Stable symmetric probability laws in quantum mechanics

Wawrzyniec Guz

Stefan Wojciechowski

Milen V. Velev, Ivo D. Dinov

Let $L_u$ be a symmetric Levy process representing an unobserved phase lift, coupling to a quantum system via charge $Q$. Averaging $e^{iL_uQ}$ gives a completely positive dephasing semigroup. If the characteristic exponent lacks an intrinsic charge scale, the Levy-Khintchine theorem forces $η(ξ)=D | ξ|^α$ ($0<α\le2$), so coherences decay at $Γ_{ab}=D | q_a-q_b |^α$. For integer charge differences, the process descends to the wrapped variable $Θ_u=L_u\bmod 2π$; for real charges, the lift is the relevant phase. Schoenberg's theorem ensures complete positivity for any finite real spectrum when $0<α\le2$, while $α>2$ yields an analytic obstruction. This boundary is exact, verified numerically here. A Gaussian integrated phase gives quadratic charge dependence, though colored Gaussian noise need not yield a Markov semigroup. Two idealized non-Gaussian mechanisms are analyzed: inverse-power Poisson shot noise ($α=d/p$) and Bochner subordination of Brownian phase by an $α/2$-stable clock. For a bounded reduction walk, we prove Born probabilities for every symmetric bounded proposal law. Simulations of truncated stable laws ($α=0.5,1,1.5,2$) confirm this law-independence. In continuous time, $α$-stable drivers fail to reduce: the exact-flow (Marcus) equation oscillates, and the naive Ito jump equation fails to preserve state. For Gaussian white noise, the Itô model collapses to Born probabilities, whereas the exact-flow (Stratonovich) model does not collapse without a threshold and yields non-Born exit probabilities. The quadratic case recovers Milburn's small-step limit, not his exact Poisson dynamics. Finally, a bi-temporal geometry is examined as a possible compact phase source, but closed timelike curves, nonunitary evolution, and an unstable mode tower prevent a consistent field-theoretic realization.

L. E. Ballentine

Bagchi, B, Quesne, C, Znojil, M

The continuity equation relating the change in time of the position probability density to the gradient of the probability current density is generalized to PT-symmetric quantum mechanics. The normalization condition of eigenfunctions is modified in accordance with this new conservation law and illustrated with some detailed examples.

Leon W. Cohen