2. The quantum potential fluctuations elicited by the stochastic gravitational background
The quantum-hydrodynamic representation of the Schrodinger equation
for the complex wave function
, are given [
3] by the conservation equation for the mass density
and by the motion equation
where
is defined, through the momentum
, where
and where
In order to introduce the metric tensor fluctuations of the space-time background, we assume that:
The fluctuations of the vacuum curvature are described by the wave function with density ;
The (dark) energy density of the gravitational waves is proportional to ;
The equivalent mass of vacuum fluctuations is defined by the identity The stochastic gravitational wrinkles originated by the big-bang and the gravitational dynamics of space-time, are approximately assumed to not interact with the physical system (gravitational interaction is sufficiently weak to be disregarded).
In this case the wave function of the overall system
reads
Moreover, by assuming that, the equivalent mass
of the dark energy of gravitational waves is much smaller than the mass of the system (i.e.,
), the overall quantum potential (2.4) reads
Furthermore, given the vacuum mass density fluctuation of wave-length
associated to the fluctuation wave-function
it follows that the overall fluctuating quantum potential energy read
where
For
, the unidimensional case leads to
In (2.11) it has been used the normalization condition
and, on large volume (
see (2.15) below), it has been used the approximation
For the three-dimensional case, (2.11) leads to
The result (2.13) shows that the mass/energy density fluctuations, increases as the inverse squared of . Being so, the quantum potential fluctuations, of very short wave length (i.e., ) can lead to unlimited large energy fluctuations even for vanishing noise amplitude . This fact, in principle, could prevent the realization of the deterministic, zero noise, limit (2.2-4) representing the quantum mechanics, if the background fluctuations would produce a white noise.
Actually, the convergence to the deterministic limit (2.2-4) of quantum mechanics for
is warranted by the fact that uncorrelated fluctuations on shorter and shorter distances are energetically unlikely so that the noise is not white. Thence, the requirement of convergence to the conventional quantum mechanics for
is warranted by the special form of the spatial correlation function of the noise as
[
4].
The calculation of the correlation function
brings a quite heavy stochastic calculation [
4]. A more simple and straight way to obtain
is through the analysis of the spectrum of fluctuations.
Since each component of spatial frequency
brings the quantum potential energy contribution (2.11), its probability of happening, reads
where
is the De Broglie length.
From (2.14) the spectrum
of the spatial frequency reads
From (2.16) we can see that the components with wave-length
smaller than
go quickly to zero. Besides, from (2.16) the spatial shape
reads
The expression (2.17) shows that uncorrelated mass density fluctuations on shorter and shorter distance are progressively suppressed by the quantum potential allowing the realization of the conventional “deterministic” quantum mechanics for systems whose physical length is much smaller than the De Broglie one .
For the sufficiently general case to be of practical interest, where the mass density noise correlation function can be assumed Gaussian with null correlation time, isotropic into the space and independent among different co-ordinates, it can be assumed of the form
that, for system whose physical length
is much smaller than the De Broglie’ one (i.e.,
), reads
On this ansatz, equation (2.3) assumes the stochastic form [
5] (see appendix A)
where the probability mass density function
is defined by the Smolukowski conservation equation stemming from (2.20) and obeys to the condition
since by (2.17-18) the convergence to the quantum mechanics is warranted.
5. The quantum path integral motion equation in presence of stochastic noise
The Markov process (2.20) obeys the Smolukowski integro-differential equation for the Markov probability transition function (PTF) [
6]
where the PTF
represents the probability that a quantity of the probability mass density (PMD)
at instant t, in a time interval
τ, in a point z, is transferred to the point
q [
6].
The conservation of the PMD
in integral form shows that the PTF generates the displacement of a vector
(q,t) – (z,0) according to the rule [
6]
5.1. Stationary eigenstates in presence of noise
Generally speaking, for the quantum case, equation (5.1) cannot be reduced to a Fokker-Planck equation (FPE), since the quantum potential
owns a functional dependence by
and the PTF
is non-Gaussian (see
Appendix B) .
Nonetheless, if the initial distribution
is stationary (e.g., quantum eigenstate [
7]) and is close to the long-time final stationary distribution
of the stochastic case, it is possible to assume the approximation
Being in this case the quantum potential not function of time, the stationary long-time solution (warranted in time independent Hamiltonian potentials by the presence of the viscous force) is given by the Fokker-Plank equation
where
leading to the final equilibrium (
) identity
In appendix C the stationary states of linear systems obeying to (5.6) in presence of small noise are shown. The results show that the quantum eigenstates are stable and maintain their shape (with a small change of their variance) when subject to fluctuations.
5.2. Evolution of quantum superposition of states submitted to noise
In order to determine the evolution of quantum superposition of states, that are not stationary, (not considering fast kinetics, large fluctuations and jumps) we have to integrate the stochastic differential equation (SDE) (2.20) that eliminating the fast variables reads
As shown below, this can be done by using the discrete approach with the help of both the Smolukowski integro-differential equation (5.1) and the associated conservation equation (5.2) for the PMD .
We integrate the SDE (5.7) by using its 2
nd order discrete expansion
where
where
has Gaussian zero mean and unitary variance whose probability function
, for
, reads
where it has been introduced the midpoint approximation
and where
and
are the solutions of the deterministic problem
By using standard manipulations [
2], from (5.12), the PTF reads
where it has been introduced the discrete PTF
that reads
Since the quantum potential is a function of the PMD
the evolution of equation (5.8) depends on the exact sequence of the noise inputs
and, therefore, also on the discrete time interval of integration. This behavior can be easily verified by performing the numerical integration of (5.8). The vagueness of the problem can be analytically identified by the fact that,
and
depend on
and
, that define the quantum potential values
,
, unknown at the time instant
.
Although there is no general solution to this problem, in the limit of small speed
and small noise amplitude, it is possible to proceed by successive steps of approximation since the existence of the deterministic limit (see appendix D)
warrants that, for sufficiently short time interval
, the speed change is small enough to have that
with
, since
, and that
Therefore, by starting from the zero order of approximation
, there exists a sufficiently small noise amplitude, as well as small diffusion coefficient
, to obtain the PTF by successive steps of approximation where the starting zero-order reads
which can be used to find the zero-order of approximation of the PMD
at the next instant
k
and to define the approximated quantum potential at the instant
k that allows to obtain
Thence, at the next order of approximation, the PTF and the associated PMD read, respectively,
and
that leads to the mean velocity
Thence, repeating the procedure, at successive
u-th order of approximation (
u=2, ,3, .......
r) we obtain
and
so that the final PTF
reads
It worth noting that the convergence of (5.32) generally depends by the chaoticity of the classical trajectories of motion of the system, by the amplitude of the noise and by the discrete time interval .
The existence of the deterministic limit of quantum mechanics warrants the existence of the basin of convergence of (5.32) in the , but its wideness depends by the specificity of each physical system.
If, for discrete values of
and
of integration ,
fluctuates and
cannot be precisely determined, the PTF
can be estimated by taking its mean values beyond the
order of approximation such as
Finally, by using
, the PMD at the
-th instant reads
leading to the velocity field
As far as it concerns the continuous limit of the PTF, it reads
Where it has been used the identity .
The general solution, given by the recursive formula (5.36), can be applied also to non-linear system that cannot be treated by standard approaches [
8].
5.3. General features of relaxation of quantum superposition of states
In the classical case, the FPE describing the Brownian process, admits the stationary long-time solution
where
leading to the canonical expression [
2]
Generally speaking, in the quantum case, (5.36) cannot be given in a closed form (5.37) since the quantum potential depends on the specific relaxation path of the system toward the steady state which significantly depends on the initial conditions , and, therefore, on the initial time when the quantum superposition of states are submitted to fluctuations.
Besides, from (5.8) we can see that depends by the exact sequence of inputs of stochastic noise since the quantum potential is not fixed, but influenced by them. This behavior, in classically chaotic systems, can lead to relevant divergences of the trajectories in a short time. Thus, in principle, different long-period stationary configurations (i.e., eigenstates described by 5.6) can be reached whenever starting from the same superposition of states. Being so, in classically chaotic systems, the Born’s rule can be applied also to the measure of the single quantum state.
Even if , it is noteworthy to observe that, in order to have finite quantum lengths and (necessary to have the quantum-stochastic dynamics of Equation (5.7) and the quantum decoupled (classical) environment and/or measuring apparatus) the non-linearity of the system-environment interaction is necessary: The quantum decoherence with the decay of the superposition of states is strongly based upon the ubiquitous classical chaoticity of real systems.
On the other hand, a perfect linear universal system would maintain as well as quantum correlations on global scale and would never allow the quantum decoupling between the system and the measuring apparatus necessary for the measure process.
Furthermore, since the connection (A6.28) between the PMD and the MDD holds only at leading order of approximation of
(i.e., slow relaxation process and small amplitude of fluctuations), in the case of large fluctuations (that can occur on time scale much longer than the relaxation one)
can make transitions not described by (5.36) even from a stationary eigenstate to a generic superposition of states (e.g., following quantum synchronization [
9]). In this case a new relaxation toward different stationary eigenstate will follow: The PMD
(5.34) describes the relaxation process in the time interval between two large fluctuations, but not the complete evolution of the system toward the statistical mixture. Due to the jumping process on long time scale, the system made by collection of a large number of particles (or independent subsystems) relaxes toward an assigned statistical mixture (whose distribution is determined by the temperature dependence of the diffusion coefficient).
5.4. Emerging of the classical behavior on large size systems
It's indeed a fact that when one manually nullifies the quantum potential in the quantum hydrodynamic equations (2.1-3), the traditional equation of motion that corresponds to classical mechanics emerges [
4]. Even though this might hold true, such an operation lacks mathematical validity as it alters the fundamental properties of the quantum hydrodynamic equations. By taking this step, the stable arrangements (referred to as eigenstates) are eliminated due to the removal of the counterbalancing effect between the quantum potential and the Hamiltonian force [
7], which is responsible for establishing the stability of the eigenstates. Therefore, even a minor quantum potential cannot be disregarded within the framework of the deterministic quantum hydrodynamic model.
Conversely, in the stochastic generalization it is possible to correctly neglect the quantum potential in (2.20) when its force is much smaller than the force noise
such as
that by (5.7) leads to
and hence, in a coarse-grained description with elemental cell side
, to
where
is the physical system length.
Besides, even if the noise
has zero mean, the mean of the quantum potential fluctuations
is not null so that the dissipative force
in (2.20) appears. In this way, the stochastic sequence of inputs of noise alters the coherent evolution of the quantum superposition of state. Moreover, by observing that the stochastic noise
grows with the size of the system, it follows that for macroscopic systems (i,e.,
), condition (5.39) is satisfied if
Actually, in order to have a large-scale description, completely free from quantum correlations, we can more strictly require
Thus, by observing that for linear systems
it immediately follows that they cannot lead to the classical macroscopic phase.
Generally speaking, stronger the Hamiltonian potential higher the wave function localization and larger the quantum potential behavior at infinity [
10]. This can be easily proven by observing that given the MDD
where
is a polynomial of order
k, in order to have a finite quantum potential range of interaction, it must result
, so that linear systems, with
, own an infinite range of action of quantum potential.
A concrete illustration can be found in solids that possess a quantum lattice structure. When observing phenomena occurring at intermolecular distances where the interaction follows the linear behavior, quantum characteristics become evident (such as in x-ray diffraction). However, when focusing on macroscopic attributes (like low-frequency acoustic waves with wavelengths significantly surpassing the linear interatomic distance range), classical behavior becomes predominant.
For instance, for systems that interact by the Lennard-Jones potential, whose long-distance wave function reads [
10]
the quantum potential reads
leading to the quantum force
so that by (5.39, 5.43), the large-scale classical behavior can appear [
10,
11] in a sufficiently rarefied phase.
It is interesting to note that in (5.47) the quantum potential reproduces the hard sphere potential model of the “pseudo potential Hamiltonian model” of the Gross-Pitaevskii equation [
12,
13] where
is the boson-boson s-wave scattering length.
By observing that, in order to fulfill the condition (5.43) we can sufficiently require that
so that it is possible to define the quantum potential range of interaction
as [
5,
10]
that gives a measure of the physical length of the quantum non-local interactions.
For L-J potentials the convergence of the integral (5.49) for
is warranted since, at short distance the L-J interaction is linear (i.e.,
) and
5.5. From micro to macro description: the coarse-grained approach
Given the PMD current
, that reads
The macroscopic behavior can be obtained by the discrete coarse-grained spatial description of (5.52), with local cell of side
, that as a function of the
j-th cell reads [
14]
where
where
where
is the spatial correlation length of the noise, where the terms
,
,
and
are matrices of coefficients corresponding to the discrete approximation of the derivatives
at the
j-th point.
Generally speaking, the quantum potential interaction stemming by the k-th cell, depends by the strength of the Hamiltonian potential .
By setting, in a system of a huge number of particles, the side length equal to the mean intermolecular distance , and is much bigger than the quantum potential length of interaction we have the realization of the classical rarefied phase.
Typically, the Lennard-Jones potential (5.48) leads to
so that the interaction of the quantum potential (stemming by the
k-
th cell) into the adjacent cells is null and
is diagonal. Thus, the quantum effects are confined into each single molecular cell domain.
Furthermore, being for classical systems , it follows that the spatial correlation length of the noise reads and the fluctuations appears spatially uncorrelated in macroscopic classical systems
Conversely, given that for stronger than linearly interacting systems
so that the quantum potential of each cell extends its interaction to the other ones, the quantum character appears on the coarse-grained large-scale description [
10,
15,
16].
5.6. Macroscopic quantum phenomena and transition to the classical behavior
By discretizing the current conservation equation (5.52) for the system of N particles [
14], it is possible to obtain the quantum hydrodynamic master equation for macroscopic system of a huge number of molecules.
Generally speaking we observe that, given the range of interaction of the quantum potential , the De Broglie length , and the system size ( represents the mean available volume per molecule in isotropic phase), we can generally distinguish in isotropic systems the cases:
In order to describe the typical phases originating by “1-4”, we observe that, typically, for L-J potential interacting molecules, the quantum potential range of interaction extends itself a little bit further the equilibrium position , in the linear zone of interaction, let’s say up to .
This can be readily checked by assuming the L-J interaction is linear for
, leading to the quantum force
while for
, by (5.48) we have that
On this ansatz
reads
that, for
(so that for ordinary microscopic mass
we have
and
), leads to
Thus, for Lennard-Jones interacting particles, under the condition
of “case 1.” we have the rarefied classic gas phases.
Case 2.
The more condensed phase of Lennard-Jones particles, with , still owns a classical behavior since, as a mean, the particles are distant each-other more than the range of interaction of the quantum potential.
In this case, since the inter-particle distance mostly lies in the non-linear range of L-J interaction (
) just beyond to the crystalline phase (staring at
), we typically have a liquid phase [
10].
Case 3.
When the neighbouring molecules lie in the linear intermolecular range of interaction at a distance smaller than the range of non-local quantum potential interaction .
The observables based on this physical length show quantum behavior (e.g., the Bragg’s diffraction of the atomic lattice).
Case 4.
When the temperature is very low (
) and the De Broglie length
becomes so large to overcomes the linear range of interaction (as well as
too) , we might have a liquid phase (i.e.,
) showing quantum behaviour. This can happen when the intermolecular interaction is so weak to maintain the liquid phase down to very low temperature (e.g.,
) that allows the De Broglie length to grow up to
. In this case the observable of fluidity shows the quantum behaviour of superfluidity [
10,
11].
Given the temperature dependence of and , we can have quantum-to-classic phase transition in the case 3 and 4, respectively:
- I.
when and , by temperature increase, we can have the solid-fluid transition with melting of crystalline lattice
- II.
when and , by temperature increase, we have the superfluid-fluid transition.
Case I.
For a system of Lennard-Jones interacting particles, the quantum potential range of interaction reads
where
is the distance up to which the interatomic force is approximately linear (
) and where
is atomic equilibrium distance.
An experimental confirmation of the physical relevance of quantum potential length of interaction comes from the quantum to classical transition in crystalline solid at melting point when the system passes from a quantum lattice to a fluid amorphous classical phase.
Assuming that, in the quantum lattice, the atomic wave-function (around ) spans itself less than the quantum coherence distance, it follows that at the melting point its variance equals .
On these assumptions, the Lindemann constant
[
10,
15] reads
and it can be theoretically calculated since
that, being
and
, leads to
More accurate evaluation, making use of the potential well approximation for the molecular interaction [
10,
11], leads to
and to the value of
for the Lindemann constant that well agrees with the measured ones, ranging between 0,2 and 0,25 [
15].
Case II.
Since the De Broglie distance
is a function of temperature, the fluid-superfluid transition can be described in monomolecular liquids at very low temperature such as for the
. The treatment of this case is detailed in ref. [
10,
11] where, for the
-
interaction, the potential well is assumed to be
where
is the Lennard-Jones potential deepness, where
and where
is the mean
-
atomic distance.
By posing that at superfluid transition the de Broglie length is of order of the
-
atoms distance so that
it follows that for
is about null the ratio of superfluid/normal
density, while for
we have almost 100% of superfluid
. Therefore, at the condition
when the superfluid/normal
density ratio is at 50%, it follows that the temperature
, for the
mass of
, reads
that well agrees with the experimental data in ref. [
16] of about
.
On the other hand, since by (5.71) for
all the couples of
falls into the quantum state, the superfluid ratio of 100% is reached at the temperature
well agreeing with the experimental data in ref. [
16] of about
.
Moreover, by utilizing the superfluid ratio of 38% at the
-point of
, the transition temperature
reads
in good agreement of the measured
superfluid transition temperature of
.
It's important to note that the weak nature of Hamiltonian interaction is what paves the way for classical behavior to arise. Indeed, when dealing with systems governed by a quadratic or stronger Hamiltonian potential, the range of interaction attributed to the quantum potential becomes infinite (as seen in equation 5.44), making the attainment of a classical phase unattainable regardless of the system's size [
5,
10,
11,
15,
17].
In this context, the complete expression of classical behavior is exclusively observed on a macroscopic scale within systems that possess sufficiently feeble interactions (weaker than linear and thus classically chaotic). This occurs due to the inability of the quantum potential to extend its non-local influence over vast distances.
Hence, classical mechanics emerges as a decoherent outcome of quantum mechanics in the presence of a fluctuating background metric within the spacetime.
5.7. Measurement process and the finite range of non-local quantum potential interaction
Throughout the process of measurement, the segment of the experimental arrangement responsible for sensing of the system, might experience quantum mechanical interaction. This interaction concludes once the measuring device is moved far away from the system being measured, at a distance significantly greater than and . Subsequently, the measuring device handles the interpretation and processing of the 'interaction output.' This usually entails a classical and irreversible procedure that follows a specific direction of time, resulting in the observable outcome of the measurement at a macroscopic scale."
However, decoherence plays a crucial role in the measurement procedure by facilitating the development of a macroscopic classical framework. This framework permits genuine separation between the measurement device and the system on a quantum level, both prior to and after the measurement event. This quantum-disconnected starting and concluding condition is vital for establishing the conclusion of the measurement process and for accumulating a set of statistical data derived from multiple independent measurement repetitions.
It's worth highlighting that, within the framework of the SQHM, simply taking the measured system to an infinite distance before and after the measurement isn't enough to ensure the separation between the system and the measuring apparatus when or .
5.8 Minimum measurements uncertainty in quantum systems submitted to stochastic noise
Any quantum theory aiming to depict the development of a physical system across a wide range of sizes must inherently clarify the process through which quantum mechanical traits transition into observable classical conduct on a grander scale. The key differentiating principles between these two explanations are quantum mechanics' minimum uncertainty principle and classical relativistic mechanics' constraint on the finite speed at which interactions and information propagate locally.
If, at a specific distance
, which is less than
, a system completely adheres to "deterministic" quantum mechanical progression, causing its individual components to lack separate identities, then for an observer to acquire data regarding the system, the observer must maintain a minimum separation from the observed system (both prior to and subsequent to the procedure) equal or bigger, at least, than the distance
. Consequently, due to the finite speed of interaction and information propagation, the procedure cannot be executed in a timeframe briefer than
Moreover, given the Gaussian noise (see 2.20, 5.7 ) (with the diffusion coefficient proportional to
), we have that the mean value of the energy fluctuation is
for degree of freedom. Thence, a non-relativistic (
) scalar structureless particle of mass
m owns an energy variance
from which it follows that
It is worth noting that the product is constant since the growing of the energy variance with the square root of is exactly compensated the equal decrease of the minimum acquisition time .
The same result is achieved if we derive the uncertainty relations between the position and momentum of a particle of mass m.
If we acquire information about the spatial position of a particle with a precision
the variance
of its relativistic momentum
due to the fluctuations reads
and the uncertainty relation reads
Equating (5.81) to the uncertainty value such as
or
it follows that
, that represents the physical length below which the quantum entanglement is fully effective and represents the minimum (initial and final) distance between the system and the measuring apparatus.
As far as it concerns the
theoretical minimum uncertainty of quantum mechanics, obtainable from the
minimum uncertainty (5.78-83) in the limit of zero noise, we observe that the quantum deterministic behavior (with
) in the low velocity limit (i.e.,
) leads to the equalities
but the products
remain finite and constitutes the
minimum uncertainty of the quantum deterministic limit.
It is interesting to note that in the relativistic limit, due to the finite light speed, the minimum acquisition time of information in the quantum limit reads
The output (5.90) shows that it is not possible to carry out any measurement in the deterministic fully quantum mechanical global system since it is endless.
Moreover, if we want to increase the system spatial precision to , we can satisfy the condition by increasing the temperature to .
In this case it follows that minimum uncertainty is unchanged since the it is independent by the temperature. Therefore, the minimum uncertainty relation (5.89) holds whatever the choice of .
Since non-locality is confined in domains of physical length of order of and information about a quantum system cannot be transferred faster than the light speed (otherwise also the uncertainty principle is violated) the local realism is established on macroscopic physics while the paradox of the “spooky action at a distance ” is limited on microscopic distance (smaller than ) where the quantum mechanics fully realize itself.
It must be noted that for the low velocity limit of quantum mechanics the conditions and are implicitly assumed into the theory and leads to (apparent) instantaneous transmission of interaction at a distance.
5.9 The stochastic quantum hydrodynamic model and the decoherence theory
In the context of the SQHM, in order to perform statistically reproducible measurement processes and to warrant that the measuring apparatus is fully independent from the measured system (free of quantum potential coupling before and after the measurement), it is necessary to have a global system with a finite length of quantum potential interaction.
In such a case, the SQHM indicates that due to the finite speed of transmission of light and information, it is possible to carry out the measurement within a finite time interval. Therefore, a finite length of quantum potential interaction, and the resulting decoherence, are necessary preconditions for carrying out the measurement process.
The decoherence theory [
18,
19,
20,
21,
22,
23,
24] does not attempt to explain the problem of measurement and the collapse of the wave function. Instead, it provides an explanation for the transition of the system to the statistical mixture of states generated by quantum entanglement leakage with the environment. Moreover, while the decoherence process may take a long time
for a microscopic system, the decoherence time for macroscopic systems, consisting of
n microscopic quantum elements, can be very short
. However, in the context of the decoherence theory, the superposition of states of the global universal wave function still exists (and remains globally coherent).
This puzzle finds its logical solution in the extensive recurrence time, a concept recently expanded to encompass quantum systems as well [
25]. Even within a universally reversible system, certain irreversible phenomena can manifest due to an exceptionally protracted recurrence interval (far surpassing the universe's lifespan). On a certain short time scale, global quantum systems can imitate classical behaviors so faithfully that distinguishing them from a genuinely classical universe becomes impossible. To illustrate, the timespan required, as calculated by Boltzmann, for a mere cubic centimeter of gas to revert to its initial state involves a staggering number of digits, reaching into the trillions, whereas the age of the universe spans merely thirteen digits.
In the context of Madelung's approach the Wigner distribution and the quantum hydrodynamic theory are closely connected and do not contradict each other [
26]. However, the interpretation of the global system as classical or quantum in nature is ultimately a matter of interpretation. Essentially, we cannot determine whether the noise from the environment is truly random or pseudo-random. In computer simulations, it is widely accepted that any algorithm generating noise will actually produce pseudo-random outputs, but this distinction is not critical in numerical simulations of irreversible phenomena.
The decoherence theory can account for the macroscopic behavior as the result of dissipative quantum dynamics. However, it falls short of specifying the prerequisites essential for establishing a genuinely classical global system. In contrast, the approach of Stochastic Quantum Hydrodynamic Model furnishes a yardstick for identifying the shift from quantum dynamics to classical behavior on a significant macroscopic level. Moreover, the potential, as revealed by SQHM, of attaining a classical global system within a space-time riddled with curvature fluctuations [
5] aligns harmoniously with the quantum-gravitational portrayal of the universe in which the gravity is seen as the catalyst for the universal decoherence [
27,
28].
Furthermore, on a conceptual level, the theory of stochastic quantum Hydrodynamic model tackles the challenging quandary of spontaneous entropy diminishment within the global quantum-reversible system. This entropy reduction is vital for the system to revert to its initial state, as stipulated by the recurrence theorem. Moreover, given that the quantum pseudo-diffusion evolution [
29] highlights the co-occurrence of entropic and anti-entropic processes in disparate domains of a quantum system, the puzzle remains unresolved: why haven't we witnessed spontaneous anti-entropic processes occurring somewhere and sometime within the universe?
5.10 The stochastic quantum hydrodynamic theory and the Copenhagen interpretation of quantum mechanics
The path-integral solution of the SQHM (5.32-5) is not general but holds in the small noise limit, before a large fluctuation occurs. It describes the “microscopic stage” of the decoherence process at De Broglie physical length scale.
Moreover, the SQHM parametrizes the quantum to classical transition by using two physical lengths, and , addressing the quantum mechanics as the asymptotical behavior for . Being so, it furnishes additional insight about the measure process.
Even if the measure process can be treated as a quantum interaction between the system and the measure apparatus, marginal decoherence effects exist for its realization due to:
real decoupling at initial and final state of the measure between the system and the measuring apparatus,
utilization of classical equipment for the experimental management, collection and treatment of the information.
The marginal decoherence is ignored or disregarded because the classical equipment is mistakenly assumed decoupled at infinity, while the assumption of perfect global quantum system (whose interaction extends itself at infinity being and ) does not allow the realization of such condition.
In order to describe the decoherence during the external interaction
, the SQHM reads
where
In principle, the marginal decoherence, with characteristic time , may affect the measurement if is comparable with the measure duration time (the absence of marginal effects is included in the treatment as the particular case of sufficiently fast measurement with ).
From the general point of view, the SQHM shows that, the steady state after the relaxation depends on its initial configuration
at the moment
, allowing the system to possibly reach whatever eigenstate of the superposition.
Since the quantum superposition of energy eigenstates possesses a cyclic evolution with recurrence time
, the probability of relaxation to the
i-
th energy eigenstate for the SQHM model reads
where
is the number of time intervals
centered around the time instants
(with
and
) in which the system is submitted to fluctuations, and
is the number of times the
i-
th energy eigenstate is reached in the final steady state..
Moreover, since the eigenstates are stable and stationary (see § 5.1) it also follows that the transition probability between the
k-
th and the
i-
th ones reads
Since the finite quantum lengths and , allowing the quantum decoupling between the system and the measuring apparatus, necessarily implies the “marginal decoherence”, it follows that the output of the measure is produced in a finite time lapse (bigger than of (5.76)) due to wave function decay time.
As far as it concerns the Copenhagen interpretation of quantum mechanics, the measurement is a process that produces the wave function collapse and the outcome (e.g., the energy value
for the state (5.92)) is described by the transition probability that reads
that for the i-
th eigenstate reads
In order to analyze the interconnetion between the wavefunction decay and the wave function collapse, we assume, as starting point, that they are different phenomena and have independent realization.
In first instance, we can assume that the wavefunction decay (with characteristic time
) happens first and then the wave function collapse (with characteristic time
) during the measure process. Without loss of generality we can assume
, and thence, in this case it follows that
On the other hand, for the Copenhagen interpretation, the measure on a quantum state with
, by (5.96-7) it follows that
The outputs (5.98-9) show that the wavefunction collapse, beyond the duration of the wavefunction decay, is ineffective in the measure. Furthermore, since after the wavefunction decay the system has already reached its final steady eigenstate, the wavefunction collapse does not happen since it does not affect the eigenstates.
Being so, we can think to shorten the measure duration
up to the wavefunction decay time
without have a change in the result (5.98) for the measure, leading the relation
On the other side, by considering in second instance that the wavefunction collapse
is much shorter than the wavefunction decoherence such as
(i.e.,
), the final output reads
showing that the wave function decay, due to the marginal quantum decoherence, does not affect the measure even if it might proceed beyond the wave function collapse. More precisely we can affirm that since the wavefunction decay does not affect the eigenstates, it does not happens after he wavefunctin collapse.
Therefore, both the wavefunction collapse as well as the wavefunction decoherence happens together only during the time of the measure
. In the case of (5.101) we might shorten the measure duration time
to
so that, being the wavefunctin decay finished at the end of the measure, identity (5.94,5.101) leads to
The proof of (5.102) can be validated by the solution of the motion equation (5.32-5) of §5.2.
The SQHM through identity (5.102), furnishes the linkage between the wave function collapse and the wave function decay generated by the marginal decoherence possibly showing that they are the same phenomenon.
5.11. Conclusion
The stochastic quantum hydrodynamic model introduces a method for elucidating the conduct of quantum systems amidst noise generated by the fluctuating metric of the physical vacuum. According to this model, the noise's spatial spectrum is not white, and its correlation length is endowed with a distinct physical extent determined by the De Broglie characteristic length. Consequently, this gives rise to a form of quantum entanglement that remains ingrained in systems whose physical dimensions are very small in comparison to this characteristic length.
The Langevin-Schrodinger equation is derived by considering the effects of fluctuations on systems whose physical length is of the order of the De Broglie length. The work elucidates the relationship between the Langevin-Schrodinger equation and the physical length of a system.
However, the range of quantum potential interaction may extend beyond the De Broglie length up to a distance that may be finite in non-linear weakly bonded systems. In this case, as the physical length of the system increases, classical physics may be achieved when the physical size of the problem is much larger than the range of interaction of the quantum potential. Under these circumstances, the quantum potential is not able to maintain coherence in the presence of fluctuations, but generates a drag force producing a relaxation process leading to the decoherence with the decay of the quantum superposition of states, while quantum system's eigenstates are stable and practically maintain the configurations of quantum mechanics. The model provides a general path-integral solution that can be obtained in recursive form. It also contains reversible quantum mechanics as the deterministic limit of the theory.
This effect can be observed in macroscopic systems, such as those made up of molecules and atoms interacting by long-range weak potentials, like in the Lennard-Jones phase. The SQHM provides a useful framework for understanding the interplay between quantum mechanics and classical behavior well explaining both the fluid-superfluid transition of and the Lindeman constant at the melting point of crystalline lattice.
The stochastic quantum hydrodynamic model also shows that the minimum uncertainty during the process of measurement asymptotically converges to the quantum uncertainty relations in the limit of zero noise. The theory shows that in an open quantum system the principle of minimum uncertainty holds only if interactions and information do not travel faster than the speed of light. This output makes compatible the relativistic macroscopic locality with the non-local quantum interactions at the micro-scale.
According to the stochastic quantum hydrodynamic model, decoherence is necessary for a quantum measurement to occur and that it also contributes to the execution, data collection, and management of the measuring apparatus. The model also shows that if reversible quantum mechanics is realized in a static vacuum, the measurement process cannot take a finite time to occur.
The outcomes derived from the stochastic quantum hydrodynamic model align harmoniously with the predictions of the decoherence theory, indicating that the interplay between a quantum system and its surroundings results in a progressive decline of coherence. The sole distinction lies in the perspective on global coherence, which is essentially a matter of interpretation.
The theory also demonstrates its congruence with the Copenhagen interpretation of quantum mechanics, illuminating the intricacies of the wavefunction collapse process and potentially offering an explanation for it as a dynamic phenomenon linked to wavefunction decay.
The portrayal provided by SQHM presents a scenario in which classical mechanics emerges on a macroscopic scale within a space-time filled with fluctuations in curvature. This depiction seamlessly corresponds with the quantum-gravitational representation of the cosmos, where gravity functions as the trigger for universal decoherence.Inizio modulo