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\par\noindent 
{\bf THE HIDDEN MEASUREMENT APPROACH}
\par\vskip 0.1 truecm\par\noindent 
{\bf TO QUANTUM MECHANICS}
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{\bf Diederik Aerts and Sven Aerts} 
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{\it TENA, Free University of Brussels}
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{\it Pleinlaan 2, B-1050 Brussels}
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\par\noindent 
 We propose an explanation for the arisal of the specific structure of quantum mechanics. More
precisely we investigate the possibility that the quantum mechanical probabilities are not
ontological but epistemological, but that they differ from Kolmogorovian probability theory
because of the nature of this lack of knowledge which is contextual rather then a lack of
knowledge about the state of the entity.
\par\ 
\par\noindent
Key words: hidden measurements, propositional structure
\par\
\par\
\par\noindent    
{\bf 1. INTRODUCTION}   
\par\
\par\noindent    
Since its advent in 1925, quantum mechanics has never ceased to present a challenge to those
who wanted to understand the contents of this theory and the relationship it holds to the
world. There are many problems for the researcher who sets out to understand quantum mechanics
such as the non-deterministic and non-local character of the theory. These dificulties are
reflected in other approaches towards quantum mechanics: the Lattice theoretic approach
initiated by Von Neumann and Birkoff  (1933) and further developed in the Geneva school (xxxxx)
shows that the quantum mechanical lattice of propostions is non-boolean.  Boole called his book
on Boolean logic: "The laws of Thought".  Even now, many researchers are convinced that a
logical stream of thought is in essence constructed out of Boolean operations, and thus the
lattice of propositions should be Boolean as well.  Is it an idle thought to assume quantum
mechanics can be understood?  Among the people that tried to investigate the possibilty of
quantum mechanics being some intermediate theory, eventually to  be supplemented or even
replaced by a more elaborate (and  hopefully more understandable) theoretical framework, we
find names like: Einstein, Schrodinger, De Broglie, Bohm.   They sought to supplement the state
of the entity with additional variables that would allow a more complete description of the
system.  We do not want to enter into the debate of whether this is a possibity or not. One
problem associated with these so called hidden variables is that, once we supplement the state
of the entity with additional variables, we restore the Kolmogorvian character of the
probabilities and we know that quantum theory contains non-Kolmogorvian elements. Should we,
as Feynmann and Bohr often stressed, abandon the attempt to find an underlying  structure of
quantum mechanics?
   
\par\ 
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\par\noindent    
{\bf 2. HIDDEN MEASUREMENTS}   
\par\
\par\noindent    
 Very short after the first theories on quantum mechanics emerged
   Bohr stressed the importance of taking into account the whole
   experimental setup because the setup would determine in an
   essential way the possible set of expected outcomes. So the
   idea that measurement is co-responsible for the outcome of an
   experiment is almost as old as quantum mechanics itself. Maybe
   one could push this idea a bit further and try to develop a
   theoretical framework wherein the probabilities of quantum
   mechanics arise not because of the indefinite character of the
   state of the entity, but because of the indefinite state of the
   measurement equipement.  This idea has been the source of
   inspiration of the Brussels group and maybe characterized by the
   following two hypotheses: 1) a measurement induces a real state
   transition  2) the probability related to an outcome of an
   experiment is due to a lack of knowledge of the interaction
   between the measurement apparatus and the entity under
   observation.  Although clear in their meaning, it may be unclear
   how to proceed from these hypotheses to a more concrete
   expample. To do so, let us introduce a model that explicitly
   incorporates these features and consitutes a model of a
   two-dimensional Hilbert space quantum entity. The sphere model/
   
   
   
   \smallskip 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 experiment
   $e_u$. 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 $e_u$.
   In Figure 2 we have represented the disk of the sphere where the
   experiment $e_u$ takes place, and we can easily calculate the
   probabilities corresponding to the two possible outcomes.
   Therefore we remark that the particle $P$ arrives in $u$ when
   the elastic breaks in a point of the interval $L_1$, and arrives
   in $-u$ when it breaks in a point of the interval $L_2$ (see
   Fig. 2). We make the hypothesis that the elastic breaks
   uniformally, which means that the probability that the particle,
   being in state $p_v$, arrives in $u$, is given by the length of
   $L_1$ (which is $1+ cos\theta$) 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:  $$P(o^u_1, p_v) = {{1+cos\theta}\over
   2} = cos^2{\theta\over 2} \eqno(1)$$ $$P(o^u_2, p_v) =
   {{1-cos\theta}\over 2} = sin^2{\theta\over 2} \eqno(2)$$  \par
   \noindent \hskip 1 cm 
   
   These transition probabilities are the same as the ones related
   to the  outcomes of a Stern-Gerlach spin experiment on a spin
   1/2  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 experiment
   $e_u$ corresponding to the spin experiment in direction $u =
   (cos\beta sin\alpha,$ $sin\beta sin\alpha, cos\alpha)$, is
   described respectively by the vector and the self adjoint
   operator $$\psi_v = (e^{-i\phi/2}cos\theta/2,
   e^{i\phi/2}sin\theta/2)\ \quad {\rm and}\quad\ H_u = {1\over 2}
   \pmatrix{cos\alpha & e^{-i\beta}sin\alpha \cr
   e^{i\beta}sin\alpha &cos\alpha \cr} \eqno(11)$$ \noindent of a
   two-dimensional complex Hilbert space. 
\par\
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\par\noindent    
{\bf 3. DOES IT WORK ONLY FOR EXPERIMENTS WITH TWO OUTCOMES?}   
\par\
\par\noindent    
The fact that the above introduced model neatly reproduces the
   quantum mechanical  probalities for a two dimensional Hilbert
   space entity, left many researchers in the field as much
   surprised as unsatsisfied.  Von Neumann and Feynmann often used
   measurements with two outcomes, such as spin 1/2 experiments
   showing the quantum mystery is already present in this extremely
   simple case. However, it is well known that Gleason's theorem
   starts only from a Hilbert space representation  with a
   dimension at least equal to three.  Also Piron's construction,
   relating the axioms of the lattice theoretic approach to quantum
   mechanics with the axioms of projective geometry and
   Vardarajan's theorem connecting the latter to Hilbert space,
   start only from a dimension at least equal to three. Also the
   possibilty of building a model for a system with only two
   outcomes with hidden variables was long known andall of  this
   did not contribute to an appreciation of the sphere model.
   However, the fact that the sphere model reproduces the quantum
   mechanical probabilities is not due to its two dimensional
   structure but rather to the fact that the hidden measurements
   are not at all the same thing as hidden variables.  If one wants
   to classify them as hidden variables, one can see that they are
   in fact contextual hidden variables, since the variables are
   only introduced upon measurement, they didn't even exist before
   the measurement.  Thus the reason why these hidden measurements
   escape the so-called non-go theorems is not due to the
   two-dimensionality of the sphere model.
   
   Indeed, as early as 1986, (Aerts D., 1986) the possibility of
   constructing the quantum probabilities as due to  hidden
   measurements for a measurement with finite possible outcomes was
   shown.  About 10 years later, as part of his Ph D thesis, Coecke
   (Coecke, 1996) showed how this can be extended to the countable
   infinite case.  For the reader who is willing to accept the
   general case from a three-dimensional model, we refer to an
   article where we constructed a mechanistic model, similar to the
   sphere model, but somewhat more involved, with a
   three-dimensional Hilbert space structure. (Aerts D., Coecke B.,
   D'Hooghe B., Valckenborgh F., 1996) 
  \par\ 
\par\
\par\noindent    
{\bf 4. FROM A PROBABILISTIC POINT OF VIEW...}   
\par\
\par\noindent    
Yet another way of approaching quantum mechanics is
   given by statistical polytopes. Starting from a set of axioms,
   we can construct an effective limit on the range of
   probabilities that the axioms allow by means of an inequality. 
   If a set of probabilities satisfy the inequality, they can be
   derived from the axioms.  The most famous, and to our knowledge
   the first in its kind, was the Bell inequality. Two Italian
   researchers ( Accardi, Fedullo, 1982) constructed the
   inequalites for a measurement with two outcomes.  In this nice
   article several intriguing results are derived.  They showed
   that for a measurement with only two outcomes, the set of 
probabilities derivable within the context of Kolmogorovian
probability theory form a subset of those derivable from a
two-dimensional Hilbert space, a result that fails to hold in
higher dimensions. The also showed that  it is possible to
distinguish between a real and a complex Hilbert space, merely
by looking at the probabilities.  
One of the interesting consequences of viewing quantum probabilities
 as being due to fluctuations of the interaction between the measurement and the measured,
is that one can easily paramterize these fluctuations.  This was done (Aerts D et al...)
with a parameter called epsilon and the model was baptized the epsilon-model.
As we said before, the hidden measurement approach consists of two main ideas: a real 
state-transition of the entity upon measurement and a measurement induced selection 
mechanism.   If we let the fluctuations vanish, is it possible to obtain a Kolmogorovian
theory?   This would be most interesting, since then we would have constructed a macroscopic 
model with an understandable structure (ie, we can see how the probabilities arise) and 
a quantum and a classical behaviour.  One of the memebers of our Brussels group, as part of 
his PhD thesis (Durt Thomas, 1996), showed that, even with zero fluctuations, we still do
 not have a Kolmogorvian structure. However, one of the authors (Aerts D., praag, 1995) 
proposed to test the polytopes for a  set of conditional probabilities .  
These calculations (Aerts S., IJTP, 1996) were done and the result was what we hoped for:
A macrsocopical model with a quantum and a Kolmogorvian limit.
\par\  
\par\ 
\par\noindent    
{\bf 5. BELL'S INEQUALITY AND NON-LOCALITY}    
\par\
\par\noindent     
In 1996 (Aerts D, Aerts S., Coecke B., Valckenborgh F.) we
focused on the question of the meaning of the bell inequality. 
A model (Aerts D, 198X) for singlet experiments like Aspect's 
based on the hidden measurments, already existed.  the
"non-local" element was introduced directly by means of a rod
that conected two sphere-models introducing  As is well known,
some authors attribute the violation of the inequality not to
non-locality, but to the non-Kolmogorovian character of the
theory.
\par\ 
\par\
\par\noindent     
{\bf 6. REFERENCES}   
\par\ 
\par\noindent     
1. D. Aerts, {\it J.
Math. Phys.}, {\bf 27}, 202 (1986).
\par\noindent 2. D. Aerts, {\it Found.
Phys.}, {\bf 24}, 1227 (1994).
\par\noindent  3. D. Aerts and S. Aerts,
"...", this book (1996). \par\noindent 4. D.
Aerts, S. Aerts, B. Coecke, B. D.Hooghe and
F. Valckenborgh, "...", this book (1996).
\par\noindent 5. B. Coecke, {\it Found.
Phys.}, {\bf 25}, 203 (1995). \par\noindent
6. B. Coecke, {\it Found. Phys. Lett.}, {\bf
8}, 437 (1995). \par\noindent 7. B. Coecke,
"Classical Representations for Quantum-like
Systems through an Axiomatics for Context
Dependence",  {\it Helv. Phys. Acta}, to
appear, (1996). \par\noindent 8. B. Coecke,
"A Classification of Classical
Representations for Quan-tum-like Systems", 
{\em Helv. Phys. Acta}, to appear, (1996).
\par\noindent 9. J.M. Jauch, {\it
Foundations of Quantum Mechanics},
Addison-Wesley, London, (1968).
\par\noindent 10. J.M. Jauch and C. Piron,
{\it Helv. Phys. Acta.}, {\bf 36}, 827
(1963). \par\noindent 11. C. Piron, {\it
Foundations of Quantum Physics}, W.A.
Benjamin, Reading, (1976).  \par\noindent
12. J. Von Neumann, {\it The Mathematical
Foundations of Quantum Mechanics}, Princeton
University Press, Princeton, (1955).  \par\
\par\ \par\noindent     {\bf 7. NOTES}   
\par\ \par\noindent     1. We remark that
'existence' is not equivalent with
'knowledge'. \par\noindent 2. This slightly
different way of formulating dispersion free
states by means of additional parameters to
every state is equivalent with the
Jauch-Piron way.  It only makes the bridge
to the hidden measurement approach easier.
\par\noindent 3. 
Due to (8),
(7) should be invariant
under unitary transformations that preserve
$e_0$ up to a permutation of the outcomes.  For a given
representation of (7), we can
always build one that fulfills this
assumption. Let ${\cal E}_0$ be all
measurements obtained through a permutation
of the outcomes of $e_0$.  If we replace
(7) by $\cup_{e\in{\cal E}_0}\Phi_{\Lambda,e}$ and $(1/_N)\sum_{e\in{\cal
E}_0}\mu_{\Lambda,e}$ (where $N$ is the number of measurements in ${\cal E}_0$),
we do respect the symmetry of $e_0$ itself. 

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