The Harmonies of the Sphere
Hans Moravec
The Robotics Institute
Carnegie-Mellon University
Pittsburgh, PA 15213
(412) 268-3829
Copyright 1986 by Hans P. Moravec
Quantum Mechanics, a cornerstone of modern physics, has
indeterminism at its heart and soul. Outcome probabilities in quantum
mechanics are predicted by summing up complex valued ``amplitude
functions'' for all the indistinguishable ways a given event might
happen, then squaring the result. The amplitudes subtract from each
other as often as they add, with the strange effect that some
otherwise possible outcomes are ruled out by the existence of other
possibilities. A excellent example is the {\it two slit} experiment.
Photons of light radiate from a pinpoint source to a screen broken by
two slits (Figure \ref{Slits}). Those that make it through the slits
encounter an array of photon detectors (often a photographic film, but
the example is clearer if we use individual immediately responding
sensors). If the light source is so dim that only one photon is
released at a time, the sensors register individually, sometimes this
one, sometimes that one. Each photon lands in exactly one place.
But, if a count is kept of how many photons have landed on each
detector, an unexpected pattern emerges. Some detectors see no
photons at all, while ones close to them on either side register many,
and a little farther away there is again a dearth. In the long run a
pattern builds that is identical with the banded interference pattern
one would see if two matched waves were being emitted from sources at
the slits.
\begin{figure}
\vspace{7in}
\caption[Two Slit Experiment]{\label{Slits}
{\bf Two Slit Experiment - }
A photon picked up by a detector at screen {\bf S} {\it might} have
come through slit {\bf A} or through slit {\bf B} - there is no way to
distinguish. In Quantum Mechanics the ``amplitudes'' for the two
cases must be added. At some points on the screen they add
constructively, making it likely that a photon will end up there, at
nearby points the amplitudes cancel, and no photons are ever found.}
\end{figure}
But waves of {\it what}? Each photon starts from one place
and lands in one place; isn't it at just one place on every part of
its flight? Doesn't it go through one slit or the other? If so, how
does the mere existence of the other slit {\it prevent} it from
landing at a certain place on the screen? For indeed, if one slit if
blocked, the total number of photons landing on the screen is halved,
but the interference pattern vanishes, and some locations that
received no photons with both slits open begin to register hits.
Quantum Mechanics' answer is that during the flight the position of
the photon is unknown, and must be modeled by a complex valued wave
describing all its possible locations. This ghostly wave passes
through {\it both} slits (though it describes the position of only a
single photon), and interferes with itself at the screen, cancelling
at some points. There the wave makes up its mind, and the photon
appears in just one of its possible locations. The undecided, wave,
condition of the photon before it hits the screen is called a {\it
mixed state} or a {\it superposition of states}. The sudden
appearance of the photon in only one detector is called the {\it
collapse of the wave function}.
This explanation profoundly disturbed some of the same
physicists who had helped formulate the theory, notably Albert
Einstein and Erwin Schr\"odinger. To formalize their intuitive
objections they constructed thought experiments that gave unlikely
results according to the theory. In some a measurement made at one
site causes the instant collapse of a wave function at a remote
location - an effect faster than light. In another, more frivolous,
example, called Schr\"odinger's Cat, a radioactive decay that may or
may not take place in a sealed box causes (or fails to cause) the
death of a cat also in the box. Schr\"odinger considered absurd the
theory's description of the unopened box as a mixed state
superimposing a live and a dead cat. He suggested that the theory
merely expressed ignorance on the part of an observer - in the box the
cat's fate was unambiguous. This is called a {\it hidden variables}
theory - the system has a definite state at all times, but some parts
of it are temporarily hidden from some observers.
The joke is on the critics. Many of the most ``absurd''
thought experimental results have been observed in mind boggling
actuality in a series of clever (and very modern) experiments carried
out by Alain Aspect at the University of Paris, and others. These
demonstrations rule out the simplest and most natural hidden variables
theories, {\it local} ones, in which, for instance the hidden
information about which slit the photon went through is contained in
the photon itself, or in which the state of health of Schr\"odinger's
cat is part of the feline.
{\it Non-local} hidden variables theories, where the
unmeasured information is distributed over an extended space, {\it
are} a possibility. It is easy to construct theories of this kind
that give results identical with ordinary Quantum Mechanics. Most
physicists find them uninteresting - why introduce a more complicated
explanation with extra variables, when the current, simpler, equations
suffice? Philosophically, also, global hidden variables theories are
only slightly less puzzling than raw Quantum Mechanics. What does it
mean that the ``exact position'' of a particle is spread out over a
large chunk of space? This question was the subject of a lively
controversy in the early part of this century among the founders of
Quantum Mechanics. It's recently become of widespread interest again.
Quantum mechanical interactions have a ``spooky'' character
clearly evident in the two slit experiment, and recently emphasized by
physical demonstrations of the Einstein-Podolsky-Rosen paradox by
Aspect and others. The ghosts can be exorcised, or at least
elucidated, by proposing underlying mechanisms for the basic
effects. These mechanisms often suggest radical new possibilities.
The ``many worlds'' interpretation developed in the 1950s by
Hugh Everett and John Wheeler at Princeton, and frequently presented
by John Gribbin in these pages (for instance in the April 1985 issue),
may be the most profligate non-local hidden variables explanation of
this puzzle. In Everett's model the two slit photon does go through
both slits, {\it in different universes}. At each decision point the
entire universe, or at least the immediate portion of it, splits into
several, like multiple pages from a copying machine. Until a
measurement is made the different ``versions'' of the universe lie in
close proximity, and interfere with each other (causing banded
patterns on screens, for instance). A measurement that can distinguish
one possibility from another causes the universes to diverge
(alternately the divergence is the definition of ``measurement'').
The interference then stops, and in each, now separate, universe a
different version of the experimenter can contemplate a different
unambiguous result.
Another possibility, outlined in the November 1986 {\bf
Analog} by John Cramer, is his ``transactional'' interpretation,
itself based on an old explanation by Feynman and Wheeler for the lack
of time-reversed waves implicit in Maxwell's equations. In it observer
and observed communicate with signals travelling both ways in time, so
the outcome of the experiment is as much part of the initial condition
as the experimental setup. Or perhaps the universe is a computation
in some kind of machine. Quantum effects might be the result of
limited accuracy and parsimony of calculation in its program. The
equations of Quantum Mechanics implicitly state that the amount of
information that can be extracted from a limited volume of spacetime
is finite. Also, with proper encoding, the undecided state of a system
contains less information than after a measurement. Only during actual
measurements must the "universe computer" bother to choose one outcome
from all the possibilities. Here, for the first time, I offer yet
another, in some ways less radical (but half baked!) mechanism for
Quantum Mechanics. I like it because it derives the spookiest
consequences from a very concrete model.
\sect{One World, not Many}
Imagine, somewhere, there is a spherical volume uniformly
filled with a gas made up of a huge but finite number of particles in
motion. Pressure waves pass through the gas, propagating at its speed
of sound, $s$. We assume no faster signal can be sent (the exact
properties required of the medium will have to be developed elsewhere
- here we deal only in generalities!). The sphere has resonances that
correspond to wave trains passing through its entire volume at
different angles and frequencies. Each combination of a particular
direction and frequency is called a {\it wave mode}. There is a
mathematical transformation called the (spatial) {\it Fourier
Transform} that arranges these wave modes very neatly and
powerfully. The Fourier transform combines the pattern of pressures
found over the original volume of the sphere ($V$) in various ways to
produce a new spherical set of values ($F$). At the center of $F$ is a
number representing the average density of $V$. Immediately
surrounding it are (complex) numbers giving the intensity of waves, in
various directions, whose wavelength is so long that one cycle spans
the diameter of $V$. Twice as far from the center of $F$ are found the
intensities of wave modes with two cycles across $V$, and so on. Each
point in $F$ describes a wave filling $V$ with a direction and a
number of cycles given by the point's orientation and distance from
the center of $F$. Another way of saying this is direction in $F$
corresponds to direction in $V$, radius in $F$ is proportional to
frequency in $V$. Since each wave is made of periodic clusterings of
gas particles, the interparticle spacing sets a lower bound on the
wavelength, thus an upper bound on frequency, and a limit on the
radius of the $F$ sphere. The closer the particles, the larger must be
$F$. A theorem about Fourier transforms states that if sufficiently
high frequencies are included, then $F$ contains about as many points
as $V$ has particles, and all the information required to reconstruct
$V$ is found in $F$. In fact $F$ and $V$ are simply alternative
descriptions of the same thing, with the interesting property that
every particle in $V$ contributes to each point in $F$, and vice
versa.
If the particles in $V$ bump into one another, or interact in
some other way (i.e. the gas is {\it nonlinear}), then energy can be
transferred from one wave mode to another - i.e. one point in $F$ can
become stronger at the expense of another. There will be a certain
amount of random transference among all wave modes. Besides this there
will be a more systematic interaction between ``nearby'' wave modes -
those very similar in frequency and orientation, thus near each other
in the $F$ space. Such waves will be in step for large fractions of
their length. Because the gas is nonlinear, the periodic bunching of
gas particles caused by one mode will influence the bunching ability
of a neighboring mode with a similar period.
Now consider the interactions viewed by a hypothetical
observer made of $F$ stuff, for whom points in $F$ are simply
locations, rather than complicated functions of another space.
Keeping as many concepts from the $V$ space as possible, we can deduce
some of this observer's ``Laws of Physics'' by reasoning about effects
in $V$, and translating back to $F$. In the following list, such
reasoning is in italics:
\begin{itemize}
\item {\bf Dimensionality:} If $V$ is three dimensional, so is
$F$. {\it Two of its dimensions correspond to angular direction of the
wavetrains, the other, the radius, corresponds to frequency.}
\item {\bf Locality:} Points near to each other in $F$ can exchange
energy in consistent, predictable ways while distant points
cannot. {\it Two wave trains in $V$ that are very similar in direction
and frequency are in step for a long portion of their length, and the
non-linear bunching effects will be roughly the same cycle after
cycle. Distant wave modes, whose crests and troughs are not
correlated, will lose here, and gain there, and in general appear like
mere random buffetings to each other.}
\item {\bf Interaction Speed:} There is a characteristic speed at each
point in $F$. Points far away from the center of $F$ interact more
quickly than those closer in. {\it An interaction is the non-random
transfer of energy from one wavetrain to another. The smallest
repeated unit in a wavetrain is a cycle. An effect which happens in a
similar way at each cycle can have a consistent effect on a whole wave
train. Effects in $V$ propagate at the speed of sound, so a whole
cycle can be affected in the time it takes sound to traverse it (which
is also the temporal frequency of the wavetrain). The outer parts of
$F$ correspond to higher frequencies, and thus to faster rates.}
\item {\bf Uncertainty Principle:} The energy of a point in $F$ can't
be determined precisely in a short time. The best accuracy possible
improves linearly with duration of the measurement. {\it The energy at
a point in $F$ is the total energy of a particular wavetrain that
spans the entire volume $V$. As no signal in $V$ can travel faster
than the speed of sound, discovering the total energy in a wavetrain
would involve waiting for signals to arrive from all over $V$, a time
much longer than the basic interaction time. If a short time the
summation is necessarily over a proportionately small volume. Since
the observer in $F$ is itself distributed over $V$, exactly {\rm
which} smaller volume is not defined - and thus the measurement is
uncertain. As the time, and the summation volume, increases, all the
possible sums converge to the average, and the uncertainty decreases.}
\item {\bf Superposition of States:} Most interactions in $F$ will
appear to be the sum of many possible ways the interaction might have
happened. {\it When two nearby wavetrains interact, they do so
initially on a cycle by cycle basis, since information from distant
parts of the wavetrain arrives only at the speed of sound. Each cycle
contains a little energy from the wavetrain in question, and a lot of
energy from many other waves of different frequency and orientation
passing through the same volume. This ``background noise'' will be
different from one cycle to the next, so the interaction at each cycle
will be slightly different. When all is said an done, i.e. if the
information from the entire wavetrain is collected, the total
interaction can be interpreted as the sum of the cycle by cycle
interactions. Sometimes energy will be transferred one way by one
cycle, and the opposite way by a distant one, so the alternatives can
cancel as well enhance one another.}
\end{itemize}
These and other properties of the $F$ world contain some of
the strangest features of Quantum Mechanics, but are the consequence
only of an unusual way of looking at a prosaic situation. There are a
few differences. The superposition of states is statistical, rather
than a perfect sum over all possibilities as in traditional Quantum
Mechanics. This makes only a very subtle difference if $V$ is very
large, but might result in a very tiny amount of ``noise'' in
measurements that could help distinguish the $F$ mechanism from other
explanations of Quantum Mechanics. The model as presented does not
model the effects of special relativity in any obvious way, and this
is a serious defect, if we hope to wrestle it into a description of
our world. There is something wrong in the way it treats time. It
does have one property that mimics the temporal effects of a general
relativistic gravitational field. Time near the center of $F$ runs
more slowly than at the extremes, since the interactions are based on
lower frequency waves. At the very center, time is stopped. The
central point of $F$ never changes its ``average energy of the whole
sphere'' value, and so is effectively frozen in time. In general
relativity the regions around a gravitating body have a similar
property: time flows slower as one gets closer. Near very dense
masses (i.e. black holes), time stops altogether at a certain
distance.
A few of modern physics' more exotic theories have possible
explanation in this model. Although energy mainly flows between wave
modes very similar in frequency and direction (i.e. between points
adjacent in $F$), non-linearities in the $V$ medium should permit some
energy to flow systematically between harmonically related wave modes,
for instance between one mode and another on the same direction, but
twice as high in frequency. Such modes of energy flow in $F$ provide
``degrees of freedom'' in addition to the three provided by nearby
points. They can be interpreted, when viewed on the small scale, as
extra dimensions (energy can move this way, that way, that way and
also {\it that} way, and {\it that} way ...). 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. Most physical theories
these days have tightly looped extra dimensions to provide a geometric
explanation the basic forces. Ten and eleven dimensions are popular,
and hinted new forces may introduce more. If something like the $F$
explanation of apparent higher dimensionality is correct, there is a
bonus. Viewed on a large scale, the harmonic ``dimensions'' are actual
links between distant regions of space, and properly exploited could
allow instantaneous communication and travel over enormous distances.
\sect{Big Waves}
Now, forget the possible implications of the idea as a
mechanism for Quantum Mechanics, and consider our universe, on the
grand scale. It is permeated by a background of microwave radiation
with a wavelength of about 1 millimeter, slowly increasing as the
universe expands. It affects and is affected by clouds of matter, and
thus interacts with itself nonlinearly. If we do a universe wide
spatial Fourier transform of this radiation (that is, treat {\it our}
world as $V$, we end up with an $F$ space with properties much like
those above. The expansion of the universe adds a new twist. As the
wavelength gets longer and longer, the subjective rate of time flow in
the $F$ world slows down. Any inhabitants of $F$ would be ideally
situated to practice the ``live forever by going slower and slower as
it gets colder and colder'' strategy proposed by Freeman Dyson. By
now they would be moving quite slow - their fastest particle
interactions would take several trillionths of a second. In past,
however, when the universe was dense and hot, the $F$ world would have
been a lively place, running millions or billions of times faster. In
the earliest moments of the universe, the speed would have been
astronomically faster.
The first microsecond of the big bang could might represent
eons of subjective time in $F$. Perhaps enough time for intelligence
to evolve, realize its situation, and seed smaller but eventually
faster life in the $V$ space. Though on the large scale $F$ and $V$
are the same thing, manipulation of one from the other, or even
communication, would be extraordinarily difficult. Any local event in
either space be diffused to non-detectability in the other. Only
massive, universe-spanning projects with long range order would work,
and these would take huge amounts of time because of the speed limits
in either universe, so real-time interaction is ruled out. Such
projects, however, could affect many locations in the other space as
easily (in many cases more easily) as one, and these could appear as
entropy violating ``miracles'' there. If lived in $F$ and wanted to
visit $V$, I would engineer such a miracle that would condense a robot
surrogate of myself in $V$, then later another one that would read out
the robot's memories back into an $F$ accessible form.
The Fourier transform that converts $V$ into $F$ is identical
except for a minus sign to the inverse transform that converts the
other way. Given just the two descriptions, it wouldn't be clear
which was the ``original'' world. In fact, the Fourier transform is
but one of an infinite class of ``orthogonal transforms'' that have
the same basic properties. Each of these is capable of taking a
description of a volume, and operating over it to produce a different
description with the same information, but with each original point
spread to every location in the result. This leads to the possibility
of an infinity of universes, each a different combination of the same
underlying stuff, each exhibiting Quantum Mechanical behavior but
otherwise having its own unique physics, each oblivious of the others
sharing its space. I don't know where to take that idea.
\end{document}