Spin Foam

Michael Redhead

  • Three Roads to Quantum Gravity: A New Understanding of Space, Time and the Universe by Lee Smolin
    Phoenix, 231 pp, £6.99, August 2001, ISBN 0 7538 1261 4

The old notions of space and time are currently being turned upside down by theoretical physicists in their attempt to reconcile the two great pillars of 20th-century physics: quantum theory and Einstein’s general theory of relativity. Lee Smolin, a major contributor to the subject, brings us right up to date with it in this book. It’s written for general readers, but is more than just another work of popular science: it is a serious attempt at clarifying the author’s own thoughts about the significance and interpretation of the highly mathematical theories he is discussing. The book belongs to a new genre of science writing, in which the author also tells the story of his own involvement in the research, so giving it a striking freshness. It’s reminiscent in this way of Brian Greene’s very successful The Elegant Universe, a book which covers some, but by no means all, of the same ground as Smolin. Indeed, the two complement each other nicely, for anyone who wants to understand, in more than a superficial way, what is currently happening to the deep conceptual foundations of physics.

At this level of enquiry, physics rapidly merges with philosophy. Neither Smolin nor Greene is a professional philosopher, but that isn’t to say that philosophers shouldn’t now be spending more of their time poring over the Physical Review and Reviews of Modern Physics. There have always been two traditions in metaphysics, characterised by Peter Strawson as ‘descriptive’ and ‘revisionary’. Descriptive metaphysics seeks to uncover how we do, or more ambitiously must, conceptualise the world we find ourselves in. Revisionary metaphysics seeks, by contrast, to show that our ordinary thinking about the world, or more generally reality, is quite wrong and needs radical revision. The great revisionists have been philosophers like Plato, Leibniz and Berkeley, weavers of fantastical metaphysical speculations. Descriptive metaphysics began in effect with Kant, and has been much in vogue ever since. Indeed, many modern analytic philosophers have rejected revisionary metaphysics altogether as wild and baseless.

Now, however, with modern theoretical physics as its conduit, revisionary metaphysics is back with a vengeance. No longer unbridled speculation, it emerges from attempts to remove glaring defects in theories that are empirically very well attested. Proof for the latest theories, as we shall see, is probably beyond experimental reach at present, but the speculations of theoretical physics might still, one day, face the ‘tribunal of sense experience’, as Quine put it. Quine argued, admittedly, that the tribunal’s verdict would never be unequivocal, but some sort of empirical check on the physicists’ speculations should be a possibility for the future. This has led some philosophers to take the view that modern physics is a new sort of ‘experimental metaphysics’, in Abner Shimony’s phrase. If that’s right, then philosophers should sit up and take notice of what physicists like Greene and Smolin are writing, the trouble being that the theories involve horrendously complicated mathematics. However, philosophers are now emerging who are equally qualified in physics and philosophy, and who aim to be taken seriously in both disciplines.

Smolin’s book begins by asking how it is that the quantum theory (QT) and the general theory of relativity (GR) in their present state are both so amazingly successful, when each is also profoundly unsatisfactory. The basic problem is that QT as currently formulated denies the most fundamental insight of GR, so that to make GR conform to QT, we must either change it to accommodate something that denies its own fundamental principles – not a very attractive prospect – or try to change radically the way we do QT.

The fundamental principle of GR is Einstein’s radical idea concerning the nature of space and time. In classical Newtonian physics, space and time are a fixed backdrop against which particles move, fields oscillate and so on. If we took away all the particles and fields we would still be left with space-time. Einstein’s special theory of relativity, which preceded GR, retained the fixed backdrop, but changed its geometry in such a way that spatial distances and temporal intervals were no longer absolute, but relativised according to the motion of the observer. An absolute notion of ‘distance’ between events in space-time was preserved, however, with a new geometry of space-time formalised by the mathematician Hermann Minkowski. Then, with GR, Einstein introduced the idea of a variable geometry in which the curvature of space-time interacts with its material contents. Geometry was now dynamic and the fixed backdrop abandoned. This, far beyond the introduction of Minkowski geometry into the earlier theory, was the true revolutionary move.

QT also introduced a very novel idea: that, in general, particles don’t have definite positions. Rather, in some mysterious way, sharply defined classical quantities ‘fluctuate’ so that ‘particles’ become probability distributions, described in terms of the famous Schrödinger wave function, which generates the probability of finding a particle at a given position if one were able to observe it. But the wave function dealt in ordinary, unquantised space and time, i.e. with a background assumed to be fixed. So how could one do QT against the variable geometry of GR? Somehow fluctuations would have to be introduced into the underlying geometry itself, which seemed to make a nonsense of QT’s formalism.

QT’s other great innovation was the idea that some physical magnitudes (though not spatial extensions) could exist only in discrete or ‘quantised’ amounts. In GR, meanwhile, the variable geometry is used to describe the force of gravity, so that to quantise GR would mean developing a theory of quantum gravity. The search for such a theory is the theme of Smolin’s book.

Of the three ‘roads’ which Smolin considers, the first is ‘string theory’. Here, gravitation is understood in terms of the vibrations of very small linear entities, the tiny loops of ‘string’ that have replaced the idea of ‘point particles’ in fundamental physics during the past twenty years. But the string is still imagined vibrating against a fixed background (typically with a larger number of spatial dimensions than Minkowski space, the extra dimensions so tightly ‘rolled up’ that their presence is revealed only at extremely short distances). So, although the empirical predictions of GR can be recovered in string theory, it does not accommodate the fundamental idea of GR, its variable geometry. There have, recently, been indications that a ‘proper’ quantum gravity might emerge from the new ideas in string theorists’ so-called ‘M’ theory (the quip is that ‘M’ stands for Mystery). But since M theory does not yet exist in any final form, Smolin is much more enthusiastic about a second road, ‘loop quantum gravity’, which he himself has been instrumental in developing.

To understand this theory we need to remember that our task is to quantise geometry. What we understand by ‘geometry’, however, is not straightforward. Smolin is convinced, for example, that the notion of a ‘point’ has no place in the space-time of GR. How could one do a geometry without points? A simple analogy: in school geometry we learn about straight lines and points, and probably think of lines as made up of points. But suppose we think of lines as fundamental entities in their own right. We could then define a point as specified by a pair of intersecting lines (if the lines are parallel their point of correspondence will be ‘at infinity’). Now suppose we move the lines about on paper, preserving their direction and hence their identity. The points will then be dragged along with the lines, so that they can’t be identified with fixed points on the paper. Their identity is constituted by the activity of the lines.

Or one can look at it in a slightly different way. The two lines can be used to specify an elementary shape, such as a small square if they are orthogonal. Different pairs of lines will specify shapes with different orientations, so our new sense of ‘point’ can be understood as a shape with a particular orientation, but no fixed location. In three dimensions we might, for example, have a collection of little cubes each with its own orientation. Now draw a diagram in which each cube is represented by a point, with lines joining the points representing the rotations or ‘spins’ that turn one cube into another. Such a diagram is called a ‘spin network’, and quantised versions of spin networks were introduced some thirty years ago by Roger Penrose to produce a geometry of ‘quantised directions’. They play a vital role in loop quantum gravity. Once again, the points are little chunks of space, but the edges of the network are associated with surfaces separating the chunks. When this whole scheme is quantised we get a theory dealing in discrete volumes and surfaces with discrete areas. The remarkable thing is that while the volumes and areas in the spin network have definite but discrete (i.e. quantised) values, there is no such thing as a definite length. To get back to a classical geometry with lengths in it, we have to look at a complicated construction known as a ‘weave’, which involves highly excited states of the spin network.

So the curved spatial geometry of classical GR is recovered in this new theory as a ‘fabric’ woven out of the loops, knots and links of spin networks. As time unfolds, the spin networks mutate, producing a structure to which the epithet ‘spin foam’ has been aptly applied. Space-time has become a spin foam.

For Smolin, to talk about space-time ‘points’ would be to intoduce unacceptable elements of ‘fixity’ into the geometry. There are events, but they don’t occur at space-time points because there aren’t any. What we call spatial distances and temporal intervals are relations between events, not between space-time points. Thus if there were no events there would be no space-time – a point of view famously expounded by Leibniz against Newton at the beginning of the 18th century.

The two roads I have discussed so far try to reconcile QT and GR by starting off from one side of the divide or the other. Smolin’s third road would introduce new concepts of space and time right at the beginning; QT and GR would then be seen to be approximately correct in some limited circumstances. This third road might begin with something like M theory or the spin foam theory, but without travelling the path of current string theory or loop quantum gravity. Smolin does not take us very far down this road, but does mention a few names of those he calls ‘the true heroes of [his] story’, such as Alain Connes and David Finkelstein. The fact is that the third road is not yet sufficiently well trodden to lend itself to popular exposition.

If there are no points in space-time, what does this imply about the nature of time in quantum gravity? To some, including Julian Barbour in The End of Time (1999), it implies that time is an illusion. But for Smolin time is nothing but the reading of a physical clock. To recite ‘the clock struck one the mouse ran down’ is to establish a correlation between two physical events, the descent of the mouse and the striking of the clock. But we mustn’t be tempted to think that the clock measures the time at which the mouse runs down. Changes and processes don’t occur in time, they are time. Needless to say, what constitutes a ‘clock’ in this sense is still the subject of much debate.

Along the way, Smolin develops many other interesting arguments. From black holes and their thermodynamic properties he deduces that area is in some sense more fundamental in physics than volume (or length). This leads, in turn, to the so-called holographic principle, a belief that the information encoded in the Universe – the totality of ways it could be in its microscopic detail – is distributed over surfaces and not throughout volumes, in the same way that a hologram represents three-dimensional information on a two-dimensional plane. In the final analysis, the history of the Universe would then be nothing but a flow of information between these holograms.

There is also an elegant account of strings, from the point of view of loop quantum gravity, as ‘embroidery’ on the space-time weave. This image is plausible because the scale of the string loops, while extremely small, is nevertheless quite a bit bigger than that of the loop structure underlying the space-time weave in loop quantum gravity.

In a final chapter, Smolin speculates as to the origin of the laws of nature themselves, outlining the view laid out in more detail in his earlier The Life of the Cosmos, that there is in fact a plurality of universes, each with its own laws, which are spawned by the collapse of black holes. And, following Darwinian principles of natural selection, the Universe is most likely to exist in a state which maximises the formation of black holes. This final chapter may be much more wildly speculative than the rest of the book, but these are remarkable and provocative ideas.

So how does the research described here fit into the overall picture of modern physics? In numerical terms there are today about a thousand string theorists in the world, while loop quantum gravity is being investigated by about a hundred, and the ‘third road’ by a handful. Sociologically, string theory is still flavour of the month, while loop quantum gravity is just beginning to gain ‘respectability’. All these theories are essentially mathematical, and interact very fruitfully with developments in pure mathematics, particularly the theory of ‘abstract knots’. Originally developed by theoretical physicists, who have a free-wheeling attitude to rigour in mathematics, they have now been taken up by the quite different discipline of mathematical physics, which tries to make proper sense of them. To a large extent the question of interpretation hinges on knowing which bits of the mathematics to take seriously as corresponding in some way to physical reality, and which belong to the ‘surplus structure’ endemic in modern mathematical physics.

For some, the new developments are too loosely tied to empirical data to count as ‘proper’ science. In particular, string theories predict lots of ‘particles’ (i.e. excitations of the strings) which have never been observed. This problem has tended to be deferred with a presumption that the mass of unobserved particles is too large to be produced in the present generation of accelerators. This is one reason for the excitement stirred by the building of a new facility, a large hadron collider, at CERN in Geneva, which might, it’s hoped, reach energies high enough to generate the new particles. There is also hope that quantum loop gravity may make itself amenable to experimental testing on the basis of the small deviations it predicts for the trajectories of photons that have travelled across a large fraction of the observable Universe.

But these theories have not been induced from experiment. They have arisen through a profound engagement with questions of mathematical consistency and more generally of mathematical elegance, and reflect the sentiment expressed four hundred years ago by Galileo that the Book of Nature is written in the language of mathematics. To some this attests a deep insight into the mind of God; to others it smacks of decadence, of a physics in real danger of losing contact with its empirical roots. To persevere with Smolin’s fascinating book is to begin to face up to the deepest issues in the methodology of modern physics.