Substantia. An International Journal of the History of Chemistry 2(1): 51-76, 2018 Firenze University Press www.fupress.com/substantia ISSN 1827-9635 (print) | ISSN 1827-9643 (online) | DOI: 10.13128/substantia-41 Citation: B.D. Hughes, B.W. Ninham (2018) A Correspondence Principle. Substantia 2(1): 51-76. doi: 10.13128/ substantia-41 Copyright: © 2018 B.D. Hughes, B.W. Ninham. This is an open access, peer- reviewed article published by Firenze University Press (http://www.fupress. com/substantia) and distribuited under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability Statement: All rel- evant data are within the paper and its Supporting Information files. Competing Interests: The Author(s) declare(s) no conflict of interest. Permission to reproduce. Repub- lished from Physica A, 2016, 443, 495- 517. With permission from Elsevier. Copyright 2016. Research Article A Correspondence Principle Barry D. Hughes1,* and Barry W. Ninham2 1 School of Mathematics and Statistics, University of Melbourne, Victoria 3010 Australia. ∗Corresponding author 2 Department of Applied Mathematics, Research School of Physical Sciences and Engi- neering, Australian National University, ACT 0200, Australia E-mail: barrydh@unimelb.edu.au (Barry D. Hughes), barry.ninham@anu.edu.au (Barry W. Ninham) Abstract. A single mathematical theme underpins disparate physical phenomena in classical, quantum and statistical mechanical contexts. This mathematical “correspond- ence principle”, a kind of wave–particle duality with glorious realizations in classical and modern mathematical analysis, embodies fundamental geometrical and physical order, and yet in some sense sits on the edge of chaos. Illustrative cases discussed are drawn from classical and anomalous diffusion, quantum mechanics of single particles and ideal gases, quasicrystals and Casimir forces. Keywords. Classical analysis, quantum mechanics, statistical mechanics, random walks and Lévy flights, quasicrystals, Casimir forces. Physics is not just Concerning the Natures of Things, but Concerning the Intercon- nectedness of all the Natures of Things [1] 1. INTRODUCTION One of the more insightful critics of relatively recent mathematics–from inside the profession–is Morris Kline, who has made the following observation [2]. “It behooves us to learn why, despite its uncertain foundations and despite the conflicting theories of mathematicians, mathematics has proved so incred- ibly effective”. The views of Wigner [3] and Hamming [4] on the “unreasonable effectiveness of mathematics” are perhaps better known, are warmer towards the mathematical profession, and have likely been better received. Philosophers of mathematics have perhaps placed undue emphasis on the apparent rightness of mathematics for the formulation of physical theories. The essential point of this article is that there is a single theme–though one which can be recast in many superficially distinct ways–that reappears in a bewildering array of mathematical and physical contexts. Its appearance is seldom in the direct formulation of models, but rather arises in the working out of the implications of those formulations. We venture to suggest, though with some diffidence, that this mathematics internal to theories may itself 52 Barry D. Hughes and Barry W. Ninham contain some measure of physical insight, and perhaps even of physical reality. Some of the ways in which the theme presents itself are collected in Table 1. It is particularly strik- ing that the formulae in Table 1 vary from highly spe- cific results about particular mathematical functions to results involving arbitrary functions, and include for- mulae that make sense in relatively elementary calculus, formulae that necessarily involve the theory of func- tions of a complex variable, and formulae that make no sense in classical real or complex analysis and need to be interpreted in the sense of generalized functions. The mathematical equivalence of the results in Table 1 has been addressed twenty years ago in a paper of Ninham, Hughes, Frankel and Glasser [5], and the reader may refer to that paper for a fuller account of the mathemati- cal inter-relations and some relevant references that are not repeated here. What we offer here is a more compel- ling case for centrality of these relations to physics, rath- er than to mathematics. In the context of physics, the “correspondence prin- ciple”, first enunciated by Bohr [6], requires quantum mechanics to be consistent with classical mechanics in an appropriate limit, initially in Bohr’s case in the limit of large quantum numbers, but now interpreted more broadly [7]. The “complementarity principle”, also due to Bohr [8], was enunciated in the context of the problem of measurement in quantum mechanics, and its conse- quence of most interest in the present paper (loosely expressed as “wave–particle duality”) is the requirement that quantum mechanical systems exhibit both wave and corpuscular characteristics, though never both at the same time. Echoes of these principles may be discerned in the discussion that follows. In Section 2.1 we discuss various perspectives on the common theme underlying the entries in Table 1, which we regard, perhaps controversially, as the deepest “cor- respondence principle” in mathematical physics. There is an elegance and a tidiness in the formulae of Table 1, but these formulae are in some sense at the edge of chaos, as we discuss in Section 2.2. Moving towards specific physical contexts, we discuss time-evolving classical and quantum processes (Section 3), before turning our atten- tion to questions of dilatational symmetry motivated by scattering data from quasicrystals (Section 4). The examples in Sections 3 and 4 all involve intrin- sically linear, non-cooperative phenomena and there is no explicit temperature dependence. In Section 5 we consider problems of quantum statistical mechanics, before concluding with perhaps the most elegant and intriguing appearance of our common theme in the con- text of Casimir forces (Section 6). A collection of useful formulae for the theta func- tions is given in Appendix A. The variety of contexts from which our examples are drawn have their own popular notations and characteristic terminologies. For the most part we are able to avoid different uses of the same symbol, however force of habit and prevailing idi- om oblige us to use τ in two different ways: as a com- plex number in the upper half-plane for the theory of theta functions and (in Section 4 and Appendix B) as the golden ratio (1 + √5)/2. For brevity we use the usual notations ℤ, ℕ, ℝ and ℂ for the integers, natural num- bers (i.e., the strictly positive integers), real numbers and complex numbers, respectively. All computations were performed with Mathematica. 2. THE MATHEMATICAL CONTEXT 2.1. Variations on a theme An infinite sum of periodically spaced delta func- tions, , corresponding to equally spaced “points” or “atoms” on a line, is one of the simplest con- ceptualizations of the atomic-scale granularity of real matter. Finite segments of such a function, stacked in two and three dimensions form visualizations of elemen- tary crystals and at large scales, where the granularity cannot be resolved, produce apparently smooth struc- tures. By purely formal Fourier analysis–though a proper derivation within the theory of generalized functions is available [9]–we shall represent as a (classically divergent) series of classical functions. As is periodic, computing its Fourier expansion in the usual manner using (1) yields (2) Although Eq. (2) is valid only in the sense of gen- eralized functions, it arises very cleanly as an extrapola- tion from a very classical result. Where (3) 53A Correspondence Principle denotes the third of the Jacobi theta functions [10] (see Appendix A) in what is now the traditional notation [11], the Jacobi theta function transformation (4) is valid for all z ∈ ℂ and for Im{τ} > 0. If we divide Eq. (4) by 2L and set τ = iε (ε > 0) and z = πx/(2L), we find that (5) For each fixed value of x, every term in the sum converges rapidly to zero as ε → 0, unless we have x = nL, in which case the nth term diverges, but we have showing that the right-hand side converges in an appro- priate sense to . We show the series (5) for ε = 1/16, 1/4, 1 and 4 in Fig. 1. The elegant identity (5) equates two conceptually distinct viewpoints: a sum of smooth waves and a sum of pulses that may be iden- tified as individual particles [12]. The particle interpreta- tion becomes increasingly more attractive as ε is reduced. The three other Jacobi theta functions have analogous transformations that connect classically divergent trigo- nometric series to periodically spaced delta functions [5]. Indeed a more general result can be obtained by writing (6) (so, for example, θ3(z|τ) = θ0,0(z/π |τ)) and noting that the generalization of Jacobi’s transformation, (7) leads to (8) The periodic delta function structures associated with the standard Jacobi theta functions θ1 θ2, θ3 and Table 1. Five essentially equivalent results, identifiable as a single theme that is central to a broad range of problems in classical and quan- tum physics. the correspondence principle or wave–particle duality theta function transformations (many equivalent or related forms) (classical) Poisson summation formula Riemann relation for the analytic continuation of transformation of Euler’s product 54 Barry D. Hughes and Barry W. Ninham θ4 are recovered by replacing (a, b) with (1/2, 1/2), (1/2, 0), (0, 0) and (0, 1/2), respectively. Some of these choices yield coefficients with alternating plus or minus signs in the string of delta functions, and so can represent microscopically charged but macroscopically neutral matter. The generalized function is sometimes called a (or the) “Dirac comb” and its impli- cations on the interpretation of diffraction data from solid crystals have received some attention [13, 14], espe- cially in the context of its invariance (up to dilation and multiplication) under Fourier transformation: (9) The generalized function identity (2) is some- times called the Poisson summation formula, a forgiv- able appropriation of terminology [15] that we shall not adopt. For us the Poisson summation formula is [16] (10) This follows immediately from the observation that Jacobi’s transformations, which we have seen pro- duce such things as generalized function identity (2), can also be regarded as consequences of Eq. (10). For exam- ple, by taking f (x) = exp(–πx2ε) in Eq. (10), we obtain θ3(0|iε) = ε−1/2θ3(0|iε−1). Riemann [17, 18] used this rela- tionship to establish his famous functional relationship (11) where the Riemann zeta function ζ(s) and the gamma function Γ(s) are defined initially by (12) and (13) and extend by analytic continuation to functions holo- morphic except for simple poles at s = 1 and at s = 0, 1, 2, …, respectively [11, 17, 19]. The Riemann zeta function is profoundly important in number theory, but surprisingly frequently encoun- tered also in physics [20]. Although a rigorous account of those of its properties that are rigorously established requires serious work [11, 19], some results fall out very simply [21]. Since inserting the binomial expansion and interchanging orders of summation (the double series is absolutely convergent for Re{s} > 1) we find after a little algebra that (14) Analy tic continuation of this result, which we obtained initially on the assumption that Re{s} > 1, shows immediately that Figure 1. The series (5) interpolates between a uniformly flat profile (ε → ∞), a continuous wave (finite ε) and a train of particles (ε → 0). We illustrate this with the cases ε = 1/16 (highest peaks), 1/4, 1 and 4 (nearly flat). −3 −2 −1 0 1 2 3 ε = 1/16 ε = 1/4 x/L 55A Correspondence Principle and so on, and the Riemann relation (11) then yields ζ(2) = π2/6, ζ(4) = π4/90, …, although closed-form elementary evaluations of ζ(3), ζ(5), … have never been found. Some of the known simple exact values of the zeta function will be needed in Section 6, as will the equation (15) that results from writing t = nx in Eq. (13) and summing over n. As noted in the Introduction, but worth emphasis- ing again, the five results collected in Table 1–four of which we have already discussed, with the fifth (an infi- nite product transformation)–are essentially a single result [5]. It is possible to obtain all of the results from any one of them, and they establish a link between many substantial fields of mathematics, including complex analysis, number theory, harmonic analysis and numeri- cal analysis [5, 22, 23]. It is the centrality of this com- mon theme to physics that we begin to address in Sec- tion 3. 2.2. Analytical irregularity There is surprising irregularity and complexity lurk- ing behind the five equivalent identities in Table 1. We illustrate this first by considering the special case of the theta function θ3(z|τ) with z = 0. If we write for brevity θ(τ) = θ3(0|τ), then θ(τ) is well-defined as a holomorphic (that is, complex-differentiable) function of the complex variable τ in the upper half plane Im{τ} > 0. From Eqs (3) and (4) we find that (16) Both of the transformations τ → τ + 1 and τ → –τ−1 are bijections of the upper half plane (that is, one-to-one correspondences between two copies of the upper half plane). These two fundamental transformations are the generators of a group of transformations of the upper half-plane known as the modular group [24]. Modu- lar transformations have the form τ ↦ (aτ + b)/(cτ + d), where a, b, c, d ∈ ℤ and ad − bc = 1. Figure 2(a) shows a subset 𝕄 of the upper half- plane known as the fundamental region for the modular group. Every point in the upper half-plane is the image of a point in 𝕄 under a modular transformation, but there is no modular transformation connecting any two points of 𝕄. Figure 2(b) shows the remarkable way in which successive applications of simple modular trans- formations carry M into regions of progressively smaller total area, located closer and closer to the real τ axis. It follows that along the line segment defined by τ = σ + iε, with –1 ≤ σ ≤ 1 and 0 < ε ≪ 1, there is enormous varia- tion in θ(σ + iε), as shown in Fig. 3. If we write q = eiπτ, the upper half-plane Re{τ} > 0 corresponds to the disk |q| < 1 and we have τ = −2 τ = 1τ = 0τ = −1 τ = 2 (a) (b) τ = −1 τ = 1 |τ| = 1e −2πi/3 e +2πi/3 Figure 2. (a) The shaded region shows the fundamental set 𝕄 for the modular group. We include only the right half of the boundary, that is, boundary points with 0 ≤ Re ≤ {τ} ≤ ½, shown as a dark curve. There are no modular transformations connecting any pair of distinct points in 𝕄. (b) The images of the fundamental set 𝕄 under the modular group tessellate the plane. We have shaded the right half of 𝕄 and all its images, while the left half of 𝕄 and all its images are left white, though their boundaries are drawn in gray. The images of 𝕄 shown here were obtained from those modular transformations with a = 0, b = –1, c = 1 and –2 ≤ d ≤ 2 (corresponding to τ' = τ + d, followed by τ'' = –1/τ'), or these transformations followed by a translation. The region close to the real axis is progressively filled as further transforma- tions of 𝕄 are considered, but it become increasingly hard to portray the images without increasing the magnification of the figure. 56 Barry D. Hughes and Barry W. Ninham (17) This is a power series in q, with the unit circle as its circle of convergence, and gaps of rapidly increasing length between powers of q with nonzero coefficients. Indeed, for fixed z, all of the theta functions θk(z|τ), k ∈ {1, 2, 3, 4}, have the form (18) where κ ∈ {0, 1/2} and either λn = n2 or λn = n(n + 1), with the series always convergent for |q| < 1 and always divergent for |q| > 1. More generally, if λn is a strictly increasing sequence of non-negative integers, then ∑n anqλn is a power series in the complex variable q. If λn/n → ∞ as n → ∞ the pow- er series is called a “lacunary series”, the name referring to the gaps between powers of q that have nonzero coef- ficients. A beautiful theorem of Fabry [25, 26] states that if ∑n anqλn is a lacunary power series with radius of con- vergence 1, then the function defined by f (q) = ∑n anqλn for |q| < 1 cannot be continued analytically beyond |q| = 1. As functions of q, the theta functions meet the condi- tions of Fabry’s Theorem. Analytic continuation across the unit circle |q| = 1 is prevented by the presence of a dense fence of singular points on this circle. Figure 3 manifests the existence of this fence. It is interesting that the five equivalent identities in Table 1, which involve either smooth functions or peri- odic functions, are the gateway to revealing dense, non- smooth behavior. 3. TIME-EVOLVING CLASSICAL AND QUANTUM PROCESSES Our point of departure in Section 2.1 was already associated with physical concepts, namely periodically spaced point masses or point charges, but no physical models or processes have really been addressed. 3.1. Classical Diffusion For –∞ < z < ∞ and Im{τ} > 0, all four Jacobi theta functions satisfy the partial differential equation (19) as indeed does the more general function θa,b(z/π |τ). If we take τ = (4D/π)it with t real, replace z by x, and write u(x, τ) = v(x, t), Eq. (19) reduces to the one-dimen- sional diffusion equation (20) The theta function transformations connect opti- mally structured short-time and long-time solutions of one-dimensional diffusion problems in finite domains, with one theta function expression corresponding to an expansion of the solution in spatial trigonometric func- tions with exponentially decaying time-dependent coef- ficients (a good solution from at long times) and the other corresponding to a “method of images” solution constructed from Gaussian propagators (a good solution at short times) [22]. For example, if we write ε = πDt/L2, then Eq. (5) equates these two solutions in the case of impenetrable reflecting boundaries (zero flux: –D∂v/∂x = 0) at x = ±L, and initial condition v(x, 0) = δ(x): Figure 3. We show the real part (blue curves) and the imaginary part (red curves) of θ3(σ + iε) for –1 ≤ σ ≤ 1: (a) ε = 0.1; (b) ε = 0.01; (c) ε = 0.001. 1 1 10 10 1 1 10 10 1 1 20 10 10 20 (a) (b) (c) 57A Correspondence Principle (21) 3.2. Anomalous diffusion In one dimension and in the absence of boundaries, the mean-square displacement for the diffusion process (20) grows linearly with time [27]: (22) The study of diffusion processes based on Eq. (20) was initiated by Fick [28] in 1855. Much more recently there has been intense interest in transport processes that are not diffusive in character [29, 30, 31, 32]. In one- dimensional unbiased non-diffusive processes the mean- square displacement may grow more slowly that linearly with time (sub-diffusive processes), or faster than line- arly (super-diffusive processes). An extreme case of one- dimensional super-diffusion has an infinite mean-square displacement and this can lead to a statistically self-sim- ilar or fractal [33] footprint structure (the set of points visited has a fractal dimension less than 1). Hughes, Shlesinger and Montroll [34] considered a random walk model in which the random displacement made at any step has the probability density function (23) with a > 1 and b > 1. Since motions on the length scale ∆bn are a times more abundant than motions on the next shortest length scale ∆bn+1, the stepping law has fractal character built in (with fractal dimension µ = ln(a)/ ln(b)), and the only question is whether fractal footprints are left visible at long times, or the legacy of the walk is smeared. If µ < 2 the mean-square displace- ment per step is infinite, the central limit theorem fails, and the continuum limit of the process does not have the standard Gaussian propagator familiar from classi- cal diffusion [30, 35]. The walk is transient if µ < 1 (any interval is visited only finitely many times with prob- ability 1) and fractal footprints are left. To analyze features of this model, it is necessary to understand the behavior near the origin of the Fourier transform of the probability density function (23), and this is equivalent to requiring the small-k behavior of (24) It is easy to see that λ(k) satisfies the rather simple- looking functional equation (25) which is reminiscent of equations obtained in real-space renormalization treatments of lattice spin systems [36, 37]. The apparent simplicity of the functional equation is illusory. Hughes et al. [34] were able to show using the Poisson summation formula (10) that for k > 0 and ln(a)/ ln(b) ∉ {2, 4, 6, · · · }, (26) where [38] (27) and we have written for brevity sn = –µ + 2nπi/ ln b. The appearance in Q(k) of “log–periodic oscillations” (peri- odic in ln k with period ln b) is striking (see Fig. 4). Similar oscillations occur in real-space renormalisation group transformations for lattice spin systems [39], in a model for the distribution of family names in a society [40] and in a variety of other systems that exhibit a form of discrete scale invariance [41]. Figure 4. The structure function (24) of the Weierstrass random walk step probability density function (23). In each case, b = 2, and we choose a values so that so that µ = ln(a)/ ln(b) takes the val- ues 0.5 [blue (most irregular) curve], 1.5 (red curve) and 2.5 (gold curve). 1 2 −0.5 0 0.5 1 μ = 2.5 μ = 1.5 μ = 0.5 k/π λ(k) 58 Barry D. Hughes and Barry W. Ninham If b is a positive integer and b ≥ 2 then we can rec- ognize λ(k) as a constant multiple of the real part of the lacunary power series evaluated on its circle of convergence (|z| = 1), so a dense set of singular points must be present and indeed for appropriate values of a and b the series for λ(k) is the celebrated nowhere-differ- entiable function of Weierstrass [42]. There is a second perspective on Eq. (23) that is also worth considering [30, 34]. By considering the contour integral of e–zzs−1 around a simple closed contour in the z-plane consisting of the arc of the circle |z| = R within the first quadrant, and straight lines along the real and imaginary axis linking the ends of the arc to the origin, it is easy to prove that for 0 < Re{s} < 1, (28) Adding this equation and its complex conjugate we find that for 0 < Re{s} < 1, (29) The definition of the Mellin transform and the asso- ciated inversion formula [16], (30) (31) with the Bromwich contour Re {s} = c placed inside a strip in which the Mellin transform integral converges, are another manifestation of the relations collected in Table 1, since they can be used to obtain both the Rie- mann relation and the theta function transformation in relatively straightforward ways. Using Eqs (29) and (31) we can write (32) for k > 0 and 0 < c < 1. Inserting this representation into Eq. (24), interchanging the order of integration and summation and recognizing a geometric series, we find [30, 34] that (33) Translating the contour of integration to the left and taking account of the residues at the poles crossed, we recover Eqs (26) and (27). The power series arises from the simple poles along the real axis at s = 0, –2, –4, …, while kµQ(k) comes from the line of poles at sn = –µ + 2nπi/ ln b. The small-k behavior, which governs the limiting behavior of the random walk, is dominated by which pole or poles the contour next encounters after we have translated it past the origin. For µ > 2 the next pole encountered is a simple pole at s = –2, so that 1 - λ(k) ∝ k2 as k → 0 (ensuring a diffusive limit). However for 0 < µ < 2 we meet the line of poles at s = sn, and this is how the term kµQ(k) arises, precluding diffusion. These kinds of calculations using Mellin trans- forms are closely connected to the powerful role of Mel- lin transforms in asymptotic analysis [43] and also give one link between several identities in Table 1. Whichever approach is used to reveal the small-k behavior of λ(k), the simplest limiting behavior is obtained as ∆ → 0 and t0 → 0 (where t0 is the time between successive steps) if we also make a → 1 and b → 1, while holding both µ = ln a/ ln b and ∆µ/t0 constant. Then if µ < 2, the evolution of the random position Xt of the moving agent satisfies (34) where c is a positive real constant, q ∈ ℝ, and E denotes mathematical expectation or averaging. The “charac- teristic function” E{exp(iqXt)} is just a spatial Fourier transform of the probability density function for the agent’s location at time t. Solving the evolution equation (34) with the initial condition X0 = 0 gives E{exp(iqXt)} = exp(-c|q|µt) and inverting the Fourier transform gives the celebrated symmetric stable densites [29, 30] of Lévy [44], (35) The borderline case µ = 2 corresponds to the Gauss- ian density, while for µ < 2, the density decays algebrai- cally rather than exponentially, with Pr{Xt > x} ∝ x–µ as x → ∞. The only other case where the symmetric stable density has a simple closed form expression [45] is µ = 1, which is the Cauchy density (c/π)(x2 + c2)–1. Super-diffusive processes, such as the stable density, are naturally formulated in unbounded space, but it may be of interest to seek solutions in finite intervals. Appro- priate boundary conditions for ref lecting boundaries are debatable (for µ < 1 the path is discontinuous), but we can use method of images arguments [cf. Eq. (21)] 59A Correspondence Principle to obtain a solution which conserves probability in the interval (–L, L). The following analysis is very much in the sense of generalised functions, as we work with clas- sically divergent series and use the identity (2): (36) It is perhaps curious that for this system, the relax- ation to the uniform density 1/(2L) on the interval is a simple exponential, rather than some form of stretched exponential, despite the transport process being highly super-diffusive. Clearly there are many subtleties that can arise when stochastic ideas intersect with self-similarity. For another manifestation of this, see Appendix B. 3.3. One quantum particle Let h denote Planck ’s constant and = h/(2π). If we write τ = –2 t/(πm) with t complex (with a negative imaginary part) and u(z, τ) = ψ(z, t), we obtain from Eq. (19) the one-dimensional Schrödinger equation in free space, (37) Thus linear combinations of theta functions with the complex time extrapolated to the real axis should be able to be used to construct non-trivial time-dependent solu- tions of Schrödinger equation. Although we are aware of no systematic study of this, for an investigation of some cognate issues the reader may consult the beautiful paper of Fulling and Güntürk [46] on the one-dimen- sional Schrödinger equation with periodic boundary conditions. Less direct applications of theta functions to solving Schrödinger’s equation have been considered by Gaveau and Schulman [47]. The formal connection between theta functions and Schrödinger’s equation (obtained by letting the artificial negative imaginary part of the time approach 0) corre- sponds to moving radially outwards towards the circle of convergence of a lacunary series, as discussed in Sec- tion 2.2. The highly irregular form of the propagator dis- cussed by Fulling and Güntürk should therefore come as no surprise. If we don’t observe the connection to theta func- tions, and instead use an energy eigenfunction approach to solve the d-dimensiona l Schrödinger equation in the box (0, L)d (with the wave function vanishing on the boundary) we obtain the general solution (38) where n = (n1, n2, … , nd) and (39) If we take the initial condition ψ(x, t') = δ(x – x') with x' ∈ (0, L)d , we obtain the (never classically conver- gent) generalized function propagator × (40) Perhaps the connection to theta functions makes the strangeness of this result easier to comprehend. Despite the irregular propagator for finite intervals, the free-space Schrödinger equation does have some comparatively simple, well-behaved normalizable solu- tions on the real line, such as the spreading Gaussian wave packet found by Darwin [48] and Kennard [49] and the Airy function solution of Lekner [50]. 4. QUASICRYSTALS Diffraction experiments probe the structure of con- densed matter using electrons, neutrons or X rays. In systems with long- range order, this order is revealed by observed intensity distributions exhibiting sharp peaks [51, 52]. Experimental realities and the finiteness 60 Barry D. Hughes and Barry W. Ninham of the atoms scattering the incident radiation broaden the peaks, but basically in an idealized but substantially correct way, the observed density in many cases is a set of delta functions, whose locations encode information about the atomic locations, which are also delta func- tions. This picture is clearest and most apt for crystals, where the diffraction data corresponds to the Fourier transform of the crystal [53]. For the one-dimensional case with equal spacing between atoms, Eq. (9) shows that the Fourier transform is simply the original lattice structure with a changed lattice spacing (the Fourier transform of the Dirac comb is a Dirac comb). Similar results hold in two and three dimensions [54]. The discovery in 1984 by Shechtman et al. [55] of a metallic phase with long-range orientational order but no translational symmetry challenged established para- digms in crystallography, which assume that crystals consist of unit cells of atoms of various species arranged periodically. Within six weeks, Levine and Steinhardt [56] had dubbed these structures “quasicrystals”–a name the structures have retained [57]–and suggested analo- gies to nonperiodic tilings of space with local pentagonal symmetry previously studied by Penrose [58]. Shecht- man received the 2011 Nobel Prize in Chemistry for the discovery of quasicrystals. The appearance in a physical context of long-range order without translational symmetry naturally motivat- ed a number of fundamental studies of a purely math- ematical character, including a careful definition of dif- fraction on aperiodic structures [59]. Many important observations concerning cognate mathematical issues can be found in Senechal [60, 61], Senechal and Taylor [62, 63] and Baake and Grimm [64]. When translational invariance breaks down in the observed crystallographic data, the unambiguous con- nection to the original lattice is lost. How is the struc- ture of a quasicrystal to be inferred? To begin, the attempt to fit the diffraction data (k) to delta func- tions with a non-zero minimal spacing and recover well- spaced delta functions for ρ(r) is doomed to failure [65]. Ninham and Lidin [66] suggested the possible rel- evance to quasicrystals of dilatational rather than trans- lational symmetry, using the following example, which has interesting historical antecedents. The gnomon (γνὡµων) is the shadow-casting blade on a sundial, but also refers to triangles or rectangles produced by inter- nal subdivision of triangles or rectangles in a special way. In particular, if an isosceles triangle with two sides of length τ > 1 and third side of length 1 is subdivided by drawing a straight line from one of the equal angles to the opposite face to create an isosceles triangle with two sides of length 1, the other triangle created in this subdivision is the gnomon. A simple argument based on similar triangles establishes that the gnomon is itself an isosceles triangle if and only if τ2 – τ – 1 = 0, (41) from which it follows that τ = (1 + √5)/2 ≈ 1.618 (42) (see Fig. 5(a), in which the gnomon is shaded in gray). For this special choice of τ, the internal angles of the tri- angles produced in the subdivision are all integer mul- tiples of π/5, as shown. In Fig. 5(b) we take the scaled replica of the original triangle produced by the subdivi- sion, subdivide it in a similar manner, and repeat this process several times, always producing isosceles trian- gles with the same angles, but with the lengths of sides diminishing by a factor of τ at each stage. The number τ is the famous golden mean, golden ratio or golden num- ber, which figures prominently in aesthetics [67] and in nature [68]. The logarithmic spirals (43) are shown as grey curves in Fig. 5(c). Their intersections generate a distribution of points with five-fold rotational symmetry about the origin. The logarithmic spiral ln r/ ln τ = θ/(3π/5) passes through these points of intersec- tion, with the distance from the origin increasing by a factor of τ between any two consecutive intersections. With suitable scaling and rotation, the inscribed trian- gles shown in Fig. 5(b) can be placed with their vertices located at the intersection points [66, 69]. We consider the Fourier-space signature of the mass distribution × (44) which places all mass on rays through the origin, with angular separation π/5 between rays, and on each ray, we have dilational invariance in the locations of the masses, with a scaling factor τ. The convergence factor a–|m| (with a > 1) is present to keep finite total mass in the system. We find that where k = (k1, k2), 61A Correspondence Principle (45) We show |ρ(k)| for a = 1.1 in Fig. 6. It is not surpris- ing that the rotational symmetry in the mass distribu- tion is reflected in the Fourier transform domain: this is clear from Eq. (45). What is more interesting, and more beautiful, is that in the Fourier transform domain, where the signal is continuous (rather than localized on lines, as in the original space domain), we see a rich structure with local intensity maxima occurring at many points in the sectors between the ten lines on which the brightest peaks are located. Also, it is by no means obvi- ous from the formula (45) where the intensity maxima on the bright lines will occur. In Fig. 7, we show |ρ(k)| on the vertical axis in the k-plane. There are many local maxima, but a sequence of locally outstanding maxi- ma can be identified at the k2 values 4.775, 7.732, 12.51, 20.25, 32.77. The successive ratios of these k2 values are all close to (but not exactly) 1.618, and we see the golden ratio from physical space recurring (to a decent approxi- mation) in intensity maxima in Fourier space. The convergence factor a|m| in Eq. (45) stops | (k)| from having exact dilatational symmetry. If we were able to set a = 1, then we would recover | (k)| = | (τk)|. Berry and Lewis [70] have considered what they call the Weierstrass–Mandelbrot fractal function (46) where ϕn represents a constant phase added onto each term. The series is convergent and, if ϕn is constant, has perfect self-similarity: W(γt) = γ2−DW(t). Ninham and Lidin [66] have considered another way of overcoming the problem of infinite mass accumulating in the neigh- borhood of the origin by using the formal series Figure 5. (a) With τ given by Eq. (42), an isosceles triangle with side lengths ratios τ : τ : 1, can be subdivided into two isosceles triangles, one of which (white interior) has side length ratios τ : τ : 1 but side lengths a factor τ smaller than those in the original triangle. (b) We can continue the process of subdivision to generate a nested set of isosceles triangles with side length ratios τ : τ : 1, but at each step of the pro- cess, the side length of the triangle just produced is reduced from that of its parent by a factor of τ. (c) The intersections of the logarithmic spirals (43) generate a distribution of points with five-fold rotational symmetry about the origin. The logarithmic spiral ln r/ ln τ = θ/(3π/5), shown in red, passes through these points of intersection, with the distance from the origin increasing by a factor of τ between any two consecutive intersections. With suitable scaling and rotation, the inscribed triangles shown in diagram (b) can be placed with their vertices located at the intersection points (figure adapted from Ninham and Lidin [66]). (b) (a) τ π/5π/5 π/5 2π/5 1 1 1/τ τ − 1/τ = 1 (c) 62 Barry D. Hughes and Barry W. Ninham for the mass distribution along a ray through the origin, where r is the distance from the origin. Quasicrystals are not the only context in which wild oscillations and apparent self-similar structure arise in the amplitude of diffracted light. Berry [71] gives a beau- tiful example, in which theta functions play a key role. 5. QUANTUM STATISTICAL MECHANICS We consider several models from quantum statisti- cal mechanics, for which we use standard notation and terminology [72, 73], so that k is Boltzmann’s constant and T is the absolute temperature. 5.1. The harmonic oscillator The free energy g(ω) associated with a harmonic oscillator of frequency ω and energy levels (n + 1/2) ω (n = 0, 1, 2, …) is given in terms of the canonical parti- tion function (ω) by (47) (48) For brevity, we have suppressed in the notation the dependence of the free energy on the temperature. Hence (49) Figure 7. We show | (k)| on the line k = (0, k2), where the Fourier transform (k), given by Eq. (45) arises from the mass distribution (44). For the convergence factor a–|m| we have taken a = 1.1. 10 5 0 −5 −10 4 2 0 −2 −4 1050−5−10 420−2−4 (a) (b) Figure 6. The k-plane is colored (with 20 levels) to show | (k)|, where the Fourier transform (k), given by Eq. (45), arises from the mass distribution (44). Lighter shades represent larger values of | (k)|. (a) –10 ≤ k1, k2 ≤ 10; (b) –4 ≤ k1, k2 ≤ 4. For the convergence factor a|m| we have taken a = 1.1. 10 20 30 40 20 40 60 80 k 2 |ρ(k)| 63A Correspondence Principle leading to the formal identification [74] (50) This superficially bizarre result connecting the oscil- lator free energy to a string of delta functions, arising from the mathematical correspondence principle, proves surprisingly useful. If the modes of oscillation of a sys- tem are given by a secular equation of the form D(ω) = 0, then the free energy can be computed as a sum over the contributions from the various modes by the contour integral (51) the contour integral being taken over a simple closed contour that surrounds all zeros of D(ω) on the posi- tive real axis. If there are infinitely many such zeros with the spacing bounded below as ω → ∞, an appropri- ate limiting construction is made. Integrating by parts, deforming the contour and making formal use of Eq. (50) enables the free energy to be computed conveniently [74, 75]. This is especially convenient in the calculation of dispersion (van der Waals) forces between dielectric media [74, 75]. 5.2. Particle in a box Using the energy eigenvalues (39), the free energy associated with a single (non-elementary [76]) particle of mass m in the d-dimensional box [0, L]d is given by (52) Here we adopt the usual notational convenience of writing θk(z|τ) = θk(z, q), where q = eiπτ. If we consider N identical non-interacting particles in the same box, Eq. (52) becomes the equation for the free energy per particle. Fixing T, the right-hand side can be evaluated asymptotically in the limit L → ∞, using the Jacobi theta function transformation (4), which in the special case z = 0 and τ = it (t ∈ ℝ, with t > 0) becomes θ3(0, it) = t–1/2θ3(0|it–1). We find that (53) (54) where for brevity in notation we have introduced the thermal wavelength λT = h(2πmkT)–1/2. The single-term approximation (54) is well known [73], but the theta function representations (52) and (53) enable us to com- pute and the associated thermodynamic observables to high precision for any value of L/λT. 5.3. Ideal gas of elementary particles Consider now ideal gases of elementary particles, which may be bosons (such as photons or mesons, for which an arbitrary number of particles can occupy any state) or fermions (such as electrons or neutrinos, for which any state may be occupied by at most one parti- cle). It is more convenient to work with the grand parti- tion function = Πn n, where n is the canonical par- tition function for occupancy of the nth state, in which each particle present has energy εn and chemical poten- tial µ. Thus we have (bosons); (fermions). If we define the fugacity as usual by z = exp[µ/(kT)] we obtain [77] (bosons); (fermions). Consider the case of fugacity z = 1. If zero point energy is neglected and we write εn = n ω as in black- body radiation, then on writing x = exp[– ω/(kT)] we find [5] that 64 Barry D. Hughes and Barry W. Ninham (55) where Euler’s product φ(x) = , intimately related to the Jacobi theta functions, is discussed in Appendix A and has the transformation formula (56) which is one of the mathematical correspondence prin- ciple relations from Table 1. The original expressions for the grand partition function are able to be used for com- putation for high temperatures, but are useless at low temperatures. However, Eq. (56) gives immediate access to low temperature expansions, since we can eliminate φ(exp[– ω/(kT)]) in favor of φ(exp[-4π2kT/( ω)]). Planat [78] has pursued the implications of the relation between Euler’s product and the theory of partitions [79] in the context of the massless Bose gas. If we retain the zero point energy and consider a general value of the fugacity, we can represent the grand partition functions in terms of the q-Pochhammer func- tion [80] (57) Building on the work of Rogers and Ramanujan [81], there is now an impressive corpus of transformations and identities related to theta functions and q-Pochham- mer functions, and they arise frequently in mathemati- cal physics [82]. 6. CASIMIR FORCES The famous prediction of Casimir [83, 84] that the zero temperature energy of interaction of two perfectly conducting plates a distance apart in vacuum provides an attractive force per unit area π2 c/(240 4) between the plates was a landmark result. Direct experimental verifi- cation was challenging, with Sparnaay [85] in 1958 find- ing that “the attractive interactions do not contradict Casimir’s theoretical prediction” (the experiments had problematically large uncertainty). Finally, in 1997, Lam- oreaux [86] effectively settled the basic issue [87]: “we have given an unambiguous demonstration of the Casimir force with accuracy of order 5%. Our data is not of suffi- cient accuracy to demonstrate the finite temperature cor- rection …”. (Casimir’s original discussion did not address either finite temperature nor limitations on conductivity.) Crudely described, the Casimir effect demonstrates the consequences of geometrically constraining free oscilla- tions of a system (here, the electromagnetic field) com- pared to the unconstrained state. Entirely classical ana- logues of the Casimir effect in macroscopic physics have been identified in a maritime context [88], and in an acoustic system suitable for lecture demonstrations [89]. The literature related to the Casimir effect is already voluminous and connections of papers with apparently cognate keywords to physics can be highly tenuous. At one extreme end of the literature [90, 91], since the eval- uations of ζ(–1) and ζ(–3) are needed in the discussion of the physical Casimir effect (depending on the geometry), the evaluation for s = –1 of the analytic continuation of a series of the form Z(s) = ∑λλ–s, where λ runs through a set of values with an interpretation related to energy levels, has been called the “Casimir energy”. The sign of Z(–1) “reflects certain dynamical and arithmetical prop- erties” [91] and formulae related to the so-called Casimir energy can be obtained for compact Riemann surfaces of genus g ≥ 2. Of greater physical interest is the embedding of the original Casimir effect in a broader context that admits predictions of interactions between more general classes of matter than perfect conductors at zero temperature [75, 92]. Profoundly important papers by Lifshitz [93, 94, 95] and his subsequent work with Dzyaloshinskii and Pitaevskii [96, 97], also appearing in a later textbook [98], replaced perfect conductors in vacuum by dielectric materials separated by an intervening dielectric mate- rial. By permitting the dielectrics to have a frequency- dependent dielectric susceptibility, a wide variety of physical (and even biological) systems could be dis- cussed, and subsequent work of Ninham and Parsegian [99, 100] showed how the required dielectric proper- ties could be determined from spectroscopic data, lead- ing to the ability to make quantitative predictions in experimentally accessible systems. The original Casimir problem arises as an extreme limit of the Lifshitz theory approach, and Lifshitz theory permits the computation of temperature-dependent effects [74, 75, 101]. Experi- mental validation of the predictions of Lifshitz theory has been obtained in many cases [102]. We focus on the original Casimir problem–plates separated by vacuum–because it exhibits most simply 65A Correspondence Principle the importance of the mathematical correspondence principle. Let F( , T ) denote the free energy of interac- tion per unit area between two infinite parallel conduct- ing plates separated by a distance , in vacuum, at finite temperature T. It is instructive to see how extensive use of results of classical analysis such as the Riemann rela- tion for the zeta function and various properties of the gamma function enable the free energy to be expressed in a highly informative way that enables dangerous issues concerning non-uniformity of asymptotic expan- sions to be dealt with [103]. Where we have already set the dielectric constant of the region between the plates to be unity, then from Lifshitz theory [75] we have (58) where the prime on the sum indicates that the n = 0 term is to be weighted with a factor of 1/2, the param- eter ξn is defined by (59) and (60) To avoid an indeterminacy [104] in the case n = 0, we evaluate I(ξn, ) for small positive real n by use of the change of variables y = 2pξn /c and then take the limit n → 0. The Riemann zeta function first arises from the n = 0 contribution from the s = 3 case of the integral (15). For convenience in the asymptotic analysis we write (61) The coupling between the temperature T and the plate spacing is very important. The limit x → 0 corresponds to the low-temperature limit, provided that the plate sep- aration is constrained, or to the small-spacing limit, pro- vided that the temperature is not too high. The analysis of Ninham and Daicic [103] to this point has (62) To evaluate the sum over n we may begin by expanding the logarithm using the series Since Eq. (13) shows that the gamma function is the Mellin transform [43] of the decaying exponential, using the Mellin inversion theorem [Eqs (30) and (31)] we have the contour integral representation with the positive constant κ that places the vertical Bro- mwich contour Re(s) = κ selected to secure convergence in the subsequent analysis based on this integral (κ > 3 suffices). We now have so we can eliminate the logarithm factor from the inte- grand in Eq. (62), evaluate the resulting elementary integral over p and recognize the sums over m and n as series for the Riemann zeta function [Eq. (12)]. In this way one arrives at the scaled free energy (63) (64) The integrand has only four singularities, namely the simple poles at s = –1, 0, 2 and 3. To see this, we note that the gamma function has simple poles of residue (–1)j/j! at s = –j(j = 0, 1, 2, 3, …), while ζ(s – 2)ζ(s + 1) has a simple pole at s = 3 and simple zeros at s = –2, –3, –4, … (noting that ζ(z) has a simple pole at z = 1 and simple zeros at z = –2, -4, –6, …). The Bromwich contour may be translated an arbi- trary finite distance, provided that we account correct- ly for the residues at poles across which the contour is dragged. If we move the contour to Re(s) = 1, then the term that must be added to account for the pole at s = 2 is easily shown to cancel with the first term on the right in Eq. (64). The pole at s = 3 leads to a term proportional to 1/x2, whose coefficient can be evaluated by recalling that ζ(0) = –1/2 and ζ(4) = π4/90. We find that 66 Barry D. Hughes and Barry W. Ninham (65) Since ζ(s – 2) = –2s−2πs−3sin(πs/2)Γ(3 – s)ζ(3 – s) from the Riemann relation (11) and (66) we deduce that (67) where the first term on the right corresponds to the Casimir formula, while (68) the change of variables s = 1 + it produces (69) Since ln(x) = −ln(1/x), Eq. (69) reveals the remark- able inversion symmetry J(x) = J(x–1). (70) The Casimir term is temperature-independent when one returns to the original variables, but the function J(x) encapsulates genuine temperature dependence. The Riemann relation, one avatar of our central theme sum- marized in Table 1, has been used to expose the inver- sion symmetry (a previously observed result [107, 108]), but is also crucial for an efficient extraction of the x dependence. It may be noted in passing that a computer algebra software (such as Mathematica) is very helpful in checking that the intricate manipulations involved are correct, but is presently (and in the foreseeable future may well continue to be) unable to offer much help in guiding the analysis. We digress for a moment. The function ζ(2 + it)ζ(2 –it) in the integrand in Eq. (69) is easily shown to be real- valued, but is by no means simple in structure: see Fig. 8. In qualitative terms, it appears roughly periodic, but the amplitudes of successive peaks and troughs and their spacing vary in an apparently random manner. More generally, for real σ and t, we have ζ(σ + it)ζ(σ – it) = |ζ(σ + it)|2. (71) and the strange behavior of |ζ(2 + it)|2 revealed in Fig. 8 arises also for other values of σ. When study- ing the response to changes in σ it is helpful to consider |ζ(σ + it)| rather than |ζ(σ + it)|2 to reduce the height of the maxima. We plot |ζ(σ + it)| in Fig. 9 for σ = 2, σ = 1 and σ = 0.5. It may be observed that the spacing of the peaks and troughs hardly changes, but the peak heights grow and the trough heights fall as σ decreases. It is relatively easy to prove that ζ(σ + it) is nonzero whenever σ > 1. That ζ(σ + it) is also never 0 for σ = 1 is much more diffi- cult to prove. Establishing this was an essential ingredient of the proofs in 1896 by Hadamard [105] and de la Vallée 20 40 60 80 100 0.5 1.0 1.5 2.0 2.5 t ζ(2+it)ζ(2−it) 10 20 30 40 50 60 70 80 90 100 1 2 3 4 5 σ = 0.5 σ = 1 σ = 2 t |ζ| Figure 8. The function ζ(2 + it)ζ(2 – it) in the integrand in Eq. (69): 0 < ζ(2 + it)ζ(2 – it) ≤ ζ(2)2 = π4/36 ≈ 2.7 for all t ∈ ℝ. Figure 9. |ζ(σ + it)| = for σ ∈ {1/2, 1, 2}, 0 < t < 100. 67A Correspondence Principle Poussin [106] that the number of prime numbers less than or equal to n has the asymptotic form n/ln n as n → ∞. In the context of our discussion, the still-unresolved Riemann hypothesis [17, 19], asserts that ζ(σ + it) ≠ 0 for σ > 1/2. It is strangely beautiful that the mathematics of the Casimir effect comes so close to such subtle matters. Returning to matters more overtly connected to physics, if FCasimir denotes the original single-term Casimir energy prediction for the energy per unit area, we have (72) In assessing the accuracy of the Casimir term, the role of the composite parameter x is crucial. We note that with measured in metres and T in Kelvin, we have (73) In principle the numerical evaluation of the integral (69) requires some care because of the rapid oscillation of the cosine factor when x ≫ 1 or 0 < x ≪ 1 and the erratic behavior of the real-valued function ζ(2 + it)ζ(2 – it) (see Fig. 8). However, the Riemann–Lebesgue Lemma ensures that J(x) → 0 as x → 0 or as x → ∞ and for the region where J(x) differs perceptibly from zero, Mathe- matica is up to the task. We show J(x) for 10–2 ≤ x ≤ 102 in Fig. 10. The fractional error η in the one-term Casimir formula, defined in Eq. (72), exceeds 1 for x > 1.14 but is less than 5 × 10–5 for x ≤ 0.03. In applications of the Casimir formula to experi- mental situations, relative errors associated with finite conductivity, departure of the real geometry from infi- nite parallel plates and other practical realities may dominate over the errors arising from the finiteness of the temperature that we have quantified through η(x). Having acknowledged that caveat, we note that the case x ≈ 0.5 arises for atomic dimensions ( ≈ 10-10m) when T ≈ 6 × 106 K, within the range of estimated temperatures for the sun (≈ 4 × 103K at the surface, 1.6 × 107K at the center [109]). The analytic structures that have been revealed with the techniques illustrated above have a number of inter- esting consequences for the Casimir problem in vacuum and for analogous problems involving dielectric or con- ducting films. Some of these, including connections with nuclear and particle physics, have been explored else- where [110, 111]. The point to be made is that viewing physical problems from a mathematical perspective in the spirit of Table 1 leads both to efficient practical anal- ysis and to new insights, though deep and subtle math- ematical exotica are seldom far away. 7. CONCLUSIONS We opened this paper with a reference to the conun- drum of the surprising effectiveness of mathematics in physics. Berry has proposed one possible explanation [112]. We are beings of finite intelligence in an infinite inscruta- ble universe. In science, our individual intelligences cooper- ate, and we can understand more. But still, we are able to comprehend only those structures in the natural world that mirror our mental constructs. And at any stage of human- ity’s development, the most sophisticated constructs are those of our mathematics. Therefore our deepest penetra- tion into the natural world is limited by our latest math- ematics. As mathematics develops, more subtle features of –2 –1 1 2 – 0.04 – 0.03 – 0.02 – 0.01 log (x) 10 J(x) 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 x η(x) Figure 10. The function J(x) computed from the integral (69) by numerical integration. Figure 11. The fractional error η(x) in the single-term Casimir for- mula, defined by Eq. (72), inferred from the numerically evaluated integral (69). 68 Barry D. Hughes and Barry W. Ninham the universe become accessible to our understanding… So, ‘The unreasonable effectiveness of mathematics in the natu- ral sciences’ is not unreasonable at all; on the contrary, it is inevitable. This is not the only possible explanation, and in some areas, there are credible alternatives. We have shown that what is essentially one grand mathemati- cal idea, which comes expressed in various ways, such as those collected in Table 1, underlies a wide range of apparently disparate physical phenomena. If one is a mathematical platonist–that is, a believer in the exist- ence of abstract mathematical objects that are independ- ent of intelligent agents and their language, thought and practices [113]–it is perhaps not such a leap of faith to conceive of a profound connection between some of these mathematical objects and physical reality. Amongst a charming collection of pithy quotes and witticisms relevant to science collected by Berry [114] one finds what he calls “three laws of discovery”. 1. Discoveries are rarely attributed to the correct per- son [115]. 2. Nothing is ever discovered for the first time [116]. 3. Everything of importance has been said before by someone who did not discover it [117]. The existence of a common underlying mathemati- cal theme in many contexts may be an explanation for the applicability of Berry’s laws in the sociology of physics. Most physicists will wisely choose to limit their pondering of metaphysical questions to the bar or cof- fee shop, but the contemplation of what may be the most natural mathematical framework for physical theory and the pursuit of the implications of the mathematics is a more defensible use of one’s office hours. While physi- cal intuition and accumulated conventional wisdom are always worthy of respect, careful analyses with appro- priate mathematical insight can yield surprising results. Recently, Lekner [118] has shown that at small separa- tions, charged conducting spheres always attract each other, even when the charges on the spheres are of the same sign, except when the spheres have charges in the ratio that would make them an equipotential sur- face on contact. This refutation of the rule that “like charges repel” in classical physics is indeed striking. In quantum mechanics, where physical intuition is a more contentious matter (and in the view of many, not even appropriate), accumulated conventional wisdom has still developed, but as noted recently by Ball [119] it is by no means a settled matter that we have either the optimal perspective on the subject or the optimal formulation. What is the appropriate mathematical training for the modern physicist may be hotly debated, but what we have styled a correspondence principle (embodied in Table 1) and the associated treasures of classical real and complex analysis have legitimate claim for inclusion. ACKNOWLEDGMENTS We are grateful to the referees for helpful comments, and for bringing several important references to our attention. APPENDIX A. JACOBI THETA FUNCTIONS Where Im{τ} > 0 to secure convergence in the sense of classical analysis and q = eiπτ (so that |q| < 1), the four Jacobi theta functions θk(z|τ) = θk(z, q) are 69A Correspondence Principle Here and Euler’s product φ(x) = (1 – xn) has the remark- able property [5] that Where 0 < arg(τ) < π, the Jacobi transformation for- mulae are We observe that in a formal sense, if not with- in classical analysis (with the limits taken under the restrictions that |q| < 1 or Re{τ} > 0, respectively) that APPENDIX B. DIFFUSION WITH INCREASING LETHARGY There is another interesting model for a kind of anomalous diffusion that raises mathematical questions that are at least as subtle as those discussed in Section 3.2 and partly related to them [120]. Consider a random walk in one dimension, for which steps to the left and right are equally likely at each stage, but the length of the nth step is cn, where cn > 0 and . The walker exhibits increasing lethargy, and asymptotical- ly comes to rest at a random position X relative to the starting location, where X = , the random vari- ables {εn} are independent, and Pr{εn = 1} = Pr{εn = –1} = 1/2. In the probability literature, the random variable X is described as an “infinite Bernoulli convolution” and it can be proved [121] that provided that < ∞, the characteristic function (Fourier transform) associated with the distribution of this random variable exists, and is given by E{exp(iqX)} = . The case cn = αn−1, where 0 < α < 1, for which E{exp(iqX)} = , is especially fascinating [122, 123, 124, 125]. This model builds in self-similarity in a way reminiscent of, but different to, Section 3.2. We have X = ε1 + αX1, where X and X1 are identically dis- tributed random variables, while ε1 and X1 are independ- ent. Since = (1 – α)−1, we know that –(1 – α)−1 ≤ X ≤ (1 – α)−1. For the case α = 1/2, it can be proved [127] that X is uniformly distributed on [–2, 2], but for many other values of α ∈ (0, 1) the distribution of X does not apportion the probability so smoothly. In general [126], the cumulative distribution func- tion F(x) = Pr{X ≤ x} of a real random variable X con- sists of either a single one, or a linear combination of both, of the following components: (i) an “absolutely continuous” component, corresponding to classical probability density function; (ii) a “singular” component, in which the probability all resides on a set of measure zero. For the increasingly lethargic walk with 0 < α < 1/2, X has a singular distribution: the nonzero probabil- ity all resides on a Cantor set of measure zero [123]. The rapid attenuation of the step lengths prevents the walker from exploring the apparent support decently. For 1/2 < α < 1, it has been proved [122] that the distribution of X for any given value of α is either entirely absolutely con- tinuous or entirely singular. The simplicity of the case α = 1/2 might suggest that absolute continuity always prevails for 1/2 < α < 1, and it has been proven that the set of values of α ∈ (1/2, 1) for which the distribution is singular has measure zero [128], but a countable number of values of α ∈ (1/2, 1) for which the distribution is sin- gular were found by Erdös [124] and the search for other anomalous α values continues. Diffusion with accumulating lethargy also has inter- esting connections to the discussion of Section 4, where the golden ratio τ = (1 + √5)/2 plays a significant role. Hu [129] has shown that for α = 1/τ = (√5 − 1)/2 and for −(1 − α)−1 < x < (1 − α)−1, the local fractal dimension d(x) = limr→0+ + log[Pr{x – r ≤ X ≤ x + r}]/ log(r) of the distribution of the random variable has maxi- mum value log(2)/log(τ) ≈ 1.4404 and minimum value log(2)/log(τ) – 1/2 ≈ 0.9404. For additional related results see Lau and Ngai [130]. 70 Barry D. Hughes and Barry W. Ninham REFERENCES [1] The quotation is from the retirement speech given by Sir Charles Frank (1911–1998) at the Univer- sity of Bristol in 1976, as recorded by M.V. Berry, “Bristol Anholonomy Calendar”, in Sir Charles Frank OBE FRS, an eightieth birthday tribute, edited by R.G. Chambers, J.E. Enderby, A. Keller, A.R. Lang, and J.W. Steeds (Adam Hilger, Bristol, 1991), pp. 207-219. [2] M. Kline, Mathematics: The Loss of Certainty (Oxford University Press, New York, 1980). [3] E.P. Wigner, “The unreasonable effectiveness of mathematics in the natural sciences”, Richard Courant lecture in mathematical sciences deliv- ered at New York University, May 11, 1959, Com- munications on Pure and Applied Mathematics 13(1), 1–14 (1960). [4] R.W. Hamming, “The unreasonable effectiveness of mathematics,” American Mathematical Monthly 87(2) 81–90 (1980). [5] B.W. Ninham, B.D. Hughes, N.E. Frankel, and M.L. Glasser, “Möbius, Mellin, and mathematical physics,” Physica A 186, 441–481 (1992). [6] N. Bohr, “Über die Serienspektra der Element,” Zeitschrift für Physik 2 423–478 (1920). English translation in L. Rosenfeld and J. Rud Nielsen (editors), Niels Bohr, Collected Works, Volume 3, The Correspondence Principle (1918–1923), pp. 241–282 (North-Holland, Amsterdam, 1976). [7] W.H. Cropper, The Quantum Physicists (Oxford University Press, New York, 1970) notes that ‘Although the correspondence principle became increasingly elaborate in later work, it was always based on one simple concept: that when the scale is suitably adjusted, classical physics and quantum physics must merge.’ [8] N. Bohr, “The quantum postulate and the recent development of atomic theory,” Nature 121, 580– 590 (1928). Bohr writes (p. 580) ‘… if in order to make observation possible we permit certain inter- actions with suitable agencies of measurement, not belonging to the system, an unambiguous descrip- tion of the state of the system is no longer possi- ble, and there can be no sense of causality in the ordinary sense of the word. The very nature of the quantum theory thus forces us to regard the space- time coordination and the claim of causality, the union of which characterises the classical theories, as complementary but exclusive features of the description…’. See also N. Bohr, “Discussions with Einstein on epistemological problems in atomic physics,” in P. Schilpp (editor), Albert Einstein: Phi- losopher-Scientist (Open Court, Chicago, 1949). [9] M.J. Lighthill, Introduction to Fourier Analysis and Generalised Functions (Cambridge University Press, Cambridge, U.K., 1958). [10] C.G.J. Jacobi, “Theorie der elliptischen Functionen aus den Eigenschaften der Thetareihen abge- leitet”, in Gesammelte Werke, Band 1, pp. 497–538 (Reimer, Berlin, 1881; reprinted Chelsea, New York, 1969). [11] E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, 4th edition (Cambridge Univer- sity Press, Cambridge, U.K.,1927). [12] The wave-particle duality aspect of our central theme can be embedded in a somewhat broader context that we only touch on here. (The authors are grateful to one of the referees for suggest- ing that we address this, and referring us to rel- evant literature). Poisson’s summation formula, in both the guises (2) and (10), relates spectral information (eigenvalues) to a discrete geometry (a regular lattice). This hints at a class of rela- tions between spectral sums and geometrical or topological sums. Perhaps the simplest manifes- tation of this is the relation between the ray and mode descriptions of waveguides, which repre- sent complementary perspectives: see C.L. Pek- eris, “Ray theory vs normal mode theory in wave propagation problems”, Proceedings of Symposia in Applied Mathematics 2, 71–75 (1950). How- ever, there are many examples, such as: (i) scatter- ing from spheres, with spectral sums over angular momentum vs rays winding different numbers of times round the scattering center (M.V. Berry and K.E. Mount, “Semiclassical approximations in wave mechanics”, Reports on Progress in Phys- ics 35, 315–397 (1972)); (ii) electron diffraction in crystals, with spectral sums over Bloch waves vs classical trajectories winding through the lattice (M.V. Berry, “Diffraction in crystals at high ener- gies”, Journal of Physics C 4, 697–722 (1971)); and (iii) complementary approximate solution meth- ods for the energy eigenvalue problem of quan- tum mechanics when separation of variables is not available (M.C. Gutzwiller, “Periodic orbits and classical quantization conditions”, Journal of Math- ematical Physics.12, 343–358 (1971)). [13] A. Córdoba, “La formule sommatoire de Poisson,” Comptes rendus de l’Académie des sciences. Série 1, Mathématique 306 (no 8) 373–376 (1988). [14] A. Córdoba, “Dirac combs”, Letters in Mathemati- cal Physics 17, 191–196 (1989). 71A Correspondence Principle [15] See P.L. Butzer, P.J.S.G. Ferreira, G. Schmeiss- er and R.L. Stens, “The summation formulae of Euler–Maclaurin, Abel–Plana, Poisson, and their interconnections with the approximate sampling formula of signal analysis,” Results in Mathemat- ics 59, 359–400 (2012). These authors note that Eq. (10) was first produced by Gauss. The results such as (4) are usually called Jacobi’s transforma- tion after their appearance in several of Jacobi’s works [C.G.J. Jacobi, “Suite des notices sur les fonctions elliptiques,” Journal für die reine und angewandte Mathematik (Crelle’s Journal) 3, 403–404 (1828); C.G.J. Jacobi, “Über die Differen- tialgleichung, welcher die Reihen 1 ± 2q ± 2q4 ± 2q9+ etc., + etc. Genüge leisten,” Journal für die reine und angewandte Mathematik (Crelle’s Journal) 36, 97–112 (1848). However if Gauss did not produce them earlier, they are cer- tainly present in work of Poisson, which Jacobi has acknowledged [S.D. Poisson, “Suite du mémoire sur les intégrales définies et sur la sommation des séries, inséré dans les précédens volumes de ce Journal”, Journal de l’École royale polytechnique 12 (19), 404–509 (1823)]. [16] E.C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 2nd edition, (Oxford University Press, Oxford, U.K., 1948; reprinted Chelsea, New York, 1986). [17] B. Riemann, “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse,” Monatsberichte der Königliche Preußische Akademie der Wis- senschaften zu Berlin aus der Jahre 1859, 671–680 (1860). [18] Riemann shows that where ψ(x) = and Re{s} > 1, we have ζ(s)π−s/2Γ(s/2) = xs/2−1ψ(x)dx. He uses the transformation formula for θ3(0|ix) to construct an analytic continuation for ζ(s) π−s/2Γ(s/2) that is symmetric under replacement of s by 1 – s, and the Riemann relation follows. [19] E.C. Titchmarsh, The Theory of the Riemann Zeta- function, 2nd edition, revised by D.R. Heath- Brown (Oxford University Press, Oxford, U.K., 1986). [20] D. Schumayer and D.A.W. Hutchinson, “Physics of the Riemann hypothesis”, Reviews of Modern Physics 83, 307–330 (2011). [21] A shorter formal argument based on a binomial expansion of (n + a)−s with a ∈ (0, 1) and sub- sequent use of the limit a → 1 yields the formal expression [5] from which the same conclusions about simple known values of ζ(–m) for m a non-negative inte- ger be drawn. The method given in the present paper is classically rigorous. For an alternative rig- orous approach, leading to the identity see §68 of E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen (1st edition, Leipzig and Berlin, Teubner, 1909; 3rd edition, Providence, Rhode Island, AMS Chelsea, 1974). [22] R. Bellman, A Brief Introduction to Theta Func- tions (Holt, Rinehart and Winston, New York, 1961). [23] Although we have produced Eq. (2) from Eq. (1), the process is quite reversible. They are essentially equivalent results. J.N. Lyness and B.W. Ninham [“Numerical quadrature and asymptotic expan- sions”, Mathematics of Computation 21, 162–178 (1967)] have shown how Eq. (1) can be used to deduce a very broad class of quadrature rules for numerical integration on finite intervals, with useful asymptotic expansions for the quadra- ture error. Further developments of this approach lead to practical techniques for computing ana- lytic continuations of functions defined by para- metrized integrals, by direct computation of the integrals even for parameter ranges where the integral diverges. See B.W. Ninham, “Generalised functions and divergent integrals”, Numerische Mathematik 8, 444–457 (1966). [24] For properties of the modular group and proofs of results discussed in Section 2.2 see the following texts: W. Magnus, Noneuclidean Tesselations and their Groups (Academic Press, New York, 1974); J.P. Serre, A Course in Arithmetic (Springer-Ver- lag, New York, 1973); and B. Schoeneberg, Elliptic Modular Functions (Springer-Verlag, Berlin, 1974). [25] E. Fabry, “Sur les pointes singuliers d’une fonc- tion donnée par son développement en série et sur l’impossibilité du prolongement analitique dans les cas très généaux”, Annales de ’École normale supé- rieure (3) 13, 107–114 (1896). [26] P. Dienes, The Taylor Series: an Introduction to the Theory of Functions of a Complex Variable (Oxford University Press, Oxford, U. K., 1931). [27] For an easy proof, multiply Eq. (20) by x2 and integrate by parts twice. We have normalized the total mass to unity, so that v(x, t) can be interpret- ed as a probability density function. 72 Barry D. Hughes and Barry W. Ninham [28] A. Fick, “Ueber Diffusion”, Annalen der Physik 94, 59–86 (1855), translated as A. Fick, “On liquid dif- fusion”, Philosophical Magazine 10, 30–39 (1855). [29] E.W. Montroll and B.J. West, “On an enriched col- lection of stochastic processes”, in E.W. Montroll and J.L. Lebowitz (editors), Fluctuation Phenomena, pp. 61–173 (North-Holland, Amsterdam, 1979). [30] B.D. Hughes, Random Walks and Random Envi- ronments, Vol. 1 (Oxford University Press, Oxford, U.K., 1995). [31] R. Kutner, A. Pękalski, and K. Sznajd-Weron (edi- tors), Anomalous Diffusion: from Basics to Applica- tions (Springer, Berlin, 1999). [32] R. Klages, G. Radons, and I.M. Sokolov (editors), Anomalous Transport: Foundations and Applica- tions (Wiley-VCH, Weinheim, Germany, 2008). [33] B.B. Mandelbrot, The Fractal Geometry of Nature (W.H. Freeman, San Francisco, 1982). [34] B.D. Hughes, M.F. Shlesinger, and E.W. Montroll, “Random walks with self-similar clusters”, Pro- ceedings of the National Academy of Sciences (U.S.A.) 78, 3287–3291 (1981). [35] When µ > 2 there is a standard diffusive limit when the time-step t0 and spatial scale ∆ are sent to zero with ∆2/t0 held constant. For µ = 2 it is necessary to take ∆2 ln(1/∆)/t0 constant to recov- er diffusion, and for 0 < µ < 2 diffusion is never attained. [36] L.P. Kadanoff, “Scaling laws for Ising models near Tc”, Physics (Long Island City, N.Y.) 2, 263–272 (1966). [37] T. Niemeijer and J.M.J. van Leeuwen, (1976). “Renormalization: Ising-like spin systems”, In C. Domb and M.S. Green (editors), Phase Transi- tions and Critical Phenomena, Vol. 6, pp. 425–505. (Academic Press, London, 1976). [38] For µ ≤ 1/2 the series for Q needs to be summed by Abelian means, with a convergence factor e−δ|m| inserted and the limit δ → 0 taken after evaluation of the sum. [39] M.F. Shlesinger and B.D. Hughes, “Analogs of renormalization group transformations in random processes”, Physica A 109, 597–608 (1981). [40] W.J. Reed and B.D. Hughes, “On the distribution of family names”, Physica A 319, 579–590 (2003). [41] D. Sornette, “Discrete-scale invariance and com- plex dimensions”, Physics Reports 297 (5), 239– 270 (1998). [42] G.H. Hardy, “Weierstrass’ non-differentiable func- tion”, Transactions of the American Mathematical Society 17, 301–325 (1916). [43] For textbook discussions of the Mellin trans- form techniques used here, see N. Bleistein and R.A. Handelsman, Asymptotic Expansions of Inte- grals (Dover, New York, 1986), B. Davies, Integral Transforms and Their Applications, 3rd edition (Springer-Verlag, New York, 2002), or Appendix 2 of Hughes [30]. [44] P. Lévy, Théorie de l’addition des variables aléatoires (Gauthier-Villars, Paris, 1937). [45] Although we have discussed here only the symmetric densities with Fourier transforms exp(–c|q|µt), there is a more general theory of sta- ble densities, containing additional parameters. In this more general context, simple closed form expressions for densities are usually not available, but there have been some interesting develop- ments since 2010. See K.A. Penson and K. Górska, “Exact and explicit probability densities for one- sided Lévy stable distributions”, Physical Review Letters 105, 210604 (2010); and K. Górska and K.A. Penson, “Lévy stable two-sided distributions: exact and explicit densities for asymmetric case”, Physical Review E 83, 061125 (2011). [46] S.A. Fulling and K. S. Güntürk, “Exploring the propagator of a particle in a box,” American Jour- nal of Physics 71, 55–63 (2003). [47] B. Gaveau and L.S. Schulman, “Explicit time- dependent Schrödinger propagators,” Journal of Physics A 19 1833–1846 (1986). [48] C.G. Darwin, “Free motion in the wave mechan- ics”, Proc. R. Soc. Lond. A 117 (1927) 258–293. [49] E.H. Kennard, “Zur Quantenmechanik einfacher Bewegungstypen”, Zeitschrift für Physik 44 (1927) 326-352. [50] J. Lekner, “Airy wavepacket solution of the Schrödinger equation,” European Journal of Phys- ics 30, L43–L46 (2009). [51] W.H. Zachariasen, Theory of X-Ray Diffraction in Crystals (Wiley, New York, 1945). [52] E. Zolotoyabko, Basic Concepts of Crystallography (Wiley-VCH, Weinheim, Germany, 2011). [53] This is a slight over-simplification from the experi- mental perspective. The amplitude of the Fourier transform can be measured experimentally, but its phase is not directly available. [54] Let be a d-dimensional lattice, that is, a set of points with position vectors , where mj ∈ ℤ and the vectors {a1, … ad} are a linearly inde- pendent set. The dual lattice or reciprocal lat- tice consists of these points whose potion vectors have integer-valued dot products with every position vector of a point in . Then (see, for example, Theorem 3.2 in Senechal[61]) the Fou- 73A Correspondence Principle rier transform of the mass distribution ρ(r) = defined by placing unit mass at each point of is given by (k) = . More general results which essentially account for all simple cases have been established by Córdoba [13], who has shown that if µ and ν are two dis- crete subsets of ℝd (this requires the existence of a nonzero lower bound for the spacing between pairs of points) and c(s) > 0 for all s ∈ µ then the relations between a mass distribution ρ(r) and its Fourier transform can hold simultaneously if and only if the follow- ing conditions hold: c(s) = 1 and there exists a linear transformation A of ℝd with determinant 1 such that µ = Aℤd and ν = (A−1)Tℤd, where T denotes the transpose. It may be noted that because the definitions of the Fourier transform and its inverse are simply complex conjugates, the factor of c(s) can be placed in either one of the sums over µ and ν without altering the result. We have used this in rephrasing Córdoba’s result to suit our notation and our application of interest, where it is more natural to assert that all masses are the same in physical space and allow position- dependent coefficients in the sum obtained on taking the Fourier transform. [55] D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “Metallic phase with long-range orienta- tional order and no translational symmetry,” Phys- ical Review Letters 53, 1951–1953 (1984). [56] D. Levine and P. J. Steinhardt, “Quasicrystals: a new class of ordered structures,” Physical Review Letters 53, 2477–2480 (1984). [57] C. Janot, Quasicrystals: a Primer, 2nd edition (Oxford University Press, Oxford, U.K., 1994). [58] R. Penrose, “The role of aesthetics in pure and applied mathematical research”, Bulletin of the Institute of Mathematics and its Applications 10, 266–271(1974). [59] A. Hof, “On diffraction on aperiodic structures”, Communications in Mathematical Physics 169, 25–43 (1995). [60] M. Senechal, “Generalizing crystallography: puz- zles and problems in dimension 1,” in I. Hargit- tai (ed.), Quasicrystals, Networks and Molecules of Fivefold Symmettry (VCH, Weinheim, Germany, 1990). [61] M. Senechal, Quasicrystals and Geometry (Cam- bridge University Press, Cambridge U.K., 1995). [62] M. Senechal and J. Taylor, “Quasicrystals: the view from Les Houches,” Mathematical Intelligencer 12 (2), 54–64 (1990). [63] M. Senechal and J. Taylor, “Quasicrystals: the view from Stockholm,” Mathematical Intelligencer 35 (2), 1–9 (2013). [64] M. Baake and U. Grimm, Aperiodic Order, Volume 1, A Mathematical Invitation (Cambridge Univer- sity Press, Cambridge, U.K., 2013). [65] In view of Córdoba’s observations [54]. [66] B.W. Ninham and S. Lidin, “Some remarks on quasicrystal structure”, Acta Crystallographica A 48, 640–650 (1992) . [67] These considerations have ancient antecedents among the Greeks and Arabs, and were certainly prominent in renaissance Italy: see, for example, Luca Pacioli, De divina proportione (Venice, 1509), available at https://archive.org/details/divinapro- portion00paci. [68] D.W. Thompson, On Growth and Form, complete revised edition (New York, Dover, 1992). [69] A.L. Loeb and W. Varney, “Does the golden spiral exist, and if so, where is its centre?”, in Spiral Sym- metry (edited by I. Hargittai and C.A. Pickover), pp. 47–61 (World Scientific, Singapore, 1992). [70] M.V. Berry and Z.V. Lewis, “On the Weierstrass– Mandelbrot fractal function”, Proceedings of the Royal Society of London, Series A 370, 459–484 (1980). [71] M.V. Berry, “A theta-like sum from diffraction phys- ics”, Journal of Physics A 32, L329–L336 (1999). [72] L.D. Landau and E.M. Lifshitz, Course of Theoreti- cal Physics: Statistical Physics, Part 1, 3rd edition, revised by E.M. Lifshitz and L.P. Pitaevski (Perga- mon, Oxford, U.K., 1980). [73] J. Honerkamp, Statistical Physics, 2nd edition (Springer, Berlin, 2002). [74] B.W. Ninham, V.A. Parsegian and G.H. Weiss, “On the macroscopic theory of temperature-dependent van der Waals forces”, Journal of Statistical Physics 2, 323–328 (1970). [75] J. Mahanty and B.W. Ninham, Dispersion Forces (Academic Press, London, 1976). [76] The approach here is relevant for a semiclassical ideal gas: see Honerkamp [73], pp. 126–127. [77] In computing thermodynamically interesting functions from ln , a multiplicative prefactor 74 Barry D. Hughes and Barry W. Ninham needs to be applied to account for the number of available spin states per particle. We do not address this here. [78] M. Planat, “From Planck to Ramanujan: a quan- tum 1/f noise in thermal equilibrium”, Journal de Théorie des Nombres de Bordeaux, Université Bordeaux 1, 14 (2002) 585–601 and https://hal. archives-ouvertes.fr/hal-00078140; and M. Planat, “Thermal 1/ f noise from the theory of partitions: application to a quartz resonator”, Physica A 318 (2003) 371–386. [79] L. Euler, Introductio in Analysin Infinitorum, Vol- ume 1, Chapter XVI (Lausanne and Geneva, Marc-Michel Bousquet, 1748); reprinted as Leon- hardi Euleri Opera Omnia, Series 1, 8 (Leipzig and Berlin,Teubner, 1922). [80] F.W.J. Olver, D.W. Lozier, R.F. Boisvert and C.W. Clark (editors), NIST Handbook of Mathemati- cal Functions (Cambridge University Press, Cam- bridge, U.K., 2010): see their Eq. (17.5.1). That the series expansion is equivalent to the product is easily deduced from the identity (1 – a)(aq; q)∞ = (a; q)∞. [81] L.J. Rogers, “Second memoir on the expansion of certain infinite products”, Proceedings of the Lon- don Mathematical Society (1) 25, 318–343 (1894); L.J. Rogers, “On two theorems of combinatory analysis and some allied identities”, Proceedings of the London Mathematical Society (2) 16, 315–336 (1917); S. Ramanujan, Collected Papers, pp. 214– 215 and pp. 344–346 (Cambridge University Press, Cambridge, U.K., 1927; reprinted AMS Chelsea, Providence, R.I, 1962). [82] A. Berkovitch and B.M. McCoy, “Rogers–Ramanu- jan identities: a century of progress from math- ematics to physics”, in Proceedings of the Inter- national Congress of Mathematicians, Vol. 3, pp. 163–172 (Berlin, 1998); available at http://www. mathunion.org/ICM/ICM1998.3/ Main/11/McCoy. MAN.ocr.pdf. [83] H.B.G. Casimir, “On the attraction between two perfectly conducting plates”, Proceedings of the Koninklijke Nederlandse Academie van Weten- schappen 51, 793–795 (1948). [84] H.B.G. Casimir, “Sur les forces van der Waals– London”, in “Colloque sur la theorie de la liaison chimique” (including a response to questions from Magar, Coulson, Prigogine, Pauling and Bauer), Journal de Chimie Physique 46, 407–410 (1949). [85] M.J. Sparnaay, “Measurement of the attractive forces between flat plates”, Physica 24 751–764 (1958). [86] S.K. Lamoreaux, “Demonstration of the Casimir force in the 0.6 to 6 µm range”, Physical Review Letters 78, 5–8 (1997). [87] Actually, Lamoreaux’s experimental system is not one based on parallel plates, but instead uses a flat plate and a sphere, with the force on this system being expressed in terms of the law to be veri- fied for parallel plates using the proximity force theorem of Blocki et al. [J. Blocki, J. Randrup, W.J. Swiatecki, and C.F. Tsang, “Proximity forces”, Annals of Physics (N.Y.) 105, 427–462 (1977)]. The proximity force theorem is essentially the famous Derjaguin approximation of colloid sci- ence [B.V. Derjaguin, “Untersuchungen über die Reibung und Adhäsion. IV. Theorie des Anhaftens kleiner Teilchen”, Kolloid-Zeitschrift 69, 155–164 (1934)]. For subsequent independent verifications of the Casimir force, with a 1% root-mean-square average deviation between theory and experi- ment at the smallest measured separations, see: U. Mohideen and A. Roy, “Precision measurement of the Casimir Force from 0.1 to 0.9µmm”, Physi- cal Review Letters 81 (1998) 4549–4552; and A. Roy, C-Y. Lin and U. Mohideen, “Improved preci- sion measurement of the Casimir force”, Physical Review D 60 111101 (1999). [88] S.L. Boersma, “A maritime analogy of the Casimir effect”, American Journal of Physics 64, 539–541 (1996); Boersma writes “The old tales were true. Rolling ships do attract each other. Two ships on a rolling sea attract each other as two atoms do in the sea of vacuum fluctuations.” [89] A. Larraza, “A demonstration apparatus for an acoustic analog to the Casimir effect”, American Journal of Physics 67, 1028–1030 (1999). [90] S. Koyama and N. Kurokawa, “Casimir effects on Riemann surfaces”, Indagationes Mathematicae 13, 63–75 (2002). [91] S. Koyama and N. Kurokawa, “Absolute zeta func- tions, absolute Riemann hypothesis and absolute Casimir energies”, In G. van Dijk and M. Wakay- ama (editors), Casimir Force, Casimir Operators and the Riemann Hypothesis: Mathematics for Innovation in Industry and Science (de Gruyter, Berlin, 2010). [92] E.M. Lifshitz and L.P. Pitaevski, Course of Theoreti- cal Physics: Statistical Physics, Part 2 (Pergamon, Oxford, U.K., 1980). [93] E . M . ( E . M . L i f s h i t z ) , “ ” (“Theory of molecular attraction between condensed bodies”), Doklady Akademii Nauk SSSR 97, 643–646 (1954). 75A Correspondence Principle [94] E . M . ( E . M . L i f sh it z ) , “ ” (“Influence of temperature on molecular attraction forces between condensed bodies”), Doklady Akademii Nauk SSSR 100, 879–881 (1955). [95] E.M. Lifshitz, “The theory of molecular attrac- tive forces between solids”, Soviet Physics JETP 2, 73–83 (1956); translation from the Russian origi- nal in Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 29, 94–110 (1955). [96] I.E. Dzyaloshinskii, E.M. Lifshitz and L.P. Pitaevs- kii, “Van der Waals forces in liquid films”, Soviet Physics JETP 10, 161–170 (1960); translation from the Russian original in Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 37, 229–241 (1959). [97] I.E. Dzyaloshinskii, E.M. Lifshitz and L.P. Pitaevs- kii, “General theory of van der Waals forces”, Soviet Physics Uspekhi 4, 153–176 (1961); transla- tion from the Russian original in Uspekhi Fizich- eskikh Nauk 73, 381–422 (1961). The same paper, prepared by a different translator, also appears as “The general theory of van der Waals forces”, Advances in Physics 10 165–209 (1961). [98] See §81–82 of Lifshitz and Pitaevski [92]. [99] B.W. Ninham and V.A. Parsegian, “Van der Waals forces across triple-layer films”, Journal of Chemi- cal Physics 52, 4578–4583 (1970). [100] B.W. Ninham and V.A. Parsegian, “Van der Waals forces: special characteristics in lipid-water sys- tems and a general method of calculation based on the Lifshitz theory”, Biophysical Journal 10, 646–663 (1970). [101] V.A. Parsegian and B.W. Ninham, “Temperature- dependent van der Waals forces”, Biophysical Jour- nal 10, 664–674 (1970). [102] E. Elizalde and A. Romeo, “Essentials of the Casimir effect and its computation”, American Journal of Physics 59, 711–719 (1991). [103] B.W. Ninham and J. Daicic, “Lifshitz theory of Casimir forces at finite temperature”, Physical Review A 57, 1870–1879 (1998). [104] It seems that all derivations of the Casimir effect require at least one formal argument to resolve an indeterminacy. [105] J. Hadamard, “Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques”, Bulletin de la Société mathématique de France 24, 199–220 (1896). [106] C. de la Vallée Poussin, “Recherches analytiques sur la théorie des nombres premiers. Première partie: La fonction ζ(s) de Riemann et les nombres premiers en général”, Annales de la Société scienti- fique de Bruxelles 20 (2), 183–256 (1896). [107] L.S. Brown and G.J. Maclay, “Vacuum stress between conducting plates: an image solution”, Physical Review 184, 1272–1279 (1969). [108] F. Ravndal and D. Tollefsen, “Temperature inver- sion symmetry in the Casimir effect”, Physical Review D 40, 4191–4192 (1989). [109] J. Christensen-Dalsgaard, W. Däppen, S.V. Ajukov, E.R. Anderson, H.M. Antia, S. Basu, V.A. Baturin, G. Berthomieu, B. Chaboyer, S.M. Chitre, A.N. Cox, P. Demarque, J. Donatowicz, W.A. Dziem- bowski, M. Gabriel, D.O. Gough, D.B. Guenther, J.A. Guzik, J.W. Harvey, F. Hill, G. Houdek, C.A. Iglesias, A.G. Kosovichev, J.W. Leibacher, P. Morel, C.R. Proffitt, J. Provost, J. Reiter, E.J. Rhodes Jr., F.J. Rogers, I.W. Roxburgh, M.J. Thompson and R.K. Ulrich, “The current state of solar modeling”, Science 272, 1286–1292 (1996). [110] B.W. Ninham and M. Boström, “Screened Casimir force at finite temperatures: a possible role for nuclear interactions”, Physical Review A 67, 030701 (2003). [111] B.W. Ninham, M. Boström, C. Persson, I. Brevik, S.Y. Buhmann and B.E. Sernelius, “Casimir forces in a plasma: possible connections to Yukawa poten- tials”, European Physical Journal D 68, 328 (2014). [112] M.V. Berry, “The arcane in the mundane”, English translation (https://michaelberryphysics.files.word- press.com/2013/07/berry405.pdf ) of a contribu- tion to Les déchiffreurs: voyage en mathématiques, edited by J.-F. Dars, A. Lesne and A. Papillault (Éditions Belin, Paris, 2008) pp. 134–135. [113] Ø. Linnebo, “Platonism in the philosophy of mathematics”, The Stanford Encyclopedia of Phi- losophy (Winter 2013 Edition), Edward N. Zalta (ed.), http://plato.stanford.edu/archives/win2013/ entries/platonism-mathematics/. [114] https://michaelberryphysics.wordpress.com/quota- tions (retrieved January 2015). [115] Attributed by Berry to Arnold, “implied by state- ments in his many letters disputing priority, usu- ally in response to what he sees as neglect of Rus- sian mathematicians”. Priority for this observation may, however, be due to S. Stigler, “Stigler’s law of epynomy”, Transactions of the New York Academy of Sciences 39, 147–158 (1980). [116] Modestly attributed by Berry to himself, though there is a certain recursively to this assertion. A more modest precursor, viz. that most discoveries of interest have significant precursors, seems wor- thy of the status of an axiom for human culture. 76 Barry D. Hughes and Barry W. Ninham [117] Quoted by M. Dresden, H.A. Kramers: between tradition and revolution (Springer, New York, 1987). [118] J. Lekner, “Electrostatics of two charged conduct- ing spheres”, Proceedings of the Royal Society of London, Series A 468, 2829–2848 (2012). [119] P. Ball, “Quantum leaps of faith”, Chemistry World (May 2013) http://rsc.org/chemistr y- world/2013/04/quantum-classical-mechanics- schrodinger-derivation; and “Quantum quest”, Nature 501, 154–156 (2013). [120] The material in this appendix has been included as a result of a suggestion made by an anonymous reviewer. [121] A. Wintner, “On analytic convolutions of Bernoul- li distributions”, American Journal of Mathematics 56, 659–663 (1934). [122] B. Jessen and A. Wintner, “On symmetric Ber- noulli convolutions”, Transactions of the American Mathematical Society 38, 48–88 (1935). [123] R. Kershner and A. Wintner, “On symmetric Ber- noulli convolutions”, American Journal of Math- ematics 57, 541–548 (1935). [124] P. Erdös, “On a family of symmetric Bernoulli convolutions”, American Journal of Mathematics 61, 974–976 (1939). [125] Y. Peres, W. Schlag and B. Solomyak, “Sixty years of Bernoulli convolutions”, in C. Bandt, S. Graf and M. Zähle (editors), Fractal Geometry and Sto- chastics. II, pp. 39–65 (Springer, Basel, 1999). [126] A. Klenke, Probability Theory (Springer-Verlag, London, 2008). [127] Jessen and Wintner [122] prove that X is uni- formly distributed on [–1, 1] when cn = 2−n, from which it follows immediately that for cn = 21−n, the random variable X is uniformly distributed on [–2, 2]. [128] B. Solomyak, “On the random series ∑±λi (an Erdös problem)”, Annals of Mathematics 142, 611–625 (1995). [129] T.-Y. Hu, “The local dimensions of the Bernoulli convolution associated with the golden number”, Transactions of the American Mathematical Soci- ety 349, 2917–2940 (1997). [130] K.-S. Lau and S.-M. Ngai, “Lq-spectrum of the Bernoulli convolution associated with the golden ratio”, Studia Mathematica 131, 225–251 (1998). Substantia An International Journal of the History of Chemistry Vol. 2, n. 1 - March 2018 Firenze University Press Why Chemists Need Philosophy, History, and Ethics Emulsion Stability and Thermodynamics: In from the cold Stig E. Friberg Finding Na,K-ATPase Hans-Jürgen Apell Mechanistic Trends in Chemistry Louis Caruana SJ Cognition and Reality F. Tito Arecchi A Correspondence Principle Barry D. Hughes1,* and Barry W. 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