aa r X i v : . [ qu a n t - ph ] F e b A model of quantum collapse induced by gravity
F. Lalo¨e ∗ LKB, ENS-Universit´e PSL, CNRS, 24 rue Lhomond, 75005 Paris, FranceFebruary 18, 2020
Abstract
We discuss a model where a spontaneous quantum collapse is induced by the gravitational in-teractions, treated classically. Its dynamics couples the standard wave function of a system withthe Bohmian positions of its particles, which are considered as the only source of the gravitationalattraction. The collapse is obtained by adding a small imaginary component to the gravitationalcoupling. It predicts extremely small perturbations of microscopic systems, but very fast collapseof QSMDS (quantum superpositions of macroscopically distinct quantum states) of a solid object,varying as the fifth power of its size. The model does not require adding any dimensional constantto those of standard physics.
Contents ********A well-known difficulty in quantum mechanics is that the dynamical equations (Schr¨odinger or vonNeumann equations) seem to predict the possible occurrence of quantum superpositions of macroscop-ically distinct states (QSMDS) [1, 2] that are never observed, for instance the creation of Schr¨odingercats [3, 4]. To solve the difficulty, von Neumann [5] suggested to introduce a quantum collapse postulate,which is nowadays part of most introductory textbooks on quantum mechanics.Several authors have proposed to relate quantum collapse to the effects of gravity. One can for instanceassume that gravity is the source of a random noise acting on the state vector, and that this noise projects ∗ laloe at lkb.ens.fr § III-B of Ref. [15] and in Ref. [16].Here we propose a model of the quantum dynamics that also provides a collapse, but with equationsthat are completely deterministic; gravity is treated as a classical field originating from the Bohmianpositions of the particles. In classical physics, gravity already plays special role, since it determinesthe curvature of space-time. In our model, we attribute to gravity another special feature, which is tointroduce small non-Hermitian component in the evolution equation of the state vector. Nothing in thismodel is stochastic; the only source of randomness is the initial randomness of the Bohmian position, asin the de Broglie-Bohm (dBB) theory [17–22]. This model is in the line of a general view where space-timeremains classical, and where the source of the curvature of space-time is the Bohmian positions of theparticles; the various quantum fields propagate inside this classical space-time frame.Combining elements from dBB and spontaneous collapse [23,24] theories is not a completely new idea.Ref. [25] proposes to localize the wave function around the Bohmian positions, but with no real change ofthe Schr¨odinger dynamics; moreover, gravity plays no role in the localization process. Refs. [26] and [27]consider a back action of the Bohmian positions on the wave function, but with a stochastic term, as instandard spontanous collapse theories [23, 24].For the sake of simplicity, here we discuss only spinless non-relativistic particles (including spinswithin a Pauli theory is nevertheless not particularly difficult). As in Refs. [28, 29], we use a dynamicsinvolving an “expanded description” of the physical system: to the standard wave function Ψ definedin the configuration space, we add (in the same space) a mathematical point Q , whose coordinates aredetermined by the Bohmian positions q n of all its particles. Incidentally, and in contrast with the usualinterpretations of the de Broglie-Bohm (dBB) theory, we make no particular assumption concerning thephysical reality of these positions; they can be seen, either as physically real, or as a pure mathematicalobject appearing in the dynamical equations.The equations of this dynamics are given in §
1. In § § We assume that the Hamiltonian H of a physical system is the sum of its internal Hamiltonian H int (including the kinetic energy of the particles and their mutual interactions) and of a gravitational Hamil-tonian H G , due to the attraction of external masses with mass density n G ( r ): H = H int + H G (1)with: H G = − gGm Z d r Ψ † ( r )Ψ( r ) Z d r ′ | r − r ′ | n G ( r ′ ) (2)In this relation, g = 1 in standard theory, G is Newton’s constant, m the mass of the particles, and Ψ( r )the quantum field operator of the particles contained in the physical system. With external sources ofgravity, this Hamiltonian is completely standard. If n G ( r ′ ) is set equal to quantum local density average < Ψ † ( r )Ψ( r ) > , we obtain the usual Schr¨odinger-Newton equation [30].2 .1 Evolution of the state vector We now leave standard quantum mechanics by making two non-standard assumptions. First, we as-sume that H G actually describes the internal gravitational attraction of the system, and that n G ( r ) isdetermined by the Bohmian positions q n of the N particles of the system: n G ( r ) = m N X n =1 δ ( r − q n ) (3)Incidentally, one could also perform a spatial average over a distance a L , as usual in GRW and CSL [23,24]theories, and write for instance: n G ( r ) = mπ / a L N X n =1 e − ( r − q n ) /α L (4)Nevertheless, in what follows, we will only use the simpler form (3). Similarly it has been proposed inRef. [31] to study chemical reactions (within standard quantum mechanics) by an approximation wherethe nuclei are treated classically, and where the backreaction of the quantum electrons on the nuclei isobtained by sampling the Bohmian positions of the electrons over their quantum distribution.Second, as in Ref. [32], we assume that the dimensionless constant g has a small imaginary part ε : g = 1 − iε (5)(one could choose any small number, for instance ε = α , the fine structure constant). This introduces anantiHermitian part in H G : H G = H G + iL (6)where H G is the Hermitian part of H G : H G = H G ( ε = 0) (7)and where L is the localization operator: L = εGm Z d r Ψ † ( r )Ψ( r ) Z d r ′ | r − r ′ | n G ( r ′ ) (8)This operator is diagonal in the position representation. It is the second quantized form of the sum of N single particle potentials taking large values in the vicinity of the Bohmian positions, in the regions ofspace where the gravitational attraction by these positions is strong. With (3), this expression becomes: L = εGm Z d r Ψ † ( r )Ψ( r ) N X n =1 | r − q n | (9)In [28] we introduced a localization operator where the quantum operator Ψ † ( r )Ψ( r ) is coupled to theBohmian positions with a Gaussian spatial average of range a L . Here the Gaussian spreading function isreplaced by a gravitational type of coupling that is proportional to the inverse distance.The state vector | Φ( t ) i evolves according to: i ~ dd t | Φ( t ) i = [ H int + H G ] | Φ( t ) i (10)If ε = 0, the norm of | Φ( t ) i does not remain constant. We can nevertheless introduce the normalized ket (cid:12)(cid:12) Φ( t ) (cid:11) : (cid:12)(cid:12) Φ( t ) (cid:11) = 1 p h Φ( t ) | Φ( t ) i | Φ( t ) i (11)3nd set: D Φ ( r ) = (cid:10) Φ( t ) (cid:12)(cid:12) Ψ † ( r )Ψ( r ) (cid:12)(cid:12) Φ( t ) (cid:11) (12)This normalized state then evolves according to: i ~ dd t (cid:12)(cid:12) Φ( t ) (cid:11) = (cid:20) H int + H G + iεGm Z d r Z d r ′ (cid:2) Ψ † ( r )Ψ( r ) − D Φ ( r ) (cid:3) | r − r ′ | n G ( r ′ ) (cid:21) (cid:12)(cid:12) Φ( t ) (cid:11) (13)To summarize, the two non-standard ingredients of our model are:- the use of the Bohmian positions to define a density of matter in ordinary space; this density is thesource of the classical gravitational field involving the usual Newton constant G .- the introduction of a small imaginary part in G , so that the dynamics becomes irreversible andcollapses QSMDS, as we see below. We assume that the Bohmian positions q n evolve according to the usual Bohmian equation of motion: d q n ( t ) dt = ℏ m −→▽ n ξ ( r , r , .., r N ) (14)where ξ ( r , r , .., r N ) is the phase of the wave function Φ ( r , r , .., r N ), and −→▽ n the gradient taken withrespect to r n = q n . Equivalently, this equation can also be written: d q n ( t ) dt = ℏ im D Φ ( r ) (cid:10) Φ( t ) (cid:12)(cid:12) Ψ † ( r ) ∇ r Ψ( r ) − ∇ r Ψ † ( r )Ψ( r ) (cid:12)(cid:12) Φ( t ) (cid:11) (15)The condition of “quantum equilibrium” means that, when averaged over many realizations of an experi-ment, the distribution of the Bohmian positions in configuration space coincides with the modulus squareof the wave function. In standard dBB theory with the usual Schr¨odinger equation, if this condition issatisfied at the initial time, it is also satisfied at any time. But this property no longer holds in our case,since we have modified the dynamics of the wave function. Nevertheless, in Ref. [29] we discuss why therelaxation process studied by Towler, Russell and Valentini [33, 34] should ensure that this condition isstill valid to an excellent approximation, except in a very short transient time during the appearance(and almost immediate collapse) of a QSMDS. We now discuss the effect of the localization term on the state vector. The situation is similar to thatalready considered in Refs. [28, 29] except that, here, the time constants of the collapse mechanism arisefrom a gravitational energy coupling the quantum particles with their Bohmian positions. We discussonly the simpler version (3) of the model, which introduces no fundamental parameter a L , but similarconclusions apply as well if a non-zero value of a L is chosen. Consider first the non-relativistic Schr¨odinger equation of the electron and proton in a Hydrogen atom,ignoring the spins for the sake of simplicity. Each of the two particles is subjected to two attractions:– the usual Coulomb attraction, which introduces the usual two-body potential in the Schr¨odingerequation for the wave function. 4 the gravitational attraction, appearing as a one-body attractive potential towards the position ofan additional variable: the electron is attracted towards the Bohmian position q p of the proton, andconversely the proton is attracted towards the Bohmian position q e of the electron.The ratio X between the Coulomb and gravitational interactions is very large: X ≃ q πε Gm e m p ≃ (16)where ε is the permittivity of vacuum, q the electronic charge, m e the mass of the electron and m p the mass of the proton. This enormous value of X ensures that the gravitational component plays norole in practice: we just recover the well-known fact that the gravitational attraction remains completelynegligible in the Hydrogen atom. The divergences of n G ( r ) when r = q n do not create any specialproblem: as in the standard theory of the Hydrogen atom, they only introduce kinks in the wave function,but these kinks are 10 times less pronounced than those introduced by the Coulomb potential; inpractice, they have no effect. Moreover, the statistical distribution of q p and q e over many realizationscoincides with the corresponding quantum distributions. Clearly, changing in this way the center ofgravitational attraction has no practical consequence. In addition to this change, the model introduces asmall imaginary component to the gravitational part of the Hamiltonian, which introduces an even morenegligible perturbation.Another example illustrates why, in most cases, the localization term has a very small effect. If (cid:12)(cid:12) Φ( t ) (cid:11) is an eigenstate of the Hamiltonian H int + H G , the average energy (cid:10) H int + H G (cid:11) remains constant:dd t (cid:10) H int + H G (cid:11) = 0 (17)More generally, if (cid:12)(cid:12) Φ( t ) (cid:11) is an eigenstate of A at time t , the localization term has no effect on the derivativeof the average value of A at time t : dd t (cid:12)(cid:12)(cid:12)(cid:12) loc (cid:10) Φ( t ) (cid:12)(cid:12) A (cid:12)(cid:12) Φ( t ) (cid:11) = 0 (18)This is because, if a is the eigenvalue of A , we have:dd t (cid:12)(cid:12)(cid:12)(cid:12) loc (cid:10) Φ( t ) (cid:12)(cid:12) A (cid:12)(cid:12) Φ( t ) (cid:11) = 2 εGm ℏ Z d r Z d r ′ (cid:10) Φ( t ) (cid:12)(cid:12) (cid:2) Ψ † ( r )Ψ( r ) − D Φ ( r ) (cid:3) A (cid:12)(cid:12) Φ( t ) (cid:11) | r − r ′ | n G ( r ′ )= 2 εaGm ℏ Z d r Z d r ′ (cid:2)(cid:10) Φ( t ) (cid:12)(cid:12) Ψ † ( r )Ψ( r ) (cid:12)(cid:12) Φ( t ) (cid:11) − D Φ ( r ) (cid:3) | r − r ′ | n G ( r ′ ) = 0(19)The situation is therefore different from that obtained with GRW and CSL theories [23, 24], where thelocalization mechanism constantly transfers energy to all particles at a small rate: in our model, if thesystem is in a stationary state, thermal equilibrium for instance, its energy remains constant. The reasonfor this difference is that, in GRW and CSL theories, the random localization process involves a noise thatis discontinuous in time, and therefore has a very broad spectrum (infinite in the case of a Wiener process);it cannot be treated as a first order perturbation and, for instance, the Ito term has to be included. Inour model, the localization term is continuous and has a limited frequency spectrum (determined by themotion of the Bohmian positions); since the coupling constant is very small, it can be treated by firstorder perturbation theory, and has a much softer effect. Assume now that the quantum state describes a QSMDS situation, for instance a measurement pointer(or any macroscopic object) in a superposition of two quantum states localized in two different regions of5pace. By contrast, the Bohmian positions remain grouped together, forming a cluster that occupies onlyone of these regions of space. Therefore, in the two branches of the state vector, a strong mismatch thenoccurs between the quantum density of particles and the Bohmian density (but with a different sign),so that the effect of the localization operator L on these branches is significantly different. To evaluateits consequences we can, in (13), ignore the normalization term in D Φ ( r ), which affects both branches inthe same way and does not change their relative amplitude. In the “full component” where the Bohmiandensity accompanies the quantum density, the localisation term in the right hand side of (10) multipliesthe wave function by a number that is of the order of (half of the absolute value of) the self-gravitationalenergy E sg of the pointer, multiplied by the constant ε ; in the “empty component”, it multiplies thewave function by an energy that is negligible with respect to this self-gravitational energy. Altogether,the differential effect takes place with a time constant of the order of: τ collapse ≃ ℏ ε | E sg | (20)with: | E sg | ≃ G M L (21)where M is the mass of the pointer and L its size (we assume that the two wave packets of the pointerare separated by approximately its size, or more). If, for instance, L = 0 . M = 10 − g, andassuming ε = 10 − , we find: τ collapse ≃ − s (22)We note that E sg varies as the fifth power of the size of the pointer (at constant density). For instance,if L = 1 µ m, we obtain a long collapse time τ collapse ≃ s. In experiments such as those of Ref. [35],very large molecules could fly on different path without being collapsed if the duration of the flight isshorter than this time. The model thus predicts a relatively sharp border between small objects that canreach and stay in a quantum superposition of remote states, and larger ones that almost immediatelyget projected onto one single location. As discussed in [29], the origin of this projection is the cohesiveinternal force of solid objects, which forces the Bohmian positions to remain clustered together; gasesthat do not have this internal cohesion do not undergo the same effect. Interestingly, in the correlatedworldline (CWL) theory of quantum gravity [36,37], fifth powers of the masses also appear in the mutualbinding energy for paths. In most situations (except, of course, during the appearance of a QSMDS), the space distribution ofBohmian variables accurately coincides with the quantum space distribution D Φ ( r ). Assuming that thegravitational attraction originates from the distribution of Bohmian positions is not very different thanassuming that the source of attraction is the quantum distribution D Φ ( r ). The effect of the localizationterm will then just be to (slowly) localize the macroscopic system inside itself, or to move towards regionof lower gravitational potential. This term should have no observable effect, except maybe on very longtime and space scales such as those considered in astrophysics; its effect is somewhat reminiscent of theattraction of the so called dark matter.We note in passing that macroscopic quantum superpositions of states that do not produce differentspatial distributions of masses are not reduced by the localization process of the model. For instance,if the flow of electrons in a superconducting ring is in a superposition of two states having rotationsin opposite directions, no significant collapse takes place. Fast collapse occurs only to resolve QSMDSinvolving different gravitational fields, as suggested by Penrose [10].6 .4 Measurements When an apparatus M is used to measure a quantum system S , both physical systems become entangledunder the effect of their mutual interaction. The state vector then splits into several branches, eachcontaining a state of S that is an eigenstate of the measured observable. During the first stages ofmeasurement, as long as the entanglement remains microscopic, the localization term plays no specialrole. But, when the entanglement involves states of M involving significantly different distributionsof masses in space, for instance different positions of a pointer, then the fast collective collapse takesplace: all branches but one of the state vector vanish. The collapse process is therefore initiated insidethe measurement apparatus, but immediately propagates back to S by a standard quantum nonlocaleffect. This is, for instance, what happens in a Bell experiment. No collapse therefore occurs before asignificant part of the measurement apparatus M is part of the entanglement. The result of measurementis determined by the initial Bohmian positions of all particles and, as discussed in [38], in some cases theresult is primarily determined by the initial Bohmian positions of the measurement apparatus.This scenario fits rather well with an old quotation by Pascual Jordan [39]): “observations not onlydisturb what has to be measured, they produce it. In a measurement of position, the electron is forcedto a decision. We compel it to assume a definite position; previously it was neither here nor there, it hadnot yet made its decision for a definite position...”. When introducing nonlinearities in quantum dynamics, one should be careful about avoiding superluminalcommunications [42–44]. In the GRW [23] and CSL [24] versions of modified Schr¨odinger dynamics,nonlinearity and stochasticity compensate each other to cancel superluminal signaling. Similarly, thenonlinear Schr¨odinger-Newton equation can be made compatible with the no-signaling requirement bychanging it to a stochastic differential equation [45]. Here, the situation is somewhat different: thenonlinearity is not introduced as a term coupling the state vector directly to itself, but by the reactionof Bohmian positions onto the wave function; the stochasticity does not arise from a random processconstantly acting on the wave function, but from the random values of the initial Bohmian positions.In order to ensure that the Hamiltonian H G ( ε = 0) is nonsignaling, we introduce a retarded potentialinto in Eq. (2): n G ( r ′ ) ⇒ n G ( r ′ , t − | r − r ′ | c ) (23)where c is the speed of light. We then just have to check that the localization term proportional to ε isalso nonsignaling.Assume that the system, described by the density operator ρ ( t ) = | Ψ( t ) i h Ψ( t ) | , is made of two remotesubsystems A and B , respectively occupying regions of space S A and S B , and described by the partialdensity operators ρ A ( t ) and ρ B ( t ). We denote {| n A i} an ensemble of states of A providing an orthonormalbasis, and {| n B i} a similar basis for system B ; for instance, n A and n B are abbreviated notations for thepositions of the N A particles that are inside S A , and N B particles inside S B , repectively. The evolutionof the matrix elements of ρ A ( t ) introduced by the localization term in ε is given by:dd t (cid:12)(cid:12)(cid:12)(cid:12) loc h n A | ρ A ( t ) | n ′ A i = 2 εGm ℏ X n B h n A , n B | Z d r Z d r ′ (cid:2) Ψ † ( r )Ψ( r ) − D Φ ( r ) , ρ ( t ) (cid:3) + n G ( r ′ , t − | r − r ′ | c ) | n ′ A , n B i (24)where [ C, D ] + denotes the anticommutator of C and D .7e now assume that system A is microscopic, but that B is macroscopic, and that at some timeit is driven to a QSMDS, for instance because a quantum measurement is performed in this region B .We are interested in the possible effects on the partial density operator ρ A ( t ) of the resolution of thisQSMDS by the localization operator. The operator in the right hand side of (24) contains the sum offour contributions: L AA , L BB , L AB and L BA . Here the first index A (or B ) indicates that the integrationvariable r ′ lies in region S A (or S B ), which determines the source of localization; the second index indicatesthat the integration variable r lies in region S A (or S B ), which determines the target of the localizationprocess. Since we assume that A is microscopic, we can ignore L AA , which is local and remains extremelysmall since A is microscopic. We are actually only interested in the terms having a macroscopic source,in other words in the effects of L BB and L BA .In fact, L BB is clearly the most important term. It looks local since it corresponds to a localizationoccurring entirely in region S B by the collective spontaneous localization process discussed in § ρ ( t ), which is a nonlocal object if systems A and B arein an entangled state, this term can introduce quantum nonlocality (in particular violations of the Bellinequalities). For instance, if the measurement is performed on two spin 1 / S B along a direction u , the localizationterm will cancel one component of the singlet state; which component is cancelled depends on the resultof measurement. In other words, in a single realization of the experiment, the spin state in region S − A will immediately be projected onto the opposite spin state on the same direction u . It is neverthelesswell-known that this nonlocatity does not imply any possible superluminal communication – this is thefamous “peaceful coexistence between quantum mechanics and relativity” [46]. Indeed, if we consider theaverage over many realizations, the density matrix of system A remains completely independent of u .Technically, while in the right hand side of (24) n G fluctuates in region B from one realization to thenext, on average it can be replaced by the local density associated with the standard (non-collapsed)solution of the Schr¨odinger equation; this provides the average effect of the localization on the densityoperator of A . So, term L BB ensures that we recover the usual nonlocal quantum correlations betweenthe remote subsystems A and B , the violation of the Bell inequalities, etc., but without any superluminalcommunication.We finally have to consider the effect of the term L BA . It also implies that the measurement resultobtained in region S B may influence the evolution of the density operator ρ A ( t ), but the effect is muchweaker that that of L BB since it tends to zero when the distance between regions S A and S B increases.Again, the average of this effect over many realizations is obtained by replacing n G by the standardquantum density in space. It is not signaling because of the delay | r − r ′ | /c appearing in the right handside of (24): whatever is done to change the Bohmian density inside subsystem B cannot affect theevolution of subsystem A at any time earlier than the minimum delay required by relativity.For the sake of simplicity, we have assumed that A is microscopic and B macroscopic, but the dis-cussion could easily be generalized to the case where both are macroscopic. Our general conclusion,therefore, is that the model is nonsignaling, at least in all situations that we have considered. In the dynamical equations of the model, we have assumed that the Bohmian position of every particleis the source of gravity acting on all other particles. This is of course necessary for the Hermitian partof the Hamiltonian (obtained with ε = 0) if one wishes to reproduce the usual effects of gravity. Butwe have also assumed that this is true for the antihermitian term (term in ε ), introducing in this way“mutual collapse terms”. As a consequence, our localization term in the dynamical equation is similarto a two-body interaction term. By contrast, the localization term of GRW or CSL theories is ratherdescribed by a single-particle potential: the state vector is subjected to the effect of random localizationterms acting on all particles independently, with a probability rule that depends on the values of thewave function at the positions of all particles. In other words, in our model the collapse is a collective8ffect, by contrast with GRW/CSL theories. This difference has several consequences.A first consequence is that, within our model, the localization rate varies roughly proportionally to thesquare of the number of particles involved in a QSMDS. Therefore, much smaller values of the collapsecoupling constant can be used, without losing a very fast collapse rate of QSMDS. In particular, thisexplains why the undesirable heating effects initially predicted in [7] with a gravitational collapse do notoccur here.Another consequence is that, as discussed in § § The Event-Enhanced-Quantum-theory (EEQT) [48] proposes a similar method to describe individualquantum systems and to explain why, in a measurement process, “potential properties of a quantumsystem become actual”. It also enhances the standard quantum description of a system by replacingthe usual space of states by a family of spaces, labelled with an index α , representing the pure stateof a classical system C . An “event”is defined by a change of the value of α . Operators are labelled bytwo indices α and α ′ , and not necessarily self-adjoint, as the non-Hermitian localization term we haveintroduced. A back action of the classical system is also introduced. Under these conditions, α plays arole in EEQT theory that is similar to the role of Bohmian positions in our model. The main differenceis that the evolution of α is not deterministic, but given by a Markov process. We have introduced two basic postulates: the source of gravitation is the Bohmian density of particles,not the quantum density; the gravitational coupling constant includes a small imaginary component.With these two assumptions, predictions that are compatible with presently known facts are obtained,including the appearance of single results in experiments. The dynamics is such that the mathematicalobjects (wave function and positions) constantly follow the physical observations closely; there is no needto update the value of the wave function in order to include new information. For instance, if a sequenceof measurements is performed on the same quantum object, its state vector automatically includes theinformation obtained in the previous measurements; there is no need to add a state vector reduction by9and, or to keep empty components of the state vector. As discussed in [29] in more detail, the modelremains compatible with a whole range of possible interpretations and ontologies.In this model, the quantum collapse is nothing but a consequence of the internal cohesion of macro-scopic objects and of their gravitational self-attraction [29]. The mutual attraction between the particlesof the object forces all Bohmian positions to remain grouped together, because they have to occupyregions of the configuration space where the many particle wave function does not vanish, a consequenceof standard dBB theory. We then assume that these positions collapse the state vector around them:in equation (13), the source of gravitational attraction is the Bohmian density, instead of the quantumdensity D Φ ( r ) appearing in the Schr¨odinger-Newton equation, discussed in detail for instance in [30].As early as in 1965, Bohm and Bub [49] proposed to introduce a collapse dynamics involving hiddenvariables (the components of a vector in the dual space of the Hilbert space in their case). As mentionedin the introduction, Penrose [10] suggested in 1996 that, when a QSMDS involving different spatial dis-tribution of masses (and therefore different space-time configurations) creates an energy fluctuation ∆ E ,the QSMDS spontaneously decays in a time of the order of ~ / ∆ E . The spontaneous collapse arisesbecause of an energy mismatch between two (or more) components of the QSMDS. In this model, theprimary origin of the collapse is a mismatch between two densities of space, the quantum density and theBohmian density; this in turn creates a mismatch of gravitational energy in different components of theQSLMDS and achieves Penrose’s scheme, but without any particular general relativistic effect. Recently,Tilloy has proposed a modification of the GRW theory where the sources of a classical gravitational fieldare the collapse space-time events of that theory [50]).Depending on one’s point of view, the role of the Bohmian positions can be seen as more, or lessimportant, than in standard dBB theory. In the dynamics, they certainly play a more active role thanin dBB theory, where the positions do not appear in the dynamical equation giving the evolution of thestate vector, but just follow the spatial variations of the wave function. Here, the Bohmian positions actas mathematical attractors of the state vector | Φ( t ) i through the gravitational term (including its smalldissipative component in ε ). This introduces a nonlinearity in the dynamics of | Φ( t ) i . Nevertheless, aswe have seen, in most situations this change has very little effect on the evolution of | Φ( t ) i – except insituations where QSMDS appear, which are then rapidly projected by this term. The model thereforeillustrates how the addition of a single additional variable to the standard equations, namely a pointposition in the configuration space, allows one to significantly enrich the dynamics and to take intoaccount collapse situations.From a purely interpretative point of view, one can see this continuous attraction as a pure mathe-matical ingredient to replace the stochastic fields of GRW and CSL theories, as well as their probabilityrule. One can then hold a view where the Bohmian positions are just mathematical objects creatingthis attraction, and where physical reality is directly represented, for instance, by the quantum density D Φ ( r ). But it is also perfectly possible to consider that all the individual Bohmian positions of theparticles provide a direct representation of reality, as usual in dBB theory.This model is in the line of calculations where gravity is treated classically, within general relativity.This remains compatible with a classical structure of space-time, in which the various quantum fields(electromagnetic for instance) propagate (semi-classical gravity [51]); such schemes are sometimes useful inquantum cosmogenesis [53,54]. The circularity of the defintion of time in quantum theory [16] is avoided.A standard approach to semi-classical gravity is to use a quantum average of the energy momentumtensor operator to construct the Einstein tensor [55–57]. Nevertheless, paradoxes may then arise: forinstance, if a body is in a quantum superposition of two locations, each localization of the body attractsthe other. Also, as discussed by Eppely and Hannah [58], one could in principle measure directly themodulus of the wave function, and therefore obtain superluminal signaling. Other arguments have beenbuilt, involving thought interference experiment, to discuss possible inconsistencies, or to plead in favorof a semi-classical theory of gravitation [59–63]. In our model, as in that of Ref. [50], the paradoxesarising from of delocalized sources of gravity disappear: in each realization of an experiment, the sourceof gravitation always remains localized in space (since it originates from the Bohmian positions). Of10ourse, in most situations (when no QSMDS occurs) it is practically equivalent to take the Bohmianpositions, or the average quantum density of particles, as the source of gravity, due to the quantumequilibrium conditions. In this sense, the predictions of this model are very similar to those of the theoryof semiclassical gravity proposed by Tilloy and Diosi [12], the major difference being that their approachis based on a stochastic spontaneous localization, while no random perturbation is invoked in the presentarticle.At this stage, the model remains very elementary, in particular because its treatment of gravity remainssimply Newtonian, not Einsteinian: for instance, it does not include gravitational waves. The hope is thatthe model could be an approximation of some more elaborate theory, compatible with general relativity.One could also speculate about a possible generalization to a quantum treatment of a gravitational field,still having its sources in the Bohmian positions of the particles. One hope could be to find a justificationof the complex value of the coupling constant by analogy with electromagnetic spontaneous emission,also taking into account the intrinsic nonlinear character of general relativity. This, of course, remainscompletely speculative. As it is, the model is definitely in the line of a semi-classical treatment of gravity. Acknowledgments : the author is grateful to Antoine Tilloy, Lajos Diosi, Philip Pearle and NicolasGisin for useful comments and suggestions.
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