Probability Distributions in Quantum Statistical Mechanics
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1148 Mark A. Kon
Probability Disfributions in Quantum Statistical Mechanics
Springer-Verlag
Author
Mark A. Kon Department of Mathematics, Boston University 111 Cummington Street Boston, MA 02215, USA
Mathematics Subject Classification (1980): 82A 15, 60E07, 60G60, 46L60 ISBN 3-540-15690-9 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-15690-9 Springer-Verlag New York Heidelberg Berlin Tokyo
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© by Springer-Verlag Berlin Heidelberg 1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
PREFACE
The purpose of this work is twofold: to provide a rigorous mathematical foundation for study of the probability distributions of cbservables in quantum statistical mechanics, and to apply the theory to examples of physical interest. Although the work will primarily interest mathematicians and mathematical physicists, I believe that results of purely physical interest (and at least one rather surprising result) are here as well. Indeed, some (§9.5) have been applied (see [JKS]) to study a model of the effect of angular momentum on the frequency distribution of the cosmic background radiation. It is somewhat incongruous that in the half century since the development of quantum statistics, the questions of probability distributions in so probabilistic a theory have been addressed so seldom. Credit is due to the Soviet mathematician Y.A. Khinchin, whose Mathematical FOuIldations of Quantum Statistics was the first comprehensive work (to my knowledge) to address the subject. Chapters 7 and 8 are a digression into probability theory whose physical applications appear in Chapter 9. These chapters may be read independently for their probabilistic content. I have tried wherever possible to make the functional analytic and operator theoretic content independent of the probabilistic content, to make it accessible to a larger group of mathematicians (and hopefully physicists). My thanks go to I.E. Segal, whose ideas initiated this work and whose work has provided many of the results needed to draw up the framework developed here. My thanks go also to Thomas Orowan, who saw the input and revision of this manuscript, using TEX, from beginning to end; his work was invariably fast and reliable. Finally I would like to express my appreciation to the Laboratory for Computer Science at M.LT., on whose DEC 10 computer this manuscript was compiled, revised, and edited.
CONTENTS
Ch. 1 Introduction
1.1 Purposes and background 1.2 The free boson and fermion fields over a Hilbert space 1.3 The state probability space Ch. 2 Value Functions on a Canonical Ens
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