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A short walk in quantum probability
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Hudson R
Philos Trans A Math Phys Eng Sci
2018[Apr]; 376
(2118
): ä PMID29555800
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This is a personal survey of aspects of quantum probability related to the
Heisenberg commutation relation for canonical pairs. Using the failure, in
general, of non-negativity of the Wigner distribution for canonical pairs to
motivate a more satisfactory quantum notion of joint distribution, we visit a
central limit theorem for such pairs and a resulting family of quantum planar
Brownian motions which deform the classical planar Brownian motion, together with
a corresponding family of quantum stochastic areas.This article is part of the
themed issue 'Hilbert's sixth problem'.