Algebraic Quantization of Causal Sets

A scheme for an algebraic quantization of the causal sets of Sorkin et al. ispresented. The suggested scenario is along the lines of a similar algebraizationand quantum interpretation of finitary topological spaces due to Zapatrin and thisauthor. To be able to apply the latter procedure to causal sets Sorkin's 'semanticswitch' from 'partially ordered sets as finitary topological spaces' to 'partiallyordered sets as locally finite causal sets' is employed. The result is the definition of'quantum causal sets'. Such a procedure and its resulting definition are physicallyjustified by a property of quantum causal sets that meets Finkelstein's requirementfor 'quantum causality' to be an immediate, as well as an algebraically represented,relation between events for discrete locality's sake. The quantum causal setsintroduced here are shown to have this property by direct use of a result fromthe algebraization of finitary topological spaces due to Breslav, Parfionov, andZapatrin.