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riemannian geometry – Morphism of non-commutative algebras retort

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riemannian geometry – Morphism of non-commutative algebras

Disclaimer: this query is a “immense portray” one which comes from my private ideas in physics. If it does not reconcile this website, delectation disclose me.

While having a stroll, I believed a bit about what made the rife of time irreversible contemplating the minus enthralling of the element $g_{00}$ of the metric tensor within the Minkowski spacetime with enthralling conference $(-+++)$.

One can simply challenge out that each time $i>0$, there exists $j$ such that $g_{jj}=sqrt{g_{ii}}$, as if the sq. roots behaved affection instructions of the parts of space-time which are freely obtainable inside a sole part of a timeline. Taking a sq. root of $g_{00}=-1$ results in a quantity that isn’t any of the $g_{ii}$ and thus we by some means depart the thought-about part. This is a bit analogous to a non common bailiwick extension that doesn’t purchase all conjugates of the uncooked element producing it.

While maxim that to mathecian buddy of mind, he informed me that in response to Alain Connes, irreversibility emerges from the non commutativity of the “accurate” quantum geometry. I heard of works suggesting space-time at a tiny scale has dimension 2, affection the dimension of $C$ over $R$ as a vector area. On the opposite hand, space-time at macroscopic scale has dimension 4 affection the quaternion skew bailiwick.

So my query is moreorless this one: can gravity breathe seen as a means of going from a non-commutative comlex operator algebra (on the quantum aspect) to a pseudoriemannian manifold domestically isomorphic to a quaternion algebra? Is some judgement of morphisms of non-commutative algebras doubtlessly pertinent right here?

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