The End of Time: The Next Revolution in Physics

CHAPTER 16

‘That Damned Equation’

HISTORY AND QUANTUM COSMOLOGY

The year 1980 was another turning point in my life. It was when Bruno Bertotti and I thought we might have found a new theory of gravitation, only to learn that the two ideas on which we had based it were already an integral part of Einstein’s theory. Karel Kuchař’s intervention rounded off our work but also brought it to an end. It was something of an anticlimax. Bruno became increasingly involved in experiments using spacecraft, aimed at detecting the gravitational waves predicted by Einstein’s theory. For a year or two I actually stopped doing physics and became politically active in the newly founded Social Democratic Party (the SDP). However, the old interests soon revived. Margaret Thatcher’s decisive general election victory in 1983 hastened the process.

Two things occupied me through the 1980s. First, I wrote the book from which I quoted the comments about Kepler. It had always been my ambition to write about absolute and relative motion, and in 1984 I signed a contract with Cambridge University Press for a book of four hundred pages covering the period from Newton to Einstein and including an account of my work with Bruno. When I embarked upon it, it occurred to me that I ought to find out why Newton had said what he had. What had given him the idea of absolute space? Might it not be an idea to look at what Galileo had said? I made a wonderful mistake by asking those questions. Before I knew what was happening, my research into Galileo dragged me ever further into past history, through the Copernican revolution to the work of Ptolemy and all the way back to the pre-Socratic philosophers. By reading the actual works of scientists such as Ptolemy, Kepler and Galileo, I found that the early history of mechanics and astronomy was far more interesting than any account of it I could find by the professional historians of science. They had missed all sorts of fascinating things, and their histories were quite inadequate. Inspired by Kepler’s comment that the ways by which men discover things in the heavens are almost as interesting as the things themselves, I started to write about all the early work. I spent from 1985 to 1988 writing a completely unplanned book: The Discovery of Dynamics. My sympathetic and understanding editor at Cambridge, Simon Capelin, agreed to publish it as the first of a two-volume work. The second volume was to be the book originally proposed and should have been completed a year or two later. However, that got badly delayed by a parallel development that turned my interest to physics that does not yet exist at the same time as I was working backward to the early history.

As I mentioned earlier, Bruno and I had been completely concerned with classical physics. We had wanted to show that Mach had been right and that his ideas could lead to new classical physics; we had given not a moment’s thought to any quantum implications they might have. Quantum cosmology was a world beyond our ken. It is strange what sparks a desire to work on something. My lack of interest in quantum gravity was particularly odd, since it was the early work done in that field which, through the remark by Dirac, quoted in the Preface, had set me on my long trek. It was the same work that had led to the work of Baierlein, Sharp and Wheeler that Bruno and I had come to see as the implementation of Mach’s ideas within general relativity. Not even working with Karel Kuchař, one of the world’s leading experts in quantum gravity, provided the stimulus I needed. Perhaps it all seemed too daunting. I needed the example and encouragement that came from a new friend, Lee Smolin.

I first met Lee a few weeks before I travelled to Salt Lake City in the autumn of 1980. It was quite a dramatic time for me since I had just narrowly escaped death through an insidious appendix that had burst without giving me any pain. My only symptoms were tiredness, slight sickness and the merest hint of stomach pain. Luckily my vigilant doctor sent me to hospital as a precaution. An X-ray proved difficult to interpret, and after quite lengthy deliberation the doctors decided to open me up. They found that any further delay could have been fatal. Seeing my state, the surgeon apparently commented that ‘this must be a very brave man’, believing I must have been in agony. In fact, I had been cheerfully reading The Times without any discomfort only half an hour before the operation. The day after I came back from hospital still convalescing, two American physicists visiting Oxford phoned to say that they had heard from Roger Penrose about my interest in Mach’s principle. Could they come and see me? They came the next day, and I greeted them in my dressing gown.

One was Lee, then a young postdoc. The meeting changed both of our lives significantly. He proved very receptive to the ideas of Leibniz and Mach to which I introduced him, while he encouraged me to see what application they might have to the problem to which he had decided to devote himself – quantum gravity. We met several times in the next few years, and collaborated on an attempt to formulate Leibniz’s philosophical system, his ‘monadology’, in mathematical form. I think we made some real progress. Lee has written about his view of things in his The Life of the Cosmos. Certain aspects of our work together were decisive in my own elaboration of the notion of time capsules and my conviction that the ultimate and only truly real things are the instants of time. As far as I am aware, Leibnizian ideas offer the only genuine alternative to Cartesian-Newtonian materialism which is capable of expression in mathematical form. What especially attracts me to them is the importance, indeed primary status, given to structure and distinguishing attributes, and the insistence that the world does not consist of infinitely many essentially identical things – atoms moving in space – but is in reality a collection of infinitely many things, each constructed according to a common principle yet all different from one another. Space and time emerge from the way in which these ultimate entities mirror each other. I feel sure that this idea has the potential to turn physics inside out – to make the interestingly structured appear probable rather than improbable. Before he became a poet, T. S. Eliot studied philosophy. He remarked, ‘In Leibniz there are possibilities.’

In 1988, when I had finished my book on the discovery of dynamics, I spent three weeks with Lee at Yale, and began to think seriously how one might make sense of the embryonic form of quantum gravity that had been developed from about the time of Einstein’s death in 1955, leading to the publication of the Wheeler-DeWitt equation in 1967. During the next four years, Lee and I had many discussions. Although we eventually followed different paths – Lee is reluctant to give up time as a primary element in physics – the ideas I want to describe in the final part of the book crystallized during those discussions. For me, their attraction stems from the inherent plausibility of Platonia as the arena of the universe and the implication of Schrödinger’s breathtaking step into a rather similar configuration space. As I see it now, the issue is simple.

A SIMPLE-MINDED APPROACH

You can play different games in one and the same arena. You can also adjust the rules of a game as played in one arena so that it can be played in a different arena. Both general relativity and quantum mechanics are complex and highly developed theories. In the forms in which they were originally put forward, they seem to be incompatible. What I found to my surprise was that it does seem to be possible to marry the two in Platonia. The structures of both theories, stripped of their inessentials, mesh. What if Schrödinger, immediately after he had created wave mechanics, had returned to his Machian paper of only a year earlier and asked himself how Machian wave mechanics should be formulated? His Machian paper implicitly required Platonia to be the arena of the universe, while any wave mechanics simply had to be formulated on a configuration space. Such is Platonia, though it is not quite the hybrid Newtonian Q he had used. But the structure of Machian wave mechanics would surely have been immediately obvious to him, especially if he had taken to heart Mach’s comments on time. As a summary of the previous chapter, here are the steps to Machian wave mechanics in their inevitable simplicity.

For a system of N particles, the Schrödinger wave function in the Newtonian case will in general change if the relative configuration is changed, if the position of its centre of mass is changed, if its orientation is changed, and if the time is changed. Mathematicians call these things the arguments of the wave function. They constitute its arena. To see what really counts, we can write the wave function in the symbolic way that mathematicians do:

ψ (relative configuration, centre of mass, orientation, time)

(1)

But if the N particles are the complete universe, there cannot be any variation with change of centre of mass, orientation or time for the simple reason that these things do not exist. The Machian wave function of the universe has to be simply

ψ (relative configuration)

(2)

Note the grander ψ. This is the wave function of the universe. It has found its home in Platonia.

I have met distinguished theoretical physicists who complain of having tried to understand canonical quantum gravity, the formalism through which the Wheeler-DeWitt equation was found, and have given up, daunted by the formalism and its seemingly arcane complexity. But, as far as I can see, the most important part boils down simply to the passage from the hybrid (1) to the holistic (2).

‘THAT DAMNED EQUATION’

This is a bold claim, but the fact is that it still remains the most straightforward way to understand the Wheeler-DeWitt equation. To conclude Part 4, I shall say something about this remarkable equation and the manner of its conception, which unlike the hapless Tristram Shandy’s was inevitable, being rooted in the structure of general relativity. You may find this section a little difficult, which is why I have just given the simple argument by which I arrive at its conclusion. Just read over any parts you find tough.

That there was a deep problem of time in a quantum description of gravity became apparent at the end of the 1950s in the work of Dirac and Arnowitt, Deser and Misner (ADM) described in Chapter 11. The existence of the problem was – and still is – mainly attributed to general covariance. The argument goes as follows. The coordinates laid down on space-time are arbitrary. Since the coordinates include one used to label space-time in the time direction and all coordinates can be changed at whim, there is clearly no distinguished label of time. This is what leads to the plethora of paths when a single space-time is represented as histories in Platonia. However, the real root of the problem lies in the deep structure of general relativity that we considered in the same chapter.

Indeed, as Dirac and ADM got to grips with the dynamics of general relativity, the problem began to take on a more concrete shape. The first fact to emerge clearly was the nature of the ‘things that change’. This was very important, since it is the ‘things that change’ that must be quantized. They turned out to be 3-spaces – everything in the universe on one simultaneity hypersurface, including the geometrical relationships that hold within it. These are the analogue of particle positions in elementary quantum mechanics. As I have mentioned, Dirac was quite startled by this discovery – it clearly surprised him that dynamics should distinguish three-dimensional structures in a theory of four-dimensional space-time. I am surprised how few theoreticians have taken on board Dirac’s comments. Many carry on talking about the quantization of space-time rather than space (and the things within it). It is as if Dirac and ADM had never done their work. Theoreticians are loath to dismantle the space-time concept that Minkowski introduced. I am not suggesting anything that he did is wrong, but it may be necessary to accommodate his insight to the quantum world in unexpected ways. One way or another, something drastic must be done.

As explained in Chapter 11, in general relativity four-dimensional space-time is constructed out of three-dimensional spaces. It turns out that their geometry – the way in which they are curved – is described by three numbers at each point of space. This fact of there being three numbers acquired a significance for quantum gravity a bit like the Trinity has for devout Christians. Intriguingly, the issue at stake is somewhat similar – is this trinity one and indivisible? Is one member of the trinity different in nature from the other two? The reason why the three numbers at each space point turned into such an issue is because it seems to be in conflict with a fact of quantum theory that I need to explain briefly.

I mentioned in Chapter 12 the ‘zoo’ of quantum particles, which are excitations of associated fields. The typical example is the photon – the particle conjectured by Einstein and associated with Maxwell’s electromagnetic field. An important property of particles is rest mass. Some have it, others do not. The massless particles must travel at the speed of light – as the massless photon does. In contrast, electrons have mass and can travel at any speeds less than the speed of light.

Now, massless particles are described by fewer variables (numbers) than you might suppose. Quantum mechanically, a photon with mass would be associated with vibrations, or oscillations, in three directions: along the direction of its motion (longitudinal vibrations) and along two mutually perpendicular directions at right angles to it (transverse vibrations). However, for the massless photon the longitudinal vibrations are ‘frozen out’ by the effects of relativity, and the only physical vibrations are the two transverse ones. These are called the two true degrees of freedom. They correspond to the two independent polarizations of light. This remark may make these rather abstract things a bit more real for the non-physicist. Humans cannot register the polarization of light, but bees can and use it for orientation.

There are many similarities between Maxwell’s theory of the electromagnetic field and Einstein’s theory of space-time. During the 1950s this led several people – the American physicist Richard Feynman was the most famous, and he was followed by Steven Weinberg (another Nobel Laureate and author of The First Three Minutes) – to conjecture that, just as the electromagnetic field has its massless photon, the gravitational field must have an analogous massless particle, the graviton. It was automatically assumed that the graviton – and with it the gravitational field – would also have just two true degrees of freedom.

From 1955 to about 1970, much work was done along these lines in studies of a space-time which is almost flat and therefore very like Minkowski space (I did my own Ph.D. in this field). In this case, the parallel between Einstein’s gravitational field and Maxwell’s electromagnetic field becomes very close, and a moderately successful theory (experimental verification is at present out of the question, gravity being so weak) was constructed for it. Within this theory it is certainly possible to talk about gravitons; like photons, they have only two degrees of freedom. However, Dirac and ADM had set their sights on a significantly more ambitious goal – a quantum theory of gravity valid in all cases. Here things did not match up. The expected two true degrees of freedom did not tally with the three found from the analysis of general relativity as a dynamical theory – as geometrodynamics.

Within the purely classical theory, the origin of the mismatch is clear: it is the criss-cross best-matching construction of space-time that I illustrated with the help of Tristan and Isolde. However, the discrepancy between the quantum expectations of well-behaved massless particles with two polarizations and the intricate interstreaming reality of relativity rapidly became the central dilemma of quantum gravity. Forty years on, it has still not yet been resolved to everyone’s satisfaction – it is that intractable. This is perhaps not surprising, for the issue at stake is the fabric of the world. Does it exist in something like that great invisible framework that took possession of Newton’s imagination, or is the world self-supporting? Do we swim in nothing? Nobody has yet been able to make quantum theory function without a framework. In fact, many people do not realize that the framework is a potential problem – Dirac’s transformation theory is in truth the story of acrobatics in a framework, and for physicists nurtured on Dirac’s The Principles of Quantum Mechanics the acrobatics is quantum theory. Acrobatics must be precise – if the trapeze is not where it should be, death can result. Such are the exigencies that led the early researchers to posit a graviton with just two true degrees of freedom.

An intriguing way to achieve this was suggested by Baierlein, Sharp and Wheeler’s paper in 1962 and its enigmatic hint that ‘time’ was somehow carried within space. This was taken literally, especially since it seemed to solve another problem of quantum acrobatics. In real acrobatics, not only location but also timing is of the essence. Nobody knew how to do quantum mechanics without an independent time external to the quantum degrees of freedom. But such a time appeared to have gone missing in gravity. Instead of time and two true degrees of freedom, there appeared to be no time but three degrees of freedom; these, moreover, were suspect. The count was all too suggestive – and many people came to the same conclusion: there is a time, but it is hidden in the three degrees of freedom.

According to this insight, the basic framework of quantum mechanics could be preserved, but the time it so urgently needed would be taken from the ‘world’ to which it was to be applied. Putting it in very figurative terms, one-third of space would become time, while the remaining two-thirds would become two true quantum degrees of freedom. Because time was to be extracted from space, from within the very thing that changes, the time that was to be found was called intrinsic time. The notion of intrinsic time was – and is – a breathtaking idea. But there was a price to be paid, and there was also a closely related problem to be overcome: which third of space is to be time?

The problem was that no clear choice could be made. Any and all 3-spaces can appear in the relations that summarize so beautifully the true essence of general relativity. What is more, any choice would ultimately amount to the introduction of distinguished coordinates on space-time. But this would run counter to the whole spirit of relativity theory, the essence of which was seen to be the complete equivalence of all coordinates. So if a choice were made, the price would be the loss of this equivalence. The price and the problem are one and the same. They presented the quantum theoreticians with a head-on collision between the basic principles of their two most fundamental theories – the need for a definite time in quantum mechanics and the denial of a definite time in general relativity. At an international meeting on quantum gravity held at Oxford in 1980, Karel Kuchař, concluding his review of the subject, stated that the problem of ‘quantum geometrodynamics is not a technical one, but a conceptual one. It consists in the diametrically opposite ways in which relativity and quantum mechanics view the concept of time’. I have added the italics. I was there to hear the talk, and Kuchaf’s comment made a deep impression on me.

The search for the third of space that would become time has been like The Hunting of the Snark, Lewis Carroll’s mythical beast that no one could find. Since the idea of intrinsic time was first clearly formulated about thirty-five years ago, the beast has not been found. Karel has done more than anyone else to try to track it down. If he cannot find it, I feel that comes quite close to a non-existence proof. My own belief is that the idea is based on an incorrect notion of time. It is a mythical beast invoked in vain to solve a titanic struggle. It does not surprise me that a special time has not been found lurking in the tapestry of space-time. All I see in that tapestry are change and differences – and the differences are measured democratically. The idea of a special intrinsic time to be extracted out of space, or out of any part of space-time or its contents, violates the democratic theory of emphemeris time that lies at the heart of general relativity.

If we look at the Newtonian parallel of the notion, it seems strange. In a world of three particles, it is like saying that one of the sides of the triangle they form is time while the other two are true degrees of freedom. Such an attempt to find time breaks up the unity of the universe. No astronomer observing a triple-star system would begin to think like that. The key property of astronomical ephemeris time is that all change contributes to the measure of duration. There has to be a different way to think about time.

I believe it was found, perhaps unintentionally, by Bryce DeWitt in 1967. John Wheeler had strongly urged him to find the fundamental equation of quantum gravity. It was Wheeler’s high priority to find the Schrödinger equation of geometrodynamics. What the theory of intrinsic time should yield is a time-dependent Schrödinger equation that – in figurative language – evolves a wave function for ‘two-thirds of space’ with respect to a ‘time’ constituted by the remaining ‘one-third of space’. Balking at the invidious task of selecting which third should be ‘time’, DeWitt fell back on a very general formalism developed fifteen years earlier by Dirac that made it possible to avoid having to make a choice.

Dirac’s method makes it possible to treat all parts of space on an equal footing, and simply defers to later the problem of time. DeWitt used Dirac’s method to write the fascinating equation that, as Kuchaf noted, he himself calls ‘that damned equation’, John Wheeler usually calls the ‘Einstein-Schrödinger equation’ and everyone else calls the ‘Wheeler-DeWitt equation’. But what is this equation, and what does it tell us about the nature of time?

The most direct and naive interpretation is that it is a stationary Schrödinger equation for one fixed value (zero) of the energy of the universe. This, if true, is remarkable, for the Wheeler-DeWitt equation must, by its nature, be the fundamental equation of the universe. I pointed out in the discussion of the structure of molecules that the ‘ball-and-strut’ models are only approximations to the quantum description, being merely the most probable configurations. The Wheeler-DeWitt equation is telling us, in its most direct interpretation, that the universe in its entirety is like some huge molecule in a stationary state and that the different possible configurations of this ‘monster molecule’ are the Instants of time. Quantum cosmology becomes the ultimate extension of the theory of atomic structure, and simultaneously subsumes time.

We can go on to ask what this tells us about time. The implications are as profound as they can be. Time does not exist. There is just the furniture of the world that we call instants of time. Something as final as this should not be seen as unexpected. I see it as the only simple and plausible outcome of the epic struggle between the basic principles of quantum mechanics and general relativity. For the one – in its standard form at least – needs a definite time, but the other denies it. How can theories with such diametrically opposed claims coexist peacefully? They are like children squabbling over a toy called time. Isn’t the most effective way to resolve such squabbles to remove the toy? We have already seen that there is a well-defined sense in which classical general relativity is timeless. That is, I believe, the deepest truth that can be read from its magical tapestry. The question then is whether we can understand quantum mechanics and the existence of history without time. That is what the rest of the book is about.