∎

11institutetext: U. Klein 22institutetext: University of Linz
Institute for Theoretical Physics
A-4040 Linz, Austria
Tel.: 0043 660 5489735
22email<EMAIL_ADDRESS>
# A reconstruction of quantum theory for nonspinning particles

U. Klein

(Received: date / Accepted: date)

###### Abstract

This work is based on the idea that the classical counterpart of quantum
theory (QT) is not mechanics but probabilistic mechanics. We therefore choose
the theory of statistical ensembles in phase space as starting point for a
reconstruction of QT. These ensembles are described by a probability density
$\rho(q,p,t)$ and an action variable $S(q,p,t)$. Since the state variables of
QT only depend on $q$ and $t$, our first step is to carry out a projection
$p\Rightarrow M(q,t)$ from phase space to configuration space. We next show
that instead of the momentum components $M_{k}$ one must introduce suitable
potentials as dynamic variables. The quasi-quantal theory resulting from the
projection is only locally valid. To correct this failure, we have to perform
as a second step a linearization or randomization, which ultimately leads to
QT. In this work we represent $M$ as an irrotational field, where all
components $M_{k}$ may be derived from a single function $S(q,t)$. We obtain
the usual Schrödinger equation for a nonspinning particle. However, space is
three-dimensional and $M$ must be described by $3$ independent functions. In
the fourth work of this series, a complete representation of $M$ will be used,
which explains the origin of spin. We discuss several fundamental questions
that do not depend on the form of $M$ and compare our theory with other recent
reconstructions of QT.

###### Keywords:

Quantum theory Projection to Configuration Space Linearization Randomization
Derivation of Schrödinger’s equation Quantum-classical relation

###### MSC:

81P05 81S99 81Q20 70H99

##  1 Introduction

Among the many different interpretations of the quantum theoretical formalism,
there is one that does not require any additional philosophical assumptions.
This is the so-called statistical interpretation or ensemble interpretation of
quantum theory (QT) preferred by Einstein ballentine:statistical . From his
point of view, the Schrödinger equation does not describe individual (one or
several) particles, but only statistical ensembles of (one or several)
particles. In fact, there is no doubt that the predictions of QT are
probabilities - even if the quantum probabilities cannot be described by
standard classical probability theory. In this minimalist interpretation there
is no unsolvable measurement problem, no controversial reduction of the wave
packet, and no mysterious nonlocality. On the other hand, one has to do
without the highly honored principle of reductionism. The more widespread
individuality interpretation is in agreement with this principle.

By the very nature of the term the truth or falseness of an interpretation
cannot be proven. Nonetheless, one can give facts that speak for or against a
particular interpretation. If it would be possible to derive QT in a
convincing manner from a classical probabilistic (ensemble) theory, then that
would be a fact favoring the ensemble interpretation of QT. On top of that,
such a derivation might lead to a better understanding of QT and eliminate
some or all of the strange features mentioned above.

The present paper is the third in a series of works which are based on
Einstein’s point of view; the previous works will be referred to as I
klein:koopman  and II klein:probabilistic . Of course, these papers are not
the first using this point of view; a number of relevant quotations may be
found in II. In I and II an attempt has been undertaken to derive QT from a
minimal number of reasonable postulates. The first postulate is that it must
be possible to derive QT from the theory of classical statistical ensembles.
From this most fundamental assumption and the fact that QT uses space-time
variables as independent variables, most of the subsequent steps in the
derivation of QT follow more or less automatically. The theory reported in II
was named “Hamilton-Liouville-Lie-Kolmogorov theory” (HLLK) quoting some of
the great scientists who made this work possible. We continue to use this
acronym for the present theory, which shares with I and II the same basic
assumptions and the same aim, but uses a different method.

Let us briefly recall the ideas on which works I, II are based. A classical
statistical ensemble in phase space is defined as the uncountable set of all
solutions of the canonical equations

$\dot{q_{k}}=\frac{\partial H(q,p)}{\partial
p_{k}},\;\;\;\dot{p}_{k}=-\frac{\partial H(q,p)}{\partial q_{k}}\mbox{.}$ (1)

The state of each member of the ensemble, at each instant of time $t$, is
described by $2n$ numbers, namely $n$ coordinates $q={q_{1},...,q_{n}}$ and
$n$ conjugate momenta $p={p_{1},...,p_{n}}$. The dependence of the state $q,p$
on time $t$ is ruled by the time-independent Hamiltonian $H(q,p)$. The
solutions of (1) are written in the form

$q_{k}=Q_{k}(t,t^{0},q^{0},p^{0}),\;\;\;p=P_{k}(t,t^{0},q^{0},p^{0})\mbox{,}$
(2)

where $(q^{0},p^{0})$ is the state at the initial time $t^{0}$; the dependence
on $t_{0}$ will often be suppressed. The ensemble is given by the infinite set
of solutions (2) with $(q^{0},p^{0})$ taking all possible values in the
$2n$-dimensional phase space
$\Omega=\mathbb{R}^{n}_{q}\times\mathbb{R}^{n}_{p}$.

The variables $q,\,p$ are “Lagrangian coordinates” describing properties of
_particles_. Such a “particle-like” description of an ensemble is unwieldy. In
the more convenient “Eulerian description”, which corresponds to the usual
field-theoretical formulation, the same symbols $q,\,p$ are used to denote
points of _space_ , in this case points of $\Omega$. The distinction is
important despite the fact that the mathematical range of values  is the same
in both cases. The relation between both descriptions is discussed in more
detail in II. The most important Eulerian dynamic variable is the probability
density $\rho(q,p,t)$ , describing the distribution of possible trajectories
in phase space, which obeys the Liouville equation

$\frac{\partial\rho}{\partial t}+\frac{\partial\rho}{\partial
q_{k}}\frac{\partial H}{\partial p_{k}}-\frac{\partial\rho}{\partial
p_{k}}\frac{\partial H}{\partial q_{k}}=0\mbox{.}$ (3)

This partial differential equation is the counterpart of Newton’s law [or its
reformulation (1)] for probabilistic ensembles of trajectories. If
$\rho(q,p,t)$ is known, all of the observable output (expectation values) of
such a probabilistic theory may be calculated.

The differences between classical mechanics (CM) and the theory of
probabilistic ensembles, which is referred to as probabilistic mechanics (PM),
are important in particular in view of a transition to QT: A state of CM (at
fixed time) is a point in $\Omega$, a state of PM is a point in an
appropriately defined function space. Observable results in a state of CM are
given by values of observables $A(q,p,t)$, observable results in a state of PM
are given by expectation values

$\bar{A}_{\rho}(t)=\int\mathrm{d}q\,\mathrm{d}p\,\rho(q,p,t)A(q,p,t)\mbox{.}$
(4)

This listing of obvious facts shows that the structural similarity between PM
and QT is much stronger than that between CM and QT. It is also clear that PM
is more realistic than CM because there are no experiments with unlimited
accuracy, as pointed out by Born born:vorhersagbarkeit . Let us also make a
comment on the limit of individual particles in the ensemble theory. It is
well-known that solutions of the Liouville equation for arbitrary initial
conditions may be constructed with the help of the solutions of the particle
equations (1). In particular, particle-like solutions of the Liouville
equation exist, where the probability is concentrated entirely on the
trajectories. The question arises as to whether the probabilistic description
and the description with individual trajectories are equivalent in this limit.
It turns out that this is _not_ the case. We refer to the literature for
arguments in favor of the probabilistic description jaffe.brumer:classical .

The dynamical variable $\rho(q,p,t)$ is all we need in order to calculate
time-dependent expectation values of arbitrary classical observables
$A(q,p,t)$; no further dynamical variables are required, in principle, to
define a “state” of PM. Nevertheless, we introduce, as in I,II, the classical
action

$S=\int_{t_{0}}^{t}\mathrm{d}t^{\prime}L\left(q(t^{\prime}),\dot{q}(t^{\prime}),t^{\prime}\right)\mbox{,}$
(5)

evaluated at the real paths $q(t^{\prime})$, as a second dynamical variable.
The action integral (5), formulated here as a function $S(q,p,t)$ on phase
space, plays an indispensable role in the transition from PM to QT. In I,II it
has been shown that $S(q,p,t)$ obeys the differential equation

$\frac{\partial S}{\partial t}+\frac{\partial S}{\partial q}\frac{\partial
H}{\partial p}-\frac{\partial S}{\partial p}\frac{\partial H}{\partial
q}=\bar{L}\mbox{,}$ (6)

which will be referred to as action equation. The quantity $\bar{L}$ (the
Lagrangian) is defined by

$\bar{L}=\bar{L}(q,p,t)=p\,\frac{\partial H(q,p)}{\partial p}-H(q,p)\mbox{.}$
(7)

The left-hand-sides of (3) and (6) may be interpreted as total derivatives
with respect to time, with a velocity field $V=(\frac{\partial H}{\partial
p},-\frac{\partial H}{\partial q})$. Arbitrary “Lagrangian” functions,
describing the distribution of certain particle properties may be transformed
to “Eulerian” functions describing corresponding fields. This is a general
method caratheodory:calculus_I  frequently applied in $3-$dimensional fluid
mechanics bennett:lagrangian  and applied here to $2n-$dimensional phase
space; in section 2.2.2 we will apply it to $n-$dimensional configuration
space $\Sigma=\mathbb{R}^{n}_{q}$.

Note that there is no coupling between the basic differential equations (3)
and (6); in particular $\rho(q,p,t)$ does not occur in the differential
equation (6) for $S(q,p,t)$. This is a triviality in classical probabilistic
physics: the dynamical laws remain intact while all the uncertainty is due to
the initial states. Adding the law (6) to PM offers the possibility to
construct an indeterministic theory (like QT), where a _coupling_ between the
corresponding variables occurs.

In I and II Schrödinger’s equation was derived from PM by means of two simple
steps. In the first step, the state variable $\rho$ was replaced by a complex-
valued state variable $\psi$ which depends in a suitable manner (Madelung
form) on $\rho$ and $S$. The resulting evolution equation for $\psi$ is the
counterpart of Schrödinger’s equation in phase space. In the second step, this
equation was projected onto $\Sigma$ . This leads to the standard quantization
rules and to Schrödinger’s equation. Furthermore, in II the structures
described by Eqs. (1), (3), (6) have be defined in an analogous way not only
for the Hamiltonian function $H(q,p)$ but for any observable $A(q,p)$. In this
way the most important relations of QT, namely the form of operators,
Schrödinger’s equation, eigenvalue equations, commutation relations,
expectation values, and Born’s rule have been derived in II. The simplicity of
the derivation in II is a consequence of the fact that mathematical structures
exist in phase space which are very similar to those of quantum theory; these
structures belong to the framework of PM and are “invisible” if CM is
considered as the classical counterpart of QT.

Let us extract now from this overview the general structure of HLLK. We start
from PM, which is a theory of ensembles of point particles with $2n+1$
independent variables $q,p,t$, and construct a theory of type QT, with $n+1$
independent variables $q,t$. Such a construction must contain at least two
essential steps, namely

  * •

a linearization, and

  * •

a projection from $\Omega$ to $\Sigma$.

Let us discuss these two requirements in more detail.

  * •

The property of _linearity_ for a fundamental law of physics (this means: a
law containing only fixed parameters like mass and universal constants like
Planck’s constant, the velocity of light, or the unit of charge) is a
necessary requirement if we want this law to describe a large number of well-
defined processes. This follows from the fact that almost all solutions of
nonlinear evolution equations become singular outside some (small) time
interval. We shall use the term “locally valid” to characterize this
restricted validity of solutions of nonlinear equations. On the other hand,
linearity implies that all solutions of the initial value problem exist not
only locally but on a global scale; this is of course a consequence of the
superposition principle, which facilitates enormously the construction of
solutions with prescribed initial values. We shall use the term “globally
valid” to characterized the almost unrestricted validity of solutions of
linear equations.

  * •

The requirement of _having space-time coordinates as independent variables_ is
also very reasonable. Space coordinates are more fundamental than momentum
coordinates; the latter can only be determined if a continuous set of the
former is available. All fundamental fields of physics represent properties
which vary as a function of space-time and not as a function of space-
momentum-time (if several particles are considered each one may have its own
space coordinates). There is of course an infinite number of coordinate
systems in $\Omega$ and $\Sigma$ , all of which are mathematically equivalent.
However, the systems $q,p,t$ in $\Omega$ and $q,t$ in $\Sigma$ are the
simplest and the only ones that allow intuitive access.

In the version of HLLK reported in I and II the linearization was performed
first, followed by a projection to $\Sigma$ as a second step. As a result, in
II we were able to derive the most important relations of QT very quickly.
There is, however, a disadvantage here because we lost the quasi-classical
theory in the course of this quick derivation. This failure may be responsible
for the fact that the phenomenon of _spin_ is not predicted by the version of
the HLLK reported in II. We know today that all structureless particles with
non-zero mass have spin one-half. The HLLK _is_ a (probabilistic) theory of
structureless (point) particles, and should therefore predict the phenomenon
of spin, in particular the universal number one-half.

In the present and the following work, to be referred to as IV klein:spin, we
carry out the transition from PM to QT in a more complete way by simply
reversing the order of the above two steps. Thus, we first carry out the
transition from $\Omega$ to $\Sigma$ \- in this way we are able to gain
insight into the structure of the momentum field - and perform afterwards the
linearization or randomization of the resulting equations. We restrict
ourselves to the study of a single observable, namely Hamilton’s function $H$.
Thus, in a first step, to be carried out in section 2, a momentum field
$M(q,t)$ is assigned to each point of $\Sigma$, at each instant of time $t$ .
In section 3 we show that $M(q,t)$ is conveniently described in terms of a
special kind of potentials originally introduced by Clebsch as a mathematical
tool of fluid mechanics clebsch:transformation . There is no reason to assume
that $M(q,t)$ is irrotational. In fact it turns out that its rotational part
is responsible for spin, as will be shown in detail in IV. However, in the
present work we cover only the very basics of the spin theory, as our primary
aim here is to understand the interface between classical physics and QT. The
essential properties of this interface as well as related questions of
interpretation become much more transparent when spin is neglected. Thus we
restrict ourselves in the present paper, despite its artificial nature, to an
irrotational momentum field, which is described by a single Clebsch potential
$S$. Using $\rho$ and $S$ as new dynamic variables we obtain a theory which
will be referred to as “quasi-quantal approximation” (QA) of PM. This theory
is closely related to the well-known quasi-classical approximation of the
Schrödinger equation for spin-less particles. In section 5 we carry out the
transition from QA to QT by means of two methods, a linearization and a
randomization. In section 6 questions of interpretation and the relation
between QT and the classical theory are discussed; we examine, in particular,
the role of Dirac’s “families of classical solutions” dirac:hamiltonian  as
seen from the point of view of the ensemble interpretation. In Section 7 we
discuss the relationship of the present theory to other recent theories based
on similar concepts.

##  2 Projection to configuration space

In this section our basic equations (1), (3), (6) will be projected to
configuration space. This first step of HLLK means defining appropriate
momentum fields which replace the independent variables $p$. The phase space
variables $\rho(q,p,t)$ and $S(q,p,t)$ will be replaced by appropriate
configuration space variables, which depend on $q,t$ only, and the equations
of motion for these new dynamical variables will be derived. The theory
obtained this way is not realized in nature but plays a very important role as
an intermediate step from PM to QT.

###  2.1 Projection of particle equations

We associate with each point $q$ of configuration space (at each instant of
time $t$) a particle momentum by replacing the variable $p$ by a $n-$component
momentum field $M$:

$p_{k}\rightarrow M_{k}(q,t)\mbox{.}$ (8)

Thus, $2n-$dimensional phase space is projected to a $n-$dimensional surface
parameterized by the configuration space coordinates $q_{k}$ mukunda:phase-
space . We may also say that we restrict our considerations from now on to a
$n$-dimensional subspace $\mathbb{M}$ of phase space, which is defined,
suppressing time for the moment, by
$\mathbb{M}=\left\\{(q,p)\in\Omega\,|\,p=M(q)\right\\}$. A large part of the
following considerations is devoted to a study of $\mathbb{M}$ and its various
transformed versions.

Of course, the momentum field $M_{k}(q,t)$ should be chosen in such a way that
the original structure of the canonical equations (1) is preserved as far as
possible. Following Rund rund:clebsch  and Kozlov kozlov:general , we
introduce fields $V(q,p,t)$, $v(q,t)$, $h(q,t)$, defined by

$\displaystyle V_{k}(q,p,t)=\frac{\partial H(q,p,t)}{\partial p_{k}}\mbox{,}$
(9) $\displaystyle v_{k}(q,t)=V_{k}\left(q,M(q,t),t\right)\mbox{,}$ (10)
$\displaystyle h(q,t)=H\left(q,M(q,t),t\right)\mbox{.}$ (11)

Then, given that the trajectory $q_{k}(t)$ is a solution of the first member
of (1),

$\dot{q}_{k}=v_{k}(q,t)\mbox{,}$ (12)

the corresponding momentum $p_{k}(t)=M_{k}(q(t),t)$ must be a solution of the
second member of (1). This requirement leads to the following equation for the
momentum field $M(q,t)$:

$\frac{\partial M_{i}(q,t)}{\partial t}+\left[\frac{\partial
M_{i}(q,t)}{\partial q_{l}}-\frac{\partial M_{l}(q,t)}{\partial
q_{i}}\right]v_{l}\left(q,t\right)=-\frac{\partial}{\partial
q_{i}}h\left(q,t\right)\mbox{.}$ (13)

This nonlinear partial differential equation (recall that $v$ and $h$ depend
on the unknown variables) will be referred to as canonical condition. Thus, we
obtain, as a result of the projection to configuration space, new equations
for the particle trajectories, namely $n$ partial differential equations (13)
for $M$ and $n$ ordinary differential equations (12) for the positions $q$.
These new equations replace (1).

These equations can still be used to calculate trajectories. One has to
perform the following steps:

  1. 1.

Assign an initial value $M_{0}(q)$, at $t=t_{0}$, to the momentum field
$M(q,t)$.

  2. 2.

Find a solution $M(q,t)$ of (13) for this initial value.

  3. 3.

Solve the ordinary differential equations (12) to obtain the trajectory
coordinates $q(t)$.

  4. 4.

Use these coordinates to obtain the corresponding momenta $p(t)=M(q(t),t)$.

While trajectories still exist they are only locally valid as a consequence of
the nonlinearity of the dynamical equations (13). A more detailed dicussion of
the modifications the concept of trajectories undergoes, when transforming
from phase space to configuration space, will be given in section 4.

Momentum fields also appear in the theory of canonical transformations in
phase space. However, these are by definition irrotational, in contrast to the
fields $M(q,t)$ above. The difference between the two types of fields 8 is
discussed in the appendix.

For $n=3$ Eq. (13) agrees with the evolution equation for the velocity field
of a barotropic ideal fluid; in this context it is sometimes referred to as
Lamb equation kozlov:general . In the present work we will encounter several
other formal similarities between our probabilistic theory and fluid
mechanics. Despite such similarities, the different physical meaning of the
basic variables should be born in mind. In particular, in fluid mechanics the
density $\rho(q,t)$ is a directly measurable “material” quantity. In the
present theory $\rho(q,t)$ is a rather abstract quantity, a probability
density which may, for example, be measured by performing a large number of
individual experiments with arbitrary large time-intervals in between.

###  2.2 Projection of ensemble equations

In this section we project the Liouville equation (3) and the action equation
(6) to the subspace defined by a solution $M(q,t)$ of the canonical condition
(13). The resulting partial differential equations represent an essential step
on our way from PM to QT.

####  2.2.1 Projection of Liouville equation

The projection of the Liouville equation to configuration space is a standard
process which is, in a three-dimensional fluid-dynamical context, sometimes
referred to as “hydrodynamical substitution”. We rewrite (3) as a continuity
equation in phase space:

$\frac{\partial\rho(q,p,t)}{\partial t}+\frac{\partial}{\partial
q_{k}}\rho(q,p,t)\frac{\partial H(q,p,t)}{\partial
p_{k}}-\frac{\partial}{\partial p_{k}}\rho(q,p,t)\frac{\partial
H(q,p,t)}{\partial q_{k}}=0\mbox{.}$ (14)

In order to make sure that $\rho(q,p,t)$ vanishes everywhere except on the
surface $p=M(q,t)$ we write $\rho(q,p,t)=\rho(q,t)\delta(p-M(q,t))$, where
$\rho(q,t)$ is the new probability density in configuration-space, and
$\delta$ is the $n$-dimensional delta function. Using this expression, the
integration with respect to the momentum variables may be performed, assuming
non-singular behavior of the relevant quantities. Then the contribution of the
last term on the l.h.s. of (14) vanishes and we obtain the desired continuity
equation in configuration-space

$\frac{\partial\rho(q,t)}{\partial t}+\frac{\partial}{\partial
q_{k}}\rho(q,t)V_{k}(q,M(q,t),t)=0\mbox{,}$ (15)

which may be interpreted as a probabilistic conservation law with $\rho
v(q,t)$ as probability current. The same result may also be obtained from the
reasonable requirement that the measure defined by $\rho(q,t)$ is invariant
under the flow defined by (12).

####  2.2.2 Projection of action equation

It is also necessary to consider the projection of the action equation (6) to
configuration space, as the (classically redundant) quantity $S(q,p,t)$ will
play an important role in the transition to QT. We define a configuration
space field $s(q,t)$ according to

$s(q,t)=S(q,M(q,t),t)\mbox{.}$ (16)

We omit an index $M$ but bear in mind that this field depends by definition on
the momentum field $M$. We may replace in (6) the partial derivatives of
$S(q,p,t)$ with respect to $q$ and $t$ by partial derivatives of $s(q,t)$ with
respect to the same variables. After performing several such manipulations,
Eq. (6) takes the form

$\frac{\partial s}{\partial t}-\frac{\partial S}{\partial
p_{k}}\Bigg{|}_{p=M}\Bigg{[}\frac{\partial M_{k}}{\partial
t}+\left(\frac{\partial M_{k}}{\partial q_{i}}-\frac{\partial M_{i}}{\partial
q_{k}}\right)v_{i}+\frac{\partial}{\partial
q_{k}}h\Bigg{]}+v_{k}\left(\frac{\partial s}{\partial
q_{k}}-M_{k}\right)+h=0\mbox{.}$ (17)

As $M_{k}(q,t)$ is a solution of (13) the bracket vanishes and Eq. (17) takes
the form

$\frac{\partial s}{\partial t}+v_{k}\left[\frac{\partial s}{\partial
q_{k}}-M_{k}(q,t)\right]+h=0\mbox{.}$ (18)

This is the projection of Eq. (6) to configuration space derived in a
straightforward way. There is an alternative derivation of this equation which
makes its physical meaning more transparent. Starting from the phase space
action (5), the derivation of (18) was performed in two steps. The first step
was a Lagrangian to Eulerian transition in phase space leading to Eq. (6). The
second step, performed above, was a projection to configuration space. It
should be possible to change the order of these steps. If we start from (5)
and project this quantity to configuration space, we obtain the “Lagrangian”
quantity

$S_{M}(q_{0},t)=\int_{t_{0}}^{t}\,\mathrm{d}t^{\prime}\left[M_{i}(q(q_{0},t^{\prime}),t^{\prime})v_{i}(q(q_{0},t^{\prime}),t^{\prime})-h(q(q_{0},t^{\prime}),t^{\prime})\right]\mbox{.}$
(19)

Performing now a Lagrangian to Eulerian transition (see II for details) for
the action integral in configuration space (19) we obtain

$\frac{\mathrm{D}s}{\mathrm{D}t}:=\frac{\partial s}{\partial
t}+v_{k}\frac{\partial s}{\partial q_{k}}=M_{k}(q,t)v_{k}-h\mbox{,}$ (20)

in agreement with (18). The left hand side of (20) is the total derivative
along the particle trajectories defined by (12) and the right-hand-side is the
projection of the Lagrangian to the subspace $\mathbb{M}$.

The projection of the canonical equations (1) to configuration space has led
us to Eqs. (12) and (13). These relations, together with the continuity
equation (15), define a theory which suffers from two shortcomings. The first,
mentioned already in section 2.1, is given by the fact that trajectories are
only locally well-defined. The second is given by the fact that the $n$
components $M_{1}(q,t),..,M_{n}(q,t)$ of the momentum field may not be
_functionally independent_ from each other.

##  3 Introducing potentials

In this section we show that the next step in the development of the HLLK has
to be a replacement of the dynamical variables $M_{i}$ by potentials. First we
consider arbitrary particle numbers $N$, but later specialize to $N=1$ and to
an irrotational momentum field, which may be described by a single scalar
potential $S$. The general (correct) treatment of the case $N=1$ requires $3$
independent potentials and is will be reported in IV.

If we consider a system of $N$ free particles in phase space, the $3N$
components of the particle momentum are not subject to any restrictions. It
seems unlikely that this freedom will be maintained for the $3N$ fields
$M_{k}$ after projection to $\mathbb{M}$, . To examine this question more
closely, we need a mathematical theory that tells us how to eliminate
redundancy from a set of fields. This question was studied by the
mathematician Pfaff in 1815 (see hawkins:frobenius  for references and
background information). Pfaff’s problem may be formulated as follows:

> Given $n$ fields $M_{1}(q,t),...,M_{n}(q,t)$ not necessarily functionally
> independent from each other. What is the minimal number $L$ of fields
> $F_{1}(q,t),..,F_{L}(q,t)$ required to derive the $M_{i}(q,t)$, and which
> relations exist between the fields $M_{i}(q,t)$ and the ”potentials”
> $F_{i}(q,t)$ ?

This problem is equivalent to a ”complete reduction” of the $1-$form, or
Pfaffian form $\omega=M_{i}(q)dq_{i}$ i.e. to a transformation of the Pfaffian
form into an equivalent form with a minimal number of terms. The number $L$ is
referred to as ”class” of the vector field $M=\\{M_{1},...,M_{n}\\}$. A
detailed treatment of Pfaff’s problem may be found, e.g. in Caratheodory’s
book caratheodory:calculus_I .

The number $L$ may be determined (by finding he rank of a certain matrix
caratheodory:calculus_I ) if the functional form of $M(q,t)$ is known. As
$M(q,t)$ is not known to us we cannot take advantage of this mathematical
result. Rather, we have to treat $L$, or the integer $m$ introduced below, as
a kind of ”adjustable parameter”. The part of Pfaff’s solution which is most
interesting for us is the form of the expansion. For momentum fields of odd
class there is an integer $m$ given by $L=2m+1$. In this case $M$ may be
written in the form

$M_{k}(q,t)=\frac{\partial S(q,t)}{\partial
q_{k}}+P_{\alpha}(q,t)\frac{\partial Q_{\alpha}(q,t)}{\partial q_{k}}\mbox{,}$
(21)

where $S(q,t),\,P_{\alpha}(q,t),\,Q_{\alpha}(q,t)$ are $2m+1$ independent
functions of $q_{1},..,q_{n},t$ (Greek indices $\alpha,\,\beta,..$ run from
$1$ to $m$ and double occurrence of these indices entails a summation from $1$
to $m$). The functions $S(q,t),\,P_{\alpha}(q,t),\,Q_{\alpha}(q,t)$ are
referred to as Clebsch potentials clebsch:transformation . For fields of even
class there is an integer $m$ given by $L=2m$ and $M$ may be written in the
form

$M_{k}(q,t)=P_{\alpha}(q,t)\frac{\partial Q_{\alpha}(q,t)}{\partial
q_{k}}\mbox{,}$ (22)

with $m$ pairs of functions $P_{\alpha}(q,t),\,Q_{\alpha}(q,t)$. We should
note that the above expansions are only locally valid. This is a serious
limitation for many tasks in physics but not for the present one. The reason
is that the global validity of the present theory has already been lost (by
projecting to configuration space) _before_ the elimination of redundancy is
implemented.

Two questions now arise, namely (i) which one of the above two expansions is
suitable for our purposes, and (ii) is there really redundancy or may we
continue to use $M_{k}$ as our dynamical variables ? Both of these questions
can be answered in a simple way, just by considering the form of the
expansions (21) and (22). We use the fact that the concept of constraints
makes only sense in macroscopic physics. This means we can identify the
coordinates of the configuration space with the Cartesian coordinates of one
or several particles. In other words, if we have $N$ particles then we have
$n=3N$ spatial degrees of freedom, and a change in $n$ is always a consequence
of a change in $N$ ; in particular $n=3,6,9,12,...$ if $N=1,2,3,4,..$.

From the fact that the structure of our (future) theory can not depend on the
number of particles, we conclude first of all that only one of the two
representations (21) and (22) can be true for all $N$. Then, only (21)
survives. In fact, the representation (22) leads to an even number of fields
and fails to describe the single particle case, $N=1$, which requires three
fields. Thus, in the three-dimensional space of nonrelativistic physics, which
is considered throughout the present study, we have to use the odd
representation, Eq. (21), with non-vanishing Clebsch potential $S(q,t)$ . On
the other hand, for the four-dimensional space-time of relativistic physics
the even representation (22) with $S(q,t)=0$ is appropriate, as has been shown
by Rund rund:clebsch , rund:clebsch_relativistic .

To answer the second question we first assume that there is no redundancy,
that is, $L=n=3N$. Since $L$ is alternately odd and even with increasing N,
this assumption leads to a contradiction. Thus, there must be redundancy, at
least for some $N$ the number $L$ of independent components of $M$ will be
less than $n$ . This leads to the important conclusion that we have to use
_potentials_ as dynamic variables instead of momentum fields. We have derived
a fundamental structural property of QT, whose origin is seldom asked for.

We know now that only (21) can be used, i.e. the class is given by $L=2m+1$.
Secondly, we know that $L$ cannot be equal to $n$ for all $n$; i.e. a relation
of the form $L=n-k$, where $1\leq k\leq n-1$, will exist at least for some $n$
($k$ is the number of functional relations between the $M_{i}$) . Combining
both relations we have

$n-k=2m+1\mbox{.}$ (23)

Since $n,k,m$ are all integers we have a countable number of solutions for $k$
and $m$. There is a solution of maximal redundancy for arbitrary $n$ which is
given by $m=0$ and $k=n-1$. For $N=1$ there is no redundancy, the relation
$3-k=2m+1$ can only be true for $k=0$ and $m=1$ (excluding here the case of
maximal redundancy). For $N=2$ we have necessarily redundancy; the relation
$6-k=2m+1$ can only be true for nonzero $k$; the possible pairs $(k,m)$ are
from the set $\\{(1,2),(3,1)\\}$. For $N=3$ the possible pairs $(k,m)$ are
from the set $\\{(0,4),(2,3),(4,2),(6,1)\\}$, etc. Inspecting the various
possible solutions for each $N$ one finds that only a single ”regular”
solution for $k$ and $m$ exists, namely the linear solution $k=N-1,\,m=N$.
Only for this ”regular” solution the increase in the total number $2m+2$ of
dynamical variables (taking $\rho$ into account) is the same for all $N$,
namely $2$. For all other solutions this number changes in an irregular way;
examples for such solutions (for $N=1,2,3,4,..$) are $2m+2=4,4,6,4,..$ or
$2m+2=4,6,4,8,..$. These irregular solutions, are, although not impossible
from a logical point of view, obviously unacceptable, and can be excluded. Let
us summarize the results of our analysis for $n=3N$.

  * •

To expand the momentum fields $M_{1},..,M_{n}$ only the odd representation,
Eq. (21) can be used, i.e. $L=2m+1$. As regards the number $m$ two
possibilities exist:

  * •

Either the case of maximal redundancy is realized, i.e. $m=0$ for all $N$. In
this case the momentum field $M$ is the gradient of a scalar function
$S(q_{1},..,q_{n})$.

  * •

Or $m=N$ and $N-1$ relations exist among the $3N$ momentum field components.
In this case $M$ can be expanded in $2m+1$ Clebsch potentials, where $m=N$.
Thus, a $N-$particle ensemble is described by a function $S$ and $N$ pairs of
dynamic variables $P_{\alpha},\,Q_{\alpha}$.

The most important conclusion of this section is that the momentum fields
_have to be replaced by potentials_. This conclusion is only correct for
theories with a variable number of particles. For theories like fluid
mechanics, where the number of degrees of freedom is always $3$ (corresponding
to $N=1$) , both the three components of the momentum (the velocity) and the
three Clebsch potentials may equivalently be used as dynamic variables. For
$N=1$ it is also possible to rewrite QT with the help of both types of
variables takabayasi:formulation ; holland:quantum . However, for our present
task of reconstructing QT for arbitrary $N$, only potentials are appropriate.

In the present work we restrict ourselves to the case $m=0$, where only a
single potential $S$ occurs. The case $m=1$, which leads to the spin of a
single particle, will be treated in IV.

###  3.1 Irrotational momentum fields

It is useful to introduce a quantity which characterizes the purely rotational
(or vortical) part of the momentum field $M$. An appropriate quantity is the
vorticity tensor $\Omega_{ik}$, defined by

$\Omega_{ik}(q,t):=\frac{\partial M_{k}(q,t)}{\partial q_{i}}-\frac{\partial
M_{i}(q,t)}{\partial q_{k}}\mbox{.}$ (24)

The evolution equation for $\Omega_{ik}$, which may be derived from the
canonical condition (13), expresses the fact that the Lie derivative of
$\Omega_{ik}$ with respect to the velocity field $v_{l}$ vanishes. Roughly
speaking the tensor $\Omega_{ik}$ moves with the flow defined by the particle
equations of motion; in particular, $\Omega_{lj}(q,0)=0$ implies
$\Omega_{ik}(q,t)=0$ for all future times rund:clebsch .

A momentum field obeying $\Omega_{ik}=0$ is referred to as irrotational
momentum field. It corresponds to the case $m=0$ and is, according to (21),
given by the $n$-dimensional gradient of a single Clebsch potential $S$,

$M_{k}(q,t)=\frac{\partial S(q,t)}{\partial q_{k}}\mbox{.}$ (25)

For such a field there are $n-1$ relations between the $M_{i}$ no matter how
large the number of particles. A physical reason for such strong dependencies
between the momentum components cannot be seen. In particular, it would be
reasonable to assume that for $N=1$ the number of independent components is
not $1$ but equal to the maximal possible number $3$, as there is no
restriction in principle for momenta in three-dimensional space. In spite of
this unsatisfactory situation, which will be discussed further in IV, the case
of the irrational momentum field is important because it leads to
Schrödinger’s equation.

For irrotational momentum fields the change of dynamical variables from
$M_{i}$ to $S$ is very simple. As a consequence of Eq. (25) the bracket in the
canonical condition (13) vanishes. The latter reduces to the Hamilton-Jacobi
equation if an unspecified function $f(t)$ is eliminated by means of a
redefinition of $S(q,t)$. If we add the particle equation (12) and the
continuity equation (15), we obtain the three basic differential equations

$\displaystyle\dot{q}_{k}=V_{k}(q,\frac{\partial S(q,t)}{\partial
q},t)\mbox{,}$ (26) $\displaystyle\frac{\partial S}{\partial
t}+H(q,\frac{\partial S}{\partial q},t)=0\mbox{,}$ (27)
$\displaystyle\frac{\partial\rho(q,t)}{\partial t}+\frac{\partial}{\partial
q_{k}}\rho(q,t)V_{k}(q,\frac{\partial S}{\partial q},t)=0\mbox{.}$ (28)

Thus, the irrotational case may be described by only two dynamical field
variables, $S(q,t)$ and $\rho(q,t)$ no matter how large $n$. These equations
have to be solved as initial value problems; the steps required to calculate
trajectories are similar to the ones listed in section 2.1.

The physical interpretation of the new dynamical variable $S(q,t)$ may easily
be found by comparing (27) with the projected action equation in the form
(18). The variable $s(q,t)$ is defined in terms of the phase space action
$S(q,p,t)$ , which is denoted by the same symbol $S$ but depends on $2n+1$
variables, as follows:

$s(q,t)=S\left(q,\frac{\partial S}{\partial q},t\right)\mbox{.}$ (29)

Inserting (25) in the projected action equation (18) and using (27) shows that
$S(q,t)$ differs from $s(q,t)$ by a function which is constant along the
trajectories defined by the velocity field $v(q,t)=V(q,\frac{\partial
S}{\partial q},t)$:

$\frac{\mathrm{D}}{\mathrm{D}t}(s-S)=0\mbox{.}$ (30)

Thus, the potential $S(q,t)$ is closely related to the projected action
integral $s(q,t)$; the simplest solution of (30) is the trivial solution
$s-S=0$.

Equations (27) and (28) represent the basic equations of the ”quasi-quantal
approximation” (QA) to PM, because (27) and (28) have been obtained here, in
the framework of HLLK, as a first step to QT starting from classical physics.
A linearization of these equations leads to Schrödinger’s equation, as will be
shown in section 5. The same equations (27) and (28) constitute also the basis
of the so-called quasi-classical approximation of QT. Although both theories
share the same basic equations, they differ in the additional conditions that
also decisively determine the form of the solutions. In the QA the
differential equations are to be solved with given initial values, as is the
case in QT. In the quasi-classical approximation the solutions are mostly
constructed with the help of complete solutions of the Hamilton-Jacobi
equation using a phase space method that goes back to van Vleck
vanvleck:correspondence ; this method will be discussed in more detail in
section 6.

##  4 The structure of the quasi-quantal approximation

The basic equations (26), (27), (28) of the QA represent the result of the
first of the two essential steps of the HLLK. They describe the simplest
version of an ensemble theory projected onto configuration space $\Sigma$. Let
us ask now whether or not this theory makes sense from a physical point of
view. If not, we ask what further steps are required to construct a physically
meaningful theory in $\Sigma$.

Our starting point was the physically meaningful theory PM of statistical
ensembles in phase space $\Omega$. The fact that this theory makes physical
sense is reflected in its mathematical structure. The Cauchy problem for
systems of first-order ordinary differential equations is well-defined for
rather weak mathematical conditions arnold:ordinary . This implies that
(almost) all solutions of PM are globally valid and that, consequently, these
solutions define an invertible map

$l_{t}:\Omega\rightarrow\Omega,\;\;\;q=Q(t,q_{0},p_{0}),\;p=P(t,q_{0},p_{0})\mbox{,}$
(31)

of phase space onto itself. Actually, a continuous family of maps (31),
parametrized by the time $t$, exists corresponding to the fact that, at each
instant of time, each point of phase space belongs to a single trajectory. The
existence of this family implies the validity of the Liouville equation (3)
which is, like all fundamental laws of nature, a _linear_ differential
equation. Linearity is a necessary and sufficient condition for ”global
validity” of a physical law as mentioned already in section 1. The global
validity of the Liouville equation and the global validity of the canonical
equations are closely related to each other.

During the transition from PM to QA the manifold of independend variables has
been reduced from $\Omega$ to $\Sigma$. An invertible map, analogous to (31)
but now for the reduced manifold of QA, would be given by

$l_{t}^{c}:\Sigma\rightarrow\Sigma,\;\;\;q=Q^{c}(q_{0},t)\mbox{.}$ (32)

If such a map existed, it would imply a globally valid, linear “Liouville
equation” in configuration space, which would then be the universal law, we
were looking for. Such a map does not exist, as mentioned already. From the
solution of the initial-value problem of the Hamilton-Jacobi equation (27) the
associated momentum field $M(q,t)$, with initial value $M_{0}(q)$, may be
calculated with the help of relation (25). Then, particle trajectories $q(t)$,
$p(t)=M(q(t),t)$ on the (irrotational) momentum surface $\mathbb{M}$ are
obtained as solutions of Eqs. (26). These trajectories are locally well-
defined, in a certain vicinity of $t_{0}$, but the momentum surface
$\mathbb{M}$ defined by $M_{0}(q)$ immediately starts deforming during its
movement through phase space. As a consequence, the map from configuration
space to the (deformed) momentum hypersurface $\mathbb{M}$ becomes multivalued
and the concept of trajectories breaks down in QA. This breakdown is closely
related to the nonlinearity of Eq. (27). Clearly, we cannot cover the whole of
phase-space by means of a single momentum surface (we would need an infinite
number as in the ”complete solutions” of the theory of canonical
transformations).

The breakdown of the concept of particle trajectories is not related to the
introduction of potentials; this also happens in the earlier theory, with
hydrodynamic variables, reported in section 2. However, in the QA, as derived
from QT, the singularities that occur when the concept of trajectories
collapses have been carefully studied, see e.g. rosenbloom:hamilton-jacobi ;
berry.balazs:evolution ; delos:catastrophes . Thus, the concept of
trajectories - which represents the ”deterministic part” of our probabilistic
theory - is still present in the QA but has been seriously ”weakened”, due to
its loss of global validity. This is the price we have to pay for the enormous
reduction in the number orf degrees of freedom brought about by the
projection.

This discussion leads us to suspect that we cannot construct a globally valid
probabilistic theory for particles in configuration space, while keeping at
the same time the concept of particle trajectories intact. This raises several
fascinating philosophical questions (concerning ”reality”, ”completeness”, and
others) which lie outside of the scope of the present paper. A probabilistic
particle theory makes predictions of a probabilistic nature (predictions to be
verified by means of statistical ensembles) about particles. This definition,
which is based on the physical principle of verification by observation, does
not require that deterministic laws for particle trajectories are part of a
probabilistic theory. Theories of this kind, where not only the initial values
are uncertain (as in PM) but also the laws of movement themselves are
“uncertain” or “unknown” (or possibly ”nonexistent”, whatever that may mean
exactly) have been called “Type 3 theories” in a recent work of the present
author klein:statistical .

This discussion also leads us to suspect that QT is a globally valid
modification of the QA. In order to test if this working hypothesis is true we
have to _linearize_ the partial differential equations (27), (28). This will
be done in section 5; an alternative way of deriving the resulting linear
equations will also be described in section 5. There is no other way to
restore the former global validity (of PM in phase space) in configuration
space. We have to accept that this step will completely eliminate the
deterministic concept of particle trajectories.

##  5 Transition to quantum theory

In this section the second essential step of the HLLK is carried out, which
leads from the QA to the final QT. Two different variants of this step are
reported which lead to the same result, but illuminate the nature of QT from
two different angles. Furthermore, we examine the relationship between
Kelvin’s theorem and the condition for uniqueness of the wave function.

###  5.1 Transition by linearization

The linearization of the spinless version of the QA, which we consider in this
work, can be carried out in a relatively simple way. As a first step we
introduce a quasiquantal wave function which is defined in an analogous way as
in II:

$\psi=\sqrt{\rho}\mathrm{e}^{\frac{\imath}{\hbar}S}\mbox{.}$ (33)

However, the quantities and $S$ and $\rho$ satisfy now (27) and (28). In order
to be able to carry out a comparison with the original equations (6) and (3)
in phase space, we use the explicit functional form of the Hamilton function
$H(q,p)=p_{k}p_{k}/2m+V(q)$ and write equations (27) and (28) in the following
form:

$\displaystyle D_{t}S=\bar{L}\mbox{,}$ (34)
$\displaystyle\hat{\mathbb{L}}_{\frac{1}{2}}\rho^{\frac{1}{2}}=0\mbox{,}$ (35)

where the Lagrangian $\bar{L}$ and the operator
$\hat{\mathbb{L}}_{\frac{1}{2}}$ are defined by
$\bar{L}=\frac{1}{2}v_{k}M_{k}-V(q)$ and
$\hat{\mathbb{L}}_{\frac{1}{2}}=\partial_{t}+v_{k}\partial_{k}+\frac{1}{2}(\partial_{k}v_{k})$.
We introduced here the abbreviations $\partial_{t},\;\partial_{k}$ for the
derivatives with respect to $t,\;q_{k}$ and the abbreviation $D_{t}$ for the
total derivative as defined in Eq. (20). Comparing (34), (35) with the phase
space relations (6) and (3) we find that the following properties that are
valid in phase space are no longer valid in configuration space: (1) the
divergence of the velocity field vanishes, (2) the probability density moves
with the flow, and (3) the operator $v_{k}\partial_{k}$ multiplied by the
imaginary unit is self-adjoint.

In the next step we evaluate the expression
$-\frac{\hbar}{\imath}\hat{\mathbb{L}}_{\frac{1}{2}}\psi$. Using the basic
Eqs. (34), (35) we obtain the relation
$(-\frac{\hbar}{\imath}\hat{\mathbb{L}}_{\frac{1}{2}}+\bar{L})\psi=0$ which is
equivalent to (27) and (28), and has some similarity to the generalized
Koopmann-von Neumann equation derived in I. We have to transform this
equation, which takes the form

$\left[-\frac{\hbar}{\imath}\partial_{t}-\frac{\hbar}{\imath}v_{k}\partial_{k}-\frac{1}{2}\frac{\hbar}{\imath}(\partial_{k}v_{k})+\frac{1}{2}v_{k}(\partial_{k}S)-V(q)\right]\psi=0\mbox{,}$
(36)

when written out, into an equation (not necessarily linear) for the complex
variable $\psi$ . According to the definition of $\psi$ it is possible to
replace the derivatives of $S$ by the derivatives of $\psi$ and $\rho$ . This
may be done with the help of the relation

$(\partial_{k}S)\psi=\frac{\hbar}{\imath}\left[\partial_{k}\psi-\frac{1}{2\rho}(\partial_{k}\rho)\psi\right]\mbox{,}$
(37)

whose validity is easily verified with the help of the definition (33). We
replace the fourth term of (36) with the help of (37) and write the third term
of (36) in the form

$-\frac{1}{2}\frac{\hbar}{\imath}(\partial_{k}v_{k})\psi=-\frac{1}{2}\frac{\hbar}{\imath}\partial_{k}v_{k}\psi+\frac{1}{2}\frac{\hbar}{\imath}v_{k}\partial_{k}\psi\mbox{.}$
(38)

Using (37) again, the equation of motion (36) takes the form

$\left[-\frac{\hbar}{\imath}\partial_{t}+\frac{\hbar^{2}}{2m}\left(\partial_{k}+\frac{1}{2\rho}(\partial_{k}\rho)\right)\left(\partial_{k}-\frac{1}{2\rho}(\partial_{k}\rho)\right)-V(q)\right]\psi=0\mbox{.}$
(39)

A few further elementary manipulations finally lead to the nonlinear
differential equation

$\left[-\frac{\hbar}{\imath}\partial_{t}+\hat{H}\right]\psi+\frac{\hbar^{2}}{2m}\frac{1}{\sqrt{\rho}}(\partial_{k}\partial_{k}\sqrt{\rho})\psi=0\mbox{,}$
(40)

where $\hat{H}=\frac{1}{2m}\hat{p}_{k}\hat{p}_{k}+V(q)$,
$\hat{p}_{k}=\frac{\hbar}{\imath}\partial_{k}$, and $\rho=\psi^{*}\psi$. The
“quantum-like” differential equation (40) for the complex variable $\psi$ is
essentially equivalent to the relations (27) and (28), apart from certain
uniqueness conditions wallstrom:inequivalence  for $\psi$ which are not
important in the present context (see however section 5.3 ). The classical
equation (40) has been reported several times before in the literature (see
e.g. schiller:quasiclassical ; rosen:classical_quantum ; klein:statistical ;
schleich_et_al:schroedinger ), what is new here is the derivation from first
principles.

Two steps have always to be carried out in the HLLK, a linearization and a
projection onto configuration space. If the vortical component of the momentum
field is neglected, as is done here, the order of these two steps does not
matter. In order to illustrate this point it is instructive to compare the
quasi-classical equation (40) with the quantum-like classical (phase space)
equation

$\Bigg{[}\frac{\hbar}{\imath}\frac{\partial}{\partial
t}-\frac{\hbar}{\imath}\frac{\partial H}{\partial
q_{k}}\frac{\partial}{\partial p_{k}}+\frac{\partial H}{\partial
p_{k}}\left(\frac{\hbar}{\imath}\frac{\partial}{\partial
q_{k}}-p_{k}\right)+H\Bigg{]}\psi^{(c)}=0\mbox{.}$ (41)

This extension of the Koopman-von Neumann equation koopman:hamiltonian  is
useful when studying the relation between classical physics and QT (see I,II)
or when investigating the problem of quantum-classical coupling
bondar_ea:koopman , majarres:projective . The classical wave function
$\psi^{(c)}$ is defined as in (33) but with $\rho$ and $S$ depending on
$q,\,p,\,t$. The classical equation (41) is the result of a suitable
linearization in phase space, which was carried out in I as a first step. The
second step in I, the projection onto the configuration space was realized by
implementing the quantization rules

$\frac{\partial}{\partial
p_{k}}=0,\;\;\;p_{k}=\frac{\hbar}{\imath}\frac{\partial}{\partial
q_{k}}\mbox{}$ (42)

which may easily be read off from (41). On the other hand, the quasi-classical
equation (40) is obtained by performing a suitable projection onto
configuration space in a first step. Roughly speaking, this first step
corresponds to the first of the above quantization rules. The linearization is
now the second step; it may be carried out very easily by omitting the
nonlinear term in (40). The second, more familiar quantization rule in (42) is
described by equation (37), where the second term on the right-hand side has
to be omitted as a consequence of the linearization. The final result is the
same in both cases, namely Schrödinger’s equation

$\Bigg{[}\frac{\hbar}{\imath}\frac{\partial}{\partial
t}+H\left(q,\frac{\hbar}{\imath}\frac{\partial}{\partial
q_{k}}\right)\Bigg{]}\psi(q,t)=0\mbox{.}$ (43)

In I we referred to (41) as classical counterpart of Schrödinger’s equation.
In the same sense, we might now refer to equation (40) as quasi-classical
counterpart of Schrödinger’s equation.

An interesting question is how the basic equations of the QA are modified in
the transition to QT. A brief calculation, first performed by Madelung
madelung:quantentheorie , shows that the continuity equation (28) remains
unchanged while the Hamilton-Jacobi equation (27) takes the following form:

$\frac{\partial S}{\partial t}+H(q,\frac{\partial S}{\partial
q})=\frac{\hbar^{2}}{2m}\frac{1}{\sqrt{\rho}}(\partial_{k}\partial_{k}\sqrt{\rho})\mbox{.}$
(44)

Thus, the transition from QA to QT leads to an additional term
$T_{Q}=(\hbar^{2}/(2m\sqrt{\rho}))(\partial_{k}\partial_{k}\sqrt{\rho})$ on
the right-hand side of (27). Due to its derivation, this term is a quantum
correction to the kinetic energy, as could have been guessed from the form of
its prefactor. Clearly, the term $T_{Q}$ introduces a coupling between the
action variable $S$ and the probability density $\rho$. What does that mean
exactly?

Let us recall that in the QA the action variable $S$ defines a particle
momentum at every point of configuration space. The solutions $S$ of the
Hamilton-Jacobi equation define, in combination with the particle equation
(26), particle trajectories - even if these trajectories are only locally
well-defined as discussed in section 4. Clearly, the coupling term $T_{Q}$
destroys this quasiclassical concept of a local particle momentum. Let us
recall at this point that equations (28) and (44) are to be solved as an
initial value problem. From the very beginning, all particle trajectories and
momenta are “mixed together”, depending on the specification of $S$ and $\rho$
at time $t=0$. As a result, the particle equation (26) looses its meaning in
QT. Only the two partial differential equations (44) and (28) survive the
transition to QT. These two relations, which are equivalent to Schrödinger’s
equation (43), determine the time-development of an object, which may be
described as a new kind of _ensemble where no separation of deterministic and
probabilistic properties is possible_.

Using the nomenclature of reference klein:statistical , we may say that,
starting from PM and arriving at QT, we performed a transition from a type 2
theory to a type 3 theory. In a type 2 theory there are deterministic laws for
trajectories, while the initial conditions must be described by probabilities
klein:statistical . In a type 3 theory only probabilities and expectation
values may be obtained, which, however, may be of an unusual form. The
peculiar (operator) form of the observables In QT indicates that a clear
separation, in a classical sense, between “observables” and “states” is no
longer possible.

In summary, it can be said that a randomization took place during the
transition from PM to QT, and that the remaining elements of determinism,
still present in PM, were eliminated. In the forthcoming paper IV it will be
shown that this basic structure is retained in the framework of a more
realistic description of the momentum field.

###  5.2 Transition by randomization

The above derivation of Schrödinger’s equation is based on the idea that a
fundamental law of nature must permit a large variety of solutions. This leads
to the requirement of linearity. The resulting theory is purely probabilistic
- without deterministic laws for particle trajectories. In this sense, the
physical effect of the formal process of linearization may be interpreted as a
randomization. In this section we try to understand the transition to the new
probabilistic theory in more detail. The following developments represent an
improved version of earlier work klein:statistical ; klein:nonrelativistic  of
the present author.

Our task is to construct a theory in which the deterministic equation (26) is
no longer included, but which is otherwise as similar as possible to the basic
equations (27) and (28) of QA. Therefore, we continue to use $\rho$ and $S$ as
our basic variables. Furthermore, we assume that the fundamental conservation
law of probability (28) remains unchanged. Another reasonable assumption is
that the particle equations of motion (1) “survive” in a statistical sense,
i.e. hold true not for particle variables but for the corresponding
expectation values,

$\frac{\mathrm{d}}{\mathrm{d}t}\overline{q_{k}}=\frac{\overline{p_{k}}}{m},\;\;\;\;\;\frac{\mathrm{d}}{\mathrm{d}t}\overline{p_{k}}=\overline{F_{k}}\mbox{.}$
(45)

Here, the expectation values in configuration space
$\overline{q_{k}},\,\overline{p_{k}},\,\overline{F_{k}}$ are defined as
follows:

$\displaystyle\overline{q_{k}}=\int\mathrm{d}q\,\rho(q,\,t)\,q_{k},$ (46)
$\displaystyle\overline{p_{k}}=\int\mathrm{d}q\,\rho(q,\,t)\,\frac{\partial
S}{\partial q_{k}},$ (47)
$\displaystyle\overline{F_{k}}=-\int\mathrm{d}q\,\rho(q,\,t)\,\frac{\partial
V}{\partial q_{k}}\mbox{.}$ (48)

Using the second of the Ehrenfest-like relations (45) and the continuity
equation (28) we derive the following integrodifferential equation:

$\int\mathrm{d}q\frac{\partial\rho}{\partial q_{k}}\left[\frac{\partial
S}{\partial t}+\frac{1}{2m}\sum_{j}\left(\frac{\partial S}{\partial
q_{j}}\right)^{2}+V\right]=0\mbox{.}$ (49)

In accordance with our general strategy of constructing a theory that is as
similar as possible to Eqs. (27) and (28) we try to replace the
integrodifferential equation (49) with a differential equation of standard
form

$\bar{L}(q,t)-L_{0}=0\mbox{,}$ (50)

which differs from (27) only by an additional term $L_{0}$ which may depend,
in principle, on the variables $S$ and $\rho$ as well as their derivatives.
The quantity $\bar{L}$ in (50) is an abbreviation for the square bracket in
Eq. (49),

$\bar{L}:=\frac{\partial S}{\partial
t}+\frac{1}{2m}\sum_{j}\left(\frac{\partial S}{\partial
q_{j}}\right)^{2}+V\mbox{.}$ (51)

We see that the statistical condition (45) is not sufficient to determine the
form of the basic equations. The unknown function $L_{0}$ must obviously [see
Eq. (49)] obey the constraint

$\int\mathrm{d}q\frac{\partial\rho}{\partial q_{k}}L_{0}=0\mbox{,}$ (52)

which is not sufficient to determine its functional form. Further constraints
of a statistical nature are required to fix $L_{0}$.

A statistical constraint that is in agreement with our general strategy is the
following. We require that the space-time average of the difference between
$\bar{L}$ and $L_{0}$ is minimal. This leads to the variational problem

$\delta\int\mathrm{d}t\int\mathrm{d}q\rho\left(\bar{L}-L_{0}\right)=0\mbox{.}$
(53)

In order to write down the Euler-Lagrange equations for
$\mathcal{L}=\rho\left(\bar{L}-L_{0}\right)$ in a compact way, we introduce
the two-component field $\chi_{\alpha}$, with $\chi_{1}=\rho$ and
$\chi_{2}=S$. If derivatives up to first order with respect to $t$ and second
order with respect to $q_{k}$ are taken into account, the Euler-Lagrange
equations of the variational problem (53) are given by

$\frac{\partial\mathcal{L}}{\partial\chi_{\alpha}}-\frac{\partial}{\partial
t}\frac{\partial\mathcal{L}}{\partial(\partial_{t}\chi_{\alpha})}-\frac{\partial}{\partial
q_{k}}\frac{\partial\mathcal{L}}{\partial(\partial_{k}\chi_{\alpha})}+\frac{\partial}{\partial
q_{k}}\frac{\partial}{\partial
q_{l}}\frac{\partial\mathcal{L}}{\partial(\partial_{k}\partial_{l}\chi_{\alpha})}=0\mbox{.}$
(54)

For the “correct” $L_{0}$, the Euler-Lagrange equations, with regard to $\rho$
and $S$, must agree with the new field equations and the function
$\rho(\bar{L}-L_{0})$) must agree with the Lagrangian density . On the other
hand, we have already determined the form of these field equations; they are
given by (50) and (28) and do _also_ depend on $L_{0}$. The field equations
are therefore overdetermined if both the conditions (50) and (28) and the
variational equations (54) are implemented. Thus, these conditions can be used
to determine, or at least restrict, the unknown function $L_{0}$, which must
be a solution of the four equations (54), (50), (28).

As for the functional form of $L_{0}$ , we assume that $L_{0}$ does not depend
on the derivatives of $\rho$ and $S$ with respect to time $t$; such a
dependency would change the basic structure of the evolution equations.
Further, if $L_{0}$ depends on $S$ and its derivatives, the validity of the
continuity equation requires, as a consequence of (54), a complicated
dependence of $L_{0}$ on the momentum distribution of the considered system.
The simplest and at the same time most general $L_{0}$ (not depending on the
considered system) can only depend on $\rho$ and its derivatives with respect
to $q_{k}$,
$L_{0}=L_{0}(\rho,\,\partial_{k}\rho,\,\partial_{k}\partial_{l}\rho)$. Then,
we find from Eqs. (54), (50), (28) that $L_{0}$ must obey the condition

$\rho\frac{\partial L_{0}}{\partial\rho}-\frac{\partial}{\partial
q_{k}}\rho\frac{\partial
L_{0}}{\partial(\partial_{k}\rho)}+\frac{\partial}{\partial
q_{k}}\frac{\partial}{\partial q_{l}}\rho\frac{\partial
L_{0}}{\partial(\partial_{k}\partial_{l}\rho)}=0\mbox{.}$ (55)

The second order derivatives in $L_{0}$ must be taken into account but should
not lead to contributions of higher order in the variational equations. The
simplest nontrivial solution of (55) that satisfies this requirement is given
by

$L_{0}=B_{0}\left[-\frac{1}{2\rho^{2}}\sum_{k}\left(\partial_{k}\rho\right)^{2}+\frac{1}{\rho}\sum_{k}\partial_{k}\partial_{k}\rho\right]\mbox{,}$
(56)

where $B_{0}$ is a constant. The second term in the square bracket, multiplied
by $\rho$, is a “Null-Lagrangian”, which means that it does not contribute
anything to the Euler- Lagrange equations [see (55)]. It is easy to verify
that this $L_{0}$ also fulfills condition (52). If $\rho$ is replaced by the
amplitude $\sqrt{\rho}$ and the constant $B_{0}$ is replaced by
$\hbar^{2}/4m$, then one obtains the simpler expression

$L_{0}=\frac{\hbar^{2}}{2m}\frac{1}{\sqrt{\rho}}(\partial_{k}\partial_{k}\sqrt{\rho})\mbox{.}$
(57)

Thus, the term $L_{0}$ agrees with the term $T_{Q}$ defined earlier, and the
randomized field equation (50) agrees exactly with the modified Hamilton-
Jacobi equation (44) obtained by linearization. This means that Schrödinger’s
equation may be derived either by linearizing equations (27), (28) according
to section 5.1, or by “randomizing” these equations in the sense of condition
(53).

If the term $\rho L_{0}$ is integrated over the configuration space, the
second term in the square brackets of (56) gives no contribution and one
obtains

$\int\mathrm{d}q\,\rho\,L_{0}=-\frac{B_{0}}{2}\,I[\rho]\mbox{,}$ (58)

where $I[\rho]$ is the Fisher information functional, defined by

$I[\rho]=\int\mathrm{d}q\,\rho\,\sum_{k}\left(\frac{\partial_{k}\rho}{\rho}\right)^{2}\mbox{.}$
(59)

While the second term in the square brackets of (56) gives no contribution to
the variation it is nevertheless indispensable, because without it condition
(52) could not be fulfilled. Fortunately, the same term is generated again
when the first term of (56), i.e. the Fisher functional, is varied. The same
final result is therefore obtained by adding the integrand of the Fisher
functional to the Lagrangian density $\rho\bar{L}$ of the original system and
performing the variation. In this way Schrödinger’s equation was derived by
Reginatto reginatto:derivation . Essentially the same method was also used by
Schrödinger in his “Erste Mitteilung” schrodinger:quantisierung_I , see also
lee.zhu:principle .

The importance of the concept of Fisher information for the construction of
the field equations of physics has been particularly emphasized by Frieden
frieden:physicsfisher ; frieden_soffer:lagrangians . The Fisher functional
(59) takes its smallest value zero for spatially constant probability
densities. Physically meaningful results are obtained if the requirement for
minimal Fisher information is combined with suitable constraints. The
situation is basically the same as with the entropy functional, which is
written in configuration space as

$S[\rho]=-\int\mathrm{d}q\,\rho(q)\ln\rho(q)\mbox{,}$ (60)

where a proportionality constant has been ommitted. If, as the simplest case,
one considers discrete probabilities and only requires as an additional
condition that the sum of all probabilities is $1$, then one obtains the most
fundamental result of probability theory from the requirement of maximum
entropy, namely that all probabilities must be equal. Other constraints lead
to the well-known probability distributions of statistical physics.

In contrast to the Boltzmann-Shannon entropy (60), the Fisher information (59)
contains the first derivatives of the probability density. While varying
$S[\rho]$ yields algebraic equations, varying $I[\rho]$ yields differential
equations. One can view the Fisher information as a local variant of the
global Boltzmann Shannon entropy. It defines what form the terms of a
differential equation must have, so that passing from one point in the state
space to the next, one is in accordance with the laws of probability theory.

This interpretation of the Fisher information is confirmed by a comparison
with the “relative entropy” or Kullback-Leibler entropy kullback:information ,
defined by

$G[\rho,\chi]=-\int\mathrm{d}q\,\rho(q)\ln\frac{\rho(q)}{\chi(q)}\mbox{,}$
(61)

where $\chi(q)$ is the initial, “prior”, probability density relative to which
$\rho$ is to be determined. We identify $\chi(q)$ with the probability
distribution $\rho(q_{1},...,q_{k}+\Delta q,...,q_{n})$ obtained from $\rho$
by shifting the $k$-th coordinate $q_{k}$ by an amount $\Delta q$, and denote
the resulting relative entropy by $\Delta G_{k}[\rho]$. For small $\Delta q$
one obtains in leading order the result

$\Delta G_{k}[\rho]=-\frac{1}{2}(\Delta
q)^{2}\int\mathrm{d}q\,\rho(q)\left(\frac{\partial_{k}\rho}{\rho}\right)^{2}\mbox{,}$
(62)

Thus, the Fisher information (59) is obtained, apart from a proportionality
constant, by adding up the contributions $\Delta G_{k}[\rho]/(\Delta q)^{2}$
from all coordinates.

The transition from the unphysical theory QA to the physically meaningful QT
may thus be performed using two methods motivated by quite different physical
ideas but leading to the same final result; either way seems plausible and
almost inevitably . We will see in IV that this duality persists if a complete
representation of the momentum field is used. The second method, the
derivation of QT by means of a randomization, is particularly interesting in
view of a comparison with other attempts to derive QT from a deeper structure
(see section 7).

###  5.3 Poincaré invariant and quantization condition

The physically meaningful solutions of the Schrödinger equation must meet two
additional conditions. First, the total probability of finding the particle(s)
anywhere must be equal to one. This condition is also present in the classical
theory from which QT was derived and needs no further explanation. Naturally,
its influence on the form of the solutions is relatively strong for confined
systems, that are located in a bounded spatial area. Second, the quantum
mechanical state variable $\psi$, as defined by (33), must be a single-valued
function in configuration space (at each instant of time). This allows for
multi-valued functions $S/\hbar$ whose values differ from each other by
multiples of $2\pi$. This single-valuedness condition is especially important
for the determination of the spectra of atomic systems and was called
“quantization condition” in the old quantum theory. The question arises
whether or not an analogous structure exists in the field of classical
physics. A more complete discussion will be given in IV where the
$\hbar$-dependence of the vortical terms of the momentum field will also be
taken into account.

A similar structure can be found in the concept of integral variants. Let us
briefly recapitulate this concept. For our purposes it is useful to consider
non-autonomous dynamical systems, with velocity field $V(x,t)$, whose basic
equations are given by

$\dot{x_{k}}=V_{k}(x,t)\mbox{.}$ (63)

Here, $x=(x_{1},...,x_{m})$ denotes a point in a $m$-dimensional space
$\mathbb{A}$ . Let us consider an arbitrary closed contour $C_{0}$ in
$\mathbb{A}$ at a fixed time $t_{0}$. The solutions of (63) transform each
point of the contour $C_{0}$ to a different point at a later (fixed) time $t$.
The totality of all transformed points forms again a closed contour $C_{t}$,
at time $t$, as a consequence of continuity. The contour may change its form
but remains a contour if transformed according to the flow; adopting a term
from fluid mechanics we may say that the contour is “moving with the fluid”.
The solutions of (63) may be written as $x_{k}(t,u)$ where $u$ is a parameter
varying, say, in the interval $[0,1]$. For fixed $t$ the points $x_{k}(t,u)$
trace out a closed contour (with $x_{k}(t,0)=x_{k}(t,1)$), for fixed $u$ the
points $x_{k}(t,u)$ describe trajectories. The totality of all trajectories is
referred to as tube.

Consider now a vector field $G(x,t)$ on $\mathbb{A}$, which may also
parametrically depend on time $t$. The contour integral

$I(t)=\oint_{C_{t}}\,G_{i}(x,t)\mathrm{d}x_{i}$ (64)

depends for given $C_{0}$ on the time $t$ and on the analytical form of the
field $G(x,t)$. Evaluating the derivative of (64) with respect to time, one
obtains the following necessary and sufficient condition for the invariance of
$I(t)$ along the considered tube pars:treatise :

$\oint_{C_{t}}\mathrm{d}x_{i}\left\\{\frac{\partial G_{i}}{\partial
t}+\left(\frac{\partial G_{i}}{\partial x_{k}}-\frac{\partial G_{k}}{\partial
x_{i}}\right)V_{k}\right\\}=0\mbox{.}$ (65)

If $G(x,t)$ fulfills condition (65) the contour integral (64) is referred to
as a (linear, relative) “integral invariant”.

Let us start in phase space and specialize this formalism to autonomous
systems of Hamiltonian type. We have $\mathbb{A}=\Omega$, $m=2n$,
$x=(q_{1},..,q_{n},p_{1},..,p_{n})=(q,p)$ and

$V(q,p,t)=\left(\frac{\partial H(q,p)}{\partial p},-\frac{\partial
H(q,p)}{\partial q}\right)\mbox{,}$ (66)

which means that (63) agrees with the canonical equations. The latter may also
be written in the form $\dot{x}=Z\frac{\partial H}{\partial x}$ where $Z$ is
an antisymmetric $2n\times 2n$ matrix with constant elements (the matrix $Z$
is sometimes referred to as symplectic matrix). It can be shown that
essentially only a single integral invariant of type (64) exists, which is
defined by $G(q,p,t)=(p_{1},..,p_{n},0,..,0)$. The corresponding line integral

$\oint_{C_{t}}\,p_{i}\mathrm{d}q_{i}\mbox{,}$ (67)

whose invariance is easily verified, is referred to as Poincaré integral
invariant. The antisymmetric matrix in (65), which is the $2n-$dimensional
counterpart of the rotation of a vector in three dimensions, agrees with the
constant matrix $Z$

$\frac{\partial G_{i}}{\partial x_{k}}-\frac{\partial G_{k}}{\partial
x_{i}}=Z_{ik}\mbox{.}$ (68)

Thus, the “vorticity tensor” of the vector field $G(q,p,t)$ is the same for
all points of phase space, as a consequence of its simple (linear in $q$ and
$p$) structure. The Poincare invariant cannot be used to characterize
individual systems, but rather describes certain structural properties of the
theory; the same applies to the conceptually similar but more general
Poincare-Cartan invariant, which may be used as an axiomatic basis for the
whole theory gantmacher:lectures .

This situation changes if the replacement $p_{k}\rightarrow M_{k}(q,t)$ is
performed. After projection to the $n$-dimensional subspace defined by $M$,
the Poincaré integral invariant takes the form

$I_{\bar{C}}(t)=\oint_{\bar{C}_{t}}\,M_{i}(q,t)\mathrm{d}q_{i}\mbox{.}$ (69)

Comparing the canonical condition (13) with the invariance condition (65) one
sees immediately that the latter is fulfilled if $h(q,t)$ is a single-valued
function. In phase space arbitrary closed paths were allowed. Now, the allowed
closed paths, denoted here by $\bar{C}_{t}$, as well as the tube formed by the
solutions of (12), must lie in the subset $(q,M(q,t))$ of $\Omega$. As a
consequence, the Poincaré invariant may now be used to characterize individual
systems, it depends not only on the considered path but also on the initial
value $M_{0}(q)$. The vorticity tensor $\Omega_{ik}(q,t)$, as defined by (24),
is not a constant anymore but varies in configuration space in a way depending
on the considered momentum surface. This means that vorticity for individual
physical systems does not exist in phase space but only arises through the
projection onto the configuration space. This is remarkable in view of the
fact that vorticity is the (quasi-)classical [more precisely: the quasi-
quantal] counterpart of quantum spin, as will be shown in IV.

The tensor $\Omega_{ik}(q,t)$ is the $n-$dimensional generalization of the
fluid-dynamical “vorticity”, which is given by $\vec{\Delta}\times\vec{u}$
where $\vec{u}$ is the velocity field. The constant $I_{\bar{C}}$ is also
referred to as “circulation”. For $n=3$ several of the above relations agree -
after a proper reinterpretation of the variables - with the dynamic equations
for an ideal barotropic fluid. In this fluid-dynamical context the invariance
of the circulation is referred to as Kelvin’s theorem.

In the next step we specialize to irrotational momentum fields (25). In this
case the circulation (69) vanishes for arbitrary $\bar{C}$ given that $S(q,t)$
is single-valued. The invariance condition (65) takes the form

$\oint_{\bar{C}_{t}}\mathrm{d}q_{i}\frac{\partial h(q,t)}{\partial
q_{k}}=0\mbox{,}$ (70)

if the derivative of $M$ with respect to time is replaced with the help of the
Hamilton-Jacobi equation (27). This condition, the single-valuedness of
$h(q,t)$, is the same as for general momentum fields. Vorticity and
circulation vanish by definition, except in the presence of topological
singularities of $S(q,t)$. In the present formalism QA there are, however, no
natural causes for such topological singularities. We may say that Kelvin’s
theorem holds true in the QA (We remind the reader at this point that the
theory QA is generally unstable, as discussed in section 4; we are considering
a possibly small period of time in which stability prevails).

If we carry out the transition to QT as a last step, we obtain an unchanged
continuity equation (28), and a modified Hamilton-Jacobi equation (44). These
two differential equations are to be solved taking the above multi-valuedness
condition for $S$ into account; if required by the considered physical system.
The particle-like Eq. (26) played no role in the transition to QT; according
to our discussion in section 5.1 this equation has no independent physical
meaning in QT anymore. The validity of Kelvin’s theorem in QA, as expressed by
the validity of Eq. (70), is an indication of the fact that the concept of
particle motion is still valid in QA. We can test Eq. (26) in this regard by
seeing whether or not a conservation law of the type of Kelvin’s theorem still
exists in the QT. Using (44) the invariance condition (65) takes the form

$\oint_{\bar{C}_{t}}\mathrm{d}q_{i}\frac{\partial}{\partial
q_{i}}\left[h(q,t)-\frac{\hbar^{2}}{2m}\frac{1}{\sqrt{\rho}}(\partial_{k}\partial_{k}\sqrt{\rho})\right]=0\mbox{.}$
(71)

At first glance, this equation looks similar to (70) , since the integrand is
given by a gradient in both cases. Due to the coupling between the phase $S$
and the probability density $\rho$ in Eq. (44), however, there is now a
natural source for singularities, since $S$ is undefined at the zeros of
$\rho$ and may become discontinuous there. This happens quite often in QT and
leads to a multivaluedness of $S$, which is characterized by an integer $n$
different from zero. The question is if this $n$ remains the same if the path
$\bar{C}_{t}$ changes as a function of time according to the differential
equation (26). Damski and Sacha have shown that this is generally not the case
by examining several simple physical systems damski:instability . Vortices in
the quantum mechanical probability fluid are rather unstable. Vortices in
many-particle systems (superfluid phases), which are described by non-linear
field equations, are more stable.

This means that Kelvin’s theorem does not hold true in the QT. There is no
support, from this point of view, for the idea that Eq. (26) describes the
motion of real particles. The analogy between Kelvin’s theorem and the
quantization condition is of a formal nature. Kelvin’s theorem is of dynamical
origin, while the quantization condition has a topological cause, namely the
uniqueness of the state variable $\psi$. The need to introduce a new state
variable is ultimately due to the requirement for _linearity_ of the basic
equations - which is one of the two fundamental requirements of the HLLK.

##  6 Concerning the interpretation

The vast number of different interpretations of the quantum theoretical
formalism is closely related to the poorly understood relationship between QT
and classical physics. In this section we examine what the theory developed in
the present series of works, the HLLK, can contribute to a better
understanding of this relationship. When we speak of the relationship between
QT and classical physics, we should more precisely distinguish the theory of
the individual classical particles CM from the probabilistic description of
classical particle ensembles PM. We speak of classical physics when a
distinction between CM and PM is not necessary.

Let us consider the following three points that seem plausible at first
glance. (1) We know that QT is suitable for describing microscopic reality
while classical physics fails in this area. (2) A concept that has evolved
over time and describes sometimes successfully the relationship between
individual areas of physics is the principle of reductionism. Put simply, it
means that the better theory - in the area under consideration - “contains”
the worse theory. It is also said that the better theory “reduces” itself to
the worse theory. In physics, this is understood to mean that the better
theory changes into the worse one if the additional structure that is present
in the better theory is “switched off” (example: let the speed of light become
infinite). (3) The additional structure that exists in QT, relative to
classical physics, is associated with a new natural constant $\hbar$. If we
consider these three points together - without a more detailed analysis - we
arrive at the conclusion that classical physics must be the limiting case of
QT in the limit of small $\hbar$.

This leads to the further question, which part of classical physics, CM or PM,
arises for $\hbar\rightarrow 0$ ? Depending on the answer, QT is either a
theory of individual particles or of statistical ensembles. This question was
one of the central points of disagreement between Bohr and Einstein
bohr:collected_vol_6 . This rhetorical battle was lost by Einstein, who
advocated the ensemble theory. This was partly due to the fact that Einstein’s
physical arguments were not distinguished clearly from his metaphysical demand
for a more complete theory. Basically, disregarding the rhetoric skills, one
can say that the question remains undecided to this day.

###  6.1 Formal versus empirical reduction

Let us now take a closer look at the above three points. What exactly does it
mean when we say that the better theory (QT) is reduced to the worse (CM or
PM)? In order to answer this question we have to fall back on some very
elementary facts. A physical theory is defined by a certain number of
fundamental _equations_ , mostly differential equations. With the help of the
solutions of the fundamental equations (of a successful theory) one can
describe a certain range of processes in nature. It is the _solutions_ that
decide whether a theory is successful or not.

The first clearly defined meaning of the term _reduction_ is obtained when one
considers the relationship between the fundamental equations. This means, for
example, that the QT is reduced (in this sense) to the CM or to the PM if
Schrödinger’s equation in the limit of small $\hbar$ merges into the canonical
equations or into the Liouville equation. An important point is, that this
kind of reduction has nothing to do with the question of the truth or falsity
of the theories in question. For example, if the Liouville equation could be
reduced to Schrödinger’s equation, that would not mean that the Liouville
equation is the better theory. Following Rosaler rosaler:formal , we refer to
this kind of reduction as _formal reduction_.

It is unsatisfactory to characterize the relationship between two theories
solely on the basis of the formal relationship between their fundamental
equations; the question which one of the two theories describes nature
correctly, is of course of paramount importance and must also be taken into
account. In order to introduce a meaningful concept of reduction from this
point of view, one must compare the solutions of the corresponding theories -
and the associated experimental results - with one another: The successful
theory A is reduced to theory B, in a certain area, if the results of B are
reproduced at least approximately in this area. Clearly, the concept of
reduction has a slightly different meaning here and, due to the large number
of possible solutions (systems), is by far not as sharply defined as in the
formal case. Using again the notation of Rosaler rosaler:formal  we refer to
this kind of reduction as _empirical reduction_.

###  6.2 The quantum-classical interface

The relationship between QT and classical physics has not been analytically
investigated for an astonishingly long period of time; one can say that the
majority approach was dominated by a philosophical principle, the principle of
reductionism. In particular, no distinction was made between formal reduction
and empirical reduction. However, we should also note that a considerable
number of works exist that deviate from the majority view; several of these
works are quoted in previous papers of the present author, see e.g.
klein:nonrelativistic .

Let us start with the question of formal reduction, which is the one studied
in the present work. The first systematic investigation of this question
klein:what  led to the conclusion that, contrary to common believe, QT is not
formally reducible to CM. A second recent study, reported in II, which took
into account all the theoretical results obtained during the past decades, led
to the stronger conclusion that QT can neither be reduced to CM nor can it be
reduced to the probabilistic version PM of classical mechanics. In II it was
shown that, to the contrary, PM may be reduced to QT. In the present work we
arrived at the same conclusion, using a different method that allows us to
study the details of the transition. This represents a radical change as
compared to the common view. The projection method used in the present work
can still be called a reduction, since phase space is larger than
configuration space. However, this type of formal reduction is not associated
with a vanishing $\hbar$, but to the contrary with the _creation_ of this new
fundamental constant. Its finite numerical value is a consequence of the
randomization, or linearization, following the projection.

Let us proceed now to the question of empirical reduction. There are several
systems in which an empirical reduction of QT to classical physics can be
observed. The historically oldest case concerns transitions between highly
excited atomic states. Here, an analogy to classical oscillators exists which
is described by Bohr’s correspondence principle. If the characteristic
dimensions of a system are much larger than the De Broglie wavelength, then a
classical approximation is often allowed. Generally speaking, the influence of
typical quantum mechanical effects decreases when a characteristic quantity of
the dimension of an action can be defined, whose numerical value is much
larger than $\hbar$. In more complex systems, high particle numbers and high
temperature have a similar effect. While all these effects have only an
approximate character, they are undoubtedly real, so it is justified to speak
of an empirical reduction of QT to classical physics.

Thus, there are two interfaces between QT and classical physics, a formal and
an empirical one. Both interfaces are associated in a natural way with a
direction. The formal interface is associated with the direction from
classical physics (PM) to QT, the empirical interface is associated with the
direction from QT to classical physics. In both cases we have transitions from
a more complicated structure to a simpler one, that is why we can speak of a
reduction in both cases. For the formal reduction, the direction points from
the phase space to the lower dimensional configuration space. For the
empirical reduction the direction points from QT to (quasi-) classical
systems, in which the fine structure of the complicated quantum effects is
“smeared out”.

###  6.3 Which systems can be described by QT

Let us return to the question, asked at the beginning of this section, whether
QT is able to describe the behavior of individual particles or whether
sensible statements can only be made about statistical ensembles. (note that
the original question, the one which played a central role in the discussions
between Bohr and Einstein, is now asked from a different point of view). With
the help of the above derivation of QT, performed in the framework of the
HLLK, we are now able to answer this question.

Our starting point was the classical probabilistic theory PM. This theory
still contains both probabilistic and deterministic elements. In a first step,
this theory was projected onto the configuration space. This did not change
the basic character of the theory, even if, as discussed in sections 2 and 4,
the deterministic element was weakened. In a second decisive step, the
deterministic relations were completely eliminated and an unavoidable
inaccuracy - not only with regard to the initial values, but also with regard
to the determination of the orbits of the particles - was introduced. This
fundamental uncertainty was indicated by the occurrence of a new natural
constant $\hbar$. In the theory derived this way there are no longer any
individual particles. This does not mean that individual particles cannot
_exist_ ; we just have no theory to describe them (and do not believe that
such a theory exists). Instead of particle trajectories, there are measurement
values, probabilities and expectation values. The answer to the above question
is therefore that QT is not a single particle theory but is only able to
describe statistical ensembles. It is well-known that the ensembles occurring
in QT cannot be described by standard probability theory but this does not
change the fundamental conclusion. Remarkably, Schrödinger already arrived at
the same conclusion when analyzing the equation he had derived: “… the true
laws of quantum mechanics do not consist of definite rules for the single path
…” schrodinger:quantisierung_II .

This conclusion is at variance with the reductionistic view formulated at the
beginning of this section, that the classical limit of QT must be CM. The
formal considerations that have been undertaken, in order to support this
historically evolved view, must be flawed somewhere if the present conclusion
is to be correct.

If one examines the behavior of Schrödinger’s equation for small $\hbar$, one
obtains the Hamilton-Jacobi equation and the continuity equation as leading
terms of an asymptotic expansion. These differential equations agree with Eqs.
(27) and (28), which were obtained by projection and subsequent introduction
of potentials. On the other hand, the Hamilton-Jacobi equation also occurs in
the theory of canonical transformations [see Eqs. (72)-(74)]. If a complete
solution $F_{1}(q_{1},..,q_{n},a_{1},..,a_{n})$ , which depends on $n$
parameters $a_{1},..,a_{n}$, is known, the trajectories of all particles can
be calculated. It was believed that this fact marked the transition from QT to
CM. A complete solution with fixed numerical values of the parameters
$a_{1},..,a_{n}$ was interpreted as (quasi-)classical counterpart of the
quantum mechanical state vector. Pauli wrote in his “General principles of
quantum mechanics” ( pauli:general , chapter VI, p. 91, bracket mine):

> If in the solution $S_{0}$ of …[the Hamilton-Jacobi equation] no further
> parameter is present, it is one mechanical path and in the general case
> giving special numerical values to parameters $a_{1},..,a_{n}$, …occurring
> in $S_{0}$ corresponds to a definite trajectory”.

In fact, a fixed choice of the parameters $a_{1},..,a_{n}$ does not correspond
to a single trajectory but to an $n$-fold infinite set - a “family” - of
trajectories, as equation (73) shows. This leads to a contradiction to the
idea that the wave function describes a single particle. Dirac expressed this
contradiction in the following frequently quoted statement dirac:hamiltonian :

> The family does not have any importance from the point of view of Newtonian
> mechanics; but it corresponds to one state of motion in the quantum theory,
> so presumably the family has some deep significance in nature, not yet
> properly understood.

This contradiction persists to this day as one of the many contradictions that
exist in the individuality interpretation of QT klein:individuality . Similar
contradictions are encountered when examining the behavior of wave packets in
the limiting case of small $\hbar$ klein:what

This line of argument is flawed as a consequence of the fact that the
solutions of Schrödinger’s equation are determined by their initial values.
Therefore, the Hamilton-Jacobi equation (27), which is obtained for small
$\hbar$, has also to be solved as an initial value problem and not as a
complete solution problem. As a consequence, it is not immediately possible to
set the quasi-classical limit of Schrödinger’s equation in correspondence to
classical mechanics. In order to make it possible, Pauli and Dirac considered
special solutions, which are obtained by assigning fixed numerical values to
the $n$ parameters $a_{1},..,a_{n}$ of a complete solution, as solutions to
the initial value problem. However, this is not mathematically consistent. As
is known from the theory of partial differential equations, one can construct
arbitrary solutions to initial value problems if a complete solution is known
evans:partial ; lanczos:variational . However, this requires the calculation
of the envelope of a whole family of solutions with different values of the
parameters $a_{1},..,a_{n}$; it is not possible to describe arbitrary initial
conditions by simply assigning numerical values to the parameters. This is the
error in the assertion that CM is the limiting case of QT for small $\hbar$.
Let us mention once again that Eqs. (27) and (28) have to be solved as initial
value problems. Thus, the quasi-classical limit of Schrödinger’s equation
fully agrees with the present theory QA.

As far as Dirac’s statement is concerned, it is in the context of the ensemble
interpretation obviously true (by construction) that a whole ensemble of
classical trajectories corresponds to a single quantum state. Dirac’s ensemble
has no special significance in the present theory, since here the ensembles
are defined by initial conditions and not by complete solutions; the “deep
significance” may of course be seen in the general validity of this
correspondence. The ensemble interpretation is seldom able to answer the
questions posed by the individuality interpretation and vice versa
landsman:between . When making the transition from one interpretation to the
other, it is the questions that are exchanged and not the answers.

##  7 Discussion

This series of works is based on the fact that there are mathematical
structures in phase space that are very similar to mathematical structures of
QT. This similarity becomes fully visible only when statistical ensembles of
particles, instead of single particles, are considered.

Many of these structures are described in Sudarshan and Mukunda’s outstanding
book on analytical mechanics sudarshan:classical . The solutions of the
canonical equations for any observable $A(q,p)$, which takes the place of
Hamilton’s function, form a one-parameter group of canonical transformations
in phase space (a realization of a corresponding one-parameter Lie group).
These transformations are the classical counterparts of the unitary
transformations in Hilbert space; the observables $A(q,p)$ correspond to the
self-adjoint operators that are the generators of these unitary
transformations. The vanishing of the Poisson bracket of two observables $A$
and $B$ implies mutual invariance under the transformations generated by the
other observable. As is well-known, the Poisson bracket of two observables is
the classical counterpart of the commutator of the corresponding operators in
QT. Each observable $A$ creates a statistical ensemble with the help of a
velocity field $V_{A}$, a probability density $\rho_{A}$, and a corresponding
Liouville equation. In this work we have identified $A$ with the Hamiltonian
function $H$. The present method may, however, be used for any observable and
leads to the same results as in II.

This summary shows that the mathematical and physical analogy between
functions in phase space (describing states of ensembles) and vectors in
Hilbert space is much stronger than the corresponding analogy for points in
phase space (describing states of particles). Accordingly, only a few simple
and natural additional assumptions have to be made, in the framework of HLLK,
in order to derive QT from the above classical probabilistic structure PM. The
first assumption is the introduction of a new dynamical variable $S(q,p,t)$
that describes the purely deterministic content of the canonical equation in a
compact way. A second fundamental assumption is that a projection to
configuration space must be performed. This is plausible from a physical point
of view; it is downright trivial if one considers the simplest formulation of
QT, on the level of differential equations, and looks at the independent
variables. The final result of the HLLK is a construction of QT that provides
not only the basic differential equation, but also most of the other
structural elements of QT, such as the probability interpretation of the wave
function, the explanation of why observables become operators, the eigenvalue
equations for measurable values of operators, and Born’s rule. A more complete
list of structural elements of QT may be found in II. In the following work IV
this list will be completed and to a certain extent closed, by showing that
the HLLK also provides an explanation for the spin of QT. Anticipating this
point, we may say that QT as a whole “emerges” from PM, if the assumptions of
the HLLK are implemented. Of course, if the transition from CM to PM is
included as first step of the theory, we may also say that QT emerges from CM.
The HLLK brings together numerous known results and some new considerations
under a common idea, which was essentially formulated by Einstein. It provides
a new conceptual foundation for QT that seems to be simpler, and at the same
time more complete, than any other attempt in this direction.

The need to find a new basis for understanding QT was felt by many physicists.
The question of whether QT is a “complete” theory plays a central role in all
of these discussions. Let us ask what the term “completeness” means exactly
(as we did in section 6.1 with the term “reduction”). In section 6 we
discussed the question of whether QT is able to describe the behavior of
individual particles, and we arrived at the conclusion that this is not the
case. In this sense, QT is incomplete. To describe this kind of
(in)completeness more precisely, one could say that QT is “empirically
(in)complete”. Does empirical incompleteness imply that there must be a more
“complete” theory, that also describes the behavior of individual particles?
This second concept of “(in)completeness” is evidently quite different from
the empirical one used above. This question cannot be decided empirically, in
the absence of empirical data it is a matter of philosophical belief. Let us
use the term “metaphysical (in)completeness” to characterize this kind of
(in)completeness.

To illustrate the difference, Einstein, Podolski and Rosen (EPR) have _shown_
in their famous work that QT is empirically incomplete; at the end of their
paper they expressed their _belief_ that QT is metaphysically incomplete
einstein.podolsky.ea:can . These authors understood perfectly the difference
between the two meanings of the term “completeness”. Most of the papers
criticizing this work try to refute EPR’s (nonexisting) claim that QT is
metaphysically incomplete. However, the fact that there are experimental
observations that we cannot explain does not automatically imply that a better
theory must exist.

###  7.1 Comparison of HLLK with the theories of Adler and t’Hooft

More recently, some authors assume that QT is metaphysically incomplete and
that a deeper deterministic (complete) theory exists, from which it can be
derived. Two theories of this kind are due to Adler adler:emergent  and ’t
Hooft thooft:quantum . Both constructions, that are otherwise quite different,
aim to enable a synthesis of QT and general relativity by means of abstract
deterministic structures, which are totally different from CM. Quantum theory
is then obtained as a statistical variant of this deterministic theory,
assuming that the initial states are not exactly known; this randomization is
of course associated with an enormous loss of information. Thus, these
theories try to confirm Einstein’s _believe_ that a determinism beneath QT
exists; just as deterministic equations of motion exist beneath classical
statistical physics. In theories of this kind, the principle of reductionism
requires that nature, when one goes to short distances, first sets up a limit
for the accuracy of measurements, which it then cancels again, when going to
even smaller distances of the order of the Planck length.

In comparison, the present theory is very simple from a conceptual point of
view. We know that QT is an empirically incomplete theory and try to derive
it, together with all its structural properties, from a number of reasonable
assumptions. We do not accept the universal validity of the principle of
reductionism and do not discuss the question of metaphysical completeness. The
present theory may also be built on the basis of a deterministic theory, but
this is the most familiar theory of its kind, namely CM; we only have to
include the transition from the canonical equations to Liouville mechanics as
the first step of HLLK. Then, QT as derived in the present work can also be
interpreted as a “statistical version” of CM. However, this concept needs to
be generalized in an appropriate manner. The statistical theory used in the
(enlarged) HLLK has to be implemented as a two-step process. In the first
step, from the canonical equations to Liouville mechanics, the uncertainty of
the initial values is taken into account. This is done with the help of the
standard formalism of probability theory. In the second step, which is
described in detail in section 5.2, the law of motion itself is randomized. In
contrast to the physics of uncertain initial values, which is ruled by the
principle of maximal entropy, the physics of uncertain dynamics is ruled by
the principle of minimum Fisher information. This second step, in which the
new constant $\hbar$ is introduced, can either be performed as a linearization
or alternatively, as the implementation of a new statistical concept. Both
steps together form, so to speak, the complete randomization. The associated
loss of information is also a two-step process.

The fact that not only the uncertainty of the initial values but also the
indeterminacy of dynamic processes may be included in the formalism represents
a fundamental generalization of the idea of statistics. Moyal called
classical statistical physics a “cryptodeterministic” theory, since the
uncertainty exists only in the initial values and not in the dynamics
moyal:quantum . He has already recognized that QT may be understood as a form
of general statistics that also includes dynamics.

###  7.2 Comparison of HLLK with the theory of Spekkens

The difference between the two types of completeness defined above has often
been overlooked. It is, however, taken into account in the classification of
possible models of QT formulated by Spekkens and Harrigan spekkens:quasi-
quantization ; harrigan_spekkens:einstein . Spekkens distinguishes between
ontic and epistemic states. Ontic states are points in phase space or
configuration space, epistemic states are probability distributions over these
spaces. In other words, CM is ontic and PM (Liouville dynamics) is epistemic;
the latter is based on the former. In recent years an increasing number of
researchers are realizing that QT is more similar to Liouville dynamics than
to CM. At the beginning of paper II a number of well-known facts supporting
this claim have been listed. This list is not complete. For example, Mauro has
shown that the Hilbert space of QT may be embedded in the Koopman-von Neumann
Hilbert space spanned by phase space functions mauro:new_quantization . Let us
also note at this point that only Liouville dynamics shares the property of
no-cloning with QT daffershofer:nocloning .

In Spekkens scheme, the HLLK belongs to the class of “$\psi$-epistemic” models
of QT. A detailed explanation of this term as well as an interesting analysis,
suggesting a reassessment of Einstein’s position, may be found in
harrigan_spekkens:einstein . The starting point of Spekkens’theory is the same
as in the present work, namely Liouville dynamics (which is, in the present
notation, PM restricted to Hamilton’s function). A constraint on the allowed
statistical distributions of Liouville dynamics is referred to as an
_epistemic restriction_. The fundamental postulate of Spekkens’ theory is the
following: if epistemic restrictions are implemented large parts of the
quantum theoretical formalism may be reproduced. A number of epistemic
restrictions are constructed, based on a classical version of the uncertainty
principle, a classical version of complementarity, and others spekkens:quasi-
quantization . If Poisson-commuting classical observables are taken into
account, one arrives at the conclusion that a $2n$-dimensional phase space can
at most be restricted to $n$-dimensional subspaces (maximally isotropic
subspaces, or Lagrangian subspaces). It is indeed possible to understand
various aspects QT by implementing such restrictions. However, it is not
possible to derive the entire formalism of QT; the theory is therefore called
a quasi-quantization method. Another disadvantage is, of course, that no
explanation is given why epistemic restrictions should be introduced.

The projection from phase space onto configuration space is the first and most
important step of the HLLK, since the further steps more or less inevitably
follow from it. The reasons for such a projection were discussed in section 1.
In this first step, the variety of possible states is restricted to an
$n$-dimensional subspace of $2n$-dimensional phase space. This projection can
therefore be interpreted as an epistemic restriction; here we find punctual
agreement with Spekken’s theory. The $n$-dimensional subspace is defined by
the momentum variable under consideration. As explained in II, the projection
must be carried out for all observables of interest, each one defining its own
momentum field. Thus, there is a multitude of epistemic restrictions in the
HLLK, but all of them have the same structure. With the exception of
Hamilton’s function which describes the dynamics, all other observables lead
to eigenvalue equations after performing the linearization (see II for
details).

###  7.3 Concluding remarks

In the first two papers of this series we gave the first coherent and complete
derivation of QT for spinless particles. In the version of HLLK reported in
the present work, the third in this series, the first step was a projection
onto configuration space followed by a linearization or randomization as
second step. The first step led to the quasi-quantal theory QA, which shares
its basic equations with the well-known quasi-classical approximation of QT.
In the papers I and II this theory QA was, so to speak, “skipped”, since we
were able to make the transition to QT in a single step, starting from a
linear classical equation of motion. The present version of HLLK allows for a
deeper insight into the nature of QT and the relationship between QT and CM.

First of all, we may say that the order chosen here reduces the number of
essential assumptions, on which this derivation of QT is based, to one, namely
to the assumption that the theory of classical probabilistic ensembles should
be formulated with the help of the independent variables $q,\,t$. The second
step, the subsequent linearization or randomization, can be interpreted as a
correction of the instability of the QA, i.e. as a necessary consequence of
the first step.

The second important point is that the present projection method allows one to
understand the properties of the momentum field $M(q,t)$. We were able to
derive an important structural property of QT, namely the fact that not the
momentum fields themselves appear as variables, but potentials have to
introduced instead, from which the momentum fields may be derived. In this
work we have exclusively used irrotational $M(q,t)$, i.e. we assumed that the
components of $M(q,t)$ are not functionally independent from each other. This
assumption is unphysical, since e.g. in three-dimensional space the three
components of $M(q,t)$ should be able to take any value in the absence of
external constraints. This consideration leads to the conclusion that massive
particles with spin $0$, as described in the present work and many others,
cannot exist. This conclusion is in agreement with experimental observations.
The correct representation of the momentum field of a single particle in
three-dimensional space - by means of three independent functions - provides
an explanation for the phenomenon of quantum spin and will be reported in the
following paper IV.

The third point in favor of the present version of the HLLK is that the
transition from QA to QT may now be formulated either with the help of a
linearization, or alternatively by means of a randomization, which leads to
exactly the same results. As a consequence, the transition from CM to QT may
be interpreted as a statistical process, although here the concept of
statistics has to be generalized appropriately, in the sense of Moyal. If one
accepts this extension one may interpret the transition from CM to QT as a
kind of realization of Einstein’s ideas.

##  8 Appendix: Comparison with phase-space momentum fields

It is interesting to study the relation between the momentum fields $M(q,t)$
and the more familiar momentum fields in phase space that occur in the theory
of canonical transformations. Let us consider a canonical transformation that
leads from old coordinates $q,p$ and the old Hamiltonian $H(q,p)$ to new
coordinates $Q,P$ and the new Hamiltonian $K(Q,P)=0$. We assume that we have a
complete set of Poisson-commuting classical observables, which is seldom the
case in practice. Furthermore, we consider non-degenerate canonical
transformation, for which the variables $q,Q$ may be used as phase space
coordinates. A generating function $F_{1}(q,Q,t)$ is then determined by the
equations

$\displaystyle p_{i}(q,Q,t)=\frac{\partial F_{1}}{\partial
q_{i}}(q,Q,t)\mbox{,}$ (72) $\displaystyle P_{i}(q,Q,t)=-\frac{\partial
F_{1}}{\partial Q_{i}}(q,Q,t)\mbox{,}$ (73) $\displaystyle
H(q,p(q,Q,t),t)=-\frac{\partial F_{1}}{\partial t}(q,Q,t)\mbox{,}$ (74)

where $Q$ stands for a set of $n$ arbitrary constants. As shown by Jacobi, the
general solution of the equations of motion (1) may be calculated if a
“complete solution” $F_{1}(q,Q,t)$ of the Hamilton-Jacobi equations is known.

The fields $p(q,Q,t)$ defined by (72) are referred to as momentum fields in
phase space. More precisely, we have a whole family of fields, each value of
the $n-$component quantity $Q$ determines a field. The crucial point is that
all these fields are irrotational; the second term on the left-hand side of
the canonical condition [the vorticity tensor (24)] vanishes by definition. On
the other hand, the momentum fields on configuration space $M(q,t)$, defined
as solution of Eq. (13), are _not_ subject to this restriction; explicit
expressions for $M(q,t)$ with non-vanishing vorticity tensor (corresponding to
spin $1/2$ ) will be reported in IV. We know that the theory of canonical
transformations provides a complete description of particle motion. How can it
be that the momentum fields $M(q,t)$, satisfying Eq. (13), are more general
than the momentum fields in phase space $p(q,Q,t)$, satisfying the formalism
of canonical transformations?

The reason is that the theory of canonical transformations describes the
detailed structure of the trajectories in phase space. There can be no
“vorticity” in the continuous set of these exactly defined trajectories.
However, this is possible for the momentum fields $M(q,t)$ that have been
forced to live in a subspace (configuration space) of phase space. The fields
$M(q,t)$ are obtained as solutions of an initial value problem, while the
fields $p(q,Q,t)$ are obtained from complete solutions of the Hamilton-Jacobi
equation. An important but sometimes overlooked point is that the solutions of
Schrödinger’s equation are defined by an _initial value problem_. The fields
$M(q,t)$ are therefore much better suited for a transition to (or from) QT
than the fields $p(q,Q,t)$ (we shall come back to this point in section 3.1 ).
As a matter of fact, the possible existence of momentum fields with vorticity
is a consequence of our projection to configuration space. In other words, it
is a consequence of the first step of HLLK in the transition from PM to QT.

The complete solutions of the Hamilton-Jacobi equation provide a most general
solution to the problem of particle motion. With their help one can construct
the general solution of the equations of motion and Hamilton’s principal
function as well as the general solution of the initial value problem of the
Hamilton-Jacobi equation conway:hamiltons ; lanczos:variational . However, the
momentum fields on configuration space $p(q,t)$ obtained in this way (by means
of the method of envelopes evans:partial ) are again irrotational, just like
the original momentum fields on phase space $p(q,Q,t)$ (the present author is
not aware of a theory that allows to solve initial value problems within the
framework of the theory of _degenerate_ canonical transformations
sudarshan:classical ).

## References

  * (1) L.E. Ballentine, Reviews of Modern Physics 42, 358 (1970)
  * (2) U. Klein, Quantum Stud.: Math. Found. 5, 219 (2018)
  * (3) U. Klein, Quantum Stud.: Math. Found. 7, 77 (2020)
  * (4) M. Born, Zeitschrift für Physik 153, 372 (1958)
  * (5) C. Jaffe, P. Brumer, J. Phys. Chem. 88, 4829 (1984)
  * (6) C. Caratheodory, _Calculus of Variations and Partial Differential Equations of the First Order, Part I_ (Holden-Day, Inc, San Francisco, 1965)
  * (7) A. Bennett, _Lagrangian fluid dynamics_ (Cambridge University Press, Cambridge, UK, 2006)
  * (8) A. Clebsch, J. reine angew. Math. 54, 293 (1857)
  * (9) P.A.M. Dirac, Can. J. Math. 3, 1 (1951)
  * (10) N. Mukunda, Proc. Indian Acad. Sci. 87, 85 (1987)
  * (11) H. Rund, Arch. Ration. Mech. Anal. 65, 305 (1977)
  * (12) V.V. Kozlov, in _Dynamical Systems X_ , _Encyclopaedia of Mathematical Sciences_ , vol. 67, (Springer, Berlin, 2003)
  * (13) T. Hawkins, Arch. Hist. Exact. Sci. 59, 381 (2005)
  * (14) H. Rund, Arch. Ration. Mech. Anal. 71, 199 (1979)
  * (15) T. Takabayasi, Progress in Theoretical Physics. 8(2), 143 (1952)
  * (16) P.R. Holland, _The quantum theory of motion_ (Cambridge University Press, Cambridge, U.K., 1995)
  * (17) J.H. Van Vleck, Proc. Natl. Acad. Sci. U.S. 14, 178 (1928)
  * (18) V.I. Arnold, _Ordinary Differential Equations_ (Springer, Berlin, 1992)
  * (19) P.C. Rosenbloom, Arch. Rational Mech. Anal. 3, 245 (1971)
  * (20) M.V. Berry, N.L. Balazs, J. Phys. A 12, 625 (1979)
  * (21) J.B. Delos, J. Chem. Phys. 86, 425 (1987)
  * (22) U. Klein, Physics Research International 2010, 18.  http://downloads.hindawi.com/journals/phys/2010/808424.pdf
  * (23) T.C. Wallstrom, Phys. Rev. A 49(3), 1613 (1993)
  * (24) R. Schiller, Phys. Rev. 125(3), 1100 (1962)
  * (25) N. Rosen, Am. J. Phys. 32, 597 (1964)
  * (26) W.P. Schleich, D. Greenberger, D.H. Kobe, M.O. Scully, Proc. Natl. Acad. Sci. USA 110(14), 5374 (2013)
  * (27) B.O. Koopman, Proc. Natl. Acad. Sci. U.S.A. 17, 315 (1931)
  * (28) D.I. Bondar, F. Gay-Balmaz, C. Tronci, Proc. R. Soc. A 475, 2018.0879 (2019)
  * (29) A.D.B. Manjarres, J. Phys. A: Math. Theor. 54, 444001 (2021)
  * (30) E. Madelung, Z. Phys. 40, 322 (1926)
  * (31) U. Klein, in _Measurements in Quantum Mechanics_ , ed. by M.R. Pahlavani (ISBN:978-953-51-0058-4, 2012), pp. 141–174.  See also arXiv:1109.6244 [quant-ph]
  * (32) M. Reginatto, Phys. Rev. A 58, 1775 (1998)
  * (33) E. Schrödinger, Annalen der Physik 79, 361 (1926)
  * (34) Y.C. Lee, W. Zhu, J. Phys. A 32, 3127 (1999)
  * (35) B.R. Frieden, _Physics from Fisher Information, a Unification_ (Cambridge University Press, Cambridge, 1998)
  * (36) B.R. Frieden, B. Soffer, Phys. Rev. E 52, 2274 (1995)
  * (37) S. Kullback, _Information Theory and Statistics_ (Wiley, New-York, 1959)
  * (38) L.A. Pars, _A Treatise on Analytical Dynamics_ (Heinemann, London, 1965)
  * (39) F. Gantmacher, _Lectures in Analytical Mechanics_ (Mir Publishers, Moscow, 1975)
  * (40) B. Damski, K. Sacha, J. Phys. A: Math. Gen. 36, 2339 (2003).  See also: https://arxiv.org/abs/quant-ph/0202137v1
  * (41) N. Bohr, _Niels Bohr Collected Works Volume 6, Foundations of Quantum Physics I (1926-1932)_ (North-Holland, Amsterdam, 1985).  See p. 99
  * (42) J. Rosaler, Topoi 34, 325 (2015)
  * (43) U. Klein, Am. J. Phys. 80, 1009 (2012)
  * (44) E. Schrödinger, Annalen der Physik 79, 489 (1926)
  * (45) W. Pauli, _General principles of quantum mechanics_ (Springer, Berlin, 1980).  Chapter VI
  * (46) U. Klein, Is the individuality interpretation of quantum theory wrong ?  ArXiv:1207.6215 [quant-ph], see also http://statintquant.net
  * (47) L.C. Evans, _Partial differential equations_ (American Mathematical Society, Providence, 1998)
  * (48) C. Lanczos, _The variational principles of mechanics_ (University of Toronto Press, Toronto, 1952)
  * (49) N.P. Landsman, in _Handbook of the Philosophy of Physics, Vol. 2_ , ed. by J. Butterfield, J. Earman (Elsevier, Amsterdam, 2005).  Available at: http://arxiv.org/abs/quant-ph/0506082
  * (50) E.C.G. Sudarshan, N. Mukunda, _Classical Dynamics: A Modern Perspective_ (Wiley, New York, 1974)
  * (51) A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 47(10), 777 (1935)
  * (52) S.L. Adler, _Quantum Theory as an Emergent Phenomenon_ (Cambridge University Press, Cambridge, New York, Melbourne, 2004)
  * (53) G. ’t Hooft, Foundations of Physics Letters 10, 105 (1997)
  * (54) J.E. Moyal, Proc. Cambridge Phil. Soc. 45, 99 (1949)
  * (55) R.W. Spekkens, _Quasi-Quantization: Classical Statistical Theories with an Epistemic Restriction_ (Springer, Dordrecht, 2016), _Fundamental Theories of Physics_ , vol. 181, pp. 83–135
  * (56) N. Harrigan, R.W. Spekkens, Foundations of Physics 40, 125 (2010)
  * (57) D. Mauro, Phys. Lett. A 315, 28 (2003)
  * (58) A. Daffertshofer, A.R. Plastino, A. Plastino, Physical Review Letters 88, 610601 (2002)
  * (59) A.W. Conway, A.J. M’Connell, Proceedings of the Royal Irish Academy, Section A 41, 18 (1932/1933)