Spacetime Stochasticity and Second Order Geometry

Kuipers, Folkert

doi:10.48550/arxiv.2203.16399

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Stochastic Mechanics1

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2023

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“…Also, the generalization of stochastic mechanics to particles moving on (pseudo-)Riemannian manifolds [23,[62][63][64][65][66] is well established. Moreover, it was recently realized [67,68] that the techniques developed in the stochastic mechanics literature to achieve such extensions can be defined in a rigorous way in the framework of second order geometry [69][70][71][72].…”

Section: Stochastic Mechanicsmentioning

confidence: 99%

Stochastic Mechanics and the Unification of Quantum Mechanics with Brownian Motion

Kuipers¹

2023

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We unify Brownian motion and quantum mechanics in a single mathematical framework. In particular, we show that non-relativistic quantum mechanics of a single spinless particle on a flat space can be described by a Wiener process that is rotated in the complex plane. We then extend this theory to relativistic stochastic theories on manifolds using the framework of second order geometry. As a byproduct, our results suggest that a consistent path integral based formulation of a quantum theory on a Lorentzian (Riemannian) manifold requires an Itô deformation of the Poincaré (Galilean) symmetry, arising due to the coupling of the quadratic variation to the affine connection.

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