# James Glimm's books

We give information about six books authored or co-authored by James Glimm. One of these books has an important Second edition and we also include information about this. We give the books in chronological order, except for the Second edition which we give immediately after the first edition of the corresponding book.

**1. Decay of solutions of systems of nonlinear hyperbolic conservation laws (1970), by James Glimm and Peter D Lax.**

**1.1. Review by: J Smoller.**

*Mathematical reviews*MR0265767

**(42 #676)**.

Perhaps the major contribution in this paper is the observation that interaction of waves usually causes some cancellation of waves. Thus, the non-linearity of the characteristic speeds causes shocks and rarefaction waves of the same family to interact with each other and thereby to cancel each other out. This qualitative observation must be made quantitative, and this is done in Section 4. Section 1 deals with shockless periodic solutions, and the result here is an upper bound on the time that such a solution can exist. In Section 2 the approximate characteristics for the difference approximations introduced by the first author are constructed, and the approximate conservation laws and decay estimates are derived. (This is the technique of the paper - derive all the relevant estimates for the difference approximations, keep control of the error terms and pass to the limit.) Section 3, as the authors point out, is "quite heavy going'' and involves the passage to the limit. In Section 5 are constructed solutions whose initial data are not of bounded variation - the a priori decay estimates, derived in the earlier sections, are essential to this regularity theorem. Also in this section is the main decay result stated above. This paper is by no means easy and involves quite delicate estimates at all stages. This delicateness is perhaps most easily illustrated by the fact that it is very important for these methods that a coordinate system consisting of Riemann invariants should exist, so that certain error terms are of third order, and not of second order, as in the general case. Thus the fact that there are just two equations is quite crucial. A reader attempting to tackle this paper should be quite familiar with the results of the first author, and the earlier work of the second author.

**2. Probability applied to physics (1978), by James Glimm and Arthur Jaffe.**

**2.1. Review by: Yu M Lomsadze.**

*Mathematical review*s MR0506159

**(80a:82024)**.

These notes are an outgrowth of the 1978 University of Arkansas Annual Lecture Series in Mathematics which convened in Fayetteville in March with Professor Glimm as the featured speaker. The notes are intended for mathematicians who are interested in the fundamental problems of statistical mechanics and quantum field theory.

**3. Quantum physics. A functional integral point of view (1981), by James Glimm and Arthur Jaffe.**

**3.1. From the Introduction.**

This book is addressed to one problem and to three audiences. The problem is the mathematical structure of modern physics: statistical physics, quantum mechanics, and quantum fields. The unity of mathematical structure for problems of diverse origin in physics should be no surprise. For classical physics is provided, for example, by a common mathematical formalism based on the wave equation and Laplace's equation. The unity transcends mathematical structure and encompasses basic phenomena as well. Thus particle physicists, nuclear physicists, and condensed matter physicists have considered similar scientific problems from complementary points of view. The mathematical structure presented here can be described in various terms: partial differential equations in an infinite number of independent variables, linear operators on infinite dimensional spaces, or probability theory and analysis over function spaces. This mathematical structure of quantisation is a generalisation of the theory of partial differential equations, very much as the latter generalises the theory of ordinary differential equations. Our central theme is the quantisation of a nonlinear partial differential equation and the physics of systems with an infinite number of degrees of freedom. Mathematicians, theoretical physicists, and specialists in mathematical physics are the three audiences to which the book is addressed.

**3.2. Review by: Paul Federbush.**

*Mathematical reviews*MR0628000

**(83c:81001)**.

Constructive Quantum Field Theory is a mathematical discipline now roughly fifteen years of age. J Glimm and A Jaffe pioneered this subject, and its development has been largely centred on an amazing sequence of difficult and inventive papers of these authors. While very essential ideas have been contributed by other mathematical physicists, and several score disciples have pursued applications and technical advances, very few fields of research have been dominated by individuals as much as the present discipline by our two authors. ... Constructive Quantum Field Theory is the mathematical (rigorous) study of quantum field theories. ... Quantum physics provides a view of Constructive Quantum Field Theory, and related fields in Statistical Mechanics, by the two researchers with the greatest insights into the theory. It comes at a pivotal time, I believe, when the fundamental development is nearly complete, and with progress into four dimensions still in the future. It offers an entrance into the field for mathematical physicists and other mathematicians. ... This book may be read through as a thorough introduction to a large segment of modern mathematical physics. Mathematicians with a background in analysis will find the book self-contained. There is a grand scope of covered material, from nonrelativistic quantum mechanics and scattering theory, through statistical mechanics, and most importantly to quantum field theory. The presentation has many pleasant surprises. Even the first steps into quantum mechanics and the treatment of the harmonic oscillator, will be of interest to the professional physicist. For the mathematical physicist the book is a good source to browse in, and a convenient reference. We found that many of the sections provide natural material to build a seminar about. It would be an ambitious but very worthwhile year's effort to attempt to read the whole book in an intense seminar.

**4. Quantum physics. A functional integral point of view (2nd Edition) (1987), by James Glimm and Arthur Jaffe.**

**4.1. Authors' Preface.**

Twenty years after its inception [Jaffe, 1965; Lanford, 1966; Glimm, 1967] constructive quantum field theory is on the threshold of achieving its major goals. This level of success, while not unprecedented in contemporary mathematics, occurs with sufficient infrequency that it is worth commenting on (a) some of the factors which contributed to this success, and (b) what the implications of this success might be for mathematics and for science.

It is easier to address the second question. We see three consequences of a satisfactory mathematical foundation for the equations of quantum field theory. First, there is the question of principle as to whether the equations are correct and are correctly formulated. Having a mathematical foundation is a necessary but not a sufficient condition to answer this question positively. This concern was the original and primary motivation for the work from which this book is drawn. Second, the equations of quantum field theory are prototypes for other equations of independent interest, which arise in statistical mechanics, turbulence, and stochastic partial differential equations. The mathematical tools and concepts used to study quantum fields will likely find use in a variety of other problems having a similar mathematical structure. In fact, it is remarkable that these field-theoretic ideas have been instrumental through the work of Donaldson, Taubes, Uhlenbeck, and others in fundamental achievements in topology, where the mathematical structures appeared to be unrelated. Third, with the increasing power of computers, problems of the type mentioned above will be increasingly amenable to numerical solution. In this case, knowledge of the mathematical structure of the solution will be a considerable help in the discovery, understanding, and analysis of numerical algorithms and of numerical solutions. At the turn of the past century, Poincaré advanced essentially the same argument (citing the increased accuracy and quantity of astronomical observations) as a reason for the development of the qualitative theory of ordinary differential equations.

It is more difficult to determine the factors contributing to the success in this subject. It appears that a dedicated and talented group of workers, strong scientific leadership, sound scientific judgments, and constructive working arrangements each had a significant role to play.

In this second edition of Quantum Physics, we have added new chapters on correlation inequalities and the cluster expansion. Included is the remarkable proof that the f4 theories are trivial in high dimensions. Nonabelian gauge theories are required on both physical and mathematical grounds, and a new chapter is devoted to this topic. Also included in this chapter are phase cell expansions. This set of ideas has provided the basis for most of the estimates and proofs in constructive field theory. They were developed independently from renormalization group theory which implements similar ideas in problems with natural scaling behaviour. An appendix on Hilbert space operators and function space integrals was added to make the book self-contained from a mathematical point of view. Certain proofs in Part I were simplified or expanded, to make them easier to follow.

**4.2. Review by: P D F Ion.**

*Mathematical review*s MR0887102

**(89k:81001)**.

In the six years between the publication of the first and second editions of this book there has been a resurgence of interest in quantum field theory on the part of mathematicians. But, as yet, the mathematical details as addressed in the programme of constructive quantum field theory have not commanded most attention. We are still in a phase of intense interest in such matters as strings, noncommutative geometry, knot theory and a great deal supposed to be related to conformal field theory. But these methods and approaches are general and geometric, or not as concerned with the details of the analysis behind the general picture as might be. When the more delicate analysis is called for, the reference book of choice is Glimm and Jaffe's volume. The book has been expanded by about 100 pages since the first edition. Partly this is due to the addition of explanatory material in the first introductory part, where, in particular, a 30-page appendix on Hilbert space and functional integrals has been added. There are other major additions such as a chapter in Part I on correlation inequalities and the Lee-Yang theorem, and a chapter in Part III on the cluster expansion, and another on constructive gauge theory and phase cell localization. ... The book was lauded by the reviewer of the first edition as "a thorough introduction to a large"of modern mathematical physics''. It remains the best one available for many purposes, and has in fact become even better.

**5. Quantum field theory and statistical mechanics. Expositions. Reprint of articles published 1969-1977 (1985), by James Glimm and Arthur Jaffe.**

**5.1. From the Introduction.**

This volume contains a selection of expository articles on quantum field theory and statistical mechanics by James Glimm and Arthur Jaffe. They include a solution of the original interacting quantum field equations and a description of the physics which these equations contain. Quantum fields were proposed in the late 1920s as the natural framework which combines quantum theory with relativity. They have survived ever since. The mathematical description for quantum theory starts with a Hilbert space H of state vectors. Quantum fields are linear operators on this space, which satisfy nonlinear wave equations of fundamental physics, including coupled Dirac, Maxwell and Yang-Mills equations. The field operators are restricted to satisfy a "locality" requirement that they commute (or anti-commute in the case of fermions) at space-like separated points. This condition is compatible with finite propagation speed, and hence with special relativity. Asymptotically, these fields converge for large time to linear fields describing free particles. Using these ideas a scattering theory had been developed, based on the existence of local quantum fields. Whenever physicists had attempted to find solutions to such equations for quantum fields, they ran into formal difficulties. The requirement of Lorentz invariance forced the quantum fields to exhibit local singularities which appeared incompatible with the possibility to interpret nonlinear functions of the fields and therefore with nonlinear equations of motion. In practical terms, calculations led to infinite answers, which physicists learned to avoid only by adhering to a set of rules known as "renormalisation." One interpretation of renormalisation is that certain constants in the quantum mechanical Hamiltonian must be infinite, as is demonstrated in the paper 'An infinite renormalisation of the Hamiltonian is necessary'.

**5.2. Review by: Paul Federbush.**

*Mathematical reviews*MR0810217

**(87c:81101)**.

Constructive quantum field theory was initiated and developed primarily by the efforts of the authors, over the past two decades. This is the mathematical discipline that has put mathematical rigor into the study of relativistic quantum field theory, the area of physics uniting quantum mechanics and special relativity. Physicists have here obtained agreements between theory and experiment to one part in $10^{12}!$ Yet, till the advent of constructive quantum field theory, there was little hope for a solid mathematical foundation for such results. In this collection of papers, mainly expository articles, which stretches over 10 years, one can trace the history of the field. ... As pleased as we are with the current collection of papers, we cannot help but feel that the collection of the basic research manuscripts will be a more satisfying monument to the authors and constructive quantum field theory.

**6. Collected papers. Vol. 2. Constructive quantum field theory. Selected papers. Reprint of articles published 1968-1980 (1985), by James Glimm and Arthur Jaffe.**

**6.1. Review by: P D F Ion.**

*Mathematical reviews*MR0947959

**(91m:81003)**.

In this second volume are collected 25 of the over 40 joint papers that the authors published during the period approximately 1968-1980. The reviewer of the first volume of collected papers regretted mildly that it held expository papers, though some were classics still consulted, whereas the monument to the authors' efforts in those days must surely be their research papers. That is what this volume contains. The authors were the prime movers in the field of constructive field theory during the period mentioned, so this collection shows its development very well. ... This is a remarkable collection of important papers which should provide a ready reference and many ideas worthy of further study to those working on understanding constructive quantum field theory.

**7. Mathematical Sciences, Technology, and Economic Competitiveness (1991), edited by James G Glimm.**

**7.1. From the Preface.**

The fundamental importance of mathematics to the U.S. technology base, to the ongoing development of advanced technology, and, indirectly, to U.S. competitiveness is well known in scientific circles. However, the declining number of U.S. high school students who decide to seek a career in science or engineering is an important indication that many people do not appreciate how central the mathematical sciences have become to our technological enterprise. The National Research Council's Board on Mathematical Sciences has prepared this report to underscore the importance of supporting mathematics instruction at all levels, from kindergarten through graduate school, to prepare our youth for successful careers in science and engineering.

**7.2. Review by: H W Pullman.**

*The Mathematics Teacher*

**85**(2) (1992), 148; 150

This report is directed to members of the professional and academic mathematics community, to corporate decision makers, and to governmental and university policymakers. The authors establish a direct and critical connection between computational modelling, mathematical modelling, and technology transfer and the economic competitiveness of the nation. Specific illustrations of this connection are given first through a focus on applications in five key American industries, second through a focus on the life cycle of products and services - from planning and financing through production and marketing to servicing - and finally through a focus on specific areas of mathematical technology. Each example includes descriptions of the commercial application, the mathematical technology applied to that application, and the link between the application and economic competitiveness. The authors urge the aforementioned groups to work cooperatively to foster the development of mathematical applications, to facilitate the transfer of mathematical knowledge to commercial applications, and to support university level programs in industrial and applied mathematics.

Last Updated April 2020