DOCSLIB.ORG

Complexified Symplectomorphisms and Geometric Quantization Master

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Complexified Symplectomorphisms and Geometric Quantization Master Complexified Symplectomorphisms and Geometric Quantization Miguel Barbosa Pereira Thesis to obtain the Master of Science Degree in Master in Mathematics and Applications Supervisor(s): Prof. João Luís Pimentel Nunes Examination Committee Chairperson: Prof. Miguel Tribolet de Abreu Supervisor: Prof. João Pimentel Nunes Member of the Committee: Prof. José Manuel Vergueiro Monteiro Cidade Mourão Member of the Committee: Prof. Sílvia Nogueira da Rocha Ravasco dos Anjos November 2017 ii This thesis is dedicated to my parents and to my brother. iii iv Acknowledgments I would like to thank my advisor, Prof. Joao˜ Nunes, for all the time he took helping me, from my first semester as a mathematics student in Riemannian geometry, up to now in my thesis. He was always available for help and was always extremely patient. I also want to express my gratitude to all the teachers of the courses I took during these last two years, for all the knowledge they gave me in class, and for helping me when I had questions about the material outside of the class. This thesis started as a project for the Centre for Mathematical Analysis, Geometry, and Dynamical Systems, that awarded me a scholarship one year and a half ago. So I would like to thank CAMGSD for this opportunity to learn and do math in a way that is closer to the professional level. I also want to thank the members of the Examination Committee, Prof. Miguel Abreu, Prof. Jose´ Mourao,˜ and Prof. S´ılvia Anjos for their time reading and evaluating my thesis, and for their helpful corrections. During the time I was in Lisbon, I had the opportunity to meet new people and spend time with many excellent friends, who enriched my life. They would be too many to list, but I want to thank my friends from aerospace engineering, especially Tiago Silva, Manuel Barjona and Telmo Pires who encouraged me to become a better student, my neighbors from B1 for making it the best floor in RDP, as I do want to thank my friends from my class Z in Braga for the past 9 years. A special mention goes to Andre´ Pereira, for helping me in many occasions. Coming to study to a different city was not an easy experience for me, and I couldn’t have done it without the continuous support of my parents. When I decided to change from engineering to maths, they gave me complete freedom to do so. They have always been there for me. A big thank you, Mom and Dad. I want to thank my brother Pedro for all the memes and for putting up with my explanations of the math I was learning. Pedro, every aspiring physicist should know what a cotangent bundle is. Thank you to all the other members of my family, uncles, and cousins for caring for me. A special thank you to my grandmother Maria, for all the meals she cooked for me, but also for being a caring, strong and inspiring person. Finally, if you are reading this, I want to thank you, and I hope that the text is clear, organized and I hope that it is helpful to your studies. v vi Resumo Nesta tese estudamos geometria complexa, simplectica´ e Kahler.¨ O cap´ıtulo1 e´ introdutorio´ e apresentamos os preliminares teoricos´ relevantes para o resto da tese. No cap´ıtulo2 apresentamos um resumo de [14]. Partindo de uma variedade K ahler,¨ atuamos na sua estrutura complexa via pullback pelo fluxo de um campo vetorial Hamiltoniano. Provamos que as metricas´ Kahler¨ resultantes formam uma curva numa variedade Riemanniana (de dimensao˜ infinita) cujos pontos sao˜ potenciais Kahler,¨ e que esta curva e´ na realidade uma geodesica´ relativamente a` metrica´ (que e´ a metrica´ de Mabuchi). No cap´ıtulo3 estudamos estruturas K ahler¨ no fibrado cotangente de um grupo de Lie G. Explicamos as estruturas Kahler¨ conhecidas de Hall e Kirwin e de Kirwin, Mourao˜ e Nunes, e relacionamos as duas. Em [12] os autores provam que e´ poss´ıvel definir estruturas Kahler¨ em T ∗G usando func¸oes˜ complexificadoras Ad-invariantes. Provamos que a condic¸ao˜ de Ad-invarianciaˆ pode ser substitu´ıda por um conjunto de equac¸oes˜ mais geral. No cap´ıtulo5, damos um exemplo de quantizac¸ ao˜ geometrica´ de T ∗G. Introduzimos um fibrado de ∗ linha L em T G e uma polarizac¸ao˜ Pτ definida por fazer um pullback da polarizac¸ao˜ vertical pelo fluxo complexificado de um campo vetorial Hamiltoniano. A partir destas estruturas, definimos um espac¸o de Hilbert cujos elementos sao˜ as secc¸oes˜ integraveis´ e Pτ -polarizadas de L. Palavras-chave: geometria Kahler,¨ quantizac¸ao˜ geometrica,´ metrica´ de Mabuchi, grupo de Lie, fibrado cotangente, difeomorfismos complexificados vii viii Abstract In this thesis, we study complex, symplectic and Kahler¨ geometry. Chapter1 is introductory and presents the relevant theoretical background for the rest of the thesis. In chapter2 we present an overview of [14]. Starting with a K ahler¨ manifold, we act on its complex structure by pulling back along the complexified flow of an Hamiltonian vector field. We prove that the resulting Kahler¨ metrics form a curve in an (infinite dimensional) Riemannian manifold whose points are Kahler¨ potentials and that this curve is actually a geodesic with respect to the metric (which is the Mabuchi metric). In chapter3 we study K ahler¨ structures on the cotangent bundle of a Lie group G. We explain the known Kahler¨ structures of Hall and Kirwin and of Kirwin, Mourao,˜ and Nunes and relate the two. In [12], the authors prove that one can define Kahler¨ structures on T ∗G using Ad-invariant complexifier functions. We prove that the condition of Ad-invariance can be replaced by a more general set of equations. In chapter5, we give an example of geometric quantization of T ∗G. We introduce a line bundle L on ∗ T G and a polarization Pτ defined by pulling back the vertical polarization along the complexified flow of an Hamiltonian vector field. From these structures, we define a Hilbert space whose elements are integrable Pτ -polarized sections of L. Keywords: Kahler¨ geometry, geometric quantization, Mabuchi metric, Lie group, cotangent bundle, complexified diffeomorphisms ix x Contents Acknowledgments...........................................v Resumo................................................. vii Abstract................................................. ix List of Tables.............................................. xiii Nomenclature.............................................. xv Glossary................................................1 1 Theoretical Background1 1.1 Symplectic Manifolds.......................................1 1.2 Moser’s Theorem.........................................3 1.3 Complex Geometry........................................3 1.3.1 Almost complex manifolds. Complex manifolds....................3 1.3.2 Splittings and forms on complex manifolds.......................5 1.3.3 When is an almost complex manifold a complex manifold?..............6 1.3.4 Compatible structures. Kahler¨ manifolds........................7 1.3.5 Cohomology results on Kahler¨ manifolds.......................8 1.4 Lie groups and Lie algebras................................... 11 1.4.1 Representations..................................... 11 1.4.2 Adjoint representations................................. 12 1.5 Structure functions and structure constants.......................... 13 2 Geodesics on the Space of Kahler¨ Metrics of a Manifold 15 2.1 Lie Series and complexified flows................................ 15 2.2 Action of a complexified analytic flow on a complex structure................ 18 2.3 Restriction to Hamiltonian flows................................. 19 2.4 Action on Kahler¨ structures................................... 19 2.5 Geodesics on the Space of Kahler¨ metrics........................... 24 2.6 Symplectic picture and complex picture............................ 25 2.7 The path of Kahler¨ metrics (generated by a complex flow of an Hamiltonian vector field) is a geodesic............................................ 26 xi 3 Kahler¨ structures on the Cotangent Bundle of a Lie Group 31 3.1 T ∗G is diffeomorphic to G × g .................................. 31 3.2 Kahler¨ structures on T ∗G .................................... 32 3.2.1 Standard Kahler¨ structure on T ∗G ........................... 32 3.2.2 The Kahler¨ structure of Hall and Kirwin........................ 33 3.2.3 The Kahler¨ structure of Kirwin, Mourao˜ and Nunes.................. 34 3.2.4 Problems to study.................................... 35 3.3 Useful coordinates based on the diffeomorphism T ∗G =∼ G × g ............... 35 3.4 Relation between theorems 3.2.2 and 3.2.4.......................... 41 3.5 The theorem of Hall and Kirwin in the case of a Lie group.................. 43 3.5.1 Attempt at proving conjecture 3.5.1........................... 43 3.5.2 !β as a pullback of !ST by a diffeomorphism..................... 45 3.6 The theorem of Kirwin, Mourao˜ and Nunes in the case of non Ad-invariant h ........ 47 τ 3.6.1 'h is a diffeomorphism. The complex structure J .................. 47 ∗ 3.6.2 A basis for T1;0(T G) .................................. 48 3.6.3 Equations for T ∗G being Kahler¨ in terms of h ..................... 49 4 A Short Overview of Geometric Quantization 53 4.1 Introduction and motivation................................... 53 4.2 General set up.......................................... 54 5 An Example of Quantization of the Cotangent Bundle of a Lie Group 55 5.1 General setup........................................... 55 5.2 A prequantum line bundle for T ∗G ............................... 56 5.3 A polarization for T ∗G ...................................... 57 5.4 T ∗G is foliated
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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
    [Show full text]
  • 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
    [Show full text]
  • 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.
    [Show full text]
  • Holomorphic Factorization for a Quantum Tetrahedron
    Holomorphic Factorization for a Quantum Tetrahedron Laurent Freidel,1 Kirill Krasnov,2 and Etera R. Livine3 1Perimeter Institute for Theoretical Physics, Waterloo N2L 2Y5, Ontario, Canada. 2 School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK. 3Laboratoire de Physique, ENS Lyon, CNRS-UMR 5672, 46 All´ee d’Italie, Lyon 69007, France. (Dated: v2: December 2009) We provide a holomorphic description of the Hilbert space j1,...,jn of SU(2)-invariant ten- sors (intertwiners) and establish a holomorphically factorizedH formula for the decomposition of identity in j1,...,jn . Interestingly, the integration kernel that appears in the decomposi- tion formula turnsH out to be the n-point function of bulk/boundary dualities of string theory. Our results provide a new interpretation for this quantity as being, in the limit of large con- formal dimensions, the exponential of the K¨ahler potential of the symplectic manifold whose quantization gives j1,...,jn . For the case n=4, the symplectic manifold in question has the interpretation of theH space of “shapes” of a geometric tetrahedron with fixed face areas, and our results provide a description for the quantum tetrahedron in terms of holomorphic coher- ent states. We describe how the holomorphic intertwiners are related to the usual real ones by computing their overlap. The semi-classical analysis of these overlap coefficients in the case of large spins allows us to obtain an explicit relation between the real and holomorphic description of the space of shapes of the tetrahedron. Our results are of direct relevance for the subjects of loop quantum gravity and spin foams, but also add an interesting new twist to the story of the bulk/boundary correspondence.
    [Show full text]
  • Tautological Tuning of the Kostant-Souriau Quantization Map 2
    Tautological Tuning of the Kostant-Souriau Quantization Map with Differential Geometric Structures T. McClain ∗ Abstract For decades, mathematical physicists have searched for a coordinate independent quan- tization procedure to replace the ad hoc process of canonical quantization. This ef- fort has largely coalesced into two distinct research programs: geometric quantization and deformation quantization. Though both of these programs can claim numerous successes, neither has found mainstream acceptance within the more experimentally minded quantum physics community, owing both to their mathematical complexities and their practical failures as empirical models. This paper introduces an alternative approach to coordinate-independent quantization called tautologically tuned quanti- zation. This approach uses only differential geometric structures from symplectic and Riemannian geometry, especially the tautological one form and vector field (hence the name). In its focus on physically important functions, tautologically tuned quantiza- tion hews much more closely to the ad hoc approach of canonical quantization than ei- ther traditional geometric quantization or deformation quantization and thereby avoid some of the mathematical challenges faced by those methods. Given its focus on stan- dard differential geometric structures, tautologically tuned quantization is also a better candidate than either traditional geometric or deformation quantization for applica- tion to covariant Hamiltonian field theories, and therefore may pave the way for the
    [Show full text]
  • GEOMETRIC QUANTIZATION Contents 1. Introduction 1 2
    GEOMETRIC QUANTIZATION ZIHNI KAAN BAYKARA Abstract. Geometric Quantization (GQ) is a program for obtaining a quan- tum mechanical system given a classical mechanical system. The object of quantization need not be a physically feasible mechanical system, symplectic manifolds (with sufficiently nice properties) are also quantizable. For exam- ple, the quantization procedure for attaining spin originates from the 2-sphere, and not a cotangent bundle of some phase space. However, quantization is not a categorical functor between symplectic manifolds and quantum mechanical spaces, therefore this procedure may not seem completely natural. This ex- pository paper assumes little background in differential geometry, and some familiarity with analytical mechanics, and quantum mechanics. Contents 1. Introduction 1 2. Classical Mechanics 2 3. Quantum Mechanics 6 4. Quantization 7 5. Prequantization 8 6. Polarization 11 6.1. Problem with Prequantization 11 6.2. Distribution and Polarization 12 6.3. K¨ahlerPolarizations 14 7. Half-form Correction 16 Acknowledgments 20 References 20 1. Introduction Geometric Quantization aims to answer the question "What is the correspond- ing quantum mechanical system for a given classical system?" mathematically. We won't have a correspondence between the states of a classical system (points of a manifold) and the states of a quantum system (complex square integrable func- tions). The objects of interest to the observer are, as the name suggests, observables, and not states. States are simply the mathematical objects from which the observ- ables, the relevant variables for physics, are derived from. For a physical theory, as long as the observables are consistent with experiment, the rest of the objects of the theory can be seen as appendices, which can later be related mathematically to other theories.
    [Show full text]
  • Arxiv:2010.14339V1 [Math.DG]
    GEOMETRIC QUANTIZATION AND QUANTUM MOMENT MAPS ON COADJOINT ORBITS AND KAHLER-EINSTEIN¨ MANIFOLDS NAICHUNG CONAN LEUNG, QIN LI, AND ZIMING NIKOLAS MA Abstract. Deformation quantization and geometric quantization on K¨ahler manifolds give the mathematical description of the algebra of quantum observables and the Hilbert spaces respectively, where the later forms a representation of quantum observables asymptotically via Toeplitz operators. When there is a Hamiltonian G-action on a K¨ahler manifold, there are associated symmetries on both the quantum algebra and representation aspects. We show that in nice cases of coadjoint orbits and K¨ahler-Einstein manifolds, these symmetries are strictly compatible (not only asymptotically). 1. Introduction ∗ n j Quantizing a classical mechanical system on X = T R ,ω = dx ∧ dpj is amount to treating j j j ℏP∂ 2 Rn position x and moment pj as operatorsx ˆ = x · andp ˆj = i ∂xj on L ( ) respectively, thus realizing the uncertainty principle xˆj, pˆ = iℏδj . In fact, this gives an action of C∞ (X) [[ℏ]] on k k L2 (Rn) with a non-commutative algebra structure ⋆ℏ on C∞ (X) [[ℏ]]. L2 (Rn) can also be realized 2 as ΓL2 (X,L) ∩ Ker (∇PR ), the space of L -sections of the trivial line bundle L, equipped with a unitary connection ∇ with curvature −iω, over X which are constant along the real polarization ∂ PR = ’s , namely those sections which are independent of p ’s. D ∂pj E j Quantizing a physical system should be independent of the choice of polarizations. If we use the ∂ n j complex polarization PC = ’s on X = C with zj = x + ipj, then D ∂z¯j E ¯ 0 ΓL2 (X,L) ∩ Ker (∇PC )=ΓL2 (X,L) ∩ Ker(∂)= HL2 (X,L) is the space of L2-holomorphic sections of the trivial line bundle L.
    [Show full text]
  • Angular Momentum Coupling, Dirac Oscillators, and Quantum Band Rearrangements in the Presence of Momentum Reversal Symmetries
    JOURNAL OF GEOMETRIC MECHANICS doi:10.3934/jgm.2020021 c American Institute of Mathematical Sciences Volume 12, Number 3, September 2020 pp. 455{505 ANGULAR MOMENTUM COUPLING, DIRAC OSCILLATORS, AND QUANTUM BAND REARRANGEMENTS IN THE PRESENCE OF MOMENTUM REVERSAL SYMMETRIES Toshihiro Iwai Department of Applied Mathematics and Physics Kyoto University Kyoto 606, Japan Dmitri´ı A. Sadovski´ı and Boris I. Zhilinski´ı∗ Department of Physics Universit´edu Littoral|C^oted'Opale 59140 Dunkerque, France Dedicated to James Montaldi Abstract. We investigate the elementary rearrangements of energy bands in slow-fast one-parameter families of systems whose fast subsystem possesses a half-integer spin. Beginning with a simple case without any time-reversal symmetries, we analyze and compare increasingly sophisticated model Hamil- tonians with these symmetries. The models are inspired by the time-reversal modification of the Berry phase setup which uses a family of quadratic spin- quadrupole Hamiltonians of Mead [Phys. Rev. Lett. 59, 161{164 (1987)] and Avron et al [Commun. Math. Phys. 124(4), 595{627 (1989)]. An explicit cor- respondence between the typical quantum energy level patterns in the energy band rearrangements of the finite particle systems with compact slow phase space and those of the Dirac oscillator is found in the limit of linearization near the conical degeneracy point of the semi-quantum eigenvalues. 1. Introduction. The interest in the geometric and topological properties of para- metric families of quantum Hamiltonians was largely stimulated by the seminal pa- per of Michael Berry [10], who analyzed the evolution of the spin states with spin 1 S = 2 ≈ kSk in the presence of the magnetic field B.
    [Show full text]