∎

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 spinning particles

U. Klein

(Received: date / Accepted: date)

###### Abstract

As part of a probabilistic reconstruction of quantum theory (QT), we show that
spin is not a purely quantum mechanical phenomenon, as has long been assumed.
Rather, this phenomenon occurs before the transition to QT takes place, namely
in the area of the quasi-classical (here better quasi-quantum) theory. This
borderland between classical physics and QT can be reached within the
framework of our reconstruction by the replacement $p\rightarrow M(q,t)$,
where $p$ is the momentum variable of the particle and $M(q,t)$ is the
momentum field in configuration space. The occurrence of spin, and its special
value $1/2$ , is a consequence of the fact that $M(q,t)$ must have exactly
three independent components $M_{k}(q,t)$ for a single particle because of the
three-dimensionality of space. In the Schrödinger equation for a “particle
with spin zero”, the momentum field is usually represented as a gradient of a
single function $S$. This implies dependencies between the components
$M_{k}(q,t)$ for which no explanation exists. In reality, $M(q,t)$ needs to be
represented by three functions, two of which are rotational degrees of
freedom. The latter are responsible for the existence of spin. All massive
structureless particles in nature must therefore be spin-one-half particles,
simply because they have to be described by $4$ real fields, one of which has
the physical meaning of a probability density, while the other three are
required to represent the momentum field in three-dimensional space. We derive
the Pauli-Schrödinger equation, the correct value $g=2$ of the gyromagnetic
ratio, the classical limit of the Pauli-Schrödinger equation, and clarify some
other open questions in the borderland between classical physics and QT.

###### Keywords:

spin derivation of pauli equation quantum-classical relation ensemble theory
quantization

###### pacs:

03.65.Ta 03.65.Sq 05.20.Gg 05.30.Ch

###### MSC:

81P05 81S99 60A99 70H99

##  1 Introduction

> ”Niels Bohr brainwashed a whole generation of theorists into thinking that
> the job (interpreting quantum theory) was done 50 years ago”
>
> Murray Gell-Mann (1929-2019)

Quantum theory (QT) is the subject of controversial discussions since its
invention more then ninety years ago. The phenomenon of spin, discovered at
about the same time, plays a particularly mysterious role, because of the
disturbing fact that we are unable to identify a classical counterpart for
this ”classically non-describable two-valuedness” pauli:coll-influence . One
suspects that a satisfying explanation of spin will not be possible without a
better understanding of QT as a whole. There are two fundamentally different
ways to interpret the formalism of QT. The first way, mainly propagated by
Bohr, claims that QT is able to describe individual particles. In contrast to
this “individuality interpretation”, Einstein’s “ensemble interpretation”
asserts that QT is only able to describe statistical ensembles
ballentine:statistical . It may seem strange that a theory whose dynamical
predictions are statistical in nature should be able to describe individual
particles. As a matter of fact, however, Bohr’s interpretation is predominant
at the moment; it is accompanied by ongoing discussions about various internal
contradictions.

The reference theory for any interpretation of QT must be part of classical
physics. In the context of the individuality interpretation, this reference
theory is classical mechanics (CM). Due to the fundamental differences in the
mathematical formulation of both theories (ordinary differential equations and
observables defined as ordinary functions in CM, as opposed to partial
differential equations and observables defined as operators in QT), the
similarities between QT and CM are limited to structural (mathematical)
similarities between both theories. A satisfactory understanding of QT in this
formal sense would be achieved if it were possible to map the structural
properties of classical observables in a one-to-one manner to corresponding
properties of quantum mechanical operators. However, such a general process
called “quantization” does not exist, as Groenewold has shown
groenewold:principles .

In the probabilistic version of CM, which we call probabilistic mechanics
(PM), the object to be examined is not one (or several) individual particles,
but a statistical ensemble of (one or several) particles. The observable
quantities are expectation values, and the fundamental dynamical variable, the
probability density $\rho$, satisfies a partial differential equation (the
Liouville equation). Thus, in contrast to CM, the mathematical formulation of
PM is very similar to that of QT. This similarity allows for a completely
different approach when studying the relation between QT and classical
physics, namely a “derivation” or “reconstruction” of one of the two theories
from the other. The relevance of the probabilistic component for a proper
understanding of QT was recognized very early by Van Vleck
vanvleck:correspondence , Schiller schiller:quasiclassical ,schiller:spinning
, Bopp bopp:mecanique ,bopp:principles  and others.

This work is the fourth in a series of papers that attempt to understand QT on
the basis of Einstein’s ensemble interpretation. These works will be referred
to as I klein:koopman , II klein:probabilistic , and III klein:nonspinning .
According to the ensemble interpretation, the natural starting point for a
reconstruction of QT is PM. In I-III it was shown that it is possible, by
choosing this starting point, to derive not only Schrödinger’s equation, but
also most other fundamental properties of QT, such as the non-commutative
structure of operators, Born’s rule and others. In the present work it will be
shown that even the strange phenomenon of spin may be derived using a refined
version (actually a corrected version) of the theory presented in III.

The mathematical similarity between both theories allows a derivation of QT
from PM that is very simple, at least in conceptual terms. The main formal
difference between PM and QT is the number of independent variables, as can be
seen immediately if one descends from Hilbert space to the most concrete
formulation of QT in terms of differential equations. Obviously, PM is a
theory in phase space with, neglecting time at the moment, $2n$ independent
variables, while QT is a theory in configuration space with $n$ independent
variables. Thus, a reconstruction of QT must necessarily contain a projection
from phase space to configuration space. As a second fundamental step a
linearization or randomization must be performed, as explained in detail in
III. A theory containing these two steps, regardless of the order, is referred
to as “Hamilton-Liouville-Lie-Kolmogorov theory” (HLLK).

In works I and II, the first version of the HLLK was used, in which the
linearization is performed first followed by the projection. In III and the
present work, the projection is performed first and then the linearization (or
randomization) follows as second step. This second version of the HLLK allows
for a deeper understanding of the relationship between QT and classical
physics. Let us also note, that in II a variety of different observables $A$
was investigated, while in III and the present work we study only the
particular sector $A=H$ of HLLK related to spin.

The present work follows the same general scheme as III, but the actual
implementation is more involved. In the following section 2 we introduce, as
in III, a general momentum field. As was shown in III, it is not possible to
use the components $M_{k}$ of the momentum field themselves as dynamical
variables because they are in general not functionally independent from each
other. Instead, one must take as dynamical variables certain independent
functions, called potentials, which may be used to represent the momenta; the
situation is similar as in QT. In III only irrotational momentum fields were
dealt with, which means that all components $M_{k}$ could derived from a
single function $S$. In this case the transition from the $M_{k}$ to the new
dynamic variable $S$ (which later becomes the quantum mechanical phase) is
obvious. In the present work we need a larger number of potentials
$S,Q_{\alpha},P_{\alpha}$ in order to represent general momentum fields, that
also have vortical components. The fields $Q_{\alpha},P_{\alpha}$ called
Clebsch potentials describe these vortical components and are ultimately
responsible for the spin. The equations of motion for the new dynamic
variables $S,Q_{\alpha},P_{\alpha}$ were derived by the mathematician H. Rund
rund:clebsch . In section 3 we give a brief overview of Rund’s theory and then
reproduce his results within the present formalism.

In the emerging quasi-quantal or quasi-classical theory, which is referred to
as QA, particle trajectories do still exist, but are only locally valid. In
the following sections we restrict ourselves to the case $N=1$ of a single
particle, where besides $S$ and $\rho$, only two additional (vortical)
variables $Q,\,P$ occur. The QA (for $N=1$) is structurally identical to the
theory of inviscid barotropic fluids; this correspondence is only of a formal
nature and it should be born in mind that the physical meaning of the
variables is quite different. We use some results from fluid mechanics
concerning vortex lines. In section 4 we implement a topological restriction
for the variables $Q,\,P$. We assume that the mapping of $\mathbb{R}^{3}$ onto
the space of the $Q,\,P$ describes linked vortex lines (the linking number is
a topological invariant). The suitable mapping is given by the famous Hopf
map, which implies the replacement of the $Q,\,P$ with new _canonical_ Clebsch
variables $\vartheta$ and $\varphi$. This map also provides the definition of
a suitable two-component state variable (spinor) $\psi$ which is of central
importance for the transition to QT and for the implementation of the minimal-
coupling rule. As shown in section 6 the special form of Hopf’s map is closely
related to the fact that the parameter space of the rotation group is doubly
connected.

Using the new state variable $\psi$, we construct in section 5 a semi-linear
form of the evolution equations, which differs from the Pauli-Schrödinger
equation (without external electromagnetic field) only by a non-linear term.
Only when using this form of the evolution equations is it possible (see
section 6) to implement the minimal-coupling-rule in a natural way and to
derive the correct g-factor of $2$ . Section 6 also contains a discussion of
the concept of the ”magnetic moment of the electron”.

The transition to QT is carried out in section 7 by linearization with respect
to the variable $\psi$. This can be done very easily by eliminating the non-
linear term in the semi-linear equation. Section 7 also examines which terms
have to be added to the original evolution equations for
$\rho,\,S,\,\vartheta,\,\varphi$ in order to generate the linearization.

The justification for performing the linearization, discussed in more detail
in III, is that a large number of globally valid solutions can only be
generated in a linear theory (because of the superposition principle). This
point is discussed in more detail in III. While this reasoning is
understandable, the process of eliminating a term in a differential equation
may seem somewhat formal, if not crude. One wonders what exactly happened
during the discontinuous transition from the quasi-classical equations to the
quantum equations. In section 8 this discontinuous process is “resolved” and a
statistical theory is constructed which is as similar as possible to the
quasi-classical theory, but which does not claim to describe individual
particles at all. This theory, whose basic assumptions are the same as in III,
represents the most detailed version of the HLLK.

In the last section 9 we list the results obtained so far, taking into account
both versions of the HLLK, both the first version used in I and II and the
second version used in III and here. Then we go into more detail on some
important points. In particular, we emphasize that quantum spin cannot be a
localizable property of individual particles, but a collective property of the
probabilistic single-particle ensemble, which is related to the possibility of
rotations in three-dimensional space. The three-dimensionality of space is
also the reason for the special value $1/2$ of the spin. Other points that
will be discussed are the classical limit of the Pauli equation and the role
of potentials in quantum theory. We conclude with a few remarks concerning the
interpretation of the quantum theoretical formalism.

##  2 Basic equations

In this section we briefly recall all those relations from III that we need
for our spin theory; detailed explanations may be found in III. The state of a
system of $N$ particles is described by $n=3N$ coordinates
$q={q_{1},...,q_{n}}$ and $n$ conjugate momenta $p={p_{1},...,p_{n}}$.
Particle trajectories $q_{k}(t),\,p_{k}(t)$ are given by the 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)

with Hamilton’s function $H(q,p)$ not depending on time $t$. A particular
trajectory in the $2n$-dimensional phase space
$\Omega=\mathbb{R}^{n}_{q}\times\mathbb{R}^{n}_{p}$ may be labeled by its
state $q^{0},\,p^{0}$ at an initial time $t_{0}$. 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 the dependence on $t_{0}$ will often be supressed. The variables $q,\,p$
are “Lagrangian coordinates” representing particle properties. A statistical
ensemble is defined as the uncountable set of all solutions (2). Describing
ensembles with the help of “Eulerian coordinates”, which are denoted by the
same symbols $q,\,p$ but represent points of $\Omega$, is much more
convenient. The most important Eulerian dynamic variable is the probability
density $\rho(q,p,t)$, ruling the time-dependent distribution of 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)

In order to perform the transition from probabilistic mechanics (PM) to QT we
need a second Eulerian variable, the action variable $S(q,p,t)$, which obeys
the differential equation

$\frac{\partial S}{\partial t}+\frac{\partial S}{\partial q}\frac{\partial
H}{\partial p}-\frac{\partial S}{\partial p}\frac{\partial H}{\partial
q}=\bar{L}\mbox{,}$ (4)

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

$\bar{L}=\bar{L}(q,p,t)=p\,\frac{\partial H(q,p)}{\partial p}-H(q,p)\mbox{.}$
(5)

As a first step on our way from PM to QT, we have to perform the projection of
the basic equations (1), (3), (4) onto configuration space. We replace the
particle momentum $p$ at each instant of time $t$ by a $n-$component momentum
field $M$:

$p_{k}\rightarrow M_{k}(q,t)\mbox{.}$ (6)

Thus, $2n-$dimensional phase space is projected to a $n$-dimensional subspace
$\mathbb{M}=\left\\{(q,p)\in\Omega\,|\,p=M(q)\right\\}$ which is parametrized
by the configuration space coordinates $q_{k}$ mukunda:phase-space .

The projection of the canonical equations (1) leads to the differential
equations rund:clebsch ; kozlov:general

$\displaystyle\dot{q}_{k}=v_{k}(q,t)\mbox{,}$ (7) $\displaystyle\frac{\partial
M_{i}(q,t)}{\partial t}+\left[\frac{\partial M_{i}(q,t)}{\partial
q_{l}}-\frac{\partial M_{l}(q,t)}{\partial
q_{i}}\right]v_{l}\left(q,t\right)=-\frac{\partial}{\partial
q_{i}}h\left(q,t\right)\mbox{,}$ (8)

where the fields $v(q,t)$, $h(q,t)$ are defined in terms of $H(q,p),\,V(q,p)$
by

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

Eq. (8) will be referred to as canonical condition. As a result of the
projection to configuration space the $2n$ ordinary differential equations (1)
are replaced by $n$ ordinary differential equations (7), for the particle
positions $q$, and $n$ partial differential equations (8), for the momentum
field $M$.

A useful quantity, characterizing the purely rotational part of the momentum
field, is the vorticity tensor $\Omega_{ik}$, defined by

$\Omega_{ik}(q,t):=\frac{\partial M_{k}(q,t)}{\partial q_{i}}-\frac{\partial
M_{i}(q,t)}{\partial q_{k}}\mbox{.}$ (12)

Its equation of motion, which may easily be derived from the canonical
condition, shows that the time-dependence of $\Omega_{ik}$ is completely
determined by the solutions of the particle equations of motion (7). As a
consequence, performing a standard calculation from fluid mechanics, one may
show that

$\Omega_{ik}(q,t)=\Omega_{lj}^{0}(q)\,\frac{\partial q_{l}^{0}(q,t)}{\partial
q_{i}}\,\frac{\partial q_{j}^{0}(q,t)}{\partial q_{k}}\mbox{.}$ (13)

Here, the functions $q_{l}^{0}(q,t)$ are obtained by inverting the solutions
$q_{l}(t,q^{0})$ of (7); the quantities $q^{0}$ and $\Omega_{lj}^{0}(q)$ are
the initial values of $q_{l}(t)$ and $\Omega_{ik}(q,t)$. This relation shows
explicitly that the time dependence of the vorticity tensor is determined by
the flow of the particle equations of motion. In particular,
$\Omega_{lj}^{0}(q)=0$ implies $\Omega_{ik}(q,t)=0$ for all future times.

The projection of the Liouville equation to configuration space leads to the
continuity relation

$\frac{\partial\rho(q,t)}{\partial t}+\frac{\partial}{\partial
q_{k}}\rho(q,t)v_{k}(q,t)=0\mbox{}$ (14)

for the probability density in configuration-space $\rho(q,t)$. The latter is
defined by the relation $\rho(q,p,t)=\rho(q,t)\delta(p-M(q,t))$, where
$\delta$ is the $n$-dimensional delta function. The projection of the action
equation to configuration space may be put in the form

$\frac{\partial s}{\partial t}+v_{k}\left[\frac{\partial s}{\partial
q_{k}}-M_{k}(q,t)\right]+h=0\mbox{, }$ (15)

where the action field on configuration space $s(q,t)$ is defined by

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

Relation (15) may also be obtained by means of a Lagrangian to Eulerian
transition of the action integral in configuration space.

As shown in III, the $n$ components $M_{1}(q,t),..,M_{n}(q,t)$ of the momentum
field are generally not functionally independent from each other. They are
therefore unsuitable as dynamic variables and must be replaced by suitable,
functionally independent quantities. The problem of finding these new
variables for arbitrary vector fields was solved by Pfaff
caratheodory:calculus_I . For the present nonrelativistic problem, we need the
part of Pfaff’s solution relating to fields with an odd number $L$ of
independent functions. In this case there is an integer $m$ given by $L=2m+1$
and $M$ may be written in the form

$M_{k}(q,t)=\frac{\partial S(q,t)}{\partial
q_{k}}+P_{\alpha}(q,t)\frac{\partial Q_{\alpha}(q,t)}{\partial q_{k}}\mbox{,}$
(17)

where the Clebsch potentials clebsch:transformation
$S(q,t),\,P_{\alpha}(q,t),\,Q_{\alpha}(q,t)$ 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 $m$).
Inserting (17) in the definition (12) we see that the components

$\Omega_{ij}=\frac{\partial P_{\alpha}}{\partial q_{j}}\frac{\partial
Q_{\alpha}}{\partial q_{i}}-\frac{\partial Q_{\alpha}}{\partial
q_{j}}\frac{\partial P_{\alpha}}{\partial q_{i}}\mbox{.}$ (18)

of the vorticity tensor will generally be different from zero. As a
consequence, if we consider a closed loop $\gamma$, lying in $\mathbb{M}$ at
time $t$, then the closed path integral

$\oint_{\gamma}\mathrm{d}q_{k}M_{k}(q,t)\mbox{,}$ (19)

will be different from zero. Thus, the components
$P_{\alpha}(q,t),\,Q_{\alpha}(q,t)$ of a general momentum field ($m>0$)
describe a _vortical_ state of motion of the probabilistic ensemble; they will
sometimes be referred to as “vortical variables”. In contrast, for the
irrotational momentum fields considered in III the integral (19) vanishes for
arbitrary $t$ (assuming that $S(q,t)$ is single-valued).

In III the case $m=0$ of an irrotional momentum field was studied. This case
is unphysical but important because it leads to Schrödinger’s equation. The
determination of the equation of motion for the single Clebsch potential $S$
in III was simple because both the canonical condition and the action equation
reduce in this case to a Hamilton-Jacobi equation. In the following, some
basic results for the general case $m>0$ are reported first; the transition to
QT is carried out later for $m=1$.

##  3 Evolution equations

Our task is to derive equations of motion for the new variables $q_{k}$,
$S(q,t)$, $P_{\alpha}(q,t)$, $Q_{\alpha}(q,t)$ from the equations of motion
(7), (8) for the old variables $q_{k}$, $M_{k}(q,t)$. This is an easy task as
regards the particle equation of motion (7); one only has to insert the
expansion (17) in Eq. (7),

$\dot{q}_{k}=V_{k}\left(q,\frac{\partial S(q,t)}{\partial
q}+P(q,t)\frac{\partial Q(q,t)}{\partial q}\right)\mbox{.}$ (20)

The less trivial problem of determining the evolution equations for the
Clebsch potentials was solved by the mathematician H. Rund more than four
decades ago rund:clebsch . Rund’s original work contained an unnecessary
restriction which was removed shortly afterwards by Baumeister
baumeister:generalized . This extension will of course be taken into account
in the present work. In this section we first give a brief overview and
discussion of Rund’s theory and then clarify the relationship between his
variables and those used in this work.

###  3.1 Outline of Rund’s theory

We give a brief outline of Rund’s theory for the convenience of the reader. If
the expansion (17) is inserted in the canonical condition (8) the resulting
relation may be put in the form

$\frac{\partial}{\partial q_{i}}T(q,t)+\frac{\partial Q_{\alpha}}{\partial
q_{i}}\frac{\mathrm{D}P_{\alpha}}{\mathrm{D}t}-\frac{\partial
P_{\alpha}}{\partial q_{i}}\frac{\mathrm{D}Q_{\alpha}}{\mathrm{D}t}=0\mbox{,}$
(21)

where

$T(q,t):=H\left(q,\frac{\partial S}{\partial q}+P\frac{\partial Q}{\partial
q},t\right)+\frac{\partial S}{\partial t}+P_{\alpha}\frac{\partial
Q_{\alpha}}{\partial t}\mbox{,}$ (22)

and the total derivatives of $P_{\alpha}$ and $Q_{\alpha}$ with respect to
time are defined by

$\frac{\mathrm{D}}{\mathrm{D}t}=\frac{\partial}{\partial
t}+V_{l}\left(q,\frac{\partial S}{\partial q}+P\frac{\partial Q}{\partial
q},t\right)\frac{\partial}{\partial q_{l}}\mbox{.}$ (23)

In fluid mechanics these total derivatives are referred to as convective
derivatives or material derivatives.

In the next step, one takes advantage of the functional independence of $S$,
$P_{\alpha}$, $Q_{\alpha}$. As a consequence a one-to-one correspondence
between the $n$ variables $S,\,P_{\alpha},\,Q_{\alpha},q_{2m+2},..,q_{n}$ and
the $n$ variables $q_{1},..,q_{n}$ must exist at each instant of time. As a
further consequence, a function $\Phi$, depending on
$S,\,P_{\alpha},\,Q_{\alpha},q_{2m+2},..,q_{n}$ must exist which fulfills the
relation

$\Phi(t,\,S,\,P_{\alpha},\,Q_{\alpha},q_{2m+2},..,q_{n})=T(q_{1},..,q_{n},t)\mbox{.}$
(24)

In two further important steps it is shown that $\Phi$ depends neither on $S$
nor on $q_{2m+2},..,q_{n}$; the latter result is required for $2m+1<n$
baumeister:generalized . The final result of Rund’s theory is given by the
dynamic equations

$\displaystyle\frac{\partial S}{\partial t}+P_{\alpha}\frac{\partial
Q_{\alpha}}{\partial t}+H\left(q,\frac{\partial S}{\partial q}+P\frac{\partial
Q}{\partial q},t\right)-\Phi(Q,\,P,\,t)=0\mbox{,}$ (25)
$\displaystyle\frac{\mathrm{D}P_{\alpha}}{\mathrm{D}t}=-\frac{\partial\Phi(Q,\,P,\,t)}{\partial
Q_{\alpha}}\mbox{,}$ (26)
$\displaystyle\frac{\mathrm{D}Q_{\alpha}}{\mathrm{D}t}=\frac{\partial\Phi(Q,\,P,\,t)}{\partial
P_{\alpha}}\mbox{,}$ (27)

where the total derivatives are defined by (23). The function
$\Phi(Q,\,P,\,t)$ is completely arbitrary.

Besides these fundamental equations we quote from Rund’s work the following
important relation which is obtained by differentiating Eq. (25) with respect
to $q_{k}$:

$\left[\frac{\mathrm{D}M_{k}}{\mathrm{D}t}+\frac{\partial H}{\partial
q_{k}}\right]-\frac{\partial Q_{\alpha}}{\partial
q_{k}}\left[\frac{\mathrm{D}P_{\alpha}}{\mathrm{D}t}+\frac{\partial\Phi}{\partial
Q_{\alpha}}\right]+\frac{\partial P_{\alpha}}{\partial
q_{k}}\left[\frac{\mathrm{D}Q_{\alpha}}{\mathrm{D}t}-\frac{\partial\Phi}{\partial
P_{\alpha}}\right]=0\mbox{.}$ (28)

Equation (25) is referred to by Rund as ”generalized Hamilton-Jacobi
equation”. Indeed, if we set $Q_{\alpha}=P_{\alpha}=\Phi=0$, Eq. (25) reduces
to the Hamilton-Jacobi equation. Equations (26), (27) are referred to as
”associated canonical equations”, as they may be interpreted as ordinary
differential equations of canonical form, with a “Hamiltonian”
$\Phi(Q,\,P,\,t)$, for the “particle variables”
$P_{\alpha}(t)=P_{\alpha}(q(t),t),\,Q_{\alpha}(t)=Q_{\alpha}(q(t),t)$.
As a first step towards a physical interpretation of these equations we note
the following points:

  * •

The “Hamiltonian” $\Phi$ is arbitrary; no physical result can depend on its
functional form. We are therefore allowed to set $\Phi=0$. Rund has already
shown, using his theory of Clebsch gauge transformations, that a gauge with
vanishing $\Phi$ may be introduced rund:clebsch ; see also sudarshan:classical
.

  * •

The relations (26), (27) reduce for $\Phi=0$ to the condition that the
potentials $Q_{\alpha},\,P_{\alpha}$ are constant if transported along the
velocity field $V\left(q,\frac{\partial S}{\partial q}+P\frac{\partial
Q}{\partial q},t\right)$.

  * •

Equation (28) shows that the generalized Hamilton-Jacobi equation (25) makes
sure that the validity of (26), (27) implies the validity of the second
canonical equation [see Eqs. (1)] and vice versa.

In view of the important role of the Hamilton-Jacobi equation for the quantum-
classical transition of the Schrödinger equation, the physical meaning of the
generalized Hamilton-Jacobi equation, in particular the meaning of the new
variables $Q_{\alpha},\,P_{\alpha}$, is of great interest. Its possible role
with regard to QT has, however, not been clarified, neither by Rund himself
nor by authors elaborating later on his theory baumeister:generalized ;
samuel:phase_space . Samuel, who constructed an elegant phase-space version of
Rund’s theory wrote samuel:phase_space :

> It is not clear at the moment whether the generalized theory has any
> relevance to quantum mechanics. It does seem safe to say however that
> analogies with quantum mechanics, if they exist, are not straightforward and
> will require some unearthing.

In section LABEL:sec:tran-quant-theor it will be shown that Rund’s theory, as
completed by the continuity equation, leads for $N=1$ to the quasi-classical
counterpart of the quantum theory of a single particle (ensemble) with spin.

###  3.2 Implementing the Clebsch potentials in the ensemble equations

We now use Pfaff’s expansion (17) to rewrite the ensemble equations in
configuration space (14), (15) in terms of the new variables
$S,\,Q_{\alpha},\,P_{\alpha}$. We expect to reproduce Rund’s results and also
to find relations between the quantities $S(q,p,t)$, $S(q,t)$, and $s(q,t)$.

The continuity equation (14) can be simply rewritten by inserting the
expansion (17),

$\frac{\partial\rho(q,t)}{\partial t}+\frac{\partial}{\partial
q_{k}}\rho(q,t)V_{k}\left(q,\frac{\partial S}{\partial q}+P\frac{\partial
Q}{\partial q},t\right)=0\mbox{.}$ (29)

The continuity equation (29), the generalized Hamilton Jacobi equation (25),
and the associated canonical equations (26), (27) represent a closed system of
$2m+2$ equations for the $2m+2$ variables
$\rho,\,S,\,Q_{\alpha},\,P_{\alpha}$. This set of extended Rund’s equations
(extended by the continuity equation) is a precursor of the quasi-quantal
approximation (QA) of PM (in the actual QA the variables $Q,\,P$ will be
replaced by other variables due to a topological restriction; see section 4).
This is of course still a classical system of equations; this means that the
equations for $S,\,Q_{\alpha},\,P_{\alpha}$ form a closed system that does not
contain the variable $\rho$. Given a solution of these “deterministic”
equations the motion of particles may be determined with the help of (20).

The projected action equation is given by Eq. (15), where $s(q,t)$ is now
defined by

$s(q,t)=S\left(q,\frac{\partial S}{\partial q}+P\frac{\partial Q}{\partial
q},t\right)\mbox{,}$ (30)

in terms of the phase space action $S(q,p,t)$. In order to rewrite (15) we use
Pfaff’s expansion (17) and introduce the symbol $\chi(q,t)=s(q,t)-S(q,t)$ for
the difference betweeen $s$ and $S$. The projected action equation may then be
written in the form

$\frac{\mathrm{D}\chi}{\mathrm{D}t}-P_{\alpha}\frac{\mathrm{D}Q_{\alpha}}{\mathrm{D}t}+\frac{\partial
S}{\partial t}+P_{\alpha}\frac{\partial Q_{\alpha}}{\partial
t}+H\left(q,\frac{\partial S}{\partial q}+P\frac{\partial Q}{\partial
q},t\right)=0\mbox{,}$ (31)

We calculate the derivative of (15) with respect to $q_{i}$ and change the
order of derivatives to obtain

$\frac{\partial}{\partial t}\frac{\partial s}{\partial
q_{i}}+\frac{\partial}{\partial q_{i}}v_{k}\left[\frac{\partial s}{\partial
q_{k}}-M_{k}(q,t)\right]+\frac{\partial}{\partial q_{i}}h=0\mbox{.}$ (32)

The canonical condition (8) and the action equation in the form (32) describe
basically the same physics. However, Eq. (8) is an initial value problem for
$M_{k}(q,t)$ while Eq. (32) describes the relation between $M_{k}(q,t)$ and
$s(q,t)$. Both equations should agree if an appropriate representation of
$M_{k}(q,t)$ is chosen. This requirement leads to the following condition for
the Clebsch potentials:

$\frac{\partial}{\partial
q_{i}}\left[\frac{\mathrm{D}\chi}{\mathrm{D}t}-P_{\alpha}\frac{\mathrm{D}Q_{\alpha}}{\mathrm{D}t}\right]-\frac{\partial
Q_{\alpha}}{\partial
q_{i}}\frac{\mathrm{D}P_{\alpha}}{\mathrm{D}t}+\frac{\partial
P_{\alpha}}{\partial q_{i}}\frac{\mathrm{D}Q_{\alpha}}{\mathrm{D}t}=0\mbox{.}$
(33)

This condition is fulfilled if $\chi,\,Q_{\alpha},\,P_{\alpha}$ are solutions
of the associated canonical equations Eqs. (26), (27) and $\chi$ fulfills the
relation

$\frac{\mathrm{D}\chi}{\mathrm{D}t}-P_{\alpha}\frac{\partial\Phi}{\partial
P_{\alpha}}+\Phi=0\mbox{,}$ (34)

where $\Phi$ is an arbitrary function of $Q,\,P,\,t$. The generalized
Hamilton-Jacobi equation follows from (31) if Eq. (34) is taken into account.
The results of Rund’s theory are thus reproduced if the ensemble equations are
combined with Pfaff’s expansion.

Equation (34) clarifies the relation between the Clebsch potential $S(q,t)$
and the projected action $s(q,t)$. The difference $\chi=s-S$ plays the role of
an action defined for a “Hamiltonian” $\Phi$ in a $2m$-dimensional phase space
with coordinates $Q_{\alpha},\,P_{\alpha}$. We may set $\Phi=0$, thereby
destroying the canonical structure of Eqs. (26), (27), (34). Condition (34)
reduces for $\Phi=0$ formally to the corresponding condition in III for the
irrotational case; note however that the velocity field is defined
differently. Then, for this most important gauge,
$\chi,\,Q_{\alpha},\,P_{\alpha}$ become quantities “moving with the flow”,
i.e. $\frac{\mathrm{D}\chi}{\mathrm{D}t}=0$,
$\frac{\mathrm{D}Q_{\alpha}}{\mathrm{D}t}=0$,
$\frac{\mathrm{D}P_{\alpha}}{\mathrm{D}t}=0$. The simplest solution of
$\frac{\mathrm{D}\chi}{\mathrm{D}t}=0$ is again $\chi=0$, just as in the
irrotational case.

###  3.3 Interpretation of Clebsch potentials

Let us ask if the new dynamical degrees of freedom
$S(q,t),\,P_{\alpha}(q,t),\,Q_{\alpha}(q,t)$ may be understood in terms of any
standard concept of physics. This question is not answered in Rund’s theory.
With regard to the Clebsch variable $S$, which is also responsible for the
$U(1)$ gauge mechanism, we can answer this question in the affirmative. We
derived in section 3.2 the relation between $S(q,t)$ and the projected action
variable $s(q,t)$. The variable $S(q,t)$ can therefore be traced back to a
standard concept in phase space, namely the action variable $S(q,p,t)$. We
mention without proof that this connection can also be established within the
framework of the theory of canonical transformations; if we are given a
complete solution of the Hamilton-Jacobi equation, we can use it to construct
a projection onto the configuration space with prescribed initial values.

The question arises if the vortical variables
$P_{\alpha}(q,t),\,Q_{\alpha}(q,t)$, may be understood in an analogous way.
Can we find a structure in phase space, using a possibly extended theory of
canonical transformations, which allows us to understand these variables in a
similar way as $S(q,t)$ ? A closer look at the theory of canonical
transformations shows that such an extension does probably not exist. This
implies that the variables $Q,\,P$ describe probably a structure which has its
origin in configuration space. A different concept, valid only in
configuration space, must be found if we want to understand the physical
meaning of the variables $Q,\,P$.

The following two hints, as to the physical origin of $Q,\,P$, may be obtained
by looking more closely at the basic equations (25)-(28) of Rund’s theory:

  * •

There are two different terms (both sums over $\alpha$) in Eq. (25) containing
the Clebsch potentials. These terms appear exactly at the positions where the
electrodynamic scalar and vector potentials are located according to the
minimal coupling rule.

  * •

Relation (28) shows that the appearance of the new variables does not lead to
new forces in the particle equations of motion. Introducing $Q,\,P$ just means
introducing new degrees of freedom, which do not interact with the particle
coordinates.

The first of these hints tells us that the vortical terms due to $Q,\,P$ may
possibly be interpreted as electrodynamic potentials, giving rise to
nonvanishing electrodynamic fields. The second hint tells us that these fields
must be constructed in such a way that they do _not_ exert any forces on the
particles. In a future publication it will be shown that such “internal
potentials” actually exist, and that the vortical Clebsch potentials may be
understood with the help of this concept.

##  4 Implementing a topological constraint

In the remaining part of this paper we will restrict ourselves to the most
important case $N=1,\,n=3$ of a single particle ensemble. The number $m$ pairs
of functions $P_{\alpha}(q,t),\,Q_{\alpha}(q,t)$ is equal to $1$ and the
particle momentum is specified at each space-time point $q,t$ by three numbers
$S(q,t),\,Q(q,t),\,P(q,t)$, corresponding to the fact that the momentum must
have three independent components at each point of three-dimensional space.
Further, the Clebsch gauge will be fixed according to $\Phi=0$ and the
Hamiltonian (LABEL:eq:IE78FA12HTA) without external electromagnetic fields
will be used (this important point will be taken into account in section NN).
The basic equations for the four fields $S(q,t),\,Q(q,t),\,P(q,t),\,\rho(q,t)$
are then given by

$\displaystyle\frac{\partial S}{\partial t}(q,t)+P(q,t)\frac{\partial
Q}{\partial t}(q,t)+H^{0}\left(q,\frac{\partial S}{\partial q}+P\frac{\partial
Q}{\partial q}\right)=0\mbox{,}$ (35)
$\displaystyle\left[\frac{\partial}{\partial
t}+V_{k}^{0}\left(q,\frac{\partial S}{\partial q}+P\frac{\partial Q}{\partial
q}\right)\frac{\partial}{\partial q_{k}}\right]Q(q,t)=0\mbox{,}$ (36)
$\displaystyle\left[\frac{\partial}{\partial
t}+V_{k}^{0}\left(q,\frac{\partial S}{\partial q}+P\frac{\partial Q}{\partial
q}\right)\frac{\partial}{\partial q_{k}}\right]P(q,t)=0\mbox{,}$ (37)
$\displaystyle\frac{\partial\rho}{\partial t}(q,t)+\frac{\partial}{\partial
q_{k}}\rho(q,t)V_{k}^{0}\left(q,\frac{\partial S}{\partial q}+P\frac{\partial
Q}{\partial q}\right)=0\mbox{.}$ (38)

The particle equations of motion $\dot{q}_{k}=v_{k}(q,t)$, with the velocity
field defined by (20), are still valid in their limited range of validity. In
this section we consider a topological property of the mapping from
$\mathbb{R}^{3}$ into the space of the $S,Q,P$, which implies a more specific
form of the $Q,P$.

For the simplest case of a single free particle considered now, the
relationship between momentum, Clebsch potentials, and velocity is given by

$M_{k}(q,t)=\partial_{k}S(q,t)+P(q,t)\partial_{k}Q(q,t)=mv_{k}(q,t)\mbox{.}$
(39)

The vorticity tensor $\Omega_{ik}$ [see Eq. (18)] may conveniently be replaced
by an axial vorticity vector
$\Omega_{i}=\frac{1}{2}\epsilon_{ikl}\Omega_{kl}=\epsilon_{ikl}\partial_{k}M_{l}$,
which is closely related to the vorticity
$\omega_{i}=\epsilon_{ikl}\partial_{k}v_{l}$ defined by the velocity field
$v$. The relationship between these vorticities and the Clebsch potentials is
given by

$\Omega_{i}(q,t)=\epsilon_{ikl}\partial_{k}P(q,t)\partial_{l}Q(q,t)=m\omega_{i}(q,t)\mbox{,}$
(40)

and the equation of motion for the three-vector $\mathbf{\Omega}$ may be
written in the form

$\frac{\partial\mathbf{\Omega}}{\partial
t}+\mathbf{\nabla}\times\left(\mathbf{\Omega}\times\mathbf{v}\right)=0\mbox{,}$
(41)

which is familiar from fluid mechanics. As mentioned already, there is a
strong formal analogy between this part of the present theory and the theory
of inviscid (ideal) barotropic fluids. This allows us to use several important
results from fluid mechanics in the present context; bearing always in mind
the completely different physical meaning of the variables.

The vorticity field $\mathbf{\Omega}$ is by definition solenoidal. Surface
effects do not exist in our theory as we are considering an infinitely
extended medium with sufficiently rapidly decreasing variables. The field
lines of $\mathbf{\Omega}$, referred to as vortex lines, are therefore closed
curves. The topology of a vector field is basically determined by the mutual
position of its field lines, and in particular by the interlinking of its
field lines. This structure remains invariant under the time evolution given
by Eq. (41). Helmholtz’s and Kelvin’s theorems, which also apply in the
present theory, can be interpreted in terms of the invariance of the
topological structure.

The case of linked vortex lines may be explained in terms of simple physical
concepts [see moffatt:degree , ranada:helicity  and references therein]. For
example, in fluid mechanics Moffatt considers two idealized vortex filaments
$C_{1}$ and $C_{2}$ with vanishing vorticity outside the filaments
moffatt:degree . Stokes’s theorem leads then immediately to the conclusion
that the closed path integral along filament $C_{1}$ differs only from zero if
filament $C_{2}$ penetrates the surface spanned by $C_{1}$. Generalizing this
consideration to more realistic distributions he arrived at the conclusion
that the quantity

$H_{\omega}=\int\mathrm{d}^{3}q\,\mathbf{v}\,\mathbf{\omega}\mbox{,}$ (42)

referred to as (total) helicity, is a temporal and topological invariant,
characterizing the degree of linkage of vortex lines.

Using the present notation it is more convenient to work with the helicity
$H_{\Omega}=m^{2}H_{\omega}$, which is defined by (42) with $v,\,\omega$
replaced by $M,\,\Omega$. In order to study the variation of $H_{\Omega}$ with
time, we need the evolution equation for the helicity density
$h_{\Omega}=M_{k}\Omega_{k}$. It is obtained by multiplying the canonical
equation (8) and the vorticity equation (41) by $\Omega$ and $M$ respectively,
and adding both equations. The result may be put in the form

$\frac{\partial}{\partial
t}h_{\Omega}+\mathbf{\nabla}\left(h\mathbf{\Omega}+h_{\Omega}\mathbf{V}-(\mathbf{M}\mathbf{V})\mathbf{\Omega}\right)=0\mbox{}$
(43)

which shows explicitly the invariance of $H_{\Omega}$. The total helicity
vanishes if linked vortex lines do nowhere exist. On the other hand, if linked
vortex lines exist $M$ cannot be single valued; in this case we expect
topological singularities of some kind. In order to examine this point more
closely we calculate $H_{\Omega}$ using Pfaff’s formula (17), expressing $M$
in terms of the Clebsch potentials $S,\,P,\,Q$. Neglecting a surface integral
at infinity the helicity may be written as

$H_{\Omega}=-\int\mathrm{d}^{3}q\,S\epsilon_{ikl}\left(\frac{\partial^{2}P}{\partial
q_{i}\partial q_{k}}\frac{\partial Q}{\partial
q_{l}}+\frac{\partial^{2}Q}{\partial q_{i}\partial q_{l}}\frac{\partial
P}{\partial q_{k}}\right)\mbox{.}$ (44)

This formula shows that $H_{\Omega}$ vanishes if $P,\,Q$ are both $C^{2}$.
Linked vortex lines are therefore, as expected, related to singular, typically
multivalued, behavior of the $P,\,Q$. In order to obtain a nonvanishing
$H_{\Omega}$ it is not required that the first order derivatives of both
Clebsch potentials do not commute; it suffices if only one of the $P,\,Q$ is
singular.

###  4.1 The Hopf map

The mapping of $\mathbb{R}^{3}$ in the space of the Clebsch variables $P,\,Q$,
which is suitable for describing the quantum mechanical ensembles occurring in
nature, belongs to a topologically non-trivial class, with linked vortex lines
and non-vanishing helicity. It is given by the so-called Hopf map or Hopf
bundle hopf:abbildungen . We use the complex form of the Hopf map (see e.g.
socolovsky:spin ), which also provides us with an appropriate definition for
the new (spinorial) state variable, which will later be used to perform the
transition to QT. Let $\mathbf{z}$ denote the elements of the two-dimensional
complex vector space $\mathbb{C}^{2}$,

$\mathbf{z}=\begin{pmatrix}z_{1}\\\ z_{2}\end{pmatrix}\mbox{,}$ (45)

with inner product $(\mathbf{z},\mathbf{w})$ defined according to
$(\mathbf{z},\mathbf{w})=\mathbf{z}^{+}\mathbf{w}=z_{1}^{*}w_{1}+z_{2}^{*}w_{2}$.
The norm $\|\mathbf{z}\|$ of $\mathbf{z}$ is defined by
$\|\mathbf{z}\|^{2}=(\mathbf{z},\mathbf{z})$. Writing $z_{1}=a+\imath
b,\,z_{2}=c+\imath d$ the points of $\mathbb{C}^{2}$ may be used to assign
coordinates $a,b,c,d$ to the points of $\mathbb{R}^{4}$. The $3-$sphere
$S_{3}$ is the subset of $\mathbb{R}^{4}$ defined by
$\|\mathbf{z}\|^{2}=a^{2}+b^{2}+c^{2}+d^{2}=1$,

$S_{3}=\left\\{\mathbf{z}\in\mathbb{C}^{2}|\,\|\mathbf{z}\|^{2}=1\right\\}\mbox{.}$
(46)

In order to characterize the topological (homotopy) class of the mapping from
$\mathbb{R}^{3}$ to our two-dimensional manifold $Q,\,P$, the subset $S_{3}$
may be identified with $\mathbb{R}^{3}$ (it is in fact a compactified version
of $\mathbb{R}^{3}$ naber:topology ). The Hopf map is a many to one map from
$S_{3}$ to the $2-$sphere $S_{2}$, defined by

$h^{i}(x)=z_{\alpha}^{*}\sigma^{i}_{\alpha\beta}z_{\beta}\mbox{,}$ (47)

where $x$ stands for $a,b,c,d$, the indices $\alpha$ and $\beta$ run from $1$
to $2$, and $\sigma^{i}$ are the Pauli matrices

$\sigma^{1}=\left(\begin{array}[]{ccc}0&1\\\
1&0\end{array}\right),\;\;\;\sigma^{2}=\left(\begin{array}[]{ccc}0&-\imath\\\
\imath&0\end{array}\right),\;\;\;\sigma^{3}=\left(\begin{array}[]{ccc}1&0\\\
0&-1\end{array}\right)\mbox{.}$ (48)

In order to verify this we use the formula
$\sigma^{i}_{\alpha\beta}\sigma^{i}_{\gamma\delta}=2\delta_{\alpha\delta}\delta_{\gamma\beta}-\delta_{\alpha\beta}\delta_{\gamma\delta}$
to obtain $h^{i}h^{i}=\left(|z_{1}|^{2}+|z_{2}|^{2}\right)^{2}$. Thus, the
image of the Hopf map is indeed a point of $S_{2}$ given that
$|z_{1}|^{2}+|z_{2}|^{2}=a^{2}+b^{2}+c^{2}+d^{2}=1$. The points of $S_{2}$
have coordinates

$\displaystyle h^{1}=2(ac+bd)\mbox{,}$ (49) $\displaystyle h^{2}=2(ad-
bc)\mbox{,}$ (50) $\displaystyle h^{3}=a^{2}+b^{2}-c^{2}-d^{2}\mbox{.}$ (51)

Two points $\mathbf{z},\,\mathbf{w}$ in $S_{3}$, which differ from each other
by a complex number of magnitude 1, are mapped to the same point of $S_{2}$.
The set of all points $\mathrm{e}^{\imath\chi}\mathbf{u}$ on $S_{3}$ that
results from a fixed $\mathbf{u}$ is called a fiber. Each fiber is the
preimage of a point of $S_{2}$. Thus $S_{3}$ has the structure of a fiber
bundle.

The points on $S_{3}$ can now be represented in the form

$\mathbf{z}=\mathrm{e}^{\imath\frac{\chi}{2}}\begin{pmatrix}a+\imath b\\\
c+\imath d\end{pmatrix},\;\;\;\;a^{2}+b^{2}+c^{2}+d^{2}=1\mbox{.}$ (52)

It will become clear soon why it is convenient to use the definition
$\frac{\chi}{2}$ in the prefactor. It makes sense to introduce new variables
that eliminate the surface constraint in (52). We use spherical coordinates,
where the angle $\varphi$ is measured from the $y$-axis in clockwise
direction, to represent the three-vector $\mathbf{h}$ in the form

$\mathbf{h}=\sin\vartheta\sin\varphi\,\mathbf{e_{1}}+\sin\vartheta\cos\varphi\,\mathbf{e_{2}}+\cos\vartheta\,\mathbf{e_{3}}\mbox{.}$
(53)

Using now Eqs. (49)-(51) the dependence of $a,b,c,d$ on $\vartheta,\,\varphi$
may be determined and we obtain

$\mathbf{z}=\begin{pmatrix}z_{1}\\\\[4.26773pt]
z_{2}\end{pmatrix}=\mathrm{e}^{\imath\frac{\chi}{2}}\begin{pmatrix}u_{1}\\\\[4.26773pt]
u_{2}\end{pmatrix}=\mathrm{e}^{\imath\frac{\chi}{2}}\begin{pmatrix}\cos\frac{\vartheta}{2}\mathrm{e}^{\imath\frac{\varphi}{2}}\\\\[4.26773pt]
\imath\sin\frac{\vartheta}{2}\mathrm{e}^{-\imath\frac{\varphi}{2}}\end{pmatrix}\mbox{,}$
(54)

as a possible representation of $\mathbf{z}$. In the literature, essentially
two different but equivalent representations are in use, which are due to
Takabayasi takabayasi:description  and Bohm bohm_schiller_tiomno:causal ,
respectively. Both were introduced in order to rewrite Pauli’s equation in
“hydrodynamic form”. The state vector derived here is associated with the
description of rotations in terms of Euler angles and agrees with Bohm’s
representation; it was already used in Ref. klein:nonrelativistic  ,

In order to better understand the physical meaning of the quantity $\chi$, we
introduce a three-vector $\mathbf{t}$, with components defined by
$t^{i}=\\{\mathbf{z},\sigma^{i}\mathbf{z}\\}$ . The curly brackets denote here
an antisymmetric product, defined by
$\\{\mathbf{z},\mathbf{w}\\}=z_{1}w_{2}-z_{2}w_{1}$. This vector fulfills the
relations $\mathbf{t}^{2}=1$ and $\mathbf{t}\mathbf{h}=0$, i.e. $\mathbf{t}$,
as a unit vector perpendicular to $\mathbf{h}$, lies in the tangential plane
determined by $\mathbf{h}$. Using the representation (54) we obtain
$\mathbf{t}$ in terms of the angles $\chi,\,\vartheta,\,\varphi$:

$\begin{pmatrix}t^{1}\\\ t^{2}\\\
t^{3}\end{pmatrix}=\begin{pmatrix}\cos\chi\cos\varphi-\sin\chi\cos\vartheta\sin\varphi\\\
-\cos\chi\sin\varphi-\sin\chi\cos\vartheta\cos\varphi\\\
\sin\chi\sin\vartheta\end{pmatrix}\mbox{.}$ (55)

This relation shows that $\chi$ may be interpreted as an angle of rotation
around an axis determined by $\vartheta$ and $\varphi$. Remarkably, the
rotation angle of the vector $\mathbf{t}$ is $2\alpha$ if the phase of
$\mathbf{z}$ changes by $\alpha$. In particular, $\mathbf{z}$ changes its sign
when $\chi$ changes by $2\pi$ and returns to its original value only when
$\chi$ changes by $4\pi$. An equivalent form of a “spinor”, as a directed
quantity that describes a rotation, was derived by Payne using intuitive
geometric methodspayne:spinor .

###  4.2 Helicity and Hopf invariant

The preimage of each point $P$ of $S_{2}$ is a circle. The topological
nontriviality of the Hopf map is given by the fact that the linking number for
every $2$ circles, that are mapped to different points $P_{1},\,P_{2}$, is not
$0$ but $1$ ; an explicit proof may e.g. be found in lyons:hopf . This linking
number is a topological invariant referred to as Hopf invariant $\gamma$ .
Whitehead whitehead:hopf  found that $\gamma$ may be written as an integral,

$\gamma=\frac{1}{16\pi^{2}}\int\,\mathrm{d}^{3}q\,C_{i}\,D_{i}\mbox{,}$ (56)

where $C$, the canonical connection of the Hopf map urbantke:hopf , is given
by $C_{i}=-2\imath z_{a}^{*}\partial_{i}z_{a}$, and
$D_{i}=\epsilon_{ijk}\partial_{j}C_{k}$. The value of the Hopf invariant
$\gamma$ is a property of the divergence-free field $D$. The field $C$ plays
the role of a vector potential for $D$ and is not gauge-invariant. Conversely,
one may choose any suitable $C$ from the class leading to $D$ in order to
calculate $\gamma$, as will be done here.

The basic fields $\chi,\,\vartheta,\,\varphi$ that determine (56) are
dimensionless quantities while the fields $C$ and $D$ have dimensions
$cm^{-1}$ and $cm^{-2}$. If one wants to use the Whitehead integral in
physics, the abstract fields $C$ and $D$ have to be replaced by suitable
physical fields (e.g. velocity and vorticity, or vector potential and magnetic
field), and appropriate dimensions and pre-factors must be introduced. In our
case we have to establish the relation between the field $C$ and the momentum
field $M$,

$C_{i}=\partial_{i}\chi+\cos\vartheta\partial_{i}\varphi\;\;\;\;\;\;\;\;M_{i}=\partial_{i}S+P\partial_{i}Q\mbox{,}$
(57)

in order to be able to find the relation between $S,\,P,\,Q$ and
$\chi,\,\vartheta,\,\varphi$. If we accept $\mathbf{z}$, as defined by (54),
as our new state variable (apart from an amplitude which we will be introduced
later) then it is obvious to associate the two-component quantity in (54),
with components $u_{1},u_{2}$, with the new rotational degrees of freedom
$Q,P$. As a consequence we identify the prefactor
$\mathrm{e}^{\imath\frac{\chi}{2}}$ with the earlier phase factor
$\mathrm{e}^{\imath\frac{S}{\hbar}}$, associated with the irrotational
momentum fields studied in III. This leads to the relations

$\frac{S}{\hbar}=\frac{\chi}{2},\;\;\;C_{i}=\frac{2}{\hbar}M_{i},\;\;\;P\partial_{i}Q=\frac{\hbar}{2}\cos\vartheta\partial_{i}\varphi\mbox{,}$
(58)

As for the identification of $P,Q$ we could, remembering that the Clebsch
potentials are canonical variables rund:clebsch , interpret $P$ as momentum
and $Q$ as position. This could be achieved by replacíng $\hbar/2$ by
$\hbar/2R$ and $\phi$ by $R\phi$ , where $R$ is a length. The numerical value
of $R$ is arbitrary since $P,Q$ always occur in pairs. Thus we may set $R=1$:

$P=\frac{\hbar}{2}\cos\vartheta,\;\;\;Q=\varphi\mbox{.}$ (59)

The topological meaning of the “canonical Clebsch potentials” defined by (59),
was clarified by Kutnetsov and Mikhailov kuznetsov:topological . These authors
studied ideal fluids which are, however, described by essentially the same
mathematical structure as the present problem. In their work the constant
$\hbar/2$ is replaced by an undetermined constant - let us recall that we were
only able to fix the value of this constant by anticipating the quantization
process.

If we replace $P,Q$ in (40) by the canonical Clebsch potentials (59) the
vorticity $\Omega_{i}$ and the helicity $H_{\Omega}$ become quantized
variables,

$\Omega_{i}=\frac{\hbar}{2}\epsilon_{ikl}\left(\partial_{k}\cos\vartheta\right)\left(\partial_{l}\varphi\right),\;\;\;\;\;\;\;H_{\Omega}=(2\pi\hbar)^{2}\gamma\mbox{.}$
(60)

The linking number $\gamma$ takes integer values in general, and is $1$ in the
present theory. The Hopf map is sometimes introduced by restricting the
vorticity according to the relation $\Omega_{i}=\frac{\hbar}{2}T_{i}$, where

$T_{i}=\frac{1}{2}\epsilon_{ijk}\epsilon_{lmn}h^{l}\left(\partial_{j}h^{m}\right)\left(\partial_{k}h^{n}\right)\mbox{.}$
(61)

The vector $\mathbf{T}$ was first introduced by Takabayasi takabayasi:vector
and later rediscovered in other contexts by Faddeev faddeev:solitons  and
Mermin and Ho mermin_ho:circulation . Topological methods have been used
successfully to classify phases in superfluid He and other many-body systems
mermin:topological . If $\mathbf{v}$ in Eq. (42) is replaced by the superfluid
velocity volovik_mineev:particle  one obtains the same $\hbar$-dependent
prefactor as in Eq. (60).

###  4.3 Invariance of circulation and quantization condition

As shown in III, after projection to the $n$-dimensional subspace defined by
$M$, the Poincaré integral invariant takes the form

$I_{\bar{C}}(t)=\oint_{\bar{C}_{t}}\,M_{i}(q,t)\mathrm{d}q_{i}\mbox{,}$ (62)

where $\bar{C}_{t}$ is a closed path in the subset $(q,M(q,t))$ of phase
space. This formula expresses the invariance of the circulation $I_{\bar{C}}$;
in a fluid-dynamical context it is referred to as Kelvin’s theorem. It remains
true if $M$ is expressed in terms of $S,Q,P$ according to (17), or in terms of
$S,\vartheta,\varphi$ according to (58).

We expect, in analogy to the helicity, that the invariant $I_{\bar{C}}$
becomes “quantized” for topological reasons. This is indeed the case as found
already by Takabayasi, who used - contrary to the present work - the basic
equations of QT as his starting pointtakabayasi:vortex . Using (57)-(61) the
circulation may be written as

$\oint_{\bar{C}_{t}}\,M_{i}(q,t)\mathrm{d}q_{i}=h\left(n+\frac{1}{4\pi}\int_{S_{t}}\mathrm{d}S_{i}T_{i}\right)\mbox{,}$
(63)

where $S_{t}$ is a surface [a cross section of the tube formed by the
solutions of (7)] with boundary $\bar{C}_{t}$. The first term on the r.h.s. is
due to the fact, that $S$ may change by $2\pi\hbar n$ (where $n$ is an
integer) when going around $\bar{C}_{t}$. This multi-valuedness, and its
associated topological singularity, does not affect the uniqueness of the
state function $\mathbf{z}$. This term corresponds to the usual quantization
condition. The second term on the r.h.s. is due to the additional non-singular
vorticity of the momentum field.

To interpret this second term, we note that the vortical part of the momentum
field may be interpreted, apart from a constant of proportionality, as a kind
of internal vector potential, with components $A_{k}^{I}(q,t)$. When expressed
in terms of the canonical Clebsch potentials, $A_{k}^{I}$ is given by

$A_{k}^{I}=-\frac{\hbar c}{2e}\cos\vartheta\partial_{k}\varphi\mbox{.}$ (64)

The associated inner magnetic field is proportional to the Takabayasi vector
or to the vorticity,
$B_{i}^{I}=\epsilon_{ikl}\partial_{k}A_{l}^{I}=\frac{c}{e}\Omega_{i}$. It is
useful to introduce, for comparison, an external vector potential with
associated magnetic field $B_{i}^{E}$ (This does not cover the complete
influence of the external magnetic field, the full theory will be given in
section 6). The circulation extended this way may be written as

$\oint_{\bar{C}_{t}}\,M_{i}(q,t)\mathrm{d}q_{i}+\frac{e}{c}\int_{S_{t}}\mathrm{d}S_{i}\left(B_{i}^{E}+B_{i}^{I}\right)=nh\mbox{.}$
(65)

We see that the vortical part of the momentum field may be interpreted as an
additional, “internal” contribution to the magnetic field. In superconducting
many-particle systems one observes the phenomenon of flux quantization, which
is described by Eq. (65) _without_ this vortical contribution. Its absence is,
however, not really unexpected, since the superconducting state is not
generated by single electrons but by spinless paired electrons (Cooper pairs).

##  5 Semilinear evolution equation

After introducing the canonical Clebsch variables, one obtains the updated set
of basic equations for the dynamical variables $\rho,S,\vartheta,\varphi$ by
replacing $P,\,Q$ in Eqs. (35)-(38) by $\cos\vartheta,\,\varphi$ according to
(59):

$\displaystyle\partial_{t}S+e\Phi+\frac{\hbar}{2}\cos\vartheta\,\partial_{t}\varphi+\frac{1}{2m}\sum_{k}\left(\partial_{k}S+\frac{\hbar}{2}\cos\vartheta\,\partial_{k}\varphi\right)^{2}=0\mbox{,}$
(66)
$\displaystyle\left[\partial_{t}+\left(\partial_{k}S+\frac{\hbar}{2}\cos\vartheta\,\partial_{k}\,\varphi\right)\partial_{k}\right]\varphi=0\mbox{,}$
(67)
$\displaystyle\left[\partial_{t}+\left(\partial_{k}S+\frac{\hbar}{2}\cos\vartheta\,\partial_{k}\,\varphi\right)\partial_{k}\right]\vartheta=0\mbox{,}$
(68)
$\displaystyle\partial_{t}\rho+\partial_{k}\rho\left(\partial_{k}S+\frac{\hbar}{2}\cos\vartheta\,\partial_{k}\,\varphi\right)=0\mbox{.}$
(69)

Instead of the two dynamical equations for $\vartheta$ and $\varphi$ the
relation $D_{t}\mathbf{h}=0$ for the unit vector $\mathbf{h}$, defined by Eq.
(53), may equivalently be used. The particle equations of motion
$\dot{q}_{k}=v_{k}(q,t)$ remain true; they define the ranges of integration in
the quantization condition (65).

The above evolution equations, as given by (66)- (69), are neither suitable
for the transition to QT by linearization, nor for the introduction of a gauge
field. To achieve these goals, these four equations must first be replaced by
a single equation for a single variable with two complex components. A
suitable quantity is provided by the Hopf map. In order to perform this
transformation, it is convenient to rewrite (66)-(69) in the following form:

$\displaystyle D_{t}S=\bar{L}\mbox{,}$ (70) $\displaystyle
D_{t}\vartheta=0\mbox{,}$ (71) $\displaystyle D_{t}\varphi=0\mbox{,}$ (72)
$\displaystyle
D_{t}\rho^{\frac{1}{2}}=-\frac{1}{2}\rho^{\frac{1}{2}}\partial_{k}v_{k}\mbox{,}$
(73)

where $D_{t}=\partial_{t}+v_{k}\partial_{k}$,
$\bar{L}=\frac{1}{2}v_{k}M_{k}-V$, and

$v_{k}=\frac{1}{m}\left(\partial_{k}S+\frac{\hbar}{2}\cos\vartheta\,\partial_{k}\varphi\right)\mbox{.}$
(74)

The basic equations of QA reformulated this way differ from the equations used
in III only with regard to the additional equations of motion (71) and (72)
and the additional vortical term in the definition (74). The structural
differences between these equations and the original phase space equations
used in I, II were mentioned in III.

Let us stress once again, that these basic equations do essentially not belong
to QT, despite the occurrence of the constant $\hbar$. The occurrence of
$\hbar$ is explained as follows: Due to topological considerations, the
introduction of a quantity with the dimension of an action became necessary.
For the sake of simplicity, we have already assigned the special numerical
value of Planck’s constant to this quantity; without this assignement the
transition to QT, to be performed later, cannot be realized. This assignment
of a numerical value may therefore be interpreted as a first step in the
quantization process. However, the very occurence of this constant is a
consequence of classical (topological) considerations.

We obtain an appropriate two-component variable $\psi$, suitable for
linearization, by multiplying the variable $\mathbf{z}$ defined by the Hopf
map [see Eq. (54)] by the factor $\sqrt{\rho}$,

$\psi_{a}=\sqrt{\rho}\,\mathrm{e}^{\frac{\imath}{\hbar}S}\,u_{a},\;\;\;a=1,2\mbox{.}$
(75)

Then, as a consequence of the physical meaning of $\rho$, the integral of
$|\psi|^{2}$ over the entire space must be $1$.

Let us write the projected Liouville equation (73) in the form
$\mathbb{L}\rho^{\frac{1}{2}}=0$, with the differential operator
$\mathbb{L}=\frac{\hbar}{\imath}\left[D_{t}+\frac{1}{2}\left(\partial_{i}v_{i}\right)\right]$.
Let us next consider the two-component quantity
$T_{a}=\mathbb{L}\sigma^{0}_{ab}\psi_{b}$. The terms in the differential
equation to be constructed are necessarily $2\times 2$ matrices. It turns out
that it is sufficient to use the $2\times 2$ identity matrix $\sigma^{0}$, at
least in the present field-free case. If we let the differential operator
$\mathbb{L}$ act on $\psi$, then $T_{a}$ takes the form

$T_{a}=\frac{\hbar}{\imath}\sigma^{0}_{ab}\psi_{b}\left[\rho^{-\frac{1}{2}}\left(D_{t}+\frac{1}{2}\left(\partial_{i}v_{i}\right)\right)\rho^{\frac{1}{2}}+\frac{\imath}{\hbar}D_{t}S+\frac{1}{u_{(b)}}D_{t}u_{(b)}\right]\mbox{,}$
(76)

whithout summation over $b$ in the bracket. If we now use the evolution
equations (70)-(73), the first and third terms vanish and $D_{t}S$ is replaced
by $\bar{L}$. If we equate the resulting expression with the original
definition of $T_{a}$, we obtain the differential equation

$\left[\frac{\hbar}{\imath}D_{t}+\frac{\hbar}{2\imath}\left(\partial_{i}v_{i}\right)-\frac{1}{2}v_{k}M_{k}+V\right]\psi_{a}=0\mbox{,}$
(77)

which is equivalent to Eqs. (70)- (73), but is already more similar to the
desired form. In order to proceed with the linearization we use the relation
$\left(\partial_{k}S\right)\psi_{b}=\frac{\hbar}{\imath}\left[\partial_{k}-\rho^{-\frac{1}{2}}\left(\partial_{k}\rho^{\frac{1}{2}}\right)+u_{(b)}^{-1}\left(\partial_{k}u_{(b)}\right)\right]\psi_{b}$
which follows directly from the definition of $\psi$ [see Eq. (75)]. Using
this last relation as well as the definition of $M_{k}$ one obtains the
equation
$v_{i}M_{i}\psi_{b}=\frac{\hbar}{\imath}v_{i}\left(\partial_{i}-f_{i(b)}\right)\psi_{b}$.
The function

$f_{kb}=\frac{1}{\rho^{\frac{1}{2}}}\left(\partial_{k}\rho^{\frac{1}{2}}\right)+\frac{1}{u_{(b)}}\left(\partial_{k}u_{(b)}\right)-\frac{\imath}{2}\cos\vartheta\,\left(\partial_{k}\varphi\right)\mbox{,}$
(78)

does not depend on the variable $S$ which carries the gauge degree of freedom.
Repeated application of this last equation leads, after a number of further
elementary rearrangements, to the following “semilinear” differential equation

$\left[\frac{\hbar}{\imath}\partial_{t}-\frac{\hbar^{2}}{2m}\partial_{k}\partial_{k}+V\right]\psi_{a}=-\frac{\hbar^{2}}{2m}\left[f_{k(a)}f_{k(a)}+(\partial_{k}f_{k(a)})\right]\psi_{a}\mbox{.}$
(79)

This differential equation, together with the particle equation of motion
$\dot{q}_{k}=v_{k}(q,t)$, represents that form of the QA which is most similar
to QT - and which accordingly enables a particularly simple transition to QT.
In I a differential equation in phase space was derived (Eq. 19 of I), which
has been called the “classical counterpart of Schrödinger’s equation”. If we
follow this naming scheme, we can call Eq. (79) the “quasiclassical
counterpart of Schrödinger’s equation”. It represents the _completion_ of the
non-linear Schrödinger equation reported in III (see Eq. 40 of III), where two
of the three possible degrees of freedom of the momentum field were
unjustifiably neglected.

##  6 Introducing a gauge field

Let us now “switch on” an electromagnetic field. This is usually done using
the principle of minimal coupling

$\frac{\hbar}{\imath}\partial_{k}\Rightarrow\frac{\hbar}{\imath}\partial_{k}-\frac{e}{c}A_{k},\;\;\;\;\;\;-\frac{\hbar}{\imath}\partial_{t}\Rightarrow-\frac{\hbar}{\imath}\partial_{t}+e\Phi\mbox{,}$
(80)

where $A,\,\Phi$ is the vector and scalar potential respectively. In the
(quasi) classical equation (79) application of this rule to the derivatives on
the right-hand side does not lead to meaningful results. We therefore use the
more fundamental version of the principle of minimal coupling formulated by
Dirac dirac:quantised  (see also kaempfer:concepts , klein:schroedingers ).
With this method, the wave function $\psi$ is multiplied by a non-integrable
phase factor:

$\exp{\bigg{\\{}-\frac{\imath}{\hbar}\frac{e}{c}\int^{x,t}\left[\mathrm{d}q_{k}^{\prime}A_{k}(q^{\prime},t^{\prime})-c\mathrm{d}t^{\prime}\Phi(q^{\prime},t^{\prime})\right]\bigg{\\}}}\mbox{,}$
(81)

This phase factor is then shifted to the left of the differential operators,
creating the potentials, and can afterwards be eliminated. The final wave
function is again single-valued as it should be. This version of the minimal
coupling rule can be applied to Eq. (79), since the right-hand side does not
contain any derivatives of $\psi$ . We obtain in this way the standard result

$\left[\frac{\hbar}{\imath}\partial_{t}-e\Phi-\frac{\hbar^{2}}{2m}\sum_{k=1}^{3}\,\left(\partial_{k}-\imath\frac{e}{\hbar
c}A_{k}\right)^{2}+V\right]\psi_{a}=-\frac{\hbar^{2}}{2m}\left[f_{k(a)}f_{k(a)}+(\partial_{k}f_{k(a)})\right]\psi_{a}\mbox{.}$
(82)

The linear part of Eqs. (79) and (82) apparently plays a decisive role in the
transition to QT. This part, in which, based on our assumptions, the kinetic
energy is proportional to the identity matrix, does not contain any coupling
between the two components of $\psi$, neither in the field-free case nor in
the presence of an electromagnetic field. Assuming that Eq. (79) is true, this
would lead to the strange conclusion that the vortical components of the
momentum field have no observable consequences at all after transition to QT.
Instead, we suspect that Eq. (79) is incomplete.

###  6.1 Two representations of Euclidean space

The geometric objects occurring in physics may be characterized by their
behavior under certain groups of transformations. The relevant group in the
present context is the rotation group $SO(3)$, the group of orthogonal
$3\times 3$ matrices with determinant $1$. A geometric object is associated
with a specific representation of $SO(3)$. For example, in the Schrödinger
equation for spinless particles studied in III, a one-component complex
quantity occurs as a geometric object. In this case, only the space of
independent variables is transformed under $SO(3)$ and the representation
associated with this geometric object is the identical representation.

The geometric object that arises in the present problem is the spinor $\psi$.
At every point of space this is an element of a Hilbert space
$\mathbb{H}^{2}$, which is defined (as in section 4.1) by $\mathbb{C}^{2}$
with the inner product
$(\psi,\phi)=\psi_{1}^{*}\phi_{1}+\psi_{2}^{*}\phi_{2}$. The group of
automorphisms of $\mathbb{H}^{2}$ is given by the linear transformations that
leave the inner product invariant. This is the group $U(2)$, but it can be
restricted to $SU(2)$, the group of two-dimensional unitary transformations
with determinant $1$. A general element $U\in SU(2)$ has the form

$U=\left(\begin{array}[]{ccc}\alpha&\beta\\\
-\beta^{*}&\alpha^{*}\end{array}\right)\mbox{,}$ (83)

where $\alpha$ and $\beta$ are complex numbers that satisfy
$|\alpha|^{2}+|\beta|^{2}=1$ . This means that the group space of $SU(2)$ is
given by the $3$-sphere $S_{3}$ (the spinorial part of $\psi$ is also
determined by a point on $S_{3}$, as was shown in section 4.1). Thus, a spinor
is the geometric object transforming according to the natural representation
of the group $SU(2)\cong S_{3}$.

Let us recall the relation between the groups $SU(2)$ and $SO(3)$. This point
is well-known from the quantum mechanical theory of spin; but of course this
relation is primarily a “classical” matter of group theory and associated
topological considerations. To find this relation we introduce a matrix
representation of Euclidean space following Eberlein eberlein:spin_model . We
start from the observation that every Hermitian operator in $\mathbb{H}^{2}$
may be represented as a linear combination, with real coefficients, of the
three Pauli matrices $\sigma^{k}$ and the identity matrix $\sigma^{0}$ . If we
associate in our non-relativistic theory the identity matrix $\sigma^{0}$ with
a time-like coordinate, then the remaining three Pauli matrices $\sigma^{k}$,
with trace $0$, may be used to represent an arbitrary three-dimensional
Hermitian $2\times 2$ matrix, say $Q$, in the form

$Q=q_{1}\sigma^{1}+q_{2}\sigma^{2}+q_{3}\sigma^{3}\mbox{,}$ (84)

where the $q_{i}$ are real numbers. Let us denote the Euclidean space with the
usual inner product by $E_{3}$ and the real vector space of the Hermitian
matrices (84), with basis elements $\sigma^{k}$, by $\mathbb{E}_{3}$. There is
a one-to-one correspondence between the vectors $q_{i}\mathbf{e}_{i}\in
E_{3}$, where $\mathbf{e}_{i}$ is an orthonormal basis, and the matrices
$q_{k}\sigma^{k}\in\mathbb{E}_{3}$,

$q_{i}\mathbf{e}_{i}\leftrightarrow q_{k}\sigma^{k}\mbox{.}$ (85)

This mapping from $E_{3}$ to $\mathbb{E}_{3}$ also preserves the inner product
$(p,q)=p_{i}q_{i}$. If $P$ and $Q$ are the elements of $\mathbb{E}_{3}$
corresponding to $p$ and $q$, then their inner product, which is denoted by
$(P\cdot Q)$, is defined by $PQ+QP=2(P\cdot Q)\sigma^{0}$. The invariance can
easily be verified with the help of the relation
$\sigma^{i}\sigma^{k}+\sigma^{k}\sigma^{i}=2\delta^{ik}\sigma^{0}$ . Thus, the
space $\mathbb{E}_{3}$ may be called the matrix model, or the spin model, of
Euclidean space naber:topology ,eberlein:spin_model .

Let us consider an element $Q\in\mathbb{E}_{3}$ and a unitary matrix $U\in
SU(2)$. The transformed element

$R_{U}(Q)=UQU^{-1}\mbox{}$ (86)

is hermitian and has vanishing trace, so it is again an element of
$\mathbb{E}_{3}$ which may be written in the form
$Q^{{}^{\prime}}=q^{{}^{\prime}}_{k}\sigma^{k}$. If we now insert $U$ as given
by (83) and evaluate Eq. (86) we find the relation

$q^{{}^{\prime}}_{i}=R_{ik}(\alpha,\beta)q_{k}\mbox{,}$ (87)

where $R_{ik}(\alpha,\beta)$ are the elements of a (real) orthogonal matrix
with determinant $1$ which is uniquely determined by the matrix $U$ defined by
(83). These elements satisfy the relationship
$R_{ik}(-\alpha,-\beta)=R_{ik}(\alpha,\beta)$. As a consequence, Eq. (86)
defines a mapping $SU(2)\rightarrow SO(3)$ which is not one-to-one, since the
two elements $\pm U$ are mapped onto the same element of $SO(3)$. One can also
show that $R_{U_{1}U_{2}}=R_{U_{1}}R_{U_{2}}$ and that every rotation in
$\mathbb{E}_{3}$ may be represented in the form (86) eberlein:spin_model
,naber:topology . The mapping $SU(2)\rightarrow SO(3)$, which is often
referred to as “spinor mapping”, is therefore a two-to-one group homomorphism.

The difference between the groups $SU(2)$ and $SO(3)$ is essentially due to
the different topological structure of their parameter spaces. The two-to-one
group homomorphism from the simply connected $SU(2)$ to the double connected
$SO(3)$ means that $SU(2)$ is the covering group (double cover) of $SO(3)$. As
already mentioned, the parameter space of $SU(2)$ is the $3$-sphere $S_{3}$.
The parameter space of $SO(3)$ may be identified with a filled sphere in
three-dimensional space with radius $\pi$; this choice of the group manifold
allows an easy intuitive understanding of the topological structure
sattinger:lie_groups . Alternatively, one may also use the surface of the
$3$-sphere $S_{3}$ with antipodal points identified, as group manifold; this
latter choice is particularly plausible in view of the two-to-one homomorphism
from $SU(2)$ to $SO(3)$.

To choose between the two representations $E_{3}$ and $\mathbb{E}_{3}$ of the
abstract Euclidean space, we note that the transformation behavior of the
Hermitian operator $Q\in\mathbb{E}_{3}$, as given by (86), is consistent with
the general condition that the action of an operator before and after a basis
transformation $U$ must be the same. The spin model $\mathbb{E}_{3}$ of
Euclidean space is therefore the appropriate choice for vectors that act as
operators on the Hilbert space $\mathbb{H}^{2}$.

The representation $E_{3}$ would be suitable if we had three-vectors as
geometric objects, in which the effect of $SO(3)$ is realized by means of the
usual rotation matrices. The spin group $SU(2)$, as discussed above, has a
richer structure compared to the $SO(3)$. Its group manifold is larger, as is
the number of its representations; any representation of $SO(3)$ can be
extended to a representation of the $SU(2)$ (by composition with the spinor
mapping), but not the other way around. [However, in the physical literature
the term “two-valued representation of $SO(3)$” is often used for $SU(2)$].
The fact that the simply connected parameter space of $SU(2)$ contains two
elements $\pm U$ for every rotation $R$ may be related to the corresponding
property of the full group $O(3)$ of the automorphisms of Euclidean space. The
statement

> The orthogonal transformations are the automorphisms of Euclidean vector
> space. Only with the spinors do we strike this level in the theory of its
> representations…

made by Hermann Weyl emphasises the fundamental role of the spin
representation weyl:classical_groups .

The spinorial geometric object $\psi$ associated with $SU(2)$ is essentially
(except for a subsequently added amplitude $\sqrt{\rho}$) given by the Hopf
map, defined in section 4.1. The introduction of the Hopf map was based on the
assumption that the statistical ensembles occurring in nature belong to a
topological class describing linked vortex lines. This choice, which may have
seemed a bit arbitrary, is now justified by the fact that the Hopf map
describes the topology of rotations in $\mathbb{H}^{2}$. The property of
linked vortex lines seems to correspond to a particular topological property
of rotations, namely the fact that $\psi$ does not return to the starting
point until it has been rotated by $4\pi$.

###  6.2 Minimal coupling in $\mathbb{E}_{3}$

According to the discussion of the last section, the transition from $E_{3}$
to $\mathbb{E}_{3}$ should be performed simultaneously with the introduction
of the spinor $\psi$. The transition is defined by the simple replacement rule
$v_{i}\mathbf{e}_{i}\rightarrow v_{i}\sigma^{i}$. This rule can only be
applied in the quantum-like version (83) of the equation of motion, and here
again only in the Laplace operator, which can be written in the form
$\partial_{i}\mathbf{e}_{i}\partial_{k}\mathbf{e}_{k}$ . One obtains

$\partial_{k}\partial_{k}=\partial_{i}\mathbf{e}_{i}\partial_{k}\mathbf{e}_{k}\rightarrow\partial_{i}\sigma^{i}\partial_{k}\sigma^{k}=\partial_{k}\partial_{k}+\imath\epsilon^{lik}\sigma^{l}\partial_{i}\partial_{k}\mbox{,}$
(88)

where the relation
$\sigma^{i}\sigma^{k}=\delta^{ik}+\imath\epsilon^{lik}\sigma^{l}$ was used to
obtain the last term on the right-hand side. Eq. (88) shows that this
replacement rule can only lead to a new contribution, as compared with the
original Laplace operator, if the first order derivatives are not
interchangeable, that is, if they operate on a a singular, typically multi-
valued $\psi$. This is the case if the minimal coupling rule (81) is applied.
The evolution equation then takes the form

$\left[\frac{\hbar}{\imath}\partial_{t}-e\Phi-\frac{\hbar^{2}}{2m}\sum_{k=1}^{3}\,\left(\partial_{k}-\imath\frac{e}{\hbar
c}A_{k}\right)^{2}+V\right]\psi_{a}+\mu_{B}\sigma_{ab}^{k}B_{k}\psi_{b}=-\frac{\hbar^{2}}{2m}\left[f_{k(a)}f_{k(a)}+(\partial_{k}f_{k(a)})\right]\psi_{a}\mbox{.}$
(89)

where $\mu_{B}=-e\hbar/2mc$ is the Bohr magneton, and $B_{k}$ are the
components of the magnetic field $\mathbf{B}=\mathbf{\nabla}\times\mathbf{A}$.
The $B-$dependent coupling term was introduced by Pauli and is sometimes
referred to as “Zeemann term”, as it played an important role in the
explanation of the anomalous Zeeman effect. In mathematical terms $\mathbf{A}$
is called connection and $\mathbf{B}$ is called curvature of the connection
naber:topology .

We brought the quasi-classical evolution Eqs (66)-(69) in the quantum-like
form (79) and then introduced an external electromagnetic field with the help
of the principle of minimal coupling. If we now go back from the result (99)
to the original form, we obtain the equations

$\displaystyle\partial_{t}S+e\Phi+\frac{\hbar}{2}\cos\vartheta\,\partial_{t}\varphi+\frac{m}{2}\sum_{k}v_{k}^{2}+\mu_{i}B_{i}=0\mbox{,}$
(90) $\displaystyle
D_{t}\varphi=\frac{e}{mc}\frac{1}{\sin\vartheta}\left(B_{3}\sin\vartheta-
B_{2}\cos\vartheta\cos\varphi-B_{1}\cos\vartheta\cos\varphi\right)\mbox{,}$
(91) $\displaystyle D_{t}\vartheta=\frac{e}{mc}\left(B_{1}\cos\varphi-
B_{2}\sin\varphi\right)\mbox{,}$ (92)
$\displaystyle\partial_{t}\rho+\partial_{k}\rho v_{k}=0\mbox{,}$ (93)

where the velocity field $v$ and the total derivative $D_{t}$ are now defined
by

$v_{k}=\frac{1}{m}\left(\partial_{k}S-\frac{e}{c}A_{k}+\frac{\hbar}{2}\cos\vartheta\,\partial_{k}\varphi\right),\;\;\;\;D_{t}=\partial_{t}+v_{k}\partial_{k}\mbox{.}$
(94)

The “magnetic moment of the electron” is defined by
$\mu_{i}=-\frac{e}{mc}s_{i}$, where $\mathbf{s}$ is a three-vector of constant
lenght $\frac{\hbar}{2}$ which is parallel to the unit vector $\mathbf{h}$
defining the Hopf map,

$\mathbf{s}=\frac{\hbar}{2}\left(\sin\vartheta\sin\varphi\,\mathbf{e_{1}}+\sin\vartheta\cos\varphi\,\mathbf{e_{2}}+\cos\vartheta\,\mathbf{e_{3}}\right)\mbox{.}$
(95)

The electromagnetic field leads to two types of new terms in Eqs. (89)-(93),
namely on the one hand to the usual extensions of the energy and momentum
fields by potentials and on the other hand to terms depending on the magnetic
field $\mathbf{B}$. The latter terms may be interpreted with the help of the
vector $\mathbf{s}$, using the fact that Eqs. (91) and (92) may be written in
the form

$D_{t}\mathbf{s}=-\frac{e}{mc}\mathbf{B}\times\mathbf{s}\mbox{,}$ (96)

which is the field-theoretical version of the equation of motion of a
classical magnetic dipole in a magnetic field.

###  6.3 The magnetic moment of the electron

According to classical electrodynamics, a rotating charge distribution with
mass $m$, total charge $q$, and angular momentum $\mathbf{L}$ leads to a
magnetic dipole moment $\boldsymbol{\mu}=\frac{q}{2mc}\mathbf{L}$. The pre-
factor $\frac{q}{2mc}$ is called the gyromagnetic ratio. The interaction with
an external magnetic field is taken into account by a term
$\boldsymbol{\mu}\mathbf{B}$ in the Hamiltonian function. The new
$B-$dependent terms in Eqs. (90) and (96) agree in form with this particle
image, differ however with regard to the gyromagnetic ratio between the
“intrinsic angular momentum” $\mathbf{s}$ and the associated magnetic moment
$\boldsymbol{\mu_{s}}$. The gyromagnetic factor $g$ defined by
$\boldsymbol{\mu_{s}}=\frac{g}{2}\frac{q}{mc}\mathbf{s}$ takes the value $g=2$
and is consequently twice as large as would be expected in the particle image.

If we want to interpret the Zeemann term in the quantum-like form (89) of the
evolution equations, we must first identify the quantum-like quantity
$\mathbf{S}$ corresponding to $\mathbf{s}$. The formula
$\rho\mathbf{s}=\psi^{*}_{a}\frac{\hbar}{2}\boldsymbol{\sigma}_{ab}\psi_{b}$
which results directly from the definitions of $\mathbf{h}$ and $\psi$,
suggests the choice $\mathbf{S}_{ab}=\frac{\hbar}{2}\boldsymbol{\sigma}_{ab}$;
the same result for the “spin operator” $\mathbf{S}$ is obtained with the help
of the angular momentum commutation relations in QT. The quantum-like
formulation (89) of the evolution equations leads of course to the same result
for $g$ as (90), (93), (96): The Zeemann term in (89) may be written in the
form $\boldsymbol{\mu_{s}^{op}}\mathbf{B}$, and the relation between
$\mathbf{S}$ and the operator of the magnetic moment
$\boldsymbol{\mu_{s}^{op}}$ is given by
$\boldsymbol{\mu_{s}^{op}}=-\frac{g}{2}\frac{e}{mc}\mathbf{S}$ with $g=2$.
This value of the gyromagnetic factor has been experimentally confirmed to a
good approximation (there are corrections to this value that are not of
interest here).

The correct gyromagnetic factor $g=2$ was obtained by Dirac in 1928, in the
course of the derivation of the relativistic wave equation that bears his name
dirac:quantum_spin . It has long been assumed that the deviation from $g=1$ is
to be regarded as a relativistic (as well as quantum) effect. About four
decades later Levy-Leblond showed, however, that the same correct value of $g$
may also be derived from Schrödinger’s equation if a _linearization_ with
regard to the differential operator $\partial_{k}$ is performed levy-
leblond:nonrelativistic . One of the assumptions made by Dirac is that the
basic quantum equations for a particle should be linear in all first order
derivatives. This assumption may be partly motivated by the relativistic
space-time structure considered by Dirac. It’s physical meaning is, however,
not invariably linked to this space-time structure. Lévy-Leblond made the same
assumption as Dirac for the non-relativistic space-time and obtained a
differential equation for a four-component spinor, just as in Dirac’s theory.
This four-component spinor is composed of two two-component spinors
$\hat{\psi}$ and $\hat{\chi}$ which obey the equations

$\displaystyle\big{[}\frac{\hbar}{\imath}\frac{\partial}{\partial
t}+V(q)\big{]}\hat{\psi}-\sigma_{k}\frac{\hbar}{\imath}\frac{\partial}{\partial
q_{k}}\hat{\chi}=0\mbox{,}$ (97)
$\displaystyle\sigma_{l}\frac{\hbar}{\imath}\frac{\partial}{\partial
q_{l}}\hat{\psi}+2m\hat{\chi}=0\mbox{.}$ (98)

The two-spinor $\hat{\chi}$, defined as a linear combination of derivatives of
$\hat{\psi}$ with respect to $q_{l}$, plays obviously the role of an auxiliary
variable. If $\hat{\chi}$ is inserted in (97) one obtains the field free
Pauli-Schrödinger equation if the derivatives of $\hat{\psi}$ with respect to
$q_{k}$ commute with each other. If, on the other hand, a gauge field is
turned on with the help of the miminal coupling rule, one obtains Pauli’s
equation with the Zeemann term including the correct factor $g=2$. Lévy-
Leblond’s derivation shows that spin is not a relativistic phenomenon. This
fact is still not widely known. You can still find statements like: “Quantum
spin arises from the combination of special relativity with quantum mechanics”
lambiase_papini:interaction .

The value $g=2$, which is “anomalous” in the sense of a deviation from the
result of the particle image, cannot therefore be explained as a relativistic
(quantum mechanical) effect. This convenient explanation breaks down. Instead,
the question now arises why the linearity in $\partial_{k}$ should be
responsible for this deviation. The answer may be found in an unjustly
forgotten work eberlein:spin_model  by Eberlein from 1962. In this work, which
provides the basis for our above considerations, the correct value $g=2$ was
found for the first time in the framework of non-relativistic QT.

As discussed in more detail in section 6.1, Eberlein introduced a matrix
representation of Euclidean space that is suitable for describing the behavior
of spinors. The deeper reason for this lies in the topology of the rotation
group. The result of these topological considerations agrees with the result
of the “linearization” (with regard to the differential operator), which was
carried out by Levy-Leblond following Dirac’s method. A comparison of these
two processes shows that this is no coincidence: The conditions for a
meaningful “linearization” lead to the same commutation relations as the
conditions for a consistent spinor representation of $\mathbb{R}^{3}$. This
means that the unclear requirement of “linearity” may be replaced by the
obvious requirement that the topological structure of the rotation group has
to be correctly taken into account. But this means that actually no assumption
at all is required in order to be able to derive the correct $g$-factor (it is
tempting to associate the factor $2$ with the ratio $4\pi/2\pi$).

This insight automatically leads to the next question. If the deviation of the
gyromagnetic factor from $1$ is a purely topological effect why should we then
need QT to derive it. The answer given by the present theory is that we
actually do _not_ need QT to derive it; according to the present derivation,
it may be classified as a semiclassical (or semiquantal) effect. The fact that
the Zeeman term is not a quantum effect may also be realized from the very
absence of $\hbar$ in the constant of proportionality between
$\boldsymbol{\mu_{s}^{op}}$ and $\mathbf{S}$ (the $\hbar$ that appears in the
definition of $\mathbf{S}$ is irrelevant, as discussed in section 4.2). It
should also be mentioned in advance that the transition to QT carried out
below does not lead to any modification of the Zeemann term. The latter is a
consequence of the minimal coupling rule if the momentum field is displayed
correctly - that is, taking into account its vortical degrees of freedom.

Let us now ask for possible interpretations of the “magnetic moment of the
electron”. In the particle picture it was suggested by Uhlenbeck and Goudsmit
uhlenbeck:spinning  that an “intrinsic” angular momentum of the electron,
called _spin_ , is responsible for the observed effects. The magnitude of the
spin vector is assumed to be a constant equal to $\frac{\hbar}{2}$. As already
mentioned, the form of the relevant coupling term is compatible with this
assumption, but not the magnitude of the prefactor ($g=2$ instead of $g=1$).
The second, much more serious shortcoming is the universally accepted fact
that such a classical intrinsic spin cannot exist because it is in conflict
with fundamental physical principles kronig:spinning . While the theories of
Dirac dirac:quantum_spin , Eberlein eberlein:spin_model , and Levy-Leblond
levy-leblond:nonrelativistic  provide an explanation for the value $g=2$, the
fundamental second difficulty remains. In a relativistic world the
construction of a classical model for a spinning electron is just as
impossible as in the non-relativistic case.Thus, while the existence of
quantum spin is experimentally extremely well confirmed, a classical
counterpart of this property - if interpreted as a property of individual
electrons - does not exist ohanian:spin . Nevertheless, the intuitive ideas of
Uhlenbeck and Goudsmit dominate our thinking about spin even today. The reason
is that there is no better explanation, at least in the framework of the
individuality interpretation (particle picture) of QT. This represents a
painful gap in our understanding of nature, especially in view of the
fundamental importance of spin for the stability of matter.

This gap in our understanding disappears if one renounces the claim to be able
to describe the behavior of individual particles with the help of QT. In the
ensemble interpretation on which the present work is based, it is possible to
explain the origins of quantum spin in a simple way: The momentum field, which
describes a collective of particles and assigns a momentum to every point of
configuration space, must have three independent components, as space is three
dimensional. This simple fact leads to the appearance of two additional fields
describing an _internal degree of freedom of the ensemble_ (so one can
dispense with the strange idea that a point particle has internal degrees of
freedom). After linearization one obtains the complete basic equation of non-
relativistic QT for a single particle, derived by Eberlein and Levy-Leblond.

It is of course also possible to introduce additional spin degrees of freedom
in phase space. The problem here, however, is that you then make assumptions
that do not correspond to reality. In contrast, the existence of spin in the
ensemble theory follows automatically from the basic assumption that the
dynamic variables of quantum theory only depend on the space-time coordinates
as independent variables.

The two difficulties of the individuality interpretation mentioned above do
not arise in the ensemble interpretation. The difficulty, or better
impossibility, to understand spin as an intrinsic angular momentum of a single
particle does not exist in the ensemble theory, since the spin in this
interpretation is a collective property that results from a vortical component
in the momentum distribution of the probabilistic ensemble of _all_ particles.
Of course, there is then no difficulty in accepting a deviation of the
gyromagnetic ratio from the single particle value.

##  7 Transition to quantum theory by linearization

Let us briefly recap at this point. Our first step was the projection from
phase space to configuration space, transforming Eqs. (1)-(4) to Eqs. (7),
(8), (14), (15). With this first, most fundamental step, the globally valid
theory PM was converted to the only locally valid theory QA. The next two
steps were the introduction of independent variables (potentials) and the
transition to canonical Clebsch potentials. Of course, these two steps, that
led to Eqs. (66)-(69) do not change anything in the fundamentally inacceptable
nature (only local validity) of the above theory. This also applies to the
next step that we have carried out, namely the rewriting of the basic
equations in the quantum-like form (79).

The form (79) of the QA, in which the quantum variable $\psi$ is already used,
is still a theory in which particle trajectories exist - albeit subject to
“local validity” as discussed in III. Using this form of the basic equations
of QA, the influence of an external electromagnetic field could be taken into
account in a simple way, with the help of the minimal coupling rule. This form
also allows a particularly simple transition to QT. This transition must be
performed in such a way that the global validity, which has been lost during
the projection, emerges again from the locally valid QA. This requires either
linearizing Eq. (79) or randomizing Eqs. (66)-(69), as discussed in detail in
III. It will be shown that both kinds of “quantization” can be carried out
quite analogously to the spinless case. This final transition to QT is a
process that is completely independent from the presence or absense of an
electromagnetic field. We therefore omit the electrodynamic terms in (89) for
the sake of clarity.

The transition to QT by linearization takes place in a simple manner by
omitting the non-linear term on the right-hand side of Eq. (79). In the
resulting quantum theoretical evolution equation

$\left[\frac{\hbar}{\imath}\partial_{t}-\frac{\hbar^{2}}{2m}\partial_{k}\partial_{k}+V\right]\psi_{a}=0\mbox{,}$
(99)

the role of $S,\,\vartheta,\,\varphi$ as functions defining the momentum field
is destroyed and the particle equations of motion $\dot{q}_{k}=v_{k}(q,t)$
become meaningless. All theorems (such as those of Helmholtz type) that are
based on deterministic laws for particle motion become invalid. This
conclusion is of course the same as in the irrotational case studied in III.
Probabilistic theories of this kind, in which probabilistic statements about
particles can be made, while no statements can be made about the orbits of the
particles, were referred to as Type 3 theories in an earlier work of the
present author klein:statistical . Thus, the restoration of the global
validity of our theory leads to a radical change in its physical meaning. Just
as radical is the change in the mathematical description that is now made
possible by linearity. The new probabilistic structure, the role of
eigenvalues and non-commuting observables, Born’s rule, and other
characteristics of QT were derived in I and II.

The equation of motion (99) is a doubling of the Schrödinger equation and at
first glance appears to be equivalent to it. But this is not the case; the new
discrete degree of freedom has important physical consequences even in the
absence of an external electromagnetic field. The latter may be introduced in
the same way as in section 6.2 and leads to Eq. (89) with vanishing right hand
side.

###  7.1 Alternative description of the linearization

Let us ask which modifications Eqs. (66)-(69) will undergo as a consequence of
the transition to QT. The answer provides us with a reformulation of
Schrödinger’s equation in terms of the variables
$S,\,\varphi,\,\vartheta,\,\rho$ which will be useful later. We assume that
the new equations take the form

$\displaystyle\frac{\partial S}{\partial
t}(q,t)+\frac{\hbar}{2}\cos\vartheta(q,t)\frac{\partial\varphi}{\partial
t}(q,t)+H^{0}\left(q,\frac{\partial S}{\partial
q}+\frac{\hbar}{2}\cos\vartheta\frac{\partial\varphi}{\partial
q}\right)=L_{S}\mbox{,}$ (100) $\displaystyle\left[\frac{\partial}{\partial
t}+V_{k}^{0}\left(q,\frac{\partial S}{\partial
q}+\frac{\hbar}{2}\cos\vartheta\frac{\partial\varphi}{\partial
q}\right)\frac{\partial}{\partial
q_{k}}\right]\varphi(q,t)=L_{\varphi}\mbox{,}$ (101)
$\displaystyle\left[\frac{\partial}{\partial
t}+V_{k}^{0}\left(q,\frac{\partial S}{\partial
q}+\frac{\hbar}{2}\cos\vartheta\frac{\partial\varphi}{\partial
q}\right)\frac{\partial}{\partial
q_{k}}\right]\frac{\hbar}{2}\cos\vartheta(q,t)=L_{\vartheta}\mbox{,}$ (102)
$\displaystyle\frac{\partial\rho}{\partial t}(q,t)+\frac{\partial}{\partial
q_{k}}\rho(q,t)V_{k}^{0}\left(q,\frac{\partial S}{\partial
q}+\frac{\hbar}{2}\cos\vartheta\frac{\partial\varphi}{\partial
q}\right)=L_{\rho}\mbox{,}$ (103)

with new terms $L_{A}$, where $A=S,\,\varphi,\,\vartheta,\,\rho$, replacing
the zeros on the right-hand sides of Eqs. (66)-(69).

We will here determine the quantum terms $L_{A}$ in a purely formal way, as a
consequence of the linearization, postponing questions concerning deeper
physical meaning. We denote the left-hand sides of (66)-(69) by $T_{A}$. The
four basic equations $T_{A}=0$ of QA are equivalent to the nonlinear equations

$\mathbb{L}\psi+\mathbb{M}=0\mbox{,}$ (104)

where $\mathbb{L}$ and $\mathbb{M}$ are given by
$\mathbb{L}=\frac{\hbar}{\imath}\partial_{t}-\frac{\hbar^{2}}{2m}\partial_{k}\partial_{k}+V$
and
$\mathbb{M}_{a}=\frac{\hbar^{2}}{2m}\left[f_{k(a)}f_{k(a)}+(\partial_{k}f_{k(a)})\right]\psi_{a}$.
On the other hand, the four basic equations $T_{A}=L_{A}$ of QT are equivalent
to the linear equations

$\mathbb{L}\psi=0\mbox{.}$ (105)

We multiply the spinor (104) from the left by $\psi^{+}$ and $\phi^{+}$, where
the spinor $\phi$, which is orthogonal to $\psi$, is defined by
$\phi=\sqrt{\rho}\,\mathrm{e}^{\frac{\imath}{\hbar}S}\,\left(u_{2}^{*},-u_{1}^{*}\right)^{T}$.
Taking advantage of the fact that all equations are evolution equations,
containing only a single first-order derivative with respect to time, we can
identify the real and imaginary parts of the resulting expressions and obtain
the relations

$\displaystyle\rho
T_{S}+\imath\frac{\hbar}{2}T_{\rho}=\psi^{+}\mathbb{M}+\psi^{+}\mathbb{L}\psi$
(106) $\displaystyle-\frac{\hbar}{2}\rho
T_{\vartheta}+\imath\frac{\hbar}{2}\rho\sin\vartheta
T_{\varphi}=\phi^{+}\mathbb{M}+\phi^{+}\mathbb{L}\psi\mbox{.}$ (107)

We proceed in the same way with the spinor (105) and obtain the relations

$\displaystyle\rho\left(T_{S}-L_{S}\right)+\imath\frac{\hbar}{2}\left(T_{\rho}-L_{\rho}\right)=\psi^{+}\mathbb{L}\psi$
(108)
$\displaystyle-\frac{\hbar}{2}\rho\left(T_{\vartheta}-L_{\vartheta}\right)+\imath\frac{\hbar}{2}\rho\sin\vartheta\left(T_{\varphi}-L_{\varphi}\right)=\phi^{+}\mathbb{L}\psi\mbox{.}$
(109)

Comparison of (106), (107) and (108), (109) shows that the terms $L_{A}$ obey
the equations $\rho L_{S}+\imath\frac{\hbar}{2}L_{\rho}=\psi^{+}\mathbb{M}$
and $-\frac{\hbar}{2}\rho L_{\vartheta}+\imath\frac{\hbar}{2}\rho\sin\vartheta
L_{\varphi}=\phi^{+}\mathbb{M}$ . It is of course reasonable that the
additional quantum terms may be determined from the nonlinear term
$\mathbb{M}$ which makes the difference between QA and QT. A longer
calculation leads to the result

$\displaystyle
L_{S}=\frac{\hbar^{2}}{8m}\Big{\\{}4\rho^{-\frac{1}{2}}\left(\partial_{k}\partial_{k}\rho^{\frac{1}{2}}\right)-\sin^{2}\vartheta\left(\partial_{k}\varphi\right)\left(\partial_{k}\varphi\right)-\left(\partial_{k}\vartheta\right)\left(\partial_{k}\vartheta\right)\Big{\\}}\mbox{,}$
(110) $\displaystyle
L_{\varphi}=\frac{\hbar}{2m}\bigg{\\{}\cos\vartheta\left(\partial_{k}\varphi\right)\left(\partial_{k}\varphi\right)-\frac{1}{\sin\vartheta}\left(\partial_{k}\partial_{k}\vartheta\right)-\frac{1}{\rho\sin\vartheta}\left(\partial_{k}\rho\right)\left(\partial_{k}\vartheta\right)\bigg{\\}}\mbox{,}$
(111) $\displaystyle
L_{\vartheta}=\frac{\hbar}{2m}\Big{\\{}2\rho^{-\frac{1}{2}}\left(\partial_{k}\rho^{\frac{1}{2}}\right)\sin\vartheta\left(\partial_{k}\varphi\right)+2\cos\vartheta\left(\partial_{k}\varphi\right)\left(\partial_{k}\vartheta\right)+\sin\vartheta\left(\partial_{k}\partial_{k}\varphi\right)\Big{\\}}\mbox{,}$
(112) $\displaystyle L_{\rho}=0\mbox{.}$ (113)

Equations (100)- (103), with the terms $L_{A}$ given by (110)- (113) are
equivalent to the spinorial Schrödinger Eq. (99). Apart from the continuity
equation (103), which remains unchanged, the new quantum terms lead to a
coupling of the variables $S,\,\rho,\,\varphi,\,\vartheta$ which makes the
concept of individual particle trajectories meaningless. The situation is
basically the same as in the case of irrotational momentum fields treated in
III; in this case only the first term of $L_{S}$ survives, which is sometimes
(misleadingly) referred to as “quantum potential”. Our next task, performed in
section 8, is to understand the terms $L_{A}$ with the help of statistical
concepts.

The form (100)- (103) of the Schrödinger equation, with the terms $L_{A}$
given by (110)- (113), was derived many years ago in pioneering work by Bohm
and co-workers bohm_schiller_tiomno:causal , bohm_schiller:causal  and
Takabayasi takabayasi:vortex . In earlier work by Takabayasi takabayasi:vector
and also in works by Bialynicki-Birula bialynicki-birula:hydrodynamic
hydrodynamic variables (velocity field or momentum field) in addition to spin
variables were used as dynamic variables instead of the three quantum
mechanical potentials used in the present theory. The basic equations of this
“hydrodynamic formulation” of QT bialynicki-birula:hydrodynamic  may be
derived from the above equations by means of a tedious differentiation. The
derivation of the above equations presented here can be seen as a continuation
and completion of the theories of Bohm and Takabayasi. It has now been
possible to derive the same equations starting from the well-understood
classical theory PM, and give also plausible reasons for the individual steps
leading from PM to QT.

##  8 Transition to quantum theory by randomization

The zeroing of the nonlinear term in Eq. (79), that creates QT is, in physical
terms, a randomization. However, it is an unusual kind of randomization. With
the standard concept of randomization, as used in classical probabilistic
physics, only the initial conditions are random while the particle movement
itself is ruled by deterministic laws. This standard type of probabilistic
theory was referred to as type 2 theory in a previous work of the present
author klein:statistical . In contrast, the above transition from (79) to (99)
makes the equations of motion themselves “random” (which means nonexistent),
while statistical predictions about the behavior of the particles are still
possible. This (quantum) type of probabilistic theory was referred to as type
3 theory klein:statistical . The question arises as to whether we can better
understand this transition from type 2 to type 3. Are there statistical
assumptions that are equivalent to the linearization process, explaining, in
an alternative way, the transition from (79) [or (66)-(69)] to (99) ? In III
it was shown that this question can be answered in the affirmative in the case
of irrotational momentum fields. Here we show that the same holds true in the
present completed theory. The following construction is an expanded and
improved version of an earlier theory of the present author
klein:nonrelativistic .

###  8.1 Definition of Ehrenfest-like relations

There should be as close a relationship as possible between the statistical
equations of type 3 that we want to construct, and the equations of classical
mechanics. We assume therefore the validity of Ehrenfest-like relations of the
form

$\displaystyle\mathrm{d}_{t}\bar{q}_{k}(t)=\frac{1}{m}\bar{p}_{k}(t)\mbox{,}$
(114)
$\displaystyle\mathrm{d}_{t}\bar{p}_{k}(t)=\overline{F_{k}(q,t)}\mbox{,}$
(115)
$\displaystyle\mathrm{d}_{t}\bar{s}_{k}(t)=\overline{T_{k}(q,t)}\mbox{,}\mbox{,}$
(116)

where $F_{k}(q,t)$ is the external force and the “macroscopic variables”
$\bar{q},\,\bar{p},\,\bar{s}$ are defined as average values

$\displaystyle\bar{q}_{k}(t)=\int\mathrm{d}q\rho(q,t)q_{k}\mbox{,}$ (117)
$\displaystyle\bar{p}_{k}(t)=\int\mathrm{d}q\rho(q,t)M_{k}(q,t)\mbox{,}$ (118)
$\displaystyle\bar{s}_{k}(t)=\int\mathrm{d}q\rho(q,t)s_{k}(q,t)\mbox{,}\mbox{,}$
(119)

of the “microscopic” field variables $q,\,M,\,s$. The “statistical conditions”
(114)-(116) ensure that the mean values of $M_{k}(q,t)$ and $s_{k}(q,t)$ obey
particle-like relations. We switch on an external electromagnetic field by
taking into account the terms containing $\Phi,\mathbf{A}$ and $\mathbf{B}$
[see Eqs. (90), (94) and (96)] which are induced by the process of minimal
coupling. We do this because in the course of the following derivation of
Schrödinger’s equation, it will be possible to establish a connection between
the “potentials” $\Phi,\mathbf{A}$ and $\mathbf{B}$ on the one hand and
associated external forces $\mathbf{F}$ on the other hand. It is instructive
to write the energy and momentum components $M_{0}$ and $M_{k}$ in the form

$M_{0}=\partial_{t}S^{{}^{\prime}}+R_{0},\;\;\;\;\;\;\;\;M_{k}=\partial_{k}S^{{}^{\prime}}+R_{k}\mbox{,}$
(120)

where

$\displaystyle\partial_{t}S^{{}^{\prime}}=\partial_{t}S+e\Phi,\;\;\;\;\;\;\;\;\;\partial_{k}S^{{}^{\prime}}=\partial_{k}S-\frac{e}{c}A_{k},$
(121) $\displaystyle
R_{0}=\frac{\hbar}{2}\cos\vartheta\partial_{t}\varphi,\;\;\;\;\;\;\;\;\
R_{k}=\frac{\hbar}{2}\cos\vartheta\partial_{k}\varphi\mbox{.}$ (122)

Both the multivalued phase $S^{{}^{\prime}}$ (in contrast to $S$ the first
derivatives of $S^{{}^{\prime}}$ with respect to $q_{k}$ and $t$ are not
interchangeable) and the terms $R_{k}$ contain rotational components of
$M_{k}$. The terms that come from $S^{{}^{\prime}}$ describe the external
field, and the terms that come from $R_{k}$ describe the internal part of the
rotational component of the momentum field. The fundamental law of
conservation of probability

$\partial_{t}\rho+\frac{1}{m}\partial_{k}\rho M_{k}=0\mbox{,}$ (123)

with the probability current $v_{k}$ defined by $mv_{k}=M_{k}$, completes the
set of our basic quations. Let us note that the forces $\mathbf{F}$ and
$\mathbf{T}$ on the right-hand sides of (115) and (116) have a completely
different character. The form of $\mathbf{T}$ is given by the minimal coupling
mechanism [see Eq. (96)]. In contrast, the form of $\mathbf{F}$ need not be
specified, as will be shown in the next section.

###  8.2 Implementing the statistical conditions

The first statistical condition (114) is automatically satisfied due to the
continuity equation and the definition of $\bar{q}_{k}(t)$ and
$\bar{p}_{k}(t)$ . Using the continuity equation again and performing some
rearrangements, the second statistical condition (115) takes the form

$\displaystyle-\int\mathrm{d}q(\partial_{k}\rho)\left[M_{0}+\frac{1}{2m}M_{i}M_{i}\right]\;\;$
$\displaystyle+\int\mathrm{d}q\rho\left[\frac{1}{m}M_{i}\Omega_{ik}+\left(\partial_{t}M_{k}-\partial_{k}M_{0}\right)\right]$
$\displaystyle=\int\mathrm{d}q\rho F_{k}\mbox{,}$ (124)

where $\Omega_{ik}$ is the vorticity tensor defined by Eq. (12). We have not
specified the external forces that appear in our statistical theory. The
reason is that the form of these forces may be _derived_ as a consequence of
the special structure of our theory. The Ehrenfest-like relations (114)-(116)
are integral equations. Our task is to derive one or more differential
equations from these integral equations. The forces $\mathbf{F}$ on the right-
hand side of (115) must be designed to _allow_ this. That means there is a
“statistical constraint” on the forces in the present theory.

The concrete form of this restriction can already be seen in Eq. (124): The
permitted forces - that are compatible with the minimal coupling rule - must
already appear on the left-hand side of (124), namely in the form of
statistical mean values. Only such forces can be real, because they may cancel
with the same forces on the right-hand side, thus disappearing from the
integral equation and making he derivation of one or more differential
equations possible.

We now use the decomposition of $M_{0}$ and $M_{k}$ according to Eq. (120).
The square bracket in the second term of Eq. (124) takes the form:

$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\left[\frac{1}{m}M_{i}\Omega_{ik}+\left(\partial_{t}M_{k}-\partial_{k}M_{0}\right)\right]=$
$\displaystyle\;\;\;\;\;\;\;\;\frac{1}{m}(\partial_{i}S^{{}^{\prime}}+R_{i})\left[\partial_{i},\partial_{k}\right]S^{{}^{\prime}}+\left[\partial_{t},\partial_{k}\right]S^{{}^{\prime}}$
$\displaystyle+\frac{1}{m}(\partial_{i}S^{{}^{\prime}}+R_{i})\left(\partial_{i}R_{k}-\partial_{k}R_{i}\right)+\partial_{t}R_{k}-\partial_{k}R_{0}\mbox{,}$
(125)

where
$\left[\partial_{i},\partial_{k}\right]S^{{}^{\prime}}=\frac{e}{c}\left(\partial_{k}A_{i}-\partial_{i}A_{k}\right)$
and
$\left[\partial_{t},\partial_{k}\right]S^{{}^{\prime}}=-\frac{e}{c}\left(\partial_{t}A_{k}+c\partial_{k}\Phi\right)$.
Because of the well-known relations $B_{k}=\epsilon_{kij}\partial_{i}A_{j}$
and $E_{k}=-\frac{1}{c}\partial_{t}A_{k}-\partial_{k}\Phi$ the first term on
the right hand side of (125) is given by the Lorentz force

$F_{k}^{(L)}=eE_{k}+\frac{e}{c}v_{i}\left(\partial_{k}A_{i}-\partial_{i}A_{k}\right)\mbox{.}$
(126)

So we write $\mathbf{F}=\mathbf{F}^{(L)}+\mathbf{F}^{(2)}$ and skip the
Lorentz force from both sides of Eq. (124). The second statistical condition
now takes the form

$\displaystyle-\int\mathrm{d}q(\partial_{k}\rho)\left[M_{0}+\frac{1}{2m}M_{i}M_{i}\right]\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle+\int\mathrm{d}q\rho\left[+\frac{1}{m}(\partial_{i}S^{{}^{\prime}}+R_{i})\left(\partial_{i}R_{k}-\partial_{k}R_{i}\right)+\partial_{t}R_{k}-\partial_{k}R_{0}\right]$
$\displaystyle=\int\mathrm{d}q\rho F_{k}^{(2)}\mbox{,}$ (127)

We now express the vortical components $R_{0},\,R_{k}$ according to (122) in
terms of the Clebsch potentials and obtain after a few rearangements the
representation

$-\int\mathrm{d}q(\partial_{k}\rho)\left[M_{0}+\frac{1}{2m}M_{i}M_{i}\right]-\frac{\hbar}{2}\int\mathrm{d}q\rho\sin\vartheta\Big{\\{}(\partial_{k}\varphi)\mathrm{D}_{t}\vartheta-(\partial_{k}\vartheta)\mathrm{D}_{t}\varphi\Big{\\}}=\int\mathrm{d}q\rho
F_{k}^{(2)}\mbox{,}$ (128)

for the second statistical condition. At this point the third statistical
condition must be taken into account.

If we identify the minimal coupling force $\mathbf{T}$ with the right hand
side of (96) and use the continuity equation (123), the integral equation
(116) takes the form

$\int\mathrm{d}q\rho\left[\mathrm{D}_{t}s_{k}+\frac{e}{mc}\epsilon_{kij}B_{i}s_{j}\right]=0\mbox{.}$
(129)

The trivial solution of this equation (vanishing of the square bracket) agrees
with the quasi-classical field equation (96). The simplest nontrivial
differential equation to be associated with the integral equation (129) has
the form

$\mathrm{D}_{t}s_{k}+\frac{e}{mc}\epsilon_{kij}B_{i}s_{j}=\frac{\hbar}{2}G_{k}\mbox{,}$
(130)

where the $G_{k}$ are three functions with vanishing average value. We split
off the factor $\frac{\hbar}{2}$ so that the functions $G_{k}$ do not depend
on the parameter defining the length of the spin vector; the quantities
$G_{k}$ should only describe the influence of the randomization. If we insert
$s_{k}$ [see Eq. (95)] in (130) we obtain the differential equations for the
two independent field variables $\vartheta$ and $\varphi$,

$\displaystyle\mathrm{D}_{t}\vartheta=\frac{e}{mc}\left(B_{1}\cos\varphi-
B_{2}\sin\varphi\right)-\frac{G_{3}}{\sin\vartheta}\mbox{,}$ (131)
$\displaystyle\mathrm{D}_{t}\varphi=\frac{e}{mc}\frac{1}{\sin\vartheta}\left(-B_{1}\cos\vartheta\sin\varphi-
B_{2}\cos\vartheta\cos\varphi+B_{3}\sin\vartheta\right)+\frac{1}{\sin\vartheta}\left(G_{1}\cos\varphi-
G_{2}\sin\varphi\right)\mbox{.}$ (132)

The form of these differential equations is still undetermined since the
functions $G_{k}$ are not known. What we do know is that the $G_{k}$ have to
obey the conditions

$\int\mathrm{d}q\rho G_{k}=0,\;\;\;\;\;\;\;\;\;G_{k}s_{k}=0\mbox{.}$ (133)

The first of these says that the average value of $G_{k}$ has to vanish, the
second is a solvability condition that takes into account the fact that
$\mathbf{s}$ is a vector of constant length. With the help of (131) and (132)
we can now eliminate the total derivatives $\mathrm{D}_{t}\vartheta$ and
$\mathrm{D}_{t}\varphi$ from the second statistical condition (128).
Performing some rearrangements and a partial integration Eq. (128) takes the
form

$\displaystyle-\int\mathrm{d}q(\partial_{k}\rho)\left[M_{0}+\frac{1}{2m}M_{i}M_{i}+\mu_{i}B_{i}\right]-\int\mathrm{d}q\rho\mu_{i}\partial_{k}B_{i}\;\;\;\;\;\;\;$
$\displaystyle+\frac{\hbar}{2}\int\mathrm{d}q\rho\Big{\\{}G_{1}\cos\varphi\partial_{k}\vartheta-
G_{2}\sin\varphi\partial_{k}\vartheta+G_{3}\partial_{k}\varphi\Big{\\}}=\int\mathrm{d}q\rho
F_{k}^{(2)}$ , (134)

We see that the vortical part of the momentum field leads to the potential
energy term $\mu_{i}B_{i}$, the force $\mu_{i}\partial_{k}B_{i}$, and to the
integral depending on the $G_{i}$. Except for the prefactor, the newly derived
force agrees with the electrodynamic force exerted by an inhomogeneous
magnetic field on a magnetic dipole. As is well known, it plays a central role
in the interpretation of the Stern-Gerlach experiment. We eliminate the new
force by setting

$F_{k}^{(2)}=-\mu_{i}\partial_{k}B_{i}+F_{k}^{(3)}\mbox{.}$ (135)

The remaining force $F_{k}^{(3)}$ can only have the form $-\partial_{k}V$.
This leads to the usual mechanical potential $V$ in the field equation. Then,
the second statistical condition takes the form

$\displaystyle-\int\mathrm{d}q(\partial_{k}\rho)\left[M_{0}+\frac{1}{2m}M_{i}M_{i}+\mu_{i}B_{i}+V\right]\;\;\;\;\;\;\;\;\;$
$\displaystyle+\frac{\hbar}{2}\int\mathrm{d}q\rho\Big{\\{}G_{1}\cos\varphi\partial_{k}\vartheta-
G_{2}\sin\varphi\partial_{k}\vartheta+G_{3}\partial_{k}\varphi\Big{\\}}=0$ ,
(136)

The second integral in Eq. (136) must be a contribution to the field
equations. This implies a relationship of the form

$\frac{\hbar}{2}\int\mathrm{d}q\rho\Big{\\{}G_{1}\cos\varphi\partial_{k}\vartheta-
G_{2}\sin\varphi\partial_{k}\vartheta+G_{3}\partial_{k}\varphi\Big{\\}}=\int\mathrm{d}q(\partial_{k}\rho)L_{0}^{{}^{\prime}}\mbox{,}$
(137)

whereby $L_{0}^{{}^{\prime}}$ is an unknown function. Then the second
statistical condition takes the form

$-\int\mathrm{d}q(\partial_{k}\rho)\left[M_{0}+\frac{1}{2m}M_{i}M_{i}+\mu_{i}B_{i}+V-L_{0}^{{}^{\prime}}\right]=0\mbox{.}$
(138)

The simplest nontrivial differential equation whose solutions solve this
integral equation is given by

$M_{0}+\frac{1}{2m}M_{i}M_{i}+\mu_{i}B_{i}+V=L_{0}\mbox{,}$ (139)

where the function $L_{0}$ obeys the conditions

$L_{0}=L_{0}^{{}^{\prime}}+\Delta
L_{0},\;\;\;\;\;\;\;\;\int\mathrm{d}q(\partial_{k}\rho)\Delta L_{0}=0\mbox{.}$
(140)

We have thus transformed the second statistical condition, originally
formulated as an integral equation, into a partial differential equation. The
latter contains, however, $4$ unknown functions $G_{i},\,L_{0}$ which are
still to be determined.

###  8.3 Determination of the quantum terms $G_{i},\,L_{0}$

The first fundamental assumption that we made when constructing our type 3
theory was the validity of the Ehrenfest-like relations (114)-(116). This
assumption led us to the field equations (123), (139), (131), (132) for our
dynamic variables $\rho,\,S,\,\vartheta,\,\varphi$. These new equations differ
from the field equations of our original type 2 theory by terms
$G_{i},\,L_{0}$, which are unknown functions of our dynamic variables and
their derivatives. The conditions listed above are not sufficient to determine
the $G_{i},\,L_{0}$. We need a second assumption, presumably of a statistical
nature. A most fundamental statistical principle says that all states that are
unknown must occur with the same probability. In statistical mechanics (a type
2 theory) this principle is implemented through the requirement for maximum
entropy. In the problem at hand, we are faced with the task of determining
certain terms in a differential equation in accordance with this principle. In
this case, the same general principle leads to the requirement for minimal
Fisher information frieden:fisher_basis ,reginatto:pauli ,klein:statistical
,klein:nonrelativistic . In the following we will derive the terms implied by
the requirement of minimal Fisher information with the help of a variational
problem. Large parts of the related calculations have already been
reportedklein:nonrelativistic . In this regard, we may be brief here.

We introduce the abbreviation $L$ for the left hand side of Eq. (139) so that
this equation takes the form $L-L_{0}=0$. We assume that the function $L_{0}$
we are looking for depends only on the variables $\rho$, $\vartheta$,
$\varphi$, and their first and second derivatives with respect to $q_{k}$
klein:nonrelativistic . Our fundamental second statistical assumption is that
the spatial and temporal average value of $L-L_{0}$ is extremal with respect
to the variation of $S,\,\rho,\,\vartheta,\,\varphi$,

$\delta\int\mathrm{d}t\int\mathrm{d}q\rho\left(L-L_{0}\right)=0\mbox{.}$ (141)

Furthermore, the four field equations (123), (139), (131), (132) must also be
fulfilled; we write this condition symbolically for the sake of brevity in the
form

$E_{a}=0,\;\;\;a=\rho,\,S,\,\vartheta,\,\varphi\mbox{.}$ (142)

The two conditions (141) and (142) lead to differential equations for
$G_{i},\,L_{0}$. As shown in detail in klein:nonrelativistic , the physically
relevant solution of these equations is given by

$\displaystyle
L_{0}=\frac{\hbar^{2}}{2m}\bigg{[}\frac{1}{\sqrt{\rho}}\frac{\partial}{\partial
q_{k}}\frac{\partial}{\partial
q_{k}}\sqrt{\rho}-\frac{1}{4}\sum_{k}\bigg{\\{}\sin^{2}\vartheta\left(\frac{\partial\varphi}{\partial
q_{k}}\right)^{2}+\left(\frac{\partial\vartheta}{\partial
q_{k}}\right)^{2}\bigg{\\}}\bigg{]}\mbox{,}$ (143) $\displaystyle\hbar
G_{1}=\frac{\hbar^{2}}{2m}\frac{1}{\rho}\frac{\partial}{\partial
q_{k}}\rho\bigg{(}\frac{1}{2}\sin
2\vartheta\sin\varphi\frac{\partial\varphi}{\partial
q_{k}}-\cos\varphi\frac{\partial\vartheta}{\partial q_{k}}\bigg{)}\mbox{,}$
(144) $\displaystyle\hbar
G_{2}=\frac{\hbar^{2}}{2m}\frac{1}{\rho}\frac{\partial}{\partial
q_{k}}\rho\bigg{(}\frac{1}{2}\sin
2\vartheta\cos\varphi\frac{\partial\varphi}{\partial
q_{k}}+\sin\varphi\frac{\partial\vartheta}{\partial q_{k}}\bigg{)}\mbox{,}$
(145) $\displaystyle\hbar
G_{3}=-\frac{\hbar^{2}}{2m}\frac{1}{\rho}\frac{\partial}{\partial
q_{k}}\big{(}\rho\sin^{2}\vartheta\frac{\partial\varphi}{\partial
q_{k}}\big{)}\mbox{.}$ (146)

A new adjustable parameter appears on the right-hand-sides of these
expressions which has been identified with $\hbar^{2}/2m$. Let us recall here
that two different adjustable parameters appeared in the course of our
developments both of which were identified with $\hbar$. The first $\hbar$ was
associated with the length of the spin vector $\mathbf{s}$. The second $\hbar$
is associated with the quantum-mechanical principle of maximal disorder. The
physical meaning of these two adjustable parameters is different, but they
must both be identified with Planck’s constant in order to enable the
transition to QT.

As may be easily checked, the conditions (133), (137), (140) for the
$L_{0},\,G_{i}$ are all fulfilled. In particular, $L_{0}=L_{0}^{\prime}+\Delta
L_{0}$, where

$L_{0}^{\prime}=-\frac{\hbar^{2}}{8m}\bigg{[}\sum_{k}\bigg{\\{}\sin^{2}\vartheta\left(\frac{\partial\varphi}{\partial
q_{k}}\right)^{2}+\left(\frac{\partial\vartheta}{\partial
q_{k}}\right)^{2}\bigg{\\}}\bigg{]},\;\;\;\;\;\Delta
L_{0}=\frac{\hbar^{2}}{2m}\frac{1}{\sqrt{\rho}}\frac{\partial}{\partial
q_{k}}\frac{\partial}{\partial q_{k}}\sqrt{\rho}\mbox{,}$ (147)

and $L_{0}^{\prime}$ fulfills (137). One can also show that the mean value of
$L_{0}$ agrees with the Fisher functional reginatto:pauli ,
klein:nonrelativistic . If the above solution for $L_{0}$ is inserted, then
the variational principle (141) leads to the correct field equations (142). We
did not use the principle of variation here in the usual way, as a
mathematical tool to derive field equations from a given Lagrangian function,
but we use it instead to construct the Lagrangian itself.

Finally, we should compare the above field theory, which is based on
statistical postulates, with the earlier one generated by linearization. The
easily verifiable relations

$\displaystyle L_{S}=L_{0},$ (148) $\displaystyle
L_{\vartheta}=-\frac{1}{\sin\vartheta}G_{3},$ (149) $\displaystyle
L_{\varphi}=\frac{\cos\varphi}{\sin\vartheta}G_{1}-\frac{\sin\varphi}{\sin\vartheta}G_{2}\mbox{.}$
(150)

show that both theories are identical. We have thus shown that one may obtain
the Pauli equation

$\left[\frac{\hbar}{\imath}\partial_{t}-e\Phi-\frac{\hbar^{2}}{2m}\sum_{k=1}^{3}\,\left(\partial_{k}-\imath\frac{e}{\hbar
c}A_{k}\right)^{2}+\mu_{B}\sigma^{k}B_{k}+V\right]\psi=0\mbox{.}$ (151)

either through the “formal” process of linearization or through the
implementation of some plausible statistical postulates. The first way is a
discontinuous process that destroys the possibility of describing particle
motion and thus has the physical meaning of a randomization. With the second
way, we have replaced this discontinuous process with a continuous one. This
enabled us to understand the detailed nature of this randomization. The second
way is longer and “less elegant” than the first, but allows a more precise
insight into the transition from QA to QT.

##  9 Discussion

The second version of the HLLK reported here starts from PM, generates the
theory QA by projection onto configuration space, and then realizes the
transition to QT by a linearization or randomization. The theory QA, appearing
only in this second version of the HLLK, represents the transition area, the
“borderland”, between classical physics and QT berry:classical . This theory
is unphysical in that the particle trajectories that occur are only locally
valid and are not realized in nature (although they may be a good
approximation in certain situations). The importance of this theory is that it
represents a clearly defined and justifiable intermediate step in the
construction of the QT.

Let us first summarize what has been achieved so far by looking at the list of
essential properties of QT already given in II:

  1. 1.

Schrödinger’s equation as fundamental dynamical law - and eigenvalues as
observable numbers.

  2. 2.

The nonstandard probabilistic structure of QT - in particular non-commuting
observables.

  3. 3.

Born’s rule - the law which tells us how to extract probabilistic predictions
from the theory.

  4. 4.

The minimal-coupling rule - the way interactions are formulated in QT.

  5. 5.

The existence of spin - a particularly mysterious phenomenon believed to
belong to QT exclusively

  6. 6.

The anomalous value of the magnetic moment of the electron - a spin related
phenomenon

  7. 7.

The spin-statistics connection - a spin related multi-particle phenomenon

Points 1, 2 and 3 were derived in papers I and II. Without going into detail,
we mention that the generalization of Born’s rule to degenerate states may be
obtained in a straightforward way (the classical counterpart of degenerate
states are non-connected level sets). In contrast to I and II, only a single
observable, namely the Hamilton function $H(q,p)$, is studied in III and the
present work. Our research in the “borderland” created a new derivation of the
Schrödinger equation, which allows for a deeper understanding of the
relationship between QT and classical physics. We found, as most important
result of the present work, that spin is not a purely quantum mechanical
phenomenon and that the value $1/2$, which is experimentally observed for the
spin of all massive structureless particles, is a consequence of the three-
dimensionality of space. An associated result is the value $g=2$ for the
gyromagnetic ratio the electron. We have thus essentially understood points
$5$ and $6$. In the course of our second derivation of the Schrödinger
equation (in which statistical assumptions were used) we were also able to
understand why in QT the influence of external force fields must be described
with the help of potentials. Of course, we do not claim to have explained all
the details related to points $1-6$. The last point $7$, the spin-statistics
connection for massive particles, remains open at the moment, but we have
little doubt that this point can also be explained in the framework of the
HLLK. In the remainder of this section, we continue to discuss some relevant
points in more detail.

###  9.1 The meaning of spin

> Pls tell me why is spin of an electron +1/2 or -1/2 and not something like
> +1 or -1?
>
> \- Tridib Banerjee (age 16) Kolkata, India tridib:spin

In this work it was possible to identify the origin of spin. This property is
not, as has long been assumed, to be regarded as a purely quantum mechanical
phenomenon. Rather, its origin may be found in the quasi-classical (or quasi-
quantum) theory QA, which represents the borderland between classical physics
and QT. As this theory is not realized in nature it would not be correct to
claim that spin has a classical origin. There is no such phenomenon in phase
space (although it is formally possible to introduce corresponding degrees of
freedom) since no self-rotation of individual particles exists in nature.

The possibility of discovering the origin of spin arises by replacing the
particle momenta $p$ by momentum fields $M(q,t)$, and thus halving the number
of degrees of freedom. A necessary prerequisite for the occurrence of spin is
the simple fact that all degrees of freedom of the momentum fields $M(q,t)$
have to be taken into account, in particular those which result in a non-
vanishing rotation. The origin of the quantum spin is thus a collective
property of the probabilistic ensemble in configuration space, which is
associated with the rotational degrees of freedom. The idea that spin should
better be understood as a non-localized phenomenon has already been expressed
several times in the literature ohanian:spin , chuu:semiclassical .

A simple and extremely important property of nature is that spin-$1/2$
particles are fermions, which means they obey Fermi-Dirac statistics. When
asked about this fact, Richard Feynman suggested that there should be a simple
explanation for such a fundamental fact. And that if we cannot find such a
simple explanation, we should admit that we have not understood the
phenomenon. We cannot give an answer to Feynman’s question in the present
work. But we can answer an even more fundamental question, a question so
fundamental that it is hardly ever asked by professionals. Namely the
question, why all massive structureless particles in nature have spin $1/2$.
Based on the present reconstruction of QT for the case of a single particle,
the reason is the three-dimensionality of space. As a consequence of this
three-dimensionality we have to introduce _three_ functionally independent
potentials in order to represent the momentum field in a correct way. Together
with the probability density, we have four independent real functions that is
a $2$-component spinor, in other words a spin $1/2$ particle. This answer is
hard to beat in terms of simplicity.

According to the current state of knowledge in the theory of elementary
particles, (almost) all structureless massive particles are actually spin
$1/2$ particles. One would justifiably object that reality has a relativistic
space-time structure that is not correctly described by our non-relativistic
theory. But it is also a fact that our conclusion is essentially based only on
the number of spatial dimensions. It is therefore not unreasonable to assume
that our conclusion remains valid even when moving to a relativistic theory.

###  9.2 The limit $\hbar\to 0$ of the Pauli-Schrödinger equation

In this section we discuss the question of which theory the Pauli-Schrödinger
theory “reduces” to in the limit $\hbar\rightarrow 0$. There are two types of
reduction which Rosaler called formal and empirical reduction rosaler:formal ;
a similar distinction was made by Berry berry:classical . The important
difference between these two concepts was discussed in detail in III. Here we
are only examining the concept of formal reduction, that is, we are examining
the behavior of the basic equations of our theory in the limit of small
$\hbar$.

The above question was decided for a long time on the basis of philosophical
postulates. Also, no distinction was made between formal and empirical
reduction. The first mathematically precise investigation of the question of
formal reduction was carried out in klein:what  and II and led to the
conclusion that QT (without spin) cannot be reduced to either CM or PM. This
conclusion meant that - quite contrary to the prevailing opinion - QT cannot
(formally) be reduced to classical physics, but, on the contrary, it must be
possible to derive it from classical physics. This conclusion thus provided
the basis for the reconstruction of QT carried out in I-III and here. The
general conclusion that QT reduces for small $\hbar$ neither to CM nor to PM
but to a (quasi-classical) probabilistic theory in configuration space, does
not depend on whether we take spin into account or not. The presence of spin
leads however to some peculiarities, which we want to summarize in the
following.

The version of the Pauli-Schrödinger equation that is best suited for
examining the limit of small $\hbar$ is given by Eqs. (139), (130), (123) for
the potentials $S,\,\vartheta,\,\varphi$ and the probability density $\rho$.
To make the dependence on $\hbar$ visible, we write Eqs. (139), (130) in the
form

$\displaystyle\partial_{t}S+e\Phi+\frac{\hbar}{2}\cos\vartheta\,\partial_{t}\varphi+\frac{1}{2m}\sum_{k}\left(\partial_{k}S-\frac{e}{c}A_{k}+\frac{\hbar}{2}\cos\vartheta\,\partial_{k}\varphi\right)^{2}-\frac{\hbar}{2}\frac{e}{mc}s_{i}B_{i}=L_{0}\mbox{,}$
(152)
$\displaystyle\left[\partial_{t}+\frac{1}{m}\left(\partial_{i}S-\frac{e}{c}A_{i}+\frac{\hbar}{2}\cos\vartheta\
(\partial_{i}\varphi)\right)\partial_{i}\right]h_{k}+\frac{e}{mc}\epsilon_{kij}B_{i}h_{j}=G_{k}\mbox{,}$
(153)

where $L_{0}$ and $G_{k}$ are of the order $\hbar^{2}$ and $\hbar$,
respectively and the mechanical potential has been ommitted. We do not write
down the continuity equation, which behaves in an obvious way. If we set
$\hbar=0$ in these equations, we obtain the equations

$\displaystyle\partial_{t}S+e\Phi+\frac{1}{2m}\sum_{k}\left(\partial_{k}S-\frac{e}{c}A_{k}\right)^{2}=0\mbox{,}$
(154)
$\displaystyle\left[\partial_{t}+\frac{1}{m}\left(\partial_{i}S-\frac{e}{c}A_{i}\right)\partial_{i}\right]h_{k}+\frac{e}{mc}\epsilon_{kij}B_{i}h_{j}=0\mbox{,}$
(155)

which, together with the continuity equation, represent the classical limit of
the Pauli-Schrödinger equation. This is a classical (deterministic) field
theory defined by the Hamilton-Jacobi equation (154) for a spinless charged
particle in an electromagnetic field and the equation of motion (155) of the
variable $h_{k}$ which is associated with the vortical component of the
momentum field. The change of $h_{k}$ with time is determined by the solution
$S$ of (154), while conversely (154) does not depend on $h_{k}$ .

The “survival” of spin variables in the case $\hbar=0$ is no surprise in our
theory, since we have identified the vortical components of the momentum field
as origin of quantum spin. In all works in which Eqs. (152), (153) were
derived so far, the starting point was the quantum mechanical Pauli-
Schrödinger equation (151), which was then rewritten, using a representation
like (75) (see takabayasi:vortex , bohm_schiller_tiomno:causal ,
yahalom:fluid_spin ). The limiting case $\hbar=0$ was rarely dealt with in
these theories, see however yahalom:fluid_spin . One reason for this might be
that this limiting case is not compatible with the prevailing interpretation
of spin as a purely quantum mechanical phenomenon. Due to this interpretation,
all spin variables (or the corresponding terms in a Lagrangian function)
should disappear from the theory in the limit $\hbar=0$. The fact that this is
not the case led Yahalom to the conclusion that the Pauli theory “has no
standard classical limit” yahalom:fluid_spin . In fact, one could have
concluded from this fact that spin cannot be a purely quantum mechanical
phenomenon.

A second difficulty is as follows. If one wants to eliminate only terms of
order $\hbar^{2}$ in Eqs. (152), (153) and keep all terms of order $\hbar$,