Simple example of quantum causal structure
This paper attempts to derive a ``quantum causal structure'' from quantum logical first principles. That is, it is assumed that the probabilities of quantum mechanics arise from a non-Boolean lattice of ``events''. The subspace lattice of a Hilbert space is therefore assumed to give the fundamental space-time structure for quantum systems. In a clear exposition, the author proceeds first to present in a brief exposition the Kronheimer-Penrose axioms for a causal structure, and then in a following section to generalize these to give a definition of a quantum causal structure which can apply in the non-Boolean case of the subspace lattice of a Hilbert space. This allows him to derive quantum ``causal relations''. Next the notion of a topology is generalized (based on the notion of set closure), in such a way that it too can be applied to the quantum definitions. The author gives examples in the paper of structures which illustrate the Boolean and quantum definitions, and remarks finally that ``There can exist many relations [on the set of atoms] satisfying the Kronheimer- Penrose axioms that cannot be obtained as a restriction of a quantum causal structure on the whole lattice. All this means of considerable restraint for the possible causal relations''. An example then follows of a Boolean and a non-Boolean lattice from the same set of atoms, in which only the Boolean can be used to derive a certain set of quantum causal relations.
scientific article; zbMATH DE number 4055616