# Has the concept of non-integer $(n+m)$-dimensional spacetime ever been investigated by theoretical physicists?

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The following image serves to aid the reader in understanding the "privileged character" of $3+1$-spacetime.

The wikipedia article on spacetime, and the sub-article "The priveleged character of $3+1$-dimensional spacetime" in particular, made me think a bit about the possibility that we might live in a non-integer amount of spatial and/or time dimensions.

The notion of attaching a non-negative real number to a metric space has at least mathematically already been described by such concepts as "Hausdorff Dimension" and "Minkowski-Bouligand Dimension".

This may sound silly/ignorant/absurd to professional practicing theoretical physicists. To me (a layman), however, it doesn't sound much stranger than the idea of wrapping up six extra dimensions (which is, from what I understand, considered to be a serious possibility by those who study $10$-dimensional String Theory) into intricate shapes called "Calabi-Yau Manifolds".

Has any research on $(a +b)$-dimensional spacetime (where $a,b \in \mathbb{R}_{\geq 0}$) ever been done? If so, what where the findings? If not, why not?

This post imported from StackExchange Physics at 2014-03-22 17:32 (UCT), posted by SE-user Max Muller
asked Jan 21, 2012
retagged Apr 19, 2014
Firstly, the six extra dimensions are not an arbitrary construction but something which follows directly from the consistency requirements of superstring theory (and thus it can be argued from the need to unify quantum field theory with general relativity). Therefore the right question is "why yes" rather than "why not". More to the point fractional spacetime dimension appears, somewhat mysteriously, in dimension regularization of QFT. It also appears in noncommutative geometry the existence of which also follows from string theory, among other reasons of interest

This post imported from StackExchange Physics at 2014-03-22 17:32 (UCT), posted by SE-user Squark
As far as I know fractional Hausdorff dimension of spacetime doesn't appear in any serious research direction

This post imported from StackExchange Physics at 2014-03-22 17:32 (UCT), posted by SE-user Squark
I don't think the string-theory tag is suitable for this question. Also, I think it's at best borderline in its appropriateness to this forum

This post imported from StackExchange Physics at 2014-03-22 17:32 (UCT), posted by SE-user Squark
The spectral dimension has been shown to be a running parameter in the model of CDT. For more on the spectral dimension in general: arxiv.org/abs/1105.6098

This post imported from StackExchange Physics at 2014-03-22 17:32 (UCT), posted by SE-user Kyle
"doesn't sound much stranger than the idea of wrapping up six extra dimensions" yes 10 dimensions is crazy, Im not saying it cant be the case, but its like talking about all the interesting things we could do if we could factor terrabit numbers... the geometric algebra for a vector space of 10 dimensions has a whopping 2^10=1024 basis blades. so the space spanned by the multivectors is R^1024, have fun trying to understand it... "just 4" dimensions gives 16 basis blades (1-4-6-4-1), and the diversity of objects spanned by them is already rich enough, lets first learn walking before running!

This post imported from StackExchange Physics at 2014-03-22 17:32 (UCT), posted by SE-user propaganda
so 10 dimensions is interesting, and factoring terrabit numbers would also be interesting, but for now its all fantasy

This post imported from StackExchange Physics at 2014-03-22 17:32 (UCT), posted by SE-user propaganda
how do you define another "time direction" in your illustration?

This post imported from StackExchange Physics at 2014-03-22 17:32 (UCT), posted by SE-user Yunlong Lian

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One example of such an approach is Ambjorn and Loll's Causal Dynamical Triangulations, which is very similar in many ways to the very old idea of Regge calculus, whereby spacetime is discretized. At small scales, non integer dimensions can emerge. For an introductory article , see

Jan Ambjørn, Jerzy Jurkiewicz and Renate Loll. The Self-Organizing Quantum Universe. Scientific American (July 2008), 299, pp. 42-49. doi:10.1038/scientificamerican0708-42, available here.

This post imported from StackExchange Physics at 2014-03-22 17:32 (UCT), posted by SE-user twistor59
answered Jan 22, 2012 by (2,490 points)
Thank you! I think the article you gave me is a very nice introduction to the subject.

This post imported from StackExchange Physics at 2014-03-22 17:32 (UCT), posted by SE-user Max Muller
No probs. I suspect that there may also be ways of thinking about some string theory states in which a spacetime manifold is not present. Maybe one of the experts could comment...

This post imported from StackExchange Physics at 2014-03-22 17:32 (UCT), posted by SE-user twistor59
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yes, in DIMENSIONAL REGULARIZATION dimension is just a parameter and after calculations you set it to $d=4-\epsilon$ with epsilon tends to 0 so the poles of the Gamma function $\Gamma (s)$ appear

curiosly enough, if we lived in a world with $4.1$ dimension, then the Gamma function wuold have no poles and the Quantum gravity would be renormalizable.

another question is could the dimension for high energies be only a 'parameter' to be fixed by experiments or by renormalization of the theory ??

This post imported from StackExchange Physics at 2014-03-22 17:32 (UCT), posted by SE-user Jose Javier Garcia
answered Jun 1, 2012 by (70 points)

could the dimension for high energies be only a 'parameter' to be fixed by experiments or by renormalization of the theory ?? .. it couldn't. There is no Hilbert space supporting field operators having a noninteger dimension.

I agree with Arnold Neumaier, there is no sequence of QFTs in non-integer dimensions constructed during dimensional regularization. Dimensional regularization is only a mnemotechnic name for a way to organize the regularization of the loop integrals. It could be called "Gamma-function regularization" as it uses the same mathematical principles as Zeta-function regularization, just with a different function.

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