The Decoherence Exponent: Stable Phase Noise and Constraints on Objective State Reduction

Abstract

Let $L_u$ be a symmetric Levy process representing an unobserved phase lift, coupling to a quantum system via charge $Q$. Averaging $e^{i{L}_{u}Q}$ gives a completely positive dephasing semigroup. If the characteristic exponent lacks an intrinsic charge scale, the Levy-Khintchine theorem forces $\eta (\xi )=D|\xi {|}^{\alpha }$ ($0<\alpha \le 2$), so coherences decay at ${\Gamma }_{ab}=D|q_a-q_b{|}^{\alpha }$. For integer charge differences, the process descends to the wrapped variable ${\Theta }_u=L_u\mathrm{mod}2\pi$; for real charges, the lift is the relevant phase. Schoenberg's theorem ensures complete positivity for any finite real spectrum when $0<\alpha \le 2$, while $\alpha >2$ 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 ($\alpha =d/p$) and Bochner subordination of Brownian phase by an $\alpha /2$-stable clock. For a bounded reduction walk, we prove Born probabilities for every symmetric bounded proposal law. Simulations of truncated stable laws ($\alpha =0.5,1,1.5,2$) confirm this law-independence. In continuous time, $\alpha$-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.