Effects and Propositions

Jorge Juan Muñoz García

This paper formulates a hypothesis concerning the logical structure of observational, temporal, causal, and geometrical propositions in timelike quantum-causal domains. In classical relativity, timelike order is invariant once a Lorentzian geometry is fixed. In quantum theory, however, propositions associated with incompatible operational contexts need not belong to a single Boolean algebra. This paper investigates the possibility that, when observers, clocks, matter sources, or effective geometrical degrees of freedom are quantum systems, the set of observer-indexed propositions describing records, clock readings, simultaneity, temporal order, causal accessibility, and effective spacetime geometry may be locally Boolean without admitting a single observer-independent global Boolean valuation. This hypothesis shows to be compatible with the postulates of quantum mechanics, allowing context-dependent propositional structures, and with relativistic causality, causal consistency is preserved within each effective Lorentzian context. The resulting framework states that temporal and causal propositions may remain well defined within compatible operational contexts, while the total structure of facts across observers, clocks, and quantum geometrical degrees of freedom need not possess a single global Boolean representation.

Peter Mittelstaedt

It has been shown by Birkhoff and v. Neumann (1936) and by Jauch and Piron (1963,1964,1968) that the subspaces of Hilbert space constitute an orthocomplemented quasi-modular lattice Lq, if one considers between two subspaces (elements) a, b the relation a⊆b and the operations a∩b, a∪b, a⊥. Furthermore, since the subspaces can be interpreted as quantum mechanical propositions, and since the operations ∩,∪ ,⊥ have some similarity with the logical operations ⋀ (and), ⋁ (or) and ⌝ (not), the question has been raised already by Birkhoff and v. Neumann, whether the lattice of subspaces of Hilbert space can be interpreted as a propositional calculus, sometimes called quantum logic. There are many kinds of lattices which can be interpreted as a propositional or logical calculus. A Boolean lattice LB of propositions corresponds to the calculus of classical logic and an implicative (Birkhoff, 1961) lattice Li has as a model the calculus of effective (intuitionistic) logic.

Vittorio Berzi, Antonio Zecca A

Günther Ludwig

Fedor Herbut

Let p(F.E, rho ) denote the probability of a yes result of the yes-no effect obtained by putting the filter of the quantum proposition F (a projection operator) after that of the proposition E in the quantum state rho (a statistical operator). It is pointed out that EF=FE implies/implied by For all rho :p(E.F, rho )=p(F.E, rho ), EF=E implies/implied by For all rho : p(E.F, rho )=p(E, rho ), and EF=0 implies/implied by For all rho :p(E.F, rho )=0. Proof is given for generalisations of these simple characterisations to arbitrary strings (products of operators) and chains (of filters). Other cases of parallelism between strings and chains and departure from it are also studied.