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Relaxation to quantum equilibrium and the Born rule in Nelson's stochastic dynamics. (arXiv:2305.04084v1 [quant-ph])

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Nelson's stochastic quantum mechanics provides an ideal arena to test how the
Born rule is established from an initial probability distribution that is not
identical to the square modulus of the wavefunction. Here, we investigate
numerically this problem for three relevant cases: a double-slit interference
setup, a harmonic oscillator, and a quantum particle in a uniform gravitational
field. For all cases, Nelson's stochastic trajectories are initially localized
at a definite position, thereby violating the Born rule. For the double slit
and harmonic oscillator, typical quantum phenomena, such as interferences,
always occur well after the establishment of the Born rule. In contrast, for
the case of quantum particles free-falling in the gravity field of the Earth,
an interference pattern is observed \emph{before} the completion of the quantum
relaxation. This finding may pave the way to experiments able to discriminate
standard quantum mechanics, where the Born rule is always satisfied, from
Nelson's theory, for which an early subquantum dynamics may be present before
full quantum relaxation has occurred.

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