Introduction To The Mathematical Structure Of Quantum Mechanics, An: A Short Course For Mathematicians: (27 Advanced Series In Mathematical Physics)

This book arises out of the need for Quantum Mechanics (QM) to be part of the common education of mathematics students. Rather than starting from the Dirac-Von Neumann axioms, the book offers a short presentation of the mathematical structure of QM using the C--algebraic structure of the observable based on the operational definition of measurements and the duality between states and observables. The description of states and observables as Hilbert space vectors and operators is then derived from the GNS and Gelfand-Naimark Theorems.For finite degrees of freedom, the Weyl algebra codifies the experimental limitations on the measurements of position and momentum (Heisenberg uncertainty relations) and Schroedinger QM follows from the von Neumann uniqueness theorem.The existence problem of the dynamics is related to the self-adjointness of the differential operator describing the Hamiltonian and solved by the Rellich-Kato theorems. Examples are discussed which include the explanation of the discreteness of the atomic spectra.Because of the increasing interest in the relation between QM and stochastic processes, a final chapter is devoted to the functional integral approach (Feynman-Kac formula), the formulation in terms of ground state correlations (Wightman functions) and their analytic continuation to imaginary time (Euclidean QM). The quantum particle on a circle as an example of the interplay between topology and functional integral is also discussed in detail.

Table of Contents:
# Mathematical Description of a Classical Physical System # Classical Observables as Abelian C*-Algebra # States as Positive Linear Functionals on the Algebra of Observables # Gelfand-Naimark Theorem: The Observables as Hilbert Space Operators # Heisenberg Uncertainty Relations # The Weyl Algebra # The Quantum Hamiltonian # Double Slit Interference and Particle Wave Duality # Quantum Harmonic Oscillator # The Hydrogen Atom # and other topics

Review :
"The structure and the style of the exposition are such that the book can be used by third- and fourth-year undergraduate students who are just beginning the study of quantum mechanics ... The structure of the book also makes it very suitable for lecturers wishing to give concise but comprehensive lectures in mathematical quantum mechanics. In spite of its briefness, the course is very informative ... The approach using the C*-algebraic formalism makes the book especially attractive, since it allows one to give a very transparent and powerful mathematical description not only of a quantum mechanical system, but a physical system in general." Mathematical Reviews "It is written in a very clear and compact style, providing relevant references along the way and proofs of the relevant steps. Applications and phenomenological aspects are kept to a minimum, being introduced only to justify and better clarify the mathematical framework ... It provides a unified and physically motivated presentation of different mathematical topics which are usually either skipped or simply ignored in physics textbooks, thus supplying the interested reader with a compact exposition of relevant mathematical structures brought about by quantum mechanics." Zentralblatt MATH "This book is written in a clear style, uses the most modern techniques, is divided into short chapters and will be a very useful tool for mathematicians to enter the not so mysterious framework of quantum theory for finite degrees of freedom. My congratulation goes to the author for this elegant and transparent book." Professor Grosse Harald University of Vienna, Austria "Strocchi's book looks like a 'tour de force', being able to cover such a large amount of difficult material in 140 pages, in a perfectly clear way ... This little book should appeal to all mathematics students, and moreover it will be of great help to their instructors, who are always in need of a concise, yet rigorous, treatment of QM, in face of a sea of textbooks that mostly seem to follow one another ..." Jean-Pierre Antoine Universite catholique de Louvain, Belgium