The marvel of the harmony of the spheres

Symmetry and the Beautiful Universe - A Different Universe
September 16, 2005

Three physicists find fearful symmetry and crystalline insights in the vast final frontier. Ian Aitchison joins the journey

Theories of everything are less in the news these days, but everything itself - the universe - remains a reliable topic for physicists interested in reaching the popular market. With one exception, the field is shared between the astrophysicists and cosmologists, who tell us about wonders that are mostly very far away in space or time, and the particle physicists, who tend to look on their subject as the bedrock of everything else.

Leon Lederman and Christopher Hill belong in the second category, but only in the final chapter of Symmetry and the Beautiful Universe do they get down to quarks and leptons. As their title suggests, they have a broader aim, which is to popularise a crucial insight of 20th-century physics: the power of symmetry (which is related to beauty) to determine so many aspects of the physical world. Their admirably accessible account deserves to be widely read.

Unlike them and others writers in the field, Robert Laughlin, who is a condensed-matter physicist, is working on a frontier closer to us all than that of the cosmologist or the particle physicist. He is interested in what happens when individual entities such as atoms are assembled in very large numbers to form aggregates such as crystals and superconductors. He is profoundly struck by the ways in which the properties of such assemblies may be qualitatively quite different from those of their constituents: a suggestive metaphor is the relation between an image created by an artist and the individual daubs of paint from which it is constructed. For Laughlin, the emergence of organisation in complex structures - such as the formation of a crystal - is of universal significance. Perhaps his book might have been better titled "An Emergent Universe"; at any rate, it is certainly "different". The particle physicists, on the other hand, following what we may loosely term a reductionist programme, pursue the goal of dissecting matter into its most fundamental parts, Laughlin - the radical emergentist - is "increasingly persuaded that all [author's italics] physical law has collective origins", emerging at many scales and in many contexts from the self-organising behaviour of large numbers of interacting sub-units.

Although Lederman and Hill betray their reductionist sympathies on occasion, their central theme largely bypasses this particular culture war. Symmetry illuminates many aspects of physical laws, whether you believe them to be emergent or not, so it is fair to turn first to Symmetry and the Beautiful Universe . Later we shall see that part, at least, of "emergence" is itself closely associated with symmetry concepts.

The book's main business begins with a fable about an inventor who discovers that g , the acceleration due to gravity in his laboratory, is not constant in time over a short period every Tuesday morning. He quickly scents a commercial opportunity: he has a free source of energy! (If you do not understand why but would like to, then this book is definitely for you.) Unfortunately for the inventor and his backers, this is not how nature works; his experiments were wrong. All (careful) experiments ever done to investigate the question have shown that the total energy we get out of a system is equal to the total energy we put into it. So exception-free is this statement that we call it a law: the law of energy conservation.

This law is surely familiar to everyone. But what the story so entertainingly teaches is something that may well be new to many readers: energy conservation will fail if the laws of nature (here, the inventor's g value) change with time. Put differently, constancy of the laws over time implies energy conservation, and vice versa. "Constancy over time" is an example of what physicists mean by a symmetry, as applied to physical laws.

In relation to an object, a symmetry is an operation that can be performed on it such as to leave its appearance unchanged. Similarly, a symmetry of a law is an operation under which it does not change.

In the present case, that operation is a shift (or "translation") in time by an arbitrary amount. We have learnt that a symmetry (time translation) is associated with a conservation law (energy).

This is a deep insight, first because it casts new light on a familiar law, the symmetry aspect going some way towards "explaining" it; and second because there are many other examples of such pairings. As Lederman and Hill explain, these include the conservation laws of linear and angular momentum, which are associated with the symmetries of translation and rotation in space.

It was not until 1918 that a general mathematical proof of the connection between symmetries and conservation laws in physics was given by Emmy Noether. Lederman and Hill devote one chapter to a summary of Noether's life and her distinguished mathematical achievements. The significance of her 1918 paper runs like a leitmotiv throughout the book, as it does throughout much of 20th-century particle physics.

Noether's work was stimulated by subtle difficulties in the very question of energy conservation in Einstein's then recently published general theory of relativity. This extraordinary theory showed, for the first time, how the existence of a force (gravity) could be understood in terms of a symmetry. The concept of symmetry thus became intimately connected with both conservation laws and dynamics. This dual role for symmetry turned out to be the key common feature possessed by the three other known "fundamental" forces, the strong, weak and electromagnetic forces between quarks and leptons, which are all based on "gauge symmetry" and which together constitute the standard model of particle physics.

These developments are attractively described by Lederman and Hill in the second half of their book, this part being evidently written more particularly from the perspective of particle physics, though still informed by the theme of symmetry.

There is, however, one piece of the standard model, not yet established experimentally, that invokes a remarkable phenomenon much studied in condensed-matter physics, called "spontaneous symmetry breaking" and that involves the essential idea of ordering, or organisation (we are now entering Laughlin's universe). An example is provided by crystal ordering.

Beautiful though it may be, when a crystal forms, it breaks the spatial translation symmetry of free space, despite the fact that the microscopic laws that govern its constituent atoms do not. This ordering emerges, quite suddenly, as the substance makes a transition from the liquid to the crystalline phase. The particular ordered state invoked by the particle physicists is, however, not a crystal but a superconductor, in which the symmetry that is broken spontaneously is the gauge symmetry of electromagnetism. Normally this symmetry requires the quanta of the force field (photons) to be massless, and the same is true in other gauge theories. This apparently rules out the possibility of describing the weak force as a gauge theory, because experiments show that its quanta (called W and Z) are definitely not massless. However, inside a superconductor photons in effect acquire mass as a consequence of the symmetry breaking.

By supposing that an analogous symmetry breaking occurs in the weak force case, massive Ws and Zs can be regarded as quanta of a gauge theory, similar to electromagnetism. This is the basis of the celebrated "unification" of these two forces.

But what is the "state" whose ordering has produced the "weak superconductor"? The answer can only be the vacuum, because that is what these particles are travelling through, after all. The idea that the vacuum might be analogous to an ordered state of matter is fundamental to much of modern particle physics, for example to hopes of further unification: there are indications that a symmetry uniting all three forces of the standard model may have been spontaneously broken very early on in the evolution of the universe.

Electro-weak (and other) unification may therefore relate to emergent properties of the vacuum. But Laughlin's programme does not stop there: he wants everything to be emergent, including the so-called fundamental particles of the standard model themselves, such as electrons and photons.

Once again, the crystal provides a model. At very low temperatures, the quantum aspects of sound waves travelling through a crystal become apparent: the vibrational energy is quantised into particulate lumps called phonons, which behave like particles. But note that it makes no (reductionist) sense to think of dividing phonons into smaller pieces; they emerge as quantised collective motions of all the atoms in the crystal.

The idea that all the standard model particles, and gravity as well, can be understood as quantised motions of a single underlying "vacuum stuff" is undoubtedly different: here is a theory of everything in which everything is reduced to emergence. Unfortunately, Laughlin tells us very little about the difficulties this ambitious programme faces, and nothing about the partially successful attempts being made to implement it.

Central to these speculations is the physics of phase transitions, which serves Laughlin as a paradigm for emergence. On the one hand, the vast number of particles involved guarantees that it will never be possible to predict exactly what happens, however well we know the inter-particle forces. On the other, there are simple models that can be solved exactly that do exhibit the characteristic ordering (or organisational) phenomena, and realistic numerical calculations with a manageable number of particles often do a pretty good job.

On this basis, we have an "in-principle" understanding of phase transitions as collective organisational phenomena. Furthermore, the emergent laws in the organised states are largely independent of the microscopic details. We can now ignore the parts and accept the emergent laws of the collectively organised whole as a reliable fact of life. By extension, we can imagine in-principle paths from particle physics to condensed-matter physics, thence to chemistry, molecular biology, cell biology and so on up the ladder of complexity, happily granting each level in the hierarchy a high degree of autonomy while preserving the conceptual linkage.

These ideas were articulated by Philip Anderson in 1972, as Laughlin acknowledges. In A Different Universe , they are applied to the determination of the fundamental constants, classical mechanics, Schrödinger's cat and life, as well as the vacuum. We also read about quantum computers, the quantum Hall effects, superconductivity, the Star Wars defence project and nanotechnology. Laughlin tries hard not to be dull, but the stream of marginally relevant anecdotes and holiday stories is a poor substitute for careful explanations of subtle, fascinating physics. Laughlin's book is a fun read for those who already know what he is talking about, but I do not recommend it for learners.

Ian Aitchison is emeritus professor of physics, Oxford University.

Symmetry and the Beautiful Universe

Author - Leon M. Lederman and Christopher T. Hill
Publisher - Prometheus
Pages - 363
Price - £19.00
ISBN - 1 59102 242 8

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