Space and time may be both digital and analog
Sampling theory provides a way for spacetime to be discrete and continuous.
The Planck length (1.616255(18)×1e−35 meters) is theoretically the smallest distance that we can measure. The reason is because the smaller the distance we want to measure, the higher frequency waves we need to use. Those higher frequency waves pack more energy in a tighter volume. The amount of energy to measure anything that small would, hypothetically, create a black hole with that diameter. More energy, bigger black hole. Since we can’t see through a black hole event horizon, we can’t measure anything smaller. (That is, if gravity works the same way at those scales as it does on the largest scales, which we don’t know.)
That does not mean, however, that nothing smaller exists. The real question is whether a smallest length exists. That is: is the universe discrete with some atom of length or is there no smallest length in which case the universe is continuous. The same goes for time as well. Is there a “tick” that time uses to move forward or does it smoothly move through all possible real times between t=0 and t=1?
Nobel laureate and bongo player Richard Feynman took issue with some features of continuous numbers in physics. One particular problem is the Banach-Tarski paradox (a consequence of the Axiom of Choice) in which a sphere can be deconstructed into a finite number of pieces and reassembled into two spheres with the same volume as the original without any stretching at all.
Pumpkin Carving
xkcd.com is best viewed with Netscape Navigator 4.0 or below on a Pentium 3±1 emulated in Javascript on an Apple IIGS…xkcd.com
The key point is that it only works if the spheres are continuous volumes because the sets into which they are deconstructed are scatterings of infinite numbers of individual points. (In less than 3 dimensions, the paradox still exists but you need an infinite number of subsets.)
Feynman scoffed at the idea saying that you couldn’t cut the atoms, so of course you couldn’t get the sets you needed. You’d end up with finite numbers of atoms, not infinite points.
Was Feynman, one of the principle authors of the manifestly continuous quantum field theory, a believer in discrete spacetime? If his invention of the Feynman checkerboard model of spacetime is any indication, he was.
But what about continuous space and time? If you can cut them up infinitely, then do notions of length, area, and volume lose meaning? Is such a model human folly?
Was the great mathematician Kronecker right when he said “God made the integers. All else is the work of man.”?
It’s all about Quantum Gravity.
Any theory of space and time has to be about gravity, the theory of gravitation, specifically, that Albert Einstein invented called General Relativity. This theory, which has been confirmed numerous times over the last 100 years, posits that space and time are a manifold, meaning a shape, like the Earth and that gravity comes from matter moving through curved space and time.
Any theory about the very small, in time or space, must be about quantum physics. Quantum physics proposes that all matter and forces are really made up of fields, not hard little nuggets, which all have wavelike properties, meaning that everything has frequency and amplitude.
Any theory about very small times or spaces must be about quantum gravity, which ultimately reduces to the wavelike nature of space and time. Unfortunately, we do not know what the theory of quantum gravity is. We just have a lot of hypotheses.
Most theories of quantum gravity take either a continuous or discrete approach. The most popular theory: string theory, takes a continuous approach, with vibrating strings moving through continuous manifolds of higher dimensional spacetime. Other theories, such as loop quantum gravity and causal set theory, assume that there is a scale at which distance loses meaning and we are instead talking about events in discrete relationships to one another.
How do these relate to the Banach-Tarski Paradox?
String theory isn’t interested in the Banach-Tarski paradox because it is a theory about strings and how they vibrate. It doesn’t care what is under the surface of that space and time. The micro spacetimes in which the strings vibrate are very specific types, e.g., Calabi-Yau manifolds. Their crystalline structure is what enables matter to exist.
It has been suggested that the Banach-Tarski paradox explains how space can expand and that it is relevant to the micro spacetimes in which strings would exist. (Banach-Tarski-Cantorian cosmology.) This hypothesis would give a physical significance to the paradox, that space and time can rearrange themselves to become larger or smaller.
Loop quantum gravity avoids the problem by reducing space and time to graph theory, so that, at the smallest possible scales, space and time stop being continuous and are instead like spider webs of inter-related points. Causal set theory reduces space and time to discrete causal sets, where how one thing causes another is what is important, nothing in between.
Feynman’s checkerboard, with its relationship with cellular automata, those little rule-based machines, seems ideal, but it has never been successfully extended to more than one dimension of space and one of time. Nevertheless, it seems to be the approach Mathematica mogul Stephen Wolfram is taking, but theories of this vein are far behind their competitors in sophistication.
Truth, like a treasure buried in a wheat field, waits to be found.
In all these discrete spacetime cases, sets cannot be cut into infinite scatterings of real numbers. Therefore, they cannot, in any way, be reconstructed to have a different length of volume than the one they imply. But there are drawbacks.
What about consciousness?
As an aside, it has been suggested (based on Searle’s Chinese Room thought experiment) that any digital brain cannot be conscious, that human consciousness fundamentally requires a non-digital implementation. The reason is because any digital brain can ultimately be reduced to a table-lookup system that is manifestly not conscious. (It is also why digital computer-based AI cannot be conscious.) If the universe were fundamentally digital, it is questionable whether you could have conscious beings at all.
Could it be that consciousness itself depends on the Banach-Tarski paradox? If so, what implication would it have on our understanding of consciousness to have a theory of space and time that is fundamentally digital? This conundrum magnifies if it rules out any continuous physical reality, that is, all matter and forces have a discrete structure.
The Best of Both Worlds
Another approach taken by Kempf at Waterloo, however, is based in sampling theory, the relationship between discrete time series samples and their corresponding signals, e.g., audio or radio signals. This approach suggests that the universe may be both continuous and discrete in the same way that information can be both continuous and discrete. Thus, it is fundamentally an information based approach to quantum gravity.
The sampling-based approach starts with Shannon’s sampling theory, which is the central theorem of information theory. Shannon’s theory says that any bandlimited signal, meaning a signal that has a limit on its frequency bandwidth, can be reconstructed perfectly if it is sampled at twice the rate of the frequency. For example, if I have a bandwidth of 10 Megahertz, then I only need to sample it at 20 Megahertz to reconstruct the signal perfectly.
What that means is that the accuracy of a digitized signal is only as accurate as the individual samples. (The samples themselves can be any real or complex numbers, but in practice they are reduced to a number of bits, which destroys complete accuracy in digitization.)
Given the limit of the Planck length (and Planck time), wavelengths smaller than that may not be allowed in a description of space and time. Therefore, space and time are bandlimited by the inverse of the wavelength. This means that space and time could be made up of discrete points but, and here’s the most important point, indistinguishable from a continuous spacetime manifold.
Such a band limitation could be observable in very high energy gamma ray bursts or gravitational waves.
Can you hear the shape of a drum?
Einstein’s theory of space and time is ultimately about the shape of that spacetime while sampling theory is about frequencies. Therefore, in order to apply sampling theory to spacetime you have to understand how shape and frequency relate to one another.
Mark Kac expounded on this idea back in the 1960s in his famous lecture “Can One Hear the Shape of a Drum?” Kac was concerned with whether two drums producing the same tones had to have the same shape. The question, when related to Einstein, is whether, given two spacetime manifolds, if they produce the same set of frequencies, are the manifolds the same?
This seems important for quantum gravity since quantum theory is concerned with waves, fields, and frequencies and gravity is concerned with shapes.
Quantum gravity has to explain how the shape of spacetime vibrates.
If there are cutoffs in either the smallest possible wavelengths around the Planck length or the largest possible around the size of the observable universe, then, because of sampling theory, spacetime may behave as if it is a continuous manifold but actually be a dense set of points. What you lose are the sets of frequencies that are possible because of the cutoffs. But you don’t have to explain how such a theory would deviate from Einstein at large scales. Sampling theory takes care of that for you. The test would be to see if you could bound the frequencies that are possible in space and time.
Could gravitational wave detectors such as LIGO help with that?
The beauty of sampling theory is that it is a meta-theory. You can apply it to anything that has spectral (meaning frequency or wave related) qualities. In quantum physics, that is everything, and everything is quantum. So any theory of space, time, or matter can, provided it is band limited, connect to sampling theory and have a representation over a discrete set of points. One thing this avoids completely, then, is the Banach-Tarski paradox. Yet, it is neither discrete nor continuous but both.
El Naschie, M. S. “Banach-Tarski theorem and Cantorian micro space-time.” Chaos, Solitons & Fractals 5.8 (1995): 1503–1508.
Feynman, R. P., and A. R. Hibbs. “Path integrals and quantum mechanics.” McGraw, New York (1965).
Kempf, Achim, and Robert Martin. “Information theory, spectral geometry, and quantum gravity.” Physical review letters 100.2 (2008): 021304.
Kempf, Achim. “Information-theoretic natural ultraviolet cutoff for spacetime.” Physical review letters 103.23 (2009): 231301.
Kempf, Achim. “Spacetime could be simultaneously continuous and discrete, in the same way that information can be.” New Journal of Physics 12.11 (2010): 115001.
Kac, Mark. “Can one hear the shape of a drum?.” The American mathematical monthly 73.4P2 (1966): 1–23.