Derivation of the wave function collapse in the context of Nelson’s stochastic mechanics

Abstract

The von Neumann collapse of the quantum mechanical wavefunction after a position measurement is derived by a purely probabilistic mechanism in the context of Nelson's stochastic mechanics.Running Title: Stochastic mechanics and measurement PACS number: 03.65.Bz

A Footnote to Nelson's Interpretation of the Two-Slit Experiment

“…We then take the latter as a reference process in a variational problem that takes into account the presence of the screen with the two slits. The solution of the variational problem, suitably extending on the results of [4,5], is the stochastic process that Nelson has analyzed (two-slit process) whose density profile is familiar in wave interference.…”

Stochastic mechanics and the Feynman integral

“…The bi-directional generator of the Nelson process ( see (III.32) for the definition) originates from a certain time-symmetric differential for finite-energy diffusions that has been used in [23]- [25] to develop elements of Lagrangian and Hamiltonian dynamics within Nelson's stochastic mechanics. Moreover, as we showed in [26], this time-symmetric kinematics permits to derive the collapse of the wave function after a position measurement through a stochastic variational principle. The connection between the operators ( …”

“…We recall now the basic facts from the time-symmetric kinematics employed in [23]- [26]. In order to develop stochastic mechanics as a generalization of classical mechanics a salient difficulty is that the finite-energy diffusion {x(t); t 0 ≤ t ≤ t 1 } representing position of the nonrelativistic particle has two natural velocities, namely the pair (β(t), γ(t)) or, equivalently, the pair (v(t), u(t)).…”

Quantum Schrödinger Bridges

“…Moreover, V denotes the family of finite-energy, C n -valued stochastic processes on [t 0 , t 1 ]. Following the same variational analysis as in [45], we get a Hamilton-Jacobi-Bellman type equation…”

“…It has been crucial in order to develop a Lagrangian and a Hamiltonian dynamics formalism in the context of Nelson's stochastic mechanics in [40,44,45]. Notice that replacing (3)- (5) with (15), we replace the pair of real drifts (v, u) by the unique complex-valued drift v − iu that tends correctly to v when the diffusion coefficient tends to zero.…”