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):

Introduction

I want to take issue with the definition of enantiomorphy that Pooley gives in his paper in this volume. His account goes something like this:

1. (a) Suppose that the relationist has an account of the dimensionality of space, according to which space is n-dimensional.

2. (b) The relations – especially the multiple relations – between the parts of a body determine whether it is geometrically embeddable in n-dimensional spaces that are either (only) orientable or (only) non-orientable.

3. (c) Then ‘an object is an enantiomorph iff, withrespect to every possible abstract [n]-dimensional embedding space, each reflective mapping of the object differs in its outcome from every rigid motion of it.’

This account depends on the truth of (b). Suppose that a body were embeddable in both orientable and non-orientable spaces of n dimensions. Then it might fail to be an enantiomorph, not because any of its possible reflections in physical space was identical to a rigid motion of the body, but because in some abstract space a reflection and a rigid motion of its image are identical. Pooley (in note 14) makes this point, but claims that the burden of proof falls on the opponent of his account to show that (b) is false.

As other contributions to this volume also testify, the notions of symmetry and equivalence are closely connected. This paper is devoted to exploring this connection and its relevance to the symmetry issue, starting from its historical roots. In fact, it emerges as an essential and constant feature in the evolution of the modern notion of symmetry: at the beginning, as a specific relation between symmetry and equality; in the end, as a general link between the notions of symmetry, equivalence class, and transformation group.

Symmetry and equality

Weyl's 1952 classic text on symmetry starts with the following distinction between two common notions of symmetry:

Bilateral symmetry is in fact a particular case of the scientific notion of symmetry, the symmetry being defined as invariance with respect to a transformation group (in the case of bilateral symmetry, the group of spatial reflections).

Introduction

Emmy Noether's greatest contributions to science were in algebra, but for physicists her name will always be remembered for her paper of 1918 on an invariance problem in the calculus of variations. The most celebrated part of this work, associated with her ‘first theorem’, has to do with the connection between continuous (global) symmetries in Lagrangian dynamics and conservation principles, though the main focus of the paper was the relationship between this and the second part of her paper, where she gives a systematic treatment of the more subtle and general case of continuous local symmetries (symmetries depending on arbitrary functions of the spacetime coordinates).

The connection between global or ‘rigid’ symmetries and conservation principles in classical mechanics was hardly news in 1918. As Kastrup (1987) discusses in his historical review, it had been appreciated in the previous century by Lagrange, Hamilton, Jacobi, and Poincaré, and an anticipation of Noether's first theorem in the special cases of the 10-parameter Lorentz and Galilean groups had been given by Herglotz in 1911 and Engel in 1916, respectively. Noether's own contribution is often praised for its degree of generality, and not without reason. But interestingly it does not cover the cases in which the symmetry transformation preserves the Lagrangian or Lagrangian density only up to a divergence term. It does not therefore cover such cases as the boost symmetry in classical pre-relativistic dynamics, although modern treatments of Noether's first theorem commonly rectify this defect.

Three attempts for an explanation of the Aharonov–Bohm effect

The Aharonov–Bohm (A–B) effect is an effect one finds in every quantum field theory book and this is so for a very good reason. The prediction and subsequent experimental verification of the effect have been crucial cornerstones in the history of physics because they suggested that the gauge potential, also known as the Aμ field, might be interpreted as a real field, rather than just a mathematical artifact. Hence, ever since its discovery, physicists have taken it for granted that Aμ does represent something at least as tangible as any matter field. However, when one examines these arguments more closely, one realizes that attributing the status of a really existent field to the Aμ field is not as straightforward as was originally thought. But first things first: we begin with an account of the effect itself, and then attempt to give some explanation for it.

The effect

The setting for the A–B effect is very similar to the two-slit experiment, with just one difference: immediately beyond the two slits and in between them is a very fine and long solenoid, ideally infinitely long, producing a magnetic field that is confined entirely within the tube of the solenoid.