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\title{The creation-discovery-view : towards a possible explanation of quantum reality}
\author{Diederik Aerts and Bob Coecke}
\date{}

\begin{document}

\maketitle 
\centerline{CLEA, Free University of Brussels, Pleinlaan 2, 1050 Brussels, Belgium.}
\centerline{e-mails: diraerts@vub.ac.be, bocoecke@vub.ac.be}


\begin{abstract}
\noindent We present a realistic interpretation for quantum mechanics that we have called the 'creation discovery
view' and that is being developed in our group in Brussels. In this view the change of state of a
quantum entity during an experiment is taken to be a 'real change' under influence of the experiment, and the
quantum probability that corresponds to the experiment is explained as due to a lack of knowledge of a deeper
deterministic reality of the measurement process. The technical mathematical theory underlying the creation
discovery view that we are elaborating we have called the 'hidden measurement formalism'. We present a
simple physical example: the 'quantum machine', where we can illustrate easily how the quantum structure arises
as a consequence of the two mentioned effects, a real change of the state, and a lack of knowledge about a deeper
reality of the measurement process. We analyze non-locality in the light of the creation discovery view, and
show that we can understand it if we accept that also the basic concept of 'space' is partly due to a creation:
when a detection of a quantum entity in a non-local state occurs, the physical act of detection itself 'creates'
partly the 'place' of the quantum entity. In this way the creation discovery view introduces a new ontology for
space: space is not the all embracing theater, where all 'real' objects have their place, but it is the
structure that governs a special type of relations (the space-like relations) between macroscopic physical
entities. We bring forward a number of elements that show the
plausibility of the approach and also analyze the way in which the presence of
Bell-type correlated events can be incorporated.  

\end{abstract}



\section{Introduction}
 
\noindent The creation discovery view and together with it its technically underlying hidden
measurement formalism has been elaborated from the early eighties on, and many aspects of it have
been exposed in different places \cite{Aerts1986, Aerts1987, Aerts1992, Aerts1993, Aerts1995,
Aerts1996, AertsCoeckeDurtValckenborgh1996a, AertsCoeckeDurtValckenborgh1996b, 
AertsDurt1994a, AertsDurt1994b, Coecke1995a, Coecke1995b, Coecke1995c, Coecke1996a, Coecke1996b,
Coecke1996c, Coecke1996d}. In this paper we give an overview of the most important of these
aspects. 

Quantum mechanics was originally introduced as a non-commutative matrix calculus of observables
by Werner Heisenberg \cite{Heisenberg1925} and parallel as a wave mechanics by Erwin
Schr{\"o}dinger \cite{Schrodinger1926}. These two structurally very different theories could explain
fruitfully the early observed quantum phenomena. Already in the same year the two theories were shown
to be realizations of the same, more abstract, ket-bra formalism by Dirac \cite{Dirac1958}. Only some
years later, in 1934, John Von Neumann put forward a rigorous mathematical framework for quantum
theory in an infinite dimensional separable complex Hilbert space \cite{VonNeumann1955}. Matrix
mechanics and wave mechanics appear as concrete realizations: the first one if the Hilbert space is
$l^2$, the collection of all square summable complex numbers, and the second one if the Hilbert space
is
$L^2$, the collection of all square integrable complex functions. The formulation of quantum
mechanics in the abstract framework of a complex Hilbert space is now usually referred to as
the 'standard quantum mechanics'.

The basic concepts  - the vectors of the Hilbert space representing the states of the system and the
self-adjoint operators representing the observables - in this standard quantum mechanics are abstract
mathematical concepts defined mathematically in and abstract mathematical space. Several
approaches have generalized the standard theory starting from more physically defined basic
concepts. John Von Neumann and Garett Birkhoff have initiated one of these approaches
\cite{BirkhoffVonNeumann1936} were they analyze the difference between quantum and
classical theories by studying the 'experimental propositions'. They could show that for a given
physical system classical theories have a Boolean lattice of experimental propositions while for
quantum theory the lattice of experimental propositions is not Boolean. Similar fundamental
structural differences between the two theories have been investigated by concentrating on different
basic concepts. The collection of observables of a classical theory was shown to be a commutative
algebra while this is not the case for the collection of quantum observables \cite{Emch1984,
Segal1947}. Luigi Accardi and Itamar Pitowski obtained an analogous result by concentrating on the
probability models connected to the two theories: classical theories have a Kolmogorovian probability
model while the probability model of a quantum theory is non Kolmogorovian \cite{Accardi1982,
Pitovski1989}. 
 
The
fundamental structural differences between the two types of theories, quantum and classical, in
different categories, was interpreted as indicating also a fundamental difference on the level of the
nature of the reality that both theories describe: the micro world should be 'very different' from
the macro world. This state of affairs was very convincing also because concrete attempts to
understand quantum mechanics in a classical way had failed as well: e.g. the many 'physical' hidden
variable theories that had been tried out \cite{Selleri1990}. The structural difference between
quantum theories and classical theories (Boolean lattice versus non-Boolean lattice of propositions,
commutative algebra versus non commutative algebra of observables and Kolmogorovian versus non
Kolmogorovian probability structure) had been investigated mostly mathematically and not much
understanding of the physical meaning of the structural differences had been gained during all these
years.

The first step that led to the creation discovery view and its underlying hidden measurement formalism was a
breakthrough in the understanding of the physical origin of these mathematical structural differences between
quantum and classical theories. Indeed, one of the authors found in the early eighties a way to 
identify the
physical aspects that are at the origin of the structural differences \cite{Aerts1982,
Aerts1986, Aerts1987}. Let us summarize these findings: it are mainly two aspects that determine the
mathematical structural differences between classical and quantum theories in the different
categories:

\smallskip
\noindent
{\it We have a quantum-like theory describing a system under investigation if the measurements needed
to test the properties of the system are such that:}

\smallskip
\noindent  {\it (1) The measurements are not just observations but provoke a real change of the
state of the system.}

\smallskip
\noindent {\it (2) There exists a lack of knowledge about the reality of what happens during the measurement
process.}

\medskip
\noindent The presence of these two aspects is sufficient to render the description of the
system under consideration quantum-like. It is the lack of knowledge (2) that is theoretically structured in a
non Kolmogorovian probability model. In a certain sense it is possible to interpret the second aspect, the
presence of the lack of knowledge on the reality of the measurement process, as the presence of 'hidden
measurements' instead of 'hidden variables'. Indeed, if a measurement is performed with the presence of such a
lack of knowledge, then this is actually the classical mixture of a set of classical hidden measurements, were
for such a classical hidden measurement there would be no lack of knowledge. In an analogous way as in a hidden
variable theory, the quantum state is a classical mixture of classical states. This is the reason why we have
called the underlying theory of the creation discovery view the hidden measurement formalism. It is possible to
illustrate the creation discovery view and the hidden measurement aspect
in a very simple way by using a mechanical model that was introduced in \cite{Aerts1985, Aerts1996,
Aerts1987} and that we have called the quantum machine. This is the subject of next section.

\section{The Quantum Machine.}

\noindent Several aspects of the quantum machine have been presented in different occasions
\cite{Aerts1986, Aerts1987, Aerts1988a, Aerts1988b, Aerts1991a, Aerts1995,
AertsCoeckeDurtValckenborgh1996a, AertsCoeckeDurtValckenborgh1996b} and we shall therefore introduce
here only the basic aspects. The machine that we consider consists of a physical entity $S$ that is a
point particle
$P$ that can move on the surface of a sphere, denoted $surf$, with center $O$ and radius $1$. The unit-vector
$v$ where the particle is located on $surf$ represents the state $p_v$ of the particle (see Fig 1,a).
For each point $u \in surf$, we introduce the following measurement $e_u$. We consider the
diametrically opposite point
$-u$, and install a piece of elastic of length 2, such that it is fixed with one of its end-points in $u$ and
the other end-point in $-u$. Once the elastic is installed, the particle $P$ falls from its original place $v$
orthogonally onto the elastic, and sticks on it (Fig 1,b). Then the elastic breaks and the particle
$P$, attached to one of the two pieces of the elastic (Fig 1,c), moves to one of the two end-points $u$ or
$-u$ (Fig 1,d). Depending on whether the particle $P$ arrives in $u$ (as in Fig 1) or in $-u$, we give the
outcome $o^u_1$ or $o^u_2$ to $e_u$. We can easily calculate the probabilities corresponding to the two
possible outcomes.

\par \noindent \hskip 0.3 cm 
{\hbox{\vrule height 4.7 true cm width 0pt \special{illustration Fig1.eps}\vrule height 0pt
width 5 true cm}} \par\vglue -4.2 true cm
\hangindent= 6 true cm \hangafter=-10 \noindent {\baselineskip= 10 pt \smallroman Fig 1 : A
representation of the quantum machine. In (a) the physical entity $\scriptstyle P$ is in state
$\scriptstyle p_v$ in the point $\scriptstyle v$, and the elastic corresponding to the measurement 
$\scriptstyle e_{u}$ is installed between the two diametrically opposed points
$\scriptstyle u$ and $\scriptstyle -u$. In (b) the particle $\scriptstyle P$ falls orthogonally onto the
elastic and stick to it. In (c) the elastic breaks and the particle $\scriptstyle P$ is pulled towards the
point $\scriptstyle u$, such that (d) it arrives at the point $\scriptstyle u$, and the measurement
$\scriptstyle e_u$ gets the outcome
$\scriptstyle o^u_1$.\par} 

\bigskip
\medskip

\noindent The particle $P$ arrives in $u$ when the  elastic breaks in a
point of the interval $L_1$ (which is the length of the  piece of the elastic between  $-u$ and the point
where the particle has arrived,  or  $1+cos\theta$), and arrives in $-u$ when it breaks in a point of
the interval
$L_2$  ($L_2=L-L_1=2-L_1$). We make the hypothesis that the elastic breaks uniformly, which means that the
probability that the particle, being in state $p_v$, arrives in $u$, is given by the length of $L_1$ divided
by the length of the total elastic  (which is 2). The probability that the particle in state $p_v$ arrives in
$-u$ is the length of $L_2$ (which is $1-cos\theta$) divided by the length of the total elastic. If we denote
these probabilities respectively by $P(o^u_1, p_v)$ and $P( o^u_2, p_v)$ we have: 
$$P(o^u_1, p_v) = {{1+cos\theta}\over 2} = cos^2{\theta\over 2} \quad \quad \quad \quad P(o^u_2, p_v) =
 {{1-cos\theta}\over 2} = sin^2{\theta\over 2} \eqno(1)$$

\noindent These transition probabilities are the same as the ones related to the  outcomes of a Stern-Gerlach
spin measurement on a spin ${1 \over 2}$  quantum particle, of which the quantum-spin-state in direction $v =
(cos\phi sin\theta,$ $ sin\phi sin\theta, cos\theta)$, denoted by $\bar {\psi_v}$, and the measurement $e_u$
corresponding to the spin measurement in direction $u = (cos\beta sin\alpha,$ $sin\beta sin\alpha,
cos\alpha)$, is described respectively by the vector and the self adjoint operator of a two-dimensional
complex Hilbert space.
$$\psi_v = (e^{-i\phi/2}cos\theta/2, e^{i\phi/2}sin\theta/2)\ \quad \quad \quad  H_u = {1\over 2}
\pmatrix{cos\alpha & e^{-i\beta}sin\alpha \cr
e^{i\beta}sin\alpha &cos\alpha \cr} \eqno(2)$$
We can easily see now the two aspects in this quantum machine that we have identified in the creation discovery
view to give rise to the quantum structure. The state of the particle $P$ is effectively changed by the measuring
apparatus ($p_v$ changes to
$p_u$ or to $p_{-u}$ under the influence of the measuring process), which identifies the first aspect, and
there is a lack of knowledge on the interaction between the measuring apparatus and the particle, namely the
lack of knowledge of were exactly the elastic will break, which identifies the second aspect. We can also
easily understand now what is meant by the term 'hidden measurements'. Each time the elastic breaks
in one specific point $\lambda$, we could identify the measurement process that is carried out afterwards as
a hidden measurement $e_u^\lambda$. The measurement $e_u$ is then a classical mixture of the collection of
all measurement $e_u^\lambda$: namely $e_u$ consists of choosing at random one of the $e_u^\lambda$ and
performing this chosen $e_u^\lambda$. We can see in the example that effectively the final states, after the
measurement $e_u$, hence the states $p_u$ and $p_{-u}$, are partly created by the measurement.
 
We remark that we
have shown in our group in Brussels that such a model can be built for any arbitrary quantum 
entity \cite{Aerts1985,
Aerts1986, Aerts1987, Aerts1994, Coecke1995a, Coecke1995b, Coecke1995c, Coecke1996c, Coecke1996d}.
However, the hidden measurement formalism is more general than standard quantum theory. Indeed, it is
very easy to produce quantum-like structures that cannot be represented in a complex Hilbert space.
For a presentation of an as general as possible mathematical framework to deal
with the hidden measurement ideas we refer to \cite{Coecke1995c, Coecke1996c, Coecke1996d}. 
An example of such quantum-like structures is presented in the following section where we define a
continuous transition from quantum to classical for the model introduced at the beginning of this
section.  Hence we claim to solve the old problem of the classical limit. 

\section{The Quantum Classical Transition.}

If the quantum structure can be explained by the  presence of a lack of knowledge on the measurement process,
we can go a step further, and wonder what types of structure arise when we consider the original models, with
a lack of knowledge on the measurement process, and introduce a variation of the magnitude of this lack of
knowledge.  We have studied the quantum machine under varying 'lack of knowledge', parameterizing  this
variation by a number $\epsilon \in [0,1]$, such that $\epsilon = 1$ corresponds to the situation of maximal
lack of knowledge, giving rise to a quantum structure, and $\epsilon = 0$ corresponds to the situation of zero
lack of knowledge, generating a classical  structure. Other values of $\epsilon$ correspond
to intermediate situations and give rise to a  structure that is neither quantum nor classical 
\cite{AertsDurt1994a, AertsDurt1994b, AertsDurtVanBogaert1992, AertsDurtVanBogaert1993}. We have
called this model the
$\epsilon$-model and introduce it shortly.


\smallskip
\par\noindent {\it a) The $\epsilon$-Model}. We start from the quantum machine, but introduce now
different types of elastic. An $\epsilon, d$-elastic consists of three different parts: one  lower
part where it is unbreakable, a middle part where it breaks uniformly, and an upper  part where it is
again unbreakable. By means of the two parameters $\epsilon \in [0,1]$  and $d \in [-1+\epsilon,
1-\epsilon]$, we fix the sizes of the three parts in the following way. Suppose that we have
installed the $\epsilon, d$-elastic between the points $-u$ and $u$ of the sphere. Then the elastic
is unbreakable in the lower part from $- u$ to $(d-\epsilon) \cdot u$, it breaks uniformly in the
part from $(d-\epsilon) \cdot u$ to $(d+\epsilon) \cdot u$, and it is again unbreakable in the upper
part from $(d+\epsilon) \cdot u$ to $u$ (see Fig 2).

\par \noindent \hskip 1 cm  {\hbox{\vrule height 3.5 true cm width 
0pt
\special{illustration Fig2.eps}\vrule height 0pt width 4 true cm}}
\par\vglue -2.8 true cm
\hangindent= 5.5 true cm \hangafter=-8 \noindent {\rightskip 1 
truecm \leftskip 0 true cm
\baselineskip= 7pt \smallroman Fig 2 :  A representation of the measurement $\scriptstyle
e^\epsilon_{u, d}$. The elastic breaks uniformly between the  points $\scriptstyle (d-\epsilon) u$
and $\scriptstyle (d +\epsilon) u$,  and is unbreakable in other points.\par} 

\vskip 1.7 true cm

\noindent An $e_u$ measurement performed by means of an $\epsilon, d$-elastic shall be denoted by
$e^\epsilon_{u,d}$. We have the following cases:
 
\smallskip
\noindent {\it (1)} $v \cdot u \le d-\epsilon$. The particle sticks to the lower part of the $\epsilon,
d$-elastic, and any breaking of the elastic pulls  it down to the point $-u$. We have $P^\epsilon(o^u_1, p_v)
= 0$ and $P^\epsilon(o^u_2, p_v) = 1$.

\smallskip
\noindent {\it (2)} $d-\epsilon < v \cdot u < d+\epsilon$. The particle falls onto the breakable part of the
$\epsilon, d$-elastic. We can easily  calculate the transition probabilities and find:
$$P^\epsilon(o_1^u, p_v) = {1 \over 2\epsilon}( v \cdot u -d +\epsilon) \quad \quad \quad P^\epsilon(o_2^u,
p_v) = {1 \over 2\epsilon}(d+\epsilon - v \cdot u) \eqno(3)$$

\smallskip
\noindent {\it (3)} $d+\epsilon \le v \cdot u$. The particle falls onto the upper part of the
$\epsilon,d$-elastic, and any  breaking of the elastic pulls it upwards, such that it arrives in $u$. We have 
$P^\epsilon(o_1^u, p_v) = 1$ and $P^\epsilon(o_2^u,  p_v) = 0$.

\smallskip
\noindent
 We are now in a very interesting situation from the point of view of the structural studies of quantum
mechanics. Since the $\epsilon$-model describes a continuous transition from quantum to classical, its
mathematical structure should be able to learn us {\it what are the structural shortcomings of the standard
Hilbert space quantum mechanics}. Therefore we have studied the $\epsilon$-model in the existing mathematical
approaches that are more general than the standard quantum mechanics: the lattice approach, the probabilistic
approach and the *-algebra approach.
\smallskip
\par\noindent {\it b) The Lattice Approach}. In this lattice approach exists a well known axiomatic
scheme that reduces the approach to standard quantum mechanics if certain axioms are fulfilled. For
intermediate values of
$\epsilon$, that is $0 <  \epsilon < 1$, we find that 2 of the 5 axioms needed in the lattice approach
to reconstruct standard quantum mechanics are violated. The axioms that are violated are the weak-modularity
and the covering law, and it are precisely those axioms that are needed to recover the vector space structure
of the state space in quantum mechanics \cite{AertsDurt1994a, AertsDurt1994b,
AertsDurtVanBogaert1993}. 
\smallskip
\par\noindent {\it  c) The Probabilistic Approach}. If we take the case of vanishing fluctuations
($\epsilon = 0$), do we obtain the Kolmogorovian theory of probability? This would be most
interesting, since then we would have constructed a macroscopic model with an understandable
structure (i.e., we can see how the probabilities arise) and  a quantum and a classical behavior. We
\cite{Aerts1995} proposed to test the polytopes for a family of conditional probabilities. The
calculations can be found in \cite{AertsS1996} and the result was what we hoped for: a macroscopic
model with a quantum and a Kolmogorovian limit. For intermediate values of the fluctuations ($0 < 
\epsilon < 1$) the resulting probability model is neither quantum nor Kolmogorovian: we have
identified here a new type of probability model, that is quantum-like, but not really isomorphic to
the probability model found in a complex Hilbert space. 
\smallskip
\par\noindent{\it d) The *-Algebra Approach}. The *-algebra provides a natural mathematical language
for quantum mechanical operators.  We applied the concepts of this approach to the epsilon model  
to find that an operator corresponding to an $\epsilon$ measurement is linear if and only if $\epsilon = 1$
\cite{AertsD'Hooghe1996}. This means that for the classical and intermediate situations the
observables cannot be described by linear operators.
\par\smallskip
\noindent {\it CONCLUSION: Quantum theory and classical theories appear as special cases ($\epsilon = 1,
\epsilon = 0$) and the general intermediate case, although quantum-like, cannot be described in standard
Hilbert space quantum mechanics.}

\smallskip \noindent {\it e) The Measurement Problem and the Schr{\"o}dingers Cat Paradox}. The
result stated in the conclusion means that all the paradoxes of standard quantum theory that are due
to the fact that quantum theory is used as a universal theory, also being applied to macroscopic
system, for example the measuring apparatus, are not present in our hidden measurement formalism. We
explicitly have in mind the 'measurement problem' and the 'Schr{\"o}dinger cat paradox'. Indeed, the
measurement apparatus should be described by a classical model in our approach, and the physical
system eventually by a quantum model. The problem of the presence of quantum correlations between
physical system and measuring apparatus, as it presents itself in the standard theory, takes a
completely different aspect. We are working now at the elaboration of a concrete description of the
measurement process within the hidden measurement formalism \cite{AertsDurt1994b}. Our result also
shows that it is possible in the hidden measurement formalism to formulate a 'classical limit',
namely as a continuous transition from quantum to classical \cite{AertsDurtVanBogaert1993}.

\section{Non-Locality: a Genuine Property of Nature.}

 \noindent The measurement problem and paradoxes equivalent to Schr{\"o}dingers cat paradox disappear in
the hidden measurement formalism, because standard quantum mechanics appears only as a special case, the
situation of maximum fluctuations. All quantum mysteries connected to the effect of 'non-locality' remain
(Einstein Podolsky Rosen paradox and the violation of the Bell inequalities). It is even so that non-locality
unfolds itself as a fundamental aspect of the creation discovery view. This is due to the fact that if we
explain the quantum structures as it is done in this view, a quantum measurement
has two concrete physical effects: (1) it changes the state of the system and (2) it produces probability
due to a lack of knowledge about the nature of this change. With the quantum machine we have
given a macroscopic model for a spin measurement of a spin ${1 \over 2}$ particle. If we apply the hidden
measurement formalism to the situation of a quantum system described by a wave function $\psi(x)$, and to a
position or a momentum measurement performed on this system, we also have the two mentioned effects. 
For example
in the case of a position measurement: the detection apparatus changes the state of the quantum system, in the
sense that it 'localizes' the quantum system in a specific place of space, and the probabilities that are
connected with this measurement are due to the fact that we have a lack knowledge about the specific
way in which this localization takes place. This means that  
'before the detection has taken place' the quantum system was in general not localized: it
was not present in a specific region of space. With other words, quantum systems are fundamentally
non-local systems: the wave function
$\psi(x)$ describing such a quantum state is not interpreted as a wave that is present in space, but
as indicating these regions of space were the particle can be localized,

\par
\noindent{\hbox{\vrule height 7.4 truecm width 0pt
\special{illustration Fig3.eps}\vrule height 0pt width
6 truecm}}\par\vglue -7.5 truecm \hangindent=6 truecm \hangafter=-17
\noindent
where $\int_R|\psi(x)|^2dx$ is the probability 
that this localization will happen in region
$R$. A similar interpretation must be given to the momentum measurement of a quantum entity: the quantum
entity has no momentum before the measurement, but the measurement creates partly this momentum.
In \cite{AertsDurtVanBogaert1993} we have calculated the $\epsilon$-situation for a quantum
system described by a wave function $\psi(x)$, element of the Hilbert space of all square integrable
complex functions, and we have found the following very simple procedure. Suppose that $\epsilon$ is
given, and the state of the quantum system is described by the wave function $\psi(x)$ and 
$\phi(x)$ is the corresponding probability distribution (hence $\phi(x) = |\psi(x)|^2$). 
We cut, by means of a
constant function $\phi_\Omega$, a piece of the function $\phi(x)$, such that the surface contained in the
cutoff piece equals $\epsilon$ (see step 1 of Fig 3). We move this piece of function to the $x$-axis
(see step 2 of Fig 3). And then we renormalize by dividing by $\epsilon$ (see step 3 of Fig 3). If we
proceed in this way for smaller values of $\epsilon$, we shall finally arrive at a delta-function for
the classical limit $\epsilon \to 0$, and the delta-function is located in the original maximum of
the quantum probability distribution. We want to point out that the state $\psi(x)$ of the physical system
is not changed by this $\epsilon$-procedure, it remains always the same state, representing the same
physical reality. It is the regime of lack of knowledge going together with the detection measurement that
changes with varying $\epsilon$. For $\epsilon =1$ this regime is one of maximum lack of knowledge on the
process of localization, and this lack of knowledge is characterized by the spread of the probability
distribution $\phi(x)$. For an intermediate value of $\epsilon$, between 1 and 0, the spread of the
probability distribution has decreased (see Fig 3) and for zero fluctuations the spread is 0. Let us
also try to see what becomes of the non-local behavior of quantum entities taking into account the
classical limit procedure that we propose. Suppose that we consider a double slit experiment, then
the state $p$ of a quantum entity having passed the slits can be represented by a probability
function $p(x)$ of the  

\par \noindent
{\hbox{\vrule height  5.8 truecm width 0pt
\special{illustration Fig4.eps}\vrule height 0pt width
6 truecm}}\par\vglue -6 truecm \hangindent= 6 truecm \hangafter=-14
\noindent
form represented in
Fig 4. We can see that the non-locality presented by this probability function gradually disappears
when
$\epsilon$ becomes smaller, and in the case where $p(x)$ has only one maximum finally disappears
completely. When there are no fluctuations on the measuring apparatus used to detect the
particle, it shall be detected with certainty in one of the slits, and always in the same one. If
$p(x)$ has two maxima (one behind slit 1, and the other behind slit 2) that are equal, the
non-locality does not disappear. Indeed, in this case the limit-function is the sum of two
delta-functions (one behind slit 1 and one behind slit 2). So in this case the non-locality remains
present even in the classical limit. If our procedure for the classical limit is a correct one, also
macroscopic classical entities can be in non-local states. How does it come that we don't
find any sign of this non-locality in the classical macroscopic world? This is due to the
fact that the set of states, representing a situation where the probability function has more
than one maximum, has measure zero, compared to the set of all possible states, and moreover
these states are 'unstable'. The slightest perturbation will destroy the symmetry of the
different maxima, and hence shall give rise to one point of localization in the classical
limit. Also classical macroscopic reality is non-local, but the local model that we use to describe it
gives the same statistical results, and hence cannot be distinguished from the non-local model.

\section{Bell Inequalities and Hidden Correlations.}

\noindent It is interesting to consider the violation of Bell's inequalities within the creation discovery view.
The quantum machine, as presented in section 3, delivers us a macroscopic model for the spin of a spin ${1 \over
2}$ quantum entity and starting with this model it is possible to construct a macroscopic situation, 
using two of
these models coupled by a rigid rod, that represents faithfully the situations of two entangled quantum
particles of spin ${1 \over 2}$ \cite{Aerts1991b}. The 'non-local' element is introduced explicitly
by means of a rod that connects the two sphere-models.  We also have studied this EPR situation of
entangled quantum systems by introducing the $\epsilon$-variation of the amount of lack of knowledge
on the measurement processes and could show that one violates the Bell-inequalities even more for
classical but non-locally connected systems, that is,
$\epsilon=0$. This illustrates that the violation of the Bell-inequalities is due to the non-locality rather
then to the indeterministic character of quantum theory. And that the quantum indeterminism (for values of
$\epsilon$ greater than 0) tempers the violation of the Bell inequalities 
\cite{AertsAertsCoeckeValckenborgh1996}. This idea has been used by one of the authors to construct
a general representation of entangled states (hidden correlations) within the hidden measurement
formalism \cite{Coecke1995c, Coecke1996a, Coecke1996b}.  More precisely, it is possible to prove that
for a collection of quantum entities described in a tensor product of Hilbert spaces there always
exists one unique hidden correlation representation, i.e., a representation as a collection of
individual entities which are correlated in such a way that a change of state of one of the
entities (due to a measurement) induces in a well-defined change of state of the
other entities (for more details we refer to \cite{Coecke1996b}).
Structurally, these correlations are definitely mechanistic.  Nonetheless, they are non-local. 
 
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\end{document} 