Quantum Reality

by Jim Baggott

  But Einstein lost his enthusiasm for this approach within a matter of weeks of formulating it. It hadn’t come out as he’d hoped. The wavefunction had taken on a significance much greater than merely statistical. It was almost sinister. Einstein thought the problem was that distant particles were exerting some kind of strange force on one another, which he really didn’t like. But the real problem was that the guiding field is capable of exerting spooky non-local influences—changing something here instantaneously changes some other thing, a long way over there. He withdrew a paper he had written on the approach before it could be published. It survives in the Einstein Archives as a handwritten manuscript.2

  We’ll be returning to this kind of ‘pilot-wave’ description in Chapter 8. This experience probably led Einstein to conclude that his initial belief—that quantum mechanics could be completed through a more direct fusion of classical wave and particle concepts—was misguided. He subsequently expressed the opinion that a complete theory could only emerge from a much more radical revision of the entire theoretical structure. Quantum mechanics would eventually be replaced by an elusive grand unified field theory, the search for which took up most of Einstein’s intellectual energy in the last decades of his life.

  This early attempt by Einstein at completing quantum mechanics is known generally as a hidden variables formulation, or just a ‘hidden variables theory’. It is based on the idea that there is some aspect of the physics that governs what we see in an experiment, but which makes no appearance in the representation. There are, of course, many precedents for this kind of approach in the history of science. As I’ve already explained, Boltzmann formulated a statistical theory of thermodynamics based on the ‘hidden’ motions of real atoms and molecules. Likewise, in Einstein’s abortive attempt to rethink quantum mechanics, it is the positions and motions of real particles, guided by the wavefunction, that are hidden.

  However, in his 1932 book The Mathematical Foundations of Quantum Mechanics, von Neumann presented a proof which appeared to demonstrate that all hidden variable extensions of quantum mechanics are impossible.3 This seemed to be the end of the matter. If hidden variables are impossible, why bother even to speculate about them?

  And, indeed, silence prevailed for nearly twenty years. The dogmatic Copenhagen view prevailed, seeping into the mathematical formalism and becoming the quantum physicists’ conscious or unconscious default interpretation. The physics community moved on and just got on with it, content to shut up and calculate.

  Then David Bohm broke the silence.

  In February 1951, Bohm published a textbook, simply called Quantum Theory, in which he followed the party line and dismissed the challenge posed by EPR’s ‘bolt from the blue’, much as Bohr had done. But even as he was writing the book he was already having misgivings. He felt that something had gone seriously wrong.

  Einstein welcomed the book, and invited Bohm to meet with him in Princeton sometime in the spring of 1951. The doubts over the interpretation of quantum theory that had begun to creep into Bohm’s mind now crystallized into a sharply defined problem. ‘This encounter had a strong effect on the direction of my research,’ Bohm later wrote, ‘because I then became seriously interested in whether a deterministic extension of quantum theory could be found.’4 The Copenhagen interpretation had transformed what was really just a method of calculation into an explanation of reality, and Bohm was more committed to the preconceptions of causality and determinism than perhaps he had first thought.

  In Quantum Theory, Bohm had asserted that ‘no theory of mechanically determined hidden variables can lead to all of the results of the quantum theory.’5 This was to prove to be a prescient statement. Bohm went on to develop a derivative of the EPR thought experiment which he published in a couple of papers in 1952 and which he elaborated in 1957 with Yakir Aharonov.6 This is based on the idea of fragmenting a diatomic molecule (such as hydrogen, H2) into two atoms.

  Now, elementary particles are distinguished not only by their properties of electric charge and mass, but also by a further property which we call spin. This choice of name is a little unfortunate, and arises because some physicists in the 1920s suspected that an electron behaves rather like a little ball of charged matter, spinning around on its axis much like the Earth spins as it orbits the Sun. This is not what happens, but the name stuck.

  The quantum phenomenon of spin is indeed associated with a particle’s intrinsic angular momentum, the momentum we associate with rotational motion. Because it also carries electrical charge, a spinning electron behaves like a tiny magnet. But don’t think this happens because the electron really is spinning around its axis. If we really wanted to push this analogy, then we would need to accept that an electron must spin twice around its axis to get back to where it started.* The electron has this property because it is a matter particle called a fermion (named for Enrico Fermi). It has a characteristic spin quantum number of ½ and two spin orientations—two directions the tiny electron magnet can ‘point’ in an external magnetic field. We call these ‘spin up’ (↑) and ‘spin down’ (↓). Sound familiar?

  The chemical bond holding the atoms together in a diatomic molecule is formed by overlapping the ‘orbits’ of the electrons of the two atoms and by pairing them so that they have opposite spins—↑↓. In other words, the two electrons in the chemical bond are entangled. Bohm and Aharonov imagined an experiment in which the chemical bond is broken in a way that preserves the spin orientations of the electrons (actually, preserving the electrons’ total angular momentum) in the two atoms. We would then have two atoms—call then atom A and atom B—entangled in spin states ↑ and ↓.

  Bohm and Aharonov brought the EPR experiment down from the lofty heights of pure thought and into the practical world of the physics laboratory. In fact, the purpose of their 1957 paper was to claim that experiments capable of measuring correlations between distant entangled particles had already been carried out. For those few physicists paying attention, Bohm’s assertion and the notion of a practical test suggested some mind-blowing possibilities.

  John Bell was paying attention. In 1964, he had an insight that was completely to transform questions about the representation of reality at the quantum level. After reviewing and dismissing von Neumann’s ‘impossibility proof’ as flawed and irrelevant, he derived what was to become known as Bell’s inequality. ‘Probably I got that equation into my head and out on to paper within about one weekend,’ he later explained. ‘But in the previous weeks I had been thinking intensely all around these questions. And in the previous years it had been at the back of my head continually.’7

  Recall from Chapter 4 that the EPR experiment is based on the creation of a pair of entangled particles, A and B, which we now assume to be atoms. Because the total angular momentum is conserved, we know that the atoms must possess opposite spin-up and spin-down states, which we will continue to write as A↑B↓ and A↓B↑. We assume that the atoms A and B separate as ‘locally real’ particles, meaning that they maintain separate and independent identities and quantum properties as they move apart.

  We further assume that making any kind of measurement on A can in no way affect the properties and subsequent behaviour of B. Under these assumptions, when we measure A to be in an ↑ state, we then know with certainty, that B must be in a ↓ state. There is nothing in quantum mechanics that explains how this can happen, the theory is incomplete, and we have a big problem. This is the essence of EPR’s original challenge.

  But is any of this really so mysterious? Bell was constantly on the lookout for everyday examples of pairs of objects that are spatially separated but whose properties are correlated, as these provide accessible analogues for the EPR experiment. He found a perfect example in the dress sense of one of his colleagues at CERN, Reinhold Bertlmann. Some years later, Bell wrote:8

  The philosopher in the street, who has not suffered a course in quantum mechanics, is quite unimpressed by Einstein–Podolsky–Rosen correlations. He c
an point to many examples of similar correlations in everyday life. The case of Bertlmann’s socks is often cited. Dr Bertlmann likes to wear two socks of different colours. Which colour he will have on a given foot on a given day is quite unpredictable. But when you see that the first sock is pink you can be already sure that the second sock will not be pink. Observation of the first, and experience of Bertlmann, gives immediate information about the second. There is no accounting for tastes, but apart from that there is no mystery here. And is not this EPR business just the same?

  This situation was sketched by Bell himself, and is illustrated in Figure 12.

  Figure 12 Bertlmann’s socks and the nature of reality.

  What if the quantum states of the atoms A and B are fixed by the operation of some local hidden variable at the moment they are formed and, just like Bertlmann’s socks, the atoms move apart in already pre-determined quantum states? This seems to be perfectly logical, and entirely compatible with our first instincts. We can’t deny an element of randomness, just as we can’t deny that Bertlmann may choose at random to wear a pink sock on his left or right foot, so the hidden variable may randomly produce the result A↑B↓ or A↓B↑. But, so long as the spins of atoms A and B are always opposed, it would seem that all is well with the laws of physics.

  How would this work? Well, we have no way of knowing what this hidden variable might be or what it might do, but here we are on the shores of Metaphysical Reality where we’re perfectly at liberty to speculate. So, let’s suppose that each atom has a further property we know nothing about, but which we presume acts to pre-determine the spins of A and B in any subsequent measurements. We could think of this property in terms of a tiny pointer, tucked away inside each atom. This can point in any direction in a sphere. When atoms A and B are formed by breaking the bond in the molecule, their respective pointers are firmly fixed in position, but they are constrained by the conservation of angular momentum always to be fixed in opposite directions.

  The atoms move apart, the pointers remaining fixed in their positions. Atom A, over on the left, passes between the poles of a magnet, which allows us to measure its spin orientation.* Atom B, on the right, passes between the poles of another magnet, which is aligned in the same direction as the one on the left. We’ll keep this really simple. If the pointer for either A or B is projected anywhere onto the ‘northern’ semicircle, defined in relation to the north pole of its respective magnet (shown as the shaded area in Figure 13), then we measure the atom to be in an ↑ state. If the pointer lies in the ‘southern’ semicircle (the unshaded area), then we measure the atom to be in a ↓ state. Figure 13 shows how a specific (but randomly chosen) orientation of the pointers leads to the measurement outcome A↑B↓.

  Figure 13 In a simple local hidden variable account of the correlation between the spins of entangled hydrogen atoms, we assume the measurement outcomes are predetermined by a ‘pointer’ in each atom whose direction is fixed at the moment the atoms are formed.

  In a sequence of measurements on identically prepared pairs of atoms, we expect to get a random series of results: A↑B↓, A↑B↓, A↓B↑, A↑B↓, A↓B↑, A↓B↑, and so on. If we assume that in each pair the pointer projections can be randomly but uniformly distributed over the entire circle, then in a statistically significant number of measurements we can see that there’s a 50% probability of getting a combined A↑B↓ result.

  This is where Bell introduces a whole new level of deviousness. Figure 13 shows an experiment in which the magnets are aligned—the north poles of both magnets lie in the same direction. But what if we now rotate one of the magnets relative to the other? Remember that the ‘northern’ and ‘southern’ semicircles are defined by the orientation of the poles of the magnet. So if the magnet is rotated, so too are the semicircles. But, of course, we’re assuming that the hidden variable pointers themselves are fixed in space at the moment the atoms are formed—the directions in which they point are supposedly determined by the atomic physics and can’t be affected by how we might choose to orientate the magnets in the laboratory. The atoms are assumed to be locally real.

  Suppose we conduct a sequence of three experiments:

  Figure 14 shows how rotating one magnet clockwise relative to the other affects the measurement outcomes for the same pointer directions used in Figure 13. In experiment #1, rotating the magnet for atom B by 135° means that the pointer for B now predetermines an ↑ state, giving the result A↑B↑. This doesn’t mean that we’ve broken any conservation laws—the hidden variable pointers for A and B still point in opposite directions. It just means that we’ve opened up the experiment to a broader range of outcomes: rotating the magnet for atom B means that both A↑B↑ and A↓B↓ results have now become permissible. And, as the probabilities for all the possible results must still sum to 100%, we can see that the probabilities for A↑B↓ and A↓B↑ must therefore fall.

  Figure 14 Bell introduced a whole new level of deviousness into the Bohm–Aharanov version of the Einstein–Podolsky–Rosen experiment by rotating the relative orientations of the magnets.

  The question I want to ask for each of these experiments is this: What is the probability of getting the outcome A↑B↓? Before rushing to find the answers, I’d like first to establish some numerical relationships between the probabilities for this outcome in each of the experiments. I don’t want to get bogged down here in the maths, so I propose to do this pictorially.9

  Imagine that we ‘map out’ the individual ↑ and ↓ results—irrespective of whether these relate to A or B—for each orientation of the magnets. For an orientation of 0°, we divide a square into equal upper and lower halves. For 135°, we divide the square into equal left and right halves. We have to be a bit more imaginative for the third 45° orientation, as we have only two dimensions to play with, so we draw a circle inside the square, such that the area of the circle is equal to the area that lies within the square, but outside the circle. This gives us

  We can now combine these into a single map, which allows us to chart the A↑B↓ results for each of our experiments. For example, in experiment #1, results in which A is measured to be ↑ and B is measured to be ↓ occupy the top right-hand corner of the map, marked below in grey. Likewise for experiments #2 and #3:

  We should note once again that this will only work if we can assume that atom A and atom B are entirely separate and distinct, and that making measurements on one can in no way affect the outcomes of measurements on the other. We must assume the atoms to be locally real.

  If it helps, think of the grey areas in these diagrams as the places where we would put a tick every time we get an A↑B↓ result in each experiment. We carry out each experiment on exactly the same numbers of pairs of atoms, and we count up how many ticks we have. The number of ticks divided by the total number of pairs we studied then gives us the probability for getting the result A↑B↓ in each experiment.

  In fact, these diagrams represent sets of numbers. So let’s have some fun with them. We can write the set for experiment #1 as the sum of two smaller subsets:

  Likewise, we can write the set for experiment #2 as

  If we now add these two expressions together, we get

  We can’t exclude the possibility that the last subset in this expression won’t have some ticks in it, but I think you’ll agree that it is perfectly safe for us to conclude that

  where the symbol ≥ means ‘is greater than or equal to’. This is Bell’s inequality. It actually has nothing whatsoever to do with quantum mechanics or hidden variables. It is simply a logical conclusion derived from the relationships between independent sets of numbers. It is also quite general. It does not depend on what kind of hidden variable theory we might devise, provided this is locally real. This generality allowed Bell to formulate a ‘no-go’ 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.’10 A complementary ‘no-go’ theorem
was devised in 1967 by Simon Kochen and Ernst Specker.11

  What this form of Bell’s inequality says is that the probability of getting an A↑B↓ result in experiment #1, when added to the probability of getting an A↑B↓ result in experiment #2, must be greater than or at least equal to the probability of getting an A↑B↓ result in experiment #3.

  Now, we can deduce these probabilities from our simple local hidden variable theory by examining the overlap between the ‘northern’ semicircles for the two magnets in each experiment, and dividing by 360°. We know from Figure 13 that complete overlap of 180° (the magnets are aligned) means a probability for A↑B↓ of 50%. In experiment #1, the overlap is reduced to 45°, and the probability of getting A↑B↓ falls to 12½%:

  In experiment #2, the overlap is 90° (25%):

  And in experiment #3 the overlap is 135° (37½%):

  These are summarized on the left in the following:

  Probability of getting A↑B↓ local hidden variables Probability of getting A↑B↓ quantum mechanics

  Experiment #1 12½% 7.3%

  Experiment #2 25% 25%

  Experiment #3 37½% 42.7%

  If we now add the probabilities for #1 and #2 together, then according to a local hidden variables theory we get a total of 37½%, equal to the probability for #3 and therefore entirely consistent with Bell’s inequality.

  So, what does quantum mechanics (without hidden variables) predict? I don’t want to go into too much detail here. Trust me when I tell you that the quantum-mechanical prediction for the probability of getting an A↑B↓ result is given by half the square of the cosine of half the angle between the magnets. The quantum-mechanical predictions are summarized in the right-hand column of the table above. If we put these predictions into Bell’s inequality, we get the result that 7.3% + 25% = 32.3% must be greater than or equal to 42.7%.