Mathematical problems of stochastic quantum mechanics: Harmonic analysis on phase space and quantum geometry

Abstract

In this paper we review the mathematical methods and problems that are specific to the programme of stochastic quantum mechanics and quantum spacetime. The physical origin of these problems is explained, and then the mathematical models are developed. Three notions emerge as central to the programme: positive operator-valued (POV) measures on a Hilbert space, reproducing kernel Hilbert spaces, and fibre bundle formulations of quantum geometries. A close connection between the first two notions is shown to exist, which provides a natural setting for introducing a fibration on the associated overcomplete family of vectors. The introduction of group covariance leads to an extended version of harmonic analysis on phase space. It also yields a theory of induced group representations, which extends the results of Mackey on imprimitivity systems for locally compact groups to the more general case of systems of covariance. Quantum geometries emerge as fibre bundles whose base spaces are manifolds of mean stochastic locations for quantum test particles (i.e., spacetime excitons) that display a phase space structure, and whose fibres and structure groups contain, respectively, the aforementioned overcomplete families of vectors and unitary group representations of phase space systems of covariance. (1980). 20C35, 81G20, 83A05. AMS (MOS) subject classifications

Quantization Methods: A Guide for Physicists and Analysts

“…The former arose, loosely speaking, from Mackey's systems of imprimitivity (U, E) (Mackey [167] -see the discussion of Borel quantization in §2.4 above), with U a unitary representation of a symmetry group and E a projection-valued measure satisfying U g E(m)U * g = E(gm) for any Borel set m, by demanding that E be not necessarily projection but only positive-operator valued (POV) measure; this leads to appearance of reproducing kernel Hilbert spaces and eventually makes contact with the prime quantization discussed in the preceding section. See Ali and Prugovečki [12]; a comparison with Berezin quantization is available in Ktorides and Papaloucas [159]. The stochastic quantization of Parisi and Wu originates in the analysis of perturbations of the equilibrium solution of a certain parabolic stochastic differential equation (the Langevin equation), and we won't say anything more about it but refer the interested reader to Chaturvedi, Kapoor and Srinivasan [62], Damgaard and Hüffel [67], Namsrai [179], Mitter [172], or Namiki [178].…”

“…It is then straightforward to verify [12] that K e,ℓ is a reproducing kernel with the usual properties, |ξ q,p ξ q,p | dp dq = I e,ℓ .…”

Probability in the formalism of quantum mechanics on phase space

“…We choose an α-admissible vector η ∈ H, and then we may define the operators A η ( f ) according to (2.21). These operators form a generalization of the stochastic quantization of Prugovečki [15,16] . They also constitute a case of 'prime quantization' [17, pp 459-65] for each choice of η.…”

Intrinsically extended stochastic particles with internal de Sitter symmetry

“…The mathematical formulation of this idea is based on the use of positive-operator-valued (POV) measures which take over the role of the projection valued (PV) measures in conventional quantum mechanics. In the presence of a group of invariance, a POV measure leads to the notion of a system of covariance which is the generalization of the system of imprimitivity for quantum systems [19]. Accordingly, the particles are stochastically extended and described by means of square integrable functions ψ(q, p) on phase space with variables (q, p).…”