Simple example of quantum causal structure
type: Item
instance of: scholarly article
title: Simple example of quantum causal structure
zbMATH Open document ID: 0647.03056
DOI: 10.1007/BF00669412
author: S. H. Smith
published in: International Journal of Theoretical Physics
publication date: 1987-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 03G12
Mathematics Subject Classification ID: 81P10
zbMATH DE Number: 4055616
zbMATH Keywords: quantum events
zbMATH Keywords: non-Boolean lattice of events
zbMATH Keywords: quantum causal structure
zbMATH Keywords: Kronheimer-Penrose axioms
MaRDI profile type: Publication
cites work: Q5551052
cites work: Q5527790
cites work: Q5541833
cites work: Empirical topology: Topologies from partially ordered sets
full work available at URL: https://doi.org/10.1007/bf00669412
OpenAlex ID: W2171702767
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description: scientific article; zbMATH DE number 4055616