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 $e^{iL_uQ}$ gives a completely positive dephasing semigroup. If the characteristic exponent lacks an intrinsic charge scale, the Levy-Khintchine theorem forces $\eta(\xi)=D | \xi |^{\alpha}$ ($0<\alpha\le2$), so coherences decay at $\Gamma_{ab}=D | q_a-q_b |^{\alpha}$. For integer charge differences, the process descends to the wrapped variable $\Theta_u=L_u\bmod 2\pi$; for real charges, the lift is the relevant phase. Schoenberg's theorem ensures complete positivity for any finite real spectrum when $0<\alpha\le2$, 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 It\^o 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.