The End of Time: The Next Revolution in Physics

CHAPTER 18

Static Dynamics and Time Capsules

DYNAMICS WITHOUT DYNAMICS

DeWitt already clearly saw the problem posed at the end of the last chapter – the crass contradiction between a static quantum universe and our direct experience of time and motion – and hinted at its solution in 1967. Quantum correlations must do the job. Somehow they must bring the world alive. I shall not go into the details of DeWitt’s arguments, since he saw them only as a first step. However, the key idea of all that follows is contained in his paper. It is that the static probability density obtained by solving the stationary Schrödinger equation for one fixed energy can exhibit the correlations expected in a world that does evolve – classically or quantum mechanically – in time. We can have the appearance of dynamics without any actual dynamics.

It may surprise you, but it was about fifteen years before physicists, and then only a few, started to take this idea seriously. The truth is that most scientists tend to work on concrete problems within well-established programmes: few can afford the luxury of trying to create a new way of looking at the universe. A particular problem in everything to do with quantum gravity is that direct experimental testing is at present quite impossible because the scales at which observable effects are expected are so small.

Something like a regular research programme to recover the appearance of time from a timeless world probably began with an influential paper by Don Page (a frequent collaborator of Stephen Hawking) and William Wootters in 1983. This was followed by several papers that concentrated on an obvious problem. In ordinary laboratory physics, the fundamental equation used to describe quantum phenomena is the time-dependent Schrödinger equation. It undoubtedly holds to an extraordinarily good accuracy for all ordinary physics: we could not even begin to understand, for example, the radiation of atoms without this equation. But if the universe as a whole is described by a stationary Schrödinger equation and time does not exist at all, how does a Schrödinger equation with time arise? This question seems to have been first addressed by the Russians V. Lapchinskii and V. Rubakov, but a paper in 1985 by the American Tom Banks did more to catch the imagination of physicists. This was followed in 1986 by a paper treating the same problem by Stephen Hawking and his student Jonathan Halliwell. Further papers on the subject appeared in the following years. The whole associated research programme has become known as the semiclassical approach, for a reason I shall explain later. The basic idea is easy to grasp.

Imagine yourself on a wide sandy beach on which the receding tide has left a static pattern of waves. As you are a free agent, nothing can stop you from laying out a rectangular grid on the beach and calling the direction along one axis ‘space’ and that along the perpendicular axis ‘time’. For each value of the ‘time coordinate’, you can examine the wave pattern along the one-dimensional line of ‘space’ at that ‘time’. When you move to the neighbouring line on the beach corresponding to ‘space’ at a slightly later ‘time’, you will find that the wave pattern has changed. Simply by laying out your grid and calling one direction ‘space’ and the other ‘time’, you have transformed – in your mind’s eye – a two-dimensional static picture into wave dynamics in one dimension. This can be done with wave patterns in spaces of any dimension N. One direction can always be called ‘time’, and this automatically creates ‘evolution’ in the remaining N – 1 dimensions.

Of course, if the original wave pattern is ‘choppy’ and has not been created by some rule, the choice of the ‘time’ direction will be arbitrary. Any choice will create the impression of evolution in the remaining N – 1 dimensions, but it will not obey any definite and simple law. In the semiclassical approach, there are two decisive differences from the arbitrary situation. First, the static wave pattern is the solution of a definite equation. Second, it is a somewhat special solution – called a semiclassical solution – in that it exhibits a more or less regular wave pattern. This assumption will be considered later. However, if the wave pattern satisfies the assumption, it automatically selects a direction that it is natural to call time. With respect to this direction, a genuine appearance of dynamics arises in a static situation (Box 14). The result is this. Two static wave patterns (in a space of arbitrarily many dimensions) can, under the appropriate conditions, be interpreted as an evolution in time of the kind expected in accordance with the time-dependent Schrödinger equation. The appearance of time and evolution can arise from timelessness.

BOX 14 The Semiclassical Approach

This box provides some necessary details about the semiclassical approach. It is important here that the quantum wave function is not one wave pattern but two (the red and green ‘mists’). I mentioned the ‘tennis’ played between them – the rate of change in time of the red mist is determined by the curvatures of the green mist, and vice versa. This leads to the characteristic form of a momentum eigenstate, in which both mists have perfectly regular wave behaviour but with wave crests displaced relative to each other by a quarter of a wavelength. If the red crests are a quarter of a wavelength ahead of the green crests, the waves propagate in one direction and the momentum is in that direction. If the red crests are a quarter of a wavelength behind, the waves travel in the opposite direction and the momentum is reversed. We can call this phase locking. In a momentum eigenstate, there is perfect phase locking.

The semiclassical approach shows how two approximately phase-locked static waves can mimic evolution described by the time-dependent Schrödinger equation. In Figure 44 each of the two-dimensional wave patterns is nearly sinusoidal, and they are approximately phase-locked. These waves, being solutions of the stationary Schrödinger equation, are static – they do not move. But there is nothing to stop us (as in the example of the waves on the beach) from calling the direction along the axis perpendicular to the wave crests ‘time’ and the direction along the crests ‘space’.

Figure 44 Two nearly sinusoidal wave patterns.

The key step now is to divide the total pattern of each wave into a regular part, corresponding to an imagined perfectly sinusoidal behaviour, and a remainder that is the difference between it and the actual (nearly sinusoidal) behaviour. Call this the difference pattern (there is one for each mist). If the condition of approximate phase locking holds, it turns out that the difference patterns satisfy with respect to our ‘space’ and ‘time’ an equation of the same form as the time-dependent Schrödinger equation, except for the appearance of one additional term. This term will have less and less importance, though, the more closely the assumptions of the semiclassical approach are satisfied.

In fact, the semiclassical approach offers the prospect of an explanation of time – in all its manifestations. It begins with a unified concept of things. Each point of Platonia is one distinct logically possible structure – it is one thing. The rules that make the structures make everything. Platonia is entire and eternal. No place in it is different from any other place, considered as something that is logically possible. But each structure is still a distinct individual. We see before us a true landscape whose every point is marked of necessity by individuality. It has striking topographic features. So there is a landscape, but nothing of a quite different nature that one might call time.

There is, though, one quite different element: a wave function. Schrödinger’s enigmatic ψ covers Platonia. Mist hovers over the eternal landscape. The static mist is a well-behaved solution – an eigenfunction – of the Wheeler-DeWitt equation. There is nothing here an unsuspecting bystander could say looks like time. You have seen mist on a landscape. Did it enter your head that such a thing could explain time? But it can, in principle. The static wave function, simply by its well-behaved response to the landscape it finds, may be induced into a regular wave-like pattern. If so, time can ‘emerge’ from timelessness. We shall see how the wave function enables the logically possible structures to interact – in a very real sense – with each other, thereby helping each other into an actual existence that seems to be deeply marked by time.

WHY DO WE THINK THE UNIVERSE IS EXPANDING?

This ‘marking with time’ brings us to the tricky part in the semiclassical approach. It is what led me to the notion of time capsules. This is a point at which my ideas part company from (comparative) orthodoxy. Two closely related difficulties convinced me that a radical step was needed. The first arises from a significant difference between the two Schrödinger equations. The complex time-dependent equation is actually two equations for two separate components – the red and the green mist. They play a kind of ‘tennis’ which tightly couples their behaviour and creates phase locking in any semiclassical solution. In contrast, the stationary equation is usually a real equation which does not couple the two components of the wave function.

The existence of two separate yet almost perfectly matched wave patterns is crucial in the semiclassical approach. The waves must be parallel, and the wave crests displaced by a quarter of a wavelength. In standard quantum mechanics this is a valid assumption. Indeed, it is imposed because the true primary equation is the time-dependent Schrödinger equation. The secondary stationary equation is just a short cut to tell us the distribution of the blue mist without having to find the red and green mists first. But they are there, and they are of necessity phase-locked.

But quantum cosmology gives us only the Wheeler-DeWitt equation. It is the primary equation, but as it stands it will give only a blue mist. We cannot assume some deeper equation hiding behind it that will give phase-locked red and green mists. The truth is that this part of the semiclassical approach assumes something that should be derived. Luckily, this difficulty threatens to undermine only that part of the semiclassical approach in which the specific structure of the time-dependent Schrödinger equation is recovered. The broad picture in which ‘time’ emerges from timelessness is not threatened. In fact, complex numbers, which appear in my account in the guise of the red and green mists, are so deeply ingrained in quantum mechanics that I feel fairly confident that this problem will be sorted out. What is needed is some independent argument which enforces the appearance of a complex wave function and a coupling between its components. That would then ensure the necessary phase locking.

Nevertheless, we must take care not to introduce inadvertently into quantum cosmology assumptions that may be valid only in ordinary quantum mechanics. This brings me to the second difficulty with the semiclassical approach. It concerns motion and our conviction that we experience it, and simultaneously the issue of where our sense of the passage of time comes from. To understand the answer to this question is to understand time. It is all very well for me to speak about static wave patterns in a mist that hangs over Platonia. Such patterns will indeed, where they are sufficiently regular, define unambiguously a direction that may be called ‘time’. But even if the wave pattern is rather regular, we could not look at it and say that it distinguishes a direction of time. The one direction at right angles to the wave crests will look the same whichever way we face. There will be no signs set up on the distant horizons saying PAST and FUTURE. This is the issue we must now address.

It will be helpful to think about what it is that determines which way quantum wave packets move. Quantum mechanics is very different from classical mechanics in this respect. A classical initial condition consists of an initial position and an initial velocity. You know which way a particle will move because it is specified in the velocity. However, in quantum mechanics the initial condition is simply the values of the two components of the wave function, the red and green mists, everywhere at the initial time. Data like this seem to correspond to giving only the position in classical mechanics. Yet wave packets move under the rules Schrödinger prescribed.

In fact, the way in which a wave packet will move is coded in the relative positioning of the crests and troughs of the red and green mists. We see this most clearly in momentum eigenstates. If the red crests are ahead of the green crests, they go one way, but if the crest positioning is reversed, they go the other way.

As we have seen, states very like momentum eigenstates play a crucial role in the semiclassical approach. In all the original papers, these states also played another role – they were used to model situations corresponding to either expanding or contracting universes. All physicists and astronomers are convinced that we live in an expanding universe. There is certainly very good evidence in many different forms to support this view. The formalism of quantum cosmology must be capable of reflecting this aspect of the observed universe. There must be something that codes expansion or contraction of the universe or, rather (in the timeless interpretation), codes the observed evidence that leads us to say the universe is expanding.

All the models of Platonia that we have considered include a dimension that we may call the ‘size’ of the universe. In fact, instead of representing Triangle Land by means of the sides of triangles, we could equally well – and more appropriately here – use two angles (the third is found by subtracting their sum from 180°) and the area of the triangles. The area is one direction, or dimension, in Triangle Land. Expansion or contraction of the universe then corresponds to motion along the line of increasing or decreasing size. The size dimension begins at the point of zero size – what I have called Alpha, or the centre of Platonia – and then proceeds all the way to infinity.

In the semiclassical approach, it was rather reasonably assumed that the regular wave pattern needed for ‘time’ to emerge from timelessness would develop along the direction of increasing or decreasing size. This is a fair working hypothesis. What worried me was the way in which expanding and contracting universes were modelled – by analogy with momentum eigenstates in ordinary quantum mechanics. Expansion or contraction were supposed to be coded in the relative positions of wave crests.

It is certainly possible to imagine two static wave patterns – our red and green mists – whose crests are perpendicular to lines that seem to emanate from Alpha. This was done by nearly all the researchers who used the semiclassical approach, and they assumed that one relative positioning of the ‘red’ and ‘green’ crests would model a universe expanding out of the Big Bang, while the opposite positioning would model a universe headed for the Big Crunch (the name given to one possible fate of the universe, in which it recollapses to a state of infinite density and zero size). Thus, momentum-like semiclassical states were used to achieve three different things at once: the emergence of ‘time’, the recovery of the time-dependent Schrödinger equation, and modelling expanding and contracting universes. I believe that only the first is soundly based. I have some concern about the second. I think the third is definitely wrong.

The point is that the position of the ‘green crests’ ahead of or behind the ‘red crests’ by itself has no significance. In ordinary quantum mechanics the wave function depends not only on the spatial position but also on the time. What really moves wave packets is the relation of the time dependence to the space dependence. It is not the case that if in some wave packet the green crests are ahead of the red crests then the wave packet is bound to move one way. This happens only because the time-dependent Schrödinger equation is written in a particular form. But this is a pure convention. All observed phenomena are described just as well by an alternative choice, analogous to changing ends in tennis. The two choices are identical in their consequences. They only differ in the relative positions of the red and green crests, but this is offset by reversing the time dependence. The real physics is unchanged. Without the time dependence, the positions of the crests cannot determine the direction of motion.

But this presents us with a real dilemma in static quantum cosmology, in which there is no external time and no time dependence to determine which way wave packets move. There is simply no motion or change at all. We have to find a different explanation for why we think there is motion in the world and that the universe expands.

One thing is clear: the origin of our belief that the universe is expanding cannot be coded in the relative positioning of the crests of the two waves, for the designations ‘red’ and ‘green’ are purely conventional. The ‘colours’ could be swapped, and nothing observable would change. The argument that mere static positioning of crests can correspond to what we call expansion of the universe is a chimera. This was clearly recognized in 1986 by my German physicist friend Dieter Zeh, who commented that it has meaning only if an absolute time exists. It really is necessary to think very differently about these things if time is abolished once and for all as an independent element of reality.

THE IDEA OF TIME CAPSULES: THE KINGFISHER

From 1988 to 1991 I was absorbed by this issue. I became more and more convinced that a decisive new idea was needed, but for a long time could find no answer that satisfied me. I formulated the problem this way. I imagined myself watching some phenomena involving motion in a very essential and vital way – a display of acrobatics, say, or the flight of a kingfisher. I then imagined being struck dead instantaneously and my ‘soul’ being carried down to a kind of Plato’s cave. Here I would find omniscient mathematicians examining a model of Platonia all covered with these red, green and blue quantum mists that I have asked you to conjure up in your mind’s eye. They are examining the solution of the Wheeler-DeWitt equation corresponding to the universe in which I had just been taken from life. I then asked myself this: what precise thing in that mysterious pattern of mists blanketing Platonia corresponds to my being aware of seeing the kingfisher in flight? Where – in a timeless static world – is the appearance of motion coded? Where can I see the kingfisher’s colours flashing in the sunlight?

As we have noted, in standard quantum mechanics the information about wave-packet motion is coded in the relative positioning of the red and green mists. This was the questionable assumption taken over in the semiclassical approach. However, there is much more to quantum mechanics than just the wave function at one instant (the pattern of red and green mists). We have already seen how time is needed if such relationships are to be translated into wave-packet motion. But even that is not enough, for the wave function acquires definite meaning only through prescriptions about the measurements that will be made on the system. These take the form of statements about the positions and construction of measuring instruments that behave classically and are external to the quantum system.

It is obvious that in quantum cosmology the whole superstructure of an external time, and of measuring instruments outside the considered system, must go. The instruments must be subsumed into the quantum system (which becomes the complete universe), and we must get to grips with a static wave function. Does this leave any scope for making a connection between actual experiences and the bare bones of embryonic quantum gravity as found by DeWitt?

I believe it does. Is not our most primitive experience always that we seem to find ourselves, in any instant, surrounded by objects in definite positions? Each experienced instant is thus of the nature of an observation, a discovery, even – we establish where we are. Moreover, what we observe is always a collection, or totality, of things. We see many things at once. In fact, most humans, indeed nearly all animals, have a wonderfully developed spatial awareness. In writing this book I have relied heavily on you possessing this gift – time and again I have asked you to imagine configurations of the universe as entities. They are all the places in Platonia.

When, therefore, I find myself in Plato’s cave and see his demesne of Platonia laid out before me, I can, using my vivid memory of the kingfisher flashing between the banks of the stream where I stood, identify the instant in which death took me. By ‘identify the instant’, I mean recognize the configuration of riverbank, sunlight and shadow, rippled water and kingfisher’s wings – all frozen in the position I last witnessed. As always, I insist that instant of time simply means configuration of the universe. This part of the problem of finding a connection between the psychical experience and the model of physical reality is relatively straightforward. There is little or no problem in the representation of position.

The real problem, then, is in the representation of motion. We seem to have exhausted all the resources of static quantum cosmology simply to put everything into place on the riverbank. Quantum mechanics does permit us to gain total information about position, but only at the expense of total loss of information about motion. We seem to have nothing left over to enable the kingfisher to fly. This is the crux of the matter. Classical physics presupposes both positions and motions, matching our experience that we see both at once. But quantum mechanics – in its present standard form – has this curious halving of the accessible data.

So how can we let the kingfisher fly? As few things delight me more than a kingfisher in flight, this is a matter of some interest to me. The answer that suddenly came to me in the summer of 1991 (which, of course, is a place in Platonia, not a time) was that the flight of the kingfisher is ultimately an illusion, though it rests on something that is very special and just as real as we take flight to be. It is flight without flight. Let me return to the imagery of the blue mist that shimmers over Platonia. It is easy to locate the instant of my death – I see the point in that great configuration space in which I stand on the bank of the stream. Now let me make an assumption in the hallowed tradition of Boltzmann: only the probable is experienced. The blue mist measures probability. Therefore, in accordance with the tradition, the blue mist must shine brightly at the point in Platonia in which I see the kingfisher frozen in flight above the water. I experienced the scene, so it must have a high probability. But there is still no motion.

I do not think there can be any. But there can be something else. As I mentioned in Part 1, nobody really knows what it is in our brains that corresponds to conscious experience. I make no pretence to any expertise here, but it is well known that much processing goes on in the brain and, employing normal temporal language, we can confidently assert that what we seem to experience in one instant is the product of the processing of data coming from a finite span of time.

This is all I need. It enables me to make the working conjecture that I outline in Part 1 – that when we think we see motion at some instant, the underlying reality is that our brain at that instant contains data corresponding to several different positions of the object perceived to be in motion. My brain contains, at any one instant, several ‘snapshots’ at once. The brain, through the way in which it presents data to consciousness, somehow ‘plays the movie’ for me.

Down in Plato’s cave, thanks to the perfect representation of everything that is, I can look more closely at the point in the model of Platonia that contains me at the point of death. I can look into my brain and see the state of all its neurones. And what do I see? I see, coded in the neuronal patterns, six or seven snapshots of the kingfisher just as they occurred in the flight I thought I saw. This brain configuration, with its simultaneous coding of several snapshots, nevertheless belongs to just one point of Platonia. Near it are other points representing configurations in which the correct sequence of snapshots that give a kingfisher in flight is not present. Either some of the snapshots are not there, or they are jumbled up in the wrong order. There are infinitely many possiblilities, and they are all there. They must be, since there is a place in Platonia for everything that is logically possible.

Now, at all the corresponding points the blue mist will have a certain intensity, for in principle the laws of quantum mechanics allow the mist to seep into all the nooks and crannies of Platonia. Indeed, the first quantum commandment is that all possibilities must be explored. But the laws that mandate exploration also say that the blue mist will be very unevenly distributed. In some places it will be so faint as to be almost invisible, even with the acuity of vision we acquire in Plato’s cave for things mathematical. There will also be points where it shines with the steely blue brilliance of Sirius – or the kingfisher’s wings. And again my conjecture is this: the blue mist is concentrated and particularly intense at the precise point in Platonia in which my brain does contain those perfectly coordinated ‘snapshots’ of the kingfisher and I am conscious of seeing the bird in flight.

As I explained in Chapter 2, a time capsule, as I define it, is in itself perfectly static – it is, after all, one of Plato’s forms. However, it is so highly structured that it creates the impression of motion. In the chapters that follow, we shall see if there is any hope that static quantum cosmology will concentrate the wave function of the universe on time capsules. As logical possibilities, they are certainly out there in Platonia. But will ψ find them?