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

n+m - dimensional spacetime diagram

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?

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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 –  Squark Jan 21 '12 at 18:47
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As far as I know fractional Hausdorff dimension of spacetime doesn't appear in any serious research direction –  Squark Jan 21 '12 at 18:49
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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 –  Squark Jan 21 '12 at 18:51
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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 –  Kyle Jan 21 '12 at 20:50
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possible duplicate of Some questions regarding $n+m$-dimensional spacetime –  Qmechanic Jan 22 '12 at 7:43

3 Answers 3

up vote 8 down vote accepted

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.

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Thank you! I think the article you gave me is a very nice introduction to the subject. –  Max Muller Jan 22 '12 at 13:34
    
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... –  twistor59 Jan 22 '12 at 15:26

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 ??

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I believe Gianluca Calcagni does some work in this area. For example, see: http://arxiv.org/abs/1209.1110

And here is a nice blog post on the topic: Dimensional Reduction

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