The End of Time: The Next Revolution in Physics

CHAPTER 15

The Rules of Creation

THE END OF CHANGE

In this chapter I am going to go into a little detail about how wave mechanics works. This means looking at two equations Schrödinger discovered in 1926 which, Dirac remarked, explained all of chemistry and most of physics. You will need to absorb enough to understand the bearing of the first part of the book on the structure of quantum cosmology. That is the goal; I hope you will find it is worth the effort. I believe it will show us how creation works. No theory can ever explain why anything is – that is the supreme mystery. But theory may be able to tell us why one thing rather than another is created and experienced. What is more, I believe that in every instant we experience creation directly. Creation did not happen in a Big Bang. Creation is here and now, and we can understand the rules that govern it. Schrödinger thought he had found the secret of the quantum prescriptions. Properly understood, what he found were the rules of creation.

Let us get down to business. We shall be considering how the wave function ψ changes. In quantum mechanics, this is all that does change. Forget any idea about the particles themselves moving. The space Q of possible configurations, or structures, is given once and for all: it is a timeless configuration space. The instantaneous position of the system is one point of its Q. Evolution in classical Newtonian mechanics is like a bright spot moving, as time passes, over the landscape of Q. I have argued that this is the wrong way to think about time. There is neither a passing time nor a moving spot, just a timeless path through the landscape, the track taken by the moving spot in the fiction in which there is time.

In quantum mechanics with time, which we are considering now, there is no track at all. Instead, Q is covered by the mists I have been using to illustrate the notion of wave functions and the probabilities associated with them. The red and green mists evolve in a tightly interlocked fashion, while the blue mist, calculated from the other two, describes the change of the probability. All that happens as time passes is that the patterns of mist change. The mists come and go, changing constantly over a landscape that itself never changes.

One of the equations that Schrödinger found governs this process. If ψ is known everywhere in Q at a certain time, you know what ψ will be slightly later. From this new value, you can go on another small step in time, and another, and so on arbitrarily far into the future. The role played here by the red and green mists, the two primary components of ψ, is quite interesting: the way the red mist varies in space determines the rate of change of the green mist in time, and vice versa. The two components play a kind of tennis. This equation is sometimes called the time-dependent Schrödinger equation because time features in it. This is not in fact the first equation that Schrödinger discovered.

The first one he found is now usually called the stationary or time-independent Schrödinger equation. This determines what happens in certain special cases in which the two components of ψ, the red and green ‘mists’, oscillate regularly, the increase of one matching the decrease of the other. This has the consequence, as we have already seen for a momentum eigenstate, that the blue mist (the probability density) has a frozen value – it is independent of time (though its value generally changes over Q). Such a state is called a stationary state. This explains the name given to the second equation – its solutions are stationary states. The standard view is that the time-dependent equation is the fundamental equation of quantum mechanics; the stationary equation is seen as a special case derived from it. This corresponds to an overall scheme in which some state of ψ is created at some time and then evolves until a measurement is made.

There are intriguing hints that in the quantum mechanics of the universe the roles of these two equations are reversed. The stationary equation (or something like it) may be the fundamental equation, from which the time-dependent equation is derived only as an approximation. We think it is fundamental because we have been fooled by circumstances that make it valid for the description of the phenomena we find around us. However, these phenomena deceive us greatly when it comes to the overall story of the universe. In particular, they lead us to believe time exists when it does not.

That this is likely to be so follows from an important property of the two Schrödinger equations. For any quantum system, we can use the time-independent equation to find all the stationary states it can have. Each of these states corresponds to a definite energy, and in each of them the red and green mists oscillate with the same fixed frequency while the blue mist remains constant. These solutions are also solutions of the time-dependent equation, though they are special, being stationary. I have mentioned linearity in quantum mechanics. Here, linearity means that two or more solutions of the time-dependent Schrödinger equation can be simply added together to give another solution. If the special stationary solutions are added, something significant results. In each solution, considered separately, the red and green mists oscillate at a fixed frequency while the blue mist remains constant. However, when we add two such solutions with different frequencies, they interfere: the added intensities of the red and green mists no longer oscillate regularly. More significantly, the blue mist varies in time.

Now this is very characteristic – indeed, it is the essence of quantum evolution. All solutions of the time-dependent equation can be found by adding stationary solutions with different frequencies. Each stationary solution on its own has regular oscillations of its red and green mists, but a constant – in fact static – distribution of its blue mist. But as soon as stationary states with different energies, and hence frequencies, are added together, irregular oscillations commence – in particular in the blue mist, the touchstone of true change. All true change in quantum mechanics comes from interference between stationary states with different energies. In a system described by a stationary state, no change takes place.

The italics are called for. We have reached the critical point. The suggestion is that the universe as a whole is described by a single, stationary, indeed static state. Why should this – with its implication that nothing happens – be so? This is where we start to make contact with the earlier part of the book. Time and change come to an end when Machian classical dynamics meets quantum mechanics. We have seen that a Machian universe should have only one value of the energy: zero. We also know (Box 2) that a quantum theory can be obtained by quantizing a corresponding classical theory. In fact, it is easy to show that whereas quantizing Newtonian dynamics, with its external framework of space and time, leads to the time-dependent Schrödinger equation, quantizing the simple Machian model considered in Chapter 7 leads to a quantum theory in which the basic equation is not the time-dependent but the stationary Schrödinger equation.

If the Machian approach to classical dynamics is correct, quantum cosmology will have no dynamics. It will be timeless. It must also be frameless.

CREATION AND THE SCHRÖDINGER EQUATION

Before I can explain how this can be achieved, I must tell you what the Schrödinger equation is like and what it can do. I believe it is even more remarkable than physicists realize. This is where – if I am right – we are getting near the secret of creation.

When Schrödinger created wave mechanics, Bohr’s was the only existing model of the atom. It suggested that atoms could exist in stationary states, each with a fixed energy, photons being emitted when the atom jumped between them. Schrödinger’s great aim was to explain how the stationary states arise and the jumps occur. De Broglie’s proposal suggested strongly that a stationary state should be described by a wave function that oscillated rapidly in time with fixed frequency, though its amplitude might vary in space. As a first step Schrödinger therefore looked for an equation for the variation in space.

It is ironic that only later did he find the time-dependent equation from which, strictly speaking, he should have derived this equation. However, he had luck and was guided by good intuition. Although it is easy for mathematicians, I shall not go into the details of how Schrödinger found his equations or how to get from one to the other. Box 13 gives the minimum about the stationary equation needed to understand the thrust of the story.

BOX 13 How Creation Works

You can think of the Schrödinger wave function in a stationary state as follows. At each point of the configuration space Q, imagine a child swinging a ball in a vertical circle on a string of length , which remains constant. As the ball whirls, its height above or below the centre of the circle changes continuously. The height is an image of the red mist, which is sometimes positive (above the centre), sometimes negative (below it). The distance sideways – to the right (positive) or the left (negative) – is an image of the green mist. The square of is the image of the constant intensity of the blue mist. A stationary state is like having children swinging such balls at the same rate everywhere in Q, all perfectly in phase – they all reach the top of the circle together. The only thing not perfectly uniform is the string length, , which can change from point to point in Q. In a momentum eigenstate, is the same everywhere. It is a very special state, but in a more general stationary state does vary over Q. The stationary Schrödinger equation governs its variation.

It does this by imposing a condition at each point of Q. The sum of two numbers, calculated in definite ways, must equal a third. The first number is the most interesting but the most difficult to find. Take a quantum system of three bodies. Its configuration space Q has nine dimensions. Each point in Q corresponds to a position of the three bodies in absolute space. Imagine holding two bodies fixed, and moving the third along a line in absolute space. This will move you along a line in Q. Suppose that along it you plot , the string length, as a curve above the line. At each point, this curve will have a certain curvature. At some places it will curve strongly, towards or away from the line, at others weakly. In the calculus, the curvature is the second derivative.

At each point of Q there are nine such curvatures because Q has nine dimensions, one for each of the three directions in absolute space in which each particle can move. The first number in the Schrödinger condition is the sum of these nine curvatures after each has been multiplied by the mass of the particle for which it has been calculated. I shall call this the curvature number.

The second number is much easier to find. Recall that any configuration of bodies has an associated potential energy. The configuration (and the nature of the bodies, their masses, etc.) determines it uniquely. For gravity, this was explained in Figure 17. The second number, which I shall call the potential number, is found simply by multiplying the potential by

The third number is also easy to find. If ω is the frequency of the state (the number of ‘rotations of the balls’ in a second), then, by the quantum rules, the energy of the state is E = hω, where h is Planck’s constant. This is the relationship Einstein found between the energy and frequency of a photon. The third number, which I shall call the energy number, is then found by multiplying the energy E by .

The condition imposed by the stationary Schrödinger equation is then

Curvature number + Potential number = Energy number

(Planck’s constant also occurs in the first number, to ensure that all three numbers have the same physical nature.)

However, finding this condition, which must hold everywhere in Q, was only half the story. Schrödinger thought that an atom in a stationary state was like a violin string vibrating in resonance. Because its two ends are fixed, the amplitude at the ends is zero. He therefore imposed on not only the above condition, but also the condition that it should tend to zero at large distances. It was this requirement that enabled him to make the huge discovery that convinced him – and very soon everyone else – that he had found the secret of Bohr’s quantum prescriptions.

This hinges on an extremely interesting property of the stationary Schrödinger equation. As yet E is a fixed but unknown number. It may be smaller or greater than the potential V, which varies over Q. The interesting thing is that the above condition forces to do very different things depending on the value of E – V. Where it is greater than zero, oscillates. As Schrödinger said rather quaintly, ‘it does not get out of control’. However, where E – V is less than zero, the condition forces an entirely different behaviour on . It must either tend rapidly to zero or else grow rapidly – exponentially in fact – to infinity. The latter would be a disaster. Schrödinger therefore commented that things become tricky and must be handled delicately. Indeed, he showed that it is only in exceptional cases, for special values of E, that does not ‘explode’ but instead subsides to zero at infinity. These are the cases he was looking for. Well-behaved solutions exist for only certain values of E, which are discrete (separated from each other) if E is less than zero.

The well-behaved solutions are called eigenfunctions, and the corresponding values of E are called (energy) eigenvalues. It is a fundamental property of quantum mechanics that any system always has at least one eigenfunction. The eigenfunction of any system that has the lowest value of its energy eigenvalue (there is often only one such eigenfunction) is called the ground state. In general, there are also eigenfunctions with higher energies, called excited states. Finally, if E is large enough for E – V to be positive everywhere, the eigenfunctions oscillate everywhere, though more rapidly where the potential is lowest. The negative eigenvalues E form the discrete spectrum, and the corresponding states are called bound states because for them has an appreciable value only over a finite region. The remaining states, with E greater than zero, are called unbound states, and their energy eigenvalues form the continuum spectrum.

Schrödinger won the 1933 Nobel Prize for Physics mainly for his wave-mechanical calculation for the hydrogen atom. He found that the energy eigenvalues of its stationary states are precisely the energies of the allowed states in Bohr’s model. This was a huge advance, since Schrödinger’s formalism had an inner unity and consistency to it completely lacking in the older model. Brilliant successes of the new wave mechanics, many achieved by Schrödinger himself, soon came flooding in, leaving no doubt about the great fruitfulness of the new scheme.

In Chapter 14 I described how molecules appear in the Schrödinger picture: as immense collections of all the configurations they could conceivably have, with the blue mist of the quantum probability strongly concentrated on the most probable configurations. These most probable configurations, generally clustered around a single point in Q, are the ones represented by the ball-and-strut models. I can now begin to make good my claim that Schrödinger found the laws of creation. His stationary equation determines the structures – indeed, creates the structures – of all these amazing atoms and molecules that constitute so much of the matter in the universe, our own bodies included. The equation does it by determining which structures are probable. But I mean creation not only in this sense of the structure of atoms and molecules, but in an even deeper one. The full explanation is still to come, but we are getting closer to our quarry.

QUANTUM MECHANICS HOVERING IN NOTHING

We must now see if we can dispense not only with time but also with absolute space in quantum mechanics. In a timeless system the energy E is zero, and the condition in Box 13 says simply that at every point of Q the sum of the curvature number and the potential number is zero. The potential number is already in the form we need. For any possible relative configuration, the potential has a unique value: it depends on nothing else. To find the potential number, we simply calculate the potential V for each configuration and then multiply by , getting This part of the calculations is pleasingly self-contained because V depends only on the relative configuration. Each structure has its own potential irrespective of how we imagine the structure to be embedded in space.

However, a lack of ‘self-containment’ shows up in the curvature number. To find it, we must know how varies from position to position in the configuration space Q. This is not a self-contained process in Schrödinger’s equation because the points of his Q are defined by the particles’ positions in absolute space, which is used crucially in Q, making it hybrid. The all-important curvatures of are ultimately determined by position differences in absolute space. As a result, in standard quantum mechanics the orientations are in general entangled with the relative data that specify the particle separations. Now, besides positions, momenta and energy there is another very important quantity in quantum mechanics – angular momentum, which, being an action, always has discrete eigenvalues. It owes its existence in quantum mechanics to absolute space. We have not yet escaped from Newton’s framework.

We are now coming to another critical point. We have seen that in classical physics the action is a kind of ‘distance’ between two configurations that are nearly but not exactly the same. Absolute space is an auxiliary device that makes it possible to define such ‘distances’. This is why angular momentum exists in classical and quantum physics. However, in Chapter 7 we found an alternative definition of ‘distances’ that works in the purely relative configuration space – in Platonia – and owes nothing to absolute space. They are defined by the best-matching procedure, which uses relative configurations and nothing else. In classical physics, this makes it possible to create a purely relative and hence self-contained dynamics. We also found that a sophisticated form of best matching lies at the heart of general relativity. Best matching would appear to be a basic rule of the world.

It is therefore very tempting to see whether it can be applied in quantum mechanics. What we would like to do is establish rules for operating on wave functions defined solely on the relative configuration space. For example, for three bodies we would want to eliminate the six dimensions associated with their position and orientation in absolute space, and work just with the sides of the triangle. We shall then have a wave function defined on a three-dimensional Platonia. For that, we shall want to calculate a curvature number and a potential number. The latter will present no difficulty, since it will be the same as in ordinary quantum mechanics. The difficulty is in the curvature number. What, after all, is curvature? For any given curve, it is the rate at which its slope changes. But the key thing about a rate of change is that it is with respect to something. That something is all-important. It is a kind of ‘distance’. The ordinary quantum-mechanical ‘distance’ is simply distance in absolute space (times the mass of the particle considered). To eliminate absolute space in classical physics, we replaced it by the Machian best-matching distance. There is no reason why we should not do the same in quantum physics.

This is where the unfolding of quantum mechanics on configuration space is so important. To retain that essential property of it – the huge step that Schrödinger took – we must pass from his hybrid Q to Platonia. If we are to succeed in formulating quantum mechanics in the new arena, there must be ‘distances’ in it. But that is precisely what the best-matching idea was developed to provide. Exactly the same ‘distances’ needed to realize Mach’s principle in classical physics can be used in a version of wave mechanics for a universe without absolute space. All we have to do is measure curvatures with respect to the Machian distances created on Platonia by best matching. We then add curvatures measured in as many mutually perpendicular directions as there are dimensions in that timeless arena, and set the sum equal to minus the potential number.

In fact, it is quite easy to see that the wave functions that satisfy the Schrödinger conditions in this Machian case are precisely the eigenfunctions of ordinary quantum mechanics for which the angular-momentum eigenvalues are zero. This exactly matches our result in classical mechanics – that the best-matching condition leads to solutions identical to the Newtonian solutions with angular momentum zero. We have already seen why they must be static solutions.

The picture that emerges is very simple. The quantum counterpart of Machian classical dynamics is a static wave function ψ on Platonia. The rules that govern its variation from point to point in Platonia involve only the potential and the best-matching ‘distance’. Both are ‘topographic features’ of the timeless arena. Surveyors sent to map it would find them. They would see that the mists of Platonia respect its topography. It determines where the mist collects.