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Quantum Mechanics for Mathematicians Leon A. Takhtajan

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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.
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