# Einstein and the old quantum theory

Alexander Afriat

###### Abstract

Objecting that Sommerfeld’s quantum conditions refer to particular
coordinates, Einstein proposes a canonically invariant rule. But even if the
invariance is canonical, Einstein may have in mind a double _configuration_
space invariance: with respect to loop deformations, and to _point_
transformations—all on a torus where features of the Liouville-Arnol’d theorem
already appear.

> > ###### Contents
>>

>>   1. 1 Introduction

>>   2. 2 The two quantum rules

>>   3. 3 Loops, _trajectoires_ , gauge

>>   4. 4 Hamilton-Jacobi theory

>>   5. 5 Einstein’s integrability theorem

>>     1. 5.1 Liouville-Arnol’d

>>     2. 5.2 Integrability in _configuration_ space

>>   6. 6 Final remarks

>>

##  1 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. Gutzwiller (1990) and
Graffi (2005) mention the peculiar history of its citations: practically half
a century of silence,111Punctuated by two important wave-mechanical citations:
Broglie (1924) pp. 63ff, and the footnote on p. 495 of Schrödinger (1926). The
story of how the _mécanique ondulatoire_ emerged—see Lanczos (1952) 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. then
rediscovery222See Keller (1958), Gutzwiller (1990) pp. 215, 282, 315, Spivak
(2010) p. 640. 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_ can be treated more as analytical mechanics than as
quantum theory. Sommerfeld’s quantum rule333Sommerfeld (1915) p. 429,
Sommerfeld (1916) p. 9; see also Epstein (1916a) p. 173, Epstein (1916b) pp.
490, 501. ((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 $q_{1},\dots,q_{l}$. Einstein proposes another rule ((4)
below), which is more invariant inasmuch as it integrates the _entire_
momentum one-form $\mathbf{p}$—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 compare theories we now know to be
empirically very inadequate.444There is no reason why the two rules should
agree on motions, ruling out the same ones. But even if such disagreement
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; and 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. 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 hypotrochoid:555Schrödinger
writes _Rosettenbahn_ ; in English one also sees _rosetta orbit_. Einstein
himself seems to have no term for such a self-intersecting orbit. We’ll see
that the exact shape of the figure is less important than its ‘topological’
features: whether it intersects itself (unlike a mere ellipse), and whether it
eventually closes. 0-0 (cumulatively 1-1). The superiority of Einstein’s rule
will only emerge on the toroidal configuration space he constructs to resolve
the self-intersections: 1-0 there (cumulatively 2-1). 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.666Even if it would make no _invariant_ sense, no _geometrical_
sense, some unfortunate _coordinate-dependent_ sense could perhaps be
salvaged.

A notable, perhaps surprising, feature of Einstein’s mechanics is its
‘Riemannian’—rather than symplectic—character: like Hamilton’s own
mechanics,777The distinction being between Hamilton’s mechanics and
_Hamiltonian mechanics_ , which is a symplectic mechanics in phase space, not
configuration space. it is firmly rooted in $l$-dimensional _configuration_
space888Einstein (1917a) p. 82: “Da diese Formulierung von der Wahl der
Koordinaten [$q_{i}$] nicht unabhängig ist, so kann sie nur bei bestimmter
Wahl der Koordinaten zutreffen.” See also p. 83, bottom; p. 84: “in dem
$l$-dimensionalen Raume der $q_{i}$ betrachten. Ziehe ich im Raume der $q_{i}$
[ …].” P. 85: “der in Betracht kommende Raum der $q_{i}$”; “im $q$-Raume”;
“ein Punkt $P$ des Koordinatenraumes mit den Koordinaten $Q_{i}$ [ …].” P. 86:
“jeden Punkt ($q_{i}$) des Koordinatenraumes”; “im Koordinatenraume der
$q_{i}$ [ …].” P. 87: “im $q_{i}$-Raume”; “in den $q_{i}$-Raum”; “des
$q_{i}$-Raumes”; and so on, page after page. There is a similar emphasis on
configuration space (as opposed to phase space) in Einstein (1917b, c).
$\mathcal{Q}$, not in the $2l$-dimensional phase space
$\mathit{\Gamma}=\mathbb{T}^{*}\mathcal{Q}$ we’re used to today. Geometric
mechanics nowadays is typically symplectic, but not always—books by Calin &
Chang (2005) and Pettini (2007) are exceptions. Today’s symplectic
abstractions would have been foreign, even unintelligible, to almost
everyone999Except a handful of mathematicians (Whittaker, Levi-Civita,
Carathéodory _etc._), and perhaps the most mathematical physicists (Born,
Schrödinger). in 1917; and Riemannian geometry is much closer to the curved
Lorentzian geometry Einstein had practically invented and was still immersed
in at the time. The presence of the Hamiltonian $\mathscr{H}$ and other
functions of position _and indeed momentum_ in Einstein’s formalism is
misleading; even such objects have to be situated in configuration space: he
thinks of the Hamiltonian as assigning _a function_

$\mathscr{H}_{\mathbf{q}}\textrm{(}\mathbf{p}\textrm{)}:\mathbb{T}^{*}_{\mathbf{q}}\mathcal{Q}\rightarrow\mathbb{R}$

_of momentum_ $\mathbf{p}$ to every $\mathbf{q}$ in configuration space
$\mathcal{Q}$—rather than a real number
$\mathscr{H}\textrm{(}\mathbf{z}\textrm{)}$ to every point
$\mathbf{z}=\textrm{(}\mathbf{q},\mathbf{p}\textrm{)}\in\mathit{\Gamma}$ of
phase space. Today we understand integrability in phase space; Einstein tries
to understand it in the configuration space in which one even has to situate
his gropings (§5 below) towards the Liouville-Arnol’d theorem.

A fixed configuration space $\mathcal{Q}$ makes Einstein’s mechanics more
Lagrangian than Hamiltonian—even if he assigns momenta
$\mathbf{p}\in\mathbb{T}^{*}_{\mathbf{q}}\mathcal{Q}$ rather than velocities
$\dot{\mathbf{q}}\in\mathbb{T}_{\mathbf{q}}\mathcal{Q}$ to the various points
$\mathbf{q}$ of $\mathcal{Q}$. The whole point of Hamiltonian mechanics being
to go beyond mere point transformations, which refer to a fixed configuration
space $\mathcal{Q}$; its added generality is contained in the exact term $dF$
in (8) and (9) below.

Einstein seems to have in mind a double _configuration_ space invariance ((6)
below): with respect to loop deformations and _point_ (as opposed to
_canonical_) transformations. Graffi (2005) extends the latter invariance from
configuration space to phase space. The integral (3) below _is_ canonically
invariant, that’s undeniable; the issue is Einstein’s _understanding_ of the
invariance—whether he (or any physicist in 1917) thought in genuinely
symplectic terms. Symplectic geometry as we know it has been around for no
more than a few decades; Einstein in any case was no symplectic geometer.

“Integrability” can mean various things. There is the purely geometrical
notion, well exemplified by the exactness of a one-form $\alpha=df$:
infinitesimal objects $\alpha\textrm{(}\mathbf{q}\textrm{)}$ assigned to every
point $\mathbf{q}$ fit together in such a way as to allow ‘derivation 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
involved101010In §5 below there will be the conserved quantities
$f=(f_{1},\dots,f_{l})\in\mathbb{R}^{l}$. acquire geometrical meaning by
identifying submanifolds, 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 known as ‘complete’ integrability: it involves confining
the motion to a one-dimensional (‘closed’) manifold, of finite length, a
loop—“die Bahn ist dann eine geschlossene, ihre Punkte bilden ein Kontinuum
von nur einer Dimension.”111111Einstein (1917a) p. 88 He seems to reduce
integrability and separability121212See Gutzwiller (1990) §3.7 on the _modern_
distinction (not Einstein’s). 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
intersects131313Intersections can only occur in configuration space
($q_{i}$-_Raum_), not phase space; see footnote 8. 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.

##  2 The two quantum rules

Sommerfeld’s quantum conditions

(1) $J_{i}^{*}=\oint p_{i}dq_{i}=n_{i}h\textrm{,}$

$i=1,\dots,l$, rule out atomic motions whose actions $J_{i}^{*}$ are not
integer multiples $n_{i}\in\mathbb{Z}$ 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 $q_{1},\dots,q_{l}$; 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) $\mathbf{p}=\sum_{i=1}^{l}p_{i}dq_{i}\textrm{,}$

now integrated over the $l$ homotopy classes
$\mathbb{H}_{1},\dots,\mathbb{H}_{l}$ 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 $\mathfrak{Q}$ bounded by circles
of radii $r_{1}$, $r_{2}$. Motions on $\mathfrak{Q}$ that never close are _in_
tractable; 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
(periodic or not, self-intersecting or not) can be re-expressed in terms of
momentum assignments to points, or rather _Stellen_ 141414To include space-
filling (non-periodic) motion in the classification Einstein broadens his
purview from the point to its immediate surroundings. (which are somewhat
larger): [1] _one_ , [2] _finitely many_ , [3] _infinitely many_. Einstein
considers an infinitesimal region$\mathfrak{r}\subset\mathfrak{Q}$ crossed by
the motion, the following cases can arise:

  1. [1]

The next time the motion crosses $\mathfrak{r}$ it assigns to
$\mathbf{q}\in\mathfrak{r}$ the same momentum $\mathbf{p}$—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. [2]

The motion eventually assigns a _finite_ number $N$ of momenta151515Each
$\mathbf{p}_{k}\in\bigwedge^{1}\mathcal{Q}$ is a one-form on $\mathcal{Q}$ and
not a coordinate $\in\mathbb{R}$. $\mathbf{p}_{1},\dots,\mathbf{p}_{N}$ to
$\mathfrak{r}$, 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.161616See Gutzwiller (1990) figures 10 & 30.

  3. [3]

The motion assigns an _infinite_ number of momenta to $\mathfrak{r}$, without
ever closing. As the motion is not periodic, it cannot be confined to a loop,
an _exakt geschlossene Bahn_.

The quantisation rules at issue here only make sense with a single momentum at
a _Stelle_. 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_ requires periodic motions.

So Einstein takes a closed hypotrochoid, which intersects itself. He chooses a
point $\mathbf{q}\in\mathfrak{Q}$ to which the motion assigns two momenta,
$\mathbf{p}_{1}$ and $\mathbf{p}_{2}$. 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
$\mathfrak{T}^{2}$ can now be quantised since it no longer intersects itself.
The topology of the torus is captured by the (nontrivial) homotopy
classes171717Graffi (2005) p. 23: “curve topologicalmente inequivalenti.”
$\mathbb{H}_{1}$ and $\mathbb{H}_{2}$, respectively made up of loops going
around the first and second circles of the torus (once).181818See Gutzwiller
(1990) figure 30. The integral

(3) $\langle\mathbf{p},\mathbb{H}_{i}\rangle=\oint_{\mathbb{H}_{i}}\mathbf{p}$

vanishes for neither $\mathbb{H}_{1}$ nor $\mathbb{H}_{2}$, whereas
$\langle\mathbf{p},\mathbb{H}_{0}\rangle$ does vanish for the trivial homotopy
class $\mathbb{H}_{0}$ of ‘contractible’ loops going around neither circle:
Einstein specifies that the one-form $\mathbf{p}$ on $\mathfrak{T}^{2}$ is
$\kappa$losed191919Since I am using “closed” in two entirely different senses,
I’ll write “$\kappa$losed” for this second sense (‘locally exact’), pertaining
to differential forms. (or perhaps even exact—see §4 below). The new quantum
rule202020See Graffi (2004) p. 175.

(4) $\langle\mathbf{p},\mathbb{H}_{i}\rangle=n_{i}h$

$i=1,\dots,l$, which in fact applies more generally to any $l$-torus212121In
the paragraph on p. 90 (1917a) containing figures 1 & 2 Einstein seems to have
in mind an $l$-dimensional _torus_ $\mathfrak{T}^{l}$ (rather than a more
general manifold with Betti number $l$). In two dimensions, _Riemannisierung_
clearly produces a torus; little generality is lost in considering
$\mathfrak{T}^{2}$; 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
$\mathfrak{T}^{l}$ even with $l>2$. $\mathfrak{T}^{l}$, requires the loop
integrals to be integer multiples $n_{i}\in\mathbb{Z}$ of Planck’s constant
$h$. The methodological, æsthetic superiority of the rule lies in the double
invariance of the expression

$\oint_{\mathbb{H}_{k}}\sum_{i}p_{i}dq_{i}:$

invariance with respect to the choice of coordinates $q_{i}$, and also with
respect to the particular loop of $\mathbb{H}_{k}$—with respect to ‘loop
deformations.’222222 _Cf_. Graffi (2005) p. 22. All ‘old quantum theories’ are
empirically too inadequate to warrant 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 $\mathfrak{T}^{l}$, where Einstein’s rule
works well and makes perfect sense. Even Sommerfeld’s conditions can be _made
to_ work on $\mathfrak{T}^{l}$—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
$\mathbb{H}_{1},\dots,\mathbb{H}_{l}$ 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 nonetheless
speaks of “ _trajectoires_ fermées” in the chapter of his _Thèse_ (1924, p.
23) he devotes to _Quantensatz_. But Broglie’s misunderstanding is in fact
more interesting than one may imagine, enough to deserve a few words.

Consider such a mechanical _trajectoire_ 232323If the Hamiltonian has no
explicit dependence on time, the _trajectoire_ is best viewed as a mere (one-
dimensional) manifold, since the progression of its (temporal) parameter would
be trivial; see §4 below. $\xi$ in the plane $\mathbb{P}$, and the action
integral

(5) $J^{*}=\oint_{\xi}(p_{1}dq_{1}+p_{2}dq_{2})$

calculated with respect to the coordinates $q_{1},q_{2}$. To simplify we can
confine the ‘source’242424The 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 $\xi$ as the
circulation $J^{*}$. to the origin $\mathbf{0}\in\mathbb{P}$; in other words
the curl $d\mathbf{p}$ of the momentum one-form

$\mathbf{p}=p_{1}dq_{1}+p_{2}dq_{2}$

vanishes everywhere else, on all of
$\bar{\mathbb{P}}=\mathbb{P}-\\{\mathbf{0}\\}.$ The integral (5) has various
interesting symmetries252525See Epstein (1916a) p. 172, Whittaker (1917) pp.
271ff, Levi-Civita & Amaldi (1974) pp. 353ff. which are worth looking at. A
diffeomorphism

$\gamma:\mathbb{P}\rightarrow\mathbb{P}\hskip 2.0pt;\hskip
4.0pt\mathbf{q}\mapsto\mathbf{Q}=\gamma\textrm{(}\mathbf{q}\textrm{)}\hskip
2.0pt;\hskip 4.0pt\xi\mapsto\Xi=\gamma\textrm{(}\xi\textrm{)}$

defined on all of $\mathbb{P}$ displaces everything— _trajectoire_ , source at
the origin, and other points $\bar{\mathbb{P}}$—so as to preserve the
relations of inclusion and exclusion, and hence the integral $J^{*}$ itself.
Even a diffeomorphism
$\bar{\gamma}:\bar{\mathbb{P}}\rightarrow\bar{\mathbb{P}}$ on
$\bar{\mathbb{P}}$ (as opposed to $\mathbb{P}$)—which only displaces points
where $d\mathbf{p}$ vanishes, and the _trajectoire_ itself, but not the source
$\mathbf{0}$—would be just as symmetric, as it could never drag $\xi$ 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 $\xi$.

Since $\gamma$ pulls a real-valued function

$Q_{i}:\mathbb{P}\rightarrow\mathbb{R}\hskip 2.0pt;\hskip
4.0pt\mathbf{Q}\mapsto Q_{i}\textrm{(}\mathbf{Q}\textrm{)}$

defined on the range $\mathbb{P}$ back262626The case is so trivial (domain and
range coincide _etc_.) that pulling back and pushing forward can be
legitimately confused; so the abuse of notation
$\gamma^{\circledast}\mathbf{p}$, for instance, is venial. A diffeomorphism
can be taken to map from the domain $\mathbb{P}$ to the range $\mathbb{P}$, or
the other way around. to the domain $\mathbb{P}$ of $\gamma$, yielding a
function

$q_{i}=\gamma^{\circledast}Q_{i}=Q_{i}\circ\gamma:\mathbb{P}\rightarrow\mathbb{R}\hskip
2.0pt;\hskip 4.0pt\mathbf{q}\mapsto
q_{i}\textrm{(}\mathbf{q}\textrm{)}=(\gamma^{\circledast}Q_{i})\textrm{(}\mathbf{q}\textrm{)}$

on the domain $\mathbb{P}$, it pulls the differential $dQ_{i}$ back
accordingly—and hence the basis $dQ_{1},dQ_{2}$, and with it the linear
combination:

$\mathbf{p}=\gamma^{\circledast}\mathbf{P}=p_{1}\gamma^{\circledast}dQ_{1}+p_{2}\gamma^{\circledast}dQ_{2}\textrm{,}$

where I have written $\circledast$ to avoid confusion with the asterisk
Einstein uses to denote the Maupertuis action $J^{*}$. We can write

$J^{*}=\oint_{\xi}\mathbf{p}=\oint_{\Xi}\mathbf{P}\textrm{,}$

or even

$J^{*}=\oint_{\Xi}\mathbf{p}=\oint_{\xi}\mathbf{P}\textrm{,}$

and

(6)
$\displaystyle\begin{split}J^{*}&=\oint_{\gamma^{\prime}\textrm{(}\xi\textrm{)}}\gamma^{\circledast}\mathbf{p}\\\
&=\oint_{\mathbb{H}}\gamma^{\circledast}\mathbf{p}=\langle\mathbf{\gamma^{\circledast}p},\mathbb{H}\rangle\end{split}$

for all diffeomorphisms

$\gamma,\gamma^{\prime}:\mathbb{P}\rightarrow\mathbb{P}$

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 $\xi\in\mathbb{H}$ as _trajectoires_ , but not an
uninteresting one.

In Hamiltonian mechanics one distinguishes between _point_ transformations
$\gamma$ (on the $l$-dimensional $q_{i}$-_Raum_) and _canonical_
transformations (on the $2l$-dimensional phase space
$\mathit{\Gamma}$).272727See Landau & Lifschitz (1970) § 45. So far we haven’t
gone beyond the rather restrictive ‘point’ condition

(7) $\mathbf{p}=\sum_{i}p_{i}dq_{i}=\sum_{i}P_{i}dQ_{i}$

of Lagrangian mechanics (where the configuration space $\mathcal{Q}$
characterised by $q_{1},\dots,q_{l}$ _maintains its identity_ , without
‘getting lost’ in the $2l$-dimensional state space); but in fact the weaker
‘symplectic’ condition

(8)
$\displaystyle\begin{split}\omega&=d\mathbf{p}=d\mathbf{p}^{\prime}=d(\mathbf{p}+dF)\\\
&=\sum_{i}dp_{i}\wedge dq_{i}=\sum_{i}dp_{i}\wedge dq_{i}+d^{2}F\end{split}$

also preserves $J^{*}$, where

(9) $\mathbf{p}\mapsto\mathbf{p}^{\prime}=\mathbf{p}+dF$

and the generating function $F$ is a zero-form. So in addition to the
diffeomorphic freedom $\gamma$ (or $\bar{\gamma}$) and the homotopic freedom
$\xi\mapsto\Xi$ (keeping clear of the source), we have the ‘gauge’ or
‘symplectic’ freedom (8) to add an exact term $dF$:

$J^{*}=\langle\gamma^{\circledast}\mathbf{p}+dF,\mathbb{H}\rangle.$

While a diffeomorphism $\gamma$ affects the momentum one-form indirectly by
first dragging points, the gauge transformation (9) is fibre-preserving and
therefore acts directly on each $\mathbf{p}\textrm{(}\mathbf{q}\textrm{)}$,
point by point. The canonical transformation generated by $F$ also corresponds
to a deformation of curves in $\bar{\mathbb{P}}$, reminiscent of the loop
deformation $\xi\mapsto\Xi$. Even if the curl at the origin (or the
corresponding circulation $J^{*}$) prevents the one-form $\mathbf{p}$ from
having a _global_ primitive, one can think of a _local_ primitive $\lambda$
satisfying $\mathbf{p}=d\lambda$ locally.282828“Locally” could mean, for
instance, _on any simply-connected region of $\bar{\mathbb{P}}$_. In much the
same way as one can take $\xi$ to be displaced by the point transformation
$\gamma$, one can take the level sets $\lambda=\textrm{const}.$ to be deformed
by $F$.

Graffi (2005) 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 _un_ symplectic, 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).

##  4 Hamilton-Jacobi theory

The old quantum theory, including _Quantensatz_ , was formulated in terms of
Hamilton-Jacobi theory,292929See Hamilton (1833, 1834), Jacobi (1884) pp.
143ff, Appell (1909) pp. 556ff, Epstein (1916b) pp. 493ff, Whittaker (1917)
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) pp. 216ff, Fasano & Marmi (2002) §11.1, Graffi (2004) p. 51,
Benci & Fortunato (2014) §1.4. which therefore deserves some attention.

The principle of least action303030See Brillouin (1938) pp. 159ff. exists in
two versions, which in Einstein’s notation would be distinguished by a star
indicating neglect of time.

  1. [i]

The ‘space-time’ Hamiltonian version determines the spatial shape of
trajectories _as well as motion along them_ by minimising the full Hamiltonian
action313131 _Cf_. Carathéodory (1937) p. 10.

$J=\int_{t_{0}}^{t}\mathscr{L}\,dt\textrm{,}$

where the ‘momentum’ or ‘covariant’ Lagrangian can be written

$\mathscr{L}=\sum_{i}p_{i}\frac{\partial\mathscr{H}}{\partial
p_{i}}-\mathscr{H}.$

  2. [ii]

The ‘spatial’ version attributed to Maupertuis only gives the spatial shape of
the trajectory, by minimising the purely spatial part

(10)
$J^{*}=\int_{\mathbf{q}_{0}}^{\mathbf{q}}\mathbf{p}=\int_{\mathbf{q}_{0}}^{\mathbf{q}}\sum_{i}p_{i}dq_{i}$

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 $\mathscr{H}$ 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 form323232See Jacobi (1884)
_Einundzwanzigste Vorlesung_ , Appell (1953) pp. 430ff, Levi-Civita & Amaldi
(1974) pp. 362ff.

$J=J^{*}-Et.$

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

$\displaystyle T=\frac{\|\mathbf{p}\|^{2}}{2m}$
$\displaystyle=\frac{1}{2}\sum_{i}p_{i}\dot{q}_{i}$
$\displaystyle=\frac{1}{2}\sum_{i}p_{i}\frac{\partial\mathscr{H}}{\partial
p_{i}}$

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 $\mathbf{q}_{0}$ as a fixed
initial position and $\mathbf{q}$ as a variable final position, we can radiate
$\mathbf{q}$ in all directions from $\mathbf{q}_{0}$ (now reminiscent of a
source in geometrical optics333333See Hamilton (1833, 1834), Whittaker (1917)
pp. 288ff.) along dynamical trajectories derived from the action function
$J^{*}\textrm{(}\mathbf{q},\mathbf{q}_{0}\textrm{)}$ satisfying the Hamilton-
Jacobi equation343434See Graffi (2004) p. 51.

(11) $\mathscr{H}(\mathbf{q},dJ^{*})=E\textrm{,}$

or rather

$\|\mathbf{p}\|^{2}=\|dJ^{*}\|^{2}\\\ =2m(E-U).$

Choosing an infinitesimal action $\delta J^{*}$, we first have the
infinitesimal sphere $\sigma(\mathbf{q}_{0},\delta J^{*})$ of radius

$\zeta(\mathbf{q}_{0},\delta J^{*})=\frac{\delta
J^{*}}{\sqrt{2m(E-U\textrm{(}\mathbf{q}_{0}\textrm{)})}}$

around $\mathbf{q}_{0}$. The momentum $\mathbf{p}$ at any $\mathbf{q}$ on the
sphere is the differential

$\mathbf{p}\textrm{(}\mathbf{q}\textrm{)}=dJ^{*}\textrm{(}\mathbf{q}\textrm{)}=\sum_{i=1}^{l}\frac{\partial
J^{*}\textrm{(}\mathbf{q}\textrm{)}}{\partial q_{i}}dq_{i}$

of $J^{*}$ viewed as a function of the final position $\mathbf{q}$. Once we
have a level surface $\sigma(\mathbf{q}_{0},\delta J^{*})$ of $J^{*}$, we
might as well generalise and propagate from an _arbitrary_ ($l-1$)-dimensional
initial surface

$\Sigma_{1}=\Sigma_{J_{[1]}^{*}}$

(which could be a sphere $\sigma$ 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_.353535See also Brillouin (1938) pp. 169ff, Arnol’d (1988) pp.
251ff. We can again take the same increment363636A smaller increment would be
even better, if secondary waves are propagated from an _already infinitesimal_
primary wavefront $\sigma(\mathbf{q}_{0},\delta J^{*})$. $\delta J^{*}$, which
provides a distance $\zeta(\mathbf{q}_{1},\delta J^{*})$ at the generic point
$\mathbf{q}_{1}\in\Sigma_{1}$. Then there are two possible constructions:
Either we repeat the above construction, now treating $\mathbf{q}_{1}$ as the
source of a ‘secondary’ wave,373737See Huygens (1690), Whittaker (1917) pp.
289ff, Carathéodory (1937) pp. 13ff, Arnol’d (1988) §9.3. a sphere
$\sigma(\mathbf{q}_{1},\delta J^{*})$ of radius $\zeta(\mathbf{q}_{1},\delta
J^{*})$ emanated from every point of the ‘primary’ wavefront $\Sigma_{1}$, in
which case the surface $\Sigma_{2}$ of action

$J_{[2]}^{*}=J_{[1]}^{*}+\delta J^{*}$

is the envelope of the secondary waves. Alternatively one lays off
$\zeta(\mathbf{q}_{1},\delta J^{*})$ normally at every point, to define
$\Sigma_{2}$ even more directly. ‘Normally’ can be understood as follows: The
($l-1$)-dimensional linear space

$\mathbb{T}_{\mathbf{q}_{1}}\Sigma_{1}\subset\mathbb{T}_{\mathbf{q}_{1}}\mathcal{Q}$

tangent to $\Sigma_{1}$ at $\mathbf{q}_{1}$ determines a ray
$\rho_{1}^{\flat}$ in the cotangent space
$\mathbb{T}^{*}_{\mathbf{q}_{1}}\mathcal{Q}$. The direction of the momentum
$\mathbf{p}_{1}$ at $\mathbf{q}_{1}$ is given by
$\mathbf{p}_{1}\in\rho_{1}^{\flat}$, the length by

$\|\mathbf{p}_{1}\|=\sqrt{2m(E-U\textrm{(}\mathbf{q}_{1}\textrm{)})}.$

The inverse $m^{\sharp}$ of the mass tensor383838See Brillouin (1938) p. 143.
$m^{\flat}$ determines a vector
$\mathbf{p}_{1}^{\sharp}=m^{\sharp}\mathbf{p}_{1}$, indeed a ray
$\rho_{1}^{\sharp}$ containing $\mathbf{p}_{1}^{\sharp}$ in the tangent space
$\mathbb{T}_{\mathbf{q}_{1}}\mathcal{Q}$. The action increment $\delta J^{*}$
fixes the length $\zeta(\mathbf{q}_{1},\delta J^{*})$ of the required vector
$\hat{\mathbf{p}}_{1}^{\sharp}\in\rho_{1}^{\sharp}$, which goes from
$\Sigma_{1}$ to the corresponding point of $\Sigma_{2}$.

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 $\mathcal{Q}$ 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
from the initial surface $\Sigma_{1}$ is determined by the local slant of the
surface itself,393939Here the normal direction of propagation is determined by
the mass tensor $m$. With a more general Hamiltonian (and Lagrangian), the
duality relations $\sharp\,\,\flat$ between level surface and motions are
given by the fibre derivatives, with components
${\partial\mathscr{H}}/{\partial p_{i}}$,
${\partial\mathscr{L}}/{\partial\dot{q}_{i}}$. 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_ $\mathcal{Q}$, but
_obliquely_ —at an ‘angle,’ relative to the surface, that varies from point to
point. Schrödinger’s orthogonal propagation, which makes the initial surface
$\Sigma_{1}$ 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 $\kappa$_losed_ and _exact_ , it should be noted, may
not have been entirely clear to Einstein (or others in 1917). Important
progress is made in Rham’s _Thèse_ (1931), where the distinction is used for
_analysis situs_. Even if the distinction is clear to Weyl (1939), he
confusingly uses the word “exact” to mean $\kappa$_losed_ : “A differential
$\omega$ whose derivative vanishes is called _exact_.” Weyl’s understanding of
$\kappa$losed is already rather modern:

> What we mean by $\omega\sim 0$ ($\omega$ homologous zero) may be explained
> in two ways: either differentially as indicating that $\omega$ is the
> derivative of a differential of next lower rank, or integrally as demanding
> that the integral of $\omega$ over any cycle vanishes. Every differential
> $\sim 0$ is exact; one readily proves this in both ways. “In the small” both
> notions, exact and $\sim 0$, 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 $\infty$ vielwertige) Funktion der
> $q_{i}$, so wird das Integral [$\int\sum p_{i}dq_{i}$] für eine solche Kurve
> im allgemeinen von Null verschieden sein.

Indeed if the one-form $\mathbf{p}$ is only

> $\mathbf{CMC}$: _closed on a multiply-connected region_ ,

it may or may not be exact, the loop integral (3) may or may not vanish. If
$\mathbf{p}$ is not exact,404040The ‘topological interpretation’ of the
Aharonov-Bohm effect shows how important it is to distinguish between _not_
exact and the above condition $\mathbf{CMC}$: not _necessarily_ exact. the
loop integral $\langle\mathbf{p},\mathbb{H}\rangle$ will not vanish, but its
_Vielwertigkeit_ will be the _denkbar einfachste_ : every point $\mathbf{q}$
on a given loop acquires an additional $\langle\mathbf{p},\mathbb{H}\rangle$
on every lap:

$\displaystyle J_{\textrm{(}n+1\textrm{)}}^{*}\textrm{(}\mathbf{q}\textrm{)}$
$\displaystyle=J_{\textrm{(}n\textrm{)}}^{*}\textrm{(}\mathbf{q}\textrm{)}+\langle\mathbf{p},\mathbb{H}\rangle$
$\displaystyle=J_{\textrm{(}1\textrm{)}}^{*}\textrm{(}\mathbf{q}\textrm{)}+n\langle\mathbf{p},\mathbb{H}\rangle\textrm{,}$

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 $\kappa$_losed_ 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
$\textbf{p}_{1}\textrm{(}\textbf{q}_{1}\textrm{)}$ to the points
$\textbf{q}_{1}$ of $\Sigma_{1}$, then radiates dynamical (Hamiltonian)
trajectories414141The almost superfluous curves $L$ and vectors $\dot{L}$ used
to extend the one-form $\mathbf{p}_{0}$ on $\Sigma_{0}$ to a one-form
$\mathbf{p}$ on the $l$-dimensional region $\mathcal{U}$ are only introduced
because Einstein seems to have such objects in mind. They can easily be
dispensed with, the one-forms are enough. $L$ satisfying

$\dot{L}\textrm{(}\textbf{q}_{1}\textrm{)}=\mathbf{p}_{1}^{\sharp}\textrm{(}\textbf{q}_{1}\textrm{)}=m^{\sharp}\mathbf{p}_{1}\textrm{(}\textbf{q}_{1}\textrm{)}$

throughout $\Sigma_{1}$. The dynamical vector field $\dot{L}$ thus determined
by the one-form $\mathbf{p}_{1}=\mathbf{p}|_{\Sigma_{1}}$ provides a one-form
$\mathbf{p}=m^{\flat}\dot{L}$ on an $l$-dimensional region
$\mathcal{U}\subset\mathcal{Q}$ covered by $\dot{L}$.

Einstein then wonders when the one-form $\mathbf{p}$ determined by the
Hamiltonian vector field $\dot{L}$ 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 $\mathbf{p}_{1}$ has to be $\kappa$losed, for then
$\mathbf{p}$ will be too; and as long as $\mathcal{U}$ is topologically
simple, $\kappa$losed means exact:

$(d\mathbf{p}_{1}=0)\Leftrightarrow(d\mathbf{p}=0)\Leftrightarrow(\exists\,J^{*}:\mathbf{p}=dJ^{*}).$

The potential $J^{*}$ is slightly underdetermined by its derivative $dJ^{*}$,
in other words

$d^{-1}\mathbf{p}=[J^{*}]=[J^{*}+\eta]_{\eta}\textrm{,}$

where $\eta$ is a constant on $\mathcal{U}$. A value
$J^{*}\textrm{(}\mathbf{q}\textrm{)}$ at a single $\mathbf{q}\in\mathcal{U}$
is enough to overcome the underdetermination and fix all of $J^{*}$. Summing
up, the theorem can be given as follows: Only a $\kappa$losed momentum field
$\mathbf{p}$ (radiated _dynamically_ from the momenta $\mathbf{p}_{1}$ freely
assigned to an initial surface $\Sigma_{1}\subset\mathcal{Q}$) can be derived
from a potential

$J^{*}=d^{-1}\mathbf{p}-\eta$

satisfying the Hamilton-Jacobi equation.

Einstein seems to change his mind in the _Nachtrag zur Korrektur_ , p. 91:

> $\mathbf{LB}$: 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 die einzelne Bahn 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
periodic (case [1] or [2]) or not (case [3]). If it is periodic, it will
assign to no point $\mathbf{q}$ (or neighbourhood) of $\mathcal{Q}$ more than
finitely many momenta $\mathbf{p}_{1},\dots,\mathbf{p}_{N}$. That’s what
Einstein means by a $p_{i}$-_Feld_ : an assignment of at most finitely many
momenta to (certain) points of $\mathcal{Q}$. If the momentum is _finitely_
multi-valued here and there, the configuration space can be enlarged to
restore single-valuedness—but not if it never closes.

There remains the issue of how a single _Bewegung_ , confined as it is to a
one-dimensional manifold, can yield a _Feld_ , a field on an $l$-dimensional
manifold. Since Einstein goes to the trouble of emphasising die einzelne Bahn
with Sperrdruck, 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 Schar, somehow or
other—despite the arresting Sperrdruck of die einzelne Bahn.

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 $\mathbf{p}$ is defined _everywhere_ on
$\mathcal{Q}$, 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 $\mathbf{LB}$, the issue of a Hamilton-Jacobi potential $J^{*}$
or of a vanishing curl $d\mathbf{p}$ does not even arise with a single
trajectory. With a whole congruence of motions radiated from an initial
surface, the curl $d\mathbf{p}$ 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 $J^{*}$.

##  5 Einstein’s integrability theorem

In the second-last paragraph of the _Nachtrag zur Korrektur_ (1917a, pp. 91-2)
Einstein formulates an integrability theorem that deserves attention:

> $\mathbf{IT}_{Q}$: Existieren $l$ Integrale der $2l$ Bewegungsgleichungen
> von der Form
>
> (12) $R_{k}(q_{i},p_{i})=\textrm{konst.,}$
>
> wobei die $R_{k}$ algebraische Funktionen der $p_{i}$ sind, so ist
> $\sum_{i}p_{i}dq$ immer ein vollständiges Differential, wenn man die $p_{i}$
> vermöge (12) durch die $q_{i}$ ausgedrückt denkt.

Or in Einstein’s letter (1917b) to Ehrenfest:

> $\mathbf{IT}_{E}$: Es liege ein Problem vor, bei dem soviel Integrale
>
> $L\textrm{(}q_{\nu},p_{\nu}\textrm{)}\hskip 1.0pt=\hskip
> 1.0pt\textrm{konst}$
>
> existieren, als Freiheitsgrade. Dann können die Impulse als (mehrwertige)
> Funktionen der $q_{\nu}$ ausgedrückt werden. Andererseits erfülle die
> Bahnkurve einen gewissen $q_{\nu}$-Raum vollständig, sodass sie jeden Punkt
> desselben beliebig nahe kommt. Dann liefert die Bahn des Systems im
> $q_{\nu}$-Raum ein Vektorfeld der $p_{\nu}$.

Here there are already elements of the theorem now attributed to
Liouville424242See Appell (1909) pp. 576ff, Appell (1953) pp. 437ff, Fasano &
Marmi (2002) §11.4. and Arnol’d.434343See Fasano & Marmi (2002) §11.5, Graffi
(2004) §1.7.1, Graffi (2005) §2, Lowenstein (2012) pp. 56ff.

###  5.1 Liouville-Arnol’d

We can begin anachronistically with (features of) the Liouville-Arnol’d
theorem, and then try to understand how much of it is already in Einstein
(1917a, b). The theorem uses $l$ functions

$F_{i}:\mathit{\Gamma}\rightarrow\mathbb{R}$

on an $2l$-dimensional phase space $\mathit{\Gamma}$ 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 $l\textrm{-dimensional}$ manifold. A
function $F$ foliates the phase space into ($2l-1$)-dimensional level surfaces
on which $F=\textrm{const}.$; 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 dynamics444444See Arnol’d (1988) §8.1.3,
Sternberg & Guillemin (1984) p. 88, Graffi (2004) p. 51.

$X_{G}=(dG)^{\sharp}=\omega^{\sharp}(dG)$

is _tangent_ to the level surfaces $F=\textrm{const}.$, rather than
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 compatibility454545See Graffi (2004) p. 50.

$\\{F,G\\}=0\textrm{,}$

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 $\mathscr{H}$, whose generic value $E$ singles out a
($2l-1$)-dimensional energy surface. The two compatibilities

$\\{\mathscr{H},F_{2}\\}=0=\\{\mathscr{H},F_{3}\\}$

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.464646In the aforementioned ‘effort to
eliminate dimensions,’ Poisson incompatibility $\\{F,G\\}\neq 0$ would be
_counter_ productive (reversing progress already made by effectively
_restoring_ an eliminated dimension), whereas linear dependence $dF=kdG$ would
be merely _un_ productive (producing neither loss nor gain, leaving the number
of dimensions unchanged). _Complete integrability_ is given by $l$ values

$f=(f_{1},\dots,f_{l})$

of the compatible, independent functions
$\mathfrak{F}_{l}=\\{F_{1},\dots,F_{l}\\}$, which eliminate $l$ of the initial
$2l$ dimensions of phase space, leaving an $l$-dimensional manifold
$\mathcal{M}_{f}$.

So far we have little more than a number $l$ of dimensions. Without
compactness, the $l$-dimensional manifold $\mathcal{M}_{f}$ could be a product
of lines and loops; compactness rules out the lines, leaving a torus
$\mathfrak{T}^{l}$, 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 $\mathcal{M}_{f}$ were a product of $l$ lines, it
would amount to $\mathbb{R}^{l}$; compactness prevents immersion in
$\mathbb{R}^{l}$ by ‘swelling’ $\mathcal{M}_{f}$. A two-dimensional torus
$\mathfrak{T}^{2}$, for instance, is an ‘enlarged’ two-dimensional
configuration space inasmuch as it cannot be embedded in the plane. Suppose
for definiteness that $\mathfrak{T}^{2}$ is not just a ‘topological torus’ (a
product of two loops) but a ‘rigid torus’ (a product of two rigid circles
$\mathbb{S}^{1}$)—literally a doughnut, with a shape and a size, embedded in
$\mathbb{R}^{3}$ parallel to the $xy$ plane. Take the simplest possible
motion, given by fixed rates of rotation around both circles of such a
$\mathfrak{T}^{2}$: 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, can actually prove quite relevant. According
to the Liouville-Arnol’d theorem, a completely integrable (and entirely
compact) dynamics can be represented as $l$ constant rates

(13) $\dot{\varphi}_{i}=\omega_{i}=\frac{\partial\mathscr{H}}{\partial
J^{*}_{i}}$

of rotation on an $l$-dimensional torus $\mathfrak{T}^{l}$, where the $l$
angles474747See Sternberg & Guillemin (1984) pp. 356ff, Lowenstein (2012) pp.
68ff. $\varphi_{i}$ with linear evolutions

$\varphi_{i}\textrm{(}t\textrm{)}=\omega_{i}t+\varphi_{i}\textrm{(}0\textrm{)}$

are canonically conjugate

(14) $\\{\varphi_{i},J^{*}_{j}\\}=\delta_{ij}$

to the actions $J^{*}_{i}$; $i,j=1,\dots,l$.

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 $\varphi_{i}$ yielding the constant frequencies $\omega_{i}$ 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 constant motions on a rigid torus are ultimately needed to
make sense of his intuitive analysis of integrability (closed _vs_. space-
filling 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 $\omega_{1}$,
$\omega_{2}$. 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 ${\omega_{1}}/{\omega_{2}}$ is rational, the motion
can be confined to a one-dimensional _Kontinuum_ which, being _geschlossen_ ,
is of finite length; whereas if $\omega_{1}/\omega_{2}$ is irrational the
motion can still be confined to a one-dimensional _Kontinuum_ , but not of
finite length.484848See Born (1933) pp. 80ff, Fasano & Marmi (2002) §11.7.
Einstein clearly understands the geometrical significance of confining motion
to intersections of level surfaces of appropriately compatible and independent
functions; what he doesn’t mention 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_.

###  5.2 Integrability in _configuration_ space

To understand Einstein’s integrability theorem(s) $\mathbf{IT}$
($\mathbf{IT}_{Q}$ & $\mathbf{IT}_{E}$) one has to bear in mind above all that
he’s in _configuration_ space $\mathcal{Q}$ and not in phase space
$\mathit{\Gamma}$. Again, throughout (1917a,b,c), Einstein is clearly and
explicitly494949See footnote 8. in the $l$-dimensional space he calls
$q$-_Raum_ ; this is especially evident at the bottom of p. 87 (1917a):

> 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
$\mathcal{Q}$—not of $\mathit{\Gamma}$:

> 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.505050 _ibid_. pp. 87-8

Once we’ve fixed $\mathbf{q}\in\mathcal{Q}$, the functions
$\mathfrak{F}_{l}=\\{F_{1},\dots,F_{l}\\}$ depend on momentum alone. If the
function $F_{k}$ is, say, quadratic in $\mathbf{p}$, a value $f_{k}$
determines an ellipsoid

$\Sigma(f_{k})\subset\mathbb{T}_{\mathbf{q}}\mathcal{Q}$

at $\mathbf{q}$ (and otherwise a more general surface). To simplify, ruling
out intractable pathologies, Einstein requires $\mathfrak{F}_{l}$ to be
algebraic (not necessarily quadratic) functions of momentum—he needs a class
of functions _that’s small enough to allow his procedure to work_. The $l$
values $f_{1},\dots,f_{l}$ of the functions $\mathfrak{F}_{l}$ of $\mathbf{p}$
at $\mathbf{q}\in\mathcal{Q}$ would confine $\mathbf{p}$ to the intersection

$\Sigma(f_{1})\cap\cdots\cap\Sigma(f_{l})\subset\mathbb{T}_{\mathbf{q}}\mathcal{Q}.$

The point of the theorem is not to express, given a set $\mathfrak{F}_{m}$,
the exact number $\\#\mathbf{p}\textrm{(}\mathbf{q}\textrm{)}$ of momenta

$\mathbf{p}\in\Sigma(f_{1})\cap\cdots\cap\Sigma(f_{l})$

at $\mathbf{q}\in\mathcal{Q}$ as a function of $m$. Einstein only wants to
know whether a certain set $\mathfrak{F}_{m}$, a certain integer $m$,
corresponds to _finitely_ many momenta at $\mathbf{q}$ or _infinitely_ many
momenta. And concerning $m$ itself, he’s mainly (perhaps even _only_)
interested in whether $m<l$ or $m=l$. Even if that simplifies matters
considerably, the logic of $\mathbf{IT}$ remains somewhat ambiguous. The clear
part of the logic is that an _incomplete_ set $\mathfrak{F}_{m}$ ($m<l$)
allows _infinitely_ many momenta $\mathbf{p}$ at the _Stelle_ —the
incompleteness rules out _finitely_ many momenta (and hence a closed, periodic
motion). A _complete_ set $\mathfrak{F}_{l}$ could in principle still allow
finitely many _or infinitely many_ momenta. Without being entirely explicit,
Einstein pretty clearly suggests that a complete set $\mathfrak{F}_{l}$ rules
out infinitely many momenta. I read him as saying:

$\textrm{\emph{complete} set
}\mathfrak{F}_{l}\Rightarrow\textrm{\emph{periodic} motion;}$

_motion is closed, periodic, ‘totally’ integrable (as opposed to ‘space-
filling’) if and only if the set of integrals is complete_.

Returning to $\mathbf{IT}_{Q}$, Einstein mentions that the momentum (2) would
be a _vollständiges Differential_. I think the point isn’t _exact _vs_.
$\kappa$losed_, but rather that once non-periodic (space-filling) motion is
ruled out, with its infinitely many momenta, the momentum just _makes sense_ ,
having only finitely-many values—or _just one_ at every point of an
appropriately enlarged configuration space.

Since $\mathbf{IT}_{E}$ is somewhat choppy, its four sentences [S1]-[S4] can
be looked at one by one. In my notation:

  1. [S1]

The set $\mathfrak{F}_{m}$ at $\mathbf{q}\in\mathcal{Q}$ is assumed complete:
$m=l$.

  2. [S2]

Since $\mathfrak{F}_{m}$ is complete, there are at most _finitely_ many
momenta $\mathbf{p}$ at $\mathbf{q}$.

  3. [S3]

Were $\mathfrak{F}_{m}$ _in_ complete ($m<l$), the motion would fill space.

  4. [S4]

Since $\mathfrak{F}_{m}$ _is_ complete, the (periodic) motion yields a
momentum (co)vector field (with _finitely_ many values) on $\mathcal{Q}$.

Einstein really does seem to say that a complete set $\mathfrak{F}_{l}$
guarantees a closed, periodic motion.

##  6 Final remarks

Einstein situates his mechanics in the $q_{i}$-_Raum_ he so often refers to;
symplectic abstractions are foreign to him. The ‘complete’ integrability one
comes across in the modern literature is hardly complete by Einstein’s
standards, being compatible with both periodic and space-filling motions. The
more extreme kind of integrability Einstein has in mind corresponds to
periodic, not space-filling 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 space-filling motion; his _Riemannisierung_ is necessarily finite.
And once on the torus, where the greater invariance of Einstein’s integrals
$\langle\mathbf{p},\mathbb{H}_{i}\rangle$ 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.

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###### Abstract

This paper presents hybrid numerical techniques for solving the Boltzmann
transport equation formulated by means of low-order equations for angular
moments of the angular flux. The moment equations are derived by the
projection operator approach. The projected equations are closed exactly using
a high-order transport solution. The low-order equations of the hybrid methods
are approximated with a finite volume scheme of the second-order accuracy.
Functionals defining the closures in the discretized low-order equations are
calculated by Monte Carlo techniques. In this study, we analyze effects of
statistical noise and discretization error on the accuracy of the hybrid
transport solution.

###### keywords:

Boltzmann transport equation, hybrid Monte Carlo methods, low-order equations,
Eddington factor, finite volume schemes

Analysis of Hybrid MC/Deterministic Methods for Transport Problems
Based on Low-Order Equations Discretized by Finite Volume Schemes

Vincent N. Novellino and Dmitriy Y. Anistratov

Department of Nuclear Engineering, North Carolina State University, Raleigh,
NC 27695
<EMAIL_ADDRESS><EMAIL_ADDRESS>
##  1 Introduction

Monte Carlo (MC) and deterministic methods for solving neutron transport
problems are widely used by modeling software to make informed design
decisions. MC is generally slow to converge when seeking a global solution.
Hybrid MC/deterministic methods have been developed for solving particle
transport problems to improve efficiency of MC simulations. They can reduce
statistical variance of the solution and accelerate MC calculations. A group
of hybrid methods for global transport problems are based on different low-
order equations for moments of the angular flux [1, 2, 3, 4].

We develop hybrid numerical techniques applying low-order equations of the
Quasidiffusion (QD) and Second Moment (SM) methods [5, 6, 7]. Low-order
equations for moments of the angular flux were derived by the projection
operator approach. Exact closures were formulated for the projected equations
using a high-order transport solution. The low-order moment equations of the
hybrid methods were discretized with a finite volume (FV) scheme. Functionals
defining the closures in the discrete moment equations are computed by MC.
There are two main factors affecting the numerical solution of hybrid methods:
(1) the discretization error related to the finite size of mesh cells and (2)
the statistical variation of functionals associated with finite number of
particle histories. In this study, we analyze effects of statistical noise and
discretization error on the accuracy of the hybrid transport solution.

##  2 Hybrid Transport Methods

We consider one-group steady-state transport problem in 1D slab geometry with
isotropic scattering and source:

$\mu\frac{\partial\psi}{\partial
x}(x,\mu)+\Sigma_{t}(x)\psi(x,\mu)=\frac{\Sigma_{s}(x)}{2}\int_{-1}^{1}\psi(x,\mu^{\prime})d\mu^{\prime}+\frac{Q(x)}{2}\,$
(1) $\psi(0,\mu)=0,\ \mu>0\,,\quad\psi(L,\mu)=0,\ \mu<0\,.$ (2)

The low-order QD (LOQD) equation in the second-order form is given by [5]

$-\frac{d}{dx}\frac{1}{\Sigma_{t}(x)}\frac{d(E\phi)}{dx}(x)+\Sigma_{a}(x)\phi(x)=Q(x)\,,$
(3)

where the QD (Eddington) factor is defined by

$E(x)=\frac{\int_{-1}^{1}\mu^{2}\psi(x,\mu)d\mu}{\int_{-1}^{1}\psi(x,\mu)d\mu}\,.$
(4)

The low-order SM (LOSM) equation has the form [6]:

$-\frac{d}{dx}\frac{1}{3\Sigma_{t}(x)}\frac{d\phi}{dx}(x)+\Sigma_{a}(x)\phi(x)=Q(x)+\frac{d}{dx}\frac{1}{3\Sigma_{t}(x)}\frac{dF}{dx}(x)\,,\vspace{-0.15cm}$
(5)

where the SM factor is given by

$F(x)=\int_{-1}^{1}\bigg{(}\frac{1}{3}-\mu^{2}\bigg{)}\psi(x,\mu)d\mu\,.$ (6)

The details on formulation of boundary conditions (BCs) for LOQD and LOSM
equations are omitted for sake of brevity (see [5, 7]). QD and SM factors
(Eqs. (4) and (6)) define the closures for the low-order equations and the
high-order transport equation (Eq. (1)).

We formulate the hybrid QD (HQD) and hybrid SM (HSM) methods for calculation
of the scalar flux as a solution of the discrete low-order equations with the
factors $E$ and $F$, respectively, computed with MC. The low-order equations
are discretized with a finite volume (FV) scheme of the second-order accuracy.
The discrete FV LOQD equations for the cell-average scalar flux $\phi_{i}$
have the following form:

$-\frac{E_{i+1}\phi_{i+1}-E_{i}\phi_{i}}{\Sigma_{t,i+1/2}\Delta
x_{i+1/2}}+\frac{E_{i}\phi_{i}-E_{i-1}\phi_{i-1}}{\Sigma_{t,i-1/2}\Delta
x_{i-1/2}}+\Sigma_{a,i}\Delta x_{i}\phi_{i}=Q_{i}\Delta x_{i}\,,$ (7)

where $i$ is the spatial cell index, the integer subscript indicates a
quantity averaged over the $i^{th}$ cell $[x_{i},x_{i+1}]$, $\Delta
x_{i}=x_{i+1}-x_{i}$,

$\Sigma_{t,i+1/2}=\frac{\Sigma_{t,i}\Delta x_{i}+\Sigma_{t,i+1}\Delta
x_{i+1}}{\Delta x_{i}+\Delta x_{i+1}}\,,\quad\Delta
x_{i+1/2}=\frac{1}{2}(\Delta x_{i}+\Delta x_{i+1})\,.$ (8)

The FV scheme for the LOSM equation is given by

$-\frac{\phi_{i+1}-\phi_{i}}{3\Sigma_{t,i+1/2}\Delta
x_{i+1/2}}+\frac{\phi_{i}-\phi_{i-1}}{3\Sigma_{t,i-1/2}\Delta
x_{i-1/2}}+\Sigma_{a,i}\Delta x_{i}\phi_{i}=Q_{i}\Delta
x_{i}+\frac{F_{i+1}-F_{i}}{\Sigma_{t,i+1/2}\Delta
x_{i+1/2}}+\frac{F_{i}-F_{i-1}}{\Sigma_{t,i-1/2}\Delta x_{i-1/2}}\,.$ (9)

Track-length tallies are used for the functional estimators to compute cell-
average factors

$E_{i}=\frac{\sum_{n=1}^{N}\sum_{m=1}^{M_{n}}\mu_{n,m}^{2}w_{n,m}\ell_{n,m}}{\sum_{n=1}^{N}\\!\\!\sum_{m=1}^{M_{n}}w_{n,m}\ell_{n,m}}\,,$
(10) $F_{i}=\frac{1}{N\Delta
x_{i}}\sum_{n=1}^{N}\sum_{m=1}^{M_{n}}\Big{(}\frac{1}{3}-\mu_{n,m}^{2}\Big{)}w_{n,m}\ell_{n,m}\,,$
(11)

where $N$ is the number of particle histories, $M_{n}$ is the number of tracks
of $n^{th}$ particle in the $i^{th}$ cell, $\ell_{n,m}$ is the $m^{th}$ track
length of the $n^{th}$ particle in the $i^{th}$ cell, $w_{n,m}$ is the weight
of this particle. A general algorithm of the HQD and HSM methods is described
by Algorithm 1.

foreach  _$n=1\dots N$_ do

Simulate particle history and collect moments of tracklength and facet
crossing tallies

Calculate functionals

Solve the low-order equations to compute $\phi_{i}$

Algorithm 1 Hybrid transport method

##  3 Numerical Results and Analysis

The errors of numerical solutions of the HQD and HSM methods are analyzed on
the transport problem in the spatial domain $0\leq x\leq 1$ with
$\Sigma_{t}=1.0$, $\Sigma_{s}=0.9$, $Q=1.0$, and vacuum BCs. The spatial mesh
with $I$ cells is uniform. The accuracy of numerical solutions is evaluated by
means of the reference deterministic solution computed on a sequence of
refined grids in the phase space with Aitken extrapolation. The obtained
benchmark solution ($\phi^{ex}$) is exact in its 6 significant digits. We
present the relative error in $L_{2}$-norm given by

$\text{RE}_{L_{2}}=\sqrt{\frac{\sum_{i=1}^{I}(\phi_{i}-\phi_{i}^{ex})^{2}\Delta
x_{i}}{\sum_{i=1}^{I}(\phi_{i}^{ex})^{2}\Delta x_{i}}}\,.$ (12)

We performed 100 calculations of the test by the HQD and HSM methods and
compare the error of each hybrid solution with one of the analog MC solution
the particle histories of which were used to compute functionals for that
hybrid algorithm calculation. Tables 1 and 2 present the fraction of times
that each hybrid method produced the solution with a lower relative $L_{2}$
error compared to the companion MC calculations. These results show the
combination of grids and number of histories where discretization error
dominates in the error in the solution of the HQD and HSM methods. The hybrid
methods produced on average a more accurate solution than direct MC
calculations for most of considered combination of $N$ and $\Delta x$’s. HSM
has a higher rate of wins vs. MC compared to the HQD method. We also compared
$L_{\infty}$ norms of error as well, though not presented in this summary,
which shows similar behavior to the integral norm results.

Table 1: HQD $L_{2}$ win ratio compared to MC as a function of $N$ particles and cell width $\Delta x$ $I$ | $\Delta x$ | $N=10^{3}$ | $N=10^{4}$ | $N=10^{5}$
---|---|---|---|---
4 | $2^{-2}$ | 0.55 | 0.13 | 0.00
8 | $2^{-3}$ | 0.71 | 0.63 | 0.17
16 | $2^{-4}$ | 0.74 | 0.72 | 0.89
32 | $2^{-5}$ | 0.70 | 0.68 | 0.70
64 | $2^{-6}$ | 0.73 | 0.77 | 0.77
Table 2: HSM $L_{2}$ win ratio compared to MC as a function of $N$ particles and cell width $\Delta x$ $I$ | $\Delta x$ | $N=10^{3}$ | $N=10^{4}$ | $N=10^{5}$
---|---|---|---|---
4 | $2^{-2}$ | 0.66 | 0.22 | 0.00
8 | $2^{-3}$ | 0.78 | 0.67 | 0.29
16 | $2^{-4}$ | 0.80 | 0.76 | 0.79
32 | $2^{-5}$ | 0.82 | 0.73 | 0.80
64 | $2^{-6}$ | 0.84 | 0.79 | 0.76

$N=10^{3}$ histories, relative errors in $L_{2}$-norm of MC and HQD

$N=10^{3}$ histories, sorted relative error in $L_{2}$-norm of MC and HQD

$N=10^{5}$ histories, relative errors in $L_{2}$-norm of MC and HQD

$N=10^{5}$ histories, sorted relative error in $L_{2}$-norm of MC and HQD

Figure 1: Relative error in $L_{2}$-norm of MC and HQD solutions, I = 8,
$\Delta x=2^{-3}$.

$N=10^{3}$ histories, relative errors in $L_{2}$-norm of MC and HSM

$N=10^{3}$ histories, sorted relative error in $L_{2}$-norm of MC and HSM

$N=10^{5}$ histories, relative errors in $L_{2}$-norm of MC and HSM

$N=10^{5}$ histories, sorted relative error in $L_{2}$-norm of MC and HSM

Figure 2: Relative error in $L_{2}$-norm of MC and HSM solutions, I = 8,
$\Delta x=2^{-3}$.

Figure 1 shows the relative $L_{2}$ errors of MC and HQD for different number
of particle histories, namely, $N=10^{3}$ and $N=10^{5}$. On the left side of
Figure 1 we present the errors of MC and the corresponding HQD solutions
computed with the same ensemble of particle histories versus the index of
simulation. These plots illustrate the performance of two methods compared to
each other in every case. The plots of sorted $L_{2}$ errors of both MC and
HQD are shown on the right sub-figure. These results illustrate the effect of
increasing the number of histories on a coarse spatial grid. An observation
from the unsorted results is the generally weak correlation between the MC and
HQD errors. Our sorted $L_{2}$ results demonstrate the average behavior of the
MC and HQD methods showing which method tends to have lower relative $L_{2}$
error. We notice that for low number of particle histories the HQD methods
performs better than MC, especially for cases with high MC statistical error.
The discretization error becomes dominant with increasing the particle number
used in calculations leading to less accurate solutions compared to MC (see
Fig. 1 for $N=10^{5}$). Figure 2 presents the same set plots for the HSM
method with MC simulations. The performance of the HSM method is similar one
of the HQD method. Figure 3 shows the sorted errors for fine mesh and large
number particle history cases for both HQD and HSM methods. We see as a trend
that the hybrid methods reduce error in the simulation and eliminate the right
tail, i.e. sharp increase in error, present in the Monte Carlo results. We
caution the reader to interpret Figure 3 in light of the win ratio shown in
Tables 1 and 2.

Table 3 presents the relative error in $L_{2}$-norm ($RE_{L_{2}}$) of the MC
solution with implicit capture, and hybrid solutions for the set of
calculations with spatial grid refinement and increasing number of particle
histories. In the cases $N=10^{2}$ and $N=10^{4}$, the HQD and HSM solutions
are slightly more accurate on all grids except for $\Delta x=2^{-2}$. Note
that the error of MC solution increases under grid refinement, since as the
tally cells shrinks the total tracks-length in a cell decreases leading to an
increase in uncertainty. The HQD and HSM methods show an increase in error
under grid refinement for low particle counts ($N=10^{2}$) due to the large
statistical uncertainty. For large number of histories ($N=10^{6}$), we notice
significant reduction in error of hybrid solutions as $\Delta x$ decreases
driven by the spatial convergence of the underlying FV schemes for LOQD and
LOSM equations. Table 4 presents the ratios of errors in $L_{2}$-norms
$RE_{L_{2}}(2\Delta x)/RE_{L_{2}}(\Delta x)$ for the solution on neighboring
grids. These results quantify the change of error in solution with grid
refinement. Figure 4 shows the relative error in $L_{2}$-norm vs. $\Delta x$
for each method for a select number of particle histories. The results
demonstrate how hybrid solutions are more accurate for small ($N=10^{2}$) and
sufficiently large number ($N=10^{4}$) of histories. We also notice that for
very high particle counts ($N=10^{6}$) discretization error is dominant for
these hybrid methods with FV spatial schemes.

Table 3: Relative error in $L_{2}$-norm ($RE_{L_{2}}$) for MC, HQD, and HSM on sequence of grids $N=10^{2}$ | MC | HQD | HSM
---|---|---|---
$\Delta x=2^{-2}$ | 1.4 $\times 10^{-1}$ | 1.5$\times 10^{-1}$ | 1.45$\times 10^{-1}$
$\Delta x=2^{-3}$ | 1.91$\times 10^{-1}$ | 1.72$\times 10^{-1}$ | 1.82$\times 10^{-1}$
$\Delta x=2^{-4}$ | 2.34$\times 10^{-1}$ | 2.10$\times 10^{-1}$ | 2.23$\times 10^{-1}$
$\Delta x=2^{-5}$ | 2.41$\times 10^{-1}$ | 2.15$\times 10^{-1}$ | 2.29$\times 10^{-1}$
$\Delta x=2^{-6}$ | 2.61$\times 10^{-1}$ | 2.33$\times 10^{-1}$ | 2.50$\times 10^{-1}$
$N=10^{4}$ | MC | HQD | HSM
$\Delta x=2^{-2}$ | 1.21$\times 10^{-2}$ | 2.30$\times 10^{-2}$ | 1.74$\times 10^{-2}$
$\Delta x=2^{-3}$ | 1.45$\times 10^{-2}$ | 1.18$\times 10^{-2}$ | 1.13$\times 10^{-2}$
$\Delta x=2^{-4}$ | 1.58$\times 10^{-2}$ | 1.26$\times 10^{-2}$ | 1.28$\times 10^{-2}$
$\Delta x=2^{-5}$ | 1.75$\times 10^{-2}$ | 1.52$\times 10^{-2}$ | 1.53$\times 10^{-2}$
$\Delta x=2^{-6}$ | 1.97$\times 10^{-2}$ | 1.76$\times 10^{-2}$ | 1.77$\times 10^{-2}$
$N=10^{6}$ | MC | HQD | HSM
$\Delta x=2^{-2}$ | 1.20$\times 10^{-3}$ | 2.30$\times 10^{-2}$ | 1.81$\times 10^{-2}$
$\Delta x=2^{-3}$ | 1.31$\times 10^{-3}$ | 6.04$\times 10^{-3}$ | 4.90$\times 10^{-3}$
$\Delta x=2^{-4}$ | 1.54$\times 10^{-3}$ | 2.20$\times 10^{-3}$ | 2.03$\times 10^{-3}$
$\Delta x=2^{-5}$ | 1.83$\times 10^{-3}$ | 1.86$\times 10^{-3}$ | 1.85$\times 10^{-3}$
$\Delta x=2^{-6}$ | 2.01$\times 10^{-3}$ | 1.97$\times 10^{-3}$ | 1.97$\times 10^{-3}$
Table 4: Ratios of errors in $L_{2}$-norms, $RE_{L_{2}}(2\Delta x)/RE_{L_{2}}(\Delta x)$ $N=10^{2}$ | MC | HQD | HSM
---|---|---|---
$\Delta x=2^{-3}$ | 0.73 | 0.87 | 0.80
$\Delta x=2^{-4}$ | 0.82 | 0.82 | 0.82
$\Delta x=2^{-5}$ | 0.97 | 0.98 | 0.98
$\Delta x=2^{-6}$ | 0.92 | 0.92 | 0.92
$N=10^{4}$ | MC | HQD | HSM
$\Delta x=2^{-3}$ | 0.83 | 1.94 | 1.55
$\Delta x=2^{-4}$ | 0.92 | 0.94 | 0.88
$\Delta x=2^{-5}$ | 0.90 | 0.83 | 0.84
$\Delta x=2^{-6}$ | 0.89 | 0.86 | 0.86
$N=10^{6}$ | MC | HQD | HSM
$\Delta x=2^{-3}$ | 0.92 | 3.80 | 3.69
$\Delta x=2^{-4}$ | 0.85 | 2.74 | 2.42
$\Delta x=2^{-5}$ | 0.84 | 1.19 | 1.10
$\Delta x=2^{-6}$ | 0.91 | 0.94 | 0.94

$N=10^{4}$ histories, MC and HQD

$N=10^{4}$ histories, MC and HSM

$N=10^{5}$ histories, MC and HQD

$N=10^{5}$ histories, MC and HSM

Figure 3: Sorted relative error in $L_{2}$-norm of MC and hybrid solutions, I
= 64, $\Delta x=2^{-6}$.

$N=10^{2}$, same histories for each method

$N=10^{4}$, same histories for each method

$N=10^{6}$, same histories for each method

Figure 4: Relative error in $L_{2}$-norm of MC and hybrid solutions vs.
$\Delta x$ under grid refinement for varying number of histories

##  4 Conclusions

The HQD and HSM methods with the FV approximation of the low-order equations
with second-order convergence generate more accurate numerical solutions than
MC on average when discretization error is small. These hybrid techniques also
showed advantages in simulations with low particle counts per tally cell. For
multi-dimensional problems, it is expensive to obtain large number of particle
histories for each phase-space elements. Therefore, average particle history
per phase space element should be considered when evaluating low-order hybrid
solvers. The HSM method shows some advantages when compared to HQD. Further
testing and analysis are needed to study effects of statistical noise and
discretization error.

##  5 Acknowledgments

The work of the first author (VNN) was supported under an University Nuclear
Leadership Program Graduate Fellowship. Any opinions, findings, conclusions or
recommendations expressed in this publication are those of the author(s) and
do not necessarily reflect the views of the Department of Energy Office of
Nuclear Energy. The work of the second author (DYA) was supported by the
Center for Exascale Monte-Carlo Neutron Transport (CEMeNT), a PSAAP-III
project funded by the Department of Energy, grant number DE-NA003967.

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