A geometric look at the objective gravitational wave function reduction
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A geometric look at the objective gravitational wave function reduction FARAMARZ RAHMANI1,2,∗ , MEHDI GOLSHANI2,3 and GHADIR JAFARI4 1 Department
of Physics, School of Sciences, Ayatollah Boroujerdi University, Boroujerd, Iran of Physics, Institute for Research in Fundamental Science (IPM), Tehran, Iran 3 Department of Physics, Sharif University of Technology, Tehran, Iran 4 School of Particles and Accelerators, Institute of Research in Fundamental Sciences (IPM), Tehran, Iran ∗ Corresponding author. E-mail: [email protected], [email protected] 2 School
MS received 17 March 2020; revised 4 July 2020; accepted 8 September 2020 Abstract. There is a famous criterion for objective wave function reduction which is derived by using the Shrödinger–Newton equation [L Diosi, Phys. Lett. A 105(4–5), 199 (1984)]. In this regard, a critical mass for the transition from quantum world to the classical world is determined for a particle or an object. In this paper, we shall derive that criterion by using the concept of Bohmian trajectories. This study has two consequences. The first is, it provides a geometric framework for the problem of wave function reduction. The second is, it represents the role of quantum and gravitational forces in the reduction process. Keywords. Gravitational reduction of the wave function; Bohmian quantum potential; Bohmian geodesic deviation equation; Bohmian trajectories. PACS Nos 03.65.Ca; 03.65.Ta; 04.20.Cv; 03.65.w
1. Introduction One of the questions that has always been raised is the boundary between quantum and classical mechanics. There is a critical mass for the transition from quantum domain to the classical world [1,2]. Such critical mass determines the macroscopicity or microscopicity of an object. By knowing the density of an object, the macroscopicity is directly determined in terms of the size of the body [2]. We expect that macroscopic bodies obey the rules of classical mechanics, i.e. definite position and momentum, determinism, etc. But, microscopic bodies obey the rules of quantum mechanics, like the uncertainty in position and momentum of the particle. One of the approaches to determine the boundary between quantum mechanics and classical mechanics is the gravitational approach. The outstanding gravitational studies for determining the boundary between the quantum world and the classical world started by Karolyhazy [2]. Diosi’s work, based on the Schrödinger–Newton equation, is a remarkable work that was done after that (see refs [1,3,4]). In that equation, there is a term due to the self-gravity of the particle or body. Here, self-gravity is due to the quantum distribution of matter and is definable even for a point-particle. According to the Born rule, 0123456789().: V,-vol
mass distribution of a particle or a body is ρ = |ψ(x, t)|2 which can be used to define self-gravity. In other words, we can consider a particle in different locations simultaneously with the distribution ρ = |ψ(x, t)|2 (see figure 1). This is a quantum mechanical
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