Stochastic Quantum Mechanics and Quantum Spacetime

Foundations of Quantum Mechanics, an Empiricist Approach

“…For a long time it has been standard practice to (successfully) correct the experimental measurement results by means of an efficiency factor so as to achieve agreement with the standard formalism. Under the influence of work by, a.o., Davies and Lewis [36,37], Holevo [34] and Ludwig [38] (see also Kraus [39], Prugovecki [40], and Busch et al [41,42]), it has only recently been realized that the generalization of the concept of a quantum mechanical observable from PVM to POVM cannot be merely understood on the basis of correction factors, but that it constitutes a fundamental extension of the domain of application of quantum mechanics. Moreover, it will become clear that even basic issues of quantum mechanics like the so-called 'thought experiments', discussed in the early days of quantum mechanics in order to try to understand such basic concepts as 'complementarity' (cf.…”

Quantization Methods: A Guide for Physicists and Analysts

“…They are actually two of the kind, the geometro-stochastic quantization of Prugovečki [206] and the stochastic quantization of Parisi and Wu [195]. 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.…”

Quantum wave equations and non-radiating electromagnetic sources

“…It does not imply that the position operator so defined has any physical significance. In fact, the position operator for the Klein-Gordon equation is rather a complicated and somewhat controversial subject [28,29,35,36]. Despite this fact, the current that we have used here for the Klein-Gordon equation is the same one that is used in standard field theory analysis of scalar particles where the localizable wave functions are of less importance.…”

“…With coworkers he has proposed a wide class of alternative wave equations which overcome these shortcomings [36,[40][41][42]. For the spinless non-relativistic case the phase-space wave function is written as [36] If for each p held fixed the functions ( , ; ) ψ t q p satisfy the regularity conditions of Theorem 1 then each these currents will all be non-radiating. If the integrals in (81) are uniformly convergent, then owing to superposition as in (12), the resulting currents will be non-radiating.…”

“…For the spinless relativistic case we have [36] (page 114, eqs. 8.11-8.13) by superposition this current will be a non-radiating source with the assumptions of Theorem 2 and assuming uniform convergence of the integrals over the momenta in this equation.…”