Probabilities in the topos approach to branching space-time

Abstract

There was previously proposed a topos unified view-point both to relativistic and quantum physics [4-5] on the basis of Belnapian branching space-time [3]. In the context of this approach, to any event of the branching space-time there can be associated a local orthocomplemented lattice (logic) of ‘physically reasonable’ propositions made by a localized observer. Elements of these local orthologics are special collections of the so-called Belnapian worlds (maximal upward directed subsets of events) which the local event belongs to. In the ordinary quantum setting, the counterpart of any local orthologic is the lattice of closed subspaces of Hilbert space of the observer’s environment. The present work suggests a natural scheme of inserting truth probabilities of the ‘physically reasonable’ propositions into the topos framework. To this end, the concept of Chu spaces is used. The role of hypothesis for the state of environment is played by an element of local ‘retro-causal’ orthologic. Sufficient condition of functorial character of the law by which Chu space is related to any event is pointed and proved, i.e. a functor is constructed from the branching space-time to the category of all Chu spaces. There presented a theorem on category nature of Bayesian reassessment of hypothesis likelihood made by the local observer.