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Bohr & Kant & Zeno

Would it not be nice if there were a simple foundation to quantum physics? Tony Wagstaff believes there is; and that the Greeks had it.

During the opening decades of the 20th century, as the secrets of quantum theory emerged, so the underlying reality it purported to describe turned odder and odder. Electrons, photons, even cats, all seem to exist in a ghostly limbo with no well-defined characteristics of their own until someone observes them. For many scientists, including Max Planck and Albert Einstein, both pioneers in the history of the quantum, things had gone too far.

To advance their case against the emerging quantum orthodoxy, Einstein devised many hypothetical experiments which tried to show that quantum theory was missing some fundamental point. It was mainly down to his old friend Niels Bohr to answer these criticisms, and answer them, without exception, he did. It is chiefly from his retorts that a picture of the reality that lies behind quantum theory appeared, a picture which came to be known, after Bohr’s home city, as the Copenhagen interpretation.

Due to the manner of its development, the Copenhagen picture is somewhat esoteric. There is no authoritative draft outlining its theses, it has no manifesto as such. Since for the most part we have only Bohr’s replies to Einstein’s puzzles, we are left without the thrust of thought that lay behind these replies.

Nonetheless the Copenhagen interpretation has become in many circles the standard interpretation of quantum theory. It has held this standing in Institutes and Universities the world over, oftentimes, one may conjecture, just to keep students from asking awkward questions. With its insistence in situation after situation that some questions are meaningless and should be left unasked, one is drawn to wondering whether the Copenhagenists are simply evading the inherent difficulties.

This is unfortunate, for behind Bohr’s dogged upholding of the all-encompassing validity of quantum theory lies, I believe, the most profound and subtle philosophical thought. Whether Bohr himself was explicitly aware of what this was is unclear, though certainly he had an assured instinctive ease with it. As Einstein said of him, “seldom has anyone possessed such an intuitive grasp of hidden things.”

At the heart of quantum theory lies the Heisenberg Uncertainty Principle, the thrust of which is that certain conjugate properties, position and momentum for instance, cannot both be measured to a totally accurate degree of precision at once. If we choose to measure perfectly the position of an electron, we then lose accuracy regarding its momentum. We could choose instead to accurately measure its momentum, but then we cannot be exactly sure of its position. This promotes the seemingly untenable situation of an electron that has neither well-defined position nor momentum until we come to measure it. In effect, what we choose to measure in some way determines the characteristics of the electron. Hardly surprising that Einstein, Planck et al found this unacceptable.

In defending quantum theory against Einstein’s many thought experiments, Bohr would repeatedly emphasise the practicality of any experiment. When Einstein proposed his famous Clock in a Box experiment [see box], it was not enough that he propose we weigh the box before and after the photon’s escape. Bohr was insistent we specify exactly how we do the weighing. Only once it is made clear that a spring or some such device must be used does it become clear how the uncertainty will manifest itself. It was not enough to argue in principle. For Bohr the practicalities had to be explicit.

Time and again Bohr would return to practicalities, and time and again his case rested on the fact that to measure position, a measuring device needs to be fixed and unmoving, like the hole (relative to the box) through which the photon escapes. Because of this any momentum is absorbed by such a measuring device, and irretrievably lost. Conversely to measure momentum the measuring device needs to be loose, like our spring. Herein lies the problem. A measuring device cannot be both loose and fixed at once.

We’ll take this problem with us back over two millennia to Zeno, a Greek thinker and puzzler, contemporary of Socrates, and one of his many paradoxes, the arrow paradox.

Picture an arrow in flight. At any instant of time, an instant lasting exactly zero seconds, is it actually moving or not? How can anything move in an instant? On the other hand how can something be moving, yet also, for every instant of its journey, be at rest?

The problem is one of incompatibility. For a moving arrow to be at a definite position, with no element of uncertainty as to where it is, we have to specify an exact instant of zero time-length. If our instant lasts any length of time, the arrow will have moved during that time, even if only a minuscule distance. Conversely to say how the arrow has moved, we have to specify a period of time during which it did move, a period which must be non-zero.

Does this help with the quantum paradox?

First we need turn to the 18th century philosopher Immanuel Kant.

Central to Kant’s philosophy is the distinction between noumena and phenomena. The literal meaning of noumena is “things that are thought”, as opposed to “things that appear” which defines phenomena. Noumena form an integral part of the work of Plato as Ideas and Forms, such as a perfect triangle, whose sides have no breadth. No actual triangle can appear that way, for when we draw a triangle it has thick lines, but we can think and abstractly manipulate perfect triangles. Kant believed this idea to be the greatest legacy bequeathed by the Ancient Greeks.

Phenomena on the other hand refers to the appearance of external reality, which should not necessarily be equated with external reality itself. It refers to our awareness of the external, that image which we would say the brain produces when for instance we observe a sunset. It doesn’t necessarily mean the sunset itself. Science teaches us that the vibrant colours of a sunset don’t actually exist ‘out there’. They are an interpretation by our brain via the optical system of the different wavelengths of light.

To illustrate this distinction we can look at an example that fired Kant’s imagination and took him from philosophical mediocrity to greatness. When he read David Hume’s argument that we have no objective knowledge of causality, Kant realised, not that causality must therefore be an illusion, but that it cannot be phenomenally portrayed as it is in-itself. Causality can be represented in our cerebral picture of the world only as one event habitually occurring after another. We can never see the causal link between the first and second event; it is phenomenally invisible to us, though we can represent it noumenally as an abstract thought. Notwithstanding it is still possible for causality to play its part noumenally within external reality.

Having marked this distinction, Kant used it to dispel what he referred to as antinomies, which are a kind of paradox. We can take our lead from him. “We must regard one and the same conception, on the one hand, in relation to experience as an object of the senses and of the understanding, on the other hand, in relation to reason, isolated and transcending the limits of experience, as an object of mere thought.” [Critique of Pure Reason] We need to ask where a particular conception rightly belongs, as noumena or phenomena. This directly mirrors Bohr’s emphasis on practicalities.

Zeno’s treat

So, we can treat Zeno to a little noumenal/phenomenal distinction. Of course conceptually we have no problem with the notion of an arrow having both a definite position and definite motion at any instant of time. The problem arises when we wonder what conscious picture the optical process can construct of such an arrow. Here we reach the logical impossibility, highlighted by Zeno, that we cannot consciously represent both at once. Any conscious representation of exact position (for a moving object) must involve an instant of time whereas any conscious representation of movement must involve a period of time, the two being mutually exclusive.

Time cannot be both an instant and a period at once.

Thus it is logically impossible to phenomenally represent both exact position and exact momentum at once.

But when we think about it, when we consider it noumenally, there is no reason that an arrow should not have exact position and momentum at once. Just that any attempt to phenomenally represent the momentum renders the position uncertain, and vice versa.

It is a credit to Kant that this consequence of his work, which in view of his Newtonian faith would certainly never have occurred to him, should come to dominate our modern understanding of the physical world.

What though does it mean to exist in this noumenal manner? We would like to think that an arrow has both welldefined motion and well-defined position at every instant of its travels. We can hold this noumenal, well-defined motionin- an-instant in mind as we consider what Bohr had to say about the unspeakable.

There is some confusion as to what Bohr meant when speaking of what can be spoken of. So much so that people grasp for some deep meaning in the hope of enlightenment when in fact what Bohr was saying is quite simple.

It is best to invoke a simple experiment in this context. We can imagine that a measuring device, designed to register some property of motion of an electron, call it its spin, has at a specified time given us a reading of +1/2. I use the term ‘electron’ without any prejudice as to what an electron actually is, or what form it takes, and what it might be doing when it is spinning. It refers to the underlying reality behind the measurement, and indeed the two terms, ‘electron’ and ‘underlying reality’ are interchangeable in what follows.

Bohr was at pains to stress the communicability of experiments. That when speaking of what has happened, we need to be unequivocal. That when we describe a particle, anybody should be able to identify that specific particle.

He realised that, with the arrival of quantum theory, the only way we can do this is with reference to specific experimental set-ups. Thus in our experiment, it is fair to say that an electron was measured as having spin of value +1/2, using the specified experimental set-up.

Furthermore we can assert that any such a statement is an objective fact about that electron. Indeed for Bohr, this is the only way to objectively describe reality. “The electron was such that it yielded a measurement of +1/2, given the experimental set-up”, is a statement about that electron. It asserts that the underlying reality was such that it produced this reading. In addition it asserts that this underlying reality could have been different – in which case we might have had a different reading – but wasn’t.

Perhaps most importantly, we can say that the electron was such that it would have registered +1/2 for that particular experimental set-up, even if we hadn’t actually measured anything; or if we’d used a different set-up to measure something else. In other words, we are entitled to say that nothing about the presence or absence, or even the orientation of any measuring device, affects the electron in any pertinent way. (Other than what we would expect to happen in a classical sense – the measuring device may halt the electron in its track for example.) The underlying reality responsible for that +1/2 reading existed at the point of measurement, irrespective of the measurement set-up.

What we are not entitled to speak of, according to Bohr, are specific identifiable electrons without reference to specific experimental set-ups. We cannot say for instance that an electron with +1/2 spin was responsible for the measurement obtained. That that electron had +1/2 spin in-itself, and would have had +1/2 spin, even if we hadn’t measured it, is quite clearly contradicted by quantum theory. That is unless we allow for some faster-than-light signalling, a view that Bohr never attempted to sustain.

This point was decisively made in a theorem published in 1964 by John Stewart Bell. Bell’s theorem shows that, if quantum theory is correct, strong correlations exist between seemingly isolated measurements, even if they are taken at opposite ends of the Universe! In a nutshell, the presence, absence or orientation of a distant measuring device might result in a different value of spin obtaining than would otherwise be the case.

So while the values obtained are dependent on the experimental set-up, Bohr is still arguing for an underlying reality that is not so dependent. It would seem that this underlying reality, our electron, is somehow valueless.

Returning to our noumenal motion-in-an-instant, we can see that this is just the case. Values of motion are expressed in terms of distance over time, yet in-an-instant there is no time, and no distance travelled. Even though the motion-in-aninstant is well-defined, it is meaningless to assign values to it.

It is in exactly this sense, I believe, that Bohr spoke of that which is meaningless.

So, there is a wealth of difference between asserting that a specific underlying reality determines the results obtained in a measurement, and asserting that specific values of an underlying reality determine the results so obtained.

Is there any way in which it is permissible to speak of this underlying reality without reference to an experimental set-up? Bohr is saying quite categorically that there is such a reality, and that it exists in a specific way regardless of any experimental set-ups we may or may not have set up to measure it. That of all the possible ways-of-being it could have, it has but one of them. The rest may be disregarded as might-have-beens. So why can we not speak of it in an identifiable way without reference to an experimental set-up?

The answer again is that we have to treat it noumenally, just like our motion-in-an-instant. For that indeed is what it is. When we examine the sub-atomic, we are on a scale so small that instantaneous motion has to be taken as a reality. Just as we cannot assign values of properties to motion-in-aninstant, neither can we assign values of properties to the underlying reality of our electron measurement. Yet in neither case can we deny that it is well-defined, that it has a specific way-of-being.

In what sense then can we say it exists as a specific way-of-being? Obviously not in the classical sense of having definite values of properties. Not in any way that can identify it, to answer our question. How then?

Thanks to the development of quantum theory, specifically to Einstein, Podolsky and Rosen who produced a series of thought experiments to challenge, albeit unsuccessfully, Bohr’s position on quantum theory, we are in a position to answer.

The Einstein- Podolsky-Rosen, or EPR, experiments, make use of paired particles, a consequence of the conservation of momentum, that have equal and opposite motion. If we consider the motion of such a pair in a noumenal sense, it is clear that in-an-instant they would still have equal and opposite motion, just as they would phenomenally over a period of time. This, even though they do not meaningfully have absolute values of motion in-an-instant, as they do over a period of time.

We can now refine our notion of noumenal motion in-an-instant, it being meaningful to say such motion has a specific relational way-of-being, even though talk of specific absolute values is meaningless. This gives justification to the assertion that motion in-an-instant is well-defined, that it is a specific way-of-being. For if one of the pair existed as a different wayof- being, as one of the might-have-beens, the motions could no longer be equal and opposite, and this would be reflected when we came to measure them.

It is important to realise that this specific relational way-ofbeing is a direct description of the underlying reality. In a classical, or phenomenal sense, particles have absolute values of properties. The way they relate to each other comes about by comparing these values after they are obtained. This relationship does not however inhere in the particles themselves, it cannot be measured directly. On the other hand, for Bohr, and in our noumenal sense, a particle is the way it relates to other particles, and nothing more. Unless we refer to experimental set-ups then there are no specific values of properties, merely ways-of-being that are the same, opposite, etc.

The fact that Einstein could not accept this when he was the author of the two great theories of Relativity, was a source of constant exasperation for Bohr.

In essence the philosophical point Bohr is making when talking about what we can talk about, is that a phenomenal description of reality is meaningless without reference to an experimental set-up. Only a noumenal description is okay without such reference, or in-itself. And by noumenal we mean valueless, relational, and determinate of, or underlying, the phenomenal.

This is about as far as Bohr took the matter. There is one problem remaining that we can give quick consideration, that of how the noumenal becomes phenomenal.

Zeno considered the overall flight of an arrow as an infinite number of motions-in-an-instant. That the motions-in-an-instant determine the overall flight – this is what is meant by the noumenal being determinate of, or underlying, the phenomenal. But how do we get from a valueless, relational, noumenal wayof- being, to its phenomenal equivalent having absolute values? Surely any values we assign to the noumenal would be arbitrary?

The answer comes via the quantum of action, which is the essence of quantum theory. Essentially, the difference between quantum theory, and classical or Newtonian theory, lies in how they do the maths.

Classical theory allows any value to be assigned to a property, to any number of decimal places. Quantum theory is not so lenient. Only discrete values are allowed such as 1/2, 1, 1 1/2 and so on. The values between these, 1/3 say, or 1.1079832, are not permissible. Generally speaking the classical values will emerge statistically with large numbers of these quantised values. This is because in any given measurement each of the permitted quantised values has a probability weighting. In turn, this probability weighting can be thought of as expressing the relational quality of the particle when considered as a noumenal entity. Thus we go from the valueless-relation, through the probabilisticquantised, to the objective-phenomenal, which has a nice ring to it! It should be noted that Bell’s strong correlations occur only if we reason in this direction.

Can this mathematical reasoning be justified in practical terms? We need to apply a little Kantian distinction to the measurement process itself.

For centuries, in the days before J.J. Thomson began his pioneering work on electrons, objects to be measured were considered in phenomenal terms, as having properties with absolute values. Yet so too was the apparatus used to measure them. The measurement process so considered is phenomenal-phenomenal. A quantum measurement is a curious hybrid. On the one hand we have a particle that is noumenal, or valueless-relational. On the other we have a classical measuring device, which has absolute phenomenal values. This is the measurement process considered as noumenal-phenomenal. This leads us to jump to the obvious conclusion that a measurement can also be considered as noumenal-noumenal. This conclusion is also suggested by the consideration that a classical measuring device, though it can be thought of as having absolute values of properties, is nonetheless made up of valueless quantum particles such as electrons and protons, as is indeed everything. We have seen above, that a classical measurement cannot measure the noumenal relationship directly, since phenomenally this relationship does not inhere in the object itself. Considered noumenally however, it does, and when we consider both particle and measuring device as noumenal devices, then what is being measured directly is this relationship. There is nothing else to measure.

The measurement process thus provides justification for the mathematical logic of quantum theory.

It should be noted that somewhere in the course of all this, the noumenal measuring device needs to run through the same process to become the phenomenal device in the hybrid quantum measurement. Those familiar with quantum theory will be aware that this problem corresponds to the problem of where the wave-function collapses. On which turf, we will tread no further!

Conclusion

Niels Bohr, the emblem of scientific integrity and good common sense, believed in a reality underlying the peculiar manifestations of quantum theory, a reality that is valueless and relational.

By considering, in the Kantian spirit, the basic property of motion from two distinct perspectives; as it is experienced, as phenomenon, and as it is thought, as noumenon, we can see that the noumenal motion-in-an-instant is just such a valueless-relational reality that underlies the phenomenal motion-over-a-period.

It gives us reason to assert that quantum theory is, after all, firmly grounded in logic, and is a necessary process in the representation of the world as-it-should-be, as the world as-it-appears-to-be.

Tony Wagstaff is the finest thinker in his field, and sponsors his philosophical musings by working as a musician.

The Clock in a Box

A box containing a clock is swimming with photons; the clock being connected to a shutter which covers a hole in the wall of the box. The apparatus is designed so that the clock will briefly open the shutter at a specified time, allowing just one photon to escape. Einstein argued that by weighing the box before and after this specified time, we can calculate the energy of the photon, since mass=energy. As we know the time the photon escaped (we specified this before the experiment) we have broken the Uncertainty Principle – the time and energy of a particle are a conjugate pair that according to quantum theory, we cannot have exact knowledge of, both at once.

All well and good in principle, came Bohr’s response, but how do we actually weigh the box? Suppose we suspend it from a spring in a gravitational field. Then obviously the box, clock and all, has to move in the gravitational field; that’s how weighing works. But according to Einstein’s own general theory of relativity, clocks run at different rates depending on their position within a gravitational field, so we cannot be exactly sure of how fast the clock is running. This uncertainty is just enough to comply with Heisenberg’s principle. Bohr is said to have been not a little amused that he had turned Einstein’s own theory against him.