11
The Only Mystery
We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way. … In reality, it contains the only mystery. We cannot make the mystery go away by ‘explaining’ how it works.
Richard Feynman1
Dirac didn’t have to wait too long to be disappointed. He had reached for the dream too soon, his proposal roundly criticized on all sides. Among other things, his equation demanded that the masses of the electron and proton should be the same. It was already well known that there is a substantial difference in the masses of these particles, the proton heavier by a factor of almost 2,000.
He finally accepted in 1931 that the other two solutions of his relativistic wave equation must describe a particle with the same mass as the electron. He went on to speculate that these extra solutions imply the existence of a positive electron: ‘a new kind of particle, unknown to experimental physics, having the same mass and opposite charge to an electron’.2 American physicist Carl Anderson found evidence for this particle, which he named the positron, in cosmic ray experiments in 1932–1933. Dirac had actually predicted anti-matter, particles with precisely the same properties as their matter counterparts, but with opposite electrical charge.
And, if that wasn’t bad enough, in February 1932 English physicist James Chadwick reported experiments on the nature of the ‘radiation’ emitted by samples of radioactive beryllium. As this radiation was unaffected by an electric field, it was initially thought to be gamma radiation (very high-energy photons). But Chadwick showed that it rather consists of a stream of neutral particles each with about the same mass as the proton. He had discovered the neutron, which quickly took its place alongside the proton inside the nuclei of atoms heavier than hydrogen.
Although what we now call ‘atoms’ had been shown to possess an inner structure (and are therefore ‘divisible’), their constituents could still be considered to be ‘elementary’ particles. These could still be viewed as ultimate, indivisible ‘lumps’ of matter, much as the Greek atomists had envisaged. Yes, particles like electrons happen to have a few properties that had eluded these early philosophers. An electron carries a unit of negative electrical charge. It possesses the rather obscure property of spin, like (but also unlike) a tiny spinning top.
But it seems there is one thing they did get right. It has mass, and we can look up the electron mass in an online catalogue maintained by the Particle Data Group, an international collaboration of some 170 physicists which regularly reviews data on elementary particles and produces an annual ‘bible’ for practitioners.3
If we do this, we discover that the mass of the electron is 9.109 × 10−31 kilograms. Now, physicists are just like ordinary human beings. (Trust me on this.) They grow weary of writing numbers with large positive or negative powers of ten, and prefer to find units that allow for a simpler representation. In this case, they associate the electron mass with an energy (using m = E/c2), and write it as 0.511 MeV/c2. Here MeV stands for mega (million) electron volts and the appearance of ‘/c2’ in the units serves to remind us of the connection with Einstein’s famous equation and indicates that this is a representation of a mass quantity.*
So, let’s now have a little fun. Let’s imagine that we form a beam of electrons (much like in an old-fashioned television tube) and pass them through a plate in which we’ve cut two small, closely spaced slits, on the same sort of scale as the wavelength of the electron as given by λ = h/p. What will we see?
Well, in the particle description we might expect that the electrons in the beam will pass through either one slit or the other. If we place a detector—such as a phosphor screen—on the other side of the slits we would anticipate that we’ll see two bright spots marking where the electrons that pass through the slits go on to strike the screen. Each spot will be brightest in the centre, showing where most of the electrons have passed straight through the corresponding slit, becoming more diffuse as we move away from the centre, representing electrons that have scattered on their way through.
But in the wave picture we would expect something rather different. Now in this picture the beam of electrons passes through the slits much like a beam of light, diffracting from each slit and forming an interference pattern of alternating bright and dark fringes, as I described in Chapter 5.
There are two ways in which we might interpret this kind of interference behaviour. We could suppose that the wave nature of the electron is the result of statistical averaging—it arises through some unknown mechanism which affects each individual electron as it passes—as a self-contained elementary particle—through one slit or the other. This is the kind of interpretation that Einstein himself tended to favour.
Or, we might suppose that the wave nature of the electron is an intrinsic behaviour. Each individual electron somehow behaves as a wave distributed through space, passing through both slits simultaneously and interfering with itself. Now this second interpretation would seem to be quite absurd. But there’s plenty of experimental evidence to suggest that this is exactly what happens.
Imagine that we now limit the intensity of the electron beam, so that at any moment only one electron passes through the slits. What happens? Well, each electron registers as a bright dot on the phosphor screen, indicating that ‘an electron struck here’. As more electrons pass through the slits—one at a time—we see an apparently random scattering of dots at the screen. But then we cross a threshold. As more and more electrons pass through, we begin to see a pattern as the individual dots group together, overlap, and merge. Eventually we see an interference pattern of alternating bright and dark fringes.
These kinds of experiments with electrons have been performed in the laboratory (an example is shown in Figure 15), and there’s no escaping the conclusion. Each electron passes through both slits. Its wavefunction on the other side is shaped into high-amplitude peaks and troughs arising from constructive interference and regions of zero-amplitude arising from destructive interference. The high-amplitude peaks and troughs represent regions where there is a high probability of ‘finding’ the electron: zero-amplitude means zero probability of finding the electron.
Figure 15. We can observe electrons as they pass, one at a time, through a two-slit apparatus by recording where they strike a piece of photographic film. Each white dot indicates the point where an individual electron is detected. Photographs (a)–(e) show the resulting images when, respectively, 10, 100, 3,000, 20,000, and 70,000 electrons have been detected. The interference pattern becomes more and more visible as the number of detected electrons increases.
When the electron wavefunction reaches the phosphor screen, the electron can in principle be found anywhere. What happens next is utterly bizarre. The interaction with the screen somehow causes the wavefunction to ‘collapse’; the location of the resulting dot is then determined by the distribution of probability derived from the amplitude of the wavefunction.* High probability yields a bright fringe. Low or zero probability yields a dark fringe. Remember that the wavefunction gives only a probability for where each electron will end up. It cannot tell us precisely where each electron will be detected.
This all seems totally ridiculous. Why don’t we just follow the paths of the electrons as they pass through the slits? Then we would surely see that electrons behave as individual, self-contained lumps of matter which pass through either one slit or the other. And when we do this, particle behaviour is indeed what we get. But the simple truth is that the interaction involved in following the path of the electron collapses its wavefunction prematurely, and so prevents any kind of interference from happening.
We’re faced with a choice. If we don’t follow the paths of the electrons, we get wave behaviour. If we look to see how we get wave behaviour, we get particle behaviour. These behaviours are complementary, and mutually exclusive. As Bohr insisted, there is no conceivable way of showing both types of behaviour simultaneously.
Einstein was very unhappy about all this. The collapse of the wavefunction and the reliance on probabilities appeared to disturb the delicate balance between cause and effect that had been an unassailable fact (and common experience) for all of human existence. In the world described by classical physics, if we do this, then that will happen. One hundred per cent, no question. But in the world described by quantum physics, if we do this then that will happen with a probability—possibly much less than 100 per cent—that can only be derived from the wavefunction.
Einstein remained stubbornly unconvinced, declaring that ‘God does not play dice’.4
He was particularly concerned about the collapse of the wavefunction. If a single electron is supposed to be described by a wavefunction distributed over a region of space, where then is the electron supposed to be prior to the collapse? Before the act of measurement, the mass (and energy) of the electron is in principle ‘everywhere’. What then happens at the moment of the collapse? After the collapse the mass of the electron is localized—it is ‘here’ and nowhere else. How does the electron get from being ‘everywhere’ to being ‘here’ instantaneously?
Einstein called it ‘spooky action-at-a-distance’.* He was convinced that this violated one of the key postulates of his own special theory of relativity: no object, signal, or influence having physical consequences can travel faster than the speed of light.
Through the late 1920s and early 1930s, Einstein challenged Bohr with a series of ever more ingenious thought experiments, devised as a way of exposing what Einstein believed to be quantum theory’s fundamental flaws—its inconsistencies and incompleteness.
Bohr stood firm. He resisted the challenges, each time ably defending the Copenhagen interpretation and in one instance using Einstein’s own general theory of relativity against him. But, under Einstein’s prosecution, Bohr’s case for the defence relied increasingly on arguments based on clumsiness: an essential and unavoidable disturbance of the system caused by the act of measurement, of the kind for which he had criticized Heisenberg in 1927. Einstein realized that he needed to find a challenge that did not depend directly on the kind of disturbance characteristic of a measurement, thereby completely undermining Bohr’s defence.
Together with two young theorists Boris Podolsky and Nathan Rosen, in 1935 Einstein devised the ultimate challenge. Imagine a physical system that emits two electrons. We can assume that the physics of the system constrains the two electrons such that they are produced in opposite spin states: one spin-up, the other spin-down.* Of course, we have no idea what the spins actually are until we perform a measurement on one, the other, or both of them.
We’ll call the electrons A and B. Suppose electron A shoots off to the left, electron B to the right. We set up our apparatus over on the left. We make a measurement and determine that electron A has a spin-up orientation. This must mean that electron B has a spin-down orientation—even though we haven’t measured it—as the physics of the system demands this. So far, so good.
Now, in a particle picture, we might assume that the spins of the electrons—whatever they may be—are fixed at the moment they are produced. The electrons move apart each with clearly defined spin properties. This is a bit like following the paths of the electrons as they pass through the two slits. Electron A could be either spin-up or spin-down, electron B spin-down or spin-up. But the measurement on electron A merely tells us what spin orientation it had, all along, and by inference what spin orientation electron B must have had, all along. According to this picture, there is no mystery.
But it’ll come as no surprise to find that quantum mechanics doesn’t see it this way at all. The physics of the two electrons is actually described in quantum theory by a single wavefunction. The two electrons in such a state are said to be ‘entangled’. The wavefunction contains amplitudes from which probabilities can be calculated for all the different possible outcomes of the measurement—spin-up for A/spin-down for B and spin-down for A/spin-up for B.
We don’t know what result we will get until we perform the measurement on electron A. At the moment of this measurement, the wavefunction collapses and the probability for one combination (say spin-up for A, spin-down for B) becomes ‘real’ as the other combination ‘disappears’. Prior to the measurement, each individual electron has no defined spin properties, as such. The electrons are in a superposition of the different possible combinations.
Here’s the rub. Although we might in practice be constrained in terms of laboratory space, we could in principle wait for electron B to travel halfway across the universe before we make our measurement on A. Does this mean that the collapse of the wavefunction must then somehow reach out and influence the spin properties of electron B across this distance? Instantaneously? Einstein, Podolsky, and Rosen (which we’ll henceforth abbreviate as EPR) wrote: ‘No reasonable definition of reality could be expected to permit this.’5
EPR argued that, despite what quantum theory says, it is surely reasonable to assume that when we make a measurement on electron A, this can in no way disturb electron B, which could be halfway across the universe. What we choose to do with electron A cannot affect the spin properties and behaviour of B and hence the outcome of any subsequent measurement we might make on it. Under this assumption, we have no explanation for the sudden change in the spin state of electron B, from ‘undetermined’ to spin-down.
We might be tempted to retreat to the particle picture and conclude that there is, in fact, no change at all. Electron B must have a fixed spin orientation all along. As there is nothing in quantum theory that tells us how the spin states of the electrons are determined at the moment they are produced, EPR concluded that the theory must be incomplete.
Bohr did not accept this. He argued that we cannot get past the complementary nature of the wave-picture and the particle-picture. Irrespective of the apparent puzzles, we just have to accept that this the way nature is. We have to deal with what we can measure, and these things are determined by the way we set up our experiment.
The EPR thought experiment pushed Bohr to drop the clumsiness defence, just as Einstein had intended. But this left Bohr with no alternative but to argue for a position that may, if anything, seem even more ‘spooky’. The idea that the properties and behaviour of a quantum wave-particle could somehow be influenced by how we choose to set up an apparatus an arbitrarily long distance away is extremely discomforting.
Physicists either accepted Bohr’s arguments or didn’t much care either way. By the late 1930s, quantum theory was proving to be a very powerful structure and any concerns about what it implied for our interpretation of reality were pushed to the back burner. The debate became less intense, although Einstein remained unconvinced.
But Irish theorist John Bell wasn’t prepared to let it go. Any attempt to eliminate the spooky action-at-a-distance implied in the EPR thought experiment involves the introduction of so-called ‘hidden variables’. These are hypothetical properties of a quantum system that by definition we can’t measure directly (that’s why they’re ‘hidden’) but which nevertheless govern those properties that we can measure. If, in the EPR experiment, hidden variables of some kind controlled the spin states of the two electrons such that they are fixed at the moment the electrons are produced, then there would be no need to invoke the collapse of the wavefunction. There would be no instantaneous change, no spooky action-at-a-distance.
Bell realized that if such hidden variables exist, then in certain kinds of EPR-type experiments the hidden variable theory predicts results that disagree with the predictions of quantum theory. It didn’t matter that we couldn’t be specific about precisely what these hidden variables were supposed to be. Assuming hidden variables of any kind means that the two electrons are imagined to be locally real—they move apart as independent entities with defined properties and continue as independent entities until one, the other, or both are detected. But quantum theory demands that the two electrons are ‘non-local’, described by a single wavefunction. This contradiction is the basis for Bell’s theorem.6
Bell was able to devise a relatively simple direct test. Local hidden variable theories predict experimental results that are constrained by something called Bell’s inequality—a range of results is possible up to, but not exceeding, a certain maximum limit. Quantum theory predicts results that are not so constrained—they can exceed this maximum, so violating the inequality.
Bell published his ideas in 1966. The timing was fortuitous. Sophisticated laser technology, optical instruments, and sensitive detection devices were just becoming available. Within a few years the first practical experiments designed to test Bell’s inequality were being carried out. The most widely known of these experiments were performed by French physicist Alain Aspect and his colleagues in the early 1980s, based on the generation and detection of entangled photons (rather than electrons). The results came down firmly in favour of quantum theory.7
There are always ‘what ifs’. More elaborate hidden variable schemes can be devised to take advantage of ‘loopholes’ in these kinds of experiments. One by one, ever more ingenious experiments have been carried out to close these loopholes. Today there is really no escaping the conclusion. Reality at the quantum level is non-local. There’s no getting around the wave description, the collapse of the wavefunction, and the spooky action-at-a-distance this seems to imply.
But the reality advocated by the proponents of hidden variable theories does not have to be a local reality. The influences of the hidden variables could be non-local. How does this help? Well, local hidden variable theories (of the kind that Bell had considered) are constrained by two important assumptions. In the first, we assume that due to the operation of the hidden variables, whatever measurement result we get for electron A can in no way affect the result of any simultaneous or subsequent measurement we make on the distant electron B.
In the second, we assume that however we set up the apparatus to make the measurement on electron A, this also can in no way affect the result we get for electron B. This is not something to which we would normally give a second thought. As English physicist Anthony Leggett put it: ‘… nothing in our experience of physics indicates that the [experimental set-up] is either more or less likely to affect the outcome of an experiment than, say, the position of the keys in the experimenter’s pocket or the time shown by the clock on the wall’.8
We could try to devise a kind of non-local hidden variable theory in which we relax the ‘set-up’ assumption but keep the ‘result’ assumption. This would mean that the outcomes of measurements are affected by how we choose to set up our apparatus, so we are obliged to accept some kind of unspecified action-at-a-distance that is somewhat spooky. Dropping this assumption means that the behaviour of the wavefunction is somehow governed by the way the apparatus is set up—it somehow ‘senses’ what’s coming and is ready for it. In such a scenario at least the results are in some sense preordained. We get rid of the collapse of the wavefunction and the inherent quantum ‘chanciness’ that this implies.
This comes down to the rather simple question of whether or not quantum wave-particles like electrons have the properties we measure them to have before the act of measurement. Do the ‘things-as-they-appear’ bear any resemblance to the ‘things-in-themselves’?
The consequences were worked out in 2003 by Leggett, who was also rather distrustful of the Copenhagen interpretation. Leggett defined a class of what he called ‘crypto’ non-local hidden variable theories. He found that keeping the result assumption but relaxing the set-up assumption is still insufficient to reproduce all the predictions of quantum theory. Just as Bell had done in 1966, Leggett now derived a relatively simple inequality that could provide a direct experimental test.
Experiments were subsequently performed in 2006 by physicists at the University of Vienna and the Institute for Quantum Optics and Quantum Information. The results were once again pretty unequivocal.9 Even more unequivocal results using different kinds of quantum states of light were reported by a team from the Universities of Glasgow and Strathclyde in 2010.10 Quantum theory rules.
The experimental tests of Leggett’s inequality demonstrate that we must abandon both the result and the set-up assumptions. The properties and behaviour of the distant electron B are affected by both the setting we use to measure electron A and the result of that measurement. It seems that no matter how hard we try, or how unreasonable the resulting definition of reality, we cannot avoid the collapse of the wavefunction.
We have to accept that the properties we ascribe to quantum particles like electrons, such as mass, energy, frequency, spin, position, and so on, are properties that have no real meaning except in relation to an observation or a measuring device that allows them to be somehow ‘projected’ into our empirical reality of experience. They are in some sense secondary, not primary properties. We can no longer assume that the properties we measure (the ‘things-as-they-appear’) necessarily reflect or represent the properties of the particles as they really are (the ‘things-in-themselves’).
At the heart of all this is what charismatic American physicist Richard Feynman declared to be quantum theory’s ‘only mystery’. Elementary particles such as electrons will behave as individual, localized particles following fixed paths through space. They are ‘here’ or ‘there’. But under other circumstances they will behave like non-local waves, distributed through space and seemingly capable of influencing other particles over potentially vast distances. They are ‘here’ and ‘there’.
These are now unassailable facts. But one question still nags: where then should we look to find the electron’s mass?