Causality and Noncommutativity, Andrzej Sitarz

In the quest for a mathematical model, which could be a suitable tool to describe the geometry of small scale physics (or quantum gravity physics) one encounters noncommutative geometry. Though mathematically very attractive and physically well motivated these models lead to serious problems when considered in the physical, pseudoriemannian setup. The main one is how to define events and causality when we are dealing with "spaces" without points and the "time" itself might be noncommutative.
This lecture was delivered during the 16th Kraków Methodological Conference "The Causal Universe", May 17-18, 2012.
More information:
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[@SamFischer1000]

I would suggest spending more time showing the slides. It is really quite difficult to take in any information with the pace at which they are being displayed here.

[@intellectualadventurers7188]

Can the slides be shared on the web.

[@StephenPaulKing]

kernel is non-zero ?
Length scale? minimal length? Does it non-commute with the Hamiltonian (energy)?

[@StephenPaulKing]

...simultaneously true propositions... iff all propositions are simultaneously true.

[@StephenPaulKing]

So is a commutative C* algebra => space related? to a Complete Atomic Boolean Algebra?

[@truebaran]

Yes, every commutative C*-algebra produces (via Gelfand Najmark theorem) an honest topological space: if the algebra is unital then this space is compact, if it is not then the space is only locally compact. The idea is to think about noncommutative C*-algebras as the algebras of ,,functions'' on some (nonexistent in fact) virtual ,,noncommutative'' space. There is no obvious link between these notions and those of Boolean algebras: Boolean algebras are always commutative from the start therefore there is no ,,noncommutative'' analogue of Boolean algebras. However the situation is notso bad: I don't know how much Stone duality type theorems do you know: there is Stone duality for general Boolean algebras and even more general Stone duality between sober topological spaces and the so called ,,spatial frames''. Every spatial frames is of the form ,,ope sets of some topological space''. One can therefore forget about spatiality of a given frame and investigate the theory of general frame as they were the open sets of some (again, nonexistent) virtual topological space. This point of view is called ,,pointfree topology'' (or sometimes ,,pointless topology'').This theory is important in the context of the so called topos theory (which is due to Grothendieck) and there are people trying to link topos theory with quantum mechanics: Isham, Butterfield, Spitters, Landsman, Heunen. The key word here is the word ,,Bohrification''