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\title{\bf Towards a framework for possible unification of quantum
and relativity theories}

\author{Diederik Aerts\thanks{Department of Theoretical Physics and Center Leo
Apostel, Vrije Universiteit Brussel, 1050 Brussels, Belgium.}}

 

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\date {} 

\maketitle

\begin{abstract}
\noindent We put forward a framework, inspired by recent axiomatic and
operational approaches to generalized quantum theories, wherein we
investigate the possibility of unifying quantum theories and relativity
theories. The framework concentrates on a detailed analysis of a
general construction of reality, that can be used in both, quantum and
relativity theories. By means of this construction of reality we clarify
some well known conceptual problems that stand in the way for a conceptual
unification of quantum and relativity theories on a more profound physical
level than the purely mathematical algebraic level on which now unification
attempts are investigated. More specifically we concentrate on the problem
of 'what is physical reality' in quantum and relativity theories.

\end{abstract}

\section {Introduction}

Nowadays physics is rooted in two well established physical
theories:  quantum theory and relativity theory. There have
been several attempts to unify quantum theory with
relativity theory, but they were only partially successful:
quantum electrodynamics for example is a well defined physical
theory describing the interaction between light and matter and
its theoretical predictions correspond to a very high degree with
the experimental results. Other attempts have encountered
conceptual as well as technical difficulties, such that it can
be stated that the program of unifying quantum theory with
relativity theory is unfinished. In this paper we want to put
forward a framework that could be used to investigate a new way
of unification. Let us first, before we introduce the framework,
explain shortly our motivation.

\par There are two aspects that would make a unification
attempt in our framework different from the
traditional approaches. The first aspect is connected to the
fact that our framework is inspired by the generalized quantum
theories that have been developed during the last decades
rather than by the standard Hilbert space quantum theory,
which is the inspiration of the classical unification
attempts. Standard quantum theory is a well defined mathematical
theory, where the state of an entity is represented by a
vector $\psi$ in a complex Hilbert space and a quantity by a
self adjoint linear operator $A$ of this Hilbert space. The
contact with experimental data is given by a rule, where the
probability that the value of the observable quantity
$A$ is in an interval $a$ when the state of the quantum
entity is $\psi$ is given by $|<\psi,P_a\psi>|^2$, where
$P_a$ is the spectral projection of the operator $A$
corresponding to the interval $a$. The ontological basis of
the standard quantum theoretical scheme is very
badly defined. The actual unification attempts (e.g. quantum
field theories) are attempts to unify quantum theory and
relativity theory starting from this standard quantum theoretical
scheme, and forcing the relativistic aspects of the theory on
the structure of the algebra of operators that
represent the observable quantities. For standard quantum
theory there is no clear meaning for the basic concepts and hence
it is not possible to reflect on a unification on a physically
deeper level than the purely mathematical algebraic level. For
relativity theory, the ontological basis was clearly introduced
from the beginning in a very careful and general way.
Meanwhile, due to new experiments on individual quantum entities
and new theoretical considerations, the ontological basis of
quantum theory has been clarified very much. Inspired by early
generalizations (Birkhoff and Von Neumann 1936; Jauch, 1968),
axiomatic approaches have been developed where the concepts are
introduced more and more in an operational way (Aerts, 1981,
1982, 1983 a,b, 1986, 1990, 1992, 1993, 1994, 1995; Gudder
1988; Piron, 1976, 1985, 1988, 1992; Randall and Foulis, 1979, 1983). The
framework that we want to present here is inspired by these
operational approaches, and more specifically by the realistic
approaches (Aerts, 1981, 1982, 1983 a,b, 1990, 1992; Piron, 1976, 1985,
1988, 1992). 

The second aspect that makes our framework especially suited for
a new attempt to unification is of a conceptual nature. Although
we have given the impression in the foregoing paragraph that the
ontological basis of relativity theory is well defined, we must
remark that in both theories, quantum mechanics as well as
relativity theory, it is not clear 'what reality is'. Therefore
in our framework we start by a subtle analysis of the nature of
reality.

In quantum theory the problem about the question 'what is
the reality of a quantum entity' presents itself in the
following way. Suppose that we consider a quantum entity $S$
in a state $\psi$. This state can mathematically be
represented by a wave function $\psi(x_1, x_2, x_3)$ element
of the complex Hilbert space $L^2(R^3)$ of square
integrable complex functions. When a measurement of position
is performed on this quantum entity in state $\psi$ no
prediction with certainty can be given of where the quantum
entity will be detected. More specifically, the probability that
the quantum entity will be located in a region $\Omega$ of space,
is given by $\int_\Omega|\psi(x_1, x_2, x_3)|^2 dx_1dx_2dx_3$,
and this is a number between 0 and 1, since
$\int_{R^3}|\psi(x_1, x_2, x_3)|^2 dx_1dx_2dx_3 = 1$. If we
consider a measurement of the momentum of the same quantum entity
$S$ in the same state $\psi$, then we find a very similar
situation. The same state $\psi$ can also be represented by the
wave function $\psi(p_1, p_2, p_3)$ in momentum space, again
an element of the complex Hilbert space $L^2(R^3)$ of square
integrable complex functions, and $\psi(p_1, p_2, p_3)$ is
the Fourier transform of $\psi(x_1, x_2, x_3)$. The
probability that the momentum of the quantum entity is
contained in the momentum region $P$ is given by
$\int_P|\psi(p_1, p_2, p_3)|^2 dp_1dp_2dp_3$, where again
$\int_{R^3}|\psi(p_1, p_2, p_3)|^2 dp_1dp_2dp_3 = 1$.
This means that for a quantum entity in state $\psi$ neither the
position nor the momentum can be predicted with certainty. This
situation has been confirmed by so many experiments on quantum
entities that it is now commonly accepted as a fact of nature.
As a consequence the question 'what is the reality of a quantum
entity?' has no obvious answer. This problem
has been in debate for more than sixty years now. It would be too
conclusive to say that the problem has been solved but some
fruitful solutions with a clearly defined ontological basis
have been put forward. Each of these solutions incorporates the
'not being localized' and 'not having a definite momentum' as a
'real effect' on the level of the ontological basis (Aerts 1990,
1992, 1993, 1994, 1995). Therefore in the framework that we
present here the quantum entity in state $\psi$ is considered to
'not have a definite position' and 'not have a
definite momentum' when it is in a state $\psi$. When a detection
takes place then the 'place' of the quantum entity is partly
created by the detection apparatus, and when a measurement of
momentum takes place, the 'momentum' of the quantum entity is
partly created by this momentum measurement. We have explained
in detail this ontological basis for quantum theory in (Aerts
1990, 1992, 1993, 1994, 1995), and have called it the
'creation discovery interpretation'. 

Let us now consider relativity theory. Perhaps it is less
well known that there are also problems with the question
'what is reality' concerning relativity theory. The
problem with this question manifests itself here in a
completely different way. Usually relativity theory is
introduced with a seemingly very well defined ontological
basis (e.g. Misner, Thorne, Wheeler 1973). The collection of
events, each event parametrized by four real numbers
$(x_0, x_1, x_2, x_3)$ is considered to be the basic
structure of the theory. For a particular observator
connected to a particular reference frame, there is no
problem of how to use this four dimensional time-space
manifold scheme to decide for what 'his personal reality'
is. His personal reality is indeed the 'space-cut' that his
reference frame makes with the four dimensional time-space
manifold. This space-cut only determines a reality that is
connected to this particular reference frame, and at first
sight it is not possible to put together the space-cuts of
different reference frames, such that they form one reality. All
this is very well known, and this problem was in fact already at
the origin of the construction of special relativity in the
original paper by Albert Einstein, namely his critique on the
concept of simultaneity (Einstein 1905). When Einstein elaborated
special relativity, he was a firm believer of an operational
approach, and that is certainly part of the reason why the
conceptual foundation of relativity theory is
very well operationally founded. As we shall see in the
next chapter, from the
introduction of our framework it follows that 'simultaneity' as
considered by Einstein is not the good concept to
define 'reality'. Reality should not be defined as 'all
that happens simultaneously'. We believe that this
conceptual mistake, concerning the definition of reality, is at
the origin of the problems with the ontological basis for
relativity theory.

Sometimes it is rather vaguely stated, and these statements are
never conceptually founded, that reality in relativity theory 'is'
the four dimensional time-space continuum. But if this position is
taken, there is another major conceptual problem: indeed then
there is no change and no evolution in time. This makes quantum
theory, which fundamentally is a theory of evolution in time,
completely irreconcilable with relativity theory.

As we see, there are a lot of conceptual as well as
structural aspects that we have to clarify, even
before we can start to work out our general framework in a
technical way. In the next chapter we shall introduce the
foundations of our general framework, and step by step
refer to the way in which the problems of which some we
have mentioned already here in the introduction, are
tackled.

\section {The construction of reality}


The foundations of the framework that we put forward here
can be found in (Aerts, 1992). To make
the paper self-contained we will repeat most of the
basic definitions. First we analyze the 'construction of
reality'. This analysis might give the impression that
we adhere to a subjective philosophy, believing perhaps that reality
is 'only' a construction of the human mind. This is not true,
we believe that reality exists, independent of the
presence of humans to participate in it. We want to put
forward this analysis of our human construction of reality,
because we think that only by understanding all the subtle steps
of this construction, we can understand the question 'what is
reality?' in relation to quantum physics and relativity theory.

\subsection {Experiences}

The basic concept of our approach is that of
an experience. An experience is the interaction between a
participator\footnote{We consciously use here the word
'participator' instead of the word 'observer' to indicate that
we consider the cognitive receiver to participate creatively in
his cognitive act.} and a piece of the world. When the
participator lives such an experience, we will say that this
experience is {\it present}, and we will call it the {\it present
experience} of the participator. When we consider a measurement
then we conceive this situation as the experimentator and his
experimental apparatus together being the participator, and the
physical entity under study to be the piece of the world that
interacts with the participator. The experiment is the
experience.

Let us give some examples of experiences. We consider the following
situation: I am inside my house in Brussels. It is night, the
windows are shut. I sit in a chair, reading a novel. I have a
basket filled with walnuts at my side, and from time to time I
take one of them, crack it and eat it. My son is in bed and
already asleep. New York exists and is busy.

Let us enumerate the experiences that are considered in such a
situation:

\noindent (1) $E_1$(I read a novel)

\noindent (2) $E_2$(I experience the inside of my house in
Brussels)

\noindent (3) $E_3$(I experience that it is night)

\noindent (4) $E_4$(I take a walnut, crack it and eat it)

\noindent (5) $E_5$(I see that my son is in bed and asleep)

\noindent (6) $E_6$(I experience that New York is busy)

The first very important remark I want to make is that obviously
I do not experience all these experiences at once. On the
contrary, in principle, I only experience one experience at once,
namely my present experience. Let us suppose that my present
experience is $E_1$(I read a novel). Then a lot of other things
happen while I am living this present experience. These things
happen in my present reality. While 'I am reading the novel' some
of the happenings that happen are the following: $H_1$(the novel
exists), $H_2$(the inside of my house in Brussels exists),
$H_3$(it is night), $H_4$(the basket and the walnuts exist, and
are at my side), $H_5$(my son is in bed and is sleeping),
$H_6$(New York exists and is busy). All the happenings, and
much more, happen while I live the present experience $E_1$(I
read a novel).

Why have I constructed reality in such a way that what I am just
saying is evident for everybody (and therefore shows that we
are not conscious of the construction that is behind this
evidence)? 

Certainly it is not because I experience also these other
happenings. My only {\it present} experience is the experience of
reading the novel. But, and this is the reason for this type of
construction, I could have chosen to live an experience
including one of the other happenings {\it in replacement} of my
present experience. Let me put down the list of these
experiences that I could have chosen to experience in
replacement of my present experience: $E_2$(I observe that
I am inside my house in Brussels), $E_3$(I see that it is
night), $E_4$(I take a walnut, crack it and eat it),
$E_5$(I go and look in the bedroom to see that my son is
asleep), $E_6$(I go to New York and see that it is busy).

This example indicates how we have started to construct reality.
First of all we have tried to identify two main aspects of an
experience. The aspect that is controlled and created by me,
and the aspect that just happens to me and can only be known by
me. Let us introduce this important distinction in a formal way.

\subsection{Creations and happenings}

To see what I mean, let us consider the experience
$E_4$(I take a walnut, crack it and eat it). In this
experience, there is an aspect that is an action  of me, the
taking and the cracking, and the eating. There is also an
aspect that is an observation of me, the walnut and the basket.
By studying how our senses work, I can indeed say that it is
the light reflected on the walnut, and on the basket, that
gives me the experience of walnut and the experience of basket.
This is an explanation that only now can be given; it is,
however, not what was known in earlier days when the first
world-models of humanity were constructed. But without knowing
the explanation delivered now by a detailed analysis, we could
see very easily that an experience contains always two aspects,
a {\it creation}-aspect, and an {\it observation}-aspect,
simply because our will can only control part of the
experience. This is the creation-aspect.

For example, in $E_1$(I read a novel) the reading is
created by me, but the novel is not created by me. In general
we can indicate for an experience the aspect that is created by
me and the aspect that is not created by me. The aspect not
created by me lends itself to my creation. We can reformulate
an experience in the following way: $E_4$(I take a
walnut, crack it and eat it) becomes $E_4$(The
walnut is taken by me, and lends itself to my cracking and
eating) and $E_1$(I read a novel) becomes $E_1$(The
novel lends itself to my reading).

The taking, cracking, eating, and reading will be called {\it
creations} or actions and will be denoted by $C_4$(I take,
crack and eat) and $C_1$(I read). The walnut and the novel
will be called {\it happenings} and will be denoted by
$H_4$(The walnut) and $H_7$(The novel).


\bigskip
\noindent {\it A creation is that aspect of an experience
created, controlled, and acted upon by me, and a happening is that
aspect of an experience lending itself to my creation, control
and action}.
\bigskip
  
An experience is determined by a description of the creation and
a description of the happening. Creations are often
expressed by verbs: to take, to crack, to eat, and to read, are the
verbs that describe my creations in the examples. The walnut and the
novel are happenings that have the additional property of being
objects, which means happening with a great stability. Often
happenings are expressed by a substantive.

\bigskip
\noindent {\it Every one of my experiences $E$ consists of one of
my creations $C$ and one of my happenings $H$, so we can write $E =
(C,H)$}.
\bigskip

A beautiful image that can be used as a metaphor for our model
of the world is the image of the skier. A skier skis downhill.
At every instant he or she has to be in complete harmony with
the form of the mountain under-neath. The mountain is the
happening. The actions of the skier are the creation. The
skier's creation, in harmony fused with the skier's happening,
is his or her experience.

\subsection{The construction of reality, present, past and
future}
 
Let us again consider the collection of experiences: $E_1$(I
read a novel), $E_2$(I observe that I am inside my house in
Brussels), E$_3$(I see that it is night), E$_4$(I take a
walnut, crack it and eat it), E$_5$(I go and look in the
bedroom to see that my son is asleep) and E$_6$(I go to New
York and see that it is busy). Let us now represent the
'construction of reality' that is made out of this small
collection of experiences.

$E_1$(I read a novel) is my present experience. In my past
I could, however, at several moments have chosen to do
something else and this choice would have led me to have
another present experience than $E_1$(I read a novel).
For example:

One minute ago I could have decided to stop reading and observe that I
am inside the house. Then $E_2$(I observe that I am inside
my house in Brussels) would have been my present experience.

Two minutes ago I could have decided to stop reading and open the windows
and see that it is night. Then $E_3$(I see that it is
night) would have been my present experience.

Three minutes ago I could have decided to stop reading, take a walnut from
the basket, crack it, and eat it. Then $E_4$(I take a
walnut, crack it and eat it) would have been my present
experience.

Ten minutes ago I could have decided to go and see in the bedroom whether
my son is asleep. Then $E_5$(I go and look in the
bedroom to see that my son is asleep) would have been my
present experience.

Ten hours ago I could have decided to take the plane and fly to New
York and see how busy it was. then $E_6$(I go to New
York and see that it is busy) would have been my present
experience.

\bigskip
\noindent
{\it Even when they are not the happening aspect of my present
experience, happenings 'happen' at present if they are the happening
aspect of an experience that I could have lived in replacement
of my present experience, if I would have decided so in my
past.}

\bigskip
\noindent The fact that a certain experience $E$ consisting
of a creation $C$ and an happening $H$ is for me a
possible present experience depends on two factors:

\medskip
\noindent
(1) I have to be able to perform the creation.

\smallskip
\noindent
(2) The happening has to be available.

\medskip
\noindent For example, the experience $E_2$(I observe that I am
inside my house in Brussels) is a possible experience for me, if:

\medskip
\noindent
(1) I can perform the creation that consists in observing the
inside of my house in Brussels. In other words, if this
creation is in my personal power.

\smallskip
\noindent
(2) The happening 'the inside of my house in Brussels' has to be
available to me. In other words, this happening has to be contained
in my personal reality.

\medskip
\noindent
{\it The collection of all creations that I can perform at the
present I will call my present personal power}.

\smallskip
\noindent
{\it The collection of all happenings that are available to me
at the present I will call my present personal reality}.

\medskip
\noindent
I define as my present personal reality the collection of these
happenings, the collection of happenings that are available to one of
my creations if I would have used my personal power in such a
way that at the present I fuse one of these creations with one
of these happenings.

\medskip
\noindent
{\it My present personal reality consists of all happenings that
are available to me at present. My past reality consists of all
happenings that were available to me in the past. My future reality
consists of all happenings that shall be available to me in the
future.}

\medskip
\noindent
{\it My present personal power consists of all creations that I
can perform at present. My past personal power consists of all
the creations that I could perform in the past. My future
personal power consists of all creations I shall be able to
perform in the future.}

\medskip
\noindent
Happenings can happen at once, because to happen, a happening
does not have to be part of my present experience. It is
sufficient that it is available, and things can be available at
once. Therefore, although my present experience is only one, my
present personal reality consists of an enormous amount of
happenings all happening at once.

This concept of reality is not clearly understood in present
physical theories. Physical theories know how to treat past,
present and future. But reality is a construction about the
possible. It is a construction about the experiences I could
have lived but probably will never live.

\subsection{Material time and material happenings}

From ancient times humanity has been fascinated by happenings
going on in the sky, the motion of the sun, the changes of the
moon, the motions of the planets and the stars. These
happenings in the sky are {\it periodic}. By means of these
periodic happenings humans started to {\it coordinate} the
other experiences. They introduced the counting of the years,
the months and the days. Later on watches were invented to be
able to coordinate experiences of the same day. And in this
sense {\it material} time was introduced in the reality of the
human species. Again we want to analyze the way in which this
material time was introduced, to be able to use it
operationally if we analyze later on the paradoxes of time and
space.

My present experience is seldom a material time experience. But
in replacement of my present experience, I always could have
consulted my watch, and in this way live a material time
experience $E_7$(I consult my watch and read the
time). In this way, although my present experience is seldom a
material time experience, my present reality always contains a
material time happening, namely the happening $H_7$(The
time indicated by my watch), which is the happening to which
the creation $C_7$(I consult) is fused to form the
experience $E_7$.

We can try to use our theory for a more
concrete description of that layer of reality that we shall
refer to as the layer of 'material or energetic happenings'. We
must be aware of the fact that this layer is a huge
one, and so first of all we shall concentrate on those
happenings that are related to the interactions between what
we call material (more generally energetic) entities. We
have to analyze first of all in which way the four
dimensional manifold that generally is referred to as the
'time-space' of general relativity theory, is related to
this layer of material or energetic reality. We shall take into
account in this analysis the knowledge that we have gathered about
the reality of quantum entities in relation with
measurements of momentum and position. 


\section {Quantum, relativity and reality}

We consider the set of all material or energetic happenings and
denote this set by ${\cal M}$. Happenings of ${\cal M}$ we shall
denote by $m, n, o$. Let us consider such a happening $m$
that corresponds to a quantum entity. Then this happening
is characterized by the fact that it is always available to
a creation of localization (consisting in localizing the
particle in a certain region of space), let us denote such
a creation of localization by $l$. Then the experience $(l,
m)$ is an experience that can be coordinated by a certain
point $(x_0, x_1, x_2, x_3)$ of the four dimensional
manifold that is referred to a time-space. 

However instead of performing a creation of localization,
one can choose to perform a creation that consists in
measuring the momentum of the quantum entity. Let us denote
this creation by $i$, then the happening $(i, m)$ can be
coordinated by a certain point $(p_0, p_1, p_2, p_3)$, that
can be interpreted as the four-momentum of general
relativity theory.

We know from quantum theory that the quantum entity can be
in different states, all corresponding to a different
statistics as related to repeated localizations and
measurements of momentum. Let us denote these states by $q,
p, ...$. The quantum entity can be in an eigenstate $q(x_0,
x_1, x_2, x_3)$ of position, which means that the creation
of localization in this eigenstate leads with certainty to
a finding of the quantum entity in the point $(x_0, x_1,
x_2, x_3)$. The quantum entity can also be in an eigenstate
$p(p_0, p_1, p_2, p_3)$ of momentum, which means that by a
measurement of momentum the entity will be found to have
the momentum $(p_0, p_1, p_2, p_3)$. But in general the quantum
entity will be in a state that is neither an eigenstate of
position nor an eigenstate of momentum. It is only after the
happening $p$ (the state of the quantum entity) has been fused
with one of the creations $l$ (the localization measurement) or
$i$ (the momentum measurement) that will be in an eigenstate
of localization (a point of time-space) or of momentum (a
point of four momentum space). This is the general situation for
material happenings.

\subsection{The construction of reality and relativity}


To show what are the problems that we can solve by
means of our framework, we will concentrate now on the
question 'what is reality in relativity theory?'. Since we have
an operational definition of reality in our framework, we can
investigate this problem in a rigorous way.

Let us suppose that
I am here and now in my house in Brussels, and it is June 1, 1996, 3
pm exactly. I want to find out 'what is the material reality for
me now?'. Let us use the definition of reality given in
the foregoing section and consider a place in New York, for example
at the entrance of the empire state building, and let us denote, the center of
this place by $(x_1, x_2, x_3)$. I also choose now a certain time,
for example June 1, 1996, 3 pm exactly, and let me denote this time by
$x_0$. I denote the happening that corresponds with the spot
$(x_1, x_2, x_3)$ located at the entrance of the empire state building,
at time $x_0$ by $m$. I can now try to investigate whether this
happening $m$ is part of my personal material reality. The
question I have to answer is, can I find a creation of
localization $l$, in this case this creation is just the
observation of the spot $(x_1, x_2, x_3)$ at the entrance of the empire
state building, at time $x_0$, that can be fused with this
happening $m$. The answer to this question can only be
investigated if we take into account the fact that I, who want
to try to fuse a creation of localization to this happening, am
bound to my body, which is also a material entity. I must specify
the question introducing the material time coordinate that I
coordinate by my watch. So suppose that I coordinate my body by
the four numbers $(y_0, y_1, y_2, y_3)$, where $y_0$ is my
material time, and $(y_1, y_2, y_3)$ is the center of mass of my
body. We apply now our operational definition
of reality. A this moment, June 1, 1996 at 3 pm exactly, my body
is in my house in Brussels, which means that $(y_0, y_1, y_2,
y_3)$ is a point such that $y_0$ equals June 1, 1996, 3 pm, and
$(x_1, x_2, x_3)$ is a point, the center of mass of my body,
somewhere in my house in Brussels. This shows that $(x_0, x_1,
x_2, x_3)$ is different from $(y_0, y_1, y_2, y_3)$, in the
sense that $(x_1, x_2, x_3)$ is different form $(y_1, y_2, y_3)$
while $x_0 = y_0$.

The question is now whether $(x_0, x_1,  x_2, x_3)$ is a point of
my material reality, hence whether it makes sense to me to claim
that now, June 1, 1996, 3 pm, the entrance of the empire state
building 'exists'. If our theoretical framework corresponds in
some way to our pre-scientific construction of reality, the
answer to the foregoing question should be affirmative. Indeed,
we all believe that 'now' the entrance of the empire state
building exists. Let us try to investigate in a rigorous
way this question in our framework. We have to verify whether
it was possible for me to decide somewhere in my past, hence
before June 1, 1996, 3 pm, to change some of my plans of action,
such that I would decide to travel to New York, and arrive
exactly at June 1, 1996, 3 pm at the entrance of the empire
state building, and observe the spot $(x_1, x_2, x_3)$. We
could give many concrete ways to realize this experiment,
and we will not give here one in detail, because we shall come
back to the tricky parts of the realization of this experiment
in the following example. But hence the answer is indeed
affirmative: I could have experienced the spot $(x_1, x_2, x_3)$
at June 1, 1996, 3 pm, if I would have decided to travel to New
York somewhere in my past. Hence $(x_0, x_1, x_2, x_3)$ is part
of my reality. It is sound to claim that the entrance of the
empire state building exists right now. And we remark that this
does not mean that I have to be able to experience this spot at
the entrance of the empire state building now, June 1, 1996, 3
pm, while I am inside my house of Brussels. I repeat again,
reality is a construction about the possible happenings that I
could have fused with my actual creation. And since I could have
decided so in my past, I could have been at the
entrance of the empire state building, now, June 1, 1996, 3 pm.

Untill this moment one could think that our framework only will
confirm our intuitive notion of reality but our next example
shows that this is certainly not the case. Indeed, let me
consider the same problem, but now consider another point of
time-space. I consider the point $(z_0, z_1, z_2, z_3)$, where
$(z_1, z_2, z_3) = (x_1, x_2, x_3)$, hence the spot we envisage
is again the entrance of the empire state building, and $z_0$ is
June 2, 1996, 3 pm exactly, hence the time that we consider is, tomorrow 3 pm. If I
ask now first, before checking rigorously by means of our operational definition
of reality, whether this point $(z_0, z_1, z_2, z_3)$ is part of my present
material reality, the intuitive answer here would be 'no'. Indeed,
tomorrow at the same time, 3 pm, is in the future and not in the
present, and hence it is not real, and hence no part of my
present material reality (this is the intuitive reasoning). If
we go now to the formal reasoning in our framework, then we can
see that the answer to this question depends on the
interpretation of relativity theory that we put forward. Indeed,
let us first analyze the question in a Newtonian conception of
the world to make things clear. Remark that in a Newtonian
conception of the world (which has been proved experimentally wrong, so here we are just
considering it for the sake of clarity), my present material reality just falls together with
'the present', namely all the points of space that have the same
time coordinate June 1, 1996, 3 pm. This means that the entrance
of the empire state building tomorrow 'is not part of my present
material reality'. The answer is clear here and in this
Newtonian conception, my present personal reality is just the
collection of all $(u_0, u_1, u_2, u_3)$ where $u_0 = y_0$ and
$(u_1, u_2, u_3)$ are arbitrary. The world is not Newtonian,
this we know meanwhile experimentally, but also if we put
forward an ether theory interpretation of relativity theory (let
us refer to such an interpretation as a Lorentz interpretation)
the answer remains the same. In a Lorentz interpretation, my
present personal reality coincides with the present reality of
the ether, namely all arbitrary points of the ether that are
at time $y_0$, June 1, 1996 3 pm, and again tomorrow the
entrance of the empire state building is not part of my present
material reality.

For an Einsteinian interpretation of relativity theory the
answer is different. To investigate this I have to ask again the
question of whether it would have been possible for me to decide
in my past such that I would have been able to make coincide
$(y_0, y_1, y_2, y_3)$ with $(z_0, z_1, z_2, z_3)$. The answer
here is that this is very easy to do, because of the well known, and
experimentally verified, effect of 'time dilatation'. Indeed, it
would for example be sufficient that I go back some weeks in my
past, let us say April 1, 1996, 3 pm, and then decide to step
inside a space ship that can move with almost the velocity of the
speed of light, such that the time when I am inside this space
ship slows down in such a way, that when I return with the space
ship to planet earth, still flying with a speed near the
velocity of light, I arrive in New York at the entrance of
the empire state building while my personal material watch
indicates June 1, 1996 3 pm, and the watch that remained at the
entrance of the empire state building indicates June 2, 1996 3
pm. Hence in this way I make coincide $(y_0, y_1, y_2, y_3)$ with
$(z_0, z_1, z_2, z_3)$, which proves that
$(z_0, z_1, z_2, z_3)$ is part of my present material reality.
First I have to remark that in practice it is not yet
possible to make such a flight with a space ship. But this is not a
crucial point for our reasoning. It is sufficient that we can
do it in principle\footnote{We have not yet made this explicit
remark, but obviously if we have introduced in our framework
an operational definition for reality, then we do not have to
interpret such an operational definition in the sense that
only operations are allowed that actually, taking into account
the present technical possibilities of humanity, can be
performed. If we would advocate such a narrow interpretation,
then even in a Newtonian conception of the world, the star
Sirius would not exist, because we cannot yet travel to it.
What we mean with operational is much wider. It must be
possible, taking into account the actual physical knowledge of
the world, to conceive of a creation that can be fused with
the happening in question, and then this happening pertains to
our personal reality}.  

\subsection {Einstein versus Lorentz: has reality four
dimensions?}

We can come now to one of the points that we want to make in
this paper, and that clarifies the paradox of time that makes the
difference between an ether interpretation of relativity
(Lorentz) and an Einsteinian interpretation of relativity. Why
would we come to a different result concerning the foregoing
question, depending on whether we advocate an Einsteinian
interpretation of relativity theory or an ether interpretation.
To see clear in this we have to come back to the essential
aspect of the construction of reality of our framework, which is
the difference between a creation and a happening. We have to
give first another example to be able to make clear what we
mean.

Suppose that I am a painter and I consider again my
present material reality, at June 1, 1996, 3 pm, as indicated on
my personal material watch. I am in my house in Brussels and let us specify: the
room where I am is my workshop, surrounded by paintings, of which
some are finished and others I am still working on. Clearly all
these paintings exist in my presents reality, June 1, 1996, 3
pm. Some weeks ago, when I was still working on a painting that
now is finished, I could certainly have decided to start to work
on another painting, a completely different one, that now does
not exist. Even if I could have decided this some weeks ago, all
of you will agree that this other painting, that I never started to
work on, does not exist now, June 1, 1996, 3 pm. The reason for
this conclusion is that the making of a painting is
a 'creation' and not a happening. It is not so that there is
some 'hidden' space of possible paintings such that my choice of
some weeks ago to realize this other painting would have made me
to detect it. If this would be the situation with paintings, then
indeed also this painting would exist now, in this hidden space.
But with paintings this is not the case. Paintings that are not
realized by the painter are potential paintings, but they do not
exist.

With this example of the paintings we can explain very well the
difference between Lorentz and Einstein. For an ether
interpretation of relativity the fact that my watch is slowing
down while I decide to fly with the space ship nearly at the speed
of light and return at the entrance of the empire state building
while my watch is indicating June 1, 1996, 3 pm and the watch
that remained at the empire state building indicates June 2,
1996, 3 pm, is interpreted as a 'creation'. It is seen as if
there is a real physical effect of creation on the material
functioning of my watch while I travel with the space ship, and
this effect of creation is generated by the movement of the
space ship through the ether. Hence the fact that I could observe
the entrance of the empire state building tomorrow June 2, 1996
3 pm, when I would have decided some weeks ago to start
traveling with the space ship, only proves that the entrance of the
empire state building tomorrow is a potentiality. Just like
the fact that this painting that I never started to paint could
have been here in my workshop in Brussels is a potentiality. This
means that as a consequence the spot at the entrance of the empire
state building tomorrow is not part of my present reality, just as
the possible painting that I did not start to paint is not part of
my present reality. If we however put forward an Einsteinian
interpretation of relativity, then the effect on my watch during
the space ship travel is interpreted in a completely different
way. There is no physical effect on the material functioning of
the watch\footnote{Certainly if we take into account that most
of the time dilatation takes place not during the accelerations
that the space ship undergoes during the trip, but during the
long periods of flight with constant velocity nearly at the
speed of light}, but the flight at the velocity nearly the speed
of light 'moves' my space ship in the time-space continuum
such that time coordinates and space coordinates get mixed. This
means that the effect of the space ship travel is an effect of a
voyage through the time-space continuum, which brings me at my
personal time of June 1, 1996, 3 pm at the entrance of
the empire state building, where the time is June 2, 1996, 3 pm.
And hence the entrance of the empire state building is a
happening, an actuality and not just a potentiality, and it can be
fused with my present creation. This means that the happening
$(z_0, z_1, z_2, z_3)$ of June 2, 1996, 3 pm, entrance of the
empire state building, is an happening that can be fused with my
creation of observation of the spot around me at June 1, 1996, 3
pm. Hence it is part of my present material reality.
The entrance of the empire state building at June 2, 1996,
3 pm exists for me today, June 1, 1996 3 pm.

If we advocate an Einsteinian interpretation of relativity theory
we have to conclude from the foregoing section that reality is
four dimensional. This conclusion will perhaps not amaze those
who always have considered the time-space continuum of
relativity representing the new reality. Now that we have however
defined very clearly what is the meaning of this, we can start
investigating the seemingly paradoxical conclusions that often
are brought forward in relation with this insight.



\subsection {The process view confronted with the geometric
view}

The paradoxical situation that we can try to solve
now is the confrontation of the process view of reality with
the geometric view. Often it is claimed that an interpretation
where reality is considered to be related to the four
dimensional time-space continuum contradicts another view of
reality, namely the one where it is considered to be of a process
like nature. By means of our framework we can now understand
exactly these two views and see that there is no contradiction.
Let us repeat now what is the meaning in our framework of the
conclusion that reality is four dimensional. It means that, at a
certain specific moment, that I call my 'present', the collection
of places that exist, and that I could have observed when I would
have decided to do so in my past, has a four dimensional
structure, well represented mathematically by the four dimensional
time-space continuum. This is indeed my present material
reality. This does not imply however that this reality is not
constantly changing. Indeed it is constantly changing. New
entities are created in it and other entities disappear,
while others are very stable and remain into existence. This is
as much the case in all of the four dimensions of this reality.
Again I have to give an example to explain what I mean. We
came to the conclusion that now, at June 1, 1996, 3 pm the
entrance of the empire state building exists for me while I am
in my house in Brussels. Then this is not a statement of
derterministic certainty. Indeed, it is very well possible that by
some extraordinary chain of events, without me knowing about
these events, the empire state building had been destructed, such
that my statement about the existence of the entrance of the
empire state building 'now', although almost certainly true, is
not deterministically certain. The reason is again the same,
namely that reality is a construction of what I would have been
able to experience, if I would have decided differently in my
past. The knowledge that I have about this reality is complex
and depends on the changes that go on continuously in it. What I
know from experience is that there do exist material objects,
and the empire state building is one of them, that are rather
stable, which means that they are into existence without changing
to much. To these stable objects, material objects but also
energetic fields, I can attach the places where I could observe
them. The set of these places has the structure of a four
dimensional continuum. At the same time all these
objects are continuously changing and moving in this four
dimensional scenery. Most of the objects that I used to shape my
intuitive model of reality are the material objects that surround
us here on the surface of the earth. They are all very fixed in
the fourth dimension (the dimension indicated by the 0 index, and
we should not call it the time dimension) while they move easily
in the other three dimensions (those indicated by the 1, 2, and 3
index). Other objects, for example the electromagnetic fields,
have a completely different way of being and changing in this
four dimensional scenery. This means that in our framework there
is no contradiction between the four dimensionality of the set of
places and the process like nature of the world. If we came to
the conclusion that the entrance of the empire state building,
tomorrow, June 2, 1996, 3 pm exists also for me now, then our
intuition reacts more strongly to this statement, because
intuitively we think that this would mean that the future exists
also and hence is determined and hence no change is possible.
This is a wrong conclusion which comes from the fact that during
a long period of time we have had an intuitive image of a
Newtonian present, that would be determined completely. We have to
be aware of the fact that it is the present, even in the newtonian
sense, which is not determined at all. We can only say that the
more stable entities in my present reality are more determined
to be there, while the places where they can be, because these
places are stable with certainty, are always there.


\subsection{The singularity of the reality construction}

We want to come back to the construction of reality in our
framework that we have confronted here with the Einsteinian
interpretation of relativity theory. Instead of wondering about
the existence of the entrance of the empire state building
tomorrow, June 2, 1996, 3 pm, I can also question the existence
of my own house at the same place of the time-space continuum.
Clearly I can make an analogous reason and come then to the
conclusion that my own house, and the chair where I am sitting
while reading the novel, and the novel itself, and the basket of
wall nuts beside me, etc..., all exist in my present reality at
June 2, 1996 3 pm, hence tomorrow. If we put it like that, we
get confronted even more with a counter-intuitive aspect of
the Einsteinian interpretation of relativity theory. But it is a
correct statement in our framework. We have to add however that
all these objects that are very close to me now June 1, 1996, 3
pm, they indeed also exist in my present reality at June 2, 1996,
3 pm, but the place in reality where I could have observed them
is of course much further away for me. Indeed, to be able to get
there, I have to fly away with a space ship at nearly the
velocity of light. We now come to a very peculiar question that
will confront us with the singularity of our reality construction.
Where do I myself exist? Do I also exist tomorrow June 2, 1996, 3
pm? If the answer to this question would be affirmative, we
would be confronted with a very paradoxical situation.
Because indeed, I myself, and this counts for all of you
also, cannot imagine me to exist at different places of time. But
indeed our framework clarifies this question very easily. It is
impossible for me to make some action in my past such that I
would be able to observe myself tomorrow June 2, 1996 3 pm.
Indeed, if I would have chosen to fly away and come back with the
space ship such that I observe now, June 1, 1996, 3 pm on my
personal watch, the inside of my house tomorrow June 2, 1996, 3
pm, then I can do this, and as we remarked already, it proves
that this inside of my house tomorrow is part of my present
personal reality. But I will not find myself in it.
Because to be able to observe my house tomorrow June 2, 1996 3
pm, I have had to move out of it. Hence, in this situation I will
enter my house, for myself being still at June 1, 1996, 3 pm, but
my house and all things in it, being at June 2, 1996, 3 pm. This
shows that there is no paradox.

\subsection {The construction of reality and quantum physics}

To be able to use this framework for an attempt of unification
between quantum mechanics and relativity theory, a lot of work
still has to be done. But what we have achieved is that we
see clear 'what reality is' as confronted with both theories. For
a more detailed analysis of quantum mechanics in our framework we
refer to (Aerts 1992). In a next article we want to try to use our
framework and both studies, the one we have made here for
relativity theory and the one we have made in (Aerts 1992) for
quantum mechanics, to put forward a theory about physical
reality and the way this reality changes.

\section {Acknowledgments}

This work was supported by the Federale Diensten voor
Wetenschappelijke, Technische en Culturele Aangelegenheden; IUAP-III
n$\circ $9. The author is a Senior Research Associate of the Belgian
National Fund for Scientific Research.   

\section {References}


\noindent Aerts, D. (1981). {\it The one and the many}, Doctoral Thesis, Free University of
Brussels, Pleinlaan 2, 1050 Brussels.
 
\smallskip
\noindent Aerts, D. (1982). {\it Description of many physical entities without the paradoxes
encountered in quantum mechanics,} Found. Phys., {\bf 12}, 1131.

\smallskip
\noindent Aerts, D. (1983 a). {\it Classical theories and non classical theories as
a special case of a more general theory,} J. math. Phys., {\bf 24}, 2441.

\smallskip
\noindent Aerts, D. (1983 b). {\it A possible explanation for the probabilities of
quantum mechanics and a macroscopic situation that violates Bell
inequalities,} in {\it Recent Developments in Quantum Logic,} eds. P.
Mittelstaedt et al., in Grundlagen der Exacten Naturwissenschaften, vol.6,
Wissenschaftverlag, Bibliographisches Institut, Mannheim, 235.



\smallskip
\noindent Aerts, D. (1986). {\it A possible explanation for the probabilities of quantum
mechanics,} J. Math. Phys., {\bf 27}, 202.
\smallskip
\par \noindent Aerts, D. (1987). {\it The origin of the non-classical character of the
quantum probability model,} in {\it Information, Complexity, and Control in Quantum Physics}, 
A. Blanquiere, et al., eds., Springer-Verlag.


\smallskip
\noindent Aerts, D. (1990). {\it  An attempt to imagine parts of the reality of the
micro-world,} in the proceedings of the conference {\it Problems in Quantum Physics ; Gdansk
'89,} World Scientific Publishing Company, Singapore, 3.



\smallskip
\noindent Aerts D., (1992). {\it The construction of reality and its influence on the 
understanding of quantum structures,} Int. J. Theor. Phys., {\bf 31} 
1815.


\smallskip
\par \noindent Aerts, D. (1994). {\it Quantum structures, separated physical entities and
probability,} Found. Phys. {\bf 24}, 1227.

\smallskip
\noindent Aerts, D. (1995). {\it Quantum structures: an attempt to explain the origin of their
appearance in  nature,} Int. J. Theor. Phys. {\bf 34}, 1165.
Physics. 





\smallskip
\noindent Birkhoff G. and Von Neumann J., (1936). {\it The logic
of quantum mechanics,} J. Ann. Math., {\bf 37}, 823.



\smallskip
\noindent Einstein A., (1905). {\it Zur elektrodynamik
bewegter K\"orper,} Ann. Phys. 17. 891.

\smallskip
\noindent Emch G.G., (1984). {\it Mathematical and conceptual 
foundations of 20th century
physics,} North-Holland, Amsterdam.

\smallskip
\noindent Gudder S.P., (1988). {\it Quantum Probability}, Academic Press, Inc.
Harcourt Brave Jovanovitch, Publishers.


\smallskip
\noindent Jauch J.M., (1968). {\it Foundations of Quantum Mechanics}, Addison Wesley.

\smallskip
\noindent Misner C. W., Thorne K. S. and  Wheeler J. A., (1973). {\it
Gravitation}, W. H. Freeman and Company, San Francisco.

\smallskip
\noindent Piron C., (1976). {\it Foundations of Quantum Physics}, Benjamin.

\smallskip
\noindent Piron C. (1985). {\it Quantum Mechanics: Fifty Years Later,} in
{Proceedings of the Symposium on the Foundations of Modern Physics,} eds.
P. Lahti and P. Mittelstaed, World Scientific, Singapore.


\smallskip
\noindent Piron C. (1988). {\it Recent Developments in Quantum Mechanics,}
Helv. Phys. Acta. {\bf 62}, 82.


\smallskip
\noindent Piron C., (1990). {\it M\'ecanique Quantique, Bases et
Applications,} Presses Polythechniques et Universitaires
Romandes, Laussane.



\smallskip
\noindent Randall C and Foulis D, (1979). {\it The 
operational approach to quantum
mechanics,} in {\it Physical theory as logico-operational 
structure,} ed. Hooker, C. A.,
Reidel.

\smallskip
\noindent Randall C. and Foulis D., (1983). {\it A mathematical 
language for quantum physics}, in {\it Les Fondements de la M\'ecanique 
Quantique}, ed. C. Gruber et al, A.V.C.P., case postale 101, 1015 Lausanne, Suisse.



\end{document}