A Geometric Look at the Objective Gravitational Wave Function Reduction

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

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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. Thus, it seems that the problem of absence of superposition in the classical domain has been resolved in this context. For such an interpretation, there is no need to modify the Schrödinger equation to explain the collapse of the wave function. Hence, the unitary evolution of the Schrödinger equation is preserved in the MWI approach. In MWI, an observer is a part of a quantum mechanical system [27]. Thus, there is no sharp cut-off between quantum and classical worlds. In the opinion of some researchers, one of the drawbacks of the MWI, is the question of what experiment can show the existence of other branches of the many-worlds [5]? But, here, we do not talk about the details of this approach or its abilities and inabilities. We want only to remember ( , ) that it is an important approach that exists along with the Figure 2. Propagation of function S x t in the configura- other approaches to explain the measurement processes tion space. Normal to it is the momentum field of the particle. of a quantum system or its classical limit. Note that in this paper we are not talking about the wave function the particle is a vector field orthogonal to S(x, t) which reduction through the measurement processes. Rather, is obtained by using the relation p =∇S(x, t) (see we discuss about the roles of mass, gravity and quan- figure 2). The energy of the particle is obtained by using tum potential or quantum force in the objective wave the relation E =−∂ S/∂t [9]. function reduction. We want to explain that the classical With the initial position x0 and the initial wave world is a world with negligible quantum potential or function ψ(x0, t0), which gives the initial velocity, the quantum force. But negligible with respect to what? In position of the particle is obtained using the guidance fact, with respect to gravity. This is what we are trying equation to explain in this paper. dx(t) ∇S(x, t) Bohmian quantum mechanics is a causal and deter- = . (5) ministic theory in which a particle has a definite trajec- dt m X=x(t) tory with definable physical quantities like in classical The initial wave function gives the initial phase S0 and mechanics. The probability density of being a parti- consequently initial velocity. With the initial velocity cle in the volume d3x is equal to ρ(x, t) = R2(x, t) ∇S0 p0 [9,28–30]. In Bohm’s own view, the departure from clas- v0 = = , sical mechanics appears in an essential entity known as m m ‘quantum potential’. It is a non-local potential with non- and the initial position x0, the evolution of the system classical features. The primary approach of Bohm was is obtained. But we are always faced with a distribution the substitution of the polar form of the wave function, of the initial positions and velocities practically. Thus, S(x, t) there is always an uncertainty in the prediction of evo- ψ(x, t) = R(x, t) exp i , lution of the system practically. The dynamics of the ¯ h system is obtained using [9] into the Schrödinger equation which leads to a quantum d2x Hamilton–Jacobi equation as m =∇(Q(x) + V(x)) . dt2 ∂ S(x, t) (∇S)2 + + V (x) + Q(x) = . The expression X = x(t) in relation (5) means that ∂ 0 (3) t 2m among all the possible trajectories, one of them is cho- The last term in the above equation is the non-relativistic sen. Due to our ignorance with respect to all the initial quantum potential which is given by data we are faced with an ensemble of trajectories. √ h¯ 2∇2 R(x, t) h¯ 2∇2 ρ(x, t) The possibility of definition of trajectories in Bohmian Q =− =− √ . (4) quantum physics provides the capability of a geometric ( , ) ρ( , ) 2mR x t 2m x t visualisation. The quantum force exerted on the particle is defined as In §2, we shall argue how an objective explanation for f =−∇Q. The function S(x, t) is the action of the the classical limit of a free particle based on Bohmian system. It propagates like a wave front in the configu- trajectories is possible. In §3, we shall derive a Poisson- ration space of the system [9]. The momentum field of like relation for the Bohmian quantum potential for 163 Page 4 of 9 Pramana – J. Phys. (2020) 94:163 the gravitational reduction of wave function based on deviation of Bohmian trajectories. For the possibility of developing and generalising the subject in future, we start with the relativistic calculations, then we shall consider non-relativistic limit of deviation equation for deriving Diosi’s formula. The effect of gravitational field of the particle is considered as a curved space– time with a fixed metric tensor gμν. The relativistic generalisation of the concept of self-gravity for a particle in the Schrödinger–Newton equation is that the particle is affected by the space–time curvature due to its mass–energy. It is natural that if we did not consider the uncertainty in position of the particle in the framework of quantum mechanics, such interpretation Figure 3. The distribution of trajectories of the particle for self-gravity either in the relativistic or in the non- which is guided by a wave packet in (1 + 1)-dimensional relativistic domain was not possible. space–time. Due to uncertainty in position of the particle, we are faced with an ensemble of trajectories.

2. Towards an objective reduction based on Bohmian trajectories in the standard quantum mechanics, the dispersion of a wave packet is explained by using the Heisenberg uncer- In Bohmian quantum mechanics, the usual condition for tainty principle. Now, we want to clarify how the concept getting classical limit is the vanishing of Bohmian quan- of Bohmian trajectories help us to get a criterion for the tum potential. Vanishing of quantum force is needed in wave function reduction of a free particle. some situations [9]. These conditions do not give an It has been demonstrated in ref. [9] that the trajec- objective criterion for the reduction of wave function or tory of the particle which is guided by a Gaussian wave the classical limit of a quantum system. In other words, packet is by vanishing the quantum potential or quantum force, ⎛ ⎞ / 2 1 2 one cannot estimate the needed mass or objective prop- ht¯ erties of an object for transition from quantum world to x(t) = x ⎝1 + ⎠ , (9) 0 σ 2 classical world. The reason is that the mass of the particle 2m 0 or body does not have any active role in its dynamical evolution. This may be achieved by considering the where x0 denotes the initial position of the particle. If effects of gravity of the particle on its dynamics. the initial position of the particle is at x0 = 0 (in the Let us ﬁrst study the trajectories of a free particle middle of the wave packet), the trajectory is a straight which is guided by a Gaussian wave packet. The ampli- line. The quantum force f =−∇Q for such a trajectory tude of the wave packet of a free particle with zero initial vanishes and its trajectory is classical. Figure 3 shows group velocity is [9] that for initial position x0 = 0, the trajectories are not = ( πσ2)−3/4 −x2/4σ 2 , straight lines and the particle is affected by a quantum R 2 e (6) force. where σ is the random mean square width of the packet So far, we found that the curved trajectories are the at time t which represents the spreading of the wave results of the quantum force. For returning them to packet [9]. The quantum potential for this system and the classical straight trajectories, we need an agent to the quantum force exerted on the particle are obtained keep the deviation between trajectories constant. The using the relations: best candidate is the self-gravity of the particle. In other words, gravity prevents from more dispersion and h¯ 2∇2 R h¯ 2 x2 Q =− = 3 − (7) consequently more uncertainty in the position of the 2mR 4mσ 2 2σ 2 particle. When the mass of the particle tends to a spe- and ciﬁc value, its self-gravity increases. Then, the quantum force and gravitational force can be equal and deviation h¯ 2 f =−∇Q = x. (8) remains constant. This is achieved when the mass of the 4mσ 2 particle affects its dynamics. Here, the mass of the parti- In Bohmian quantum mechanics, the quantum force is cle has an active role, not a passive one. In the following, responsible for the spreading of wave packet [9]. But, we explain this idea further. Pramana – J. Phys. (2020) 94:163 Page 5 of 9 163

3. Bohmian trajectories and wave function reduction

For the possibility of developing and generalising the subject in future, we obtain the deviation between trajectories for a relativistic particle. On the other hand, the Bohmian dynamics of relativistic matter can be described as a conformal transformation of the background metric [19–21]. Hence, for the richness of the Figure 4. The distribution of time-like trajectories associ- topic, we start from the relativistic case. Karolyhazy, in ated with a wave packet on a space-like hybersurface t . his famous paper [2], started from a relativistic action, Vector η represents the deviation vector between two neigh- because he was searching for answer in a more fun- bour trajectories. damental way, such as the properties of space–time. Penrose, too, relates the gravitational reduction to the fundamental principles of the space–time dynamics, Constant deviation implies that the width of the wave such as general covariance and equivalence principle packet remains constant. In other words, we shall face [6–8]. with a wave packet in the form The generalisation of self-gravity to the relativistic iEt case is as follows: The quantum distribution of the par- ψ = R(x) exp − , ticle ρ =|ψ|2 makes the space–time curved. Then the h¯ dynamics of the particle is affected by this curvature. In where R(x) is the amplitude in relation (6). An obvi- other words, the particle moves in its own gravity. This ous difference between these two types of forces is can be justified in the quantum mechanical framework that gravitational force is defined in a real space, while where there is uncertainty in the position of the particle. the quantum force is defined in the configuration space We represent the metric of the space–time with the sym- of the particle or system. This type of view brings up bol gμν. By this imagination, we are going to calculate this picture in our mind that in the reduction process, the deviation between trajectories. quantum effects do not vanish absolutely; rather the Here, in addition to gravity, the quantum force is self-gravity of the particle as an attractive force, does present. Thus, we first derive Bohmian acceleration for a not allow the quantum force to appear. While the wave spinless relativistic particle. The Hamilton–Jacobi equa- packet evolves according to the Schrödinger equation, tion in a curved space is in the form [31]: the particle can be at different locations in the space, with ∇ ∇μ = μ = M2, different probability densities ρ(x) = R2(x) which is μS S pμ p (11) determined by the wave function. By spreading the wave where packet, the deviation between trajectories varies with time. On the other hand, the self-gravity of the particle M2 = m2(1 + Q). (12) would localise the trajectories. This leads to the equal- Also, ity of the two forces. Figure 4 schematically represents μ the distribution of an ensemble of time-like trajectories h¯ 2 ∇μ∇ R Q = (13) associated with a wave packet. m2 R Fortunately, it can be proved that Bohmian trajectories do not cross each other. Thus, these trajectories can be is the relativistic Bohmian quantum potential of a spin- considered as a congruence in space. In fact, the single- less particle in a curved background [32]. Here, R stands valuedness of the wave function for a closed loop leads for the amplitude of the relativistic wave function and to S is the action of the particle or the phase of its associated wave function [9,19]. The explicit form of Q is S = dS = ∇S · dx = p · dx = nh. (10) not needed, because we need the non-relativistic form of c c c the Bohmian quantum potential (4) for obtaining Diosi’s Then, the net phase difference around a closed loop formula. Four-momentum of the particle is defined as is constant and for every instant at the position x,the follows: =∇ μ μ momentum p S is single-valued [9]. p = Mu . (14) In the next section, we demonstrate that Diosi’s formula is obtained through the investigation of the devi- In the presence of quantum force, the geodesic equa- μ ν μ μ ation η between Bohmian trajectories. tion u ∇νu = 0, with four-velocity u , is not valid 163 Page 6 of 9 Pramana – J. Phys. (2020) 94:163 anymore. It will be replaced by the equation ν μ 1 μ ν 1 μ u ∇νu =− u u ∇ν ln(1 + Q) + ∇ ln(1 + Q). ν∇ μ = μ , 2 2 u νu a(B) (15) (27) μ where a(B) refers to the Bohmian acceleration of the The left-hand side of this equation is the Bohmian accel- particle due to the quantum force. For deriving a rela- eration (15). Thus, we get Bohmian acceleration in the tion for the Bohmian acceleration, we start from relation following form: (11). By differentiation from both sides of relation (11) with respect to the parameter τ which parametrises the μ 1 μ ν 1 μ a =− u u ∇ν ln(1 + Q) + ∇ ln(1 + Q). trajectories of the particle, we have (B) 2 2 (28) d μ d 2 (pμ p ) = M . (16) dτ dτ As the quantum potential Q is very small with respect ( 2/ 2) Then, to the classical energies, ( h¯ m 1 in relation (13), μ specially for a particle with significant mass and self- d p dM Q ( +Q) Q 2pμ = 2M . (17) gravity), we can assume that 1andln 1 . dτ dτ Then, the last relation reduces to On the other hand, we have μ 1 μ ν 1 μ μ ν a( ) =− u u ∇νQ + ∇ Q. (29) d p dx μ ν μ B 2 2 = ∇ν p = u ∇ν p , (18) dτ dτ This problem is studied in a fixed background metric where we have used the replacement: with the signature (+1, −1, −1, −1), i.e. we do not con- ν sider the back-reaction effects of matter on space–time. d dx ν μ μ → ∇ν = u ∇ν. Four-vector field u = ∂x /∂τ is tangent to the τ τ (19) d d trajectories. The deviation vector between two neigh- Now, by substituting (18)and(19)into(17), we get bouring trajectories is defined as ημ = ∂xμ/∂s,in ν μ μ μ which the parameter s parametrises the curves between pμ(u ∇ν p ) = Muμ∇ M = Mu ∇μM. (20) μ neighbouring trajectories so that η is tangent to them. μ μ Substitution of relation (14)into(20), leads to Also, η and u are orthogonal. The velocity field for the deviation vector between two neighbourhood trajec- M ( ν∇ (M μ)) = M ∇μM uμ u ν u uμ (21) tories is defined as μ or μ dη ν μ v = = u ∇νη , (30) ν μ μ τ u ∇ν(Mu ) =∇ M (22) d where we have used relation (19). The acceleration of which gives the deviation vector is μ ν∇ M + M ν∇ μ =∇μM. u u ν u νu (23) μ 2 μ dv d η d ν μ λ ν μ = = (u ∇νη ) = u ∇λ(u ∇νη ), By multiplying both sides of the above equation by dτ dτ 2 dτ 1/M, we obtain (31) μ ν μ μ ν ∇νM ∇ M u ∇νu =−u u + . (24) where we have used relation (19) again. According to M M the definitions μ μ Now, we should express the right-hand side of the above μ ∂x μ ∂x η = and u = equation in terms of quantum potential. For this purpose, ∂s ∂τ we start from eq. (12)toget and independence of parameters s and τ,wehave ∇ M 2∇ Q 2 ν 1 m ν μ μ μ μ 2M∇νM = m ∇νQ ⇒ = . (25) ∂u ∂ ∂x ∂ ∂x ∂η M 2 M2 = = = . (32) ∂s ∂s ∂τ ∂τ ∂s ∂τ By using relation (12) again in the above relation we get But this is the same result that can be reached through ∇νM 1 m2∇νQ 1 ∇νQ 1 the definition of the Lie derivative. In other words, = = = ∇ν ( + Q). 2 ln 1 μ μ M 2 m (1 + Q) 2 1 + Q 2 ∂u ∂η μ μ = ⇒ L η = Lηu = 0 (26) ∂s ∂τ u ⇒ ν∇ ημ = ην∇ μ. Now, we substitute this result into relation (24)toget u ν νu (33) Pramana – J. Phys. (2020) 94:163 Page 7 of 9 163 2 μ Now, we substitute this result into relation (31)toget d η λ 1 μ ν 1 μ = η ∇λ − u u ∇νQ + ∇ Q 2 2 μ dτ 2 2 d η λ ν μ = u ∇λ(η ∇νu ). (34) + μ ρηλ ν, dτ 2 Rρλνu u (43) In fact, substituting relation (33) into relation (31), gives where we have used relation (29). Now, we need its the acceleration of the deviation vector in terms of veloc- non-relativistic limit to demonstrate the correctness μ ity field derivative ∇νu . Now, we expand relation (34). of our previous arguments about the role of quantum and gravitational forces in the objective gravitational 2 μ collapse of the wave function. d η λ ν μ λ ν μ μ = (u ∇λη )∇νu + u η (∇λ∇νu ). (35) i δ τ 2 At the non-relativistic limit, we can take u 0 , d τ → ∇ → ∂ μ = ∂μ∂ ϕ( ) ϕ( ) t, μ μ and R0ρ0 ρ x ,where x For the first term of the above equation, we use the result is the Newtonian gravitational potential. Also the non- (33). This gives relativistic Bohmian quantum potential has no explicit 2 μ dependence on time. In other words, ∂0 Q = 0 (see rela- d η λ ν μ λ ν μ = (η ∇λu )∇νu + u η (∇λ∇νu ). (36) tion (4)). Then, eq. (43) takes the form dτ 2 ∂2ηi ∂i Q For the second term, we use the curvature formula = η j ∂ − ∂i ϕ(x) . 2 j (44) μ μ ρ ∂t m [∇λ∇ν −∇ν∇λ] A = Rρλν A . One of the possibilities for having constant deviation μ Here, Rρλν denotes the curvature due to the mass– or parallel trajectories in the ensemble in the non- energy of the particle. For the field, uμ,wehave relativistic domain, is that in the above equation we take μ μ ρ [∇λ∇ν −∇ν∇λ] u = Rρλνu (37) ∂i Q = m∂i ϕ(x), i = 1, 2, 3or∇ Q = m∇ϕ. or (45) μ μ μ ρ ∇λ∇νu =∇ν∇λu + Rρλνu . (38) This relation represents the equivalence of Bohmian By substituting this result into the second term of (36) quantum force and self-gravitational force of the par- we get ticle for constant deviation between trajectories. Thus, 2 μ the validity of our physical argument, i.e. the equality d η λ ν μ λ ν μ μ ρ = (η ∇λu )∇νu + u η (∇ν∇λu + Rρλνu ) of self-gravitational force with the quantum force in the dτ 2 transition regime is confirmed here. Originally, the evo- λ ν μ λ ν μ = (η ∇λu )∇νu + u η (∇ν∇λu ) lution of the wave packet is not stationary but if the mass μ ρ λ ν +Rρλνu u η . (39) of the particle is large enough to produce enough gravity, then the equality of the two forces makes the wave The second term of the above equation is equal to packet stationary. For obtaining an objective criterion for λ ν μ ν λ μ the transition from the quantum world to the classical u η (∇ν∇λu ) = η ∇ν(u ∇λu ) world, we act as follows. For simplicity, we do calcu- −(ην∇ λ)(∇ μ). νu λu (40) lations for a one-dimensional stationary wave packet, σ The second term of this result cancels out the first term with the width 0. If we calculate the average quantum of (39), after substituting (40)into(39). Note that ν and potential for a stationary one-dimensional wave packet λ are dummy indices and the replacement ν ↔ λ is 2 −1/4 −x2/4σ 2 iEt/h¯ ψ(x, t) = (2πσ ) e 0 e allowed. Thus, relation (39) reduces to the relation 0 2 μ with d η λ ν μ μ ρ λ ν = η ∇λ(u ∇νu ) + Rρλνu u η . (41) − / − 2/ σ 2 2 ( ) = ( πσ2) 1 4 x 4 0 , dτ Rs x 2 0 e The expression in parentheses is equal to eq. (15). Then, we get we have +∞ 2ημ = 2 d λ μ μ ρ λ ν Q s Rs Qs dx = η ∇λa + R u η u . (42) −∞ dτ 2 (B) ρλν +∞ h¯ 2 ∇2 R h¯ 2 Relation (42) can be expressed in terms of quantum = R2 − s dx ∼ (46) s σ 2 potential: −∞ 2m Rs 2m 0 163 Page 8 of 9 Pramana – J. Phys. (2020) 94:163 which is the average quantum potential of the particle, As in those approaches in which gravity is responsible when it is described by a stationary wave packet with for reducing wave function, so is in our approach. But, the width σ0. In Bohmian quantum mechanics, for a there are technical differences between our approach stationary wave packet we have p =∇S(t) = 0, and their’s. They consider uncertainty in the structure because for the stationary wave functions, the phase of space–time (an ensemble of metrics). This has been of the wave is a function of time only. Therefore, the considered specially in the works of Karolyhazy and kinetic energy of the particle vanishes, and the energy Penrose [1–3,5–8]. But, we work in a single space– of the particle is due to the quantum potential com- time. Furthermore, we use the definition and ability of pletely, while in standard quantum mechanics it is due the concept of Bohmian trajectories which is not pos- to the kinetic term p2/2m. The result of (46), i.e. sible in the standard quantum mechanics. Instead of ¯ 2/ σ 2 h 2m 0 , is exactly the same as kinetic term in standard an ensemble of metrics (dispersion in metric) we are quantum mechanics. There, it is obtained through the faced with an ensemble of trajectories which leads to relation a mass distribution in the configuration space of the +∞ 2 2 particle and consequently provides a self-gravitational ∗ pˆ h¯ K = ψ ψ dx ∼ . (47) force. When the self-gravitational force is equal to the s s s σ 2 −∞ 2m 2m 0 quantum force, then we have parallel trajectories. At This result has been obtained in ref. [1]. this moment, the particle is at the threshold of classical The gravitational self-energy for a particle with mass world. This equilibrium leads to a criterion (characteris- m is defined as tic width (50)) for transition from the quantum domain to the classical domain. This criterion also gives a crit- |ψ(x , t)|2 2 1/3 2 3 ical mass mc = (h¯ /Gσ0) for which the transition Ug(x) =−Gm d x , (48) |x − x| occurs. Naturally, for masses grater than the critical |ψ( , )|2 = 2( ) mass, the gravitational force overcomes the quantum where x t Rs x . The average gravitational force and classical behaviour appears. Specially when self-energy of the particle in one dimension, with width m →∞,σ0 → 0wehave σ0,is +∞ 1 − 2/σ 2 = 2 ψ = x 0 → δ( ). Ug Rs Ug dx lim 1/4 e x (51) −∞ σ0→0 σ 0 +∞ +∞ 2( ) 2( ) =− 2 Rs x Rs x Gm dx dx −∞ −∞ |x − x| Then, the wave packet tends to a Dirac delta function and we have more certainty in the position of the Gm2 ∼− . particle (classical domain-wave function reduction). In σ (49) 0 contrast, for masses less than the critical mass, the quan- It is obvious from relations (48)and(49) that the average tum force overcomes the gravitational force and the quantum potential and average self-gravitational energy deviation between trajectories increases (wave func- are functions of σ0. Because we are actually dealing with tion is dispersive in the usual language) and quantum average values, we obtain the critical width for which the features become significant. In fact, we have shown average gravitational force and quantum force become the existence of a dynamical procedure for the wave equal: function collapse which is real and is not due to the environmental effects like in decoherence interpreta- U ¯ 2 d Q s = d g ⇒ (σ ) ∼ h , tion. In all these approaches, the superposition breaks 0 critical 3 (50) dσ0 dσ0 Gm down through the gravitational effects. Thus, the system does not have a unitary (but norm-preserving) where the derivative has been taken with respect to the evolution [5]. probabilistic width σ . The above relation is the famous 0 By imposing operator (∇·) on both sides of condition result of Diosi that has been obtained here from a geo- (45), we get the Poisson-like equation for the quantum metrical approach related to Bohmian trajectories (see potential as ref. [1]). This study reveals the role of gravity in the wave function reduction in the context of Bohmian quantum 2 mechanics [2,5]. ∇ Q = 4πGmρ It is necessary to mention that our approach is in the or √ field of actual collapse similar to those of the Penrose, h¯ 2 ∇2 ρ Karolyhazi and Diosi approaches, not in the context of − ∇2 √ = 4πGρ. (52) 2m2 ρ decoherence approach. Let us make the point more clear. Pramana – J. Phys. (2020) 94:163 Page 9 of 9 163

This is a non-linear differential equation which [12] J Von Neumann, Mathematical foundations of quantum expresses that, in the transition from quantum world to mechanics (Princeton University Press, 1996) the classical world, quantum information (Q) is replaced [13] J A Wheeler and W H Zurek, Quantum theory and mea- by gravitational information (ϕ). Equation (52) can be surement (Princeton University Press, 1983) solved for different distributions ρ to get further results. [14] F J Belinfante, A survey of hidden-variables theories We have derived this relation in ref. [35] from another (Elsevier Science and Technology, 1973) [15] G C Ghirardi, A Rimini and T Weber, Quantum proba- point of view. Then we investigated the behaviour of ρ bility and applications II (1984) pp. 223–232 in the wave function reduction with respect to the [16] G C Ghirardi, A Rimini and T Weber, Phys. Rev. D 34, mass variations. Relation (52) is the consequence of 470 (1986) gravitational wave function reduction in Bohm’s causal [17] V Allori, S Goldstein, R Tumulka and N Zanghi, On quantum theory. This may be a starting point for further the common structure of Bohmian mechanics and the studies in this context. Ghirardi–Rimini–Weber theory, quant-ph/0603027 [18] N Gisin, Phys. Lett. A 143(1–2) 1 (1990) [19] R W Carrol, Fluctuations, information, gravity and the quantum potential (Springer, Netherlands, 2006) 4. Conclusion [20] F Shojai and M Golshani, Int. J. Mod. Phys. A 13(4), 677 (1998) In this study, we argued how it is possible to obtain a [21] I Licata and D Fiscaletti, Quantum potential: Physics, criterion for the gravitational objective wave function geometry and algebra (Springer International Publish- reduction in the Bohmian context. It was done based on ing, 2014) Bohmian trajectories and geometrical concepts. Finally, [22] F Rahmani, M Golshani and M Sarbishei, Pramana – in addition to the famous result of ref. [1], an inter- J. Phys. 86(4), 747 (2016) esting nonlinear equation, i.e. eq. (52), was obtained. [23] Bryce Seligman Dewitt and Neill Graham, The many It represents a Poisson-like equation for the Bohmian worlds interpretation of quantum mechanics (Princeton quantum potential in the reduction process. The solu- Series in Physics, 2015) [24] Max Tegmark, Fortsch. Phys. 46, 855 (1998) tions of eq. (52) can be investigated numerically or [25] H Everett, Rev. Mod. Phys. 29, 454 (1957) analytically. In fact, we have stated that quantum infor- [26] J A Wheeler, Rev. Mod. Phys. 29, 463 (1973) mation reduces to the gravitational information in the [27] David Wallace, The emergent multiverse: Quantum the- reduction process. It is displayed as the balance between ory according to the Everett interpretation (Oxford average self-gravitational force of the particle and its University Press, 2014), ISBN 978-0-19-954696-1 average quantum force. Now, we have a geometrical [28] D Bohm, Phys. Rev. 85(2), 166 (1952) and a dynamical notion about the objective gravitational [29] D Bohm, Wholeness and the implicate order (Routledge wave function reduction. This geometrical study has & Kegan Paul, 1980) never been done before. [30] D Bohm and B J Hiley, The undivided universe: An onto- logical interpretation of quantum theory (Routledge, 1993) [31] The Bohmian quantum potential for the relativistic References electron has been derived recently by Hiley and co- workers through complicated methods of Clifford alge- 0 [ ] [1] L Diosi, Phys. Lett. A 105(4–5), 199 (1984) bra C3 33 . There, the quantum potential is considered [2] F Karolyhazy, Nuovo Cimento A 42, 390 (1966) as additional term in the Hamilton–Jacobi equation of [3] L Diosi, Phys. Lett. A 120(8), 377 (1987) the electron and is obtained by comparing with the μ [4] L Diosi, Phys. Rev. A 40, 1165 (1989) original equation pμ p = m2. In other words, in μ [5] A Bassi, K Lochan, S Satin, T P Singh and H Ulbricht, that case too the relation pμ p = M2 holds, where 2 2 Rev. Mod. Phys. 85, 471 (2013) M = m + QDirac [6] R Penrose, Gen. Relativ. Gravit. 28(5), 581 (1996) [32] The polar form substitution of the wave function into [7] R Penrose, J. Phys.: Conf. Ser. 174012001 (2009) the Klein–Gordon equation is not the only way to get [8] R Penrose, Found. Phys. 44, 557 (2014) the Hamilton–Jacobi equation (11) [9] P R Holland, The quantum theory of motion (Cambridge [33] B J Hiley and R E Callaghan, Found. Phys. 42, 192 University Press, Cambridge, 1993) (2012) [10] J J Sakurai and J Napolitano, Modern quantum mechan- [34] F Rahmani and M Golshani, Int. J. Theor. Phys. 56, 3096 ics (Cambridge University Press, 2017) (2017) [11] K Landsman, The measurement problem, in: Foun- [35] F Rahmani, M Golshani and Gh Jafari, Int. J. dations of quantum theory. Fundamental theories of Mod. Phys. A 33(22) (2018), https://doi.org/10.1142/ physics (Springer, Cham, 2017) Vol. 188 S0217751X18501294