Dissipative extension of the GhirardiRiminiWeber model
Abstract
In this paper we present an extension of the GhirardiRiminiWeber model for the spontaneous collapse of the wavefunction. Through the inclusion of dissipation, we avoid the divergence of the energy on the long time scale, which affects the original model. In particular, we define new jump operators, which depend on the momentum of the system and lead to an exponential relaxation of the energy to a finite value. The finite asymptotic energy is naturally associated to a collapse noise with a finite temperature, which is a basic realistic feature of our extended model. Remarkably, even in the presence of a low temperature noise, the collapse model is effective. The action of the new jump operators still localizes the wavefunction and the relevance of the localization increases with the size of the system, according to the socalled amplification mechanism, which guarantees a unified description of the evolution of microscopic and macroscopic systems. We study in detail the features of our model, at the level of both the trajectories in the Hilbert space and the master equation for the average state of the system. In addition, we show that the dissipative GhirardiRiminiWeber model, as well as the original one, can be fully characterized in a compact way by means of a proper stochastic differential equation.
pacs:
03.65.Ta, 03.65.Yz, 02.50.EyI Introduction
Collapse models were formulated to describe in a unified framework the behavior of microscopic systems, as accounted for by quantum mechanics, and the emergence of the objective macroscopic world, described by classical mechanics. After the pioneering works by Pearle Pearle1976 (), the first consistent collapse model was put forward by Ghirardi, Rimini and Weber (GRW) Ghirardi1986 (); see Bassi2003 (); Bassi2013 () for more details and a list of references about the historical development of collapse models.
The crucial feature of the GRW model is that the wavefunction associated with the state of a physical system undergoes sudden and random localization processes. The latter do not practically affect microscopic systems, while they become relevant already on very short time scales for macroscopic systems, by virtue of the amplification mechanism. The localization processes prevent macroscopic systems from being in a superposition of states centered around macroscopically distinct positions. In addition, since any measurement process consists in an interaction between a microscopic system and a macroscopic measurement apparatus, the localization processes give a dynamical explanation of the collapse of the wavefunction to one of the eigenstates of the measured observable, without the need of introducing an adhoc reduction postulate Bassi2007b (). The nonlinear and stochastic nature of the reduction postulate in standard quantum mechanics is replaced within collapse models by a modification of the Schrödinger equation which includes proper nonlinear and stochastic terms.
One of the main advantages of collapse models is that, besides the relevant conceptual differences with respect to the standard theory, they provide experimentally testable predictions which depart from those of quantum mechanics Nimmrichter2011 (); RomeroIsart2011 (); see Bassi2013 () for a detailed list of references. In particular, collapse models call into question the universal nature of the superposition principle, as they set a fundamental limit above which no physical system can exhibit position superpositions, except for a negligibly small time. The investigation of collapse models thus plays a significant role in the experimental tests on the boundaries between the classical and the quantum description of reality at the mesoscopic and macroscopic scale. Will it be possible to prepare quantum superpositions for more and more complex systems with the future advances of the experimental techniques, or is there an intrinsic limit which will prevent from this, as predicted by collapse models?
The renewed interest in collapse models and the demand to compare their predictions with actual experimental data coming from very different setups has further motivated the formulation of more realistic models. The initial idea that the nonlinear and stochastic modification of the Schrödinger equation represents an intrinsic property of Nature, was then superseded by the view that the collapse of the wavefunction is induced by a physical field filling space, which acts as a universal noise. Indeed, the precise definition of such a field needs the formulation of a new fundamental theory going beyond standard quantum mechanics Bassi2013 (). Hence, collapse models should be understood as phenomenological models, which capture the main effects of the interaction with the abovementioned noise. Basic physical principles set some general constraints on the features of the admissible models. This is the case, e.g., for the absence of fasterthanlight signaling Gisin1990 (); Bassi2013b () or the principle of energy conservation. An important drawback of the GRW model is the violation of the energy conservation: the stochastic action of the noise induces larger and larger fluctuations in the momentum space, so that the energy of the system diverges for asymptotic times, although with a rate which is very small Ghirardi1986 (). This is a common feature of the first collapse models Ghirardi1986 (); Diosi1989 (); Ghirardi1990 () and it traces back to the absence of any dissipative mechanism within the interaction between the system and the noise Bassi2005a (); Bassi2005 (); Vacchini2007 (), as also witnessed by the structure of the master equations which can be associated with these models.
In this paper, we modify the GRW model in order to avoid the divergence in time of the energy, thus making an important step toward the reestablishment of the energy conservation within the model. We explicitly show that this can be achieved via the introduction of new localization operators, without changing the other defining features of the model. This provides us with a more realistic collapse model, while we keep the original effectiveness and physical transparency of the GRW model. The convergence of the energy of the system to a finite value allows us to associate the noise with a finite temperature , such that the GRW model is recovered in the high temperature limit . We follow a strategy similar to that exploited for a simplified collapse model Bassi2005 (); Vacchini2007 () and based on the formal analogy with a Lindblad master equation Lindblad1976 () including dissipation. Besides discussing in detail the physical consequences of the extension of the model, we also express both the original and the generalized GRW models in terms of a stochastic differential equation. This analysis, which relies on the framework of the stochastic differential equations of jump type in Hilbert spaces Barchielli1991 (); Barchielli1994 (); Barchielli1995 (), fills a gap with respect to collapse models with localization continuous in time Bassi2013 ().
The paper is organized as follows. In Sec. II, we briefly recall the main features of the GRW model. In Sec. III, we introduce the dissipative GRW model, characterizing the evolution in the Hilbert space induced by the new jump operators, in the same spirit as it was done for the original GRW model Ghirardi1986 (). In Sec. IV, we show how the extended GRW model, and the original one as well, can be formulated in terms of a stochastic differential equation and we study some relevant features of its solutions, focusing on the occurrence of the localization. In Sec. V, we derive the master equation associated with the model, which is shown to be equivalent to an equation exploited in the description of collisional decoherence Vacchini2000 (); Vacchini2001 (); Hornberger2006 (). After studying the solution of the equation, we prove explicitly, in Sec. VI, that it implies dissipation and an exponential relaxation of the energy to a finite value. In Sec. VII, we study the amplification mechanism in the presence of dissipation: we first deal with a macroscopic rigid body and show that its center of mass behaves for all the practical purposes as a classical object, then we discuss the difficulties which are unavoidably encountered when more general situations are considered. Finally, the conclusions and final remarks are given in Sec. VIII.
Ii The GRW model
ii.1 General structure of the model
Let us start by briefly recalling the main features of the collapse model introduced by Ghirardi, Rimini and Weber Ghirardi1986 (). The GRW model can be formulated in terms of discrete jumps of the wavefunction which represents the state of the system taken into account. For the sake of simplicity, we consider a particle in one dimension and we neglect spin and other internal degrees of freedom, so that the wavefunction is an element of the Hilbert space . The dynamics of the particle is then characterized through the following assumptions:

At random times the particle experiences a sudden jump described by
(1) where is the state immediately before the jump, which occurs at time and position , and is the selfadjoint contractive linear operator defined as
(2) with the position operator of the particle and a new parameter of the model.

The overall number of jumps is distributed in time according to a Poisson process with rate , which is the second new parameter of the model. The standard value of the rate is , while a higher rate was proposed more recently Adler2007 (), .

If there is a jump at time , the probability density that it takes place at the position is
(3) 
In the time interval between two consecutive jumps, the state vector evolves according to the usual Schrödinger equation.
As will be shown in the following, see Sec.IV.1, this dynamics can be equivalently formulated through a stochastic differential equation. Moreover, the jump operators satisfy the relation
(4) 
which corresponds to the normalization of the probability distribution defined in Eq.(3).
The jump operator describes a localization process around the position and with width . Consider a gaussian wavefunction ,
(5) 
where is the mean position, the mean momentum, while determines the position variance and the momentum variance ; is the normalization constant. The wavefunction after the jump, see Eq.(1), is still gaussian: apart from an irrelevant global phase, one has , where
(6) 
with
(7) 
The mean value of the momentum does not change, while the mean value of the position is shifted toward the localization position . The position variance decreases and hence after the jump the particle is more localized: the reciprocals of and sum up and give the reciprocal of the new variance . Importantly, the latter does not depend on the position of the localization process. For large gaussian wave packets, such that , one has and . The effects of the localization process on the gaussian wavefunction are summarized in Fig.1(a). In addition, the probability that the localization process occurs around is, according to Eq. (3),
(8) 
The role of the localization operators can be further understood by means of this simple example Bassi2003 (), which clarifies how the action of the localization operators can prevent the system from being in a position superposition. Consider a particle which is in the state given by the superposition of two gaussian wavefunctions with null mean momentum and the same variance, one centered around the position , the other around :
(9) 
where and is a normalization constant. The state after the localization around ,
(10) 
is still the superposition of two gaussian functions; the normalization constant depends on where the localization takes place. Now, consider a localization process around and assume that the distance between the two gaussians is much greater than the localization amplitude, while their width is much smaller than it, i.e. . The previous formula directly gives
(11) 
so that the gaussian centered around the localization position is left almost unchanged, while the gaussian centered around is suppressed by a factor . The localization process practically destroys the superposition between the two gaussian wavefunctions, leading to a single gaussian state localized around , see Fig.1(b). In addition, by further exploiting , one finds that the probability of a localization in a neighborhood of is given by : the definitions in Eqs. (2) and (3) allow to recover the usual Born’s rule for the probability distributions Bassi2007b (); see also Appendix A.
ii.2 Master equation associated with the model
The GRW model is fully determined by the stochastic evolution of the wavefunction previously presented. However, it is often convenient to deal with the dynamics of the statistical operator which describes the state of the system averaged over all the possible trajectories built up by the different combinations of Schrödinger evolutions and localization processes, see also Sec. V. The equation of motion of the statistical operator , i.e., the master equation for the GRW model reads
(12)  
The first term describes the standard quantum evolution induced by the Hamiltonian , while the second term accounts for the occurrence of the localization processes. Equation (12) establishes a semigroup evolution Lindblad1976 () with pure decoherence in position: the offdiagonal terms in the position representation are suppressed, and distant superpositions are suppressed faster than closer ones.
The master equation associated with the collapse model allows to investigate relevant features, such as the extension to an particle system and then the amplification mechanism, as well as the asymptotic behavior of the energy of the system.
ii.2.1 Amplification mechanism
Now, consider an particle system in which the localization processes occur individually for each constituent, so that the master equation associated with the particle statistical operator is simply
(13) 
where is a shorthand notation for , being the identity operator on the Hilbert space associated with the th particle, and is the position operator of the th particle, see Eq.(2). As we will see in Sec. VII, the hypothesis of individual localization processes has to be considered with a certain caution. It is useful to introduce the center of mass coordinates through the invertible linear transformation
(14) 
with , where is the mass of the th particle and the total mass,. Accordingly
(15) 
is the centerofmass coordinate, while are the relative coordinates. The position of the th particle can be expressed as
(16) 
One can then easily prove the relation
(17) 
where denotes the partial trace with respect to the relative degrees of freedom. Hence, if we assume that the total Hamiltonian is the sum of a term associated with the center of mass and a term associated with the internal motion, i.e.
(18) 
we find that the state of the center of mass, satisfies the same master equation as that in Eq.(12), with the one particle Hamiltonian replaced by and, most importantly, the localization rate replaced by . This is a direct manifestation of the amplification mechanism, which, along with localization, is the crucial feature of the GRW model. It explains why the model describes both microscopic and macroscopic systems. The localization rate (assume for the sake of simplicity) of microscopic systems is negligible and therefore the predictions of the GRW model about the wavefunction of microscopic systems reproduce for all practical purposes the predictions of standard quantum mechanics. On the other hand, if we consider a macroscopic object, the rate , with of the order of the Avogadro’s number, induces a localization of the center of mass on very short time scales: the wavefunctions of macroscopic objects are almost always welllocalized in space, so that their centers of mass behave, for all the practical purposes, according to classical mechanics.
ii.2.2 Energy divergence
A wellknown drawback of the GRW model is that it predicts an infinite increase of the energy of the system. This energy divergence is due to larger and larger fluctuations of the momentum induced by the interplay between the Schrödinger evolution and the localization mechanism, as will be explicitly discussed in Sec.IV.2. Nevertheless, the infinite increase of the energy can be inferred directly from the master equation (12). In fact, this master equation predicts a linear increase of the mean value of the energy of the system with a rate Ghirardi1986 ()
(19) 
This rate of the energy increase is actually very small, for a nucleon, even if one considers the particle case Ghirardi1986 (). However, it is clear that from a fundamental point of view one would like to avoid the energy divergence and point to a reestablishment of the energy conservation principle within the model. This forces us to put forward a more realistic description of the interaction between the system and the noise, also in order to test whether and how possible mechanisms excluding infinite energy increase modify the testable predictions of the model Bassi2013 ().
The use of the master equation formalism suggests a way out from the problem of the energy divergence. Despite the deep conceptual differences between collapse models and the notion of decoherence, the same master equation associated with the GRW model can be also derived in a specific model of collisional decoherence Vacchini2007 (). This correspondence clarifies that the origin of the energy divergence in the GRW model can be ascribed to the lack of a dissipation mechanism, which would account for the energy loss of the system due to the action of the noise. By taking into account the master equation which generalizes Eq.(12) to include dissipation Vacchini2000 (); Vacchini2001 (); Hornberger2006 (), we have thus been led to a possible structure of a new jump operator replacing that in Eq.(2) and excluding the energy divergence. Indeed, this was a preliminary benchmark, but the choice of the jump operator in a collapse model is subjected to further constraints, the most relevant being the induction of localization. In addition, the definition of a collapse model in terms of the different trajectories within the Hilbert space can be done without reference to any subsequent master equation or decoherence model. For these reasons, we present our results by first postulating a new localization operator and, as a consequence, a new collapse model and only after that we derive the corresponding master equation, see Sec.V.
Iii Extended GRW model
In this paper, we propose the following extension of the GRW model: the jump operators defined in Eq.(2) are replaced with
(20) 
where is the momentum operator of the system, the mass of the particle and
(21) 
a new parameter of the model, which is related with the temperature of the noise inducing the localization. This will be explicitly shown in Sec.VI, where the specific choice of , as well as its peculiar role in the definition of , will be discussed. Note that the jump operators of the model are no longer selfadjoint. An equivalent way to express is given by
(22) 
by which one immediately sees how the original GRW jump operator , see Eq.(2), is obtained in the limit . On the other hand, the general structure provided by items 14 in the previous section is left untouched. Explicitly, our collapse model can be formulated as follows:

The sudden jumps are now described by
(23) 
The overall number of jumps is still distributed in time according to a Poisson process with rate .

If there is a jump at time , the probability density that the jump takes place at the position is now given by
(24) 
Still, between two consecutive jumps the state vector evolves according to the Schrödinger equation.
It is important to observe that the jump operators satisfy a normalization condition as in Eq.(4); explicitly,
(25)  
This property guarantees that the probability distribution associated with the localization position is properly normalized, see Eq.(24), and its role within the model will be further discussed at the end of Sec. V.1. As will be shown extensively in the following, the replacement of with preserves all the desired features of the resulting collapse model, while the dependence on the momentum operator prevents the infinite energy increase and thus leads to a more realistic description of the action of the noise.
The role of the jump operator is illustrated directly by evaluating its action on a gaussian wavefunction , see Eq.(5). The gaussian structure of the wavefunction is preserved and, specifically, one has that the state after the jump, see Eq.(23), is , with
(26) 
where we introduced the adimensional quantity
(27) 
which will be crucial in the following analysis, as well as
(28) 
The jump shifts the mean value of the position toward , and, now, it also damps the mean value of the momentum. The variance after the jump does not depend on where the jump takes place and it is given by the reciprocal of the sum of the reciprocals of and . The contribution due to the dependent term causes a slight increase of the wavefunction width, which partially counterbalances the decrease due to the usual GRW contribution. For gaussian wavefunctions such that is smaller than the threshold value
(29) 
the jump induced by will increase overall the position variance. Contrary to the original GRW model, the repeated action of the jump operators does not induce an unlimited contraction of the wavefunction: there is a lower threshold under which the jump processes cease to be localization processes. In realistic situations, this threshold value is not reached by the evolution, see Eq.(42) and the following discussion. However, plays a crucial role in fixing the asymptotic finite value of the energy, see Sec. VI. Let us note that, as in the original GRW model, the wavefuction after the localization process is nonvanishing over the whole space. This is the socalled ’problem of the tails’ in collapse models Lewis1997 (); Clifton1999 (); Bassi1999 (); Bassi2003 (); Wallace2008 () and, indeed, the introduction of dissipation leaves it unaltered.
The probability density for a jump to take place at the position is, see Eq.(24),
(30) 
Finally, by taking into account the action of on a superposition of two gaussian wavefunctions as in Eq.(9), with , one can show along the same lines as for the original GRW model, see Appendix A, that also in this case the localization process destroys the superposition and selects a single gaussian wavefunction, see Fig.1 (b), with a probability which corresponds to the usual Born’s rule.
Iv Trajectories in the Hilbert space
In the previous section, we have extended the GRW model by introducing new jump operators, while leaving the general structure of the collapse model untouched. The dynamics of the wavefunction consists in a unitary evolution interrupted by sudden discontinuous transformations (jumps) at random and separated times. The stochastic differential equations which usually define collapse models are governed by Wiener processes Ghirardi1990 (); Bassi2013 (), so that they do not supply the piecewise deterministic evolution now recalled. However, we will show in this section how also the generalized GRW model can be formulated by postulating a stochastic differential equation. The latter determines the trajectories in the Hilbert space of the system through a random field, i.e. a family of stochastic processes, one for each point of space . As for the other collapse models, this equation has to be understood as a phenomenological equation, whose fundamental motivation has to be looked for by some underlying theory beyond standard quantum mechanics Bassi2013 (). In the proper limit, we will also get a stochastic differential equation for the original GRW model.
The reader is referred to Barchielli1991 (); Barchielli1994 (); Barchielli1995 () for further details and a rigorous treatment of the stochastic differential equations with jumps in Hilbert spaces.
iv.1 Stochastic differential equation
First, let us introduce a family of stochastic processes such that the counting process counts the jumps taking place at a position within and . The stochastic processes are defined on a common probability space and indicates the statistical mean with respect to the probability . Furthermore, we denote as a generic sequence of instants and positions in which the jumps occur up to time ; indeed, this corresponds to specifying the trajectories of the counting processes up to time . We assume that the processes are independent and satisfy
(31)  
(32)  
(33) 
where is the increment of in a time . We introduced the shorthand notation to indicate the state of the system depending on the trajectory up to time : the wavefunction is itself a stochastic process, which has values in the Hilbert space associated with the system and is determined by the sequences of jumps, i.e. by the trajectories of the counting processes. Equations (31) and (32) tell us that the probability of one count in a time interval is of order , while the probability of more than one count is of higher order Barchielli1994 (). Equation (33) yields the expected value of the increment of the counting processes conditioned on the occurrence of the sequence of jumps up to time Barchielli1991 (); Barchielli1994 (). This conditional expected value depends both on the (stochastic) wavefunction at time and on the (deterministic) jump operator . Finally, the wavefunction is fixed by the following nonlinear stochastic differential equation:
(34) 
The evolution of the wavefunction in a time interval has a deterministic contribution due to the Hamiltonian and a stochastic contribution due the jumps described by Eq.(23); a jump around the position corresponds to a nonzero increment of the counting process . The solution of Eq.(34) can be represented straightforwardly: given the sequence of jumps and the initial condition , the corresponding trajectory in the Hilbert space is
(35) 
where is the normalization factor. This equation formally characterizes all the possible evolutions of the system’s state within our model. The deterministic evolution of the wavefunction induced by the group of unitary operators is interrupted by the jumps described by the operators ; the dynamics introduced in Sec. III is then recovered, compare with items and .
Furthermore, all the other features of the collapse model can be retrieved by the properties of the stochastic processes in Eqs.(31)(33). Let us say that the system is in the state at time and recall that the probability of more than one jump in a time interval is negligible. Hence, the probability density of a jump at the position and a time between and is simply given by the conditional expectation of the increment of the corresponding process , i.e., Barchielli1991 ()
(36) 
Now, as the jump operators satisfy the normalization condition in Eq.(25), the probability to have a jump within and at any position is simply
(37) 
i.e. the overall jump rate does not depend on the state of the system and is given by , according to item As a matter of fact, this corresponds to the rate of the Poisson process
(38) 
which counts the total number of jumps up to time . Finally, the probability density that, if there is a jump at a time between and , it takes place at the position is , so that Eq.(24) and item are recovered.
The results of this paragraph apply to the original GRW model in the limit , i.e. . For example, we can associate the GRW model with the nonlinear stochastic differential equation
(39) 
By following Barchielli1991 (); Barchielli1994 (); Barchielli1995 (), one can also introduce a linear equation equivalent to Eq.(34) after a proper change of probability on the measurable space .
iv.2 Position and momentum localization
In Appendix B, we study in detail the gaussian solutions of the stochastic differential equation (34). Here, we focus on the evolution of the position variance, thus confirming the effectiveness of the localization mechanism ruling the collapse model. We further characterize the finite values of the position and the momentum variances in the asymptotic time limit.
Localization of gaussian wavefunctions
Given a gaussian solution of Eq.(34), i.e. as in Eq.(5), with and the normalization in Eq.(97), the position variance is defined as, compare with Eq.(104),
(40) 
As shown in Appendix B, depends on the instants of the jumps, but not on their position, and hence it is a function of the trajectories of the Poisson process . To illustrate in a compact way the evolution of the position variance, we deal with its statistical mean. For a Poisson process with rate , the probability that there is one count between and , , one count between and and no other counts up to time is Barchielli1994 () : the probability density associated with only depends on the overall time and number of jumps . Thus, the expected value of the position variance reads
(41) 
where is defined in Eq.(100), and the integrand expresses the position variance at time on the trajectory , see Eqs. (99) and (106).
In Fig. 2, we can observe the evolution of for different values of the initial variance and for both the microscopic and the macroscopic regime. The former refers to the evolution of a single particle with a mass of the order of the nucleon mass , while the latter describes the evolution of the center of mass of a system composed by an Avogadro’s number of particles. As will be shown in Sec. VII, we can apply our collapse model to an particle system by simply replacing the jump rate with and referring to the total mass of the system, at least as long as a rigid body is considered. In the microscopic regime, Fig. 2.(a) and (b), the evolution of the expected value of the position variance strictly follows the deterministic unitary evolution up to very long time scales and then saturates to a finite value, see the next paragraph. The jump rate is and therefore the probability to have a jump will be negligible up to, say, : the action of the noise does not induce any observable localization process on the microscopic systems. On the other hand, when macroscopic systems are taken into account, Fig. 2.(c) and (d), the evolution described by Eq.(41) strongly departs from the unitary one from the very beginning of the dynamics. The repeated occurrence of the jumps rapidly reduces the position spread of the wavefunction, so that the localization mechanism is clearly manifested. The time scale of the wavefunction localization is and it is the same as that for the GRW model. The total rate of events in our extended GRW model is in fact the same as in the original one, compare items 2 of Secs. III and II.1. We conclude that the modification of the jump operators put forward with Eq.(20) does not introduce any significant change in the localization mechanism compared to the original GRW model, as also shown by the results of the next paragraph.
Asymptotic values of position and momentum variances
The trajectories of the model are made up of a sequence of deterministic unitary evolutions and random jumps. These two transformations have opposite effects on the wavefunction, as long as the position variance is concerned. The free evolution induces a spread of the position variance, which is the faster the narrower the wavefunction. On the contrary, the jumps shrink the wavefunction, at least as long as , see Eq.(29). At some point of the evolution the two opposite effects balance each other and thus the position variance reaches a finite and nonzero equilibrium value Ghirardi1986 (); Bassi1999 (). As shown in Appendix C, the asymptotic value of the position variance can be evaluated via the relation
(42) 
with
(43) 
This asymptotic value is in general much higher than the value , which would correspond to the threshold in Eq.(29). For a macroscopic system, with , one has , while , which is also in agreement with the estimate given in Ghirardi1986 () for . As a matter of fact, due to the specific choice of , the threshold value is very small, so that the free evolution and the jumps balance each other before the spread of the wavefunction can reach it.
Analogously, see Appendix C, the asymptotic value of the momentum variance is given by
(44) 
For a macroscopic system with , one gets , still perfectly compatible with the value for Ghirardi1986 (), so that is approximately twice the minimum value allowed by the uncertainty relation.
It is worth noting that, as the position variance, also the momentum variance reaches a finite asymptotic value, both for and for , which naturally leads to the following remark. The reason for the energy divergence in the original GRW model is quite a subtle one. It is often understood by saying that the jump operator in Eq.(2) induces an indefinite contraction of the width of the wavefunction, so that and therefore, in accordance with the uncertainty relation, , implying the divergence of the energy. However, it is clear how this picture is not the end of the story and it is, to some extent, misleading. A crucial role here is played by the Schrödinger evolution between the jumps. As now recalled, the balance between unitary evolution and jumps implies a finite asymptotic value of the momentum variance, also for . This means that the energy divergence in the GRW model is actually due to fluctuations of the mean value of the momentum, as . To be more explicit, consider an initial gaussian wavefunction, . The unitary evolution up to the first jump at time does not modify the mean value of the momentum and shifts the mean value of the position as , see Eqs.(95), (105) and (107) for . Moreover, the unitary evolution introduces an imaginary component in , according to . Because of such an imaginary component, the jump at time and position actually modifies the mean value of the momentum, which after the jump will be:
(45) 
where has reduced to , see Eq.(7), since we are now considering the limit , i.e., the original GRW model. The system varies its momentum proportionally to the distance between the position of the jump and the mean value of the position before the jump. This shift of the momentum, in turn, contributes to the change in position after the jump: between the first and the second jump the mean value of the position evolves as . The iteration of these two transformations, according to the different spatial distribution of the jumps, will generate some trajectories such that both the mean position and the mean momentum will diverge to , and some other trajectories where they will diverge to : in both cases the mean kinetic energy will asymptotically diverge. Due to the symmetric probability distribution of the location of each jump, see for example Eq.(8), the effect on the mean momentum now described will be on average null, and then . However, the statistical average of the squared mean value of the momentum will diverge, , and thus the average of the mean energy will diverge with it.
The introduction of a dissipative mechanism, through a small , only slightly modifies the action of the jump operators, see Sec. III and Appendix B. Nevertheless, this tiny modification is enough to damp the longtime momentum fluctuations, thus leading to an asymptotic finite value of the energy, as will be shown and discussed in Sec.VI.
V Master equation
Up to now, we have dealt with the stochastic evolution of the wavefunction, as fixed by Eq.(34). The latter provides a complete characterization of the collapse model, as it yields all the possible piecewise deterministic trajectories which can be obtained according to items 14 in Sec. III. Nevertheless, it is often convenient to study the predictions of the model related with the statistical mean of relevant physical quantities, i.e., compare with Eq.(103),
(46) 
Here, we have introduced
(47) 
which is by construction a statistical operator on the Hilbert space associated with the system. Incidentally, since the stochastic wavefunction is uniquely determined by the trajectories , the statistical mean in Eq.(47) corresponds to the mean over the different trajectories , each one weighted with its probability density. In the following, we will focus on the evolution of the statistical operator , which will allow us to describe the evolution of relevant physical quantities, as well as to further characterize the dissipation and the amplification mechanism in the model.