The role of the zero-point fluctuations in the classical stochastic Schrödinger like equation

L. Fritsche, M. Haugk

We present a modified version of Nelson's seminal paper on the derivation of the time-dependent Schrödinger equation which draws on the equation of motion of a particle that moves under the influence of a classical force field and additional stochastic forces. The emphasis of our elaboration is focused on the implication of allowing stochastic forces to occur, viz. that the energy E of the particle is no longer conserved on its trajectory in a conservative force field. We correlate this departure Δ E from its classical energy with the energy/time uncertainty relation where Δ t is the average time for Δ E to persist. The stability of atoms, the zero-point energy of oscillators, the tunneling effect and the diffraction at slits are shown to be directly connected with the occurrence of such energy fluctuations. We discuss and rederive Nelson's theory entirely from this point of view and generalize his approach to systems of N particles which interact via pair forces. Achieving reversibility in a description of particle motion that is akin to Brownian motion, represents a salient point of the derivation. We demonstrate that certain objections raised against Nelson's theory are without substance. We also try to put the particular worldview of this version of stochastic quantum mechanics into perspective with regard to the established Copenhagen interpretation.

Jay Gambetta, Howard M. Wiseman

Jonathan Oppenheim, Emanuele Panella

Theories of gravity in which the metric is fundamentally classical predict stochastic fluctuations in the gravitational field. In this article, we study the stochastic Klein-Gordon equation as a starting point to understand the phenomenology of linearised classical-quantum hybrid gravity. In particular, we describe how to compute the non-equilibrium two point function of the scalar field, showing explicitly the role of the initial state in regulating divergences. To do so, we use a "mod-squared-retarded" pole-prescription and find that the covariance in the field is non-zero only outside the lightcone, scales inversely with the spatial distance of the spacetime points and grows linearly in time. The energy has a contact divergence similar to that found in the quantum case. We conclude by discussing possible implications of anomalous diffusion for hybrid theories of gravity, especially looking at the energy density in the predicted gravitational waves background, which can be inferred from the scalar covariances.

Андрей Валерьевич Чуркин

Wilfried Grecksch, Hannelore Lisei

L. Fritsche, M. Haugk

Abstract We present a modified version of Nelson's seminal paper on the derivation of the time‐dependent Schrödinger equation which draws on the equation of motion of a particle that moves under the influence of a classical force field and additional stochastic forces. The emphasis of our elaboration is focused on the implication of allowing stochastic forces to occur, viz. that the energy E of the particle is no longer conserved on its trajectory in a conservative force field. We correlate this departure Δ E from its classical energy with the energy/time uncertainty relation where Δ t is the average time for Δ E to persist. The stability of atoms, the zero‐point energy of oscillators, the tunneling effect and the diffraction at slits are shown to be directly connected with the occurrence of such energy fluctuations. We discuss and rederive Nelson's theory entirely from this point of view and generalize his approach to systems of N particles which interact via pair forces. Achieving reversibility in a description of particle motion that is akin to Brownian motion, represents a salient point of the derivation. We demonstrate that certain objections raised against Nelson's theory are without substance. We also try to put the particular worldview of this version of stochastic quantum mechanics into perspective with regard to the established Copenhagen interpretation.