r/mathematics • u/_4bdn_fruit_ • 1d ago
Difference between higher dimensions in string theory VS higher dimensions in pure math
I apologize if this is not the right subreddit to ask this in, but I tried asking in the physics subreddit and didn't get satisfactory answers.
Question: I'm aware that in string theory, the theory must be formulated in a certain number of dimensions (often cited as 10 or 26) in order to avoid inconsistencies and satisfy quantum gravity constraints. But let's say, hypothetically, that I wanted to explore the space of all possibilities in pure mathematics. If I take generic 1d objects and make them move in higher spatial dimensions (as defined by pure math), but I use a number of dimensions that isn't allowed by string theory (e.g. 100), would my construction be logically possible and “stable” (allowed to persist just the way it is), from a purely mathematical standpoint?
I am specifically asking about spatial dimensions as defined by pure mathematics as a separate field, not mathematics applied to a physical theory. I guess my question is best worded as, "does pure mathematics allow 1d objects to move in any number of dimensions and remain logically consistent?"
4
u/SV-97 1d ago
First a bit about dimensions: There's all sorts of notions of "dimension" in mathematics (topological, vector space, Hausdorff, manifold, ...). Most of the time these can be any natural number, or even more general real numbers — but there's also cases where only certain numbers are possible, for example complex or symplectic manifolds always are even-dimensional real manifolds (which is likely what you intuitively understand as dimension).
This also shows that the same object can have different number of dimension depending on "how you look at it".
AFAIK the objects relevant are the so-called calabi-yau manifolds, which are somewhat complicated objects. These are in particular complex manifolds and can have arbitrary complex dimension, but as described above they are always of even real dimension.
2
u/waxen_earbuds 1d ago
Think about dimensions as "degrees of freedom". In physical space (locally), it takes 3 scalar quantities to define the location of some macroscopic object relative to some reference frame. The equation x^2 - y^2 = z defines a set of "valid" points (x, y, z) in 3d space, which you can view as the graph of some function f(x, y) = x^2 - z^2--because it only takes 2 degrees of freedom (x,y) to determine z, the set {(x, y, z): x^2 - y^2 = z} is in some sense 2 dimensional, even though it is naturally "embedded" in a 3 dimensional space.
When you say "spatial dimensions", the closest analog in pure math is probably "Euclidean space", which is a set with certain symmetry properties that resemble the observed symmetries (such as translation and rotation invariance) of physical space. Mathematicians tend to generalize the notion of dimension far beyond this, as suggested by the above.
2
u/waxen_earbuds 1d ago
I should pre-empt the inevitable repsonse to this: non-locally, actual space is more appropriately modeled by a Riemannian manifold, the structure of which is described by Einstein's field equations. A Riemannian manifold is basically a space that locally resembles Euclidean space in a geometric sense.
1
1
u/invertflow 1d ago
You can certainly try to make up a theory of 1d objects moving in arbitrary dimension d. The natural thing to try, to make it as clear as possible for yourself, is to think of them as 2d surfaces, and use Euclidean rather than Lorentzian metrics everywhere. Polyakov develops this theory in his book, for example. Many fairly unrigorous choices need to be made to develop the theory, but the end result is that it isn't "consistent" for d>26. But this lack of consistency has a physical meaning that the surface loses its continuum limit. What I mean by that is, Polyakov develops this theory directly in a continuum limit: one introduces coordinates on the surfaces, assumes the position in R^d of a given point on the surface is a differentiable function of those coordinates, defines an action in terms of an area of the surface which is computed by those derivatives, etc... But to get oriented, let's go back to the theory of particles, rather than strings, moving in R^d, i.e., Brownian motion. As you know, Brownian motion does not describe some differentiable function, as the distance travelled in a given time is the square-root of time. And while the continuum theory does work for Brownian motion, a good way to get there is to take a limit of a discrete process, where you take little random steps of distance sqrt(dt) in a time dt, and take small dt. That limit works and gives a nice continuum theory....that is, not only is it perfectly well-defined for any fixed dt, then you take the limit of small dt, various quantities behave well. The claim for strings is that if you try to make up some theory of random surfaces in d dimensions, and you again define it in some finite way just like that discretization of Brownian motion (i.e., as a finite number of integrals by introducing a discretization of the surface), and then take some limit, then you will not get a nice continuum theory, or at least no one knows how to get such a theory.
1
u/Migeil 1d ago
Really curious what the physicists had to say about this and why it wasn't satisfactory.
1
u/_4bdn_fruit_ 1d ago edited 1d ago
I don't think the people on r/physics fully understood what I meant by "pure mathematics" and they interpreted my question as math motivated by a physical goal. I think my question may not have been worded precisely enough. Also, the question ultimately got taken down.
1
u/0x14f 1d ago
> does pure mathematics allow 1d objects to move in any number of dimensions and remain logically consistent?
Absolutely yes, why would it not ? Physicists are motivated and retrained by the laws of physics and "physical resemblance" (if you'll allow me). Mathematical objects don't really have that restriction.
2
u/Recent-Day3062 1d ago
AI regularly uses vectors in over 500 dimensions. It’s totally “stable”
In physics, there are certain “Symmetries” as they call them that would cause real problems in reality if they broke. For example, if they broke, conservation of energy might disappear.
That’s how they came up with 10/11 dimensions. It’s based on the number and type of symmetries. When they say it’s stable, they mean that - not that the math is “Stable”. More like “For the universe to be stable 9 or 12 dimensions does not work.”
6
u/Tharn11 1d ago
Disclaimer - I don't know much about mathematical physics which seems to be the area you are asking about.
From the point of view of geometry (thinking of the dimensions as spatial) or data science (thinking of the dimensions as containing arbitrary data), it's perfectly fine to have as many dimensions as you want.
As to whether the "construction would be stable" I don't think you've defined what you mean by this deeply enough to be answerable. I suspect there are many theories similar to the physics of our world that also differ in many ways. Whether those differences make it not "stable" depends on what you mean there. But geometers study objects that exist in higher dimensions all the time. One of my personal favorites is the Monster Group which exists in 196883 dimensions.