Operators and Representation Theory: Canonical Models for Algebras of Operators Arising in Quantum Mechanics

Operators and Representation Theory

Canonical Models for Algebras of Operators Arising in Quantum Mechanics

THIRD EDITION

PALLE E. T. JORGENSEN

The University of Iowa

Dover Publications, Inc.

Mineola, New York

Copyright

Copyright © 1988, 2008, 2017 by Palle E. T. Jorgensen

Foreword Copyright © 2008 by William Klink

All rights reserved.

Bibliographical Note

This Dover edition, first published in 2017, is a revised and updated republication of the work originally published as Volume 147 in the North-Holland Mathematics Studies series by the North-Holland Publishing Company, Amsterdam and New York, in 1988. The 2008 Dover edition added a new Foreword by William Klink and a new Preface by the author.

International Standard Book Number

ISBN-13: 978-0-486-81572-5

ISBN-10: 0-486-81572-2

Manufactured in the United States by LSC Communications

81572201   2017

www.doverpublications.com

Foreword

by Professor William Klink

Einstein was one of the first to realize the importance of symmetry in grounding physical theories. His analysis of the contrasting symmetry structures of Maxwell’s equations for electromagnetism and Newton’s mechanics led directly to the special theory of relativity.

Symmetry has played an even more important role in the development of quantum theory. For example, early on in the development of quantum theory it was realized that the quantization of spin and orbital angular momentum was a direct consequence of rotational symmetry. Representations of the rotation group are used routinely to couple angular momentum for many-electron systems. Other compact groups, such as the unitary groups, have played an important role in nuclear physics, and with so called internal symmetries dealing with quantum numbers such as charge, baryon number and strangeness.

Although the earliest formulations of quantum theory did not explicitly draw on group theory, it was soon realized that many of the grounding principles of quantum theory could naturally be expressed in group theoretical language. This was made particularly clear when trying to combine the symmetry structure of the special theory of relativity with quantum theory. Here the underlying group is the Poincaré group, the group of transformations leaving the structure of space-time invariant. Wigner’s analysis of the representations of the Poincaré group form today the basis of all versions of relativistic quantum theory, including quantum field theory. What is particularly striking is that the invariant operators of the Poincaré group are mass and spin operators, exactly the observables that characterize elementary particles such as electrons and protons.

Surprisingly, the symmetry properties of what is now called nonrelativistic quantum theory, in which the transformations in space-time called collectively the Galilei transformations, those that leave the form of Newton’s equations unchanged, were not investigated until after the work by Wigner on the Poincaré group. Bargmann and Wigner, among others, investigated the representation structure of the Galilei group and found that many of the grounding principles of the earliest formulations of quantum theory could be understood in group theoretical language. In particular, the necessity of infinite-dimensional Hilbert spaces for describing even single-particle systems was seen to be a direct consequence of unitary irreducible representations of a noncompact group, the Galilei group.

Along with developments in atomic and nuclear physics, it also became clear in the 1930s that for processes involving the creation and annihilation of particles such as photons, the quantum of light, that larger groups and Lie algebras than the compact and locally compact groups were needed. The earliest such examples included the bosonic and fermionic algebras generated by the commutation and anticommutation relations of creation and annihilation operators. Large classes of representations of these algebras were discovered, and work with these and other infinite-dimensional algebras continues to be an area of active research today, in both the mathematics and mathematical physics communities.

The mathematical expression of symmetry is group theory in all of its ramifications, including in particular Lie algebras, and representations of groups and Lie algebras on Hilbert spaces. There are many ways in which group theory is used in quantum theory, going well beyond the applications mentioned above. It is therefore of interest to have a mathematician develop in modern mathematical language some of the newer developments occurring in the intersection between group theory (in its most generalized sense) and quantum theory. In this monograph, Palle Jorgensen begins with some of the mathematical machinery needed to understand modern developments. He introduces the notions of projective representations and central extensions and finishes part 1 with the imprimitivity theorem, which grounds in more mathematical language the work of Wigner on representations of the Poincaré and Galilei groups.

Part 2 of the monograph deals with algebras of operators on Hilbert spaces, broadening the mathematics used in earlier versions of quantum theory. There are many examples where the Hamiltonian, the operator that translates a quantum system in time, can be written as a polynomial in elements of an underlying Lie algebra. This part of the monograph deals with properties of such operators, for example their spectral properties.

Part 3 deals with infinite-dimensional Lie algebras, a topic of increasing importance in a variety of different approaches to quantum theory, including string theory, theories of gravitation and gauge quantum field theories. The structures of interest here include the gauge (map) groups, Diff¹ groups and infinite-dimensional groups such as the infinite unitary groups. New mathematical machinery is needed to deal with these structures, and this is done by looking at several different examples; in particular, the Virasoro algebra is analyzed in some detail.

This monograph is an excellent example of the interplay between mathematics and mathematical physics. Most of the mathematical topics are motivated by physical systems, and then developed in their own right. Anyone interested in the mathematical formulation of modern quantum theory will benefit from this monograph.

William Klink

Department of Physics

The University of Iowa

¹Diff groups is short for (infinite-dimensional) groups of diffeomorphisms on suitable manifolds.

Preface to the 2017 Dover Edition

The 2017 Dover edition has benefitted from numerous improvements, both in form and substance: The original typescript has been converted into by Dr. Feng Tian. Dr. Tian also made improvements during the conversion process. Moreover, the book itself has been updated to make it both more user friendly and more current. The updates and improvements include the following: In the front matter, and in each chapter, corrections and refinements of the presentation have been made. In the period since the first Dover edition (2008), we have benefitted from feedback from students and colleagues – from classroom use. This feedback led to numerous improvements too; all improvements with a view to making the new edition more reader friendly and up to date. An effort is made to make it easier for students to get started with the subject.

Among the recent exciting new uses of noncommutative harmonic analysis on infinite groups and infinite-dimensional groups is the following: applications of infinite-dimensional unitary groups to determinantal point-processes. A determinantal point-process is a stochastic process, realized on an infinite state space, whose probability distribution is prescribed by a certain system of determinants. Although they generalize Poisson processes, they are quite different from Gaussian processes. Nonetheless, they serve as important new tools in the study of random matrix theory, in combinatorics, and in statistical physics. While the literature is vast, we refer readers to [BO05a, BBO15, BO05b] and the papers cited there.

In the present 2017 edition, we have added new sections with applications, with new hands-on examples, and with motivation. We updated the bibliography and the citations. We explained and motivated choice of notation and conventions. We added a list of terminology for easy reference. We added more background material, making it easier for students getting started with the themes and topics. We included new topics, and we added a historical appendix.

Palle E. T. Jorgensen

Iowa City, 2017

Preface to the 2008 Dover Edition

… the desire to know nature has had the most constant and the happiest influence on the development of mathematics. — Henri Poincaré.

This book grew out of seminars and courses we taught covering a core of mathematics used in quantum mechanics, and in other applications.

Motivated by quantum theory, we begin with Hilbert space, and with the corresponding linear transformations (linear operators). A central theme is the use of representations of groups and algebras by operators in Hilbert space: differential operators, wave operators, Schrödinger operators, etc. The questions we ask, both in the context of mathematics and physics, relate to these operators: Are they selfadjoint? What is the spectrum: the spectral resolution? Where does the Hilbert space come from? How do we use symmetries in computations of spectrum? What is the appropriate notion of equivalence in infinite dimensions?

In the teaching of mathematics and physics, there is a core of ideas with a good amount of permanence: principles that are more timeless than others. For quantum mechanics, on the mathematical side, they center around our use of Hilbert space , and of transformations (linear operators) transforming the vectors in . This is the language used in asking questions about spectral lines, scattering and the like.

This amounts to a generalization of linear analysis to infinite dimensions (operator theory). It entails an infinite-dimensional extension of the fundamentals from undergraduate courses in linear algebra: matrix, transformation, vectors, eigenvector, eigenvalue, eigenspace, basis, subspace and the spectral theorem; i.e., the radical idea of asking for a diagonalized version of a given normal linear transformation.

At its inception in the 1920s, quantum theory seemed perhaps radical. Furthermore, at the time, it appeared somewhat artificial to select axioms and constructs from mathematics with a view to making precise sense of such physical realities as, say, state, observable, measurement, spectrum, transition probability, symmetry, energy, momentum, position, electron, photon, elementary particle, entanglement, quantum fields, quantum communication, etc. But in the seven decades following the pioneering discoveries of Heisenberg, Born, Dirac, Schrödinger, Pauli, etc., the mathematical framework has now become common place, and a permanent fixture.

Despite this state of affairs, there is still a divide between how students are taught the subject in math and in physics. In the mathematics and physics departments the emphasis differs; and what is more, the use of lingo, terminology and definitions continues to diverge. Interdisciplinary communication has not become easier. Rather the other way around. Some talk of a divorce between math and physics. Take a look, for example, at such a basic notion as state (as in state of a particle, ground state, excited state). Comparing physics and math texts, you will find that in fact this fundamental notion of a state is defined differently depending on which side of the divide you are.

True, the authors of the two texts might both have been motivated by Paul Dirac’s vision, but the product in current textbooks came out differently, or at least so it would seem.

The present book aims to organize this material on a common ground. A recurrent theme is the use of mathematics in understanding symmetry, and the use of groups and algebras in computations of spectra. This framework is part of what has become known as operator theory, the use of linear operators in Hilbert space in computations of mathematical physics. Here spectrum refers to some underlying selfadjoint operator in Hilbert space. And the Spectral Theorem for selfadjoint operators allows us to use operator theory in creating a precise mathematical description of measurements in quantum mechanics; in giving an answer to questions such as these: "Given some interval J on the real axis, and some specific observable A, what is the probability of measuring values of the observable A in the interval J when an experiment is prepared in a specific state? And if there are two states, what is the corresponding transition probability?"

Since the work of J. von Neumann, E. Wigner, V. Bargmann, and H. Weyl (see Appendix B), we now take for granted that groups and algebras allow us to use symmetry in computations, but the distinction between the groups on the one hand, and their representations on the other is occasionally blurred in textbooks. While the groups G and their isomorphism classes may be easy entities, based only on a small number of coordinates, their representations (for non-compact G) are typically infinite-dimensional; i.e., their irreducible pieces cannot be broken down into finite-dimensional constituents. Since the irreducible representations of the appropriate symmetry group G correspond to elementary particles from physics, infinite-dimensional Hilbert space is inescapable. Starting with infinite-dimensional Hilbert space , we get raising and lowering operators, particle number, groups of unitary operators acting on , and much more. It isn’t just a generalization for generalization’s sake, dreamed up by some crazy mathematician.

On occasion, in the time period since the first edition, I was asked to explain one point. It is about the following two facts, and choices I made:

(1) The variety of types and classes of groups that are needed in physics applications make up for a vast and diverse gallery: continuous vs discrete, finite-dimensional vs infinite-dimensional, Abelian vs non-Abelian, central extensions vs more general extensions, nilpotent vs semisimple.

(2) For each of the possibilities for the group G under consideration, we are interested in understanding the associated unitary representations of G, as well as covariant systems of operators in Hilbert space.

And for each class of the groups and their representations, the literature discussing analysis of the unitary representations and the equivalence classes of irreducibles is vast. In fact, for a number of groups, the answers to natural questions about classification are not even within reach.

Because of (1) we have adopted a terminology that is a bit more expansive that is perhaps customary; for example, in our use of the term short exact sequence. Because of (2), our list of references is more extensive than would be the case in a more specialized monograph.

A more detailed overview of the book: The idea of symmetry is central in physics, especially in quantum physics, where symmetries manifest themselves in subtle ways. Quantization of symmetries is fundamental in relativistic quantum physics via superselection sectors, anomalies, and spontaneous symmetry breaking.

The mathematics of symmetries was pioneered in the work of Herman Weyl, Eugene Wigner, George Mackey, Valentin Bargmann, Harish-Chandra, Edward Nelson… Some of the mathematical subtleties in the subject include a transition from ray (projective) representations to unitary representations (in Hilbert space.)

Powerful tools for probing quantum manifestations of symmetry are induced representations, Mackey’s systems of imprimitivity, central extensions of Lie algebras, and representations of Lie algebras by unbounded operators with dense domain in a Hilbert space, coupled with their associated spectral theory (energy levels, spectral type, continuous vs discrete). Stressing physical motivation, this is presented with an emphasis on Hilbert space geometry, and relying on basics in functional analysis, homological algebra, Lie groups and Lie algebras, and in especially representation theory.

We probe cutting-edge topics such as representations of Virasoro algebras (conformal field theory, string theory) and noncommutative differential geometry, e.g., noncommutative tori in the form of specific C*-algebras.

The reader should ideally have some acquaintance with basic functional analysis (Banach and Hilbert space, measure theory), Lie groups, and some idea of quantum mechanics. Yet, the required preliminaries are relatively minor, and are taught in most beginning courses.

In writing the book, the author had in mind both students in math courses and physics students (e.g., quantum theory courses). One isn’t favored over the other.

Use of citations. In our inclusion of citations, inside the book, we adopted the following dual approach. Inside each of the chapters, as the material is developed, we have included citations to key sources, books and papers, that we rely on; but this is done sparingly so as not to interrupt the narrative too much. To remedy the sparsity of citations inside chapters, and, to help the reader orient herself in the literature, each of the eight chapters concludes with a short bibliographical section, summarizing papers and books of special relevance to the topic inside the text. Thus there is a separate list of citations that concludes each chapter. So if a reader might not find a needed citation in connection with results from inside the chapter itself, it is very likely that the citation will instead be in the end-of-chapter list.

Comments to the literature. In Chapter 2, we outline the fundamentals needed for the applications inside the book: Hilbert space, linear operators, with emphasis on unbounded operators, groups, algebras, Lie theory (Lie groups and Lie algebras), and representations. These basic notions and tools will be further developed and applied inside the book as follows:

Chapter 3 deals with fundamentals of linear operators in Hilbert space. Chapter 4 deals with unitary representations, especially with the notion of induced representations. Chapters 5-6 explore the approach to global representations that starts with an infinitesimal setting; for example with a given Hermitian symmetric operator, or with a representation of a Lie algebra. The analogous issues arise for derivations with dense domain in C*-algebras. The interplay between the two settings is at the root of Heisenberg’s approach vs the approach via the Schrödinger equation. The two are often referred to as the Heisenberg picture vs the Schrödinger picture. Matrix mechanics vs the Schrödinger equation. In the first, the observables are dynamical quantities, and transform as a function of the time-variable. By contrast, in the Schrödinger equation we are transforming wave functions, i.e., states in a Hilbert space. So the states are then the dynamical quantities.

Palle E. T. Jorgensen

Iowa City, 2007

Preface to the Original Edition

Professor Marshall Stone (see Appendix B) told me, some time ago, that when he began the work on his book [Sto32a], he had in mind unitary representations of groups on Hilbert space. Of course, at the time, quantum mechanics, operator theory, and representation theory were emerging at a rapid pace and taking separate forms. As we now know, it was left to others to carry out what Stone may have had in mind (with regard to group representations).

Such books have indeed been written, and covering different time periods, reflecting the different stages in the development of the subjects.

In the intervening time, since Stone’s book, the number of subspecialties in this area of mathematics and mathematical physics has grown (see, for example, the list of references below), and the diversity of subjects has increased as well. In the meantime, connections to developments in operator algebras have come to play an important role.

During the 1980s, there seem to have been tendencies for some of the related subjects to develop along separate paths, and the potential for fruitful interaction may not have been fully realized and used. It is the aim of the present monograph to improve on this state of affairs. We have picked certain subjects from the theory of operator algebras, and from representation theory, and showed that they may be developed starting with Lie algebras, extensions, and projective representations. In some cases, this Lie algebra approach to familiar problems in C*-algebra theory is new. Furthermore, we have picked problems from the theory of representations of infinite-dimensional Lie algebras, and demonstrated that C*-algebraic methods may be used in nontraditional ways with some success.

The book is addressed to graduate students who wish to get an introduction to some of the more recent developments in these interacting subject areas. It is also addressed to researchers with specialized knowledge in some, but not all, of the research fields. Connections to mathematical physics have been stressed throughout, and the book is written also with the mathematical physics community in mind. Our group theoretic treatment of curved magnetic field Hamiltonians is a case in point.

Acknowledgements

A portion of the research reported in the present monograph has been done in various collaborative projects. My co-authors in the joint research projects include (in alphabetical order): Daniel Aplay, Ola Bratteli, Joachim Cuntz, Dorin Dutkay, George A. Elliott, David E. Evans, Frederick M. Goodman, Akitaka Kishimoto, William H. Klink, David W. Kribs, Robert T. Moore, Paul S. Muhly, Gestur Olafsson, Judith A. Packer, Anna Paolucci, Steen Pedersen, Costel Peligrad, Wayne Polyzou, Robert T. Powers, Geoffrey L. Price, Derek W. Robinson, Yury Stefanovich Samoĭlenko, Feng Tian, and Reinhard Werner. I am pleased to acknowledge their contributions. When the picture emerges, it is difficult, and often impossible, to say who did exactly what. The pleasant fact about collaborations is that the outcome is better, and larger, than the sum of the individual components.

I am also pleased to thank colleagues W.H. Klink, S. Pedersen, and my student H. Prado for help with proofreading.

Special thanks are due to Mrs. Ada Burns for the typing (and production) of the first edition of the final manuscript. Her artistic skill is pleasantly visible, and greatly appreciated.

The research reported in the present monograph was supported in part by the National Science Foundation.

Notation

Operators in Hilbert space.

Operations on subspaces of Hilbert spaces J%?.

⊥⊥ = span

Normal or not! It depends:

•An operator T (bounded or not) is normal if and only if (Def.)

•A state s on a *-algebra is normal if it allows a representation ( , ρ) where is a Hilbert space, and p is a positive trace-class operator in s.t. trace (ρ) = 1, and

•A random variable X on a