Introduction

Abstract

“Once we have the fundamental equation (Urgleichung ¹) we have the theory of everything” is the creed of some physicists. They go on to say that “then physics is complete and we have to seek other employment”. Fortunately, other scientists do not subscribe to this credo, for they believe that it is not the few Greek letters of the Urgleichung that are the essential physics but rather that physics consists of all the consequences of the basic laws that have to be unearthed by hard analysis. In fact, sometimes it is the case that physics is not so much determined by the specific form of the fundamental laws but rather by more general mathematical relations. For instance, the KAM-theorem that determines the stability of planetary orbits does not depend on the exact 1/r law of the gravitational potential but it has a number theoretic origin. Thus a proper understanding of physics requires following several different roads: One analyzes the general structure of equations and the new concepts emerging from them; one solves simplified models which one hopes render typical features; one tries to prove general theorems which bring some systematics into the gross features of classes of systems and so on. Elliott Lieb has followed these roads and made landmark contributions to all of them. Thus it was a difficult assignment when Professor Beiglböck of Springer-Verlag asked me to prepare selecta on one subject from Lieb’s rich publication list². When I finally chose the papers around the theme “stability of matter” I not only followed my own preference but I also wanted to bring the following points to the fore:

• (a) It is sometimes felt that mathematical physics deals with epsilontics irrelevant to physics. Quite on the contrary, here one sees the dominant features of real matter emerging from deeper mathematical analysis.

• (b) The Urgleichung seems to be an ever receding mirage which leaves in its wake laws which describe certain more or less broad classes of phenomena. Perhaps the widest class is that associated with the Schrödinger equation with 1/r-potentials, which appears to be relevant from atoms and molecules to bulk matter and even cosmic bodies. Thus the papers reproduced here do not deal with mathematical games but with the very physics necessary for our life.

• (c) In mathematics we see a never ending struggle for predominance between geometry and analysis, the fashions swinging between extremes. Not too long ago the intuitive geometrical way in which most physicists think was scorned by mathematicians and only results from abstract analysis were accepted. Today the pendulum has swung the other way and the admired heroes are people with geometric vision, whereas great analysts like J. von Neumann tend to be thought of as degenerate logicians. As a physicist one should remain neutral with regard to internal affairs of mathematics, but it is always worthwhile to steer against the trend. Stability of matter illustrates beautifully that the great masters have forged for us the very analytical tools which we need to extract the physics from the fundamental equations.