# Old Game, New Rules: Rethinking The Form of Physics

C. Baumgarten  5244 Birrhard, Switzerland<EMAIL_ADDRESS>
###### Abstract

We investigate the modeling capabilities of sets of coupled classical harmonic
oscillators (CHO) in the form of a modeling game. The application of simple
but restrictive rules of the game lead to conditions for an isomorphism
between Lie-algebras and real Clifford algebras. We show that the correlations
between two coupled classical oscillators find their natural description in
the Dirac algebra and allow to model aspects of special relativity, inertial
motion, electromagnetism and quantum phenomena including spin in one go. The
algebraic properties of Hamiltonian motion of low-dimensional systems can
generally be related to certain types of interactions and hence to the
dimensionality of emergent space-times. We describe the intrinsic connection
between phase space volumes of a 2-dimensional oscillator and the Dirac
algebra. In this version of a phase space interpretation of quantum mechanics
the (components of the) spinor wavefunction in momentum space are abstract
canonical coordinates, and the integrals over the squared wave function
represents second moments in phase space. The wave function in ordinary space-
time can be obtained via Fourier transformation. Within this modeling game,
3+1-dimensional space-time is interpreted as a structural property of
electromagnetic interaction. A generalization selects a series of Clifford
algebras of specific dimensions with similar properties, specifically also 10-
and 26-dimensional real Clifford algebras.

Hamiltonian mechanics, Coupled Oscillators, Lorentz transformation, Dirac
equation

###### pacs:

45.20.Jj, 47.10.Df, 41.75, 41.85, 03.65.Pm, 05.45.Xt, 03.30.+p,
03.65.-w,29.27.-a

##  I Introduction

D. Hestenes had the joyful idea to describe physics as a modeling game
Modeling0 . We intend to play a modeling game with (ensembles of) classical
harmonic oscillators (CHO). The CHO is certainly one of the most discussed and
analyzed systems in physics and one of the few exactly solveable problems. One
would not expect any substantially new discoveries related to this subject.
Nevertheless there are aspects that are less well-known than others. One of
these aspects concerns the transformation group of the symplectic
transformations of $n$ coupled oscillators, $Sp(2n)$. We invite the reader to
join us playing “a modeling game” and to discover some fascinating features
related to possible reinterpretations of systems of two (or more) coupled
oscillators. We will show that special relativity can be reinterpreted as a
transformation theory of the second moments of the abstract canonical
variables of coupled oscillator systems 111 The connection of the Dirac
matrices to the symplectic group has been mentioned by Dirac in Ref. Dirac63 .
For the connection of oscillators and Lorentz transformations (LTs) see also
the papers of Kim and Noz KimNoz1981 ; KimNoz1982 ; KimNoz2003  and references
therein. The use of CHOs to model quantum systems has been recently described
- for instance - by Briggs and Eisfeld BriggEis .. We extend the application
beyond pure LTs and show that the Lorentz force can be reinterpreted by the
second moments of two coupled oscillators in proper time. Lorentz
transformations can be modeled as symplectic transformations KimNoz1982 . We
shall show how Maxwell’s equations find their place within the game.

The motivation for this game is to show that many aspects of modern physics
can be understood on the basis of the classical notions of harmonic
oscillation if these notions are appropriately reinterpreted.

In Sec. II we introduce the rules of our game, in Sec. III we introduce the
algebraic notions of the Hamilton formalism. In Sec. IV we describe how
geometry emerges from coupled oscillator systems, in Sec. V we describe the
use of symplectic transformations and introduce the Pauli- and Dirac algebra.
In Sec. VI we introduce a physical interpretation of oscillator moments and in
Sec. VIII we relate the phase space of coupled oscillators to the real Dirac
algebra. Sec. IX contains a short summary.

##  II The Rules Of The Game

The first rule of our game is the principle of reason (POR): No distinction
without reason \- we should not add or remove something specific (an
asymmetry, a concept, a distinction) from our model without having a clear and
explicite reason. If there is no reason for a specific asymmetry or choice,
then all possibilities are considered equivalently. The second rule is the
principle of variation (POV): We postulate that change is immanent to all
fundamental quantities in our game. From these two rules, we take that the
mathematical object of our theory is a list (n-tuple) of quantities
(variables) $\psi$, each of which varies at all times. The third rule is the
principle of objectivity (POO): Any law within this game refers to
measurements, defined as comparison of quantities (object properties) in
relation to other object properties of the same type (i.e. unit). Measurements
require rulers to enable for measurements. A measurement standard (ruler) has
to be objective, i.e. based on properties of the objects of the game. This
apparent self-reference is unavoidable, as it models the real situation of
physics as experimental science. Since all fundamental objects (quantities) in
our model vary at all times, the only option to construct a constant quantity
that might serve as a ruler, is given by constants of motion (COM). Hence the
principle of objectivity means that measurement standards are based on
constants of motion.

This third rule implies that the fundamental variables can not be directly
measured, but only functions of the fundemantal variables of the same
dimension (unit) of a COM. Thus the model has two levels: The level of the
fundamental variable list $\psi$, which is experimentally not directly
accessible and a level of observables which are (as we shall argue) even
moments of fundamental variables.

###  II.1 Discussion of the Rules

E.T. Jaynes wrote that “Because of their empirical origins, QM and QED are not
physical theories at all. In contrast, Newtonian celestial mechanics,
Relativity, and Mendelian genetics are physical theories, because their
mathematics was developed by reasoning out the consequences of clearly stated
physical principles from which constraint the possibilities”. And he continues
“To this day we have no constraining principle from which one can deduce the
mathematics of QM and QED; […] In other words, the mathematical system of the
present quantum theory is […] unconstrained by any physical principle” Jaynes
. This remarkably harsh criticism of quantum mechanics raises the question of
what we consider to be a physical principle. Are the rules of our game
physical principles? We believe that they are no substantial physical
principles but formal first principles, they are preconditions of a sensible
theory. They contain no immediate physical content, but they define the form
or the idea of physics.

It is to a large degree immanent to science and specifically to physics to
presuppose the existence of reason: Apples do not fall down by chance - there
is a reason for this tendency. Usually this believe in reason implies the
believe in causality, i.e. that we can also (at least in principle) explain
why a specific apple falls at a specific time, but practically this latter
believe can rarely be confirmed experimentally and therefore remains to some
degree metaphysical. Thus, if, as scientists, we postulate that things have
reason, then this is no physical principle but a precondition, a first
principle.

The second rules (POV), is specific to the form (or idea) of physics, e.g.
that it is the sense of physics to recognize the pattern of motion and to
predict future. Therefore the notion of time in the form of change is indeed
immanent to any physical description of reality. The principle of objectivity
is immanent to the very idea of physics: A measurement is the comparison of
properties of objects with compatible properties of reference objects, e.g.
requires “constant” rulers. Hence the rules of the game are to a large degree
unavoidable: They follow from the very form of physics and therefore certain
laws of physics are not substantial results of a physical theory. For instance
a consistent “explanation” of the stability of matter is impossible as we
presumed it already within the idea of measurement. More precisely: if this
presumption does not follow within the framework of a physical theory, then
the theory is fundamentally flawed, since it can not even reproduce it’s own
presumptions.

Einstein wrote with respect to relativity that “It is striking that the theory
(except for the four-dimensional space) introduces two kinds of things, i.e.
(1) measuring rods and clocks, (2) all other things, e.g., the electromagnetic
field, the material point, etc. This, in a certain sense, is inconsistent;
strictly speaking, measuring rods and clocks should emerge as solutions of the
basic equations […], not, as it were, as theoretically self-sufficient
entities.” EinsteinBio . The more it may surprise that the stability of matter
can not be obtained from classical physics as remarked by Elliott H. Lieb: “A
fundamental paradox of classical physics is why matter, which is held together
by Coulomb forces, does not collapse” Lieb . This single sentence seems to
rule out the possibility of a fundamental classical theory and uncovers the
uncomfortable situation of theoretical physics today: Despite the overwhelming
experimental and technological success, there is a deep-seated confusion
concerning the theoretical foundations. Our game is therefore a meta-
experiment. It is not the primary goal to find “new” laws of nature or new
experimental predictions, but it is a conceptional “experiment” that aims to
further develop our understanding of the consequences of principles: which
ones are really required to derive central “results” of contemporary physics.
In this short essay final answers can not be given, but maybe some new
insights are possible.

###  II.2 What about Space-Time?

A theory has to make the choice between postulate and proof. If a 3+1
dimensional space-time is presumed, then it cannot be proven within the same
theoretical framework. Or at least the value of such proof remains
questionable. This is a sufficient reason to refuse any postulates concerning
the dimensionality of space-time. Another - even stronger - reason to avoid a
direct postulate of space-time and its geometry has been given above: The
fundamental variables that we postulated above, can not be directly measured.
This excludes space-time coordinates as primary variables (which can be
directly measured), but with it almost all other apriori assumed concepts like
velocity, acceleration, momentum, energy and so on. At some point these
concepts certainly have to be introduced, but we suggest an approach to the
formation of concepts that differs from the Newtonian axiomatic method. The
POR does not allow to introduce distinctions between the fundamental variables
into coordinates and momenta without reason. Therefore we are forced to use an
interpretational method, which one might summarize as function follows form.
We shall first derive equations and then we shall interpret the equations
according to some formal criteria. This implies that we have to refer to
already existing notions if we want to identify quantities according to their
appearance within a certain formalism. The consequence for the game is, that
we have to show how to give rise to geometrical notions: If we do not
postulate space-time then we apparently have to construct it.

A consequence of our conception is that both, objects and fields have to be
identified with dynamical structures, as there is simply nothing else
available. This nicely fits to the framework of structure preserving
(symplectic) dynamics that follows from the described principles.

##  III Theory of Small Oscillations

In this section we shall derive the theory of coupled oscillators from the
rules of our game. According to the POO there exists a function (COM) ${\cal
H}(\psi)$ such that 222 Let us first (for simplicity) assume that
${\partial{\cal H}\over\partial t}=0$.:

${d{\cal H}\over dt}=\sum\limits_{k}\,{\partial{\cal
H}\over\partial\psi_{k}}\,\dot{\psi}_{k}=0\,,$ (1)

or in vector notation

${d{\cal H}\over dt}=(\nabla_{\psi}\,{\cal H})\,\cdot\dot{\psi}=0\,.$ (2)

The simplest solution is given by an arbitrary skew-symmetric matrix ${\cal
X}$:

$\dot{\psi}={\cal X}\,\nabla_{\psi}\,{\cal H}\,.$ (3)

Note that it is only the skew-symmetry of ${\cal X}$, which ensures that it is
always a solution to Eqn. 2 and which ensures that ${\cal H}$ is constant. If
we now consider a state vector $\psi$ of dimension $k$, then there is a
theorem in linear algebra, which states that for any skew-symmetric matrix
${\cal X}$ there exists a non-singular matrix ${\bf Q}$ such that we can write
MHO1 :

${\bf Q}^{T}\,{\cal X}\,{\bf
Q}=\textrm{diag}(\eta_{0},\eta_{0},\eta_{0},\dots\,,0,0,0)\,.$ (4)

where $\eta_{0}$ is the matrix

$\eta_{0}=\left(\begin{array}[]{cc}0&1\\\ -1&0\\\ \end{array}\right)\,.$ (5)

If we restrict us to orthogonal matrices ${\bf Q}$, then we may still write

${\bf Q}^{T}\,{\cal X}\,{\bf
Q}=\textrm{diag}(\lambda_{0}\,\eta_{0},\lambda_{1}\,\eta_{0},\lambda_{2}\,\eta_{0},\dots\,,0,0,0)\,.$
(6)

In both cases we may leave away the zeros, since they correspond to non-
varying variable, which is in conflict with the second rule of our modeling
game. Hence $k=2n$ must be even and the square matrix ${\cal X}$ has the
dimension $2n\times 2n$. As we have no specific reason to assume asymmetries
between the different degrees of freedom (DOF), we have to choose all
$\lambda_{k}=1$ in Eqn. 6 and return to Eqn. 4 without zeros and define the
block-diagonal so-called symplectic unit matrix (SUM) $\gamma_{0}$:

${\bf Q}^{T}\,{\cal X}\,{\bf
Q}=\textrm{diag}(\eta_{0},\eta_{0},\dots,\eta_{0})\equiv\gamma_{0}\,.$ (7)

These few basic rules thus lead us directly to Hamiltonian mechanics: Since
the state vector has even dimension and due to the form of $\gamma_{0}$, we
can interpret $\psi$ as an ensemble of $n$ classical DOF - each DOF
represented by a canonical pair of coordinate and momentum:
$\psi=(q_{1},p_{1},q_{2},p_{2},\dots\,,q_{n},p_{n})^{T}$. In this notation and
after the application of the transformation ${\bf Q}$, Eqn. 3 can be written
in form of the Hamiltonian equations of motion (HEQOM):

$\begin{array}[]{rcl}\dot{q}_{i}&=&{\partial{\cal H}\over\partial p_{i}}\\\
\dot{p}_{i}&=&-{\partial{\cal H}\over\partial q_{i}}\\\ \end{array}$ (8)

The validity of the HEQOM is of fundamental importance as it allows for the
use of the results of Hamiltonian mechanics, of statistical mechanics and
thermodynamics - but without the intrinsic presupposition that the $q_{i}$
have to be understood as positions in real space and the $p_{i}$ as the
corresponding canonical momenta. This is legitimate as the theory of canonical
transformations is independent from any specific physical interpretation of
what the coordinates and momenta represent physically. The canonical pairs are
coordinates $q_{i},p_{i}$ in an abstract phase space and they can be
interpreted as canonical coordinates and momenta due to the form of the HEQOM.
The choice of the specific form of $\gamma_{0}$ is for $n>1$ DOF not unique.
It could for instance be written as

$\gamma_{0}\equiv\eta_{0}\,\otimes\,{\bf 1}_{n\times n}\,,$ (9)

which corresponds a state vector of the form

$\psi=(q_{1},\dots,q_{n},p_{1},\dots,p_{n},)^{T}\,,$

or by

$\gamma_{0}\equiv{\bf 1}_{n\times n}\,\otimes\eta_{0}\,,$ (10)

as in Eq. 7. Therefore we are forced to make an arbitrary choice 333But we
should keep in mind, that other “systems” with a different choice are
possible. If we can not exclude their existence, then they should exist as
well. With respect to the form of the SUM, we suggest that different
“particle” types (different types of fermions for instance) have a different
SUM.. But in all cases the SUM $\gamma_{0}$ must be skew-symmetric and have
the following properties:

$\begin{array}[]{rcl}\gamma_{0}^{T}&=&-\gamma_{0}\\\ \gamma_{0}^{2}&=&-{\bf
1}\,,\end{array}$ (11)

which also implies that $\gamma_{0}$ is orthogonal and has unit determinant.
Note also that all eigenvalues of $\gamma_{0}$ are purely imaginary. However,
once we have chosen a specific form of $\gamma_{0}$, we have specified a set
of canonical pairs $(q_{i},p_{i})$ within the state vector. This choice fixes
the set of possible canonical (structure preserving) transformations.

Now we write the Hamiltonian ${\cal H}(\psi)$ as a Taylor series, we remove
the rule-violating constant term and cut it after the second term. We do not
claim that higher terms may not appear, but we delay the discussion of higher
orders to a later stage. All this is well-known in the theory of small
oscillations. There is only one difference to the conventional treatment: We
have no direct macroscopic interpretation for $\psi$ and following our first
rule we have to write the second-order Hamiltonian ${\cal H}(\psi)$ in the
most general form:

${\cal H}(\psi)=\frac{1}{2}\,\psi^{T}\,{\cal A}\,\psi\,,$ (12)

where ${\cal A}$ is only restricted to be symmetric as all non-symmetric terms
do not contribute to ${\cal H}$. Since it is not unlikely to find more than a
single constant of motion in systems with multiple DOFs, we distinguish
systems with singular matrix ${\cal A}$ from those with a positive or negative
definite matrix ${\cal A}$. Positive definite matrices are favoured in the
sense that they allow to identify ${\cal H}$ with the amount of a substance or
an amount of energy 444It is immanent to the concept of substance that it is
understood as something positive semidefinite..

Before we try to interprete the elements in ${\cal A}$, we will explore some
general algebraic properties of the Hamiltonian formalism. If we plug Eqn. 12
into Eqn. 3, then the equations of motion can be written in the general form

$\dot{\psi}=\gamma_{0}\,{\cal A}\,\psi={\bf F}\,\psi\,.$ (13)

The matrix ${\bf F}=\gamma_{0}\,{\cal A}$ is the product of the symmetric
(positive semi-definite) matrix ${\cal A}$ and the skew-symmetric matrix
$\gamma_{0}$. As known from linear algebra, the trace of such products is
zero:

$\mathrm{Tr}({\bf F})=0\,.$ (14)

It is obvious that pure harmonic oscillation of $\psi$ is given for matrices
${\bf F}$ that have purely imaginary eigenvalues. Furthermore these are the
only stable solutions MHO . Note that Eq. 13 may represent a tremendous amount
of different types of systems - all linearily coupled systems in any
dimension, chains or $d$-dimensional lattices of linear coupled oscillators
and wave propagation 555However the linear approximation does not allow for
the description of the transport of heat..

One quickly derives from the properties of $\gamma_{0}$ and ${\cal A}$ that

${\bf F}^{T}={\cal A}^{T}\,\gamma_{0}^{T}=-{\cal
A}\,\gamma_{0}=\gamma_{0}^{2}\,{\cal A}\,\gamma_{0}=\gamma_{0}\,{\bf
F}\,\gamma_{0}\,.$ (15)

Since any square matrix can be written as the sum of a symmetric and a skew-
symmetric matrix, it is nearby to also consider the properties of products of
$\gamma_{0}$ with a skew-symmetric real square matrices ${\cal B}$. If ${\bf
C}=\gamma_{0}\,{\cal B}$, then

${\bf C}^{T}={\cal B}^{T}\,\gamma_{0}^{T}={\cal
B}\,\gamma_{0}=-\gamma_{0}^{2}\,{\cal B}\,\gamma_{0}=-\gamma_{0}\,{\bf
C}\,\gamma_{0}\,.$ (16)

Symmetric $2n\times 2n$-matrices contain $2n\,(2n+1)/2$ different matrix
elements and skew-symmetric ones $2n\,(2n-1)/2$ elements, so that there are
$\nu_{s}$ linear independent symplices

$\nu_{s}=n\,(2n+1)$ (17)

and $\nu_{c}$ cosymplices with

$\nu_{c}=n\,(2n-1)\,.$ (18)

In the theory of linear Hamiltonian dynamics, matrices of the form of ${\bf
F}$ are known as “Hamiltonian” or “infinitesimal symplectic” and those of the
form of ${\bf C}$ as “skew-Hamiltonian” matrices. This convention is a bit odd
as ${\bf F}$ does not appear in the Hamiltonian and it is in general not
symplectic. Furthermore the term “Hamiltonian matrix” has a different meaning
in quantum mechanics - more in analogy to ${\cal A}$. But it is known that
this type of matrix is closely connected to symplectic matrices as every
symplectic matrix is a matrix exponential of a matrix ${\bf F}$ MHO . We
consider the matrices as defined by Eqn. 15 and Eqn. 16 as too important and
fundamental to have no meaningful and unique names: Therefore we speak of a
symplex (plural symplices), if a matrix holds Eqn. 15 and of a cosymplex if it
holds Eqn. 16.

###  III.1 Symplectic Motion and Second Moments

So what is a symplectic matrix anyway? The concept of symplectic
transformations is a specific formulation of the theory of canonical
transformations. Consider we define a new state vector (or new coordinates)
$\phi(\psi)$ \- with the additional requirement, that the transformation is
reversible. Then the Jacobian matrix of the transformation is given by

${\bf J}_{ij}=\left({\partial\phi_{i}\over\partial\psi_{j}}\right)\,,$ (19)

and the transformation is said to be symplectic, if the Jacobian matrix holds
MHO

${\bf J}\,\gamma_{0}\,{\bf J}^{T}=\gamma_{0}\,.$ (20)

Let us see what this implies in the linear case:

$\begin{array}[]{rcl}{\bf J}\,\dot{\psi}&=&{\bf J}\,{\bf F}\,{\bf
J}^{-1}\,{\bf J}\,\psi\\\ \tilde{\psi}&=&{\bf J}\,\psi\\\
\dot{\tilde{\psi}}&=&{\bf J}\,{\bf F}\,{\bf J}^{-1}\,\tilde{\psi}\\\
\dot{\tilde{\psi}}&=&{\bf\tilde{F}}\,\tilde{\psi}\\\ \end{array}$ (21)

and - by the use of Eqn. 20 one finds that ${\bf\tilde{F}}$ is still a
symplex:

$\begin{array}[]{rcl}{\bf\tilde{F}}^{T}&=&({\bf J}^{-1})^{T}\,{\bf
F}^{T}\,{\bf J}^{T}\\\ {\bf\tilde{F}}^{T}&=&({\bf
J}^{-1})^{T}\,\gamma_{0}\,{\bf F}\,\gamma_{0}\,{\bf J}^{T}\\\
{\bf\tilde{F}}^{T}&=&-\gamma_{0}^{2}\,({\bf J}^{-1})^{T}\,\gamma_{0}\,{\bf
F}\,{\bf J}^{-1}\,\gamma_{0}\\\ {\bf\tilde{F}}^{T}&=&-\gamma_{0}\,{\bf
J}\,\gamma_{0}^{2}\,{\bf F}\,{\bf J}^{-1}\,\gamma_{0}\\\
{\bf\tilde{F}}^{T}&=&\gamma_{0}\,{\bf J}\,{\bf F}\,{\bf J}^{-1}\,\gamma_{0}\\\
{\bf\tilde{F}}^{T}&=&\gamma_{0}\,{\bf\tilde{F}}\,\gamma_{0}\\\ \end{array}$
(22)

Hence a symplectic transformation is first of all a similarity transformation,
but secondly, it preserves the structure of all involved equations. Therefore
the transformation is said to be canonical or structure preserving. The
distinction between canonical and non-canonical transformations can therefore
be traced back to the skew-symmetry of $\gamma_{0}$ and the symmetry of ${\cal
A}$ \- both of them consequences of the rules of our physics modeling game.

Recall that we argued that the matrix ${\cal A}$ should be symmetric because
skew-symmetric terms do not contribute to the Hamiltonian. Let us have a
closer look what this means. Consider the matrix of second moments $\Sigma$
that can be build from the variables $\psi$:

$\Sigma\equiv\langle\psi\,\psi^{T}\rangle\,,$ (23)

in which the angles indicate some (yet unspecified) sort of average. The
equation of motion of this matrix is given by

$\begin{array}[]{rcl}\dot{\Sigma}&=&\langle\dot{\psi}\,\psi^{T}\rangle+\langle\psi\,\dot{\psi}^{T}\rangle\\\
\dot{\Sigma}&=&\langle{\bf
F}\,\psi\,\psi^{T}\rangle+\langle\psi\,\psi^{T}\,{\bf
F}^{T}\rangle\,.\end{array}$ (24)

Now, as long as ${\bf F}$ does not depend on $\psi$, we obtain

$\begin{array}[]{rcl}\dot{\Sigma}&=&{\bf F}\,\Sigma+\Sigma\,{\bf F}^{T}\\\
\dot{\Sigma}&=&{\bf F}\,\Sigma+\Sigma\,\gamma_{0}\,{\bf F}\,\gamma_{0}\\\
(\dot{\Sigma}\,\gamma_{0})&=&{\bf
F}\,(\Sigma\,\gamma_{0})-(\Sigma\,\gamma_{0})\,{\bf F}\,,\end{array}$ (25)

so that with the matrix ${\bf S}\equiv\Sigma\,\gamma_{0}$ this yields:

$\begin{array}[]{rcl}{\bf\dot{S}}&=&{\bf F}\,{\bf S}-{\bf S}\,{\bf
F}\,.\end{array}$ (26)

This equation is a key to our theory. Firstly it contains only observables:
though we started with unmeasurable abstract quantities, we reach here a stage
in which the original quantities $\psi$ are completely hidden. In accelerator
physics this equation is called envelope equation as it describes the dynamics
of the so-called “envelope” of the beam, which is nothing but a
parametrization by statistical (here: second) moments.

For completeness we introduce the “adjunct” spinor
$\bar{\psi}=\psi^{T}\,\gamma_{0}$ so that we may write

${\bf S}=\langle\,\psi\bar{\psi}\,\rangle\,.$ (27)

Note that ${\bf S}$ is also a symplex. The matrix ${\bf S}$ (i.e. all second
moments) is constant, iff ${\bf S}$ and ${\bf F}$ commute.

Now we define an observable to be an operator ${\bf O}$ with a (potentially)
non-vanishing expectation value, defined by:

$\langle{\bf O}\rangle\equiv\langle\bar{\psi}{\bf
O}\psi\rangle=\langle\psi^{T}\,\gamma_{0}\,{\bf O}\psi\rangle\,.$ (28)

Thus, if the product $\gamma_{0}\,{\bf O}$ is not skew-symmetric, i.e.
contains a product of $\gamma_{0}$ with a symmetric matrix ${\cal B}$, then
the expectation value is potentially non-zero:

$\langle{\bf O}\rangle\equiv\langle\psi^{T}\,\gamma_{0}\,(\gamma_{0}\,{\cal
B})\psi\rangle=-\langle\psi^{T}\,{\cal B}\,\psi\rangle\,.$ (29)

This means that only the symplex-part of an operator is “observable”, while
cosymplices yield a vanishing expectation value. Hence Eq. 26 delivers the
blueprint for the general definition of observables. Furthermore we find in
the last line the constituting equation for Lax pairs Lax . Peter Lax has
shown that for such pairs of operators ${\bf S}$ and ${\bf F}$ that obey Eqn.
26 there are the following constants of motion

$\mathrm{Tr}({\bf S}^{k})=\mathrm{const}\,.$ (30)

for arbitrary integer $k>0$. Since ${\bf S}$ is a symplex and therefore by
definition the product of a symmetric matrix and the skew-symmetric
$\gamma_{0}$, Eqn. 30 is always zero and hence trivially true for $k=1$. The
same is true for any odd power of ${\bf S}$, as it can be easily shown that
any odd power of a symplex is again a symplex (see Eq. LABEL:eq_cosy_algebra),
so that the only non-trivial general constants of motion correspond to even
powers of ${\bf S}$, which implies that all observables are functions of even
powers of the fundamental variables.

To see the validity for $k>1$ we have to consider the general algebraic
properties of the trace operator. Let $\lambda$ be an arbitrary real constant
and $\tau$ be a real parameter, then

$\begin{array}[]{rcl}\mathrm{Tr}({\bf A})&=&\mathrm{Tr}({\bf A}^{T})\\\
\mathrm{Tr}(\lambda\,{\bf A})&=&\lambda\,\mathrm{Tr}({\bf A})\\\ {d\over
d\tau}\mathrm{Tr}({\bf A}(\tau))&=&\mathrm{Tr}({d{\bf A}\over d\tau})\\\
\mathrm{Tr}({\bf A}+{\bf B})&=&\mathrm{Tr}({\bf A})+\mathrm{Tr}({\bf B})\\\
\mathrm{Tr}({\bf A}\,{\bf B})&=&\mathrm{Tr}({\bf B}\,{\bf A})\,.\end{array}$
(31)

It follows that

$\begin{array}[]{rcl}0&=&\mathrm{Tr}({\bf A}\,{\bf B}-{\bf B}\,{\bf A})\\\
0&=&\mathrm{Tr}({\bf A}^{n}\,{\bf B}-{\bf A}^{n-1}\,{\bf B}\,{\bf A})\\\
0&=&\mathrm{Tr}\left[{\bf A}^{n-1}\,({\bf A}\,{\bf B}-{\bf B}\,{\bf
A})\right]\\\ \end{array}$ (32)

From the last line of Eqn. LABEL:eq_lax0 follows for ${d{\bf A}\over
d\tau}=\lambda\,({\bf A}\,{\bf B}-{\bf B}\,{\bf A})$

${d\over d\tau}\mathrm{Tr}({\bf A}^{n})=0$ (33)

Remark: This conclusion is not limited to symplices.

However for single spinors $\psi$ and their second moments ${\bf
S}=\Sigma\,\gamma_{0}=\psi\psi^{T}\,\gamma_{0}$ we find:

$\begin{array}[]{rcl}\mathrm{Tr}({\bf
S}^{k})&=&\mathrm{Tr}[\psi\,\psi^{T}\,\gamma_{0}\cdots\psi\,\psi^{T}\,\gamma_{0}]\\\
&=&\mathrm{Tr}[\psi\,(\psi^{T}\,\gamma_{0}\cdots\psi\,\psi^{T}\,\gamma_{0})]\\\
&=&\mathrm{Tr}[(\psi^{T}\,\gamma_{0}\cdots\psi\,\psi^{T}\,\gamma_{0})\,\psi]\\\
&=&\mathrm{Tr}\left[(\psi^{T}\,\gamma_{0}\,\psi)\cdots(\psi^{T}\gamma_{0}\psi)\right]=0\\\
\end{array}$ (34)

since each single factor $(\psi^{T}\,\gamma_{0}\,\psi)$ vanishes due to the
skew-symmetry of $\gamma_{0}$. Therefore the constants of motion as derived
from Eqn. 30 are non-zero only for even $k$ and after averaging over some kind
of distribution such that ${\bf S}=\langle\psi\psi^{T}\,\gamma_{0}\rangle$ has
non-zero eigenvalues as in Eq. 35 below.

The symmetric matrix $2n\times 2n$-matrix $\Sigma$ (and also ${\cal A}$) is
positive definite, if it can be written as a product $\Sigma=\Psi\Psi^{T}$,
where $\Psi$ is a non-singular matrix of size $2n\times m$ with $m\geq 2n$.

For $n=m/2=1$, the form of $\Psi$ may be chosen as

$\begin{array}[]{rcl}\Psi&=&{1\over\sqrt{q^{2}+p^{2}}}\,\left(\begin{array}[]{cc}q&-p\\\
p&q\\\ \end{array}\right)={1\over\sqrt{q^{2}+p^{2}}}\,({\bf
1}\psi,\eta_{0}\,\psi)\\\ \Rightarrow&&\Sigma=\Psi\Psi^{T}=\Psi^{T}\,\Psi={\bf
1}\\\ {\bf S}&=&\gamma_{0}\end{array}$ (35)

so that for $k=2$ the average of two “orthogonal” column-vectors $\psi$ and
$\eta_{0}\,\psi$ gives a non-zero constant of motion via Lax pairs as
$\gamma_{0}^{2}=-{\bf 1}$.

These findings have numerous consequences for the modeling game. The first is
that we have found constants of motion - though some of them are physically
meaningful only for a non-vanishing volume in phase space, i.e. by the
combination of several spinors $\psi$. Secondly, a stable state
${\bf\dot{S}}=0$ implies that the matrix operators forming the Lax pair have
the same eigenvectors: a density distribution in phase space (as described by
the matrix of second moments) is stable if it is adapted or matched to the
symplex ${\bf F}$. The phase space distribution as represented by ${\bf S}$
and the driving terms (the components of ${\bf F}$) must fit to each other in
order to obtain a stable “eigenstate”. But we also found a clear reason, why
generators (of symplectic transformations) are always observables and vice
versa: Both, the generators as well as the observables are symplices of the
same type. There is a one-to-one correspondence between them, not only as
generators of infinitesimal transformations, but also algebraically.

Furthermore, we may conclude that (anti-) commutators are an essential part of
“classical” Hamiltonian mechanics and secondly that the matrix ${\bf S}$ has
the desired properties of observables: Though ${\bf S}$ is based on
continuously varying fundamental variables, it is constant, if it commutes
with ${\bf F}$, and it varies otherwise 666In accelerator physics, Eqn. 26
describes the envelope of a beam in linear optics. The matrix of second
moments $\Sigma$ is a covariance matrix - and therefore our modeling game is
connected to probability theory exactly at the stage where we define
observables..

Hence it appears sensible to take a closer look on the (anti-) commutation
relations of (co-) symplices and though the definitions of (co-) symplices are
quite plain, the (anti-) commutator algebra that emerges from them has a
surprisingly rich structure. If we denote symplices by ${\bf S}_{k}$ and
cosymplices by ${\bf C}_{k}$, then the following rules can quickly be derived:

$\begin{array}[]{ccc}\left.\begin{array}[]{c}{\bf S}_{1}\,{\bf S}_{2}-{\bf
S}_{2}\,{\bf S}_{1}\\\ {\bf C}_{1}\,{\bf C}_{2}-{\bf C}_{2}\,{\bf C}_{1}\\\
{\bf C}\,{\bf S}+{\bf S}\,{\bf C}\\\ {\bf S}^{2\,n+1}\\\
\end{array}\right\\}&\Rightarrow&\mathrm{symplex}\\\ &&\\\
\left.\begin{array}[]{c}{\bf S}_{1}\,{\bf S}_{2}+{\bf S}_{2}\,{\bf S}_{1}\\\
{\bf C}_{1}\,{\bf C}_{2}+{\bf C}_{2}\,{\bf C}_{1}\\\ {\bf C}\,{\bf S}-{\bf
S}\,{\bf C}\\\ {\bf S}^{2\,n}\\\ {\bf C}^{n}\\\
\end{array}\right\\}&\Rightarrow&\mathrm{cosymplex}\\\ \end{array}$ (36)

This Hamiltonian algebra of (anti-)commutators is of fundamental importance
insofar as we derived it in a few steps from first principles (i.e. the rules
of the game) and it defines the structure of Hamiltonian dynamics in phase
space. The distinction between symplices and cosymplices is also the
distinction between observables and non-observables. It is the basis of
essential parts of the following considerations.

##  IV Geometry from Hamiltonian Motion

In the following we will demonstrate the geometrical content of the algebra of
(co-)symplices (Eqn. LABEL:eq_cosy_algebra) which emerges for specific numbers
of DOF $n$. As shown above pairs of canonical variables (DOFs) are the a
direct consequence of the abstract rules of our game. Though single DOFs are
poor “objects”, it is remarkable to find physical structures emerging from our
abstract rules at all. This suggests that there might be more structure to
discover when $n$ DOF are combined, for instance geometrical structures. The
following considerations obey the rules of our game, since they are based
purely on symmetry considerations like those that guided us towards
Hamiltonian dynamics. The objects of interest in our algebraic interpretation
of Hamitonian dynamics are matrices. The first matrix (besides ${\cal A}$)
with a specific form that we found, is $\gamma_{0}$. It is a symplex:

$\gamma_{0}^{T}=-\gamma_{0}=\gamma_{0}\,\gamma_{0}\,\gamma_{0}\,.$ (37)

According to Eq. 17 there are $\nu_{s}=n\,(2\,n+1)$ (i.e. $\nu_{s}\geq 3$)
symplices. Hence it is nearby to ask if other symplices with similar
properties like $\gamma_{0}$ exist - and if so, what the relations between
these matrices are. According to Eq. LABEL:eq_cosy_algebra the commutator of
two symplices is again a symplex, while the anti-commutator is a cosymplex.
Hence - as we are interested in observables and components of the Hamiltonians
(i.e. symplices), respectively, we would like to find other symplices that
anti-commute with $\gamma_{0}$ and with each other. In this case, the product
of two such matrices is also a symplex, i.e. another potential contribution to
the general Hamiltonian matrix ${\bf F}$.

Assumed we had a set of $N$ mutually anti-commuting orthogonal symplices
$\gamma_{0}$ and $\gamma_{k}$ with $k\in\,[1\dots N-1]$, then a Hamiltonian
matrix ${\bf F}$ might look like

$\begin{array}[]{rcl}{\bf
F}=\sum\limits_{k=0}^{N-1}\,f_{k}\,\gamma_{k}+\dots\,.\end{array}$ (38)

The $\gamma_{k}$ are symplices and anti-commute with $\gamma_{0}$:

$\gamma_{0}\,\gamma_{k}+\gamma_{k}\,\gamma_{0}=0\,.$ (39)

Multiplication from the left with $\gamma_{0}$ gives:

$-\gamma_{k}+\gamma_{0}\,\gamma_{k}\,\gamma_{0}=-\gamma_{k}+\gamma_{k}^{T}=0\,.$
(40)

Hence all other possible symplices $\gamma_{k}$ that anticommute with
$\gamma_{0}$ are symmetric - and hence they square to ${\bf 1}$ as they are
also assumed to be orthogonal. This is an extremely important finding for what
follows, as it can (within our game) be interpreted as a classical proof of
the uniqueness of (observable) time-dimension: Time is one-dimensional as
there is no other skew-symmetric symplex that anti-commutes with $\gamma_{0}$.
We can choose different forms for $\gamma_{0}$, but the emerging algebra
allows for no second “direction of time”.

The second order derivative of $\psi$ is (for constant ${\bf F}$) given by
$\ddot{\psi}={\bf F}^{2}\,\psi$ which yields:

$\begin{array}[]{rcl}{\bf
F}^{2}=\sum\limits_{i=0}^{N-1}\,f_{i}^{2}\,\gamma_{i}^{2}+\sum\limits_{i\neq
j}\,f_{i}\,f_{j}\,(\gamma_{i}\,\gamma_{j}+\gamma_{j}\,\gamma_{i})\,.\end{array}$
(41)

Since the matrices on the right anticommute, we are left with:

$\begin{array}[]{rcl}{\bf
F}^{2}=\left(\sum\limits_{k=1}^{N-1}\,f_{k}^{2}-f_{0}^{2}\right)\,{\bf
1}\,.\end{array}$ (42)

Thus - we find a set of (coupled) oscillators, if

$f_{0}^{2}>\sum\limits_{k=1}^{N-1}\,f_{k}^{2}\,,$ (43)

such that

$\ddot{\psi}=-\omega^{2}\,\psi\,.$ (44)

Given such matrix systems exist - then they generate a Minkowski type “metric”
as in Eq. 42 777 Indeed it appears that Dirac derived his system of matrices
from the this requirement DiracMatrices .. The appearance of such a metric may
guide us towards further aspects of physics to be modeled. It should be
possible to construct a Minkowski type geometry from the driving terms of
oscillatory motion. This is indeed possible - at least for symplices of
certain dimensions as we will show below. The first thing needed is some kind
of measure to define the length of a “vector”. Since the length is a measure
that is invariant under certain transformations, specifically under rotations,
we prefer to use a quantity with certain invariance properties to define a
length. The only one we have at hand is given by Eqn. 30. Accordingly we
define the (squared) length of a matrix representing a “vector” by

$\|{\bf A}\|^{2}\equiv\frac{1}{2\,n}\,\mathrm{Tr}({\bf A}^{2})\,.$ (45)

The division by $2\,n$ is required to make the unit matrix have unit norm.
Besides the norm we need a scalar product, i.e. a definition of orthogonality.
Consider the Pythagorean theorem which says that two vectors $\vec{a}$ and
$\vec{b}$ are orthogonal iff

$(\vec{a}+\vec{b})^{2}=\vec{a}^{2}+\vec{b}^{2}\,.$ (46)

The general expression is

$(\vec{a}+\vec{b})^{2}=\vec{a}^{2}+\vec{b}^{2}+2\,\vec{a}\cdot\vec{b}\,.$ (47)

The equations are equal, iff $\vec{a}\cdot\vec{b}=0$. Hence the Pythagorean
theorem yields a reasonable definition of orthogonality. However, we had no
method yet to define vectors within our game. Using matrices ${\bf A}$ and
${\bf B}$ we may then write

$\begin{array}[]{rcl}\|{\bf A}+{\bf
B}\|^{2}&=&\frac{1}{2\,n}\,\mathrm{Tr}\left[({\bf A}+{\bf B})^{2}\right]\\\
&=&\|{\bf A}\|^{2}+\|{\bf B}\|^{2}+\frac{1}{2\,n}\,\mathrm{Tr}({\bf A}{\bf
B}+{\bf B}{\bf A})\,.\end{array}$ (48)

If we compare this to Eqn. 46 and Eqn. 47, respectively, then the obvious
definition of the inner product can be defined as follows:

${\bf A}\cdot{\bf B}\equiv{{\bf A}\,{\bf B}+{\bf B}\,{\bf A}\over 2}$ (49)

Since the anticommutator does in general not yield a scalar, we have to
distinguish between inner product and scalar product:

$({\bf A}\cdot{\bf B})_{S}\equiv\frac{1}{4\,n}\,\mathrm{Tr}({\bf A}\,{\bf
B}+{\bf B}\,{\bf A})\,,$ (50)

where we indicate the scalar part by the subscript “$S$”. Accordingly we
define the exterior product by the commutator

${\bf A}\wedge{\bf B}\equiv{{\bf A}\,{\bf B}-{\bf B}\,{\bf A}\over 2}\,.$ (51)

Now that we defined the products, we should come back to the unit vectors. The
only “unit vector” that we explicitely defined so far is the symplectic unit
matrix $\gamma_{0}$. If it represents anything at all then it must be “the
direction” of change, the direction of evolution in time as it was derived in
this context and is the only “dimension” found so far. As we have already
shown, all other unit vectors $\gamma_{k}$ must be symmetric, if they are
symplices. And vice versa: If $\gamma_{k}$ is symmetric and anti-commutes with
$\gamma_{0}$, then it is a symplex. As only symplices represent observables
and are generators of symplectic transformations, we can have only a single
“time” direction $\gamma_{0}$ and a yet unknown number of symmetric unit
vectors 888 Thus we found a simple answer to the question, why only a single
time direction is possible, a question also debated in Ref. BN2000 . However,
for $n>1$, there might be different equivalent choices of $\gamma_{0}$.
Whatever the specific form of $\gamma_{0}$ is, we will show that in
combination with some general requirements like completeness, normalizability
and observability it determines the structure of the complete algebra. Though
we don’t yet know how many symmetric and pairwise anti-commuting unit vectors
$\gamma_{k}$ exist - we have to interpret them as unit vectors in “spatial
directions” 999The meaning of what a spatial direction is, especially in
contrast to the direction of time $\gamma_{0}$, has to be derived from the
form of the emerging equations, of course. As meaning follows form, we do not
define space-time, but we identify structures that fit to the known concept of
space-time.. Of course unit vectors must have unit length, so that we have to
demand that

$\begin{array}[]{rcl}\|\gamma_{k}\|^{2}={1\over
2\,n}\mathrm{Tr}(\gamma_{k}^{2})=\pm 1\,.\end{array}$ (52)

Note that (since our norm is not positive definite), we explicitely allow for
unit vectors with negative “length” as we find it for $\gamma_{0}$. Note
furthermore that all skew-symmetric unit vectors square to $-{\bf 1}$ while
the symmetric ones square to ${\bf 1}$ qed_paper .

Indeed systems of $N=p+q$ anti-commuting real matrices are known as real
representations of Clifford algebras $Cl_{p,q}$. The index $p$ is the number
of unit elements (“vectors”) that square to $+1$ and $q$ is the number of unit
vectors that square to $-1$. Clifford algebras are not necessarily connected
to Hamiltonian motion, rather they can be regarded as purely mathematical
“objects”. They can be defined without reference to matrices whatsoever. Hence
in mathematics, sets of matrices are merely “representations” of Clifford
algebras. But our game is about physics and due to the proven one-
dimensionality of time we concentrate on Clifford algebras $Cl_{N-1,1}$ which
link CHOs in the described way with the generators of a Minkowski type metric.
Further below it will turn out that the representation by matrices is - within
the game - indeed helpful, since it leads to an overlap of certain symmetry
structures. The unit elements (or unit “vectors”) of a Clifford algebra, ${\bf
e}_{k}$, are called the generators of the Clifford algebra. They pairwise
anticommute and they square to $\pm{\bf 1}$ 101010 The role as generator of
the Clifford algebra should not be confused with the role as generators of
symplectic transformations (i.e. symplices). Though we are especially
interested in Clifford algebras in which all generators are symplices, not all
symplices are generators of the Clifford algebra. Bi-vectors for instance are
symplices, but not generators of the Clifford algebra.. Since the inverse of
the unit elements ${\bf e}_{k}$ of a Clifford algebra must be unique, the
products of different unit vectors form new elements and all possible products
including the unit matrix form a group. There are $\left({N\atop k}\right)$
possible combinations (products without repetition) of $k$ elements from a set
of $N$ generators. We therefore find $\left({N\atop 2}\right)$ bi-vectors,
which are products of 2 generators, $\left({N\atop 3}\right)$ trivectors) and
so on. The product of all $N$ basic matrices is called pseudoscalar. The total
number of all k-vectors then is 111111We identify $k=0$ with the unit matrix
${\bf 1}$.:

$\sum\limits_{k=0}^{N}\,\left({N\atop k}\right)=2^{N}\,.$ (53)

If we desire to construct a complete system, then the number of variables of
the Clifford algebra has to match the number of variables of the used matrix
system:

$2^{N}=(2n)^{2}\,.$ (54)

Note that the root of this equation gives an even integer $2^{N/2}=2\,n$ so
that $N$ must be even. Hence all Hamiltonian Clifford algebras have an even
number of dimensions. Of course not all elements of the Clifford algebra are
symplices. The unit matrix (for instance) is a cosymplex. Consider the
Clifford algebra $Cl_{1,1}$ with $N=2$, which has two generators, say
$\gamma_{0}$ with $\gamma_{0}^{2}=-{\bf 1}$ and $\gamma_{1}$ with
$\gamma_{1}^{2}={\bf 1}$. Since these two anticommute (by definition of the
Clifford algebra), so that we find (besides the unit matrix) a fourth matrix
formed by the product $\gamma_{0}\,\gamma_{1}$:

$\begin{array}[]{rcl}\gamma_{0}\,\gamma_{1}&=&-\gamma_{1}\,\gamma_{0}\\\
(\gamma_{0}\,\gamma_{1})^{2}&=&\gamma_{0}\,\gamma_{1}\,\gamma_{0}\,\gamma_{1}\\\
&=&-\gamma_{0}\,\gamma_{0}\,\gamma_{1}\,\gamma_{1}={\bf 1}\,.\end{array}$ (55)

The completeness of the Clifford algebras as we use them here implies that any
$2\,n\times 2\,n$-matrix ${\bf M}$ with $(2n)^{2}=2^{N}$ can be written as a
linear combination of all elements of the Clifford algebra:

${\bf M}=\sum\limits_{k=0}^{4n^{2}-1}\,m_{k}\,\gamma_{k}\,.$ (56)

The coefficients can be computed from the scalar product of the unit vectors
with the matrix ${\bf M}$:

$m_{k}=(\gamma_{k}\cdot{\bf M})_{S}={s_{k}\over
4n}\,\mathrm{Tr}(\gamma_{k}\,{\bf M}+{\bf M}\,\gamma_{k})\,.$ (57)

Recall that skew-symmetric $\gamma_{k}$ have a negative length and therefore
we included a factor $s_{k}$ which represents the “signature” of $\gamma_{k}$,
in order to get the correct sign of the coefficients $m_{k}$.

Can we derive more properties of the constructable space-times? One
restriction results from representation theory: A theorem from the theory of
Clifford algebras states that $Cl_{p,q}$ has a representation by real matrices
if (and only if) Lounesto

$p-q=0\textrm{ or }2\textrm{ mod }8\,.$ (58)

The additional requirement that all generators must be symplices so that
$p=N-1$ and $q=1$ then restricts $N$ to

$N-2=0\textrm{ or }2\textrm{ mod }8\,.$ (59)

Hence the only matrix systems that have the required symmetry properties
within our modeling game are those that represent Clifford algebras with the
dimensions $1+1$, $3+1$, $9+1$, $11+1$, $17+1$, $19+1$, $25+1$, $27+1$ and so
on. These correspond to matrix representations of size $2\times 2$, $4\times
4$, $32\times 32$, $64\times 64$, $512\times 512$ and so on. The first of them
is called Pauli algebra, the second one is the Dirac algebra. Do these two
have special properties that the higher-dimensional algebras do not have? Yes,
indeed.

Firstly, since dynamics is based on canonical pairs, the real Pauli algebra
describes the motion of a single DOF and the Dirac algebra decribes the
simplest system with interaction between two DOF. This suggests the
interpretation that within our game, objects (Dirac-particles) are not located
“within space-time”, since we did not define space at all up to this point,
but that space-time can be modeled as an emergent phenomenon.

Secondly, if we equate the number of fundamental variables ($2\,n$) of the
oscillator phase space with the dimension of the Clifford space $N$, then Eqn.
54 leads to

$2^{N}=N^{2}\,,$ (60)

which allows for $N=2$ and $N=4$ only. But why should it be meaningful to
assume $N=2\,n$? The reason is quite simple: If $2\,n>N$ as for all higher-
dimensional state vectors, there are not enough generators of the algebra as
there are variables. This discrepancy increases with $n$. Hence the described
objects can not be pure vectors anymore, but must contain tensor-type
components ($k$-vectors) 121212For a deeper discussion of the dimensionality
of space-time, see Ref. qed_paper  and references therein..

But before we describe a formal way to interprete Eqn. 60, let us first
investigate the physical and geometrical implications of the game as described
so far.

###  IV.1 Matrix Exponentials

We said that the unit vectors $\gamma_{0}$ and $\gamma_{k}$ are symplices and
therefore generators of symplectic transformations. All symplectic matrices
are matrix exponentials of symplices MHO . The computation of matrix
exponentials is in the general case non-trivial. However, in the special case
of matrices that square to $\pm{\bf 1}$ 131313E.g. along the “axis”
$\gamma_{k}$ of the coordinate system., the exponentials are readily
evaluated:

$\begin{array}[]{rcl}\exp{(\gamma_{a}\,\tau)}&=&\sum\limits_{k=0}^{\infty}\,{(\gamma_{a}\,\tau)^{k}\over
k!}\\\
\exp{(\gamma_{a}\,\tau)}&=&\sum\limits_{k=0}^{\infty}\,s^{k}\,{\tau^{2k}\over(2\,k)!}+\gamma_{a}\,\sum\limits_{k=0}^{\infty}\,s^{k}\,{\tau^{2k+1}\over(2\,k+1)!}\,,\end{array}$
(61)

where $s=\pm 1$ is the sign of the matrix square of $\gamma_{a}$. For $s=-1$
($\gamma_{a}^{2}=-{\bf 1}$), it follows that

${\bf
R}_{a}(\tau)=\exp{(\gamma_{a}\,\tau)}=\cos{(\tau)}+\gamma_{a}\,\sin{(\tau)}\,,$
(62)

and for $s=1$ ($\gamma_{a}^{2}={\bf 1}$):

${\bf
B}_{a}(\tau)=\exp{(\gamma_{a}\,\tau)}=\cosh{(\tau)}+\gamma_{a}\,\sinh{(\tau)}\,,$
(63)

We can indentify skew-symmetric generators with rotations and (as we will show
in more detail below) symmetric generators with boosts.

The (hyperbolic) sine/cosine structure of symplectic matrices are not limited
to the generators but are a general property of the matrix exponentials of the
symplex ${\bf F}$ 141414These properties are the main motivation to choose the
nomenclature of “symplex” and “cosymplex”.:

${\bf M}(t)=\exp{({\bf F}\,t)}={\bf C}+{\bf S}\,,$ (64)

where the (co-) symplex ${\bf S}$ (${\bf C}$) is given by:

$\begin{array}[]{rcl}{\bf S}&=&\sinh{({\bf F}\,t)}\\\ {\bf C}&=&\cosh{({\bf
F}\,t)}\,,\end{array}$ (65)

since (the linear combination of) all odd powers of a symplex is again a
symplex and the sum of all even powers is a cosymplex. The inverse transfer
matrix ${\bf M}^{-1}(t)$ is given by:

${\bf M}^{-1}(t)={\bf M}(-t)={\bf C}-{\bf S}\,.$ (66)

The physical meaning of the matrix exponential results from Eqn. 13, which
states that (for constant symplices ${\bf F}$) the solutions are given by the
matrix exponential of ${\bf F}$:

$\psi(t)={\bf M}(t)\,\psi(0)\,.$ (67)

A symplectic transformation can be regarded as the result of a possible
evolution in time. There is no prove that non-symplectic processes are
forbidden by nature, but that only symplectic transformations are structure
preserving. Non-symplectic transformations are then structure defining. Both
play a fundamental role in the physics of our model reality, because
fundamental particles are - according to our model - represented by dynamical
structures. Therefore symplectic transformations describe those processes and
interactions, in which structure is preserved, i.e. in which the type of the
particle is not changed. The fundamental variables are just “carriers” of the
dynamical structures. Non-symplectic transformations can be used to transform
the structure. This could also be described by a rotation of the direction of
time. Another interpretation is that of a gauge-transformation newlook .

##  V The Significance of (De-)Coupling

In physics it is a standard technique to reduce complexity of problems by a
suitable change of variables. In case of linear systems, the change of
variables is a linear canonical transformation. The goal of such
transformations is usually to substitute the solution of a complicated problem
by the solution of multiple simpler systems. This technique is known under
various names, one of these names is decoupling, but it is also known as
principal component analysis or (as we will later show) transformation into
the “rest frame”. In other branches of science one might refer to it as
pattern recognition.

In the following we investigate, how to compute (or recognize) the
eigenvectors and eigenvalues of a general $2n\times 2n$-dimensional symplex.
Certainly it would be preferable to find a “physical method”, i.e. a method
that matches to the concepts that we introcuded so far and that has inherently
physical significance. Or at least significance and explantory power with
respect to our modeling game. Let us start from the simplest systems, i.e.
with the Pauli and Dirac algebras which correspond to matrices of size
$2\times 2$ and $4\times 4$, respectively.

###  V.1 The Pauli Algebra

The fundamental significance of the Pauli algebra is based on the even
dimensionality of (classical) phase space. The algebra of $2\times 2$ matrices
describes the motion of a single (isolated) DOF. Besides $\eta_{0}$, the real
Pauli algebra includes the following three matrices:

$\begin{array}[]{rcl}\eta_{1}&=&\left(\begin{array}[]{cc}0&1\\\ 1&0\\\
\end{array}\right)\\\
\eta_{2}&=&\eta_{0}\,\eta_{1}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)\\\ \eta_{3}&=&{\bf 1}=\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right)\\\ \end{array}$ (68)

All except the unit matrix $\eta_{3}$ are symplices. If $\eta_{0}$ and
$\eta_{1}$ are chosen to represent the generators of the corresponding
Clifford algebra $Cl_{1,1}$, then $\eta_{2}$ is the only possible bi-vector. A
general symplex has the form:

$\begin{array}[]{rcl}{\bf F}&=&a\,\eta_{0}+b\,\eta_{1}+c\,\eta_{2}\\\
&=&\left(\begin{array}[]{cc}c&a+b\\\ -a+b&-c\\\
\end{array}\right)\,.\end{array}$ (69)

The characteristic equation is given by $\mathrm{Det}({\bf F}-\lambda\,{\bf
1})=0$

$\begin{array}[]{rcl}0&=&(c-\lambda)(-c-\lambda)-(a+b)(-a+b)\\\
\lambda&=&\pm\sqrt{c^{2}+b^{2}-a^{2}}\\\ \end{array}$ (70)

The eigenvalues $\lambda_{\pm}$ are both either real for $a^{2}<c^{2}+b^{2}$
or both imaginary $a^{2}>c^{2}+b^{2}$ (or both zero). Systems in stable
oscillation have purely imaginary eigenvalues. This case is most interesting
for our modeling game.

Decoupling is usually understood in the more general sense to treat the
interplay of several (at least two) DOF, but here we ask, whether all possible
oscillating systems of $n=1$ are isomorphic to normal form oscillators. Since
there are $3$ parameters in ${\bf F}$ and only one COM, namely the frequency
$\omega$, we need at least two parameters in the transformation matrix. Let us
see, if we can choose these two transformations along the axis of the Clifford
algebra. In this case we apply subsequentially two symplectic transformations
along the axis $\eta_{0}$ and $\eta_{2}$. Applying the symplectic
transformation matrix $\exp{(\eta_{0}\,\tau/2)}$ we obtain:

$\begin{array}[]{rcl}{\bf F}_{1}&=&\exp{(\eta_{0}\,\tau/2)}\,{\bf
F}\,\exp{(-\eta_{0}\,\tau/2)}\\\
&=&a^{\prime}\,\eta_{0}+b^{\prime}\,\eta_{1}+c^{\prime}\,\eta_{2}\\\
\end{array}$ (71) 151515The “half-angle” argument is for convenience.

The transformed coefficients $a^{\prime}$, $b^{\prime}$ and $c^{\prime}$ are
given by

$\begin{array}[]{rcl}a^{\prime}&=&a\\\
b^{\prime}&=&b\,\cos{\tau}-c\,\sin{\tau}\\\
c^{\prime}&=&c\,\cos{\tau}+b\,\sin{\tau}\\\ \end{array}$ (72)

so that - depending on the “duration of the pulse”, we can chose to transform
into a coordinate system in which either $b^{\prime}=0$ or $c^{\prime}=0$. If
we choose $t=\arctan{(-c/b)}$, then $c^{\prime}=0$, so that

$\begin{array}[]{rcl}{\bf
F}^{\prime}&=&a\,\eta_{0}+\sqrt{b^{2}+c^{2}}\,\eta_{1}=a^{\prime}\,\eta_{0}+b^{\prime}\,\eta_{1}\\\
\end{array}$ (73)

If we chose the next generator to be $\eta_{2}$, then:

$\begin{array}[]{rcl}a^{\prime\prime}&=&a^{\prime}\,\cosh{\tau}-b^{\prime}\,\sinh{\tau}\\\
b^{\prime\prime}&=&b^{\prime}\,\cosh{\tau}-a^{\prime}\,\sinh{\tau}\\\
\end{array}$ (74)

In this case we have to dinstinguish between the case, where
$a^{\prime}>b^{\prime}$ and $a^{\prime}<b^{\prime}$. The former is the
oscillatory system and in this case the transformation with
$\tau=\mathrm{artanh}{(b^{\prime}/a^{\prime})}$ leads to the normal form of a
1-dim. oscillator:

$\begin{array}[]{rcl}a^{\prime\prime}&=&\sqrt{a^{2}-b^{2}-c^{2}}\\\
b^{\prime\prime}&=&0\\\ c^{\prime\prime}&=&0\\\ \end{array}$ (75)

and the matrix ${\bf F}^{\prime\prime}$ has the form

${\bf F}^{\prime\prime}=\sqrt{a^{2}-b^{2}-c^{2}}\,\eta_{0}\,.$ (76)

If the eigenvalues are imaginary, then $\lambda=\pm\,i\,\omega$ and hence

${\bf F}^{\prime\prime}=\omega\,\eta_{0}\,,$ (77)

so that the solution is - for constant frequency - given by the matrix
exponential:

$\begin{array}[]{rcl}\psi(t)&=&\exp{(\omega\,\eta_{0}\,t)}\,\psi(0)\\\
&=&\left({\bf
1}\,\cos{(\omega\,t)}+\eta_{0}\,\sin{(\omega\,t)}\right)\,\psi(0)\,.\end{array}$
(78)

This shows that in the context of stable oscillator algebras - the real Pauli
algebra can be reduced to the complex number system: This becomes evident, if
we consider possible representations of the complex numbers. Clearly we need
two basic elements - the unit matrix and $\eta_{0}$, i.e. a matrix that
commutes with the unit matrix and squares to $-{\bf 1}$. If we write “$i$”
instead of $\eta_{0}$, then it is easily verified that 161616 See also Refs.
Lounesto ; Dyson  and Eqn. 35 in combination with Ref. Ralston1989 .:

$\begin{array}[]{rcl}z&=&x+i\,y={\bf Z}=\left(\begin{array}[]{cc}x&y\\\
-y&x\end{array}\right)\\\ \bar{z}&=&x-i\,y={\bf Z}^{T}=x\,{\bf
1}+\eta_{0}^{T}\,y\\\ \exp{(i\,\phi)}&=&\cos{(\phi)}+i\,\sin{(\phi)}\\\
\|z\|^{2}&=&{\bf Z}\,{\bf Z}^{T}=z\,\bar{z}=x^{2}+y^{2}\\\ \end{array}$ (79)

The theory of holomorphic functions is based on series expansions and can be
equally well formulated with matrices. Viewed from our perspective the complex
numbers are a special case of the real Pauli algebra - since we have shown
above that any one-dimensional oscillator can be canonically transformed into
a system of the form of Eqn. 77. Nevertheless we emphasize that the complex
numbers interpreted this way can only represent the normal form of an
oscillator. The normal form excludes a different scaling of coordinates and
momenta as used in classical mechanics, i.e. it avoids intrinsically the
appearance of different “spring constants” and masses 171717 There have been
several attempts to explain the appearance of the complex numbers in quantum
mechanics StB60 ; Strocchi ; Hest75 ; Baylis92 ; MEH2012 ; Gibb2012 ;
Schiff2012 . A general discussion of the use of complex numbers in physics is
beyond the scope of this essay, therefore we add just a remark. Gary W.
Gibbons wrote that “In particular there can be no evolution if $\psi$ is real”
Gibb2012 . We agree with Gibbons that the unit imaginary can be related to
evolution in time as it implies oscillation, but we do not agree with his
conclusion. Physics was able to describe evolution in time without imaginaries
before quantum mechanics and it still is. The unconscious use of the unit
imaginary did not prevent quantum mechanics from being experimetally
successful. But it prevents physicists from understanding its structure..

###  V.2 The Dirac Algebra

In this subsection we consider the oscillator algebra for two coupled DOF, the
algebra of $4\times 4$ matrices. In contrast to the real Pauli algebra, where
the parameters $a$, $b$ and $c$ did not suggest a specific physical meaning,
the structure of the Dirac algebra bears geometrical significance as has been
pointed out by David Hestenes and others STA1 ; STA2 ; GLD . The (real) Dirac
algebra is the simplest real algebra that enables for a description of two DOF
and the interaction between them. Furthermore the eigenfrequencies of a
general symplex ${\bf F}$ may be complex, while the spectrum of the Pauli
matrices does not include complex numbers off the real and imaginary axis. The
spectrum of general $2n\times 2n$-symplices has a certain structure - since
the coefficients of the characteristic polynomial are real: If $\lambda$ is an
eigenvalue of ${\bf F}$, then its complex conjugate $\bar{\lambda}$ as well as
$\lambda$ and $-\bar{\lambda}$ are also eigenvalues. As we will show, this is
the spectrum of the Dirac algebra and therefore any $2n\times 2n$-system can
at least in principle be block-diagonalized using $4\times 4$-blocks. The
Pauli algebra is therefore not sufficient to cover this general case.

The structure of Clifford algebras follows Pascal’s triangle. The Pauli
algebra has the structure $1-2-1$ (scalar-vector-bivector), the Dirac algebra
has the structure $1-4-6-4-1$, standing for unit element (scalar), vectors,
bi-vectors, tri-vectors and pseudoscalar. The vector elements are by
convention indexed with $\gamma_{\mu}$ with $\mu=0\dots 3$, i.e. the
generators of the algebra 181818 According to Pauli’s fundamental theorem of
the Dirac algebra, all possible choices of the Dirac matrices are, as long as
the “metric tensor” $g_{\mu\nu}$ remains unchanged, physically equivalent
Pauli . :

$\begin{array}[]{rclp{4mm}rcl}\gamma_{0}&=&\left(\begin{array}[]{cccc}0&1&0&0\\\
-1&0&0&0\\\ 0&0&0&1\\\ 0&0&-1&0\\\
\end{array}\right)&&\gamma_{1}&=&\left(\begin{array}[]{cccc}0&-1&0&0\\\
-1&0&0&0\\\ 0&0&0&1\\\ 0&0&1&0\\\ \end{array}\right)\\\
\gamma_{2}&=&\left(\begin{array}[]{cccc}0&0&0&1\\\ 0&0&1&0\\\ 0&1&0&0\\\
1&0&0&0\\\
\end{array}\right)&&\gamma_{3}&=&\left(\begin{array}[]{cccc}-1&0&0&0\\\
0&1&0&0\\\ 0&0&-1&0\\\ 0&0&0&1\\\ \end{array}\right)\\\ \end{array}\,.$ (80)

We define the following numbering scheme for the remaining matrices 191919 The
specific choice of the matrices is not unique. A survey of the different
systems can be found in Ref.( rdm_paper ).:

$\begin{array}[]{rclp{4mm}rcl}\gamma_{14}&=&\gamma_{0}\,\gamma_{1}\,\gamma_{2}\,\gamma_{3};&&\gamma_{15}&=&{\bf
1}\\\
\gamma_{4}&=&\gamma_{0}\,\gamma_{1};&&\gamma_{7}&=&\gamma_{14}\,\gamma_{0}\,\gamma_{1}=\gamma_{2}\,\gamma_{3}\\\
\gamma_{5}&=&\gamma_{0}\,\gamma_{2};&&\gamma_{8}&=&\gamma_{14}\,\gamma_{0}\,\gamma_{2}=\gamma_{3}\,\gamma_{1}\\\
\gamma_{6}&=&\gamma_{0}\,\gamma_{3};&&\gamma_{9}&=&\gamma_{14}\,\gamma_{0}\,\gamma_{3}=\gamma_{1}\,\gamma_{2}\\\
\gamma_{10}&=&\gamma_{14}\,\gamma_{0}&=&\gamma_{1}\,\gamma_{2}\,\gamma_{3}&&\\\
\gamma_{11}&=&\gamma_{14}\,\gamma_{1}&=&\gamma_{0}\,\gamma_{2}\,\gamma_{3}&&\\\
\gamma_{12}&=&\gamma_{14}\,\gamma_{2}&=&\gamma_{0}\,\gamma_{3}\,\gamma_{1}&&\\\
\gamma_{13}&=&\gamma_{14}\,\gamma_{3}&=&\gamma_{0}\,\gamma_{1}\,\gamma_{2}&&\\\
\end{array}\,.$ (81)

According to Eq. 17 we expect $10$ symplices and since the $4$ vectors and $6$
bi-vectors are symplices, all other elements are cosymplices. With this
ordering, the general $4\times 4$-symplex ${\bf F}$ can be written as (instead
of Eq. 56):

${\bf F}=\sum\limits_{k=0}^{9}\,f_{k}\,\gamma_{k}\,.$ (82)

In Ref. rdm_paper  we presented a detailed survey of the Dirac algebra with
respect to symplectic Hamiltonian motion. The essence of this survey is the
insight that the real Dirac algebra describes Hamiltonian motion of an
ensembles of two-dimensional oscillators, but as well the motion of a “point
particle” in 3-dimensional space, i.e. that Eqn. 26 is, when expressed by the
real Dirac algebra, isomorphic to the Lorentz force equation as we are going
to show in Sec. VI.3. Or, in other words, the Dirac algebra allows to model a
point particle and its interaction with an electromagnetic field in terms of
the classical statistical ensemble of abstract oscillators.

##  VI Electromechanical Equivalence (EMEQ)

The number and type of symplices within the Dirac algebra (81) suggests to use
the following vector notation for the coefficients rdm_paper ; geo_paper  of
the observables:

$\begin{array}[]{rcl}{\cal E}&\equiv&f_{0}\\\
\vec{P}&\equiv&(f_{1},f_{2},f_{3})^{T}\\\
\vec{E}&\equiv&(f_{4},f_{5},f_{6})^{T}\\\
\vec{B}&\equiv&(f_{7},f_{8},f_{9})^{T}\,,\end{array}$ (83)

where the “clustering” of the coefficients into 3-dimensional vectors will be
explained in the following. The first four elements ${\cal E}$ and $\vec{P}$
are the coefficients of the generators of the Clifford algebra and the
remaining symplices are $3$ symmetric bi-vectors $\vec{E}$ and skew-symmetric
bi-vectors $\vec{B}$. As explained above, the matrix exponentials of pure
Clifford elements are readily evaluated (Eq. 62 and Eq. 63). The effect of a
symplectic similarity transformation on a symplex

$\begin{array}[]{rcl}\tilde{\psi}&=&{\bf R}(\tau/2)\,\psi\\\
{\bf\tilde{F}}&=&{\bf R}(\tau/2)\,{\bf F}\,{\bf R}^{-1}(\tau/2)\\\ &=&{\bf
R}(\tau/2)\,{\bf F}\,{\bf R}(-\tau/2)\\\ \end{array}$ (84)

can then be evaluated component-wise as in the following case of a rotation
(using Eq. 82):

$\begin{array}[]{rcl}{\bf\tilde{F}}&=&\sum\limits_{k=0}^{9}\,f_{k}\,{\bf
R}_{a}\,\gamma_{k}\,{\bf R}_{a}^{-1}\\\ {\bf R}_{a}\,\gamma_{k}\,{\bf
R}_{a}^{-1}&=&\left(\cos(\tau/2)+\gamma_{a}\,\sin(\tau/2)\right)\,\gamma_{k}\\\
&\times&\left(\cos(\tau/2)-\gamma_{a}\,\sin(\tau/2)\right)\\\
&=&\gamma_{k}\,\cos^{2}(\tau/2)-\gamma_{a}\,\gamma_{k}\,\gamma_{a}\,\sin^{2}(\tau/2)\\\
&+&(\gamma_{a}\,\gamma_{k}-\gamma_{k}\,\gamma_{a})\,\cos(\tau/2)\,\sin(\tau/2))\\\
\end{array}$ (85)

Since all Clifford elements either commute or anti-commute with each other, we
have two possible solutions. The first ($\gamma_{k}$ and $\gamma_{a}$ commute)
yields with $\gamma_{a}^{2}=-{\bf 1}$:

${\bf R}_{a}\,\gamma_{k}\,{\bf
R}_{a}^{-1}=\gamma_{k}\,\cos^{2}(\tau/2)-\gamma_{a}^{2}\,\gamma_{k}\,\sin^{2}(\tau/2)=\gamma_{k}\,,$
(86)

but if ($\gamma_{k}$ and $\gamma_{a}$ anti-commute) we obtain a rotation:

$\begin{array}[]{rcl}{\bf R}_{a}\,\gamma_{k}\,{\bf
R}_{a}^{-1}&=&\gamma_{k}\,(\cos^{2}(\tau/2)-\sin^{2}(\tau/2))\\\
&+&\gamma_{a}\,\gamma_{k}\,2\,\cos(\tau/2)\,\sin(\tau/2))\\\
&=&\gamma_{k}\,\cos(\tau)+\gamma_{a}\,\gamma_{k}\,\sin(\tau)\,.\end{array}$
(87)

For $a=9$ ($\gamma_{a}=\gamma_{1}\,\gamma_{2}$) for instance we find:

$\begin{array}[]{rcl}\tilde{\gamma}_{1}&=&\gamma_{1}\,\cos(\tau)+\gamma_{1}\,\gamma_{2}\,\gamma_{1}\,\sin(\tau)=\gamma_{1}\,\cos(\tau)-\gamma_{2}\,\sin(\tau)\\\
\tilde{\gamma}_{2}&=&\gamma_{2}\,\cos(\tau)+\gamma_{1}\,\gamma_{2}\,\gamma_{2}\,\sin(\tau)=\gamma_{2}\,\cos(\tau)+\gamma_{1}\,\sin(\tau)\\\
\tilde{\gamma}_{3}&=&\gamma_{3}\,,\end{array}$ (88)

which is formally equivalent to a rotation of $\vec{P}$ about the “z-axis”. If
the generator $\gamma_{a}$ of the transformation is symmetric, we obtain:

$\begin{array}[]{rcl}{\bf R}_{a}\,\gamma_{k}\,{\bf
R}_{a}^{-1}&=&(\cosh(\tau/2)+\gamma_{a}\,\sinh(\tau/2))\,\gamma_{k}\\\
&\times&(\cosh(\tau/2)-\gamma_{a}\,\sinh(\tau/2))\\\
&=&\gamma_{k}\,\cosh^{2}(\tau/2)-\gamma_{a}\,\gamma_{k}\,\gamma_{a}\,\sinh^{2}(\tau/2)\\\
&+&(\gamma_{a}\,\gamma_{k}-\gamma_{k}\,\gamma_{a})\,\cosh(\tau/2)\,\sinh(\tau/2))\,,\end{array}$
(89)

so that (if $\gamma_{a}$ and $\gamma_{k}$ commute):

$\begin{array}[]{rcl}\tilde{\gamma}_{k}&=&\gamma_{k}\,\cosh^{2}(\tau/2)-\gamma_{a}^{2}\,\gamma_{k}\,\sinh^{2}(\tau/2)\\\
\tilde{\gamma}_{k}&=&\gamma_{k}\,(\cosh^{2}(\tau/2)-\sinh^{2}(\tau/2))=\gamma_{k}\\\
\end{array}$ (90)

and if $\gamma_{a}$ and $\gamma_{k}$ anticommute:

$\begin{array}[]{rcl}\tilde{\gamma}_{k}&=&\gamma_{k}\,(\cosh^{2}(\tau/2)+\sinh^{2}(\tau/2))\\\
&+&2\,\gamma_{a}\,\gamma_{k}\,\cosh(\tau/2)\,\sinh(\tau/2))\\\
&=&\gamma_{k}\,\cosh(\tau)+\gamma_{a}\,\gamma_{k}\,\sinh(\tau)\,,\end{array}$
(91)

which is formally equivalent to a boost, as it allows for the following
parametrization of “rapidity” $\tau$:

$\begin{array}[]{rcl}\tanh{(\tau)}&=&\beta\\\ \sinh{(\tau)}&=&\beta\,\gamma\\\
\cosh{(\tau)}&=&\gamma\\\ \gamma&=&{1\over\sqrt{1-\beta^{2}}}\,.\end{array}$
(92)

A complete survey of these transformations and the (anti-) commutator tables
can be found in Ref. rdm_paper . However this formalism corresponds exactly to
the relativistic invariance of a Dirac spinor in QED as described for instance
in Ref. Schmueser . The “spatial” rotations are generated by the bi-vectors
associated with $\vec{B}$ and Lorentz boosts by the components associated with
$\vec{E}$. The remaining $4$ generators of symplectic transformations
correspond to ${\cal E}$ and $\vec{P}$. They where named phase-rotation
(generated by $\gamma_{0}$) and phase-boosts (generated by
$\vec{\gamma}=(\gamma_{1},\gamma_{2},\gamma_{3})$) and have been used for
instance for symplectic decoupling as described in Ref. geo_paper .

It is nearby (and already suggested by our notation) to consider the
possibility that the EMEQ (Eq. 83) allows to model a relativistic particle as
represented by energy ${\cal E}$ and momentum ${\bf P}$ either in an external
electromagnetic field given by $\vec{E}$ and $\vec{B}$ or - alternatively - in
an accelerating and/or rotating reference frame, where the elements $\vec{E}$
and $\vec{B}$ correspond to the axis of acceleration and rotation,
respectively. We assumed at beginning, that all components of the state vector
$\psi$ are equivalent in meaning and unit. Though we found that the state
vector is formally composed of canonical pairs, the units are unchanged and
identical for all elements of $\psi$. From Eq. 13 we take, that the simplex
${\bf F}$ (and also ${\cal A}$) have the unit of a frequency. If the
Hamiltonian ${\cal H}$ is supposed to represent energy, then the components of
$\psi$ have the unit of the square root of action.

If the coefficients are supposed to represent electromagnetic field, then we
need to express these fields in units of frequency. This can be done, but it
requires to involve natural conversion factors like $\hbar$, charge $e$,
velocity $c$ and a mass, for instance the electron mass $m_{e}$. The magnetic
field (for instance) is related to a “cyclotron frequency” $\omega_{c}$ by
$\omega_{c}\propto{e\over m_{e}}\,B$.

However, according to the rules of the game, the distinction between particle
properties and “external” fields requires a reason, an explanation. Especially
as it is physically meaningless for macroscopic coupled oscillators. In Refs.
rdm_paper ; geo_paper , we used this nomenclature in a merely formal way,
namely to find a descriptive scheme to order the symplectic generators, so to
speak an equivalent circuit to describe the general possible coupling terms
for two-dimensional coupled linear optics as required for the description of
charged particles beams.

Here we play the reversed modeling game: Instead of using the EMEQ as an
equivalent circuit to describe ensembles of oscillators, we now use ensembles
of oscillators as an equivalent circuit to describe point particles. The
motivation for Eqn. 83 is nevertheless similar, i.e. it follows the formal
structure of the Dirac Clifford algebra. The grouping of the coefficients
comes along with the number of vector- and bi-vector-elements, $4$ and $6$,
respectively. The second criterium is to distinguish between generators of
rotations and boost, i.e. between symmetric and skew-symmetric symplices,
which separates energy from momentum and electric from magnetic elements.
Third of all, we note that even 202020Even $k$-vectors are those with even
$k=2\,m$, where $m$ is a natural number. elements (scalar, bi-vectors,
$4$-vectors etc.) of even-dimensional Clifford algebras form a sub-algebra.
This means that we can generate the complete Clifford algebra from the vector-
elements by matrix multiplication (this is why we call them generators), but
we can not generate vectors from bi-vectors by multiplication. And therefore
the vectors are the particles (which are understood as the sources of fields)
and the bi-vectors are the fields, which are generated by the objects and
influence their motion. The full Dirac symplex-algebra includes the
description of a particle (vector) in a field (bi-vector). But why would the
field be external? Simply, because it is impossible to generate bi-vectors
from a single vector-type object, since any single vector-type object written
as ${\cal E}\,\gamma_{0}+\vec{P}\cdot\vec{\gamma}$ squares to a scalar. Dirac
aimed for this result when he invented the Dirac matrices. Therefore, the
fields must be the result of interaction with other particles and hence we
call them “external”. This is in some way a “first-order” approach, since
there might be higher order processes that we did not consider yet. But in the
linear approach (i.e. for second-order Hamiltonians), this distinction is
reasonable and hence a legitimate move in the game.

Besides the Hamiltonian structure (symplices vs. co-symplices) and the
Clifford algebraic structure (distinguishing vectors, bi-vectors, tri-vectors
etc) there is a third essential symmetry, which is connected to the real
matrix representation of the Dirac algebra and to the fact that it describes
the general Hamiltonian motion of coupled oscillators: To distinguish the even
from the odd elements with respect to the block-diagonal matrix structure. We
used this property in Ref. geo_paper  to develop a general geometrical
decoupling algorithm (see also Sec. VI.2).

Now it may appear that we are cheating somehow, as relativity is usually
“derived” from the constancy of the speed of light, while in our modeling
game, we did neither introduce spatial notions nor light at all. Instead we
directly arrive at notions of quantum electrodynamics (QED). How can this be?
The definition of “velocity” within wave mechanics usually involves the
dispersion relation of waves, i.e. the velocity of a wave packet is given by
the group velocity $\vec{v}_{gr}$ defined by

$\vec{v}_{gr}\equiv\vec{\nabla}_{\vec{k}}\,\omega(\vec{k})\,,$ (93)

and the so-called phase velocity $v_{ph}$ defined by

$v_{ph}={\omega\over k}\,.$ (94)

It is then typically mentioned that the product of these two velocities is a
constant $v_{gr}\,v_{ph}=c^{2}$. By the use of the EMEQ and Eq. 30, the
eigenvalues of ${\bf F}$ can be written as:

$\begin{array}[]{rcl}K_{1}&=&-\mathrm{Tr}({\bf F}^{2})/4\\\
K_{2}&=&\mathrm{Tr}({\bf F}^{4})/16-K_{1}^{2}/4\\\
\omega_{1}&=&\sqrt{K_{1}+2\,\sqrt{K_{2}}}\\\
\omega_{2}&=&\sqrt{K_{1}-2\,\sqrt{K_{2}}}\\\
\omega_{1}^{2}\,\omega_{2}^{2}&=&K_{1}^{2}-4\,K_{2}=\mathrm{Det}({\bf F})\\\
K_{1}&=&{\cal E}^{2}+\vec{B}^{2}-\vec{E}^{2}-\vec{P}^{2}\\\ K_{2}&=&({\cal
E}\,\vec{B}+\vec{E}\times\vec{P})^{2}-(\vec{E}\cdot\vec{B})^{2}-(\vec{P}\cdot\vec{B})^{2}\\\
\end{array}$ (95)

Since symplectic transformations are similarity transformations, they do not
alter the eigenvalues of the matrix ${\bf F}$ and since all possible
evolutions in time (which can be described by the Hamiltonian) are symplectic
transformations, the eigenvalues (of closed systems) are conserved. If we
consider a “free particle”, the we obtain from Eq. LABEL:eq_eigenfreq:

$\omega_{1,2}=\pm\sqrt{{\cal E}^{2}-\vec{P}^{2}}\,.$ (96)

As we mentioned before both, energy and momentum, have (within this game) the
unit of frequencies. If we take into account that $\omega_{1,2}\equiv m$ is
fixed, then the dispersion relation for “the energy” ${\cal E}=\omega$ is

${\cal E}=\omega=\sqrt{m^{2}+\vec{P}^{2}}\,.$ (97)

which is indeed the correct relativistic dispersion. But how do we make the
step from pure oscillations to waves? 212121 The question if Quantum theory
requires Planck’s constant $\hbar$, has been answered negative by John P.
Ralston Ralston2012 ..

###  VI.1 Moments and The Fourier Transform

In case of “classical” probability distribution functions (PDFs) $\phi(x)$ we
may use the Taylor terms of the characteristic function
$\tilde{\phi}_{x}(t)=\langle\exp{i\,t\,x}\rangle_{x}$, which is the Fourier
transform of $\phi(x)$, at the origin. The $k$-th moment is then given by

$\langle x^{k}\rangle=i^{k}\,\tilde{\phi}^{(k)}(0)\,.$ (98)

where $\phi^{(k)}$ is the $k$-th derivative of $\tilde{\phi}_{x}(t)$.

A similar method would be of interest for our modeling game. Since a (phase
space-) density is positive definite, we can always take the square root of
the density instead of the density itself: $\phi=\sqrt{\rho}$. The square root
can also defined to be a complex function, so that the density is
$\rho=\phi\phi^{\star}=\|\phi\|^{2}$ and, if mathematically well-defined
(convergent), we can also define the Fourier transform of the complex root,
i.e.

$\tilde{\phi}(\omega,\vec{k})=N\,\int\phi(t,\vec{x})\,\exp{(i\,\omega\,t-i\,\vec{k}\vec{x})}\,dt\,d^{3}x\,.$
(99)

and vice versa:

$\tilde{\phi}(t,\vec{x})=\tilde{N}\,\int\phi(\omega,\vec{k})\,\exp{(-i\,\omega\,t+i\,\vec{k}\vec{x})}\,d\omega
d^{3}k\,.$ (100)

In principle, we may define the density no only by real and imaginary part,
but by an arbitrary number of components. Thus, if we consider a four-
component spinor, we may of course mathematically define its Fourier
transform. But in order to see, why this might be more than a mathematical
“trick”, but physically meaningful, we need to go back to the notions of
classical statistical mechanics. Consider that we replace the single state
vector by an “ensemble”, where we leave the question open, if the ensemble
should be understood as a single phase space trajectory, averaged over time,
or as some (presumably large) number of different trajectories. It is well-
known, that the phase space density $\rho(\psi)$ is stationary, if it depends
only on constants of motion, for instance if it depends only on the
Hamiltonian itself. With the Hamiltonian of Eq. 12, the density could for
example have the form

$\rho({\cal H})\propto\exp{(-\beta\,{\cal H})}=\exp{(-\beta\,\psi\,{\cal
A}\,\psi)}\,,$ (101)

which corresponds to a multivariate Gaussian. But more important is the
insight, that the density exclusively depends on the second moments of the
phase space variables as given by the Hamiltonian, i.e. in case of a “free
particle” it depends on ${\cal E}$ and ${\vec{P}}$. And therefore we should be
able to use energy and momentum as frequency $\omega$ and wave-vector
$\vec{k}$.

But there are more indications in our modeling game that suggest the use of a
Fourier transform as we will show in the next section.

###  VI.2 The Geometry of (De-)Coupling

In the following we give a (very) brief summary of Ref. geo_paper . As already
mentioned, decoupling is meant - despite the use of the EMEQ - first of all
purely technical-mathematical. Let us delay the question, if the notions that
we define in the following have any physical relevance. Here we refer first of
all to block-diagonalization, i.e. we treat the symplex ${\bf F}$ just as a
“Hamiltonian” matrix. From the definition of the real Dirac matrices we obtain
${\bf F}$ in explicit $4\times 4$ matrix form:

$\begin{array}[]{rcl}{\bf
F}&=&\left(\begin{array}[]{cccc}-E_{x}&E_{z}+B_{y}&E_{y}-B_{z}&B_{x}\\\
E_{z}-B_{y}&E_{x}&-B_{x}&-E_{y}-B_{z}\\\
E_{y}+B_{z}&B_{x}&E_{x}&E_{z}-B_{y}\\\
-B_{x}&-E_{y}+B_{z}&E_{z}+B_{y}&-E_{x}\\\ \end{array}\right)\\\
&+&\left(\begin{array}[]{cccc}-P_{z}&{\cal E}-P_{x}&0&P_{y}\\\ -{\cal
E}-P_{x}&P_{z}&P_{y}&0\\\ 0&P_{y}&-P_{z}&{\cal E}+P_{x}\\\ P_{y}&0&-{\cal
E}+P_{x}&P_{z}\\\ \end{array}\right)\,,\end{array}$ (102)

If we find a (sequence of) symplectic similarity transformations that would
allow to reduce the $4\times 4$-form to a block-diagonal form, then we would
obtain two separate systems of size $2\times 2$ and we could continue with the
transformations of Sec. V.1.

Inspection of Eqn. 102 unveils that ${\bf F}$ is block-diagonal, if the
coefficents $E_{y}$, $P_{y}$, $B_{x}$ and $B_{z}$ vanish. Obviously this
implies that $\vec{E}\cdot\vec{B}=0$ and $\vec{P}\cdot\vec{B}=0$. Or vice
versa, if we find a symplectic method that transforms into a system in which
$\vec{E}\cdot\vec{B}=0$ and $\vec{P}\cdot\vec{B}=0$, then we only need to
apply appropriate rotations to achieve block-diagonal form. As shown in Ref.
geo_paper  this can be done in different ways, but in general it requires the
use of the “phase rotation” $\gamma_{0}$ and “phase boosts” $\vec{\gamma}$.
Within the conceptional framework of our game, the application of these
transformations equals the use of “matter fields”. But furthermore, this shows
that block-diagonalization has also geometric significance within the Dirac
algebra and - with respect to the Fourier transformation, the requirement
$\vec{P}\cdot\vec{B}=0$ indicates a divergence free magnetic field, as the
replacement of $\vec{P}$ by $\vec{\nabla}$ yields
$\vec{\nabla}\cdot\vec{B}=0$. The additional requirement
$\vec{E}\cdot\vec{B}=0$ also fits well to our physical picture of e.m. waves.
Note furthermore, that there is no analogous requirement to make
$\vec{P}\cdot\vec{E}$ equal to zero. Thus (within this analogy) we can accept
$\vec{\nabla}\vec{E}\neq 0$.

But this is not everything to be taken from this method. If we analyze in more
detail, which expressions are required to vanish and which may remain, then it
appear that $\vec{P}\cdot\vec{B}$ is explicitely given by

$\begin{array}[]{l}P_{x}\,B_{x}\,\gamma_{1}\,\gamma_{2}\,\gamma_{3}+P_{y}\,B_{y}\,\gamma_{2}\,\gamma_{3}\,\gamma_{1}+P_{z}\,B_{z}\,\gamma_{3}\,\gamma_{1}\,\gamma_{2}\\\
=(\vec{P}\cdot\vec{B})\,\gamma_{10}\\\
E_{x}\,B_{x}\,\gamma_{4}\,\gamma_{2}\,\gamma_{3}+E_{y}\,B_{y}\,\gamma_{5}\,\gamma_{3}\,\gamma_{1}+E_{z}\,B_{z}\,\gamma_{6}\,\gamma_{1}\,\gamma_{2}\\\
=(\vec{E}\cdot\vec{B})\,\gamma_{14}\\\
P_{x}\,E_{x}\,\gamma_{1}\,\gamma_{4}\,\gamma_{3}+P_{y}\,E_{y}\,\gamma_{2}\,\gamma_{5}\,\gamma_{1}+P_{z}\,E_{z}\,\gamma_{3}\,\gamma_{6}\,\gamma_{2}\\\
=-(\vec{P}\cdot\vec{E})\,\gamma_{0}\,.\end{array}$ (103)

That means that exactly those products have to vanish which yield cosymplices.
This can be interpreted via the structure preserving properties of symplectic
motion. Since within our game, the particle type can only be represented by
the structure of the dynamics, and since electromagnetic processes do not
change the type of a particle, then they are quite obviously structure
preserving which then implies the non-appearance of co-symplices. Or - in
other words - electromagnetism is of Hamiltonian nature. We will come back to
this point in Sec. VII.1.

###  VI.3 The Lorentz Force

In the previous section we constituted the distinction between the
“mechanical” elements ${\bf P}={\cal E}\,\gamma_{0}+\vec{\gamma}\cdot\vec{P}$
of the general matrix ${\bf F}$ and the electrodynamical elements ${\bf
F}=\gamma_{0}\,\vec{\gamma}\cdot
E+\gamma_{14}\,\gamma_{0}\,\vec{\gamma}\cdot\vec{B}$. Since the matrix ${\bf
S}=\Sigma\,\gamma_{0}$ is a symplex, let us assume to be equal to ${\bf P}$
and apply Eqn. 26. We then find (with the appropriate relative scaling between
${\bf P}$ and ${\bf F}$ as explained above):

${d{\bf P}\over d\tau}=\dot{\bf P}={q\over 2\,m}\left({\bf F}\,{\bf P}-{\bf
P}\,{\bf F}\right)\,,$ (104)

which yields written with the coefficients of the real Dirac matrices:

$\begin{array}[]{rcl}{d{\cal E}\over d\tau}&=&{q\over
m}\,\vec{p}\cdot\vec{E}\\\ {d\vec{p}\over d\tau}&=&{q\over m}\,\left({\cal
E}\,\vec{E}+\vec{p}\times\vec{B}\right)\\\ \end{array}$ (105)

where $\tau$ is the proper time. If we convert to the labe frame time $t$
using $dt={d\tau\over\gamma}$ EQ. 104 yields (setting $c=1$):

$\begin{array}[]{rcl}\gamma\,{d{\cal E}\over
dt}&=&q\,\gamma\,\vec{v}\cdot\vec{E}\\\ \gamma\,{d\vec{p}\over dt}&=&{q\over
m}\,\left(m\,\gamma\,\vec{E}+m\,\gamma\,\vec{v}\times\vec{B}\right)\\\
{dE\over dt}&=&q\,\vec{v}\cdot\vec{E}\\\ {d\vec{p}\over
dt}&=&q\,\left(\vec{E}+\vec{v}\times\vec{B}\right)\,,\end{array}$ (106)

which is the general form of the Lorentz force. Therefore the Lorentz force
acting on a charged particle in 3 spatial dimensions can be modeled by an
ensemble of 2-dimensional CHOs. The isomorphism between the observables of the
perceived 3-dimensional world and the second moments of density distributions
in the phase space of 2-dimensional oscillators is remarkable.

In any case, Eq. 104 clarifies two things within the game. Firstly, that both,
energy ${\cal E}$ and momentum $\vec{p}$, have to be interpreted as mechanical
energy and momentum (and not canonical), secondly the relative normalization
between fields and mechanical momentum is fixed and last, but not least, it
clarifies the relation between the time related to mass (proper time) and the
time related to $\gamma_{0}$ and energy, which appears to be the laboratory
time.

##  VII Collective Motion and the “Spin”

Eqn. 26 describes the motion of the second moments of the phase space
distribution. Since the equation is linear, it is possible to bring it into a
form $\dot{f}={\bf B}\,f$ with a ten-component “vector” $f$ rdm_paper . The
frequencies $\Omega_{k}$ of the modes of collective motion are given by the
ten eigenvalues $\lambda_{k}=i\,\Omega_{k}$ of the $10\times 10$ matrix ${\bf
B}$. Two of these eigenvalues are zero and correspond to a matched phase space
distribution. The frequencies of the other modes are given by

$\begin{array}[]{rcl}\Omega_{1}&=&\pm\sqrt{K1+2\,\sqrt{K2}}\\\
\Omega_{2}&=&\pm\sqrt{K1-2\,\sqrt{K2}}\\\
\Omega_{3}&=&\pm{1\over\sqrt{2}}\,\sqrt{K1+\sqrt{K1^{2}-4\,K2}}\\\
\Omega_{4}&=&\pm{1\over\sqrt{2}}\,\sqrt{K1-\sqrt{K1^{2}-4\,K2}}\\\
\end{array}$ (107)

The frequencies $\Omega_{1}$ and $\Omega_{2}$ are identical to the frequencies
of motion of a single phase space point (Eqn. LABEL:eq_eigenfreq). If they are
excited, the averages of the complete distribution oscillate like single
spinors. We might call them coherent modes, as the complete ensemble
coherently oscillates in these modes, much like the coherent Glauber states or
like the betatron oscillation of a non-centered beam in a particle
accelerator. A matched beam (or density distribution) is centered, if all
first moments of $\psi$ are constantly zero. If we add a fixed centering error
$\delta\psi$ to a matched $\psi$, then the combined distribution ${\bf S}$
yields:

$\begin{array}[]{rcl}{\bf\tilde{S}}&=&\langle(\psi+\delta\psi)\,(\psi+\delta\psi)^{T}\rangle\,\gamma_{0}\\\
&=&\langle\psi\psi^{T}+\delta\psi\delta\psi^{T}+\delta\psi\,\psi^{T}+\psi\,\delta\psi^{T}\rangle\,\gamma_{0}\\\
&=&{\bf S}+(\delta\psi\delta\psi^{T})\,\gamma_{0}\\\ &=&{\bf S}+\delta{\bf
S}\\\ \end{array}$ (108)

The envelope which can be represented by the second moments oscillates as a
whole about the center. However, the oscillation generates a symplex
$\delta{\bf S}$ with vanishing determinant. As we will show in the next
section this can be interpreted as the generation of a lightlike spinor. At
the same time, this is a possible Ansatz for the development of a “mechanical”
model of how the oscillatory motion of ${\bf S}={\bf P}$ generates
electromagnetic waves, since we can describe oscillations by positions, but
also as an oscillation in momentum space. In the latter case, we may use the
proper (absolute time in the comoving frame) time.

The intrinsic collective modes are those related to the frequencies
$\Omega_{3}$ and $\Omega_{4}$. There is a high frequency mode with frequency
$\Omega_{3}$ and a low frequency mode$\pm\Omega_{4}$. Note further, that in
the limit $K_{2}\to 0$, these modes degenerate: the low frequency mode
vanishes while the high frequency mode has the same frequency as the modes of
individual motion. However, the collective modes depend on the same two
invariants $K_{1}$ and $K_{2}$ as the single modes. Since these are invariant
under (symplectic) transformations, we can describe collective motion as well
in normal coordinates, where the matrix ${\bf F}$ has the form:

$\begin{array}[]{rcl}{\bf F}&=&\left(\begin{array}[]{cccc}0&\omega_{1}&0&0\\\
-\omega_{1}&0&0&0\\\ 0&0&0&\omega_{2}\\\
0&0&-\omega_{2}&0\end{array}\right)\\\ &=&{\cal
E}\,\gamma_{0}+B_{y}\,\gamma_{8}\\\ \end{array}$ (109)

These are characterized by $\vec{E}=\vec{P}=0$ and
$\vec{B}=B_{y}\,\vec{e}_{y}$. Hence only mass, energy and a gyroscopic
quantity $|\vec{B}|$ are non-zero. Then we have $K_{1}={\cal E}^{2}+B_{y}^{2}$
and $K_{2}={\cal E}^{2}\,B_{y}^{2}$, so that the frequencies are:

$\begin{array}[]{rcl}\Omega_{1}&=&{\cal E}+B_{y}=\omega_{1}\\\
\Omega_{2}&=&{\cal E}-B_{y}=\omega_{2}\\\ \Omega_{3}&=&{\cal
E}={\omega_{1}+\omega_{2}\over 2}\\\
\Omega_{4}&=&B_{y}={\omega_{1}-\omega_{2}\over 2}\\\ \end{array}$ (110)

The first two states - the stationary states - show the frequency spectrum of
a spin-$\frac{1}{2}$-particle in an external magnetic field. The last two, the
collective modes, correspond to the average frequency of the two oscillators
and half of the frequency difference, respectively. Hence the energy of the
spin in an external magnetic field is proportional to the frequency difference
between the two oscillators.

The energy levels - and hence the measurable frequencies $\tilde{\omega}_{k}$
\- of the modes described by Eqn. LABEL:eq_normfreq, are given by:

$\begin{array}[]{rcl}m^{2}&=&{\cal E}^{2}+B_{y}^{2}\\\
\tilde{\omega}_{1}&=&\sqrt{m^{2}-B_{y}^{2}}+B_{y}\\\
\tilde{\omega}_{2}&=&\sqrt{m^{2}-B_{y}^{2}}-B_{y}\\\ \end{array}$ (111)

However, up to now the mass $m$ (and the energy) are understood as properties
of the particle while $B_{y}$ is an external magnetic field, for which
$B_{y}\ll m$222222The magnetic frequency $\omega_{b}={e\over 2\,m}\,B$ equals
the “eigenfrequency” of the electron $\omega_{e}={m_{e}\,c^{2}\over\hbar}$ in
case that $B\approx{10^{10}\,\mathrm{T}}$. Magnetic fields of this size would
- according to our model - significantly reduce the electron self-energy -
though the energy difference between “spin up” and “spin down” remains
unchanged. so that the measurable frequencies are simply ${\cal E}\pm B_{y}$.

If the oscillator energy has a contribution that is proportional to an
external field, then we say that the system has a magnetic moment that is
usually interpreted classically by a current loop and hence by an intrinsic
angular momentum called spin $\vec{S}$. In section VI.3 we found the correct
scale for the Lorentz force, if the fields $\vec{E}$ and $\vec{B}$ are scaled
by ${q\over 2\,m}$, so that

${\cal E}_{magn}=g_{e}\,{e\over 2\,m_{e}}\,B_{y}\,S_{y}\,,$ (112)

where $g_{e}$ is the so-called $g$-factor of the electron. Since the two
energy levels are interpreted by the “spin component” parallel to the field
${\bf S}_{y}=\pm\,{\hbar\over 2}$, we may conclude that $g_{e}=2$, as
expected. But the idea of a “charge” that really “rotates” in space-time thus
generating a magnetic moment is - in view of our oscillator model - at best a
metaphor.

The “classical” picture in spacetime requires either a distributed charge
density in some rotary motion or a point particle in circular motion to
explain the magnetic moment. The former implies a finite radius of the charge
density, the latter fails since an oscillating charge must - according to
electrodynamics - radiate and therefore continuously loose energy. Both
pictures are intrinsically inconsistent while there are no contradictions
within our modeling game with respect to “spin”. As spacetime is only defined
by the interaction of several “objects” involving a “distance”. In our
abstract oscillator model “spin” is a “tune-shift” between coupled oscillators
by the “gyroscopic force” $B_{y}$. But the oscillation itself is a motion in
an abstract phase space - it is a consequence of classical oscillator algebra
and does not imply any “real” motion in any “real” space-time.

###  VII.1 The Maxwell Equations

As we already pointed out, waves are (within this game) the result of a
Fourier transformation (FT). But there are different ways to argue this. In
Ref. qed_paper  we argued that Maxwell’s equations can be derived within our
framework by (a) the postulate that space-time emerges from interaction, i.e.
that the fields $\vec{E}$ and $\vec{B}$ have to be constructed from the
4-vectors. ${\bf X}=t\,\gamma_{0}+\vec{x}\cdot\vec{\gamma}$, ${\bf
J}=\rho\,\gamma_{0}+\vec{j}\cdot\vec{\gamma}$ and ${\bf
A}=\Phi\,\gamma_{0}+\vec{A}\cdot\vec{\gamma}$ with (b) the requirement that no
co-symplices emerge. But we can also argue with the FT of the density (see
Sec. VI.1).

If we introduce the 4-derivative

$\partial\equiv-\partial_{t}\,\gamma_{0}+\partial_{x}\,\gamma_{1}+\partial_{y}\,\gamma_{2}+\partial_{z}\,\gamma_{3}\,.$
(113)

The non-abelian nature of matrix multiplication requires to distinguish
differential operators acting to the right and to the left, i.e. we have
$\partial$ as defined in Eq. 113, $\overset{\rightarrow}{\partial}$ and
$\overset{\leftarrow}{\partial}$ which is written to the right of the operand
(thus indicating the order of the matrix multiplication) so that

$\begin{array}[]{rcl}{\bf
H}\overset{\leftarrow}{\partial}&\equiv&-\partial_{t}\,{\bf
H}\,\gamma_{0}+\partial_{x}\,{\bf H}\,\gamma_{1}+\partial_{y}\,{\bf
H}\,\gamma_{2}+\partial_{z}\,{\bf H}\,\gamma_{3}\\\
\overset{\rightarrow}{\partial}{\bf H}&\equiv&-\gamma_{0}\,\partial_{t}\,{\bf
H}+\gamma_{1}\,\partial_{x}\,{\bf H}+\gamma_{2}\,\partial_{y}\,{\bf
H}+\gamma_{3}\,\partial_{z}\,{\bf H}\,.\end{array}$ (114)

The we find the following general rules (see Eq. LABEL:eq_cosy_algebra) that
prevent from non-zero cosymplices:

$\begin{array}[]{rcl}\frac{1}{2}\,\left(\overset{\rightarrow}{\partial}\textrm{
vector }-\textrm{ vector
}\overset{\leftarrow}{\partial}\right)&\Rightarrow&\textrm{ bi-vector}\\\
\frac{1}{2}\,\left(\overset{\rightarrow}{\partial}\textrm{ bi-vector
}-\textrm{ bi-vector
}\overset{\leftarrow}{\partial}\right)&\Rightarrow&\textrm{ vector}\\\
\frac{1}{2}\,\left(\overset{\rightarrow}{\partial}\textrm{ bi-vector
}+\textrm{ bi-vector
}\overset{\leftarrow}{\partial}\right)&\Rightarrow&\textrm{ axial vector}=0\\\
\frac{1}{2}\,\left(\overset{\rightarrow}{\partial}\textrm{ vector }+\textrm{
vector }\overset{\leftarrow}{\partial}\right)&\Rightarrow&\textrm{
scalar}=0\\\ \end{array}$ (115)

Application of these derivatives yields:

$\begin{array}[]{rcl}{\bf
F}&=&\frac{1}{2}\,\left(\overset{\rightarrow}{\partial}{\bf A}-{\bf
A}\overset{\leftarrow}{\partial}\right)\\\ 4\,\pi\,{\bf
J}&=&\frac{1}{2}\,\left(\overset{\rightarrow}{\partial}{\bf F}-{\bf
F}\overset{\leftarrow}{\partial}\right)\\\
0&=&\overset{\rightarrow}{\partial}{\bf F}+{\bf
F}\overset{\leftarrow}{\partial}\\\
0&=&\frac{1}{2}\,\left(\overset{\rightarrow}{\partial}{\bf A}+{\bf
A}\overset{\leftarrow}{\partial}\right)\\\
0&=&\frac{1}{2}\,\left(\overset{\rightarrow}{\partial}{\bf J}+{\bf
J}\overset{\leftarrow}{\partial}\right)\,,\end{array}$ (116)

The first row of Eq. 116 corresponds to the usual definition of the bi-vector
fields from a vector potential ${\bf A}$ and is (written by components) given
by

$\begin{array}[]{rcl}\vec{E}&=&-\vec{\nabla}\phi-\partial_{t}\vec{A}\\\
\vec{B}&=&\vec{\nabla}\times\vec{A}\,.\end{array}$ (117)

The second row of Eq. 116 corresponds to the usual definition of the 4-current
${\bf J}$ as sources of the fields and the last three rows just express the
impossibility of the appearance of cosymplices. They explicitely represent the
homogenuous Maxwell equations

$\begin{array}[]{rcl}\vec{\nabla}\cdot\vec{B}&=&0\\\
\vec{\nabla}\times\vec{E}+\partial_{t}\vec{B}&=&0\,,\end{array}$ (118)

the continuity equation

$\begin{array}[]{rcl}\partial_{t}\rho+\vec{\nabla}\vec{j}=0\,.\end{array}$
(119)

and the so-called “Lorentz gauge”

$\begin{array}[]{rcl}\partial_{t}\Phi+\vec{\nabla}\vec{A}=0\,.\end{array}$
(120)

The simplest idea about the 4-current within QED is to assume that it is
proportional to the “probability current”, which is within our game given by
the vector components of ${\bf S}=\Sigma\,\gamma_{0}$.

##  VIII The Phase Space

Up to now, our modeling game referred to the second moments and the elements
of ${\bf S}$ are second moments such that the observables are given by
(averages over) the following quadratic forms:

$\begin{array}[]{rcl}{\cal
E}&\propto&\psi^{T}\,\psi=q_{1}^{2}+p_{1}^{2}+q_{2}^{2}+p_{2}^{2}\\\
p_{x}&\propto&-q_{1}^{2}+p_{1}^{2}+q_{2}^{2}-p_{2}^{2}\\\
p_{y}&\propto&2\,(q_{1}\,q_{2}-p_{1}\,p_{2})\\\
p_{z}&\propto&2\,(q_{1}\,p_{1}+q_{2}\,p_{2})\\\
E_{x}&\propto&2\,(q_{1}\,p_{1}-q_{2}\,p_{2})\\\
E_{y}&\propto&-2\,(q_{1}\,p_{2}+q_{2}\,p_{1})\\\
E_{z}&\propto&q_{1}^{2}-p_{1}^{2}+q_{2}^{2}-p_{2}^{2}\\\
B_{x}&\propto&2\,(q_{1}\,q_{2}+p_{1}\,p_{2})\\\
B_{y}&\propto&q_{1}^{2}+p_{1}^{2}-q_{2}^{2}-p_{2}^{2}\\\
B_{z}&\propto&2\,(q_{1}\,p_{2}-p_{1}\,q_{2})\\\ \end{array}$ (121)

If we analyze the real Dirac matrix coefficents of ${\bf
S}=\psi\,\psi^{T}\,\gamma_{0}$ in terms of the EMEQ and evaluate the quadratic
relations between those coefficients, then we obtain:

$\begin{array}[]{rcl}\vec{P}^{2}&=&\vec{E}^{2}=\vec{B}^{2}={\cal E}^{2}\\\
0&=&\vec{E}^{2}-\vec{B}^{2}\\\ {\cal
E}^{2}&=&\frac{1}{2}\,(\vec{E}^{2}+\vec{B}^{2})\\\ {\cal
E}\,\vec{P}&=&\vec{E}\times\vec{B}\\\ {\cal
E}^{3}&=&\vec{P}\cdot(\vec{E}\times\vec{B})\\\ m^{2}&\propto&{\cal
E}^{2}-\vec{P}^{2}=0\\\
\vec{P}\cdot\vec{E}&=&\vec{E}\cdot\vec{B}=\vec{P}\cdot\vec{B}=0\\\
\end{array}$ (122)

Besides a missing renormalization these equations describe an object without
mass but with the geometric properties of light as decribed by
electrodynamics, e.g. by the electrodynamic description of electromagnetic
waves, which are $\vec{E}\cdot\vec{B}=0$,
$\vec{P}\propto\vec{E}\times\vec{B}$, $\vec{E}^{2}=\vec{B}^{2}$ and so on.
Hence single spinors are light-like and can not represent massive particles.

Consider the spinor as a vector in a four-dimensional Euclidean space. We
write the symmetric matrix ${\cal A}$ (or $\Sigma$, respectively) as a product
in the form of a Gramian:

${\cal A}={\cal B}^{T}\,{\cal B}\,,$ (123)

or - componentwise:

$\begin{array}[]{rcl}{\cal A}_{ij}&=&\sum_{k}\,({\cal B}^{T})_{ik}\,{\cal
B}_{kj}\\\ &=&\sum_{k}\,{\cal B}_{ki}\,{\cal B}_{kj}\\\ \end{array}$ (124)

The last line can be read such that matrix element ${\cal A}_{ij}$ is the
conventional 4-dimensional scalar product of column vector ${\cal B}_{i}$ with
column vector ${\cal B}_{j}$.

From linear algebra we know that Eqn. 123 yields a non-singular matrix ${\cal
A}$, iff the column-vectors of the matrix ${\cal B}$ are linearily
independent. In the orthonormal case, the matrix ${\cal A}$ simply is the pure
form of a non-singular matrix, i.e. the unit matrix. Hence, if we want to
construct a massive object from spinors, we need several spinors to fill the
columns of ${\cal B}$. The simplest case is the orthogonal case: the
combination of four mutual orthogonal vectors. Given a general 4-component
Hamiltonian spinor $\psi=(q_{1},p_{1},q_{2},p_{2})$, how do we find a spinor
that is orthogonal to this one? In 3 (i.e. odd) space dimensions, we know that
there are two vectors that are perpendicular to any vector $(x,y,z)^{T}$, but
without fixing the first vector, we can’t define the others. In even
dimensions this is different: if we fine a non-singular skew-symmetric matrix
like $\gamma_{0}$, then we have a general expression for a vector that is
perpendicular to $\psi$, namely $\gamma_{0}\,\psi$. As in Eq. 3, it is the
pure form of the matrix that ensures the orthogonality. Any third vector
$\gamma_{k}\,\psi$, which is perpendicular to $\psi$ and to
$\gamma_{0}\,\psi$, must then be skew-symmetric and must fulfil
$\psi^{T}\gamma_{k}^{T}\,\gamma_{0}\,\psi=0$, which means that the product
$\gamma_{k}^{T}\,\gamma_{0}$ must be skew-symmetric and hence that
$\gamma_{k}$ must anti-commute with $\gamma_{0}$:

$\begin{array}[]{rcl}(\gamma_{k}^{T}\,\gamma_{0})^{T}&=&\gamma_{0}^{T}\,\gamma_{k}=-\gamma_{k}^{T}\,\gamma_{0}\\\
\Rightarrow&&\gamma_{0}^{T}\,\gamma_{k}+\gamma_{k}^{T}\,\gamma_{0}=0\\\
0&=&\gamma_{0}\,\gamma_{k}+\gamma_{k}\,\gamma_{0}\,.\end{array}$ (125)

Now let us for a moment return to the question of dimensionality. There are in
general $2\,n\,(2\,n-1)/2$ non-zero independent elements in a skew-symmetric
square $2n\times 2n$ matrix. But how many matrices are there in the considered
phase space dimensions, i.e. in $1+1$, $3+1$ and $9+1$ (etc) dimensions which
anti-commute with $\gamma_{0}$? We need at least $2\,n-1$ skew-symmetric anti-
commuting elements to obtain a diagonal ${\cal A}$. However, this implies at
least $N-1$ anticommuting elements of the Clifford algebra that square to
$-{\bf 1}$. Hence the ideal case is $2n=N$, which is only true for the Pauli
and Dirac algebra. For the Pauli algebra, there is one skew-symmetric element,
namely $\eta_{0}$. In the Dirac algebra there are $6$ skew-symmetric
generators that contain two sets of mutually anti-commuting skew-symmetric
matrices: $\gamma_{0}$, $\gamma_{10}$ and $\gamma_{14}$ on the one hand and
$\gamma_{7}$, $\gamma_{8}$ and $\gamma_{9}$ on the other hand. The next
considered Clifford algebra with $N=9+1$ dimensions requires a representation
by $2n=32=\sqrt{2^{10}}$ -dimensional real matrices. Hence this algebra may
not represent a Clifford algebra with more than $10$ unit elements - certainly
not $2\,n$. Hence we can not use the algebra to generate purely massive
objects (e.g. diagonal matrices) without further restrictions (i.e.
projections) of the spinor $\psi$.

But what exactly does this mean? Of course we can easily find $32$ linearily
independent spinors to generate an orthogonal matrix ${\cal B}$. So what
exactly is special in the Pauli- and Dirac algebra? To see this, we need to
understand, what it means that we can use the matrix ${\cal B}$ of mutually
orthogonal column-spinors

${\cal B}=(\psi,\gamma_{0}\,\psi,\gamma_{10}\,\psi,\gamma_{14}\,\psi)\,.$
(126)

This form implies that we can define the mass of the “particle” algebraically,
and since we have $N-1=3$ anticommuting skew-symmetric matrices in the Dirac
algebra, we can find a multispinor ${\cal B}$ for any arbitrary point in phase
space. This does not seem to be sensational at first sight, since this appears
to be a property of any Euclidean space. The importance comes from the fact
that $\psi$ is a “point” in a very special space - a point in phase space. In
fact, we will argue in the following that this possibility to factorize $\psi$
and the density $\rho$ is everything but self-evident.

If we want to simulate a phase space distribution, we can either define a
phase space density $\rho(\psi)$ or we use the technique of Monte-Carlo
simulations and represent the phase space by (a huge number of random)
samples. If we generate a random sample and we like to implement a certain
exact symmetry of the density in phase space, then we would (for instance)
form a symmetric sample by appending not only a column-vector to ${\cal B}$,
but also its negative $-\psi$. In this way we obtain a sample with an exact
symmetry. In a more general sense: If a phase space symmetry can be
represented by a matrix $\gamma_{s}$ that allows to associate to an arbitrary
phase space point $\psi$ a second point $\gamma_{s}\,\psi$ where $\gamma_{s}$
is skew-symmetric, then we have a certain continuous linear rotational
symmetry in this phase space. As we have shown, phase-spaces are intrinsically
structured by $\gamma_{0}$ and insofar much more restricted than Euclidean
spaces. This is due to the distinction of symplectic from non-symplectic
transformations and due to the intrinsic relation to Clifford algebras: Phase
spaces are spaces structured by time. According to the rules and results of
the game, they appear to be the only physically fundamental spaces at all.

We may imprint the mentioned symmetry to an arbitrary phase space density
$\rho$ by taking all phase space samples that we have so far and adding the
same number of samples, each column multiplied by $\gamma_{s}$. Thus, we have
a single rotation in the Pauli algebra and two of them in the Dirac algebra:

$\begin{array}[]{rcl}&&{\cal B}_{0}=\psi\\\ \gamma_{0}&\to&{\cal
B}_{1}=(\psi,\gamma_{0}\,\psi)\\\ \gamma_{14}&\to&{\cal
B}_{2}=(\psi,\gamma_{0}\,\psi,\gamma_{14}\,\psi,\gamma_{14}\,\gamma_{0}\,\psi)\\\
&&=(\psi,\gamma_{0}\,\psi,\gamma_{14}\,\psi,\gamma_{10}\,\psi)\\\ \end{array}$
(127)

or:

$\begin{array}[]{rcl}&&{\cal B}_{0}=\psi\\\ \gamma_{7}&\to&{\cal
B}_{1}=(\psi,\gamma_{7}\,\psi)\\\ \gamma_{8}&\to&{\cal
B}_{2}=(\psi,\gamma_{7}\,\psi,\gamma_{8}\,\psi,\gamma_{8}\,\gamma_{7}\,\psi)\\\
&&=(\psi,\gamma_{7}\,\psi,\gamma_{8}\,\psi,-\gamma_{9}\,\psi)\\\ \end{array}$
(128)

Note that order and sign of the column-vectors in ${\cal B}$ are irrelevant \-
at least with respect to the autocorrelation matrix ${\cal B}\,{\cal B}^{T}$.
Thus we find that there are two fundamental ways to represent a positive mass
in the Dirac algebra and one in the Pauli-algebra. The $4$-dimensional phase
space of the Dirac algebra is in two independent ways self-matched.

Our starting point was the statement that $2\,n$ linear independent vectors
are needed to generate mass. If we can’t find $2\,n$ vectors in the way
described above for the Pauli and Dirac algebra, then this does (of course)
not automatically imply that there are not $2\,n$ linear independent vectors.

But what does it mean that the dimension of the Clifford algebra of
observables ($N$) does not match the dimension of the phase space ($2\,n$) in
higher dimensions? There are different physical descriptions given.
Classically we would say that a positive definite $2\,n$-component spinor
describes a system of $n$ (potentially) coupled oscillators with $n$
frequencies. If ${\cal B}$ is orthogonal, then all oscillators have the same
frequency, i.e. the system is degenerate. But for $n>2$ we find that not all
eigenmodes can involve the complete $2\,n$-dimensional phase space. This
phenomenon is already known in $3$ dimensions: The trajectory of the isotropic
three-dimensional oscillator always happens in a $2$-dimensional plane, i.e.
in a subspace. If it did not, then the angular momentum would not be
conserved. In this case the isotropy of space would be broken. Hence one may
say in some sense that the isotropy of space is the reason for a
$4$-dimensional phase-space and hence the reason for the $3+1$-dimensional
observable space-time of objects. Or in other words: higher-dimensional spaces
are incompatible with isotropy, i.e. with the conservation of angular
momentum. There is an intimate connection of these findings to the
impossibility of Clifford algebras $Cl_{p,1}$ with $p>3$ to create a
homogeneous “Euclidean” space: Let $\gamma_{0}$ represent time and
$\gamma_{k}$ with $k\in[1,\dots,N-1]$ the spatial coordinates. The spatial
rotators are products of two spatial basis vectors. The generator of rotations
in the $(1,2)$-plane is $\gamma_{1}\,\gamma_{2}$. Then we have $6$ rotators in
$4$ spatial dimensions:

$\begin{array}[]{cccccc}\gamma_{1}\,\gamma_{2},&\gamma_{1}\,\gamma_{3},&\gamma_{1}\,\gamma_{4},&\gamma_{2}\,\gamma_{3},&\gamma_{2}\,\gamma_{4},&\gamma_{3}\,\gamma_{4}\,.\end{array}$
(129)

However, we find that some generators commute and while others anticommute and
it can be taken from combinatorics that only sets of $3$ mutual anti-commuting
rotators can be formed from a set of symmetric anti-commuting $\gamma_{k}$.
The $3$ rotators

$\begin{array}[]{ccc}\gamma_{1}\,\gamma_{2},&\gamma_{2}\,\gamma_{3},&\gamma_{1}\,\gamma_{3}\\\
\end{array}$ (130)

mutually anticommute, but $\gamma_{1}\,\gamma_{2}$ and
$\gamma_{3}\,\gamma_{4}$ commute. Furthermore, in $9+1$ dimensions, the
spinors are either projections into 4-dimensional subspaces or there are non-
zero off-diagonal terms in ${\cal A}$, i.e. there is “internal interaction”.

Another way to express the above considerations is the following: Only in $4$
phase space dimensions we may construct a massive object from a matrix ${\cal
B}$ that represents a multispinor $\Psi$ of exactly $N=2n$ single spinors and
construct a wave-function according to

$\Psi=\phi\,{\cal B}\,,$ (131)

where $\rho=\phi^{2}$ is the phase space density.

It is easy to prove and has been shown in Ref. qed_paper  that the elements
$\gamma_{0}$, $\gamma_{10}$ and $\gamma_{14}$ represent parity, time reversal
and charge conjugation. The combination of these operators to form a
multispinor, may lead (with normalization) to the construction of symplectic
matrices ${\bf M}$. Some examples are:

$\begin{array}[]{rcl}{\bf M}&=&({\bf
1}\,\psi,\gamma_{0}\,\psi,-\gamma_{14}\,\psi,-\gamma_{10}\,\psi)/\sqrt{\psi^{T}\,\psi}\\\
{\bf M}\,\gamma_{0}\,{\bf M}^{T}&=&\gamma_{0}\\\ &&\\\ {\bf M}&=&({\bf
1}\,\psi,-\gamma_{14}\,\psi,-\gamma_{10}\,\psi,\gamma_{0}\,\psi)/\sqrt{\psi^{T}\,\psi}\\\
{\bf M}\,\gamma_{10}\,{\bf M}^{T}&=&\gamma_{10}\\\ &&\\\ {\bf
M}&=&(\gamma_{10}\,\psi,-{\bf
1}\,\psi,-\gamma_{14}\,\psi,\gamma_{0}\,\psi)/\sqrt{\psi^{T}\,\psi}\\\ {\bf
M}\,\gamma_{14}\,{\bf M}^{T}&=&\gamma_{14}\\\ \end{array}$ (132)

Hence the combination of the identity and CPT-operators can be arranged such
that the multispinor ${\bf M}$ is symplectic with respect to the directions of
time $\gamma_{0}$, $\gamma_{10}$ and $\gamma_{14}$, but not with respect to
$\gamma_{7}$, $\gamma_{8}$ or $\gamma_{9}$. As we tried to explain, the
specific choice of the skew-symmetric matrix $\gamma_{0}$ is determined by a
structure defining transformation. Since particles are nothing but dynamical
structures in this game, the $6$ possible SUMs should stand for $6$ different
particle types. However, for each direction of time, there are also two
choices of the spatial axes. For $\gamma_{0}$ we have chosen $\gamma_{1}$,
$\gamma_{2}$ and $\gamma_{3}$, but we could have used
$\gamma_{4}=\gamma_{0}\,\gamma_{1}$, $\gamma_{5}=\gamma_{0}\,\gamma_{2}$ and
$\gamma_{6}=\gamma_{0}\,\gamma_{3}$ as well.

Thus, there should be either 6 or 12 different types of structures (types of
fermions) that can be constructed within the Dirac algebra. The above
construction allows for $3$ different types corresponding to $3$ different
forms of the symplectic unit matrix, further $3$ types are expected to be
related to $\gamma_{7}$, $\gamma_{8}$ and $\gamma_{9}$:

$\begin{array}[]{rcl}{\bf M}&=&({\bf
1}\,\psi,-\gamma_{9}\,\psi,-\gamma_{8}\,\psi,-\gamma_{7}\,\psi)/\sqrt{\psi^{T}\,\psi}\\\
{\bf M}\,\gamma_{7}\,{\bf M}^{T}&=&\gamma_{7}\\\ &&\\\ {\bf M}&=&({\bf
1}\,\psi,-\gamma_{8}\,\psi,-\gamma_{7}\,\psi,-\gamma_{9}\,\psi)/\sqrt{\psi^{T}\,\psi}\\\
{\bf M}\,\gamma_{8}\,{\bf M}^{T}&=&\gamma_{8}\\\ &&\\\ {\bf
M}&=&(\gamma_{7}\,\psi,-{\bf
1}\,\psi,-\gamma_{8}\,\psi,-\gamma_{9}\,\psi)/\sqrt{\psi^{T}\,\psi}\\\ {\bf
M}\,\gamma_{9}\,{\bf M}^{T}&=&\gamma_{9}\\\ \end{array}$ (133)

These matrices describe specific symmetries of the 4-dimensional phase space,
i.e. geometric objects in phase space. Therefore massive multispinors can be
described as volumes in phase space. If we deform the figure by stretching
parameters $a,b,c,d$ such that

${\bf\tilde{M}}=(a\,{\bf
1}\,\psi,-b\,\gamma_{0}\,\psi,-c\,\gamma_{14}\,\psi,-d\,\gamma_{10}\,\psi)/\sqrt{\psi^{T}\,\psi}\,,$
(134)

then one obtains with $f_{k}$ taken from Eq. LABEL:eq_qforms:

$\begin{array}[]{rcl}{\bf\tilde{M}}\,{\bf\tilde{M}}^{T}\,\gamma_{0}&=&\sum\limits_{k=0}^{9}\,g_{k}\,f_{k}\,\gamma_{k}/\sqrt{\psi^{T}\,\psi}\\\
g_{0}&=&a^{2}+b^{2}+c^{2}+d^{2}\\\
g_{1}&=&-g_{2}=g_{3}=a^{2}-b^{2}+c^{2}-d^{2}\\\
g_{4}&=&-g_{5}=g_{6}=a^{2}-b^{2}-c^{2}+d^{2}\\\
g_{7}&=&g_{8}=g_{9}=a^{2}+b^{2}-c^{2}-d^{2}\\\ \end{array}$ (135)

This result reproduces the quadratic forms $f_{k}$ of Eq. LABEL:eq_qforms, but
furthermore the phase space radii $a$, $b$, $c$ and $d$ reproduce the
structure of the Clifford algebra, i.e. the classification into the 4 types of
observables ${\cal E}$, $\vec{P}$, $\vec{E}$ and $\vec{B}$. This means that a
deformation of the phase space “unit cell” represents momenta and fields, i.e.
the dimensions of the phase space unit cell are related to the appearance of
certain symplices:

$\begin{array}[]{rcl}(a=b)\,\mathrm{AND}\,(c=d)&\Rightarrow&\vec{P}=\vec{E}=0\\\
(a=c)\,\mathrm{AND}\,(b=d)&\Rightarrow&\vec{E}=\vec{B}=0\\\
(a=d)\,\mathrm{AND}\,(b=c)&\Rightarrow&\vec{P}=\vec{B}=0\,,\end{array}$ (136)

while for $a=b=c=d$ all vectors but ${\cal E}$ vanish. Only in this latter
case, the matrix ${\bf M}$ is symplectic for $a=b=c=d=1$. These relations
confirm the intrinsic connection between a classical 4-dimensional Hamiltonian
phase space and Clifford algebras in dimension 3+1.

##  IX Summary and Discussion

Based on three fundamental principles we have shown that the algebraic
structure of coupled classical degrees of freedom is (depending on the number
of the DOFs) isomorph to certain Clifford algebras that allow to explain the
dimensionality of space-time, to model Lorentz-transformations, the
relativistic energy-momentum relations and even Maxwell’s equations.

It is usually assumed that we have to define the properties of space-time in
the first place: “In Einstein’s theory of gravitation matter and its dynamical
interaction are based on the notion of an intrinsic geometric structure of the
space-time continuum”ES1950 . However - as we have shown within this “game” -
it has far more explanatory power to derive and explain space-time from the
principles of interaction. Hence we propose to reverse the above statement:
The intrinsic geometric structure of the space-time continuum is based on the
dynamical interaction of matter. A rigorous consequence of this reversal of
perspective is that “space-time” does not need to have a fixed and unique
dimensionality at all. It appears that this dimensionality is relative to the
type of interaction. However, supposed higher-dimensional space-times (see
Ref. qed_paper ) would emerge in analogy to the method presented here, for
instance in nuclear interaction, then these space-times would not simply be
Euclidean spaces of higher dimension. Clifford algebras - especially if they
are restricted by symplectic conditions by a Hamiltonian function, have a
surprisingly complicated intrinsic structure. As we pointed out, if all
generators of a Clifford algebra are symplices, then in $9+1$ dimensions, we
find $k$-vectors with $k\in\,[0..10]$ but $k$-vectors generated from symplices
are themselves symplices only for $k\in\,[1,2,5,6,9,10,\dots]$. However, if
space-time is constraint by Hamiltonian motion, then ensembles of oscillators
may also clump together to form “objects” with $9+1$ or $25+1$-dimensional
interactions, despite the fact that we gave strong arguments for the
fundamentality of $3+1$-dimensional Hamiltonian algebra.

There is no a priori reason to exclude higher order terms - whenever they
include constants of motion. However, as the Hamitonian then involves terms of
higher order, we might then need to consider higher order moments of the phase
space distribution. In this case we would have to invent an action constant in
order to scale $\psi$.

Our game is based a few general rules and symmetry considerations. The math
used in our derivation - taken the results of representation theory for
granted - is simple and can be understood on an undergraduate level. And
though we never intended to find a connection to string theory, we found -
besides the $3+1$ \- dimensional interactions a list of possible higher-
dimensional candidates, two of which are also in the focus of string theories,
namely $9+1=10$-dimensional and $25+1=26$-dimensional theories StringTheory .

We understand this modeling game as a contribution to the demystification (and
unification) of our understanding of space-time, relativity, electrodynamics
and quantum mechanics. Despite the fact that it has become tradition to write
all equations of motion of QED and QM in a way that requires the use of the
unit imaginary, our model seems to indicate that it does not have to be that
way. Though it is frequently postulated that evolution in time has to be
unitary within QM, it appears that symplectic motion does not only suffice,
but is superior as it yields the correct number of relevant operators. While
in the unitary case, one should expect $16$ ($15$) unitary (traceless)
operators for a $4$-component spinor, while the natural number of generators
in the corresponding symplectic treatment is $10$ as found by Dirac himself in
QED Dirac49 ; Dirac63 . If a theory contains things which are not required,
then we have added something arbitrary and artificial. The theory as we
described it indicates that in momentum space, which is used here, there is no
immediate need for the use of the unit imaginary and no need for more than
$10$ fundamental generators. The use of the unit imaginary however appears
unavoidable when we switch via Fourier transform to the “real space”.

There is a dichotomy in physics. On the one hand all causes are considered to
inhabit space-time (local causality), but on the other hand the physical
reasoning mostly happens in energy- or momentum space: There are no Feyman-
graphs, no scattering amplitudes, no fundamental physical relations, that do
not refer in some way to energy or momentum (-conservation). We treat problems
in solid state physics as well as in high energy physics mostly in Fourier
space (reciprocal lattice).

We are aware that the rules of the game are with their rigour difficult to
accept. However, maybe it does not suffice to speculate that the world might
be a hologram 232323As t’Hooft suggested tHooft  and Leonard Susskind sketched
in his celebrated paper, Ref. Susskind .  \- we really should play modeling
games that might help to decide, if and how it could be like that.

##  Appendix A Microcanonical Ensemble

Einstein once wrote that “A theory is the more impressive the greater the
simplicity of its premises, the more different kinds of things it relates, and
the more extended its area of applicability. Hence the deep impression that
classical thermodynamics made upon me. It is the only physical theory of
universal content concerning which I am convinced that, within the framework
of the applicability of its basic concepts, it will never be overthrown […]”
EinsteinBio . We agree with him and we will try to show in the following that
this holds also for the branch of thermodynamics that is called statistical
mechanics.

By the use of the EMEQ it has been shown, that the expectation values

$f_{k}={\mathrm{Tr}(\gamma_{k}^{2})\over 16}\,\bar{\psi}\,\gamma_{k}\,\psi$
(137)

can be associated with energy ${\cal E}$ and momentum $\vec{p}$ of and with
the electric (magnetic) field $\vec{E}$ and $\vec{B}$ as seen by a
relativistic charged particle. It has also been shown that stable systems can
always be transformed in such a way as to bring ${\cal H}$ into a diagonal
form:

${\bf F}=\left(\begin{array}[]{cccc}0&\omega_{1}&0&0\\\ -\omega_{1}&0&0&0\\\
0&0&0&\omega_{2}\\\ 0&0&-\omega_{2}&0\end{array}\right)\,,$ (138)

In the following we will use the classical model of the microcanonical
ensemble to compute some phase space averages. Let the constant value of the
Hamiltonian be ${\cal H}=U$ where $U$ is some energy, the volume in phase
space $\Phi^{\star}$ that is limited by the surface of constant energy $U$ is
given by Becker :

$\Phi^{\star}=\int\limits_{{\cal H}<U}\,dq_{1}\,dp_{1}\,dq_{2}\,dp_{2}\,,$
(139)

and the partition function $\omega^{\star}$ is the derivative

$\omega^{\star}={d\Phi^{\star}\over dU}\,,$ (140)

which is the phase space integral over all states of constant energy $U$. The
average value of any phase space function $\overline{f(p,q)}$ is then given by

$\overline{f(p,q)}={1\over\omega^{\star}}{d\over dU}\int\limits_{{\cal
H}<U}\,f(p,q)\,dq_{1}\,dp_{1}\,dq_{2}\,dp_{2}\,.$ (141)

In case of a 2-dimensional harmonic oscillator, for instance, we may take the
following parametrization of the phase space:

$\begin{array}[]{rcl}q_{1}&=&r\,\cos{(\alpha)}\,\cos{(\beta)}\\\
p_{1}&=&r\,\cos{(\alpha)}\,\sin{(\beta)}\\\
q_{2}&=&r\,\sin{(\alpha)}\,\cos{(\gamma)}\\\
p_{2}&=&r\,\sin{(\alpha)}\,\sin{(\gamma)}\\\ \end{array}$ (142)

Note that Eqn. LABEL:eq_par2 describes a solution of the equations of motion

$\dot{\psi}={\bf F}\,\psi$ (143)

when we replace

$\begin{array}[]{rcl}\beta&\to&-\omega_{1}\,t\\\
\gamma&\to&-\omega_{2}\,t\,.\end{array}$ (144)

This means that the (normalized) integration over $\beta$ and $\gamma$ is
mathematically identical to an integration over all times (time average). From
Eqn. (141) one would directly conclude

$\overline{f(p,q)}={1\over\omega^{\star}}{d\over dU}\int\limits_{{\cal
H}<U}\,f(p,q)\,\sqrt{g}\,dr\,d\alpha\,d\beta\,d\gamma\,,$ (145)

where $g$ is the Gramian determinant. However the relative amplitude
controlled by the parameter $\alpha$ can not be changed by symplectic
transformations and hence remains constant in a closed system. Therefore the
phase space trajectory of the oscillator can not cover the complete 3-dim.
energy surface, but only a 2-dim. subset thereof. This is known very well in
accelerator physics as the emittance preservation of decoupled DOF. And we
have shown in Ref. geo_paper  that all stable harmonic oscillators systems are
symplectically similar to a decoupled system. Consequently $\alpha$ has to be
excluded from the integration of a “single particle” average and has to be
treated instead as an additional parameter or “boundary condition”:

$\overline{f(p,q)}={1\over\omega^{\star}}{d\over dU}\int\limits_{{\cal
H}<U}\,f(p,q)\,\sqrt{g}\,dr\,d\beta\,d\gamma\,,$ (146)

The Gramian determinant hence is given by:

$\begin{array}[]{rcl}g&=&\mathrm{Det}({\bf G}^{T}\,{\bf G})\\\ {\bf
G}&=&\left(\begin{array}[]{ccc}{\partial q_{1}\over\partial r}&{\partial
q_{1}\over\partial\beta}&{\partial q_{1}\over\partial\gamma}\\\ {\partial
p_{1}\over\partial r}&{\partial p_{1}\over\partial\beta}&{\partial
p_{1}\over\partial\gamma}\\\ {\partial q_{2}\over\partial r}&{\partial
q_{2}\over\partial\beta}&{\partial q_{2}\over\partial\gamma}\\\ {\partial
p_{2}\over\partial r}&{\partial p_{2}\over\partial\beta}&{\partial
p_{2}\over\partial\gamma}\\\ \end{array}\right)\,,\end{array}$ (147)

so that one finds

$\sqrt{g}=r^{2}\,\cos{(\alpha)}\,\sin{(\alpha)}\,.$ (148)

Accordingly, Eqn. (139) has to be written as

$\Phi^{\star}=\int\limits_{H<U}\,\sqrt{g}\,dr\,d\beta\,d\gamma\,.$ (149)

We use the abbreviations

$\begin{array}[]{rcl}{\bar{\omega}}&=&{\omega_{1}+\omega_{2}\over 2}\\\
{\Delta\omega}&=&{\omega_{1}-\omega_{2}\over 2}\\\
\Omega&=&\bar{\omega}+\Delta\omega\,\cos{(2\,\alpha)}\,.\end{array}$ (150)

The Hamilton function is given in the new coordinates by

${\cal H}={r^{2}\,\Omega\over 2}\,,$ (151)

so that the condition ${\cal H}<U$ translates into

$r\leq\sqrt{2\,\varepsilon}\,,$ (152)

where $\varepsilon={U/\Omega}$. The integration over $\beta$ and $\gamma$ is
taken from $0$ to $2\,\pi$. The complete integration results in

$\begin{array}[]{rcl}\Phi^{\star}&=&{2\,\pi^{2}\over
3}\,\sin{(2\,\alpha)}\,\left({2\,U\over\Omega}\right)^{3/2}\\\
\omega^{\star}&=&\frac{3}{2\,U}\,\Phi^{\star}\\\ \end{array}$ (153)

The following average values are computed from Eqn. (146):

$\begin{array}[]{rcl}\overline{\cal H}&=&U\\\ \overline{{\cal
H}^{2}}&=&U^{2}\\\
\overline{f_{k}}&=&\left\\{\begin{array}[]{lcl}\varepsilon&\mathrm{for}&k=0\\\
\varepsilon\,\cos{(2\,\alpha)}&\mathrm{for}&k=8\\\
0&\mathrm{for}&k\in\\{1-7,9\\}\\\ \end{array}\right.\\\
\overline{f_{k}^{2}}&=&\left\\{\begin{array}[]{lcl}\overline{f_{k}}^{2}&\mathrm{for}&k=0,8\\\
\frac{1}{4}\,\varepsilon^{2}\,(1+\cos^{2}{(2\,\alpha)})&\mathrm{for}&k\in\\{1,3,4,6\\}\\\
\frac{1}{2}\,\varepsilon^{2}\,\sin^{2}{(2\,\alpha)}&\mathrm{for}&k\in\\{2,5,7,9\\}\\\
\end{array}\right.\,.\end{array}$ (154)

Hence we find that $f_{0}$ (energy), $f_{8}$ (one spin component) and ${\cal
H}$ (mass) are “sharp” (i.e. operators with an eigenvalue), while the other
“expectation values” have a non-vanishing variance. The fact that spin always
has a “direction of quantization”, i.e. that only one single “sharp”
component, can therefore be nicely modelled within our game. It is a
consequence of symplectic motion. Note also that the squared expectation
values of all even ($\gamma_{1}$, $\gamma_{3}$, $\gamma_{4}$ and $\gamma_{6}$,
except $\gamma_{0}$ and $\gamma_{8}$) and all odd ($\gamma_{2}$, $\gamma_{5}$,
$\gamma_{7}$ and $\gamma_{9}$) operators are equal 242424 The even Dirac
matrices are block-diagonal, the odd ones not. There are six even symplices
and four odd ($\gamma_{2}$, $\gamma_{5}$, $\gamma_{7}$ and $\gamma_{9}$) ones
rdm_paper . Obviously this pattern is the reason for the grouping in Eq. 154.
.

Consider the coordinates are given by the fields ($\vec{E}\propto\vec{x}$)
$\vec{Q}=(f_{4},f_{5},f_{6})^{T}$ and the momenta as usual by
$\vec{P}=(f_{1},f_{2},f_{3})^{T}$, then the angular momentum $\vec{L}$ should
be given by $\vec{L}=\vec{Q}\times\vec{P}$. We obtain the following
expectation values from the microcanonical ensemble:

$\begin{array}[]{rcl}\overline{L_{x}}&=&\overline{L_{z}}=0\\\
\overline{L_{x}^{2}}&=&\overline{L_{z}^{2}}=\frac{1}{2}\,\varepsilon^{4}\,\sin^{2}{(2\,\alpha)}\\\
\overline{L_{y}}&=&\varepsilon^{2}\,\cos{(2\,\alpha)}\\\
\overline{L_{y}^{2}}&=&\varepsilon^{4}\,\cos^{2}{(2\,\alpha)}\\\ \end{array}$
(155)

That is - up to a common scale factor of $\varepsilon$ (or $\varepsilon^{2}$,
respectively) - we have the same results as in Eq. 154. Consider now the
quantum mechanical postulates that the spin component of a fermion is
$s_{z}=\pm s=\pm\frac{1}{2}$ and
$|\vec{s}|^{2}=s_{x}^{2}+s_{y}^{2}+s_{z}^{2}=s(s+1)=\frac{3}{4}$. We can
“derive” this result (up to a factor) from an isotropy requirement for the 4th
order moments, i.e. from the condition that $\langle P_{x}^{2}\rangle=\langle
P_{y}^{2}\rangle=\langle P_{z}^{2}\rangle$:

$\sin^{2}{(2\,\alpha)}=2\,\cos^{2}{(2\,\alpha)}$ (156)

so that

$\alpha=\frac{1}{2}\,\arctan{\sqrt{2}}=27.3678^{\circ}\,,$ (157)

or equivalently with

$1+3\,\cos^{2}{(2\,\alpha)}=0$ (158)

we obtain

$\cos{(2\,\alpha)}=\pm\frac{1}{\sqrt{3}}\,.$ (159)

With respect to the symplex ${\bf F}$ as defined in Eq. 138, we have

${\bf F}=\bar{\omega}\,\gamma_{0}+\Delta\omega\,\gamma_{8}\,,$ (160)

so that with Eq. 154 one finds

${f_{8}/f_{0}}={\overline{f_{8}}/\overline{f_{0}}}=\cos{(2\,\alpha)}$ (161)

The total spin would then be given by the 4th order moments as
$\vec{S}^{2}=\varepsilon^{2}$, so that for a spin-$\frac{1}{2}$-particle we
would have to normalize to
$\varepsilon^{2}=(s(s+1))\hbar^{2}=\frac{3}{4}\,\hbar^{2}$ and hence
$f_{8}=\sqrt{3\over 4}\,\hbar\,{1\over\sqrt{3}}={\hbar\over 2}$. However, the
mass formula (Eq. LABEL:eq_eigenfreq), refers to the second moments, so that
in linear theory we would have

${\cal
H}=m=\omega_{0}\,\sqrt{f_{0}^{2}+f_{8}^{2}}=\omega_{0}\,\sqrt{\varepsilon^{2}+\frac{1}{3}\,\varepsilon^{2}}=\omega_{0}\,\sqrt{\frac{4}{3}}\,\varepsilon=\omega_{0}\,\hbar$
(162)

In order to relate this to a frequency difference, we use Eq. 161:

$\begin{array}[]{rcl}{f_{8}\over
f_{0}}&=&{\Delta\omega\over\bar{\omega}}={\omega_{1}-\omega_{2}\over\omega_{1}+\omega_{2}}=\cos{(2\,\alpha)}=\frac{1}{\sqrt{3}}\\\
\Rightarrow&&\\\ {\omega_{1}\over\omega_{2}}&=&2+\sqrt{3}\\\
\Omega&=&4\,\Delta\omega=\frac{4}{\sqrt{3}}\,\bar{\omega}\,.\end{array}$ (163)

Then from $r=\sqrt{2\,\varepsilon}$ and Eq. 151 we find

$\begin{array}[]{rcl}{\cal H}&=&{r^{2}\,\Omega\over
2}=\varepsilon\,\Omega=\sqrt{3\over 4}\,\hbar\,\Omega=\sqrt{3\over
4}\,\hbar\,\frac{4}{\sqrt{3}}\,\bar{\omega}\\\
&=&2\,\hbar\,\bar{\omega}=\hbar(\omega_{1}+\omega_{2})\,.\end{array}$ (164)

To conclude, classical statistical mechanics allows for a description of spin,
if the rules of symplectic motion are taken into account. This alone is
remarkable. Secondly, assumed that the microcanonical ensemble is the right
approach, then the isotropy of the emergent 3+1-dimensional space-time (with
respect to 4th-order moments) apparently requires a certain ratio between the
frequencies and amplitudes of the two coupled oscillators, i.e. an asymmetry
on the fundamental level.

###  A.1 Entropy and Heat Capacity

The entropy ${\cal S}$ of the microcanonical ensemble can be written as Becker
:

${\cal S}=k\,\log{\Phi^{\star}}\,.$ (165)

The temperature $T$ of the system is given by

$\begin{array}[]{rcl}{\partial{\cal S}\over\partial U}&=&{1\over
T}={\partial(k\,\log{\Phi^{\star}})\over\partial
U}=k\,{\omega^{\star}\over\Phi^{\star}}\\\ &=&{3\,k\over 2\,U}\\\ \end{array}$
(166)

so that energy as a function of temperature is

$U={3\over 2}\,k\,T\,,$ (167)

and the heat capacity $C_{V}={\partial U\over\partial T}$ is (per particle)

$C_{V}={3\over 2}\,k\,.$ (168)

This important result demonstrates – according to statistical mechanics – the
3-dimensionality of the “particle” as the energy per DOF is ${k\,T\over 2}$: a
two-dimensional harmonic oscillator of fundamental variables is equivalent to
an free 3-dimensional “point particle”. To our knowledge this is the first
real physical model of a relativistic point particle.

## References

  * (1)

## References

* (2) D. Hestenes; Am. J. of Phys., Vol. 60, No. 8 (1992), pp. 732-748.
* (3) P.A.M. Dirac; Rev. Mod. Phys. Vol. 21, No. 3 (1949), pp. 392-399.
* (4) P.A.M. Dirac; J. Math. Phys. Vol. 4 (1963), pp. 901-909.
* (5) Y.S. Kim and M.E. Noz; J. Math. Phys. Vol. 22(10) (1981), 2289-2293.
* (6) Y.S. Kim and M.E. Noz; Am. J. Phys. Vol. 50(8) (1982), 721-724.
* (7) Y.S. Kim and M.E. Noz; J. Opt. B: Quantum Semiclass. Opt. 7 (2005), S458-S467.
* (8) John Briggs and Alexander Eisfeld; 1) Phys. Rev. E 83 (2011), 051911, 2) Phys. Rev. A 85 (2012), 052111 and 3) Phys. Rev. A 88 (2013), 062104.
* (9) F. Strocchi; Rev. Mod. Phys. 38, Issue 1 (1966), pp. 36-40.
* (10) E.T. Jaynes in: “The Electron”, Eds. D. Hestenes and A. Weingartshofer. Kluwer Academic Publishers (1991).
* (11) A. Einstein, Autobiographical Notes, Trans. by Paul Arthur Schilpp, Open Court Publ. 1979.
* (12) Elliott H. Lieb: “The stability of matter”, Rev. Mod. Phys. Vol. 48, No. 4 (1976), pp. 553-569.
* (13) E.C.G. Stückelberg; Helv. Phys. Acta 33 VIII (1960), pp. 727-752.
* (14) See theorem 3.2.1. in Ref. MHO .
* (15) K.R. Meyer, G.R. Hall and D. Offin: Introduction to Hamiltonian Dynamical Systems and the N-Body Problem; 2nd. Ed., Springer, New York, 2000.
* (16) C. Baumgarten; Phys. Rev. ST Accel. Beams. 14, 114002 (2011).
* (17) W. Pauli; Ann. de l’Inst. Henry Poincare, Vol. 6, No. 2 (1936), pp. 109-136.
* (18) Peter D. Lax; Courant Inst. (N.Y. Univ.), Rep. NYO-1480-87 (1968); also in: Comm. Pure Appl. Math. Vol. 21, No. 5 (1968), pp. 467-490.
* (19) P.A.M. Dirac; Proc. R. Soc. Lond. A 117 (1928), pp. 610-624.