Pramana – J. Phys. (2020) 94:163 © Indian Academy of Sciences https://doi.org/10.1007/s12043-020-02032-6 A geometric look at the objective gravitational wave function reduction FARAMARZ RAHMANI1,2,∗, MEHDI GOLSHANI2,3 and GHADIR JAFARI4 1Department of Physics, School of Sciences, Ayatollah Boroujerdi University, Boroujerd, Iran 2School of Physics, Institute for Research in Fundamental Science (IPM), Tehran, Iran 3Department of Physics, Sharif University of Technology, Tehran, Iran 4School of Particles and Accelerators, Institute of Research in Fundamental Sciences (IPM), Tehran, Iran ∗Corresponding author. E-mail: [email protected], [email protected] 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 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, One of the questions that has always been raised is the we can consider a particle in different locations simulta- boundary between quantum and classical mechanics. neously with the distribution ρ =|ψ(x, t)|2 (see figure There is a critical mass for the transition from quantum 1). This is a quantum mechanical concept which refers domain to the classical world [1,2]. Such critical mass to the uncertainty in the position of the particle and is determines the macroscopicity or microscopicity of an not obvious in our classical world. object. By knowing the density of an object, the macro- The Schrödinger–Newton equation for a single parti- scopicity is directly determined in terms of the size of cle with distribution ρ =|ψ(x, t)|2 is the body [2]. We expect that macroscopic bodies obey ∂ψ( , ) ¯ 2 |ψ( , )|2 the rules of classical mechanics, i.e. definite position and x t = − h ∇2 − 2 x t 3 ih¯ GM d x momentum, determinism, etc. But, microscopic bodies ∂t 2M |x − x| obey the rules of quantum mechanics, like the uncer- × ψ(x, t). (1) tainty in position and momentum of the particle. One iEt/h¯ of the approaches to determine the boundary between By using the stationary state ψ = ψ(x)e ,forthe quantum mechanics and classical mechanics is the grav- classical limit, which satisfies the above equation, and itational approach. The outstanding gravitational studies using the variational method, a relation between the for determining the boundary between the quantum mass of a particle and the width of its associated sta- world and the classical world started by Karolyhazy [2]. tionary wave packet is obtained. That relation, which is Diosi’s work, based on the Schrödinger–Newton equa- h¯ 2 σ( ) = tion, is a remarkable work that was done after that (see min 3 (2) refs [1,3,4]). In that equation, there is a term due to the Gm self-gravity of the particle or body. Here, self-gravity is provides a criterion for the transition from the quantum due to the quantum distribution of matter and is definable world to the classical world or the breakdown of quan- even for a point-particle. According to the Born rule, tum superposition in terms of universal constants and 0123456789().: V,-vol 163 Page 2 of 9 Pramana – J. Phys. (2020) 94:163 be explained by taking into account the self-gravity of the particle (object, body). We shall prove this in the Bohmian context. After Diosi, the most significant work which has been done, is the gravitational approach of Penrose which is Figure 1. An imagination for the self-gravity of a particle or body in the context of quantum mechanics. The particle has based on two essential concepts in physics: the princi- no exact location due to the uncertainty principle. ple of equivalence in general relativity and the principle of general covariance [6–8]. The overview of Penrose’s the mass of the particle (objective property) [1]. For a work is as follows: Consider a body in two different body with radius R, the minimum wave packet width is locations with their associated states |φi , i = 1, 2. Each (R) (1/4) 3/4 σ = (σ(min)) R . The critical size of a body for state satisfies the Schrödinger equation separately as a the transition from quantum domain to classical domain stationary state, with a unique Killing vector. The super- −5 |ψ=α|φ +β|φ is about Rc = 10 cm. For details, see [2,4]. posed state 1 2 is also a stationary The works of Karolyhazy and Diosi are conceptually state with the unique Killing vector K = ∂/∂t.When the same. In fact, in the both works, the main con- the self-gravity of the body (the curvature of the space– cept is the existence of uncertainty in the structure of time due to the mass of the object itself) is considered, space–time. But their different results (different charac- the quantum state of the gravitational field of the body teristic width, collapse time etc.) for extensive objects, at different locations, i.e., |Gi , i = 1, 2, must also be are due to the different mathematical approaches [5]. taken into account. This changes the state |ψ to the state For a point particle, their results are identical. We shall |ψG=α|φ1|G1+β|φ2|G2 which is not a stationary show that our Bohmian analysis is also consistent with state in the sense that it does not have a unique Killing their results, i.e., in obtaining the characteristic width vector. Thus, the total state decays to one of the states similar to the work of Karolyhazy and Diosi for a point to get a stationary state with definite Killing vector of particle. We shall not do Bohmian investigation for an space–time. In this approach, the decay time for tran- extended object in this paper. sition from the quantum domain to classical domain is The self-gravity of a body or a particle would localise obtained [6]. the position distribution of the particle or body. It The gravitational considerations of Penrose which reduces the uncertainty in the position of the particle. It were stated above can be used for justifying the wave seems that our classical world is the product of a gravi- function reduction through the measurement process. tational reduction [6–8]. Usually, in a measurement process we have an appa- Wave function reduction also takes place through a ratus and a microscopic system with their associated measurement process. In a measurement process, the wave functions. The quantum state of the apparatus, wave function of a quantum system reduces instan- as a macroscopic body, is entangled with the micro- taneously to one of its eigenvectors (wave function scopic or quantum system (an electron for example) collapse). This is one of the postulates of orthodox or during the measurement [6,9,12]. As the apparatus is standard quantum mechanics, known as collapse postu- a macroscopic body, its self-gravity is significant. Then, late. Thus, there is no room for justifying the collapse according to the previous statements, the total entangled phenomena in the standard quantum mechanics. In a state (|ψG) is not stationary and it decays to a specific measurement process, a pure state which is governed state for having a definite unique Killing vector. Conse- by the linear Schrödinger equation, evolves to a mixed quently, the microscopic system which is entangled with state after measurement, through a non-unitary evolu- the apparatus also goes to a specific eigenstate. Here, the tion [9–14]. Some physicists modify the Schrödinger role of gravity is obvious [6–8]. equation by adding non-Hamiltonian terms for justify- Why do we want to study this topic in the Bohmian ing the non-unitary evolution of the wave function and framework? Because, relativistic Bohmian quantum its reduction process [5,15–18]. mechanics can be represented geometrically [19–22]. In this paper, we do not study what is happening in On the other hand, gravity is described geometrically. the measurement processes. Our aim is to study the Thus, we were persuaded to study the objective gravita- objective gravitational wave function reduction in the tional wave function reduction in the Bohmian context. Bohmian context. Against the collapse theories, there is a different If quantum mechanics is universal, our classical world approach known as ‘many-words interpretation’ or in should be in a superposition. But, it seems that our short ‘MWI’. In this approach, the wave function is a classical world is not in a superposition. For example, real entity and it does not collapse through a measure- a chair in a room is not in its different states of its ment [23–26]. Rather, any outcome of the measurement different degrees of freedom simultaneously. This can is in a different real word. These different real worlds Pramana – J. Phys. (2020) 94:163 Page 3 of 9 163 or branches of our Universe do not interfere with each other.