Alexander Afriat

Einstein and the old quantum theory

come

Introduction

Einstein's role in quantum theory is well known: one associates him with the foundational debate in the twenties and thirties, with the photoelectric effect, perhaps with the quantum theory of gases-or even statistical mechanics-in general; but less with the old quantum theory of Bohr and Sommerfeld (from 1913), or with analytical mechanics for that matter. "Zum Quantensatz von Sommerfeld und Epstein" (1917a), which I propose to consider, has an unusual place in the history of science, characterised by neglect, limited attention then unexpected, retarded recognition. [START_REF] Gutzwiller | Chaos in classical and quantum mechanics[END_REF] and [START_REF] Graffi | Alcuni aspetti matematici della meccanica quantistica, Quaderni dell'Istituto nazionale di alta matematica, Gruppo nazionale di fisica matematica Graffi[END_REF] mention the peculiar history of its citations: practically half a century of silence,1 then rediscovery 2 in the more mathematical context of analytical mechanics, dynamical systems. Even if analytical mechanics lay outside of Einstein's main mechanical interests-statistical mechanics, relativistic mechanics, foundations of mechanics-Quantensatz should be treated more as analytical mechanics than as quantum theory. Sommerfeld's quantum rule 3 ((1) below), referred to in the title, becomes little more than a point of departure, almost a pretext for the development or at least adumbration of a futuristic, topological, highly invariant analytical mechanics; that at any rate is what it amounts to, however Einstein himself saw it.

In a nutshell, Einstein (1917a,b) objects that Sommerfeld's rule refers to particular coordinates, in the sense that there's a condition for each of the l coordinates q 1 , . . . , q l . Einstein proposes another rule ((4) below), which is more invariant inasmuch as it integrates the entire momentum one-form p-not one component p i at a time-over the l homotopy classes that characterise the topology of the torus he eventually introduces ( §2 below). Which raises the issue of why Einstein's rule should be any better than Sommerfeld's. Empirical superiority is often important in physics. Not here: Einstein never even brings it up, being only concerned with formal invariance; and even for us, favoured as we are by hindsight, empirical adequacy can hardly be invoked to discriminate theories we now know to be empirically very inadequate. 4 But perhaps there are cases where Einstein's rule makes more sense than Sommerfeld's, or works better. For an elliptical Kepler motion, Einstein's rule is no better than Sommerfeld's, both work well: the score there is one all. And neither rule can really handle the self-intersections of a more complicated Rosettenbahn: nil nil (cumulatively one all). The superiority of Einstein's rule will only emerge on the toroidal configuration space he constructs to resolve the self-intersections: one nil there (cumulatively two one). But even on the torus, the advantage is only qualified: all one can say is that Einstein's rule makes more sense and is more invariant, not that Sommerfeld's rule would make no sense at all. 5The peculiarities of Quantensatz, which is paradoxically both primitive6 and decades ahead of its time, give rise to peculiar historiographical difficulties which require a peculiar, ad hoc methodology. Some of the mathematics is so futuristic that anachronisms are needed to make any sense of it at all; but there one's on a slippery slope that can lead to the worst exaggerations. Anachronisms, however indispensable in such cases, have to be kept under control, it is too easy to get carried away.

Much of my analysis will concern three central (and indeed related) themes: (i) mechanics in configuration space Q (as opposed to phase space Γ ); (ii) integrability, ergodicity; (iii) Liouville-Arnol'd:

(i) Even if Einstein uses the Hamiltonian function of position and momentum (and confusingly makes a reference to Phasenraum7 ), his mechanics seems to be firmly rooted in the q i -Raum he so often mentions. 8 It is not a symplectic mechanics, in phase space.9

(ii) Einstein's treatment of integrability and ergodicity is undeniably modern, perhaps decades ahead of its time. But does it all make sense? Is mere configuration space enough? Would phase space be needed after all?

(iii) One can easily identify features of the Liouville-Arnol'd theorem ( §5 below) in Quantensatz. But is it all there? Does Einstein really have a steady motion on a rigid torus? Or just a shapeless, merely 'topological' torus?

Is it possible that Einstein, with little or no symplectic geometry, anticipated so much modern, geometrical mechanics: integrability, ergodicity, Liouville-Arnol'd? "Integrability" can mean various things. There is the purely geometrical notion, well exemplified by the exactness of a one-form α = df : infinitesimal objects α(q) assigned to every point q fit together in such a way as to allow 'derivation [df ] from a potential f ,' as Einstein would put it. But that's abstract; other notions are more concretely arithmetical, and have to do with the simplicity or even the very possibility of coordinate representation. The numbers involved10 acquire geometrical meaning by identifying manifolds, which can simplify the representation of (integrated) motion by adaptation-by 'following it so as to eliminate it,' absorbing its twists & turns into their adapted shapes, thus enclosing and therefore representing it.

Einstein has his own notion of integrability, which is so strong it goes well beyond what is now called 'complete' integrability: it involves confining the motion to a one-dimensional closed manifold-"die Bahn ist dann eine geschlossene, ihre Punkte bilden ein Kontinuum von nur einer Dimension." 11 He seems to reduce integrability and separability12 ( §6 below) to two simple 'topological' distinctions: whether the motion eventually closes, or never does (which would pose an irremediable problem for quantisation); and if it does close, whether it intersects itself (which poses a remediable, indeed most welcome, problem for quantisation-so welcome that Einstein's whole invariant, topological agenda rests on it). Again, self-intersections are remedied by Riemannisierung, in other words enlargement of the configuration space, considered in the next section.

The two quantum rules

Sommerfeld's quantum conditions (1)

J * i = p i dq i = n i h, i = 1, .
. . , l, rule out atomic motions whose actions J * i are not integer multiples n i ∈ Z of Planck's constant h. Having spent the previous years immersed in the tensorial covariance of general relativity, Einstein sees a problem here: Sommerfeld's rule refers to the specific coordinates q 1 , . . . , q l ; each one of the l conditions concerns a particular momentum component p i . Einstein replaces them with l conditions, each one of which involves the entire momentum one-form

(2) p = l i=1 p i dq i ,
now integrated over the l homotopy classes H 1 , . . . , H l or 'topological features' of the space-there being a quantum condition for every such feature. To understand the construction we can begin with an annulus. Einstein considers the example of an annulus Q bounded by circles of radii r 1 , r 2 . Motions on Q that never close are intractable; those that close without intersecting themselves are too tractable-for Sommerfeld's rule in particular, thereby giving Einstein no advantage. Einstein's whole strategy relies on closed motions that intersect themselves: the only motions combining the two virtues of being somehow tractable, but not so much as to preclude the welcome difficulties in whose solution lies Einstein's real edge over Sommerfeld, and which represent the chief interest of Quantensatz.

To see the problem, the above 'topological' classification of motions can be reexpressed in terms of momentum assignments to (or right around) points: [1] one, [2] finitely many, [3] infinitely many. Einstein considers an infinitesimal region13 r ⊂ Q crossed by the motion, the following cases can arise:

[1] The next time the motion crosses r it assigns to q ∈ r the same momentum p-and hence every time thereafter. This is the simplest kind of periodicity: the motion is closed (and hence confined to a loop, a one-dimensional manifold of finite length) and never intersects itself.

[2] The motion eventually assigns a finite number N of momenta 14 p 1 , . . . , p N to r, finally closing on the N -th lap. The Bahn is still eine geschlossene, ihre Punkte bilden ein Kontinuum von nur einer Dimension; but it intersects itself. 15 [3] The motion assigns an infinite number of momenta to r, without ever closing.

Being ergodic, the motion cannot be confined to a loop, an exakt geschlossene Bahn.

The quantisation rules at issue here only make sense with a single momentum at (or right around) a point. To make the momentum assignment amenable to quantisation, Einstein enlarges the configuration space, thus restoring single-valuedness. But the enlargement procedure he adopts is finite, and cannot be repeated infinitely many times (for infinitely many momentum values). Riemannisierung cannot be applied to ergodic motions.

So Einstein takes a closed but nontrivial Rosettenbahn, which intersects itself. He chooses a point q ∈ Q to which the motion assigns two momenta, p 1 and p 2 . In order to restore the Einwertigkeit needed for quantisation he superposes a second annulus on the first, identifying the delimiting circles, and stipulating that whenever the motion reaches either one it changes annulus. The motion on the resulting two-torus T 2 can now be quantised since it no longer intersects itself. The topology of the torus is captured by the (nontrivial) homotopy classes 16 H 1 and H 2 , respectively made up of loops going around the first and second circles of the torus (once). 17

H k i p i dq i :
14 Each p k ∈ 1 Q is a one-form on Q and not a coordinate ∈ R. 15 See [START_REF] Gutzwiller | Chaos in classical and quantum mechanics[END_REF] figures 10 & 30. 16 Graffi (2005) p. 23: "curve topologicalmente inequivalenti." 17 See [START_REF] Gutzwiller | Chaos in classical and quantum mechanics[END_REF] figure 30. 18 Since I am using "closed" in two entirely different senses, I'll write "κlosed" for this second sense ('locally exact'), pertaining to differential forms. 19 See Graffi (2004) p. 175. 20 In the paragraph on p. 90 containing figures 1 & 2 Einstein seems to have in mind an l-dimensional torus T l (rather than a more general manifold with Betti number l). In two dimensions, Riemannisierung clearly produces a torus; little generality is lost in considering T 2 ; and one wonders how the scheme can work in general if the enlarged configuration space is not toroidal. So I'll speak of a torus T l even with l > 2.

invariance with respect to the choice of coordinates q i , and also with respect to the particular loop of H k -with respect to 'loop deformations.' 21 All 'old quantum theories' are empirically too inadequate to justify an empirical preference of Einstein's rule over Sommerfeld's.

Again, both quantum rules work in case [1], neither one in cases [2] and [3]; but at least in case [2], the multi-valuedness (being finite) can be resolved on a larger configuration space T l , where Einstein's rule works well and makes perfect sense. Even Sommerfeld's conditions can be made to work on T l -which, however, strongly favours the double invariance of Einstein's rule.

3 Loops, trajectoires, gauge Einstein's loops are meaningless on their own and only acquire significance collectively as elements of the homotopy classes H 1 , . . . , H l which capture the topological peculiarities of the enlarged configuration space. Even if Einstein goes to the trouble of writing "irgendeine geschlossene Kurve, welche durchaus keine "

Bahnkurve" des mechanischen Systems zu sein braucht," Broglie (1924) p. 63 nonetheless speaks of "trajectoires fermées." But Broglie's misunderstanding is in fact more interesting than one may imagine, interesting enough to deserve a few words.

Consider such a mechanical trajectoire 22 ξ in the plane P, and the action integral

(5)

J * = ξ (p 1 dq 1 + p 2 dq 2 )
calculated with respect to the coordinates q 1 , q 2 . To simplify we can confine the 'source' 23 to the origin 0 ∈ P; in other words the curl dp of the momentum one-form

p = p 1 dq 1 + p 2 dq 2
vanishes everywhere else, on all of P = P-{0}. The integral (5) has various interesting symmetries 24 which are worth looking at. A diffeomorphism

γ : P → P ; q → Q = γ(q) ; ξ → Ξ = γ(ξ)
defined on all of P displaces everything-trajectoire, source at the origin, and other points P-so as to preserve the relations of inclusion and exclusion, and hence the integral J * itself. Even a diffeomorphism γ : P → P on P (as opposed to P)which only displaces points where dp vanishes, and the trajectoire itself, but not the source 0-would be just as symmetric, as it could never drag ξ over the source. For a diffeomorphism to alter the topological relations on which (5) depends it would have to displace selectively, telling apart origin and points of ξ.

21 Cf. Graffi (2005) p. 22. 22 If the Hamiltonian has no explicit dependence on time, the trajectoire is best viewed as a mere (onedimensional) manifold, since the progression of its (temporal) parameter would be trivial; see §4 below. 23 The term "source" comes from the 'divergence' version of Stokes's theorem (attributed to Gauß or Ostrogradsky or perhaps Green), here it is more metaphorical. Here the 'source' at the origin produces a turbulence which by the theorem manifests itself on the loop ξ as the circulation J * .

24 See Epstein (1916a) p. 172, Whittaker (1917) pp. 271ff, Levi-Civita & Amaldi (1974) pp. 353ff.

Since γ pulls a real-valued function

Q i : P → R ; Q → Q i (Q)
defined on the range P back25 to the domain P of γ, yielding a function

q i = γ Q i = Q i • γ : P → R ; q → q i (q) = (γ Q i )(q)
on the domain P, it pulls the differential dQ i back accordingly-and hence the basis dQ 1 , dQ 2 , and with it the linear combination:

p = γ P = p 1 γ dQ 1 + p 2 γ dQ 2 ,
where I have written to avoid confusion with the asterisk Einstein uses to denote the Maupertuis action J * . We can write

J * = ξ p = Ξ P,
or even

J * = Ξ p = ξ P,
and

J * = γ (ξ) γ p = H γ p = γ p, H (6)
for all diffeomorphisms γ, γ : P → P leaving the source inside the trajectoire. Broglie's trajectoires thus become highly transformable, invariant entities-little more than deformable loops expressive of topological properties. It remains a mistake to think of Einstein's loops ξ ∈ H as trajectoires, but not an uninteresting one.

In Hamiltonian mechanics one distinguishes between point transformations γ (on the l-dimensional q i -Raum) and canonical transformations (on the 2l-dimensional phase space Γ ).26 So far we haven't gone beyond the rather restrictive 'point' condition

(7) p = i p i dq i = i P i dQ i
of Lagrangian mechanics (where the configuration space Q characterised by q 1 , . . . , q l maintains its identity, without 'getting lost' in the 2l-dimensional state space); but in fact the weaker 'symplectic' condition

ω = dp = dp = d(p + dF ) = i dp i ∧ dq i = i dp i ∧ dq i + d 2 F (8) also preserves J * , where (9)
p → p = p + dF and the generating function F is a zero-form. So in addition to the diffeomorphic freedom γ (or γ) and the homotopic freedom ξ → Ξ (keeping clear of the source), we have the 'gauge' or 'symplectic' freedom (8) to add an exact term dF :

J * = γ p + dF, H .
While a diffeomorphism γ affects the momentum one-form indirectly by first dragging points, the gauge transformation ( 9) is fibre-preserving and therefore acts directly on each p(q), point by point. The canonical transformation generated by F also corresponds to a deformation of curves in P, reminiscent of the loop deformation ξ → Ξ.

Even if the curl at the origin (or the corresponding circulation J * ) prevents the oneform p from having a global primitive, one can think of a local primitive λ satisfying p = dλ locally. 27 In much the same way as one can take ξ to be displaced by the point transformation γ, one can take the level sets λ = const. to be deformed by F . [START_REF] Graffi | Alcuni aspetti matematici della meccanica quantistica, Quaderni dell'Istituto nazionale di alta matematica, Gruppo nazionale di fisica matematica Graffi[END_REF] has rightly pointed out that Einstein's rule ( 4) is canonically invariant. The symplectic freedom (8) is undeniably available; but Einstein's mechanics is so unsymplectic, so firmly rooted in the q i -Raum he so often mentions, that I doubt he had in mind anything beyond the two genuinely q i -Raum symmetries represented in (6).

Hamilton-Jacobi-Einstein

The old quantum theory, including Quantensatz, was formulated in terms of Hamilton-Jacobi theory, 28 which therefore deserves some attention.

The principle of least action 29 exists in two versions, which in Einstein's notation would be distinguished by a star indicating neglect of time.

[i] The 'space-time' Hamiltonian version determines the spatial shape of trajectories as well as motion along them by minimising the full Hamiltonian action 30

J = t t0
L dt, 27 "Locally" could mean, for instance, on any simply-connected region of P.

28 See [START_REF] Hamilton | On a general method of expressing the paths of light, and of the planets, by the coefficients of a characteristic function[END_REF][START_REF] Hamilton | On the application to dynamics of a general mathematical method previously applied to optics[END_REF], Jacobi (1884) pp. 143ff, [START_REF] Appell | Traité de mécanique rationnelle (tome premier)[END_REF] pp. 556ff, Epstein (1916b) pp. 493ff, [START_REF] Whittaker | A treatise on the analytical dynamics of particles and rigid bodies[END_REF]) pp. 288ff, Carathéodory (1937) pp. 66ff, Brillouin (1938) pp. 168ff, Appell (1953) pp. 429ff, Levi-Civita & Amaldi (1974) pp. 355ff, Arnol'd (1988) §9.4, Jacobi (1996) where the 'momentum' or 'covariant' Lagrangian can be written

L = i p i ∂H ∂p i -H .
[ii] The 'spatial' version attributed to Maupertuis only gives the spatial shape of the trajectory, by minimising the purely spatial part (10)

J * = q q0 p = q q0 i p i dq i of the action.
Hamilton's principle is in a sense more general; but the generality it adds to Maupertuis' is only of any interest if the Hamiltonian H depends explicitly on time. Since Einstein takes it not to, the motion is confined to a level surface where the Hamiltonian remains equal to some constant E; and the action assumes the degenerate additive form 31 J = J * -Et.

Let us write H = T + U (and L = T -U ), where the potential U depends only on position and the kinetic energy

T = p 2 2m = 1 2 i p i qi = 1 2 i p i ∂H ∂p i
is quadratic in the momenta. Only the spatial shape of the trajectory would then remain interesting. To confine our attention to that shape, ignoring the trivial time evolution given by the term -Et, we just take the spatial 'Maupertuis' part of J, namely (10).

Viewing q 0 as a fixed initial position and q as a variable final position, we can radiate q in all directions from q 0 (now reminiscent of a source in geometrical optics 32 ) along dynamical trajectories derived from the action function J * (q, q 0 ) satisfying the Hamilton-Jacobi equation 33 ( 11)

H (q, dJ * ) = E, or rather p 2 = dJ * 2 = 2m(E -U ).
Choosing an infinitesimal action δJ * , we first have the sphere σ(q 0 , δJ * ) of radius

ζ(q 0 , δJ * ) = δJ * 2m(E -U (q 0 ))
31 See Jacobi (1884) Einundzwanzigste Vorlesung, Appell (1953) pp. 430ff, Levi-Civita & Amaldi (1974) pp. 362ff.

32 See [START_REF] Hamilton | On a general method of expressing the paths of light, and of the planets, by the coefficients of a characteristic function[END_REF][START_REF] Hamilton | On the application to dynamics of a general mathematical method previously applied to optics[END_REF], [START_REF] Whittaker | A treatise on the analytical dynamics of particles and rigid bodies[END_REF] pp. 288ff. 33 See [START_REF] Graffi | Alcuni aspetti matematici della meccanica quantistica, Quaderni dell'Istituto nazionale di alta matematica, Gruppo nazionale di fisica matematica Graffi[END_REF] p. 51.

around q 0 . The momentum p at any q on the sphere is the differential

p(q) = dJ * (q) = l i=1
∂J * (q) ∂q i dq i of J * viewed as a function of the final position q. Once we have a level surface σ(q 0 , δJ * ) of J * , we might as well generalise and propagate from an arbitrary (l -1)dimensional initial surface

Σ 1 = Σ J * [1]
(which could be a sphere σ or not), viewed as a level surface of action J * [1] . I'll follow Schrödinger (1926, pp. 492-3) on the way to his Wellenmechanik. 34 We can again take the same increment 35 δJ * , which provides a distance ζ(q 1 , δJ * ) at the generic point q 1 ∈ Σ 1 . Then there are two possible constructions: Either we repeat the above construction, now treating q 1 as the source of a 'secondary' wave, 36 a sphere σ(q 1 , δJ * ) of radius ζ(q 1 , δJ * ) emanated from every point of the 'primary' wavefront Σ 1 , in which case the surface Σ 2 of action J *

[2] = J * [1] + δJ * is the envelope of the secondary waves. Alternatively one lays off ζ(q 1 , δJ * ) normally at every point, to define Σ 2 even more directly. 'Normally' can be understood as follows: The (l -1)-dimensional linear space T q1 Σ 1 ⊂ T q1 Q tangent to Σ 1 at q 1 determines a ray ρ 1 in the cotangent space T * q1 Q. The direction of the momentum p 1 at q 1 is given by p 1 ∈ ρ 1 , the length by

p 1 = 2m(E -U (q 1 )).
The inverse m of the mass tensor 37 m determines a vector p 1 = m p 1 , indeed a ray ρ 1 containing p 1 in the tangent space T q1 Q. The action increment δJ * fixes the length ζ(q 1 , δJ * ) of the required vector p 1 ∈ ρ 1 , which goes from Σ 1 to the corresponding point of Σ 2 .

So that's one way (specificially a rather Hamiltonian way) of understanding the Hamilton-Jacobi equation ( 11)-which is so central to the old quantum theory, and even to wave mechanics: as an infinitesimal condition governing the orthogonal propagation in configuration space Q of a solution J * from an initial surface (which could be as small as a point). But the orthogonal construction also serves to elucidate, by opposition, the somewhat different logic of the 'oblique' construction Einstein proposes in § 3 of Quantensatz. In Schrödinger's construction, the direction of propagation 34 See also Brillouin (1938) pp. 169ff, Arnol'd (1988) pp. 251ff. 35 A smaller increment would be even better, if secondary waves are propagated from an already infinitesimal primary wavefront σ(q 0 , δJ * ).

36 See Huygens (1690), [START_REF] Whittaker | A treatise on the analytical dynamics of particles and rigid bodies[END_REF] from the initial surface Σ 1 is determined by the local slant of the surface itself,38 not by initial data freely assigned to it, point by point. Einstein also propagates from an (l -1)-dimensional surface in the q i -Raum Q, but obliquely-at an 'angle,' relative to the surface, that varies from point to point. Schrödinger's orthogonal propagation, which makes the initial surface Σ 1 a level surface of J * , also makes the propagated field exact, indeed consistent with a Hamilton-Jacobi potential J * ; Einstein's 'oblique' construction has to require exactness independently, as it is not provided by orthogonality.

The difference between κlosed and exact, it should be noted, may not have been entirely clear to Einstein (or others in 1917). Important progress is made in [START_REF] Rham | Quantisierung als Eigenwertproblem: zweite Mitteilung[END_REF], where the distinction is used for analysis situs. Even if the distinction is clear to [START_REF] Weyl | The classical groups[END_REF], he confusingly uses the word "exact" to mean κlosed: "A differential ω whose derivative vanishes is called exact." Weyl's understanding of κlosed is already rather modern:

What we mean by ω ∼ 0 (ω homologous zero) may be explained in two ways: either differentially as indicating that ω is the derivative of a differential of next lower rank, or integrally as demanding that the integral of ω over any cycle vanishes. Every differential ∼ 0 is exact; one readily proves this in both ways. "In the small" both notions, exact and ∼ 0, coincide, but not in the large.

But one can imagine how roughly, if at all, the distinction was understood in 1917. In (1917c) Einstein seems to treat his equations ( 7) and (7a) as equivalent; in Quantensatz he writes "bzw." between his equations 10) and 10a)-but then does draw a related distinction, between einwertig and vielwertig, on the next page:

Ist aber der in Betracht kommende Raum der q i ein mehrfach zusammenhängender, so gibt es geschlossene Bahnen, welche nicht durch stetige Änderung auf einen Punkt zusammengezogen werden können; ist dann J * keine einwertige (sondern eine ∞ vielwertige) Funktion der q i , so wird das Integral [ p i dq i ] für eine solche Kurve im allgemeinen von Null verschieden sein. Indeed if the one-form p is only CMC: closed on a multiply-connected region, it may or may not be exact, the loop integral (3) may or may not vanish. If p is not exact,39 the loop integral p, H will not vanish, but its Vielwertigkeit will be the denkbar einfachste: every point q on a given loop acquires an additional p, H on every lap:

J * (n+1) (q) = J * (n) (q) + p, H = J * (1) (q) + n p, H ,
where the integer n stands for the lap. In terms of the Ein/Vielwertigkeit of the primitive J * , exact means has a single-valued global primitive J * ; merely κlosed means exact "in the small"-the primitive J * is locally einwertig but may be globally vielwertig.

Returning to Einstein's construction, he freely assigns momenta p 1 (q 1 ) to the points q 1 of Σ 1 , then radiates dynamical (Hamiltonian) trajectories40 L satisfying L(q 1 ) = p 1 (q 1 ) = m p 1 (q 1 ) throughout Σ 1 . The dynamical vector field L thus determined by the one-form p 1 = p| Σ1 provides a one-form p = m L on an l-dimensional region U ⊂ Q covered by L.

Einstein then wonders when the one-form p determined by the Hamiltonian vector field L will also satisfy the Hamilton-Jacobi equation; for the radiated congruence can, confusingly, be 'Hamiltonian' (in other words made up of dynamical trajectories) without being 'Hamilton-Jacobi'-consistent with a potential J * satisfying the Hamilton-Jacobi equation. It turns out that p 1 has to be κlosed, for then p will be too; and as long as U is topologically simple, κlosed means exact:

(dp 1 = 0) ⇔ (dp = 0) ⇔ (∃ J * : p = dJ * ).
The potential J * is slightly underdetermined by its derivative dJ * , in other words

d -1 p = [J * ] = [J * + η] η ,
where η is a constant on U. A value J * (q) at a single q ∈ U is enough to overcome the underdetermination and fix all of J * . Summing up, the theorem can be given as follows: Only a κlosed momentum field p (radiated dynamically from the momenta p 1 freely assigned to an initial surface Σ 1 ⊂ Q) can be derived from a potential

J * = d -1 p -η
satisfying the Hamilton-Jacobi equation.

Einstein seems to change his mind in the Nachtrag zur Korrektur, p. 91:

LB: Liefert eine Bewegung ein p i -Feld, so besitzt dieses notwendig ein Potential J * .

Whatever he means has to do with § 4 of Quantensatz: "die zweite der in § 4 angegebenen Bedingungen für die Anwendbarkeit der Formel 11) stets von selbst erfüllt sein muß [ . . . ]"; at the end of § 4 we discover that Die Anwendung der Quantenbedingung 11) verlangt, daß derartige Bahnen existieren, daß d i e e i n z e l n e B a h n ein p i -Feld bestimmt, für welches ein Potential J * existiert.

To try to make sense of this we can start with the primitive notion of Bewegung: a motion, a trajectory in configuration space-however it may be defined or generated. Returning to the above classification, the motion can be closed (case [1] or [2]) or ergodic (case [3]). If it is closed, it will assign to no point q (or neighbourhood) of Q more than finitely many momenta p 1 , . . . , p N . That's what Einstein means by a p i -Feld: an assignment of at most finitely many momenta to (certain) points of Q. If the momentum is finitely multi-valued here and there, the configuration space can be enlarged to restore single-valuedness-but not if it is ergodic.

There remains the issue of how a single Bewegung, confined as it is to a onedimensional manifold, can yield a Feld, a field on an l-dimensional manifold. Since Einstein goes to the trouble of emphasising d i e e i n z e l n e B a h n with S p e rr d r u c k, he really does seem to mean a single trajectory. He may simply be unaware of the problem; or perhaps it somehow doesn't bother him and he just ignores it; or perhaps he has a way of dealing with it. But if he does, it would surely have to involve a congruence, a S c h a r, somehow or other-despite the arresting S p e r r d r u c k of d i e e i n z e l n e B a h n. Indeed Einstein's use of Hamilton-Jacobi theory is most peculiar, perhaps even contradictory or downright wrong. It is a theory that undeniably involves congruences of trajectories. In § 3 of Quantensatz and in (1917c), Einstein considers the entire congruence; but elsewhere in Quantensatz he seems to use the same theory to produce a single trajectory. The integral (3) only makes invariant sense if p is defined everywhere on Q, not just on a single trajectory-Einstein accordingly speaks of an entire p i -Feld. But Einstein's whole analysis and classification of trajectories (our cases [1]-[3])-which would get confused, perhaps even undermined, by congruences-seems to depend on single trajectories. We may simply have a case of Einstein wanting to have his cake and eat it.

Returning to LB, the issue of a Hamilton-Jacobi potential J * or of a vanishing curl dp does not even arise with a single trajectory. With a whole congruence of motions radiated from an initial surface, the curl dp automatically vanishes if the propagation is orthogonal, as in Schrödinger's construction. If the propagation is only transversal, as in § 3 of Quantensatz, the curl has to vanish for the motions to admit a Hamilton-Jacobi potential J * .

Liouville-Einstein-Arnol'd?

In the second-last paragraph of the Nachtrag Einstein formulates an integrability theorem that deserves attention:

IT Q : Existieren l Integrale der 2l Bewegungsgleichungen von der Form (12) R k (q i , p i ) = konst.,
wobei die R k algebraische Funktionen der p i sind, so ist i p i dq immer ein vollständiges Differential, wenn man die p i vermöge (12) durch die q i ausgedrückt denkt.

The formulation is misleading, he may be groping towards the theorem now attributed to Liouville 41 and Arnol'd. 42 We can begin anachronistically with (features of) the Liouville-Arnol'd theorem, and then try to understand how much of it is already in Quantensatz. The theorem uses l functions

F i : Γ → R
on an 2l-dimensional phase space Γ to reduce the number of dynamically relevant dimensions from 2l to l; in the sense that by l -1 intersections of level surfaces of appropriately compatible and independent functions it confines the dynamics to an l-dimensional manifold. A function F foliates the phase space into (2l-1)-dimensional level surfaces on which F = const.; specifying a value f of F already eliminates one dimension by fixing a level surface. But the theorem concerns dynamics; the issue is whether a given dynamics 43

X G = (dG) = ω (dG)
is tangent to the level surfaces F = const., or transversal to them. If the dynamics X G were transversal to the level surfaces of F , the dimension lost by choosing a level surface would be thus restored, with no net progress in the effort to eliminate dimensions. The relevant notion is Poisson compatibility 44

{F, G} = 0, which can either be understood as compatibility between the dynamics X F generated by F and the level surfaces of G, or the other way around. {F, G} vanishes if the vector field X F is tangent to the level surfaces of G, in other words if the graph of each integral curve of X F is confined to a level surface of G. We can take the first function F 1 to be the Hamiltonian H , whose generic value E singles out a (2l -1)-dimensional energy surface. The two compatibilities

{H , F 2 } = 0 = {H , F 3 }
and values F 2 = f 2 , F 3 = f 3 only eliminate both dimensions (and not just one) if dF 2 and dF 3 are independent; if dF 2 and dF 3 were parallel, F 2 and F 3 would have the same level surfaces, which would be redundant. 45 Complete integrability is given by l

values f = (f 1 , . . . , f l )
of the compatible, independent functions F 1 , . . . , F l , which eliminate l of the initial 2l dimensions of phase space, leaving an l-dimensional manifold M f . Einstein's integrability theorem IT Q suggests he may have got this far. But the two cases at the bottom 41 See [START_REF] Appell | Traité de mécanique rationnelle (tome premier)[END_REF]) pp. 576ff, Appell (1953) [START_REF] Sternberg | Symplectic techniques in physics[END_REF] p. 88, [START_REF] Graffi | Alcuni aspetti matematici della meccanica quantistica, Quaderni dell'Istituto nazionale di alta matematica, Gruppo nazionale di fisica matematica Graffi[END_REF] p. 51. 44 See Graffi (2004) p. 50. 45 In the aforementioned 'effort to eliminate dimensions,' Poisson incompatibility {F, G} = 0 would be counterproductive (reversing progress already made by effectively restoring an eliminated dimension), whereas linear dependence dF = kdG would be merely unproductive (producing neither loss nor gain, leaving the number of dimensions unchanged). of p. 87 of Quantensatz give the impression that Einstein wants to reduce the dynamically available dimensions of q i -Raum, not phase space; that he wants to confine the Bahnkurve to a lower-dimensional manifold of configuration space:

Die Bahnkurve läßt sich ganz in einem Kontinuum von weniger als l Dimensionen unterbringen.

The kind of integrability he has in mind, taken as far as possible, would confine the Bahnkurve to a one-dimensional submanifold of Q, not Γ : Hierzu gehört als spezieller Fall derjenige der Bewegung in exakt geschlossener Bahn. [ . . . ] die Bahn ist dann eine geschlossene, ihre Punkte bilden ein Kontinuum von nur einer Dimension. This brings us to the next part of the Liouville-Arnol'd theorem. So far we have little more than a number l of dimensions. Without compactness, the l-dimensional manifold M f could be a product of lines and loops; compactness rules out the lines, leaving a torus T l , a product of l topological circles. It is worth noting that Arnol'd and Einstein obtain their tori in different ways: Arnol'd by imposing compactness, Einstein by Riemannisierung, to eliminate self-intersections. The two ways, however different, are not unrelated: if the l-dimensional manifold M f were a product of l lines, it would amount to R l ; compactness prevents immersion in R l by 'swelling' M f . A two-dimensional torus T 2 , for instance, is an enlarged two-dimensional configuration space inasmuch as it cannot be embedded in the plane. Suppose for definiteness that T 2 is not just a 'topological torus' (a product of two loops) but a 'rigid torus' (a product of two rigid circles S 1 )-literally a doughnut, with a shape and a size, embedded in R 3 parallel to the xy plane. Take the simplest possible motion, given by fixed rates of rotation around both circles of such a T 2 : projected onto the xy plane it would intersect itself (at regular intervals). That's how compactness is related, albeit indirectly, to Einstein's Riemannisierung.

The distinction between rigid and topological tori, introduced above for mere definiteness of representation, will actually prove quite relevant. According to the Liouville-Arnol'd theorem, a completely integrable (and compact) dynamics can be represented as l constant rates

(13) φi = ω i = ∂H ∂J * i
of rotation on an l-dimensional torus T l , where the l angles46 ϕ i with linear evolutions

ϕ i (t) = ω i t + ϕ i (0) are canonically conjugate (14) {ϕ i , J * j } = δ ij
to the actions J * i ; i, j = 1, . . . , l.

Einstein seems to construct a topological, l-dimensional torus (topologically equivalent to a rigid torus) to resolve self-intersections; his l integrals (3) are undeniably action integrals; but that's not enough to provide the 'symplectic' rigidity that turns a topological torus into a rigid one. To do so, Einstein would have needed something along the lines of ( 13) or ( 14)-which were by no means obvious in 1917, especially to a physicist, and cannot be taken for granted. To obtain a rigid torus one has to recognise that the angles ϕ i yielding the constant frequencies ω i are canonically conjugate to the actions J * i ; but nowhere does Einstein betray such symplectic awareness. The rigid torus of the Liouville-Arnol'd theorem is really quite different from Einstein's merely topological torus: it has a metrically definite shape and is twice as big, with 2l degrees freedom (l areas and l angles), not just l (l arbitrary parameters along l loops). Even if steady motions on a rigid torus are ultimately needed to make sense of his intuitive differentiation of ergodic and closed motions, that distinction alone hardly warrants the attribution of so much definite structure to Einstein's loose, highly topological constructions.

But let us nonetheless consider a rigid torus, which with little loss of generality can be taken to be two-dimensional, with frequencies ω 1 , ω 2 . The motion can in any case be confined to a one-dimensional manifold; the issue is its length (finite or not), its topology (closed or open)-whether we have a Bewegung in exakt geschlossener Bahn, whose Punkte bilden ein Kontinuum von nur einer Dimension. The relevant criterion is rational dependence: if ω 1 /ω 2 is rational, the motion can be confined to a one-dimensional Kontinuum which, being geschlossen, is of finite length; whereas if ω 1 /ω 2 is irrational the motion can still be confined to a one-dimensional Kontinuum, but not of finite length. 47 Einstein clearly understands the geometrical significance of confining motion to intersections of level surfaces of appropriately compatible and independent functions; what may be less clear to him is the numerical-rational vs. irrational-rather than set-theoretical (or 'manifold-theoretical') character of the last step, needed to bring the (appropriately finite) dimensions down to one. Indeed in the very last paragraph of the Nachtrag he seems to suggest that ergodic motion is only possible with an incomplete set R 1 , . . . , R j (j < l). We now know that ergodicity can be a merely numerical matter, of irrational dependence-compatible with a 'complete' set of l functions R 1 , . . . , R l .

Returning to Einstein's integrability theorem IT Q , he mentions that the momentum (2) would be a vollständiges Differential. I think the point is not exact vs. κlosed, but that once ergodic motion is ruled out, with its 'infinite-valued' momentum, the momentum just makes sense, being single-valued on an appropriate configuration space. He seems to understand the theorem as follows: his l conserved quantities R 1 , . . . , R l are numerous enough to provide a well-defined momentum p by confining the motion to an exakt geschlossene Bahn whose Punkte bilden ein Kontinuum von nur einer Dimension, thus ruling out ergodic motion which, being 'infinite-valued,' is unmanageable. Einstein's letter (1917b) to Ehrenfest puts it all pretty clearly:

IT E : Es liege ein Problem vor, bei dem soviel Integrale L(q ν , p ν ) = konst existieren, als Freiheitsgrade. Dann können die Impulse als (mehrwertige) Funktionen der q ν ausgedrückt werden. Andererseits erfülle die Bahnkurve einen gewissen q ν -Raum vollständig, sodass sie jeden Punkt desselben beliebig nahe kommt. Dann liefert die Bahn des Systems im q ν -Raum ein Vektorfeld der p ν .

Andererseits refers to the number of integrals: if there aren't enough conserved quantities, the motion is ergodic (case [3] above Even if it could be enough to say that integrals, like (1) or (3), only make sense for single-valued integrands, Einstein seems to view it as a matter of separability:48 quantisation relies on separability, which in turn depends on single-valuedness.

It may be best to think of separability backwards, starting with canonical coordinates (J * , ϕ) = (J * 1 , . . . , J * l , ϕ 1 , . . . , ϕ l ) which are separable by construction, in the sense that the total action can be written as a sum (15)

J * = l i=1 J * i of the l actions J i = J * i (H i ).
The Hamiltonian breaks up accordingly:

H (J * 1 , . . . , J * l ) = l i=1 H i (J * i ),
where the decomposition into l pieces corresponds to the constitution of the torus

T l = T l (H 1 , . . . , H l ) = l i=1 S 1 l
itself, a product of l circles S 1 . A general canonical transformation (J * 1 , . . . , J * l , ϕ 1 , . . . ϕ l ) → (p 1 , . . . , p l , q 1 , . . . , q l ) from (J * , ϕ) cuts across the decomposition by producing new canonical coordinates (p, q) whose lines are transversal to those of (J * , ϕ): the entangled general transformation entangles the unentangled coordinates. But the entanglement of coordinates is avoided if the canonical transformation itself is unentangled, in the sense that it breaks up into l canonical transformations (J * 1 , ϕ 1 ) → (p 1 , q 1 ), . . . , (J * l , ϕ l ) → (p l , q l ), one for every part H i of the torus, where the symplectic two-form

ω = l i=1 ω i = l i=1 dJ * i ∧ dϕ i = l i=1 d(J * i dϕ i + dF ) = l i=1 dp i ∧ dq i .
If the canonical transformation is generated by an unentangled function

F (ϕ 1 , . . . , ϕ l , q 1 , . . . , q l ) = l i=1 F i (ϕ i , q i )
that decomposes accordingly, the new coordinates (p, q) will be just as separable as (J * , ϕ). Being even-dimensional, each two-dimensional submanifold determined by (J * i , ϕ i )-by fixing the other 2l -2 coordinates of phase space-is a symplectic manifold Γ i in its own right, with its own symplectic two-form

ω i = dJ * i ∧ dϕ i = d(J * i dϕ i + dF i ) = dp i ∧ dq i
and its own canonical transformations

(J * i , ϕ i ) → (p i , q i )
produced by its own generating functions

F i . If the covector dF i (z) ∈ T * z Γ i is in the coplane span{dJ * i (z), dϕ i (z)} ⊂ T * z Γ i spanned by the covectors dJ * i (z)
and dϕ i (z), it will put both dp i (z) and dq i (z) in the same coplane. If that holds at every z ∈ Γ i , the new pair (p i , q i ) will determine the same symplectic submanifold Γ i as the action-angle pair (J * i , ϕ i ). That at any rate is a modern notion of separability, which is worth bearing in mind to understand Einstein's. He of course sees things differently. Again, he may not even have the idea of an angle ϕ i canonically conjugate to an action J * i . He has l actions J * 1 (q 1 , . . . , q l ), . . . , J * l (q 1 , . . . , q l ), or perhaps J * 1 (p 1 , . . . , p l , q 1 , . . . , q l ), . . . , J * l (p 1 , . . . , p l , q 1 , . . . , q l ),

where each p i = p i (q 1 , . . . , q l ) can in general be a function of all l coordinates q 1 , . . . , q l of q i -Raum. Separability holds when "jedes p ν nur von dem zugehörigen q ν abhängt,"49 in other words

J * = l i=1 J * i (p i (q i ), q i ).
The lines50 of the coordinates q i then turn the Hamilton-Jacobi PDE into l ODEs, each on its own one-dimensional manifold51 U i ; so that the Unabhängigkeit on which separability rests is encoded into the very framework (U 1 , . . . , U l ) of the ODEs. Rather than as a 'symplectic alignment' of each (p i , q i ) along each (J * i , ϕ i ), Einstein sees separability as a matter of adapting the l coordinates q 1 , . . . , q l of q i -Raum to the shape of the solution J * ; as a condition τ [J * ; (q 1 , . . . , q l )] relating the coordinates-or the Kontinua (U 1 , . . . , U l )-to the level surfaces of J * . Locally, around a point q ∈ U in Q, the coordinates can always be adapted to satisfy the condition: since the infinitesimal behaviour dJ * of J * is captured by l derivatives in l independent directions, one can always find l terms J * i such that (15) and

dJ * i dq i = ∂J * ∂q i .
The real problem is global, not local. As Einstein explains in the Nachtrag, J * may eventually52 reach the same q from different directions; coordinates that satisfy the relationship τ on the first pass may not satisfy it on the next: coordinates can be adapted to a single J * , but not to two different passes53 J * (1) , J * (2) . Which is why Einstein has to enlarge the configuration space to resolve self-intersections; in the larger toroidal configuration space, not only are self-intersections eliminated, but separability is restored, since the coordinates can be adapted everywhere to the single-valued trajectory. One can see the problem around a point q ∈ U ⊂ Q on the annulus: suppose the self-intersecting motion, the propagation of J * , first assigns one momentum p(q, t 1 ) = p 1 (q) = dJ(q, t 1 ) = dJ *

(1) (q)

to q, then a second p(q, t 2 ) = p 2 (q) = dJ(q, t 2 ) = dJ * (2) (q); separabilty would require one coordinate system q 1 , q 2 for the first pass J * (1) , another Q 1 , Q 2 for the second J *

(2) . Once there are two annuli Q 1 , Q 2 forming a torus T 2 , everything gets split: q becomes two points

q 1 ∈ U 1 ⊂ Q 1 q 2 ∈ U 2 ⊂ Q 2 ,
with p(q 1 , t 1 ) = p 1 (q 1 ) = dJ(q 1 , t 1 ) = dJ *

(1) (q 1 ) = ∂J *

(1) ∂q 1 dq 1 (q 1 ) + ∂J *

(1) ∂q 2 dq 2 (q 1 ) at q 1 and p (q 2 , t 2 ) = p 2 (q 2 ) = dJ(q 2 , t 2 ) = dJ * (2) (q 2 ) = ∂J *

(2)

∂Q 1 dQ 1 (q 2 ) + ∂J *

(2)

∂Q 2 dQ 2 (q 2 ) at q 2 . Separability is restored by adapting the coordinates

q 1 , q 2 : U 1 → R Q 1 , Q 2 : U 2 → R
to J * , thus satisfying τ . An example would be alignment of q 1 along the level curves of J * (1) , with q 2 perpendicular and appropriately normalised, so that ∂J * (1) /∂q 1 vanishes and p 1 = dJ * (1) = dq 2 ; and likewise on the second pass:

p 2 = dJ * (2) = dQ 2 .
The double condition ∂J *

(1)

∂q 1 = 0 = ∂J * (2)
∂q 1 would be very restrictive and hard to satisfy.

Final remarks

Einstein situates his mechanics pretty firmly in the q i -Raum he so often refers to; symplectic abstractions seem rather foreign to him. But one wonders how to make sense of the configuration space integrability referred to in the two cases at the bottom of p. 87. Integrability is closely related to conservation, which surely depends on position and momentum. Einstein appears to have the right modern geometrical intuitionsenclosing motion in intersections of level sets of appropriate functions-but in the wrong space. His symplectic ignorance makes Quantensatz all the more impressive; a real feat of economy, efficiency: making relatively little go a very long way. The 'complete' integrability one comes across in the modern literature is hardly complete by Einstein's standards, being compatible with both ergodic and periodic motions. The more extreme kind of integrability Einstein has in mind corresponds to closed, not ergodic motions: tertium non datur. He needs a self-intersecting motion assigning more than one momentum here and there-to justify the enlarged configuration space whose topological peculiarities are captured by the homotopy classes he integrates over-but not the infinitely many momenta of ergodic motion; his Riemannisierung is necessarily finite. And once on the torus, where the greater invariance of Einstein's integrals p, H i is evident, the score against Sommerfeld is (an admittedly mathematical, nonempirical) 2-1. I thank Ermenegildo Caccese, Sandro Graffi, Stefano Marmi and Nic Teh for many useful conversations; and audiences in Urbino, at Paris Diderot and Notre Dame for valuable feedback.

  ). The implication being that if there are enough functions L, we have a Bewegung in exakt geschlossener Bahn, whose Punkte bilden ein Kontinuum von nur einer Dimension ([1] or [2]); l integrals are enough to provide a finitely-many-valued one-form p (on configuration space Q), which can be made single-valued by Riemannisierung.6 Einwertigkeit, separability, quantisation Einstein's imaginative, idiosyncratic treatment of integrability and ergodicity is of the greatest interest; as is the torus he constructs to restore Einwertigkeit, not to mention what he puts on it. For us the point is more what Einstein does to restore singlevaluedness, than why single-valuedness should ever be important in the first place; but a few words on the matter may nonetheless not go amiss.

Punctuated by two important wave-mechanical citations: Broglie (1924) pp. 63ff, and the footnote on p. 495 of[START_REF] Rham | Quantisierung als Eigenwertproblem: zweite Mitteilung[END_REF]. The story of how the mécanique ondulatoire emerged-see[START_REF] Lanczos | The variational principles of mechanics[END_REF] pp.

277ff-as if by mistake from Louis de Broglie's remarkable misunderstandings of Quantensatz deserves to be told, but elsewhere; and Schrödinger suggests, in the footnote, that Quantensatz was behind his Wellenmechanik as well.2 See Keller (1958),[START_REF] Gutzwiller | Chaos in classical and quantum mechanics[END_REF]) pp. 215, 282, 315, Spivak (2010) p. 640.3 Sommerfeld (1915) p. 429, Sommerfeld (1916) p. 9; see alsoEpstein (1916a) p. 173, Epstein (1916b) pp. 490, 501. 4 There is no reason why the two rules should agree on motions, ruling out the same ones. But even if that would make the rules significantly inequivalent, it would only give the preference to one if an independent (empirical) way of checking the selections were available. We now know that an undulatory quantum theory-with superposition, coherence, interference, resonance etc.-is needed to make any sense at all of atomic mechanics, and select the right motions.

Even if it would make no invariant sense, no geometrical sense, some unfortunate coordinate-dependent sense could perhaps be salvaged.

Symplectically at any rate; certainly compared to[START_REF] Whittaker | A treatise on the analytical dynamics of particles and rigid bodies[END_REF], for instance.

Einstein (1917a) p. 90

Einstein (1917a,b,c)

Even today, geometric mechanics can be symplectic or Riemannian; seeCalin & Chang (2005),[START_REF] Pettini | Geometry and topology in Hamiltonian dynamics and statistical mechanics[END_REF]. Needless to say, Riemannian geometry is much closer to the curved Lorentzian geometry Einstein had practically invented and was thoroughly used to.

In §5 below there will be the conserved quantities f = (f 1 , . . . , f l ) ∈ R l .

Einstein (1917a) p. 88

See[START_REF] Gutzwiller | Chaos in classical and quantum mechanics[END_REF] §3.7 on the modern distinction (not Einstein's).

To include ergodic motion in the classification Einstein broadens his purview from the point to its immediate surroundings.

The case is so trivial (domain and range coincide etc.) that pulling back and pushing forward can be legitimately confused; so the abuse of notation γ p, for instance, is venial. A diffeomorphism can be taken to map from the domain P to the range P, or the other way around.

See Landau & Lifschitz (1970) § 45.

Here the normal direction of propagation is determined by the mass tensor m. With a more general Hamiltonian (and Lagrangian), the duality relations between level surface and motions are given by the fibre derivatives, with components ∂H /∂p i , ∂L /∂ qi .

The 'topological interpretation' of the Aharonov-Bohm effect shows how important it is to distinguish between not exact and the above condition CMC: not necessarily exact.

The almost superfluous curves L and vectors L used to extend the one-form p 0 on Σ 0 to a one-form p on the l-dimensional region U are only introduced because Einstein seems to have such objects in mind. They can easily be dispensed with, the one-forms are enough.

See Sternberg & Guillemin (1984) pp. 356ff,[START_REF] Lowenstein | Essentials of Hamiltonian dynamics[END_REF] pp. 68ff.

SeeBorn (1933) pp. 80ff,[START_REF] Fasano | Meccanica analitica[END_REF] §11.7.

See Appell (1909) pp. 576ff, Epstein (1916a) pp. 170ff, Epstein (1916b) pp.

494ff, Born (1925) §14, Appell (1953) pp. 437ff, Levi-Civita & Amaldi (1974) pp. 405ff, Fasano & Marmi (2002) §11.2,[START_REF] Graffi | Alcuni aspetti matematici della meccanica quantistica, Quaderni dell'Istituto nazionale di alta matematica, Gruppo nazionale di fisica matematica Graffi[END_REF]) p. 175, Lowenstein (2012) pp. 57ff.

Einstein (1917b)

The lines of q i can be understood as the integral curves of the vector field ∂/∂q i .

In fact U is partitioned into l independent congruences of one-dimensional manifolds U i .

Even if we are in a configuration space Q in which time has no more than an implicit role, Einstein himself uses explicitly temporal language: "im Laufe der Zeit."

In an (l + 1)-dimensional configuration space Q × R, enlarged to include time, the two passes would of course take place at different times t 1 and t 2 .


Luc Ardaillon
email: luc.ardaillon@ircam.fr

Axel Roebel
email: axel.roebel@ircam.fr

Fully-Convolutional Network for Pitch Estimation of Speech Signals

Keywords: speech analysis, F0 estimation, CNN

The estimation of fundamental frequency (F0) from audio is a necessary step in many speech processing tasks such as speech synthesis, that require to accurately analyze big datasets, or realtime voice transformations, that require low computation times. New approaches using neural networks have been recently proposed for F0 estimation, outperforming previous approaches in terms of accuracy. The work presented here aims at bringing some more improvements over such CNN-based state-of-theart approaches, especially when targeting speech data. More specifically, we first propose to use the recent PaN speech synthesis engine in order to generate a high-quality speech database with a reliable ground truth F0 annotation. Then, we propose 3 variants of a new fully-convolutional network (FCN) architecture that are shown to perform better than other similar datadriven methods, with a significantly reduced computational load making them more suitable for real-time purposes.

Introduction

Fundamental frequency (F0) estimation is a necessary step in many speech processing tasks such as TTS or singing synthesis, for which it is often necessary to analyze big databases. However, analysis errors are likely to induce undesirable artifacts and manual correction is difficult for such big datasets. Until recently, all approaches used to be based on digital signal processing and heuristics, including methods based on the auto-correlation function like YIN [START_REF] De Cheveigné | YIN, a fundamental frequency estimator for speech and music[END_REF] or PYIN [START_REF] Mauch | PYIN: A fundamental frequency estimator using probabilistic threshold distributions[END_REF], or template matching with the spectrum of a sawtooth signal as in SWIPE [START_REF] Camacho | A sawtooth waveform inspired pitch estimator for speech and music[END_REF], among many others.

But recently, new data-driven approaches using various kinds of neural networks have been proposed for monophonic F0 estimation [START_REF] Verma | Frequency estimation from waveforms using multi-layered neural networks[END_REF], [START_REF] Han | Neural network based pitch tracking in very noisy speech[END_REF], [START_REF] Kim | CREPE: A Convolutional Representation for Pitch Estimation[END_REF], [START_REF] Kato | Waveform to Single Sinusoid Regression to Estimate the F0 Contour from Noisy Speech Using Recurrent Deep Neural Networks[END_REF], and for multi-pitch [START_REF] Bittner | Deep Salience Representations for F0 Estimation in Polyphonic Music[END_REF] or main melody estimation [START_REF] Basaran | Main melody extraction with source-filter nmf and crnn[END_REF], [START_REF] Doras | On the use of u-net for dominant melody estimation in polyphonic music[END_REF], outperforming previous results. A recent trend, since the success of the wavenet model [START_REF] Van Den | Wavenet: a generative model for raw audio[END_REF], is towards end-to-end systems using the raw signal's waveform as input to the network. Following this trend, several neural networks for monophonic F0 analysis have been proposed, like the CREPE model [START_REF] Kim | CREPE: A Convolutional Representation for Pitch Estimation[END_REF], or the ones proposed in [START_REF] Verma | Frequency estimation from waveforms using multi-layered neural networks[END_REF] and [START_REF] Kato | Waveform to Single Sinusoid Regression to Estimate the F0 Contour from Noisy Speech Using Recurrent Deep Neural Networks[END_REF], using the signal's waveform as input to a CNN, an MLP, or a RNN, respectively. A particular advantage of such end-to-end approaches is that once the model is trained, no parametrisation is required from the user.

A particular use case for F0 estimation is in real-time applications and audio plugins for voice transformations [START_REF] Flux | Ircam Trax v3 audio plugin[END_REF] requiring low computation times. But recent approaches like [START_REF] Kim | CREPE: A Convolutional Representation for Pitch Estimation[END_REF] are too computationally heavy, and some more improvements are thus still necessary in order to make such methods compatible with real-time constraints, without reducing the accuracy.

Building up on previous studies, this paper aims at bringing some improvements over data-driven state-of-the-art approaches for monophonic pitch estimation of speech sound, both in terms of accuracy and efficiency. Following the methodology described in [START_REF] Kim | CREPE: A Convolutional Representation for Pitch Estimation[END_REF], summarized in section 2, we propose a new fully-convolutional neural network architecture that will be described in Section 3. Then, in order to train this model we propose to use the recent PaN speech synthesis engine [13, Section 3.5.2] to build a high-quality speech database with a reliable ground truth, as detailed in Section 4. Finally, we conducted some evaluations to compare the performances of our models to that of the CREPE and SWIPE algorithms, both in terms of accuracy and computational efficiency, that will be detailed in Section 5.

Method

For our purposes, we adopt a similar approach to the one described in [START_REF] Kim | CREPE: A Convolutional Representation for Pitch Estimation[END_REF] for CREPE, with a few adaptations. In this approach, the F0 estimation is seen as a classification task, using a CNN that takes a raw waveform as input and outputs a vector of probabilities of the F0 to belong to each possible output pitch classes. In our case, those 486 pitch classes correspond to a division of the [30-1000] Hz range into steps of 12.5 cents, assuming that such a range covers all possible pitch values for vocal sounds (including soprano singing and para-linguistic sounds like baby cries). Note that 30Hz is very low for voice, but might occur occasionally (e.g. in fry mode). The resolution has been increased compared to [START_REF] Kim | CREPE: A Convolutional Representation for Pitch Estimation[END_REF] (12,5 cents instead of 25), assuming that it might help to get more accurate results.

For training, the target vector is gaussian-blurred according to equation 1, with a standard deviation of 25 cents, such that the activation values yi of the bins of pitch ci (in cents) surrounding a ground truth frequency ctrue decay accordingly (similarly to [START_REF] Kim | CREPE: A Convolutional Representation for Pitch Estimation[END_REF]).

yi = exp(- (ci -ctrue) 2 2 • 25 2 ) ( 1
)
The network is then trained to minimize the binary crossentropy between the target and predicted vectors.

From the predicted vector ŷ, the resulting pitch estimate, in cents, is then computed following equation 2, similarly to what is done in CREPE 1 .

ĉ = I+4 i=I-4 ŷici I+4 i=I-4 ŷi , I = argmaxi(ŷi) (2)
This value can then be converted back from cents to Hz to obtain the final F0.

Network architecture

The CREPE architecture proposed in [START_REF] Kim | CREPE: A Convolutional Representation for Pitch Estimation[END_REF] is a rather simple CNN architecture composed of 6 convolutional layers followed each by max pooling and relu activations, and a final dense layer with sigmoid activations that produces the prediction vector. However, this architecture has several drawbacks which will be discussed in the following. A first drawback is related to the zero-padding applied to the input of each convolutional layer (mode "same" in keras), which is necessary in order to keep the same size between the input and output of the layer. This may appear as a minor issue when using small filters compared to the size of the input (as is often the case in image processing for instance). But in the CREPE architecture, the filters used are quite long (e.g. 512 samples in the first layer for an input of size 1024), and this border effect is thus not negligible. In particular, in the layers 3 to 6 of the CREPE architecture, the size of the filters (64) is superior to the temporal dimension of their inputs (8 to 64, depending on the layer). Thus, most of the convolution operations in these layers results in multiplications with zeros, from which one would not expect the model to benefit, and which also raises the number of computations. To overcome this problem, a solution is to use only 'valid' convolutions, for which no padding is applied. However, in this case the output of a convolution is smaller than its input, and the network architecture needs to be adapted to take this into account.

A second limitation is due to the final dense layer of CREPE which can only accept fixed-sized inputs, which implies to do the prediction on a frame basis. For this purpose, the input sound is split into overlapping frames that are fed individually to the network which predicts one F0 value for each frame (the degree of overlap depending on the expected time resolution of the produced F0 curve). Each convolution is thus computed several times on the overlapping parts of each succeeding frames sharing the same values, which implies some redundancy in the computation.

Instead of doing the prediction on a frame basis, we thus propose here to use a Fully-Convolutional Network (FCN) architecture. For this purpose, it is possible to replace the final dense layer by an equivalent convolutional one whose length is equal to the temporal dimension of the input tensor, and whose number of filters is equal to the size of the expected ouput vector (in our case 486) [START_REF] Long | Fully Convolutional Networks for Semantic Segmentation[END_REF]. Since a convolutional layer does not require a fixed-sized input, this configuration allows running the convolutions only one time on the whole input signal instead of frame-wise, thereby saving a lot of calculation. Then, the network will not output a single vector, but a 3D tensor containing the output prediction vectors at each temporal step, with the pitch classes arranged along the depth dimension (as illustrated in figure 1).

Beside this fully-convolutional approach, one way to reduce the computation load of a CNN is to reduce the number of parameters (i.e. the number of layers, the number of filters in each layer, or the size of those filters). In the implementation of CREPE, it is suggested to use less filters to reduce the size of the model. But it may also be beneficial to try reducing the size of the filters and number of layers. In the CREPE architecture, the filters in the first layer are particularly long (512), such that they cover at least 1 full period of the signal (1 period of a 30Hz sinusoid at 16 kHz is 500 samples). However, there is no evidence that this is necessary, and the network might be able to learn as well using smaller filters. In addition to reducing the number of filters in each layer, we thus also propose to reduce their size.

Finally, the CREPE model is trained on 16 kHz audio. However, only a few harmonics are necessary to infer a pitch. Reducing the sampling rate may also help to reduce the necessary computation, by getting read of unnecessary details in the signal. As we limit the target pitch range to [30-1000] Hz, we chose to reduce the sampling rate to 8 kHz, which still allows to preserve at least the 4 first harmonics (including the fundamental).

Based on all those considerations, we propose here a new fully-convolutional architecture, described in figure 1, that incorporates all the previously-exposed proposals. The proposed model is composed of 7 convolutional layers. Max pooling is included only after the first 3 convolutional layers (including more max pooling layers after the 3 rd one would require to use a much bigger input size, which is undesirable for latency). Each layer is followed by batch normalization [START_REF] Ioffe | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift[END_REF], and its output is passed through relu activations, except for the last layer for which sigmoid activations are used. The number of filters in each layer and their sizes are detailed in figure 1. All convolutional layers are applied with a stride of 1. The minimal input to the network to obtain 1 F0 value is a 1953-samples excerpt from an audio signal sampled at 8 kHz. Note that because no padding is applied before the convolutions, the temporal dimension is reduced after each convolutional layer even without max pooling. The relation between the output size lout and input size lin of a convolutional layer and the size of the filters l f il is lout = lin -l f il + 1. The choice of the input size of the network has thus been determined according to this relation, after the network architecture (number and positions of the convolutional and max pooling layers, and size of the filters) has been fixed. Because of the size reduction implied by the 3 pooling layers, the time step between each output prediction is equal to 8 samples (= 2 3 ). The temporal resolution of the prediction (1ms) is thus greatly increased compared to a frame-based prediction.

However, this model imposes a minimum latency of a half input size duration, i.e. 1953/2 8000 ≈ 0.122s, which is quite big for real-time purposes. In order to reduce this latency, we propose two alternative models based on the same architecture. The first proposed alternative is to remove the 2 nd convolutional layer in figure 1 (including the max pooling). This alternative model requires an input of size ≥ 929, which reduces the latency to 929/2 8000 ≈ 0.058s. The second alternative is to keep the same architecture as in figure 1, but reducing the size of the filters in all layers to 32. In this case, the minimum input size is 993 (latency of 993/2 8000 ≈ 0.062s). We call those 3 models respectively FCN-1953, FCN-993, and FCN-929 according to their respective input sizes.

With the proposed architectures, the number of parameters has been reduced from about 22.2 • 10 6 for CREPE to about 12.3 • 10 6 for FCN-1953 and FCN-929, and 6.7 • 10 6 for FCN-993, while getting similar or better results in our evaluations, as will be described in section 5. Moreover, the proposed fullyconvolutional approach allows to significantly reduce the computation time. The proposed architectures have been implemented in keras, and a python implementation with pre-trained models are made available online 2 .

Database

Data generation

For the purpose of proper training and objective evaluations, we need a database with a perfect ground truth F0 annotation. Though many databases of voice recordings are available, obtaining a reliable F0 ground truth is less straightforward, as ex- isting estimators are likely to make errors, and manual correction is not precise and not applicable to big databases.

For this purpose, a first approach, proposed in [START_REF] Kato | Waveform to Single Sinusoid Regression to Estimate the F0 Contour from Noisy Speech Using Recurrent Deep Neural Networks[END_REF], [START_REF] Kato | A Regression Model of Recurrent Deep Neural Networks for Noise Robust Estimation of the Fundamental Frequency Contour of Speech[END_REF], consists in using a specific database, such as the PTDB-TUG database [START_REF] Pirker | A Pitch Tracking Corpus with Evaluation on Multipitch Tracking Scenario[END_REF], that includes laryngograph recordings on which one of the already available F0 estimators is applied to obtain a good estimation of the ground truth. (However, our experience with one such database revealed that the laryngograph recordings where sometimes too corrupted to obtain a reliable ground truth).

A second solution, proposed in [START_REF] Kim | CREPE: A Convolutional Representation for Pitch Estimation[END_REF], [START_REF] Salamon | An Analysis/Synthesis Framework for Automatic F0 Annotation of Multitrack Datasets[END_REF], consists in using synthesis to have a perfect control over the F0 of the resulting signals. In this case, a database of real recordings is first analyzed using a pre-existing estimator to get annotations that will be used as a ground truth, but that are likely to contain errors. Then, instead of correcting these annotations, the sound is resynthesized based on this ground truth F0 annotation, such that the resynthesis perfectly matches the corresponding F0 track, including the potential errors.

Since proper laryngograph recordings are rarely available and given that the synthesis approach allows more control over the ground truth, we chose this second approach. However, the synthesis method used in [START_REF] Kim | CREPE: A Convolutional Representation for Pitch Estimation[END_REF] is purely harmonic, using a bank of oscillators, and thus doesn't include any unvoiced component. Though this approach may be acceptable for some musical instruments, it seems oversimplistic for vocal sounds, and is likely to result into too homogeneous and unrealistic data that doesn't account well for all the complexity and variability of vocal sounds.

Many available vocoders allow to synthesize speech with a good quality. It seems thus benefitial to use one of them in order to improve the quality of the dataset for speech. We propose here to use the recent PaN vocoder for this purpose [13, Section 3.5.2]. Basically, the PaN model uses the target F0 values to generate a sequence of pulses based on the LF model of glottal source [START_REF] Fant | A four-parameter model of glottal flow[END_REF]. In order to model various voice qualities, a constant random Rd value has been chosen in the range [0.3-2.7] for each synthesized file. Then, the vocal tract filter is applied on the pulses. Finally, the unvoiced component, separated from the original sound, is added to the voiced source to generate the complete signal. Since the separation of the voiced and unvoiced components is not perfect, some sinusoids may sometimes remain in the unvoiced residual. In order to avoid this problem, the phases in the residual signal are randomized.

Dataset description

To create our own dataset, we first merged the BREF [START_REF] Gauvain | Design Considerations and Text Selection for BREF, a large French Read-Speech Corpus[END_REF] and TIMIT [START_REF] Zue | Speech Database Development at MIT : TIMIT and Beyond[END_REF] datasets. Then, the CREPE algorithm [START_REF] Kim | CREPE: A Convolutional Representation for Pitch Estimation[END_REF] with the provided pre-trained model has been used to estimate the F0 accross the whole database. Finally, each sound has been resynthesized using the PaN synthesis engine. This ends up into a total of 11616 short sentences spoken by many male and female voices in English and French languages. Then, in order to better span the targeted range of F0 values ([30-1000] Hz), the same dataset is also generated one octave lower and one octave higher. Files that contain F0 values outside of the target range are excluded from the training and evaluation data. The sounds have been first generated at a 16 kHz sampling rate, and then subsampled to 8 kHz, as required by our models.

Evaluation

Methodology

For evaluation purposes, we split our database into 3 subsets for training, validation and evaluation (with the proportions 60/20/20). Since the same synthesis are replicated at 3 different sets of transpositions in the database, the same split is applied to them, so that the content of the samples present in the 3 subsets are completely different, independently of their pitch, and a similar pitch distribution is selected for the 3 subsets.

With this methodology, we compare the results of our models against that of the CREPE and the SWIPE algorithms. For the CREPE model, we evaluate both the pre-trained model provided by the authors ("CREPE"), and another version re-trained in similar conditions than our models ("CREPE-speech"). Note that the CREPE and SWIPE algorithm are trained and evaluated on the 16 kHz database, while 8 kHz is used for our models. In [START_REF] Kim | CREPE: A Convolutional Representation for Pitch Estimation[END_REF], the CREPE algorithm has been shown to outperform both the SWIPE and pYIN algorithms. However, this evaluation has only been conducted on synthetic data. In order to attest that the results are not biased by the synthetic quality of our database, we thus also evaluate the results on a manually-annotated subset of 100 real recordings taken from the test split.

The results are evaluated in terms of Raw Pitch Accuracy (RPA) using the mir eval [START_REF] Raffel | mir eval: A Transparent Implementation of Common MIR Metrics[END_REF] python package, with 25, 50, and 200 cents thresholds. Note that 50 cents corresponds to about only 3Hz for a F0 of 100Hz, while the width of a sinusoid on the spectrogram display used for manual correction is ≥ 15Hz.

Training procedure

All models are trained using mini-batch gradient descent with the Adam optimizer [START_REF] Kingma | Adam: A Method for Stochastic Optimization[END_REF], with mini-batches composed of 32 examples randomly selected from the training set. An initial learning rate of 0.0002 is used, and a reduction by a factor 0.75 is applied when the validation loss doesn't decrease for 5 epochs, where an epoch consists of 500 batches, with a minimum value 0.0000025. The training is stopped when the validation accuracy has not improved for 32 epochs. Note that while an FCN can accept an input of arbitrary length, the input segments used for the training stage remain of fixed size, such that the temporal dimension of the output tensor is always equal to 1 during training. The input segments are normalized by their mean and variance prior to be fed to the network for training. (However, during inference, since the size of the input is not fixed, each sample is normalized individually by the mean and variance computed over a sliding window). Unlike in [START_REF] Kim | CREPE: A Convolutional Representation for Pitch Estimation[END_REF], we did not use dropout for training our models.

Results

The obtained RPA values (with mean and variance) are shown in table 1, both for the synthetic speech test split ("PaN-synth") and the manually-corrected subset of real recordings ("manual"). As can be seen, our models always perform better than the SWIPE and CREPE algorithms (except for the manual 50 cents case where the variance is the largest). The difference is particularly important with the 25 cents threshold, wich is probably partly due to the adapted target range and finer output resolution compared to CREPE. Regarding CREPE, the important difference of results between the default model and the one trained on our database ("CREPE-speech") outlines the importance of using an appropriate database according to the targeted data for training. The 3 variants of the proposed architecture gave relatively similar results in all cases.

Besides the accuracy, we also compared the computation time for the different approaches. The mean computation time over 100 analysis of a sound of duration 2.51s for each algorithm, both on a GPU and on a single core CPU, are reported in table 2, along with the minimum latency (half of input length divided by sampling rate. The biggest window size used by the SWIPE algorithm is 4096). We can see that both our model FCN-1953 and FCN-993 could be applied in real-time even on a single core CPU and FCN-993 allows reducing the computation time by a factor 16 compared to the CREPE algorithm. However, although the FCN-929 model has the lowest latency, it is also about 2 times slower than FCN-1953. This is due to the removal of the 2 nd max pooling layer in this case, which implies a finer output time resolution of 4 samples instead of 8, and thus about twice more computations. While both models obtain similar accuracy, it is thus more efficient to use smaller filters rather than removing one layer of the network to reduce the latency. (However, complementary experiments showed that results start to degrade when the filters get too small.) About the manual database, one can observe that the results are much lower than on the synthetic one for all algorithms (including swipe which is not data-driven), when using a threshold of 50 cents for the evaluation. However, they are already much better when evaluating with a 200 cents threshold. This may be explained by the inaccuracy of the manual correction which doesn't allow obtaining a very precise ground truth. Another possible explanation for those differences is that some files contain rough voice quality which is not present in the training database, and for which the ground truth F0 is ambiguous. Finally, those results reveal that our FCN models (along with "CREPE-speech") tend to give much more consistent result across all the evaluation data, compared to SWIPE that is better for high-pitched sounds than for low-pitched ones, according to complementary analysis.

In overall, the best alternative in terms of accuracy, speed and latency would be the FCN-993 model.

Conclusion

We presented a new Fully-Convolutional Network (FCN) architecture for monophonic F0 estimation. An evaluation showed that our model ouperformed the CREPE and SWIPE approaches on synthetic and real speech data, in terms of accuracy. Furthermore, thanks to the fully-convolutional approach and the reduction of the model size, the computation time of our proposed models is significantly reduced compared to CREPE, making it applicable in real-time. However, there is still room for improvements to further reduce the latency. Further investigations would be necessary in order to evaluate the robustness of our model to different types of noise, and compare its results with other state-of-the-art models like [START_REF] Kato | Waveform to Single Sinusoid Regression to Estimate the F0 Contour from Noisy Speech Using Recurrent Deep Neural Networks[END_REF]. While a voiced/unvoiced decision could be made by setting a threshold on the maximum value of the output vector, this approach did not work robustly in our experiments. It would thus be better if the network could output directly a voicing decision additionally to the f0 values, which should be investigated in future work. Finally, another remaining problem is in the case of rough sounds containing subharmonics, where the ground truth F0 is ambiguous. One possible approach to avoid this ambiguity would be to detect the pulse positions instead of the F0 directly.

Figure 1 :

 1 Figure 1: Architecture of the proposed fully-convolutional network FCN-1953. All convolutions are valid with stride 1. For the alternative model FCN-929, the 2nd layer is removed and the input size is 929. For the alternative model FCN-993, all filters are of size 32 except in the last layer, and the input size is 993.

Table 1 :

 1 Raw Pitch Accuracy values (mean and std) on the synthesized and manually corrected databases

		FCN-1953	FCN-993	FCN-929	CREPE	CREPE-speech	SWIPE
	PaN-synth (25 cents)	93.62 ± 3.34%	94.31 ± 3.15%	93.50 ± 3.43%	77.62 ± 9.31%	86.92 ± 8.28%	84.56 ± 11.68%
	PaN-synth (50 cents)	98.37 ± 1.62%	98.53 ± 1.54%	98.27 ± 1.73%	91.23 ± 6.0%	97.27 ± 2.09%	93.1 ± 7.26%
	PaN-synth (200 cents)	99.81 ± 0.64%	99.79 ± 0.65%	99.77 ± 0.73%	95.65 ± 5.17%	99.25 ± 1.07%	97.51 ± 4.9%
	manual (50 cents)	88.32 ± 6.33%	88.57 ± 5.77%	88.88 ± 5.73%	87.03 ± 7.35%	88.45 ± 5.70%	85.93 ± 7.62%
	manual (200 cents)	97.35 ± 3.02%	97.31 ± 2.56%	97.36 ± 2.51%	92.57 ± 5.22%	96.63 ± 2.91%	95.03 ± 4.04%

Table 2 :

 2 Minimum latency and mean computation time over 100 trials for a 2.51s sound, on a GPU and single core CPU

		FCN-1953 FCN-993 FCN-929 CREPE SWIPE
	latency	0.122s	0.062s	0.058s	0.032s	0.128s
	GPU	0.016s	0.010s	0.021s	0.092s	X
	CPU	1.65s	0.89s	3.34s	14.79s	0.63s

https://github.com/marl/crepe

https://github.com/ardaillon/FCN-f0