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Fortschritte der Physik

Algebraic Methods in Ordinary Quantum Mechanics Revisited

G. C. Ghirardi

Published Jan 1, 1973• 7 citations
Publisher

Abstract

Abstract We reconsider the problem of giving an algebraic treatment to quantum problems within the framework of ordinary quantum mechanics. After a concise review of the irreducible representations of the relevant Lie algebra, we formulate the general scheme within which all previous approaches can be easily fitted. We then show which are the explicit connexions between this general scheme and the specific approaches found in the literature. The whole paper constitutes a concise and synthetic summary of algebraic techniques in ordinary quantum mechanics.

DOI
10.1002/prop.19730211202

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Abstract We reconsider the problem of giving an algebraic treatment to quantum problems within the framework of ordinary quantum mechanics. After a concise review of the irreducible representations of the relevant Lie algebra, we formulate the general scheme within which all previous approaches can be easily fitted. We then show which are the explicit connexions between this general scheme and the specific approaches found in the literature. The whole paper constitutes a concise and synthetic summary of algebraic techniques in ordinary quantum mechanics.
The number of exactly solvable eigenvalue problems has been rapidly increasing since the advent of quantum mechanics and there are many efficient ways to treat them. As a general rule, it is possible to write exactly solvable abstract eigenvalue equations in terms of the generators of finite-dimensional Lie algebras and solve them by means of purely algebraic methods. Matrix elements are commonly necessary for the application of perturbation theory and variational principles, among other approximate methods. The chapter discusses examples of three-dimensional Lie algebras with well-known quantum-mechanical realizations. Typically, exactly solvable quantum-mechanical problems are either onedimensional models or lead to separable equations in many dimensions. In the coordinate representation both cases reduce to ordinary differential equations of second order that determine the eigenvalues and separation constants compatible with the physical boundary conditions. In its purest form, a Lie algebraic method is representation-independent. Extensive application of algebraic methods to selected physical problems is already available elsewhere.
In Part I of this series we presented the general ideas of applying group-algebraic methods for describing quantum systems. The treatment was there very "ascetic" in that only the structure of a locally compact topological group was used. Below we explicitly make use of the Lie group structure. Basing on differential geometry enables one to introduce explicitly representation of important physical quantities and formulate the general ideas of quasiclassical representation and classical analogy.
The goal of this book is to present classical mechanics, quantum mechanics, and statistical mechanics in an almost completely algebraic setting, thereby introducing mathematicians, physicists, and engineers to the ideas relating classical and quantum mechanics with Lie algebras and Lie groups. The book emphasizes the closeness of classical and quantum mechanics, and the material is selected in a way to make this closeness as apparent as possible. Much of the material covered here is not part of standard textbook treatments of classical or quantum mechanics (or is only superficially treated there). For physics students who want to get a broader view of the subject, this book may therefore serve as a useful complement to standard treatments of quantum mechanics. Almost without exception, this book is about precise concepts and exact results in classical mechanics, quantum mechanics, and statistical mechanics. The structural properties of mechanics are discussed independent of computational techniques for obtaining quantitatively correct numbers from the assumptions made. The standard approximation machinery for calculating from first principles explicit thermodynamic properties of materials, or explicit cross sections for high energy experiments can be found in many textbooks and is not repeated here.
This book provides a comprehensive treatment of quantum mechanics from a mathematics perspective and is accessible to mathematicians starting with second-year graduate students. It addition to traditional topics, like classical mechanics, mathematical foundations of quantum mechanics, quantization, and the Schrodinger equation, this book gives a mathematical treatment of systems of identical particles with spin, and it introduces the reader to functional methods in quantum mechanics. This includes the Feynman path integral approach to quantum mechanics, integration in functional spaces, the relation between Feynman and Wiener integrals, Gaussian integration and regularized determinants of differential operators, fermion systems and integration over anticommuting (Grassmann) variables, supersymmetry and localization in loop spaces, and supersymmetric derivation of the Atiyah-Singer formula for the index of the Dirac operator. Prior to this book, mathematicians could find these topics only in physics textbooks and in specialized literature. This book is written in a concise style with careful attention to precise mathematics formulation of methods and results.Numerous problems, from routine to advanced, help the reader to master the subject. In addition to providing a fundamental knowledge of quantum mechanics, this book could also serve as a bridge for studying more advanced topics in quantum physics, among them quantum field theory. Prerequisites include standard first-year graduate courses covering linear and abstract algebra, topology and geometry, and real and complex analysis.
Contrary to the usual view that quantum mechanics in configuration space has, in general, only one algebaic representation( the Heisenberg algebraic representation), we have proved that quantum mechanics in configuration space has, in general, alternative algebraic representations: (i) In a finite domain of configuration space, it can be expressed in terms of su(2) algebra; (ii) in an infinite domain of configuration space, it can be expressed in terms of su(l,l) algebra.The above results open a new possibility to reformulate quantum mechanics and provide more mathematical tools to solve diverse physical problems.Nonlinear relations of different Lie algebas may imply a unification of Lie algebras and their physical implications ( the quantum motion modes ) in a deeper nonlinear domain.This observation raises challenging mathematical and physical questions.