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Quantum Spontaneous Stochasticity

The quantum wave-function of a massive particle with small initial uncertainties (consistent with the uncertainty relation) is believed to spread very slowly, so that the dynamics is deterministic. This assumes that the classical motions for given initial data are unique. In fluid turbulence non-uniqueness due to "roughness" of the advecting velocity field is known to lead to stochastic motion of classical particles. Vanishingly small random perturbations are magnified by Richardson diffusion in a "nearly rough" velocity field so that motion remains stochastic as the noise disappears, or classical spontaneous stochasticity, . Analogies between stochastic particle motion in turbulence and quantum evolution suggest that there should be quantum spontaneous stochasticity (QSS). We show this for 1D models of a particle in a repulsive potential that is "nearly rough" with $V(x) \sim C|x|^{1+α}$ at distances $|x|\gg \ell$ , for some UV cut-off $\ell$, and for initial Gaussian wave-packet centered at 0. We consider the WKB limit with $\hbar/m \to 0$, then position-spread $σ\to 0$. The limit of the position density is non-deterministic, with equal probabilities of th