In this note, a one-dimensional system composed of an assembly of interacting particles, is considered. Each particle in the assembly moves in a well-defined trajectory on a line with a velocity of fixed magnitude, which randomly reverses direction. The intrinsic forces in the system are assumed to induce stochastic transitions between velocity states in such a way that the average dynamics of the assembly is Newtonian. It is shown that there is a close analogy between collective oscillations in the model system and the propagation of a free quantum particle in one dimension.
In this note, a one-dimensional system composed of an assembly of interacting particles, is considered. Each particle in the assembly moves in a well-defined trajectory on a line with a velocity of fixed magnitude, which randomly reverses direction. The intrinsic forces in the system are assumed to induce stochastic transitions between velocity states in such a way that the average dynamics of the assembly is Newtonian. It is shown that there is a close analogy between collective oscillations in the model system and the propagation of a free quantum particle in one dimension.
Recently, Plyukhin and Schofield have shown that there is a close analogy between collective oscillation of a system composed of an assembly of interacting particles and the propagation of a free quantum particle in one dimension. They could not have interpreted some of the equations involved in this analogy. We show here that the arbitrary field, they have introduced, satisfies the Klein-Gordon equation, too. Moreover, the arbitrary field is proportional to the local electric field. This local electric filed may induce a transition between the two-particle states; one moving to the left and the other to the right with the speed of light.
The Klein-Gordon equation is often created by taking the deterministic relativistic invariant EE = pp+ momo (c=1), replacing E with E-V(x), E=id/dt partial and p with -id/dx partial and having the derivatives act on a function W(x,t). This seems to suggest that the starting point is deterministic classical mechanics (special relativistic form) E,p equation with stochasticity somehow introduced by having average E and p follow from the action of the operators id/dt partial and -id/dx partial on a stochastic function W(x,t). It also makes it somewhat difficult to understand, from first principles, why one would create a linear equation in E,p (compatible with EE=pp+momo), i.e. the Dirac equation. We suggest that the entire approach to Klein-Gordon equation is stochastic, with the EE=pp+momo equation appearing as an invariant expression of average values which describes motion as seen from different frames moving with constant average velocity v. In such a case, one is not limited to the form EE=pp+momo and there is nothing unusual about trying to create an equation linear in E and p. In particular, we argue that the free particle form exp(-iEt) exp(ipx) follows from two separate maximizations of Shannon’s entropy subject to complex constraints (for periodicity). As a result, free motion with a constant E and p is already stochastic as seen from a photon moving in a region with an index of refraction n or a particle moving in a region with constant potential. Changing to a potential V(x) leads to the notion of an average kinetic energy because a constant p is associated with a distribution in x i.e. exp(ipx). One may think in terms of these average values in order to consider frames moving with constant v. This leads to both -Et+px -EE+pp=momo being invariants. If a V(x) is present, one may replace E with E-V(x), but then needs a statistical average Sum over p a(p)exp(ipx) to describe pp.. The time portion may still be exp(-iEt) if one has a bound system characterized by a single average energy. This yields the Klein-Gordon equation.There is, however, no particular reason why one could not use an equation linear in E and p (consistent with -EE+pp=momo), because the latter follows form the an average v (deterministic view) of the stochastic exp(-iEt+ipx) two-dimensional probability.
Abstract This work presents an alternative approach to obtain the quantum field equations in curved spacetime, considering that sufficiently small particles follow stochastic trajectories around geodesic. Our proposal is based on a stochastic differential equation in which the noise term experienced by the quantum particles is a consequence of the stochastic background in spacetime. This fact allows the particles to describe erratic movements and locally the universe exhibits characteristics akin to a lake with gentle ripples rather than a flat unyielding surface. Building upon this foundational understanding, we investigate the influence of this background on quantum-scale particles without considering the metric to be stochastic, rather we let test particles move randomly around the geodesic of macroscopic particles. Their behavior aligns with solutions to the Klein-Gordon (KG) equation specific to this curved spacetime. As the KG equation, in its non-relativistic limit within a flat spacetime, reduces to the Schrödinger equation, consequently, we propose a compelling connection: the Schrödinger equation may emerge directly from a spacetime lacking local smoothness.
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.
The issue of non-locality in quantum mechanics can potentially be resolved by considering relativistically covariant diffusion in four-dimensional spacetime. Stochastic particles described by the Klein-Gordon equation are shown to undergo a classical diffusion process in spacetime coordinates, which is seen by transforming the quantum Cauchy-momentum equations to a Lagrangian frame of reference. Since the quantum potential term is removed under this transformation, the equations for momentum propagation along particle trajectories assume a classical form. A local stochastic de Broglie-Bohm interpretation for the Klein-Gordon system can subsequently be derived. We also introduce the concept of momentum equivariance to replace the second-order Bohm-Newton equations of motion, which break down due to non-linear terms of the stochastic Lagrangian derivative.
We apply stochastic quantization to a system of N interacting identical bosons in an external potential Φ, by means of a general stationary-action principle. The collective motion is described in terms of a Markovian diffusion on , with joint density and entangled current velocity field , in principle of non-gradient form, related to one another by the continuity equation. Dynamical equations relax to those of canonical quantization, in some analogy with Parisi–Wu stochastic quantization. Thanks to the identity of particles, the one-particle marginal densities ρ, in the physical space , are all the same and it is possible to give, under mild conditions, a natural definition of the single-particle current velocity, which is related to ρ by the continuity equation in . The motion of single particles in the physical space comes to be described in terms of a non-Markovian three-dimensional diffusion with common density ρ and, at least at dynamical equilibrium, common current velocity v . The three-dimensional drift is perturbed by zero-mean terms depending on the whole configuration of the N -boson interacting system. Finally, we discuss in detail under which conditions the one-particle dynamical equations, which in their general form allow rotational perturbations, can be particularized, up to a change of variables, to the Gross–Pitaevskii equations.