Quantum curl forces

Classical nonhamiltonian dynamics, driven by external ‘curl forces’ (which are not the gradient of a potential) is extended to the quantum domain. This is a generalisation of the two-stage Madelung procedure for the quantum Hamiltonian case: (i) considering not individual trajectories but families of them, characterised by their velocity and density fields (both functions of position and in general time); and (ii) adding the gradient of the quantum potential to the external curl force. Curl forces require the velocity field to have nonzero vorticity, so there is no underlying singlevalued wavefunction. Two explicit examples are presented. A possible experiment would be the motion of small particles with complex polarisability, influenced by forces from optical fields.


Introduction
Newtonian mechanics is more general than Hamiltonian or Lagrangian mechanics.An example is the dynamics of particles driven by 'curl forces' [1]: forces F = F (r), depending on position r but not velocity u, which are not the gradient of a potential.In the familiar case where there is a potential V, then F = −∇V and ∇ × F = 0.For 'curl forces' this is not the case; the classical dynamics is (1.1) * Author to whom any correspondence should be addressed.
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Curl forces contrast with forces derivable from a potential in several ways [1].Although nonconservative, motion is nondissipative: volume is conserved in the state space r, u; therefore there are no attractors.Noether's theorem does not apply: the link between symmetries and conservation laws is broken.And there can be dynamical evolutions with no conserved quantity, even though F is independent of time: for N-dimensional configuration space r, trajectories can explore a 2N-dimensional region in r, u space [2].
Curl forces do not occur in fundamental classical physics, where forces are associated with potentials.But they are fundamental dynamical mathematics, in the sense that Newton's equation (1.1) is more general than the special case for potential forces.To avoid confusion, we emphasise that the classical curl forces we consider are velocity-independent; as explained earlier [1], this excludes magnetic forces, where F depends on u as well as r and there is a familiar Hamiltonian description.Curl forces appear as emergent forces: approximate descriptions, in engineering [3][4][5], and as a class of forces exerted by optical fields on small dielectric particles with complex polarisability [6].
Here we explore the natural question: can the concept of curl forces be extended to quantum mechanics?At first thought, the answer is no, because in its standard formulation quantum physics is fundamentally Hamiltonian.But we will propose a natural generalisation of quantum mechanics that does incorporate curl forces.This is based on the Madelung picture of quantum mechanics [7] described later, developed from ideas of de Broglie [8], and formally equivalent to 'Bohmian mechanics' [9,10].Extending Madelung mechanics to include velocity fields with vorticity has been considered before [11][12][13][14]; our emphasis here is the connection with curl forces.
The paper is structured as follows.Section 2 contains the quantum curl force formalism, emphasising its differences from familiar quantum mechanics.Section 3 describes a quantum curl force where ∇ × F is nonzero only at the origin, and suitably adapted techniques from conventional quantum mechanics can be applied if the origin is excluded.Section 4 describes a quantum velocity field with streamlines on cylinders.The concluding section 5 includes a suggestion for possible experiments involving quantum curl forces.
Appendix 1 shows that Madelung quantum mechanics, and therefore our curl force generalisation, also apply where the kinetic energy is an anisotropic function of velocity.Appendix 2 supplies two checks on the calculations in section 3. Appendix 3 elaborates and generalises an issue arising in section 3, concerning angle variables in the plane and 'unwrapped' on a helicoid.Appendix 4 explores the continuity relation between the flow velocity and its density for steady flows.
We emphasise that we are not proposing quantum curl force dynamics as a fundamental alternative to well-established standard quantum physics.Rather, we regard it as a formalism worth exploring, which could have application as a higher-level approximate ('effective', or 'emergent') theory, as for classical curl forces [6].We are also agnostic on questions of interpretation, for example concerning the physical meaning of the trajectories in conventional quantum mechanics or our generalisation to curl forces; we employ Madelung's formulation, and do not discuss the philosophical aspects commonly considered in 'Bohmian mechanics' [15].Nor will we be concerned with the mathematical subtleties of the precise relation between the Madelung and Schrödinger pictures, well explored elsewhere [16][17][18].

Formulation
We will simplify writing by indicating the arguments of functions only where necessary to avoid confusion, and will use units where the particle mass, and also ℏ (except in section 3), are unity.
We need to review the Madelung (hydrodynamic) picture [7,15] of familiar quantum mechanics.Waves satisfying the Schrödinger equation are written in the polar form with ρ regarded as a density at position r and time t, and the phase gradient as a quantum flow velocity.There are two fundamental differences from classical Hamiltonian particle dynamics.First, the flow describes a family of trajectories: u is a flow field, depending on r and t, not the velocity of an isolated single trajectory.Second, the dynamics is supplemented by the force from a quantum potential.Thus Introducing V Q is fundamental because it incorporates nonlocality (holism) into the Madelung formulation (trajectories in the family represented by the velocity field u (r) are connected by the underlying Schrödinger equation, in contrast with Newtonian trajectories, where the velocity can be specified separately at each point).Temporarily reinstating physical units, √ ρ , i.e. proportional to ℏ 2 .Recently, there has been a revival of interest in the quantum potential, with applications to thermodynamics and information theory [19,20], superoscillations [21,22], and trajectories in optical beams [23].
The flow u and density ρ are connected by the continuity equation The fact that u is a phase gradient implies that the flow is irrotational, i.e. the vorticity ω vanishes: except at zero lines of ψ , around which, to ensure that ψ is singlevalued [11,16], This completes the reprise of the Madelung formalism.
Our proposed quantum curl force generalisation retains the picture of a flow field ρ, u and the continuity equation (2.4), which for steady flows is explored further in appendix 4. We also retain (2.3), with the fundamental difference that now F can be a curl force as in (1.1), i.e.F ̸ = −∇V.We also discard the circulation condition (2.6).This is important: in the Madelung formulation of conventional quantum (and more generally wave) physics: velocity fields associated with singlevalued wavefunctions can possess vorticity in the form of isolated vortex lines [13,[24][25][26][27][28][29], but these are constrained by (2.6); by contrast, although the velocity fields we consider here can possess isolated vortex lines (as in section 3), their circulation is not so constrained.
To understand an important implication of quantum curl force dynamics, we use the identity (2.7) Thus the external force that would generate a given flow is i.e.
This involves only the velocity u and its vorticity ω, not the density ρ.
The important implication is that quantum curl forces cannot generate irrotational flows: for quantum curl forces, (2.5) is violated.This further implies that u is not a phase gradient as in (2.2), so there is no phase and therefore no singlevalued wavefunction satisfying a Schrödinger equation: all that survives is the flow.As in the Madelung picture of standard quantum mechanics, the field u = u (r, t) determines 'quantum trajectories' r (t) by For the important class of steady (i.e.time-independent) flows, the underlying particle trajectories are the streamlines of the velocity field u, along which particles move with speed |u|.In this case, it follows from (2.9) that Beltrami flows [30], in which the vectors u and ω are parallel, cannot be generated by curl forces.
Quantum curl dynamics can be regarded as the fourth level in a hierarchy: Level 1. Classical conservative Newtonian dynamics.This is represented by the first of equation (1.1) with F = −∇V, and quantum potential V Q = 0. Level 2. Classical curl forces.This is both equations in (1.1), also with V Q = 0. Level 3. Quantum Madelung dynamics.This is (2.3), i.e. (1.1) with F = −∇V and V Q ̸ = 0. Level 4. Quantum curl dynamics.This is the new level: (2.3) with V Q ̸ = 0 and also F ̸ = −∇V.
There is a fundamental difference between the quantum trajectories generated by (2.10) and the corresponding classical trajectories.Classical trajectories in the family representing a quantum state can cross in spacetime; in the semiclassical description, such crossings underly quantum interference.But in families of quantum trajectories, including those generated by curl forces, such crossings are avoided, because the velocity u is uniquely defined for each r, t.Instead of crossing, quantum trajectories typically represent interference by exhibiting undulations.This will be illustrated in section 3. (To avoid confusion, we note that if the velocity field is time-dependent the tracks of the quantum orbits can cross in r space; this simply corresponds to orbits reaching the same point at different times, at which u is different; this too will be illustrated in section 3.) The undulation phenomenon is ubiquitous in the physics of waves, not restricted to quantum mechanics, and occurring also in quantum curl force dynamics.Undulations were conjectured by Newton [31] in connection with the intensity oscillations associated with optical edge diffraction [32], with rays described as having '…a motion like that of an eel' [33,34] (for undulations in 2-slit interference, or waves from two sources, see [15], figure 3 of [32], and section 2 of [12]).In modern terminology, these different interpretations of interferencesemiclassical superposition, and undulations of quantum trajectories-can be expressed in standard wave physics as follows: phase is a nonlinear function of its wavefunction, so the phase gradient of a superposition is different from the superposition of phase gradients [32].) A cautionary remark.The first equality in (2.3), combined with (2.7), can be written (2.11) involving two fields E and B and their potentials, connected by the vector field u: (2.12) As has been noted [12,13], this superficially resembles electromagnetism.The analogy is misleading, for several reasons.First, the pair of equations (2.11) and (2.12) constitutes a tautology: it is satisfied for any u and so cannot determine the (classical or quantum) velocity field or its dynamics.To determine u, the contribution from F − ∇V Q must be included.Second, the fields E and B do not satisfy Maxwell's equations.Third, the seemingly natural Hamiltonian generates dynamics different from (2.11).

Formulation
This is the two-dimensional force (in plane polar coordinates r, θ) in which here and hereafter e will denote unit vectors.(A coefficient modifying the strength of F can be eliminated by scaling r and t in the Schrödinger equations to follow.)F is a curl force with circular symmetry (i.e.invariant under rotations).The curl is nonzero only at r = 0; the circulation around any loop surrounding the origin is constant, so, from Stokes's theorem, Excluding the origin, F can be written as the minus the gradient of the scalar potential whose domain is the full line −∞ < θ < ∞, i.e. including windings, not the plane angle 0 ⩽ θ < 2π .For this case, in contrast to general curl forces, there is a Hamiltonian; excluding r = 0, this is Earlier [1], we considered the classical particle dynamics generated by H 1 , and will return to this later.Our main concern here is the quantum dynamics.For this particular curl force, where ∇ × F is concentrated at a point, there is an underlying Schrödinger equation, namely (for particle mass defined as unity) in which we now write ℏ explicitly.But the Hamiltonian H 1 is not singlevalued in θ, i.e. the potential violates the circular symmetry of the force, reflecting its nonzero curl.The solutions of (3.5) are not plane-periodic but are smooth and singlevalued on the helicoid, with windings specified by the integer (θ − θ (mod 2π )) / ( 2π ); see appendix 3 for more discussion.Circular symmetry can be restored by the time-dependent gauge transformation where Ψ satisfies Physically, the time-independent curl force Hamiltonian H 1 , whose solutions are singlevalued on the helicoid but not the plane, has been replaced by the time-dependent Hamiltonian H 2 (3.7); this is periodic in θ and its solutions are singlevalued on the punctured plane.
The Hamiltonian H 2 represents an intriguing variant of the Aharonov-Bohm effect: it corresponds, as noted earlier [1], to the motion of a charged quantum particle in the presence of a line of magnetic flux whose strength increases linearly with time, located at the origin.The relation between solutions of the 'unwrapped' Hamiltonion H 1 and those of the periodic H 2 is the Poisson sum formula, elaborated and described in a more general context in appendix 3.
With circular symmetry restored and θ eliminated, the Hamiltonian H 2 is separable in r and θ, so Ψ can be written as a superposition of partial waves.Reinstating the variables, where the radial wavefunction satisfies In the representation H 1 , the angular momentum is Thus the circulation condition (2.6) is violated.There is no vector potential, so the kinetic and canonical angular momenta are equal.In the representation H 2 , ℏ∂ θ arg ψ = mℏ is the conserved canonical angular momentum, and by adding −rA • e θ = t the gauge-invariant kinetic angular momentum (3.10) is regained, and the circulation condition (2.6) restored.The gaugeinvariant angular momentum (3.10) is not conserved; as noted earlier in the classical case, this increases linearly in time, illustrating the non-applicability of Noether's theorem in the presence of the curl force (3.1) which possesses rotational symmetry.Since the angular quantum number appears only in the combination t + mℏ, we need consider only m = 0; the amplitudes of the other partial waves in the superposition (3.8) can be obtained simply by time translation.
The gauge transformation (3.6) has replaced the Hamiltonian H 1 , whose Schrödinger equation (3.5) is separable in r, t but not r, θ, by the Hamiltonian H 2 , whose Schrödinger equation (3.7) is separable in r, θ but not r, t.This is a simplification because the radial equation (3.9) is first-order in t, which can be more easily solved numerically for any initial state R 0 (r) ≡ R (r, 0).
The quantities of interest in our quantum curl theory are the density ρ and the velocity u.For the partial wave m, these are with t → t + mℏ.The velocity u is not irrotational; its vorticity ω, reflecting its origin in the curl force (3.1), is concentrated at the origin: (cf. the corresponding formula (3.2) for ∇ × F).Concentrated vorticity (i.e.vortex lines in three-dimensional space) has been studied [13] for the special case where its strength is timeindependent, in contrast to (3.12).

Radial dynamics
To illustrate these results, we invoke separability and consider first the r dynamics, and incorporate θ later.As a convenient normalised initial state, we choose a semiclassically narrow Gaussian at r = r 0 , approximating δ (r − r 0 ).In the position and momentum (=velocity) representation, and with generic constants K, This choice has the advantage that the velocity intensity distribution does not involve ℏ, so the initial state corresponds classically to a distribution concentrated on the v axis in state space r, v, with velocity (momentum) width Π .Scaling of (3.9) enables any r 0 to be transformed to r 0 = 1.We will compare the classical and quantum trajectories.We showed [1] that the classical Newtonian trajectories r (t) are determined by This represents a family of trajectories, parameterised by v 0 .The quantum radial velocity at the spacetime point r, t is (cf (3.11))where R (r, t) is determined by (3.9) and the first of (3.13).Then the quantum trajectories r (t) are the solutions of for different values of r 0 .(The choice of small but nonzero t 0 reflects the very rapid spread of the wave away from r = 1; the results are not sensitive to t 0 .) Figure 1 illustrates the quantum and classical evolutions.Figure 1(a) shows classical paths, each calculated from (3.14) for a given value of the parameter v 0 .Through each spacetime point in the wedge-shaped region in figure 1(a), bounded below by a caustic and above by a cutoff in v 0 , two classical paths cross.These crossings give rise to the Moiré pattern, whose details depend on the choice of v 0 values; we chose −15 ⩽ v 0 ⩽ 15 with values separated by 1 /4.The crossings are clearly visible in the magnification in figure 1(c).
The quantum trajectories are entirely different: as described earlier, they do not cross.As figure 1(b) shows, they gather into bunches as t increases, reflecting the quantum oscillations of R (r, t).The magnification (figure 1(d)) shows that the bunching is associated with undulations of the quantum trajectories (Newton's eels).Appendix 3 describes two numerical checks of these calculations.

Dynamics in the x,y plane
Again invoking separability, we now reinstate θ and look at classical and quantum trajectories in the coordinate plane r = {x, y} = r {cos θ, sin θ}, corresponding to the r trajectories in figure 1.For the classical trajectories, we choose the family parameterised by v 0 as in (3.14) with r 0 = 1, ∂ t r (0) = v 0 , and θ (0) = 0, ∂ t θ (0) = 0, i.e. x (0) = 1, y (0) = 0, ∂ t x (0) = v 0 , ∂ t y (0) = 0. Thus, from the Hamiltonian (3.4) (conserved energy E), with angular momentum written as t, we fix E = v 2 0 /2 and get (3.18) Figure 2 shows this family in r space.Over the smaller range (figure 2(a)), they do not cross, but over much larger ranges (figure 2(b)) they do (as discussed in section 2).As we showed earlier [1], for vastly longer times, of order t ∼ exp(4π 2 n 2 ) ∼ (1.4 × 10 17 ) n 2 , the trajectories will wind n times, with more crossings.The full family of classical trajectories corresponding to the quantum states we are describing are the patterns in figure 2, rotated, but it is clearer to show the unrotated family.The quantum trajectories r (t) are generated by (2.10) with the quantum velocity u (r, t) in (3.11).Figure 3 shows the intensity, with the instantaneous velocity field superimposed, at a particular time.The twirl of u near the origin reflects the underlying curl force, even for this mode with m = 0, for which Ψ is independent of θ.
Figure 4 shows the quantum trajectories corresponding to figure 1, on three different scales.On the largest scale (figure 4(a)), the undulations (Newton's eels) are not discernable.Zooming in (figure 4(b)), the undulations can be glimpsed for positive θ; and the crossing phenomenon described earlier can be seen: more than one trajectory can wind and reach the same r at different times, when u is different.Zooming further (figure 4(c)), Newton's eels are clearer; they form a hierarchy, with finer-scale undulations closer to the positive x axis.

More general states
The initial state (3.13), concentrated near r = r 0 , generates only one of the infinitely many solutions of the radial equation (3.9).Another interesting class of solutions is the set of instantaneous eigenstates of the operator in (3.9).These satisfy with eigenvalue k 2 , and are the Bessel functions possessing the unusual feature that the order increases linearly with time.These instantaneous eigenstates (which are not square-integrable, concordant with the unbound nature of the corresponding classical curl force dynamics [1]), are the building-blocks of the adiabatic approximation to the time-dependent solutions of (3.9).

Quantum curl force with velocity streamlines on cylinders
In this section, we will illustrate several features of quantum curl dynamics by contriving a specified steady three-dimensional velocity field with nonzero vorticity, and 'reverse-engineering' to determine the curl force that would generate it.The field is The parameter ε can be regarded as a perturbation, switching on the vorticity.Figure 5(a) illustrates one of the streamlines for the case The streamlines oscillate on cylinders centred on the x axis, and the complete field of streamlines consists of copies, translated along x and rotated and expanded about the x axis.Figure 5(b) shows a few rotated streamlines.
The vorticity is non-zero for finite ε: 3) The field is not Beltrami: In addition, the helicity field is nonzero: A simple argument shows that this implies that u cannot rescaled to make F curl-free; such 'decurling' would lead to a velocity field with the same pattern of trajectories, differently traversed.
The particular field u in (4.1) is divergenceless: and so describes an incompressible flow.This is helpful in calculating the density ρ because the continuity equation (2.4) simplifies to i.e. the gradient of the density is perpendicular to the flow.The calculation of densities compatible with velocity fields that are not incompressible is described in appendix 4. For the field (4.1) and (4.7) is satisfied by any density of the form A convenient choice is representing a localised quantum beam driven by a curl force centred on the x axis.
The corresponding quantum potential is This generates a quantum force repulsive away from the x axis, which might seem puzzling.But note that, in the 'reverse-engineered' total external force (2.8), ∇V Q occurs with a + sign, attracting the quantum state near the x axis.(A helpful non-curl, i.e. potential force, example is the ground state of the 1D quantum simple harmonic oscillator.The wavefunction is real, so the velocity u is zero: the Madelung particles are stationary, so the quantum potential must be repulsive in order to cancel the attractive external potential.)A similar example, involving a linear potential, has also been interpreted in terms of the force from the quantum potential [35].) The external force (2.8), that would generate particles moving along the streamlines of (4.1), is correctly attracting the waves.And it is a curl force: Finally, the explicit form of the streamlines, solving for the field (4.1) and the special case (4.2), is (using where the amplitude c and phase γ are real. The contrived velocity field (4.1) lacks the 'Newton's eel' undulations characteristic of Madelung streamlines, described in section 2 and illustrated in section 3; this demonstrates that there are quantum curl forces without these undulations.

Concluding remarks
The Madelung picture of quantum mechanics does not explicitly refer to a Schrödinger equation or a wavefunction.It has been argued [36] that in a counterfactual history, in which the wave equation had been discovered later, the Madelung representation (with the Bohmian interpretation) might be today's default picture.But of course there is a wave underlying the Madelung picture; its presence is betrayed by the singlevaluedness condition (2.6).
In our proposed generalisation to quantum curl forces, by adding the quantum potential to classical curl dynamics by analogy with the Madelung procedure for potential forces, there is in general no underlying singlevalued wavefunction.(The exceptional case studied in section 3 does involve a wavefunction, but before the gauge transformation it is not singlevalued in the plane.)In our exploration of this generalisation, we gave examples where it was not necessary to solve the nonlinearly coupled continuity and velocity equations (2.4) and (2.8) for a general curl force.Instead, in section 3 we were able to adapt standard quantum techniques to the unusual situation where the curl of the force was concentrated at a point, interpreting this as an unfamiliar time-dependent Aharonov-Bohm system.And in section 4 we specified the velocity field and by reverse-engineering determined the curl force that would generate it.The general case, for which these strategies are unavailable, seems difficult; it deserves further study.
A possible experimental implication of the quantum generalisation involves the forces exerted by light waves on small dielectric polarisable particles with complex polarisability α.Classically, these forces are of two kinds [37,38].There is the gradient force employed for trapping the particles; this is proportional to Reα.And there is a curl force (in optical terminology it is the 'scattering force'), proportional to Imα.As discussed elsewhere [6], there is no contradiction in the fact that curl force dynamics is nondissipative even though Imα represents absorption by the particle.
For a monochromatic electric field represented by a complex position-dependent function E, this curl force is proportional to (5.1) so its curl is proportional to where the scalar product • links E * and E and the vector product × links ∇ and ∇. (We note that (5.1) and (5.2) have the same forms as the geometric phase one-form and two-form respectively [39,40].)Quantum curl dynamics would correspond not to a single orbit of the polarisable particle but to a family of trajectories (without a curl force, this would correspond to a wavepacket).The family would depend on the initial conditions.Where the particles are detected would depend on which trajectory in the family is selected randomly.also involving v rather than the canonical u.
The real part of (A1.2) leads (after some calculation) to in which the natural anisotropic Madelung quantum potential is The r.h.s. of (A1.7) (energy) is constant, so the derivatives of the l.h.s with respect to x and y are zero.A crucial step depends on the identity whose proof depends on the irrotationality of u (cf (A1.3)).It leads to the main result, that the Madelung force (acceleration) of particles moving along the streamlines of v is This result gives the Madelung trajectories for general quantum states governed by an anisotropic Hamiltonian.The quantum curl force version would correspond simply to adding a curl force to F in (A1.10).

Appendix B Two checks on calculations in section 3
The first check is a comparison of the solution of the radial equation (3.9) with its semiclassical approximation.For each r, t, the semiclassical contributions are found by evolving points with initial radial velocity v 0 on the line r = 1, for time t, as a curve in the r, v, plane.Figure 6 shows the curve.The explanation of the hairpin shape is that points on the initial curve with v 0 > 0 move towards increasing r, and for points with v 0 < 0 r initially decreases before turning and then increasing (i.e.v (t) is initially negative, then positive).
For each position r and time t, standard semiclassical theory [42,43] indicates that there are interfering contributions from the two branches v 0 = v 0± (r, t) , which are the solutions of The intensity of each contribution is 1/ |∂ v0 r (t, v 0± )| weighted by the modulus of the initial momentum distribution in (3.13), and its phase γ ± is the time integral of the Lagrangian (kinetic-potential energy, cf (3.9)), in units of ℏ: Phase space curve at t = 2, evolved from r = 1 at t = 0 for different values of initial radial velocity v 0 ; the branches meet at the caustic singularity (black square at r ≈ 1.811).
Thus the semiclassical intensity is in which the factor i is the Stokes multiplier [43] between the contributions, and the factor 1/r arises from the polar coordinates.Figure 7 illustrates the accuracy of the approximation.As always, the semiclassical approximation fails at the caustic singularity where the branches in figure 6 meet.This is easily fixed by the uniform approximation [44] in terms of the Airy function [45,46]-now standard (section 36.12 in [47]), so it is not necessary to give details.
The second check is the continuity equation (2.4) associated with the one-dimensional r dynamics (3.9) and (3.15), namely Figure 8 shows the two members of this equation as a function of r for fixed t, showing comfortable agreement.

Appendix C Wrapped and unwrapped angles
This refers to section 3 and the potential (3.3), in which the domain of θ is the full line −∞ < θ < ∞, i.e. including windings, not the plane angle 0 ⩽ θ < 2π .Solutions of the Schrödinger equation (3.5) are singlevalued on the helicoid −∞ < θ < ∞, 0 < r < ∞, and solutions of the gauge-transformed Schrödinger equation (3.7) are smooth and periodic on the punctured plane  0 ⩽ θ < 2π , 0 < r < ∞.Connecting the two interpretations of the angle is the Poisson sum formula [47,48].Any periodic function can be expressed as a superposition of the windings of an unwrapped (i.e.not periodic) function: where and c (µ) is any reasonable interpolation of c (m).
In physics, Poisson summation describes a general phenomenon: representing a sum over discrete eigenvalues as a sum over topologies.In the present case, the eigenvalues are angular momenta and the topologies are windings.Related examples of this association are the 'manywhirls' representation of the Aharonov-Bohm wavefunction [49,50] and potential scattering by spheres [43,51,52].In waveguides [53], the eigenvalues are modes, and the topologies are bouncings of rays at the walls.In electron microscopy of crystals [54], the eigenvalues denote Bloch waves and the topologies are undulations of classical paths.In the quantum mechanics of classically integrable systems [55], the eigenvalues are energies and the topologies are classical periodic orbits.In the optical Talbot effect [56,57], the eigenvalues label diffracted beams and the topologies are image repetitions.For divergenceless flows, ρ can be any function constant along streamlines of u, as in the example of section 4. The more interesting case is ∇ • u ̸ = 0. Then (A4.1) describes the variation of ρ along u but says nothing about the variation of ρ across the streamlines of u.Any solution of (A4.1) can be multiplied by any solution ρ 0 of u • ∇ρ 0 = 0. To get the contribution that depends on ∇ • u, we note that where s is distance along a streamline.Therefore the solution of (A4.1), after writing it in the form u • ∇ log ρ = u∂ s log ρ = −∇ • u, (A4.3)

Figure 2 .
Figure 2. Family of classical trajectories in {x, y} space for 0 ⩽ t ⩽ 2, over (a) a smaller range, (b) a much larger range.