DOCSLIB.ORG

Quantum Mechanics for Mathematicians Leon A. Takhtajan

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Quantum Mechanics for Mathematicians Leon A. Takhtajan Quantum Mechanics for Mathematicians Leon A. Takhtajan Department of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651, USA E-mail address: [email protected] To my teacher Ludwig Dmitrievich Faddeev with admiration and gratitude. Contents Preface xiii Part 1. Foundations Chapter 1. Classical Mechanics 3 1. Lagrangian Mechanics 4 § 1.1. Generalized coordinates 4 1.2. The principle of the least action 4 1.3. Examples of Lagrangian systems 8 1.4. Symmetries and Noether’s theorem 15 1.5. One-dimensional motion 20 1.6. The motion in a central field and the Kepler problem 22 1.7. Legendre transform 26 2. Hamiltonian Mechanics 31 § 2.1. Hamilton’s equations 31 2.2. The action functional in the phase space 33 2.3. The action as a function of coordinates 35 2.4. Classical observables and Poisson bracket 38 2.5. Canonical transformations and generating functions 39 2.6. Symplectic manifolds 42 2.7. Poisson manifolds 51 2.8. Hamilton’s and Liouville’s representations 56 3. Notes and references 60 § Chapter 2. Basic Principles of Quantum Mechanics 63 1. Observables, states, and dynamics 65 § 1.1. Mathematical formulation 66 vii viii Contents 1.2. Heisenberg’s uncertainty relations 74 1.3. Dynamics 75 2. Quantization 79 § 2.1. Heisenberg commutation relations 80 2.2. Coordinate and momentum representations 86 2.3. Free quantum particle 93 2.4. Examples of quantum systems 98 2.5. Old quantum mechanics 102 2.6. Harmonic oscillator 103 2.7. Holomorphic representation and Wick symbols 111 3. Weyl relations 118 § 3.1. Stone-von Neumann theorem 119 3.2. Invariant formulation 125 3.3. Weyl quantization 128 3.4. The ?-product 136 3.5. Deformation quantization 139 4. Notes and references 146 § Chapter 3. Schr¨odinger Equation 149 1. General properties 149 § 1.1. Self-adjointness 150 1.2. Characterization of the spectrum 152 1.3. The virial theorem 154 2. One-dimensional Schr¨odinger equation 155 § 2.1. Jost functions and transition coefficients 155 2.2. Eigenfunction expansion 163 2.3. S-matrix and scattering theory 170 2.4. Other boundary conditions 177 3. Angular momentum and SO(3) 180 § 3.1. Angular momentum operators 180 3.2. Representation theory of SO(3) 182 4. Two-body problem 185 § 4.1. Separation of the center of mass 185 4.2. Three-dimensional scattering theory 186 4.3. Particle in a central potential 188 5. Hydrogen atom and SO(4) 193 § 5.1. Discrete spectrum 193 5.2. Continuous spectrum 197 5.3. Hidden SO(4) symmetry 199 6. Semi-classical asymptotics – I 205 § 6.1. Time-dependent asymptotics 206 Contents ix 6.2. Time independent asymptotics 209 6.3. Bohr-Wilson-Sommerfeld quantization rules 212 7. Notes and references 213 § Chapter 4. Spin and Identical Particles 217 1. Spin 217 § 1.1. Spin operators 217 1.2. Spin and representation theory of SU(2) 218 2. Charged spin particle in the magnetic field 221 § 2.1. Pauli Hamiltonian 221 2.2. Particle in a uniform magnetic field 223 3. System of identical particles 224 § 3.1. The symmetrization postulate 225 3.2. Young diagrams and representation theory of SymN 229 3.3. Schur-Weyl duality and symmetry of the wave functions 232 4. Notes and references 234 § Part 2. Functional Methods and Supersymmetry Chapter 5. Path Integral Formulation of Quantum Mechanics 239 1. Feynman path integral 239 § 1.1. The fundamental solution of the Schr¨odinger equation 239 1.2. Feynman path integral in the phase space 242 1.3. Feynman path integral in the configuration space 245 1.4. Several degrees of freedom 247 2. Symbols of the evolution operator and path integrals 249 § 2.1. The pq-symbol 249 2.2. The qp-symbol 250 2.3. The Weyl symbol 251 2.4. The Wick symbol 252 3. Feynman path integral for the harmonic oscillator 255 § 3.1. Gaussian integration 255 3.2. Propagator of the harmonic oscillator 257 3.3. Mehler identity 259 4. Gaussian path integrals 260 § 4.1. Gaussian path integral for a free particle 261 4.2. Gaussian path integral for the harmonic oscillator 264 5. Regularized determinants of di↵erential operators 268 § 5.1. Dirichlet boundary conditions 268 5.2. Periodic boundary conditions 274 5.3. First order di↵erential operators 278 x Contents 6. Semi-classical asymptotics – II 280 § 6.1. Using the Feynman path integral 281 6.2. Rigorous derivation 282 7. Notes and references 285 § Chapter 6. Integration in Functional Spaces 289 1. Gaussian measures 289 § 1.1. Finite-dimensional case 289 1.2. Infinite-dimensional case 291 2. Wiener measure and Wiener integral 293 § 2.1. Definition of the Wiener measure 293 2.2. Conditional Wiener measure and Feynman-Kac formula 296 2.3. Relation between Wiener and Feynman integrals 299 3. Gaussian Wiener integrals 301 § 3.1. Dirichlet boundary conditions 301 3.2. Periodic boundary conditions 302 4. Notes and references 305 § Chapter 7. Fermion Systems 307 1. Canonical anticommutation relations 307 § 1.1. Motivation 307 1.2. Cli↵ord algebras 311 2. Grassmann algebras 314 § 2.1. Realization of canonical anticommutation relations 315 2.2. Di↵erential forms 317 2.3. Berezin integral 319 3. Graded linear algebra 324 § 3.1. Graded vector spaces and superalgebras 324 3.2. Examples of superalgebras 326 3.3. Supertrace and Berezinian 328 4. Path integrals for anticommuting variables 330 § 4.1. Wick and matrix symbols 330 4.2. Path integral for the evolution operator 335 4.3. Gaussian path integrals over Grassmann variables 338 5. Notes and references 340 § Chapter 8. Supersymmetry 343 1. Supermanifolds 343 § 2. Equivariant cohomology and localization 345 § 2.1. Finite-dimensional case 345 2.2. Infinite-dimensional case 349 Contents xi 3. Classical mechanics on supermanifolds 354 § 3.1. Functions with anticommuting values 354 3.2. Classical systems 356 4. Supersymmetry 359 § 4.1. Total angular momentum 359 4.2. Supersymmetry transformation 360 4.3. Supersymmetric particle on a Riemannian manifold 362 5. Quantum mechanics on supermanifolds 364 § 6. Atiyah-Singer index formula 369 § 7. Notes and references 370 § Bibliography 373 Index 383 Preface This book is based on graduate courses taught by the author over the last fourteen years in the mathematics department of Stony Brook University. The goal of these courses was to introduce second year graduate students with no prior knowledge of physics to the basic concepts and methods of quantum mechanics. For the last 50 years quantum physics has been a driv- ing force behind many dramatic achievements in mathematics, similar to the role played by classical physics in the seventeenth to nineteenth cen- turies. Classical physics, especially classical mechanics, was an integral part of mathematical education up to the early twentieth century, with lecture courses given by Hilbert and Poincar´e.Surprisingly, quantum physics, es- pecially quantum mechanics, with its intrinsic beauty and connections with many branches of mathematics, has never been a part of a graduate math- ematics curriculum. This course was developed to partially fill this gap and to make quantum mechanics accessible to graduate students and research mathematicians. L.D. Faddeev was the first to develop a course in quantum mechanics for undergraduate students specializing in mathematics. From 1968 to 1973 he regularly lectured in the mathematics department of St. Petersburg State University in St. Petersburg, Russia1, and the author enjoyed the opportu- nity to take his course. The notes for this book emerged from an attempt to create a similar course for graduate students, which uses more sophisticated mathematics and covers a larger variety of topics, including the Feynman path integral approach to quantum mechanics. 1At that time in Leningrad, Soviet Union. xiii xiv Preface There are many excellent physics textbooks on quantum mechanics, starting with the classic texts by P.A.M. Dirac [Dir47], L.D. Landau and E.M. Lifshitz [LL58], and V.A. Fock [Foc78], to the encyclopedic trea- tise by A. Messiah [Mes99], the recent popular textbook by J.J. Sakurai [Sak94], and many others. From a mathematics perspective, there are clas- sic monographs by J. von Neumann [vN96] and by H. Weyl [Wey50], as well as a more recent book by G.W. Mackey [Mac04], which deal with the basic mathematical formalism and logical foundations of the theory. There is also a monumental project [DEF+99], created with the purpose of in- troducing graduate students and research mathematicians to the realm of quantum fields and strings, both from a mathematics and a physics perspec- tive. Though it contains a very detailed exposition of classical mechanics, classical field theory, and supersymmetry, oriented at the mathematical au- dience, quantum mechanics is discussed only briefly (with the exception of L.D. Faddeev’s elegant introduction to quantum mechanics in [Fad99]). Excellent lecture notes for undergraduate students by L.D. Faddeev and O.A. Yakubovskiˇi[FY80] seems to be the only book on quantum mechanics completely accessible to mathematicians2. Recent books by S.J. Gustafson and I.M. Sigal [GS03] and by F. Strocchi [Str05] are also oriented at math- ematicians. The latter is a short introductory course, while the former is more an intermediate level monograph on quantum theory rather than a textbook on quantum mechanics. There are also many specialized books on di↵erent parts of quantum mechanics, like scattering theory, the Schr¨odinger operator, C⇤-algebras and foundations, etc. The present book gives a comprehensive treatment of quantum mechan- ics from a mathematics perspective and covers such topics as mathematical foundations, quantization, the Schr¨odinger equation, the Feynman path in- tegral and functional methods, and supersymmetry. It can be used as a one-year graduate course, or as two one-semester courses: the introductory course based on the material in Part 1, and a more advanced course based on Part 2.
Load more

Recommended publications
  • Quantum Groups and Algebraic Geometry in Conformal Field Theory
    QUANTUM GROUPS AND ALGEBRAIC GEOMETRY IN CONFORMAL FIELD THEORY DlU'KKERU EI.INKWIJK BV - UTRECHT QUANTUM GROUPS AND ALGEBRAIC GEOMETRY IN CONFORMAL FIELD THEORY QUANTUMGROEPEN EN ALGEBRAISCHE MEETKUNDE IN CONFORME VELDENTHEORIE (mrt em samcnrattint] in hit Stdirlands) PROEFSCHRIFT TER VERKRIJGING VAN DE GRAAD VAN DOCTOR AAN DE RIJKSUNIVERSITEIT TE UTRECHT. OP GEZAG VAN DE RECTOR MAGNIFICUS. TROF. DR. J.A. VAN GINKEI., INGEVOLGE HET BESLUIT VAN HET COLLEGE VAN DE- CANEN IN HET OPENBAAR TE VERDEDIGEN OP DINSDAG 19 SEPTEMBER 1989 DES NAMIDDAGS TE 2.30 UUR DOOR Theodericus Johannes Henrichs Smit GEBOREN OP 8 APRIL 1962 TE DEN HAAG PROMOTORES: PROF. DR. B. DE WIT PROF. DR. M. HAZEWINKEL "-*1 Dit proefschrift kwam tot stand met "•••; financiele hulp van de stichting voor Fundamenteel Onderzoek der Materie (F.O.M.) Aan mijn ouders Aan Saskia Contents Introduction and summary 3 1.1 Conformal invariance and the conformal bootstrap 11 1.1.1 Conformal symmetry and correlation functions 11 1.1.2 The conformal bootstrap program 23 1.2 Axiomatic conformal field theory 31 1.3 The emergence of a Hopf algebra 4G The modular geometry of string theory 56 2.1 The partition function on moduli space 06 2.2 Determinant line bundles 63 2.2.1 Complex line bundles and divisors on a Riemann surface . (i3 2.2.2 Cauchy-Riemann operators (iT 2.2.3 Metrical properties of determinants of Cauchy-Ricmann oper- ators 6!) 2.3 The Mumford form on moduli space 77 2.3.1 The Quillen metric on determinant line bundles 77 2.3.2 The Grothendieck-Riemann-Roch theorem and the Mumford
    [Show full text]
  • Non-Self-Adjoint Toeplitz Matrices Whose Principal Submatrices Have Real Spectrum
    Constr Approx (2019) 49:191–226 https://doi.org/10.1007/s00365-017-9408-0 Non-Self-Adjoint Toeplitz Matrices Whose Principal Submatrices Have Real Spectrum Boris Shapiro1 · František Štampach1 Received: 29 March 2017 / Revised: 28 August 2017 / Accepted: 5 October 2017 / Published online: 7 December 2017 © The Author(s) 2017. This article is an open access publication Abstract We introduce and investigate a class of complex semi-infinite banded Toeplitz matrices satisfying the condition that the spectra of their principal submatri- ces accumulate onto a real interval when the size of the submatrix grows to ∞.We prove that a banded Toeplitz matrix belongs to this class if and only if its symbol has real values on a Jordan curve located in C\{0}. Surprisingly, it turns out that, if such a Jordan curve is present, the spectra of all the principal submatrices have to be real. The latter claim is also proved for matrices given by a more general symbol. The special role of the Jordan curve is further demonstrated by a new formula for the limiting density of the asymptotic eigenvalue distribution for banded Toeplitz matrices from the studied class. Certain connections between the problem under investigation, Jacobi operators, and the Hamburger moment problem are also discussed. The main results are illustrated by several concrete examples; some of them allow an explicit analytic treatment, while some are only treated numerically. Keywords Banded Toeplitz matrix · Asymptotic eigenvalue distribution · Real spectrum · Non-self-adjoint matrices · Moment problem · Jacobi matrices · Orthogonal polynomials Mathematics Subject Classification 15B05 · 47B36 · 33C47 Communicated by Serguei Denissov.
    [Show full text]
  • Quantum Group and Quantum Symmetry
    QUANTUM GROUP AND QUANTUM SYMMETRY Zhe Chang International Centre for Theoretical Physics, Trieste, Italy INFN, Sezione di Trieste, Trieste, Italy and Institute of High Energy Physics, Academia Sinica, Beijing, China Contents 1 Introduction 3 2 Yang-Baxter equation 5 2.1 Integrable quantum field theory ...................... 5 2.2 Lattice statistical physics .......................... 8 2.3 Yang-Baxter equation ............................ 11 2.4 Classical Yang-Baxter equation ...................... 13 3 Quantum group theory 16 3.1 Hopf algebra ................................. 16 3.2 Quantization of Lie bi-algebra ....................... 19 3.3 Quantum double .............................. 22 3.3.1 SUq(2) as the quantum double ................... 23 3.3.2 Uq (g) as the quantum double ................... 27 3.3.3 Universal R-matrix ......................... 30 4Representationtheory 33 4.1 Representations for generic q........................ 34 4.1.1 Representations of SUq(2) ..................... 34 4.1.2 Representations of Uq(g), the general case ............ 41 4.2 Representations for q being a root of unity ................ 43 4.2.1 Representations of SUq(2) ..................... 44 4.2.2 Representations of Uq(g), the general case ............ 55 1 5 Hamiltonian system 58 5.1 Classical symmetric top system ...................... 60 5.2 Quantum symmetric top system ...................... 66 6 Integrable lattice model 70 6.1 Vertex model ................................ 71 6.2 S.O.S model ................................. 79 6.3 Configuration
    [Show full text]
  • Branched Hamiltonians and Supersymmetry
    Branched Hamiltonians and Supersymmetry Thomas Curtright, University of Miami Wigner 111 seminar, 12 November 2013 Some examples of branched Hamiltonians are explored, as recently advo- cated by Shapere and Wilczek. These are actually cases of switchback poten- tials, albeit in momentum space, as previously analyzed for quasi-Hamiltonian dynamical systems in a classical context. A basic model, with a pair of Hamiltonian branches related by supersymmetry, is considered as an inter- esting illustration, and as stimulation. “It is quite possible ... we may discover that in nature the relation of past and future is so intimate ... that no simple representation of a present may exist.” – R P Feynman Based on work with Cosmas Zachos, Argonne National Laboratory Introduction to the problem In quantum mechanics H = p2 + V (x) (1) is neither more nor less difficult than H = x2 + V (p) (2) by reason of x, p duality, i.e. the Fourier transform: ψ (x) φ (p) ⎫ ⎧ x ⎪ ⎪ +i∂/∂p ⎪ ⇐⇒ ⎪ ⎬⎪ ⎨⎪ i∂/∂x p − ⎪ ⎪ ⎪ ⎪ ⎭⎪ ⎩⎪ This equivalence of (1) and (2) is manifest in the QMPS formalism, as initiated by Wigner (1932), 1 2ipy/ f (x, p)= dy x + y ρ x y e− π | | − 1 = dk p + k ρ p k e2ixk/ π | | − where x and p are on an equal footing, and where even more general H (x, p) can be considered. See CZ to follow, and other talks at this conference. Or even better, in addition to the excellent books cited at the conclusion of Professor Schleich’s talk yesterday morning, please see our new book on the subject ... Even in classical Hamiltonian mechanics, (1) and (2) are equivalent under a classical canonical transformation on phase space: (x, p) (p, x) ⇐⇒ − But upon transitioning to Lagrangian mechanics, the equivalence between the two theories becomes obscure.
    [Show full text]
  • Operators, Functions and Linear Systems
    Operators, Functions and Linear Systems Organizers: S. ter Horst and M.A. Kaashoek Daniel Alpay, Ben-Gurion University of the Negev Title: Schur analysis in the setting of slice hyper-holomorphic functions Abstract: In the setting of Clifford algebras and quaternions, there are at least two extensions of the notion of analyticity, Fueter series and slice hyper-holomorphic functions. This last notion is of particular importance since it relates to the functional calculus of N possibly non-commuting oper- ators on a real Banach space. In the talk we report on recent joint work (see [1]–[4]) with Fabrizio Colombo and Irene Sabadini (Politecnico Milano), where the notion of functions analytic and con- tractive in the open unit disk are replaced by functions slice hyper-holomorphic in the open unit ball of the quaternions and bounded by one in modulus there. We outline the beginning of a Schur analysis in this setting. [1] D. Alpay, F. Colombo and I. Sabadini. Schur functions and their realizations in the slice hyperholomorphic setting. Integral Equations and Operator Theory, vol. 72 (2012), pp. 253-289. [2] D. Alpay, F. Colombo and I. Sabadini. Krein-Langer factorization and related topics in the slice hyperholomorphic setting. Journal of Geometric Analysis. To appear. [3] D. Alpay, F. Colombo and I. Sabadini. Pontryagin de Branges-Rovnyak spaces of slice hyper- holomorphic functions. Journal d’Analyse Mathematique.´ To appear. [4] D. Alpay, F. Colombo and I. Sabadini. On some notions of convergence for n-tuples of opera- tors. Mathematical Methods in the Applied Sciences. To appear. Joseph A. Ball, Virginia Tech, Blacksburg Title: Convexity analysis and integral representations for generalized Schur/Herglotz function classes Abstract: We report on recent work with M.D.
    [Show full text]
  • Non-Commutative Phase and the Unitarization of GL {P, Q}(2)
    Non–commutative phase and the unitarization of GLp,q (2) M. Arık and B.T. Kaynak Department of Physics, Bo˘gazi¸ci University, 80815 Bebek, Istanbul, Turkey Abstract In this paper, imposing hermitian conjugate relations on the two– parameter deformed quantum group GLp,q (2) is studied. This results in a non-commutative phase associated with the unitarization of the quantum group. After the achievement of the quantum group Up,q (2) with pq real via a non–commutative phase, the representation of the algebra is built by means of the action of the operators constituting the Up,q (2) matrix on states. 1 Introduction The mathematical construction of a quantum group Gq pertaining to a given Lie group G is simply a deformation of a commutative Poisson-Hopf algebra defined over G. The structure of a deformation is not only a Hopf algebra arXiv:hep-th/0208089v2 18 Oct 2002 but characteristically a non-commutative algebra as well. The notion of quantum groups in physics is widely known to be the generalization of the symmetry properties of both classical Lie groups and Lie algebras, where two different mathematical blocks, namely deformation and co–multiplication, are simultaneously imposed either on the related Lie group or on the related Lie algebra. A quantum group is defined algebraically as a quasi–triangular Hopf alge- bra. It can be either non-commutative or commutative. It is fundamentally a bi–algebra with an antipode so as to consist of either the q–deformed uni- versal enveloping algebra of the classical Lie algebra or its dual, called the matrix quantum group, which can be understood as the q–analog of a classi- cal matrix group [1].
    [Show full text]
  • Non-Commutative Dual Representation for Quantum Systems on Lie Groups
    Loops 11: Non-Perturbative / Background Independent Quantum Gravity IOP Publishing Journal of Physics: Conference Series 360 (2012) 012052 doi:10.1088/1742-6596/360/1/012052 Non-commutative dual representation for quantum systems on Lie groups Matti Raasakka Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am M¨uhlenberg 1, 14476 Potsdam, Germany, EU E-mail: [email protected] Abstract. We provide a short overview of some recent results on the application of group Fourier transform to quantum mechanics and quantum gravity. We close by pointing out some future research directions. 1. Introduction The group Fourier transform is an integral transform from functions on a Lie group to functions on a non-commutative dual space. It was first formulated for functions on SO(3) [1, 2], later generalized to SU(2) [2–4] and other Lie groups [5, 6], and is based on the quantum group structure of Drinfel’d double of the group [7, 8]. The transform provides a unitarily equivalent representation of quantum systems with a Lie group configuration space in terms of non- commutative algebra of functions on the classical dual space, the dual of the Lie algebra. As such, it has proven useful in several different ways. Most importantly, it provides a clear connection between the quantum system and the corresponding classical one, allowing for a better physical insight into the system. In the case of quantum gravity models this has turned out to be particularly helpful in unraveling the geometrical content of the models, because the dual variables have an intuitive interpretation as classical geometrical quantities.
    [Show full text]
  • BRST APPROACH to HAMILTONIAN SYSTEMS Our Starting Point Is the Partition Function
    BRST APPROACH TO HAMILTONIAN SYSTEMS A.K.Aringazin1, V.V.Arkhipov2, and A.S.Kudusov3 Department of Theoretical Physics Karaganda State University Karaganda 470074 Kazakstan Preprint KSU-DTP-10/96 Abstract BRST formulation of cohomological Hamiltonian mechanics is presented. In the path integral approach, we use the BRST gauge fixing procedure for the partition func- tion with trivial underlying Lagrangian to fix symplectic diffeomorphism invariance. Resulting Lagrangian is BRST and anti-BRST exact and the Liouvillian of classical mechanics is reproduced in the ghost-free sector. The theory can be thought of as a topological phase of Hamiltonian mechanics and is considered as one-dimensional cohomological field theory with the target space a symplectic manifold. Twisted (anti- )BRST symmetry is related to global N = 2 supersymmetry, which is identified with an exterior algebra. Landau-Ginzburg formulation of the associated d = 1, N = 2 model is presented and Slavnov identity is analyzed. We study deformations and per- turbations of the theory. Physical states of the theory and correlation functions of the BRST invariant observables are studied. This approach provides a powerful tool to investigate the properties of Hamiltonian systems. PACS number(s): 02.40.+m, 03.40.-t,03.65.Db, 11.10.Ef, 11.30.Pb. arXiv:hep-th/9811026v1 2 Nov 1998 [email protected] [email protected] [email protected] 1 1 INTRODUCTION Recently, path integral approach to classical mechanics has been developed by Gozzi, Reuter and Thacker in a series of papers[1]-[10]. They used a delta function constraint on phase space variables to satisfy Hamilton’s equation and a sort of Faddeev-Popov representation.
    [Show full text]
  • 1.25. the Quantum Group Uq (Sl2). Let Us Consider the Lie Algebra Sl2
    55 1.25. The Quantum Group Uq (sl2). Let us consider the Lie algebra sl2. Recall that there is a basis h; e; f 2 sl2 such that [h; e] = 2e; [h; f] = −2f; [e; f] = h. This motivates the following definition. Definition 1.25.1. Let q 2 k; q =6 ±1. The quantum group Uq(sl2) is generated by elements E; F and an invertible element K with defining relations K − K−1 KEK−1 = q2E; KFK−1 = q−2F; [E; F] = : −1 q − q Theorem 1.25.2. There exists a unique Hopf algebra structure on Uq (sl2), given by • Δ(K) = K ⊗ K (thus K is a grouplike element); • Δ(E) = E ⊗ K + 1 ⊗ E; • Δ(F) = F ⊗ 1 + K−1 ⊗ F (thus E; F are skew-primitive ele­ ments). Exercise 1.25.3. Prove Theorem 1.25.2. Remark 1.25.4. Heuristically, K = qh, and thus K − K−1 lim = h: −1 q!1 q − q So in the limit q ! 1, the relations of Uq (sl2) degenerate into the relations of U(sl2), and thus Uq (sl2) should be viewed as a Hopf algebra deformation of the enveloping algebra U(sl2). In fact, one can make this heuristic idea into a precise statement, see e.g. [K]. If q is a root of unity, one can also define a finite dimensional version of Uq (sl2). Namely, assume that the order of q is an odd number `. Let uq (sl2) be the quotient of Uq (sl2) by the additional relations ` ` ` E = F = K − 1 = 0: Then it is easy to show that uq (sl2) is a Hopf algebra (with the co­ product inherited from Uq (sl2)).
    [Show full text]
  • Antibrackets and Supersymmetric Mechanics
    Preprint JINR E2-93-225 Antibrackets and Supersymmetric Mechanics Armen Nersessian 1 Laboratory of Theoretical Physics, JINR Dubna, Head Post Office, P.O.Box 79, 101 000 Moscow, Russia Abstract Using odd symplectic structure constructed over tangent bundle of the symplec- tic manifold, we construct the simple supergeneralization of an arbitrary Hamilto- arXiv:hep-th/9306111v2 29 Jul 1993 nian mechanics on it. In the case, if the initial mechanics defines Killing vector of some Riemannian metric, corresponding supersymmetric mechanics can be refor- mulated in the terms of even symplectic structure on the supermanifold. 1E-MAIL:[email protected] 1 Introduction It is well-known that on supermanifolds M the Poisson brackets of two types can be defined – even and odd ones, in correspondence with their Grassmannian grading 1. That is defined by the expression ∂ f ∂ g {f,g} = r ΩAB(z) l , (1.1) κ ∂zA κ ∂zB which satisfies the conditions p({f,g}κ)= p(f)+ p(g) + 1 (grading condition), (p(f)+κ)(p(g)+κ) {f,g}κ = −(−1) {g, f}κ (”antisymmetrisity”), (1.2) (p(f)+κ)(p(h)+κ) (−1) {f, {g, h}1}1 + cycl.perm.(f, g, h) = 0 (Jacobi id.), (1.3) l A ∂r ∂ where z are the local coordinates on M, ∂zA and ∂zA denote correspondingly the right and the left derivatives, κ = 0, 1 denote correspondingly the even and the odd Poisson brackets. Obviously, the even Poisson brackets can be nondegenerate only if dimM = (2N.M), and the odd one if dimM =(N.N). With nondegenerate Poisson bracket one can associate the symplectic structure A B Ωκ = dz Ω(κ)ABdz , (1.4) BC C where Ω(κ)AB Ωκ = δA .
    [Show full text]
  • The Lagrangian and Hamiltonian Mechanical Systems
    THE LAGRANGIAN AND HAMILTONIAN MECHANICAL SYSTEMS ALEXANDER TOLISH Abstract. Newton's Laws of Motion, which equate forces with the time- rates of change of momenta, are a convenient way to describe mechanical systems in Euclidean spaces with cartesian coordinates. Unfortunately, the physical world is rarely so cooperative|physicists often explore systems that are neither Euclidean nor cartesian. Different mechanical formalisms, like the Lagrangian and Hamiltonian systems, may be more effective at describing such phenomena, as they are geometric rather than analytic processes. In this paper, I shall construct Lagrangian and Hamiltonian mechanics, prove their equivalence to Newtonian mechanics, and provide examples of both non- Newtonian systems in action. Contents 1. The Calculus of Variations 1 2. Manifold Geometry 3 3. Lagrangian Mechanics 4 4. Two Electric Pendula 4 5. Differential Forms and Symplectic Geometry 6 6. Hamiltonian Mechanics 7 7. The Double Planar Pendulum 9 Acknowledgments 11 References 11 1. The Calculus of Variations Lagrangian mechanics applies physics not only to particles, but to the trajectories of particles. We must therefore study how curves behave under small disturbances or variations. Definition 1.1. Let V be a Banach space. A curve is a continuous map : [t0; t1] ! V: A variation on the curve is some function h of t that creates a new curve +h.A functional is a function from the space of curves to the real numbers. p Example 1.2. Φ( ) = R t1 1 +x _ 2dt, wherex _ = d , is a functional. It expresses t0 dt the length of curve between t0 and t1. Definition 1.3.
    [Show full text]
  • Jacobi Operators and Completely Integrable Nonlinear Lattices
    http://dx.doi.org/10.1090/surv/072 Selected Titles in This Series 72 Gerald Teschl, Jacobi operators and completely integrable nonlinear lattices, 2000 71 Lajos Pukanszky, Characters of connected Lie groups, 1999 70 Carmen Chicone and Yuri Latushkin, Evolution semigroups in dynamical systems and differential equations, 1999 69 C. T. C. Wall (A. A. Ranicki, Editor), Surgery on compact manifolds, second edition, 1999 68 David A. Cox and Sheldon Katz, Mirror symmetry and algebraic geometry, 1999 67 A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, second edition, 2000 66 Yu. Ilyashenko and Weigu Li, Nonlocal bifurcations, 1999 65 Carl Faith, Rings and things and a fine array of twentieth century associative algebra, 1999 64 Rene A. Carmona and Boris Rozovskii, Editors, Stochastic partial differential equations: Six perspectives, 1999 63 Mark Hovey, Model categories, 1999 62 Vladimir I. Bogachev, Gaussian measures, 1998 61 W. Norrie Everitt and Lawrence Markus, Boundary value problems and symplectic algebra for ordinary differential and quasi-differential operators, 1999 60 Iain Raeburn and Dana P. Williams, Morita equivalence and continuous-trace C*-algebras, 1998 59 Paul Howard and Jean E. Rubin, Consequences of the axiom of choice, 1998 58 Pavel I. Etingof, Igor B. Frenkel, and Alexander A. Kirillov, Jr., Lectures on representation theory and Knizhnik-Zamolodchikov equations, 1998 57 Marc Levine, Mixed motives, 1998 56 Leonid I. Korogodski and Yan S. Soibelman, Algebras of functions on quantum groups: Part I, 1998 55 J. Scott Carter and Masahico Saito, Knotted surfaces and their diagrams, 1998 54 Casper Goffman, Togo Nishiura, and Daniel Waterman, Homeomorphisms in analysis, 1997 53 Andreas Kriegl and Peter W.
    [Show full text]