Society
ISSN 0739-4934 NEWSLETTER I IISTORY OF SCIENCE SUPPLEMENTVOLUM-E18 NU-MBER-41989 .__L,___SOCIETY - WELCOME TO GAINESVILLE HSS EXECUTIVE BY FREDERICK GREGORY COMMITTEE "A SOPI-llSTICATED SLICE of small-town south": so wrote Jonathan Lerner PRESIDENT about Gainesville for Washington Post readers this past spring. Like the majority MARY JO NYE, University of Oklahoma of visitors to Gainesville, Lerner was impressed with the topography of the city, VICE-PRESIDENT which forms a hammock-a dry area, relatively higher than its surroundings, STEPHEN G. BRUSH, University of Maryland that can support hardwood trees. Residents of Gainesville are enormously proud of the extensive canopy that covers 46 percent of their town, the highest per EXECUTIVE SECRETARY MICHAEL M. SOKAL, Worcester centage of any Florida city. In addition to the majestic live oaks, the southern Polytechnic Institute pine, and a variety of palm trees, dogwoods and magnolias are also plentiful. TREASURER Unfortunately the HSS Annual Meeting is held at a time of year that misses the MARY LOUISE GLEASON, New York City blossoms of our giant azaleas, some older ones of which are as high as roof tops. EDITOR f obvious interest to historians of science is nearby Paynes Prairie, an 18,000- RONALD L. NUMBERS, University of acre wildlife preserve whose zoological and botanical life was described in vivid Wisconsin-Madison detail by William Bartram after his travels through the region in 1774. Meeting sessions will be held on the campus of the University of Florida, which, at least in this part of the country, is never to be mixed up with Florida State University in Tallahassee. -
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. -
Geometric Quantization
Geometric Quantization JProf Gabriele Benedetti, Johanna Bimmermann, Davide Legacci, Steffen Schmidt Summer Term 2021 \Quantization is an art, not a functor." { Folklore Organization of the seminar When: Thursdays at 2:00 pm sharp (First talk on 15.4.) Where: Online Language: English Presentation: Participants will give a 90-minutes talk (including 10 minutes of time for questions) Online talk: Write on a tablet in real time (preferred option) or prepare slides using Latex beamer - if you need help with this, ask us. We will also reserve a room in the Mathematikon if the speaker wants to give the talk from there. Evaluation: Give a presentation, write notes or slides of the talk that will be uploaded to the homepage of the seminar, actively participate during the seminar talks. Meet us: 1 or 2 weeks before your talk to discuss your plan and to clarify questions. Please contact the respective organiser of the talk via mail. E-mail adresses: Davide : [email protected] Gabriele : [email protected] Johanna : [email protected] Steffen : Schmidt-Steff[email protected] 1 List of Topics Topic 1: The Mathematical Model of Classical Mechanics (Davide) We introduce symplectic manifolds (M; !), the natural setting where classical Hamil- tonian systems induced by a smooth function H : M ! R can be defined. Beyond symplectic vector spaces, the main examples we will consider are cotangent bundles T ∗Q of a configuration manifold Q and K¨ahlermanifolds such as S2 and, more in general, CPn. The symplectic structure induces a Poisson bracket on the space of ob- servables C1(M) satisfying crucial algebraic properties and determining the dynamics of Hamiltonian systems. -
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.