Studies in Mathematical Physics

Princeton Series in Physics

edited by Arthur S. Wightman and John I. Hopfield

Quantum Mechanics for Hamiltonians Defined as Quadratic Forms by Barry Simon

Lectures on Current Algebra and Its Applications by Sam B. Treiman, Roman Jackiw, and David J. Gross

Physical Cosmology by P. J. E. Peebles

The Many-Worlds Interpretation of Quantum Mechanics edited by B. S. DeWitt and N. Graham

The Ρ(Φ)2 Euclidean (Quantum) Field Theory by Barry Simon

Homogeneous Relativistic Cosmologies by Michael P. Ryan, Jr., and Lawrence C. Shepley

Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann edited by Elliott H. Lieb, B. Simon, and A. S. Wight- man WBM

Valentine Bargmann Studies in Mathematical Physics Essays in Honor of Valentine Bargmann edited by E. H. Lieb, B. Simon, and A. S. Wightman

Princeton Series in Physics

Princeton University Press Princeton, New Jersey 1976 Copyright © 1976 by Princeton University Press Published by Princeton University Press, Princeton, New Jersey IN THE UNITED KINGDOM: Princeton University Press, Guildford, Surrey

All Rights Reserved

Library of Congress Cataloging in Publication Data will be found on the last printed page of this book

Printed in the United States of America by Princeton University Press Princeton, New Jersey CONTENTS

Introduction vii Publications x

The Inverse r-Squared Force: An Introduction to Its Symmetries by Henry D. I. Abarbanel 3

Certain Hilbert Spaces of Analytic Functions Associated With the Heisenberg Group by Donald Babbitt 19

Lower Bound for the Ground State Energy of the Schrodinger Equation Using the Sharp Form of Young's Inequality by John F. Barnes, Herm Jan Brascamp and Elliott H. Lieb 83

Alternative Theories of Gravitation by Peter G. Bergmann 91

Generalized Wronskian Relations: A Novel Approach to Bargmann-Equivalent and Phase-Equivalent Potentials by F. Calogero 107

Old and New Approaches to the Inverse Scattering Problem by Freeman J. Dyson 151

A Family of Optimal Conditions for the Absence of Bound States in a Potential by V. Glaser, A. Martin, H. Grosse and W. Thirring 169

Spinning Tops in External Fields by Sergio Hojman and Tullio Regge 195

Measures on the Finite Dimensional Subspaces of a Hilbert Space: Remarks to a Theorem by A. M. Gleason by Res Jost 209

The Froissart Bound and Crossing Symmetry by N. N. Khuri 229

ν vi CONTENTS

Intertwining Operators for SL(n, R) by A. W. Knapp and Ε. M. Stein 239

Inequalities for the Moments of the Eigenvalues of the Schrodinger Hamiltonian and Their Relation to Sobolev Inequalities by Elliott H. Lieb and Walter E. Thirring 269

On the Number of Bound States of Two Body Schrodinger Operators — A Review by Barry Simon 305

Quantum Dynamics: From Automorphism to Hamiltonian by Barry Simon 327

Semiclassical Analysis Illuminates the Connection Between Potential and Bound States and Scattering by John Archibald Wheeler 351

Instability Phenomena in the External Field Problem for Two Classes of Relativistic Wave Equations by A. S. Wightman 423 INTRODUCTION

This volume is dedicated to Valentine Bargmann on the occasion of his retirement as Professor of Mathematical Physics at Princeton University. Valentine Bargmann was born in Berlin, Germany on April 6, 1908. He studied at the University of Berlin from 1926 to 1933. He moved to Zurich on Hitler's rise to power and wrote his doctor's thesis at the University under the guidance of Gregor Wentzel. On the completion of his degree, Bargmann emigrated to the United States. (That flat statement is correct, but does not evoke the temper of the times. Bargmann received a five-year German passport in 1931, before the National Socialists came to power, and used it to go to Switzerland to study. After Hitler took office, adminis­ trative regulations were issued withdrawing the citizenship of persons of the wrong "race." For that reason, if the German government had succeeded in finding Bargmann, it would have invalidated his passport. Nevertheless, the passport was accepted by the United States government as a valid basis for an immigration visa. The passport expired two days after he reached the United States in 1937.) On his arrival in the United States, Bargmann looked up I. I. Rabi, who recommended he attend the symposium on physics at the University of Michigan in the summer of 1937. There he met a number of theoretical physicists, in particular, George Uhlenbeck and Gregory Breit. Breit sug­ gested that he look into the Institute for Advanced Study as a possible place to work. After conversations with John von Neumann, Bargmann was accepted at the Institute. He was soon drawn into the work that Albert Einstein was carrying out on unified field theories of gravitation and electromagnetism. For several years he and Peter Bergmann were Einstein's scientific assistants and coworkers in this enterprise. This work continued viii INTRODUCTION until 1943, when he undertook war work in collaboration with John von Neumann. After the war, he joined von Neumann's computer project, work­ ing with von Neumann and Deane Montgomery on the inversion of matrices of large dimension. From 1941 on, Bargmann taught graduate courses at Princeton University, but it was only in 1946 that he received a regular faculty appointment as visiting lecturer in physics. Apart from one term spent at the University of Pittsburgh in 1948, he has been at Princeton ever since. Beginning with ASTP (Army Specialized Training Program) and V12 (Navy) courses, and the aforementioned graduate courses during the war, Bargmann has taught physics and mathematics to generations of graduate and undergraduate students. His courses were noted for their clarity and polish. However, for connoisseurs of the post-war period, it was the set of specialized lectures on his own research that were the gems: the lectures on the Lorentz group and its representations of 1948-1949, the lectures on ray representations of Lie groups in 1953-1954, the lectures on second quanti­ zation of 1946-1947. These last lectures were recorded in the elegant calligraphy of Oscar Goldman and deposited on the reference shelf in Fine Hall Library. They served several generations of graduate students until the advent of modern library customs, when they were stolen. Bargmann's interests in mathematical physics have been broad. As students at the University of Berlin, he and Carl Hempel had a common interest in the philosophical problems at the foundations of physics, an interest they shared with Hans Reichenbach, then Professor at Berlin. The reader of Reichenbach's book Philosophic F oundations of Quantum Mechanics, University California Press, Berkeley, 1944, will see some typical results of conversations with Bargmann, the sharpening and clari­ fication of ideas by the construction of examples and counterexamples. One of the undersigned (A. S. Wightman) can vouch for another charac­ teristic example that occurred a decade later. In the year-long efforts that went into the production of what is sometimes called the Bargmann-Hall- Wightman Theorem, Bargmann's remarks played an important role. Yet Bargmann did not want to put his name on the paper as a co-author. INTRODUCTION ix

Bargmann's papers are not numerous by the standards of productivity of our day, but many of them have started industries. It suffices to mention his work on the representations of the group SL(2, R) or on the inverse scattering problem. For that reason the contents of this book are somewhat different from those of the typical Festschrift. The Editors made a list of "Bargmann industries" and sought authors who could give a good account of present knowledge or recent developments. A majority of the articles in this volume fall into this category. Bargmann's name is synonymous in mathematical physics with depth and lucidity. His work is an inspiration to all those who work in the sub­ ject. The authors of the present volume join in wishing Valja and Sonja Bargmann many more happy and fruitful years.

ELLIOTT LIEB

BARRY SIMON ARTHUR WIGHTMAN PUBLICATIONS

Uber eine Verallgemeinerung des Einsteinschen Raumtyps, Zeitschrift fiir Physik 65, 830, 1930.

Bemerkungen zur allgemein-relativistischen Fassung der Quantentheorie, Preussische Akademie der Wissenschaften (Sitzungsberichte) 1932, p. 346.

Uber den Zusammenhang zwischen Semivektoren and Spinoren, Helvetica Physica Acta 7, 57, 1934.

Zur Theorie des Wasserstoffatoms, Zeitschrift fiir Physik 99, 576, 1936.

Uber die durch Elektronenstrahlen in Kristallen angeregte Lichtemission, Helvetica Physica Acta 10, 361, 1937.

(with A. Einstein and P. G. Bergmann) Five-dimensional representation of gravitation and electricity, Theodore von Karman Anniversary Volume, p. 212, Pasadena, California Institute of Technology, 1941.

(with A. Einstein) Bivector field I, Ann. of Math. 45, 1, 1944.

On the Glancing Reflection of Shock Waves, Applied Mathematics Panel Report No. 108, 2R, 1945.

(with D. Montgomery and L. von Neumann) Solution of Linear Systems of High Order, Report to the Bureau of Ordinance, Navy Department, October 1946. Reported in J. von Neumann, Collected Works Vol. V. p. 421, Macmillan Company, New York, 1963.

(with H. F. Ludloff) Elastic Limit for Dynamic Loading, Jour. Applied Physics 17, 63, 1946.

Irreducible Unitary Representations of the Lorentz Group, Annals of Math. 48 , 568, 1947.

(with E. P. Wigner) Group Theoretical Discussion of Relativistic Wave Equations, Natl. Acad, of Sci. (USA), Proceedings 34, 211, 1948.

χ INTRODUCTION xi

Remarks on the Determination of a Central Field of Force from the Elastic Scattering Phase Shifts, Phys. Rev. 75, 301, 1949.

On the Connection between Phase Shifts and Scattering Potential, Reviews of Modern Physics 21, 488, 1949.

On the Number of Bound States in a Central Field of Force, Natl. Academy of Science (USA) Proceedings 38, 961, 1952.

On Unitary Ray Representations of Continuous Groups, Annals of Math. 59, 1, 1954.

Relativity, Reviews of Mod. Phys. 29, 161, 1957.

(with L. Michel and V. Telegdi) Precession of the Polarization of Particles Moving in a Homogeneous Electromagnetic Field, Phys. Rev. Letters 2, 435, 1959.

Relativity, in "Theoretical Physics in the Twentieth Century" (Memorial Volume to Wolfgang Pauli, edited by M. Fierz and V. F. Weisskopf), p. 187. Interscience Publishers, New York, 1960.

(with M. Moshinsky) Group Theory of Harmonic Oscillators I: The Collect­ ive Modes, Nuclear Physics 18, 697, 1960.

(with M. Moshinsky) Group Theory of Harmonic Oscillators II: The Integrals of Motion for the Quadrupole-quadrupole Interaction, Nuclear Phys. 23, 177, 1961.

On a Hilbert Space of Analytic Functions and an Associated Integral Transform, Part I, Communications on Pure and Applied Mathematics 14, 187, 1961.

On the Representations of the Rotation Group, Reviews of Modern Physics 34, 829, 1962.

Remarks on a Hilbert Space of Analytic Functions, Proc. Natl. Acad. Sci., U.S.A., 48, 1962, 19-22, 2204.

Note on Wigner's Theorem on Symmetry Operations, Jour. Math. Phys. 5, 862, 1964.

On a Hilbert Space of Analytic Functions and an Associated Integral Transform, Part II, Communications on Pure and Applied Mathematics 20, 1, 1967. xii INTRODUCTION

Group Representations on Hilbert Spaces of Analytic Functions, in "Analytic Methods of Mathematical Physics," edited by R. P. Gilbert and R. G. Newton, p. 27, Gordon and Breach, New York 1970.

Group Representations in Mathematics and Physics (Battelle Seattle 1969 Rencontres) Edited by V. Bargmann Lecture Notes in Physics, Vol. 6, Springer-Verlag, Berlin-Heidelberg-New York (published Dec. 1970).

(with P. Butera, L. Girandello and J. R. Klauder) On the Completeness of Coherent States. Reports on Math. Physics 2, 221, 1971.

Notes on Some Integral Inequalities, Helvetica Physica Acta 45, 249, 1972.

(with I. T. Todorov) Spaces of Analytic Functions on a Complex Cone as Carriers for the Symmetric Tensor Representations of SO(n), Preprint 1975.