Causality in the quantum world

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Einstein's causality is one of the fundamental principles underlying modern physical theories. Whereas it is readily implemented in classical physics founded on Lorentzian geometry, its status in quantum theory has long been controversial. It is usually believed that the quantum nature of spacetime at small scales induces the breakdown of causality, although there is no empirical evidence that would support such a view. In my talk, I will argue that one can have a sound notion of causality even in a `quantum spacetime' -- understood as the space of states of an abstract algebra of observables. To this end I will draw from the mathematical richness of noncommutative geometry a la Connes. I will illustrate the general concept with an `almost commutative' toy-model and discuss the potential emprical consequences.