Lie Groups And Quantum Mechanics

Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich Series: Mathematisches Institut der Universit~t Bonn • Adviser: F. Hirzebruch 52 III D. J. Simms Trinity College, Dublin Lie Groups and Quantum Mechanics 1968 Springer-Verlag- Berlin-Heidelberg-New York All rights reserved. N o part of this b o o k may be translated or reproduced in any form without written p e r m i s s i o n from Springer Verlag. © by Springer-Verlag Berlin • Heidelberg 1968 Library of C o n g r e s s Catalog Card N u m b e r 68 - 24468 Printed in Germany. Title No. 7372 Preface These notes are based on a series of twelve lectures given at the University of Bonn in the Wintersemester 1966-67 to a mixed group of mathematicians and theoretical physicists. I am grateful to Professors H~rzebruch, Bleuler and Klingenberg for giving me the opportunity of s p e a k ~ a t their seminar. These notes are written primarily for the mathematicianwho has an elementary acquaintance with Lie groups and Lie algebras and who would like an account of the ideas which arise from the concept of relativistic invariance in ouantum mechanics. They may also be of interest to the theoretical physicist who wants to see familiar material presented in a f o ~ w h i c h uses standard concepts from other I areas of mathematics. The presentation owes very much to the lecture notes of Mackey [24] and [25] and of Hermann [15] . Many of the original ideas are due to Wigner and Bargmann [37] , [1] and [39] . A useful collection of reprints is contained in Dyson [11] . I should like to thank Professor F. Hirzebruch for his help and stimulation, Dr. D. Arlt for useful criticisms, and the Mathematisches Institut Bonn for support during this time and for the ty