An Introduction To The Mathematical Structure Of Quantum Mechanics Pdf

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The second printing contains a critical discussion of Dirac derivation of canonical quantization, which is instead deduced from general geometric structures. 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 Schrödinger 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 θ sectors.

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Chapter 0: Introduction (88 KB)

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Contents:

Mathematical Description of a Physical System
Mathematical Description of a Quantum System
The Quantum Particle
Quantum Dynamics. The Schrö;dinger Equation
Examples
Quantum Mechanics and Stochastic Processes

Readership: Academics, mathematicians, advanced undergraduate and graduate students in mathematics and mathematical physics.

Makes the structure of quantum mechanics natural and logically simple to mathematicians, at the level of an undergraduate one-semester course, by arguing the C*-algebraic structure of the observables from their operational definition, rather than starting from the Dirac–von Neumann (or from Mackey) axioms
Representation of states and observables by Hilbert space vectors and operators, and Schrödinger quantum mechanics, as tight logical consequences of the book's approach on the subject, with non-abelianess being implied by the experimental occurrence of complementary observables
Gives a concise account of the functional integral approach to quantum mechanics according to the modern ideas developed by quantum field theory, in terms of analytic continuation to imaginary times
All sections have been updated and improved in this edition. In particular, the discussion of the Principles of Quantum Mechanics has been expanded

Review of the First Edition:

"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, as it includes not only all standard topics in mathematical quantum mechanics, but also gives ideas of such issues as quantum logic, Feynman path integrals, Feynman–Kac formula, Euclidean quantum mechanics, and functional integral and its applications … 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 … The book … can be of interest to both mathematicians and physicists. 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

"Within the Glimm–Jaffe spirit the Feynman–Kac formula, Nelson positivity and the analytically continuing from Schrödinger quantum mechanics to the Euclidean framework is nicely given … The last application is the most interesting one: He studies the θ vacuum situation within the von Neumann algebra approach. This explains the occurrence of superselection rules in the very simple quantum mechanical particle setting … This book is written in a clear style, uses the most modern techniques, … 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."

H Grosse
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 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 …"

J-P Antoine
Université catholique de Louvain, Belgium