How_to_Un_Quantum_Mechanics-article1.pdf

arXiv:1810.06981v1
[physics.gen-ph]
12
Oct
2018
How to (Un-) Quantum Mechanics
C. Baumgarten
5244 Birrhard, Switzerland∗
(Dated: October 17, 2018)
When compared to quantum mechanics, classical mechanics is often depicted in a specific meta-
physical flavour: spatio-temporal realism or a Newtonian “background” is presented as an intrinsic
fundamental classical presumption. However, the Hamiltonian formulation of classical analytical
mechanics is based on abstract generalized coordinates and momenta: It is a mathematical rather
than a philosophical framework.
If the metaphysical assumptions ascribed to classical mechanics are dropped, then there exists a
presentation in which little of the purported difference between quantum and classical mechanics
remains. This presentation allows to derive the mathematics of relativistic quantum mechanics on
the basis of a purely classical Hamiltonian phase space picture. It is shown that a spatio-temporal
description is not a condition for but a consequence of objectivity. It requires no postulates.
This is achieved by evading spatial notions and assuming nothing but time translation invariance.
I. INTRODUCTION
Uncountable articles and books have been written on
the interpretation of quantum theory and most share
a number of assertions, briefly summarized as follows:
Firstly, yes, quantum mechanics is by far the most pre-
cise and successful theory ever formulated and secondly,
no, there is no general agreement on what it tells us about
the world. Thirdly, it is usually asserted, that classical
mechanics (CM) is intuitive and clear while quantum me-
chanics (QM) is counter-intuitive, weird and somehow
raises questions like “is the moon there when nobody
looks?” [1], questions that no layman should expect to
be raised by physicists.
Even if there is no agreement concerning the interpre-
tation of QM, there seems to be consensus that QM and
CM are very, very different. We shall prove this wrong.
From a mathematical perspective, there is no fundamen-
tal difference between CM and QM. The mathematics of
QM can be obtained from classical notions in a straight-
forward manner, without axioms or postulates. Our view
contradicts the standard presentation of QM (SPQM),
which stresses the profound mathematical differences be-
tween CM and QM. But the truth is that there are none.
This article is dedicated to those physicists who share
the author’s intuition, that both, CM as well as QM, have
just been presented wrongly and that the real difference
between CM and QM is not mathematical.
A. What is Quantum in Quantum Mechanics?
A typical assertion of the SPQM is the following: “In
fact, the basic concepts such as observable, ensemble,
state, or yes-no measurement, employed in the ”usual”
interpretation of quantum mechanics, are themselves not
explainable by known pretheories.” [2]
∗ christian-baumgarten@gmx.net
Of course, the “pretheories” are those theories that
do not require anything but “classical” concepts. Then
it can be precisely defined what should be regarded as
“quantum”, namely what can not be explained using
classical physics. Hence if, in what follows, we claim
that some equation or fact is not “quantum”, then it is
not intended to say that it does not belong to the tech-
niques usually employed when “doing” QM, but that it
can indeed be derived on the basis of classical concepts.
Therefore, according to the SPQM, it can not be “quan-
tum”.
Axioms are only required for things that can not be
established otherwise, and the reader may judge for him-
self, how much “quantumness” eventually remains that
requires an “axiomatic” foundation.
B. No Commandments
Victor Stenger wrote that “most laypeople think of the
laws of physics as something like the Ten Commandments
– rules governing the behavior of matter imposed by some
great lawgiver in the sky. However, no stone tablet has
ever been found upon which such laws were either nat-
urally or supernaturally inscribed” [3]. But if laypeople
have this understanding, then because the usual presen-
tation of physics suggests this view.
Commandment-like-laws (CLLs) can be descriptive,
obtained from a fit to experimental data and as such they
are, of course, acceptable. But such CLLs are also preli-
menary, at least concerning their presentation. The prob-
lem of how to “interpret” QM arises, because the SPQM
is a collection of CLLs (axioms) which do not originate in
experimental facts. This is often presented wrongly: Of
course there are experimental facts that awaited a phys-
ical explanation and QM is the only theory found so far
that is able to predict and describe those facts correctly.
But while the law of free fall might be obtained by a fit
to the data, the SPQM can not. It cannot be a fit to data
since it is declared to be a completely new theory based
on completely different and alien principles. Since these

2
principles are described as unexplainable from pretheo-
ries, they are established as ad hoc commandments. In
short, the SPQM tells us what to do but not why.
Descriptive CLLs are usually regarded as pieces of a
puzzle, implying that even if it is yet unknown how the
bits can be combined, one expects that eventually all
CLLs ought to fit seamless into a more encompassing
deductive approach. The probably best known exam-
ple for this process of unification are Maxwell’s equa-
tions. Hence scientists are obliged to presume that the
presentation of CLLs must eventually be adapted once a
larger theoretical framework has been established. This
is sometimes called “Occham’s razor”.
However leading pioneers of quantum physics recom-
mended to accept the theory and its interpretation as
commandments. Heisenberg, for instance, wrote: “We
must keep in mind this limited range of applicability of
the classical concepts while using them, but we cannot
and should not try to improve them.” [4], or Dirac: “One
must not picture this reality as containing both the waves
and particles together and try to construct a mechanism,
acting according to classical laws, which shall correctly
describe their connexion and account for the motion of
the particles. Any such attempt would be quite opposed
to the principles by which modern physics advances.” [5].
It is an oddity (to say the least) that quantum scientists
dared to call for loyalty to a theory and its interpretation
and to demand that one should not try to improve.
One might say, that these quotes are old and outdated,
which is true and false, since the authoritative habitus
has not disappeared [6]. As Mermin expressed it: “If I
were forced to sum up in one sentence what the Copen-
hagen interpretation says to me, it would be ”Shut up
and calculate!”” [7].
But in science there is no place for an authoritative
attitude. Humans will always ask for reasons and scien-
tists ought to do so for professional reasons: “The prob-
lem with the standard textbook formulations of quantum
theory is that the postulates therein impose particular
mathematical structures without providing any funda-
mental reason for this choice” [8].
To refuse commandments therefore serves as our main
guiding principle. This principle seems to be both, rea-
sonable and scientifically sensible, but also unrealistic.
The SPQM claims that about a dozen postulates are re-
quired to establish the basics of QM and we claim that
none is really needed?
In a preceeding paper we argued that the only logical
possibility to elude commandments is to derive the “laws
of physics” from a definition of what is essentially meant
by “objective physical reality”, a world composed of real
physical objects (RPOs) [10]. And indeed we found, to
our own surprize, that it is possible to derive major parts
of QM from nothing but a definition of physicality –
from a single fundamental constraint which is our “re-
ality condition”. And this constraint is not even excep-
tionally deep or profound. It is simple, evident, well-
known and straightforward. Gerard t’Hooft expressed it
as follows [9]: “[...] in particular string theorists expect
that the ultimate laws of physics will contain a kind of
logic that is even more mysterious and alien than that of
quantum mechanics. I, however, will only be content if
the logic is found to be completely straightforward.”
The question of a definition of physicality is also raised
by multiverse theories that have been suggested in the
past decades, for instance by Tegmark, who conjectured
the possibility of many different physical worlds, not only
different copies or versions of the same kind of world, but
worlds that obey very different physical laws [11, 12].
If not only the known physical world, but many differ-
ent worlds are conjectured, then these worlds must have
something in common that allows to call them “physi-
cal”. Hence a simple and straight definition of physicality
is required, something that must hold in any thinkable
physical world, even in hypothetical worlds which might,
according to this idea, be ruled by different and alien
physical laws, that might have, for instance, a different
number of spatial dimensions or, who knows, no spatial
dimensions at all.
II. SETUP
Physics creates models of (parts of) reality. These
models allow to “simulate reality”, or, in simple cases,
to directly calculate results. Hence it is arguable, that,
whatever is physical in a world, should allow for a descrip-
tion by algorithms that predict (probabilities for possible)
evolutions of physical quantities in time. Hence the basis
of a physical model of reality is a (possibly very long) list
of quantities ψ(τ), that depend on time τ. This is the
raw material for a general physical model of any think-
able physical reality.
A. Symmetries and Quantities
One of the first facts children learn about real objects
is called object permanence [13], namely that the moon is
still there, even if nobody looks. Object permanence does
not seem to be general enough to serve as the desired
constraint that defines physicality, because macroscopic
objects can be disassembled and destroyed, one can break
tea cups and burn wooden chairs. But matter can be ma-
nipulated only within specific constraints. Objects are
made of other objects. Insofar as one can disassemble
objects, they can be destroyed, but chemistry found that
the amount of matter remains unchanged, even if objects
are burned. And even though it is theorectially possible
to destroy all individual microscopic objects (particles)
that a macroscopic thing is made of by annihilation with
a perfect copy made of anti-matter – still the energy re-
mains. This, eventually, is an insight that is as simple as
it is not trivial.
The impossibility for a perpetuum mobile that produces
net energy is the fundamental constraint for any known

3
closed physical system or process. According to Einstein
“The most satisfactory situation is evidently to be found
in cases where the new fundamental hypotheses are sug-
gested by the world of experience itself. The hypothesis
of the nonexistence of perpetual motion as a basis for
thermodynamics affords such an example for a funda-
mental hypothesis suggested by experience.” [14].
As Einstein rightly remarked, this principle is sug-
gested by experience, but once it’s depth has been rec-
ognized, physicists understood that it has the strength
and status of a definition of physical realness itself. If
a theory fails to provide conservation of energy then it
is unphysical by definition. But if it is possible to define
un-physicality on the basis of a conservation “law”, why
then should it not be possible to also define physicality?
But what exactly is a conservation law?
Emmy Noether, in 1928, discovered the math fact [15],
as Stenger puts it, “[...] that coordinate independence
was more than just a constraint on the mathematical
form of physical laws. She proved that some of the
most important physics principles are, in fact, nothing
more than tautologies that follow from space-time co-
ordinate independence: energy conservation arises from
time translation invariance, linear momentum conserva-
tion comes from space translation invariance, and angular
momentum conservation is a consequence of space rota-
tion invariance. These conserved quantities were simply
the mathematical generators of the corresponding sym-
metry transformation.” [3] Hence it is a math fact that a
conservation law is nothing but a continuous symmetry,
the generator of which is a conserved quantity.
The concepts of energy as well as of action can only
be defined on the basis of an already elaborated phys-
ical theory 1
. This would of course be a theory of the
known physical world and not necessarily valid in any
hypothetical physical world. How should one know a pri-
ori whether these notions are releveant and meaningful
in any thinkable physical world? Therefore, if arbitrary
physical worlds are considered, these notions are too spe-
cific to be used from the start.
However, it is not required to specify the type of the
conserved quantity at this point. It suffices to formally
refer to some positive definite constant of motion (PD-
COM) which serves as a measure of object permanence,
because it is not the object itself that is permanent, but
some abstract quantity that objects are “charged with”.
We can anticipate that this quantity will turn out to be
a possible measure of the amount of substance. This re-
quires no postulate: If correct then it should be a math-
ematical consequence of our approach.
The very idea of a real physical object logically re-
quires at least one positive definite constant of motion
(PDCOM). Little more than this will be used to a num-
ber of basic “laws” of physics. Without commandments.
1 It is known to be a non-trivial problem to find non-circular
definitions of the central notions of physics.
B. Time
There is no a priori reason to introduce more than a
single symmetry, namely constancy in time, aka perma-
nence. Or, to be more precise: If more than a single
symmetry would be presumed, then one would need to
specify how many and why not one more or less.
Time is a primary quantity that remains basically “un-
defined”. Stanley Goldberg wrote that “Either you know
or you don’t know what I mean when I use a phrase like
”time passes.”” [16]. The “dimension” of time is different
from spatial dimensions insofar as it is unique. One can
discuss the dimensionality of physical space and one can,
within the classical framework, imagine physical worlds
with more or less than 3, maybe even zero, spatial dimen-
sions. But it is questionable if it is possible to imagine
a physical world without the unique dimension of time:
“[...] only one true integer may occur in all of physics.
The laws of physics refer to one dimension of time” [17].
Hamilton wrote that “the notion or intuition of order
in time is not less but more deep-seated in the human
mind, than the notion of intuition of order in space; and
a mathematical science may be founded on the former,
as pure and as demonstrative as the science founded on
the latter. There is something mysterious and transcen-
dent involved in the idea of Time; but there is also some-
thing definite and clear: and while Metaphysicians med-
itate on the one, Mathematicians may reason from the
other.” [18].
We shall show that, even though Hamilton’s idea to
derive algebra as the science of pure time failed [19],
there is nonetheless an algebra of pure (aka proper) time.
It reveals the possibility to formulate (central parts of)
physics as a tautology [10]. This has absolutely no
negative connotation, as Goldberg explained: “Differ-
ent branches of mathematics have different rules but in
all branches, since the rules are predetermined, the con-
clusion is actually a restatement, in a new form, of the
premises. Mathematics, like all formal logic, is tautolog-
ical. That is not to say that it is uninteresting or that
it doesn’t contain many surprises.“ [16] If it is possible
to derive (essential parts of) physics from a definition of
physicality, then the result is a tautology in Goldberg’s
sense.
C. Reason
Generality is maintained by presuming nothing spe-
cific, neither about the “nature” of the dynamical vari-
ables nor about the “nature” of the conserved quantity.
This attitude has been summarized by Hamilton un-
der the name principle of sufficient reason (PSR): “Infer
nothing without a ground or reason.” [20]. In his form it

4
might also be called the principle of insufficient reason 2
.
Of course, the PSR contains little more than Stenger’s
claim that there are no commandments.
If a no-assumption-approach allows for any conclusion
about the nature of the conserved quantity, then it must
emerge from the form of equations or symmetries. Only
if familiar structures emerge, if equations suggest spe-
cific interpretations as nearby, we are authorized to map
quantities of the theory to known physical observables,
aka to interprete.
The PSR has a bias towards symmetry since nothing-
ness (the void) is the most symmetric state: To assume
nothing specific about a number of things or quantities
has to be understood as assuming no asymmetry and the
PSR forbids to introduce asymmetries, distinctions and
classifications without a ground or reason.
Hence the raw material for the simplest physical object
contains an arbitrary number ν of dynamical variables
ψ = (ψ1, . . . , ψν) (quantities) that depend on a time, the
evolution of which is constrained by a PDCOM H(ψ) =
H0 = const.
With the condition that H(ψ) is a constant of motion,
it is implied that ψ itself does not contain any other con-
stant, i.e. all variables in the list ψ depend on time so
that no linear combination of the elements of ψ may pre-
sumed to be constant.
D. Structure of the Paper
In Sec. III we shall firstly show that in any classical
dynamical system, which can be derived from the as-
sumed PDCOM, the number of true dynamical variables
is even, i.e. the variables come in pairs and secondly
that one can always describe the dynamics, after an ap-
propriate change of variables, by Hamilton’s laws of mo-
tion. Hamiltonian dynamics has maximal generality and
is not negotiable. If the SPQM suggests that it requires
modifications then we shall show that this is wrong.
In Sec. IV we introduce the phase space distribution
as the fundamental mathematical representation of phys-
ical objects. We show that it suffices to consider the sim-
plest possible description of phase space distributions,
namely the matrix of second moments (sloppily called
auto-correlation matrix), to derive Heisenberg’s equation
of motion for operators. We show that it can be made
“quantum” merely by notation.
In Sec. V we shall derive the basic algebras of phase
space, namely the algebra of (skew-) Hamiltonian matri-
ces. We explain the necessity to describe stable phase
space distributions by second and higher even moments,
and why this implies that it is impossible to measure ψ
directly.
2 Ariel Caticha identified the principle of insufficient reason in
Quantum mechanics [21].
In Sec. VI we explain the meaning and the role of Clif-
ford algebras in low-dimensional phase spaces. We criti-
cize the specific notational convention concerning the use
of complex numbers in QM in general and specifically in
Dirac’s theory. We explain the general conditions that
Hamiltonian physics imposes on the dimensionality of
phase spaces.
In Sec.VII we use simple group-theoretical considera-
tions, that, when applied to the Dirac algebra, suggest
an interpretation in terms of relativistic electrodynam-
ics. We show that this interpretation directly yields the
Lorentz force law, the Lorentz transformations and the
relativistic energy-momentum-relation. We demonstrate
that this framework also enables to derive the Zeeman ef-
fect, the spin, and the physics of adiabatic high frequency
transitions (Breit-Rabi-model).
In Sec. VIII we explain what is meant by “un-
quantization”: Since the so-called “canonical quantiza-
tion” can be derived and explained on the basis of clas-
sical notions, we simultaneously un-quantize QM in the
sense explained above and “quantize” CM. Then we dis-
cuss Born’s rule and explain why classicality is compati-
ble with background independence but nonetheless leads
with necessity to 3 + 1-dimensional geometrical notions.
III. THE “LAW” OF MOTION
Our inventory consists of a number ν of dynamical
variables ψ, subject to change in time τ and a PDCOM
H(ψ). With the prelimenary simplifying assumption that
H does not explicitely depend on time ∂H
∂t = 0, the phys-
icality constraint can be formulated as follows:
Ḣ =
ν
X
k=1
∂H
∂ψk
ψ̇k = 0 (1)
where the overdot indicates the temporal derivative.
Eq. 1 can be written in vectorial notation as
(∇ψH) · ψ̇ = 0 (2)
where the “·” indicates a scalar product. The solution is
given by
ψ̇ = J (∇ψH) (3)
with some arbitrary ν × ν skew-symmetric matrix J. In-
serted into Eq. 2 the condition for constancy of H is ful-
filled by the skew-symmetry of J alone.
It is a math fact that if λ is an eigenvalue of a real
square skew-symmetric matrix J, then −λ is also an
eigenvalue. Hence any skew-symmetric matrix of size
ν×ν has at least one vanishing eigenvalue, unless ν = 2 n
is even. A vanishing eigenvalue corresponds to a hidden
constant in ψ. Since this was excluded by definition, J
has full rank and ν = 2 n is even or can be reduced to
even dimension by an appropriate coordinate transforma-
tion. In both cases we can restrict ourselves to an even

5
number of dynamical variables without loss of generality:
In any physical world, the number of dynamical variables
required to describe an RPO, is even.
We use the prelimenary simplifying assumption that
H(ψ) can be written as a Taylor series of ψ and initially
concentrate on the terms of lowest order. For this case
of small oscillations one may skip higher than quadratic
terms and translate by ψ0 such that linear terms vanish,
without loss of generality. The constant term can be
excluded as trivial. Then H(ψ) can be written as
H(ψ) =
1
2
ψT
A ψ (4)
with a positive definite symmetric 3
matrix A of size
2 n × 2 n. The linearized law of motion (LOM) Eq. 3
then is
ψ̇ = J A ψ , (5)
where ψ is a vector of 2 n components.
According to a theorem of linear algebra for every non-
singular skew-symmetric matrix J of size 2n × 2n there
exists a non-singular matrix Q such that [22]:
QT
J Q = diag(λ0 η0, λ1 η0, . . . , λn η0) (6)
where λk are real non-zero constants (the modulus of two
eigenvalues) and
η0 =

0 1
−1 0

. (7)
Since there is no reason to assume anything specific about
the eigenvalues, beyond being non-zero, the PSR recom-
mends the most symmetric case, i.e. all λk are equal to
unity 4
. In this case Q is an orthogonal transformation
so that
QT
J Q = 1n ⊗ η0 ≡ γ0 (8)
Note that this transformation is only required to ob-
tain the symplectic unit matrix in a simple form that
allows to recognize firstly that J2
= γ2
0 = −1 and sec-
ondly that the dynamical variables can formally be re-
garded as canonical pairs qi and pi. One may write
ψ = (q1, p1, q2, p2, . . . , qn, pn)T
. A canonical pair rep-
resents the smallest thinkable dynamical system with a
PDCOM and is called a degree of freedom (DOF).
Eq.3 can then be written, without loss of generality, in
the form of Hamilton’s equations of motion
q̇ = ∂H
∂p
ṗ = −∂H
∂q
(9)
3 Skew-symmetric components don’t contribute and are therefore
irrelevant.
4 For n = 1 there is only a single pair of eigenvalues ±λ which
gives just a factor in Eq. 5 and can therefore be dropped by the
use of the suitable time unit. The commitment to the PSR does
not permit to introduce different eigenvalues for n 1 without
reason.
or, using the linear approximation:
ψ̇ = γ0 A ψ = H ψ , (10)
where γ0 A has been replaced by a single matrix H. Since
nothing but physicality is assumed, Hamilton’s equations
of motion must pop up in any thinkable physical reality
in some form.
The matrix γ0 is the so-called symplectic unit ma-
trix (SUM). A matrix that can be written in the form
H = γ0 A is called Hamiltonian. The transpose of a
Hamiltonian matrix is
HT
= A γT
0 = γ0 H γ0 . (11)
We define the “adjunct spinor” ψ̄ = ψT
γT
0 so that the
Hamiltonian (Eq. 4) can be written as
H = 1
2 ψT
A ψ
= 1
2 ψ̄ H ψ
(12)
since γT
0 γ0 = 1.
We stress again that no assumptions were used to ar-
rive at Hamilton’s equations of motion (EQOM) and no
assumption about the meaning of qi and pi are implied
by notation. The use of the symbols “q” and “p” is just
the convention of Hamiltonian theory. They represent
arbitrary pairs of conjugate dynamical variables. And
we stress again that they are “classical” in the sense that
q p − p q = 0.
Time, an arbitrary number of dynamical variables, a
constant of motion and the PSR are the only required
ingredients for the concept of a 2 n-dimensional phase
space. Hence the concept of phase space has no intrin-
sic connection to spatial coordinates or mechanical mo-
menta, but is purely abstract. It is the basis of any phys-
ical world.
IV. PHASE SPACE
Almost all classical presentations of quantum mechan-
ics as given by Born and Heisenberg [25] as well as by
Schrödinger [26], Dirac [27] or von Neumann [28], empha-
sized the Hamiltonian nature of Quantum theory. Even
if the SPQM postulates that in QM the classical Pois-
son brackets have to be replaced by the commutator of
conjugate operators, Birkhoff and von Neumann wrote
that there “[...] is one concept which quantum theory
shares alike with classical mechanics and classical electro-
dynamics. This is the concept of a mathematical ”phase-
space.”” [29].
But if CM and QM share the concept of phase space,
then the purported fundamental differences between CM
and QM must be due to the interpretation, due to the
assumed relation between phase space variables and mea-
surable (“observable”) quantities. As we shall demon-
strate, this is indeed the only fundamental difference be-
tween CM and QM.

6
In presentations of QM, CM is often reduced to mass
point dynamics, which implies a direct identification of
the dynamical variables (i.e. elements of ψ) to measur-
able positions and mechanical momenta. But analytical
mechanics, as formulated by Lagrange, Hamilton, Jacobi
and others, is a set of abstract mathematical principles
that underlie any dynamical system. There is no law
in classical physics that limits the applicability of these
concepts to mass points. The methods of Hamilton and
Lagrange are applied in all fields of physics and any kind
of dynamical variables, also in those that do not refer to
a spatio-temporal description in the first place. The the-
ory of canonical transformations, which allows any kind
of transformation that preserves the Hamiltonian equa-
tions of motion, is the core concept of analytical mechan-
ics and it is incompatible with a limitation of CM to mass
point dynamics.
The variable list ψ is formally a “coordinate” in some
2 n-dimensional phase space and according to what has
been said before, this does not imply or suggest any spe-
cific interpretation. Then real physical objects (RPOs)
are, in the first place, inhabitants of phase spaces. As
a single classical mass point makes no tangible object
in “physical” space, a single coordinate in phase space
makes no sensible object as well. Some kind of distribu-
tion is needed, a droplet in phase space 5
.
Since a general distribution ρ(ψ) in phase space im-
plies an infinite amount of information 6
, it is convenient
and required to reduce the complexity of the description.
A common way to describe phase space distributions is
to characterize their respective size in either direction.
The size is usually measured by the (square root of) the
second moments 7
. The matrix of second moments Σ is
given by
Σij ≡ hψiψji = hψψT
i , (13)
where the embracing angles indicate some (yet unspeci-
fied) average 8
. Without loss of generality, one can write
this as a matrix product of some 2 n×m matrix K of the
form
Σij = K KT
(14)
where m ≥ 2 n [33].
5 It is well-known that Hamiltonian motion in phase space corre-
sponds to the flow of an incompressible fluid [30].
6 That the “quantum state” contains infinite information is the
content of a Hardy’s theorem [31, 32].
7 Mathematically it is well-known that reasonable distributions
with finite moments are completely and uniquely determined by
their moments.
8 We leave aside subtleties of possible interpretations of how to
obtain and understand this average. At this point it suffices
to agree that one can average over an ensemble of phase space
points or some phase space volume. If the system is presumed
to be ergodic, the average might also be obtained by integration
over time.
From Eq. 10 one finds the (linearized) equation of the
motion of the autocorrelation matrix:
Σ̇ = hψ̇ ψT
+ ψ ψ̇T
i
= hH ψ ψT
i + hψ ψT
HT
i
= H Σ + Σ HT (15)
Multiplication from the right of both sides with γT
0 gives:
Σ̇γT
0 = H Σ γT
0 + Σ HT
γT
0 (16)
Now we define another Hamiltonian matrix S by S ≡
ΣγT
0 and with γT
0 = −γ0, γT
0 γ0 = 1 and Eq. 11 one
obtains
Ṡ = H S − S H ≡ [H, S] , (17)
which is known as Heisenberg’s equation of motion for
operators 9
.
It is still missing the quantum look and feel, namely
the unit imaginary and ~, which are both absent from
Eq. 17. But it is wrong to think that these factors are
valid indicators for the quantumness of equations. We
prove this by simply introducing both from void.
Since all variables in ψ are treated equally they all
have the same unit. According to Eq. 10 the elements of
the Hamiltonian matrix H have the unit of frequency 10
.
No one can prevent us from giving H the unit of energy
by multiplication with some conversion factor ~ with the
dimension of action. We then obtain with H̃ = ~ H:
Ṡ =
1
~
[H̃, S] (18)
Any Hamiltonian matrix that represents stable dynam-
ical systems has purely imaginary eigenvalues. Further-
more, if λ is eigenvalue of a Hamiltonian matrix, then
−λ, as well the complex conjugates ±λ̄ are also eigenval-
ues [23]. Since H̃ is by definition a stable non-degenerate
Hamiltonian matrix, it can be written as
H̃ = E Diag(iǫ1, −iǫ1, iǫ2, −iǫ2, . . . , −iǫn) E−1
(19)
where E is the matrix of eigenvectors and ǫi = ~ωi are
real energy eigenvalues. We introduce another matrix H̆
by multiplication with −i:
H̆ = −i H̃ = E Diag(ǫ1, −ǫ1, ǫ2, −ǫ2, . . . , ǫn, −ǫn) E−1
(20)
which has now real energy eigenvalues so that the unit
imaginary appears explicitely:
Ṡ =
i
~
[H̆, S] (21)
9 We shall use the term auto-correlation matrix not only for Σ
but also for S.
10 The autocorrelation matrix S can, up to this point, be given an
arbitrary unit.

7
The introduction of factors that otherwise cancel out can
not add anything physical to an equation 11
. Therefore
Heisenberg’s operator equation is as such not quantum:
we just derived it from classical Hamiltonian mechan-
ics. Without further assumptions it is simply an equation
that describes the linearized equations of motion of sec-
ond moments in some classical phase space. Furthermore
Eq. 17 proves that commutators are just an algebraic re-
sult of considering the evolution of second moments in
time. Hence commutators are, as such, not quantum ei-
ther.
A. Unitary vs. Symplectic Motion
However, the matrix H is Hamiltonian (not Hermi-
tian), therefore it generates symplectic (and not unitary)
evolution in time. Symplectic motion is more general
than unitary motion since firstly, unitary motion is al-
ways symplectic 12
but secondly, symplectic motion al-
lows for complex eigenvalues 13
which are excluded in
unitary motion. But no law of the universe and no
commandment forbids complex eigenvalues. Such a law
would be superfluous anyway as complex eigenvalues and
stability are incompatible. Complex eigenvalues may ap-
pear (for a limited time) in nature, for instance in case of
resonance [34–36], but they are incompatible with long-
term stability.
Bender and others have shown that unitarity is not
universally required, not even in the case of real eigen-
values [37–40]. The corresponding postulate of QM is
therefore not universally valid and can be dropped.
This is a simple math fact and requires no postulates:
the description of stable states (of motion) requires the
corresponding mathematical form of eigenvalues. If one
insists on the unit imaginary as an indispensable QM
factor, then the eigenvalues of a stable system must be
real. In stable symplectic motion, which is considered to
be classical, the unit imaginary is not written explicitely
and the eigenvalues of stable motion are purely imagi-
nary. In the former case one uses the unit imaginary
explicitely and writes the frequency as ω = E/~, in the
latter case the unit imaginary is used implicitely as the
eigenvalues have the form ± i ω with a real valued fre-
quency ω. But neither nature nor mathematical reason
cares much about notational conventions: Neither the
explicite use of the unit imaginary nor unitary evolution
are quantum.
The derivation of Eq. 17 from a PDCOM (the Hamil-
tonian) suffices to generate a structure preserving sym-
11 The idea that ~ represents a quantum of action in the sense that
classical mechanics is restored in the limit of ~ → 0, has been
falsified experimentally [24].
12 See App. A.
13 Complex here means truely complex, points in the complex plane
that are neither on the real nor on the imaginary axis.
plectic framework in which “probability current conser-
vation” pops up automatically, since symplectic motion
is known to conserve the occupied volume of phase space
as we shall derive in the next section.
V. PHASE-SPACE ALGEBRA
Let us mention some math facts about Hamiltonian
matrices and symplectic motion that are known, but
maybe not well-known. We begin with the fact that
Eq. 17 constitutes a so-called Lax pair. As Peter Lax has
shown, if a pair S and H of operators obeys Eq. 17, then
the trace of any power of S is a constant of motion [41]:
Tr(Sk
) = const , (22)
for all k ∈ N. This also holds for non-linear operators.
Within our approach, both matrices are by definition the
product of the skew-symmetric SUM γ0 and a symmetric
positive definite matrix A. According to linear algebra
they have a vanishing trace and it can be shown that all
odd powers of H and S share this property:
Tr(S2k+1
) = 0 , (23)
for k ∈ N, so that only the even powers in Eq. 22 are
“non-trivial” constants of motion (COMs). It has been
shown elsewhere that the eigenvalues of the autocorrela-
tion matrix are a measure of the occupied phase space
volume [42, 43]. Again: This is a math fact and requires
no postulate. Furthermore, in statistical mechanics, a
phase space density is as close as can be to a probability
density.
As already mentioned, both matrices, the driving ma-
trix H and the matrix S, have the same structure, namely
both are Hamiltonian (Eq. 11). It is well known that such
matrices are generators of symplectic motion by the fact
that the solution of Eq. 10 is the matrix exponential
ψ(τ) = exp (H τ) ψ(0) = M(τ) ψ(0) . (24)
It is straightforward to show that a symplectic matrix
M = exp (Fτ) holds:
M γ0 MT
= γ0 . (25)
A matrix N is skew-symplectic if
N γ0 NT
= −γ0 . (26)
One finds after few steps
S(τ) = M(τ) S(0) M−1
(τ) = M(τ) S(0) M(−τ) . (27)
The result of Hamiltonian evolution in time, the result of
motion, is a symplectic similarity transformation (SST).
And since similarity transformations do not change eigen-
values, the eigenvalues are COMs.

8
A. Eigenvectors and Eigenvalues
It is yet another math fact that commuting matrices
share a system of eigenvectors. According to Eq. 17 the
matrix S (and hence the second moments) is constant, iff
H and S commute. Only diagonal matrices always com-
mute, so that commuting matrices must have the same
matrix of eigenvectors E:
DS DF = DF DS
(E S E−1
) (E H E−1
) = (E H E−1
) (E S E−1
)
S H = H S
(28)
where DS = Diag(i ε1, . . . , −i εn) is the diagonal matrix
containing the eigenvalues of S.
Therefore eigen-vectors and -values play an important
role in physics. This is a consequence of constructing
observables from second moments on the basis of Hamil-
tonian mechanics. Again this requires no postulates, nei-
ther quantum nor otherwise. Oscillatory systems have
eigenvalues – the frequencies – and eigenvectors 14
. It
is a math fact that strongly stable systems must have
purely imaginary eigenvalues and complex eigenvectors.
And since the eigenvalues come in pairs (or quadruples,
if complex), the eigenvectors also come in complex con-
jugate pairs. This is a math fact about (classical) cou-
pled oscillating systems, subject to linear Hamiltonian
motion. No commandment is required.
B. Symplectic Motion is Structure Preserving
Since similarity transformations do not change eigen-
values, this also holds for linear symplectic motion, i.e.
SSTs. SSTs also preserve the structure of Hamilton’s
equations of motion, i.e. the form of the matrix γ0. With
respect to an RPO it is specifically the dynamical struc-
ture which determines the properties, or the type of these
objects. The fact that evolution in time is a SST guaran-
tees that Hamiltonian matrices remain Hamiltonian. The
exponential of a Hamiltonian matrix is symplectic and
the logarithm of a symplectic matrix is Hamiltonian [23].
A skew-Hamiltonian matrix C is a product of the SUM
γ0 and an arbitrary skew-symmetric matrices B = −BT
:
C = γ0 B (29)
such that
CT
= −γ0 C γ0 . (30)
Accordingly the number of linear independent elements
νs in a Hamiltonian matrix of size 2 n × 2 n is
νs = n (2 n + 1) (31)
14 Note that S commutes with analytical functions of H: if it
commutes with H, it commutes with M = exp (Hτ).
and in a skew-Hamiltonian matrix it is νc:
νc = n (2 n − 1) (32)
It is a straightforward exercise to show that the (anti-)
commutators of Hamiltonian (Si) and skew-Hamiltonian
(Cj) matrices have (anti-) commutators of the following
type:
S1 S2 − S2 S1
C1 C2 − C2 C1
C S + S C
S2 n+1





⇒ Hamiltonian
S1 S2 + S2 S1
C1 C2 + C2 C1
C S − S C
S2 n
Cn









⇒ skew − Hamiltonian
(33)
Note that the unit matrix 1 is skew-Hamiltonian. It is
remarkable that it is possible to derive the complex struc-
ture 33 from nothing but symmetry arguments, i.e. from
pure physical logic.
C. Observables and Generators
Quantum mechanics is not the first physical theory
that requires a reflection about the meaning of a mea-
surement. Also special relativity is a theory that strug-
gles with the meaning of time and length measurements.
Our considerations are based on a definition of phys-
ical realness and are therefore incompatible with claims
that ψ is somehow “unreal”. In the contrary, ψ was
the only ontic thing we presumed at all. This view is
confirmed by the no-go-theorem of Pusey, Barrett and
Rudolph in which the authors claim that “if a quantum
state merely represents information about the real phys-
ical state of a system, then experimental predictions are
obtained which contradict quantum theory” [44]. There
is a minority report of physicists that do not subscribe the
dogma of unreality. Roger Penrose, for instance, wrote
“if we are to believe that any one thing is in the quan-
tum formalism is ’actually’ real,[...], then I think it has
to be the wavefunction [...]” [46], or Lev Vaidman: “The
only fundamental physical ontology is the quantum wave
function” [47].
But though we defy to regard ψ as somehow unreal,
it can not be denied that the meaning of ψ is not self-
evident. So far we did not consider the physical unit of
the variables ψ. What type of quantity do these vari-
ables represent? The answer could be “Since it’ s part of
nature, we don’ t really know.” [75].
We can only stress again that formally ψ is a coordi-
nate in phase space. Classical phase spaces coordinates
have no fixed units, only the product of canonical pairs
is fixed to the unit of action, or angular momentum, re-
spectively.

9
Hence we can say that a 2n-dim. volume (n ≥ 1) of
phase space has a unit, single coordinate values don’t.
From the PSR it follows that both, the canonical coor-
dinates and momenta forming ψ, have the same unit.
Though one might formally say that
√
~ would be the
nearby unit for a phase space coordinate, this has little
practical value.
A unit requires not only a name and a symbol. For
a direct measurement it is necessary to have a reference
artifact that has a constant property of the same type:
a certain weight, length, clock frequency or voltage. But
since the variables ψ are supposed to be fundamental,
how should such an artifact emerge from a more funda-
mental level, if there is none, by definition?
Furthermore, ψ is by construction a list of dynamical
variables in the literal sense. By definition we required
that none of these variables (and no linear combination
thereof) can be considered a non-zero constant. Hence
there is no constant reference and therefore ψ can not
be directly measured [33, 48]. Only available constant
quantities can provide a reference, i.e. second or higher
even moments like the Hamiltonian H(ψ). Linear Hamil-
tonian theory is based on a quadratic form, the Hamil-
tonian, which is constant by construction and provides
the reference for all second (and possibly higher even)
moments.
Then the use of second (or higher even) moments and
correlations to describe the phase space distribution is
not only a convenient and natural choice, it is the only
possible choice.
There is no need to postulate that ψ can not be mea-
sured: unless someone presents a solution to the refer-
ence problem, we doubt that it has a solution. Mermin
asserted that “[...] the proper subject of physics [are] cor-
relations and only correlations” [49]. Here is the reason
why.
Humans are inhabitants of a physical world and have
the perspective of insiders. One can not prevent anyone
from considering the possibility that some supernatural
being, some kind Maxwellian demon, might have a dif-
ferent perspective and is in the posession of a reference
that enables to “measure” ψ. But from within the physi-
cal world, a direct measurement is hardly possible. Some
schools of philosophy deny the possibility to presume ex-
istence of unmeasurable entities. But we do not suggest
that the entities are unmeasurable, we just doubt that
the values of the variables in ψ can be directly measured
at some time τ.
If one regards this as a reason to exclude the wave-
function from classical physics, then classicality would
have to be limited to observable physical quantities as
well. However, such a limitation of classicality would be
historically untenable: Newton based the fundament of
his theory on the existence of something unmeasurable,
namely absolute space. Furthermore he suggested a cor-
puscular theory for light, before any such corpuscle was
experimentally detected. Boltzmann used atoms in the
kinetic theory of gases before there was sufficient evi-
dence that atoms exist at all. A sober view of physics
reveals that there are plenty of entities which can not be
observed “directly”. This is due to the very method of
physics: Physics proceeds by presenting simple but not
directly observable common causes. If the cause would
be directly observable, it would not need a theory for it.
The reference problem explains many of the difficul-
ties to understand and accept un-quantum physics for
inhabitants of physical worlds, if these inhabitants try
to establish a physical theory on the basis of measurable
quantities alone. This also holds for human beings.
The matrix of second moments S = ΣγT
0 , and the
spinor ψ have very different LOMs. In contrast to the
dynamical variables in ψ, that can by construction not be
constant, the variables in S (“observables”) can be con-
stant, if S and H commute. Otherwise they (for instance)
oscillate with some frequency and amplitude. The Hamil-
tonian as a PDCOM is available as a reference quantity
so that the correlations of S can always be measured.
Hence there are (at least) two different levels of reality,
the “spinor” ψ and its auto-correlation matrix S.
The matrix S is, like H, a Hamiltonian matrix. Since
skew-Hamiltonian matrix components do not contribute
to the Hamiltonian, they cannot be generators of possible
evolutions in time. Correspondingly the autocorrelation
matrix Σ is symmetric and skew-Hamiltonian matrices
have zero “expectation” values [48]. This means that
there are further “parameters” emergent in the theory
that necessarily vanish and are in this sense “unmeasur-
able” or “hidden”.
A phase space density is constant if it is exclusively a
function of COMs. Since only even moments can mathe-
matically generate COMs, a stable phase space density is
an even function of ψ: ρ(ψ) = ρ(−ψ). Classical statisti-
cal mechanics is concerned with many DOFs and in this
case, only positive definite values do not cancel by averag-
ing over some thermal ensemble, namely the known PD-
COM, so that eventually in this case one finds ρ = ρ(H).
The Boltzmann distribution ρ(H) ∝ exp (−β H) is such
a case and corresponds, using Eq. 4, to a multivariate
normal distribution in ψ, up to a normalization.
The constraint that only Hamiltonian “operators”, pa-
rameters of the Hamiltonian matrix, represent observ-
ables, might be regarded as the true origin of the Her-
miticity condition for complex Dirac matrices 15
. But out
classical approach is clearer, more straightforward and
also more stringent. Our analysis suggests that funda-
mental dynamical quantities can not be classical observ-
ables and that, instead, second (or higher even) moments
are required to obtain observables 16
. Then of course,
some strange effects concerning the statistical properties
15 Given by γ0 γµ γ0 = γ†
µ [50]. In the standard presentation of the
Dirac electron theory, the adjunct spinor is defined as ψ̄ ≡ ψ†γ0,
but the multiplication with γ0 is not explained.
16 This implies that our approach denies the possibility that clas-
sical physics could be fundamental at all.

10
of observables are unavoidable 17
.
D. The Pauli Matrices from Hamiltonian
Symmetry
DOFs are the basic elements of the dynamical descrip-
tion of real objects. If one considers a single DOF, the
matrices γ0, H, S and M are of size 2 × 2. Consider an
arbitrary real 2 × 2 matrix K:
K =

a b
c d

(34)
This parametrization is simple and (sort of) nearby but
it does not fit the needs of Hamiltonian theory. Any ma-
trix is the sum of a Hamiltonian and a skew-Hamiltonian
matrix. The chosen parameters should belong to either
of them. Since Hamiltonian matrices have zero trace, we
can easily identify the skew-Hamiltonian part as a mul-
tiple of the unit matrix η3 = 12:
K =

a b
d −a

+ c

1 0
0 1

. (35)
Since the definition of Hamiltonian matrices uses matrix
transposition, it is required to distinguish between the
purely symmetric, purely skew-symmetric, and the diag-
onal parts. One thus arrives at
K = s0 η0 + s1 η1 + s2 η2 + c η3 , (36)
where
η0 =

0 1
−1 0

η1 =

0 1
1 0

η2 =

1 0
0 −1

η3 =

1 0
0 1

= 1
(37)
are the real Pauli matrices (RPMs). η0 = (γ0)2×2 is
the SUM for a single DOF. Hence the RPMs provide a
parameterization that matches the symmetries relevant
in Hamiltonian dynamics:
K =

c + s2 s0 + s1
s1 − s0 c − s2

. (38)
An analysis of the properties of these matrices reveals
that
(η0)2
= −1 (η1)2
= 1
(η2)2
= 1 (η3)2
= 1
(39)
Furthermore one finds that the three non-trivial matrices
mutually anti-commute, i.e. for i, j ∈ [0, 1, 2]:
ηiηj + ηjηi = 2 Diag(−1, 1, 1) . (40)
17 D.N. Klyshko considered that many if not all “quantum para-
doxes” have a common origin, namely the “failure to find a so-
lution to a certain moments’ problem” [51].
All individual RPMs are either symmetric or skew-
symmetric, they either pairwise commute or anti-
commute, they square to ±12 and they are either Hamil-
tonian or skew-Hamiltonian, symplectic (Eq. 25) or skew-
symplectic (Eq. 25). Their trace vanishes except for the
unit matrix. The relevant symmetries of the Pauli alge-
bra are given by
ηi ηj = ± ηj ηi
η2
i = ± 1
ηi = ±ηT
i
Tr(ηi) = 0 unless ηi = 1
(41)
Note the math fact that skew-symmetric matrices ηi
square to −1 while symmetric matrices ηi square to
1 [33]. The signature (the sign of the trace of the square)
of the Pauli matrices corresponds to their symmetry un-
der matrix transposition.
The type of transformation that these matrices gen-
erate (Eq. 24) is the matrix version of Euler’s formula
eiφ
= cos φ + i sin φ:
exp (ηi φ) = cos φ + ηi sin φ for η2
i = −1
exp (ηi φ) = cosh φ + ηi sinh φ for η2
i = +1 (42)
Formally trigonometric functions belong to rotations
while the hyperbolic functions belong to boosts [52, 53].
Hence Eq. 42 suggests that it is thinkable to obtain an ac-
count of the Lorentz transformations directly on the ba-
sis of Hamiltonian algebras: Without considering spatio-
temporal notions the Hamiltonian algebra of proper time
automatically generates the mathematical means to de-
scribe Minkowski type space-times.
Note that only the transformation matrix for rotations
is symplectic and orthogonal, while for boosts it is only
symplectic. The matrix algebra of a single DOF is the
real Pauli algebra. Since we derived the significance of
the real Pauli matrices (RPMS) from classical Hamilto-
nian theory, the Pauli algebra can not be quantum.
E. The Kronecker Product and Hamiltonian
Clifford Algebras
Two methods to generalize the Pauli algebra are pos-
sible: One can either add more DOFs and analyze the
properties of Hamiltonian systems with two, three, four
DOF and so on, or one may use a multiplicative approach
based on the Kronecker product. The next system, con-
structed from an additive approach has two DOF and
requires the use of the real 4 × 4 matrices, i.e. the real
Dirac algebra. Three DOF would require 6 × 6-matrices
and one can anticipate that the natural symmetries in-
herited from the real Pauli matrices will be broken.
The multiplicative generalization is based on Kro-
necker (or tensor) products. The Kronecker product of

11
two Pauli matrices A = {aij} and B = {bkl} is given by:
C = A ⊗ B =

a11B a12B
a21B a22B

=



a11b11 a11b21 a12b11 a12b12
a11b12 a11b22 a12b21 a12b22
a21b11 a21b12 a22b11 a22b12
a21b21 a21b22 a22b21 a22b22


 ,
(43)
i.e. the Kronecker product is a method to systematically
write down all possible products between all elements of
A and B, respectively. The general rules of Kronecker
matrix products are [54]:
(A ⊗ B)T
= AT
⊗ BT
A ⊗ (B + C) = A ⊗ B + A ⊗ C
(A ⊗ B) (C ⊗ D) = A C ⊗ B D
Tr(A ⊗ B) = Tr(A) Tr(B)
(A ⊗ B)−1
= A−1
⊗ B−1
(44)
It is straightforward to verify that Kronecker multiplica-
tion preserves the symmetries of Eq. 41.
The Kronecker product allows to construct all Clifford
algebras (CAs) with real matrix representations from the
real Pauli matrices, i.e. all Hamiltonian Clifford algebras
(HCA) 18
. Apparently there is exactly one algebra that
follows both rules, which is the only additive and multi-
plicative generalization of the Pauli algebra, namely the
real Dirac algebra.
Before we discuss the Dirac algebra, we shall first give
a (very) brief introduction to CAs as they are usually pre-
sented, i.e. without reference to Hamiltonian theory, and
explain our motivation to restrict us to Clifford algebras
with (irreducible) real matrix representation.
VI. CLIFFORD ALGEBRAS
Mathematically Clifford algebras can be defined as gen-
erated by a list ek , k ∈ [0 . . . N − 1] of N pairwise an-
ticommuting elements that hold e2
k = ±1. These are
called the generators of the Clifford algebra. If one con-
siders p generators that square to 1 and q = N − p
generators that square to −1, then the algebra is de-
noted by Cl(p, q) and it has a signature (or metric ten-
sor) gµν = Diag(1, . . . , 1, −1, . . ., −1) with p positive and
q negative entries in the diagonal.
The N generators can be used to obtain new elements
by multiplication since products of two (or more) differ-
ent generators eiej are unique elements, different from
18 As we started out from a conservation law, we are specifically
interested in Hamiltonian Clifford algebras, and in Clifford alge-
bras mainly as they preserve the dynamical symmetries of sym-
plectic theory, but we are not specifically interested in Clifford
algebras as such.
the unit element and different from each factor, which
square to ±1. It follows from combinatorics that there
are

N
k

products of k generators, called k-vectors, so
that one has
X
k

N
k

= 2N
(45)
elements in total. Hence m × m matrix representations
require at least the same number of independent param-
eters, so that m2
≥ 2N
or m ≥ 2N/2
. An isomorphism
between a given CA and some corresponding matrix al-
gebra is only possible for even N with m = 2N/2
.
Clifford algebras are used in various branches of
physics, but the Dirac algebra is of special interest since
it matches the geometry of 3 + 1-dimensional space-
time as David Hestenes has shown in many publica-
tions (see for instance Ref. [55]). The commutator ta-
ble of the Dirac algebra also determines the form of the
electromagnetic field tensor [56]. Furthermore the anti-
commutation properties, enable to naturally explain the
vector cross product and therefore to describe the hand-
edness of space [53], a product that has otherwise to be
introduced by some commandment.
Certainly CAs are an interesting mathematical topic in
themselves and Hestenes has shown, that it is possible to
give a presentation of Dirac’s theory without matrix rep-
resentations [55]. Indeed, from a general mathematical
point of view, this question can be outsourced into a spe-
cial branch of mathematics called representation theory,
which is then another interesting topic in its own right. It
is a fairly common approach to abstract CAs from their
respective matrix representations. And of course CAs do
not need Hamiltonian theory to be interesting and use-
ful. They generate geometric spaces even if they are not
considered in our specific context of Hamiltonian phase
spaces.
But in the dynamical context of our presentation, ma-
trix transposition is an indispensable element that allows
to distinguish Hamiltonian from skew-Hamiltonian ele-
ments and to analyze their algebra (Eq. 33). It is es-
sential for the motivation to consider Clifford algebras at
all. 19
.
A. The Complex Numbers
Consider for instance the case N = 1 in which we have,
besides the unit element 1 only one single non-trivial el-
ement e0, which gives the algebra Cl(0, 1).
19 For a discussion of real CAs in the context of linear Hamilto-
nian theory see [33, 53, 56, 57]. We would have prefered to also
cite other authors in this context, but haven’t found much. In
Refs. [58, 59], for instance, the name “Hamilton” is mentioned
exclusively in context of quaternions. That CAs might be useful
in the context of classical Hamiltonian dynamics, is not consid-
ered.

12
If this element squares to −1, then the corresponding
“Clifford number” (CNs) x has the form x = a 1 + b e0.
The multiplication of two CNs is then
x y = (ax 1 + bx e0)(ay 1 + by e0)
= (ax ay − bx by) 1 + 2 (ax by + bx ay) e0 .
(46)
This is the product of two complex numbers, which are
hence almost identical to the Clifford algebra Cl(0, 1).
We say “almost”, because the theory of complex num-
bers knows the operation of complex conjugation, which
has no correspondence in Cl(0, 1) unless we refer to real
matrix representations.
The operation of complex conjugation however can be
naturally obtained from representation theory, namely if
we use the real Pauli matrices. Then e0 = η0 is the
(only) skew-symmetric element and complex conjugation
is identical to matrix transposition, so that in the con-
ventional notation z = x + i y is
z =

x y
−y x

(47)
and
z⋆
= x − i y = zT
=

x −y
y x

. (48)
This means that transposition and complex conjugation
can not be properly distinguished with full logical rigour.
One might also substitute matrix transposition by a mul-
tiplication with the signature of the corresponding Clif-
ford k-vectors. But again, the signature is only fixed, if
exclusively real representations are used.
It also means that the complex numbers are, regarded
from this perspective, a special case of (the algebra of)
2 × 2-matrices. As we have shown in Ref. [33], the re-
duction of the real Pauli algebra to the algebra of the
complex numbers corresponds to the reduction to the
general LOM of a DOF to normal form, to an harmonic
oscillator. It is well known in many branches of physics,
for instance in accelerator physics, that the unit circle in
the complex plane is the normal form trajectory of the
motion of a single DOF so that the complexity of the
actual state of affairs can be reduced to a single number,
namely the “phase advance” (i.e. time) [60].
The analysis of normal forms is of course a useful math-
ematical technique, but one should always keep in mind,
that it describes the system in a special coordinate sys-
tem and that coordinate transformations have a two-fold
meaning. They can be understood as passive transfor-
mations and, regarded this way, they just concern our
mathematical methods to describe a given physical pro-
cess. But symplectic transformations also describe the
full space of physical possibilities, of possible evolutions
in time. This space of possibilities is substantially nar-
rowed if we restrict our math to the use of normal forms
only.
B. The Unit Imaginary And The Dirac Equation
Foreclosing what we are going to argue below, let us
remark that it is not as nearby as often suggested to
consider representations “over” the complex numbers as
something fundamental 20
. In fact we suspect that the
incautious use of complex (or quaternionic) “numbers”
substantially contributes to the scrambling of the quan-
tum omelette 21
. To understand this point correctly is
indispensable for a successful unscrambling. Let us there-
fore spend a few words for it.
It is part of the logic of quantum theory that the
Schrödinger equation is the non-relativistic approxima-
tion of the Dirac equation [63]. Hence the Dirac equa-
tion must be regarded as more fundamental than the
Schrödinger equation and is the logical basis of quantum
theory. Nonetheless, discussions about the interpreta-
tion of QM rarely refer to Dirac’s theory. As Hestenes
noted, “[it] has long puzzled me is why Dirac theory is al-
most universally ignored in studies on the interpretation
of quantum mechanics, despite the fact that the Dirac
equation is widely recognized as the most fundamental
equation in quantum mechanics” [64].
Dirac’s theory is required to provide Lorentz covari-
ance, to explain the spin, the gyromagnetic ratio and es-
sential parts of the hydrogen spectrum. The Dirac equa-
tion is the basis of QED and QFT. In the current ap-
proach it is Dirac’s theory that underlies most of modern
physics and it is therefore annoying that students must
first and sometimes exclusively undergo the brainwash-
ing 22
of textbook QM before they have the chance to
understand that this formalism is essentially classical 23
.
It is annoying that most textbooks on quantum theory
do not treat the Dirac equation at all or just briefly as
a kind of addendum, in the last chapter or the second
volume.
The Dirac equation, however, allows for but it does not
require the explicite appearance of the unit imaginary.
This becomes quite obvious by the fact that the complex
Clifford algebra Cl(1, 3) can be directly replaced with
20 See also Ref. [61]: “[...] complex numbers are not required in or-
der to describe quantum mechanical systems and their evolution
[...]”.
21 ”Our present QM formalism is a peculiar mixture describing in
part laws of Nature and in part incomplete human information
about Nature-all scrambled up together by Bohr into an omelette
that nobody has seen how to unscramble.” [62]
22 Murray Gell-Mann is quoted with the following words: “The fact
that an adequate philosophical presentation [of quantum physics]
has been so long delayed is no doubt caused by the fact that Niels
Bohr brainwashed a whole generation of theorists.” [65].
23 The author heart about the fact that the dynamics governing
the time evolution of Quantum theory is indeed classical for the
first time during his PhD, 1998 or 1999, in a talk given by John
Ralston at DESY in Hamburg. The title of the talk was “Spin
and the well-dressed Quark”. Like F. Strocchi in 1966, Ralston
argued that the time-dependent part of Schrödinger’s equation
is identical to Hamilton’s equations of motion [66, 67].

13
the real Clifford algebra Cl(3, 1), just by letting the γ-
matrices “absorb” the unit imaginary. Due to Pauli’s fun-
damental theorem of the Dirac matrices [68] any choice
of Dirac matrices that generates the same metric, can be
obtained by similarity transformations from each other
and is hence physically equivalent. Hence one can al-
ways use the (purely imaginary) Majorana matrices. If
we denote the generating elements of the real algebra
Cl(3, 1) by γµ and those of a complex CA Cl(1, 3), using
the Majorana basis, by Γµ, then i Γµ = γu and we are
done. The “complex” version:
(i Γµ∂µ ± m)ψ = 0 (49)
becomes
(γµ∂µ ± m)ψ = 0 (50)
with purely real matrices γµ. Since this is just a nota-
tional issue, it can by no means imply different physics 24
.
The algebraic form that is obtained in energy-
momentum space is the eigenvalue equation
(γ0 E + px γ1 + pyγ2 + pz γz ± i m)ψ = 0 , (51)
where λ = ± i m is an eigenvalue of the matrix
H = γ0 E + px γ1 + pyγ2 + pz γ3 , (52)
such that a positive mass corresponds to a purely imag-
inary eigenvalue λ, as required in stable linear Hamilto-
nian systems [23]. Moreover, the matrix H is a Hamil-
tonian matrix and the Dirac equation is therefore just
a special case of Eq. 10, hence it is but classical linear
Hamiltonian theory, applied to fundamental variables.
This is the proof that the Dirac equation is as such
not quantum, as it can be obtained classically. It will
be discussed in more detail below. The use of the unit
imaginary is pure notation and does not make it quan-
tum either. The use of scaling factor like ~ also can not
be quantum [73]. Since the spin is an original result of
Dirac’s theory, also spin is not quantum. Hence, if the
more general equation, namely the relativistic equation
of Dirac, is not quantum, then the non-relativistic ap-
proximation, Schrödinger’s equation, can’t be quantum
either.
24 Of course, the use of Cl(3, 1) instead of Cl(1, 3) is accompanied
with the use of the of the so-called “east coast metric” (ECM)
instead of the “west coast metric” (WCM) [69]. But though the
WCM is used more often, both choices are physically equiva-
lent. As mentioned by Woit, Weinberg preferred the ECM in
his presentation of QFT [70]. A third notational convention that
writes time as a kind of imaginary fourth coordinate in the form
of ds2 = dx2
1 + dx2
2 + dx2
3 + dx2
4 with dx4 = i c dt has also been
used, for instance by Sommerfeld in Ref. [71] and by Einstein in
Ref. [72].
C. Math and Physics
In our view it is important to understand that, though
physics requires the use of math (and mathematical
logic), mathematics does not require physics, i.e. even
if many tools developed in pure mathematics turned out
later to be useful in physics, they usually have not been
designed for this purpose. Physicists have to re-design
and to select the mathematical tools that serves their
needs best. It is obvious that nature restricts the mathe-
matics that is useful for physics and the approach of this
work is based on a restricted use of the math.
The PSR suggests the equivalence of all variables in
ψ, until it turned out to be inevitable to break this
equivalence formally with the introduction of a skew-
symmetric matrix γ0, the symplectic unit matrix. The
skew-symmetry of γ0 suggested to formally introduce
canonical pairs. This is the reason why Schrödinger’s
non-relativistic equation must be complex: It needs to
implement a canonical pair to generate a constant of
motion [66, 67]. We insisted on real-valued dynamical
variables ψ and PDCOM H, but this does not imply to
abandon the use of the complex numbers as such.
As stressed before, stable Hamiltonian systems have
purely imaginary eigenvalues and complex eigenvectors.
This is accepted and known from classical mechanics and
is nowhere regarded as an argument for suspicious con-
clusions about the realness of the dynamical quantities.
What is unacceptable however in a classical setting is
the a priori use of complex numbers for the coordinates
ψ or the conserved quantity H due to insouciance or by
a commandment. We have shown it is not required by
the math of QM: the unit imaginary does not generate
quantumness.
We can not prevent anyone from using a suggestive
notation, but we can doubt that a specific notation is
physically relevant. The only logical constraint for ψ is an
even number of variables. This alone does not require the
use of complex numbers, even if it might be convenient
to use them in specific problems.
Though the use of complex numbers for ψ and H is
not wrong per se, but in the context of Dirac’s theory
it wrongly suggests the physical equivalence of all six-
teen matrices, while the restriction to the reals enables
to properly distinguish between ten Hamiltonian and six
skew-Hamiltonian components. Also Dirac found and
discussed only ten generators (and not sixteen) [74].
If intended or not, it seems that the complex nota-
tion mainly serves the purpose of scrambling the quan-
tum omelette. But as Ralston argued, there is little in
quantum theory that proves the non-reality of the wave-
function [75]: “Bohr and Heisenberg had made up their
minds about a philosophy of unreality before the actual
quantum theory existed.”
In his book “Der Teil und das Ganze”, published 1969,
Heisenberg frankly admitted that already in 1926 he had
made up his mind, that nature must be discontinuous.
Even 43 years later, he was unable (or unwilling) to pro-

14
vide arguments in support of this conviction. Further-
more his writings suggests that for him, objectivity was
identical to a spatio-temporal description [76]. A spatio-
temporal description however is not a condition for but
a consequence of objectivity.
D. Hamiltonian Clifford Algebras
The usefulness of Clifford algebras in the context of
Hamiltonian theory is due to Eqs. 33.
Given we have a set of N anti-commuting matrices
γk , k ∈ [1..N − 1] and the SUM γ0 generated by (re-
peated) Kronecker multiplication of the real Pauli ma-
trices, then the matrix system has a dimension 2N
=
2m
× 2m
. 25
: It follows that also all real Dirac matrizes
are either Hamiltonian or skew-Hamiltonian:
γ0 γk γ0 = ± γ0 (γ0 γk)
= ± γk
= ± γT
k
(53)
Then any matrix γµ that anti-commutes with γ0, holds
γ0 γµ γ0 = γ0 (−γ0 γµ)
= γµ (54)
(since γ2
0 = −1) and is therefore either Hamiltonian and
symmetric or skew-Hamiltonian and skew-symmetric.
This connection between the different symmetries is of
severe importance for the theory of Hamiltonian Clifford
algebras and has consequences for the general description
of n DOF. It is specifically the mixture of the properties
of Clifford algebras and Hamiltonian constraints (Eq. 33),
that produces new and complex structures.
As the generators of Clifford algebras all anti-commute
(by definition), Eq. 54 is such a constraint: In sys-
tems in which all generators of the Clifford algebra
are also generators of symplectic motion (i.e. Hamilto-
nian), the metric tensor necessarily has the form gµν =
Diag(1, 1, . . ., 1, −1) and the Clifford algebra has dimen-
sion Cl(p, q) with q = 1 and p = N −1. This implies that
the formalism reproduces the fact that time is unique.
The use of Hamiltonian Clifford algebras (and not only
Hamiltonian algebras) for the parametrization of even
moments of phase provides maximal symmetry with re-
spect to the individual variables in ψ as well as with
respect to the individual DOFs 26
. This naturally con-
forms the requirement of the PSR to treat all elements on
equal grounds: the matrix representation of all k-vector
25 Latin indices γk denote a range k = [1, . . . , N − 1], greek indices
γµ a range µ = [0, 1, . . . , N − 1].
26 The Hamiltonian algebra of 6 × 6-matrices as it is usually asso-
ciated with the classical motion of particles in “physical” space,
has less symmetry.
Type m 0 1 2 3 4 5
Pauli N = 8 m + 2 2 10 18 26 34 42
p+q 1+1 9+1 17+1 25+1 33+1 41+1
Dirac N = 8 m + 4 4 12 20 28 36 44
p+q 3+1 11+1 19+1 27+1 35+1 43+1
TABLE I. Possible dimensionalities of HCAs in which all
generators of the Clifford algebra are Hamiltonian matrices
and therefore correspond to non-vanishing (and hence “ob-
servable”) auto-correlations of Hamiltonian spinors ψ.
elements γA of any real Clifford algebra has one (and only
one) entry of ±1 in each row and each column while all
other elements are identically zero.
Hamiltonian Clifford algebras Cl(p, q), i.e. CAs with
real representations, exist only for [77]
p − q = 0, 1, 2 mod 8 , (55)
which, since N = p+ q must be even, reduces in our case
to
p − q = 0, 2 mod 8 . (56)
From Eq. 54 we derived that, if all generators of Cl(p, q)
are Hamiltonian (i.e. “observable”), then one has q = 1
and p = N − 1 so that
N − 2 = 0, 2 mod 8 . (57)
This selects the dimensionalities listed in Tab. I as candi-
dates of special interest within Hamiltonian theory. The
simplest algebras are the real Pauli algebra Cl(1, 1), and
the real Dirac algebra Cl(3, 1).
According to Eq. 57 there are two sequences of HCAs,
given by
N = 8 m + 2 (58)
which we call a HCA of the Pauli type and
N = 8 m + 4 (59)
which we call a HCA of the Dirac type where m ∈ N. If
the dynamical significance of CAs and hence the require-
ment of a real representation is ignored, then CAs can
be defined for practically any dimensionality. In a purely
mathematical setting, this might be an interesting gen-
eralization, but in a physical context it is the easy road
to dynamical misconceptions.
In order to understand the logical and dynamical prop-
erties of the individual elements of HCAs of the men-
tioned dimensionalities, it is important to notice that all
generators except the SUM γ0 are symmetric real matri-
ces. Since we have no specific argument to prefer any of
them, it is nearby to consider the role of matrices that
play a special role by their formal position within the
CA. Besides the SUM γ0 and the unit matrix 1, any CA

15
has two unique elements, the first being the N-vector γπ,
which is the product of all generators
γπ =
N−1
Y
µ=0
γµ (60)
called pseudo-scalar.
Since N must be an even integer, the pseudo-scalar
anti-commutes with all generators (vector elements), it
therefore commutes with all 2-vectors and anti-commutes
again with all 3-vectors and so forth: The pseudo-scalar
distinguishes even from odd k-vectors. Therefore the
pseudoscalar of the Dirac algebra induces charge con-
jugation, namely a change of sign of the bi-vectors only
(see Sec. VI E below).
As derived in App. D, the pseudoscalar of Pauli type
HCAs is Hamiltonian and symmetric while in Dirac type
algebras it is skew-Hamiltonian and skew-symmetric.
Furthermore, as shown in App. D, in both, the Pauli
type and the Dirac type algebras, only k-vectors with
k = 1, 2, 5, 6, 9, 10, . . . are Hamiltonian while k-vectors
with k = 3, 4, 7, 8, . . . are skew-Hamiltonian (App. D).
Another special element is the product γ0 γπ, which
is the product of all Clifford generators except γ0. This
operator anti-commutes with the SUM and γπ, but com-
mutes with all other generators of Cl(N − 1, 1). It can
hence distinguish between the two types of Clifford gen-
erators and part of the CPT-theorem [48].
E. The real Dirac algebra
Since the smallest system with some kind of internal
dynamics, with interaction, is composed of two DOF and
described by the Dirac algebra, it is as fundamental as
the real Pauli algebra.
The usefulness of the real Dirac algebra in classi-
cal Hamiltonian theory has been described in previous
works [56, 57]. It was shown, for instance, that a general
block-diagonalization of stable Hamiltonian matrices can
be achieved with a Jacobi type iterative algorithm: In
each step, two DOF are blockdiagonalized based on sym-
plectic similarity transformations using the real Hamilto-
nian Dirac matrices as generators [57, 78]. Hence the real
Hamiltonian Dirac algebra suffices to describe all possible
linear interactions between two Hamiltonian DOF.
In the previous section we derived the conditions for
a possible isomorphism between real matrix reps of CAs
Cl(p, q) = Cl(N − 1, 1) and Hamiltonian algebras. Note
that the size of the spinor that corresponds to Cl(N−1, 1)
is 2 n = 2N/2
so that Cl(9, 1) corresponds to a spinor
of size 2 n = 25
= 32 and an algebra with 2N
= 1024
elements, n(2n + 1) = 528 Hamiltonian and 496 skew-
Hamiltonian elements. These numbers alone clearly in-
dicate that Cl(9, 1) can not represent the simplest pos-
sible RPO. But there are more reasons why a RPO
must be composed of two DOF, which are discussed else-
where [33, 48]. Here we restrict us to a short summary of
the main points: we stress again that the Dirac algebra
with 4 × 4-matrices is the minimal size required to repre-
sent the general case of complex eigenvalues. But there is
no fundamental reason for nature to exclude those types
of dynamical processes that require, maybe for a short
time, complex eigenvalues; they belong to the full scope
of possibilities.
Secondly, if the number of variables in the spinor is
supposed to correspond to the number of variables rep-
resenting the RPO, then
2 n = N = 2N/2
(61)
which has only two solutions, namely N = 2 or N =
4. And thirdly, as we shall elaborate now, the system
of Clifford generators should determine the structure of
the algebra and hence provide the basic web of physical
notions uniquely, without ambiguity.
Regarding the real Dirac algebra Cl(3, 1), one has
the following unique elements: The SUM γ0, the pseu-
doscalar (Eq. 60) γπ = γ0γ1γ2γ3 and the product of both
(γ10).
While there is only a single skew-symmetric element
in the real Pauli algebra, the Dirac algebra contains six
of them. By Pauli’s fundamental theorem of the Dirac
matrices [68] it is allowed to select any of the skew-
symmetric matrices to represent the SUM γ0. Above
we have chosen the form 12 ⊗ η0. If one choses to use a
different skew-symmetric matrix as SUM, this is equiva-
lent to a permutation of the order of the elements in ψ.
One obtains ψ = (q1, q2, p1, p2)T
in case of γ0 = η0 ⊗ 12.
Hence the real Dirac matrices have their meaning relative
to the initial choice of the SUM.
Next one has to select one of nine symmetric matri-
ces 27
, however it is a math fact that γ0 anticommutes
only with six of them. Hence one has to choose again one
out of six matrices 28
and fix it as γ1. This choice is again
arbitrary insofar as all choices give the same physics [56].
But the selection of these two matrices suffices to decide
about the type of all remaining matrices, i.e. whether
they are vectors, bi-vector and so forth.
According to Eq. 31 and Eq. 32 there are 10 Hamil-
tonian elements in the Dirac algebra, but we identi-
fied only 4 of them, namely the generators (called 1-
vectors or simply vectors). In the previous section we
have shown that in HCAs in which all generators of
the Clifford algebra are Hamiltonian, only k-vectors for
k ∈ [1, 2, 5, 6, 9, 10, . . .] are Hamiltonian. Since the high-
est k-vector of the Dirac algebra is the pseudo-scalar with
k = 4, a Dirac type Hamiltonian may contain only vector
and bi-vector elements. We use γ14 = γ0γ1γ2γ3 to denote
the pseudo-scalar.
27 The unit matrix can not be a generator of a CA since it com-
mutes with all others.
28 We can’t tell if “god throws dices” or not. Here dices are an
option.

16
The first (symmetric) bi-vectors are given by
γ4 = γ0 γ1
γ5 = γ0 γ2
γ6 = γ0 γ3 ,
(62)
and the second (skew-symmetric) set by
γ7 = γ14 γ4 = γ2 γ3
γ8 = γ14 γ5 = γ3 γ1
γ9 = γ14 γ6 = γ1 γ2
(63)
The skew-Hamiltonian 3-vector elements are
γ10 = γ14 γ0 = γ1 γ2 γ3
γ11 = γ14 γ1 = γ0 γ2 γ3
γ12 = γ14 γ2 = γ0 γ3 γ1
γ13 = γ14 γ3 = γ0 γ1 γ2 .
(64)
The last element is the scalar γ15 = 1, e.g. the unit
matrix.
Any 4 × 4-matrix M can be written as a linear combi-
nation of the real Dirac matrices (RDMs):
M =
15
X
k=0
mk γk (65)
Where a sequential index k ∈ [0, . . . , 15] is used instead
of the multi-index convention γµγν.
This means that the Dirac algebra enable, as the real
Pauli algebra, for a re-parametrization of the elements
of 4 × 4-matrices, suited to symmetries relevant in ab-
stract Hamiltonian dynamics. This is usually presented
in wrong order: It is true that Dirac introduced his matri-
ces with heuristic arguments from the relativistic energy-
momentum relation. But it is not forbidden to prefer a
logical presentation instead of a historical one.
Since all RDMs besides the unit matrix are orthogonal
and have zero trace, one obtains the coefficients mk by
mk =
1
4
Tr(M γT
k ) . (66)
The general form of the symplex H that couples two de-
grees of freedom is a linear combination of ten symplices,
of 4 vectors and 6 bi-vectors:
H =
9
P
k=0
fk γk
S =
9
P
k=0
sk γk
(67)
In order to symplify the calculation one may use Eq. 42
to analyze the result of a SST (compare Eq. 27):
S(τ) = exp (γa τ) S(0) exp (−γa τ) . (68)
so that, using the convention of from Eq. 67, the sk(τ)
are functions of sk(0) and τ. According to Eq. 42 the ma-
trix exponentials yield, (hyperbolic) trigonometric func-
tions 29
.
VII. FUNCTION FOLLOWS FORM
We promised that the classical notions of mass, energy
and momentum would follow from the logic of the im-
posed dynamical constraint. Of course it is impossible to
provide a logical proof for an interpretation. Interpreta-
tions can be consistent and plausible, but not logical or
illogical.
As well known in classical mechanics, the generators of
canonical transformations correspond to physical quan-
tities. We started with a single PDCOM and apparently
this suffices to explain the emergence of 10 quantities
that may act as generators of SSTs within the Dirac al-
gebra waiting for an interpretation, four vector compo-
nents and 3 + 3 bi-vector components. The tri-vectors,
the scalar and pseudo-scalar are skew-Hamiltonian, they
do not correspond to non-zero correlations, and require
no interpretation, at least at this point.
One can construct k-vectors with even k from products
of k-vectors for k even or odd, but one can not obtain
k-vectors with odd k from products of even k-vectors:
bi-vectors can be obtained multiplicatively from vectors
but not vice versa. In other words: the even elements,
namely the 0-vectors (scalar), the six bi-vectors, and the
4-vector element (pseudoscalar) form the even subgroup.
This holds for all even-dimensional CAs, i.e. with N =
2 M, M ∈ N.
Hence the algebra forces us to distinguish between
the set of quantities associated with the vector elements
(γ0, γk) and two sets of 3 bi-vectors each, namely the
symmetric elements γ4, γ5, γ6 and the skew-symmetric el-
ements γ4, γ5, γ6.
This structure suggest to interpret vector components
as representing the particle (RPO) and bi-vector com-
ponents as fields: objects are the sources of fields, fields
act on objects. Hence, it is nearby to interpret the six
bi-vectors, 3 skew-symmetric and 3 symmetric, as gener-
ators of the Lorentz transformations. This is indeed the
case.
For a detailed account of the Lorentz transformations
as they naturally emerge from Cl(3, 1), see Refs. [53, 56,
57]. Here we just mention the result, namely that the
skew-symmetric bi-vectors γ7, γ8 and γ9 are generators
of spatial rotations while the symmetric bi-vectors γ4,
γ5 and γ6 generate Lorentz boosts in the corresponding
directions, both a mathematical consequence of Eq. 42
or Eq. 68, respectively. This interpretation is completely
29 This holds for exponentials of non-singular matrices. The gen-
eral case is described for instance in Ref. [79].

17
determined by the structure of the Hamiltonian Dirac al-
gebra and by the transformation properties of the quan-
tities under canonical transformations (SSTs).
Hence the structure of the real Dirac algebra is iso-
morphic to relativistic electrodynamics, it is allowed to
interpret the parameters accordingly. The vector param-
eters of the auto-correlation matrix S, are then identified
with energy and momentum:
s0 ≡ E
(s1, s2, s3)T
≡ ~
P (69)
and the bi-vectors with the fields:
(s4, s5, s6)T
≡ ~
E
(s7, s8, s9)T
≡ ~
B (70)
Due to the fact that the electromagnetic fields appear
in this context as based on pure dynamical notions, we
named this interpretation the electro-mechanical equiv-
alence (EMEQ) in preceeding papers Ref. [56, 57]. In
App. E 2 it is shown that this interpretation is consistent
with both, the Dirac equation as well as with Maxwell’s
equations.
We write the vector quantities as a matrix
P = E γ0 + ~
p · ~
γ (71)
and the fields in a second matrix:
F = γ0
~
E · ~
γ + γ14 γ0
~
B · ~
γ (72)
where the notation using the dot “·” for the scalar prod-
uct is purely formal. Using the pseudo-scalar this can be
written as
P = (S + γ14 S γ14)/2
F = (S − γ14 S γ14)/2
(73)
Since both, the density matrix and the Hamiltonian ma-
trix have the same structure, the EMEQ applies to both.
In isolated equilibrium systems, the Hamiltonian ma-
trix H can only be a function of quantities produced by
the RPO itself: H = f(S). Consider that f is analytical
so that it can be written as a Taylor series 30
. Obvi-
ously then H and S commute and we find from Eq. 17
that Ṡ = 0. Even though the spinor oscillates and has
eigenfrequencies, the observables are ensemble properties
which are in this case static.
Hence, in order to obtain observable change from
Eq. 17, we must add some external Hamiltonian Hx:
Ṡ = (H + Hx) S − S (H + Hx)
= Hx S − S Hx
(74)
30 Then, since H and S are Hamiltonian, only odd terms can
contribute, since only odd powers of a Hamiltonian matrix are
Hamiltonian.
Hence, self-interaction is, within this linear approxima-
tion, unobservable.
Then, given the RPO is in interaction with external
fields, one obtains:
Ṗ + Ḟ = Fx (P + F) − (P + F) Fx
= Fx P − P Fx + Fx F − F Fx
(75)
It follows from the commutator table of the Dirac alge-
bra, that this can be splitted into:
Ṗ = Fx P − P Fx , (76)
which is the Lorentz force equation as we shall show in
Sec. VII D and secondly
Ḟ = Fx F − F Fx , (77)
which describes spin precession (see Sec. VII D below).
A. Units: The Schwinger Limiting Fields
In Eq. 67 different physical quantities like electromag-
netic fields, energy and momentum are added. This is
allowed if one uses appropriate natural units. Modern
physics identified a number of scaling factors, namely the
“speed of light” c for the scale between mass, energy and
momentum and between electric and magnetic fields, ~
for the scale between energy and frequency and the unit
charge to scale fields relative to mechanical quantites.
A detailed account of how physical constants are under-
stood has been given in Ref. [10].
As we shall show below, according to the EMEQ, the
eigenvalues of the Hamiltonian matrix correspond to the
mass of a particle. Hence, if the RPO has the mass of the
electron me, then this scales the electromagnetic fields
automatically relative to the so-called Schwinger limiting
fields ES and BS, which were first derived by Sauter [80–
82] ES and BS
31
. Hence the scaling factor between a
magnetic field B in SI-units and in units of frequency
is of the order e
m and for electric fields E of order e
m c .
The fields as they appear here, are scaled relative to the
properties of the RPO, e and m.
B. The Eigenvalues of Dirac Hamiltonian
Let us first have a look at the eigenvalues of the Hamil-
tonian matrix “operator” that follows the parametriza-
tions Eq. 69 and Eq. 70, separately and combined. The
trace of a matrix equals the sum of its eigenvalues, the
31 These fields are, given in SI-units: ES = m2
c3
e ~
= 1.323 ·
1018 V/m and BS = m2 c2
e ~
= 4.414 · 109 T. These values are be-
yond any technical scale. Only the largest modern pulsed lasers
might allow to generate fields of this strength [83].

18
trace of the squared matrix equals the sum of the squared
eigenvalues and so on. From Eq. 23 we know that the
trace of any odd power of some Hamiltonian matrix van-
ishes. Hence only even powers are left, i.e. the second
and fourth power:
Tr(H2
) =
P
k
λ2
k
Tr(H4
) =
P
k
λ4
k
(78)
which allows to compute the eigenfrequencies [33, 57, 79].
The result is given by:
K1 = −Tr(H2
)/4
K2 = Tr(H4
)/16 − K2
1 /4
ω1 =
p
K1 + 2
√
K2
ω2 =
p
K1 − 2
√
K2
ω2
1 ω2
2 = K2
1 − 4 K2 = Det(H)
K1 = E2
+ ~
B2
− ~
E2
− ~
P2
K2 = (E ~
B + ~
E × ~
P)2
− ( ~
E · ~
B)2
− (~
P · ~
B)2
(79)
Hamiltonian matrices of stable systems have purely imag-
inary eigenvalues, corresponding to real frequencies ωi, so
that for stable systems one has K2 0 and K1 2
√
K2.
From this we find that K2 = 0 when ~
E = ~
B = 0, i.e.
for pure vectors (Eq. 69)
ω = ±
q
E2 − ~
P2 , (80)
and for pure bi-vectors, where E = 0 = ~
P, the frequencies
are (Eq. 70)
ω = ±
r
~
B2 − ~
E2 ± 2
q
−( ~
E · ~
B)2 . (81)
The frequencies are invariants under SSTs and hence are
Lorentz scalars, i.e. invariant quantities. We therefore
know that pure bi-vectors have two relativistic invari-
ants, namely ~
B2
− ~
E2
and ~
E · ~
B and we know this with-
out any reference to Maxwell’s equations. Furthermore
we directly know that a stable bi-vector type oscillation
is only possible if ~
E · ~
B = 0, since only under this condi-
tion one obtains real frequencies (aka purely imaginary
eigenvalues) 32
.
In the theory of electromagnetic wave propagation one
finds that ~
B2
− ~
E2
= 0 so that, if this is inserted into
Eq. 81, apparently electromagnetic fields have no eigen-
frequency. This is generally known to be true, it nev-
ertheless leads, in our approach, to a degenerate matrix
H. This can be understood if we consider the frequency
32 Note that, if one uses the metric of Cl(1, 3) instead of Cl(3, 1),
the terms representing e.m. fields, receive a factor i and the signs
of the squares are reversed.
of the RPO Eq. 80, which apparently provides a con-
stant and invariant frequency, which equals the mass of
the RPO. As is well-known, the frequency of the Dirac
spinor that describes a particle at rest is (up to constant
scaling factors ~ and c) identical to the mass. Hence
the time variable τ must be identified with proper time,
the time of a co-moving observer. Then it is clear why
the electromagnetic bi-vector has a vanishing frequency:
electromagnetic waves, regarded from the perspective of
a (hypothetical) comoving observer, are indeed static.
Expressed in language of special relativity we would say
that we can not transform into the co-moving frame of
an electromagnetic wave, as it moves with the speed of
light. However, this requires no commandment concern-
ing space-time, but is a math fact about boost transfor-
mations.
C. Special Relativity in a Nutshell
The analysis of the Dirac algebra leads to the in-
sight that an RPO is essentially described by the vector
type elements that are associated with the 4-momentum
(E, ~
p):
H = E γ0 + γ1 px + γ2 py + γ3 pz (82)
The square of this matrix is
H2
= −E2
+ ~
p2
= −m2
(83)
so that
ψ̈ = H2
ψ = −m2
ψ , (84)
i.e. the mass m is proportional to the oscillation fre-
quency, an eigenvalue of H. It is a constant of motion
and a scalar, a 0-vector of the Clifford algebra. Nonethe-
less we have no unique state of affairs of the RPO. We
just selected a “mass shell”. The structure of the Dirac
algebra given in the previous section suggests that the
RPO as described by Eq. 82 is not in interaction, it is a
free “particle”.
Is it possible not only to formally derive rotations and
boosts, i.e. Lorentz transformations (Ref. [53]), but also
a space-time interpretation? This requires to switch to
the next level of description, to use another emergent
constant of motion as Hamiltonian. The original Hamil-
tonian described the motion of the spinor ψ. But spinors
are not directly measurable. In order to (re-) construct
classical physics, we need a relation between observables
in which the mass is just a constant “parameter”. It
is nearby to reinterpret the equations of motion for ob-
servables (Eq. 17) in a Hamiltonian context. Then one
obtains the “classical” (relativistic) Hamiltonian of a free
RPO. The only unique choice for the Hamiltonian of the
RPO is the parameter E, which then depends on the vec-
tor components of the momentum. Hence one obtains the
classical relativistic energy-momentum-relation (EMR):
E = H(~
p) =
p
m2 + ~
p2 . (85)

19
The use of Hamilton’s equation of motion for the velocity
β is then given by
~
β = ˙
~
q = ~
∇~
pH(~
p) =
~
p
E
(86)
so that inserting the result into Eq. 85 yields
E = γ m (87)
where γ ≡ 1
√
1−~
β2
and
~
p = m γ~
β (88)
Hence, within our approach, it is just another math fact
that the velocity |~
β| is limited to 1. This is a property of
space-time valid by construction. It is not a consequence
of the constancy of the speed of light. Both, a maximal
speed for massive objects as well as the Lorentz transfor-
mations, emerge from the same Hamiltonian formalism.
Even the very concept of “speed” itself can be regarded
as a result of this formalism.
There is no commandment and no a priori exis-
tent space-time required that determines the energy-
momentum relation, but vice versa: Minkowski space-
time emerges as a consequence of the energy-momentum
relation (EMR), which is itself a consequence of classical
Hamiltonian theory and the algebra of proper time.
Since |~
β| ≤ 1, one may write β = tanh (ε), where ε
is the so-called “rapidity”, and then one obtains γ =
cosh (ε), E = m cosh (ε) and p = m sinh (ε) [53].
Our approach is based on the Lorentz transformations
and is therefore mathematically equivalent with the stan-
dard presentation of relativity theory (SPRT). Nonethe-
less it modifies the SPRT insofar as both, Lorentz trans-
formations and “inertial frames” are notions that require
no direct reference to space-time at all. We introduced
and explained them in a purely Hamiltonian context.
This came out almost automatically and we could not
possibly have done otherwise. But relativity is not the
central issue of this article and hence we can not elab-
orate in more detail. We refer to the introduction in
Ref. [84].
D. The Lorentz Force
The three skew-symmetric Hamiltonian elements
f7, f8, f9 act as generators of rotations in a 3-dimensional
parameter space and are therefore gyroscopic quantities
as for instance magnetic fields or angular momenta. The
three symmetric elements f4, f5, f6 act as generators of
boosts in a 3 + 1-dimensional parameter space and are
hence associated with a linear accelerating quantity like
the electric field. In other words, the parametrization by
the use of Clifford algebras uncovers a unique structure
and establishes certain transformation classes for other-
wise uninterpretable elements of the Hamiltonian matri-
ces S and H. If the real physical object is represented
by the second moments of S, then the matrix H contains
the driving terms of the symplectic motion, which must
then be called (self-) fields.
Eq. 76 written explicitely using the EMEQ, yields:
Ė = ~
P · ~
E
˙
~
P = E ~
E + ~
P × ~
B (89)
which are the Lorentz force equations formulated in
proper time τ [33, 48, 53, 57]. With d
dτ = γ d
dt one finds
(assuming c = 1):
γ dE
dt = m γ ~
v · ~
E
γ d ~
P
dt = m γ ~
E + m γ ~
v × ~
B
(90)
and hence
dE
dt = m~
v · ~
E
d ~
P
dt = m ~
E + m~
v × ~
B
(91)
As explained in Sec. VII A also the bi-vector fields, like
all elements of H, have a unit of frequency and require a
re-scaling by e/m to obtain their values in SI units 33
:
dE
dt = q ~
v · ~
E
d ~
P
dt = q ~
E + q ~
v × ~
B (92)
The second part, Eq. 77, describes the precession of
the remaining correlations. We denote the internal bi-
vectors by ~
a and ~
s to distinguish them from the external
fields ~
E and ~
B, so that
F = γ0 ~
a · ~
γ + γ14 γ0 ~
s · ~
γ , (93)
and one obtains (Eq. 77):
˙
~
a = ~
a × ~
B + ~
s × ~
E
˙
~
s = −~
a × ~
E + ~
s × ~
B (94)
If one uses a complex notation ~
σ = ~
s + i~
a and ~
F =
~
B + i ~
E, then
˙
~
σ = ~
σ × ~
F (95)
which is the equation that describes the precession of the
spin in an external field.
33 As we argued in Ref. [10], scaling factors like these can not be
derived logically since they depend on a historical, hence arbi-
trary, choice of units. They have to be introduced in an ad hoc
fashion if equations are to be aligned to the MKS system. See
also Sec. VII.

20
E. Electromagnetic Waves and Spin (-flips)
The Lorentz force that we derived from Eq. 17 using
the EMEQ, refers to static, or at least slowly varying,
electromagnetic fields. The relevant frequency scale is
given by the mass of the RPO, i.e. 0.511 MeV in case of
electrons. Visible light belongs to frequencies of order of
eV, i.e. several orders of magnitude below the typical de
Broglie frequency of electron waves. Still, the wave length
of visible light is in the order of nm, while technical fields,
for instance in undulators or spectrometer magnets, vary
with macroscopic “wave-length”, i.e. order of mm up
to m, again several orders of magnitude larger than the
wavelength of light. Hence it is legitimate to assume that
the variation of the fields is slow.
How do we treat the case in which (the electromag-
netic part of) H varies, slowly compared to the de Broglie
wave? Of course this depends on the type of variation.
For electro-magnetic waves, we know from Maxwell’s
equations 34
that polarized e.m. waves, as seen can be
described by a rotating “Dreibein”. If ~
E(τ = 0) = E ~
ex
and ~
B(τ = 0) = B ~
ey, then, for an observer in some
“inertial reference frame”, these vectors rotate with fre-
quency Ω around the z-axis. The generator for rotations
around the z-axis is γ9. Hence the time dependency can
be written, according to Eq. 17 as 35
Ḟ =
Ω
2
(γ9 F − F γ9) . (96)
It is then possible (see App. B) to represent the time
dependency of F by adding the term Ω/2 γ9 to F, effec-
tively the same as a magnetic field component Bz = Ω/2.
Hence a circular polarized electromagnetic wave can be
described in this approach by effective electromagnetic
terms which give ~
E · ~
B = 0 and ~
B2
− ~
E2
= Ω2
/4, i.e.
with an additional energy term that is proportional to
frequency Ω/2 (times ~, in MKS-units), in agreement
with Eq. 81.
Assuming that we describe the RPO in its “rest
frame”, then ~
P = 0 and the eigenfrequencies, given by
Eq. 79, are:
K1 = E2
+ B2
+ Ω2
/4
K2 =
p
E2 ~
B2
= E2
(B2
+ Ω2
/4)
ω = ±
q
E2 + B2 + Ω2/4 ± 2 E
p
B2 + Ω2/4
= ±
q
(E ±
p
B2 + Ω2/4)2
(97)
so that with ω = m = const, assumed here to be positive,
one finds two possible eigen-frequencies
m = E ±
p
B2 + Ω2/4 , (98)
34 We have shown how to derive Maxwell’s equations on the basis
of this approach in Ref. [48].
35 The factor 1
2
is required to generate a spatial rotation frequency
Ω and is a peculiarity of spinors [53].
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
B (a.u.)
E−m
(a.u.)
|1i
|2i
|1i |2i
∆E(0) = Ω
FIG. 1. Finestructure (energy levels) of an RPO in the
presence of a static magnetic field B and a circular polarized
electromagnetic wave of frequency Ω.
so that
E − m = ∆E/2 = ±
p
B2 + Ω2/4 . (99)
Even if one can estimate that the amplitude B of the
magnetic part of the wave is small B2
≪ Ω2
and can
usually be neglected, it is of course possible to increase
this field component by some external static field with-
out any other change in the calculation. Then the fine-
splitting of the energy in dependence of B and Ω is shown
in Fig. 1. Eqn. 99 is for instance used to describe the
mechanism of adiabatic RF transitions between spin-
states [85]; as shown in Fig. 1, if the field B is slowly
raised from negative to positive values, the rf-field, indi-
cated by Ω causes, using “quantum” language, a mixture
of the “pure” states |1i and |2i and a level splitting Ω.
When the field, as indicated by the arrows, is raised even
further, an object that was originally in state |1i is flipped
into state |2i and vice versa.
This example provides further insights. Firstly, en-
ergy and frequency of electromagnetic fields have, on the
basic level, the exact same meaning and secondly, the
two oscillator states are mixed around B = 0 and it
is thus thinkable to explain un-quantum leaps by res-
onances [75]. Furthermore, without any reference to
Hilbert spaces or “quantum principles”, we found sep-
arate energy levels which are usually postulated to be
pure “quantum” effects. Everything so far suggests that
“quantization” indeed is an eigenvalue problem as sug-
gested by Schrödinger [26], and can therefore be described
by classical notions.
It is sometimes claimed that discrete eigenfrequencies
or the “superposition” of different states of motion with
different frequencies would be weird and could not be
understood classically [86]. This is wrong. A classical
Hamiltonian system with two DOF has in general two

21
eigenfrequencies and two (pairs of complex conjugate)
eigenvectors. The general state of motion is a super-
position of these eigenvectors. This is purely classical
Hamiltonian physics.
VIII. “CANONICAL” (UN-) QUANTIZATION
We did not yet show how to obtain the so-called canon-
ical quantization, i.e. why a spatial derivative represents
the momentum operator and a time derivative an energy
operator. This requires to use a second method to con-
struct space-time, “complementary” to the one used in
Sec. VII C.
Again this requires almost zero steps. We introduced
the phase space density ρ and the matrix of second mo-
ments Σ of this density. It is nearby and well-known in
the theory of probability distributions to use the Fourier
transform to represent the moments of a distribution.
In Sec. VII C we introduced velocities (and hence space-
coordinates by the option to integrate the velocity β over
time) by the eigenvalue equation. The Fourier transform
is mathematically rigorous and directly yields the me-
chanics of waves. However it requires that the phase
space density and the spinor are functions of energy and
momentum.
The phase space density ρ(H) of some stable state de-
pends on the constant parameters E and p, that describe
a particle (RPO) so that ρ(H) = ρ(E, p). Equivalently,
the spinor ψ is, in case of stable oscillations, also a func-
tion of energy and momentum. Hence we define the four-
component spinor Ψ = Ψ(E, p) = ψ
√
ρ so that the matrix
of second moments can be written as 36
:
Σ =
Z
ψ ψT
ρ(ψ) d4
ψ =
Z
Ψ ΨT
d4
ψ , (100)
which is the so-called “density matrix” in the SPQM.
Hence the spinor Ψ is square integrable and therefore
has a Fourier transform, which can be written as
Ψ̃(t, ~
x) ∝
Z
Ψ(E, ~
p) exp (−i E t + i ~
p · ~
x)) d4
p . (101)
This requires no postulate. It is just the Fourier trans-
form of a phase space function and as such not quantum,
so that also the operator rule, the so-called “canonical
quantization”,
hΨ̃†
(t, ~
x) E Ψ(t, ~
x)i = hΨ†
(t, ~
x) i ∂t Ψ(t, ~
x)i
hΨ̃†
(t, ~
x) ~
p Ψ(t, ~
x)i = −hΨ†
(t, ~
x) i ~
∇ Ψ(t, ~
x)i
(102)
often shortly written as
E = i ∂t
~
p = −i ~
∇ (103)
36 See also Ref. [87].
is not quantum. It is but a special way to compute av-
erages, aka statistical mechanics. By construction, the
parametric space-time, represented by t and ~
x, matches
to the framework of SSTs that has been developed, if t
and ~
x are vector components in a Dirac algebra. In this
case, the phase of the Fourier transform is an invariant
quantity, i.e. a scalar.
Inspection of the (anti-) commutator tables of the
Dirac algebra [56] shows, that, if P = Eγ0 + ~
p · ~
γ is a
vector and X = t γ0 + ~
x · ~
γ is also a vector, then the
anticommutator (P X + X P)/2 is a scalar. Hence the
anticommutator is generalization of the scalar (“inner”)
product. The commutator is, no big surprise, a general-
ized vector (“outer”) product, which suggests the follow-
ing convention:
P · X ≡ (P X + X P)/2 = (−E t + ~
x · ~
p) 1
P ∧ X ≡ (P X − X P)/2 = (~
p × ~
x) · (γ14γ0~
γ)
P X = P · X + P ∧ X
(104)
This is just a matter of convenient notation.
The “physical” space, defined this way, is no more the
reified fundamental container of everything as in New-
tonian physics, but is recognized, as it should be, as a
non-entity [88].
However, the Fourier transform requires that energy E
and momentum p are real-valued, which excludes reso-
nances. Even if we can not elaborate here in detail, but
the use of the Fourier transform which allows to obtain a
spatio-temporal image of the phase space process, seems
unproblematic only in specific circumstances, namely in
eigenstates of the energy.
The general autocorrelation matrix requires 10 param-
eters, while the Fourier transform uses only 4 and and
hence ignores spin. Hence there are variables and corre-
lations that have no spatio-temporal “location”.
Furthermore, Eq. 69 defines energy and momentum as
(linear combinations of) second moments of ψ, i.e. energy
and momentum depend on ψ, while the Fourier trans-
form is formulated as if ψ was a function of energy and
momentum: the dependency is reversed. However, the
dependency is not (always) bijectiv. The eigen-spinors
of free Dirac particles are in many textbooks expressed
as functions of (E, p), but this is not the case if one uses
the corresponding eigenstates. Or, in other words, there
are very likely “loopholes” and it is not far-fetched to as-
sume that these might allow to explain the findings that
required to introduce the projection postulate. But in
any case it is clear that the most “mysterious” features
of QM are, if locality is not presumed to be fundamental,
merely technical or mathematical issues. In our presenta-
tion they neither suggest nor do they suffice to establish
a philosophy of unreality.
Space-time geometry and electromagnetism carry the
signature of the simplest possible description of Hamil-
tonian interaction. This can be taken literally: as shown
in Ref. [56, 57], the parametrization of a general 4 × 4
Hamiltonian matrix that describes the coupling of two

22
DOF by the use of Dirac matrices and the EMEQ allows
a straightforward analysis and transformation to normal
forms. Hence the physical notions of the Dirac alge-
bra, provide the mathematical means to solve the general
problem of diagonalizing Hamiltonian matrices. This is
remarkable insofar as usually one expects that math is
used to solve physical problems and not vice versa.
If physical notions are useful to solve a general math
problem, then the two are isomorphic. All possible terms
that are allowed by the physicality constraint are param-
eters of the Hamitonian matrix and have physical signif-
icance. Math and physics are isomorphic, the theory is
saturated.
The Dirac electron is described by a wave-function,
i.e. by a charge distribution. In App. E 2 it is shown
that the “fields” generated by the Dirac current density
obey Maxwell’s equations. Hence this picture is hence
self-consistent. The difference to the “classical” picture
of a 4-current density is a matter of the order of the
presentation.
Classical metaphysics presumes that space-time is fun-
damental. A charge distribution can hence be split into
infinitesimal parts that are distinguishable by their po-
sitions in space. Then these fractions should be able to
move independently and, according to the Lorentz force,
the parts must repel each other. Hence they can not
give rise to some stable distribution. Classical (meta-)
physics escaped by postulating point charges. This how-
ever implies infinite self-energy so that electromagnetism
and space-based physics has a renormalization problem,
even if this is rarely explicitely mentioned.
But as we have shown, if the “classical” metaphysical
presumptions are dropped, the Lorentz force naturally
emerges, but it is a force relevant for statistical averages,
namely second moments. It does not refer to the motion
of single phase space points, but to changes of second
moments of phase space distributions under the influence
of external fields. The math used is still classical, but the
presumption of space as a fundamental notion has been
removed.
In our presentation, classicality is based on a logico-
mathematical and not to a metaphysical framework: not
space but real physical objects (RPOs) are fundamen-
tal. Without having established objects first, we can not
meaningfully refer to spatial notions. It would be in-
consistent to cut an RPO into different “parts” and to
locate them independently in space-time. It makes even
less sense to presume that they repel each other by a force
that can only be established by the second moments of
the complete RPOs phase-space ensemble. This demon-
strates that un-quantum physics can not be consistently
understood, as long as space is regarded as fundamental
and this is the core of “how to” un-quantum mechanics:
to accept that space, though real, is not fundamental.
Furthermore, the description of an electron, which is
mostly defined by its talent for electromagnetic interac-
tion, must somehow provide the mathematical means to
explain this talent. The standard presentation simply
postulates that electrons “carry” charge. If we aim to
avoid commandments, this is not satisfying. In our pre-
sentation, the talent for electromagnetic interaction is
provided by the bi-vector elements of the Hamiltonian
matrix parameterized by the Dirac algebra.
A. Uncertainty Relations
Heisenberg’s “uncertainty relations” are, depending on
the presentation, a consequence of Eq. 103 or of the
Fourier transform. They describe yet another math
fact [75]: A certain width of a distribution in momen-
tum space determines a minimal width of the Fourier-
transformed distribution in physical space and vice versa.
And yet again, since it is a math fact about Fourier trans-
forms, it requires no postulates and is therefore not quan-
tum.
Griffith writes: “This principle is often discussed in
terms of measurements of a particle’s position or mo-
mentum, and the difficulty of simultaneously measuring
both of these quantities. While such discussions are not
without merit [...] – they tend to put the emphasis in
the wrong place, suggesting that the inequality some-
how arises out of peculiarities associated with measure-
ments.” [89]. The so-called “uncertainty relations” are
not due to measurement uncertainties. Nowhere in the
derivation of the uncertainty relations is it required to re-
fer to “distortions by measurements”. There is not even
an intrinsic need to speak of “uncertainties” at all, as
the width of a distributions is not uncertain. It is just a
width.
B. The Born Rule
We still have to analyze Born’s Rule, namely that
Ψ†
(~
x, t)Ψ(~
x, t) can be regarded as a probability density
to “find” a particle at position ~
x and time t [90]. Or,
in other words, the transfer of a phase space density
ρ into the parametric space-time, might appear some-
what unclear, since ρ and not ΨT
Ψ = ψT
ψ ρ is - in our
presentation - a density in phase space. So why should
the latter now be a valid density in space-time? This
question is legitimate and nearby. It has a simple an-
swer: In agreement with (the mathematical principles
of) spatio-temporal logic, the continuity equation, which
follows from Dirac’s equation 37
, suffices to validate the
consistency of the spatio-temporal “image” and Born’s
rule. Not more and not less. It is not in the scope of the
continuity equation to guarantee the absence of other,
non-spatial, correlations between objects.
The mathematical correctness of an image, however,
does not change the fact that the true postal address
37 See App. E.

23
of RPOs is phase space. This means that we can take
the mathematical form of QM seriously: The squared
“amplitude” merely generates the mathematical form of
a substance inhabiting space-time. What we perceive
and measure in space-time, is a Fourier spectrum. The
Fourier transform is a reversible unitary transformation,
an isomorphism with respect to the transformed prop-
erties: Constraints imposed in either space have conse-
quences in the other. This justifies to regard the image
appearing in space-time as real, but it does not suffice
to make it fundamental. In our presentation, the spatio-
temporal image of an RPO is like an avatar.
It is sometimes claimed that QM and specifically the
Born rules requires some kind of non-classical probability
theory. Leon Cohen has shown that this is just another
myth [91]. We found no convincing reason to think oth-
erwise.
C. Space-Time: The Arena of Avatars
As explained in the previous section, Minkowski’s
space-time is, from the perspective of a single RPO, a
kind of a holographic screen, it is less fundamental than
supposed “classically” - nonetheless it is more than a
mere image of reality. For many physicists it seems sim-
ply not imaginable to weaken local realism 38
, but from a
logical perspective a merely commandment-based space-
time concept is scientifically unsatisfactory. Even Ein-
stein, in later years, wrote: “Spacetime does not claim
existence on its own, but only as a structural quality of
the field” [93].
Also the experimental tests of Bell’s theorem suggest
that nature is on the fundamental level non-local [94].
As Maudlin expressed it: “What Bell proved, and what
theoretical physics has not yet properly absorbed, is that
the physical world itself is non-local” [95].
De Haro and de Regt argued that physical theories
without the primary assumption of space-time are indeed
able, contrary to other claims, to “provide scientific un-
derstanding” [96], though they are - of course - difficult
to visualize.
Mariani and Truini suggested exceptional Lie algebras
might be at the “foundations of space and time”. They
suggest the basic principle that “there is no way of defin-
ing spacetime without a prelimenary concept of interac-
tion” [97]. We agree with them concerning the emergence
of space-time from interaction, but we doubt it requires
a principle. In this work, such a principle does not ap-
pear, since the definition of physicality did not require
to specify “where” some object is. The question “where”
38 Einstein wrote 1947 in a letter to Born: “I cannot seriously
believe in it [quantum mechanics] because the theory cannot be
reconciled with the idea that physics should represent a reality
in time and space, free from spooky actions at a distance.” [92]
a thing “is”, can only have relevance in the presence of
other objects, i.e. by inter-action.
Modern physics invented the notion of background in-
dependence: “[...] a classical field theory is background-
independent if the structure required to make sense of
its equations is itself subject to dynamical evolution,
rather than being imposed ab initio. [...] a theory is
fully background-independent relative to an interpreta-
tion if each physical possibility corresponds to a distinct
spacetime geometry; and it falls short of full background-
independence to the extent that this condition fails” [98].
Regarding these criteria we think that our physico-logical
approach is, though “classical”, fully background inde-
pendent. It directly and inevitably leads to the simplest
possible physical objects and the first order interaction
of these generate the 3+1 dimensional parameter space,
that human inhabitants of the constructed physical world
call space-time.
The Lorentz transformations are not primarily re-
quired to describe “coordinate” transformations between
“inertial reference frames”, they are (also) active canon-
ical transformations, changing the physical state of the
system under consideration. The issues that many - also
renowned - physicists had with special relativity [99],
might also be due to an unclear attitude (of Einstein,
but also others) towards the ontological status of space-
time. Though Einstein suggested in his theory of special
relativity that the assumption of a material substance,
an aether, is dispensible, he was not able or willing to
dispense the hegemony of the Newtonian heritage of ab-
solute space 39
.
According to that view it does not suffice to formulate
a theory that allows for the derivation of geometrical no-
tions, aka an emergent space-time, but spatio-temporal
notions must inevitably be the most fundamental ones.
This philosophy might be called space-time fundamen-
talism (STF) and Einstein frequently, but not always,
appeared as a proponent of STF. STF almost requires a
reification of space, or space-time, respectively.
Mermin warned us that the reification of mathematical
abstractions used in physics is a “bad habit” [102]. He ar-
gues mainly from a pragmatic point of view (which is wel-
come). However, viewed pragmatically: Does the dogma
of unreality of wave-functions enable students to under-
stand QM or does it lead to unnecessary confusion? Is it
required and justified to claim that “the ”orbit” is created
by the fact that we observe it” as Heisenberg claimed 40
?
Do we really need to accept an interpretation that ques-
tions object permanence, despite the fact that it is the
fundament of physicality. Does it at least correspond to
any practice in physics? Do accelerator physicists pro-
vide any measure to “observe” particle beams in order to
39 For Einstein, local causality was a fundamental requirement.
This (and not determinism) is the core of his concerns regarding
QM [100, 101].
40 “Die ”Bahn” entsteht erst dadurch, dass wir sie
beobachten” [103].

24
establish the existence of particle orbits? Of course they
don’t. Of course there is no need to do so. As far as we
can tell, orbits are established by electro-magnetic fields,
not by observation.
We believe that Heisenberg’s philosophy of unreality
went beyond any reasonable requirement and his ar-
guments suspiciously oscillate between the “uncertainty
principle”, commutators of conjugate pairs and consid-
erations about the limitations of measurement precision,
enriched with claims of positivistic nature [103]. What
he did, is exactly what Mermin (should have) criticized:
He overrated mathematical abstractions, not in support
of reification, but of un-reification. Heisenberg scrambled
the quantum omelette with ingredients of a theory that
wasn’t finalized or established yet. And this is certainly
a bad habit.
It is rarely emphasized, but the most important “quan-
tum” effects are macroscopic and not microscopic: The
stability of matter, the properties of thermal radia-
tion, ferro-magnetism, the properties of chemical bonds,
metallic states (Fermi surfaces), superconductivity, su-
perfluidity and so on. If you carefully think it through,
it is far more difficult to find examples of physical systems
where “quantum” effects can be safely ignored than oth-
erwise. Nonetheless students are taught that quantum
effects are specifically microscopic and somehow weird.
But again: it would be completely wrong to conclude
that emergence per se implies a questionable reality sta-
tus. Temperature is an emergent notion and we think
there is consensus that temperature is real. Chemistry
emerges from solutions of the Schrödinger equation, i.e.
from wave-functions and orbitals. This alone does not
suggest that chemical bonds are not real. Thus, even if
our presentation of classical un-quantum mechanics re-
fuses to regard space-time as apriori given, this does not
mean that we regard space-time as being less real. We
just regard it as less fundamental.
Many, maybe most, of the alleged mysteries of QM
have been debunked before [75, 104], or their non-
classicality has been critically reviewed [32]. We went
beyond a mere critic of the standard approach: as we
have shown there is little in the mathematical formalism
of quantum theory that can not be obtained from classi-
cal Hamiltonian mechanics. We stress again that classi-
cality is often misrepresented as some kind of STF: If the
objective is to describe motion, then, according to STF it
must be motion in space and the classical canonical vari-
ables have to be understood literally as space-time coor-
dinates and momenta. But this is not part of the classical
physical but part of metaphysical presuppositions.
Classical analytical mechanics is but a mathematical
framework. The applicability of the notion of generalized
coordinates in the sense of dynamical variables in an ab-
stract phase space is, in principle, unlimited. Abstract
variables like those forming the wave function have not
been invented by QM. We explained what exactly distin-
guishes wave functions them from observables and why
this is a consequence of fundamentality, using classical
logic.
Though the Hamiltonian formalism has, with respect
to Eq. 9, a perfect (skew-) symmetry between coordi-
nates and momenta, the only classical Hamiltonian that
accounts for this symmetry, is the harmonic oscillator.
In the Hamiltonian of a “classical” free particle, only
the momenta appear, but there is no classical system,
in which only coordinates are used. The derivation of
Compton’s scattering formula demonstrates the irrele-
vance of coordinates in the actual treatment. It requires
only the EMR, energy and momentum conservation and
the de Broglie scaling relations E = ~ ω and ~
p = ~~
k for
it’s derivation. It is the same picture in many branches of
physics: Positions obtain their physical relevance exclu-
sively from fields, i.e. from interactions, while the con-
straints that allow to draw physical conclusions and to
make real calculations, are derived in energy-momentum-
space. It is only the size of the human visual cortex that
underlies the human preference for spatio-temporal (i.e.
geometrical) notions.
There is, within the standard presentation of classical
physics, no explanation why the symmetry of coordinates
and momenta should be broken. Max Born wrote: “This
lack of symmetry seems to me very strange and rather
improbable. There is strong formal evidence for the hy-
pothesis, which I have called the principle of reciprocity,
that the laws of nature are symmetrical with regard to
space-time and momentum-energy [...]” [105]. Born was
absolutely right, but with respect to the fundamental
level of Hamiltonian theory, namely the wavefunction.
We explained why and how the Dirac algebra breaks this
symmetry between spatial coordinates and mechanical
momenta. Born spells out what we mentioned above:
the principle of sufficient reason. And his intuition was
correct, but for spinorial phase space: here the (skew-)
symmetry of the canonical pair is fully valid. But yet
again, this requires no independent principle: the PSR
completely suffices.
We started without any specific assumptions about any
“background”. We did not even ask for it in the first
place. The structure of space-time is, in our presenta-
tion, a consequence of the unique properties of the Hamil-
tonian Dirac algebra. Our presentation of un-quantum
mechanics is entirely based on simple math facts and
straightforward logic. By definition it holds in the most
general physical world. Hence, without commandments,
the celebrated conjecture that many different physical
worlds should be possible [12], looses much of it’s plausi-
bility.
The intrinsic non-locality of un-quantum mechanics ex-
plains why it makes only limited sense to ask where an
electron or a photon “really” is in space: The electron
itself is not located at some specific position in space at
all. Because physical ontology is not primarily defined by
spatial notions, it is meaningless to ask if it can simulta-
neously “be” at different positions. Surely it can, since
projected into space-time, the electron has no definite
location, but “is” a wave. However, the location of the

25
constraining principles that define its dynamical charac-
ter, is energy-momentum space. This energy-momentum
space emerges from (auto-) correlations that originate
elsewhere: in an abstract classical phase space.
IX. PROSPECTS: HIGHER DIMENSIONS
The distinction between Hamiltonian and skew-
Hamiltonian elements and the fact that skew-
Hamiltonian elements have always, by definition,
vanishing expectation values, could be useful to discuss
higher dimensional “spaces” as well. As mentioned
above, our approach suggests to consider not only the
Pauli and Dirac algebras alone, but two series of HCAs
given in Tab. I.
Two CAs of this list, namely Cl(9, 1) and Cl(25, 1) are
also regarded as interesting from the perspective of string
theory [106]. Hence, by the way, we already succeeded to
un-string these two algebras, unintentionally 41
. As de-
rived above, Hamilton’s algebra of proper time allows, in
principle, to consider many, arbitrarily large and compli-
cated, algebras on the basis of classical phase spaces. But
furthermore, HCAs have another feature, namely that
they allow to “hide” dimensions. How that?
We found Clifford algebras from symmetries originat-
ing in Hamiltonian theory. The analysis of the Hamilto-
nian Dirac algebra Cl(3, 1) generates a system of quan-
tities and relations that precisely fits to relativistic elec-
trodynamics. It forces us to introduce a 3+1 dimensional
parametric space-time. Without the use of Hamiltonian
notions however, the Dirac matrices do not uniquely se-
lect a specific Clifford algebra: Instead of Cl(3, 1), the
algebra of real 4 × 4-matrices is also a “representation”
of Cl(2, 2). It was shown, that, if all Clifford genera-
tors are supposed to be Hamiltonian, then we are re-
stricted to the algebras listed in Tab. I, namely to a sin-
gle time-like vector element and N − 1 spatial elements.
Then it seems, that Cl(9, 1) inevitably leads to a 10-
dimensional space-time. But as we argued in Ref. [48],
geometric spaces with more than 3 dimensions, when de-
rived from Clifford algebras, have some problematic fea-
tures. Consider some N-dimensional space-time is sup-
posed to emerge from a HCA. Then there are N − 1
generators of boosts γ0 γ1, γ0 γ2, γ0 γ3, . . . , but there are

N−1
2

= (N −1)(N −2)/2 generator of rotations. While
all generators of boosts mutually anti-commute, this does
not hold for all rotators. N − 1 = 3 is the largest num-
ber of spatial dimensions which is “homogeneous” in this
respect [48]. Hence, from the perspective of HCAs, 3
spatial dimensions are the optimal case.
This suggests to consider HCAs with the dimensional-
ity Cl(3, N − 3). From Bott’s periodicity (Eq. 56) one
41 To un-string string theory means to keep findings of the theory,
but without “strings”.
obtains the condition
p − q = 6 − N = 0, 2 mod 8 . (105)
which yields two sequences for m ∈ N:
N = 6 + 8 m
N = 4 + 8 m
(106)
The sequence N = 6 + 8 m has no real representation in
which all generators of the Clifford algebra are Hamilto-
nian, but HCAs of the Dirac type (N = 4 + 8 m) allow
for a reinterpretation in 3-dimensional space. In Ref. [48]
we scetched an interpretation of the algebra of real 8×8-
matrices Cl(3, 3), which lead to the introducing an “in-
ternal” degrees of freedom.
The real representation of the Cl(11, 1) is based of the
same set of matrices as Cl(3, 9) - they are just ordered
and interpretated differently. All Dirac type HCAs with
m ≥ 1, i.e. Cl(3, N − 3) = Cl(3, 1 + 8 m), are alge-
bras that can be obtained from multiple Kronecker prod-
ucts of Dirac matrices. Indeed it has been claimed by
Sogami that triple tensor products of Dirac spinors are
able to reproduce much of the Standard model of particle
physics [107–109], an approach that fits seamlessly to our
presentation of un-quantum mechanics (see App. F).
As scetched in App. F, the Dirac sequence also
emerges in a generalization with higher moments of de-
gree 2 M, M ∈ [3, 5, 7, . . .] or cross-correlation of multi-
ple Dirac particles. Cl(3, 9), for instance, has 3 symmet-
ric generators and 9 skew-symmetric generators. One of
the 9 skew-symmetric generators is the symplectic unit
matrix γ0, which is Hamiltonian by construction. Then
the remaining 8 skew-symmetric Clifford generators are
skew-Hamiltonian and neither represent observables nor
are they generators of SSTs. They are “hidden”.
Given γa, a 6= 0 is a skew-symmetric generator of
Cl(3, 9), then γa and γ0 anti-commute (by definition
of CA generators) γ0 γa = −γa γ0 so that γa is skew-
Hamiltonian:
γ0 γT
a γ0 = −γ0 γa γ0
= γ2
0 γa
= −γa
(107)
It is therefore neither a generator of a canonical trans-
formation nor an observable. Furthermore, γa is skew-
symplectic:
γa γ0 γT
a = −γ0 γa γT
a
= −γ0
(108)
since γa γT
a = 1. These are features known from the
pseudo-scalar of the Dirac-algebra. In context of the
Dirac algebra, the pseudo-scalar represents the charge
conjugation operator. This suggests that the skew-
Hamiltonian γa might be interpreted in a similar way,
namely as representing discrete (instead of continuous)
symmetry-transformations.

26
X. CONCLUSIONS AND OUTLOOK
It was shown that it is classical metaphysics rather
than classical mathematics that prevents from the insight
that QM is classical. The SPQM attacks the problem by
constructing a metaphysical rather than a physical para-
dox: It asks whether electrons are particles or waves, only
in order to demonstrate that the electron can’t be either.
The typical conclusion however, that, since it is neither,
QM can not possibly be understood, is untenable.
Why should these two metaphors be the only possi-
ble options? As has been shown, it is indeed possible
to give a logical account of the “wave-particle duality”,
without the use of ad hoc metaphysical assumptions and
without changing anything else but notation and presen-
tation. The math is literally the same. The only victim of
our presentation is the metaphysical presupposition that
space is fundamental. This however is in agreement with
the experimental tests of Bell’s theorem; it is a price we
have to pay anyhow.
A. Approximations
While there are examples of phase space distributions
that are exclusively parametrized by their second mo-
ments, for instance (multivariate) Gaussians, there is no
reason to presume that this is the only possible case.
Hence we have no reason to believe that the second mo-
ments alone are sufficient to fully characterize the phys-
ical situation. The same holds with respect to the form
of the Hamiltonian: While it is often possible and legit-
imate to use a truncated Taylor serie approximation as
a simplifying assumption, this alone is no reason to be-
lieve that a limitation to second order is necessarily an
intrinsic feature of nature. However we believe it is re-
markable what can be obtained on the basis of logic and
some prelimenary simplifying assumptions.
The fact that we needed only second moments to de-
rive many essentials of QM might be taken in support of
the position that QM can not be complete, but this does
not mean that one has to presume “hidden” variables.
It might suffice to consider higher moments and higher
order terms in the Hamiltonian. A first, very incomplete,
look at possible generalizations for higher moments, rep-
resented by Kronecker products, is given in App. F.
B. Classicality
It was shown that Dirac’s theory is mathematically
classical 42
, that QM altogether is mathematically clas-
42 A hint by Res Jost, pointing in this direction, even found its
way into a celebrated paper of Dirac [74]: “It has been pointed
out to me by R. Jost that this group is just the 4-dimensional
sical: No anti-commutation rules need to (and may) be
presumed for the fundamental variables (ψ).
Since observables are, in our presentation, nothing but
(auto-) correlations of dynamical variables, followers of a
purely information-theoretic approach of QM might feel
confirmed. Though it is an intriguing idea to think that
“correlations have physical reality; that which they cor-
relate does not” [49], it is difficult to see, what exactly
this claim explains that cannot be explained otherwise.
Two methods to introduce “physical” space were con-
sidered: the first obtains the “velocity” directly by the
use of Hamiltonian mechanics applied to the observables
and secondly by a statistical description of moments
based on the Fourier transform. The former case sug-
gests properly defined trajectories, and given the land-
scape of electric and magnetic fields is known, one can
integrate the trajectory of the particle (ignoring Heisen-
berg’s claim) by the Lorentz force. There is little in this
approach that seems to suggest quantum features. The
second moments are correlations and they are in this
sense mathematically exact 43
. The matrix of second
moments defines energy and momentum precisely.
The latter approach, the Fourier transform, is the ba-
sis of wave mechanics and it is required when the spa-
tial extent of structures closed to the particles trajec-
tory is in the same order of magnitude as the de-Broglie
wavelength. Provided one has sympathy for the so-called
“wave-particle duality” or Bohr’s “complementarity prin-
ciple”, one might take the duality of methods as a con-
firmation of his ideas. However, if a principle can be
mathematically derived, then there is few reason to call
it a “principle” at all. And of course there is a bridge
“principle” connecting both accounts, namely the rela-
tivistic energy momentum relation which can likewise be
regarded as dispersion relation, where the “group veloc-
ity” is given by
~
vg = ~
∇~
k ω(~
k) = ~
β = ~
∇~
p E(~
p) . (109)
Is all this still classical physics? This depends on the
point of view. However, we have shown, as promised,
that the difference between CM and QM is not mathe-
matical.
According to our definition, real physical objects are
characterized by their permanence which is, translated
into the language of physics, symmetry in time. This
implies not more and not less than a positive definite
constant of motion (PDCOM). We have shown that the
primary PDCOM of RPOs is mass, i.e. a form of energy.
No commandments so far – just a single constraint.
simplectic group, which is equivalent to the 3+2 de Sitter group.”
The mentioned group is the group of the Dirac matrices, and
what Res Jost remarked is that this group is a classical group,
subject to classical Hamiltonian equations of motion.
43 No one prevents us from interpreting
p
hx4i − hx2i2 as a mea-
sure for the “uncertainty” of x2. However, we doubt that the
same quantity can be interpreted as an “uncertainty” of hx2i.

27
Maybe this is what Feynman had in mind when saying
“...you know how it always is, every new idea, it takes a
generation or two until it becomes obvious that there’s
no real problem” [112].
ACKNOWLEDGMENTS
Mathematica R
has been used for part of the symbolic
calculations. XFig 3.2.4 has been used to generate the
figure, different versions of L
A
TEXand GNU c
-emacs for
editing and layout.
Appendix A: Unitary Motion is Symplectic
Linear symplectic motion is due to Eq. 5 where A is
real symmetric, ψ is real and J is a symplectic unit ma-
trix. Linear unitary motion is given by
i ψ̇ = H ψ , (A1)
where H is hermitian and ψ complex.
If we split a Hermitian matrix H and a complex spinor
ψ into its respective real and imaginary parts H = A +
i B, such that A = AT
and −B = BT
, and ψ = φ + i χ,
then Eqn. A1 can be written as follows:
i (φ̇ + i χ̇) = (A + i B) (φ + i χ)
i φ̇ − χ̇ = A φ + i B φ + i Aχ − B χ
φ̇ = B φ + Aχ
χ̇ = −A φ + B χ
(A2)
Thus, if we compose a real spinor Ψ =
φ
χ
!
by the real
and imaginary parts of the spinor ψ, then unitary motion
has the form:
Ψ̇ =
B A
−A B
!
Ψ
=
0 1
−1 0
!
A −B
B A
!
Ψ
=
0 1
−1 0
!
A BT
B A
!
Ψ
= γ0 A Ψ
(A3)
where γ0 is a SUM and A is symmetric. In other words,
any unitary law of motion can always be expressed by
symplectic motion with specific restrictions for the ma-
trix H as given by Eq. A3.
Appendix B: Periodic Time-Dependent Hamiltonian
In the case of a time-dependent Hamiltonian matrix,
the condition for a PDCOM Eq. 1 requires a modifica-
tion:
dH
dt = ∂H
∂t + ∇ψH ψ̇ = 0
0 = 1
2 ψT
Ȧψ + ψT
A ψ̇ = 0 (B1)
We introduce an additional Hamiltonian matrix G and
use the Ansatz
ψ̇ = (γ0 A + G) ψ , (B2)
Inserted into Eq. B1 this gives:
0 = 1
2 ψT
Ȧψ + ψT
A (γ0 A + G) ψ
0 = ψT

1
2 Ȧ + A γ0 A + A G

ψ
(B3)
The due to the skew-symmetry of A γ0 A, it follows
(as before) ψT
A γ0 Aψ = 0. The remaining matrix
1
2 Ȧ + A G must then also be skew-symmetric to fulfill
this condition. Since Ȧ = ȦT
and Ḟ = γ0 Ȧ, the condi-
tion Ḣ = 0 requires that [48]:
0 = (1
2 Ȧ + A G)T
+ 1
2 Ȧ + A G
0 = Ȧ + γ0 G γ0 A + A G
(B4)
Multiplication with γ0 from the left yields:
0 = γ0 Ȧ − G γ0 A + γ0 A G (B5)
so that
Ḟ = G F − F G . (B6)
Thus, if the time dependence of F can be obtained from
Eq. 17, then then we have a kind of level transparency
for the driving term G: Concerning the original problem,
Eq. B2 suggests that G can be directly added to F.
Appendix C: (Multi-) Spinors in Electrodynamics?
There are different possibilities to represent phase
space densities. One possibility has been used so far,
namely a density function ρ(ψ). There is another
approach, specifically useful in numerical simulations,
namely phase space sampling. This implies to uses not a
single spinor ψ, but several, i.e. the column vector ψ is
replaced by a multi-column vector with m columns, aka
matrix a 4 × m-matrix. This approach can also be used
to impose a symmetry onto the phase space density [33].
The matrix of second moments Σ is then given in the
form of Eq. 14.
Our approach so far concentrated on the description of
the simplest RPOs, i.e. matter fields. Electromagnetic
waves appeared only as terms that act on RPOs, but
not as objects in themselves. Even worse, we found that
vector components can not be generated from bi-vectors
by Eq. 17. This still holds, but raises the question of how
to define electromagnetic energy and momentum within

28
our approach. The electromagnetic fields, written in the
Dirac matrix formalism as a bi-vector, is given by
F = γ0 ( ~
E · ~
γ) + γ14 γ0 ( ~
B · ~
γ) . (C1)
Eq. 14 gives for an arbitrary Hamiltonian matrix:
F FT
γT
0 = F γ0 F . (C2)
For pure bi-vectors, this expression yields pure vector
components for free electromagnetic fields, aka “pho-
tons”:
1
2 F γ0 F = 1
2 ( ~
E2
+ ~
B2
) γ0 + ( ~
E × ~
B) · ~
γ . (C3)
This suggests that there is at least some formal similarity
between the spinors (phase space coordinates) that we
used to model RPOs and electromagnetic fields.
The matrix F has,written explicitely, the form [56]:
F =





−Ex By + Ez −Bz + Ey Bx
−By + Ez Ex −Bx −Bz − Ey
Bz + Ey Bx Ex −By + Ez
−Bx Bz − Ey By + Ez Ex





(C4)
In case of free electromagnetic waves, we can choose a
coordinate system such that the wave propagates along
the z-axis so that Ez = Bz = 0 and hence:
F =





−Ex By Ey Bx
−By Ex −Bx −Ey
Ey Bx Ex −By
−Bx −Ey By Ex





(C5)
If we define a spinor φ = (−Ex, −By, Ey, −Bx)T
as the
first column (vector) of F, then the other columns are
given by −γ6 φ, −γ9 φ and γ14 φ, respectively, so that F
can be written as a “multispinor” [33]:
F = (φ, −γ6 φ, −γ9 φ, γ14 φ) . (C6)
Then one obtains a density matrix F FT
γ0 of the form
F FT
γ0 = 2
1 0
0 1
!
⊗
−Pz E
−E Pz
!
, (C7)
with E = (B2
x +B2
y +E2
x+E2
y)/2 and Pz = Ex By −Ey Bx.
That is, the matrix F is block-diagonal or decoupled.
However, it must be kept in mind that F is not a mul-
tispinor, but a Hamiltonian matrix. This becomes clear
if one considers the transformation properties: Spinors
transform (like conventional vectors) according to the
rule
Ψ′
= R Ψ (C8)
while Hamiltonian matrices transform according to
F′
= R F R−1
(C9)
so that
(Fγ0F)′
= R F (R−1
γ0 R) F R−1
. (C10)
Then the matrix F FT
γ0 is only a proper Hamiltonian,
if R−1
γ0 R equals γ0. According to Eq. 25, this is the
case, if the matrix R is not only symplectic but also or-
thogonal, i.e. in case of rotations. But it is not correct
in case of boosts (see Eq. 42). Hence the electromag-
netic energy density is not a Minkowski 4-vector. It is
part of an object called “stress-energy-tensor” in classi-
cal electrodynamics [110]. This is due to the fact that
the volume element in Minkowski space-time is not an
invariant quantity, while the volume element of a phase
space is an invariant quantity.
The almost obscene complexity of many spatio-
temporal descriptions in physics contrasts with the sim-
plicity of the underlying phase space. Since equations
should be the simpler the more fundamental, phase space
and not space-time must be regarded as fundamental.
Appendix D: Pseudoscalars
The pseudo-scalar is defined by Eq. 60. Is the pseudo-
scalar Hamiltonian? The transpose of γπ is
γT
π =
0
Y
µ=N−1
γT
µ = γT
N−1 γT
N−2 . . . γT
0 . (D1)
If all generators are Hamiltonian, this gives:
γT
π = (γ0γN−1γ0) (γ0γN−2γ0) . . . (γ0γ0γ0)
= (−1)N−1
γ0(γN−1γN−2 . . . γ0) γ0 (D2)
A re-sorting of the order of the bracketed product re-
quires a certain number of commutations of the factors
and each commutation is accompanied the a reversal of
the sign. The number of permutations required to reverse
the order of N matrix factors is N (N − 1)/2, so that
γT
π = (−1)N−1+N(N−1)/2
γ0 γπ γ0 . (D3)
The sign is hence positive and γπ is Hamiltonian, if the
exponent is an even integer. Since N = 2 m is even,
N − 1 + N(N − 1)/2 = m − 1 + 2 m2 (D4)
the even term 2 m2
can be skipped, so that γπ is Hamil-
tonian if m − 1 = N/2 − 1 is even, i.e. of the Pauli type
(Eq. 58):
N/2 − 1 = 4 m . (D5)
The pseudo-scalar of Dirac type algebras (Eq. 59) is skew-
Hamiltonian, since N/2 − 1 is odd:
N/2 − 1 = 4 m + 1 . (D6)

29
This means that the Pauli algebra provides no criterium
to distingush between the even generator of Cl(1, 1) and
the pseudo-scalar: The real Pauli algebra is not uniquely
defined by the Hamiltonian properties.
According to Eq. D2 general k-vectors are Hamiltonian
if the exponent k − 1 + k(k − 1)/2 = k2
+k−1
2 is even. It
is quickly verified that this condition can be written as
k2
+k−2
2 = 2 m
k2
+ k = 4 m + 2
(D7)
and has a solution for integer m for k = 4 j + 1 and k =
4 j+2, but not for k = 4 j and k = 4 j+3. Hence in HCAs
that are generated from Hamiltonian generators, only k-
vectors with k = 1, 2, 5, 6, 9, 10, . . . are Hamiltonian while
k-vectors with k = 3, 4, 7, 8, . . . are skew-Hamiltonian.
Appendix E: Real Dirac Theory
1. Dirac Current Conservation
The real Dirac equation (Eq. 50) is:
0 = (∂µγµ ± m) ψ
0 = (γ0 ∂t + ~
γ · ~
∇) ψ ± m ψ
(E1)
Since here we discuss spinors in “physical space” in-
stead of energy-momentum space, spinors are complex
(Eq. 101) and the “adjunct” spinor is ψ̄ = ψt
γt
0 where
the superscript t stands for the transposed complex con-
jugate. Matrix transposition gives:
0 = (∂µψt
) γt
µ ± m ψt
0 = (∂t ψ̄) + (~
∇ψ̄) · (γ0 ~
γ) ± m ψ̄ γ0
(E2)
where we used the fact that γk = γt
k, γ0 = −γt
0 and the
anti-commutation rules. Multiplication with γ0 from the
right yields
0 = (∂tψ̄) γ0 + (~
∇ψ̄) · ~
γ ∓ m ψ̄ , (E3)
so that, using ψ̄ψ = 0:
0 = ψ̄(γ0 ∂t + ~
γ · ~
∇)ψ
0 = (∂t ψ̄) γ0ψ + (~
∇ψ̄) · ~
γ ψ
(E4)
The sum of both equations then yields the conserved cur-
rent:
0 = ∂t (ψ̄γ0ψ) + ~
∇ · (ψ̄~
γψ) . (E5)
One obtains an electric 4-current (ρe,~
je) by multiplica-
tion with the scaling factor ± e:
ρe = ± e ψ̄γ0ψ
~
je = ± e ψ̄~
γψ (E6)
2. Maxwell’s Equations From Dirac Theory
This section provides evidence that the EMEQ (Eq. 70)
is consistent, i.e. that it is sensible to identify the bi-
vector elements of the Dirac algebra with electric and
magnetic fields, respectively. We will show this by show-
ing that the bi-vector elements ~
E and ~
B obey Maxwell’s
equations.
a. Gauss Law
In Ref. [48] we derived Maxwell’s equations from the
Hamiltonian Dirac algebra. But it is also possible to use
Dirac’s equation in order to show, that the Dirac current
(Eq. E6) is compatible with Maxwell’s equations:
0 = −∂t ψ + γ0 ~
γ · ~
∇ ψ ± m γ0 ψ
0 = ∂tψ̄ + (~
∇ψ̄) · γ0 ~
γ ± m ψ̄ γ0 ,
(E7)
so that
0 = −ψ̄ ∂t ψ + ψ̄ γ0 ~
γ · ~
∇ ψ ± m ψ̄ γ0 ψ
0 = (∂tψ̄) ψ + (~
∇ψ̄) · γ0 ~
γ ψ ± m ψ̄ γ0 ψ ,
(E8)
The sum yields
0 = (∂tψ̄) ψ − ψ̄ ∂t ψ + ~
∇ · (ψ̄ γ0 ~
γ ψ) ± 2 m ψ̄ γ0 ψ (E9)
From Eq. 103 we take that the first term can be written
as 2 i E ψ̄ ψ, which vanishes due to the algebraic identity
ψ̄ψ = 0. Then one has
0 =
e
2 m
~
∇ · (ψ̄ γ0 ~
γ ψ) ± e ψ̄ γ0 ψ , (E10)
which gives Gauss’ law (for electron and positrons):
~
∇ · ~
E = ∓ρe , (E11)
where ~
E = e
2 m (ψ̄ γ0 ~
γ ψ) is the electric field.
b. Gauss Law for Magnetism
In order to show that the magnetic field is free of
sources, we multiply with the pseudo-scalar:
0 = −γ14 ∂t ψ + γ14 γ0 ~
γ · ~
∇ ψ ± m γ14 γ0 ψ
0 = ∂tψ̄ γ14 + (~
∇ψ̄) · γ14 γ0 ~
γ ∓ m ψ̄ γ14 γ0 ,
(E12)
so that
0 = −ψ̄ γ14 (∂t ψ) + ψ̄ γ14 γ0 ~
γ · ~
∇ ψ ± m ψ̄ γ14 γ0 ψ
0 = (∂tψ̄) γ14 ψ + (~
∇ψ̄) · γ14 γ0 ~
γ ψ ∓ m ψ̄ γ14 γ0 ψ ,
(E13)
The terms containing the time derivatives again vanish
by Eq. 103 as in case of Gauss Law. However, this time

30
the “mass term” also vanishes when the equations are
added:
~
∇ · (ψ̄ γ14 γ0 ~
γ ψ) = 0 , (E14)
which gives
~
∇ · ~
B = 0 . (E15)
where ~
B = e
2 m (ψ̄ γ14 γ0 ~
γ ψ) is the magnetic field.
c. Ampere’s Law
If we regard the time derivative of the electric field, we
obtain:
2 m
e ∂t
~
E = (∂t ψ̄) γ0 ~
γ ψ + ψ̄ γ0 ~
γ (∂t ψ)
=

∓ m ψ̄ γ0 − (~
∇ ψ̄) · γ0~
γ

γ0 ~
γ ψ
+ ψ̄ γ0 ~
γ (γ0 ~
γ · ~
∇ψ ± m γ0 ψ)
= ± 2 m
e
~
je − [(~
∇ ψ̄) · ~
γ]~
γ ψ
+ ψ̄ ~
γ (~
γ · ~
∇ψ)
(E16)
Let us consider the x-component of the remaining terms
of the right side:
ψ̄ γ1 (γ1∂x + γ2∂y + γ3∂z)ψ
− [∂xψ̄γ1 + ∂yψ̄γ2 + ∂zψ̄γ3] γ1 ψ
= ψ̄ ∂xψ + ψ̄ γ1γ2∂yψ + ψ̄ γ1γ3∂z ψ
− (∂xψ̄)ψ − (∂yψ̄)γ2γ1ψ − (∂zψ̄)γ3γ1 ψ
= ψ̄ ∂xψ − (∂xψ̄)ψ + 2 m
e (∂y Bz − ∂z By)
(E17)
Once again, it follows from Eq. 103, that the first term
vanishes and one obtains Ampere’s law:
~
∇ × ~
B − ∂t
~
E = ∓~
je . (E18)
We leave Faraday’s Law as an exercise.
Appendix F: Higher Even Moments
Second moments are averages of quadratic forms and
can either be represented in the form of the Σ-matrix or
alternatively by the use of Kronecker products ψ ⊗ ψ.
The simplest spinor ψ = (q, p)T
for instance generates
a spinor ψ ⊗ ψ = (q2
, q p, p q, p2
)T
, two different spinors
give ψ1 ⊗ ψ2 = (q1 q2, q1 p2, p1 q2, p1 p2)T
.
The rules for the Kronecker product “⊗” are given in
Eq. 44. If we define the second order spinor according
to ψ2 = ψ ⊗ ψ, the 4th-order moments can be written in
matrix form according to
Σ4 = hψ2ψT
2 i . (F1)
The spinor ψ2 and the matrix Σ4 are not free of redun-
dancy, since q and p commute. However, the use of
Kronecker products allows to stay within the algebraic
framework as described for the case of simple spinors.
For (skew-) Hamiltonian matrices S (C) one finds:
(S1 ⊗ S2)T
= (γ0 S1 γ0) ⊗ (γ0 S2 γ0)
= (γ0 ⊗ γ0) (S1 ⊗ S2) (γ0 ⊗ γ0)
(C1 ⊗ C2)T
= (−γ0 C1 γ0) ⊗ (−γ0 C2 γ0)
= (γ0 ⊗ γ0) (C1 ⊗ C2) (γ0 ⊗ γ0)
(F2)
As mentioned above and explained in Refs. [33, 48], the
constitutive properties of the SUM γ0 are, that it must be
skew-symmetric, orthogonal and that it squares to −1,
which is not fulfilled by γ0⊗γ0, but by γ0⊗γ0⊗γ0, or more
general: The moments of order D = 2 d for d odd, lead
automatically to symplectic motion, if the basic spinors
is subject to symplectic EQOMs.
1. Fourth Order Moments
Given that the fundamental EQOM are linear, e.g. are
given by Eq. 3, one finds the simple generalization, start-
ing with ψ̇1 = F ψ1 and ψ̇2 = G ψ2:
φ ≡ ψ1 ⊗ ψ2
φ̇ = ψ̇1 ⊗ ψ2 + ψ1 ⊗ ψ̇2
= (F ⊗ 1 + 1 ⊗ G) (ψ1 ⊗ ψ2)
= H φ ,
(F3)
such that the EQOM for the second moments are linear
as well with the driving matrix H given by
H = F ⊗ 1 + 1 ⊗ G ≡ F ⊕ G (F4)
which is called Kronecker sum. As well known from linear
algebra, the matrix exponential holds:
exp (F ⊕ G) = exp (F) ⊗ exp(G) . (F5)
From Eq. 17 one finds:
d
dt (S1 ⊗ S2) = (Ṡ1 ⊗ S2) + (S1 ⊗ Ṡ2)
= (F ⊕ G) (S1 ⊗ S2)
− (S1 ⊗ S2) (F ⊕ G)
(F6)
such that with S = S1 ⊗ S2 we may again write:
Ṡ = H S − S H . (F7)
The transpose of the driving matrix HT
is given by:
HT
= FT
⊗ 1 + 1 ⊗ GT
= γ0 F γ0 ⊗ 1 + 1 ⊗ γ0 G γ0
= −(γ0 ⊗ γ0) (F ⊗ 1 + 1 ⊗ G) (γ0 ⊗ γ0)
= −(γ0 ⊗ γ0) H (γ0 ⊗ γ0) .
(F8)
Obviously H obeys a new criterium and is neither obvi-
ously Hamiltonian nor skew-Hamiltonian, since the ma-
trix (γ0 ⊗ γ0) is not skew-symmetric and can hence not

31
be interpreted as a symplectic unit matrix in the above
sense. Therefore H is not a (higher order) Hamiltonian
matrix, though the trace of H is zero. Nevertheless the
EQOM are of a form that constitutes a Lax pair. The
corresponding constants of motion are then again
Tr(Sk
) = Tr hS1 ⊗ S2ik

= const . (F9)
Also Eq. F3 can be derived within the framework of
Hamiltonian motion as we will show in the following. If
we write γ̃0 = γ0 ⊗ γ0 (where γ̃2
0 = 1 and γ̃T
0 = γ̃0) and
the Hamiltonian H(φ) according to
H = φT
γ̃0 H φ , (F10)
then we obtain
Ḣ = φ̇T
γ̃0 H φ + φT
γ̃0 H φ̇
= φT
HT
γ̃0 H φ + φT
γ̃0 H2
φ
= φT
(−γ̃0 H γ̃0 γ̃0 H + γ̃0 H2
)φ
= φT
(−γ̃0 H2
+ γ̃0 H2
)φ
= 0
(F11)
So that H is conserved. However, we find that the prod-
uct γ̃0 H is skew-symmetric
(γ̃0 H)T
= HT
γ̃T
0
(γ̃0 H)T
= −γ̃0 H γ̃0 γ̃T
0
(γ̃0 H)T
= −(γ̃0 H)
(F12)
and the Hamiltonian function H of the fourth order mo-
ments vanishes, if the first order motion is symplectic.
The transfer matrix is given by Eq. F5 and is given by
M = M1 ⊗ M2 . (F13)
Since M1 and M2 are symplectic, it follows that
M γ̃0 MT
= (M1 ⊗ M2)(γ0 ⊗ γ0)(MT
1 ⊗ MT
2 )
= (M1 γ0 MT
1 ) ⊗ (M2 γ0 MT
2 )
= γ0 ⊗ γ0 = γ̃0
(F14)
Hence, though γ̃0 is not a symplectic unit matrix (since
γ̃2
0 = +1) and though M is not symplectic, nonetheless
M obeys an equation that is equivalent to Eq. 25.
2. Eigenvalues of Kronecker Sums
It is a known result in matrix analysis that the eigen-
values of the Kronecker sum of F and G are sums of
eigenvalues of F and G. More precisely, if f is eigen-
vector of F with eigenvalue f and g is eigenvector of G
with eigenvalue g, then f ⊗ g is eigenvector of F ⊗ G
with eigenvalues f + g [111]. F and G are Hamiltonian
matrices and for such matrices it is known that if f is
an eigenvalue of F, then −f, ¯
f and − ¯
f are also eigen-
value of F [23]. Thus the 4-th order moments contain the
frequencies f + g and f − g, −f + g and −f − g.
In case of two single degees of freedom, the normal
forms are:
F = ω1 γ0 = ω1
0 1
−1 0
!
G = ω2 γ0
(F15)
so that
H = F ⊕ G =





0 ω2 ω1 0
−ω2 0 0 ω1
−ω1 0 0 ω2
0 −ω1 −ω2 0





(F16)
The eigenvalues of H are ±|ω1 + ω2| and ±|ω1 − ω2|.
Eq. F8 seemingly suggests the introduction of complex
numbers and of a symplectic unit matrix (γ0)2 = i (γ0 ⊗
γ0), but there is no way to derive the EQOM from a
non-zero real-valued Hamiltonian function.
3. Sixth Order Moments
It is quite obvious that the next even order φ ≡ ψ1 ⊗
ψ2 ⊗ ψ3, based on the definitions
φ̇1 = F ψ1 φ̇2 = G ψ2
φ̇3 = H ψ3 (γ0)3 ≡ γ0 ⊗ γ0 ⊗ γ0
(F17)
again leads to
J = F ⊗ 1 ⊗ 1 + 1 ⊗ G ⊗ 1 + 1 ⊗ 1 ⊗ H
= F ⊕ G ⊕ H
S = S1 ⊗ S2 ⊗ S3
(F18)
which again are symplectic laws of motion
φ̇ = J φ
(γ0)2
3 = (−)3
1 ⊗ 1 ⊗ 1 = −13
(γ0)T
3 = −(γ0)3
JT
= (γ0)3 J (γ0)3
Ṡ = J S − S J ,
(F19)
with the Lax pair S and J and the respective constants of
motion. The generalization of these findings is obvious:
All spinors ψk =
k
Q
i=0
⊗ψi with k odd that are composed
of equal sized spinors ψi, each of which subject to sym-
plectic motion, are again subject to symplectic motion.
Spinors with k even produce constants of motion, but the
linearized Hamiltonian from which they can be derived,
is identically zero.
4. Symplectic High Order Moments
As we argued above, the simplest non-trivial and hence
fundamental algebra is the real Dirac algebra and the

32
size of the corresponding spinor is 2 n = 4. Hence if
spinors for higher moments are composed as a Kronecker
product from an uneven number k = 2 m + 1 of simple
spinors, then they fulfill the constraints for symplectic
motion, if all individual spinors do. For the fundamen-
tal spinor size of 2 n = 4 this means that spinors com-
posed from Kronecker-products corresponding to these
moments have the size 4k
= 42 m+1
and hence the cor-
responding matrices have the size (4k
)2
= 44 m+2
=
28 m+4
. This matrix size corresponds to Clifford alge-
bras Cl(N − 1, 1) with N = 8 m + 4, i.e. HCAs or the
Dirac type.
Real Dirac spinors, that are constructed from even
Kronecker products with an even number 2 m of Dirac
spinors, have the size 42 m
= 24 m
, which corresponds to
real matrix reps with 28 m
independent elements. How-
ever there exists no real Clifford algebra Cl(N − 1, 1) of
size N = 8 m. And vice versa, the case N = 2 + 8 m has
no correspondence in higher order moments or higher or-
der correlations. One could say that HCAs of the Pauli
type can not be Kronecker-decomposed.
5. Higher Order Hamiltonian
Given a Hamiltonian is of higher than 2nd order, then
it can be written as a Taylor series according to
H(ψ) =
1
2!
ψT
A ψ +
1
3!
Bijk ψi ψj ψk + . . . (F20)
One can argue that in stable (static) systems all terms of
odd order vanish, i.e. that the Hamiltonian is invariant
under ψ → −ψ, then:
H(ψ) =
1
2!
ψT
A ψ +
1
4!
Cijkl ψi ψj ψk ψl + . . . (F21)
Since the SUM is orthogonal γT
0 γ0 = 1, the second order
term H2 can be written as
H2 (ψ) =
1
2!
ψ̄T
F ψ (F22)
where ψ̄T
≡ ψT
γT
0 . Using the Kronecker product we
define γ2 = ψ ⊗ ψ and γ̄T
2 = ψT
γT
0 ⊗ ψT
γT
0 such that
the fourth order term is
H4(ψ2) =
1
4!
ψ̄T
2 F2 ψ2 (F23)
where F2 = (γ0 ⊗ γ0) A2 with a symmetric matrix A2.
The sixth order term is then written as
H6(ψ3) =
1
6!
ψ̄T
3 F3 ψ3 , (F24)
and so on. If the spinor ψ is of Dirac size 2 n = 4, (or,
more generally, if (2 n)2
= 2N
), then there exists a com-
plete real matrix system that represents some Clifford
algebra with elements Γk , k ∈ [0, . . . , 2N
− 1] such that
any matrix F can by written as F =
P
k fk Γk and the
tensor products can be written as
F2 =
P
jk fjk Γj ⊗ Γk
F3 =
P
ijk fijk Γi ⊗ Γj ⊗ Γk ,
(F25)
so that
H4 = 1
4!
P
jk fjk (ψ̄T
⊗ ψ̄T
) (Γj ⊗ Γk) (ψ ⊗ ψ)
= 1
4!
P
jk fjk (ψ̄T
Γjψ) (ψ̄T
Γkψ)
H6 = 1
6!
P
ijk fijk (ψ̄T
⊗ ψ̄T
⊗ ψ̄T
)
× (Γi ⊗ Γj ⊗ Γk) (ψ ⊗ ψ ⊗ ψ)
H6 = 1
6!
P
ijk fijk (ψ̄T
Γiψ) (ψ̄T
Γjψ) (ψ̄T
Γkψ)
(F26)
such that all terms (of the last line) vanish unless (γT
0 Γi)
and γT
0 Γj and γT
0 Γk are symmetric matrices, respec-
tively.
The number of symmetric matrices of size 2 n × 2 n is
2 n (2 n+1)/2, so that the number of non-vanishing terms
in the Hamiltonian H2k has an upper limit of
[n (2 n + 1)]k
(F27)
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