# Could there be an infinite number of dimensions?

Is it possible?

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mathman
In physics, string theory has 11 dimensions. In mathematics, there is no limit - for example, Hilbert space is infinite dimensional.

In physics, string theory has 11 dimensions. In mathematics, there is no limit - for example, Hilbert space is infinite dimensional.
Actually, this is something I wonder about. Technically, what string theory requires is that the central charge be zero. Right?

Is it the case that space must be 11 dimensions in order for the central charge to be zero? Or is it just that 11 dimensions is the minimum possible number of dimensions for a vanishing central charge? I.E. is there any number of dimensions greater than 11 where superstring theory is anomaly-free, but we don't care about it because it's impractical?

What is the central charge in infinite-dimensional Hilbert space?

Haelfix
If spacetime has more than 11dimensions and includes both fermions and bosons, there are fairly general theorems that state that the hilbert space will contain multiple gravitons (bimetric theories) and what seems to be particle states with spin >2, so a priori not consistent with quantum mechanics. This isn't completely robust, and people have found ways to avoid the theorems, but its hard to get around and leads to problems of its own.

And string theory and later Mtheory are only anomaly free in exactly D =10 and D = 11 respectively from a straightforward calculation found in most textbooks.

Haelfix, thanks much for the explanation.

Post #4 nicely describes what little I might suggest about this universe...Keep in mind NOTHING about string theories has been proven experimentally..hence I take 11 dimensional spacetime limits as theoretically possible but hardly an ironclad finding...as suggested in the referenced post.

But if your question applies universally, then parallel universes, the multiverse, considerations make it appear rather likely more dimensions exist. In fact parallel universes might just be partitions of an infinite number of dimensions with different boundary conditions....whatever the case, if there are an infinite number of universes (hard to imagine, I know) then it's not impossible to imagine some of them possibly having infinite dimensions....

Is it possible?
I think that question may be linked to the other question as to whether there is an infinite regression into smaller and smaller particles. As I recall there is a theorem which states that there is a limit to discreteness of particles. I think it has something to do with entropy, event horizons, and number of microstates. Maybe someone here can give more info on that. But anyway, if there were an infinite number of dimensions, then wouldn't that mean there were an infinite number of degrees of freedom, and you'd have the same issues as with an infinite number of infinitesimal particles. Right?

apeiron
Gold Member
The standard tools of thermodynamics could not help you here, although thermodynamics as a more general way of thought might help you in framing some kind of selection mechanism that averages across the situation.

But the most salient point here is that "particles" would be as much like zero-D entities as possible - discrete points. An infinity of dimensions would be at the other end of the spectrum - infinite-D.

We think of dimensions as spatial - flat euclidean space in fact - but that is a special case. So again, the question would be rather different depending on whether you meant infinite flat space-like dimensions or just the more abstract thing of an infiinty of degrees of freedom.

There are many people who have tried to argue that "only three extended and flat dimensions" are physically possible. So if we could prove that - and I feel it is likely to be the case - then an infinity of dimensions could not crisply exist. It would be a imaginary thing.

On the somewhat broader question of what would be the story for an infinity of degrees of freedom, this is also enlightening to consider. I would argue that an infinity would actually be continuous - unbroken. So it would be total freedom with, in effect, no orderly directions. A perfect symmetry that in certain logics would be called a vagueness.

I have started a thread on vagueness at...