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}