QUANTUM STATISTICAL MECHANICS

This book had its origin in a graduate course in statistical mechanics given by
Professor W. C. Schieve in the Ilya Prigogine Center for Statistical Mechanics at
the University of Texas in Austin.
The emphasis is quantum non-equilibrium statistical mechanics, which makes
the content rather unique and advanced in comparison to other texts. This was
motivated by work taking place at the Austin Center, particularly the interaction
with Radu Balescu of the Free University of Brussels (where Professor Schieve
spent a good deal of time on various occasions). Two Ph.D. candidate theses at
Austin, those of Kenneth Hawker and John Middleton, are basic to Chapters 3
and 4, where the master equations and quantum kinetic equations are discussed.
The theme there is the dominant and fundamental one of quantum irreversibil
ity. The particular emphasis throughout this book is that of open systems, i.e.
quantum systems in interaction with reservoirs and not isolated. A particularly
influential work is the book of Professor A. McLennan of Lehigh University,
under whose influence Professor Schieve first learned non-equilibrium statistical
mechanics.
An account of relatively recent developments, based on the addition in the
Schrödinger equation of stochastic fluctuations of the wave function, is given in
Chapter 13. These methods have been developed to account for the collapse of the
wave function in the process of measurement, but they are deeply connected as
well with models for irreversible evolution.
The first six chapters of the present work set forth the theme of our book, par
ticularly extending the entropy principle that was first introduced by Boltzmann,
classically. These, with equilibrium quantum applications (Chapters 7, 8, 9 and
possibly also Cha
As frequently pointed out in the text, quantum mechanics introduces special
problems to statistical mechanics. Even in Chapter 1, written by the coauthor of this
work, Professor Lawrence P. Horwitz of Tel Aviv, the idea of a density operator is
required which is not a probability distribution, as in the classical case. The idea of
the density operator lies at the very foundations of the quantum theory, providing
a description of a quantum state in the most general way. Statistical mechanics
requires this full generality. We give a proof of the Gleason theorem, stating that
in a Hilbert space of three or more real dimensions, a general quantum state has a
representation as a density operator, based on an elegant construction of C. Piron.
This structure gives the quantum H theorem, a content which is essentially different
from the classical one. This makes the subject surely interesting and important, but
difficult.
Quantum entanglements are quite like magic, so to speak. It is necessary and
important to see these modern developments; they are described in Chapter 15.
This is one chapter that might be used in the extension of the course to a second
semester. One- and two-time Green’s functions, introduced by Kadanoff and Baym,
might be included in the extended treatment, since they are popular but difficult.
This is included in Chapter 16 with an application in Chapter 19.
Anextension to special relativity is described in Chapter 10. This is a new deriva
tion of a many-body covariant kinetic theory. The Boltzmann-like kinetic equation
outlined here was derived in collaboration by the authors. The covariant picture is
an event dynamics controlled by an abstract time variable first introduced by both
Feynman and Stueckelberg and obtains a covariant scalar many-body wave func
tion parameterized by the new time variable. The results of this event picture are
outlined in Chapter 10.
Another arena of activity utilizing quantum kinetic equations for open systems
is the extensive development in quantum optics. This has been a personal interest
of one of the authors (WCS). This interest was a result of a Humboldt Founda
tion grant to the Max Planck Institute in Munich and later to Ulm, under the
direction of Professors Herbert Walther, Marlon Scully and Wolfgang Schleich.
The particular area of interest is described in the results outlined in Chapter 11.
This material can be included as an introduction to quantum optics in an extended
two-semester course.
The idea of spontaneous decay in a quantum system goes back to Gamov
in quantum mechanics. This irreversible process seems intrinsic, introducing the
notion of the Gel’fand triplet and rigged Hilbert spaces states. The coauthor (LPH)
has made personal contributions to this fundamental change in the wave function
picture. It is very appropriate to include an extensive discussion of this, which is
the content of Chapter 17, describing, among other things, the Wigner–Weisskopf
method and the Lax–Phillips approach to enlarging the scope of quantum wave

functions. All of this requires a more advanced mathematical approach than the
earlier discussions in this book. However, it is necessary that a well-grounded
student of quantum mechanics know these things, as well as acquire the mathe
matical tools, and therefore it is very appropriate here in a discussion of quantum
statistical mechanics.
Chapter 18 is in many ways an extension of Chapter 17. It is an outline of what
has been called extended statistical mechanics. Ilya Prigogine and his colleagues
in Brussels and Austin, in the past few years, have attempted to formulate many
body dynamics which is intrinsically irreversible. In the classical case this may
be termed the complex Liouville eigenvalue method. As an example, the Pauli
equation is derived again by these nonperturbative methods. This is not an open
system dynamics but rather, like the previous Chapter 17 discussion, one of closed
isolated dynamics. This effort is not finished, and the interested student may look
upon this as an introductory challenge.
The final chapter of this book is in many ways a diversion, a topic for personal
pleasure. The remarkable objects of our universe known as black holes apparently
exist in abundance. These super macroscopic objects obey a simple equilibrium
thermodynamics, as first pointed out by Bekenstein and Hawking. Remarkably,
the area of a black hole has a similarity to thermodynamic entropy. More remark
able, the S-matrix quantum field theoretic calculation of Hawking showed that the
baryon emission of a black hole follows a Planck formula. Hawking introduced a
superscattering operator which is analogous to the extended dynamical theory of
Chapter 18.
To complete these comments, we would like to thank Florence Schieve for sup
port and encouragement over these last years of effort on this work. She not only
gave passive help but also typed into the computer several drafts of the book as well
as communicating with the coauthor and the editorial staff of the publisher. The
second coauthor wishes also to thank his wife Ruth for her patience, understanding,
and support during the writing of some difficult chapters.
We also acknowledge the help of Annie Harding of the Center here in Austin.
Three colleagues at the University of Texas—Tomio Petrosky, George Sudarshan
and Arno Bohm—also made valuable technical comments. WCS also thanks the
graduate students who, over many years of graduate classes, made enlightened
comments on early manuscripts.
We recognize the singular role of Ilya Prigogine in creating an environment in
Brussels and Austin in which the study of non-equilibrium statistical mechanics
was our primary goal and enthusiasm.
Finally, WCS thanks the Alexander von Humboldt Foundation for making pos
sible extended visits to the Max Planck Institute of Quantum Optics in Garching
and later in Ulm. LPH thanks the Center for Statistical Mechanics and Complex

Systems at the University of Texas at Austin for making possible many visits over
the years that formed the basis for his collaboration with Professor Schieve, and the
Institute for Advanced Study at Princeton, particularly Professor Stephen L. Adler,
for hospitality during a series of visits in which, among other things, he learned of
the theory of stochastic evolution, and which brought him into proximity with the
University of Texas at Austin