Reverse-Engineering Quantum Mechanics, V.

‘The Duckmole Problem’

Quantum theory has been plagued since its formulation by a conception known as ‘wave-particle duality’, a throwback to two distinct and incommensurable concepts from classical physics, forced together into an uneasy shotgun marriage. It was natural for the founders of quantum theory to try to understand and make sense of the new physics in terms of what they were familiar with. But 100 years later, perhaps we can dispense with such unhelpful constructs and seek to develop a nomenclature that moves beyond such dualities and treats quantum entities on their own terms. This may help dispel some of the contortions necessary to accommodate concepts that are not only outmoded but counterproductive. Luckily, some progress has been made on this front.

‘Neither waves, nor particles, but quantons’

This section heading is the title of an article (in fact, all of them are) in Nature by Jean-Marc Lévy-Leblond (1988), which is part of a longer program to re-cast the language of quantum theory (Lévy-Leblond 1977, 1981, 1999, 2003). The purpose is to avoid the use of classical concepts like ‘particle’ and ‘wave’ which are unsuited to the quantum realm and instead to treat the quantum entities on their own terms. In particular, he has championed the term “quanton”, proposed earlier by Mario Bunge (1967, p.235):

the things with which the quantum theories are concerned cannot be pictured as classical entities. They are sui generis entities deserving a special name — say quantons — this being why the quantum theories themselves are sui generis. True, the words ‘particle’ and ‘field’ are retained even in the names of some quantum theories, but they designate concepts differing from the classical ones.

Lévy-Leblond has also suggested that, since quantum mechanics is concerned with quantons, it should instead be called simply “quantics”, in direct analogy with acoustics, electronics, optics, thermodynamics, etc. This was no idle statement, since he went on to co-author a university-level textbook which adopts the proposed terminology and uses it consistently throughout (Lévy-Leblond and Balibar 1990).

The 1977 chapter (note its somewhat pointed title) emerged from a symposium marking 50 years since the development of quantum mechanics. In it, he observes that the terminology used is (still) expressed in terms of concepts which are unhelpful for the proper understanding of quantics. Indeed (1977, p.180):

The choice of the terminology … is a very delicate affair. If adequate, it may greatly help the understandability of the crucial points as it may hinder it in the contrary case. Now, the difficulty is that such a choice necessarily relies on abuses of language or metaphors.

In that chapter he critiques a number of widely-used terms that were then (and still are!) found in books on quantum physics. The reader would do well to read it in detail for the full-blown run-down of (often-pointed) criticism (like the title!), of which I can only present some snippets here. The fact that it is now another five decades since this call for terminological change was made suggests that it is well past time to pay proper heed to the basis of these criticisms.

The term “observables” comes into his cross-hairs very early on (p.181):

The word is a direct imprint in quantum physics of the positivism advocated by its founders. To call ‘observable’ any self-adjoint operator associated to a physical property of a system is a multiple nonsense. To start with, … no one will ever actually observe or measure such a highly complex mathematical being.

This refers to the fact that physical properties are represented in widely-used quantum formalism by a certain type of mathematical object (self-adjoint operator) that lives in an infinite-dimensional complex-number-valued mathematical space, known as Hilbert Space. Even if you are not familiar with the specific technical meanings of the terms, you can already get a sense of how the abstract representation is divorced from the 3D reality it is supposedly meant to represent. That such (un-observable!) things were used to describe physical properties called ‘observables’ is due to the positivist dogma advanced by the Copenhagen school of thought which claimed that it makes no sense to assume any physical quantity has a meaningful value or even existence unless and until it is measured or ‘observed’. In this view, anything that cannot be, or isn’t being, ‘observed’ is literally without meaning. Recall from the previous post that John Bell famously rebelled against this term, to the degree that he coined the counter-term “beable” (i.e., be-able) to push back against this stance, and instead used a description that insists that entities or quantities do have such values or existence, even when not being observed. Lévy-Leblond proposes the more general (and, frankly, much more meaningful) term physical properties.

The next term critiqued is “observer”, which he asserts is “without any real theoretical function. It may be suppressed and, along with it, the whole sentence that contains it, without damage.” In other words, it is totally dispensable. Instead, the terms experimenter or measuring apparatus are suggested. This is an important point because the whole notion of an observer being required to complete an observation or measurement leads directly to the (hard to stomach) metaphysical claim that the moon is not there when no-one is looking at it (see previous post). A “measurement” should simply be just another specific type of quantum interaction process between two systems, not requiring the “shifty split” between observer and observed in order to occur.

The term “uncertainty” also gets a serve. In quantics, unlike classical physics, physical properties are generally not subject to being “characterised by a sharp numerical value” (p.182), but by a spectrum of their possible values (“proper values”, which in quantics are generally called eigenvalues – “inherent” or “characteristic” values). In classical physics, where in principle physical properties are assumed to have single sharp values, the fact that these values cannot be measured to arbitrary accuracy (i.e., to a very high number of decimal places, say) introduces what are technically named uncertainties (or, sometimes much less helpfully, errors).¹ The fact that quantum values are not similarly sharp but are characterised by a spectrum of values was conflated with this uncertainty of (classical) measurement. Lévy-Leblond suggests several better terms, such as “spread”, “spectrum width”, “dispersion” or “extension”, and points out that even in classical wave theory it is known that a wave generally has a spectrum of values, yet nobody calls the width of the pulsation spectrum an “uncertainty”. He notes (p.183):

the failure comes from not taking the quantum theory seriously enough, by keeping stuck to classical ideas irrelevant in the quantum domain.

Therefore, the (Heisenberg) Uncertainty Principle or the Uncertainty Relations are recast as simply the Heisenberg Inequalities (see also Lévy-Leblond and Balibar 1998), since they stem from the spectrum widths or dispersions of two non-compatible physical properties inherent in quantum entities, such as, for example, the position $q$ and momentum $p$ mentioned in Part II, to wit, $\Delta q \Delta p \gtrsim\hbar$. There is a similar comparable inequality for energy and time:
$$
\Delta E \Delta t \gtrsim \hbar \; .
$$
The term which comes in for the greatest amount of discussion, however, is “wave-particle duality”, which is revisited in many of the subsequent works cited above, as are most of the others. The two main concepts in classical physics are those of particle (something discrete and localised), and wave (something continuous and extended). These are fundamentally distinct and non-commensurable concepts. While it was natural for the scientists trying to make sense of the newly-emerging physics to use these two well-known foundational concepts, owing to the experimental results that were being found, nonetheless (p.185):

we should come to realize today, that the basic quantum objects are not either waves, or particles, but neither waves, nor particles. They must be described by some new concept, which, furthermore, turns out to be a unique one; several names have been proposed for such a concept, for instance ‘wavicle’ or ‘quanton’. [Here he cites Bunge as his source].

He then gives the example of a cylinder to make the point. A cylinder, when viewed from one perspective (i.e., along its axis) looks like a circle. When viewed from another (i.e., from transverse to the axis) looks like a rectangle. The cylinder is the underlying reality; the circle or rectangle aspects are but partial and incomplete views of it, wholly dependent upon and wholly arising from the context of the observation. Thus (p.186):

‘wave-particle’ duality is no more a correct way to analyze a quanton, than a ‘rectangle-circle’ duality would be to analyze a cylinder.

‘On the nature of quantons’

This incorrect duality thinking is later taken up and extended in the 1999 and 2003 papers via the highly memorable example of the “duckmole”. When the early European settlers in Australia first came across a creature they had never seen before—a semi-aquatic, egg-laying mammal, with a duck’s bill and webbed feet, and a furry body and tail—they named it according to the two concepts that it seemed to them to embody; hence “duck-mole”. It is said that the first preserved specimen sent back to a London museum was initially thought to be an elaborate hoax and that the naturalist who examined it searched for signs of it having been stitched together from different animal parts. And yet, the platypus is not a stitched-together amalgam of a duck and a mole (although in some Aboriginal Dreamtime legends it is the offspring of a duck and a water rat), but rather a legitimate creature in and of itself with a long and well-established fossil history. That it is both egg-laying and a mammal is another way in which it defied these two distinct categories used by 18th-Century naturalists. In much the same way, then, to speak of wave-particle duality is as meaningless as to speak of rectangle-circle duality, or to speak of the platypus as a duckmole. I therefore refer to this meaninglessness arising from the use of incorrect terminology as “the duckmole problem”, which is present whenever one uses classical concept words for the quantum quanton reality.

The main characteristics of classical-mechanical and quantic entities are laid out in the following table (cf. Lévy-Leblond 2003, p.497):

Lévy-Leblond then observes (ibid.):

This double nature of quantons … is the very lesson of quantum physics.

That said, however, depending on the precise form of the experimental setup, one or other of these characteristics may be brought into primary focus while the other becomes secondary. Therefore, in some instances, quantons may be approximately described as particles, while in others approximately as waves, in precisely the same way that a cylinder may be viewed approximately as a circle or approximately as a rectangle depending on the observational orientation. In both cases, the entity is neither of these basic types but something more fundamental whose partial apparent characteristic is an artifact of the method of observation.

‘Quantum words for a quantum world’

In order to counter the tendency to use ill-suited nomenclature, Lévy-Leblond advocates for a number of neologisms in the two later papers (1999, 2003) which I will discuss here in brief (he also revisits some of the terminology we have already examined above). I can only surmise that they are intended to give the existing rectangle-circle duckmole terminology of quantum physics the cylindrical platypus treatment!

First up, since quantum mechanics is now to be called quantics, the appellation “classical” can be dropped from (Newtonian) classical mechanics and it simply becomes mechanics, which he considers a more suitable usage given the etymology of the term.

We found in the previous post on Bell’s Theorem that one of the major experimental discoveries of science is that nature is inherently “non-local” at the quantum level. This gives rise to “entanglement”, which is manifested in the mathematics as the “non-separability” of the individual state-vectors of interacting quantons. Lévy-Leblond suggests that these terms have an unfortunately negative formulation in that they don’t describe quantics by what it is, but rather by what it is not, which is to say, in contrast to what is generally possible in mechanics. He therefore suggests a positive formulation for each characteristic.

Instead of non-locality, the term pantopy is suggested, indicating the property of being in all places at once, from the Greek pan (all) + topoi (places). This accounts for the continuous spatial extension that is a hallmark of quantons.² A quanton, then, is not a point-like entity like a classical particle, but is instead “pantopic”.

In a similar vein, an alternative is suggested for the non-separability of quantons, first named by Schrödinger in a three-part paper (1935) as Verschränkung, and usually translated as “entanglement”; however, the German has cognate meanings of interlocking or overlapping, which seem much more apposite for quantics. For this Lévy-Leblond proposes the (delightful!) term implexity (from the Greek plekó rendered as emplexis, intertwined or knitted), which carries a nice connotation of complexity (and perhaps even perplexity), while also paying homage to David Bohm’s concept of the implicate order that he believed was so inherent in quantics (Bohm [1980] 2002).

The collective behaviour of quantons is described by two different types of statistics, depending on the type of quanton. Bose-Einstein statistics describe bosons, while Fermi-Dirac statistics describe fermions. Mathematically, the distinction has to do with their permutational symmetry properties when multi-quanton states are represented in Hilbert space. Bosons are completely symmetric with respect to permutations and fermions completely anti-symmetric. This latter property of fermions gives rise to the famous Pauli exclusion principle—and, incidentally, makes it possible for stable matter to exist! (Dyson and Lenard 1967; Lenard and Dyson 1968). Given that these properties are present even in systems containing very few (or even just two) quantons, Lévy-Leblond considers it “preposterous” to use the term statistics, which has the connotation of large numbers. Instead (2003, p.499),

a more appropriate wording would seem useful, referring to a specific physical property of the quantons; one could for instance speak of their “permutancy”, even or odd, according to the symmetrical (for bosons) or antisymmetrical (for fermions) character of a collective state under permutation.

Finally, as I was using Feynman’s path integral approach to try to visualise the behaviour of quantons, it became apparent to me that the classical term “trajectory” would simply not cut it for Lévy-Leblond-approved use in quantics. In (classical) mechanics, a zero-dimensional, idealised point-particle traces out a 1-dimensional path in space(time), which is the trajectory. However, a quanton possesses pantopy (i.e., continuous extension) so that the classical concept of an infinitely-thin trajectory is clearly entirely inappropriate.

In the path integral approach, the classical trajectory, obtained in the limit as $\hbar\to0$, is a sort of “spine” around which an infinite number of trajectories all contribute to the propagator $K$, each of which being weighted by the specific phase $e^{iS/\hbar}$ for that individual trajectory (see Part II under Level 2 – Lagrangian). For phases that are ‘close’ to the classical (i.e., stationary) value of the action, the phases will interfere constructively; those that are not will oscillate rapidly and interfere destructively, essentially cancelling each other out far from the classical ‘spine’ of the propagator. This suggests there is a kind of “fuzzy-cylinder-with-no-hard-edge-that-trails-off” of near-stationary trajectories that contribute coherently to the propagator. I wanted a term that captured this idea for the individual path of a single quanton. Seeing as how Lévy-Leblond had not suggested one, I asked Proton’s Lumo AI-assistant, which looked at Lévy-Leblond’s terminology, noted how he chose the words, and proposed several possibilities. The best of these, by far, was perihodon, from peri- (around), + hodos (path), with the -on suffix for quantum entities. Thus, a quanton does not follow a “trajectory”, since this is an idealised zero-width-line classical concept, but rather it follows a perihodon, a fuzzy pantopic “path” through space(time), that may more or less be centred around the classical ‘spine’.

Now, let’s collect these terms together and see how our terminology will change hereafter.

We are now equipped with more precise and better-suited terms to circumvent—or at least minimise—the pervasive duckmole problem which still obtains in modern-day quantum physics. We are now ready, then, to return to the main arc of this series and see how well our old friend the de Broglie-Bohm theory has helped the Coyote come down again towards the ground. We’ll see in the next post that it came awfully close to, but not quite, all the way…

Next: Part VI – The Coyote Has (Almost) Landed – ‘The Fenchurch Problem’