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\centerline{Indeterminacy, Time Dilation and Higher Dimensionality}
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\centerline{from a Classical Model}
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\centerline{\sl Hans P. Moravec}
\centerline{\sl Robotics Institute}
\centerline{\sl Carnegie-Mellon University}
\centerline{\sl Pittsburgh, PA 15213}
\centerline{\sl (412) 421-6441}
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\centerline{March 15, 1985}
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\proclaim ABSTRACT.  We consider wave modes in a medium behaving according
to classical statistical mechanical laws.  A mathematical operation converts
the usual space domain description of the medium into an equivalent
description in the spatial frequency domain.  Each position in the frequency
domain description represents a particular wave mode in the space domain.
In addition to random statistical transfers of energy between wave modes, we
assume that non-linearities in the medium permit energy to flow
systematically between wave modes near each other in frequency and
orientation.  The locations in the frequency domain description may be
interpreted as a three dimensional space with a physics of its own.
Interactions in the frequency domain are characterized by quantum
indeterminacy, variable local time rates and sometimes a complex topology
that mimics extra dimensions of restricted size.  The resemblance to modern
physical theories of the real world is tantalizing. The suggestion is
that the world as we know it may be a global transform of an underlying
space with different, possibly deterministic, local interactions. Our
fundamental small scale laws would be statistical consequences of large
scale behavior in this space.
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\def\ispace/{{\bf I}-{\it space}}
\def\uspace/{{\bf U}-{\it space}}
\def\c{{\bf c}}
\def\k{{\bf k}}

\beginsection Introduction

This paper examines a class of physical models where intuitively apparent
local interactions are explained by a non-intuitive underlying space that
can be derived from the intuitive one only by a globally applied orthogonal
transformation.  The spatial Fourier transform is a well developed example
of this kind of operation and will be used as a prototype for the class.
Under the Fourier model each point in the intuitive space (called \ispace/)
corresponds to a wave train of a particular frequency and orientation that
spans the underlying space (called \uspace/). For concreteness imagine that
we are discussing waves of density.  If \uspace/ is three dimensional,
\ispace/ will have three dimensions, one corresponding to the frequency of
the waves in \uspace/ and two depending on their orientation. \ispace/ may be
thought of as spherical, with the zero frequency mode at the center, with
successive concentric shells of ever higher frequency modes.

	Let \uspace/ have a wave speed and information speed limit of $\c$
and a size described by a diameter $D$, and the equilibrium characteristics
of classical thermodynamics.  \ispace/ will then exhibit indeterminacies
resembling those of quantum mechanics.  For instance, the energy at a point
in \ispace/ is the energy in a particular wave mode in \uspace/, found by an
integration over the entire \uspace/ volume. Since \uspace/ has an
information speed limit, determining this energy to maximum accuracy
requires a time $D/c$.  Because of the thermodynamic noise in \uspace/ even
this value will be accurate only to a certain precision. If the integration
is carried out over a shorter time, involving a smaller volume of \uspace/,
the energy uncertainty will be higher in proportion. The \ispace/
measurement thus exhibits an energy/time uncertainty duality, as in quantum
mechanics.  Curiously the ``Planck's constant'' in \ispace/ is inversely
proportional to the ``speed of light'', $\c$, in \uspace/.

	There is an even more interesting resemblance to quantum
descriptions. When two wave modes in \uspace/ are very similar to each other
in frequency and orientation we assume they may exchange energy in a
systematic way through non-linearities in the medium.  This corresponds to
energy moving from a point to a nearby one in \ispace/.  The interaction is
systematic because the wave trains are correlated for long spatial and
temporal periods. Conversely, wave modes differing greatly in frequency and
orientation show no such correlation, and appear to each other only as
random noise. The characteristic interaction time in the nearby (in
\ispace/), correlated (in \uspace/) case is proportional to the temporal
period of the waves. Under our assumptions if the wavelength in \uspace/ is
$\lambda$, this time is $\lambda/c$. There is no time in so small an
interval for information about the interaction in one cycle along the
wavetrain to communicate itself with any other.  If more than one kind of
interaction is possible, this independence will allow different interactions
to happen at different points along the wavetrain, encouraged by the high
thermal noise level seen at such a small scale.  This is a model for
quantum-mechanical superposition of states, with the possible interactions
being enumerated in a statistical fashion, as opposed to the systematic
integrations or summations found in present quantum theories.  If $D$ is
sufficiently small this difference should be experimentally discernable in
the second order statistics of some experiments.

	The characteristic time of interactions at particular points in
\ispace/ depends on the period of the corresponding wave mode in \uspace/.
This period is longer at the low frequencies near the center of the \ispace/
than at the higher frequency regions farther out. Since all interactions,
including information transfer, in \ispace/ are paced by this constant,
there should be an increase in the subjective time rate with distance from
the center.  This may be a model for gravitational time dilation effects.

	Although energy mainly flows between wave modes very similar in
frequency and direction (i.e. between points adjacent in \ispace/),
non-linearities in the \uspace/ medium should permit some energy to flow
systematically between harmonically related wave modes.  Thus, energy in
\ispace/ will appear to be free to move not only with three degrees of
freedom among contiguous points, but in a few extra selected directions as
well. Globally the effect is to complicate the topology of \ispace/, but
locally it can be interpreted as the existence of extra dimensions.  Since a
circumnavigation from harmonic to harmonic will cover the available space\
in fewer steps than a move along adjacent wave modes, these extra dimensions
will appear to have a much smaller extent than the basic three.  The greater
the energy involved, the more harmonics are activated, and the higher the
dimensionality. These properties are similar to ones required in
Kaluza-Klein theories of the basic forces.

	Although our \uspace/ model has an explicit ennumeration of the
various possibilities in an uncollapsed wave function as in the many-worlds
interpretation of quantum mechanics, and the higher dimensionality required
by the Kaluza-Klein theories, it does not share the profligacy of those two
theories.  The multiple universes of the many-worlds interpretation are
neatly folded on top of one another, so that the ennumeration of the
possible states of one particle becomes the ennumerating background noise of
another.  Similarly the extra, but diminutive, dimensions of the
Kaluza-Klein theories are folded into the basic three (or four, if you count
time), also participating in the noise.

	It can also be noted that our construction can be generalized to
more than one orthogonalization of \uspace/, opening the possibility of many
overlapping universes, each appearing to the others primarily as noise.
It may, of course, be moot to ask which one is the {\it real} \uspace/.
\bye