Introduction

I think that there is interest in introducing into the study of physical phenomena the symmetry arguments familiar to crystallographers.

For example, an isotropic body may have a rectilinear or rotational motion. If liquid, it may have turbulence. If solid, it may be compressed or twisted. It may be in an electric or magnetic field. It may carry an electric or thermal current. It may be traversed by unpolarized or linearly, circularly, elliptically, etc. polarized light. In each case a certain characteristic asymmetry is necessary at each point of the body. These asymmetries are even more complex if one assumes that several phenomena coexist in the same medium or if they take place in a crystallized medium, which already possesses, by its constitution, a certain asymmetry.

Physicists often utilize symmetry conditions, but generally neglect to define the symmetry in a phenomenon, for sufficiently often these symmetry conditions are simple and quite obvious a priori.

When teaching physics, it would be better to state these questions openly: In the study of electricity, for example, one should state almost immediately the characteristic symmetry of the electric field and of the magnetic field. One can then use these notions to simplify proofs.

From a general point of viewthe idea of symmetry can be linked to the concept of dimension: These two fundamental concepts are respectively characteristic of the medium in which a phenomenon occurs and the quantity which serves to evaluate its intensity.

The following text is part of a talk entitled ‘Cross Fertilization in Theoretical Physics: the Case of Condensed Matter and Particle Physics’ given at the Young Researchers Symposium held on the occasion of the XIII International Congress on Mathematical Physics, London, 17–22 July 2000. One of the purposes of the symposium was the communication of personal experiences and points of view on the part of older researchers and this is reflected in the style of exposition.

The full text appears in Highlights in Mathematical Physics, edited by A. Fokas, J. Halliwell, T. Kibble, and B. Zegarlinski, and published in 2002 by the American Mathematical Society.

Breaking gauge and chiral invariance

Spontaneous breakdown of symmetry (SBS) is a concept that is applicable only to systems with infinitely many degrees of freedom. Although it pervaded the physics of condensed matter for a very long time – magnetism is a prominent example – its formalization and the recognition of its importance has been an achievement of the second half of the twentieth century. Strangely enough, the name was adopted only after its introduction in particle physics: it is due to Baker and Glashow (1962). I think that concepts acquire a proper name only when they attain their full maturity: this emphasizes the relevance of this case that took place over forty years ago.

Introduction

In our contribution to this volume we deal with discrete symmetries: these are symmetries based upon groups with a discrete set of elements (generally a set of elements that can be enumerated by the positive integers). In physics we find that discrete symmetries frequently arise as ‘internal’, non-spacetime symmetries. Permutation symmetry is such a discrete symmetry, arising as the mathematical basis underlying the statistical behaviour of ensembles of certain types of indistinguishable quantum particle (e.g. fermions and bosons). Roughly speaking, if such an ensemble is invariant under a permutation of its constituent particles (i.e. permutation symmetric) then one doesn't ‘count’ those permutations which merely ‘exchange’ indistinguishable particles; rather, the exchanged state is identified with the original state.

This principle of invariance is generally called the ‘indistinguishability postulate’ (IP), but we prefer to use the term ‘permutation invariance’ (PI).

Introduction

Mathematically, gauge theories are extraordinarily rich – so rich, in fact, that it can be all too easy to lose track of the connections between results, and become lost in a mass of beautiful theorems and properties: indeterminism, constraints, Noether identities, local and global symmetries, and so on.

One purpose of this short article is to provide some sort of a guide through the mathematics, to the conceptual core of what is actually going on. Its focus is on the Lagrangian, variational-problem description of classical mechanics, from which the link between gauge symmetry and the apparent violation of determinism is easy to understand; only towards the end will the Hamiltonian description be considered.

The other purpose is to warn against adopting too unified a perspective on gauge theories. It will be argued that the meaning of gauge freedom in a theory such as general relativity is (at least from the Lagrangian viewpoint) significantly different from its meaning in theories such as electromagnetism. The Hamiltonian framework blurs this distinction, and orthodox methods of quantization obliterate it; this may, in fact, be genuine progress, but it is dangerous to be guided by mathematics into conflating two conceptually distinct notions without appreciating the physical consequences.

The price paid by this article for abandoning the mathematics of gauge theory as far as possible is an inevitable loss of rigour. Virtually nothing will be ‘proved’ below; at most, the shape of proofs will be gestured at and strong plausibility-arguments advanced.

Symmetry considerations dominate modern fundamental physics, both in quantum theory and in relativity. Philosophers are now beginning to devote increasing attention to such issues as the significance of gauge symmetry, the role of symmetry breaking, the empirical status of symmetry principles, and so forth. These issues relate directly to traditional problems in the philosophy of science, including the status of the laws of nature, the relationships between mathematics, physical theory, and the world, and the extent to which mathematics dictates physics.

In January 2001 the first philosophy of physics workshop on symmetries in physics was held in Oxford. It became clear from the success of the workshop, the enthusiasm and sense of shared work-in-progress, that the time is right for a collection of papers in philosophy of physics on the subject of symmetry. As the organizers of the workshop, we decided to bring together in one book the current philosophical discussions of symmetry in physics, and to do so in a format that would provide a point of entry into the subject for non-experts, including students and philosophers of science in general. As such, the book is intended to be accessible and of interest to a wide audience of physicists and philosophers. It is appropriat for courses in foundations of physics, philosophy of physics, and advanced courses in philosophy of science. Some of the papers in this collection originated from papers presented at the Oxford workshop, but most have been written expressly for this book.

Initial conditions, laws of nature, invariance

The world is very complicated and it is clearly impossible for the human mind to understand it completely. Man has therefore devised an artifice which permits the complicated nature of the world to be blamed on something which is called accidental and thus permits him to abstract a domain in which simple laws can be found. The complications are called initial conditions; the domain of regularities, laws of nature. Unnatural as such a division of the world's structure may appear from a very detached point of view, and probable though it is that the possibility of such a division has its own limits, the underlying abstraction is probably one of the most fruitful ones the human mind has made. It has made the natural sciences possible.

The possibility of abstracting laws of motion from the chaotic set of events that surround us is based on two circumstances. First, in many cases a set of initial conditions can be isolated which is not too large a set and, in spite of this, contains all the relevant conditions for the events on which one focuses one's attention….

However, the possibility of isolating the relevant initial conditions would not in itself make possible the discovery of laws of nature. It is, rather, also essential that, given the same essential initial conditions, the result will be the same no matter where and when we realize these.

Introduction

Two views…

When Einstein formulated his General Theory of Relativity, he presented it as the culmination of his search for a generally covariant theory. That this was the signal achievement of the theory rapidly became the orthodox conception. A dissident view, however, tracing back at least to objections raised by Erich Kretschmann in 1917, holds that there is no physical content in Einstein's demand for general covariance. That dissident view has grown into the mainstream. Many accounts of general relativity no longer even mention a principle or requirement of general covariance.

What is unsettling for this shift in opinion is the newer characterization of general relativity as a gauge theory of gravitation, with general covariance expressing a gauge freedom. The recognition of this gauge freedom has proved central to the physical interpretation of the theory. That freedom precludes certain otherwise natural sorts of background spacetimes; it complicates identification of the theory's observables, since they must be gauge invariant; and it is now recognized as presenting special problems for the project of quantizing of gravitation.

…that we need not choose between

It would seem unavoidable that we can choose at most one of these two views: the vacuity of a requirement of general covariance or the central importance of general covariance as a gauge freedom of general relativity. I will urge here that this is not so; we may choose both, once we recognize the differing contexts in which they arise.

Introduction

Symmetry has an undeniable heuristic value in physics, as is demonstrated throughout this volume. To see if the value is more than instrumental, that is, to see if the symmetries are somehow in nature itself, we should ask a transcendental question: what must the world be like such that symmetry would be so effective in understanding it?

I will describe two apparent conditions of symmetry: objectivity and design. It will turn out that only one of these, objectivity, can be securely linked to symmetry. But symmetry does not serve as evidence for design in nature. In fact, the very aspects of symmetry that link it to objectivity suggest that it is not the result of design. The two apparent implications of symmetry are incompatible, and there is clear reason to retain the notion of objectivity and give up design.

Symmetry and objectivity

The accomplishment of knowledge involves keeping track of relations between the permanent and the ephemeral. Sensations keep changing while the relevant categories to describe them stay the same, and we have empirical knowledge of the world when we can accurately associate the fleeting sensations with their more permanent concepts. Coordinate systems remain fixed as an object's position changes, and the science of kinematics is useful insofar as it can describe the variable positions in terms of the stable reference frame. In general, knowledge is intimately involved in the interplay between what changes and what doesn't.

Introduction

Like moths attracted to a bright light, philosophers are drawn to glitz. So in discussing the notions of ‘gauge’, ‘gauge freedom’, and ‘gauge theories’, they have tended to focus on examples such as Yang–Mills theories and on the mathematical apparatus of fibre bundles. But while Yang–Mills theories are crucial to modern elementary particle physics, they are only a special case of a much broader class of gauge theories. And while the fibre bundle apparatus turned out, in retrospect, to be the right formalism to illuminate the structure of Yang–Mills theories, the strength of this apparatus is also its weakness: the fibre bundle formalism is very flexible and general, and, as such, fibre bundles can be seen lurking under, over, and around every bush. What is needed is an explanation of what the relevant bundle structure is and how it arises, especially for theories that are not initially formulated in fibre bundle language.

Here I will describe an approach that grows out of the conviction that, at least for theories that can be written in Lagrangian/Hamiltonian form, gauge freedom arises precisely when there are Lagrangian/Hamiltonian constraints of an appropriate character. This conviction is shared, if only tacitly, by that segment of the physics community that works on constrained Hamiltonian systems.

Introduction

Arguments regarding the ontological status of symmetries typically involve questions such as the following: how does the mathematics of symmetry relate to the matter of the physical world and do we have good reasons for thinking that the symmetries inherent in the mathematical structure of our theories have a counterpart in the physical world? In cases where there seems to be a corresponding relation between the symmetries present in the physical system (e.g. rotational and translational symmetries) and the symmetries in the equations that govern this system, one might think the relation is relatively straightforward and that the former is simply an empirical manifestation of the latter. But our questions are complicated by the fact that spontaneous symmetry breaking (SSB) is also a crucial feature of modern physics. In cases such as these the physical system displays none of the symmetry present in the equations that govern it. This symmetry is sometimes referred to as a hidden symmetry so the question, then, becomes one of determining whether the symmetry of the equation should be interpreted in a realistic way given that it seems to have no empirical manifestation.

But perhaps this notion of a ‘hidden’ symmetry should not raise philosophical worries, especially given that SSB lies at the foundation of some of the most successful theories in physics – superconductivity and quantum field theory (QFT) to name just two.

Symmetries can be a potent guide for identifying superfluous theoretical structure. This topic provides a revealing illustration of the power of formal methods for illuminating the contents of our theories, and bears potentially on some very old philosophical problems. The philosophical and scientific literature contains a good many discussions of individual cases, but the treatment is rarely general and tends to be technically involved in a way that may bury the basic physical insight as well as making it inaccessible to philosophers. We wish to identify the sorts of symmetry that signal the presence of excess structure, and do so in a completely general way, applicable to all theories and all genres of theory.

What is superfluous structure?

For any entity whether concrete or abstract we distinguish its elements and its structure; the latter is specified by listing relations between the elements (equivalently, features of sets or sequences of elements). Whether or not some of its structure is superfluous is clearly an interest-relative question. A sowing machine has superfluous structure if some features of or relations between its elements are dispensable for sowing, although these may be quite relevant to it from an aesthetic or antique collectors' point of view. Each of two features may be dispensable for the given purpose, but they may not be both dispensable at once, namely if the machine has multiple features which can play each other's roles.

Leibniz's third letter to Clarke, paragraph 5

I have many demonstrations, to confute the fancy of those who take space to be a substance, or at least an absolute being. But I shall only use, at the present, one demonstration, which the author here gives me occasion to insist upon. I say then, that if space was an absolute being, there would something happen for which it would be impossible there should be a sufficient reason. Which is against my axiom. And I prove it thus. Space is something absolutely uniform; and, without the things placed in it, one point of space does not absolutely differ in any respect whatsoever from another point of space. Now from hence it follows, (supposing space to be something in itself, besides the order of bodies among themselves,) that 'tis impossible there should be a reason, why God, preserving the same situations of bodies among themselves, should have placed them in space after one certain particular manner, and not otherwise; why every thing was not placed the quite contrary way, for instance, by changing East into West. But if space is nothing else, but that order or relation; and is nothing at all without bodies, but the possibility of placing them; then those two states, the one such as it now is, the other supposed to be the quite contrary way, would not at all differ from one another.

This book is about the various symmetries at the heart of modern physics. How should we understand them and the different roles that they play? Before embarking on this investigation, a few words of introduction may be helpful. We begin with a brief description of the historical roots and emergence of the concept of symmetry that is at work in modern physics (section 1). Then, in section 2, we mention the different varieties of symmetry that fall under this general umbrella, outlining the ways in which they were introduced into physics. We also distinguish between two different uses of symmetry: symmetry principles versus symmetry arguments. In section 3 we change tack, stepping back from the details of the various symmetries to make some remarks of a general nature concerning the status and significance of symmetries in physics. Finally, in section 4, we outline the structure of the book and the contents of each part.

The meanings of symmetry

Symmetry is an ancient concept. Its history starts with the Greeks, the term συμμετρíα deriving from σύν (with, together) and μέτρоν (measure) and originally indicating a relation of commensurability (such is the meaning codified in Euclid's Elements, for example). But symmetry immediately acquired a further, more general meaning, with commensurability representing a particular case: that of a proportion relation, grounded on (integer) numbers, and with the function of harmonizing the different elements into a unitary whole (Plato, Timaeus, 31c):