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arXiv:2012.04606 · International Journal of Modern Physics A

On the dynamics of gravity-induced wave function reduction

Faramarz Rahmani, Mehdi Golshani

Published Sep 20, 2021• 3 citations
arXivHTMLPDF

Abstract

In this study, we use the concept of Bohmian trajectories to present a dynamical and deterministic interpretation for the gravity-induced wave function reduction. We shall classify all possible regimes for the motion of a particle, based on the behavior of trajectories in the ensemble and under the influence of quantum and gravitational forces. In the usual approaches all the information is obtained from the wave function evolution. But, on the basis of Bohm’s deterministic quantum theory, we can investigate the motion of the particle during the reduction processes. This leads to analytical and numerical results for the reduction time and equation of motion of the particle. In this regard, a new visualization will be provided for the reduction time.

DOI
10.1142/s0217751x21501815

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In this study, we use the concept of Bohmian trajectories to present a dynamical and deterministic interpretation for the gravity-induced wave function reduction. We shall classify all possible regimes for the motion of a particle, based on the behavior of trajectories in the ensemble and under the influence of quantum and gravitational forces. In the usual approaches all the information is obtained from the wave function evolution. But, on the basis of Bohm’s deterministic quantum theory, we can investigate the motion of the particle during the reduction processes. This leads to analytical and numerical results for the reduction time and equation of motion of the particle. In this regard, a new visualization will be provided for the reduction time.
In this research, we study the gravitational reduction of the wave function, which has been investigated in standard quantum mechanics. As a new look at the problem, we investigate the reduction of the wave function by studying the dynamics of the particle motion, which can be defined in the quantum Bohmian framework. In this regard, quantities such as the critical mass of reduction, reduction time, and reduction temperature, which are similar to the Unruh temperature, are systematically obtained.
In objective gravitational reduction of the wave function of a quantum system, the classical limit of the system is obtained in terms of the objective properties of the system. On the other hand, in Bohmian quantum mechanics the usual criterion for getting classical limit is the vanishing of the quantum potential or the quantum force of the system, which suffers from the lack of an objective description. In this regard, we investigated the usual criterion of getting the classical limit of a free particle in Bohmian quantum mechanics. Then we argued how it is possible to have an objective gravitational classical limit related to the Bohmian mechanical concepts like quantum potential or quantum force. Also we derived a differential equation related to the wave function reduction. An interesting connection will be made between Bohmian mechanics and gravitational concepts.
Development of a general approach to construction of efficient high-dimensional bases is an outstanding challenge in quantum dynamics describing large amplitude motion of molecules and fragments. A number of approaches, proposed over the years, utilize Gaussian bases whose parameters are somehow-usually by propagating classical trajectories or by solving coupled variational equations-tailored to the shape of a wave function evolving in time. In this paper we define the time-dependent Gaussian bases through an ensemble of quantum or Bohmian trajectories, known to provide a very compact representation of a wave function due to conservation of the probability density associated with each trajectory. Though the exact numerical implementation of the quantum trajectory dynamics itself is, generally, impractical, the quantum trajectories can be obtained from the wave function expanded in a basis. The resulting trajectories are used to guide compact Gaussian bases, as illustrated on several model problems.
Bohmian mechanics is a causal interpretation of quantum mechanics in which particles describe trajectories guided by the wave function. The dynamics in the vicinity of nodes of the wave function, usually called vortices, is regular if they are at rest. However, vortices generically move during time evolution of the system. We show that this movement is the origin of chaotic behavior of quantum trajectories. As an example, our general result is illustrated numerically in the two-dimensional isotropic harmonic oscillator. © EDP Sciences.
Bohmian mechanics, developed by Bohm in 1952, provides an alternative interpretation to nonrelativistic quantum mechanics [1-3]. In the hydrodynamic formulation of quantum mechanics, the continuity equation and the quantum Hamilton-Jacobi equation (QHJE) are obtained by substituting the wave function expressed in terms of the real amplitude and the real action function into the time-dependent Schrödinger equation. Bohm’s analytical approach has been used to compute and interpret realvalued quantum trajectories from a precomputed wave function for a diverse range of physical processes [3-8]. In the synthetic approach, the quantum trajectory method has been developed as a computational tool to generate the wave function by evolving ensembles of real-valued quantum trajectories through the integration of the hydrodynamic equations on the fly [9]. Remarkable progress has been made in the use of T&F Cat # K11224, Chapter 18, Page 284, 15-9-2010 real-valued quantum trajectories for solving a wide range of quantum mechanical problems [10].
Abstract The recent experimental proposals by Bose et al. and Marletto et al. (BMV) outline a way to test for the quantum nature of gravity by measuring gravitationally induced differential phase accumulation over the superposed paths of two ∼ 10−14 kg masses. These authors outline the expected outcome of these experiments for semi-classical, quantum gravity and collapse models. It is found that both semi-classical and collapse models predict a lack of entanglement in the experimental results. This work predicts the outcome of the BMV experiment in Bohmian trajectory gravity - where classical gravity is assumed to couple to the particle configuration in each Bohmian path, as opposed to semi-classical gravity where gravity couples to the expectation value of the wave function, or of quantized gravity, where the gravitational field is itself in a quantum superposition. In the case of the BMV experiment, Bohmian trajectory gravity predicts that there will be quantum entanglement. This is surprising as the gravitational field is treated classically. A discussion of how Bohmian trajectory gravity can induce quantum entanglement for a non superposed gravitational field is put forward.