# Structure of spacetime as a non-commutative algebra

Disclaimer: this question is a cross-posting from mathoverflow.net, as what I expect there is a link to mathematical concepts, while here I'd like an examination of how meaningful it is physically.

While having a walk, I thought a bit about what made the flow of time irreversible considering the minus sign of the component $$g_{00}$$ of the metric tensor in the Minkowski spacetime with sign convention $$(-+++)$$.

One can easily figure out that whenever $$i>0$$, there exists $$j$$ such that $$g_{jj}=\sqrt{g_{ii}}$$, as if the square roots behaved like directions of the components of space-time that are freely available inside a single section of a timeline. Taking a square root of $$g_{00}=-1$$ leads to a number that is not any of the $$g_{ii}$$ and thus we somehow leave the considered section. This is a bit analogous to a non normal field extension that does not contain all conjugates of the primitive element generating it.

While saying that to mathematician friend of mine, he told me that according to Alain Connes, irreversibility emerges from the non commutativity of the "true" quantum geometry. I heard of works suggesting space-time at a tiny scale has dimension 2, like the dimension of $$C$$ over $$R$$ as a vector space. On the other hand, space-time at macroscopic scale has dimension 4 like the quaternion skew field.

So my question is moreorless this one: can gravity be seen as a process consisting in going from a non-commutative complex operator algebra (on the quantum side) to a pseudoriemannian manifold locally isomorphic to a quaternion algebra (on the relativistic one)? Is some notion of morphisms of non-commutative algebras potentially relevant here? Is it anyhow related to a scale factor increasing?

Edit: I also expect space-time to be $$\mathcal{C}^{0}$$ at tiny scale in the purely quantum realm, then $$\mathcal{C}^{1}$$ with a smooth fractal structure (see computer rendering of Nash's embedding theorem) at a scale where quantum effects and gravity are of similar importance, and finally $$\mathcal{C}^{2}$$ as scale keeps increasing together with the intensity of gravity with regard to quantum effects.