# Minkowski Spacetime and QED from Ontology of Time

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

Classical mechanics, relativity, electrodynamics and quantum mechanics are
often depicted as separate realms of physics, each with its own formalism and
notion. This remains unsatisfactory with respect to the unity of nature and to
the necessary number of postulates. We uncover the intrinsic connection of
these areas of physics and describe them using a common symplectic Hamiltonian
formalism. Our approach is based on a proper distinction between variables and
constants, i.e. on a basic but rigorous ontology of time and on a simple
analysis of the conditions for measurements in physics. The result put the
measurement problem of quantum mechanics and the Copenhagen interpretation of
the quantum mechanical wavefunction into perspective. Based on this (onto-)
logic spacetime can not be fundamental and we show how a geometric
interpretation of symplectic dynamics emerges from the isomorphism between
corresponding Lie algebra and the representation of a Clifford algebra. We
derive the dimensionality of spacetime, the form of Lorentz transformations
and fundamental laws of physics as the Lorentz force and with some mild
additional assumptions Maxwell electrodynamics and finally the Dirac equation.

Philosophy of physics, Lorentz transformation, Electrodynamics, Dirac equation

###### pacs:

01.70.+w, *43.10.Mq, 05.45.Xt, 03.50.De, 03.65.Pm

##  I Introduction

Schrödinger wrote that “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 . In this article we conjecture
the reverse statement, i.e. that the intrinsic geometric structure of
spacetime is based on the very notion of matter and its dynamical interaction.
The idea that spacetime is not fundamental but emergent has been proposed in
the past by several authors KW1985 ; Pir03 ; Pir04 ; Pir05 ; ARM ; VBF ;
ACFKGS ; Smilga . However we start from pure logical reasoning based on the
distinction of variables and constants, i.e. from the (onto-)logic of time.

Our motivation results from typical claims in standard textbooks like (for
instance) the following concerning the Schrödinger equation Messiah : “It is
quite clear that no deductive reasoning can lead us to that equation. Like all
equations in mathematical physics, it must be postulated and its only
justification lies in the success of the comparison of its predictions with
the experimental results”. Apparently this statement suggests that internal
coherence in physics as a whole is impossible and not worth to be considered.
Another example was found in an older (outdated) german edition of Goldstein’s
Classical Mechanics, where he remarks that “there is no intrinsic connection
between quantum mechanics and relativity” and further “one can discuss the
consequences of the one theory without referring to the other” 111 Translated
to english by the author Goldstein . To be fair, the author did not find a
similar statement in the third edition of Classical Mechanics by Goldstein,
Poole and Safko GPS .. Again, the unity of physics and the unity of nature
went casually overboard. Both cites contain (independent of their truth)
philosophical rather than physical statements. They make ontological claims
but don’t explicitely mark them 222More on the role of implicit ontological
commitments in physics can be found in Ref. (PER2014 )..

Such statements suggest that nature is essentially not intelligible (i.e. laws
have to be postulated) and that physics is unable to yield fundamental
understanding. If this was correct, then the postulates of one area of physics
are not intrinsically connected to other areas of physics. However there are
counterexamples to this attitude. The most famous is certainly Newton’s law of
gravitation. It’s kernel is unification: He invented a common description of
falling apples and planetary motion and thus revealed the unity of nature
underlying the apparent variety of phenomena. We take this as our guiding
idea: To reveal a formalism that connects classical and quantum mechanics as
well as relativity and electrodynamics. And ideally we find a basic but
logically consistent explanation for this formalism with a minimal number of
postulates.

Thus - at startup - we reject all merely postulated concepts such as mass,
point particles, wave functions and even spacetime. We intend to derive and
explain the concepts instead of just postulating them. But what then is left
as starting point to do physics with? It is the very idea of physics itself:
It is essential to physics to observe and measure quantities as functions of
time. A physical theory is a mathematical description of observations called
physical law. This concept is meaning- and useful because of repeatability.
Originating in astronomy physics intends to recognize the pattern of
repetitions, i.e. that under similar (or the same) conditions we observe
similar (or the same) facts. A physical law is (in combination with the
starting conditions) the mathematical form of a prediction - based on the
analysis of experiments. Though this is a strongly simplified description of
physics, it adresses fundamental ontological commitments: the possibility of
measurements, the existence of quantities in time, repeatability, the
possibility of experimental preparation with all its implications and last but
not least pattern recognition. Therefore physics is irresolvably committed to
time, to (measurements of) quantities and to the experience that there are
quantifyable aspects of reality that can be described by physical laws in
mathematical form. This is not a postulate, it is a conditio sine qua non.
Even if we leave all other philosophical questions away - physics as an
experimental science is founded on these ontological commitments333 Sir
Hamilton was aware of the intimate connection between physics and time. He
even had the vision to develop algebra as the science of pure time Hamilton
and it was his deep belief that “that the intuition of time is more deep-
seated in the human mind than the intuition of space” Ohrstrom ..

The idea of this essay is to investigate how far this concept reaches, i.e.
what physics can be derived from these first principles. And it may surprise
how far this Ansatz reaches: It allows to draw conclusions about the
dimensionality of spacetime, Lorentz transformations, the Lorentz force,
Maxwell’s equations, central statements of quantum mechanics (i.e. that mass
is a frequency) and the Dirac equation. Most equations that we use should be
well-known and the math used is simple. However we haven’t found the story
that we are going to tell elsewhere in this form.

The unity of nature is our first and main premise: To be part of the world
implies (direct or indirect) possibility in principle of interaction with all
other things that are part of the world and therefore we (have to) presume the
existence of a common denominator, a fundamental level. Physical models are
based on a quantitative description of reality. Hence whatever a physical
theory considers to be fundamental must be representable by quantities.
Secondly the physical description of fundamental entities has to reflect that
these entities exists in time. Since variation is the only method to express
existence in time, any variable that represents a fundamental entity has to
vary at all times. However quantities and their developement in time are not
sufficient to make up a physical theory. We also need an interpretation that
maps quantities of the model to measurable quantities in the world. This is
what makes up physical meaning: Physics as an experimental science can only
assign meaning to measurable entities. The possibility of a measurement
presupposes the existence of two quantities of the same dimension, one that
varies (called the observable) and second that stays constant - the ruler or
the “reference”. If we measure a length, there must be some fixed length
(historically it was the prototype meter) that we use as reference. This means
that for a measurement we have to find both, varying and constant quantities
(of the same type or unit) on this fundamental level. That is, if all
fundamental entities vary, then there is no constant quantity that might serve
as a measurement ruler. Hence on the fundamental level quantities can not be
measured. However if quantities can not be measured, they cannot be physically
interpreted. Of cause there might be functions of the fundamental variables
that are constant in time: such constants all usually called constants of
motion (COMs). But these functions do not include linear terms - otherwise it
is easy to show that the functions can not be constants. Hence the dimension
(unit) of these functions is different from the dimension (unit) of the
fundamental variables and they do not serve as rulers.

According to quantum mechanics (QM) the fundamental level is identical to (the
components of) the wavefunction. This fits surprisingly well to what we just
said about fundamental variables and it yields a simple but fundamental
explaination why the components of wave functions vary at all times. It also
explains why the quantummechanical wave function implies an interpretational
problem and why QM has to postulate that measurement devices must be
“classical” or “macroscopic”.

There is no doubt that we can measure distances and angles. Hence space (or
spacetime) is not fundamental, but has to be constructed. This is not only a
consequence of our presumptions but it is also desireable: Only if it is
possible to derive the properties of spacetime, we can explain its properties
from first principles. If we instead consider spacetime to be fundamental we
are obliged to postulate its properties. As we are going to argue it is
possible to construct spacetime from the isomorphism of a Lie algebra with
(reps of) real Clifford algebras $Cl(n,1)$. Hence the perception of spacetime
is a result of pattern recognition in analogy to the perception of sound as
language and music. We should always be aware that physically there just
sound, i.e. a density fluctuation with time.

The paper is organized as follows: In Sec. II we describe our understanding of
fundamental entities in physics. In Sec. III we derive Hamilton’s equations of
motion (EQOM) and the basic properties of the symplectic unit matrix. In Sec.
IV constants of motion are introduced with the help of Lax Pairs. We review
important algebraic relations that indicate the construction principles of
spacetime: The possible congruence of the algebraic structure of (skew-)
Hamiltonian matrices with the basic elements (i.e. generators) of Clifford
algebras. We describe the basic measurable entities in a world based on time:
(Second) moments of fundamental variables. Since second moments can be
represented by expectation values of matrix operators, the relations between
these matrices are the relations between the observables (i.e. the laws of
physics).

In Sec. V we derive the conditions for the emergence of a geometric space from
symplectic dynamics. Observables are expectation values of matrix operators
and hence spacetime geometry should be representable by a system of basic
matrices with certain properties. Such matrix systems indeed exist and are
known as (representations of) Clifford algebras.

In Sec. VI we describe symplectic transformations as structure preserving
transformations. These transformations are isomorph to Lorentz transformations
and are the basis for the apparent geometry of spacetime. We explore the
meaning of structure defining transformations.

In Sec. VII we give a short overview over the basic properties of Clifford
algebras in general and especially of $Cl_{N-1,1}(\mathbb{R})$. We derive
conditions that limit the possible dimensionality of an emergent spacetime.
Further analysis of the properties of these Clifford algebras allows to
restrict spacetime to 3+1 dimensions. Hence the appropriate algebra is the
algebra of the real Dirac matrices.

In Sec. VIII we describe Lorentz transformations - boosts and rotations - as
structure preserving (symplectic) transformations. We present an
interpretation of the Dirac matrix system called the “electromechanical
equivalence” (EMEQ). Guided by the EMEQ we derive the Lorentz force and
Maxwell’s equations (MWEQs). We give arguments why momentum and energy should
be related to spatial- and time-derivatives and finally we scetch the path
towards the Dirac equation.

Sec. IX finalizes the discussion and a summary in given in Sec. X.

##  II Setup

In a fundamental theory the variables are supposed to represent entities of
physical reality and hence they are entities that are characterized by their
existence in time. The only way to physically describe existence in time is by
variation: The brickstones of the physical aspects of reality are quantities
that continuously vary 444Fundamental physical constants are not existent
(“material”) objects but (if at all) relations. Neither the fine-structure
constant nor the speed of light (nor $\pi$) are “objects” of the physical
world, but objects of a physical (mathematical) theory.. Since quantum theory
postulates that the wavefunction is fundamental, it is obvious that the
components of the wavefunction are identical to what we call fundamental
variable. A meaningful answer to the question what varies requires some
constant entity to compare, i.e. a reference for a measurement. If we say that
an entity represented by a variable in our equation is a length then we need
to have, in principle, a unit (for instance the meter) to refer to. Not only
that this reference has to be of the same type (i.e. dimension), but it also
has to be macroscopic as “meaning” can only be derived from everyday
(macroscopic) experience.

The Copenhagen interpretation avoids the reference problem by assigning an
only probabilistic meaning to the wavefunction, i.e. by postulating that a)
there is no direct physical meaning and b) the wavefunction gives a complete
description of reality, i.e. is fundamental. Most discussions of the
Copenhagen interpretation are focused on the question, if the probability
interpretation is correct and if the probability is “classical” or “quantum”
in nature. We do not adress this question. Our claim is, that if the
wavefunction is truely fundamental, then a meaningful physical interpretation
of the wavefunction is impossible 555This is no reason to assume that it is
less real than macroscopic quantities.. Physical meaning depends on a constant
quantity that could serve as a reference. And the lack of reference is a
logical consequence of fundamentality666 Maybe Einstein had something like
this in mind when he made the following remark in the context of measurements
by the use of rigid rods: “One should always be aware that the presupposition
of the existence in principle of rigid rods is a presupposition suggested by
approximate experience but is, in principle, arbitrary”EinsteinBio . . There
is no meaningful answer to the question what varies, but only how the
fundamental variables vary. This is a logical limit in physics: A measurement
requires a reference, i.e. something constant of the same physical dimension.
We measure length in “meter” and a measurement of length is a comparison of an
unknown and varying quantity with a constant quantity (“the meter”). But since
existence is essentially being in time, constant quantities that could serve
as a reference have to be constructed from a process committed to time, i.e.
by variation. Hence the only possible physically real constants are constants
of motion: the prototype meter is an object composed of atoms and molecules
which consist of “elementary particles”. The motion of these components is
stable and the length of the reference object “meter” is a constant of motion
of these particles. But at the level of these particles - or at the level of
their constituents - at some level we find pure variation. Acuminated we
conclude that if it is possible to directly measure (the value of) a variable,
it can not be fundamental.

Thus if physical entities that are considered to be fundamental, they can not
be directly measured, since this would require constants of motion from “more”
fundamental entities. But then the more fundamental variable could not be
measured etcpp. It is impossible to assign physical meaning to truely
fundamental dynamical variables. In this respect fundamental variables are not
identical to degrees of freedom (DOFs) in classical mechanics as DOFs are
usually assumed to be measurable at least in principle. But in most other
aspects fundamental variables are similar to (DOFs) in classical mechanics and
this includes the possibility of a classical description of the dynamics of
these variables.

All fundamental variables are of the same type. As long as we have no forceful
argument to introduce different types of variables, Ockham’s razor requires
them to be all ontologically identical. We might compare this to the bits or
bytes in a computer RAM: They are all equal. They only differ by their
address, i.e. by their position in address space. Hence all fundamental
variables are either all real or all imaginary. The latter might be preferable
to reflect their unmeasurability, however as long as there is no need to
introduce the concept of complex numbers, Ockham’s razor demands to stay with
reals imag ; imag2 .

One might think of fundamental variables as pure pieces of information but we
prefer the term “variables” as this name best expresses the fundamental
commitment of existence to variation. And in this respect variables are
intrinsically different from “bits”: fundamental variables do not stop
varying.

It follows from this logical conception that fundamental “objects” (like
electrons) have to be understood by their structure. The only structure at
hand is the structure of the variations of the variables - the structure of
the dynamics. Fundamental objects (elementary particles) are not directly
represented by the variables, but by the dynamical structure of these
variables (the elements of a spinor). If objects do not change their structure
(i.e. their “identity”) in some interaction, then the involved physical
processes apparently must be structure preserving: If a system like an
elementary particle is defined and understood by its structure, then its
continuous existence requires - besides a continuous variation of its
constituents (the variables) - that the dynamical structure must be preserved.

##  III Variables, Constants and the Hamiltonian

###  III.1 Variables

We suggest the following axioms for an ontologically proper basis of physics
are:

  1. 1.

Existence happens in time. The “time” that we have in mind does not have to be
identical to the time that an observer would measure using a clock, but we
assume that there is a monotonic functional relationship.

  2. 2.

Time is manifest by variation. Any fundamental physical entity varies and is
therefore a “variable”.

  3. 3.

Measurements require constant references (rulers), i.e. a physical world
requires constants.

From these axioms it follows that

  1. 1.

There are no other (physical) constants than constants of motion (COMs).

  2. 2.

Fundamental variables can not be directly measured.

The value of a single variable varies. Hence a single variable can not
generate constants of motion. Therefore we start with an arbitrary number
$k>1$ of dynamical variables 777Ockham’s razor commits us to determine the
minimum number of variables required to generate spacetime and objects..
According to the axioms there exist one (or a set of) constant function(s)
${\cal H}(\psi_{1},\psi_{2},\dots\psi_{k})=\mathrm{const}$, where ${\bf\psi}$
is a list of variables. Hence we may write (with the presumption ${\cal
H}=\mathrm{const}$ we imply here ${\partial{\cal H}\over\partial t}=0$):

${d{\cal H}\over dt}={\partial{\cal
H}\over\partial\psi_{1}}\,\dot{\psi}_{1}+{\partial{\cal
H}\over\partial\psi_{2}}\,\dot{\psi}_{2}+\dots+{\partial{\cal
H}\over\partial\psi_{k}}\,\dot{\psi}_{k}=0\,,$ (1)

or in vector notation:

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

The (simplest) general solution is given by

$\dot{\bf\psi}={\bf S}\,(\nabla_{\psi}\,{\cal H})\,,$ (3)

where ${\bf S}$ is a $k\times k$ skew-symmetric real matrix, i.e. ${\bf
S}^{T}=-{\bf S}$. Then there exists a non-singular matrix ${\bf Q}$ such that
888See for instance Ref. (MHO1 ) and App. (C.3):

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

where

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

This change of variables is used to find (or recognise or define) the
“natural” variables. We call such transformations structure defining (see
below and App. (C)). According to the axioms the matrix has full rank
$2\,n\leq k$, i.e. all variables vary and hence we remove all constants from
the state vector such, that

${\bf Q}^{T}\,{\bf S}\,{\bf
Q}=\textrm{diag}(\eta_{0},\eta_{0},\eta_{0},\dots\,,\eta_{0})\equiv\gamma_{0}\,,$
(6)

and rewrite Eq. 2 accordingly

$\begin{array}[]{rcl}{d{\cal H}\over dt}&=&({\bf Q}^{-1}\,\nabla\,{\cal
H})^{T}\,{\bf Q}^{T}\,{\bf S}\,{\bf Q}({\bf Q}^{-1}\,\nabla\,{\cal H})=0\\\
&=&({\bf Q}^{-1}\,\nabla\,{\cal H})^{T}\,\gamma_{0}\,({\bf
Q}^{-1}\,\nabla\,{\cal H})=0\,.\end{array}$ (7)

The matrix $\gamma_{0}$ has the even dimension $2n\times 2n$: in any time-like
physical world the dynamical variables come in pairs 999 The unit imaginary
has indeed significance in quantum mechanics as it ensures an even number of
fundamental variables and a continuous variation of these variables. However
our approach contradicts the frequently expressed opinion that QM inherently
requires the use of complex numbers in the sense that it could not be
formulated without the unit imaginary. There is nothing mysterious about the
unit imaginary in quantum mechanics.. It is therefore sensible to refer to a
pair of variables when we speak of a degree of freedom. We call
$\psi_{2j}=q_{j}$ the j-th canonical coordinate and $\psi_{2j+1}=p_{j}$ the
j-th canonical momentum, i.e. ${\bf\psi}=(q_{1},p_{1},\dots,q_{n},p_{n})^{T}$.
But this nomenclature is purely formal, if the variables in $\psi$ are
fundamental. The matrix $\gamma_{0}$ is called the symplectic unit matrix and
the equations of motion (EQOM) have Hamilton’s form:

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

or, in vector notation:

$\dot{\psi}=\gamma_{0}\,\nabla_{\psi}\,{\cal H}\,.$ (9)

Whenever we have constants (of motion), we can derive Hamilton’s EQOM in some
way. Up to this point we did not refer to space at all. Instead we introduced
the phase space of the abstract and intrinsically unmeasurable basic variable
list $\psi$. Just formally we call them “coordinates” and “momenta”. However
the Hamiltonian formalism itsself does not require that variables are
fundamental and therefore this formalism can be applied to any system with
dynamical constants. The difference to the “classical” Hamiltonian formalism
is solely that the classical state vector $\psi$ describes the average
properties of macroscopic systems. Thus the components of the classical $\psi$
can - at least in principle - be measured. But if the state vector represents
fundamental quantities, then we first have to constructs constants and
observables.

###  III.2 The Hamiltonian

We introduced the “Hamiltonian” ${\cal H}$ as (an arbitrary) constant of
motion. Typically there are several constants of motion and hence the function
${\cal H}$ is not yet well-defined. In the following we assume that ${\cal H}$
is positive (semi-) definite with respect to the variables $\psi$. This
restriction is neither arbitrary nor weak: If the constructed constant of
motion ${\cal H}$ is a reference of existing measurable things, then there
must be a Hamiltonian function ${\cal H}$ that reflects the amount of
something, for instance the amount of substance. In a physical theory that is
free of ad-hoc postulates the constant existence of a substance must be
represented by a positive (semi-) definite Hamiltonian.

We write the “Hamiltonian” ${\cal H}$ as a Taylor series of the $2\,n$
variables:

${\cal H}(\psi)={\cal
H}_{0}+\varepsilon^{T}\,\psi+\frac{1}{2!}\,\psi^{T}\,{\bf
A}\,\psi+\frac{1}{3!}\,{\bf B}_{ijk}\,\psi_{i}\,\psi_{j}\,\psi_{k}+\dots\,,$
(10)

where $\varepsilon$ is a $2\,n$-dimensional vector and ${\bf A}$ is a
symmetric $2\,n\times 2\,n$ matrix and ${\bf B}$ is a tensor that is symmetric
in all indices. We assume that ${\cal H}$ has a local minimum somewhere. If we
truncate to second order, then an offset $\tilde{\psi}=\psi-\psi_{0}$ of
$\psi_{0}=-{\bf A}^{-1}\,\varepsilon$ ($\tilde{\psi}=\psi-\psi_{0}$) enables
to get rid of the linear term. If higher orders are kept, the calculation of
the shift $\psi_{0}$ is more involved. But in any (regular) case, if a local
minimum exist, then we can choose the mimimum to coincide with $\psi=0$. The
Hamiltonian EQOM (9) can then be written as a product of a matrix ${\bf
F}(\psi)$ and the vector $\psi$:

$\dot{\psi}=\gamma_{0}\,\nabla_{\psi}\,{\cal H}={\bf F}(\psi)\,\psi\,.$ (11)

##  IV Constants of Motion

The matrix of second moments $\sigma$ is defined as follows:

$\sigma=\psi\psi^{T}\,,$ (12)

so that for the matrix ${\bf S}$ defined by ${\bf S}=\sigma\,\gamma_{0}$
holds:

$\begin{array}[]{rcl}\dot{\bf
S}&=&\dot{\sigma}\,\gamma_{0}=\dot{\psi}\,\psi^{T}\,\gamma_{0}+\psi\,\dot{\psi}^{T}\,\gamma_{0}\\\
&=&{\bf F}(\psi)\,\psi\,\psi^{T}\,\gamma_{0}+\psi\,\psi^{T}\,{\bf
F}^{T}(\psi)\,\gamma_{0}\\\ &=&{\bf F}(\psi)\,{\bf S}-{\bf
S}\,\gamma_{0}\,{\bf F}^{T}(\psi)\,\gamma_{0}\,,\end{array}$ (13)

where we use the fact that $\gamma_{0}^{2}=-{\bf 1}$ and
$\gamma_{0}=-\gamma_{0}^{T}$. We are especially interested in processes that
allow for the construction of references, i.e.

$\gamma_{0}\,{\bf F}^{T}(\psi)\,\gamma_{0}={\bf F}(\psi)\,,$ (14)

so that

$\dot{\bf S}={\bf F}(\psi)\,{\bf S}-{\bf S}\,{\bf F}(\psi)$ (15)

holds. That is, if ${\bf F}$ and ${\bf S}$ commute, then the matrix ${\bf S}$
is constant, while $\psi$ varies. If also the matrix ${\bf F}$ is constant
(i.e. bilinear Hamiltonian in $\psi$), then the symmetry of ${\bf A}$ results
in a matrix ${\bf F}=\gamma_{0}\,{\bf A}$, for which Eq. (14) holds. In case
of validity of Eq. (15) it is well known that ${\bf S}$ and ${\bf F}$ form a
so-called “Lax Pair” so that

$Tr({\bf S}^{k})=\mathrm{const}\,,$ (16)

holds for any natural number $k$ Lax ; Lax2 . Hence the bilinear form (12) is
the basis for a set of constants as given by (16). The average of the bilinear
form equals a matrix of second moments, if summed over an ensemble of $N$
systems:

$\sigma={1\over N}\,\sum\limits_{k=1}^{N}\,\psi_{k}\,\psi_{k}^{T}\,,$ (17)

or - which is a complementary description - if we use a “density” $\rho(\psi)$
to describe the distribution of states in the phase space of $\psi$. In this
case the matrix of second moments is given by

$\sigma={\int\,\rho(\psi)\,\psi\,\psi^{T}\,d^{n}\psi\over\int\,\rho(\psi)\,d^{n}\psi}\,.$
(18)

Eq. (15) holds also for a single “phase space trajectory” $\psi(t)$. However,
the eigenvalues of the matrix of second moments $\sigma$ depend on whether
they are computed for a single vector $\psi$ (which results in a matrix
$\sigma$ with vanishing determinant and vanishing eigenvalues), for two linear
independent vectors $(\psi_{1},\psi_{2})$ ($\sigma$ has vanishing determinant,
but two non-zero eigenvalues) or $\geq 2\,n$ linear independent vectors
$\psi_{k}$ as in Eq. (17). In the latter case, $\sigma$ is non-singular
101010In accelerator physics the eigenvalues of ${\bf S}=\sigma\,\gamma_{0}$
are the “emittances” (times the unit imaginary). It is evident, that the
position of a single particle at time $t$ does not define a phase space area
rdm_paper ; geo_paper ..

Since ${\bf S}$ is by definition the product of a symmetric matrix $\sigma$
and the skew-symmetric matrix $\gamma_{0}$, it follows that

$Tr({\bf S})=0\,.$ (19)

so that the simplest meaningful constants are given as

$Tr({\bf S}^{2})=\mathrm{const}\,.$ (20)

In the following we restrict ourselves to linear systems (or systems that are
linear in good approximation), where the Hamiltonian is second order in the
fundamental variables $\psi$:

${\cal H}(\psi)\approx\frac{1}{2}\,\psi^{T}\,{\bf A}\,\psi\,.$ (21)

Eq. (21) describes an $n$-dimensional harmonic oscillator. The truncation of
the Hamiltonian to second order is not arbitrary - it guarantees stability. It
is well-known, that non-linearities yield (in many or even most cases)
instable or chaotic behavior. Therefore in order to establish a system of
stable references linear systems are preferable. This restriction is per se
not problematic, since the equations that we are going to “derive” are also
linear 111111 Furthermore we refer to the Kustaanheimo-Stiefel (KS)
transformation and appropriate generalizations that allow to map (for
instance) the Kepler problem to the harmonic oscillator KS . This shows that
the harmonic oscillator is a surprisingly versatile model for stable systems..
We might also interpret this restriction as a low-energy approximation as in
Ref. (KW1985 ).

The constants that can be used as references are quadratic forms (16). If a
reference is based on a quadratic form, the variable compared with the
reference (the measured variable) also must be a quadratic form. Therefore the
“dynamical variables” $q_{i}$ and $p_{i}$ cannot be measured directly, but
only (functions of) quadratic forms based on these variables. We call these
(functions of) quadratic forms observables.

If we plug this result into Eq. 3, we obtain:

$\dot{\psi}=\gamma_{0}\,{\bf A}\,\psi={\bf F}\,\psi\,,$ (22)

Since ${\bf A}$ is symmetric and $\gamma_{0}$ is skew-symmetric, the matrix
${\bf F}=\gamma_{0}\,{\bf A}$ has zero trace. Furthermore one finds that Eq.
(14) is fulfilled. Such matrices are called “Hamiltonian” or sometimes
“infinitesimally symplectic”. We believe that these names are misleading, the
former mainly because the matrix ${\bf F}$ does not appear in the Hamiltonian
and the latter since ${\bf F}$ is neither symplectic nor infinitesimal.
Therefore we use the term symplex (plural symplices) rdm_paper ; geo_paper . A
symplex is a matrix (operator) ${\bf S}$ that holds

${\bf S}^{T}=\gamma_{0}\,{\bf S}\,\gamma_{0}\,.$ (23)

A cosymplex (or “antisymplex” or “skew-Hamiltonian” matrix) is a matrix ${\bf
C}$ that holds 121212The nomenclature of (co-) symplices combines the terms
“symplectic”, “(co-) sine” and “matrix”. The connection to (co-) sine will
become obvious in Sec. VI.1. Furthermore, in geometry a “1-simplex” is a line,
a “2-simplex” is a triangle and a “3-simplex” is a volume (tetrahedron).
Equivalently the elements of a Clifford algebra (for instance the Dirac
algebra) include vectors (vertices), bi-vectors (edges) and tensors (faces).
Pascal’s triangle can be used for both - the number of k-simplices of an
n-simplex ($k<=n$) and to the number of traceless k-(co-) symplices of an
$N$-dimensional Clifford algebra. Hence the Clifford algebra $Cl_{3,1}$
represented by the real Dirac-matrices can be compared to a regular 3-simplex
(tetrahedron). See Fig. 3 in the appendix.:

${\bf C}^{T}=-\gamma_{0}\,{\bf C}\,\gamma_{0}\,.$ (24)

A cosymplex ${\bf C}$ can always be written as a product of $\gamma_{0}$ and a
skew-symmetric matrix. The sums of (co-) symplices are (co-) symplices, i.e.
the superposition principle holds. Hence (co-) symplices form a linear vector
space and any (co-) symplex can be written as a linear combination of “basic”
(co-) symplices. The algebra of (co-) symplices is the Lie algebra $sp(2\,n)$.

Denoting symplices by ${\bf S}$ and cosymplices by ${\bf C}$ (optionally with
subscript) it is easily shown that the anticommutator of two symplices is a
cosymplex:

$\begin{array}[]{rcl}({\bf S}_{1}\,{\bf S}_{2}+{\bf S}_{2}\,{\bf
S}_{1})^{T}&=&{\bf S}_{2}^{T}\,{\bf S}_{1}^{T}+{\bf S}_{1}^{T}\,{\bf
S}_{2}^{T}\\\ &=&\gamma_{0}\,{\bf S}_{2}\,\gamma_{0}\,\gamma_{0}\,{\bf
S}_{1}\,\gamma_{0}+\gamma_{0}\,{\bf S}_{1}\,\gamma_{0}\,\gamma_{0}\,{\bf
S}_{2}\,\gamma_{0}\\\ &=&-\gamma_{0}\,({\bf S}_{2}\,{\bf S}_{1}+{\bf
S}_{1}\,{\bf S}_{2})\,\gamma_{0}\,.\end{array}$ (25)

The following general rules are obtained:

$\begin{array}[]{ccc}\left.\begin{array}[]{c}{\bf S}_{1}\,{\bf S}_{2}-{\bf
S}_{2}\,{\bf S}_{1}\\\ {\bf C}_{1}\,{\bf C}_{2}-{\bf C}_{2}\,{\bf C}_{1}\\\
{\bf C}\,{\bf S}+{\bf S}\,{\bf C}\\\ {\bf S}^{2\,n+1}\\\
\end{array}\right\\}&\Rightarrow&\mathrm{symplex}\\\ &&\\\
\left.\begin{array}[]{c}{\bf S}_{1}\,{\bf S}_{2}+{\bf S}_{2}\,{\bf S}_{1}\\\
{\bf C}_{1}\,{\bf C}_{2}+{\bf C}_{2}\,{\bf C}_{1}\\\ {\bf C}\,{\bf S}-{\bf
S}\,{\bf C}\\\ {\bf S}^{2\,n}\\\ {\bf C}^{n}\\\
\end{array}\right\\}&\Rightarrow&\mathrm{cosymplex}\\\ \end{array}$ (26)

The relations Eq. (LABEL:eq_cosy_algebra) define the structure of a specific
algebra, called the Lie algebra $sp(2\,n)$ of the symplectic group, denoted by
$Sp(2\,n)$. As the matrix of generators ${\bf F}$, also the matrix of
observables ${\bf S}$ of Eq. (16) is a symplex. According to Eq.
(LABEL:eq_cosy_algebra) odd powers of ${\bf S}$ are symplices with zero trace.
Hence the non-zero (i.e. useful) constants of motion given by Eq. (16) are of
even order.

Since the matrix ${\bf A}$ is symmetric (Eq. 21), the maximal number $\nu$ of
free parameters of a symplex for $n$ degrees of freedom (DOF) is

$\nu_{s}={2\,n\,(2\,n+1)\over 2}\,.$ (27)

If the matrix ${\bf F}$ can be expressed as a sum of basic matrices, the basis
$\nu_{s}$ elements. A basis for all real $2n\times 2n$ matrices is completed
by $\nu_{c}$ cosymplices:

$\nu_{c}={2\,n\,(2\,n-1)\over 2}\,.$ (28)

###  IV.1 Measurable Quantities: Observables

Usually the second moments are assumed to be average values of $N$ identical
systems as in Eq. (17), but the equations hold equivalently for a single
system. However it should be noted that for a vector of $2\,n$ fundamental
variables, $2\,n$ linear independent “samples” are required, if the matrix
${\bf S}=\sigma\,\gamma_{0}$ is supposed to have a non-vanishing determinant.
Later we will come back to this point.

We can define the $\sigma$-matrix as well using a (normalized) probability
density $\rho$:

$\sigma=\int\,\rho(\psi)\,\psi\,\psi^{T}\,d^{2n}\psi\equiv\langle\psi\,\psi^{T}\rangle\,,$
(29)

with the normalization according to:

$1=\int\,\rho(\psi)\,d^{2n}\psi\,,$ (30)

i.e. the density $\rho(\psi)$ is defined as a function of the fundamental
variables or phase space variables.

We introduced the (“S-matrix”) ${\bf S}=\sigma\gamma_{0}$ and derived Eq.
(15). Let the adjunct vector $\bar{\psi}$ be defined by

$\bar{\psi}\equiv\psi^{T}\,\gamma_{0}\,,$ (31)

so that ${\bf S}=\langle\psi\bar{\psi}\rangle$ and the expectation value of an
operator ${\bf O}$:

$\langle{\bf O}\rangle\equiv\langle\bar{\psi}\,{\bf O}\,\psi\rangle\,.$ (32)

Now consider that ${\bf O}$ is a cosymplex, i.e. can be written as
$\gamma_{0}\,{\bf B}$ with some skew-symmetric matrix ${\bf B}$, then the
expectation value of ${\bf O}=\gamma_{0}\,{\bf B}$ vanishes:

$\begin{array}[]{rcl}\langle{\bf
O}\rangle&=&\langle\psi^{T}\,\gamma_{0}^{2}{\bf B}\,\psi\rangle\\\
&=&-\langle\psi^{T}\,{\bf B}\,\psi\rangle=0\,.\end{array}$ (33)

Hence only symplices represent measurable properties of (closed) systems.
Cosymplices ${\bf C}$ only yield a non-vanishing result when they appear
between different (i.e. linearily independent) states:

$\bar{\phi}\,{\bf C}\,\psi\neq 0\Rightarrow\phi\neq\,\lambda\,\psi\,,$ (34)

with some arbitrary (non-zero) factor $\lambda$. Vice versa - if two
(normalized) states $\phi$ and $\psi$ may not “generate” non-vanishing
expectation values of the cosymplex type, then is follows that:

$\phi=\pm\psi\,.$ (35)

Since we assume that physical reality can be described by the totality of all
fundamental variables, the coefficients of the matrix ${\bf F}$ are functions
of (internal or external) measurable quantities, i.e. of some symplex ${\bf
S}$ generated from fundamental variables. And since cosymplices have a
vanishing expectation value, they can not contribute. We will use this
important result in Sec. VIII.3 in the derivation of MWEQ. Furthermore, if a
structure that represents a “particle” is preserved in a physical process,
then the involved driving terms must be symplices. Otherwise the evolution in
time would not be symplectic, i.e. structure preserving. And vice versa: If
the particle structure (particle type) is transformed in a process, then there
must be a contribution from cosymplices as for instance axial vector currents
as in the V-A theory of weak interaction.

This means that if we focuse on a closed system, then (the expectation values
of) all cosymplices have to vanish.

The properties of “open” and “closed” have a relative meaning which depends on
the context. If we consider two separate “particles” described by state
vectors $\psi$ and $\phi$ and let them get in contact, then we would for
instance combine the state vectors into a larger state vector
$\Psi=(\psi,\phi)^{T}$. In App. (A.2) we give the explicit form of general
$2\,n\times 2\,n$-symplices. It is shown that non-diagonal sub-blocks may per
se have arbitrary terms. Hence the question, if a (sub-) matrix is a symplex
or a cosymplex depends on the context. However all quantities - including the
matrix ${\bf F}$ \- must by some means result from “fundamental variables”,
i.e. from state vectors. Also a static electromagnetic field has its origin in
charges and currents of “particles” that are described by state vectors. If
reality is described by the totality of the fundamental variables, then also
the symplex ${\bf F}$ must somehow be generated from $\psi$. Since the matrix
of observables ${\bf S}=\sigma\,\gamma_{0}$ is a symplex as well, the
functional relationship might be simple: consider (for instance) a Hamiltonian
of the form

${\cal H}\propto\psi^{T}\,\sigma^{-1}\,\psi\,,$ (36)

which implies ${\bf F}\propto\,\gamma_{0}\,\sigma^{-1}$. In this special case
the Boltzmann distribution $\rho(\psi)\propto\exp{(-{{\cal H}\over k\,T})}$ is
identical to a multivariate Gaussian density distribution stat_paper . In
general the functional relationship between ${\bf F}$ and ${\bf S}$ might be
more involved, but obviously there is a kind of duality between the matrix of
observables ${\bf S}=\sigma\,\gamma_{0}$ and the matrix of generators ${\bf
F}=\gamma_{0}\,{\bf A}$ that can be illustrated by Eq. (36). More details on
the duality of generators and observables can be found in Grgin .

The time derivative of an expectation value is (assuming ${\bf O}$ does not
explicitely depend on time):

$\begin{array}[]{rcl}\langle{\bf\dot{O}}\rangle&=&\langle\dot{\bar{\psi}}\,{\bf
O}\,\psi\rangle+\langle\bar{\psi}\,{\bf O}\,\dot{\psi}\rangle\\\
&=&\langle\dot{\psi}^{T}\,\gamma_{0}\,{\bf
O}\,\psi\rangle+\langle\bar{\psi}\,{\bf O}\,{\bf F}\,\psi\rangle\\\
&=&\langle\psi^{T}\,{\bf F}^{T}\,\gamma_{0}\,{\bf
O}\,\psi\rangle+\langle\bar{\psi}\,{\bf O}\,{\bf F}\,\psi\rangle\\\
&=&\langle\psi^{T}\,\gamma_{0}\,{\bf F}\,\gamma_{0}^{2}\,{\bf
O}\,\psi\rangle+\langle\bar{\psi}\,{\bf O}\,{\bf F}\,\psi\rangle\\\
&=&\langle\bar{\psi}\,({\bf O}\,{\bf F}-{\bf F}\,{\bf O})\,\psi\rangle\\\
\end{array}$ (37)

This general law connects the commutator of an operator ${\bf O}$ and ${\bf
F}$ with its time derivative. This is the linear form of the Poisson bracket
and it is purely classical. One conclusion from Eqn. LABEL:eq_eqom in
combination with Eqn. LABEL:eq_cosy_algebra is that the time evolution is such
that (co-)symplices stay (co-)symplices. They belong to different domains.

##  V From Lie Algebra to Spacetime Geometry

In the following we consider conditions for the emergence of spacetime
structure from $sp(2\,n)$. I.e. we derive conditions for which the spinors
$\psi$ \- which are composed of fundamental variables - are able to produce
the relations between observables that represent the structure of Minkowski
spacetime. In order to do this we have to identify the structures that
represent (unit) vectors for the space and time directions (or - equivalently
- the energy and momentum directions). For this purpose we need a definition
of length (a norm) and of orthogonality. So far we only introduced time and a
single symplex: $\gamma_{0}$. Hence $\gamma_{0}$ is the natural choice of the
“unit vector” in the “time direction”.

From the (anti-) commutation properties described by Eq. LABEL:eq_cosy_algebra
it follows that any fundamental system of (co-) symplices requires that the
basic (co-) symplices either commute or anticommute: Only in this case the
product is a pure symplex or cosymplex, i.e. either observable or not. If we
consider pure (co-) symplices ${\bf A}$ and ${\bf B}$, then

$({\bf A}\,{\bf B})^{T}={\bf B}^{T}\,{\bf A}^{T}=\pm\,\gamma_{0}\,{\bf
B}\,{\bf A}\,\gamma_{0}\,.$ (38)

It follows that the product of pure (co-) symplices is again a pure (co-)
symplex, iff ${\bf A}\,{\bf B}=\pm{\bf B}\,{\bf A}$.

Usually vector norms in Euclidean spaces represent the length of a vector,
i.e. a property that is invariant under rotations and translations. Hence we
prefer an invariant quantity for the desired definition of a norm. Besides the
Hamiltonian the only invariants we introduced so far are given by Eq. (16).
The simplest non-trivial $k$-norm is given for $k=2$, i.e.:

$\|{\bf A}\|=\left(\frac{1}{2\,n}\,\mathrm{Tr}({\bf
A}^{2})\right)^{\frac{1}{2}}\,.$ (39)

where the division by $2\,n$ is introduced to let the unit matrix have unit
norm. The matrix $\gamma_{0}$ hence has a norm of $-1$, i.e. the norm (39) is
not positive definite.

Two elements of an algebra are said to be orthogonal, if their inner product
vanishes. Since we have no inner product defined (yet), we use an alternative
geometric definition which we consider to be equivalent in flat spacetime \-
the Pythagorean theorem. Two vectors ${\vec{v}}$ and ${\vec{w}}$ are
orthogonal, iff napier

$\|{\vec{v}}+{\vec{w}}\|^{2}=\|{\vec{v}}\|^{2}+\|{\vec{w}}\|^{2}\,,$ (40)

which implies (in usual vector notation) that the inner product vanishes:

$({\vec{v}}+{\vec{w}})^{2}={\vec{v}}^{2}+{\vec{w}}^{2}+2\,\vec{v}\cdot\vec{w}\,.$
(41)

For two elements ${\bf A}$ and ${\bf B}$ of the Lie algebra $sp(2\,n)$ the use
of Eq. 39 then yields

$\begin{array}[]{rcl}\|{\bf A}+{\bf
B}\|^{2}&=&\frac{1}{2\,n}\,\mathrm{Tr}\left[({\bf A}+{\bf B})^{2}\right]\\\
&=&\frac{1}{2\,n}\,\mathrm{Tr}\left[{\bf A}^{2}+{\bf B}^{2}+{\bf A}{\bf
B}+{\bf B}{\bf A}\right]\\\ &=&\|{\bf A}\|^{2}+\|{\bf
B}\|^{2}+\frac{1}{2\,n}\,\mathrm{Tr}\left[{\bf A}{\bf B}+{\bf B}{\bf
A}\right]\,.\end{array}$ (42)

As we argued above, all elements of the algebra either pairwise commute or
anticommute. Two anticommuting elements ${\bf A}$ and ${\bf B}$ are therefore
orthogonal:

$\mathrm{Tr}\left({\bf A}\,{\bf B}+{\bf B}\,{\bf A}\right)=0\,.$ (43)

Hence we identify the anticommutator with the inner product 131313 Since the
anticommutator does not necessarily result a scalar, one may distinguish
between inner product and scalar product. The scalar product can then be
defined by the trace of the inner product:  $({\bf A}\cdot{\bf
B})_{S}\equiv{1\over 2\,n}\mathrm{Tr}({{\bf A}\,{\bf B}+{\bf B}\,{\bf A}\over
2})\,,$ (44) where the index “S” indicates the scalar part. :

${\bf A}\cdot{\bf B}\equiv{{\bf A}\,{\bf B}+{\bf B}\,{\bf A}\over 2}\,.$ (45)

Given the anticommutator of $\gamma_{k}$ and $\gamma_{0}$ vanishes, then we
find (using $\gamma_{0}^{2}=-1$):

$\begin{array}[]{rcl}\gamma_{k}\,\gamma_{0}+\gamma_{0}\,\gamma_{k}&=&0\\\
\gamma_{k}\,\gamma_{0}^{2}+\gamma_{0}\,\gamma_{k}\,\gamma_{0}&=&0\\\
\gamma_{k}&=&\gamma_{0}\,\gamma_{k}\,\gamma_{0}\,,\end{array}$ (46)

so that in accordance with Eqs. 23 and 24 $\gamma_{k}$ is a symplex, iff it is
symmetric and a cosymplex, iff it is skew-symmetric. Using

$\mathrm{Tr}({\bf A}\,{\bf A}^{T})=\sum\limits_{i,j}\,a_{ij}^{2}$

it follows, that (with appropriate normalization) symmetric unit “vectors”
${\bf A}$ square to $+{\bf 1}$ and skew-symmetric ${\bf B}$ square to $-{\bf
1}$:

$\begin{array}[]{rcl}\mathrm{Tr}({\bf A}^{2})&=&\mathrm{Tr}({\bf A}\,{\bf
A}^{T})=\sum\limits_{i,j}\,a_{ij}^{2}\geq 0\\\ \mathrm{Tr}({\bf
B}^{2})&=&-\mathrm{Tr}({\bf B}\,{\bf
B}^{T})=-\sum\limits_{i,j}\,b_{ij}^{2}\leq 0\,.\end{array}$ (47)

Therefore there are two types of measurable unit vectors: $\gamma_{0}$ that
represents time and squares to $-{\bf 1}$ and a number of symmetric symplices
$\gamma_{k}$ (anticommuting with $\gamma_{0}$ and with each other) that square
to $+{\bf 1}$. The latter (if existent) are the unit vectors in space-like
directions. Skew-symmetric elements that are orthogonal to $\gamma_{0}$ are
cosymplices and have vanishing expectation values. It follows that there is
only a single measurable time direction, as any other antisymmetric (time-
like) unit element that is orthogonal to $\gamma_{0}$ is a cosymplex. In
summary: if a (flat) spacetime emerges from dynamics, all symplices that
represent orthogonal directions have to anticommute. Therefore the obvious
choice of the inner product or scalar product of two elements of a dynamically
generated geometric algebra is the anticommutator. The unit vector in the
time-direction squares to $-{\bf 1}$ and all vector-type symplices
$\gamma_{x}$ that are orthogonal to $\gamma_{0}$ are symmetric and square to
$+{\bf 1}$.

Given that we can find a (sub-) set of $N$ orthogonal symplices
$\gamma_{k},\,\,k\in\,[0,1,\dots,N-1]$ in $sp(2\,n)$, then the product of two
of them holds:

$\begin{array}[]{rcl}\gamma_{i}\,\gamma_{j}&=&{\gamma_{i}\,\gamma_{j}+\gamma_{j}\,\gamma_{i}\over
2}+{\gamma_{i}\,\gamma_{j}-\gamma_{j}\,\gamma_{i}\over 2}\\\
&=&{\gamma_{i}\,\gamma_{j}-\gamma_{j}\,\gamma_{i}\over 2}\,,\end{array}$ (48)

for $i\neq j$. It is easy to prove that the product $\gamma_{i}\,\gamma_{j}$
with $i\neq j$ is orthogonal to both - $\gamma_{i}$ and $\gamma_{j}$. Hence we
identify the exterior product with the commutator of ${\bf A}$ and ${\bf B}$:

${\bf A}\,\wedge{\bf B}={{\bf A}\,{\bf B}-{\bf B}\,{\bf A}\over 2}\,.$ (49)

A system of $N$ pairwise anti-commuting real matrices (that square to $\pm{\bf
1}$) represents a Clifford algebra $Cl_{p,q}$, $N=p+q$. $p$ is the number of
unit vectors that square to $+{\bf 1}$ and $q$ the number of unit vectors that
square to $-{\bf 1}$. We found that in our case $q=1$. If (as we presumed)
spacetime emerges from the dynamics of fundamental variables, then the
apparent geometry of spacetime is described by the (representation of) a
Clifford algebra $Cl_{N-1,1}$. The unit vectors are called generators of the
Clifford algebra.

There are $\left({N\atop k}\right)$ possible combinations (without repetition)
of $k$ elements from a set of $N$ generators. Then there are $\left({N\atop
2}\right)$ products of 2 basic matrices (named bivectors), $\left({N\atop
3}\right)$ products of 3 basic matrices (named trivectors) and so on. The
product of all $N$ basic matrices is called pseudoscalar. The $N$ anti-
commuting generators are the (unit) vectors. The total number of all k-vectors
then is 141414The case $k=0$ can be identified with the unit matrix.:

$\sum\limits_{k=0}^{N}\,\left({N\atop k}\right)=2^{N}\,.$ (50)

As we have no reason to allow more elements in the algebra than $2^{N}$, the
vector space $sp(2\,n)$ and $Cl_{N-1,1}$ have the same dimension:

$2^{N}=(2n)^{2}\,,$ (51)

which - if fulfilled - is nothing but the condition for completeness.

If such a basis of anticommuting matrices exists, we know that

$\begin{array}[]{rcll}\gamma_{k}^{2}&=&\pm\,{\bf 1}&\textrm{ for all
}k\,\in\,[0,\dots,\,4n^{2}-1]\\\ \gamma_{k}&=&\pm\,\gamma_{k}^{T}&\textrm{ for
all }k\,\in\,[0,\dots,\,4n^{2}-1]\\\
\gamma_{k}\,\gamma_{j}&=&\pm\,\gamma_{j}\,\gamma_{k}&\textrm{ for all }j\neq
k,\,\in\,[0,\dots,\,4n^{2}-2]\,.\end{array}$ (52)

From Eqs. 52 above we readily conclude:

$\begin{array}[]{rcl}\gamma_{0}\,\gamma_{k}\,\gamma_{0}&=&\pm\,\gamma_{k}^{T}\\\
\gamma_{k}\,\gamma_{0}\,\gamma_{k}^{T}&=&\pm\,\gamma_{0}\,,\end{array}$ (53)

so that all $\gamma_{k}$ are either symplices or cosymplices and they are
either symplectic ($\gamma_{k}\,\gamma_{0}\,\gamma_{k}^{T}=\gamma_{0}$) or
“cosymplectic” ($\gamma_{k}\,\gamma_{0}\,\gamma_{k}^{T}=-\gamma_{0}$). Since
$\mathrm{Det}({\bf A}{\bf B})=\mathrm{Det}({\bf A})\mathrm{Det}({\bf B})$ and
since $\gamma_{k}^{2}=\pm\,{\bf 1}$ it follows that all basic matrices have a
determinant of $\pm 1$.

Given such a matrix system exists for some $n$ and $N$, then it has remarkable
symmetry properties. Note that we derived these properties without reference
to the specific form (or size) of the matrices. If (as we demand) Eq. (51)
holds, then the system is a complete basis of the vector space of real
$2n\times 2n$-matrices and therefore these matrices ${\bf M}$ can be written
as a linear combination of the $\gamma_{k}$ according to:

${\bf M}=\sum\limits_{k=0}^{(2n)^{2}-1}\,m_{k}\,\gamma_{k}\,,$ (54)

so that with the signature $s_{j}$ defined by
$s_{j}\equiv\mathrm{Tr}(\gamma_{j}^{2})/(2n)$ we have

$\begin{array}[]{rcl}\frac{1}{4n}\,\mathrm{Tr}({\bf
M}\,\gamma_{j}+\gamma_{j}\,{\bf
M})&=&\frac{1}{4n}\,\mathrm{Tr}\left(\sum\limits_{k=0}^{2n-1}\,m_{k}\,(\gamma_{j}\,\gamma_{k}+\gamma_{k}\,\gamma_{j})\right)\\\
&=&s_{j}\,m_{j}\,.\end{array}$ (55)

Given that $\gamma_{k}\,k\in\,[0,\dots,N-1]$ are the generating elements of
the algebra, then the vector ${\bf
A}=\sum\limits_{k=0}^{N-1}\,a_{k}\,\gamma_{k}$ squares to a scalar:

${\bf
A}^{2}=\sum\limits_{k=0}^{N-1}\,s_{k}\,a_{k}^{2}=\sum\limits_{k=1}^{N-1}\,a_{k}^{2}-a_{0}^{2}\,.$
(56)

A dynamical system generates a flat Minkowski spacetime with proper
observables if the corresponding real matrix algebra is a representation of a
Clifford algebra in which all generating elements are symplices. The idea to
relate Clifford algebras and symplectic spaces is not new (see for instance
Ref. (BGB ; BB ) and references therein), however (to the knowledge of the
author) it has not been shown yet that the concept of a geometric (Clifford)
algebra can be derived from the idea of a dynamically emergent spacetime.

##  VI Structure Preservation

###  VI.1 Transfer Matrizes and Symplectic Transformations

The solution of the Hamiltonian EQOM (Eq. 22) for the linearized Hamiltonian
(with constant matrix ${\bf A}$) are given by the matrix exponential:

$\psi(t)=\exp{({\bf F}\,(t-t_{0}))}\,\psi(t_{0})={\bf
M}(t,t_{0})\,\psi(t_{0})\,.$ (57)

The matrix ${\bf M}(t,t_{0})=\exp{({\bf F}\,(t-t_{0}))}$ is usually called
transfer matrix and it is well-known to be symplectic if ${\bf F}$ is a
symplex (“Hamiltonian”), that is, ${\bf M}$ fulfills the following
(equivalent) equations MHO :

$\begin{array}[]{rcl}{\bf M}\,\gamma_{0}\,{\bf M}^{T}&=&\gamma_{0}\\\ {\bf
M}^{-1}&=&-\gamma_{0}\,{\bf M}^{T}\,\gamma_{0}\\\ ({\bf
M}^{-1})^{T}&=&-\gamma_{0}\,{\bf M}\,\gamma_{0}\\\ {\bf
M}^{T}&=&-\gamma_{0}\,{\bf M}^{-1}\,\gamma_{0}\,.\end{array}$ (58)

Since the matrix exponential of ${\bf F}$ is quite complicated in the general
form of Eq. 81, it is more convenient to consider transformations generated by
basic symplices $\gamma_{j}$, since all $\gamma_{j}$ square to $\pm{\bf 1}$:

$\begin{array}[]{rcl}{\bf R}_{j}&=&\exp{(\gamma_{j}\,\varepsilon)}\\\
&=&\left\\{\begin{array}[]{lcl}{\bf
1}\,\cos{(\varepsilon)}+\gamma_{j}\,\sin{(\varepsilon)}&\mathrm{for}&\gamma_{j}^{2}=-1\\\
{\bf
1}\,\cosh{(\varepsilon)}+\gamma_{j}\,\sinh{(\varepsilon)}&\mathrm{for}&\gamma_{j}^{2}=1\\\
\end{array}\right.\\\ {\bf R}_{j}^{-1}&=&\exp{(-\gamma_{j}\,\varepsilon)}\\\
&=&\left\\{\begin{array}[]{lcl}{\bf
1}\,\cos{(\varepsilon)}-\gamma_{j}\,\sin{(\varepsilon)}&\mathrm{for}&\gamma_{j}^{2}=-1\\\
{\bf
1}\,\cosh{(\varepsilon)}-\gamma_{j}\,\sinh{(\varepsilon)}&\mathrm{for}&\gamma_{j}^{2}=1\\\
\end{array}\right.\\\ \end{array}$ (59)

The solutions are the sum of the unit matrix and $\gamma_{j}$ with
(hyperbolic) (co-) sine coefficients. From Eqs. LABEL:eq_sine_cosine we find
that skew-symmetric symplices (those that square to $-{\bf 1}$) are generators
of orthogonal transformations, since

$\begin{array}[]{rcl}{\bf R}_{j}&=&{\bf
1}\,\cos{(\varepsilon)}+\gamma_{j}\,\sin{(\varepsilon)}\\\ {\bf
R}_{j}^{T}&=&{\bf 1}\,\cos{(\varepsilon)}-\gamma_{j}\,\sin{(\varepsilon)}\\\
\Rightarrow&=&{\bf R}_{j}\,{\bf R}_{j}^{T}={\bf 1}\,.\end{array}$ (60)

Hence skew-symmetric symplices are generators of (symplectic) rotations. The
symplectic transformations generated by symmetric symplices are essentially
boosts. As we will show in more detail in Sec. VIII.1, it is legitimate to
call the structure preserving property of symplectic transformations the
principle of (special) relativity since Lorentz transformations of “spinors”
$\psi$ in Minkowski spacetime are a subset of the symplectic transformations
for two DOFs.

More generally the structure of the matrix exponential of a symplex ${\bf F}$
is:

${\bf M}(t)=\exp{({\bf F}\,t)}={\bf C}+{\bf S}\,,$ (61)

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

$\begin{array}[]{rcl}{\bf S}&=&\sinh{({\bf F}\,t)}\\\ {\bf C}&=&\cosh{({\bf
F}\,t)}\,,\end{array}$ (62)

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

${\bf M}^{-1}(t)={\bf M}(-t)={\bf C}-{\bf S}\,.$ (63)

For the matrix of second moments $\sigma$ and ${\bf S}=\sigma\,\gamma_{0}$ Eq.
(57) and Eq. (58) yield:

$\begin{array}[]{rcl}\sigma(t)&=&\psi(t)\,\psi^{T}(t)={\bf
M}(t,t_{0})\,\psi(t_{0})\,\psi(t_{0})^{T}\,{\bf M}^{T}(t,t_{0})\\\
\sigma(t)\,\gamma_{0}&=&-{\bf M}(t,t_{0})\,\sigma(t_{0})\,\gamma_{0}^{2}\,{\bf
M}^{T}(t,t_{0})\,\gamma_{0}\\\ {\bf S}(t)&=&{\bf M}(t,t_{0})\,{\bf
S}(t_{0})\,{\bf M}^{-1}(t,t_{0})\,,\end{array}$ (64)

so that the evolution of ${\bf S}$ in time is a symplectic similarity
transformation. Hence all eigenvalues of ${\bf S}$ are constants.

From Eq. 57 it is obvious that the column vectors of the transfer matrix are
the solutions for initial conditions $\psi(0)$ that are equal to unit vectors.
The first column $\psi_{1}(t)$ of ${\bf M}(t,t_{0})$ is given by:

$\psi_{1}(t)={\bf M}(t,t_{0})\,(1,0,0,0)^{T}\,.$ (65)

Furthermore we note that if a symplex ${\bf F}$ is symplectically similar to
$\gamma_{0}$, then it follows from Eq. (58)

${\bf F}\propto{\bf M}\,\gamma_{0}\,{\bf M}^{-1}={\bf M}\,{\bf
M}^{T}\,\gamma_{0}\,,$ (66)

so that ${\bf F}$ equals the matrix ${\bf S}=\sigma\,\gamma_{0}$ if the second
moments are the average over the orthogonal unit vectors as exemplified by Eq.
(65):

$\sigma={\bf M}\,{\bf M}^{T}=\sum_{k}\,\psi_{k}(t)\psi_{k}^{T}(t)\,.$ (67)

This is another example for the mentioned duality between observables and
generators (see Eq. (36) and Ref. (Grgin )).

###  VI.2 Cosymplices as Generators of Transformations

The definition of what we consider to be a symplex or a co-symplex depends on
the choice of the symplectic unit matrix $\gamma_{0}$. Eq.
(LABEL:eq_sine_cosine) holds also for (skew-) symmetric cosymplices as it
holds for symplices. Consider a skew-symmetric cosymplex $\gamma_{j}$ (which
we know to anticommute with $\gamma_{0}$), then the matrix exponential is an
orthogonal similarity transformation, so that the transformed matrix
$\gamma_{0}$ is skew-symmetric and squares to the negative unit matrix:

$\begin{array}[]{rcl}\tilde{\gamma}_{0}^{T}&=&({\bf R}_{j}\,\gamma_{0}\,{\bf
R}_{j}^{T})^{T}={\bf R}_{j}\,\gamma_{0}^{T}\,{\bf R}_{j}^{T}\\\ &=&-{\bf
R}_{j}\,\gamma_{0}\,{\bf R}_{j}^{T}=-\tilde{\gamma}_{0}\\\
\tilde{\gamma}_{0}^{2}&=&({\bf R}_{j}\,\gamma_{0}\,{\bf R}_{j}^{T})^{2}={\bf
R}_{j}\,\gamma_{0}^{2}\,{\bf R}_{j}^{T}\\\ &=&-{\bf R}_{j}\,{\bf
R}_{j}^{T}=-{\bf 1}\\\ \end{array}$ (68)

The transformed matrix differs from $\gamma_{0}$, but has the same properties.

If we consider on the other hand a symmetric cosymplex $\gamma_{j}$ (which we
know to commute with $\gamma_{0}$), then the matrix exponential is also
symmetric and commutes with $\gamma_{0}$ so that

$\begin{array}[]{rcl}({\bf R}_{j}\,\gamma_{0}\,{\bf R}_{j}^{T})^{T}&=&{\bf
R}_{j}\,\gamma_{0}^{T}\,{\bf R}_{j}^{T}=-{\bf R}_{j}^{2}\,\gamma_{0}\\\ ({\bf
R}_{j}\,\gamma_{0}\,{\bf R}_{j}^{T})^{2}&=&{\bf
R}_{j}^{4}\,\gamma_{0}^{2}=-{\bf R}_{j}^{4}\,{\bf 1}\\\ \end{array}$ (69)

The transformed matrix is still antisymmetric but (in general) it does not
square to the (negative) unit matrix.

One may conclude that skew-symmetric cosymplices are generators of time-like
rotations: They change the “orientation” of $\gamma_{0}$, the system is
rotated towards a new “time direction” $\tilde{\gamma}_{0}={\bf
R}\,\gamma_{0}\,{\bf R}^{T}$, but remains structurally similar (though not the
same).

The physical meaning of such non-symplectic transformations can be related to
a variety of phenomena. In Ref. (newlook ) for instance non-symplectic
transformation have been shown to be related to the appearance of vector
potentials and to the transformation into curvilinear coordinate systems. In
App. C we give an example which is directly related to the structure of a
system. Furthermore we explicitely give a matrix that transforms between the
six possible choices for $\gamma_{0}$ of the Dirac algebra in App. C.3.

##  VII The Dimensionality of Spacetime

###  VII.1 Clifford Algebras

Sets of anticommuting nonsingular matrices are well-known as generators of
Clifford algebras. The inner products of the basic elements are the elements
of the metric tensor. The vector space constructed in this way is Euclidean
only for diagonal metric tensors with signatures $s_{1}=s_{2}=\dots=s_{N}=1$,
i.e. Euclidean unit vectors square to $+1$. Therefore generators that square
to $+1$ are spatial or space-like and generators that square to $-1$ time-like
151515 This is only meaningful for real representations, since the case of
complex representations allows to switch the sign of matrices by the
multiplication with the unit imaginary.. A Clifford algebra with $p$ space-
like and $q$ time-like generator matrices is indicated by $Cl_{p,q}$.

From the dimensional requirement

$2^{N}=(2n)^{2}=4\,n^{2}\,,$ (70)

it follows that such matrix systems can only be complete for even $N=2\,M$
161616 A Clifford algebra also can be defined for odd $N$, but these algebras
are not isomorphic to a representation in form of $2n\times 2n$ real
matrices.. Then:

$n=2^{M-1}\,$ (71)

Any real matrix system that fulfills the above requirements is based on a
number of DOFs which is a power of $2$. The simplest system hence is the one
for which $M=1$($N=2$, $n=1$) holds, i.e. it consists of $2\times 2$-matrices.

As mentioned in Sec. IV only those elements $\gamma_{k}$ that are symplices
have a non-vanishing expectation value $\bar{\psi}\,\gamma_{k}\,\psi$ and are
therefore observables. Hence we restrict ourselves to Clifford algebras with
generators that can be represented by symplices. This includes $\gamma_{0}$ as
the direction of time. Since any generator that anticommutes with $\gamma_{0}$
is symmetric and therefore squares to $+1$, we have $q=1$, i.e. the Clifford
algebras are of the type $Cl_{p,1}(\mathbb{R})$.

If $\gamma_{0},\gamma_{1},\dots,\gamma_{N-1}$ are the generators of a Clifford
algebra $Cl_{N-1,1}$ with metric

$\gamma_{\mu}\,\gamma_{\nu}+\gamma_{\nu}\,\gamma_{\mu}=2\,\textrm{Diag}(-1,1,\dots,1)\,,$
(72)

then $\gamma_{0}\,\gamma_{k}$ with $k=1,2,\dots\,N-1$ are the generators of
boosts (“boosters”) and $\gamma_{j}\,\gamma_{k}$ with $j\neq k$ and $j,k\neq
0$ are the generators of spatial rotations (“rotators”). We come back to this
in the next subsection.

###  VII.2 The Number of Space Dimensions

We have shown that it is possible to construct spacetime on the basis of
(Hamiltonian) dynamics and on the geometric interpretation of the matrix
operators that represent the observables. The possible numbers of spacetime
dimensions $N=p+q$ in this approach is given by the representation theory of
real matrices Okubo :

$p-q=0\textrm{ or }2\textrm{ mod }8\,.$ (73)

As we have derived above, there is only one single skew-symmetric (time-like)
symplex-generator of the Clifford algebra (i.e. $\gamma_{0}$), i.e. we have to
set $q=1$ and therefore there are $p=N-1$ space dimensions so that:

$N-2=0\textrm{ or }2\textrm{ mod }8\,.$ (74)

Hence dynamically emergent spacetimes are algebraically limited to $1+1$,
$3+1$, $9+1$, $11+1$, $17+1$, $19+1$, $25+1$, $27+1$, $\dots$ dimensions,
corresponding to symplices of size $2\times 2$, $4\times 4$, $32\times 32$,
$64\times 64$, $\dots$.

Symplices of size $2\times 2$ can be shown to have real or pure imaginary
eigenvalues and are therefore not able to represent the general (i.e. complex)
case. Therefore the $2\times 2$ case can not be generalized and is rejected as
too simple. Certainly there are multiple arguments, why spacetime should have
$3+1$ dimensions. In the framework described here, we give the following
reasons (besides simplicity):

  1. 1.

The eigenvalues of $2n\times 2n$ matrices over the reals are (for $n>1$) pairs
of complex conjugate eigenvalues. Since $N$ is even, all representations for
$N>1$ have matrix dimensions that are multiples of $4$ and can therefore
always be block-diagonalized to $4\times 4$-blocks. Therefore higher
dimensional cases based on $4^{M}\times 4^{M}$-matrices can be split into $M$
objects with representation $4\times 4$, i.e. into $M$ objects in $Cl_{3,1}$.

  2. 2.

The Pauli and the Dirac algebra are the only cases where the state vector
$\psi$ has the same dimension as the spacetime constructed from it, i.e. the
only case where $2\,n=N$ so that

$2^{N}=N^{2}\,.$ (75)

The number of variables of the spinor should be equal of higher than the
dimension of the object constructed from it - certainly not less.

  3. 3.

In 3 dimensions the number of rotation axes equals the number of spatial
directions since one can form $3$ rotator pairs $\gamma_{j}\,\gamma_{k}$ (both
space-like vectors) without repetition from 3 spatial directions. Both -
rotators and boosters - mutually anti-commute with each other and hence
generate an orthogonal space of the same dimension. In more than 3 space
dimensions this changes: The number of boosts stays the same, but the number
of (symplectic, i.e. binary) rotators grows quadratically with the number of
spatial directions $N-1$ since one can select $\left(N-1\atop
2\right)=(N-1)(N-2)/2$ rotators. For $N-1=4$ space dimensions this gives $6$
rotators and not all of them are orthogonal to each other. In more than $3$
space dimensions some rotators commute, for instance in a 4-dimensional space
the rotators $\gamma_{1}\gamma_{2}$ and $\gamma_{3}\gamma_{4}$ commute Nibart
. This means that (dynamically generated) spaces with more than 3 dimensions
are topologically not homogeneous. More precisely: $D=3$ is the only
considered space dimension in which all axis have the same relation to all
others, i.e. the only possible dynamically generated Euklidean space.

Assumed that spacetime is emergent, the only possible and generally useful
dimensionality of “objects” is $3+1$. Maybe each given argument alone
suffices.

In Sec. IV we raised the question of measurability. We emphasized that all
unit vectors have to be symplices in order to be measurable. Since all basic
elements (unit vectors) anticommute, also the bi-vectors - i.e. the products
of two unit vectors - are symplices. Geometrically bi-vectors are “oriented
faces”. However products of 3 orthogonal symplices \- which would correspond
to a volume - are cosymplices and therefore have vanishing expectation values
(see also App. A.1)). It follows that in dynamically emerging spaces “objects”
can have positions (or directions), but no measurable volume. I.e. fundamental
(measurable) objects are point-like particles.

###  VII.3 The Real Dirac Matrices

In what has been scetched above, we convinced ourselves that the case of real
$4\times 4$-matrices that represent a geometric (Clifford) algebra is unique.
It is the simplest and most general algebra that allows for construction of
spacetime. And it is the only algebra that generates a spacetime as we
experience it. With respect to the basic variables $\psi$ it represents
systems with 2 DOFs. It can also be regarded as “fundamental” as it describes
the smallest possible oscillatory system with (internal) coupling or
“interaction” 171717 The role of the Dirac matrices as generators of the
symplectic group $Sp(4,R)$ has been described before, see for instance Ref.
(DGL )..

An appropriate choice of the basic matrices of the Clifford algebra
$Cl_{3,1}\mathbb{(}R)$ are the well known Dirac matrices. Ettore Majorana was
the first who described a system of Dirac matrices that contains exclusively
imaginary (or exclusively real) values Majorana . We call these matrices the
“real Dirac matrices” (RDMs) and since we are committed to the reals, we will
use the RDMs instead of the conventional system 181818 According to the
fundamental theorem of the Dirac matrices by Pauli, we are free to choose any
system of Dirac matrices, since all systems are algebraically similar, i.e.
can be mapped onto each other via similarity transformations Pauli . The
transformation to the conventional system is given in App. C.4. The
isomorphism of this matrix group to the 3+2-dimensional de Sitter group has
been pointed out by Dirac in Ref. Dirac . .

Often the term “Dirac matrices” is used in more restrictive way and designates
only four matrices, i.e. the generators of the Clifford algebra 191919 Note
that we define $\gamma_{14}=\gamma_{0}\,\gamma_{1}\,\gamma_{2}\,\gamma_{3}$,
which is labeled $\gamma_{5}$ in QED.. We define the RDMs as a complete system
of matrices and include all elements of the Clifford algebra generated by
$\gamma_{\mu}$:

$\begin{array}[]{cccp{5mm}ccc}\gamma_{14}&=&\gamma_{0}\,\gamma_{1}\,\gamma_{2}\,\gamma_{3};&&\gamma_{15}&=&{\bf
1}\\\
\gamma_{4}&=&\gamma_{0}\,\gamma_{1};&&\gamma_{7}&=&\gamma_{14}\,\gamma_{0}\,\gamma_{1}=\gamma_{2}\,\gamma_{3}\\\
\gamma_{5}&=&\gamma_{0}\,\gamma_{2};&&\gamma_{8}&=&\gamma_{14}\,\gamma_{0}\,\gamma_{2}=\gamma_{3}\,\gamma_{1}\\\
\gamma_{6}&=&\gamma_{0}\,\gamma_{3};&&\gamma_{9}&=&\gamma_{14}\,\gamma_{0}\,\gamma_{3}=\gamma_{1}\,\gamma_{2}\\\
\gamma_{10}&=&\gamma_{14}\,\gamma_{0}&=&\gamma_{1}\,\gamma_{2}\,\gamma_{3}&&\\\
\gamma_{11}&=&\gamma_{14}\,\gamma_{1}&=&\gamma_{0}\,\gamma_{2}\,\gamma_{3}&&\\\
\gamma_{12}&=&\gamma_{14}\,\gamma_{2}&=&\gamma_{0}\,\gamma_{3}\,\gamma_{1}&&\\\
\gamma_{13}&=&\gamma_{14}\,\gamma_{3}&=&\gamma_{0}\,\gamma_{1}\,\gamma_{2}&&\\\
\end{array}$ (76)

There are $2\,n!=24$ possible permutations of the variables in the state
vector $\psi$, but since a swap of the two DOFs does not change the form of
$\gamma_{0}$, there are $12$ possible basic systems of RDMs rdm_paper . Since
$\gamma_{0}$ is antisymmetric, there are only six possible choices for
$\gamma_{0}$. Each of them allows to chose between two different sets of the
“spatial” matrices 202020 Given one system is known as $\gamma_{\mu}$ then the
alternative system is
$\gamma_{0},\,\gamma_{0}\gamma_{1},\,\gamma_{0}\gamma_{2},\,\gamma_{0}\gamma_{3}$..
The explicit form of the RDMs is given in Ref. (rdm_paper ; geo_paper ). The
four basic RDMs are anti-commuting

$\gamma_{\mu}\,\gamma_{\nu}+\gamma_{\nu}\,\gamma_{\mu}=2\,g_{\mu\nu}\,{\bf
1}\,,$ (77)

and the “metric tensor” $g_{\mu\nu}$ has the form 212121 The convention of QED
($g_{\mu\nu}=\mathrm{Diag}(1,-1,-1,-1)$) cannot be represented with real Dirac
matrices Okubo ; Scharnhorst .

$g_{\mu\nu}=\mathrm{Diag}(-1,1,1,1)\,.$ (78)

Any real $4\times 4$ matrix ${\bf A}$ can be written as a linear combination
of RDMs according to

${\bf A}=\sum\limits_{k=0}^{15}\,a_{k}\,\gamma_{k}\,,$ (79)

where a quarter of the trace $\mathrm{Tr}({\bf A})$ equals the scalar
component $a_{15}$ \- since all other RDMs have zero trace. The RDM-
coefficients $a_{k}$ are given by the scalar product

$a_{k}={\bf A}\,\cdot\,\gamma_{k}={\mathrm{Tr}(\gamma_{k}^{2})\over
4}\,\mathrm{Tr}({{\bf A}\,\gamma_{k}+\gamma_{k}\,{\bf A}\over 8})\,.$ (80)

Any real $4\times 4$ Hamiltonian matrix (symplex) ${\bf F}$ can be written as
a linear combination in this basis

${\bf F}=\sum\limits_{k=0}^{9}\,f_{k}\,\gamma_{k}\,.$ (81)

If Minkowski spacetime is interpreted as a physical reality without intrinsic
connection to quantum mechanics, then it appears to be a jolly contingency
that the (real) Dirac matrices have the geometric properties that exactly fit
to the needs of spacetime geometry Hestenes ; JCA . In Ref. JCA  the authors
write: “Historically, there have been many attempts to produce the appropriate
mathematical formalism for modeling the nature of physical space, such as
Euclid’s geometry, Descartes’ system of Cartesian coordinates, the Argand
plane, Hamilton’s quaternions, Gibbs’ vector system using the dot and cross
products. We illustrate however, that Clifford’s geometric algebra (GA)
provides the most elegant description of physical space.” If spacetime emerges
from dynamics then there is deeper reason for this elegance. Geometric algebra
(GA) - if introduced by postulating abstract mathematical anticommuting
objects - leave a mark of “artificial construction” that has few explanatory
power. But if spacetime emerges from dynamics in the form of the Dirac
algebra, then we have the missing link, the physical reason for the congruence
of GA with $3+1$-dimensional Minkowski spacetime.

##  VIII Lorentz Transformations are Symplectic Transformations

Structure preservation refers to Eq. (4). That is: the transformation of the
skew-symmetric matrix ${\bf S}$ to $\gamma_{0}$ is the exact contrary of
structure preserving - it is structure defining. Only transformations that
preserve the form of $\gamma_{0}$ are called structure preserving. And since
$\gamma_{0}$ represents the form of Hamilton’s equations, structure preserving
transformations are canonical. Linear structure preserving transformations are
called symplectic.

Using Eq. 58 it is easy to show that symplectic transformations are structure
preserving, i.e. a transformation of the dynamical variables $\psi$ that does
not change the form of the equations, neither it changes the form of
$\gamma_{0}$. For the matrix of observables ${\bf S}=\sigma\gamma_{0}$ we
expect a similarity transformation. The linear transformation ${\bf R}$ is
given as

$\tilde{\psi}={\bf R}\,\psi\,.$ (82)

Hence

$\begin{array}[]{rcl}{\bf S}&=&\langle\psi\psi^{T}\rangle\,\gamma_{0}\\\
{\bf\tilde{S}}&=&\langle\tilde{\psi}\tilde{\psi}^{T}\rangle\,\gamma_{0}\\\
&=&\langle{\bf R}\,\psi\psi^{T}\,{\bf R}^{T}\rangle\,\gamma_{0}\\\ &=&-{\bf
R}\,\langle\psi\psi^{T}\rangle\,\gamma_{0}\,\gamma_{0}\,{\bf
R}^{T}\gamma_{0}\\\ &=&-{\bf R}\,{\bf S}\,\gamma_{0}\,{\bf
R}^{T}\gamma_{0}\,,\end{array}$ (83)

so that the conditions are fulfilled if ${\bf R}$ is symplectic (fulfills Eq.
58):

$\begin{array}[]{rcl}-\gamma_{0}\,{\bf R}^{T}\gamma_{0}&=&{\bf R}^{-1}\\\ {\bf
R}\,\gamma_{0}\,{\bf R}^{T}&=&\gamma_{0}\,.\end{array}$ (84)

Then it is easy to show that symplectic transformations are structure
preserving, i.e. a symplex ${\bf F}$ transformed by a symplectic similarity
transformation ${\bf R}$ remains a symplex:

$\begin{array}[]{rcl}{\bf\tilde{F}}&=&{\bf R}\,{\bf F}\,{\bf R}^{-1}\\\
\Rightarrow{\bf\tilde{F}}^{T}&=&({\bf R}^{-1})^{T}\,{\bf F}^{T}\,{\bf
R}^{T}\\\ &=&(-\gamma_{0}\,{\bf R}\,\gamma_{0})\,(\gamma_{0}\,{\bf
F}\,\gamma_{0})\,(-\gamma_{0}\,{\bf R}^{-1}\,\gamma_{0})\\\
&=&\gamma_{0}\,{\bf R}\,{\bf F}\,{\bf R}^{-1}\,\gamma_{0}\\\
&=&\gamma_{0}\,{\bf\tilde{F}}\gamma_{0}\,.\end{array}$ (85)

If we compare this result with Eq. (64), then we recognize that a
transformation is symplectic if it is possible to find a (constant)
Hamiltonian from which the transformation can be derived, i.e. a
transformation is symplectic, if it can be expressed as the result of a
possible evolution in time.

As we show in the following, it is legitimate to call the structure preserving
property of symplectic transformations the principle of (special) relativity
since Lorentz transformations (LTs) of spinors in Minkowski spacetime are a
subset of the possible symplectic transformations for two DOFs. It is well-
known that the matrix exponential of a “Hamiltonian matrix” (i.e. a symplex)
is a symplectic transformation. It remains to be shown which symplices
generate LTs. Consider we transform a matrix ${\bf X}={\cal
E}\,\gamma_{0}+P_{x}\,\gamma_{1}+P_{y}\,\gamma_{2}+P_{z}\,\gamma_{3}$ using
${\bf R}_{4}=\exp{(\gamma_{4}\,\varepsilon/2)}$, then we obtain after
decomposition of the transformed matrix ${\bf X}^{\prime}$ into the RDM-
coefficients:

$\begin{array}[]{rcl}{\bf X}^{\prime}&=&{\bf R}_{4}\,{\bf X}\,{\bf
R}_{4}^{-1}\\\ {\bf X}^{\prime}&=&{\cal
E}^{\prime}\,\gamma_{0}+P_{x}^{\prime}\,\gamma_{1}+P_{y}^{\prime}\,\gamma_{2}+P_{z}^{\prime}\,\gamma_{3}\\\
{\cal E}^{\prime}&=&{\cal
E}\,\cosh{(\varepsilon)}+P_{x}\,\sinh{(\varepsilon)}\\\
P_{x}^{\prime}&=&P_{x}\,\cosh{(\varepsilon)}+{\cal E}\,\sinh{(\varepsilon)}\\\
P_{y}^{\prime}&=&P_{y}\\\ P_{z}^{\prime}&=&P_{z}\\\ \end{array}$ (86)

Using the usual parametrization of special relativity where $\varepsilon$ is
the “rapidity” ($\beta=\tanh{(\varepsilon)}$, $\gamma=\cosh{(\varepsilon)}$,
$\beta\gamma=\sinh{(\varepsilon)}$), we find

$\begin{array}[]{rcl}{\cal E}^{\prime}&=&\gamma\,{\cal
E}+\beta\,\gamma\,P_{x}\\\ P_{x}^{\prime}&=&\gamma\,P_{x}+\beta\,\gamma\,{\cal
E}\,,\end{array}$ (87)

which is a Lorentz boost along $x$. Same transformation, but ${\bf
X}=E_{x}\,\gamma_{4}+E_{y}\,\gamma_{5}+E_{z}\,\gamma_{6}+B_{x}\,\gamma_{7}+B_{y}\,\gamma_{8}+B_{z}\,\gamma_{9}$:

$\begin{array}[]{rcl}{\bf X}^{\prime}&=&{\bf R}\,{\bf X}\,{\bf R}^{-1}\\\
E_{x}^{\prime}&=&E_{x}\\\ B_{x}^{\prime}&=&B_{x}\\\
E_{y}^{\prime}&=&E_{y}\,\cosh{(\varepsilon)}+B_{z}\,\sinh{(\varepsilon)}\\\
E_{z}^{\prime}&=&E_{z}\,\cosh{(\varepsilon)}-B_{y}\,\sinh{(\varepsilon)}\\\
B_{y}^{\prime}&=&B_{y}\,\cosh{(\varepsilon)}-E_{z}\,\sinh{(\varepsilon)}\\\
B_{z}^{\prime}&=&B_{z}\,\cosh{(\varepsilon)}+E_{y}\,\sinh{(\varepsilon)}\\\
\end{array}$ (88)

which is again a Lorentz boost along $x$, but now it shows the transformation
behavior of electromagnetic fields. Obviously the algebraic structure of the
Dirac matrices in combination with symplectic dynamics results directly in
Lorentz transformations. Indeed the transformations above are identical to the
Lorentz transformations of the Dirac equation222222Most textbooks on
relativistic QM contain inadequately short and/or incomprehensible
descriptions of the covariance of the Dirac equation. The best depiction found
by the author is given in Ref. (Schmueser )..

This derivation did not require any reference to the speed of light, to time
dilation or with alike. The usual approach that derives LTs from the
“constancy of the speed of light” has certainly a historical legitimization,
but is (in our opinion) logically misleading: the “constancy” of the speed of
light is neither the cause nor the reason for, but a consequence of (special)
relativity. Logically electrodynamics and hence the constancy of the speed of
light has to be derived from relativity and not vice versa - unless we can
show how and why the LTs can be derived from electrodynamics and
electrodynamics independently from first principles. However - as often
declared - it is not possible to derive electrodynamics from relativity alone
FLS . But if spacetime emerges from (symplectic) dynamics, then there are
additional restrictions and MWEQs can indeed be “derived”. It is an
experimental fact that electromagnetic interactions are structure preserving.
No electromagnetic process known to the author is able to transform a fermion
into another. There is pair production and particle-antiparticle annihilation,
but it is conventional wisdom to identify the antiparticle with the particle
going backwards in time Schmueser . This only strengthens the argument, that
electromagnetic fields are structure preserving.

###  VIII.1 The Electromechanical Equivalence (EMEQ)

Since we started our considerations with abstract entities, it is unavoidable
to present an interpretation of the terms (i.e. the observables) derived from
these entities. The transformation properties (LABEL:eq_lt1) and
(LABEL:eq_lt2) suggest the introduction of the electromechanical equivalence
(EMEQ), as presented in Refs. (rdm_paper ,geo_paper ). In those papers the
EMEQ was merely used as a formal tool that allows to obtain a descriptive
interpretation of symplectic transformations. Here we argue substantially and
present a derivation of MWEQs.

In Sec. VI.1 we have shown that the fundamental solution of the EQOM is given
by a symplectic transfer matrix. In order to produce constants of motion, the
transfer matrix should represent a strongly stable system, i.e. all of its
eigenvalues lie on the unit circle in the complex plane. In this case the
symplex ${\bf F}$ has exclusively purely imaginary eigenvalues. We analyzed
the structure of ${\bf F}$ by introducing the RDMs which are a representation
of the Clifford algebra $Cl_{3,1}$. Therefore we have four basic elements and
six bi-vectors which are symplices. The other $6$ members of the group of the
RDMs are cosymplices. The analysis of the transformation properties indicates
how to interpret the coefficients of the algebra: we identify the $f_{k}$
((Eq. 81) as follows:

$\begin{array}[]{rclr}f_{0}&\equiv&{\cal E}&\textrm{Energy}\\\
(f_{1},f_{2},f_{3})^{T}&\equiv&{\vec{P}}&\textrm{Momentum}\\\
(f_{4},f_{5},f_{6})^{T}&\equiv&{\vec{E}}&\textrm{electric field}\\\
(f_{7},f_{8},f_{9})^{T}&\equiv&{\vec{B}}&\textrm{magnetic field}\\\
\end{array}$ (89)

With this nomenclature the eigenvalues
$\pm\,i\,\omega_{1},\,\pm\,i\,\omega_{2}$ of ${\bf F}$ are given as:

$\begin{array}[]{rcl}\omega_{1}&=&\sqrt{K_{1}+2\,\sqrt{K_{2}}}\\\
\omega_{2}&=&\sqrt{K_{1}-2\,\sqrt{K_{2}}}\\\ K_{1}&=&-\mathrm{Tr}({\bf
F}^{2})/4={\cal E}^{2}+\vec{B}^{2}-\vec{E}^{2}-\vec{P}^{2}\\\
K_{2}&=&\mathrm{Tr}({\bf F}^{4})/16-K_{1}^{2}/4\\\ &=&({\cal
E}\,\vec{B}-\vec{P}\times\vec{E})^{2}-(\vec{E}\cdot\vec{B})^{2}-(\vec{P}\cdot\vec{B})^{2}\\\
\mathrm{Det}({\bf F})&=&\omega_{1}^{2}\,\omega_{2}^{2}=K_{1}^{2}-4\,K_{2}\\\
\end{array}$ (90)

Bi-vectors form a so-called even subalgebra of the Dirac algebra232323 We
define elements to be even, if they do not change sign when all basic elements
$\gamma_{0},\dots,\gamma_{3}$ change sign. They are odd otherwise. I.e. bi-
vectors are even, vectors and tri-vectors are odd.. We have shown that any
Clifford algebra $Cl_{p,q}$ which is able to represent spacetime, must have an
even number of generators $N$. This holds also for the Dirac algebra. Even
dimensional algebras have even subalgebras, which means that even elements
generate exclusively even elements, whereas odd elements can be used to
generate the full algebra: Fermions can generate electromagnetic fields, but
photons can not generate fermions. Consequently there should be
(differential-) equations that allow to derive the bi-vector fields $\vec{E}$
and $\vec{B}$ from vectors, but not vice versa. This means, that the
description of a free particle without external fields should not refer to bi-
vectors. Hence it is natural to distinguish the vector and bi-vector
components and for vanishing bi-vectors ($\vec{E}=\vec{B}=0$) one finds:

$\begin{array}[]{rcl}K_{1}&=&{\cal E}^{2}-\vec{P}^{2}\\\ K_{2}&=&0\\\
\omega_{1}&=&\omega_{2}=\sqrt{K_{1}}=\sqrt{{\cal
E}^{2}-\vec{P}^{2}}\,.\end{array}$ (91)

At the end of Sec. IV we argued that a constant reference can be constructed
as quadratic forms. Here we have an example, since $\sqrt{K_{1}}=m=\sqrt{{\cal
E}^{2}-\vec{P}^{2}}$: The mass is a constant of motion and has the physical
meaning and unit of frequency. This is an important result and a significant
step towards quantum mechanics. It follows that the “time” parameter as we
introduced it in Eqs. (8), is identical to the eigentime. The eigenfrequency
that defines the scale of the eigentime is the mass and hence a massive
particle is described by an inertial frame of reference in combination with an
oscillator of constant frequency (i.e. a clock). This is the deeper meaning of
Einstein’s clock attached to all inertial frames.

For vanishing vector components (${\cal E}=0=\vec{P}$) we have

$\begin{array}[]{rcl}K_{1}&=&\vec{B}^{2}-\vec{E}^{2}\\\
K_{2}&=&-(\vec{E}\cdot\vec{B})^{2}\,,\end{array}$ (92)

which are the well-known relativistic invariants of the electromagnetic field.
It follows that the frequencies are only real-valued for
$\vec{E}\cdot\vec{B}=0$. An analysis of the transformations generated by the
bi-vectors yields that the elements $\gamma_{4},\gamma_{5}$ and $\gamma_{6}$
are responsible for Lorentz boosts, while $\gamma_{7},\gamma_{8}$ and
$\gamma_{9}$ are generators of (spatial) rotations. This corresponds to our
physical intuition since (charged) particles are accelerated by electric
fields but their “trajectories” are bended (i.e. rotated) in magnetic fields.
It is therefore no surprise that - using the EMEQ - the derivation of the
Lorentz force is straightforward (see Sec. VIII.2).

Furthermore we note that there are vector-elements associated with ${\cal E}$
and $\vec{P}$, but no elements associated directly with the spacetime
coordinates. For a representation of a single elementary particle this makes
sense insofar as the dynamics of a single particle may not refer to absolute
positions (unless via the bi-vector fields). Spacetime coordinates do not
refer to particle properties, but to the relativ position (relation) of
particles. Hence as long as we refer to a single object based on fundamental
variables, we observe the momentum and energy of a particle as it behaves in
an external electromagnetic field.

As we already mentioned in Sec. (IV), the symplex ${\bf S}=\sigma\,\gamma_{0}$
is singular, if we use only a single real spinor $\psi$ to define

$\sigma=\langle\,\psi\,\psi^{T}\,\gamma_{0}\,\rangle\,,$ (93)

where the angles imply some (unspecified) sort of average. Since the
eigenvalues of the symplex ${\bf S}$ yield the “frequency” or “mass”
$m=\sqrt{{\cal E}^{2}-\vec{P}^{2}}$ of the structure which we identify with a
particle, we need at least $N\geq 4$ linear independent spinors to obtain a
massive fermion. In the classical Dirac theory, the electron spinor is a
linear combination of two complex spinors, i.e. also requires 4 real spinors.
From this point of view inertial mass might be interpreted statistically - as
a property that can only be derived using averaging - either over samples or
in time. The idea that mass might be a statistical phenomenon is neither new
nor extraordinarily exotic, see for instance Ref. (Verlinde ; MvR1 ) and
references therein. But here we do not refer to entropy but rather to
dynamical or algebraic properties of real spinors.

###  VIII.2 The Lorentz Force

The 4-momentum of a particle (fermion) is defined by vectors. The vector
components of ${\bf S}$ are associated with the 4-momentum:

${\bf P}={\cal
E}\,\gamma_{0}+P_{x}\,\gamma_{1}+P_{y}\,\gamma_{2}+P_{z}\,\gamma_{3}={\cal
E}\,\gamma_{0}+\vec{P}\,\vec{\gamma}\,,$ (94)

where ${\cal E}$ is the energy and $\vec{P}$ the momentum. Accordingly fields
are associated with the bi-vectors of the “force” matrix ${\bf F}$. Then Eq.
(13) is (in appropriate units) the Lorentz force equations

${d{\bf P}\over d\tau}=\dot{\bf P}={q\over 2\,m}\left({\bf F}\,{\bf P}-{\bf
P}\,{\bf F}\right)\,,$ (95)

where $\tau$ is the proper time and ${q\over 2m}$ is a relative scaling
factor. In the lab frame time $dt=\gamma\,d\tau$ EQ. 95 yields (setting
$c=1$):

$\begin{array}[]{rcl}{d{\cal E}\over d\tau}&=&{q\over m}\,\vec{P}\,\vec{E}\\\
{d\vec{P}\over d\tau}&=&{q\over m}\,\left({\cal
E}\,\vec{E}+\vec{P}\times\vec{B}\right)\\\ \gamma\,{d{\cal E}\over
dt}&=&q\,\gamma\,\vec{v}\,\vec{E}\\\ \gamma\,{d\vec{P}\over dt}&=&{q\over
m}\,\left(m\,\gamma\,\vec{E}+m\,\gamma\,\vec{v}\times\vec{B}\right)\\\
{dE\over dt}&=&q\,\vec{v}\,\vec{E}\\\ {d\vec{P}\over
dt}&=&q\,\left(\vec{E}+\vec{v}\times\vec{B}\right)\,.\end{array}$ (96)

(Note that Eqn. 95 allows to add arbitrary multiples of ${\bf P}$ to ${\bf F}$
without an effect on ${\bf\dot{P}}$.)

###  VIII.3 Electrodynamics from Symplectic Spacetime

The space- and time- coordinates are now written as a 4-vector

${\bf X}=\gamma_{0}\,t+\vec{\gamma}\cdot\vec{x}\,,$ (97)

and we will try to express the fields $\vec{E}$ and $\vec{B}$ \- which were
originally understood as functions of the eigentime $\tau$, are now
parameterized by $\vec{x}$ and $t$:

$\begin{array}[]{rcl}\vec{E}(\tau)&\to&\vec{E}(\vec{x},t)\\\
\vec{B}(\tau)&\to&\vec{B}(\vec{x},t)\,.\end{array}$ (98)

We extend the dynamics of (single) variables to the dynamics of “fields”,
depending on vector type parameters. We denote the coefficients of these
vectors $(t,\vec{x})$ and keep the question open, how these vectors are
related to $({\cal E},\vec{P})$. We can’t resist to cite Einstein here:
“Spacetime does not claim existence on its own, but only as a structural
quality of the field” Einstein . Following his comprehension we introduce
spacetime as a property of fields.

There have been several attempts to “derive” MWEQs in the past Krefetz ; Kobe1
; Kobe2 ; Dyson ; Gersten ; DWGS . None of these succeeded in finding general
acceptance. Most textbooks treat just the Lorentz covariance of MWEQs. A
derivation of MWEQs from the covariance condition alone is considered
impossible FLS . But our ansatz is different insofar as we constructed an
algebraic framework that allows for additional symplectic constraints.
Symplectic constraints are well-known in different context and led (for
instance) to the non-squeezing theorem, i.e. to the metapher of the
“symplectic egg” Gromov ; Gosson . We have shown that Lorentz transformations
as well as the action of the Lorentz force are symplectic. Furthermore it is
an experimental fact that electromagnetic interaction is structure preserving.
Hence the appearance of cosymplices as generators is excluded for
electromagnetic theory.

In order to keep track of the transformation properties with respect to
symplectic transformations the bi-vectors fields can only be expressed (for
instance in form of a Taylor series) by algebraic expressions that respect the
appropriate transformation properties. According to Eq. (LABEL:eq_emeq) the
field matrix is given by

${\bf
F}=E_{x}\,\gamma_{4}+E_{y}\,\gamma_{5}+E_{z}\,\gamma_{6}+B_{x}\,\gamma_{7}+B_{y}\,\gamma_{8}+B_{z}\,\gamma_{9}\,.$
(99)

If we write the fields as a Taylor series, the first must have the following
form in order to be linear in ${\bf X}$ and to yield a bi-vector:

${\bf F}={\bf F}_{0}+({\cal D}{\bf F})\,{\bf X}-{\bf X}({\cal D}{\bf
F})+\cdots\,,$ (100)

where ${\cal D}{\bf F}$ is an appropriate derivative taken at ${\bf X}=0$ and
must be a vector. Alternative forms of the linear term either include an
“axial” vector (cosymplex) ${\bf V}$ or a pseudoscalar, both having a
vanishing expectation value:

$\begin{array}[]{rcl}{\bf
V}&=&V_{t}\,\gamma_{10}+V_{x}\,\gamma_{11}+V_{y}\,\gamma_{12}+V_{z}\,\gamma_{13}\\\
{\bf F}_{1}&=&({\bf V}\,{\bf X}+{\bf X}{\bf V})/2\\\ \end{array}$ (101)

Since we can exclude the appearance of cosymplices in structure preserving
interactions, the first order term has the form of Eq. 100.

In the following we analyze first and second order expressions that can be
constructed with respect to the transformation properties. We skip constant
terms for the moment and analyze the relations of the partial derivatives.
Linear terms can exclusively be expressed by the commutator of two vector
quantities, i.e. ${\bf X}$ and another (constant) 4-vector ${\bf
J}=\rho_{0}\,\gamma_{0}+\vec{j}_{0}\cdot\vec{\gamma}$:

${\bf
F}_{1}=(\gamma_{0}\,\vec{E}+\gamma_{14}\,\gamma_{0}\vec{B})\cdot\vec{\gamma}={4\pi\over
3}\,\frac{1}{2}\,({\bf X}\,{\bf J}-{\bf J}\,{\bf X})\,,$ (102)

so that we obtain

$\begin{array}[]{rcl}\vec{E}&=&{4\pi\over
3}\,(\vec{j}_{0}\,t-\rho_{0}\,\vec{x})\\\ \vec{B}&=&{4\pi\over
3}\,(\vec{x}\times\vec{j}_{0})\\\ \end{array}$ (103)

and find that these linear terms fulfill MWEQs:

$\begin{array}[]{rcl}\vec{\nabla}\cdot\vec{E}&=&4\pi\,\rho_{0}\\\
\vec{\nabla}\cdot\vec{B}&=&0\\\
\vec{\nabla}\times\vec{E}+\partial_{t}\vec{B}&=&0\\\
\vec{\nabla}\times\vec{B}-\partial_{t}\vec{E}&=&{4\pi\vec{j}_{0}}\\\
\end{array}$ (104)

We continue with a second order term 242424In first order the two partials of
the induction law vanish separately, i.e
$\vec{\nabla}\times\vec{E}=0=\partial_{t}\vec{B}$.:

${\bf F}_{2}=\frac{1}{4}\,({\bf X}\,{\bf J}\,{\bf X}\,{\bf J}-{\bf J}\,{\bf
X}\,{\bf J}\,{\bf X})\,,$ (105)

and it also fulfills MWEQs:

$\begin{array}[]{rcl}\vec{\nabla}\cdot\vec{E}&=&4\,\pi\,\rho_{1}\\\
\vec{\nabla}\cdot\vec{B}&=&0\\\
\vec{\nabla}\times\vec{E}+\partial_{t}\vec{B}&=&0\\\
\vec{\nabla}\times\vec{B}-\partial_{t}\vec{E}&=&4\,\pi\,\vec{j}_{1}\,,\end{array}$
(106)

where

$\begin{array}[]{rcl}\rho_{1}&=&-\vec{j}_{0}^{2}\,t-3\,t\,\rho_{0}^{2}+4\,(\vec{j}_{0}\cdot\vec{x})\,\rho_{0}\\\
\vec{j}_{1}&=&4\,(\vec{j}_{0}\cdot\vec{x}-\rho_{0}\,t)\vec{j}_{0}+(\rho_{0}^{2}-\vec{j}_{0}^{2})\,\vec{x}\,.\end{array}$
(107)

Both orders fulfill the continuity equation:

$\partial_{t}\,\rho+\vec{\nabla}\cdot\vec{j}=0\,.$ (108)

So far this is not a derivation of MWEQs, but it shows that the transformation
properties of bi-vectors and the construction of spacetime imply MWEQs. There
is a method to express these ideas mathematically by the use of differential
operators. Since the aim is the construction of spacetime, the appropriate
generalized derivative must by of vector-type, i.e. we define the covariant
derivative by

$\partial\equiv-\partial_{t}\,\gamma_{0}+\partial_{x}\,\gamma_{1}+\partial_{y}\,\gamma_{2}+\partial_{z}\,\gamma_{3}\,.$
(109)

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. 109, $\overset{\rightarrow}{\partial}$ and
$\overset{\leftarrow}{\partial}$ which is written to the right of the operand
(thus indicating the order of the matrix multiplication) so that

$\begin{array}[]{rcl}{\bf
F}\overset{\leftarrow}{\partial}&\equiv&-\partial_{t}\,{\bf
F}\,\gamma_{0}+\partial_{x}\,{\bf F}\,\gamma_{1}+\partial_{y}\,{\bf
F}\,\gamma_{2}+\partial_{z}\,{\bf F}\,\gamma_{3}\\\
\overset{\rightarrow}{\partial}{\bf F}&\equiv&-\gamma_{0}\,\partial_{t}\,{\bf
F}+\gamma_{1}\,\partial_{x}\,{\bf F}+\gamma_{2}\,\partial_{y}\,{\bf
F}+\gamma_{3}\,\partial_{z}\,{\bf F}\,.\end{array}$ (110)

In order to keep the transformation properties of derivative expressions
transparent, we distinguish between the commutative

$\partial\wedge{\bf
A}\equiv\frac{1}{2}\,\left(\overset{\rightarrow}{\partial}{\bf A}-{\bf
A}\overset{\leftarrow}{\partial}\right)$ (111)

and the anti-commutative

$\partial\cdot{\bf
A}\equiv\frac{1}{2}\,\left(\overset{\rightarrow}{\partial}{\bf A}+{\bf
A}\overset{\leftarrow}{\partial}\right)$ (112)

derivative. Then we find:

$\begin{array}[]{rcl}\frac{1}{2}\,\left(\overset{\rightarrow}{\partial}\textrm{
vector }-\textrm{ vector
}\overset{\leftarrow}{\partial}\right)&\Rightarrow&\textrm{ bi-vector}\\\
\frac{1}{2}\,\left(\overset{\rightarrow}{\partial}\textrm{ vector }+\textrm{
vector }\overset{\leftarrow}{\partial}\right)&\Rightarrow&\textrm{
scalar}=0\\\ \frac{1}{2}\,\left(\overset{\rightarrow}{\partial}\textrm{ bi-
vector }-\textrm{ bi-vector
}\overset{\leftarrow}{\partial}\right)&\Rightarrow&\textrm{ vector}\\\
\frac{1}{2}\,\left(\overset{\rightarrow}{\partial}\textrm{ bi-vector
}+\textrm{ bi-vector
}\overset{\leftarrow}{\partial}\right)&\Rightarrow&\textrm{ axial vector}=0\\\
\end{array}$ (113)

Since (in case of the Dirac algebra) any symplex is the sum of a vector and a
bivector, we find (for this special case):

$\begin{array}[]{rcl}\frac{1}{2}\,\left(\overset{\rightarrow}{\partial}\textrm{
symplex }+\textrm{ symplex
}\overset{\leftarrow}{\partial}\right)&=&\partial\cdot\textrm{ symplex }=0\\\
\end{array}$ (114)

Now we return to Eq. 100 and find that ${\cal D}{\bf F}$ must be a vector and
therefore has (in “first order”) the form

${\cal D}{\bf F}=\frac{1}{2}\,\left(\overset{\rightarrow}{\partial}{\bf
F}-{\bf F}\overset{\leftarrow}{\partial}\right)=4\,\pi\,{\bf J}\,,$ (115)

which is nothing but a definition of the vector current

${\bf
J}=\rho\,\gamma_{0}+j_{x}\,\gamma_{1}+j_{y}\,\gamma_{2}+j_{z}\,\gamma_{3}\,.$
(116)

Written explicitely in components, Eq. 115 is given by

$\begin{array}[]{rcl}\vec{\nabla}\cdot\vec{E}&=&4\,\pi\,\rho\\\
\vec{\nabla}\times\vec{B}-\partial_{t}\vec{E}&=&4\,\pi\,\vec{j}\\\
\end{array}$ (117)

We define a vector-potential ${\bf
A}=\phi\,\gamma_{0}+A_{x}\,\gamma_{1}+A_{y}\,\gamma_{2}+A_{z}\,\gamma_{3}$
which has (to first order) the form 252525 The commutative derivative of the
trivial first order form ${\bf A}\propto{\bf X}$ vanishes and is therefore
useless in this context.:

${\bf A}_{1}=\frac{1}{4}\,\left({\bf X}\,{\bf F}_{0}-{\bf F}_{0}\,{\bf
X}\right)\,,$ (118)

where ${\bf F}_{0}$ is the constant term of Eq. 100. It is easily verified
that ${\bf F}_{0}=\frac{1}{2}\,\left(\overset{\rightarrow}{\partial}{\bf
A}_{0}-{\bf A}_{0}\overset{\leftarrow}{\partial}\right)$ and hence the
electromagnetic (bi-vector-) fields of the general symplex ${\bf F}$ is given
by

${\bf F}=\frac{1}{2}\,\left(\overset{\rightarrow}{\partial}{\bf A}-{\bf
A}\overset{\leftarrow}{\partial}\right)\,,$ (119)

or explicitely in components:

$\begin{array}[]{rcl}\vec{E}&=&-\vec{\nabla}\phi-\partial_{t}\vec{A}\\\
\vec{B}&=&\vec{\nabla}\times\vec{A}\,.\end{array}$ (120)

It is well-known that the homogeneous MWEQs are a direct consequence of Eq.
119:

$\begin{array}[]{rcl}\overset{\rightarrow}{\partial}{\bf F}+{\bf
F}\overset{\leftarrow}{\partial}&=&\frac{1}{2}\,\left(\overset{\rightarrow}{\partial}^{2}\,{\bf
A}-{\bf A}\overset{\leftarrow}{\partial}^{2}\right)=0\\\
\vec{\nabla}\cdot\vec{B}&=&0\\\
\vec{\nabla}\times\vec{E}+\partial_{t}\vec{B}&=&0\\\ \end{array}$ (121)

since the squared operators are scalars and commute with ${\bf A}$.
Accordingly the continuity equation is a direct consequence of Eq. 115:

$\begin{array}[]{rcl}\overset{\rightarrow}{\partial}{\bf J}+{\bf
J}\overset{\leftarrow}{\partial}&=&\frac{1}{8\pi}\,\left(\overset{\rightarrow}{\partial}^{2}{\bf
F}-\overset{\rightarrow}{\partial}{\bf
F}\overset{\leftarrow}{\partial}+\overset{\rightarrow}{\partial}{\bf
F}\overset{\leftarrow}{\partial}-{\bf
F}\overset{\leftarrow}{\partial}^{2}\right)\\\
&=&\partial_{t}\rho+\vec{\nabla}\vec{j}=0\,.\end{array}$ (122)

If one assumes the validity of the Lorentz gauge according to Eq.
LABEL:eq_difftypes

$\frac{1}{2}\,\left(\overset{\rightarrow}{\partial}{\bf A}+{\bf
A}\overset{\leftarrow}{\partial}\right)=\partial_{t}\,\phi+\vec{\nabla}\vec{A}=0\,,$
(123)

then one obtains the wave equation of the vector potential ${\bf A}$:

$\begin{array}[]{rcl}4\,\pi\,{\bf
J}&=&-\overset{\rightarrow}{\partial}^{2}{\bf
A}=(\partial_{t}^{2}-\vec{\nabla}^{2})\,{\bf A}\,.\end{array}$ (124)

In a “current free region” the fields have to fulfill the homogeneous wave
equation:

$\overset{\rightarrow}{\partial}^{2}\,{\bf
F}=(\vec{\nabla}^{2}-\partial_{t}^{2}){\bf F}=0\,.$ (125)

It follows that plane electromagnetic fields “in vacuum” hold
$\omega^{2}=\vec{k}^{2}$ so that the fields have no “eigenfrequency” and no
dispersion.

Note that Eqs. (LABEL:eq_MWhomo) and (122) are of the same form. The
continuity equation (122) describes charge as something with the properties of
a substance in spacetime.

In summary: We have shown that Lorentz transformations can indeed be
understood as symplectic transformations. The generators of these
transformations are electromagnetic fields. Since electrodynamics is a
structure preserving interaction, the description of electromagnetic fields
requires that exclusively structure preserving terms appear. With these
“additional conditions” that we were able to derive MWEQs if we interpret the
bi-vectors of the matrix ${\bf F}$ as functions of vector-type spacetime
coordinates of $Cl_{3,1}(\mathbb{R})$. The bi-vector fields in ${\bf F}$
fulfill a wave equation that describes the propagation of electromagnetic
waves, i.e. of light. One finds that electromagnetic waves “in vacuum” fulfill
the relation $\omega^{2}=\vec{k}^{2}$. From the constant group velocity
${\partial\omega\over\partial k}=1$ we obtain the constancy of the speed of
light, Therefore we did not derive MWEQs from special relativity, but both
theories are the result of the construction of spacetime from symplectic
dynamics.

Above equations are not restricted to $3$ space dimensions. However the
connections between commutators and anti-commutators are way more complex in
$10$ or $12$ dimensions. We had to consider additional “channels”: In
3-dimensional space, the symplex ${\bf F}$ is composed exclusively of
components of the vector and bi-vector type. In higher-dimensional space (10-
or 12-dimensional spacetime), the general symplex ${\bf F}$ consists (besides
vectors and bi-vectors) of n-vectors with $n\in[5,6,9,10,\dots]$ (see App.
A.1). Accordingly we had to include higher-order terms into the Hamiltonian
and likely we would have trouble with the stability of the described objects.

###  VIII.4 The Density

From the combination of Eq. 98 and Eq. 124 it follows that also the (current-)
density has also to be constructed as a function of space and time:

$\begin{array}[]{rcl}\rho&=&\rho(\vec{x},t)\\\
\vec{j}&=&\vec{j}(\vec{x},t)\,.\end{array}$ (126)

and it is clear that we have to normalize the density by

$\int\,\rho(\vec{x},t)\,d^{3}x=const=Q\,,$ (127)

where $Q$ is the “charge”. Especially the relation between the density
$\rho(\vec{x},t)$ and the “phase space density” $\tilde{\rho}(\psi)$ according
to Eq. 29 requires some attention. It is clear that the naive assumption

$\rho(\vec{x},t)=\tilde{\rho}(\psi(\vec{x},t))$ (128)

can not be applied directly. According to Eq. 29 we have

$1=\int\,\tilde{\rho}(\psi)\,d^{4}\psi=\int\,\tilde{\rho}(\psi(\vec{x},t))\,\sqrt{g}\,d^{3}x\,,$
(129)

where $g$ is the appropriate Gramian determinant. We assume in the following
that the normalization has been adjusted accordingly.

If spacetime was no construction, we could be sure that the wave function
$\psi(\vec{x},t)$ is well-defined and single valued for any coordinate of
Minkowski spacetime. However, if spacetime emerges from dynamics, then
spacetime coordinates might be functions of the phase space position
$t(\psi),\vec{x}(\psi)$. This includes the possibility that a) different phase
space positions are mapped to the same spacetime-position and b) that space
and time coordinates are not unique, i.e. particles appear “instantaneously”
at different “locations” of spacetime. Finally this might indeed be the reason
why $\psi$ has to be interpreted as a “probablity density”.

###  VIII.5 The Wave Equations

According to Eq. 124 the vector potential fulfills a wave equation - and also
the (electromagnetic) fields can be described by waves. Solutions of wave
equations are usually analyzed with the help of the Fourier transformation,
i.e. the solutions can be written as superpositions of plane waves

$\begin{array}[]{rcl}\phi(\vec{x},t)&=&\int\,\tilde{\phi}(\vec{k},\omega)\,e^{i(\vec{k}\cdot\vec{x}-\omega
t)}\,d^{3}k\\\
\tilde{\phi}(\vec{k},\omega)&=&\int\,\phi(\vec{x},t)\,e^{-i(\vec{k}\cdot\vec{x}-\omega
t)}\,d^{3}x\,,\end{array}$ (130)

where we skipped normalization constants for simplicity. One important feature
of the Fourier transform is the replacement of nabla operator $\vec{\nabla}$
with the wave vector $i\vec{k}$ and of the time derivative with the frequency
$\partial_{t}\to-i\,\omega$. The Fourier transformed MWEQs are algebraic
conditions for wave functions (we skip the tilde as it is usually clear from
the context, if we refer to the fields or their Fourier transform):

$\begin{array}[]{rcl}i\,\vec{k}\cdot\vec{B}&=&0\\\
i\,\vec{k}\times\vec{E}-i\,\omega\vec{B}&=&0\\\
i\,\vec{k}\cdot\vec{E}&=&4\pi\,\rho\\\
i\,\vec{k}\times\vec{B}+i\,\omega\vec{E}&=&{4\pi\vec{j}}\\\ \end{array}$ (131)

The second of Eq. LABEL:eq_maxwellfourier implies that all (single) solutions
also fulfill $\vec{E}\cdot\vec{B}=0$. The homogeneous parts of Eq.
LABEL:eq_maxwellfourier are structurally an exact copy of the expressions that
we obtain from Eq. LABEL:eq_eigenfreq with the condition $K_{2}=0$:

$\begin{array}[]{rcl}{\cal E}\,\vec{B}-\vec{P}\times\vec{E}&=&0\\\
\vec{P}\cdot\vec{B}&=&0\\\ \vec{E}\cdot\vec{B}&=&0\\\ \end{array}$ (132)

In Ref. geo_paper  we made use of the condition $\vec{P}\cdot\vec{B}=0$ and
$\vec{E}\cdot\vec{B}=0$ which have to be fulfilled in order to block-
diagonalize the matrix ${\bf F}$. But block-diagonalization (“decoupling”) did
not require the third term ${\cal E}\,\vec{B}-\vec{P}\times\vec{E}$ to vanish.
However, if the corresponding MWEQ would not yield zero, then we had to deal
with a non-zero but divergence-free magnetic current density
${4\pi\vec{j}_{m}}=\partial_{t}\,\vec{B}+\vec{\nabla}\times\vec{E}$ \- which
has not been found to date. There is another more abstract reason, why we may
restrict the solutions to the case of $K_{2}=0$: Only with this condition
being fulfilled the symplex ${\bf F}$ is a symplectic similarity
transformation of the pure time direction as described by Eq. (66) 262626 For
$K_{2}>0$ there are two different frequencies while for $K_{2}=0$ the
eigenfrequencies are identical as in case of $\gamma_{0}$.. The third scalar
product $\vec{P}\cdot\vec{E}$ (and the corresponding field equation
$\vec{\nabla}\cdot\vec{E}$) is in the general case non-zero and we emphasize
that this is another correspondence to the symplectic decoupling formalism
(Ref. geo_paper ). It is also remarkable, that the so-called duality rotations
272727 Concerning the electro-magnetic duality rotation see for instance the
short theoretical review article of J.A. Mignaco Duality1  and the latest
experimental results Duality2 . of the electromagnetic field do not fit into
this approach rdm_paper : Electric and magnetic fields can not be exchanged or
mixed. Though both are bi-vector fields (forming a “second rank” tensor), the
special role of the time coordinate breaks the suspected symmetry between
$\vec{E}$ and $\vec{B}$. Furthermore the only generator that might be used for
such a rotation is the pseudo-scalar, i.e. a cosymplex. Insofar the symplectic
foundation of electrodynamics has a higher explanatory power than the
“conventional” formalism.

If we summarize (and extend) the stability conditions $K_{2}\geq 0$ and
$K_{1}\geq 2\,\sqrt{K_{2}}$ (Ref. geo_paper ) by the condition $K_{2}=0$, then
it is appropriate (or even mathematically inevitable) to use this striking
structural similarity and to postulate the equivalence of $\vec{k}$ and
$\vec{P}$ ($\omega$ and ${\cal E}$). That is - we claim that
$\vec{P}\propto\vec{k}$ (${\cal E}\propto\omega$) and hence that the momentum
and energy equal via the Fourier transform the spatial and time derivatives,
respectively. There is still an optional proportionality factor. But for the
same reason that we did not refer to the speed of light, we also skip the
proportionality factor $\hbar$, since it has no physical significance, but
depends on the choice of units Ralston ; Hsu ; Duff :

$\begin{array}[]{rcl}{\cal E}&=&\omega=i\,\partial_{t}\\\
{\vec{P}}&=&\vec{k}=-i\,\vec{\nabla}\\\ \end{array}$ (133)

This means that the constant $K_{2}$ in Eq. LABEL:eq_eigenfreq vanishes for
every plane wave solution since all single terms forming $K_{2}$ vanish
separately:

$\begin{array}[]{rcl}K_{2}&=&({\cal
E}\,\vec{B}-\vec{P}\times\vec{E})^{2}-(\vec{E}\cdot\vec{B})^{2}-(\vec{P}\cdot\vec{B})^{2}\\\
{\cal
E}\,\vec{B}-\vec{P}\times\vec{E}&\to&\omega\,\vec{B}-\vec{k}\times\vec{E}=0\\\
\vec{P}\cdot\vec{B}&\to&\vec{k}\cdot\vec{B}=0\\\ \vec{E}\cdot\vec{B}&=&0\\\
\end{array}$ (134)

It is easy to show that “in vacuum” we also have $\vec{E}^{2}-\vec{B}^{2}=0$
so that also $K_{1}=0$ and there are no eigenfrequencies of electromagnetic
waves. Furthermore they do not generate inertial frames of reference.

As we projected the bivector elements (fields) of the general symplex ${\bf
F}$ into spacetime, we need to do the same with the phase space variables
$\psi$ and thus obtain the “wave function” or spinor $\psi(\vec{x},t)$. A
normalization might be written as

$\int\,\psi(\vec{x},t)^{T}\psi(\vec{x},t)\,d^{3}x=1\,.$ (135)

The phase space density function $\rho$ (and optionally the Gramian, see Eq.
129) might be “included” into the spinor function, such that

$\tilde{\psi}=\sqrt{\rho}\,\psi\,.$ (136)

Since the electric current must be a vector, the simplest Ansatz for the
current density of a “particle” described by $\psi$ is 282828 This definition
looks quite similar to the definition of the momentum by the EMEQ: Mass and
charge density are proportional in case of “point particles”.

$\begin{array}[]{rcl}\rho&=&-\bar{\psi}\,\gamma_{0}\,\psi\\\
j_{x}&=&\bar{\psi}\,\gamma_{1}\,\psi\\\
j_{y}&=&\bar{\psi}\,\gamma_{2}\,\psi\\\
j_{z}&=&\bar{\psi}\,\gamma_{3}\,\psi\\\ \end{array}$ (137)

The EQOM of $\psi(\vec{x},t)$ must result in a current density (Eq.
LABEL:eq_fourcurrent) that fulfills the continuity equation. The simplest
possible solution is known to be the Dirac equation. At first sight Eq. (22)
does not appear to be similar to the Dirac equation. But with the help of the
Fourier transform, the equivalence becomes obvious. The eigenvalues of ${\bf
F}$ for a “particle” in a field-free region are given by $\pm\,i\,\sqrt{{\cal
E}^{2}-\vec{P}^{2}}=\pm\,i\,m$. Together with Eq. (LABEL:eq_QMop) we can now
flip the “account” of Eq. (22) by the replacement of operators and eigenvalues
(and vice versa):

$\begin{array}[]{rcl}({\cal
E}\,\gamma_{0}+P_{x}\,\gamma_{1}+P_{y}\,\gamma_{2}+P_{z}\,\gamma_{3})\,\psi&=&\dot{\psi}={d\psi\over
d\tau}\\\
(i\,\partial_{t}\,\gamma_{0}-i\,\partial_{x}\,\gamma_{1}-i\,\partial_{y}\,\gamma_{2}-i\,\partial_{z}\,\gamma_{3})\,\psi&=&\pm\,i\,m\,\psi\\\
(\partial_{t}\,\gamma_{0}-\partial_{x}\,\gamma_{1}-\partial_{y}\,\gamma_{2}-\partial_{z}\,\gamma_{3})\,\psi&=&\pm\,m\,\psi\,.\end{array}$
(138)

Apart from “missing” unit imaginary, which is “hidden” in the definition of
the Dirac matrices $\gamma_{\mu}$, Eq. 138 is the Dirac equation. A unitary
matrix which transforms between RDMs and the conventional Form of the Dirac
matrices ($\tilde{\gamma}_{\mu}$) is explicitely given in App. C.4. However -
unitary transformations do not change the signs of the metric tensor. If we
desire to have $\gamma_{0}^{2}={\bf 1}$, we still need to multiply the
transformed matrices by the unit imaginary, so that:

$(-i\,\partial_{t}\,\tilde{\gamma}_{0}+i\,\partial_{x}\,\tilde{\gamma}_{1}+i\,\partial_{y}\,\tilde{\gamma}_{2}+i\,\partial_{z}\,\tilde{\gamma}_{3})\,\psi=\pm\,m\,\psi\,.$
(139)

##  IX Anything Else?

###  IX.1 Once More: Cosymplices

Much of what has been derived above is a consequence of the distinction
between symplices and cosymplices (or “Hamiltonian” and “Skew-Hamiltonian”)
matrices and their expectation values. One central argument was that the
generators of the considered Clifford algebras have to be symplices since only
simplices have non-vanishing expectation values. Only symplices are generators
of symplectic transformations and hence only symplices represent forces with
measurable effects in time. One might argue that some entity might exist
though its expectation value and its obvious consequences remain zero. Maybe
it has hidden or indirect effects? For instance the algebra of $2\,n\times
2\,n=8\times 8$ (co-) symplices represents a Clifford algebra $Cl_{3,3}$ that
has - compared to the Dirac algebra - two additional cosymplectic time-like
generators. Let the generators $\zeta_{\mu}$ with $\mu\in\,[0,\dots,5]$ (with
the real Pauli matrices $\eta_{\nu}$ defined in App. (A.2) be defined by:

$\begin{array}[]{rcl}\zeta_{0}&=&\eta_{3}\otimes\eta_{3}\otimes\eta_{0}\\\
\zeta_{1}&=&\eta_{3}\otimes\eta_{3}\otimes\eta_{1}\\\
\zeta_{2}&=&\eta_{3}\otimes\eta_{1}\otimes\eta_{2}\\\
\zeta_{3}&=&\eta_{3}\otimes\eta_{2}\otimes\eta_{2}\\\
\zeta_{4}&=&\eta_{1}\otimes\eta_{0}\otimes\eta_{2}\\\
\zeta_{5}&=&\eta_{2}\otimes\eta_{0}\otimes\eta_{2}\,.\end{array}$ (140)

The first four elements correspond to (a variant of) the Dirac algebra, and
the last two represent “hidden” dimensions corresponding to additional energy
contributions $\varepsilon_{1}$ and $\varepsilon_{2}$, but are cosymplices, if
$\zeta_{0}$ is the symplectic unit matrix. All $\zeta_{\mu}$ are pairwise
anticommuting and the metric tensor is given by
$g_{\mu\nu}=\textrm{diag}(-1,1,1,1,-1,-1)$. The commutator table of the first
four generators then is (as in case of the Dirac matrices) the electromagnetic
field tensor rdm_paper . But since (according to Eq. LABEL:eq_cosy_algebra)
the commutator of a symplex and a cosymplex is a cosymplex, the mixed terms
between symplices and cosymplices have vanishing expectation values:

$\langle[\zeta_{\mu},\zeta_{\nu}]\rangle=\left(\begin{array}[]{cccccc}0&E_{x}&E_{y}&E_{z}&0&0\\\
-E_{x}&0&B_{x}&-B_{y}&0&0\\\ -E_{y}&-B_{x}&0&B_{z}&0&0\\\
-E_{z}&B_{x}&-B_{y}&0&0&0\\\ 0&0&0&0&0&\lambda\\\ 0&0&0&0&-\lambda&0\\\
\end{array}\right)$ (141)

Hence the two additional elements are (in average) “decoupled” from the first
four (and all other symplices). The commutator of the two extra cosymplices is
a symplex and hence yields a non-zero field value $\lambda$, which acts only
between the two additional time-like dimensions. Considerations like these are
the background for the claim that time might indeed be multidimensional - but
it is unlikely that we would be able to notice.

###  IX.2 Higher-Dimensional Spaces

If spacetime is an emergent phenomenon, then the question arises whether other
(high-dimensional) spacetimes may not emerge “in parallel”. Let’s ignore the
theoretical arguments for $3+1$ dimensions for a moment: Indeed the strongest
evidence for a 3+1-dimensional spacetime that we have at hand, is based on
light and electrodynamics. Chemistry is based almost exclusively on
electromagnetic interactions: our world is the world of electrons. What we
see, is light emitted, reflected and absorbed by electrons. If we touch a
solid object, then electrons are “touching” electrons. We can experimentally
investigate weak and strong forces, gravitation and so on - but whatever we
“see” with our own eyes, is to any degree of approximation light emitted by
electrons.

Assume that the $9+1$-dimensional spacetime emerges in parallel by the
properties of other (high-dimensional) fields \- then it still remains
questionable, if and how observers would interpret the $9+1$-dimensional
spacetime. Certainly we can not expect that $9+1$-dimensional is just “3 times
of the same”, since the algebraic features (for instance of rotations) are
different: An essential feature of 3 dimensions is that the spatial rotations
do not commute. In more than 3 dimensions we can form commuting rotators, say
$\gamma_{1}\,\gamma_{2}$ and $\gamma_{3}\,\gamma_{4}$. Hence in 9 dimensions
we sort out 3 sets of 3 non-commuting rotators. Consider we use

$\begin{array}[]{rlll}1)&\gamma_{2}\,\gamma_{3},&\gamma_{3}\,\gamma_{1},&\gamma_{1}\,\gamma_{2}\\\
2)&\gamma_{5}\,\gamma_{6},&\gamma_{6}\,\gamma_{4},&\gamma_{4}\,\gamma_{5}\\\
3)&\gamma_{8}\,\gamma_{9},&\gamma_{9}\,\gamma_{7},&\gamma_{7}\,\gamma_{8}\\\
\end{array}$ (142)

We then find that any two rotators from different groups commute 292929
Dynamically emerging spacetimes have their own properties, which are
determined by the structure of the corresponding symplectic Clifford algebra.
One can not simply add another dimension as in arbitrary dimensional Euklidean
spaces.. On the level of observables we derive from Eq. 15 that if two
observables commute they are decoupled. Presumably observers “grown up” in a
3+1 dimensional world would not interprete the $9+1$ dimensional spacetime
correctly. Instead of “seeing” a $9+1$-dimensional spacetime “parallel” to our
“electronic” $3+1$-dimensional spacetime, an observer might see $3$ objects
located in $3+1$ dimensions. The ontologically motivated idea that spacetime
dimensionality must be a unique property of the world (and may not depend on
the type of the interaction), might have guided physics into the wrong
direction after all.

##  X Summary and Conclusion

Based on the ontology of time, we introduced pure variables, which are defined
exclusively by the property of variation in time. Measurability requires
reference to constants. Since pure constants are by definition not available
on the fundamental level they have to be constructed from dynamics as
constants of motion. Looking back we can identify pure variables with the
components of the Dirac spinor, i.e. with the components of the quantum
mechanical wavefunction. If we include a density function, then the spinors
can be interpreted as “probability amplitudes”.

Main thesis of our gedankenexperiment is the emergence of spacetime through
symplectic dynamics: spacetime is an interpretation of dynamics and we showed
that the algebraic construction of geometry based on symplices requires
matrices that are representations of Clifford algebras. Most (if not all)
arguments that we used in the derivation are related to specific (algebraic)
symmetries. Apparently these symmetries are less obvious in the conventional
form of the Dirac matrices. Furthermore the use of complex numbers in the
conventional form of the Dirac theory wrongly suggests that we are in a non-
classical domain. We have shown that some of the apparent differences between
quantum and classical mechanics are of ontological nature and loose much of
their significance in the light of the ontology of existence in time. Non-
classicality is in this scheme neither connected to $\hbar$ or the unit
imaginary but to the fact that all quantities of the fundamental level vary at
all times and to the consequences thereof.

The idea that the elements of Clifford algebras are related to Minkowski
spacetime is well-known and has been described by D. Hestenes and others in
various publications on spacetime algebra Hestenes ; STA : “The Dirac matrices
are no more and no less than matrix representations of an orthonormal frame of
spacetime vectors and thereby they characterize spacetime geometry”. However
(to our knowledge) Hestenes never argued why Minkowski spacetime should have
these properties. According to our ansatz Clifford algebras are the optimal
mathematical representation of spacetime because spacetime emerges by a
pattern of special symmetry properties which can be expressed by the
isomorphism of the Hamiltonian dynamical structure to Clifford algebras. The
emergence of spacetime is the reason why the Dirac algebra is the appropriate
mathematical tool to represent spacetime RV .

Furthermore we gave arguments for the (apparent) dimensionality of spacetime
and we have shown that Lorentz transformations are (in this framework)
structure preserving (i.e. symplectic) transformations. 3 central arguments
were given in preparation of the derivation of Lorentz force and
electrodynamics in form of MWEQs: The first argument is based on the
difference between even and odd elements of the algebra, the second on the
fact that the expectation values of cosymplices must vanish in symplectic
dynamics (in order be structure preserving) and the last one on the
transformation properties. The wave equations follow from MWEQs and the
comparison of the Fourier transformed MWEQs with the structure of decoupling
then let us identify momentum (energy) with space (time) derivatives and hence
to the Dirac equation.

We do not claim that our “derivation” is rigorous in a strict mathematical
sense. And it can not be: our abstract ansatz may only include abstract
objects and becomes a physical theory only with an appropriate interpretation.
Interpretations can not be proven, they can only be adequat of not. Our
central “interpretation” is the EMEQ which is inducted by the isomorphism of
certain classical quantities (energy, momentum, electric and magnetic field
components) with the abstract observables of the 3+1 dimensional Clifford
algebra. The fact that we can derive the Lorentz force equations from it, is
not a mathematical “proof” of this interpretation. However it seems remarkable
to the author that a single 3-dimensional orbit is mathematically isomorph to
the average envelope of a 2-dimensional ensemble. In this sense we join the
view of Mark Van Raamsdonk: “Everything around us — the whole three-
dimensional physical world — is an illusion born from information encoded
elsewhere, on a two-dimensional chip” MvR . Most authors that analyzed the
structure of the symplectic group $Sp(4,R)$ interpreted this space to be 2+1
dimensional. However - as we argued above - the space of observables that
corresponds to $Sp(4,R)$ is a $3+1$ dimensional Minkowski spacetime.
Mathematically it is evident that this view implies restrictions on the $3+1$
dimensional space - for instance the “quantization” of the angular momentum.

But even if we could present a rigorous derivation of the (fundamental)
relativistic equation of motion of quantum mechanics - the Dirac equation - it
would not automatically imply a “derivation” of quantum mechanics as a whole.
Though it is rarely explicitely mentioned, it is known (though not well-known)
that the equations of motion used in quantum mechanics are (taken as such)
classical RalstonQM . This has been confirmed in what we carved out in this
essay. We started with the classical Hamilton function, we were guided to
interprete the fundamental variables as the quantum mechanical wave function
in momentum space. But mathematically the alleged antagonism between classical
and quantum mechanics seems to evaporate in the light of what we described
above.

On the basis of the presented ontology of time we found an explanation, why
the quantummechanical wavefunction has some features that seem to be
mysterious or “non-classical”: Impossible to be directly measured and without
a well-defined dimensional unit and physical meaning. However these are
exactly the properties of fundamental variables that we derived from the
ontology of time and a proper logic of measurement. Supposed our ontology is
“correct”, it is impossible to know what the wave function is. But at least we
have an explanation why: The question has no meaningful physical answer - it
can not. The consequences of our ontology are certainly difficult to accept:
On the fundamental level, the world is apparently very different from our
every day experience: loosely speaking it is flat. The apparent 3+1
dimensional spacetime is “in reality” a four-dimensional phase space. However
there are interpretations of quantum mechanics (and general relativity) which
are way more esoteric than this.

###### Acknowledgements.

Mathematica® has been used for part of the symbolic calculations. Additional
software has been written in “C” and compiled with the GNU©-C++ compilers on
different Linux distributions. XFig 3.2.4 has been used to generate the
figures, different versions of LaTeXand GNU©-emacs for editing and layout.

##  Appendix A (Co-) Symplices for higher-dimensional Spacetimes

###  A.1 Which $k$-Vectors are (Co-) Symplices

Given that we have a set of $N$ pairwise anticommuting (co-) symplices ${\bf
S}_{i}$ (${\bf C}_{i}$), which can be regarded as generators of real Clifford
algebras, then the question if the $k$-products (i.e. bivectors, trivectors
etc.) are symplices or cosymplices, depends on a sign and can be calculated as
follows:

$\begin{array}[]{rcl}({\bf S}_{1}\,{\bf S}_{2}\,{\bf S}_{3}\,\dots\,{\bf
S}_{k})^{T}&=&{\bf S}_{k}^{T}\,{\bf S}_{k-1}^{T}\,{\bf
S}_{k-2}^{T}\,\dots\,{\bf S}_{1}^{T}\\\ &=&(-1)^{k-1}\,\gamma_{0}\,{\bf
S}_{k}\,{\bf S}_{k-1}\,{\bf S}_{k-2}\,\dots\,{\bf S}_{1}\,\gamma_{0}\\\
\end{array}$ (143)

The number of permutations that is required to reverse the order of $k$
matrices is $k\,(k-1)/2$, so that

$({\bf S}_{1}\,{\bf S}_{2}\,{\bf S}_{3}\,\dots\,{\bf
S}_{k})^{T}=(-1)^{s}\,\gamma_{0}\,({\bf S}_{1}\,{\bf S}_{2}\,{\bf
S}_{3}\,\dots\,{\bf S}_{k})\,\gamma_{0}\,,$ (144)

with $s$ given by:

$s=k-1+k\,(k-1)/2={k^{2}+k-2\over 2}\,.$ (145)

If we consider $k$ cosymplices ${\bf C}_{i}$ instead, we obtain $k$ more sign
reversals, so that

$({\bf C}_{1}\,{\bf C}_{2}\,{\bf C}_{3}\,\dots\,{\bf
C}_{k})^{T}=(-1)^{c}\,\gamma_{0}\,({\bf C}_{1}\,{\bf C}_{2}\,{\bf
C}_{3}\,\dots\,{\bf C}_{k})\,\gamma_{0}\,,$ (146)

with $c$ given by:

$c=2\,k-1+k\,(k-1)/2={k^{2}+3\,k-2\over 2}\,.$ (147)

If $s$ and $c$ are even (odd), respectively, then the products are (co-)
symplices.

k | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
---|---|---|---|---|---|---|---|---|---|---|---|---
s | 0 | 2 | 5 | 9 | 14 | 20 | 27 | 35 | 44 | 54 | 65 | 77
$(-1)^{s}$ | + | + | - | - | + | + | - | - | + | + | - | -
c | 1 | 4 | 8 | 13 | 19 | 26 | 34 | 43 | 53 | 64 | 76 | 89
$(-1)^{c}$ | - | + | + | - | - | + | + | - | - | + | + | -
Table 1: Signs for products of $k$ anti-commuting (co-) symplices according to
Eq. (145) and (147). The “plus” signs correspond to symplices: Products of
$2,5,6,9,10,\,\dots$ symplices are again symplices.

Tab. 1 lists the resulting signs for $k=2\dots 10$. Clifford algebras with
more than $4$ generators include penta- and hexavectors as “observables”.
Furthermore one may conclude from Tab. 1: In $9+1$-dimensional spacetime, the
pseudoscalar is the product of all $10$ generators and therefore a symplex.
This allows (in principle) for a scalar field. Since spatial rotators are
combinations of two spatial generators, in $9+1$ dimensional spacetime there
are $\left(9\atop 2\right)=36$ spatial rotations. A derivation of the
corresponding MWEQs for such higher-dimensional spaces (if possible or not),
lies beyond the scope of this paper, but if all bivector boosts have a
corresponding electric field component and all bivector rotations a
corresponding magnetic field component, then we should expect $9$ “electric”
and $36$ “magnetic” field components in $9+1$ dimensional spacetime.

###  A.2 The real Pauli matrices and the structure of (co-) symplices for $n$
DOFs

For one DOF and an appropriate treatment of the $2\times 2$-blocks of the
general case we introduce the real Pauli matrices (RPMs) according to

$\begin{array}[]{rclp{10mm}rcl}\eta_{0}&=&\left(\begin{array}[]{cc}0&1\\\
-1&0\\\ \end{array}\right)&&\eta_{1}&=&\left(\begin{array}[]{cc}1&0\\\ 0&-1\\\
\end{array}\right)\\\ \eta_{2}&=&\left(\begin{array}[]{cc}0&1\\\ 1&0\\\
\end{array}\right)&&\eta_{3}&=&{\bf 1}=\left(\begin{array}[]{cc}1&0\\\ 0&1\\\
\end{array}\right)\\\ \end{array}$ (148)

$\eta_{0}$ is the symplectic unit matrix (i.e. corresponds to $\gamma_{0}$ of
the general case). $\eta_{0}$, $\eta_{1}$ and $\eta_{2}$ are symplices. If we
consider general $2\,n\times 2\,n$ Hamiltonian matrix ${\bf F}$ composed of
$n^{2}$ $2\times 2$ blocks ${\bf A}_{ij}$, then $\gamma_{0}$ has the form
given in Eq. (6). The $2\,n\times 2\,n$-symplex ${\bf F}$ that fulfills Eq. 14
has the form

${\bf F}=\left(\begin{array}[]{ccccc}{\bf D}_{1}&{\bf A}_{12}&{\bf
A}_{13}&\dots&{\bf A}_{1n}\\\ -{\bf\tilde{A}}_{12}&{\bf D}_{2}&{\bf
A}_{23}&\dots&{\bf A}_{2n}\\\ -{\bf\tilde{A}}_{13}&-{\bf\tilde{A}}_{23}&{\bf
D}_{3}&\dots&{\bf A}_{3n}\\\ \vdots&&\vdots&&\vdots\\\
-{\bf\tilde{A}}_{1n}&-{\bf\tilde{A}}_{2n}&-{\bf\tilde{A}}_{3n}&\dots&{\bf
D}_{n}\\\ \end{array}\right)$ (149)

where the $2\times 2$ matrices ${\bf D}_{k}$ on the diagonal must be $2\times
2$-symplices and can therefore be written as

${\bf D}_{k}=d_{0}^{k}\,\eta_{0}+d_{1}^{k}\,\eta_{1}+d_{2}^{k}\,\eta_{2}\,.$
(150)

If the blocks ${\bf A}_{ij}$ above the diagonal have the general form

${\bf A}=\sum\limits_{k=0}^{3}\,a_{k}\,\eta_{k}\,,$ (151)

then Eq. 14 fixes the form of the corresponding blocks below the diagonal
${\bf\tilde{A}}_{ij}$ to

$\begin{array}[]{rcl}-{\bf\tilde{A}}_{ij}&=&\eta_{0}\,{\bf
A}_{ij}^{T}\,\eta_{0}\\\
&=&-\eta_{0}\,(a_{ij}^{0}\,\eta_{0}+a_{ij}^{1}\,\eta_{1}+a_{ij}^{2}\,\eta_{2}+a_{ij}^{3}\,\eta_{3})^{T}\,\eta_{0}\\\
&=&\eta_{0}\,(-a_{ij}^{0}\,\eta_{0}+a_{ij}^{1}\,\eta_{1}+a_{ij}^{2}\,\eta_{2}+a_{ij}^{3}\,\eta_{3})\,\eta_{0}\\\
&=&a_{ij}^{0}\,\eta_{0}+a_{ij}^{1}\,\eta_{1}+a_{ij}^{2}\,\eta_{2}-a_{ij}^{3}\,\eta_{3}\\\
\end{array}$ (152)

From this one can count that each subblock ${\bf A}_{ij}$ has two
antisymmetric and two symmetric coefficients. We count $n(n-1)/2$ sublocks
${\bf A}_{ij}$. Together with $n$ diagonal subblocks with each having one
antisymmetric coefficients, we should have

$\nu_{s}^{a}=n\,(n-1)+n=n^{2}$ (153)

antisymmetric symplices. The number of independent antisymmetic parameters is
$\nu^{a}=\nu_{s}^{a}+\nu_{c}^{a}=n\,(2\,n-1)=2\,n^{2}-n$. Therefore we have

$\nu_{c}^{a}=n\,(2\,n-1)-n^{2}=n^{2}-n$ (154)

antisymmetric cosymplices. Since we have as many cosymplices as we have
antisymmetric matrix elements $\nu_{c}=\nu^{a}$, the number of symmetic
cosymplices $\nu_{c}^{s}$ is:

$\nu_{c}^{s}=\nu_{c}-\nu_{c}^{a}=2\,n^{2}-n-(n^{2}-n)=n^{2}$ (155)

The number $\nu_{s}^{s}$ of symmetric symplices is:

$\nu_{s}^{s}=\nu_{s}-\nu_{s}^{a}=2\,n\,(2\,n+1)/2-n^{2}=n^{2}+n\,.$ (156)

${\bf\tilde{A}}_{ij}$ is called the symplectic conjugate of ${\bf A}_{ij}$. If
${\bf S}$ and ${\bf C}$ are the symplex and cosymplex-part of a matrix ${\bf
A}={\bf C}+{\bf S}$, then the symplectic conjugate is

${\bf\tilde{A}}=-\gamma_{0}\,{\bf A}^{T}\,\gamma_{0}={\bf C}-{\bf S}\,.$ (157)

As well-known one can quickly derive that

${\bf\tilde{AB}}={\bf\tilde{B}}\,{\bf\tilde{A}}\,.$ (158)

If ${\bf A}$ is written “classically” as

${\bf A}=\left(\begin{array}[]{cc}a_{11}&a_{12}\\\ a_{21}&a_{22}\\\
\end{array}\right)$ (159)

then $\eta_{0}\,{\bf A}^{T}\,\eta_{0}$ is given by:

$\eta_{0}\,{\bf A}^{T}\,\eta_{0}=\left(\begin{array}[]{cc}-a_{22}&a_{12}\\\
a_{21}&-a_{11}\\\ \end{array}\right)\,,$ (160)

so that

${\bf A}\,(\eta_{0}\,{\bf
A}^{T}\,\eta_{0})=(a_{12}\,a_{21}-a_{11}\,a_{22})\,{\bf 1}=-\mathrm{Det}({\bf
A})\,{\bf 1}\,.$ (161)

A matrix ${\bf A}$ is symplectic, if ${\bf A}\,{\bf\tilde{A}}={\bf 1}$. But we
can say in any case, that ${\bf A}\,{\bf\tilde{A}}$ is a co-symplex:

$\begin{array}[]{rcl}{\bf A}\,{\bf\tilde{A}}&=&({\bf C}+{\bf S})\,({\bf
C}-{\bf S})\\\ &=&{\bf C}^{2}-{\bf S}^{2}+{\bf S}\,{\bf C}-{\bf C}\,{\bf S}\\\
\end{array}$ (162)

since squares of (co-) symplices as well as the commutator of symplex and
cosymplex are cosymplices.

A $2\,n\times 2\,n$-cosymplex ${\bf C}$ has according to Eq. 24 the form

${\bf C}=\left(\begin{array}[]{ccccc}{\bf E}_{1}&{\bf B}_{12}&{\bf
B}_{13}&\dots&{\bf B}_{1n}\\\ {\bf\tilde{B}}_{12}&{\bf E}_{2}&{\bf
B}_{23}&\dots&{\bf B}_{2n}\\\ {\bf\tilde{B}}_{13}&{\bf\tilde{B}}_{23}&{\bf
E}_{3}&\dots&{\bf B}_{3n}\\\ \vdots&&\vdots&&\vdots\\\
{\bf\tilde{B}}_{1n}&{\bf\tilde{B}}_{2n}&{\bf\tilde{B}}_{3n}&\dots&{\bf
E}_{n}\\\ \end{array}\right)$ (163)

where the $2\times 2$ matrices ${\bf E}_{k}$ on the diagonal must be $2\times
2$-cosymplices and are hence proportional to the unit matrix

${\bf E}_{k}=e_{k}\,{\bf 1}\,.$ (164)

###  A.3 Which $k$-Vectors of $Cl_{N-1,1}$ are (Anti-) Symmetric

Possible dimensionalities for emergent spacetimes are given by Eq. 74. Since
all but one generator of $Cl_{N-1,1}$ are symmetric, the analysis of how many
$k$-vectors are (anti-) symmetric, is reasonably simple. The total number of
$k$-vectors is given by $\left({N\atop k}\right)$, the number $\mu_{k}$ of
$k$-vectors generated only with spatial elements, hence is $\left({N-1\atop