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\title{Equations Of Physics Become Consistent 
If We Take Measurement Uncertainty Into Consideration}
\author{A. B. Korlyukov$^1$ and V. Kreinovich$^2$}

\pagestyle{myheadings}

\markright{APIC'95, El Paso,
Extended Abstracts,
A Supplement to the international journal of {\rm Reliable
Computing}\ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }

\maketitle

\auffil{The authors are from $^1$Grodno, Belarus, and from the 
$^2$Department of Computer
Science, University of Texas at El Paso, El Paso, TX 79968, email
vladik@cs.utep.edu. V. Kreinovich is thankful to A. Neumaier for
inspiring discussions. This work was partially 
supported by NSF grant No. CDA-9015006 and NASA Research Grant No.
9-757.}


\section{Classical Physics: Idealized Description}
In classical physics, the world consists of {\it particles} and {\it fields}.
Therefore, to describe the current state of the world, we must
describe:
\begin{itemize}
\item the current positions $\vec x^{(1)}, \vec x^{(2)}, ...$ and velocities
$\vec v^{(1)}, \vec v^{(2)}, ...$ of all the particles, and 
\item the current
values $f(\vec x)$ of all the fields $f$ in all spatial points $\vec x$.
\end{itemize}
Physical equations describe how a state changes with time. 
So, if we can measure a current state, we will be able to predict the
future states (this idea is called {\it Laplace determinism}). 

\section{In Classical Physical, We Can (Potentially) Measure With
Arbitrary Accuracy}
Of course, we cannot measure the state with absolute accuracy, but
nothing in classical physics prevents us from knowing it with better and
better accuracy. For example, a realistic field probe has non-zero
thinness, and thus, it measures not the value of the field in one spatial
point, but an average over all the points covered by this probe. 
However, we can imagine thinner and thinner probes
that can compute the averages over the smaller and smaller areas
surrounding a given point $\vec x$, and therefore, produce the
measurement results that are closer and closer to the desired value
$f(\vec x)$. 

\section{Quantum Physics: Uncertainty Principle}
The major change that has been introduced by quantum physics is related
to the so-called {\it uncertainty principle}. According to this principle, the
better spatial accuracy we want to achieve, the more energy we have to
pump into the system. If we pump lots of energy into a small area of
space, then we will sure disturb the very process that we want to
measure. 

As a result, if we use thinner and thinner field probes in the
vicinity of a given spatial point $\vec x$, we are measuring the more
and more disturbed field. Therefore, if we want to follow the idea 
of classical mechanics and define the ``true'' value $f(\vec x)$ of a
physical field $f$ at a point $\vec x$ as a limit of the values that we
get when we apply thinner and thinner probes, then, instead of the
sequence of values that converge to a 
physically meaningful finite limit, we get a divergent process that
does not have a finite limit.

\section{Divergence Problem}
This phenomenon leads to 
a serious problem of quantum physics: Indeed, quantum equations
are usually obtained by modifying the classical ones, and therefore,
these equations are usually formulated in terms of point values of the
fields. If we try to integrate these differential equations, we thus get
senseless infinite results for $f(\vec x)$ (called {\it divergences}).

\section{The Existing Solutions to This Problem}
There exist several semi-formal tricks that help to overcome this problem
and come up with finite predictions for experimentally observable
quantities. It would be nice, however, to use some physically valid
and mathematically idea
instead of the heuristic tricks.

\section{Our Main Idea}
From our viewpoint, however, the most reasonable approach will be
not to use any mathematical formal tricks, but to nip this problem in
the bud by avoiding physically meaningless point values of the fields,
and thus, to describe the fields in terms of observable quantities only. 

In other words, we want to describe a physical field by the values that
are measured by different field probes. 

\section{This Idea Leads to 9-Dimensional Space}
As we have already mentioned, a
probe cannot be located exactly in a point $\vec x$. No matter how
accurately we locate it, there is always an inevitable location error.
This error is unpredictable and can be, thus, viewed as random. 

There can be many different
sources of a location error. If we have one prevailing error source,
depending on one or few parameters, then
usually, we can compensate for this particular error. So, if a field
probe has been carefully designed, all dominating errors have been
eliminated, and the remaining error is the result of the joint action of
several small independent error components of similar size. According to
the limit theorems of mathematical statistics (see, e.g., 
\cite{Gnedenko 1954}, \cite{Arak 1988}, \cite{Wardsworth 1990}, 
pp. 2.17,
6.5, 9.8, and references therein),
this
error is thus distributed according to the Gaussian law, with the
density $$\rho(x_1,x_2,x_3)=const\cdot \exp(-{1\over 2}\sum_{ij}
A_{ij}\Delta x_i\Delta x_j),$$ where:
\begin{itemize}
\item $\Delta x_i=x_i-a_i$ is the
location error, i.e., the difference between the actual coordinates
$x_i$ and the desired coordinates $a_i$, and 
\item $A_{ij}$ is a symmetric
matrix that describes how small the locations errors are. 
\end{itemize}
So, to describe a probe, we
need to know 9 parameters: 
\begin{itemize}
\item 3 parameters $a_i$ that describe the
desired location, and 
\item 6 parameters $A_{ij}$ that characterize the
location error. 
\end{itemize}
Let us denote this set of parameters by $$\vec
p=(a_1,a_2,a_3,A_{11},A_{12},A_{13},A_{22},A_{23},A_{33}).$$

To describe the quantum field, we need to know the values measured by
every possible probe. In other words, for each set of 9 parameters
$\vec p$, we must know the ``average''
value $\bar f(\vec p)$ of the field that is measured by the
corresponding probe.
So:
\begin{itemize}

\item if we describe a classical field in observable terms, 
it is sufficient to describe a
function of 3 spatial variables. 

\item To describe a quantum field, we need
a function of 9 variables. 
\end{itemize}

Mathematically, it is equivalent to saying
that we consider arbitrary fields in a {\it 9-dimensional space}. 

\section{9-D space Means 10-D Space-Time}
In the above calculations, we considered the state of the field at a
given moment of time. If we take time into consideration, we will
arrive at a conclusion that to describe how a field evolves, we need
a function $\bar f(\vec p,t)$ 
of 10 parameters (time is an additional parameter). Mathematically,
this is equivalent to saying that we have to consider a {\it 10-dimensional}
space-time. 

\section{In 10-D Space-Time, Quantum Field Theory Is Consistent}
It is interesting to mention that 10 is indeed the smallest dimension
of space-time for which a consistent (``{\it renormalizable}'') theory
is possible (see, e.g., \cite{Brink 1988,Siegel 1988}). 

\section{Connection Between Our Idea And String Theory}
Another proof
that the above idea makes sense is as follows: 
\begin{itemize}

\item In our interpretation,
instead of points in space, we consider distributed space areas. 

\item The
mathematical formalism that underlies the correspondent consistent
theory is a so-called {\it string theory}, in which the basic objects
are also not point particle, but spatially distributed objects
(``strings''). 
\end{itemize}

So, our idea can be viewed as a (natural) generalization of a string
theory: instead of 1-D objects, we have 3-D ones. 

\noindent{\it Comment.} Another application of interval approach to
foundations of physics is described in \cite{Korlyukov 1992}.

\begin{thebibliography}{99}

\bibitem{Arak 1988}
T. V. Arak and A. Yu. Zaitsev, {\bf Uniform limit theorems for
sums of independent random variables},
(Proceedings of the Steklov Institute of
Mathematics, Vol. 174), American Mathematical Society,
Providence, RI, 1988.

\bibitem{Brink 1988}
L. Brink and M. Henneaux, {\bf Principles of string theory}, Plenum Press,
N.Y., 1988. 

\bibitem{Gnedenko 1954}
B. V. Gnedenko and A. N. Kolmogorov, {\it Limit distributions
for sums of independent random variables}, Addison-Wesley, Cambridge,
1954. 

\bibitem{Korlyukov 1992}
A. V. Korlyukov, 
``Introduction to interval field theory'',
{\it Proceedings of
the International Conference on Interval and Stochastic Methods in
Science and Engineering INTERVAL`92}, Moscow, 1992, Vol. 1, pp. 71--73
(in Russian; English abstract Vol. 2, p. 46). 

\bibitem{Siegel 1988}
W. Siegel, {\bf Introduction to String Filed Theory}, World
Scientific, Singapore, 1988.

\bibitem{Wardsworth 1990}
H. M. Wardsworth, Jr. (editor), {\bf Handbook of statistical
methods for engineers and scientists}, McGraw-Hill Publishing Co.,
N.Y., 1990. 

\end{thebibliography}

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