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?
Answer
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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