It seems surprising that space is three-dimensional. Why so few dimensions?
An anthropic answer seems implausible. Anthropic considerations might explain why we don’t have one or two dimensions—perhaps it’s hard to have life in one or two dimensions, Planiverse notwithstanding—but thye don’t explain why don’t have thirty or a billion dimensions.
A simplicity answer has some hope. Maybe it’s hard to have life in one and two dimensions, and three dimensions is the lowest dimensionality in which life is easy. But normally when we do engage in simplicity arguments, mere counting of things of the same sort doesn’t matter much. If you have a theory on which in 2050 there will be 9.0 billion people, your theory doesn’t count as simpler in the relevant sense than a theory on which there will be 9.6 billion then. So why should counting of dimensions matter?
There is something especially mathematically lovely about three dimensions. Three-dimensional rotations are neatly representable by quaternions (just as two-dimensional ones are by complex numbers). There is a cross-product in three dimension (admittedly as well as in seven!). Maybe the three-dimensionality of the world suggests that it was made by a mathematician or for mathematicians? (But a certain kind of mathematician might prefer an infinite-dimensional space?)
14 comments:
Quaternions are just as neat for four-dimensional rotations. The cross-product thing is of no real interest, it seems to me, as it's just an inferior stopgap for the wedge product, which works in all dimensions.
Three dimensional space is provably the only one in which waves propagate without distortion, which strikes me as a much more likely explanation.
Do you think intuitively ugly mathematical results constitute any evidence against explanations like this?
e.g. https://x.com/Winterflan/status/1834308016226939295?t=hpULvnHwjHbQzqf6SrEWtg&s=19
Shadow: Fair enough about quaternions. The thing that puzzles me is the smallness of the number of dimensions, and four is still pretty small. (Yeah, everything is small relative to infinity, but we still have some intuition that 1-12 is VERY small.)
I don't know about wave propagation and distortion. It looks like three is optimal, but one can still get signal transmission in other odd dimensions: https://repository.ias.ac.in/1079/1/347.pdf
That twitter link leads to a page that doesn't exist. Do you have any backups of that?
About everything being small relative to infinity, that's true, but I think one could also say infinity has an interesting equalising effect for finite numbers - any finite number, although it can be smaller than other finite numbers, is only ever infinitely smaller than infinity. And all the other finite numbers are also infinitely smaller than infinity.
In the Newtonian approximation to general relativity with d > 2 space dimensions, the gravitational force falls off like 1/r^(d-1), because the force lines spread out over the area of a sphere. However, the centrifugal force always falls off like 1/r^3, regardless of dimensionality. As a result, orbits are not stable if d > 3; with even the slightest deviation from a circular orbit, planets would crash into stars or escape into space. On the other hand, in d=2 GR, there is no gravitatational force at all; instead point masses produce conical singularities of space.
For a related but quantum reason---forces become comparatively stronger at short distances when d is larger---it is impossible to find renormalizable QFT interactions when the number of spacetime dimensions D > 4 (although there are some exotic theories that work in up to D = 6, they are quite different from the Standard Model).
Matrices of quaternions are useful for describing rotations whenever the number of space dimensions MINUS the number of time dimensions, modulo 8, is: 3, 4, or 5. For 2 or 6 you can use complex-valued matrices. For 0,1, or 7, the corresponding matrices are real-valued. Key words include "Clifford algebras" and "Bott periodicity".
So, for example, the Lorentz group in 3 space dimensions and 1 time dimension, can be described by 2x2 complex matrices with unit determinant.
What exactly are additional time dimensions though? Especially anything above the first? Sorry if this is a bit of a layman's question, but do you know of a way to break it down to understand it better?
Regarding Wesley's query, in the spacetime geometry of special relativity, time differs from the other dimensions in that it contributes with a minus sign to the Pythagorean distance formula. See here for an explanation:
http://www.wall.org/~aron/blog/the-geometry-of-spacetime-i-distance/
I would note that, when I was talking about the nature of rotations in different numbers of m space and n time dimensions, I am purely talking about the mathematics of transformations that preserve a distance formula where m dimensions have + signs while n dimensions have - signs. This can be studied without regard to whether it would really make sense (philosophically or physically) to have multiple time dimensions. In particular, nothing in my comment about Bott periodicity requires introducing concepts such as causality, thermodynamics, etc. It's just a purely geometric claim. (Unlike my other comment about gravitional orbits, which concerns actual physics.)
Not an exact backup but the same idea, the optimal way to pack 17 squares into the least amount of space: https://www.reddit.com/r/math/s/UnVI3ouMVW
Dr Wall:
I once said that a difference between space and time is the sign in the metric, and Nic Teh corrected me by observing that you can't use that to differentiate space from time in two-dimensional relativity where you have one space and one time dimension. (I am guessing that he was thinking that the choice of whether timelike distances get a + or - is purely conventional--it's only the difference between + and - that has physical significance.)
If that's right, then we may need something more for the difference between space and time than the signs in the metric. I am inclined to think that the crucial ingredient is causation, which is (at least typically) timelike but not spacelike.
Yes, the choice of sign for space vs time is purely conventional---in fact there are physicists who use the opposite convention from the one I said!
Regarding the actual difference between space and time, I think it is a subtle question. I agree that time has something to do with causality, but there is a question of exactly how to cash this out in physics models. In QFT you have the property that observables always commute at spacelike separation, but can fail to commute for timelike and spacelike separation. (Although, if we restrict attention to time ordered correlators maybe we couldn't detect this problem, however I think in a typical 1+1 dimensional theory there may be other things that go wrong with unitarity.)
Another difference is that switching the roles of space and time would replace massive particles with tachyons (and vice versa).
In the case where you have only massless particles in 1+1 dimensions, which propagate at the speed of light, the considerations above may not suffice to distinguish the two cases, and it might come down to the question of, in which direction we impose the "initial conditions".
@Aron If time is uniquely causal while space isn't, then even in the massless, 1+1 dimension, lightspeed particles case, there'd still be a difference between time and space - it's just that we can't easily tell without knowing or imposing initial conditions.
In fact, if time is uniquely about causation, then any other potential dimensions of time would also have something to do with causation at least.
@Wesley, the principle of "Microcausality" states that failure to commute can only occur at timelike OR lightlike separation.
But, in some theories involving massless particles only, this in fact happens ONLY at lightlike separation. Not at either timelike or spacelike separation. So, if we restrict attention to this special type of theory, it would seem truer to say that causation is associated with lightlike directions. And then, this particular formalization of causality does not seem to discriminate between space and time.
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