Sedimental symbols

«In our description of nature, the purpose is not to disclose the real essence of phenomena but only to track down as far as possible relations between the multifold aspects of our experience.»
Niels Bohr, 1939

Physicists refer to properties of Nature, alias observables, by means of symbols which differ from purely mathematical notations since they must be experimentally validated, i.e. a definite procedure exists that assigns numbers to them. Nevertheless, phenomena may sometimes appear to be quite counterintuitive or rest on pure abstraction. Let it suffice to quote the well celebrated case of the elusive wave-particle dualism.
In Hamlet and his problems, Eliot wrote: «The only way of expressing emotion in the form of art is by finding an “objective correlative”; in other words, a set of objects, a situation, a chain of events which shall be the formula of that particular emotion; such that when the external facts, which must terminate in sensory experience, are given, the emotion is immediately evoked.»[1] Laying the scene thus addresses the expectation of immediacy and concreteness of emotion.
It may be stimulating to rephrase this “chain of events” in terms of the quantum principle of superposition: «the way of expressing any state of any microscopic system in the form of quantum physics is by finding a “superposition state”; in other words, a set of complex numbers and a maximal set of commuting observables which, combined together, shall conveniently define the formula of any state of that particular system; such that when the external devices, which must terminate in an experimental result, are given, the microscopic system is immediately set into a particular state». While not actually solving any potential excess of abstraction, it may help bridge the gap between symbols and “reality”.
Several maximal sets of observables often exist, each giving a complete, yet qualitatively different, description of the system. This demands a subtler philosophical attitude, not to ascribe what would appear as an unacceptable paradox to the lack of accuracy in the measuring apparatus. Misconceptions especially occur when phenomena exceed conventional logic and the will of figuring everything out in the usual terms is confronted with the fairly non-pictorial character of modern physics.
Classical and quantum approach are, to a certain degree, mutually exclusive conceptual structures. Each observation involves two confronting systems at least: the phenomenon under study and the measuring devices, including investigators. According to classical physics, a distinction between the two terms can always be drawn and any reciprocal interference be confined to our accuracy. It thus makes perfectly sense to think that the contents of outer experiences are recorded rather than determined. Quite the opposite, as expressed by Bohr, «the quantum theory is characterized by the acknowledgement of a fundamental limitation in the classical physical ideas when applied to atomic phenomena. […] Its essence may be expressed in the so-called quantum postulate, which attributes to any atomic process an essential discontinuity, or rather individuality, completely foreign to classical theories and symbolized by Planck’s quantum of action. […] The quantum postulate implies that any observation of atomic phenomena will involve an interaction with the agency of observation not to be neglected. Accordingly, an independent reality in the ordinary physical sense can neither be ascribed to the phenomena nor to the agencies of observation. After all, the concept of observation is in so far arbitrary as it depends upon which objects are included in the system to be observed. Ultimately, every observation can, of course, be reduced to our sense perceptions»[2].
As a first example, let electrons, coming from very far away, pass through a diffraction slit, whose size is comparable with their de Broglie wavelength \lambda=h/p [3]. Several difficulties arise in the interpretation of the experimental results. By shooting them one at a time and allowing for a certain delay between two shots, the position of each one is collected on a screen past the slit. The diffraction pattern which goes forming point by point is identical to that obtained in the case of several electrons shot simultaneously. In addition to the principal maximum at the centre, a number of secondary maxima appear, whose separation depends on the size of the slit and on the electron’s wavelength. A similar pattern is obtained if photons are used in place of electrons, i.e. electromagnetic waves.
If electrons strictly behaved as classical particles fired in a straight line, a pattern, roughly corresponding to the size and shape of the slit, would be expected on the screen, not a diffraction-blurred one. It would be incorrect to interpret the results by depicting each electron as a tiny wave of energy split/spread by the slit, with one part going in one direction and the other(s) in a different one. Each point collected on the screen corresponds to one whole electron: points are recorded one at a time.
In the case of many electrons, a statistical interpretation of the solution \Psi of Schrödinger’s equation would naturally lead to treat |\Psi|^2 as the distribution function depicting the observed diffraction. The number of recorded hits is maximum where |\Psi|^2 attains its maximum value and minimum correspondingly.
Were the experiment repeated on a macroscopic scale, using gum bullets for instance, each hit scored might be accidental, hence resulting in a Gaussian distribution with only one maximum at the centre, seemingly without fringes. Wavelengths of macroscopic objects are so negligible owing to their mass, that diffraction curves are smoothed as a consequence of some kind of average performed over finest fringes by the recording device.
How to interpret the cumulative effect of the electrons individually fired? Does each electron know in advance where it should go if it were part of a quantum-mechanical ensemble?
Classical physics postulates that each elementary particle can, at any instant of time, be located at some definite point x along a certain direction labelled as x-axis and have a definite velocity v_x, hence a definite momentum p_x along the same direction. Provided that the external forces are known, its trajectory of motion can be exactly and uniquely predicted, given certain initial conditions. According to quantum mechanics, only the probability that the particle will move in a certain direction with a certain velocity/momentum can instead be calculated, the accuracy of this prediction being limited by Heisenberg’s uncertainty relation: \Delta p_x\Delta x \approx h[4]. Since the numerical value of h is considerably small, these effects become noticeable only on an atomic scale.
Since p_x and x are canonical conjugate variables, their quantum-mechanical operators do not commute. Bohr’s complementarity principle (1928) postulates that two such variables form a complementary pair, i.e. either they are both undetermined or one of them is “perfectly” determined only. A trade-off between conjugate quantities must be faced: each act of measurement selects reality. Any experiment designed to measure one variable will inevitably destroy information about the other. «Within the scope of classical physics, all characteristic properties of a given object can in principle be ascertained by a single experimental arrangement, although in practice various arrangements are often convenient for the study of different aspects of the phenomena. In fact, data obtained in such a way simply supplement each other and can be combined into a consistent picture of the behaviour of the object under investigation. In quantum mechanics, however, evidence about atomic objects obtained by different experimental arrangements exhibits a novel kind of complementary relationship. Indeed, it must be recognized that such evidence which appears contradictory when combination into a single picture is attempted, exhaust all conceivable knowledge about the object. Far from restricting our efforts to put questions to nature in the form of experiments, the notion of complementarity simply characterizes the answers we can receive by such inquiry, whenever the interaction between the measuring instruments and the objects form an integral part of the phenomena.»[5]
Let the vertical direction in the above-mentioned experiment be the x-axis and the slit be A wide. After being fired, each electron travels horizontally, along a straight line, towards the target with a certain momentum. If initially p_x=0 “exactly” (\Delta p_x \rightarrow 0), nothing (\Delta x \rightarrow \infty) about the vertical position x is known: the particle will either hit somewhere about the slit, or pass through it and reach the screen. As soon as a score is registered, then an electron must have passed through the slit and its position, at some instant of time, must, in principle, have been known with uncertainty \pm A/2. Out of Heisenberg’s relation: \Delta x \approx A/2 \rightarrow \Delta p_x \approx 2h/A. Owing to this amount of information about the particle’s vertical position, the original accuracy (\Delta p_x \rightarrow 0) on px is lost and since \Delta p_x \approx 2h/A \ne 0, the trajectory has spread out. So, the smaller A, the wider the diffraction pattern. Physicists refer to this phenomenon as the chance of a particle to travel from the source to the screen, passing through the slit.
If Heisenberg’s principle were not at work in the Universe, the fact that the detected electron has passed through the slit (i.e. its position has become known with a given uncertainty at that time) would not effect its momentum and the scores would approximately collect within an A-wide range. «Sometimes people say quantum mechanics is all wrong. When the particle arrived from the left, its vertical momentum was zero. And now that it has gone through the slit, its position is known. Both position and momentum seem to be known with arbitrary accuracy. It is quite true that we can receive a particle, and on reception determine what its position is and what its momentum would have had to have been to have gotten there. That is true, but that is not what the uncertainty relation refers to. It refers to the predictability of a situation, not remarks about the past. It does no good to say “I knew what the momentum was before it went through the slit, and now I know the position,” because now the momentum knowledge is lost. The fact that it went through the slit no longer permits us to predict the vertical momentum. We are talking about a predictive theory, not just measurements after the fact. So we must talk about what we can predict.»[6]
The whole set of observables any phenomenon consists of cannot be investigated as such to an arbitrary degree of accuracy. Contrary classical expectations, objects do not posses definite properties at all times. The maximal knowledge of all of its parts is not allowed. As previously discussed in the case of x and p_x, canonical conjugate variables affect each other. This is a fact in itself not a technological factor. Symbolism serves to elaborate predictions, but the conventional picture of reality as a still bedrock at full disposal cannot be restored and falls beyond language.
Space-time offers the natural framework for all the relationships scaling up from atoms to planets. But how to visualise it? «Someone might call it ether, but for you/ the light at the end of the tunnel is never quite air,// and breath is a shape that sails out over the rooftops,/ into the lights off the quay and the tethered yawls.»[7] Again, the drawback of any visualisation resides in the peculiarity of the terms under comparison. The joyful idea of electrons spinning like toy tops is, in many ways, misleading and so was it for the atomic planetary model. An accurate approach requires to get acquainted with abstract signs and to manipulate them.
To illustrate this point with a last example, let a proton and an electron be in the neighbourhood of each other. The relative orientations of their spins, which can be either up or down, are usually described with the help of the handy ket notation: |+ +>; (electron up, proton up), |+ ->; (electron up, proton down) and so forth. These symbols act as a base in the vector space of the states of the system, hence any state can be described as a linear combination of them. Once the properties of vector spaces are known and the existence of an entity called spin is verified, complex problems may be rationalised and worked out without even the idea of how to depict the spin actually. Kets do not stand for shortened sentences or words. They extend language at a level at which phenomena cannot be figured out otherwise. Operating with kets serves to decompose systems into fewer elements to be experimentally questioned. «The conception of the objective reality of the particles has thus evaporated in a curious way, not into the fog of some new, obscure, or not yet understood reality concept, but into the transparent clarity of a mathematics that represents no longer the behaviour of the elementary particles but rather our knowledge of this behaviour.»[8]
Do physics and poetry disturb the Universe that tacitly works? Their repeated attempts to trace and record numbers and words are beside the point, for, while doing so, they are part of the whole work.
«If you think of a poem, you can think of it as the most super-charged kind of language; the way you compress meaning into a very, very brief area on the page. Dirac was producing equations that had that kind of concision. And you can then unpack them just as you re-read a Shakespeare sonnet, see more and more in it, see more and more elegance. Same with the Dirac equation. You find an equation there, and you keep finding things that were not obvious on first reading. In fact Dirac actually said that the equation was smarter than he was, because it actually gave more stuff out than he put into it.»[9]

Federico Federici

Footnotes

[1] Eliot, Thomas Stearns: Hamlet and his problems, in: The Sacred Woods: essays on poetry and criticism, London 1920, p. 92.
[2] Bohr, Niels: Atomic theory and the description of nature, Cambridge 1934, p. 54.
[3] h=6.6\times 10^{-34} \ Js is Planck’s constant; p=\gamma mv is the momentum, \gamma\rightarrow 1 if v/c \rightarrow 0, being c=3.0\times 10^8 \ m/s the speed of light in vacuum.
[4] \Delta p_x and \Delta x: uncertainty in momentum and position along x-axis respectively.
[5] Bohr, Niels: Essays 1958-1962 on atomic physics and human knowledge, New York 1963, p. 4.
[6] Feynman, Richard: The Feynman Lectures on Physics vol. 3, New York 2010, p. 2-5.
[7] Burnside, John: The hunt in the forest, London 2009, p. 3.
[8] Heisenberg, Werner: The representation of nature in contemporary physics, in: Deadalus n.87, Cambridge 1958, p. 95.
[9] Farmelo, Graham: The strangest man: the hidden life of Paul Dirac, quantum genius, London 2009.

5 thoughts on “Sedimental symbols

  1. Excellent reading and expression… I especially understand your philosophies on poetry in amongst your thinking – “meaning compression” (powerful) – the strength of poetry. And I’m not a physical science major!

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  2. I have browsed some Online Etymological Dictionary and they all state that the term quark was coined by Gell-Mann himself: not any hint at some other usage of the term.
    Finally, I have found this, which may glue the puzzle:
    “In 1963, when I assigned the name “quark” to the fundamental constituents of the nucleon, I had the sound first, without the spelling, which could have been “kwork”. Then, in one of my occasional perusals of Finnegans Wake, by James Joyce, I came across the word “quark” in the phrase “Three quarks for Muster Mark”. Since “quark” (meaning, for one thing, the cry of the gull) was clearly intended to rhyme with “Mark”, as well as “bark” and other such words, I had to find an excuse to pronounce it as “kwork”. But the book represents the dream of a publican named Humphrey Chimpden Earwicker. Words in the text are typically drawn from several sources at once, like the “portmanteau” words in “Through the Looking-Glass”. From time to time, phrases occur in the book that are partially determined by calls for drinks at the bar. I argued, therefore, that perhaps one of the multiple sources of the cry “Three quarks for Muster Mark” might be “Three quarts for Mister Mark”, in which case the pronunciation “kwork” would not be totally unjustified. In any case, the number three fitted perfectly the way quarks occur in nature.”
    Can you quote some other 19C writer who used “quark” as the sound of birds? I would like to embody these information in the second darft of my paper.
    Thanks in advance if your help.
    F.

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  3. Nice annotation! I will correct the second line in my citation according to what you suggest and will explore the reasons why Gell-Mann chose that name (and why he eventually tricked the true explanation…).
    In my Oxford Dictionary there’s no trace of any old meaning of the term “quark”. I am going to check it in some ethimological one.
    I have much appreciated your contribution.
    Thanks again
    F.

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  4. Contrary to Gell-Mann’s amusing but idiosyncratic view, ‘quark’ is not a Joycean invention, but an imitative English word, used by 19C writers for the cry of an animal (most of the OED citations give birds). The second line of your citation should read ‘Sure he hasn’t got much of a bark’. For what it’s worth, I should add that the Penguin edition retains all the worst errors of the pre-corrected 1939 first printing of FW, some of which were subsequently corrected in the Faber/Viking editions of 1950 and 1964.

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