Stochastic microcausality in relativistic quantum mechanics

Geometro-stochastically quantized fields with internal spin variables

“…(2.13), is, for small stochastic smearing characterized by the fundamental length parameter l [see Sect. II A], indeed "close" to the Feynman propagator i∆ F (q ′ − q) which is known to be nonzero for spacelike separation of the points q ′ and q (compare the discussion presented in [36]). For finite (small) nonzero l the stochastic phase space description using generalized wave functions is formulated in terms of spread out quantum events (at the scale of l) and, correspondingly, the propagation of wave functions describing such events is only "stochastically causal" and not deterministically causal in the strict sense as in the yes-no manner realized in classical relativistic physics with strictly zero influences on points outside the future light cone of an idealized pointlike event localized at q.…”

“…For finite (small) nonzero l the stochastic phase space description using generalized wave functions is formulated in terms of spread out quantum events (at the scale of l) and, correspondingly, the propagation of wave functions describing such events is only "stochastically causal" and not deterministically causal in the strict sense as in the yes-no manner realized in classical relativistic physics with strictly zero influences on points outside the future light cone of an idealized pointlike event localized at q. In the stochastic setting used here one has the result, obtained first for the flat space case in [36], that the probability for a particle of propagating outside the future light cone of a certain point tends to zero with τ → ∞. Thus no events violating Einstein causality do occur in the infinite future in this stochastic formalism.…”

“…Sloppily stated the measuring procedure is the following: Produce a pure measuring or detection state at x ′ on the hypersurface σ(τ ′ ) and let it interfere with the state propagated to x ′ from all x on the hypersurface σ(τ ). The analyzing or detection field ψ As was also discussed in the previous section, the quantum propagation on H [m,s] is not causal in a classical sense (Einstein causality) but is "stochastically causal" [36]. Furthermore, we remarked at the end of the section that the path integral representation of the probability amplitude associated with a section on H [m,s] (a generalized geometro-stochastic one-particle or one-antiparticle wave function) satisfies a certain invariant second order wave equation on H [m,s] with certain restrictions imposed on the curvature invariants appearing in the term denoted by β: The β-term in (4.2) had to vanish by itself compensating thus mass and possibly spin dependent terms against invariant curvature contributions.…”

“…[29] as well as [30]) for the quantum propagation on H Before we continue our construction of a g-s propagator in the presence of gravitation, let us inject here some brief remarks concerning the so-called Einstein causality, observed to hold for macroscopic distances in space-time, arising in the present context as the re-sult of the superposition and destructive interference of probability amplitudes originating from classically forbidden space-time regions. The property of stochastic microcausality in the framework of the stochastic phase space formulation of quantum mechanics has been investigated in detail by Greenwood and Prugovečki [36] using the concept of "asymptotic causality" [37], i.e. the causal features arising in the limit τ → ∞.…”

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

“…71-83 and 214-217), which culminate in general mathematical proofs [52,53] of violations of Einstein causality. Thus, a more radical reconsideration of the localizability question is mandatory, that has to be then coupled with a stochastic reformulation [55] of the microcausality problem. Thus, a more radical reconsideration of the localizability question is mandatory, that has to be then coupled with a stochastic reformulation [55] of the microcausality problem.…”