Structure of Probabilistic Information and Quantum Laws\\ {This paper was given as a talk at the conference {Foundations of Probability %%@ and Physics} organized by A. Khrennikov at Vaxjo-University, Vaxjo, Sweden, from Nov.27 to %%@ Dec.1, 2000. Proceedings will appear sometime in 2001.}
Johann Summhammer
The acquisition and representation of basic experimental information %%@ under the probabilistic paradigm is analysed. The multinomial probability distribution is %%@ identified as governing all scientific data collection, at least in principle. For this distribution %%@ there exist unique random variables, whose standard deviation becomes asymptotically %%@ invariant of physical conditions. Representing all information by means of such random %%@ variables gives the quantum mechanical probability amplitude and a real alternative. For %%@ predictions, the linear evolution law (Schr\"odinger or Dirac equation) turns out to be the %%@ only way to extend the invariance property of the standard deviation to the predicted %%@ quantities. This indicates that quantum theory originates in the structure of gaining pure, %%@ probabilistic information, without any mechanical underpinning.
Vxquanph.tex
%% Paper for proceedings of conference in Vaxjo, 
%% organized by A. Khrennikov
%% Nov 27 - Dec 1, 2000.
%% This version is for the quant-ph archive.

\documentstyle[aps,epsf]{revtex}

\twocolumn
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}


\begin{document}

\title{Structure of Probabilistic Information and Quantum Laws\\
{\small This paper was given as a talk at the conference {\it Foundations of Probability %%@
and Physics} organized by A. Khrennikov at Vaxjo-University, Vaxjo, Sweden, from Nov.27 to %%@
Dec.1, 2000. Proceedings will appear sometime in 2001.} }

\author{Johann Summhammer}
\address{Atominstitut der \"{O}sterreichischen Universit\"{a}ten,\\
Stadionallee 2, A--1020 Wien, Austria\\
Email: summhammer@ati.ac.at}
\maketitle
\begin{abstract}The acquisition and representation of basic experimental information %%@
under the probabilistic paradigm is analysed. The multinomial probability distribution is %%@
identified as governing all scientific data collection, at least in principle. For this distribution %%@
there exist unique random variables, whose standard deviation becomes asymptotically %%@
invariant of physical conditions. Representing all information by means of such random %%@
variables gives the quantum mechanical probability amplitude and a real alternative. For %%@
predictions, the linear evolution law (Schr\"odinger or Dirac equation) turns out to be the %%@
only way to extend the invariance property of the standard deviation to the predicted %%@
quantities. This indicates that quantum theory originates in the structure of gaining pure, %%@
probabilistic information, without any mechanical underpinning.
\end{abstract}

\section{Introduction}
The probabilistic paradigm proposed by Born is well accepted for comparing experimental %%@
results to quantum theoretical predictions \cite{Born}. It states that only the probabilities %%@
of the outcomes of an observation are determined by the experimental conditions. 
In this paper we wish to place this paradigm first. We shall investigate its consequences %%@
without assuming quantum theory or any other physical theory. We look at this paradigm as %%@
defining the method of the investigation of nature. This consists in the collection of %%@
information in probabilistic experiments performed under well controlled conditions, and in %%@
the efficient representation of this information. Realising that the empirical information is %%@
necessarily finite permits to put limits on what can at best be extracted from this information %%@
and therefore also on what can at best be said about the outcomes of future experiments. At %%@
first, this has nothing to do with {\em laws of nature}. But it tells us how optimal laws look %%@
like under probability. Interestingly, the quantum mechanical probability calculus is found as %%@
almost the best possibility. It meets with difficulties only when it must make predictions from %%@
a low amount of input information. We find that the quantum mechanical way of prediction %%@
does nothing but take the initial uncertainty volume of the representation space of the finite %%@
input information and move this volume about, without compressing or expanding it. However, %%@
we emphasize, that any mechanistic imagery of particles, waves, fields, even space, must be %%@
seen as what they are: The human brain's way of portraying sensory impressions, mere %%@
images in our minds. Taking them as corresponding to anything in nature, while going a long %%@
way in the design of experiments, can become very counter productive to science's task of %%@
finding laws. Here, the search for {\em invariant structures} in the empirical information, %%@
{\em without any models}, seems to be the correct path. Once embarked on this road, the %%@
old question of {\em how nature really is}, no longer seeks an answer in the muscular %%@
domain of mass, force, torque, and the like, which classical physics took as such %%@
unshakeable primary notions (not surprisingly, considering our ape nature, I cannot help %%@
commenting).  Rather, one asks: Which of the structures principally detectable in probabilistic %%@
information, are actually realized?

In the following sections we shall analyse the process of scientific investigation of nature %%@
under the probabilistic paradigm. We shall first look at how we gain information, then how %%@
we should best capture this information into numbers, and finally, what the ideal laws for %%@
making predictions should look like. The last step will bring the quantum mechanical time %%@
evolution, but will also indicate a problem due to finite information.

\section{Gaining experimental information}
Under the probabilistic paradigm basic physical observation is not very different from tossing %%@
a coin or blindly picking balls from an urn. One sets up specific conditions and checks what %%@
happens. And then one repeats this many times to gather statistically significant amounts of %%@
information. The difference to classical probabilistic experiments is that in quantum %%@
experiments one must carefully monitor the conditions and ensure they are the same for each %%@
trial. Any noticeable change constitutes a different experimental situation and must be %%@
avoided.\footnote{Strictly speaking, identical trials are impossible. A deeper analysis of %%@
why one can neglect {\em remote} conditions, might lead to an understanding of the notion %%@
of spatial distance, about which relativity says nothing, and which is badly missing in todays %%@
physics.}

Formally, one has a probabilistic experiment in which a single trial can give $K$ different %%@
outcomes, one of which happens. The probabilities of these outcomes, $p_1,...,p_K$, %%@
($\sum p_j = 1$), are determined by the conditions. But they are unknown. In order to find %%@
their values, and thereby the values of physical quantities functionally related to them, one %%@
does $N$ trials.
Let us assume the outcomes $j=1,...,K$ happen $L_1,...,L_K$ times, respectively ($\sum L_j %%@
=N$). The $L_j$ are random variables, subject to the {\em multinomial probability %%@
distribution}. Listing $L_1,...,L_K$ represents the complete information gained in the $N$ %%@
trials. The customary way of representing the information is however by other random %%@
variables, the so called relative frequencies $\nu_j \equiv L_j/N$. Clearly, they also obey %%@
the multinomial probability distribution.

{\em Examples:} 

\noindent
*  A trial in a spin-1/2 Stern-Gerlach experiment has two possible outcomes. This experiment %%@
is therefore goverend by the binomial probability distribution. 

\noindent
*  A trial in a GHZ experiment has eight possible outcomes, because  each of the three %%@
particles can end up in one of two detectors \cite{GHZ}. Here, the relative frequencies %%@
follow the multinomial distribution of order eight. 

\noindent
*  Measuring an intensity in a detector, which can only fire or not fire, is in fact an experiment %%@
where one repeatedly checks whether a firing occurs in a sufficiently small time interval. %%@
Thus one has a binomial experiment. If the rate of firing is small, the binomial distribution can %%@
be approximated by the Poisson distribution.

We must emphasize that the multinomial probability distribution is of utmost importance to %%@
physics under the probabilistic paradigm. This can be seen as follows: The conditions of a %%@
probabilistic experiment must be verified by auxiliary measurements. These are usually coarse %%@
classical measurements, but should actually also be probabilistic experiments of the most %%@
exacting standards. The probabilistic experiment of interest must therefore be done by %%@
ensuring that for each of its trials the probabilities of the outcomes of the {\it auxiliary} %%@
probabilistic experiments are the same. Consequently, empirical science is characterized by a %%@
succession of data-takings of multinomial probability distributions of various orders. The laws %%@
of physics are contained in the relations between the random variables from these different %%@
experiments. Since the statistical verification of these laws is again ruled by the properties of %%@
the multinomial probability distribution, we should expect that the {\it inner structure} of %%@
the multinomial probability distribution will appear in one form or another in the fundamental %%@
laws of physics.
In fact, we might be led to the bold conjecture that, under the probabilistic paradigm, basic %%@
physical law is no more than the structures implicit in the multinomial probability distribution. %%@
There is no escape from this distribution. Whichever way we turn, we stumble across it as the %%@
unavoidable tool for connecting empirical data to physical ideas.

The multinomial probability distribution of order $K$ is obtained when calculating the %%@
probability that, in $N$ trials, the outcomes 1,...,K occur $L_1,...,L_K$ times, respectively %%@
\cite{Feller}:
\be
Prob(L_1, ..., L_K|N, p_1, ..., p_K) = \frac{N!}{L_1!...L_K!}p_1^{L_1}...p_K^{L_K}.
\ee
The expectation values of the relative frequencies are 
\be
\bar{\nu}_j = p_j
\ee
and their standard deviations are
\be
\Delta \nu_j = \sqrt{\frac{p_j(1-p_j)}{N}}.
\ee

\section{Efficient representation of probabilistic information}
The reason why probabilistic information is most often represented by the relative frequencies %%@
$\nu_j$ seems to be history: Probability theory has originated as a method of estimating %%@
fractions of countable sets, when inspecting all elements was not possible (good versus bad %%@
apples in a large plantation, desirable versus undesirable outcomes in games of chance, etc.). %%@
The relative frequencies and their limits were the obvious entities to work with. But the %%@
information can be represented equally well by other random variables $\chi_j$, as long as %%@
these are one-to-one mappings $\chi_j(\nu_j)$, so that no information is lost. The question %%@
is, whether there exists a most efficient representation.

To answer this, let us see what we know about the limits $p_1,...,p_K$ {\em before} the %%@
experiment, but having decided to do $N$ trials. Our analysis is equivalent for all $K$ %%@
outcomes, so that we can pick out one and drop the subscript. We can use Chebyshev's %%@
inequality\cite{Chebyshev} to estimate the width of the interval, to which the probability %%@
$p$ of the chosen outcome is pinned down.\footnote{Chebyshev's inequality states: For %%@
any random variable, whose standard deviation exists, the probability that the value of the %%@
random variable deviates by more than $k$ standard deviations from its expectation value is %%@
less than, or equal to, $k^{-2}$. Here, $k$ is a free confidence parameter greater 1.}

 If $N$ is not too small, we get
\be
w_{p} = 2k \sqrt{\frac{\nu(1-\nu)}{N}},
\ee
where $k$ is a free confidence parameter. (Eq.(4) is not valid at $\nu$=0 or 1.) Before the %%@
experiment we do not know $\nu$, so we can only give the upper limit, 
\be
w_{p} \le \frac{k}{\sqrt{N}}.
\ee
But we can be much more specific about the limit $x$ of the random variable %%@
$\chi(\nu)$, for which we require that, at least for large $N$, the standard deviation %%@
$\Delta \chi$ shall be {\em independent} of $p$ (or of $x$ for that matter, since %%@
there will exist a function $p(x)$),
\be
\Delta \chi = \frac{C}{\sqrt{N}},
\ee
where $C$ is an arbitrary real constant. A straightforward analysis reveals
\be
\chi = C \arcsin\left(2\nu-1\right) + \theta, 
\ee
where $\theta$ is an arbitrary real constant.\cite{Wootters} For comparison with %%@
$\nu$ we confine $\chi$ to [0,1] and thus set $C =\pi^{-1}$ and $\theta=.5$. %%@
Then we have $\Delta \chi =1/(\pi \sqrt{N})$, and upon application of %%@
Chebyshev's inequality we get the interval $w_{x}$ to which we can pin down the %%@
unknown limit $x$ as
\be
w_{x} =\frac{2}{\pi}\frac{k}{\sqrt{N}}.
\ee
Clearly, this is narrower than the upper limit for $w_{p}$ in eq.(5). Having done no %%@
experiment at all, we have better knowledge on the value of $x$ than on the value of $p$, %%@
although both can only be in the interval [0,1]. And note that, the actual experimental data %%@
will add nothing to the accuracy with which we know $x$, but they may add to the accuracy %%@
with which we know $p$. Nevertheless, even with data, $w_p$ may still be larger than %%@
$w_x$, especially when $p$ is around 0.5. Figure 1 shows the relation of $\nu$ and %%@
$\chi$, and how the probability distribution of $\nu$ for various values of $p$ gets %%@
squeezed and stretched when plotting it for $\chi$.

For the representation of information the random variable $\chi$ is {\em the} proper %%@
choice, because it disentangles the two aspects of empirical information: The number of trials %%@
$N$, which is determined by the experimenter, not by nature, and the actual data, which are %%@
only determined by nature. The experimenter controls the accuracy $w_x$ by deciding $N$, %%@
nature supplies the data $\chi$, and thereby the whereabouts of $x$. In the real domain %%@
the only other random variables with this property are the linear transformations afforded by %%@
$C$ and $\theta$. From the physical point of view $\chi$ is of interest, because its %%@
standard deviation is an {\em invariant of the physical conditions} as contained in $p$ or %%@
$x$. The random variable $\chi$ expresses empirical information with a certain efficiency, %%@
eliminating a numerical distortion that is due to the structure of the multinomial distribution, %%@
and which is apparent in all other random variables. We shall call $\chi$ an {\em efficient %%@
random variable} (ER). More generally, we shall call any random variable an ER, whose %%@
standard deviation is asymptotically invariant of the limit the random variable tends to, eq.(6).

A graphical depiction of the relation between $\nu$ and $\chi$ can be given by drawing %%@
a semicircle of diameter 1 along which we plot $\nu$ (Fig.2a). By orthogonal projection %%@
onto the semicircle we get the random variable $\zeta=[\pi+2\arcsin(2\nu-1)]/4$ %%@
and thereby $\chi$, when we choose different constants. The drawing also suggests a %%@
simple way how to obtain a {\it complex} ER. We scale the semicircle by an arbitrary real %%@
factor $a$, tilt it by an arbitrary angle $\varphi$, and place it into the complex plane as %%@
shown in Fig.2b. This gives the random variable 
\be
\beta = a \left(\sqrt{\nu\left(1-\nu\right)} + i\nu\right) 
e^{-i\varphi} + b
\ee
where $b$ is an arbitrary complex constant. We get a very familiar special case by setting %%@
$a=1$ and $b = 0$:
\be
\psi = \left(\sqrt{\nu\left(1-\nu\right)} + i\nu\right) 
e^{-i\varphi}.
\ee
For large $N$ the probability distribution of $\nu$ becomes gaussian, but also that of any %%@
smooth function of $\nu$, as can be seen in Fig.1. Therefore the standard deviation of %%@
$\psi$ is obtained as 
\be
\Delta \psi=\left|\frac{\partial \psi}{\partial \nu }\right| %%@
\Delta \nu = \frac{1}{2\sqrt{N}}.
\ee
Obviously, the random variable $\psi$ is an ER. It fulfills %%@
$\left|\psi\right|^2=\nu$, and we recognize it as the {\it probability %%@
amplitude} of quantum theory, which  we would infer from the observed relative frequency %%@
$\nu$. Note, however, that the intuitive way of getting the quantum mechanical probability %%@
amplitude, namely, by simply taking $\sqrt{\nu}\exp(i \alpha)$, where $\alpha$ %%@
is an arbitrary phase, does {\it not} give us an ER.

We have now two ways of representing the obtained information by ERs, either the real %%@
valued $\chi$ or the complex valued $\beta$. Since the relative frequency of each of the %%@
$K$ outcomes can be converted to its respective efficient random variable, the result of a %%@
general probabilistic experiment is efficiently represented by the vector %%@
($\chi_1$,...,$\chi_K$), or by the vector ($\beta_1$,...,$\beta_K$). The latter is %%@
equivalent to the quantum mechanical state vector, if we normalize it: %%@
($\psi_1$,...,$\psi_K$).

At this point it is not clear, whether fundamental science could be built solely on the real ERs %%@
$\chi_j$ or whether it must rely on the complex ERs $\beta_j$, and for practical reasons %%@
on the normalized case $\psi_j$, as suggested by current formulations of quantum theory. %%@
We cannot address this problem here, but mention that working with the $\beta_j$ or %%@
$\psi_j$ can lead to nonsensical predictions, while working with the $\chi_j$ never %%@
does, so that the former are more sensitive to inconsistencies in the input data %%@
\cite{more.in.quantph}. Therefore we use only the $\psi_j$ in the next section, but will %%@
not read them as if we were doing quantum theory.

\section{Predictions}
Let us now see whether the representation of probabilistic information by ERs suggests %%@
specific laws for predictions. A prediction is a statement on the expected values of the %%@
probabilities of the different outcomes of a probabilistic experiment, which has not yet been %%@
done, or whose data we just do not yet know, on the basis of auxiliary probabilistic %%@
experiments, which have been done, and whose data we do know. We intend to make a %%@
prediction for a probabilistic experiment with $Z$ outcomes, and wish to calculate the %%@
quantities $\phi_s$, ($s=1,...,Z$), which shall be related to the predicted probabilities %%@
$P_s$ as $P_s = |\phi_s|^2$. We do not presuppose that the $\phi_s$ are ERs. 

We assume we have done $M$ different auxiliary probabilistic experiments of various %%@
multinomial order $K_m$, $m=1,...,M$, and we think that they provided all the input %%@
information needed to predict the $\phi_s$, and therefore the $P_s$. With (10) the %%@
obtained information is represented by the ERs $\psi_j^m$, where $m$ denotes the %%@
experiment and $j$ labels a possible outcome in it ($j=1,...,K_m$). Then the predictions are
\be
\phi_s = \phi_s(\psi_1^1,...,\psi_{K_1}^1, ... ... ..., %%@
\psi_1^M,...,\psi_{K_M}^M)
\ee
and their standard deviations are, by the usual convolution of gaussians as approximations of %%@
the multinomial distributions,
\be
\Delta \phi_s =\sqrt{ \sum_{m=1}^M \frac{1}{4 N_m}\left( %%@
\sum_{j=1}^{K_m} \left| \frac{\partial \phi_s}{\partial %%@
\psi_j^m}\right|^2 \right)},
\ee
where $N_m$ is the number of trials of the $m^{th}$ auxiliary experiment. 
If we wish the $\phi_s$ to be ERs, we must demand that the $\Delta \phi_s$ depend %%@
only on the $N_m$. (A technical requirement is that in each of the $M$ auxiliary %%@
experiments one of the phases of ERs $\psi_j^m$ cannot be chosen freely, otherwise the %%@
second summations in (13) could not go to $K_m$, but only to $K_m-1$.)
Then the derivatives in (13) must be constants, implying that the $\phi_s$ are {\em %%@
linear} in the $\psi_j^m$. 
However, we cannot simply {\em assume} such linearity, because (12) contains the laws of %%@
physics, which cannot be known {\em a priori}. But we want to point out that a linear %%@
relation for (12) has very exceptional properties, so that it would be nice, if we found it %%@
realized in nature. To be specific, if the
$N_m$ are sufficiently large, linearity would afford predictive power, which no other %%@
functional relation  could achieve: It would be sufficient to know the number of trials of each %%@
auxiliary probabilistic experiment in order to specify the accuracy of the predicted %%@
$\phi_s$. No data would be needed, only a decision how many trials each auxiliary %%@
experiment will be given! Moreover, even the slightest increase of the amount of input %%@
information, by only doing one more trial in {\em any} of the auxiliary experiments, would %%@
lead to better accuracy of the predicted $\phi_s$, by bringing a definite decrease of the %%@
$\Delta \phi_s$. This latter property is absent in virtually all other functional relations %%@
conceivable for (12). In fact, most nonlinear relations would allow {\em more} input %%@
information to result in {\em less} accurate predictions. This would undermine the very idea %%@
of empirical science, namely that, by observation our knowledge about nature can only %%@
increase, never just stay the same, let alone decrease. For this reason we assume linearity %%@
and apply it to a concrete example.

We take a particle in a one dimensional box of width $w$.
Alice repeatedly prepares the particle in a state only she knows. At time $t$ after the %%@
preparation Bob measures the position by subdividing the box into $K$ bins of width $w/K$ %%@
and checking in which he finds the particle. In $N$ trials Bob obtains the relative frequencies %%@
$\nu_1,...,\nu_K$, giving a good idea of the particle's position probability distribution at %%@
time $t$. He represents this information by the ERs $\psi_j$ of (10) and wants to use it to %%@
predict the position probability distribution at time $T$ ($T>t$).

First he predicts for $t+dt$. With (12) the predicted $\phi_s$ must be linear in the %%@
$\psi_j$ if they are to be ERs,
\be
\phi_s(t+dt)=\sum_{j=1}^K a_{sj}\psi_j.
\ee
Clearly, when $dt\rightarrow 0$ we must have $a_{sj}=1$ for $s=j$ and %%@
$a_{sj}=0$ otherwise, so we can write 
\be
a_{sj}(t)=\delta_{sj}+g_{sj}(t)dt,
\ee
where $g_{sj}(t)$ are the complex elements of a matrix ${\bf G}$ and we included the %%@
possibility that they depend on $t$. Using matrix notation and writing the $\phi_s$ and %%@
$\psi_j$ as column vectors we have
\be
\vec{\phi}(t+dt) = \left[{\bf 1}+{\bf G}(t)dt\right] \vec{\psi}.
\ee
For a prediction for time $t+2dt$ we must apply another such linear transformation to the %%@
prediction we had for $t+dt$, 
\be
\vec{\phi}(t+2dt) =\left[{\bf 1}+{\bf G}(t+dt)dt\right] %%@
\vec{\phi}(t+dt).
\ee
Replacing $t+dt$ by $t$, and using 
$\vec{\phi}(t+dt)=\vec{\phi}(t)+\frac{\partial\vec{\phi}(t)}
{\partial t}dt$,
we have
\be
\frac{\partial\vec{\phi}(t)}{\partial t}= {\bf G}(t) \vec{\phi}(t).
\ee
With (10) the input vector was normalized, $|\vec{\psi}|^2=1$. We also demand this %%@
from the vector $\vec{\phi}$. This results in the constraint that the diagonal elements %%@
$g_{ss}$ must be imaginary and the off-diagonal elements must fulfill $g_{sj}=-%%@
g_{js}^*$. And then we have obviously an evolution equation just as we know it from %%@
quantum theory.

For a quantitative prediction we need to know ${\bf G}(t)$ and the phases $\varphi_j$ %%@
of the initial $\psi_j$. We had assumed the $\varphi_j$ to be arbitrary. But now we see %%@
that they influence the prediction, and therefore they attain physical significance. ${\bf %%@
G}(t)$ is a unitary complex $K \times K$ matrix. For fixed conditions it is independent of %%@
time, and with the properties found above, it is given by $K^2-1$ real numbers. The initial %%@
vector $\vec{\psi}$ has $K$ complex components. It is normalized and one phase is %%@
free, so that it is fixed by $2K-2$ real numbers. Altogether $K^2+2K-3 = (K+3)(K-1)$ %%@
numbers are needed to enable prediction. Since one probabilistic experiment yields $K-1$ %%@
numbers, Bob must do $K+3$ probabilistic experiments with different delay times between %%@
Alice's preparation and his measurement to obtain sufficient input information. But neither %%@
Planck's constant nor the particle's mass are needed. It should be noted that this analysis %%@
remains unaltered, if the initial vector $\vec{\psi}$ is obtained from measurement of %%@
joint probability distributions of several particles. Therefore, (18) also contains entanglement %%@
between particles.


\section{Discussion}
This paper was based on the insight that under the probabilistic paradigm data from %%@
observations are subject to the multinomial probability distribution. For the representation of %%@
the empirical information we searched for random variables which are stripped of numerical %%@
artefacts. They should therefore have an invariance property. We found as unique random %%@
variables a real and a complex class of {\em efficient random variables} (ERs). They capture %%@
the obtained information more efficiently than others, because their standard deviation is an %%@
asymptotic invariant of the physical conditions. The quantum mechanical probability %%@
amplitude is the normalized case of the complex class. It is natural that fundamental %%@
probabilistic science should use such random variables rather than any others as the %%@
representors of the observed information, and therefore as the carriers of meaning. 

Using the ERs for prediction has given us an evolution prescription which is %%@
equivalent to the quantum theoretical way of applying a sequence of infinitesimal rotations to %%@
the state vector in Hilbert space\cite{Lande.Peres.Fivel}. It seems that simply analysing %%@
how we gain empirical information, what we can say from it about expected future %%@
information, and not succumbing to the lure of the question what is behind this information, %%@
can give us a basis for doing physics. This confirms the operational approach to science.
And it is in support of Wheeler's {\em It-from-Bit} hypothesis\cite{Wheeler}, %%@
Weizs\"acker's {\em ur-theory}\cite{Weizsacker}, Eddington's idea that information %%@
increase itself defines the rest\cite{Eddington}, Anandan's conjecture of {\em absence %%@
of dynamical laws}\cite{Anandan}, Bohr and Ulfbeck's hypothesis of {\em mere %%@
symmetry}\cite{Bohr.Ulfbeck} or the recent {\em 1 Bit --- 1 Constituent} hypothesis of %%@
Brukner and Zeilinger\cite{Caslav}. 

In view of the analysis presented here the quantum theoretical probability calculus is an %%@
almost trivial consequence of probability theory, but not as applied to 'objects' or anything %%@
'physical', but as applied to the naked data of probabilistic experiments. If we continue this %%@
idea we encounter a deeper problem, namely whether the space which we consider physical, %%@
this 3- or higher dimensional manifold in which we normally assume the world to unfurl %%@
\cite{Mohrhoff}, cannot also be understood as a peculiar way of representing data. Kant %%@
conjectured this - in somewhat different words - over 200 years ago \cite{Kant}. And %%@
indeed it is clearly so, if we imagine the human observer as a robot who must find a compact %%@
memory representation of the gigantic data stream it receives through its senses %%@
\cite{Jaynes}. That is why our earlier example of the particle in a box should only be seen %%@
as illustration by means of familiar terms. It should not imply that we accept the naive %%@
conception of space or things, like particles, 'in' it, although this view works well in everyday %%@
life and in the laboratory --- as long as we are not doing quantum experiments. We think that a %%@
full acceptance of the probabilistic paradigm as the basis of empirical science will eventually %%@
require an attack on the notions of spatial distance and spatial dimension from the point of %%@
view of optimal representation of probabilistic information.

Finally, we want to remark on a difference of our analysis to quantum theory. We have %%@
emphasized that the standard deviations of the ERs $\chi$ and $\psi$ become %%@
independent of the limits of these ERs only when we have infinitely many trials. But there is a %%@
departure for finitely many trials, especially for values of $p$ close to 0 and close to 1. With %%@
some imagination this can be noticed in Fig.1 in the top and bottom probability distributions of %%@
$\chi$, which are a little bit wider than those in the middle. But as we always have only %%@
finitely many trials, there should exist random variables which fulfill our requirement for an ER %%@
even better than $\chi$ and $\psi$. This implies that predictions based on these %%@
unknown random variables should also be more precise! Whether we should see this as a %%@
fluke of statistics, or as a need to amend quantum theory is a debatable question. But it %%@
should be testable. We need to have a number of different probabilistic experiments, all of %%@
which are done with only very few trials. From this we want to predict the outcomes of %%@
another probabilistic experiment, which is then also done with only few trials. Presumably, %%@
the optimal procedure of prediction will {\em not} be the one we have presented here (and %%@
therefore not quantum theory). The difficulty with such tests is however that, in the usual %%@
interpretation of data, statistical theory and quantum theory are treated as separate, while %%@
one message of this paper may also be that under the probabilistic paradigm the bottom level %%@
of physical theory should be equivalent to optimal representation of probabilistic information, %%@
and this theory should not be in need of additional purely statistical theories to connect it to %%@
actual data. We are discussing this problem in a future paper\cite{long.paper}.


\section*{Acknowledgments}
This paper is a result of pondering what I am doing in the lab, how it can be that in the %%@
evening I know more than I knew in the morning,  and discussing this with G. Krenn, K. Svozil, %%@
C. Brukner, M. Zukovski and a number of other people.


\section*{References}
\begin{thebibliography}{99}
\bibitem{Born} M. Born,
%%{\em Zur Quantenmechanik der Stossvorg"ange},
Zeitschrift f. Physik {\bf 37}, 863 (1926); 
%%{\em The Interpretation of Quantum Mechanics},
Brit. J. Philos. Science {\bf 4}, 95 (1953).
\bibitem{GHZ} D. Bouwmeester et al.,
%%{\em Observation of Three-Photon Greenberger-Horne-%%@
%%Zeilinger Entanglement},
Phys. Rev. Lett. {\bf 82}, 1345 (1999) and references therein.
\bibitem{Feller} W. Feller, {\em An Introduction to Probability Theory and its %%@
Applications}, (John Wiley and Sons, New York, 3rd edition, 1968), Vol.I, p.168.
\bibitem{Chebyshev} ibid., p.233.
\bibitem{Wootters} The connection of this relation to quantum physics was first stressed %%@
by W. K. Wootters, Phys. Rev. {\bf D 23}, 357 (1981).
\bibitem{more.in.quantph} We give the example in quant-ph/0008098.
\bibitem{Lande.Peres.Fivel} Several authors have noted that probability theory itself %%@
suggests quantum theory: 
A. Lande,
%%{\em Albert Einstein and the Quantum Riddle},
Am. J. Phys. {\bf 42}, 459 (1974);
A. Peres, {\em Quantum Theory: Concepts and Methods}, (Kluwer Academic Publishers, %%@
Dordrecht, 1998);
D. I. Fivel, 
%%{\em How interference effects in mixtures determine the %%@
%%rules of quantum mechanics},
Phys. Rev. {\bf A 50}, 2108 (1994).
\bibitem{Wheeler} J. A. Wheeler in {\em Quantum Theory and Measurement}, eds. J. %%@
A. Wheeler and W. H. Zurek (Princeton University Press, Princeton, 1983) 182.
\bibitem{Weizsacker}C. F. von Weizs\"acker, {\it Aufbau der Physik} (Hanser, %%@
Munich, 1985). Holger Lyre, Int. J. Theor. Phys., {\bf 34}, 1541 (1995). Also quant-%%@
ph/9703028.
\bibitem{Eddington} C. W. Kilmister, {\em Eddington's Search for a Fundamental %%@
Theory} (Cambridge University Press, 1994).
\bibitem{Anandan} J. Anandan,
%%{\it Are there dynamical laws?},
Found. Phys. {\bf 29}, 1647 (1999).
\bibitem{Bohr.Ulfbeck} A. Bohr and O. Ulfbeck,
%%{\it Primary manifestation of symmetry, Origin of quantal %%@
%%Indeterminacy.}
Rev. Mod. Phys. {\bf 67}, 1 (1995). 
\bibitem{Caslav} C. Brukner and A. Zeilinger, 
%%{\it Operationally Invariant Information in Quantum %%@
%%Measurements},
Phys. Rev. Lett. {\bf 83}, 3354 (1999).
\bibitem{Mohrhoff} A penetrating analysis of the view of space implied by quantum %%@
theory is given by U. Mohrhoff, Am. J. Phys. {\bf 68} (8), 728 (2000).
\bibitem{Kant} Immanuel Kant, {\em Critik der reinen Vernunft} (Critique of Pure %%@
Reason), Riga (1781). There should be many English translations.
\bibitem{Jaynes} E.T. Jaynes introduced the 'reasoning robot' in his book {\em %%@
Probability Theory: The Logic of Science} in order to eliminate the problem of subjectivism %%@
that has been plaguing probability theory and quantum theory alike. The book is freely %%@
available at http://bayes.wustl.edu/etj/prob.html
\bibitem{long.paper} J. Summhammer (to be published).
\end{thebibliography}

\begin{figure}[ht]
\begin{center}
\epsffile{Fc100.eps}
\caption[Fig.1:]{Functional relation between random variables $\nu$ and $\chi$, %%@
and their respective probability distributions as expected for $N=100$ trials, plotted for five %%@
different values of $p$: .07, .25, .50, .75 and .93. The bar above each probablity distribution
indicates twice its standard deviation. Notice that the standard deviations of $\nu$ differ %%@
considerably for different $p$, while those of $\chi$ are all the same, as required in %%@
eq.(6)}
\end{center}
\end{figure}

\begin{figure}[ht]
\begin{center}
\epsffile{Nu_psi.eps}
\caption[Fig.2:]{{\bf (a)} Graphical construction of efficient random variable %%@
$\zeta$ (and thereby of $\chi$) from the observed relative frequency $\nu$. %%@
$\zeta$ is measured along the arc. {\bf (b)} Similar construction of the efficient random %%@
variable $\beta$. It is given by its coordinates in the complex plane. The quantum %%@
mechanical probability amplitude $\psi$ is the normalized case of $\beta$, obtained by %%@
setting $a=1$ and $b=0$.}
\end{center}
\end{figure}

\end{document}

quant-ph0102099.extracted_bib
\begin{thebibliography}{99}
\bibitem{Born} M. Born,
%%{\em Zur Quantenmechanik der Stossvorg"ange},
Zeitschrift f. Physik {\bf 37}, 863 (1926); 
%%{\em The Interpretation of Quantum Mechanics},
Brit. J. Philos. Science {\bf 4}, 95 (1953).
\bibitem{GHZ} D. Bouwmeester et al.,
%%{\em Observation of Three-Photon Greenberger-Horne-%%@
%%Zeilinger Entanglement},
Phys. Rev. Lett. {\bf 82}, 1345 (1999) and references therein.
\bibitem{Feller} W. Feller, {\em An Introduction to Probability Theory and its %%@
Applications}, (John Wiley and Sons, New York, 3rd edition, 1968), Vol.I, p.168.
\bibitem{Chebyshev} ibid., p.233.
\bibitem{Wootters} The connection of this relation to quantum physics was first stressed %%@
by W. K. Wootters, Phys. Rev. {\bf D 23}, 357 (1981).
\bibitem{more.in.quantph} We give the example in quant-ph/0008098.
\bibitem{Lande.Peres.Fivel} Several authors have noted that probability theory itself %%@
suggests quantum theory: 
A. Lande,
%%{\em Albert Einstein and the Quantum Riddle},
Am. J. Phys. {\bf 42}, 459 (1974);
A. Peres, {\em Quantum Theory: Concepts and Methods}, (Kluwer Academic Publishers, %%@
Dordrecht, 1998);
D. I. Fivel, 
%%{\em How interference effects in mixtures determine the %%@
%%rules of quantum mechanics},
Phys. Rev. {\bf A 50}, 2108 (1994).
\bibitem{Wheeler} J. A. Wheeler in {\em Quantum Theory and Measurement}, eds. J. %%@
A. Wheeler and W. H. Zurek (Princeton University Press, Princeton, 1983) 182.
\bibitem{Weizsacker}C. F. von Weizs\"acker, {\it Aufbau der Physik} (Hanser, %%@
Munich, 1985). Holger Lyre, Int. J. Theor. Phys., {\bf 34}, 1541 (1995). Also quant-%%@
ph/9703028.
\bibitem{Eddington} C. W. Kilmister, {\em Eddington's Search for a Fundamental %%@
Theory} (Cambridge University Press, 1994).
\bibitem{Anandan} J. Anandan,
%%{\it Are there dynamical laws?},
Found. Phys. {\bf 29}, 1647 (1999).
\bibitem{Bohr.Ulfbeck} A. Bohr and O. Ulfbeck,
%%{\it Primary manifestation of symmetry, Origin of quantal %%@
%%Indeterminacy.}
Rev. Mod. Phys. {\bf 67}, 1 (1995). 
\bibitem{Caslav} C. Brukner and A. Zeilinger, 
%%{\it Operationally Invariant Information in Quantum %%@
%%Measurements},
Phys. Rev. Lett. {\bf 83}, 3354 (1999).
\bibitem{Mohrhoff} A penetrating analysis of the view of space implied by quantum %%@
theory is given by U. Mohrhoff, Am. J. Phys. {\bf 68} (8), 728 (2000).
\bibitem{Kant} Immanuel Kant, {\em Critik der reinen Vernunft} (Critique of Pure %%@
Reason), Riga (1781). There should be many English translations.
\bibitem{Jaynes} E.T. Jaynes introduced the 'reasoning robot' in his book {\em %%@
Probability Theory: The Logic of Science} in order to eliminate the problem of subjectivism %%@
that has been plaguing probability theory and quantum theory alike. The book is freely %%@
available at http://bayes.wustl.edu/etj/prob.html
\bibitem{long.paper} J. Summhammer (to be published).
\end{thebibliography}