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Help Wanted: Philosopher required to sort out Reality
Apply to Mike Alder or any school of physics.
If you travelled to a different universe and found out that things were weird, well, that wouldn’t be too surprising. Going, possibly via the back of a wardrobe, into a universe where magic worked, or where the animals could talk, or both, would undoubtedly require a bit of mental flexibility. But none of these begin to compare with the weirdness of what actually happens in laboratories studying light and electrons. The terrible truth is that the wildest imaginings of fantasy writers don’t come within coo-ee of what actually happens in a physics lab.
Recently, some physicists have decided on the basis of the results of quantum experiments that the idea of a reality out there existing independent of our minds has to go. They say that the consciousness of the observer is crucial to determining the results of measurements: that the coming into my consciousness of the outcome of a measurement determines the answer. If so, what if I’m drunk and barely have any consciousness? This suggests an interesting experiment involving huge quantities of beer. Other physicists have suggested that everything that might happen does happen, in some alternative universe, and infinitely many alternative universes are being spawned all the time. The quasi-mystical view that our universe only exists because we are here to observe it, has been proposed with all seriousness. You can see that sharp-minded and properly-informed philosophers are badly needed. Indeed, some already exist, but more would be useful.
We now know heaps about the structure of matter at the very small (quantum) scale, and it turns out to be beyond our capacity to understand it. The observed phenomena of the quantum world are mind-boggling. We observe it, we can describe it, we can predict its behaviour to astonishing precision. What we can’t do is get any visualisable picture of what is going on. And it is possible that our brains can never understand it, because they work the wrong way, being evolved for the human scale, where things happen very differently.
The scariest and most incomprehensible part of the whole business is a little, seemingly trivial, mathematical result called Bell’s Inequality. I shall explain it shortly, and convince you that it is blindingly obvious and has to be true, and is bound to work. Before that I shall describe a physical set-up which violates it. This will therefore show that the impossible happens, and indeed it happens all the time, as a matter of routine.
Little Coloured Balls?
Suppose we have some coloured balls, but we’re all blind and so can’t see the colours. This is a fair caricature of the quantum world, where there are many properties we cannot detect directly. We also have some filters; three of them. One is called a ‘Black-White’ filter, one is called a ‘Red-Green’ filter and one a ‘Blue-Orange’ filter. We don’t know if this describes the quantum balls in any way, because we don’t even know what ‘colour’ is, but we do get consistent behaviour from the filters: if a ball is dropped into the Black-White filter and comes out on the Black side, then if we drop it back in, it still comes out the Black side. Likewise for the other filters. Any ball, dropped repeatedly in any filter, is bound to always come out the same side.
The balls come in twinned pairs. If one of the pair is dropped into a filter and comes out the Red side, its twin always comes out the Green side; and similarly for both the other two filters. Any ball always comes out one side or the other side of any filter, and the twin always comes out the other side if dropped in the same filter. (In quantum mechanics, this conjoining of pairs is called ‘quantum entanglement’.)
For all we know, the filters might actually be measuring something quite different from colours, but the consistent behaviour of the filters makes it clear they are measuring some property of the balls. We could certainly picture a ball as having three colours on it, for example Black, Green and Blue – for which its twin would have to have the colours White, Red and Orange. In this set-up there are eight possible combinations of colours: there are Black-Red-Blue balls and White-Red-Orange balls, and so on for the other six possibilities. Nevertheless, if we take a squillion balls and drop them in the Black-White filter, we find that half come out the Black side and the other half come out the White side. The same holds for the other two filters. This can give us information about the numbers of balls of each colour combination. Any scientist worth his salt would like to know the relative frequencies of each colour combination, and it is easy to devise experiments to count them.
This is where the trouble starts.
First for some mild weirdness. You would expect the ‘three-coloured balls’ theory to give you the result that if a ball has just gone through the Black side of a Black-White filter and you just drop it in the filter again, it will come out the Black side again. This indeed happens. You would also expect that if it comes out the Black side of a Black-White filter and then you put it through the Blue-Orange filter, if you then put it in the Black-White filter again, it will still come out the Black side. Unfortunately, this is not the case for quantum balls. Only half of them do, and the other half come out on the White side. This is how things happen in the world of quantum measurement. Time for a quick brood, to see if you have enough imagination to explain this.
Finished brooding? It’s clear that if the colour labels stayed the same, and if the filter is reading them and making a choice based on the fixed colour label, then this change in behaviour couldn’t happen. But conversely, if the filters just made random choices, then we couldn’t get the twinned pairs always behaving as they do when they go through just one filter – so the filters must be making some kind of deterministic choice based on some property of the balls. The only conclusion left is that the filters themselves must be changing the labels.
This situation with the balls and filters is characteristic of quantum effects: making a measurement changes the state of the system being measured. The state of the ball is specified by some label we’re calling colour, and the colour-measurement process of being dropped through a filter can alter it. It works exactly the way I’ve described when the balls represent electrons or photons, and the filters are systems designed to measure a positive or negative component of the so-called spin of the particle (equivalent to the Black or White of the coloured balls). You don’t have to know what quantum spin is to get the idea (it’s not the same as spin in the ordinary world): you just need to know that there are binary filters for it, and that there are subatomic things which get through one side or the other of them, that physicists can find out which way they went and count them, and that the filters behave as I’ve described.
So far this is mildly weird but not really weird. You might even imagine doing a lot of experiments to work out something about the different labels and what the filter does when it comes across each type, bearing in mind that it can pass a ball either of two ways, and if it passes it then it may (sometimes) change the label in some way. It has to do whatever changing of the quantum ball it does according to some definite rule, or we could get subsequent non-opposite behaviour from the twin ball, and we don’t ever see this. But by doing some careful counting of what happens as we try different sequences of putting balls and their twins through filters of different types, we should be able to work out something of what the colour labels on the balls are, and what is being done to them by the filters.
An even easier measurement would be to find the numbers of Black-and-Green balls, and likewise for all other pairs of colour combinations. The Black-and-Green balls include both Black-Green-Blue and Black-Green-Orange balls, and so we wouldn’t know how many of the Black-Green ones were also Blue rather than Orange, but we’d have some useful information.
This is where the really weird comes in. The counts we get are impossible. They can’t happen.
But they do.
And so we come to Bell’s Theorem. This is a simple result telling us that counts of things happening must satisfy a certain inequality. It works for finite sets of things – any things at all. The fact that this inequality is violated in the quantum world has been plausibly called the most profound discovery in all Science. Which by some accounts would make it the most important discovery of the entire human race. It’s certainly worth knowing about.
We can think of finite sets – that is, sets with a definite number of things in them. Well, those of an abstract cast of mind can; the rest of us can think of a simple and clear example, say a herd of cows in a field. Some of the cows are brown, and some are other colours. Some of the cows have tails longer than a metre, and some don’t. And some are facing towards the northern half of the field at one time, and others are not. Let the set of cows which are brown be called A, the set of cows with tails longer than a metre be called B, and the set of cows facing in a generally northern direction be called C.
Now if ||A & ~B|| is the number of cows in set A but not in set B (‘ ~’ means ‘not’ in algebra), and ||B & ~C|| is the number of cows in set B but not in set C, and ||A & ~C|| is the number of those cows in set A and not C, then Bell’s Inequality says:
||A & ~B|| + ||B & ~C|| ≥ ||A & ~C||
where the ‘≥’ sign means ‘greater than or equal to’.
In the case of our example this translates as “The number of brown cows with tails less than a metre long, plus the number of cows not facing north with tails more than a metre long, is always greater than or equal to the number of brown cows not facing north.” If your brain seizes up at the sight of mathematical symbols, beat it into submission by drawing a field and some cows, and label each of the cows for colour, tail length and facing north. Then count the number of cows in your picture which are brown and have tails of length a metre or less, and write this number down. Now write just in front of it “||A & ~B|| =” because this is what you have just worked out. Do the same for the other properties of the cows and you can easily verify that Bell’s Inequality holds for your herd of paper cows. Now draw a different collection of cows in a different field and do the counting all over again. After you have repeated this about twenty times, your brain will start working again and you will strongly suspect that Bell’s Inequality will always hold.
The theorem indeed works for cows, and it also works for any collection of finite sets A, B and C. It is mathematically provable. Some of the counts may be zero, but it still works.
To prove it holds for everything, I need only two observations. The first is that if one set, U is a subset of another set, V, then U can’t have more things in it than in V (otherwise it’s not a sub set of V at all). It’s a bit hard to argue with this logic. The second observation is that if we have two sets which have no members in common, the number of things in the union of the sets (ie in both of them) is the sum of the things in each individual set. If there are three cows in one half of a field and four cows in the other half of the same field then there are seven cows in the field. It is pretty hard to argue with this one as well. The only constraint on this is that the sets be finite, so that we can add up the numbers of things in them.
Now look at the sets ||A & ~B & ~C|| and ||A & B & ~C||. The first contains those things in ||A & ~C|| which are not in B, and the second contains precisely those things in ||A & ~C|| which are in B. Since anything is either in B or not in B, we see that the union of those two sets is ||A & ~C||. We can write this as || A & ~B & ~C|| + || A & B & ~C|| = || A & ~C||.
Translate this back into English or cow, and you can convince yourself that it is true whatever the sets A or B or C may be, so long as they are finite and can be counted.
All we need to do now is to observe that ||A & ~B & ~C|| is a subset of ||A & ~B||, so ||A & ~B|| ≥ ||A & ~B & ~C||. Similarly ||A & B & ~C|| is a subset of ||B & ~C||, so we must have ||B & ~C|| ≥ ||A & B & ~C||. Put these two results together and Bell’s Inequality follows immediately. Don’t take my word for it – try it with simple examples or check the logic carefully. Incidentally, Bell himself called this the Wigner-D’Espagnat inequality.
Now let’s see how this mathematical logic deals with our quantum balls. Logically, Bell’s Inequality must apply to our resulting counts. But we make the measurements, and it doesn’t.
We take a ball and it comes out on the Black side of a Black-White filter. We can put the twin through the Red-Green filter, and if it comes out of the Green side, we know that the original ball would have come out the Red side if we had put it through the Red-Green filter instead of the Black-White one. And if the twin comes out of the Red side of the Red-Green filter instead, then the original ball could not have been able to come out the Red side, had we put it into the Red-Green filter instead of the Black-White one.
So we take a squillion balls in colour-twinned pairs, put one of the twins in the Black-White filter and the other in the Red-Green filter. Then we count the fraction where the first ball comes out of the Black side and its twin comes out of the Red side. This number we can call the ‘Black and not Red’ value. This is the same as the ‘Black and Green’ proportion (Green means ‘not Red’ here). Consistently around seven percent of all the balls show this colour scheme. Similarly we can count the ‘Red and not Blue’ values, and the ‘Black and not Blue’ values for different collections of balls. Since there is nothing to distinguish any one squillion balls from another squillion, we assume that had we made the measurements in a different order it wouldn’t affect the fractions more than a tiny bit due to random variation of the amounts of different balls we have.
It’s possible to set up three filters which work as described, on electrons or photons instead of balls. We can do this in such a way that we find that about seven and a third percent of electrons are in the spin equivalent of the ‘Black and not Red’ category. This fraction gets more precise the more electrons we test. In fact, the fraction can be calculated from quantum theory to be the number 0.073223... or ‘one minus the square root of one half, divided by four.’ This is 7.3223 percent, as advertised. Testing lots of electrons is possible and can be done quite quickly. The count of electrons which are in the spin-equivalent ‘Red and not Blue’ category also gives roughly the same fraction of all electrons presented; just over seven percent. Applying Bell’s Inequality reasoning to this situation as we did with our cows, if A is Black and B is Red and C is Blue, we would expect that the ‘Black and not Red’ plus ‘Red and not Blue’ proportion would be greater than or equal to the proportion of balls which are ‘Black and not Blue’. Yet of all electrons in our sample of a squillion, statistically one quarter of a squillion will turn out to be ‘Black and not Blue’ by measurement. But about seven percent plus about seven percent is about fourteen percent, which is not bigger than twenty-five percent. Bell’s Inequality tells us these counts can’t happen. But in quantum measurement they do. The numbers do not add up.
Bell’s Not Ringing True
Now the search for the fatal flaw in the reasoning has to start. Something is horribly wrong. The fact that Bell’s Inequality is so obviously true by the simplest considerations (it works for cows!) tells us that what’s wrong has to be something deep. If Bell’s Inequality depended on a huge number of delicate assumptions and was hardly ever found to work in the real world nobody would worry too much if it failed. But it always works (except at the quantum level); and we can see exactly why it has to work everywhere, because it’s just a statement of the bleeding obvious. But the actual counts of electron spin states, and many other types of quantum measurement do not.
At this point a pause for thought, and maybe refreshment, is in order. If you have followed the argument closely, you may be in need of a stiff drink. Conversely if you are not severely gob-smacked, you have missed the point. The fact that the experiments merely count things is part of the awfulness of the result. Mostly, counting things works.
Of course, you can solve the problem of Bell’s Inequality by not thinking about it, just as you can solve all the problems of life by being dead. The systematic quantum violation of Bell’s Inequality won’t strike you as the least bit scary in either case. But if you do think about it, it gives you a strong sense of total insecurity about everything, and you have to face up to the fact that the universe just doesn’t work the way you thought it did. It is weirder than you can imagine. Measurement can change the states of physically-unconnected twin quantum particles
Does this have anything to do with the infamous quantum fact that making a measurement changes the state? No, because we only make one measurement on a ball/electron, and one on its twin, so we don’t even measure to see what sort of change our first measurement made to the ball/electron itself. It wouldn’t be so surprising if measuring which electrons go through a Black filter and subsequently fail to go through a Red filter gave us weird answers, but we aren’t doing that.
Could we be getting these answers because the filters are sending messages to each other saying what has happened to the latest ball and changing the rules for the other filter? Or could a filter looking at one ball broadcast to change other balls? One could maybe devise a protocol for filters to communicate with each other, but conversely, to counter any possibility of communication you can do your filtering experiments a long way away from each other in sealed lead-lined rooms. That anybody would even consider the possibility of the filters (or the balls) sending out broadcasts about the results of putting one in the other tells you how desperate things are. The sealed lead-lined rooms will rule out any conventional signal, but so will being far apart, and the latter is cheaper. In reality we still get the same apparently illogical results even when the places doing the counting get the answers long before a light-speed message between them could mess things up. Measurements have been made that show the anti-Bell results happen faster than any information travelling at light speed could get from one filter to the other. This is the spooky ‘action at a distance’ which upset Einstein so much and made him believe quantum mechanics was fundamentally flawed. He was right to be unhappy. You should be feeling pretty unhappy too.
If common sense and counting screw up so badly with quantum results, what does the theory of quantum mechanics have to say? At least the theory generates the observed answers. Unfortunately for your peace of mind, part of the theory is that there is a fundamental randomness about measurement results. Quantum theory also asserts that the way a electron will go through a filter is not determined until it actually happens and is measured. Believing that the electrons or photons are not really making random choices, but are instead following some complicated non-random rules we don’t know about, is not how quantum mechanics could work, and is not an option given Bell’s Inequality, unless you believe that information can be sent faster than light from twin to twin or from filter to filter. And while the result of whether a electron will go through the ‘Black’ side or the ‘White’ side is not predictable in advance in quantum theory, the twin always does the opposite. Even if it is the other side of the Universe at the time.
We have no coherent picture of what is going on here. We have a theory which we cannot understand. We can do the sums and we get the right answers – but what we cannot do is form a picture of what is happening. This, nearly everyone agrees, is absolutely maddening. Sane people like a picture of what is happening and use it when doing the sums; the algebra should describe the picture. In classical mechanics it does. In relativity it does too, although the pictures are more complicated. But in quantum mechanics, the pictures evaporate. Quantum results mean you are not allowed to say that an electron or photon in motion has a definite trajectory; it has an infinite number of possible trajectories, all at once. Until you measure it – then it has one. It is hard to visualise this.
The quantum theory explanation is as incomprehensible as the things it tries to explain. In fact quantum theory isn’t so much an explanation or a picture of anything as a compact summary of what will happen if you make certain measurements. The standard ‘Copenhagen interpretation’ of the theory tells you that if you expect more than this then tough luck, you ain’t gonna get it.
This is the bad news. The good news is that as a consequence of the laws of quantum physics we can in principle build quantum computers which can do a huge number of computations simultaneously. A small one has already been built. And it is just possible we might get some sort of understanding of quantum theory out from them after all. But that’s another story.
© Dr Mike Alder 2007
Mike Alder is a mathematician at the University of Western Australia. He has published in the philosophy of science although he is currently working on pattern recognition. He holds degrees in Physics, Pure Mathematics and Engineering Science.
John Stewart Bell
John Stewart Bell (1928-1990) was a physicist from Belfast, Northern Ireland. He worked in particle physics at Harwell and later at CERN. In a paper in 1964, he showed that the behaviour of particles as predicted by quantum mechanics violates a simple mathematical rule (Bell’s Inequality). A series of experiments from 1972 onwards confirmed that this was so. This is now one of the most-discussed problems in quantum theory.