Integration in Hilbert Space and Quantum Theory

Zameddin I. İsmailov, Elif Otkun Çevik

In this work, a connection between some spectral properties of direct integral of operators in the direct integral of Hilbert spaces and their coordinate operators has been investigated.

Alexander Bach

Ryszard Paweł Kostecki

We discuss new approach to mathematical foundations of quantum theory, which is completely independent of Hilbert spaces and measure spaces. New kinematics is defined by nonlinear geometry of spaces of integrals on abstract non-commutative algebras. New dynamics is defined by constrained maximisation of quantum relative entropy. We recover Hilbert space based approach (including unitary evolution and the von Neumann-Lüders rule) and measure theoretic approach to probability theory (including Bayes' rule) as special cases of our approach.

Dave Applebaum

Alfred Rieckers

Abstract A representation of symmetry transformations motivated by the functional formulation of quantum field theory is rigorously discussed in a functional Hilbert space. The set of generating functionals is equipped with an inner product by means of the Friedrichs-Shapiro-integral and completed to an Hilbert space. Unitarity, continuity, and reducibility are investigated for the symmetry operations in this space. Also non-unitary transformations are considered.