by Jim Baggott
Heisenberg eventually bowed to the pressure. He accepted Bohr’s view and the Copenhagen interpretation was born.
The uncertainty principle is not limited to position and momentum. It applies to other pairs of physical properties, called conjugate properties, such as energy and time. It also applies to the different spin orientations of quantum particles.
For example, photon polarization can be ‘vertical’ or ‘horizontal’, which implies some kind of reference frame against which we judge these orientations. ‘Vertical’ must mean vertical with regard to some co-ordinate axis. When applied to polarization, the uncertainty principle tells us that certainty in one co-ordinate axis means complete uncertainty in another. If in the laboratory I fix a piece of Polaroid film so that its transmission axis lies along the z axis (say), and I measure photons passing through this film, then I have determined that these photons have vertical polarization measured along the z axis, with a high degree of certainty.* This implies a high degree of uncertainty for polarizations oriented along either the x or y axis.
Quantum fluctuations of the vacuum
The science fiction writer Arthur C. Clarke famously formulated three laws of prediction. The third law, a guide for aficionados of ‘hard science fiction’ (characterized by its emphasis on scientific accuracy), declares that any sufficiently advanced technology is indistinguishable from magic.**
So, here’s an interesting bit of quantum physics that at first glance looks indistinguishable from magic.
Take two small metal plates and place them side by side a few millionths of a metre apart in a vacuum, insulated from any external electric and magnetic fields. There is no force between these plates, aside from an utterly insignificant gravitational attraction between them which, for the purposes of this experiment, can be safely ignored.
Now here comes the magic. Although there can be no force between them, the plates are actually pushed very slightly together.
This is an effect first identified by the Dutch physicist Hendrik Casimir in 1948. Experiments conducted in 2001 by a team of physicists at Italy’s National Institute of Nuclear Physics and the Department of Physics at the University of Padua demonstrated the existence of a ‘Casimir force’ in this specific arrangement of two closely spaced metal plates.
What’s going on?
Heisenberg’s uncertainty principle sets a fundamental threshold which nothing in nature can cross. We might be used to thinking about the uncertainty principle in relation to constraints placed on the position and momentum of a quantum particle, or on its energy and the rate of change of this energy with time. We might not think to apply the uncertainty principle to the vacuum; to ‘empty’ space. What would be the point?
Suppose we create a perfect vacuum, completely insulated from the external world. We might be tempted to argue that there is ‘nothing’ at all in this vacuum. But what does this imply? It implies that the energy of an electromagnetic field in the vacuum is zero. It also implies that the rate of change of the amplitude of this field is zero too. But the uncertainty principle denies that we can know precisely the energy of an electromagnetic field and its rate of change. They can’t both be exactly zero.
What happens is that the vacuum suffers a bad case of the jitters. It experiences fluctuations of the electromagnetic field which average out to zero, in terms of both energy and rate of change, but which are nevertheless non-zero at individual points in space and time.
Fluctuations in a quantum field are equivalent to quantum particles. The vacuum fluctuations of the electromagnetic field can be thought of as virtual photons; ‘virtual’ not because they are not ‘real’, but because they are not directly perceived.
In the experiment devised by Casimir, the space between the plates constrains the types of vacuum fluctuations that can persist. Only fluctuations that ‘fit’ in this space can contribute to the vacuum energy. Alternatively, we can imagine that the narrow space between the plates reduces the number of virtual photons that can persist there. The density of virtual photons between the plates is then lower than the density of virtual photons elsewhere. The end result is that the plates experience a kind of virtual ‘radiation pressure’; the higher density of virtual photons on the outsides of the plates pushes them closer together.
The Italian physicists’ painstaking measurements showed that the magnitude of the Casimir force between the plates was precisely as Casimir himself had predicted.
Spooky action-at-a-distance
Einstein was far from satisfied with the Copenhagen interpretation. Having sowed the seeds of the quantum revolution with his breathtakingly speculative paper in 1905, by the mid-1920s he was fast becoming quantum theory’s most determined critic.
He was particularly concerned about the collapse of the wavefunction. If a single photon is supposed to be described by a wavefunction distributed over a region of space, where then is the photon supposed to be prior to the collapse? Before the act of measurement, the energy of the photon is in principle ‘everywhere’. What then happens at the moment of the collapse? After the collapse, the energy of the photon is localized — it is ‘here’ and nowhere else. How does the photon 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, he challenged Bohr with a series of ever more ingenious ‘thought experiments’. These were experiments carried out only in the mind 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 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 that he had criticized Heisenberg for. Einstein realized that he needed to find a challenge that did not depend directly on the kind of disturbance characteristic of a measurement, thus undermining Bohr’s defence.
In 1935, together with two young theorists, Boris Podolsky and Nathan Rosen, Einstein devised the ultimate challenge. Imagine a physical system that produces two photons. We assume that the physics of the system constrains the two photons such that they are both produced in identical states of linear polarization.* We have no idea what these orientations are until we impose a reference frame by performing a measurement. According to the Copenhagen interpretation, until the measurement, the actual orientations are ‘undetermined’ — all orientations are possible in the wavefunction, just as the single photon in the two-slit experiment can be found anywhere on the photographic film prior to the collapse.
For the sake of clarity, we’ll call these photons A and B. Photon A shoots off to the left, photon B to the right. We set up our polarizing film over on the left. We make a measurement and determine that photon A has vertical polarization along our laboratory z axis.
What does this mean for photon B? Obviously, we have not yet made any measurement on photon B, yet we can deduce that it, too, must have vertical polarization along this same axis. The physics of the process that produced the photons demands this. The polarization state of photon B appears to have suddenly changed, from ‘undetermined’ to vertical, even though we have made no measurement on it. And although we might in practice be constrained in terms of laboratory space, we could in principle wait for photon B to travel halfway across the universe before we make our measurement on A.
All this talk of photons and their polarization states might seem rather esoteric, and it might be a little difficult to follow precisely what’s going on. But it’s important that we understand the nature of Einstein, Podolsky and Ros
en’s challenge and so fully appreciate what’s at stake here.
Let’s try the following (imperfect) analogy. Suppose I toss a coin. This happens to be a coin with some special properties. As it spins in the air, it splits into two coins. The operation of a ‘law of conservation of coin faces’ means that if the coin that lands over on the left (coin A) gives ‘heads’, then the coin that lands over on the right (coin B) gives ‘heads’ too. If coin A gives ‘tails’, then coin B also gives ‘tails’. There are no circumstances under which we would expect to observe the results ‘heads’—‘tails’ or ‘tails’—‘heads’.
Now suppose I toss the coin and we look to see what result we got for coin A. We see that it lands ‘heads’. We know that coin B must also give the result ‘heads’ — the law of conservation of coin faces demands it.
Our instinct is to assume that the properties of the two coins are established at the instant they split from the original coin. The coins split apart and separate in mid-air, in some way that ensures that we eventually get correlated results — ‘heads’—‘heads’ or ‘tails’—‘tails’. But assuming this implies that there are other variables involved of which we are ignorant, and that if this is all described by quantum theory, then the theory is in some sense incomplete.
In the photon example, the two particles are said to be ‘entangled’. Because of the way they are produced, both photons are described by a single wavefunction. When we make a measurement on photon A, the wavefunction collapses and the polarization properties of both photons mysteriously become ‘real’. If photon B is halfway across the universe, then the collapse must reach out across this distance. Instantaneously.
Likewise, quantum theory insists that our two coins don’t possess fixed properties the instant they’re created. It’s almost as though they each oscillate back and forth from ‘heads’ to ‘tails’ as they spin to the ground. But the moment we see that coin A has landed ‘heads’, then coin B must also land ‘heads’, even though the two coins may fall far apart from each other.
Einstein, Podolsky and Rosen wrote: ‘No reasonable definition of reality could be expected to permit this.’12
Despite what quantum theory says, Einstein, Podolsky and Rosen argued that it is surely reasonable to assume that when we make a measurement on photon A, this can in no way disturb photon B. What we choose to do with photon A cannot affect the 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 polarization state of photon B, from ‘undetermined’ to vertical.
We conclude that there is, in fact, no change at all. Photon B must have vertical polarization all along. As there is nothing in quantum theory that tells us how the polarization states of the photons are determined at the moment they are produced, Einstein, Podolsky and Rosen concluded that the theory is incomplete.
‘This onslaught came down upon us as a bolt from the blue,’ wrote Belgian theorist Léon Rosenfeld, who was working with Bohr at his institute when news of this latest challenge reached Copenhagen.13 The eminent English theorist Paul Dirac declared: ‘Now we have to start all over again, because Einstein proved that it does not work.’14
Bohr’s response was simply to restate the Copenhagen interpretation. He argued that we simply cannot get past the wave shadows and the particle shadows. Irrespective of the apparent puzzles caused by the need to invoke a collapse of the wavefunction, we just have to accept that that’s the way it is. We have to deal with what we can measure and perceive. And these things are determined by the way we set up our experiment.
Bohr argued that it does not matter that the polarization state of photon B can be inferred from measurements we make on photon A. By setting up an experiment to measure the polarization of photon A with certainty (along the z axis, say), we deny ourselves the possibility of measuring the polarization along any other axis (x or y). And if we cannot exercise a choice, then the actual properties and behaviour of photon B are really rather moot. Even though there is no mechanical disturbance of photon B (no clumsiness), its properties and behaviour are nevertheless defined by the way we have set up the measurement on photon A.
The Einstein—Podolsky—Rosen thought experiment pushed Bohr to drop the clumsiness defence, just as Einstein had intended. But this left him 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 particle could somehow be influenced by how we choose to set up an apparatus an arbitrarily long distance away is very discomforting. Many years later the English physicist Anthony Leggett summarized this as follows:
But in physics we are normally accustomed to require some positive reason before we accept a particular part of the environment as relevant to the outcome of an experiment. Now the [distant] polarizer … is nothing more than (e.g.) a calcite crystal, and nothing in our experience of physics indicates that the orientation of distant calcite crystals 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.15
For whom the Bell tolls
Physicists either accepted Bohr’s arguments or didn’t much care either way. 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 stubbornly unconvinced.
But Irish theorist John Bell continued to feel uncomfortable. Any attempt to eliminate the spooky action-at-a-distance implied in the Einstein, Podolsky and Rosen thought experiment involved the introduction of so-called ‘hidden variables’. These are hypothetical properties of a quantum system that by definition are not accessible to experiment (that’s why they’re ‘hidden’) but which nevertheless govern those properties that we can measure. If, in the Einstein—Podolsky-Rosen experiment, hidden variables of some kind controlled the polarization states of the two photons such that they are fixed at the moment the photons 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 were assumed to exist, then in certain kinds of Einstein—Podolsky—Rosen-type experiments the hidden variable theory would predict results that disagreed 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 photons are imagined to be locally real — they move apart as independent entities and continue as independent entities until one, the other or both are detected.
Going back to our coin analogy, a hidden variables extension would have the properties of the two coins fixed at the moment they split apart and separate. The coins are assumed to be locally real.
This seems perfectly reasonable, but quantum theory, in contrast, demands that the two photons or the two coins are non-local and entangled; they are described by a single wavefunction. They continue to be non-local and entangled until one, the other or both are detected, at which point the wavefunction collapses and the two photons or the two coins become localized, replete with the properties we measure.
This is Bell’s theorem: ‘If the [hidden variable] extension is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local. This is what the theorem says.’16
Bell was able to devise a relatively simple direct test. Hidden variable theories that establish some form of local reality predict experimental results that conform to something called Bell’s inequality. Quantum theory does not.
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 d
esigned 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. These made use of two high-powered lasers to produce excited calcium atoms, formed in an atomic ‘beam’ by passing gaseous calcium from a high temperature oven through a tiny hole into a vacuum chamber. Calcium atoms excited in this way undergo a ‘cascade’ emission, producing two photons in quick succession. The physics of the atom demands that angular momentum is conserved in this process, and the two photons are emitted in opposite states of circular polarization. This means that when they are passed through linear polarization filters, both photons will have the same linear polarization state, either both vertical or both horizontal.* The photons are entangled.
The two photons have different energies, and hence different frequencies (and different colours). The physicists monitored green photons (which we will call photons A) on the left and blue photons (photons B) on the right. Each polarizing filter was mounted on a platform which allowed it to be rotated about its optical axis. Experiments could therefore be performed for different relative orientations of the two filters, which were placed about 13 metres apart.
Imposing this separation distance meant that any kind of ‘spooky’ signal passing between the photons at the moment the wavefunction collapsed, ‘informing’ photon B of the fate of photon A, for example, would need to travel at about twice the speed of light.
The results came down firmly in favour of quantum theory. The physicists performed four sets of measurements with four different orientations of the two polarizing filters. This allowed them to test a generalized form of Bell’s inequality. For the specific combination of orientations of the polarizing filters chosen, the generalized form of the inequality demands a value that cannot be greater than 2. Quantum theory predicts a value of 2.828.** The physicists obtained the result 2.697±0.015.17 In other words, the experimental result exceeded the limit predicted by Bell’s inequality by almost fifty times the experimental error, a powerful, statistically significant violation.