INTROD TO THE MATH'L STRUCTURE (2E)(V28): A Short Course for Mathematicians (Second Edition) (Advanced Series In Mathematical Physics) - Hardcover

F STROCCHI

Hardcover

ISBN 10: 9812835229 ISBN 13: 9789812835222

Publisher: World Scientific, 2008

Synopsis

This book arises out of the need for Quantum Mechanics (QM) to be part of the common education of mathematics students. The mathematical structure of QM is formulated in terms of the C*-algebra of observables, which is argued on the basis of the operational definition of measurements and the duality between states and observables, for a general physical system. The Dirac-von Neumann axioms are then derived. The description of states and observables as Hilbert space vectors and operators follows from the GNS and Gelfand-Naimark Theorems. The experimental existence of complementary observables for atomic systems is shown to imply the noncommutativity of the observable algebra, the distinctive feature of QM; for finite degrees of freedom, the Weyl algebra codifies the experimental complementarity of position and momentum (Heisenberg commutation relations) and Schrodinger QM follows from the von Neumann uniqueness theorem.The existence problem of the dynamics is related to the self-adjointness of the Hamiltonian and solved by the Kato-Rellich conditions on the potential, which also guarantee quantum stability for classically unbounded-below Hamiltonians. 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), to the formulation in terms of ground state correlations (the quantum mechanical analog of the Wightman functions) and their analytic continuation to imaginary time (Euclidean QM). The quantum particle on a circle is discussed in detail, as an example of the interplay between topology and functional integral, leading to the emergence of superselection rules and sectors.

About this title

Review

... 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. -- Jean-Pierre Antoine, Université catholique de Louvain, Belgium

... written in a very clear and compact style ... provides a
unified and physically motivated presentation of different mathematical
topics. -- Zentralblatt MATH

A very useful tool for mathematicians to enter the not so mysterious framework of quantum theory for finite degrees of freedom. -- Professor Grosse Harald, University of Vienna, Austria

Edition details

• Publisher: World Scientific
• Publication date: 2008
• ISBN 10: 9812835229
• ISBN 13: 9789812835222
• Binding: Hardcover
• Language: English
• Edition number: 2
• Number of pages: 194