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Stochastic quantum hydrodynamic model from the dark matter of vacuum fluctuations: The Langevin-Schr\"odinger equation and the large-scale classical limit

The work derives the quantum evolution in a fluctuating vacuum by introducing the related (dark) mass density noise into the Madelung quantum hydrodynamic model. The paper shows that the classical dynamics can spontaneously emerge on the cosmological scale allowing the realization of the classical system-environment super system. The work shows that the dark matter-induced noise is not spatially white and owns a well defined correlation function with the intrinsic vacuum physical length given by the De Broglie one. The resulting model, in the case of microscopic systems, whose dimension is much smaller than the De Broglie length, leads to the Langevin-Schrodinger equation whose friction coefficient is not constant. The derivation puts in evidence the range of application of the Langevin-Schrodinger equation and the approximations inherent to its foundation. The work shows that the classical physics can be achieved in a description whose length scale is much bigger both than the De Broglie length and the quantum potential range of interaction. The model shows that the quantum-to-classical transition is not possible in linear systems, and defines the long-distance characteristics as well as the range of interaction of the non-local quantum potential in order to have a coarse-grained large-scale classical phase. The theory also shows that the process of measurement (by a large-scale classical observer) satisfies the minimum uncertainty conditions if interactions and information do not travel faster than the light speed, reconciling the quantum entanglement with the relativistic macroscopic locality.


Introduction
The conflict between the quantum mechanics and the classical ones attracts the interest of many researchers of the noways physics [1][2][3]. This lack of knowledge has lead to many logical paradoxes that contrast with our sense of reality [1][2][3]. A quantitative tentative to investigate the problem was given by Bell [3] in response to the so called EPR paradox [2] a critical analysis of the quantum non-locality respect to the notion of the macroscopic classical freedom and local relativistic causality. The Copenhagen interpretation of quantum mechanics [3][4][5] treats the wave function as representing 'the probability' of finding a particle at some location' [12]. However, such a treatment leads to the non-intuitive conclusion that the physical state is just a probability wave until observed. The absence of an analytical link with the pre-measure physical state fights against the common sense of reality and the existence of a real world independent by the observer and the measure process [5]. If the Copenhagen probabilistic connection with the pre-measure world is strictly assumed, the conclusion that the real state is not physically defined before the measure is unavoidable. Actually, the completeness and self-consistency of this logical result cannot be achieved since the process of the observation is out the Hamiltonian description of quantum mechanics. The need of having a classical environment in order both to perform the measure and to define the quantum eigenstates, indeed, leads to a great theoretical loophole: Is the classical world necessary to the quantum mechanics or the quantum evolution is the fundamental law? Besides, the unavailability of the theoretical connection between the quantum and the classical mechanics, that would explain how the laws of physics pass from the deterministic quantum behavior to the classical one (even irreversible), leaves open many questions about how concepts of the classical experience such as, measure, principle of causality, locality, physical state of the external reality, wave and particle behaviors, can be compatible or related to the quantum mechanics. The connection between the quantum state and the statistical (classical) process of measure is defined by a postulate that, is a matter of fact, makes the quantum mechanics a semi-empirical theory without a self-consistent theoretical framework. On the other hand, if the wave function is something physically real, then, there must exist a defined mechanism (e.g., the so-called wave function collapse, out of the canonical law of quantum mechanics) expressing the interaction with the observer embedded into a classical universe. In this case, there also exists the problem about how the Schrödinger equation can be generalized [13,14] or derived in the frame of such a more general quantum theory [15]. In order to fill this theoretical lack, there exist various interpretations of quantum mechanics like the many-worlds interpretation [17], the Bohmian mechanics [18,19], the modal interpretation [20], the relational interpretation [21], the consistent histories [22], the transactional interpretation [23,24], the QBism [25], the Madelung quantum hydrodynamics [26][27][28] and the decoherence approach [29] . The Madelung approach (that is a particular case of the Bohmian mechanics [30]) owns the important peculiarities to be both mathematically equivalent to the Schrödinger one [31] and to treat the wave function evolution . In this way it introduces the concept of trajectories of motion and naturally hosts the notion of physical reality before the measure. The Madelung description has the advantage to disembogues into the classical mechanics as soon as  and the so-called quantum pseudo-potential are set to zero.
Nevertheless, if we wipe out (by hand) the quantum potential from the quantum hydrodynamic equations in order to obtain the classical mechanics, we also cancel the stationary quantum eigenstates where the total force exerted by the Hamiltonian potential and the quantum one (on the mass density distribution 2 | |  ) is null. Doing that, we change the nature of the equation of motion. Thence, a more correct and analytic mechanism is needed to pass from the quantum nonlocal description to the classical one in the frame of the hydrodynamic approach. Others characteristics of the quantum to classical transition are captured by the decoherence approach that investigates the possibility of obtaining the classical state through the lost of quantum coherence generated by the presence of the environment. The decoherence is shown to be produced into the system by treating it as a sub-part of the overall system, comprehending the environment whose interaction is semi-empirically defined by non unitary interaction [29]. However, this approach is not able to explain how, by having a quantum overall system, the observer can perform the irreversible processes of the statistical measure (and to be quantum decoupled with the measured system). To overcome this problem, the relational quantum mechanics introduces the super-observer that is not entangled with the overall system [21]. Actually, this "ad hoc" postulate, is unsatisfactory and brings logical contradictions. From the experimental and numerical simulation point of view, there exist the important evidence that the decoherence and the localization of quantum states come from the interaction with the stochastic fluctuations of the environment [32][33][34][35] and/or dark matter. In this work the authors generalize the Madelung quantum hydrodynamic approach to its stochastic version, where the noise, due to the quantum-mechanical properties of a fluctuating vacuum (in term of curvature associated to dark matter), owns a non-white spectrum showing the emergence of the intrinsic De Broglie physical length into the vacuum. The work also shows that the stochastic Langevin-Schrodinger equation is derived from the theory for systems whose physical length is much smaller than the De Broglie length. In the final section the authors analyze how the classical mechanics can be achieved, under appropriate conditions, on a large scale description. The uncertainty principle in the measure process is investigated in the frame of the stochastic quantum hydrodynamic model (SQHM). The paper analyzes how the measure in a classical large-scale system can satisfy both the uncertainty principle and the finite velocity of transmission of light and information.

The quantum hydrodynamic equation in presence of vacuum dark mass density fluctuations
In the present work we go beyond the flat static solution as for classical matter vacuum, and assume that there is still energy and momentum within the spacetime due to the possible presence of gravitational waves that can give a contribution . Solution to such equations has been introduced by de Sitter that illustrates that matter may not be the only source of gravity and thus wrinkles in the space-time may not be due to matter only. By considering the vacuum as a fluctuating background, we define the stochastic generalization of the quantum-hydrodynamic equations [26][27][28]31] where H is the classical Hamiltonian of the system and where 1 2 leads to the fluctuating force (2.7) Being the mass density give rise to an additional mass that, owning just the gravitational interaction, is dark matter.
For the purpose of this work, we assume that the vacuum dark matter (DM) does not interact with the physical system (the gravity interaction is disregarded for its weak constant and it is not included in H ).
As far as concerning the dark matter evolution, it is defined by additional (gravitational) motion equation

Spectrum and correlation function of the dark-matter-induced mass density fluctuations
In deriving the characteristics of the quantum potential fluctuations (and hence of its force), we use the condition that the vacuum dark matter, described by the wave function vac  , does not interact with the physical system (this due to the weak gravity constant). In this case the wave function tot  of the overall, system reads tot vac at small wave length ( where it has been used the normalization condition tot( q ,t ) tot V n dq n V   and where, on large volume, it has been used the approximation The result (2.1.7) shows that the energy, due to the mass density fluctuations, increases as the inverse squared of  . Being so, the associated quantum potential fluctuations, on very short distance (i.e., The derivation of conditions on the noise correlation function shape G( )  , brings a quite heavy stochastic calculations [36]. A more simple and straight derivation of G( )  can be obtained by considering the spectrum of the fluctuations.
is the De Broglie length.
From (2.1.10) it comes out that the spatial frequency spectrum S( k ) is not white and the components with wave-length  smaller than c  go quickly to zero.
Thence, given the mass density noise correlation function, that for the sufficiently general case, to be of practical interest, can be assumed Gaussian with null correlation time, isotropic into the space and independent among different co-ordinates such as The expression (2.1.14) shows that uncorrelated MDD fluctuations on shorter and shorter distance are progressively suppressed by the quantum potential allowing the deterministic quantum mechanics to realize itself for systems whose physical length is much smaller than the De Broglie one.

The stochastic potential approach
The characteristics of the stochastic force noise induced by the fluctuations of the quantum potential (due to the vacuum mass density fluctuations) can be derived by assume the quantum potential as composed by a regular part Moreover, given the force noise it is possible to show (see appendix A) that it owns the correlation function where the spatial shape F( )  is connected to G( )  of the dark matter and where T is the fluctuation amplitude parameter (of DM). Thence, the motion equation acquires the form is the boson-boson s-wave scattering length for Lennard-Jones interacting particles (see (3.6) in section 3).
By using the variance (2.3.1), for systems, whose physical length L is much smaller than the De Broglie where the spatial probability mass density (PMD) ( q ,t ) where, as shown in Appendix B, at first order of approximation in ( t ) and where Moreover, the relation between the friction coefficient can generally read where  is a numerical parameter that measures how the quantum hydrodynamic trajectories of motion are perturbed by fluctuations (leading to quantum decoherence and to energy dissipation). This parameter is specific for each considered system since the sensibility of the system to fluctuations is related to the Lyapunov exponents of their classical trajectories of motion. This aspect goes beyond the purpose of this work and (2.3.10) is semi-empirically assumed here.
For 0   we have a system that maintains the quantum coherence with no dissipation (e.g., as happens in the deterministic limit).
that holds for the so called "dust matter" (the mass density, constituted by monomolecular dust, representing the classical MDD limit of the Madelung quantum hydrodynamics [31]). In the general case of a system submitted to fluctuations, neither in quantum deterministic limit nor in the classical one, we assume that In order the SQHM theory comprehends the canonical quantum mechanics in the deterministic , where L is the physical length of the system), it must hold both that That, by posing Moreover, by by assuming in the limit of small fluctuations amplitude so that (2.3.11-12) are satisfied by For instance, for

The generalized Langevin-Schrödinger equation for De Broglie-length systems
The quantum-hydrodynamic equation (2.3.7) for the complex field leading to the partial stochastic differential equation

Introducing the environment
The presence of unnoticeable small dark matter fluctuations, even negligible on the ordinary scale systems, is suffice to lead to a finite De Broglie length that is much smaller than the cosmological scale allowing the quantum decoherence and the emergence of the classical behavior into the universe.
The possibility of dividing the universe in classical sub-parts, allows to correctly introduce the existence of the environment. Besides, given that the action of force noise of the environment on the MDD derivatives generates an increase of the energy of the quantum potential in the same way as the dark matter, it follows that the spatial shape of the correlation function of the force noise of the environment owns the same form of that one of the dark matter. Thence, in presence of the physical environment it is possible to assume the stochastic interaction from which it follows that 1 2 1 2 1 2 It must be noted that, in absence of dark matter fluctuations, the quantum-decoupling of the environment cannot be assumed and that ext st V cannot be formulated in the form of (2.4.1.1) .
Thence, (2.4.1.5-6) can be assumed only in a physical vacuum that, following the general relativity, is constituted by a fluctuating geometrical background.

Robust quantum systems and the canonical LSE
Moreover, since for a quantum system that owns 0 0   and hence D    , able to strongly maintain its quantum coherence (we name it "robust" quantum systems), it holds As already observed, the sensibility of the system to fluctuations is related to the Lyapunov exponents of its classical trajectories of motion. This aspect goes beyond the purpose of this work and is not analyzed here. It is suffice to say that for linear systems (non classically chaotic) 0 0   and the (LSE) can be applied to them. Generally speaking, for the case of classically chaotic systems, the LSE (2.4.1.10) has to be considered. Moreover, it must be noted that the SLE description is made possible by the integrability of the velocity field 1 S q m q       that can be warranted in small scale (slightly perturbed) quantum system but it may fall in macroscopic large-scale system whose velocity field can be non-integrable.

The SQHM with adiabatic elimination of fast variables
For slow kinetics with the characteristic time ch  satisfying the condition For a quantum system with 30 10 Kg

Emerging of the classical behavior on coarse-grained large scale
Is matter of fact that, if the quantum potential is canceled by hand in the quantum hydrodynamic equations of motion (2.1-3), the classical mechanical equation of motion emerges [28]. Even if this is true, this operation is not mathematically correct since it changes the characteristics of the QHA equations. Doing so, the stationary configurations (i.e., eigenstates) are wiped out because we cancel the balancing of the quantum potential force against the Hamiltonian force [37] that generates the stationary condition. Thence, an even small quantum potential cannot be neglected into the deterministic QHA model. On the contrary, in the SQHM it is possible to correctly neglect the quantum potential (at least in classically chaotic systems) when its force is much smaller than the noise  such as When the non-local force generated by the quantum potential is quite small (respect to the fluctuations amplitude) so that 1 2 1 2 its effect can be disregarded in (2.3.7) .
Besides, even if the noise ( q ,t ,T )  has zero mean, the mean of the quantum potential fluctuations is not zero, and the stochastic sequence of inputs of noise alters the coherent reconstruction of the quantum superposition of state by the dissipative force Moreover, by observing that grows with the scale of the system (i,e., (3.4) and the classical behavior can emerge in systems of sufficiently large physical length. Actually, in order to have a large-scale description, completely free from quantum effects, we can more strictly require By observing that for linear systems from (105) the SQHM shows that they do never have a macroscopic classical phase. Generally speaking, stronger the Hamiltonian potential higher the wave function localization and larger the quantum potential behavior at infinity [38]. Given the MDD where k ( q ) P is a polynomial of order k, in order to have a finite quantum potential range of interaction it must be 3 2 k  (it results 2 k  for uni-dimensional linear interaction). Actually, since the linear interaction is not maintained up to infinity (for energetic reason, a finite bound energy requires a weaker than linear interaction such as there exists a largescale classical description when the physical length of the system is much larger than the range of linear interaction. A physical example comes from solids owning a quantum lattice. If we look at the intermolecular features where the interaction is linear, the behavior is quantum (such as the xray diffraction shows), but if we look at their macroscopic properties (e.g., low-frequency acoustic wave propagation) the classical behavior is shown.
For instance, systems that interact by the Lennard-Jones potential for which the long distance wave function reads [39] the large scale classical behavior can appear [38,40] in a sufficiently rarefied phase (see section 4.4).
It is interesting to note that in (3.6) the quantum potential acquires the form of the hard sphere potential of the pseudo potential Hamiltonian model of the Gross-Pitaevskii equation [15,41] where 4 a  is the boson-boson s-wave scattering length.
By observing that, in order to fulfill the condition (3.5) a sufficient condition reads (3.11) it is possible to define the quantum potential range of interaction [38] 1 0 that gives a measure of the physical length of the quantum non-local effects.
The convergence of the integral (3.11) for 0 r  is warranted for L-J type potentials since, near the equilibrium point ( 0 r  ), the L-J interaction is linear and being (3.13)

From micro to macro description
By discretizing the phase space conservation equation given by the current equation (2.2.5-6, 2.4, 2.3.4, 2.3.8) for the system of N coupled particles [42], it is possible to obtain the quantum hydrodynamic master equation for a macroscopic system of a huge number of molecules. In order to obtain the macroscopic description, we may procede by discretization of the stochastic quantum hydrodynamic equations For the purpose of this work, we can exemplify the to the simpler case kinetics with a characteristic time ch  larger than Given the conserved equation in the local j-th cell of side l with the current j( q ,t ) and where the terms jk D , jk D' and jk D'' are matrices of coefficients that corresponding to the discrete approximation of the derivaties q  and Generally speaking, the form of the overall interaction (classicl plus quantum) steming by the k-th cell, depends by the physical system and its evolution.
For instance, by assuming L is the available volume per molecule), for a system of sufficiently rarefied phase of L-J interacting particles with a asymptotically vanishing quantum potential (3.9-10) , the quantum potential interaction between adjacent cells is null and, hence, the classical master equation is obtained.
Here, generally speaking we observe that, given the range of interaction of the quantum potential qu  , the De Broglie length c  , and the system size L ( 3  L the mean available volume per molecule), we can generally distinguish the cases: Typically, for the L-J potential the quantum potential range of interaction qu  extends itself a little bit further than the linear zone around the equilibrium position 0 r (let' say up to 0 r   ) .
By using this approximation for the L-J interaction, so that for  L refers to dense phases (e.g., fluid phase) that still own a classical behavior since, as a mean, the particle are distant each-other more than the range of interaction of the quantum potential. The inter-particle distance mostly lies in the non-linear range of L-J interaction [38].
applies when the neighboring molecules lie in the linear range of the intermolecular potential at a distance smaller than the non-local quantum potential interaction qu  .
The observables on such physical scale show quantum behavior (e.g., the Bragg's diffraction of the atomic lattice).
In the case "iv" c  > L , when the condensed fluid phase (i.e., 0 r    L ) persists down to a very low temperature so that the De Broglie length becomes larger than the mean intermolecular distance ( 4 T k    for typical intermolecular distance of order of 9 10 m  ), the fluid shows an extreme decrease of molecular viscosity [38]. The super-fluidity is induced by the quantum potential interaction between the molecules [40] By changing the temperature and, accordingly, both c  and the mean inter-molecular distance L , we can have quantum-to-classic phase transition in the case iii and iv, respectively: i.

Measurement process and quantum decoherence
The SQHA model shows that the measure is not necessarily a decoherent process by itself: The sensing part of the measuring apparatus (the pointer) and the measured system may have a canonical quantum interaction that, after the measurement when the measuring apparatus is brought to the infinity (at a distance much beyond c  ), ends. Then the reading and the treatment of the "pointer" state is done by the measurement apparatus: This process is practically a classic irreversible process (with a defined arrow of time) leading to the macroscopic output of the measure. On the other hand, the decoherence is necessary for the measurement process in order to have, both before the initial time and after the final one, quantum-decoupling between the measurement apparatus and the system in order to collect a statistical ensemble of data from repeated measures.

Minimum measurements uncertainty
If for physical length much smaller than c  any system approaches the quantum deterministic behavior and behave as a wave so that its sub-parts are not independent each-other, it follows that in order to perform the measurement (with independence between the measuring apparatus and the measured system) it is necessary that they are far apart (at least) more than c  and hence, for the finite speed of propagation of interactions and information, the measure process must last longer than the time It is worth noting that the product   E is constant since the growing of the energy variance with the square root of T is exactly compensated the decrease of the minimum time  of measurement The same result is achieved if we derive the experimental uncertainty between the position and momentum of a particle of mass m in the quantum fluctuating hydrodynamic model.
If we measure both the spatial position of a particle with a precision c L    (so that we are able to not perturb the quantum configuration of the measured system) and the variance p  of the modulus of its relativistic momentum due to the fluctuations that reads we obtain the experimental uncertainty If we measure the spatial position with a precision c L    , we have to perturb the quantum state. Due to the increase of the spatial confinement of the wave function (by increasing the environmental temperature or by an external potential), the increase of both the quantum potential energy and its fluctuations are generated so that the final particle momentum gets a variance p  higher than (3.3.5).
It is worth mentioning that the SQHM leads to the minimum measurements uncertainty as a consequence of the relativistic postulate of finite speed of light and information.
Even if the quantum deterministic behavior ( c    ) in the low velocity limit ( c   ) leads to the undetermined inequalities remains defined and constitutes the minimum uncertainty of the quantum deterministic limit. Beside, (3.3.6) in the relativistic limit shows that the duration of the measurement process in the deterministic limit becomes infinite. Being it endless, it is not possible to perform it in the canonical quantum mechanical universe.
Moreover, since non-locality is confined in domains of physical length smaller than c  and information cannot be transferred faster than the light speed (otherwise also the uncertainty principle will be violated) the local realism is obtained in the coarse-grained large scale physics and the paradox of a "spooky action at a distance [43]" is limited on a distance of order of c  or of qu  .
The above result holds for particles with rest mass different from zero, while for determining the length of non local interaction (entanglement) of the photon ( c    ?) the relativistic generalization of the SQHM is required.

Field of Application of the SQHM
The theory that describes how the quantum entanglement is maintained up to a certain distance and how it can be maximized, can lead to important improvements in the development of materials for high-temperature superconductors and Q bits systems. Moreover, the theory owing a self-defined quantum correlation distance can be also very important in defining different regimes of chemical kinetics in complex reactions and phase transitions. Besides, the SQHM can furnish an analytical self-consistent theoretical model for mesoscale phenomena and quantum irreversibility.

Conclusions
The SQHM describes how the quantum dynamics realizes itself in a vacuum whose metric fluctuates. In this scenario the canonical quantum mechanics is the limiting description achieved in a flat static vacuum. Figuratively, in a 3-dimensional space time, where the space can be represented by the surface of a see with very small ripples (instead by a flat static plane) the non local interaction of quantum mechanics breaks down on large scale and, in huge systems of weakly bounded particles, the classical mechanics emerges. The SQHM, shows that in the physical fluctuating vacuum, the spatial spectrum of the noise is not white and it owns the De Broglie characteristic length. Due to this fact, the quantum entanglement is effective in systems whose physical length is much smaller than such a length. The model shows that the non-local quantum interactions may extend themselves up to a finite distance in the case of non-linear weakly bonded systems. The SQHM shows that the Schroedinger equation can be derived by taking into account, at the first order of approximation, the effect of fluctuations on microscopic systems. The derivation of the LSE from the general SQHM allows to define its basic assumptions and its range of applicability limited to microscopic systems whose physical length is much smaller than the De Broglie one. The SQHM shows that the minimum uncertainty condition is satisfied during the process of measurement in a fluctuating environment and that it can have a finite duration. The theory shows that the minimum uncertainty in the measurement process is satisfied if, and only if, interactions and information do not travel faster than the speed of light, making compatible the relativistic postulate (at the base of the large scale locality) with the non-local quantum interactions at the micro-scale. The SQHM makes compatible the hydrodynamic description of quantum mechanics with the decoherence approach showing that the quantum potential is not able to maintain the quantum coherence in presence of fluctuations, generating a frictional force leading to a relaxation process (decoherence). The theory shows that the superposition of states does not physically exist in macroscopic systems made up of molecules and atoms interacting by long-range weak potentials such as the Lennard-Jones one. condition) and it owns (irrotational [31]) stationary states (i.e., quasi-eigenstates) so that it is still quantum and. the action function S (as integral of the momentum field) exists, we have that that owns an infinite number of terms due to the presence of the quantum potential.
If on one hand, the continuity of the Hamiltonian potential warrants that velocities   Thence, being the cumulants higher than two non-vanishing, the PTF P q,z | ,t )  ( is not Gaussian and. and equation (C.1.1) does not reduce to the FPE.

C.3. The motion equation for the spatial densities
By integrating over the momenta, the conservation equation