The Hilbert Space Formulation of Quantum Physics

Sergio Cecotti

Jaromir Tosiek, P. Brzykcy

M. Revzen

An hidden variable (hv) theory is a theory that allows globally dispersion free ensembles. We demonstrate that the Phase Space formulation of Quantum Mechanics (QM) is an hv theory with the position q, and momentum p as the hv. Comparing the Phase space and Hilbert space formulations of QM we identify the assumption that led von Neumann to the Hilbert space formulation of QM which, in turn, precludes global dispersion free ensembles within the theory. The assumption, dubbed I, is: "If a physical quantity $\mathbf{A}$ has an operator $\hat{A}$ then $f(\mathbf{A})$ has the operator $f(\hat{A})$". This assumption does not hold within the Phase Space formulation of QM. The hv interpretation of the Phase space formulation provides novel insight into the interrelation between dispersion and non commutativity of position and momentum (operators) within the Hilbert space formulation of QM and mitigates the criticism against von Neumann's no hidden variable theorem by, virtually, the consensus.

T. R. Govindarajan

Quantum theory as formulated in conventional framework using statevectors in Hilbert spaces misses the statistical nature of the underlying quantum physics. Formulation using operators [Formula: see text] algebra and density matrices appropriately captures this feature in addition leading to the correct formulation of particle identity. In this framework, Hilbert space is an emergent concept. Problems related to anomalies and quantum epistemology are discussed.

Enrico G. Beltrametti, Gianni Cassinelli, Peter A. Carruthers

Walter E Thirring