Some questions regarding $n+m$-dimensional spacetime [duplicate]

Question

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

The following image:

serves to aid the reader in understanding the "privileged character" of $3+1$-spacetime. It is however, incomplete, as the wikipedia sub-article also mentions String Theory, which also considers having 10 or even 26 spatial dimensions. My first question is: are these the only notable exceptions to all the "unstable" versions of spacetime in the second row?

Furthermore, I would like to know if the concept of having a negative amount of dimensions has been considered, or even adequately mathematically described.

Lastly, I am wondering whether or not (mathematical) physicists have considered the possibility of having a non-integer amount of time and/or space dimensions. The notion of having a non-integer amount of dimensions is at least mathematically defined by concepts such as the "Hausdorff Dimension" and the "Minkowski Dimension".

Answer 1

Perhaps there is a philosophical issue of whether mathematical structure should accommodate or define. Since Clifford algebra can accommodate any number of dimensions it can not explain why space is three dimensional. The default position in physics is to accommodate - but then you lose the ability to say that things have to be as they are. But then nobody is quite sure exactly how they are anyway, and much that seems definitive turns out to be wrong. Of course one might accommodate so much that it makes no sense why the world has any particular structure, and that might be wrong too.

Answer 2

Let me take a whack at this. The question in the wiki page is why the Minkowski signature is "privileged" and as usual one looks at the other possibilities, as did Synge in 'Special Relativity' p 17. Perhaps one says Privileged because experiment agrees with Minkowski. But if the world makes algebraic sense we should expand the search space to include Octonions and observe that complex quaternions are even subalgebras of complex octonions, and thus are Lorentz operators, which makes the signature of complex octonions +--- or -+++ (or some variant). I argue that the Minkowski signature is privileged because Complex Octonions are privileged, at least if one accepts that alternativity restricts the algebras to consider. Dixon does not - using OxHxC, which is not alternative, but is like SU(3)xSU(2)xU(1). [G.M. Dixon 'Division Algebras ... and the Algebraic Design of Physics']. One feature indicating Complex octonions are priviliged is that one is not running thru signatures, but rather has a pair of opposite signatures, which happen to be Minkowski, so it is not put in by hand, or chosen to agree with experiment. The wiki page would appear to be missing something here. So Complex Octonions appear to Define rather than merely Accommodate spacetime, as would be the case with Clifford Algebra. In fact Clifford has +--- in a different algebra from -+++. To push the case that Complex Octonions are 'really privileged' one would hope to see quanta defined as well as spacetime structure, but one might say the parentheses allow us to define duality oscillators, like the photon, and generalize to all quanta. Perhaps this is what is missing from the Standard Model. Anyway, this is rather unlike Tegmark's Platonistic view, and no mention of anything Anthropic is needed. Minkowski doesn't just agree with experiment, it also 'makes algebraic sense'. Back to your questions: Four generators looks like a natural number to me, and I can't make any sense of a 'negative amount of dimensions'. As to 10d etc, if the LHC finds supersymmetric boson-fermion pairing then experiment would invalidate the alternativity assumption above, and Complex Octonions would be only 'partly privileged'.