Richard Feynman
Page 33
This wasn’t just an afterthought by Feynman. In Gravitation, he had already said that the Lagrangian that emerges from Einstein’s General Theory of Relativity is merely an ‘effective Lagrangian’ that describes the low-energy behaviour of a more fundamental theory, in much the same way that Newtonian gravity is an effective theory that describes the behaviour of objects under even less extreme conditions where we do not even need to take account of general relativistic effects. The more fundamental theory that underpins both the General Theory and Newtonian gravity would operate, he suggested, on the tiniest scale, the so-called ‘Planck length’.
The Planck length is a number, with the dimensions of length, that can be derived from the three fundamental constants of physics (the constant of gravity, the speed of light and Planck’s constant), which respectively relate to gravitation, electromagnetism, and the quantum world. There is only one length that can be derived from a combination of these numbers, and it has a value of about 10–33 centimetres. The Planck length is the length scale on which gravity, electromagnetism and quantum phenomena are on an equal footing, and it is in a sense the smallest possible distance that can exist, the ‘quantum of length’.
One of the most dramatic developments in theoretical physics came in the mid-1980s, when, almost by accident, theorists found a theory which describes what is going on at these astonishingly small length scales, and which automatically gives rise to gravity as we know it, in just the way that Feynman predicted. It is known as superstring theory, and is still the best all-embracing theory of the origins of particles and gravity that we have.
The central idea of all string theories is that the fundamental entities of the physical world are not point-like objects, the way we are used to thinking of leptons and quarks, but have some extension in one dimension, like a line drawn on a piece of paper. The extension is very small, comparable to the Planck length, but it is definitely not zero. Even so, there is no prospect of ever being able to detect one of these strings – it would take 100 billion billion of them, laid end to end, to stretch across the diameter of a proton. This means that the size of such a string, compared to the nucleus of an atom, is equivalent to the size of a nucleus, compared to the size of the Sun. In the 1970s, some mathematicians dabbled with calculations describing the behaviour of such strings, but this was more because they were interested in the mathematics involved for its own sake, rather than through any suspicion that the equations they were playing with might describe the real world. Then, in the 1980s, they began playing with an improved version of the idea, called superstring theory, and came up with results that made the physicists begin to sit up and take notice.
In superstring theory, the fundamental entities are thought of as little lengths or loops of vibrating string, with the kinds of properties we associate with ‘fundamental particles’ (such as the charge on the electron) either tied to the ends of the open strings or associated with the way the strings vibrate. A closed loop of string – like a tiny, vibrating elastic band – is fundamentally different from an open string, but any theory that describes open strings automatically includes closed loops as well. To their surprise, when mathematical physicists calculated the properties of these closed loops of string in the mid-1980s, they found that they were equivalent to massless particles with two units of quantum spin. In other words, gravitons. Superstring theory predicts the existence of gravitons, and Feynman had shown, twenty years before, that gravitons are all you need to produce a theory of gravity identical, on the appropriate energy and distance scale, to the General Theory of Relativity.
It gets better. The infinities that plagued earlier attempts to develop a quantum theory of gravity do not arise in superstring theory, which is both mathematically self-consistent and finite. It has all the characteristics of the new theory required to describe what goes on at very short distance scales that Feynman alluded to.
Feynman’s fascination with Mach’s Principle also came into the lectures on gravitation, and also provides a direct link with current developments in physics. The idea that the inertia of an object arises as a result of gravitational interactions with very distant objects clearly has a family resemblance to Feynman’s old idea that the radiation resistance experienced by a charged particle – a kind of electrical inertia – arises as a result of electromagnetic interactions with very distant charged particles. In his gravity lectures, Feynman stops short of invoking a role for advanced gravitational interactions to account for inertia in the way that he and Wheeler had once invoked a role for advanced electromagnetic interactions to describe the forces acting between charged particles. Rather, he concludes his discussion of Mach’s Principle with another memorable piece of his philosophy of science:
The answer to all these questions may not be simple. I know there are some scientists who go about preaching that Nature always takes on the simplest solutions. Yet the simplest solution by far would be nothing, that there should be nothing at all in the universe. Nature is far more inventive than that, so I refuse to go along thinking it always has to be simple.14
It is easy to imagine, in a speculative way, a kind of ‘explanation’ of Mach’s Principle involving advanced and retarded gravitational interactions criss-crossing the Universe in the way that electromagnetic interactions move forwards and backwards in time in the Wheeler–Feynman theory of radiation. But it was only in 1993 that this kind of approach was put on a secure footing, by the work of Shu-Yuan Chu, of the University of California. Chu has developed a model of how to do quantum mechanics in the presence of gravity, which combines some of the latest ideas in particle physics (including superstrings) with a time-symmetric Wheeler–Feynman description of gravity and inertia.
Following Feynman’s example, Chu does away with the concept of a ‘field’, and works entirely in terms of particles (photons, gravitons and the like) being exchanged between other particles in a time-symmetric way. He suggests that this continuous feedback, on the smallest scale, builds up what we think of as continuous fields (such as gravity) as the average over all the interactions involving little pieces of matter. The averaging takes place on a scale that is large compared with the size of a string – but that still means that it happens on a scale far smaller than the size of a proton, so that our instruments are quite incapable of detecting it directly, and we only perceive a smooth field. Chu says that the effect would be like admiring a superbly woven tapestry from across the room, where it seems to make a smoothly continuous picture; only when you look at it up close could you see the individual threads that go together to make up the tapestry. And the kind of averaging you have to do, to get the smooth picture that we are familiar with, is, of course, the averaging involved in Feynman’s path integral approach to quantum physics.
This approach explains the origin of inertia, and Mach’s Principle, in the context of superstring theory, using exactly the mathematical formalism of Wheeler–Feynman electrodynamics. It also implicitly includes the Wheeler–Feynman theory of electrodynamics and the origin of radiation resistance. It’s a rather nice bonus that there is now good evidence that, as Feynman suspected, the Universe does contain the critical density of matter, making it flat, which ensures that there is enough matter in the future to provide the ‘echoes’ needed for the advanced and retarded waves to match up in the required way, without having to introduce any extra bells and whistles into the theory. Chu confessed to feeling more than a little nervous at going public with such an outrageous idea, that advanced interactions (‘messages from the future’) might play a fundamental part in determining the structure of the world as we perceive it.15 But what he didn’t know at the time he developed his model was that this outrageous idea had already been revived, in the context of ‘ordinary’ quantum mechanics (without strings) back in 1986, by John Cramer, of the University of Washington, Seattle.16
Cramer picked up on a rather peculiar feature of the Schrödinger equation itself, a feature which has long been known about and largely be
en ignored. Way back in 1927, in the early days of quantum mechanics, the pioneering astrophysicist Arthur Eddington pointed out that the quantum probabilities which are so important in making calculations of the behaviour of entities in the quantum world are ‘obtained by introducing two symmetrical systems of waves travelling in opposite directions of time’.17 The situation is very similar to the way in which there are two sets of solutions to Maxwell’s equations of electromagnetism, but with an important difference. With Maxwell’s equations, you can carry out the calculations either by using just one set of solutions, and completely ignoring the other set of solutions; or you can choose to use a mixture of half advanced and half retarded waves. With Schrödinger’s equation, you have no choice. You always have to use a mixture of advanced and retarded waves to calculate the probabilities.
It happens like this. Schrödinger’s wave equation involves what mathematicians refer to as complex numbers, in which the square root of –1, denoted by i, appears. In spite of their name, there is nothing particularly difficult about handling complex numbers; as we mentioned earlier, the name actually indicates that they are made up of two components, typically with a form like (x + it), instead of ‘simply’ being made up of either ordinary numbers (like x) or so-called ‘imaginary’ numbers (like it). And the need for two components to describe these numbers can be pictured in terms of the little arrows described in Chapter 4. As in this example, the ‘imaginary part’ of a complex equation describing the behaviour of a wave is linked to the time, denoted by t. The whole thing describes what is known as the amplitude for a particular interaction, or, say, for one route which might be taken by an electron through one hole in the experiment with two holes. But remember that in order to calculate the probability of a particular quantum event, you have to take the square of the amplitude; and this is where things get interesting.
Everyone knows how to make the square of an ordinary number, like x. You simply multiply it by itself, x × x. But this is not the way you make the square of a complex number, like (x + it). Instead, you multiply it by something called its complex conjugate, in which you change the sign in front of the imaginary part of the number, making it (x – it), so that for the square you get (x + it) × (x – it). Schrödinger’s equation is just a little more complicated than this simple example, but the principle is the same. And by reversing the sign in front of the t in Schrödinger’s equation, you have automatically selected the opposite version of the equation, describing a wave moving backwards in time. Extending the analogy used earlier, the rotating arrow that defines the phase of the wave is rotating in the opposite direction. Every time any physicist has ever calculated a quantum probability using Schrödinger’s equation in this way, they have automatically been taking account of both the advanced and the retarded waves in their calculation.
So, as Cramer pointed out in 1986, the quantum world can be described along exactly the lines of the Wheeler–Feynman theory of radiation, in which the advanced and retarded waves combine to produce an effective ‘action at a distance’ which takes no time at all. The way to picture this is to imagine standing outside of time, and watching what goes on as if it were happening sequentially, but remembering that it is really all happening at once. On this picture, a particle which has the potential to get involved in a quantum interaction sends out what Cramer calls an ‘offer’ wave, moving symmetrically in both directions of time, into the past and into the future. There is no distinction between the roles of past and future in this picture, but for our peace of mind just concentrate on the wave going out, in all directions, into the future. Out in the Universe at large, the wave triggers a response – indeed, it may trigger many responses, from many other particles. In each case, the triggered particle sends out a ‘confirmation’ wave, also into the past and into the future, indicating its ability to take part in the interaction. All of the confirmation waves travelling back in time arrive at the originating particle at the same instant that it made the original offer, and it ‘chooses’ one of the confirmation waves, in accordance with the familiar rules of quantum probability, to take part in the transaction. Everywhere else, all of the waves cancel each other out, leaving a completed transaction between two particles (see Figure 16), made up from both solutions to Schrödinger’s equation and forming a firm handshake across spacetime. From the ‘point of view’ of the waves themselves, the whole thing takes zero time.
Figure 16. John Cramer has developed the idea of waves moving forwards and backwards in time (see Figure 5) to describe quantum interactions. Because the offer wave travels forwards in time and the confirmation wave travels backwards in time, the transaction takes no time at all to complete. This provides an explanation of quantum mysteries such as how electrons ‘know in advance’ whether one or both holes are open in the experiment with two holes (see Figure 4). E = emitter; A = absorber.
The classic example of the experiment with two holes makes the situation clear. In this case, the offer wave goes out through both holes, before the electron ever sets out on its journey. The confirmation wave also comes back through both holes – indeed, confirmation waves come back by any possible route from anywhere the electron could possibly go, just like the crazy way in which light bouncing off a mirror behaves in QED. Just one confirmation wave is accepted by the electron, so the electron itself actually goes by one route to its destination on the detector screen. But its position on the detector screen – the point where it makes a blob of light – is determined by the whole structure of the experiment, taking account of both holes to create an interference pattern as more and more electrons are faced with the equivalent choices. Crucially, if one of the holes is covered up, then this theory predicts that the behaviour of the electrons and the pattern they make on the detector screen will change in exactly the way it is seen to change in experiments. As well as making full use of both advanced and retarded waves, nature really is, on this picture, carrying out a ‘sum over histories’ to determine where the electron ends up.
This view resolves the famous ‘Schrödinger’s cat paradox’. At the beginning of the experiment, advanced waves from the future present the quantum system with a ‘choice’ of a dead cat or a live cat, and the transaction confirming which choice will become real is made, on a 50:50 basis, before anything happens. The fate of the cat is indeed sealed by quantum probabilities, but it is sealed from the outset, with no need for a ‘superposition of states’, and no need for the role of an observer in creating reality in the way that Feynman ridiculed. And all of the puzzles and mysteries of the quantum world dissolve away into transparency in this way, because any quantum entity involved in any quantum experiment really does ‘know’ about the entire structure of the experiment, and the entity’s ultimate fate, before anything happens at all in a human timeframe. As Cramer puts it:
If there is one particular link in [the] event chain that is special, it is not the one that ends the chain. It is the link at the beginning of the chain when the emitter, having received various confirmation waves from its offer wave, reinforces one of them in such a way that it brings that particular confirmation wave into reality as a completed transaction. The atemporal transaction does not have a ‘when’ at the end.18
Cramer called this the ‘transactional interpretation’ of quantum mechanics. In a sense, it is ‘only’ an interpretation – this way of looking at things makes no predictions about the behaviour of the quantum world that differ from the predictions made by the Copenhagen Interpretation, or by Feynman’s path integral formalism. But this, of course, is a strength of Cramer’s interpretation, because that means that his picture, like the others, agrees with all the thousands of experimental results concerning the quantum world that have been obtained in the past 70 years and more. The great thing about the transactional interpretation is that it provides you with an easy way to get a picture of what is going on in the quantum world, without mysteries such as cats that are dead and alive at the same time, or electrons that go through two
holes at once, at the cost of accepting the reality of the advanced waves. But since physicists have been implicitly accepting the reality of advanced waves every time they have used Schrödinger’s equation to calculate quantum probabilities since 1926 (and some of them, like Eddington, even knew what they were doing), that seems a pretty small price to pay!
This is an example of the direct influence of Richard Feynman on modern physics, with researchers picking up on his ideas and developing them in new ways – in this case, half a century after he first got involved with describing the behaviour of the world with the aid of advanced waves. It is a good note on which to end our discussion of modern science, because it combines one of Feynman’s earliest pieces of scientific research with one of the latest ideas in thinking about the quantum world, to resolve what he himself called the central mystery of the experiment with two holes; and, through the work of Chu, it provides a possible explanation of the physics behind one of the mysteries of the Universe, Mach’s Principle, that both puzzled and intrigued Feynman for decades.