Comments by Laurie M. Brown

I shall assume that I was invited to contribute to this symposium-workshop as an historian of recent physics; any ‘philosophical’ inferences will be unintended. In order to comment on Steve Weinberg's talk, I expected to have the text a few weeks in advance, which did not happen; but Steve kindly supplied a list of some writings on which he would base his talk:

1. Chapter 1 of the recently published (1995) first volume of ‘The Quantum Theory of Fields’ (QFT).

2. The well-known article in Daedalus (1977) called ‘The search for unity: Notes for a history of quantum field theory.’

3. ‘The ultimate structure of matter,’ in A Passion for Physics - Essays in Honor of Geoffrey Chew (1985).

4. The (1993) book ‘Dreams of a Final Theory,’ especially Chapters 5 and 8.

5. ‘Night thoughts of a quantum physicist,’ Bull Amer. Acad. Arts and Sciences 69, No. 3, p. 51 (1995).

6. ‘Nature itself,’ an essay in Chapter 27 of Twentieth Century Physics, a recently (1995) published compendium. The same chapter has parallel essays by Philip Anderson and John Ziman.

7. ‘Conference summary,’ in Proc. XXVI Intern. Conf. on High Energy Physics (AIP 1993).

In general, I did not detect appreciable differences in the writings I consulted, which extend from 1977 to 1995, over nearly two decades. This may say something about the progress (or lack thereof) in fundamental physics during that time interval.

Introduction

Recently, the following question appeared in the American Journal of Physics (Andereck, 1995): how can gravity be unified with the other fundamental forces when it is not a force at all, but an aspect of the geometry of space-time? In general relativity (GR) the gravitational field is the space-time metric, but in the standard model of quantum field theory the electro-weak and strong forces are manifestations of massless vector particles - of gauge fields. If our most complete contemporary theories treat forces so differently how can they be unified?

Roughly, there are two kinds of answers; either all four fundamental forces are aspects of the geometry of space-time, so the vector fields are also absorbed into the metric. Or, gravity is a force field like the other fundamental forces, so GR is formulated somehow as a gauge theory with curved space-time as an effective low energy approximation. Both proposals put all the forces on a similar footing.

But the primary interest of this paper is not unification, in any of its senses. Rather we want to consider the other questions raised here. What is gravity? What are forces more generally? How does gravity relate to the other forces? These questions are as old as physics, and have been reposed and reinterpreted after every revolution in physical thought. In the present century, remarkable progress has been made in finding potentially definitive anwers to them, and this paper is intended as an account of this work.

One of the purposes of this conference is consideration of the historical background to the topics under discussion, so I shall open with a historical introduction to ‘quantum gravity’.

Once he completed work on the general theory of relativity at the end of 1915, Einstein began to concern himself with its implications for microphysics. In his first (1916) paper on gravitational radiation, Einstein argued that quantum effects must modify the general theory:

He reiterated this conclusion in his second (1918) paper:

Although thinkers had been groping for two hundred years for the idea of a field - something that would transmit forces across the vastness of interstellar space - it was not until Faraday and Maxwell that field theory in the modern sense was born. Not really in the modern sense, of course, because it was not initially decided whether fields are merely mathematical descriptions of a deeper reality or are themselves physical entities that occupy the absolute space of Newton. Maxwell certainly regarded the electromagnetic field as something physical, but he was curiously unwilling to ascribe to it an independent existence as a ‘nonmechanical’ entity in its own right.

Even Einstein, in the early years of this century, was unwilling to ascribe an independent existence to the gravitational field. Instead he made the bold advance of expressing it, not as something that occupies space-time, but as a property, curvature, of space-time itself, and he tried to develop a geometrical theory of the electromagnetic field as well.

The question ‘Are fields merely mathematical descriptions of a deeper reality, or are they independent physical entities?’ and questions like it, recur over and over in different guises. As titillating as such questions may be to some philosophers, the modern response to them is surely that they are irrelevant.

Introduction

Cantor, famously, defined a set as ‘… collections into a whole of definite, distinct objects of our intuition or of our thought’ (Cantor 1955, p. 85, our emphasis). On this basis the standard formulations of set theory, and consequently much of contemporary mathematics, are erected. Reflecting upon this definition, and the underlying ontological presuppositions, the question immediately arises, how are we to treat, mathematically, collections of objects which are not distinct individuals? This question becomes particularly acute in the quantum mechanical context, of course. As Yu. I. Manin noted at the 1974 American Mathematical Society Congress on the Hilbert Problems,

We should consider possibilities of developing a totally new language to speak about infinity [that is, axioms for set theory]. Classical critics of Cantor (Brouwer et al.) argue that, say, the general choice is an illicit extrapolation of the finite case.

I am presumably here to give you my perspective on quantum field theory from the point of view of a condensed matter theorist. I must begin with a disclaimer, a warning that I may not be really representing anyone but myself, since I find myself today working in condensed matter after following a fairly tortuous route. I must begin with a few words on this, not only since it will allow you to decide who else, if any, I represent, but also because my past encounters with field theory will parallel that of many others from my generation.

I started life in electrical engineering, as Professor Schweber said in the introduction. As an electrical engineering student, the only field theorists I knew about were Bethe and Schwinger, who had done some basic work on wave guides. When I graduated, I switched to physics and was soon working with Geoff Chew at Berkeley on particle physics. In those days (sixties and early seventies), the community was split into two camps, the field theorists and the S-matricists, and Geoff was the high priest of the latter camp. The split arose because, unlike in QED, where everyone agreed that the electron and the proton go into the Lagrangian, get coupled to the photon and out comes the hydrogen atom as a composite object, with strong interactions the situation seemed more murky.

Quantum field theory (QFT) is a powerful language for describing the subatomic constituents of the physical world (quarks, leptons, gauge bosons, Higgs scalars, and so on) and the laws and principles that govern them. Not only has it provided a framework for understanding the hierarchical structure that can be built from these constituents, it has also profoundly influenced the development of contemporary cosmology and deeply penetrated into the current conception and imagination of the origin and evolution of the universe. For this reason, it has justifiably long been taken to be the foundation of fundamental physics: elementary particle physics and cosmology.

QFT reached its most spectacular success in the early 1970s in the standard model, which is able to describe the fundamental interactions of nature within a unified theoretical structure: the non-Abelian gauge theory. Ironically enough, however, this success brought about a long period of stagnation in its conceptual development: virtually nothing of physical importance occurred after the construction of the standard model except for its detailed experimental verification. This situation can be assessed in different ways, depending on what perspective we start with. For example, the stagnation could be taken to be an indication that the theoretical development of fundamental physics has come to an end.

Philosophical introduction

The following seemingly trivial story raises important questions which lead to the philosophical issues essential for our discussion.

My cat is alive and well. I can describe her by size, color, behavior, etc. I can watch her, touch her, play with her, and feed her. I know this animal is really there. But my biologist friend came to visit and informed me that my cat is really nothing but a very complicated collection of different kinds of cells in mutual interaction, and he can prove it to me by having me look through his microscope, and, if I would only let him, by dissecting my cat. But the biochemist next door has a quite different opinion: he claims that my cat is really just a very complex assemblage of organic compounds, proteins, nucleic acids, etc., etc., all in mutual interaction. He, too, can prove his claim.

Of course, physicists know better: a cat, like all matter living or dead, is in last analysis really nothing but atoms bonded together to form the complicated molecules my biochemist friend is talking about. And they can prove that too. But, finally, one cannot ignore the elementary particle people who tell us that atoms are just composed of quarks, gluons, and electrons, so that a cat is really, really just a very complicated assembly of those few types of elementary particles in mutual interaction.

Knowledge of Fisher information is not part of the educational background of most physicists. Why should a physicist bother to learn about this concept? Surely the (related) concept of entropy is sufficient to describe the degree of disorder of a given phenomenon. These important questions may be answered as follows.

1. (a) The point made about entropy is true, but does not go far enough. Why not seek a measure of disorder whose variation derives the phenomenon? The concept of entropy cannot do this, for reasons discussed in Sec. 1.3. Fisher information will turn out to be the appropriate measure of disorder for this purpose.

2. (b) Why should a physicist bother to learn this concept? Aside from the partial answer in (a): (i) Fisher information is a simple and intuitive concept. As theories go, it is quite elementary. To understand it does not require mathematics beyond differential equations. Even no prior knowledge of statistics is needed: this is easy enough to learn ‘on the fly’. The derivation of the defining property of Fisher information, in Sec. 1.2.3, is readily understood. (ii) The subject has very little specialized jargon or notation. The beginner does not need a glossary of terms and symbols to aid in its understanding. (iii) Most importantly, once understood, the concept gives strong payoff – one might call it ‘phenomen-all’ – in scope of application. It's simply worth learning.