This paper was originally meant to be my private, personal response to a letter by David Nettleingham upon a certain relationship between Dirac’s symbolism and poetry. It has actually spread out on a far more complex trip. Though quite a lot of time is required to read it through, I hope someone will find it enjoyable.
«Three quarks for Muster Mark!
Sure he has not got much of a bark
And sure any he has it’s all beside the mark.»
James Joyce, 1939
«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, 1934
The idea of visualising “things” by means of symbols is one of the main point in physics. What makes this different from any purely mathematical notation is that those “things” are, at least in principle, “measurable” and are referred to as “quantities”, “observables”. On the one hand, experimental investigation consists of gathering numbers within the model under study, on the other of eventually discovering new observables, thus widening the available amount of knowledge. The former approach is rather pertaining to the verification of a well established theory, the latter to the extension to/formulation of a new one. Any symbol is accepted as real if and only if it refers to a measurable quantity, that is if it is possible to assign numbers to it. Fingers are to hands like numbers are to physics, therefore the whole Universe must be seized in a finest net of numbers. In this regard, experimental physics is drastically different from experimental literature, being blind symbols (alias invented words) unacceptable for the mere sake of conveying the playful, maybe unconscious, attitude of some universal mind. I personally agree with what Dirac wrote in 1939: «The research worker, in his effort to express the fundamental laws of Nature in mathematical form, should strive mainly for mathematical beauty. It often happens that the requirements of simplicity and beauty are the same, but where they clash the latter must take precedence». This does not seem to fit to some kind of writing actually. Whether God plays dice or not, Finnegans won’t easily be his prophet, and this by no ways infringes the beauty of questioning Nature by either physics or poetry itself. It is thus quite curious that what was first a funny crasis (qu
estion mark) in Joyce’s Finnegans wake was brilliantly chosen as the name for one particular class of subatomic particles (quark) by Murray Gell-Mann in 1963. In most recent times, we read in a poem: «He, pondered on the nature of infinity./ She, tried to paint infinity on a large canvas./ He dwelt in a singular world of quarks and mesons./ She lived in a physical world of paints and colour./ When he started speaking of ‘charm’ and ‘beauty’/ she thought they’d found at last a common interest,/ until he explained they were ever smaller particles of matter.» (Kate Edwards, Parallel universe)
Physicists often deal with objective correlatives of some abstract schemes, as well as poets do with emotion. Let it suffice to quote the well celebrated case of the elusive wave-particle dualism. In his essay Hamlet and his problems (1919), T. S. Eliot writes: «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.» It may be stimulating to try to interpret this “chain of events” as some sort of poetic rendering 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, a maximal set of commuting observables, a combination of the two which shall be 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», slightly changing Eliot’s words and overlooking some subtle formal details. The contrivance of laying a proper scene of objects is only one further attempt to address the expectation of immediacy and concreteness of emotion. From a purely physical point of view instead, there may exist several maximal sets of observables that give complete (though qualitatively different) descriptions of the same system. A subtler sense of reality is thus requested to account for this weird side of Nature, often regarded as an unacceptable paradox, or superficially ascribed to the lack of sufficient accuracy in our measuring apparatus. «Nature operates in the shortest way possible» (Aristotle, Book V, Physics), not in the most classical way though. It’s probably worth recalling here the old Platonic and Aristotelic theory of mimesis (μίμησις, which shows) contrasted with diegesis (διήγησις, which tells), and stress that “showing” needs a careful selection within the fine frame of experience to find the most proper terms of imitation. In this regard, the risk of misleading the pure symbolism of language is especially run when facts seem to exceed reality, being part of something more intrinsic instead.
In the following pages I am taking up a few features of modern physics, with particular regard to the fairly “non-pictorial character” of its symbolic schemes and of the drawbacks of appeasing our classical moods as well.
Classical and quantum approach are, to a certain degree, mutual exclusive conceptual structures. Each observation involves two confronting systems at least: the phenomenon under study and the measuring devices (including ourselves). According to classical physics we can clearly draw a distinction between the two terms, any disturbance effect being confined to our accuracy. That’s why it makes sense to think that we barely record the contents of outer experience without determining it, nor taking part in it. On the contrary, as it is well explained by Bohr himself (1934): «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.»
I’ll try to discuss this into further detail within a concrete framework, showing what occurs when electrons, coming from very far away with a certain energy, pass through a diffraction slit (whose size is comparable with their de Broglie’s wavelength), since a number of difficulties arise in the interpretation of the experimental results. By shooting electrons one at a time and allowing for the proper delay between two shots, we can collect the position of each one past the slit by placing a suitable screen behind it. The diffraction pattern which goes forming point by point turns out to be identical to the one 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 depend on the size of the slit and the de Broglie’s wavelength. A similar pattern is obtained if photons are used in place of electrons, that is the pattern formed by electromagnetic waves. If electrons strictly behaved as classical particles fired in a straight line through the slit a pattern corresponding to the size and shape of the slit would be expected on the screen, not a diffraction-blurred one. Moreover, even while accepting de Broglie’s point of view, it would be incorrect to interpret this 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, for points are recorded one at a time.
In the case of many electrons, a statistical interpretation of the solution of the Schrödinger’s equation naturally leads to treat as the distribution function, that is the number of recorded hits is maximum where attains its maximum value and minimum where it attains its minimum value, thus accounting for diffraction fringes. If we should repeat the same experiment on a macroscopic scale, shooting gum bullets or whatever else, each hit scored might be “accidental”. This would result in a barely Gaussian distribution with only one maximum lying at the centre, seemingly without fringes: wavelengths of macroscopic objects are so negligible that diffraction curves appear to be smoothed as a consequence of some kind of average performed by the recording device over finest fringes. But how to interpret the cumulative effect of the individual electrons fired? Are we supposed to think that each electron “knows” in advance where it would go if it were in a pure quantum-mechanical ensemble?
Classical physics postulates that each elementary particle (in this case each electron) can be located at some definite point x along a certain direction labelled as x-axis and have a definite velocity (hence a definite momentum ) along the same direction at each instant of time, so that, once the values of external forces are known, its trajectory of motion can be exactly and uniquely predicted, given certain initial conditions. From a quantum-mechanical point of view instead, only the probability that the particle will move in a certain direction with a certain velocity or momentum can be calculated, the accuracy of this prediction being limited by Heisenberg’s uncertainty relation: . Since the numerical value of h is considerably small , these effects become noticeable only in microsystems on an atomic scale.
In classical physics and x are canonical conjugate variables, but the quantum-mechanical operators associated to canonical conjugate variables do not commute. Bohr’s complementarity principle (1928) postulates that two such variables form a complementary pair, namely either both variables are undetermined or one of them is perfectly determined only. We are constrained by the trade-off between conjugate quantities: while performing any act of measurement we, under many aspects, “choose” reality. An experiment designed to measure one variable will inevitably destroy information about the other and vice versa. Broadly speaking, despite head and cross are the two sides of the same coin, the coin rests on one of them at a time. As Bohr wrote in Quantum physics and Philosophy: Causality and Complementarity (1958), «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.»
If we now go back to our experiment, we can properly figure out this quantum weirdness at work. All of the electrons travel towards the target with a certain horizontal momentum p, that is the vertical component of p is “perfectly” known in a classical sense, namely it is zero before the passage through the slit. At this stage we know nothing about the vertical position of each particle: we have fired it from some source very far away, so all we can guess is that it moves along a “straight line” either to hit somewhere, or to pass through the hole and reach the screen. But when some particle registers a score, it informs us about its position with uncertainty ±A/2 at a certain instant of time, having gone through a slit A wide. According to Heisenberg’s relation, we can’t state that the particle’s momentum is perfectly horizontal any more: the uncertainty in the vertical position is inversely proportional to that in the vertical component of the momentum by a factor h. Upon gathering a certain amount of information about the particle’s vertical position, we have lost our original accuracy on the vertical component of its momentum. According to the wave theory, diffraction results from this spreading out, that’s why we prefer to speak of the chance of a particle to travel from the source to the screen, passing through the slit. The smaller A is, the wider the diffraction pattern actually gets.
One final remark is worth before proceeding further in the matter: «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.», Feynman (1966).
By the light of this, the perpetual solitude of things is half put in safety. Nature is well-disposed to answer our questions, but we’re not allowed to put them all at once. Reality is bound up with our probing but its most intimate secrecy is not a technological drawback, it’s rather a fact of Nature in itself. Any phenomenon, no matter how complex it may be, is described by a certain set of quantities to be measured, but, and this is some kind of hardest step forward, we are not allowed to choose and arbitrarily measure the quantities in the set. What is then reality in itself? What is a “thing” from its point of view? Reality seems to escape us and tend to lie partly out of any language, leaving us to handle purely symbolic schemes which surprisingly gives consistent prediction among the many hints and guesses. The quest of the ultimate still point of the whole will probably remain forever as such, and provide the artists of all times with the most evocative inspiration. From a John Burnside’s poem (2009): «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.»
Physicists are not only concerned with numbers, as well as poets are not with words. One bare number can say very little about the bit of reality it refers to, although there are special ones (universal constants, such as h) which often come into play in quite a lot of meaningful occurrences and are regarded as almost landmarks. In this mess of seemingly random values, complicated relationships are to be worked out and symbolised to account for the progress from elementary entities to aggregates and further, scaling up from atoms to planets. The subtle role of the space-time framework, wherein each single point in the Universe seems to be well informed about the Universe as a whole, is the underneath. But how to visualise this all? How to render it through a language other than that of physics? Despite its being fascinating indeed, the drawback of any visualisation may reside in the peculiarity of the terms of the comparison. The joyful idea of an electron spinning like a toy top may be in many ways misleading, as well as the atomic planetary model was. A more accurate approach requires to get acquainted with some abstract sign and use it as the outer tip of the most obscure bulk of the matter. «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.», Graham Farmelo on Paul Dirac (2009).
To illustrate this point of view, let us consider a proton and an electron in the neighbourhood of each other. The relative orientations of their spins can be described with the help of this common handy notation, the first sign referring to the electron and the second one to the proton: ; (electron up and proton up); ; (electron up and proton down); ; (electron down and proton up); ; (electron down and proton down).
These symbols act as vectors in the vector space of the states of the system and any state of the system can be described as a linear combination of these base states, as well as each single word in a line bears different levels of meaning. We have to learn how this particular vector space works and to accept that there’s a certain entity (spin) which can be either up (+) or down (―) for both proton and electron. Everything looks smart then, though we have no idea of how to depict the spin otherwise. Symbols rationalise complex ideas making them writable and handleable. It’s not barely like learning a new language, for these symbols point far beyond our common experience and the more we advance through this landscape, the more reality gets blurred and we tend to identify it with the symbols themselves. Our logic reacts against this merging. We’re so used to give things a name, that we can’t accept names/symbols pointing at “no/thing”, at least in the common sense. Is everything in the Universe part of some language? I sometimes think that there’s much more stuff out there than we can put in any language. For sure the beauty of such a complex construction remains untouched and resides in the simplicity of how it empirically works at first glance.
Finally, I want to spend a few words about the Dirac delta δ, which is not strictly a function but a distribution, whose value is zero everywhere but in one point and whose integral from −∞ to +∞ is equal to one. It is often used whenever the idea of some tall narrow interaction and other abstractions as such come into play (point charges, point masses etc.).
In many applications, the δ is regarded as a kind of limit of a sequence of functions with a spike at the origin. Here again we find the same idea at work: to sum up elementary “things” in order to build something different. In any case, the δ symbol enjoys certain properties of its own, which let us handle it again in a pure abstract manner. As I was informally taught once at University, distributions are such functions which «exist but cannot be “written”» (Gianni Cassinelli). Maybe you can even assume this process of limit to be a further hermeneutic attempt to unfold the ultra-condensed ideas rationalised in the symbol.
If a poem can bring together two seemingly contradictory ideas to show that they are related (thesis, antithesis and synthesis), natural things can be even more interleaved as such: «The opposite of a correct statement is a false statement. But the opposite of a profound truth may well be another profound truth» (Niels Bohr). Poetry does most of its work by accident, whereas the Universe works on its own, perpetually. The Universe is not just the addition of what we see to what is yet unseen, it is also what we can sometimes write, something arcane hence unconfined, something which we have neither words, nor numbers, nor symbols for. A quantum leap in the dark. How magnificent it would be to balance the nothing with the whole! The pattern is in our mind though. Between one line and the next one, one word and silence, one atom and a planet we miss the meaning.
Bohr N., Atomic theory and the description of nature, Cambridge University Press, (1934)
Bohr N., Essays on Atomic Physics and Human Knowledge, Ox Bow Press, (1963/1987)
Burnside J., The Hunt in the Forest, Cape poetry, 2009
Eliot T. S., The Sacred Wood: Essays on Poetry and Criticism, London Menthuen, 1950.
Farmelo G., The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius, Faber and Faber (2009)
Feynman R., The Feynman Lectures on Physics vol. 3, Addison Wesley Longman, (1966)
Joyce J., Finnegans Wake, Penguin Books (1939)