Gianfausto Dell’Antonio Lectures on the Mathematics of Quantum Mechanics February 12, 2015 Mathematical Department, Universita’ Sapienza (Rome) Mathematics Area, ISAS (Trieste) 2 A Caterina, Fiammetta, Simonetta Whether our attempt stands the test can only be shown by quantitative calculations of simple systems Max Born, On Quantum Mechanics Z. fur Physik 26, 379-395 (1924) Contents Presentation. 11 Volume I – Basic elements. 12 Volume II – Selected topics.
13 Bibliography for volumes I and II. Elements of the history of Quantum Mechanics I 19 1.2 Birth of Quantum Mechanics. The early years .3 Birth of Quantum Mechanics 1. The work of de Broglie .4 Birth of Quantum Mechanics 2.
Schrödinger’s formalism .5 References for Lecture 1. Elements of the history of Quantum Mechanics II .1 Birth of Quantum Mechanics 3. Born, Heisenberg, Jordan .2 Birth of Quantum Mechanics 4. Heisenberg and the algebra of matrices .3 Birth of Quantum Mechanics 5.4 Birth of Quantum Mechanics 6.
Pauli; spin, statistics .5 Further developments: Dirac, Heisenberg, Pauli, Jordan, von Neumann .7 Quantum Field Theory .9 Algebraic structures of Hamiltonian and Quantum Mechanics. Pauli’s analysis of the spectrum of the hydrogen atom .11 References for Lecture 2. Axioms, states, observables, measurement, difficulties .2 The axioms of Quantum Mechanics .3 States and Observables .4 Schrödinger’s Quantum Mechanics .5 The quantization problem .6 Heisenberg’s Quantum Mechanics .7 On the equivalence .10 Information-theoretical analysis of Born’s rule .11 References for Lecture 3. 77 4 Lecture 4: Entanglement, decoherence, Bell’s inequalities, alternative theories .6 References for Lecture 4.
Automorphisms; Quantum dynamics; Theorems of Wigner, Kadison, Segal; Continuity and generators .1 Short summary of Hamiltonian mechanics .3 Automorphisms of states and observables .4 Proof of Wigner’s theorem .5 Proof of Kadison’s and Segal’s theorems .6 Time evolution, continuity, unitary evolution .7 Time evolution: structural analogies with Classical Mechanics .8 Evolution in Quantum Mechanics and symplectic transformations .9 Relative merits of Heisenberg and Schrödinger representations .10 References for Lecture 5. Operators on Hilbert spaces I; Basic elements .1 Characterization of the self-adjoint operators .3 Spectral theorem, bounded case .4 Extension to normal and unbounded self-adjoint operators .6 Convergence of a sequence of operators .8 References for Lecture 6 .1 Relation between self-adjoint operators and quadratic forms .2 Quadratic forms, semi-qualitative considerations .3 Further analysis of quadratic forms .4 The KLMN theorem; Friedrichs extension .5 Form sums of operators .6 The case of Dirichlet forms .7 The case of −∆ + λ|x|−α , x ∈ R3 .8 The case of a generic dimension d .9 Quadratic forms and extensions of operators .11 References for Lecture 7. Properties of free motion, Anholonomy, Geometric phase .1 Space-time inequalities (Strichartz inequalities) .2 Asymptotic analysis of the solution of the free Schrödinger equation .3 Asymptotic analysis of the solution of the Schrödinger equation with potential V .5 The role of the resolvent .8 Anholonomy and geometric phase in Quantum Mechanics .9 A two-dimensional quantum system .10 Formal analysis of the general case .13 References for Lecture 8. Elements of C ∗ -algebras, GNS representation, automorphisms and dynamical systems .1 Elements of the theory of C ∗ −algebras .4 The Gel’fand-Neumark-Segal construction .5 Von Neumann algebras .6 Von Neumann density and double commutant theorems.7 Density Theorems, Spectral projecton, essential support .8 Automorphisms of a C ∗ -algebra.9 Non-commutative Radon-Nikodim derivative .10 References for Lecture 9.
Derivations and generators. Elements of modular structure.2 Derivations and groups of automorphisms .4 Two examples from quantum statistical mechanics and quantum field theory on a lattice .10References for Lecture 10. Semigroups and dissipations. Quantum dynamical semigroups I .1 Semigroups on Banach spaces: generalities .3 Markov approximation in Quantum Mechanics .4 Quantum dynamical semigroups I .5 Dilation of contraction semigroups .6 References for Lecture 11.
Positivity preserving contraction semigroups on C ∗ -algebras.2 Completely positive semigroups.4 Properties of dissipations .6 General form of completely dissipative generators .7 References for Lecture 12. Weyl system, Weyl algebra, lifting symplectic maps. Magnetic Weyl algebra .1 Canonical commutation relations .5 Construction of the representations .6 Lifting symplectic maps.7 The magnetic Weyl algebra .8 Magnetic translations in the magnetic Weyl algebra .9 References for Lecture 13. A Theorem of Segal.
Representations of Bargmann, Segal, Fock.2 Complex Bargmann-Segal representation .3 Berezin-Fock representation .6 Non-constant magnetic field .7 Real Bargmann-Segal representation .8 Conditions for equivalence of representations under linear maps312 14.10The formalism of quantization .12Quantization of a Poisson algebra .13Deformation quantization, ∗-product .14Strict deformation quantization .15Berezin-Toeplitz ∗-product .18Bohr-Sommerfeld quantization .19References for Lecture 14. Semiclassical limit; Coherent states; Metaplectic group .1 States represented by wave functions of class A .2 Qualitative outline of the proof of 1), 2), 3), 4) .3 Tangent flow, quadratic Hamiltonians .6 Semiclassical limit through coherent states: one-dimensional case .7 Semiclassical approximation theorems .8 N degrees of freedom.9 Linear maps and metaplectic group.10References for Lecture 15. 346 8 Contents 16 Lecture 16: Semiclassical approximation for fast oscillating phases. Semiclassical quantization rules .1 Free Schrödinger equation .2 The non-stationary phase theorem .3 The stationary phase theorem .5 Transport and Hamilton-Jacobi equations .6 The stationary case .8 Semiclassical quantization rules .1 One point of inversion .2 Two points of inversion .9 Approximation through the resolvent .10References for Lecture 16.
Kato-Rellich comparison theorem. Rollnik and Stummel classes.2 Rollnik class potentials .3 Stummel class potentials .4 Operators with positivity preserving kernels .5 Essential spectrum and Weyl’s comparison theorems .6 Sch’nol theorem .7 References for Lecture 17. Weyl’s criterium, hydrogen and helium atoms .2 Coulomb-like potentials. spectrum of the self-adjoint operator .3 The hydrogen atom.
Group theoretical analysis .5 Pauli exclusion principle, spin and Fermi-Dirac statistics .3 Pauli exclusion principle .6 Helium-like atoms .8 Two-dimensional hydrogen atom .9 One-dimensional hydrogen atom .11References for Lecture 18. Estimates of the number of bound states. The Feshbach method .2 Estimates depending on Banach norms .3 Estimates for central potentials .1 the physical problem .6 References for Lecture 19. Self-adjoint extensions.
Relation with quadratic forms. Laplacian on metric graphs.1 Self-adjoint operators: criteria and extensions .2 Von Neumann theorem; Krein’s parametrization .3 The case of a symmetric operator bounded below .4 Relation with the theory of quadratic forms .5 Special cases: Dirichlet and Neumann boundary conditions .6 Self-adjoint extensions of the Laplacian on a locally finite metric graph .7 Point interactions on the real line .8 Laplacians with boundary conditions at smooth boundaries in R3 .9 The trace operator .12Interaction localized in N points .13References for Lecture 20. 466 Presentation These are notes of lectures that I have given through many years at the Department of Mathematics of the University of Rome, La Sapienza, and at the Mathematical Physics Sector of the SISSA in Trieste. The presentation is whenever possible typical of lectures: introduction of the subject, analysis of the structure through simple examples, precise results in the form of Theorems.
I have tried to give a presentation which, while preserving mathematical rigor, insists on the conceptual aspects and on the unity of Quantum Mechanics. The theory which is presented is Quantum Mechanics as formulated in its essential parts by de Broglie and Schrödinger and by Born, Heisenberg and Jordan with important contributions by Dirac and Pauli. For editorial reason the volume of Lecture notes is divided in two parts. The first part , lectures 1 to 20, contains the essential part of the conceptual and mathematical foundations of the theory and an outline of some of the mathematical techniques that are most useful in the applications.
Some parts of these lectures are about topics that are at present subject of active research. The second volume consists of Lectures 1 to 17. The Lectures in this second part are devoted to specific topics, often still a subject of advanced research. They are chosen among the ones that I regard as most interesting.
Since ”interesting” is largely a matter of personal taste other topics may be considered as more significant or more relevant. At the end of the introduction of both volumes there is a list of books that may be help for further studies. At the end of each Lecture references are given for self-study. A remark on the lengths of each Lecture: by and large, each of them is gauged on a two-hours presentation but the time may vary in relation with the level of preparation of the students.
They can also use in self-study, and in this case the amount of time devoted to each Lecture may be vastly different. I want to express here my thanks to the students that took my courses and to numerous colleagues with whom I have discussed sections of this book for 12 Presentation – vol.I comments, suggestions and constructive criticism that have much improved the presentation. In particular I want to thank Giuseppe Gaeta and Domenico Monaco for the very precious help in editing and for useful comments and Sergio Albeverio, Alessandro Michelangeli and Andrea Posilicano for suggestions. Volume I – Basic elements Some details of the contents of the Lectures in Volume I: • Lectures 1 and 2.
These lectures provide an historical perspective on the beginning of Quantum Mechanics, on its early developments and on the shaping of present-day formalism. An analysis of the mathematical formulation of Quantum Me- chanics and of the difficulties one encounters in relating this formalism to the empirical word, mainly for what concerns the theory of measurement. Entanglement and the attempts to describe the mathematics of decoherence. An analysis of Bell’s inequalities and brief outline of a formalism, originated by de Broglie, in which material points are guided by a velocity field defined by the solution of Schrödinger’s equation.
Groups of transformations of the fundamental quantities in Quantum Mechanics: states and observables.Theorems of Wigner, Kadison and Segal on implementability with unitary or anti-unitary maps. Conti- nuity of the maps and the basis of Quantum Dynamics. Basic facts from the theory of operators in a Hilbert space. Since in Quantum Mechanics these operator represent observables a good control of this formalism is mandatory.
Elements of the theory of quadratic forms. Quadratic forms are an important tool in the theory of operators on Hilbert spaces and they play a major role in the theory of extensions. Friedrich’s extension of a semi-bounded symmetric operator. Analytic study of the solutions of the Schödinger equation, be- ginning with the simple but instructive case of free motion.
Propagation inequalities and their relation to the description of the asymptotic prop- erties of a quantum mechanical system. The problem of anholonomy and the geometric phase. Elements of the theory of C ∗ algebras and von Neumann alge- bras. This lecture provides some elements of the theory of automorphisms of C ∗ -algebras and the description of the dynamics of quantum systems.
Generators, derivations and in particular the KMS condition for a group of automorphims of a C ∗ -algebra. Implementation of a group of automorphism by a group of unitary operators. Modular structure of a representation and standard form of a von Neumann algebra. Basic elements of the theory of semigroups in Banach spaces and of the theory of dissipations.
Markov approximation and conditions for its validity. Elements of a converse problem, the dilation of a Markov semigroup. Role of positivity and complete positivity in the theory of con- traction semigroups on C ∗ -algebras. Elements of the theory of dissipations and basic facts in the theory of Quantum Dynamical semigroups.
The problem of quantization. Weyl system and Weyl algebra, uniqueness theorem of von Neumall and Weyl. Formalism of second quan- tization. Magnetic Weyl algebra.
Various representations of the Weyl algebra (real and com- plex representations of Bargmann and Segal, representations of Fock and Berezin). The case of an infinite number of degrees of freedoms,. The real representation and the quantization of the free relativistic field (Se- gal). van Hove’s theorem.
Brief outline of deformation quantization and of geometric quantization.