THE MEANINGS OF PLANCK’S CONSTANT JEAN-MARC LEVY-LEBLOND Physique théorique, University of Nice Abstract. This paper purposes to discuss the many roles played by Planck’s constant in our understanding of quantum theory. It starts by retracing the rise of h to the status of a universal constant, which was certainly not obvious to every one at the beginning. The rather advanced views of Planck himself in that respect are stressed, particularly concerning the meaning of h as a ‘quantum of action’. It is shown that this point of view, applied to statistical considerations, leads to a very interesting expression of the Pauli principle (Heisenberg-Pauli inequalities). The orthodox Copenhagen interpretation, which sees h as a ‘gauge of indeterminism’, is then challenged and reformulated in terms of a standard of quanticity. But the deep meaning of Planck’s constant (in keeping with that of any fundamental constant), is to underly the synthesis leading to the new and specific concepts of quantum theory. It is then pointed out that the change from Planck’s constant h to Dirac’s  , is much more than a convenient shortcut, as it enables one to obtain correct quantum order-ofmagnitude estimates by heuristic considerations. Finally, it is stressed that the role of Planck’s constant is by no means limited to the microscopic domain, and examples are given of its manifestations in the macroscopic (human scale) and megascopic (astronomic scale) domains. Contents 1 2 3 4 5 6 7 Introduction The rise of h to universality The quantum of action Gauge of indeterminism or standard of quanticity? The concept synthetizer From Planck’s to Dirac’s constant The macroscopic level JMLL/ ‘The meanings of h’, Pavia 2000 1 1 Introduction The value of h is 6,…×10-34 in the SI unit system — or 2π in the trade system (  = 1). The history of h starts with Planck’s (quantum) jump in 1900, comes into full light in 1905 with Einstein, and takes momentum in 1924 with de Broglie. The role of h is crucial in many areas of physics, from particle physics to solid state. But what is to be said of its meaning, or rather, as I wish to stress the plural, of its meanings ? Questions of meaning are often dismissed in a rather patronizing way by working physicists as “mere philosophy”, as opposed to serious research, that is, theoretical formalism and experimental results. Yet, the purpose of science is (or should be) not only to predict or verify numbers and facts, but to assess and elaborate ideas, without which there is no real “understanding”. It is crucial in that respect to recognize that almost any theoretical notion of physics, and certainly each important one, is prone to various interpretations and is endowed, in the course of time, with widely different meanings, or even with conflicting ones. This can be seen most clearly in the terminological diversity associated with some of these notions, and the ensuing confusion. A worse situation still is that of ancient ways of speaking, kept just out of mental laziness, while their initial strict meaning has long been forgotten. Think for instance of the so-called ‘displacement current’ in Maxwell equations (which was indeed for Maxwell — at the beginning — a true electric current in the ethereal substance, while we now think of it as a genuine field term), or consider the so-called ‘velocity of light’ (the fundamental role of which as a scale standard for the structure of space-time would amply justify a less specific name — for instance, Einstein constant). The importance of these remarks lies in the obvious but overlooked fact that neglecting the diversity of meanings and failing to discuss it explicitly may have strong adverse effects, not only on the spreading of knowledge (in teaching — see the example of the displacement current, or popularizing science — see the example of the velocity of light), but on its development as well; one does not know in advance which of the competing views, if any, will show the greatest fecundity. I now come to the case of Planck’s constant, for which I will discuss, without any claim to completenesss, a handful of the very different meanings it has been given. The argument will be set up in the framework I have proposed for understanding and classifying physical constants in general1. One has first to distinguish between fundamental constants, which are to be taken as basic elements of our knowledge, and derived, or phenomenological ones, which we know to be explainable (in priciple at least) from the fundamental ones; the electron charge or Newton’s constant clearly belong to the first category, while Rydberg’s constant or the proton mass (related to the quark masses and coupling constants) belong to the second one. The fundamental constants in turn may be classified under three headings: 1) specific properties of particular objects (say, the electron mass), 2) characteristics of whole classes of phenomena (say, the elementary electric charge which measures the strength of electromagnetic interactions), 3) universal constants which enter universal theories, ruling all physical phenomena (say, the limit velocity). I should stress that these distinctions should by no means be taken as giving a closed and atemporal classification; quite the contrary, they permit a detailed discussion of the historical changes in the role and meaning of the constants, as will now be seen. 2 The rise of h to universality Planck’s constant nowadays is clearly taken as a fundamental constant, and even a universal one, since quantum theory is thought to be universally valid; accordingly, there is no doubt that h in principle enters the treatment of any physical phenomenon. In many cases, a classical approximation leading us to forget its underlying presence is possible, but that is 1 Jean-Marc Lévy-Leblond, ‘On the Conceptual Nature of the Physical Constants’, Riv. Nuovo Cimento 7 (1977), 187. JMLL/ ‘The meanings of h’, Pavia 2000 2 another question (and a difficult one at that, with some surprises in store as will be seen at the end of this paper). Now, this universality was far from characterizing h at the beginning of its life. The constant h first appeared in Planck’s paper on the blackbody spectrum. It is perhaps not without interest to ask first why Planck chose to denote his new constant by this very letter; such questions, concerning the choices of symbols in the formalism of physics, certainly deserve more attention than they usually receive, as the answers could shed some light on the intellectual (and sometimes psychological) processes at the core of theoretical innovation. In the present case, it must be remarked that in its first printed occurrence (in october 1900), Planck’s formula is written in terms of the wavelength : (1) eλ = Cλ −5 (e c' / λT − 1) −1 , with constants C and c’ (taken from the previously known Wien’s formula) having a purely empirical meaning. It will take a few weeks for Planck to discover a theoretical explanation of this formula2. He considers the entropy of a ‘resonator’, starting from the statistical formula S = k lnN (2) where the now so-called ‘Boltzmann constant’ appears in physics for the first time, and proceed to compute S under the hypothesis that the exchange of energy between radiation and matter is quantized with an ‘Energieelement’ given by: ε = hν , (3) leading at last to the formula (4) uν = 8π hc −3v3 (e hν / kT − 1)−1 . It is to be stressed that Planck introduced the two constants h and k . Later on, he would repeatedly complain that ‘Boltzmann’s constant’ was in fact the other Planck’s constant. In any case, the simultaneous appearance of h and k sheds some light, although a rather trivial one, upon the process of their denomination; it looks as if Planck (who seems never to have given any explanation whatsoever on this point) just chose the two first (and related) letters which did not yet bear too heavy a symbolic role in the accepted conventions of physics. What is more important for the theme of our discussion, is that Planck, commenting upon the expression (3) of the ‘energy element’ and the general formula (2) for the entropy, explicitly stated that “Hierbei sind h und k universelle Constante.” Contrarily to the usual view of Planck as rather old-minded and reluctant towards the modern aspects of quantum theory (an assessment which could be argued to reason considering his positions in the following decades), this statement about the universality of h and k certainly proves a more advanced stand than that of most of his contemporaries, for whom the fundamental nature of h was far from obvious, not to speak of its universality. Indeed, it must be remembered that for quite a few years, h appeared only in considerations related to radiation theory, from the blackbody spectrum (Planck, 1899-1900) to the photoelectric effect (Einstein, 1905). It was only natural, then, to think of h as specifically ruling electromagnetic phenomena. In fact, not until Einstein’s 1907 paper on the specific heat of solids, giving a first example of quantum statistical theory, did extend effectively its realm beyond radiation theory. It is all the more interesting to note, first, that Einstein himself, in his 1905 paper3, uses neither the expression ‘Planck’s constant’ (which was used for the first time rather late, probably by Millikan around 1915), nor even a specific symbol! Although he of course refers to this original paper by Planck, he does not write the ‘Plancksche formel’ (4), but expresses it in the more empirical form: (5) ρν = αν 3 (e βν / T − 1)−1 where neither h nor k appear. It is even more striking to look at Einstein’s formula for the photoelectric effect, since he writes: 2 Max Planck, ‘Ueber das Gesetz der Energieverteilung in Normaspektrum’, Ann. d. Phys. 4 (1901), 553. 3 Albert Einstein, ‘Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristichen Gesichtspunkt’, Ann. d. Phys. 17 (1905), 132. JMLL/ ‘The meanings of h’, Pavia 2000 3 “Die kinetische Energie solcher Elektronen ist (R / N)βν − P .” (where P is the extraction potential). This hiding away of h can certainly be taken as displaying at least some skepticism concerning the relevance of the theoretical derivation by Planck, as well as about the fundamental role of the constants h and k . The idea was rather common in these days that h was not a fundamental constant, expressing the inception of a radically new theory, and that it could be explained away by some mechanical model leading to a more or less classical explanation of the quantization of energye exchanges. Born later on recalled the ‘apple-tree model of Planck’s quantization formula’, an admittedly farcical model, which was discussed at the time4. Imagine an appletree with the property that the stems of the apples decrease with their height above ground; more specifically, let us suppose that the length l of the stems is inversely proprtional to the −2 square of their height H, that is, l ∝ H . Then, the frequency of free oscillations of the apples considered as pendula is ν = (2π ) −1 (g / l)1 / 2 ∝ H . Suppose now that the tree is shaking in the wind. Pressure waves with frequency ν will excite specifically the oscillations of those apples which lie at the corresponding height H ∝ ν . These apples only will fall to the ground, transferring to it an energy E = mgH ∝ ν ; in other words, the exchange of energy between the tree and the ground is quantized in apple-units with an energy E = hν , where the constant h may be expressed in terms of more fundamental quantities (the apple mass, the acceleration of gravity, and the constant of the length-height relationship). In a more serious vein, there were several quite explicit attempts to relate h with suposedly more fundamental magnitudes of the atomic realm, like the Haas model based on the Thomson atom (which was favorably quoted by Sommerfeld as late as 1911). Einstein himself made a start in the same direction in 1909. He remarked that h had the same physical dimensionality as the combination e 2 / c and looked upon this coincidence as indicating the possibility of a specifically electromagnetic mechanism5 (note that this happened the very same year when he himself showed, through his theory of the specific heat of solids, h to have a general relevance, beyond purely electromagnetic phenomena!). Lorentz, in a letter dated 6 May 1909, expressed serious doubts, based upon the numerical gap (three orders of magnitudes) between the two expressions, writing “I could imagine to have a factor of 4π or so intervening, but a factor 900, that is really too much.” To what Einstein carefreely answered that “a factor like 6(4π )2 is not so extraordinary.” But unless someone produces a theory of the fine structure constant α yielding its numerical value, thus giving h as equal to 2πα (e 2 / c) , but above all explaining its seemingly ubiquitous role (beyond electromagnetism), we have better think of Planck’s constant, in the very terms of Planck himself, as a universal constant. However, within this general conception, there remains a variety of possible views, as will now be seen. 3 The quantum of action A first concrete understanding of h as a universal constant was put forward by Planck himself. He had for long noticed that h had the dimensionality of classical ‘action’. In his contribution to the 1911 Solvay conference, he introduced the very fecund idea that h in fact defined the magnitude of irreducible quantities of action, “elementaren Wirkungsquanten” in Planck’s original terms. There was born the wording ‘quantum of action’ for h . This new vision of quantization was in turn expressed as the existence of elementary areas in phase space, that is, cells A with finite extension: (6) ∫∫A dpdq ≅ h . Planck was clearly aware that this point of view called for a renunciation to all attempts at classical interpretations of h (of the type mentioned above). He wrote, in his Solvay paper: “The framework of classical dynamics, even if combined with the Lorentz-Einstein principle 4 5 See Max Jammer, The Conceptual Development of Quantum Mechanics (McGraw Hill, 1966). Albert Einstein, ‘Zum gegenwärtigen Stand der Strahlungsproblem’, Phys. Zeits. 10 (1909), 185. JMLL/ ‘The meanings of h’, Pavia 2000 4 of relativity, is too narrow to account for all those physical phenomena which are not directly accessible to our coarse senses.”6 Planck’s idea was seized upon by Sommerfeld, who, inspired by what he called the “most fortunate” naming of h as a quantum of action, related it more precisely to Hamilton’s action function which enabled him to develop the Old Quantum Theory. Later on, the formal development of quantum theory was to be built upon another, more general, meaning of h (see below). Nevertheless, despite the limitations of the ‘quantum of action’ point of view, it has not lost its fecundity, and can be put to good use, at least as a heuristic tool for understanding some aspects of quantum behviour. As an example, it leads to a very picturesque and useful way of expressing the Pauli principle, as I now proceed to show. The Heisenberg-Pauli inequalities Consider a system of N one-dimensional particles, classical ones to start with. The state of the system can be represented at a given time, by a collection of N points in the twodimensional one-particle phase-space (p,q). Contemplate now a system of N quantons (quantum ‘particles’). In accordance with Planck’s idea about the quantum of action, an individual state can no longer be represented by a point, but is to be associated to an extended cell with an area of order h , or rather  ; indeed, as will stressed below, one should definitely use the Dirac constant  := h / 2π as soon as numerical evaluations are contemplated. Denoting by Δp and Δq respectively the extension of an individual cell in momentum and position respectively (in order of magnitude), the expression (6) for the minimal area of the cell constitures but a rewriting (and an interpretation) of the Heisenberg inequality: (7) ΔpΔq   (where the symbol  denotes inequality up to a constant numerical factor of order unity). The collective state then consists of an ensemble of N such cells. Let us suppose now that the quantons are identical fermions. The effect of Fermi statistics may be expressed by the Pauli exclusion principle which requires that no two individual states be identical. It is traightforward to translate this constraint as requiring the N cells now to be disjoint (Figure 2c). One sees vividly how the Pauli principle requires the collective state of the system to have a much greater minimal extension in phase-space. Indeed, for a system with N quantons not obeying the Pauli principle, the inequality (7) for the individual regions in phase-space applies as well to the total region, since nothing prevents the individual states to be one and the same. For identical fermions however, the total region, consisting of nonoverlapping cells, must have a minimal area at least equal to N times that of an individual cell, leading to an inequality characterizing the collective extensions in position and momentum of a system with N identical fermions in one dimension: (1-D) (8) ΔpΔq  N  . I call it the ‘Heisenberg-Pauli inequality’. It can easily be generalized to three (or more…) 3 dimensions. The individual cells now have a minimal volume  and the collective state with N identical fermions must occupy a region of volume at least N3 . Hence the HeisenbergPauli inequality in three dimensions: (3-D) (9) ΔpΔq  N 1/ 3  . It should be emphasized that this heuristic derivation may be given a fully rigorous form, and that the inequality (9) may be proved formally7. The best way to understand the HeisenbergPauli inequalities (8) and (9), is to compare them to the standard Heisenberg inequality (7), and to consider that the effect of the Pauli principle is to replace the fundamental quantum 1/ d constant  by an effective fermionic one,  (d)  , where d is the dimensionality of f (N) = N space (which shows, by the way, that the effect of the exclusion principle is all the more powerful, the lower is the dimensionality of space). 6 Max Jammer, op. cit. Jean-Marc Lévy-Leblond, ‘Generalized Uncertainty Relations for Many-Fermion Systems’, Physics Letters 26A, 540 (1968). 7 JMLL/ ‘The meanings of h’, Pavia 2000 5 Let us put to use the Heisenberg-Pauli inequality in a simple example, namely, the evaluation of the ground-state energy of an N electron atom. We start with the hamiltonian N N p2 Ze2 N k −1 e 2 H = ∑ k −∑ +∑∑ , (10) k =1 2m k =1 rk k =1 l =1 | rk − rl | where the notations are obvious enough. Denote by p˜ and r˜ respectively the average, orderof-magnitude wise, of the momentum and radial distance of the electrons in a stationary sate of the atom. The energy of the state may then be written as: 2 ˜p2 2 e , (11) E≈Z −Z 2m r˜ the terms in the right-hand side being taken up to a numerical constant [the potential term results from the combined coulombic attraction of the electrons by the nucleus and the mutual repulsion of the electrons, both being, within a constant, of order Z 2e 2 / r˜ , but the first one dominating, as shown by the very existence of atoms, hence the negative sign in (11)]. If quantum theory is now invoked but no supplementary assumption is made, its effects may be simulated by the Heisenberg inequality (7), leading to: p˜ 2 E2 (12) E Z − Z2 p˜ . 2m  Minimizing this expressions with respect to p˜ , we obtain an evaluation of the ground-sate energy: ? me 4 (13) E0 ≈ − Z3 2 .  This is wrong, as shown for instance by the fact that it predicts an average size for the atoms (corresponding to the value of the electronic radial distance in the ground state (13)) ? (14) r˜0 ≈ Z −1  2 me2 , which would make the atoms shrink very fast with the number of the electrons, contrariwise to evidence. The reason of the discrepancy clearly is our neglect of the fermionic nature of the electrons. We now simply account for the Pauli principle by replacing Planck’s constant in 1/ 3 the result (13) by its effective fermionic value  (3)  , which lead to the correct estimate: f = Z 4 me (15) E0 ≈ − Z 7/ 3 2 .  The rather unexpected exponent 7/3 usually is obtained from elaborated calculations, for instance in the Thomas-Fermi approximation. The present derivation is both simpler and more transparent. The same reasoning may be applied to give a heuristic discussion of the all important question of the saturation of Coulombic forces in macroscopic matter (namely, the constancy of the binding energy per particle with respects to the size of the system)8, shedding some light on the very sophisticated discussion by Dyson, Lenard, Lieb, and Thirring. This type of argument also finds applications in the discussion of gravitationally bound systems, explaining the transition from rocks (dominated by Coulomb cohesive forces) to planets (dominated by Newton cohesive forces), and even to white dwarfs9,10. We will return later on to the macroscopic manifestations of Planck’s constant. 4 Gauge of indeterminism or standard of quanticity? 8 7. Jean-Marc Lévy-Leblond & Françoise Balibar, Quantics (Rudiments) (North-Holland,1990), chapter 9 Jean-Marc Lévy-Leblond, ‘Mécanique quantique des forces de gravitation et stabilité de la matière’, J. de Phys. 30 (1969), C3-43. 10 Jean-Marc Lévy-Leblond, ‘Quantum Theory at Large’, in E. Beltrametti & JMLL eds, Adv. in Quantum Phenomena, (Plenum, 1996), pp. 281-295. JMLL/ ‘The meanings of h’, Pavia 2000 6 With the advent of the new quantum mechanics, and in the wake of its discussions by the Copenhagen school, a novel meaning was attributed to Planck’s constant. At the 1927 Solvay conference, Bohr and Heisenberg presented a paper on matrix mechanics and the probability interpretation, in which they delivered the following pronouncement: “The real meaning of Planck’s constant h is this: it constitutes a universal gauge of the indeterminism inherent in the laws of nature owing to the wave-particle duality.” 11 I have no time here to delve into a detailed criticism of the notions of indeterminism and duality. Let us only note that they carry a rather pessimistic view of quantum mechanics, giving an essentially negative role to Planck’s constant. Indeed, it is still rather common, in discussions of the Heisenberg inequalities, formulated in terms of alleged ‘uncertainties’, to view h as a quantitative measure of the limits imposed to our knowledge of nature. This assessment does not fit well, to say the least, with the positive use of the Heisenberg inequalities in heuristic approaches (as exemplified above), which, far from leading us to stumble against some alleged intrinsic limitations of our knowledge, enable us to gain some intuitive feeling of quantum phenomena12. More generally, the subtle, specific and constructive roles of h in many areas of modern quantum physics, should relegate the emphasis on a supposed quantum indeterminism to the historical record. For it is only from a classical point of view that the ‘wave-particle duality’ and its associated ‘indeterminism’ make sense. Once we fully accept quantum ideas, the whole terminology loses its meaning. A modern formulation would rather consider h as a ‘standard of quanticity’, a universal standard against which to assess the necessity of putting quantum theory to work, a signpost of the limits of validity of classical approximations. The criterion is the following : when considering some phenomenon, evaluate the relevant quantities with the dimension of an 2 -1 action ( ML T ) and compare them with Planck’s constant ; if they are much larger than h , then a classical theory will yield a good approximation, if not, the recourse to quantum theory is compulsory. Note that this criterion by no means is equivalent to setting up a distinction between microscopic and macroscopic quantities. This is clearly emphasized by Feynman13, when he discusses how the so-called Heisenberg principle (not a principle, in fact, since it derives from the basic formalism of quantization) ‘protects’ quantum theory by virtue of the universality of h . Indeed, he shows that a contradiction would appear if one were to apply the Heisenberg inequalities only in the microscopic world (say, to the electron in a two-slit experiment) and not in the macroscopic one (say, to the screen in the same experiment). And we now know many macroscopic quantum effects, which make the point obvious (see also below). 5 The concept synthetizer Traditionally, the Planck-Einstein relationship E = hν (16) has been interpreted in the spirit of the so-called ‘wave-particle duality’, as associating an energy (mechanical quantity linked to the particle aspect) with a frequency (vibratory quantity linked to the wave aspect). From a modern point of view, it rather appears that this relationship in fact expresses the emergence of a new and original concept, which trancends both the classical concepts of energy and frequency. This is the role generally played by universal contants, which may be best characterized as concept synthetizers 14. Consider for instance the Joule constant J appearing in the formula W = JQ expressing the mechanical equivalent of heat. Its role goes far beyond that of a simple unit conversion coefficient to which it is often unduly relegated; indeed, this very formula upholds the general concept of energy, surpassing the previously unrelated notions of work and heat. The same could be said 2 of the Einstein constant c2, which, through the formula E = c m (purposely written here in an unfamiliar guise), gives rise to a new concept of ‘relativistic’ energy, beyond the previously 11 See Max Jammer, The Philosophical Development of Quantum Mechanics (Wiley, 1994), p. 114. Ref. 8, chapter 3 13 The Feynman Lectures in Physics, vol. 3, chapter 2, (Addison-Wesley). 14 See ref. 1. 12 JMLL/ ‘The meanings of h’, Pavia 2000 7 separate notions of mass and energy. From that point of view, the relationship (16) is not to be interpreted as linking two classical concepts, but rather as transcending them through their synthesis, and establishing a new concept with a broader scope. In fact, a new name could profitably have been given to this new concept, stressing its originality. That the quantum energy thus constructed differs from the classical energy is sufficiently shown, for instance, by the fact that a quantum system is not, in general, characterized by a single and well-defined value of its energy, but rather by a whole numerical spectrum. In other words, energy and frequency, through the relationship (16) appear as but two particular facets of a more general notion, each of which being the only visible one from either one of two specific viewpoints (the wave and the particle aspects, respectively). The present interpretation reverses the traditional one, as it stresses, instead of a duality of classical objects (wave vs particle), the unity of quantum objects — which would deserve a specific name, ‘quantons’ (as suggested by M. Bunge) probably being the best proposal15. A universal constant in general does not synthetizes a mere pair of notions, but unifies whole theoretical structures, and brings about several syntheses. This is certainly true for Planck’s constant, since, besides bringing together energy and frequency as time-related notions, it also ties the two space-related notions of momentum and wave-number according to the de Broglie relationship: (17) p = hk By the way, from this modern point of view, one may (naively) wonder why it has taken twenty years to go from the formula (16) to its homolog (17). A further step may be taken, remembering the existence of yet another conserved quantity in classical physics, besides energy and momentum, namely angular momentum. In the same way as, through the constant h , energy is linked to time frequency and momentum to space frequency, it is most natural to conjecture the relationship (18) L = hµ where L is some component of the angular momentum, and µ the angular frequency of a rotationally periodic (or rather harmonic) phenomenon. That is to say, µ = 1/ α if α is the angular period. Now comes the bonus. Indeed, contrarily to space and time, which are represented by the real line, angles define a compact set, the circle. This means that an angular period cannot be arbitrary, but has to be a submultiple of the full-turn (daisies have an integral number of petals), i.e. α = 2π / m where m is some integer. Finally: (19). L = m , m integer This line of reasoning may be extended to allow for half-integer values as well16. That the quantization of the angular momentum may be obtained from such a simple and deep argument clearly shows the interest of recognizing the nature of h as the universal quantum synthetizer. The present point of view gives a solid and clear basis to the now customary choice of units in quantum theory, whereby Planck’s constant, or rather the Dirac’s constant  (see below), is taken as unity. Far from being a mere convenience, this choice, exhibitng the identification of energy and frequency, entails the acknowledgement that there is but one single quantal notion of energy-frequency. It is only from a macroscopic and limited point of view that our dealing with objects approximately described as particles or waves has led us to set up two separate notions, energy and frequency, which we now recognize as particular aspects of a more general concept. 6 From Planck’s to Dirac’s constant It seems that the introduction of the constant  := h / 2π is due to Dirac, in his book, Principles of Quantum Mechanics. By that time in effect, the dividing factor 2π had made its appearance in various expressions, from the rigorous form of the Heisenberg inequalities to 15 Jean-Marc Lévy-Leblond, ‘Quantum Words for a Quantum World’, in D. Greenberger & al., Epistemological and Experimental Perspectives on Quantum Physics (Kluwer, 1999), pp. 75-87. 16 Jean-Marc Lévy-Leblond, ‘Quantum Heuristics of Angular Momentum’, Am. J. Phys. 44 (1976), 719 , and ref. 8, chapter 2. JMLL/ ‘The meanings of h’, Pavia 2000 8 the quantization of angular momentum, so that the new notation brought a welcome simplification. What I want to stress here is that there is more to it than a handy convention. A time periodic phenomenon is specified by its period T. However, this duration is rather too large if one is to define a characteristic time for the phenomenon, that is, the order of magnitude of a time interval during which the phenomenon changes ‘notably’, passing from small to large, for instance. For a harmonic phenomenon, the most natural choice corresponds to a change of phase of the order of 1 radian — which is indeed a characteristic angle, neither too small (a few degrees) nor too large (a full turn or so). The associated characteristic time then is τ = T / 2π , which leads to use the pulsation ω = 1/ τ = 2π / T = 2πν in theoretical expressions rather than the frequency ν = 1/ T (the latter, on the other hand, being better adapted to experimental measurements, as it is more natural to count full cycles rather than radians). The same argument can be developed for a space periodic phenomenon, leading to the use of the ‘reduced wave number’  = λ / 2π , as the characteristic length of the phenomenon, and the inverse quantity κ = 1/  = 2π / λ in the reciprocal space. This quantity unfortunately lacks an accepted appellation; it could well be named ‘undulation’, in perfect analogy with the time pulsation. The extension to angular periodicity, with period α , is immediate, leading to the ‘angulation’ m = 2π / α Using these characteristic quantities, the fundamental quantum relationship now read : E = ω (time) (20) p = κ (space) L = m (angle) with  now appearing in the role of the quantum synthetizer. The importance of these considerations lies in the fact that it is by using Dirac’s  , and not Planck’s h , that reliable order of magnitudes estimates can be obtained by using dimensional analysis or other heuristic methods. For instance, dimensional analysis alone is sufficient to write the ionization energy of the hydrogen atom as A me4 / 2 (with obvious notations) or its size as A! 2 / me 2 , where A and A’ are dimensionless constants. In effect, these constants are 2 of order unity, while they would be of order 40 ( = 4π ) if we had chosen h instead of  in these expressions. The basic principle of physics according to Wheeler, namely: “All dimensionless constants are of order unity, IF the characteristic magnitudes are correctly chosen (with allowance for exceptions)” is indeed satisfied, but only if one is careful enough to systematically use Dirac’s constant as the ‘good’ quantum constant. 7 The macroscopic level The status of Planck’s constant as the keystone of quantum theory by no means imply that its role is confined to the microscopic domain, and to atomic or subatomic phenomena. Despite the smallness of its numerical value in the SI unit system, many macroscopic phenomena, on the human scale, are governed by it. As major examples, consider: — the blackbody radiation, indeed one of the problems at the origin of quantum theory ; Stefan’s constant, which governs the amount of energy radiated by a heated body is (within a numerical constant) σ ≈ k 4 / 3 c 3 . Planck’s constant thus rules the baker’s or potter’s owen, and, first of all, the amount of energy radiated by the Sun and received by the Earth, that is, the very possibility of life. — the density of ordinary matter (a few grams per cubic centimeter) is in fact a microscopic quantity as well, since the average atomic volume is given by the size of Bohr’s radius ; the density of matter in bulk, thanks to the saturation of Coulombic forces (a major quantum effect), is the same as that of individual atoms, to wit ρ ≈ M /(2 / me2 )3 , where M is a representative atomic mass. JMLL/ ‘The meanings of h’, Pavia 2000 9 — the typical voltage of ordinary electrochemical batteries (a few volts17), is governed by the ionization energy of atoms, so that V ≈ q-1e me 4 / 2 ; in other terms, one volt is worth one electron-volt per electron… In fact, the role of Planck’s constant extends beyond the macroscopic domain (human scale) well into the megascopic domain (astronomic scale). Indeed, cold enough stellar objects, such as white dwarfs or neutron stars, are governed by the quantum Fermi pressure of their constituents (electrons or neutrons respectively); it turns out that there is an upper bound to their size, which is known as the Chandrasekhar limit, corresponding to a critical number of nuclei of order Nc ≈ (c / GM 2 )3 / 2 — which is, and not by chance, also the order of 57 magnitude of the number of atoms ( ≈ 10 ) in a typical star such as the Sun. Less well-known 18 is the fact that the size of a life-bearing planet may be shown to be necessarily of order R ≈ (e 2 / GM 2 )1/ 2 (2 / me 2 ) and the maximal size of the living beings it hosts of order l ≈ (e 2 / GM2 )1/ 4 (2 / me2 ) — at least for ground animals. It is appropriate to conclude with the following quotation by Maxwell (1870): “If we wish to obtain standards of length, time and mass which shall be absolutely permanent, we must seek them not in the dimensions or the motion or the mass of our planet, but in the wavelength, the period of vibration and the absolute mass of these unperishable and unalterable and perfectly similar molecules.” Maxwell was indeed right to call for a redefinition of our metrological sandards based on atomic (and therefore quantum) magnitudes. But we now know that the dimensions of our planet and our own are not arbitray, but closely linked to those of the ‘molecules’ — thanks to Planck’s constant. 17 It is particularly appropriate to mention this aspect here and now, as we celebrate in 2000 the bicentenary of the discovery of the battery by Alessandro Volta, who spent quite a time living and workin here in Pavia. 18 Ref. 10. JMLL/ ‘The meanings of h’, Pavia 2000 10 

Institute Vienna Circle “Epistemological & Experimental Perspectives on Quantum Physics” Vienna, September 1998 Quantum Words for a Quantum World Jean-Marc Lévy-Leblond* Université de Nice Départements de physique et de philosophie Abstract. One of the tenets of the conventional view on quantum physics, put forward with considerable insistence by Bohr, is that all statements referring to the quantum world ought to be ultimately couched in classical language, in order to make sense with respect to our common experience. It will be argued that such a requirement, although it certainly had a liberating effect on the emergence of quantum theory, can no longer be accepted at face value to-day. On the one hand, it is very difficult to see why this epistemological dogma, bearing on the overtaking of an old theory by a new one, should hold solely at the crossing of the frontier between classical and quantum physics, and not, for instance, when going from (classical) mechanics to (classical) field theory, or from galilean to einsteinian chronogeometry. As for another contemporary bohrian view, namely, the correspondence principle, the argument can have but a heuristic and temporary value, its consistent enforcement being impossible — and counter-productive. On the other hand, more than half a century of quantum practice, both experimental and theoretical, has led to a new awareness of the quantum world and a genuine intuition of quantum behaviour, the expression, development and diffusion of which is seriously hampered by the lack of an adequate and specific terminology. The delay in the emergence of a quantum vocabulary is to be contrasted with the rich and fruitful linguistic creativity of nineteenth century physics. Although it seems founded in theory on the Copenhagen position, this delay find its ultimate roots in the changes of scientific practice during the twentieth century (specialization and separation of tasks). It has had devastating effects on the conceptual understanding of quantum physics, both by physicists themselves and by philosophers, not to speak of laymen. It is all the more necessary, if one wishes to reconcile quantum views of the world with contemporary culture, to exert a voluntary terminological activity, and to establish new ways of speaking for no-longer-so-new ways of thinking. Some proposals in this direction are discussed. * Physique théorique, Parc Valrose, F-06108 Nice cedex, France. E-mail: <jmll@math.unice.fr>. JMLL « Quantum Words » /Vienna (DRAFT) 25/04/06 1 A little-known movie by Alfred Hitchcock, Torn Curtain (1966) — admittedly not one of his best — tells a story of spying and science. It features a strange scene, where two physicists confront one another on some theoretical question. Their “discussion”, if it may be so called, consists solely in one of them writing some equations on the blackboard, only to have the other angrily grabbing the eraser and wiping out the formulas to write new ones of his own, etc., without ever uttering a single word. This picture of theoretical physics as an aphasic knowledge entirely consisting of mathematical symbols, as common as it may be in popular representations, we know to be wrong, of course, and we have to acknowledge that, far from being mute, we are a very talkative kind; physics is made out of words. What I wish to question here, however, is the very nature of our relationship with language, particularly as concerns quantum theory. My thesis will be that we have been somewhat offhand and rather indifferent with respect to the words we use, or rather without respect for them, and that this attitude has reinforced, and sometimes perhaps even produced some of the persisting epistemological and pedagogical difficulties in our field — not to speak of the new cultural problems that we are facing. 1. Quantum physics and ordinary language It is obviously impossible to discuss this question without going back to Bohr and his famous argument on the use of language in quantum physics, which he stated again and again. His position, as we all know, was that there is, and can be, no specific way of expressing quantum physics. He wrote, for instance: “(…) it is decisive to recognize that, however far the phenomena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms. The argument is simply that by the word ‘experiment’ we refer to a situation where we can tell others what we have done and what we have learned and that, therefore, the account of the experimental arrangement and of the results of the observations must be expressed in unambiguous language with suitable application of the terminology of classical physics.”1 Beyond the traditionally acknowledged obscurity of Bohr’s writings, it has not been sufficiently remarked, to my mind, that his standpoint here lacks not only clarity but JMLL « Quantum Words » /Vienna (DRAFT) 25/04/06 2 consistency as well. The deep ambiguities it contains spring up if one compares the preceding quotation with an apparently very similar one: “It lies in in the nature of physical observation (…) that all experience must ultimately be expressed in terms of classical concepts. The unambiguous interpretation of any measurement must be essentially framed in terms of the classical physical theories, and we may say that in this sense the language of Newton and Maxwell will remain the language of physicists for all time. (…) Even when the phenomena transcend the scope of classical physical theories, the account of the experimental arrangement and the recording of observations must be given in plain language, suitably supplemented by technical physical terminology. This is a clear logical demand, since the very word ‘experiment’ refers to a situation where we can tell others what we have done and what we have learned.”2 It is in fact quite unclear in what sense Bohr uses the word “language”. If he refers to language in general, then his position is uncontroversial and verges on triviality; as a matter of fact, human language (beyond the variety of specific languages) is one, and there can indeed be no spoken or written communication outside of it, whether in physics or elesewhere. The impossibility of creating ex nihilo a novel language, with syntactic structures previously unheard of, has nothing to do with quantum physics as such, and simply derives from the necessary continuity and commonality of all human experience. Indeed, it has often been stressed that scientific language differs from ordinary speech mainly by the use of specialized nouns and adjectives, specific verbs being very rare and relating mainly to direct actions. As to general expressions, the caution, often advocated by Bohr, to be exercised in using the inescapable verbs “to be” or “to have” is not more imperative when referring to quantum objects than to human beings, for instance… The difficulties arise when going from syntactic to semantic considerations, and, more specifically, to the “terms” we use for describing the world. One might certainly argue that science should not introduce any particular term and stick to the words of everyday language, in order to avoid creating a gap between common experience and scientific practice. But the whole of science should then be rewritten anew, since right from it beginnings it has taken to create new words or coopt old ones in order to express its specific notions. Bohr of course is quite aware that it is impossible to do science without using special words. He nevertheless considers scientific terminology as an “unessential convention”, JMLL « Quantum Words » /Vienna (DRAFT) 25/04/06 3 which does not lead to a break with “common langage”, but only in so far as classical physics is concerned: “From a logical standpoint, we can by an objective description understand a communication of experience to others by means of a language which does not admit ambiguity as regards the perception of such communications. In classical physics, this goal was secured by the circumstance that, apart from unessential conventions of terminology, the description is based on pictures and ideas embodied in common language, adapted to our orientation in dailylife events. The exploration of new fields of physical experience has, however, revaled unsuspected limitations of such approach and has demanded a radical revision of the foundations for the unambiguous application of our most elementary concepts (…).”3 These “new fields” are of course those of quantum physics. It seems, however, very hard to justify setting up a boundary within science itself and to decree that it was all right to invent new terms up to, say, 1900, but not afterwards. In fact, the unintuitive character of the new physics, which is invoked by Bohr to justify sticking to classical terms, is by no means a specificity of the quantum domain. Newtonian physics, consisting of corpuscular mechanics, may in some sense be thought of as a “natural” extension of ordinary “pictures and ideas” about a world of pebbles and bullets; but such a standpoint already is rather limited and contrived, though, since the newtonian concept of a mass point, that is, a material object deprived of any spatial extension, stretches to the extreme the link between intuitive representations and scientific notions. It does not do justice to the heroic efforts of the founders of physics to consider, as put by a follower of Bohr, that “classical mechanics is a straightforward mathematical idealization of the part of ordinary language treating the external world, and the reality of this world is directly transmitted to classical objects.”4. In any case, beyond classical mechanics, the whole of nineteenth century physics has witnessed a continuous and manifold departure from the common understanding of the world. Thermodynamics as well as electromagnetism go so far beyond our usual ideas that they could not be dealt with in “plain language”; there is no such thing as “the language of Newton and Maxwell”, to use Bohr’s words. Indeed, most of the classical terms he seems to take for granted as having a clear meaning, were introduced in physics during the nineteenth century and certainly did not belong to JMLL « Quantum Words » /Vienna (DRAFT) 25/04/06 4 ordinary speech. Consider for instance “energy” — a term foreign to the language of Newton: the very concept was not clarified before the middle of the last century, and the word was certainly not used in common parlance, as it has come to be in the past decades. A stronger argument yet could be made around “entropy”. Even the apparently elementary idea of “potential”, although formally introduced by Laplace for gravitation and Poisson for electricity, was named only later on by Green, and at the end of the century was still considered as very abstract and introduced in academic courses with much caution5. The clearest case in point is perhaps the notion of “field” which took many decades to emerge, from Faraday to Maxwell — who, apparently, was the first to use the word in his famous 1865 paper6, although the idea as we understand it today was not yet clear to him; he still thought of the field as a certain dynamical state of a mechanically conceived ether. The electromagnetic field would not obtain its ontological status before the 1900’s and the final demise of the ether. In some sense then, the word preceded the idea and prepared its full extension; any forbidding pronouncement as to the introduction of non-(archeo)classical (i. e. mechanistic) terms would certainly have delayed and hampered the developments of electromagnetism, and beyond of the whole of physics as we understand it today. For it may be argued that the deepest revolution in modern physics did not take place in 1905 (relativity) or 1905-1913 (quanta), but precisely in 1865, with the advent of a completely new physical entity, of a non-mechanistic nature, continuous and non-localized: “Before Maxwell, people thought of physical reality — in so far as it represented events in nature — as material points (…). After Maxwell, they thought of physical reality as represented by continuous fields, not mechanically explicable (…). This change in the conception of reality is the most profound and the most fruitful that physics have experienced since Newton.”7 So said Einstein at the centenary of Maxwell’s birth, that is, in 1931 — well beyond the advent of quantum theory. Although this is not what Bohr had in mind, we may perhaps take quite literally his assertion that it is “the exploration of new fields of physical experience [which] has demanded a radical revision (…) of our most elementary concepts”. JMLL « Quantum Words » /Vienna (DRAFT) 25/04/06 5 Yet, one may feel some uneasiness in Bohr’s position when he admits that the use of plain language is to be “suitably supplemented by technical physical terminology”. Does Bohr consider such “technical” terms as not properly belonging to language and fulfilling, perhaps, a purely formal purpose, similar to symbolic formulas, the only function of such terms being to label without ambiguities theoretical procedures or designate specific pieces of experimental apparatus? But, as hinted at on the above examples, even technical terms are true words and carry a heavy load of historical connotations and conceptual associations. As such, it is hard to imagine what could be a criterion to separate and discriminate this “technical physical terminology” from the englobing matrix of “plain language”. Many of these technical terms are borrowed from lay vocabulary and given a restricted and specific meaning, which nevertheless cannot suffice to cut them from their deep vernacular roots in the fields of nonscientific practice (of course, “root” — in algebra — precisely is an example, as well as “field”…). And many words initially created for and to be found in professional scientific discourse slowly leak out to find their way in common parlance, where they take on original meanings which cannot but come back within the scientific discourse to give it new colours (“energy” and “entropy”, “electricity” and “magnetism” are cases in point). If one is to follow Catherine Chevalley when she relates Bohr’s ideas on language to the German philosophical tradition, and, specifically, to the work of Wilhelm von Humboldt8, one cannot help noting that this work, even if it received a renewed attention in the 1920’s, was accomplished at the very beginning of the nineteenth century, that is, in the context of a scientific world-view completely dominated by the Newtonian mechanical paradigm. It is no surprise, then, that these linguistic theories could not give full justice to the deep conceptual and terminological revolution witnessed by nineteenth century physics. It is intriguing though to realize that Bohr belonged to the first generation of physicists for whom the concept of field could be taken for granted; one may wonder how he could consider the then very recent ideas and words of field theory as belonging to an indefinite and undifferentiated “classical physics”. The explanation is probably that an impending breakthrough is necessarily thought of as much more difficult that an already accomplished one — for those who do have to make the break. Confronted by the extraordinary difficult task of setting up a new theory, it is conceivable that Bohr did not want or need to apply distinctions within what was by then the established JMLL « Quantum Words » /Vienna (DRAFT) 25/04/06 6 science. His strategy (contrarily to Heisenberg’s, for instance) was to rely as heavily as possible on this classical physics and to try using it as far as possible — farther than its a priori domain of validity. It has often been remarked that Bohr, a true Moses-like figure, did not really enter the Promised Land of quantum theory. He used with an admirable dexterity the Principle of correspondence and the notion of complementarity to supplement classical physics with the lightest possible quantal touch able to open new vistas on the quantum domain9. One can then understand the deep heuristic role of his insistance on classical descriptions — and admire the extent of the results he obtained from such an a priori entrenched position. This being said in earnest, and lest I would still be accused of lese-majesty, let me remind you that other people dared to voice rather strong disagreement with Bohr’s epistemological positions: “I know that it is not N. B.’s fault, he has just not found the time to study philosophy. But I deeply deplore that by his authority the brains of one or two or three generations will be upset and prevented from thinking about problems that ‘He’ pretends to have solved.”10 What is not so easy to understand his that, while the followers of Bohr bravely entered the new land and started developing a genuine conceptual framework, going well beyond the limited frontier regions accessible with the help of the sole Principle of correspondence, they did not produce as well a specific terminology, at least on the same scale and with the same determination. This is all the more perplexing since Heisenberg, for instance, entertained a view of the role of language in science quite different from Bohr’s and much more dynamical11. It is time to reaffirm that the creation of new words is a constitutive process of science, which should accompany the emergence of its new ideas, as it has done for most of the history of science — except during the century just ending, where linguistic inventivity has been drastically reduced, at least in physics (mathematics and biology certainly fare much better in that respect). It is even a double paradox that physicists have never produced so many new ideas and so few new words, and that they have used common and concrete words all the more so since their ideas became more esoteric and abstract (see “quarks” and their “colors”, “big bang” and “chaos”, etc.). One cannot help to think that this linguistic weakness is in some way linked to the domination of a single language, which moreover has been used for decades by non-native speakers — I refer here to most of the great names of physics JMLL « Quantum Words » /Vienna (DRAFT) 25/04/06 7 in the thirties who experienced a certainly rather painful and perhaps inhibitory transition from German to English. In more recent times, it is probably the overtaking of science communication proper by mere advertising that explains the simplistic borrowing of picturesque but misleading common words, as in expressions like “big bang”, “coloured quarks” or “butterfly effect”. 2. Quantum physics and extraordinary language The past decades have seen a tremendous extension in our capacities for manipulating and exploiting quantum phenomena from the individual atomic scale (single electron electronics, single atoms optics, nanotechnologies) to the macroscopic (lasers, superconductors, superfluids, etc.). It is worth emphasizing that such feats not only were unforeseen by our great predecessors, but even declared unattainable in principle. In any case, our growing familiarity with quantum phenomena has led us to a new intuition and, necessarily, to new ways of expression. Accordingly, we, quantum physicists, do have created quite a few words; in effect, we do not obey Bohr’s rule restricting our language to the ordinary classical one — even those among us who continue paying lip service to the alleged Copenhagen orthodoxy. Any paper in the field is witness to this statement. Consider, for instance, the first page of a typical article, in which all nonclassical expressions have been underlined (Fig. 1). So, we do cultivate new flowers in our terminological garden. But my contention is that we do not take good care of it; by the way, I would not object to your interpreting this opinion as reflecting the old opposition between the apparently free growth of English gardens and the meticulously controlled planning of French ones. Be that as it may, any amateur gardener knows that some upkeep is necessary, and that it consists as well in the weeding of obsolete vegetation as in the tending of young plants and in the sowing of new varieties. a) Weeding Quantum physics is by now old enough — almost a century — to have known the complex processses of internal recasting and reshuffling of ideas which naturally lead quite a few expressions to obsolescence. However, we too often keep using uncritically such terms although they have lost most of their meaning. This is not a process particular to quantum physics of course (think of the maxwellian JMLL « Quantum Words » /Vienna (DRAFT) 25/04/06 8 ”displacement current“*), but it is certainly more marked here. Here is a short list of terms we could well dispense with: — ”Complementarity“. As already alluded to, this bohrian notion is but a sort of safe-conduct allowing a denizen of the classical domain to make some incursions into the quantum realm without running into trouble. Deep inside this quantum realm, the conflicting classical views that complementarity is supposed to keep apart cease to hold altogether. It is certainly too simplistic to hope that a new reality can be fully described by the mere use, however ingenious, of previously conflicting ideas. Let us not forget, after all, that complementarity was not so convincing an idea, even to Bohr’s closest co-workers, such as Heisenberg or Pauli, not to speak of Schrödinger who, privately and lately at least, did not hesitate to speak of “…that silly and entirely unphilosophical twaddle from Kopenhagen about subject and object and complementarity and what not.”12 — ”Wave-particle duality“. Closely linked to the preceding argument, the description of quantum objects by a duality between two classical aspects is in fact of limited validity. While it is a very useful point of view for the first contacts with these strange objects, it is by far not sufficient to take into account all the subtleties of their behaviour. Australian settlers, on their first observations of a weird animal, named it “duckmole” on account of its mole-like fur and form and duck-like beak and feet; but duck-mole duality certainly is insufficient for a full appreciation of the specificities of the platypus (which, by the way, already had a name of its own before European scientists came to study it — namely, “mallingong”). In fact, the expression ”wave-particle duality“ offers not an answer to the question of the nature of physical entities, but asks a question — and a nontrivial one: how is it that quantum objects do appear at the classical approximation either as waves or as particles? Or, more precisely, what are the conditions of validity of these two (exclusive but non complementary!) approximate descriptions? In any case, as an epistemological solution to the general problem of the nature of reality, ”waveparticle duality“ falls very short of its goal; Schrödinger again: “I believe the problem of ‘the real world around us’ to be much older, much deeper and more difficult to put in order than that old particle-wave duality about which there is at present so much ado, and its palliative — * From now on, I will use the awkard looking reversed quotation marks to single out the terms I question. JMLL « Quantum Words » /Vienna (DRAFT) 25/04/06 9 complementarity — which, to be honest, has not yet got beyond the rhetoric stage and, in my opinion, never will.”13 — ”Indeterminism“. Here also, this term, far from elucidating a concrete problem, does in fact hides it under a veil of abstract generality. Quantum theory per se is quite deterministic, as the time development of states is governed by a strict evolution equation. The difficulty is that this determinism is incompatible with the one of classical physics, as they do not refer to the same physical magnitudes! Here also, indeterminism appears when the quantum realm is considered from the outside, or, at least, from its problematic and fuzzy borderline with the classical domain. — ”Uncertainty (principle)“. First of all, there is no “principle” here; the Heisenberg inequalities nowadays clearly appears as a consequence of the fullfledged quantum theory, and do not stand as one of its basic independent assumptions. But above all, the very idea of ”uncertainties“ once again is an undue importation from a domain into another; the margin of ignorance necessarily associated to any experimental measurement (uncertainty proper) was likened to the margin of indefiniteness of a quantum magnitude. The confusion partly resulted from Heisenberg having operationally introduced his inequalities through the analysis of a gedanken experiment (the famous ‘microscope’), before they could be shown to derive from a more general theoretical reasoning14. The situation was still muddled by a complex story of hesitation and mistranslation, Heisenberg himself using ”Unsicherheit“, ”Ungenauigkeit“ (uncertainty), but finally settling for “Unbestimmtheit”, that is “indeterminacy”, which is certainly a better wording, and had some success, before being, alas, eclipsed15. Nothing puts into a clearer light the unbecoming character of the ”uncertainty“ terminology than the comparison of the (classical) undulatory inequalities with the (quantum) Heisenberg inequalities. In the first case, the spectral bandwidth of the frequency spectrum of some signal, Δν, is linked with its characteristic time extension Δt by Δν. Δt > 1 ; mere multiplication by the Planck constant h leads to ΔE. Δt > h . How comes that the “spectral widths”, “extensions” or “dispersions” — we are not in want of terms!16 — of the first case become ”uncertainties“ in the second? The situation is in fact very ironical, for the term ”uncertainty“, although it is used in the most orthodox presentations of quantum mechanics, would seem to express much better the reservations of its neoclassical opponents; if one does believe that the electron in fact is somewhere, that is, has a definite position, according to some hidden variable theory, then its JMLL « Quantum Words » /Vienna (DRAFT) 25/04/06 10 dispersion in position is but a provisional and superficial character, due to our ignorance — an uncertainty, indeeed. — ”Observables“. The founders of quantum physics developed a critical analysis of experiment and measurement in the quantum domain, which greatly assisted them in getting rid of some classical prejudices and building the new theory. In so doing, the emphasis put on the act of observation and its limits led them to insist on the observability — or not — of physical quantities. Accordingly, the term ”observable“ came to substitute for that of physical properties or magnitudes. There is absolutely no reason to stick to this quite misleading terminology, indiscriminately applied to all formally defined properties (hermitian operators in the hilbertian formalism) although very few of them can actually be observed, not to speak of the fact that we still lack, even in the simplest cases of effectively observable quantities, a thorough analysis of the concrete and complex process of measurement. The epistemological weakness of the term is put into full light when one remembers how it made its first appearance in the seminal 1925 paper by Heisenberg introducing matrix mechanics, the abstract of which read: “The present paper seeks to establish a basis for theoretical quantum mechanics founded exclusively upon relationships between quantities which in principle are observables”17. Heisenberg then went on to exclude from such quantities the very position of an electron — that is, the simplest and commonest of today ”observables“. — ”Interpretation“. It has not been sufficiently remarked how strange is the rather sudden appearance, within quantum physics, of the idea that the theory should be ”interpreted“. This idea certainly does not belong to the classical tradition, that “of Newton and Maxwell”, where the challenge of a new doctrine was simply to understand it. The notion of interpretation echoes the logical-positivist standpoint that theoretical contents is to be equated to its mathematical framework; the formalism, taken at face value, is meaningless and subject only to logical consistency internal criteria, so that it asks for external semantic rules establishing its correspondences with empirical data. But such a point of view certainly was not Bohr’s; in fact, the very term of ”interpretation“ did appear only in connection with the late and mainly polemical invention of ”the Copenhagen interpretation“, in the fifities18. As such, it does not even belong to the original epistemological corpus of quantum theory, and apparently was not that much missed during the tense discussions of the thirties. JMLL « Quantum Words » /Vienna (DRAFT) 25/04/06 11 In order to assess the relevance of the preceding criticisms, as well as to convince oneself that it is in fact possible to get rid of obsolete terminology, one may compare the standard written formulations of quantum physics, especially its very repetitive textbooks, where the above expressions are plentiful, with the spoken communication in the labs or conferences; real shop talk makes in fact very little use of ”complementarity“, ”wave-particle duality“, ”indeterminism“, ”uncertainties“ or ”interpretation“ — which means that their necessity is, to say the least, open to question. b) Sowing I would now like to propose a few neologisms, the purpose of their introduction being to emphasize the specific and intrinsic character of quantum physics. In inventing new words, there are good reasons to follow the traditional method of scientific terminology, that is, to rely on Greco-Latin roots. While this strategy may run counter to the temptation of public advertising and media attention, it has the merit of not capturing too easily the lay minds by the use of concrete and pseudointuitive wordings and to stress the real difficulty of new scientific concepts. I will come back to this point in my conclusions. A second argument for not shying away from scholarly and literate terms is that they offer a better prospect for mutual understanding between different languages, as such words usually may be adapted (and adopted) with very little changes — at least in Indo-European languages. — Let us start by considering the terme ”quantum mechanics“ itself. I will not make a quixotic attempt at questioning the reference to the (latin) quanta, although its emphasis on the discrete aspects of the new physics is certainly overdone. While the discretization of energy clearly was the most conspicuous and revolutionary phenomenon for Planck and his contemporaries at the beginning of the century, de Broglie showed us twenty years later that entities previously thought as belonging to the realm of spatial discreteness (classical particles) did in fact exhibit continuous characteristics as well. Quantum theory eventually is not more discrete or continuous that classical theory; it is only much more subtle as to the interplay of continuity and discreteness, for both these notions now relate to the same (quantum) entities instead of bearing upon different ones (classical waves or particles). It would nevertheless be preposterous to call into question the root “quantum” in our terminology. However, the term “mechanics” which usually accompanies it, certainly is much more obnoxious as it defaces a theory which has nothing to do JMLL « Quantum Words » /Vienna (DRAFT) 25/04/06 12 with the machines (mechanê) at the origin of classical mechanics. Furthermore, quantum theory historically is legatee (and epistemologically is donee) not only to (classical) mechanics, but to (classical) wave theory as well; stressing the first aspect yields a distorted view. So why not drop altogether the mechanical connotation, and resort to the word “quantics” to designate this branch of physics? Such a nominalized adjective follows the very general model of standard terms as acoustics, thermodynamics, electronics, etc. — and, for that matter, physics itself. I cannot see any objection to the use of such a simple and natural wording. Note that it would also clarify the relationship between ”quantum mechanics“ and ”quantum field theory“ (further obscured by the alleged ”second quantization“); it would be enough to call the first one “galilean quantics” while the second one would go by the name of “einsteinian quantics”. A side benefit of adopting the term “quantics” is that it renders obsolete the need to specify “mechanics” by the epithet “classical” — “mechanics” alone would suffice. — I now come to the two main ideas which have emerged lately — much too late, indeed! — as the most genuine and profound characteristics of quantics. I refer to ”non-locality“ and ”non-separability“, as they have come to be called. The trouble with these terms is their negative formulation: they depict quantics by what it is not, thus failing to properly spell out the theory for itself (an sich). If one does believe that this world is a quantum one, and that its classical description is but an approximation (as poorly understood as it may be), it would seem fitting to positively describe its very nature. This is an opportunity to use the traditional resources of Greek and Latin. Instead of ”non-locality“, let me then propose the term “pantopy”, the property of being in all (pan) places (topoi) at once, the construction of which is rather familiar (see utopy, etc.). Of course, we could have used an already existing word, namely “ubiquity”, but I tend to prefer a new one, as their shades of meaning are not exactly identical (in botany, for instance, ubiquity refers to the possibility of finding a given species in almost any place, implying the simultaneous existence of different individuals in many places). Let me mention, only for the fun of it, yet another posibility. Besides ubique (= anywhere), root of the term ubiquity, there is another Latin adverb, with the same general meaning, namely undique; the new term “undiquity” would then have the same sense, and contain as well a nice implicit pun on the undulatory aspects of quantics. The term ”non-separability“, besides its negative connotation, has the drawback of being simultaneously rather vague and too concrete, as the idea of separation in common parlance is closely JMLL « Quantum Words » /Vienna (DRAFT) 25/04/06 13 linked to spatiality in ordinary space, while here, in the quantum domain, it refers to the more abstract state (Hilbert) space. It is sobering to note that our present understanding of ”non-separability“ in fact only expresses our realizing, at last!, the very nature of a tensor product of vector spaces as being much larger than the set of its tensor factored vectors. A neologism then should be based on the more specific concept of ”entanglement“ which has come to express this idea. The corresponding notion of folding, intertwining, interlacing, etc. is rendered in Greek by the word emplexis, from the verb plekô. It is thus straightforward to propose the term “implexity” which has the double advantage of paying regards to David Bohm’s insistance on “implicate order” of quantics, and to take a natural place in an long and familiar series of words, like complexity and perplexity (both of which, after all, already properly characterizing the context of quantum theory…). One could then go on to replace ”entanglement“ itself with its too concrete connotations, which, furthermore are rather different in other languages (the original Schrödinger’s ”Verschränkung“ and the French ”enchevêtrement“ are certainly not exact equivalents), by the related “implexion”, and, instead of an ”entangled“ state, speak of an “implexed” one. — As a last example of the terminological reform I am pleading for, let me consider one more important and usual quantum term. The behaviour of identical ”quantum particles“ (see below for a better name) is usually described by their socalled ”statistics“ (Fermi-Dirac or Bose-Einstein), even when one deals with but a few of them and no proper statistical argument is relevant. As a matter of fact, the statistical description of a large number of such identical objects is just a consequence of a deeper property, which manifests itself as soon as we have two identical ones and consider the transformation of their collective state under permutation. I thus advocate the term “permutability” (or perhaps the shorter but less familiar “permutancy”) to indicate this property. c) Tending Whatever may be the impact of the deliberate creation of such new terms for quantum physics and their fate, it is worthwhile noticing and hailing the apparition already of some rather adequate words, more or less commonly used. We will be content with a few examples, which will enable us to strengthen our previous arguments. JMLL « Quantum Words » /Vienna (DRAFT) 25/04/06 14 — I have stressed in the above critique of ”wave-particle duality“, that there is indeed a classical dualism, but a quantum monism. A quantum entity is neither a wave, nor a particle, so that it seems fitting to find a generic name for all these things which, according to Feynman, “all behave in a crazy way, but at least in one and the same way”: protons, electrons, photons (which, by the way, were christened by Lewis in 1926, more than 20 years after their invention by Einstein; this at least is an example of a successful neologism!), positrons (which, incidentally, should better be called “positons”, as in French), phonons, rotons, gluons, etc. The general acceptance and fecundity of the suffix “-on” for designating the specific quantum entities make it quite natural to follow Bunge, who some decades ago proposed for them the generic neologism “quantons”. Since then, it has made in the literature a slow but continuous progression, which, to my mind, is to be determinedly supported19. Note that the unquestioned use of the (totally nonclassical!) terms “fermions” and “bosons” to denote the two categories of quantum objects cries for a common term of the same kind. — In the same vein, one must greet the advent of “qubits” in quantum communication theory (what about “quommunication”?), which, after all follow the old and perhaps too neglected nowadays Dirac’s terminology, contrasting “cnumbers” and “q-numbers” (certainly a better term than the more common ”observables“). — One of the best recent terminological innovations surely is “decoherence” about which it must be noted that it is built in exactly the opposite way as ”nonlocality“ and ”non-separability“; namely, a phenomenon which characterizes the process leading from quantum to classical theory is rightly designated by a negation of the former. 3. Words matter Lest I should be misunderstood, let me make clear that I do not believe in the possibility of a ”scientifically pure“ and ”epistemologically correct“ terminology, settling the matter of words once and for all. Quite on the contrary, the idea is that we should recognize the inescapable historicity and cultural embedding of our formulations, and keep working on them. Sisyphus, along with his stone, had to carry words, and to cope with language for accomplishing his never ending task. JMLL « Quantum Words » /Vienna (DRAFT) 25/04/06 15 I am personally convinced that a permanent critical re-examination of our vocabulary, leading to the elimination of obsolete terms and introduction of new ones, would help us in getting a better grasp on our ideas. I find it pityful that so many wonderfully delicate and subtle experiments keep being framed in a formulation which does not render full justice to their novelty20. A second purpose of the advocated reform is educational. As a matter of fact, several of the proposals made here find their origin in my teaching experience, and have already been thoroughly tested on these grounds21. But our responsibility in these matters goes well beyond our professional tasks, be them research or teaching. It is widely recognized today that, in the very interest of the scientific enterprise, we have to share our knowledge with the lay public. But how can we expect the sophisticated concepts we forge and the experiments we run to be correctly understood by the unitiated if we are so careless in expressing them that our own understanding is hampered? Against naive expectations (or fears), the use of a more esoteric vocabulary, provided it is suitably tailored and explained, does not run counter to the effort of promoting a better public understanding of science. Quite on the contrary, by helping to point out the specifics and novelties of scientific concepts, and their differences with respect to common notions, this linguistic demand goes into the right direction. Efficient popularization is not achieved by blurring the distinctions between scientific and ordinary knowledge, and pretending in the existence of a continuous transition, but in pointing out the gaps which separate them, and assesssing their real width and depth. In any case, we have little right to criticize and patronize those in other fields, scientific or not, if they take in an admittedly simplistic or naive way the very words we deliver onto them; let us not forget the parabola of the beam and the moat. I am here alluding to the so-called ”Sokal hoax“ and the ongoing discussions it has started22. Beyond the details of this affair, it is no less than the very place of physics within contemporary culture which is at stakes. It is a pleasure to acknowledge the help of Catherine Chevalley with the original Bohr’s writings, and to thank Kurt Gottfried for his comments and lingusitics advice. JMLL « Quantum Words » /Vienna (DRAFT) 25/04/06 16 For a proper quantum speech Do not say: Say: complementarity —> — wave-particle duality —> — indeterminism —> — interpretation —> — uncertainty principle —> Heisenberg inequalities uncertainties —> extensions, spectral widths observables —> (quantum) properties quantum mechanics —> quantics particles —> quantons non-locality —> pantopy, undiquity non-separability entanglement, entangled —> implexity —> implexion, implexed statistics —> permutability, permutancy JMLL « Quantum Words » /Vienna (DRAFT) 25/04/06 17 1 . Niels Bohr, in P. A. Schilpp (Ed.), Albert Einstein: Philosopher-Scientist, Evanston: The Library of Living Philosophers 1949, pp. 200-241. 2 . Niels Bohr, as reported by Petersen (1968), in J. A. Wheeler and W. H. Zurek (Eds), Quantum Theory and Measurement, Princeton: Princeton University Press 1983, p.7. 3 . Niels Bohr, “Physical Science and the Study of Religions”, Studia Orientalia Ioanni Pedersen septuagenario A.D. VII id. Nov. Anno MCMLIII, Copenhagen: Ginar Mimles-Gaard 1953, pp. 385-390. 4 . T. Bergstein, Quantum Physics and Ordinary Language, London: Macmillan 1972. 5 . See the remembrances by Paul Langevin, in “La notion de corpuscule et d’atome”, Actualités scientifiques, Paris: Hermann 1934, pp. 44-46. 6 . James Clerk Maxwell, “A Dynamical Theory of the Electromagnetic Field”, in: Phil. Trans. R. Soc. 155, 1865, pp. 459-512. 7 . John Hendry, James Clerk Maxwell and the Theory of the Electromagnetic Field, Adam Hilger, 1986. 8 . Catherine Chevalley, notes and comments in Niels Bohr, Physique atomique et connaissance humaine, Paris: Gallimard 1991; see especially pp. 480-502. 9 . The role of “safeguard” played by ”complementarity“ and its relation with the question of the limits of the ordinary language has been analyzed by Catherine Chevalley, “Complémentarité et langage dans l’interprétation de Copenhague”, Rev. Hist. Sci. XXXVIII-3/4, 1985, pp. 251-292. 10 . Erwin Schrödinger, letter to Léon Brillouin, Bozen, 6 November 1959, New-York: American Institute of Physics, Niels Bohr Library-Brillouin Archives (unpublished, courtesy of Rémi Mosseri). 11 . See the recently published “manuscript of 1942”: Werner Heisenberg, Ordnung der Wirklichkeit, Munich: Piper 1989. The question is discussed by Catherine Chevalley in the presentation of her French translation, Werner Heisenberg, Philosophie, Le manuscrit de 1942, Paris: Seuil 1998, pp. 153-187. 12 . Ref. 9. 13 . Erwin Schrödinger, letter to Léon Brillouin, Bozen, 16 October 1959, New-York: American Institute of Physics, Niels Bohr Library-Brillouin Archives (unpublished, courtesy of Rémi Mosseri). 14 . Such a general theoretical derivation was given by Roberston as soon as 1929, that is, two years only after Heisenberg’s seminal paper; H. P. Robertson, “The Uncertainty Principle”, Physical Review 34, 1929, pp. 163-164. It is interesting to note that, as was going to happen over and over again, the title and terminology of the paper do not reflect its contents. 15 . Jean-Marc Lévy-Leblond & Françoise Balibar, “When did the indeterminacy principle become the uncertainty principle?” (Answer to Query #62), American Journal of Physics (under press). 16 . For instance, Schrödinger, in his essential 1935 paper, while speaking of the “Heisenberg Ungenauigkeitsbeziehung”, calls “Toleranz- oder Variationsbreiten” the quantities involved; Erwin Schrödinger, “Die gegenwärtige Situation in der Quantenmechanik”, Die Naturwissenschaften 1935, pp. 807-812, 823-828, 844-849. 17 . Werner Heisenberg, Zeitschrift für Physik 33, 1925, pp. 879-, English translation in B. L. Van der Waerden (Ed.), Sources of Quantum Mechanics, Amsterdam: North-Holland 1967. 18 . See the contribution of Catherine Chevalley at this Symposium, “Why do we find Bohr obscure?”. 19 . See Jean-Marc Lévy-Leblond, “Classical Apples and Quantum Potatoes”, Eur.J. Phys. 2, 1981, pp.44, and “Neither Waves, nor Particles, but Quantons”, Nature 334, 1988, p. 6177. 20 . As a very recent example, see S. Dürr, T. Nonn & G. Remp, “Origin of Quantum-mechanical Complementarity probed by a ‘which-way’ Experiment in an Atom Interferometer”, Nature 395, 1998, pp. 33-37. In this paper, the whole epistemological discussion of the results of a beautiful experiment is marred by a fake opposition between ”complementarity“ ans ”uncertainty“, both unsuitable. 21 . The textbook by Jean-Marc Lévy-Leblond & Françoise Balibar, Quantics (t.1, Rudiments), Amsterdam: North-Holland 1984, puts in practice some of these recommendations. 22 . For a collective analysis and rebuttal, see B. Jurdant ed., Impostures scientifiques, Paris:La Découverte 1996 and Alliage n°35-36, Nice: Anais 1998. JMLL « Quantum Words » /Vienna (DRAFT) 25/04/06 18 

Quantum entanglement, a macroscopic feature Vlatko Vedral (Nature 425, 28-29; 2003) asks « …do quantum correlations affect anything of importance in the real, macroscopic world? », and refers on on recent experiments by Ghosh & al. (Nature 425, 48-51; 2003) to answer that « amazingly, they do. » The crucial role of quantum entanglement in shaping the world on our scale may well be considered as amazing, but it should have been recognized for a long time. Indeed, the existence and properties of gross matter as we know it (and thus our very existence) rely crucially on the exclusion principle, which entails a fully entangled electronic state. The variety and specificities of atoms, as expressed by Mendeleev table, cannot be understood without appealing to the fermionic nature of electrons. But the role of the Pauli priniciple goes far beyond the atomic level, as it constitutes a necessary condition for the extensivity of energy in the macroscopic realm, or, in other words, the saturation of the Coulomb forces which ensure the cohesion of matter. The cost of removing an atom from a piece of matter is the same (a few electron-volts) whether this piece is a small pebble or a huge mountain; in explaining this fact of nature, the antisymmetrisation of the global electronic state plays an essential role. This is a most deep and difficult result, first proved by F. J. Dyson & A. Lenard (J. Math. Phys. 8, 423-434; 1967, and 9, 698-711; 1968), improved by E.H. Lieb & H. Thirring (Phys. Rev. Lett. 35, 687-689; 1975) and reviewed by E. Lieb (Bull. Am. Math. Soc. 22, 1-49; 1990). In order to appreciate its full strength, consider what would happen if the Lord, wishing to punish humankind for its sins, rather than provoking floods or plagues, decided to commute the symmetrization switch for electrons, from - (fermions) to + (bosons). All of a sudden, any body with a mass of, say, a few tens of kilograms (our body), would collapse from a size of the order of the meter to some 10-9 Å, releasing an energy around 1048 eV, the equivalent of 1011 gigatons of TNT: we are all Pauli bombs (see J.-M. Lévy-Leblond in Advances in Quantum Phenomena, E. Beltrametti & J.-M. Lévy-Leblond eds, Plenum Press, New-York 1995, pp. 281-295). Now, what is antisymmetrisation, if not a maximal form of quantum correlation? The electronic state of any macroscopic morsel of the world thus bears incontrovertible witness to that deepest and most original aspect of quantum theory, entanglement. From this perspective, the problem is not so much to appreciate the role of quantum correlations in the macroscopic world, than to understand why they can be neglected when dealing with apparently separated objects: assessing the validity of the classical approximation is the deepest difficulty of quantum theory. Jean-Marc Lévy-Leblond Physique théorique, Université de Nice F-06108 Nice cedex, France JMLL—>Nature (corresp.)/oct. 2003 

Answer to Question #62. When did the indeterminacy principle become the uncertainty principle? Jean-Marc Lévy-Leblond and Françoise Balibar Citation: American Journal of Physics 66, 279 (1998); doi: 10.1119/1.18873 View online: http://dx.doi.org/10.1119/1.18873 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/66/4?ver=pdfcov Published by the American Association of Physics Teachers This article is copyrighted as indicated in the abstract. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.59.1.109 On: Sat, 23 Nov 2013 17:52:27 This article is copyrighted as indicated in the abstract. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.59.1.109 On: Sat, 23 Nov 2013 17:52:27 This article is copyrighted as indicated in the abstract. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.59.1.109 On: Sat, 23 Nov 2013 17:52:27 

On the Nature of Quantons JEAN-MARC LÉVY-LEBLOND Physique théorique, Université de Nice F-06108 Nice cedex, France jmll@unice.fr It is now well over a century since quantum theory was born with the seminal work of Planck on blackbody radiation. It took only a quarter of a century for quantum theory to develop a consistent formalism and show an amazing power of explanation and prediction. Yet, strange as it may seem, the conceptual status of fundamental quantum notions still is under dispute, despite decades of arguments. Most elementary textbooks and popularisation works about quantum physics remain plagued by archaic wordings and formulations. Mario Bunge for long has advocated the necessity of confronting these misunderstandings. His paper exposes some of his proposals in this direction. It seems as if times were now ripe for such views to gain acceptance, as several recent publications on the epistemological problems of quantum theory bear witness (Auletta 2000). My purpose here is to contribute to the same endeavour, by concentrating on the concepts used for describing the basic entities of quantum theory, while stressing the terminological questions they rise (Lévy-Leblond 1999). It is hoped that the ideas (and words) presented here may contribute to clarify and simplify the teaching of quantum physics (Lévy-Leblond & Balibar 1990). Before entering the discussion, it may be worth to offer a brief comment upon the roots of the curious paradox which sees the epitome of modern physics, quantum theory, hampered by a long tardiness. Of course, no new idea, whatever its domain, is born fully grown. Initial formulations of novel conceptions, being still tributary to the old views they are to replace, of necessity are awkward and inappropriate. Any scientific theory, following its inception, then has to undergo a recasting process through which its notions are clarified and its terms improved. An example in point is offered by maxwellian electromagnetism, which, within a few decades, evolved from a mechanistic presentation (dealing with stresses and motions in a material JMLL, ‘On the Nature of Quantons’, Science & Education, 2001 1 medium, the ether) to a radically novel theory of fields. The question then is that of the reasons for the delays in the recasting process of quantum theory (and much the same remarks could be developed for relativity theory). One cannot help relating this situation to the deep changes in the organisation of scientific activity during the twentieth century. As in most domains of social life, technicisation, industrialisation and commercialisation have extended their empire to the scientific world; short term productivity and practical applications have been privileged over critical thinking and intellectual demands. In spite of its apparent triumphs, it may well be the case that contemporary science is not in such a good health and could even show signs of senescence (Lévy-Leblond 1996b). Neither waves, nor particles, but quantons! That the true nature of quantum objects has long been misunderstood is proved by their still all too common description in terms of an alleged “wave-particle duality”. It must be remarked first of all that this formulation is at best ambiguous. For it may be understood as meaning either that a quantum object is at once a wave and a particle, or that it is sometimes a wave and sometimes a particle. Neither one of these interpretations in fact make sense. “Wave” and “particle” are not things but concepts, and incompatible ones; as such, they definitely cannot characterise the same entity. While it is true that quantum objects may in some cases look like waves, and in other cases like particles, it is truer still that in most situations, particularly the ones explored by the elaborate modern experiments, they resemble neither one nor the other. The situation here is reminiscent of that encountered by the first explorers of Australia, when they discovered strange animals dwelling in brooks. Viewed from the forefront, they exhibited a duckbill and webbed feet, while, seen from behind, they showed a furry body and tail. They were then dubbed “duckmoles”. It was later discovered that this “duck-mole duality” was of limited validity, and that the zoological specificity of these beasts deserved a proper naming, which was chosen as “platypus”1. Much in the same way, we thus can (and must) safely assert that quantum objects are neither waves, nor particles, but are to be described by a specific and novel concept, which certainly deserve a name of its own (Lévy-Leblond 1981, 1988). Bunge’s proposal to call them “quantons”, building on the common terminology (electrons, photons, nucleons, etc.) and extending it to a common 1. Of course, the aboriginal people already had a specific and appropriate name for the animals, that is, “mallingong”. JMLL, ‘On the Nature of Quantons’, Science & Education, 2001 2 categorisation, is most to the point, and it is to be hoped that this terminology gradually gains ground. For indeed, quantons are novel entities! The best way, perhaps, to stress the originality of the notion is to examine it from the point of view of the discrete/continuous dichotomy. Quantons show discreteness in that they come in units, and can be counted: an atom has an integer number of electrons, and a photographic plate registers the individual impacts of photons. Nevertheless, electrons as well as photons (and all quantons) do show continuous essence as well, since they can be subjected to interferences, superposition, etc. In fact, it should be realised that a physical object must be characterised through the consideration of two discrete/continuous dichotomies; one has to consider separately the question of the number of objects and the question of their extension (spatiotemporal properties). Within classical physics, these two questions merge. Classical particles are discrete under both aspects; they come in discontinuous counts and are discretely localised. Classical fields are continuous under both aspects; they have continuous amplitudes and continuous spatial extensions. But quantons exhibit the original combination of discreteness in number and continuity in extension, as shown by the following table: Number Extension Particles discrete discrete Fields continuous continuous Quantons discrete continuous This double nature of quantons (not a contradictory one, since discreteness and continuity do not refer to the same notions) is the very lesson of quantum physics 2. The unity of quantics It is now easier to understand the two partial classical appearances of quantons; if, in a given experimental set-up, the discrete character of their number is preponderant and the continuous character of their extension secondary, they can be approximately described as particles. Conversely, if, in another experimental set-up, the discrete character of their number is secondary and the continuous character of their extension preponderant, they can be approximately described as waves. The latter case 2. We let it open here the question of the possible consistence and existence of entities showing continuity as to their number and discreteness as to their extension. JMLL, ‘On the Nature of Quantons’, Science & Education, 2001 3 is mostly met for macroscopic systems comprising a large number of quantons, which often may be reasonably treated by a continuous description (as, in classical physics, a flow of sand or grain may be assimilated to a fluid)3. But in most cases, especially in the very sophisticated modern quantum experiments, quantons certainly look neither as waves nor as particles, and must be accounted for through their intrinsic and unique conceptualisation. As Feynman is supposed to have said, “quantum objets are crazy, but they all have the same craziness”. In fact, it is not quantum physics, but classical physics which does exhibit a wave-particle duality (Lévy-Leblond 1977). It is perhaps worthwhile pointing out here that the deepest revolution in the theorisation of physical objects probably took place, not at the beginning of the twentieth century with the advent of quanta, but earlier, by the second half of the nineteenth century, when there emerged the notion of field, which Maxwell brought to fruition. Initially conceived as the description of a disturbance in an underlying medium (such as water for ordinary waves), electromagnetic waves were first supposed to travel in ether. The demise of this medium led to the idea of electromagnetic fields existing per se, in the vacuum. The novelty of this ontological status probably was not realised and assimilated deeply enough; yet the very existence of physical entities without bulk substantiality, nor spatial localisation or form, radically challenged the mechanistic legacy of Newtonian physics. It may be argued that a part of the misunderstandings which have plagued quantum theory up to now, is due to the older failure to fully assimilate the (classical) notion of field. This is reflected, for instance, in the surviving of the appellation “quantum mechanics”, despite the symmetrical status of the classical notions of particle and field with respect to their (unique) successor, the notion of quanton. A simple alternative exists, following the well-established tradition of substantivation for naming the fields of physics; as for acoustics, thermodynamics, electronics (and physics itself!), why do we not simply use the term “quantics” to denote the whole field4? Quantum properties Thus, quantons are, by their very nature, spatially extended objects. The mental difficulty experienced in reconciling this idea with that of their numerical discreteness accounts for the negative characterisation commonly given of these 3. Further considerations about the collective properties of quantons show that only bosons (obeying a gregarity principle) and not fermions (obeying the Pauli exclusion principle) may give rise to such an approximate macroscopic description in terms of waves or fields. 4 In adopting this title for our textbook (Lévy-Leblond and Balibar 1990), we have in fact followed a terminological innovation spontaneously initiated by our students. JMLL, ‘On the Nature of Quantons’, Science & Education, 2001 4 objects. So it has become customary to speak of the “non-locality” of quantons, as if they were deprived of the ‘normal’ property of locality. A better strategy would be to try taming the epistemological difficulty by adopting a more assertive and more intrinsic terminology. A rather natural neologism could be introduced, naming “pantopy” this spatial extensiveness of quantons. It must be stressed that the continuous nature of quantons is not limited to their spatial localisation; it holds as well for all physical magnitudes associated to space-time, such as speed, momentum, and energy. While for classical entities, the physical properties take on unique and determined numerical values, for quantons they are characterised by numerical spectra, extended sets of numerical values. The possible discretisation of some of these spectra, for instance the energy levels of a bound system, is but a particular case, linked to the spatial confinement of the system (in close analogy with the quantisation of the frequencies of vibrating strings, as noted by Bunge). A physical magnitude of a quanton then is characterised by the spectrum of its possible numerical values (“proper values”) and the set of particular states which are associated with a single such value (“proper states”). It may then be shown that both these aspects can be encompassed under the mathematical form of a linear operator in the state space. In other terms, the classical description of a physical property by a unique numerical value must be replaced in quantum theory by this more general mathematical notion. If such a formalisation is accepted, there is no longer any difficulty in attributing objective physical properties to quantons (Lévy-Leblond and Balibar 1990, chapter 2, also Paty 1999). Of course, this new mathematisation has important original consequences. For “incompatible” couples of physical magnitudes, that is, properties the representative operators of which do not commute, e. g. the spatial position and the momentum, there exists a correlation between the widths of their respective spectra, such that, for instance, the narrower is the spatial distribution, the wider is the momentum distribution. Here again, the usual terminology, that of the (in)famous “uncertainty relations” does not give justice to the situation: the width (or dispersions) of the distributions are not “uncertainties”, in that they have nothing to do with some lack of precision in the description of the state of the quanton; they are objective characterisations of this state5. Indeed, far from expressing a limitation in our knowledge, the Heisenberg inequalities (as they could more soberly be called) are of utmost value for our understanding and heuristic treatment of many quantum problems (Lévy-Leblond and Balibar 1990, chapter 3). 5. The story of the term “uncertainty” is by itself rather entertaining. It results from the combination of epistemological misunderstandings with doubtful translations ; the original German wording by Heisenberg, was closer in meaning to “indeterminacy”, which is certainly better, if not yet fully adequate (Lévy-Leblond and Balibar,1998). JMLL, ‘On the Nature of Quantons’, Science & Education, 2001 5 Collective quantum behaviour If the quantum specificities pertaining to the description of individual quantons cannot be too strongly stressed, it must now be added that systems of quantons exhibit even more original behaviour. It had already been realised in the early days of quantum theory that identical quantons should show up statistical properties of a totally non-classical nature. Quantons were proved to belong to two mutually exclusive categories, fermions and bosons; while bosons have an intrinsic tendency to gregarity, fermions obey the Pauli exclusion principle. There is absolutely no classical equivalent of this property. Indeed, the respective gregarious tendencies of bosons and exclusive tendencies of fermions have nothing to do with mutual interaction forces between the quantons; it is not a dynamical and contingent property, but an essential and ontic one. Any system comprising several identical quantons is strongly determined by this property; for instance, the fermionic nature of electrons is crucial for the architecture of atoms and molecules. Let us note in passing that for systems with few quantons, the identity of the components has physical consequences which manifest themselves at the individual level, and do not require or imply statistical considerations. It thus seem rather preposterous to describe the fermionic or bosonic nature of quantons in terms of their “statistics”. Here again, 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. It remains to stress that, far from being confined to the microscopic realm, the collective nature of identical quantons is crucial to the existence and properties of macroscopic matter as we know it. It has taken an unduly long time to recognise (and the proof still requires quite sophisticated methods) that the very possibility of applying ordinary classical thermodynamics, relying on the extensivity of energy, is ensured only in virtue of the fermionic nature of electrons (Lieb 1984, and, for a pedagogical discussion, Lévy-Leblond and Balibar 1990, chapter 7). More generally, we must now realise that macroscopic is not synonymous with classical. Not only do we build more and more specifically quantum objects of paramount technical importance operating on our ordinary human scale (laser, superconductors, superfluids, etc.), but deep common properties of matter cannot be understood except in quantum terms (Lévy-Leblond 1996a). JMLL, ‘On the Nature of Quantons’, Science & Education, 2001 6 Implexity, the quantum essence A full realisation of the specificities pertaining to the very nature of quantons has been rather slow to emerge at the individual level — that of quantons considered one by one. But the delay has been much worse for our understanding of the collective level — beyond, or rather, before the problem of identical quantons. Indeed, the fermion/boson duality is but the expression of the permutational invariance of multiquanton states, abstractly formulated at the level of the Hilbert state space; these states are either totally symmetric (bosons) or totally antisymmetric (fermions). In other words, a collective state cannot be considered as a mere collection of individual states, but shows a peculiar wholeness. The situation contrasts with the classical one, for the state of a system consisting of many classical particles is completely described by the individual states of the particles. Reciprocally, given such a classical collective state, it is always possible to attribute a well-defined state to any of its constituent particles. Such is no longer the case for quantum systems. In the perceptive words of Schrödinger, who was certainly the first to stress what is perhaps the deepest and most original specificity of quantum theory, a full description of a compound system does not entail a full description of its components (Schrödinger 1935). This highly counter-intuitive situation is intrinsically linked to the superposition principle, that is, to the linear structure of quantum theory, according to which the states of a quanton are described by vectors in a Hilbert space. The states of a multiquanton system then define a new Hilbert space, which is the tensor product of the individual state spaces; now, if (tensor) products of individual states do generate the collective space, a collective state in general is not a product of such individual states. It cannot be too strongly stressed how the very notion of tensor product, while mathematically elementary, is heuristically opaque; the reason is probably that, contrarily to the metric structure of a Hilbert space of which we have a rather good intuition starting from our experience in ordinary Euclidean space, the tensor structure has no analogue that we are familiar with (the smallest nontrivial tensor product is four-dimensional…). In any case, it took several decades to realise the full extent of the consequences of the tensor structure and the non-factorisability of generic vector states. Schrödinger coined the word “Verschränkung”, usually translated as “entanglement”, to describe this ‘nonseparability’ of general quantum states. Yet it may be judged that the use of a word with all too familiar connotations, risks yielding concrete and false pictures. A neologism would seem, here again, a better solution. Since the notion of entangling JMLL, ‘On the Nature of Quantons’, Science & Education, 2001 7 or intertwining is rendered in Greek by the word emplexis, from the verb plekô, it is tempting to propose the term “implexity” which has the double advantage of paying regards to David Bohm’s insistence on what he called the “implicate order” of quantics, and to take a natural place in an long and familiar series of words, like complexity and perplexity (both of which, after all, already properly characterising the context of quantum theory…). It is all the more surprising that it took so long to face the implications of implexity since the states of identical fermions exhibit it to a maximal degree, which, as has been stressed, is the precise reason why they play an essential role in accounting for the properties of atoms as well as condensed matter. In any case, it is clear by now that, whatever word is used for it, this idea is indeed the core of the argument behind the Einstein-Podolsky-Rosen (pseudo)paradox, its re-exploration by Bell and his ensuing inequalities, the experimental vindication of the quantum predictions by Aspect and others, and the recent flurry of amazing experiments on these basic quantum phenomena, which nowadays lead to practical perspectives in quantum computing and quantum cryptography (Greenberger & al. 1993, Ghose 1999, Zeilinger 1999, Macchiavello & al. 2000). JMLL, ‘On the Nature of Quantons’, Science & Education, 2001 8 References Auletta, G.: 2000, Foundations and Interpretation of Quantum Mechanics, Singapore, World Scientific Ghose, P.: 1999, Testing Quantum Mechanics on New Grounds, Cambridge University Press Greenberger, D. M., Horne, M. A., and Zeilinger, A.: 1993, “Multiparticle Interferometry and the Superposition Principle’, Physics Today 46, n°8, 22-29 Lévy-Leblond, J.-M.: 1977, ‘The Picture of the Quantum World: from Duality to Unity’, Int. J. Quant.Chem. 12, supp. 1, 415 Lévy-Leblond, J.-M.: 1981, ‘Classical Apples and Quantum Potatoes’, Eur.J. Phys. 2, 44 (1981) Lévy-Leblond, J.-M.: 1988, ‘Neither Waves, nor Particles, but Quantons’, Nature 334, 6177 Lévy-Leblond, J.-M.: 1996a, ‘Quantum Theory at Large’ in Adv. in Quantum Phenomena, Beltrametti, E., & Lévy-Leblond, J.-M., eds., Plenum Lévy-Leblond, J.-M.: 1996b, La pierre de touche (la science à l’essai…), Gallimard Lévy-Leblond, J.-M.: 1999, ‘Quantum Words for a Quantum World’, in Greenberger, D. M., & al. eds., Epistemological and Experimental Perspectives on Quantum Physics,Vienna Circle Institute Yearbook, Springer Lévy-Leblond, J.-M., and Balibar, F.: 1990, Quantics (Rudiments), North-Holland Lévy-Leblond, J.-M., and Balibar, F.: 1998, ‘When did the Indeterminacy Principle become the Uncertainty Principle?’, Am. J. Phys. 66, 278-279 Lieb, E. H: 1990, ‘The Stability of Matter: From Atoms to Stars’, 1989 Gibbs Lecture, Bull. Amer. Math. Soc. 22, 1-49 Macchiavello, C., Palma, G. M., and Zeilinger, A., eds.: 2000, Quantum Computation and Quantum Information Theory, World Scientific, Singapore Paty, M.: 1999, ‘Are Quantum Systems Physical Objects with Physical Properties?’, Eur. J. Phys. 20, 373-388. Schrödinger, E.: 1935, ‘Die gegenwärtige Situation in der Quantenmechanik’, Die Naturwissenschaften 1935, pp. 807-812, 823-828, 844-849 Zeilinger, A.: 1999, ‘Experiment and the Foundation of Quantum Physics’, Rev. Mod. Phys. 71(2), S288-S297 JMLL, ‘On the Nature of Quantons’, Science & Education, 2001 9