Quantum Measure Theory, 2004
Fundamental Theories of Physics Series, Vol. 134

This book is the first systematic treatment of measures on projection lattices of von Neumann algebras. It presents significant recent results in this field. One part is inspired by the Generalized Gleason Theorem on extending measures on the projection lattices of von Neumann algebras to linear functionals. Applications of this principle to various problems in quantum physics are considered (hidden variable problem, Wigner type theorems, decoherence functional, etc.). Another part of the monograph deals with a fascinating interplay of algebraic properties of the projection lattice with the continuity of measures (the analysis of Jauch-Piron states, independence conditions in quantum field theory, etc.). These results have no direct analogy in the standard measure and probability theory. On the theoretical physics side, they are instrumental in recovering technical assumptions of the axiomatics of quantum theories only by considering algebraic properties of finitely additive measures (states) on quantum propositions.

1 Introduction.- 2 Operator Algebras.- 3 Gleason Theorem.- 4 Completeness Criteria.- 5 Generalized Gleason Theorem.- 6 Basic Principles of Quantum Measure Theory.- 7 Applications of Gleason Theorem.- 8 Orthomorphisms of Projections.- 9 Restrictions and Extensions of States.- 10 Jauch-Piron States.- 11 Independence of Quantum Systems.