Quantum Theory Without Hilbert Spaces

Stephen Bruce Sontz

In a recent paper it was shown that all the Hilbert space formulas for quantum probabilities can be realized as functions of geometric properties of the associated projective space, but those functions were expressed using the structures of the associated Hilbert space. In this paper a direct description of all these probabilities is given as formulas involving only the geometric properties of the projective space itself without referring to the associated Hilbert space theory. In large part this depends on a projection theorem for complex projective space which is analogous to the projection theorem for Hilbert spaces. The importance of this is that this exhibits quantum probability in terms of the geometry of a Riemannian metric in a non-linear Kähler manifold without any reference to a linear Hilbert space. As such this is a part of a larger program of the geometrization of physics. This opens the possibility of generalizations of quantum theory in other similar geometric settings. The theory presented includes projective spaces of both finite and infinite dimension. Some comments explain how quantum theory based on a von Neumann algebra is compatible with this approach.

Bogdan Mielnik

Seungyoung Choi

This paper proposes a mathematical approach for applying a single Hilbert space (HS) in QTIP Cosmology. Based on the mathematical structure of composite systems in quantum mechanics (QM) and quantum information theory (QIT), it examines the correspondence of superposition, normalization and the Born Rule, entanglement, subsystem measurement statistics, and generalization to N-particle systems. Rather than constructing a new composite Hilbert space through tensor products of independent Hilbert spaces, QTIP Cosmology proposes defining a relational subset HS within a single Hilbert space based on the boundary conditions among QIF_Is. This approach does not exclude the mathematical structures of QM/QIT, but proposes an alternative framework for applying them within a single Hilbert space.

Jean-Pierre Antoine

Matthew Gagne

We explore the basic mathematical physics of quantum mechanics. Our primary focus will be on Hilbert space theory and applications as well as the theory of linear operators on Hilbert space. We show how Hermitian operators are used to represent quantum observables and investigate the spectrum of various linear operators. We discuss deviation and uncertainty and briefly suggest how symmetry and representations are involved in quantum theory.

Charles R. Giardina