A Half-Century of Physical Asymptotics and Other Diversions

The following sections are included:

• Preface
• Contents

From the beginning, I was fascinated by waves, especially their characteristic property, namely phase. My first paper [B1], published in 1965, when I was still a graduate student in St Andrews, concerned the phase difference between two waves. It had been claimed that this was different for moving observers. We showed that this is wrong: the phase difference is invariant under Lorentz (also Galilean) transformations…

When an ultrasonic pulse, containing, say, ten quasi-sinusoidal oscillations, is reflected in air from a rough surface, it is observed experimentally that the scattered wave train contains dislocations, which are closely analogous to those found in imperfect crystals. We show theoretically that such dislocations are to be expected whenever limited trains of waves, ultimately derived from the same oscillator, travel in different directions and interfere – for example in a scattering problem. Dispersion is not involved. Equations are given showing the detailed structure of edge, screw and mixed edge-screw dislocations, and also of parallel sets of such dislocations. Edge dislocations can glide relative to the wave train at any velocity; they can also climb, and screw dislocations can glide. Wavefront dislocations may be curved, and they may intersect; they may collide and rebound; they may annihilate each other or be created as loops or pairs. With dislocations in wave trains, unlike crystal dislocations, there is no breakdown of linearity near the centre. Mathematically they are lines along which the phase is indeterminate; this implies that the wave amplitude is zero.

We study the wavefronts (i.e. the surfaces of constant phase) of the wave discussed by Aharonov and Bohm, representing a beam of particles with charge q scattered by an impenetrable cylinder of radius R containing magnetic flux Ф. Defining the quantum flux parameter by α=qФ/h. we show that for the case R = 0 the wave ψ_AB possesses a wavefront dislocation on the flux line, whose strength (i.e. the number of wave crests ending on the dislocation) equals the nearest integer to α. When α passes through half-integer values, the strength changes, by wavefronts unlinking and reconnecting along a nodal surface. In quantum mechanics this phase structure is unobservable, but we devise an analogue where surface waves on water encounter an irrotational ‘bathtub’ vortex; in this case α depends on the frequency of the waves and the circulation of the vortex. Experiments show dislocation structures agreeing with those predicted. ψ_AB is an unusual function. in which incident and scattered waves cannot be clearly separated in all asymptotic directions; we discuss its properties using a new asymptotic method.

Often, light can be represented, approximately or exactly, by a complex scalar wave ψ, smoothly varying in space and/or time. The field ψ could be a cartesian component of the electromagnetic field or of the vector potential, or one of several scalar potentials appropriate to different circumstances. This meeting has been concerned with the line singularities of the phase of ψ. Here, I wish to make some general remarks about these lines. Except where stated, all the remarks are independent of the particular wave equation that ψ Satisfies…

In complex scalar fields, singularities of the phase (optical vortices, wavefront dislocations) are lines in space, or points in the plane, where the wave amplitude vanishes. Phase singularities are illustrated by zeros in edge diffraction and amphidromies in the heights of the tides. In complex vector waves, there are two sorts of polarization singularity. The polarization is purely circular on lines in space or points in the plane (C singularities); these singularities have index ±1/2. The polarization is purely linear on lines in space for general vector fields, and surfaces in space or lines in the plane for transverse fields (L singularities); these singularities have index ±1. Polarization singularities (C points and L lines) are illustrated in the pattern of tidal currents.

Exact solutions of the Helmholtz equation are constructed, possessing wavefront dislocation lines (phase singularities) in the form of knots or links where the wave function vanishes (‘knotted nothings’). The construction proceeds by making a non-generic structure with a strength n dislocation loop threaded by a strength m dislocation line, and then perturbing this. In the resulting unfolded (stable) structure, the dislocation loop becomes an (m, n) torus knot if m and n are coprime, and N linked rings or knots if m and n have a common factor N; the loop or rings are threaded by an m-stranded helix. In our explicit implementation, the wave is a superposition of Bessel beams, accessible to experiment. Paraxially, the construction fails.

A quantal system in an eigenstate, slowly transported round a circuit C by varying parameters R in its Hamiltonian Ĥ(R), will acquire a geometrical phase factor exp {iγ(C)} in addition to the familiar dynamical phase factor. An explicit general formula for y(C) is derived in terms of the spectrum and eigenstates of Ĥ(R) over a surface spanning C. If C lies near a degeneracy of Ĥ, γ(C) takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy. As an illustration γ(C) is calculated for spinning particles in slowly-changing magnetic fields; although the sign reversal of spinors on rotation is a special case, the effect is predicted to occur for bosons as well as fermions, and a method for observing it is proposed. It is shown that the Aharonov–Bohm effect can be interpreted as a geometrical phase factor.

The moment of conception of the geometric phase can be pinpointed precisely, but related ideas had been formulated before, in various guises. Not less varied were the ramifications that became clear once the concept was identified formally.

The quantum mechanics of two identical particles with spin S in three dimensions is reformulated by employing not the usual fixed spin basis but a transported spin basis that exchanges the spins along with the positions. Such a basis, required to be smooth and parallel-transported, can be generated by an ‘exchange rotation’ operator resembling angular momentum. This is constructed from the four harmonic oscillators from which the two spins are made according to Schwinger’s scheme. It emerges automatically that the phase factor accompanying spin exchange with the transported basis is just the Pauli sign, that is (−1)^2S. Singlevaluedness of the total wavefunction, involving the transported basis, then implies the correct relation between spin and statistics. The Pauli sign is a geometric phase factor of topological origin, associated with non-contractible circuits in the doubly connected (and non-orientable) configuration space of relative positions with identified antipodes. The theory extends to more than two particles.

We review the formulation of quantum mechanics for identical spinning particles with wavefunctions that are singlevalued when permuted configurations are identified. The identification requires the spins to be smoothly permuted along with position variables, so spin is represented in a position-dependent ‘transported basis’, rather than the usual fixed basis. The simplest transported basis, constructed in terms of spins represented as pairs of commuting harmonic oscillators, gives the correct connection between spin and statistics. More complicated constructions can give the wrong exchange sign. The theory is generalized to incorporate additional properties such as isospin, colour and strangeness. Some remarks about the relation between this approach and those based on relatitivity and/or field theory are given.

My first teaching assignment, while still a Ph.D. student at St Andrews, was to give the graduate course in general relativity. In this daunting task, I was greatly assisted by the eloquent presentations in the books by J. L. Synge, so I seized the chance to hear his lectures in London the following year. But his topic was not relativity; instead, he spoke about the Hamiltonian theory of rays and waves. This subject, and Synge’s presentation of it, enchanted me, and I immediately conceived a postgraduate research project: to marry the largely geometric descriptive approach of Synge with the refined asymptotics of my St. Andrews supervisor Bob Dingle (see Chapter 4 and [7.8]). The resulting ‘physical asymptotics’ has animated much of my research…

We consider scattering from a corrugated hard surface Σ with random moving perturbations (a ‘rippling mirror’). Kirchhoff’s approximation enables the classical limit, diffraction effects and incoherence to be treated within the same framework. The classical rainbow is a curve ℓ in the two-dimensional space of deflections G; we study the topology of ℓ and show that it has cusps whose positions are sensitive to the form of Σ. Classically the scattering is singular on Σ but diffraction softens the singularities; we give the diffraction functions to be used near and on smooth parts and cusps of ℓ, and derive criteria for the observability of rainbow structure (taking account of surface periodicity which quantizes G). Random thermal perturbations of Σ blur the diffracted beams; we introduce a simple approximation for the blurring function, and this suggests a simple method for inverting experimental data to obtain the ‘surface phonon spectrum’, even in cases where ‘multiphonon processes’ dominate.

Short-wave fields can be well approximated by families of trajectories. These families are dominated by their singularities, i.e. by caustics, where the density of trajectories is infinite. Thom’s theorem on singularities of mappings can be rigorously applied and shows that structurally stable caustics—that is those whose topology is unaltered by ‘ generic ’ perturbation—can be classified as ‘ elementary catastrophes ’. Accurate asymptotic approximations to wave functions can be built up using the catastrophes as skeletons : to each catastrophe there corresponds a canonical diffraction function. Structurally unstable caustics can be produced by special symmetries, and the detailed form of the caustic that results from symmetry-breaking can often be determined by identifying the structurally unstable caustic as the special section of a higher-dimensional catastrophe. Sometimes it is clear that the unstable caustic is a special section of a catastrophe of infinite co-dimension ; these fall outside the scope of Thom’s theorem and suggest new directions for mathematical investigation. The discussion is illustrated with numerous examples from optics and quantum mechanics.

With passion and poetry, David Park sets out to get behind the optical science we are familiar with as physicists. He tells us how attempts to understand light have been at the heart of people’s efforts to make sense of the world ever since they began to reflect on it (note how the natural choice of metaphor reflects this). It is a fascinating story, beginning with the “immense fact [that] we can see”, and ending … well, it has not ended yet, as we will “see”…

Transparent overhead-projector foil is an anisotropic material with three different principal refractive indices. Its properties can be demonstrated very simply by sandwiching the foil between crossed polarizers and looking through it at any diffusely lit surface (e.g., the sky). Coloured interference fringes are seen, organized by a pattern of rings centred on two ‘bullseyes’ in the directions of the two optic axes. The fringes are difference contours of the two refractive indices corresponding to propagation in each direction, and the bullseyes are degeneracies where the refractive-index surfaces intersect conically. Each bullseye is crossed by a black ‘fermion brush’ reflecting the sign change (geometric phase) of each polarization in a circuit of the optic axis. Simple observations lead to the determination of the three refractive indices, up to an ordering ambiguity.

In the 2 × 2 matrices representing retarders and ideal polarizers, the eigenvectors are orthogonal. An example of the opposite case, where eigenvectors collapse onto one, is matrices M representing crystal plates sandwiched between a crossed polarizer and analyser. For these familiar combinations, M² = 0, so black sandwiches can be regarded as square roots of zero. Black sandwiches illustrate physics associated with degeneracies of non-Hermitian matrices.

The following sections are included:

• Introduction
• Preliminaries: electromagnetism and the wave surface
• The diabolical singularity: Hamilton’s ray cone
• The bright ring of internal conical refraction
• Poggendorff’s dark ring, Raman’s bright spot
• Belsky and Khapalyuk’s exact paraxial theory of conical diffraction
• Consequences of conical diffraction theory
• Experiments
• Concluding remarks
• Acknowledgements
• Appendix 1: Paraxiality
• Appendix 2: Conical refraction and analyticity
• References

In 1836 Henry Fox Talbot, an inventor of photography, published the results of some experiments in optics that he had previously demonstrated at a British Association meeting in Bristol (figure 1a). “It was very curious to observe that though the grating was greatly out of the focus of the lens…the appearance of the bands was perfectly distinct and well defined…the experiments are communicated in the hope that they may prove interesting to the cultivators of optical science.”…

The following sections are included:

• Introduction
• The appearance of the images
• The structure of the images
• Conclusions
• Acknowledgments
• References

The pattern embossed on the back of an oriental magic mirror appears in the patch of light projected onto a screen from its apparently featureless reflecting surface. In reality, the embossed pattern is reproduced in low relief on the front, and analysis shows that the projected image results from pre-focal ray deviation. In this interesting regime of geometrical optics, the image intensity is given simply by the Laplacian of the height function of the relief. For patterns consisting of steps, this predicts a characteristic effect, confirmed by observation: the image of each step exhibits a bright line on the low side and a dark line on the high side. Laplacian-image analysis of a magic-mirror image indicates that steps on the reflecting surface are about 400 nm high and laterally smoothed by about 0.5 mm.

Optical phenomena visible to everyone have been central to the development of, and abundantly illustrate, important concepts in science and mathematics. The phenomena considered from this viewpoint are rainbows, sparkling reflections on water, mirages, green flashes, earthlight on the moon, glories, daylight, crystals and the squint moon. And the concepts involved include refraction, caustics (focal singularities of ray optics), wave interference, numerical experiments, mathematical asymptotics, dispersion, complex angular momentum (Regge poles), polarisation singularities, Hamilton’s conical intersections of eigenvalues (‘Dirac points’), geometric phases and visual illusions.

One of the great sights of the Bristol area in the UK is the tidal bore on the River Severn – a wave that steepens and grows as the tide advances inland towards the city of Gloucester. Marking the beginning of the incoming tide, this giant wave is pulled up-river by the Moon’s gravity. I like to see the bore at night. As it approaches, the effect of the distant but then growing roar is magical – and there is plenty of light from Gloucester (and from the Moon when it is visible) to watch the bore as it passes by…

The earliest phase of quantum asymptotics, described in the previous chapter, culminated in a wide-ranging review of semiclassical mechanics [B23], written with Kate Mount in 1972. I now regard this as ‘prehistoric semiclassical mechanics’, because it was written before two important developments. The first, already described in Chapter 2, was the catastrophe classification of stable caustics. The second, brought to my attention by Ian Percival, was the then new area of deterministic chaos in classical dynamics. Percival insisted that new insights, beyond our existing semiclassical formulations, were required to understand quantum phenomena where the corresponding classical motion is chaotic (i.e. irregular or non-integrable)…

The form of the wavefunction ψ for a semiclassical regular quantum state (associated with classical motion on an N-dimensional torus in the 2N-dimensional phase space) is very different from the form of ψ for an irregular state (associated with stochastic classical motion on all or part of the (2N − 1)-dimensional energy surface in phase space). For regular states the local average probability density π. rises to large values on caustics at the boundaries of the classically allowed region in coordinate space, and ψ exhibits strong anisotropic interference oscillations. For irregular states π falls to zero (or in two dimensions stays constant) on ‘anticaustics’ at the boundary of the classically allowed region, and ψ appears to be a Gaussian random function exhibiting more moderate interference oscillations which for ergodic classical motion are statistically isotropic with the autocorrelation of ψ given by a Bessel function.

The spectral rigidity Δ(L) of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of L mean level spacings. In the semiclassical limit (ℏ→0), formulae are obtained giving Δ(L) as a sum over classical periodic orbits. When L≪L_max where L_max ∼ ℏ ^−(N−1) for a system of N freedoms, Δ(L) is shown to display the following universal behaviour as a result of properties of very long classical orbits: if the system is classically integrable (all periodic orbits filling tori), $\text{Δ}\left(L\right)=\frac{1}{15}L$ (as in an uncorrelated (Poisson) eigenvalue sequence); if the system is classically chaotic (all periodic orbits isolated and unstable) and has no symmetry, Δ(L) = ln L/2π² + D if 1 ≪ L ≪ L_max (as in the gaussian unitary ensemble of random-matrix theory); if the system is chaotic and has time-reversal symmetry, Δ(L) = ln L/π² + E if 1 ≪ L ≪ L_max (as in the gaussian orthogonal ensemble). When L ≫ L_max, Δ(L) saturates non-universally at a value, determined by short classical orbits, of order ℏ^−(N−1) for integrable systems and ln (ℏ⁻¹) for chaotic systems. These results are obtained by using the periodic-orbit expansion for the spectral density, together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere. For two examples Δ(L) is studied in detail: the rectangular billiard (integrable), and the Riemann zeta function (assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic).

Bounded or driven classical systems often exhibit chaos (exponential instability that persists), but their quantum counterparts do not. Nevertheless, there are new régimes of quantum behaviour that emerge in the semiclassical limit and depend on whether the classical orbits are regular or chaotic, and this motivates the following definition…

The following sections are included:

• INTRODUCTION
• RIEMANN ZEROS AND RANDOM MATRICES
• OUROBOROLOGY
• THE RIEMANN-SIEGEL FORMULA: A RULE FOR QUANTIZING CHAOS?
• ACKNOWLEDGMENT
• REFERENCES

By pretending that the imaginary ports E_m of the Riemann zeros are eigenvalues of a quantum Hamiltonian whose corresponding classical trajectories are chaotic and without time-reversal symmetry, it is possible to obtain by asymptotic arguments a formula for the mean square difference V(L; x) between the actual and average number of zeros near the xth zero in an interval where the expeded number is L. This predicts that when $L\ll L_{\max }=\text{ln}\left(E/2\pi \right)/2\pi \ln 2\left(\text{where}x=\left(E/2\pi \right)\left(\ln \left(E/2\pi \right)-1\right)+\frac{7}{8}\right),$V is the variance of the Gaussian unitary ensemble (GUE) of random matrices, while when L≪L_max, V will have quasirandom oscillations about the mean value π^-2(ln ln(E/2π) + 1.4009). Comparisons with V(L; x) computed by Odlyzko from 10⁵ zeros E_m near x = 10¹² confirm all details of the semiclassical predictions to within the limits of graphical precision.

Comparison between formulae for the counting functions of the heights t_n of the Riemann zeros and of semiclassical quantum eigenvalues E_n suggests that the t_n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H_cl. Many features of H_cl are provided by the analogy; for example, the “Riemann dynamics” should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the t_n have a similar structure to those of the semiclassical E_n; in particular, they display random-matrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the t_n can be computed accurately from formulae with quantum analogues. The Riemann-Siegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian H_cl = XP.

While in St Andrews in the early 1960s, I learned from Bob Dingle about his theory of the high orders of divergent series. This inspired an enthusiasm for asymptotics, but I did not then realise how seminal his achievements were. Indeed, for two decades afterwards, I declared myself as a ‘first terms asymptotist’, arguing that high orders are unnecessary: getting the leading order right, as in the uniform approximations near caustics (Chapter 2) is accurate enough for most applications…

Across a Stokes line, where one exponential in an asymptotic expansion maximally dominates another, the multiplier of the small exponential changes rapidly. If the expansion is truncated near its least term the change is not discontinuous but smooth and moreover universal in form. In terms of the singulant F – the difference between the larger and smaller exponents, and real on the Stokes line – the change in the multiplier is the error function

Integrals involving exp {−kf(z)}, where |k| is a large parameter and the contour passes through a saddle of f(z), are approximated by refining the method of steepest descent to include exponentially small contributions from the other saddles, through which the contour does not pass. These contributions are responsible for the divergence of the asymptotic expansion generated by the method of steepest descent. The refinement is achieved by means of an exact ‘resurgence relation’, expressing the original integral as its truncated saddle-point asymptotic expansion plus a remainder involving the integrals through certain ‘adjacent’ saddles, determined by a topological rule. Iteration of the resurgence relation, and choice of truncation near the least term of the original series, leads to a representation of the integral as a sum of contributions associated with ‘multiple scattering paths’ among the saddles. No resummation of divergent series is involved. Each path gives a ‘hyperseries’, depending on the terms in the asymptotic expansions for each saddle (these depend on the particular integral being studied and so are non-universal), and certain ‘hyperterminant’ functions defined by integrals (these are always the same and hence universal). Successive hyperseries get shorter, so the scheme naturally halts. For two saddles, the ultimate error is approximately ε^2.386, where ε (proportional to exp (–A|k|) where A is a positive constant), is the error in optimal truncation of the original series. As a numerical example, an integral with three saddles is computed hyperasymptotically.

Despite the denunciations of the mathematician Abel, if the devil did invent divergent series it was because his creator counterpart chose to build our physical universe so that they are among the more useful ways to describe its finite properties…

The following sections are included:

• Introduction
• The Classical Period
• The Neoclassical Period
• The Modern Period
• The Postmodern Period
• Further Reading

On the critical line $s=\frac{1}{2}+it$(t real), Riemann’s zeta function can be calculated with high accuracy by the Riemann–Siegel expansion. This is derived here by elementary formal manipulations of the Dirichlet series. It is shown that the expansion is divergent, with the high orders r having the familiar ‘factorial divided by power’ dependence, decorated with an unfamiliar slowly varying multiplier function which is calculated explicitly. Terms of the series decrease until r = r* ≈ 2πt and then increase. The form of the remainder when the expansion is truncated near r* is determined; it is of order exp(–πt), indicating that the critical line is a Stokes line for the Riemann-Siegel expansion. These conclusions are supported by computations of the first 50 coefficients in the expansion, and of the remainders as a function of truncation for several values of t.

By analytic continuation of the Dirichlet series for the Riemann zeta function ζ(s) to the critical line $s=\frac{1}{2}+\text{i}t$ (t real), a family of exact representations, parametrized by a real variable K, is found for the real function $Z(t)=\zeta \left(\frac{1}{2}+it\right)\exp \left\{\text{i}\theta \left(t\right)\right\},$where θ is real. The dominant contribution Z₀(t, K) is a convergent sum over the integers n of the Dirichlet series, resembling the finite ‘main sum’ of the Riemann–Siegel formula (RS) but with the sharp cut-off smoothed by an error function. The corrections Z₃(t, K),Z₄(t, K)… are also convergent sums, whose principal terms involve integers close to the RS cut-off. For large K,Z₀ contains not only the main sum of RS but also its first correction. An estimate of high orders m ≫ 1 when $K<t^{\frac{1}{6}}$shows that the corrections z_k have the ‘factorial/power’ form familiar in divergent asymptotic expansions, the least term being of order exp $\left\{-\frac{1}{2}{K}^{2}t\right\}$…

In 1990, 30 years after he and David Bohm had discovered the AB effect in Bristol (Chapter 1), Yakir Aharonov visited again. He likes to frame his discoveries as paradoxes, and during the visit, he told me: “I can imagine opening a window in a box containing only red light, and out would come a gamma ray”. Paradoxical indeed! It was several years before I understood what he meant: a band-limited function (‘red light’) can oscillate arbitrarily faster (‘a gamma ray’) than its fastest Fourier component, over arbitrarily long intervals…

Band-limited functions f(x) can oscillate for arbitrarily long intervals arbitrarily faster than the highest frequency they contain. A class of integral representations exhibiting these ‘superoscillations’ is described, and by asymptotic analysis the origin of the phenomenon is shown to be complex saddles in frequency space. Computations confirm the existence of superoscillations. The price paid for superoscillations is that in the infinitely longer range where f(x) oscillates conventionally its value is exponentially larger. For example, to reproduce Beethoveen’s ninth symphony as superoscillations with a 1Hz bandwidth requires a signal exp{10¹⁹} times stronger than with conventional oscillations.

Weak values, resulting from the action of an operator on a preselected state when measured after postselection by a different state, can lie outside the spectrum of eigenvalues of the operator: they can be ‘superweaky’. This phenomenon can be quantified by averaging over an ensemble of the two states, and calculating the probability distribution of the weak values. If there are many eigenvalues, distributed within a finite range, this distribution takes a simple universal generalized lorentzian form, and the ‘superweak probablility’, of weak values outside the spectrum, can be as large as $1-1\sqrt{2}=\mathrm{...}$ By contrast, the familiar expectation values always lie within the spectral range, and their distribution, although approximately gaussian for many eigenvalues, is not universal.

The association between large shifts of a pointer in a weak measurement and fast oscillations in an associated function involving the pre- and post-selected states has been clarified in a recent paper (Aharonov et al 2011 J. Phys. A: Math. Theor. 44 365304). Here we explore the association further for the case of an observable with N discrete eigenvalues, by calculating and illustrating how the supershift emerges, even for N = 2, as the uncertainty in the pointer position increases. This happens if the initial pointer wavefunction is Gaussian or Lorentzian but not if it is exponential.

My research themes of asymptotics and singularities led to a contribution to the philosophical problem of theory reduction: how a theory (e.g. classical mechanics) can be a special case of a more general theory (e.g. quantum mechanics) that has a completely different conceptual basis. The new insight [6.1] stemmed from the recognition that in physics the emergence of a theory occurs as a limiting case when a parameter in the more general theory (e.g. a dimensionless combination proportional to Planck’s constant) tends to zero. And in almost every case (quantum to classical, waves to rays, statistical mechanics to thermodynamics…), the reduction is both complicated and enriched by the fact that the limits are singular [6.2]…

The following sections are included:

• Introduction
• Singular limits and emergent phenomena
• Quantum and classical mechanics
• Divergent series
• Concluding remarks
• References

Biting into an apple and finding a maggot is unpleasant enough, but finding half a maggot is worse. Discovering one-third of a maggot would be more distressing still: The less you find, the more you might have eaten. Extrapolating to the limit, an encounter with no maggot at all should be the ultimate bad-apple experience. This remorseless logic fails, however, because the limit is singular: A very small maggot fraction(f ≪ 1) is qualitatively different from no maggot (f = 0). Limits in physics can be singular too – indeed they usually are – reflecting deep aspects of our scientific description of the world…

Newton’s third law does not apply to the interaction between philosophers (‘them’) and physicists (‘us’). It has usually been asymmetrical, with ‘us’ influencing ‘them’, without ‘them’ acting on ‘us’. In a way this is natural, because the raw material that philosophers study are the discoveries and theories of science and the interactions between scientists, while the primary preoccupation of physicists is not the study of philosophy or philosophers. I do not deny that there have been eminent scientists (Einstein, Poincare, Bohr…) who have pondered on the philosophical significance of the scientific picture of the world, and much of what they said has been immediately appreciated by practicing scientists. But their wise intellectual interventions have usually been outside the philosophical mainstream…

Quantum mechanics and classical mechanics are magnificent structures, each with vast explanatory reach. In their usual formulations, the two theories look very different. Classical physics describes the motion of particles in terms of their positions and velocities, influenced by forces acting between them and from outside. In quantum physics, dynamical variables are operators, acting on states in a Hilbert space of vectors, with evolution determined by a Hamiltonian operator. Nevertheless, there is considerable overlap in the phenomena they describe. Although nobody planning an extraterrestrial mission would use Schrödinger’s equation as a starting point for programming the trajectories of rockets, few physicists doubt that quantum mechanics applies to planets and spacecraft as well as atoms. The subtle and intricate relations between the classical and quantum worlds are the subject of this very welcome book by Alisa Bokulich…

These thirteen contributions record my appreciation of some scientists I greatly admire. With the exception of Dirac [7.5], they are not on everyone’s all-time favourites list (Galileo, Newton, Maxwell, Einstein…), but they have influenced the content and style of my science. My debt to my supervisor Dingle [7.8] should be clear from Chapter 4. John Ziman ([7.7], see also [B394]) was the intellectual attractor who brought me to Bristol…

The past two years have seen centenaries of the birth of several of India’s most brilliant citizens: Nehru, founder of the modern state; Ramanujan, diviner of amazing mathematical formulae; and Raman, the prolific and original physicist whom this book celebrates. Raman’s life and work were complex and many-sided, causing great difficulties of judgement and emphasis for any biographer. Venkataraman (who in spite of his name does not share with several other talented Indian scientists the distinction of being a relative of Raman) solves these problems by anchoring his story firmly in the science. He has produced a model of scientific biography, written with respect and affection for its subject but with a clear-eyed perception of his faults, in a relaxed and gently witty style…

Sivaraj Ramaseshan graciously invited me to write an essay review of the collected works of S. Pancharatnam. As a partial response to this invitation, I am happy to show my admiration of Pancharatnam by providing the following comments on three of his papers.

Mr Vice-Chancellor:

When Yakir Aharonov began studying physics in the 1950s at the Technion in Haifa, Israel, one of his teachers was Nathan Rosen. Twenty years earlier, Rosen had made an important contribution to the interpretation of the (then rather new) quantum mechanics. But by the time the young Aharonov became entranced by the fundamentals of that theory, its study was unfashionable, and Rosen advised him to concentrate on applications…

We mourn the death of Charles Frank, but that unsentimental man would have wished us rather to celebrate his life — a life that spanned most of this century, was shaped by its main events, and, at a crucial moment, helped to determine those events, to the benefit of all of us…

Each day, I walk past the road where Paul Adrien Maurice Dirac lived as a child. It is pleasant to have even this tenuous association with one of the greatest intellects of the 20th century. Paul Dirac was born at 15 Monk Road in Bishopston, Bristol, on 8 August 1902, and educated at the nearby Bishop Road Primary School. The family later moved to Cotham Road, near the University of Bristol, and in 1914 the young Dirac joined Cotham Grammar School, formerly the Merchant Venturers…

Philip Morrison’s review of the first English Edition [22] of Benoit Mandelbrot’s book hit a precise resonance with me. For the previous few years I had been studying waves reflected from irregular surfaces, motivated by an application to geophysics. All existing theories assumed random surfaces where asperities with a single length scale perturbed a plane; this disturbed me, because it created an artificial distinction between ‘roughness’ and ‘geography’ (in this case the flat earth). Before fractals, I had no idea how to convert this unease into physics…

To illuminate some aspects of John Ziman’s intellectual character and his years in Bristol, I have to be a little personal…

Robert (‘Bob’) Dingle was born on March 26, 1926 in Manchester. He studied at Cambridge University (Tripos Part I 1945, Part II 1946) and began research in theoretical physics under the supervision of D R Hartree, earning a Ph.D from Cambridge in 1952 after spending the year 1947-1948 visiting Bristol under the supervision of Professors Mott and Fröhlich. Following research positions in Delft in the Netherlands and Ottawa in Canada, he was appointed to a Readership at the University of Western Australia. In June 1960 he arrived in St Andrews as the first occupant of the Chair of Theoretical Physics. He was elected to the Royal Society of Edinburgh in 1961. After a sabbatical period in Canada, California and Western Australia, he remained in St Andrews until his early retirement through ill-health in 1987…

It’s an honour to be asked to speak at this afternoon of remembering, this celebration for our dear colleague and friend Balazs Györffy. But it’s an honour I never wanted. By rights — and I mean statistically — it should have been the other way round: the men in my family die young. But Balazs was so strong, so superfit, that, as one of our former colleagues wrote, if anyone would be immortal it would be Balazs…

Vladimir Arnold, an eminent mathematician of our time, passed away on June 3, 2010, nine days before his seventy-third birthday. This article, along with one in the next issue of the Notices, touches on his outstanding personality and his great contribution to mathematics…

In the 1970s, physicists were made aware, largely through the efforts of the late Joseph Ford, that classical hamiltonian mechanics was enjoying a quiet revolution. The traditional emphasis had been on exactly solvable models, with as many conserved quantities as degrees of freedom, in which the motion was integrable and predictable. Examples are the Kepler ellipses of planetary motion, and the simple pendulum: ‘as regular as clockwork’. The new research, incorporating Russian analytical mechanics and computer simulations inspired by statistical mechanics, revealed that most (technically, ‘almost all’) dynamical systems behave very differently. There are few conserved quantities, and motion, in part or all of the phase space, is nonseparable and unpredictable, that is, unstable: initially neighbouring orbits diverge exponentially. This is classical chaos…

My parents came from the same part of London as Frank Olver, so I recognized his London ways: smart dry wit, and of course the accent, unchanged after more than sixty years in the USA…

Three of Tonomura’s fundamental quantum physics experiments are discussed from a personal perspective…

In my family, nobody was scientifically-educated. My father was a taxi-driver (see [9.13]) and my mother a dressmaker; both left school at 14. When the romance of scientific discovery captivated me as a youngster, I had no idea that the life scientific would involve travelling, to a degree then enjoyed only by royalty, politicians, business people and sports teams. What attracts me about such travels — so far, to more than 60 countries — are the contrasts between the universality of the science as practised in different parts of the world (albeit with minor differences of style), and the diversity of cultures, scenery and (especially!) food…

Some of my Israeli colleagues urged me not to visit occupied Palestine, or, if I insisted on going, to take a gun. They cited examples of well-meaning visitors who had been attacked or even killed. I took comfort from the fact that these were always the same few examples, and decided to go anyway (unarmed), trusting the hospitality of my hosts to protect me from possible assailants. However, prudence did suggest that driving through Palestine in an Israeli rental car might be risky, so my wife dropped me by the checkpoint at Tulkaram. My host was waiting in a Palestinian -registered car on the other side, to drive me to Nablus, a few miles away in the West Bank. It was unnerving to stop in a back street just moments later and be told “Now we have to change to another car”, but this turned out to be connected more with the mysteries of insurance than the beginning of an abduction…

“This is Hizbollah country”, our guide reassured us as we approached the Roman temple complex of Baalbeck (Heliopolis), with the world’s largest standing columns back-lit by snow-covered mountains. Indeed there were checkpoints every few miles — Syrian and Lebanese, carefully separated — and posters of President Assad and Ayatollah Khomeini (also, incongruously, the Restaurant Lady Diana). By two days we missed a huge rally in Baalbeck, celebrating ‘Jerusalem day’…

I was curious to see Odessa — Ukraine’s principal port on the northern shore of the Black Sea — because that was the city from where my grandparents emigrated to London in 1906. A few weeks earlier, I had been in Little Odessa, otherwise known as Brighton Beach, Brooklyn, New York City (and reputed money-laundering centre of the USA). Little Odessa is populated almost entirely with emigrants from Ukraine, and the ambience is markedly unamerican: shop signs in Cyrillic, bad-tempered waiters, no credit cards accepted…

This ragbag of a chapter starts [9.1] with what is becoming a new research theme, being developed with Pragya Shukla, whose significance has yet to emerge. Curl forces are classical forces depending only on position and not derivable from a scalar potential: their curl is non-zero. Although non-conservative, these forces are not dissipative, and, because there is (usually — but see [B475]) no underlying Hamiltonian or Lagrangian, Noether’s theorem does not apply: a curl force can possess a symmetry without a corresponding conservation law, and there can be a conservation law without a corresponding symmetry. Although largely unfamiliar to physicists, curl forces describe [B466] the effective dynamics of polarizable particles in fields of light…

This is a theoretical study of Newtonian trajectories governed by curl forces, i.e. position-dependent but not derivable from a potential, investigating in particular the possible existence of conserved quantities. Although nonconservative and nonhamiltonian, curl forces are not dissipative because volume in the position–velocity state space is preserved. A physical example is the effective forces exerted on small particles by light. When the force has rotational symmetry, for example when generated by an isolated optical vortex, particles spiral outwards and escape, even with an attractive gradient force, however strong. Without rotational symmetry, and for dynamics in the plane, the state space is four-dimensional, and to search for possible constants of motion we introduce the Volume of section: a numerical procedure, in which orbits are plotted as dots in a three-dimensional subspace. For some curl forces, e.g. optical fields with two opposite-strength vortices, the dots lie on a surface, indicating a hidden constant of motion. For other curl forces, e.g. those from four vortices, the dots explore clouds, in an unfamiliar kind of chaos, suggesting that no constant of motion exists. The curl force dynamics generated by optical vortices could be studied experimentally.

Diamagnetic objects are repelled by magnetic fields. If the fields are strong enough, this repulsion can balance gravity, and objects levitated in this way can be held in stable equilibrium, apparently violating Earnshaw’s theorem. In fact Earnshaw’s theorem does not apply to induced magnetism, and it is possible for the total energy (gravitational + magnetic) to possess a minimum. General stability conditions are derived, and it is shown that stable zones always exist on the axis of a field with rotational symmetry, and include the inflection point of the magnitude of the field. For the field inside a solenoid, the zone is calculated in detail; if the solenoid is long, the zone is centred on the top end, and its vertical extent is about half the radius of the solenoid. The theory explains recent experiments by Geim et al, in which a variety of objects (one of which was a living frog) was levitated in a field of about 16 T. Similar ideas explain the stability of a spinning magnet (Levitron^TM) above a magnetized base plate. Stable levitation of paramagnets is impossible.

When he was a child, Appleyard tells us, he was astonished by the ability of his father (an engineer) to calculate the volume of water in a water tower, but “sensed something dangerous and ominous in this wisdom”. During a later career as a journalist, in which he was often concerned with scientists and technological issues related to science, this unease matured and has now become the mainspring of an intricately connected, wide-ranging and passionate attack on science. I will try to summarise it…

It has been said that while the optimist believes that we live in the best of all worlds, the pessimist knows that we do. David Deutsch’s book is profoundly optimistic, but in a way that carefully avoids these extremes. His optimism is based not on Panglossian reassurance, but on a principled and passionate confidence in people’s constructive inventiveness. He thinks that the potential for unlimited creativity has arisen only once, as the culmination of a process that began in the Enlightenment…

For the past seven years, I have been the steward of the journal where some of the world’s best scientists published their seminal work: Maxwell, Kelvin, Marconi, the Braggs, Ramanujan, Dirac, Raman, Crick and Watson—to name a few. As Editor-in-Chief, I have felt the weight of this tradition as a responsibility to be taken very seriously. But it has also been an agreeable responsibility; the staff at the Royal Society, and the members of the Editorial Board, have been unfailingly helpful. And I have enjoyed negotiating the challenges and contrasts that come with the job: trying to keep the quality high, dealing with authors and referees who are brilliant, conscientious, argumentative, irascible, careless, eccentric, etc…

It’s unreal, isn’t it? I’ve always had a strong sense of the ridiculous, of the absurd, and this is working overtime now. Many of you must have been thinking, ‘Why Berry?’ Well, since I am the least knightly person I know, I have been asking the same question, and although I haven’t come up with an answer I can offer a few thoughts…

I am a quantum mechanic. So is Yakir Aharonov. A technical term in our subject is the entangled state. Anyone who has a conversation with Yakir gets into an entangled state, with contradictions, digressions and interruptions all mixed up — Talmudic, I suppose. But now, here, Yakir can’t interrupt me, as I declare what an honour and delight it is to share this occasion with such a quick, deep and subtle man. There is only one sadness that I’m sure he shares with me: that David Bohm, with whom he did some of his seminal research, is no longer living; if he were, he would surely be here tonight…

This morning I had the pleasure of listening to several of your talks, and I’ve seen the programme of your activities today. Very impressive it is too. In my department — physics — at the University of Bristol, England, we have encouraged undergraduate research for many years, but there ’s not a university-wide organization like yours…

I’ve known this abode of passionate rationality for nearly twenty years. Passionate rationality? One example: the inspiration that comes from startling connections — like last week, in the journal Nature, a picture from the neurophysiology department here, representing what a cat sees with its eyes shut — identical to pictures we have been making in the physics of random waves. The sciences seem very different, but a mathematical theme links them. Another: when the colour of gold is calculated from the quantum mechanics of electrons, the predicted colour is silver! But gold is heavy, so its electrons move fast, and to get the colour right it’s necessary to incorporate Einstein’s relativity: gold is relativistic silver. Such delights, such passionate rationality…

The H H Wills Physics Laboratory is the highest point in the centre of Bristol, so it can be seen from many places across the city and is easy to find (‘Keep going up’). When I arrived in 1965, damp behind the ears with a new PhD from St Andrews, I was impressed not only by the location of the building but also by its imposing 1927 design as a mock castle. Then, the physics department was clamorous with builders constructing the ‘new wing’. Four decades later, as my retirement looms, the builders are back, renovating the whole laboratory…

If I’m run over crossing the road tomorrow, I’ll be stereotyped in the newspapers as “Pensioner and grandfather”. Until the last moment, you think it will never happen. I’m still in denial…

For more than forty years, I have been able to make physics with delight and enthusiasm. In large measure, I owe this privilege to John Ziman’s protection and mentoring during my early years in Bristol. We never worked on the same physics, and what I did then was unappreciated for a long time, so it remains a mystery how he could divine that I might have a spark of intellect, and shield this from potentially hostile winds so that it had the chance to grow into a flame…

How delightful to discover our abstractions clothing Nature’s realities:

Singularities of smooth gradient maps in rainbows and tsunamis

The Laplace operator in oriental magic mirrors

Elliptic integrals in the polarization pattern of the clear blue sky

Geometry of twists and turns in quantum indistinguishability

Matrix degeneracies in overhead-projector transparencies

Gauss sums in the light beyond a humble diffraction grating…

The following sections are included:

• Chapter 10: Awards, Honorary Degrees, etc.
• Chapter 11: Publications