Lectures on the Geometry of Quantization
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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. -
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]. -
University of California Santa Cruz on An
UNIVERSITY OF CALIFORNIA SANTA CRUZ ON AN EXTENSION OF THE MEAN INDEX TO THE LAGRANGIAN GRASSMANNIAN A dissertation submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in MATHEMATICS by Matthew I. Grace September 2020 The Dissertation of Matthew I. Grace is approved: Professor Viktor Ginzburg, Chair Professor Ba¸sakG¨urel Professor Richard Montgomery Dean Quentin Williams Interim Vice Provost and Dean of Graduate Studies Copyright c by Matthew I. Grace 2020 Table of Contents List of Figuresv Abstract vi Dedication vii Acknowledgments viii I Preliminaries1 I.1 Introduction . .2 I.2 Historical Context . .8 I.3 Definitions and Conventions . 21 I.4 Dissertation Outline . 28 I.4.1 Outline of Results . 28 I.4.2 Outline of Proofs . 31 II Linear Canonical Relations and Isotropic Pairs 34 II.1 Strata of the Lagrangian Grassmannian . 35 II.2 Iterating Linear Canonical Relations . 37 II.3 Conjugacy Classes of Isotropic Pairs . 40 II.4 Stratum-Regular Paths . 41 III The Set H of Exceptional Lagrangians 50 III.1 The Codimension of H ............................... 51 III.2 Admissible Linear Canonical Relations . 54 IV The Conley-Zehnder Index and the Circle Map ρ 62 IV.1 Construction of the Conley-Zehnder Index . 63 IV.2 Properties of the Circle Map Extensionρ ^ ..................... 65 iii V Unbounded Sequences in the Symplectic Group 69 V.1 A Sufficient Condition for Asymptotic Hyperbolicity . 70 V.2 A Decomposition for Certain Unbounded Sequences of Symplectic Maps . 76 V.2.1 Preparation for the Proof of Theorem V.2.1................ 78 V.2.2 Proof of Theorem V.2.1.......................... -
Ellipticity of the Symplectic Twistor Complex
ARCHIVUM MATHEMATICUM (BRNO) Tomus 47 (2011), 309–327 ELLIPTICITY OF THE SYMPLECTIC TWISTOR COMPLEX Svatopluk Krýsl Abstract. For a Fedosov manifold (symplectic manifold equipped with a symplectic torsion-free affine connection) admitting a metaplectic structure, we shall investigate two sequences of first order differential operators acting on sections of certain infinite rank vector bundles defined over this manifold. The differential operators are symplectic analogues of the twistor operators known from Riemannian or Lorentzian spin geometry. It is known that the mentioned sequences form complexes if the symplectic connection is of Ricci type. In this paper, we prove that certain parts of these complexes are elliptic. 1. Introduction In this article, we prove the ellipticity of certain parts of the so called symplectic twistor complexes. The symplectic twistor complexes are two sequences of first order differential operators defined over Ricci type Fedosov manifolds admitting a metaplectic structure. The mentioned parts of these complexes will be called truncated symplectic twistor complexes and will be defined later in this text. Now, let us say a few words about the Fedosov manifolds. Formally speaking, a Fedosov manifold is a triple (M 2l, ω, ∇) where (M 2l, ω) is a (for definiteness 2l dimensional) symplectic manifold and ∇ is a symplectic torsion-free affine connection. Connections satisfying these two properties are usually called Fedosov connections in honor of Boris Fedosov who used them to obtain a deformation quantization for symplectic manifolds. (See Fedosov [5].) Let us also mention that in contrary to torsion-free Levi-Civita connections, the Fedosov ones are not unique. We refer an interested reader to Tondeur [18] and Gelfand, Retakh, Shubin [6] for more information. -
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. -
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. -
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 -
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. -
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. -
Complex of Twistor Operators in Symplectic Spin Geometry
Complex of twistor operators in symplectic spin geometry Svatopluk Kr´ysl ∗ Charles University of Prague, Sokolovsk´a83, Praha, Czech Republic. † October 22, 2018 Abstract For a symplectic manifold admitting a metaplectic structure (a symplectic ana- logue of the Riemannian spin structure), we construct a sequence consisting of differential operators using a symplectic torsion-free affine connection. All but one of these operators are of first order. The first order ones are symplectic analogues of the twistor operators known from Riemannian spin geometry. We prove that under the condition the symplectic Weyl curvature tensor field of the symplectic connection vanishes, the mentioned sequence forms a complex. This gives rise to a new complex for the so called Ricci type symplectic manifolds, which admit a metaplectic structure. Math. Subj. Class.: 53C07, 53D05, 58J10. Keywords: Fedosov manifolds, metaplectic structures, symplectic spinors, Kostant spinors, Segal-Shale-Weil representation, complexes of differential op- erators. 1 Introduction In the paper, we shall introduce a sequence of differential operators acting on symplectic spinor valued exterior differential forms over a symplectic manifold (M,ω) admitting the so called metaplectic structure. To define these operators, arXiv:0904.0763v1 [math.SG] 6 Apr 2009 we make use of a symplectic torsion-free affine connection on (M,ω). Under certain condition on the curvature of the connection , described∇ bellow, we prove that the mentioned sequence forms a complex. ∇ Let us say a few words about the metaplectic structure. The symplectic group Sp(2l, R) admits a non-trivial two-fold covering, the so called metaplectic group, which we shall denote by Mp(2l, R).