Can fluctuating quantum states acquire the classical behavior on large scale?
11 Can fluctuating quantum states acquire the classical behavior onlarge scale?
Piero Chiarelli
National Council of Research of Italy, Area of Pisa, 56124 Pisa, Moruzzi 1, Italy
Interdepartmental Center “E.Piaggio” University of Pisa
Phone: +39-050-315-2359Fax: +39-050-315-2166Email: [email protected].
Abstract:
The quantum hydrodynamic analogy (QHA) equivalent to the Schrödinger equation isderived as a deterministic limit of a more general stochastic version. On large scale, the quantumstochastic hydrodynamic analogy (SQHA) shows dynamics that under some circumstances mayacquire the classical evolution. The SQHA puts in evidence that in presence of spatially distributednoise the quantum pseudo-potential restores the quantum behavior on a distance shorter than thecorrelation length (named here c ) of fluctuations of the quantum wave function modulus. Thequantum mechanics is achieved in the deterministic limit when c tends to infinity with respect tothe scale of the problem. When, the physical length of the problem is of order or larger than c , thequantum potential (QP) may have a finite range of efficacy maintaining the non-local behavior ona distance L (named here “quantum non - locality length”) depending both by the noise amplitude and by the inter-particle strength of interaction. In the deterministic limit (quantum mechanics) the model shows that the “quantum non - locality length” L also becomes infinite. The SQHA unveilsthat in linear systems fluctuations are not sufficient to break the quantum non-locality showing that L is infinite even if c is finite. PACS:
Keywords:
Quantum hydrodynamic analogy; quantum to classical transition; quantumdecoherence; quantum dissipation; noise suppression; open quantum systems; quantum dispersivephenomena; quantum irreversibility
1. Introduction
The emergence of classical behavior from a quantum system is a problem of interest in manybranches of physics [1]. The incompatibility between the quantum and classical mechanics comesmainly from the impossibility to manage the non local character of the quantum mechanics. Evenif this is a great theoretical problem, from the empirical point of view, the solution seems to beachievable. It has been shown by may authors that fluctuations may destroy quantum coherenceand elicit the emergence of the classical behavior [1-6]. By using the alternative approach of thequantum hydrodynamic analogy (QHA) [7] in this paper we investigate how the fluctuationsinfluence the quantum non locality. The goal is to propose a satisfying model that theoreticallygives analytical details about the pathway that brings to the quantum decoherence and possibly tothe large-scale classical evolution.
The motivation of using the quite unknown QHA can be really appreciated once the overalldescription is achieved. By now, we can observe that even if the Schrödinger equation is widelyknown and more manageable than the QHA, it owns some incompatibilities with the large scalelocal physics: In the Schrödinger approach is not clear how the non-local behavior can be managedto make it compatible with the local character of the classical behavior: non-locality is built-in inthe theory and has an infinite range of application. On the other hand, the QHA is practicallyintractable for any physical problem but it owns a classical-like structure that makes it suitable forthe achievement of a comprehensive understanding of quantum and classical phenomena.The suitability of the classical-like theories in explaining open quantum phenomena is a matter offact and is confirmed by their success in the description of the dispersive effects in semiconductors[8,9] multiple tunneling [10], mesocopic and quantum Brownian oscillators [11], criticalphenomena [12-14], and theoretical regularization procedure of quantum field [15-16].Since the introduction by Schrödinger of the quantum wave equation, the QHA was presented byMadelung [7] as an alternative equivalent approach to quantum mechanics that gives rise to aninteresting logical approach to it.The interest for the quantum hydrodynamic analogy (QHA) of quantum mechanics had neverinterrupted since nowadays. It has been studied and extended by many authors as Jánossi [17]resulting useful in the numerical solution of the time-dependent Schrödinger equation [18-20].More recently it has been used for modeling quantum dissipative phenomena in semiconductorsthat cannot be described by the semi-classical approximation [8,9].Moreover, compared to others classical-like approaches (e.g., the stochastic quantizationprocedure of Nelson [21-23], the mechanics given by Bohm [24-27] and those proposed byTakabayasi [28], Guerra and Ruggiero [29], Parisi and Wu [30] and others [31-32] ) the QHA hasthe precious property to be exactly equivalent to the Schrödinger equation (giving rise to the sameresults [18-20]) and it is free from problems such as the unclear relation between the statistical andthe quantum fluctuations as in the Nelson theory [21-23] or the undefined variables of theBohmian mechanics. Concerning the last point, it must be noted that the QHA has not to beconfused with the Bohmian mechanics. Even there exist a great similarity between the twotheories, as clearly shown by Tsekov [33-34], the Bohmian model seem to be more a mean-fieldlimit of quantum theory than a real punctual model with the defect to possess undefined variables.On the other hand, the QHA has no undefined variables and is perfectly equivalent to theSchrödinger mechanics. The QHA is constituted by two coupled first order differential equationfor two real variables: the wave function modulus (WFM) and it phase. By the variablesubstitution, such a system of equation can be reduced to a single second order differentialequation of a complex variable (i.e., the wave function) that is the Schrödinger one.Among the objectives that could benefit from the present work there are: The clarification of thehierarchy between the classical and quantum mechanics [35-36]; The achievement of a consistenttheory of quantum gravity [37-41]; The quantum treatment of chaotic dynamical systems andirreversibility[42-57].Actually, to describe critical dynamics kinetic Langevin equations are assumed on aphenomenological point of view where it is decided a priori what is pertinent to the approximateddynamics. In this context it is really difficult to have a rigorous Langevin description.
The achievement of a theory that on a “small length scale” preserves the standard quantum mechanics while on a large one self-consistently disembogues into the classical one, gives thechance to describe by means of a coarse-grained Langevin equation the connection amongirreversibility, chaos and quantum dynamics in a systematic manner by passing from themicroscopic scale to the macroscopic one.
In this paper, the standard quantum mechanics (represented in the QHA) is derived as adeterministic limit of a more general stochastic QHA (SQHA). The goal of the work is to showhow the quantum mechanics is retrieved in the frame of such a more general theory and underwhich conditions its non-local character is maintained or modified in the stochastic case. Thework investigates how the non-local quantum character (that in the QHA is given by the range ofinteraction of the quantum pseudo-potential) is restored when the amplitude of the noise convergesto zero.In the case of a small non-zero value of the fluctuations amplitude the work inspects in details: (1)if exists a scale below which the standard quantum mechanics is still achieved (2) what is therange of interaction of the quantum pseudo-potential, (3) how it depends by the fluctuationamplitude and by the inter-particle strength of interaction.
2. The SQHA phase space equation of motion
In this section we analyze the QHA in the case of spatially distributed noise. The QHA-equationsare based on the fact that the Schrödinger equation, applied to a wave function (q,t) = A (q,t) exp[i S (q,t) / ], is equivalent to the motion of a fluid with particle density n (q,t) = A and avelocity mSq )t,q(q , governed by the equations [ 7] )q( (q,t)q(q,t)t , (1) Hq p , (2) )VH(p quq , (3) with )p,....,p( np (4) )q,....,q( nq (5) where )q( VmppH (6) is the Hamiltonian of the system of n structureless particles of mass m and V qu is the quantumpseudo-potential that reads nn2 // qqqu )m(V . (7) For the purpose of this paper, it is useful to observe that equations (1-3) can be derived by thefollowing phase-space equations ))xx(( qut),(q,pt),(q,pt H (8) where pd nt),(q,p(q,t) n . ( ) )H,H(x qpH (10) )V,(x quqqu (11) once equation (8) is integrated over the momentum p with the sufficiently general condition that t),(q,p|p| lim (12) and the phase space quantum field has the form )Sp( q(q,t)t),(q,p n , (13)where tt qu)q( )VVmpp(dtS . (14)Due to the fact that the ensemble of solutions of equations (8-11) is wider than that one of theQHA-equations (1-3), the accessory condition mSq q (15) (namely the wave-particle equivalence) warranted by the -function in (6) must be applied to (8)-Generally speaking, for a solutions of the problem (8) we have mSpdqq q(q,t) nt),(q,pt),(q,p(q,t) n ( ) This is an important point since the satisfaction of condition (15) is necessary to pass back fromthe QHA equations to the Schr ödinger one [7,55].
For the more general case of spatially distributed stochastic noise, the stochastic-PDE (SPDE),whose zero noise limit can lead to the deterministic PDE (4), reads )Sp())xx(( q),t,q(qut),(q,pt),(q,pt H , ( ) where is a measure of the noise amplitude and where the accessory condition (see Appendix A) )Sp( q(q,t)t),(q,p n , (18)is held in order to warrants the wave particle equivalence in the deterministic limit. Since in the limit of zero noise the ensemble of solution of the SPDE (17) is wider than that one ofthe deterministic equation (8) (see for instance Ref. [56]), it needs to enucleate the conditions thatwarrant the establishing of the quantum mechanics (i.e., the PDE (1)) for that goes to zero .To this end, we investigate (17) in the limit of small noise amplitude for the sufficientlygeneral case to be of great interest of a Gaussian random noise.In order to perturbingly investigate (17) near the deterministic limit, we re-write it as )Sp(*))Fxx(( q),t,q()(qut),(q,pt),(q,pt H , (19)where is the solution of the PDE (8) and where F *(x,t), containing the QP fluctuations due tothe SPDE field (that can be very large even in presence of a vanishing noise) reads *I*F , (20) )(qu)(qu qqqq VV }){m(*I //// nn 002 nnnn2 . (21)and where pd nt),(q,p(q,t) n . (22) ))xx(( qut),(q,pt),(q,pt H (23)Equation (21) without the term *)F( t),(q,p is a flow equation with spatially distributednoise that has been already extensively studied in the form a Fokker-plank equation [57] thatconverges to the deterministic limit for going to zero.Actually, the derivative structure of the term *)F( t),(q,p , in principle can lead to a finitecontribution even if the noise amplitude is vanishing. This is quite evident since a small abruptvariation of the quantum field t),(q,p can give a large output on its derivative in the quantumpotential expression.In principle, nothing makes the solution expressing the quantum mechanics in (19) privilegedexcept the physical constraints that introduced into the abstract plane of the equations will explainwhy all alternative solutions cannot happen but those of quantum mechanics.Following this logical pathway, we observe that in order the fluctuating states of the SPDE (21)can realizes themselves, it is necessary that their energy is finite, that is: The energy gap betweenthe quantum deterministic states of (1) and the corresponding fluctuating one has to be finite.Given that the energy of the QP is a real energy for the system [58], if we impose that the energyof the fluctuating state is finite, we have also to impose that the energy increase introduced by thefluctuations of the QP is finite, that is )VV(lim )(ququ (25)(more precisely, we impose that the root mean square of QP energy fluctuations in (20) are finite).As far as it concerns this point, in presence of spatially distributed noise, the derivative structure ofthe QP (3) immediately shows to play an important role since the fluctuations of the quantum field , on shorter and shorter distance, will produce higher and higher QP values. Therefore, sincewhite quantum field fluctuations would lead to an infinite value of the QP energy fluctuations,they are not possible for the problem (19). Moreover, the fact that spatially non-white quantumfield fluctuations are the consequence of the action of the QP, means that it acts to suppress themon shorter and shorter distance. This QP cut-off of high spatial quantum field fluctuationsfrequencies clearly means that exists a distance c > 0 (we name it here “quantum coherencelength”), below which the noise is damped and the standard quantum mechanics (deterministic limit) are approached. For sake of completeness, we have to observe that if we want to build up a perturbing expansionaround the deterministic field solution (23), making a true perturbing expansion parameter inthe sense that )(lim t),(q,pt),(q,p , (26) actually, we have to warrant that the overall term *)F( t),(q,p converge to zero in thesense *)F(lim q (27)A s shown in the appendix B, the energy requirement given by (25) is a sufficient condition towarrant (26).On the condition (to be defined) that the root mean square of the QP energy fluctuations is finite(for any finite), so that equation (19) converges in the sense of (27) to the deterministic limit for going to zero, it is possible to approximate the field solution (whose integral over momentagives the WFM) as the sum in the following ................. n (28)so that the m-th order solution reads mm .......... . (29) Moreover, given the general initial conditions )t,p,q(m (30)it will hold )t,p,q(mt lim m (31)and hence, under the condition of Gaussian field fluctuations (to be check at the end) in thevanishing noise amplitude , for a sufficiently short interval of time t after the initial instant,the solution m will remain close to m in the sense that )(lim t),(q,pmt),(q,pmt . (32)In force of that, in such an interval of time t, the equation of motion (19) can be approximated bythe following system of coupled equations ))xx(( )(qu)t,p,q()t,p,q(t H , (33) N))xx(( )(qu)t,p,q()t,p,q(t H (34) m)(qu)t,p,q(m)t,p,q(mt N))xx(( H (35)and so o n; where )Sp()*F(N q),t,q(mt),(q,pmm (36) )VV(*F )(qu)(qum m nn (37) pd nt),(q,pm(q,t)m n ( ) tt )(qu)q(m )VVmpp(dtS m n (39) and where the approximation (32) has been introduced into (37).Furthermore, given that from (37) it follows that F * = 0, equation (34) at first order ofapproximation reads )Sp())xx(( q),t,q()(qu)t,p,q()t,p,q(t H (39)We note that the approximation in equation (39) F *
0, steaming by (37), is correct providingthat in the small noise amplitude the condition (25) and hence (26) are satisfied.
After the interval of time t is passed by, the final condition ( q,p, t t ) is assumed as theinitial condition for the subsequent interval of time t. The repetition of such a procedure willallow to find the solution at an arbitrary instant of time into the future for sufficiently close tozero.If we consider the sufficiently general case to be of practical interest that N is Gaussian, in orderto determine it, we need to define its mean and correlation function. If we assume that: (1) thenoise has null mean; (2) The noise has null correlation time; (3) The space is isotropic; (4) thenoises on different co-ordinates are independent; ( q,t , ) generally reads )()(G)(g, )t,q()t,q( (40)At lowest order the SQHA problem reduces to find the noise shape G( ) that warrants conditions(25,26). As shown in Appendix B, this is obtained by requiring that the QP energy does notdiverge when the correlation distance of the WFM fluctuations tends to zero. This can beimplemented in the small noise approximation, by using the WFM fluctuations at the zero order ofapproximation (see (C.31) in appendix C). Here we derive the conditions, on the correlation length of the zero order Gaussian WFMfluctuations, under which the QP root mean square energy fluctuations become vanishing and theconvergence to the deterministic PDF (1) is warranted for vanishing.Once the (lowest order) WFM fluctuations are defined by (57-40) (see (C.31) in appendix C) wecan correspondingly evaluate the QP fluctuations.Given the zero order Gaussian WFM fluctuations (C.31), we can find the form of )(G byimposing that the root mean square of QP fluctuations do not diverge as goes to zero. This ispractically implemented by operating in the discrete approach and then by passing to thecontinuous limit (see appendix C and D).Since in the discrete approach the correlation length of fluctuations cannot be smaller than thediscrete cell dimension , the non-diverging QP condition is practically obtained by imposing thatthe root mean square of QP fluctuation, in the discrete form, remains limited when the celldimension goes to zero.By using the QP expression that after simple manipulation reads ])[m(V qqqqqu nnn-nn2 , (41)and by writing the spatial quantum field derivatives as the limit of the corresponding discretequantity as nnnn ][lim )i()i( q(q(qq (42)and ][lim )i()i()i( q(q(q(qq nn2nn , (43)we are able to calculate the variance of the QP fluctuations (41) by means of the followingequalities [see Appendix D] )](G)[(gtA })](G[)(gt{ qqqq
20 222400 ,(44) )](G)(G)[(gt qqqq (45)and qqqq , (46)where )q()q( ,)(g , (47) )(g,)(G )q()q( (48) )(g,)(G )q()q( . (49) qqA nn ttt and where in (69) has been used the identity )q( lim .If we require that the root mean square of the QP energy fluctuations (see Appendix D) satisfiesthe condition ][)m(lim V,Vlim qqqqqqqqququ nnn,n-dnn,d2 (50)when and d are weighted mean particle densities given in Appendix D), it mustnecessarily follow that )](G[lim (51) ]G[lim )( (52) ]GG[lim )()( (53)Developing G ( ) for small in series expansion as a function of / c , where c is definedfurther on, we obtain jj cjcccc)( )(a)(a)(a)(aaaGlim ,(54)from where it follows that (72-3 51-53) are verified if a a
1= 0, and a a a j with j )(G reads jj cjcc)( )(a)(a)(aGlim . (55)where without a leaking of generality we can put a by a re-definition of the spatial cellside such as / a' . Introducing (55) into (51-53) for a check, we obtain c a)](G[lim (56) c)( a]G[lim (57) c)()( a]GG[lim (58). >> c ) As shown further on in this paper, since the quantum coherence length c results by thegeometrical mean of the Compton length l C and the stochastic length ck , the SQHA model isanyway linked to the Compton length as a reference scale.Therefore, for a macroscopic system whose dimensions are huge compared to the quantumcoherence length c (for instance, for as small as 1°K, it results c = ( l C c / k )½ = (2.2 )½ = 4.7 cm for a particle of proton mass) the continuum macroscopic limit can be achieved for that numerically goes to zero but that is still very large compared to c . In this caseit is not enough to know the series expansion (55) of the correlation function . Given the physical values of the Compton’s length l C, very small noise amplitude (of order ofone or tens of degree Kelvin) still leads to a very small value of c compared with the standardlength units of the macroscopic physics. Therefore in this case, the Gaussian small noiseapproximation can hold without solution of continuity from the micro-scale to the macro-scaleapproaches.In order to obtain a model holding also for a large-scale approach, hence, we utilize the correlationfunction of the Gaussian fluctuations in the form of an exponential law in agreement with (73 55).To this end, we investigate in detail the model with a (that warrants the ergodicity) in theparticular case where the shape of the correlation function reads ])(exp[G c)( . (59)that both satisfies (55) and leads to a large-scale -correlated spatial fluctuations. The model with Gaussian quantum field fluctuations that owns ( ) as correlation function doesnot exclude others macroscopic Gaussian noises that are present at the large-scale level [59 ].Given that the Gaussian processes ( q ) with the correlation function )(])(exp[)(g, c)t,q()t,q( , (60)where (see appendix E ) )(g and c respectively read c)q()q( k,)(g , (61) //// )k cl()()mk()( Cc (62)where mcl C is the Compton’s length, the large scale correlation function reads )()(k )(])(exp[klim,lim c cc/)t,q()t,q(/ cc -(63)In the Appendix E, the quantum coherence length c is calculated by imposing that the root meansquare of the system energy fluctuation at equilibrium calculated by the SQHA are the same to thatone obtained by statistical mechanics. The expression of c is obtained with the convention thatthe vacuum fluctuations amplitude (in the small value limit) is measured in a scale by which itequals the temperature of an ideal gas at equilibrium (placed in such a space region).By using (62), finally, (62) reads
23 2 )k(m,)(g )q()q( (64) where the “form“ factor for an ideal gas confined in a vessel of side L reads (see AppendixE) L ( ) where L ( ) has the dimension of a mobility constant. Thence, the motion equation in the small noiseapproximation reads ),t,q()((q,t)q(q,t)t )q( n nn ( ) )(])(exp[)k(m, c)t,q()t,q(
223 2 ( ) // )mk()( c ( ) mpq )( n , ( ) )VV(p )(qu)q(q n0 , ( ) )q( )((q,t)q(q,t)t n00 nn ( ) tt )(qu)q(tt )(qu)q( *)IVVmpp(dt)VVmpp(dtS nn ( ) sttt )(qu)q(qq pp}*)IVVmpp(dt{Spqm (74)where }dt*I{p ttqst , ( ) In addition to the large-scale noise limit, to obtain the macro-scale form of equations (67-75) weneed to investigate the large-scale limit of the quantum force quqqu Vp in (71). T he behavior of the WFM determines the QP through the term n / of (3) . For sake of simplicitywe discuss here the one-dimensional case of localized state that a t large distance n / goes like ]Pexp[limm-s )q(h|q| / n (76)where )q(h P is a polynomial of degree equal to h , z q =
1q is the macroscopic variable (where = q / L and q is the macro-scale resolution cell size) and L is the range (to be defined) ofinteraction of the QP.The QP (3) at large scale reads hqh)h(q)h(qu z)h(hzhlimVlim (77) Changing the exponent variable by = 3 – 2 h , we obtain /)(q).(qqu z)h(hzhlimVlim (78)and therefore the quantum force quq V at large scale reads ½23½2 q)(qqquq z)z)(h)(h(h)z)(h(hlimVlim (79)that for > 0 (i.e., h < 3/2) z q
012 2112 q)h(h zq)h)(h(hq)h(hlimVlim q)()z(qquq)z(q finiteqfiniteq (80)
Moreover, since the following integral dq|q|dq|Vq| quq (81) converges for > 0 (i.e., h < 3/2), the requisite (81) tells us whether or not the QP force isnegligible on large scale as given by (80).It is worth mentioning that condition (81) is not satisfied for linear system whose eigenstates have h = 2 so that they cannot admit the classical limit. It is also worth noting that condition ( ), obtained for a WFM owing the form ( ), holds alsoin the case of oscillating wave functions that at large distances are of type ]Pexp[Mlim||lim )q(h)q(|q||q| / n ( ) with n )q(pnnm)q(|q| ]iAexp[aqMlim ( ) where )q(pn A are polynomials of degree equal to p . In this case, in addition to the requisite h ( ) the conditions m and p are required (see appendix F). Moreover, given that for regular continuous and derivable Hamiltonian forces that are attractiveat large distance and whose zero potential can be put to infinity (already sufficiently general tobe of great practical interest such as the L-J type potentials) it holds qAlim )q()q(pn|q| (85)so that in this case p = 1 and hence (81) is still a valid condition in order to warrant the existenceof a finite L .Finally it must be observed that even the convergence (76) is satisfied, there can exist short lengthquantum wave modulus oscillations that can leads to very high QP values (that in principle cannotbe disregarded). Respect to this we observe that: (1) since any curvature of the quantum wave modulus generates a quantum potential force that opposes itself to it like an “elastic type” response, such short-length oscillations were smoothed out in a short interval of time. Since in alarge – scale description we are not interested in microscopic details as well as in a very short timebehavior, they can be disregarded. (2) the integral condition (81) is not influenced by this micro-scale oscillations since it mediates over such oscillations so that it is able to discriminate if theglobal behavior of the QP is relevant at large distances (bigger than c ). L By considering ( ) as a measure of the quantum potential force at large distance, the quantumpotential range of interaction can be obtained as the mean weighted distance of the integrand of(81) that for the unidimensional case reads )q(quc qu cL |dqdV| dq|dqdVq|
10 1 . (86)where the origin (0,0) is the mean position of the particle. When L , with c finite, so that dq|dqdVq| qu (87)(e.g., it happens for Gaussian states of linear systems), the quantum potential is not vanishing atinfinite and the system evolution is affected by the quantum non-local forces even on large scaledynamics . Given the 3n-dimension generalization of ( ) leading to the scalar parameter ),....,,r(qu n F measuring the non-local strength of the quantum force qdq|Vq)qq(|...)n( |q|d|Vn)qq(|...)n(F nr /r nr /rn quq quq),....,,r(qu
001 001 (88) where |q| qn , the quantum non-locality length L can be defined as })VV( Flimmax{ ),...,,r(quqquq ),...,,r(qurc nc/ nL (89)The expression ( ) for a system of a large number of particles is quite complex, nevertheless forthe interaction of a couple of particles (e.g., mono-dimensional case, real gas or a chain ofneighbors interacting atoms) the expression (86) is quite manageable [60]. Since the SPDE ( ) depends by two lengths c and L , various limiting cases are possible. Inorder to correctly name and identify them, in the following we use adjectives according to the rule: q << c “microscopic”. (2) q >> c “macroscopic”. (3) L << L “non - local”. (4) L << q “local”. (5) k = 0, “ deterministic” . (7) = q “continuum limit”, where q is the resolution length in the scale of the problem and L (with L >> q ) is the physical length of the system. Non-local deterministic dynamics (i.e., the standard quantum mechanics). The present case isidentified by the following values of the parameters k = 0 (i.e., c = , L = ).Given by (81) that )(])(exp[klim,lim cc)t,q()t,q( it follows that the SPDE (67-74) assumes the deterministic form )q( (q,t)q(q,t)t nn ( ) qutt )(qu)q( S)VVmpp(dtS n ( ) pS})VVmpp(dt{Sp quqtt )(qu)q(qq ( ) mpq , ( )5 )VV(p )(qu)q(q n , ( ) Local (classic) deterministic dynamics.
Even if the local deterministic dynamics requires L =0 being c = , it is interesting for confined particles (i.e., lim|q| = 0 lim|q| / xj = 0)to discuss this case.Given that the deterministic dynamics requires k = 0 ( c = ) and that the local characterrequires L = 0, by (93) it is straightforward to demonstrate that for a system of confined particlesthis can happen if and only if = 0.The result is immediately achieved by observing that in order to have L = 0 (for c tending to ),it must indeed result dq|dqdVq| qu and hence |dqdV| qu rigorously null over all thespace.For confined particles for which it holds lim|q| = 0 and lim|q| / xj = 0, a rigorouslyconstant QP (i.e., V qu ( q ) q = 0) can be obtained from (3) just for the meaningless case = 0over all the space if is a not null physical constant, in the SQHA model the classicalbehavior cannot happen in the deterministic manner for localized particles but only in thestochastic mode. Macroscopic non-local stochastic dynamics . The present case is identified by the followingvalues of the parameters k c << q << L << L or even L = (e.g., Gaussianstates of a linear system). In this case the system of equations (67-74) read ),t,q()((q,t)q(q,t)t )q( n nn ( ) )()(k, c)t,q()t,q( ( ) mpq )( n , ( ) )VV(p )(qu)q(q n0 , ( ) )q( )((q,t)q(q,t)t n00 nn ( ) stqutt )(qu)q(tt )(qu)q( SS*)IVVmpp(dt)VVmpp(dtS nn ( )6 ttst dt*IS ( ) stqucltt )(qu)q(qq ppp}*)IVVmpp(dt{Spqm n ( )where tt )q(qcl dtVp ( ) tt )(quqqu dtVp n ( ) ttqst dt*Ip , ( ) Macroscopic local stochastic dynamics.
This case is defined by the conditions k c L << q << L .Given the condition L >> q >> c for sufficiently small , by (25, 27) we can set / *I*,I (106) as well as / *F*,F ( )and given the condition L << q << L so that it holds n )(quqq Vlim (108)the SPDE of motion acquires the form ),t,q((q,t)q(q,t)t )q( nn ( ) )()(k, c)t,q()t,q( ( ) mp)}Vmppdt{m }*)IVVmpp(dtmlim{mSlimmpq cltt )q(q tt )(qu)q(/qq/q LL n0 (111) )q(qcl Vp , ( )
3. Discussion
The realization of condition (89) allows fluctuations, as small as we like, to overcome the quantumforce on large distance so that the quantum non-locality can only be maintained on a finite distanceof order of L . Since condition (81) is satisfied in a large number of real non-linear potentials, while thecase of an infinite quantum non-locality length (such as in the linear systems) actually seemsto be an exception, the universe behaves classic on its huge scale.
Generally speaking, it must be observed that even thought fluctuations are present, we may havesystems characterized by an infinite quantum non-locality length L (e.g., linear systems owingGaussian states with h =2) so that, in principle, fluctuations are not sufficient to break the quantummechanics and to lead to the classical one.Under this light, the macro-scale description is not sufficient to obtain the classical behavior if notcoupled to a finite range of quantum non-local interaction about whose the realization of (81) is asufficient condition. On the contrary, fluctuations may break quantum non-locality in non-linear systems (satisfyingcondition ( )) because, in this case, the quantum pseudo-potential decreases with distance and,beyond L , it becomes negligible with respect the fluctuations.It must be noted that, in the large-scale description a vanishing small quantum force can becorrectly neglected in presence of fluctuations but it cannot be taken out by the deterministic PDE(1) because in such a case this operation will change the structure of the equation. This can be easily shown by noting that the presence of the QP is needed to the realization ofquantum stationary states (i.e., eigenstates) that happen when the force steaming by the QP exactlybalances the Hamiltonian one; if we remove the QP in (1) we also cancel the eigenstates anddeeply change the structure of the QHA equation.Moreover, the SQHA approach shows that the large-scale classical character can only emerge inthe stochastic case while the deterministic classical mechanics is only a conceptual abstraction thatcan be achieved for = 0. This result, even seeming strange, is quite interesting since it canfurnish the explanation why fluctuations are so wide-spread in nature and cannot be eliminated inthe classical reality. The classical localization of states as a fluctuation-mediated phenomenon innon-linear systems is at glance with the current outcomes of numerical simulation on the classicalto quantum transition [46].
Moreover, by observing that the QHA is constituted by two coupled first order differentialequation for two real variables: the WFM and it phase and that by a substitution reduces to theSchrödinger second order differential equation of a complex variable, it follows that any solutionof the Schrödinger problem is also a solution for the QHA one but not vice versa. Since in order topass from the system of two first order differential equations of the QHA to the correspondentsecond order differential equation of the Schrödinger problem the wave-particle equivalence isnecessary [55], this does not appear to be an important point in the deterministic limit where thewave particle equivalence holds.On the contrary, since in presence of noise some solutions of the SQHA problem may not satisfythe wave-particle equivalence mSq quq (i.e., for the classical states mSq clq ). suchSQHA states do not have their corresponding ones in the Schrödinger representation [61].In figure 1 the traditional point of view of quantum mechanics and the SQHA one are compared.The quantum stochastic mechanics contains as a particular case the standard quantum mechanics,the classical mechanics, with h=0, and others cases as for instance the Brownian harmonicoscillator. The SQHA model shows that the problems of the standard quantum mechanics as wellas of the Brownian harmonic oscillators have their own corresponding representations [62-64] butthe classical mechanics with = 0 has not.Alternatively, the SQHA model shows that the stochastic classical mechanics can be achievedeven if Figure 1. The correspondence among the quantum mathematical models.Finally, it is noteworthy to note that the justification of spatially distributed noise as a consequenceof an external environment or thermostat, can also seen as a consequence of the initial conditionof the system. From the relativistic point of view this fact is not so unexpected since in the fourdimensional space-time the initial condition (on the time co-ordinate) and the boundary conditions(on the space co-ordinates) can be treated on equal foot.
To elucidate this point, let's consider a closed system (universe) with a finite volume at the initialtime. The light cones coming out from two different spatial points (not correlated at time t=0since they are at finite distance and the light speed (propagation of interactions and information) isfinite) will enlarge themselves on different spatial domains (see figure 2).Figure 2. The expansion of cone lights from two points of a finite volume initial universe.The interaction propagating from this two points deals with two different ensembles of vacuumoscillators. Then, after a small time increment, the vacuum oscillator that have already interactedwith the force coming from one point will interact with an uncorrelated input of force comingfrom the other one. Therefore, since this process is endless (in an open universe), any point of the space issubsequently reached by uncorrelated inputs coming from each point of the universe. In such asystem there will always be a background of un-correlated retarded force that act as a booster.If we acknowledge this background of infinitesimal noise as the manifestations of the lack ofknowledge of the retarded effects of the interaction potentials (quantum one included) determinedby the initial state of the universe (due to the finite speed of propagation of interactions togetherwith the finite volume of the universe at the initial time) the stochastic approach (i.e., the SQHA)becomes logically self-standing and the consequence of the initial state of the universe.
4. Conclusion
In this paper, the standard quantum mechanics is derived as a deterministic limit of a more generalstochastic QHA. The work investigate the features of the quantum behavior for a vanishing non-zero value of the fluctuations amplitude. The standard quantum mechanics in a noisy environmentis always achieved on a scale much smaller than the theory-defined quantum coherence length.This is allowed by the fact that spatial fluctuations of the WFM are energetically suppressed by thequantum potential .
More analytically, the SQHA shows that in presence of spatial noise the QP modifies the shape ofthe fluctuations of the quantum field (whose spatial density in the deterministic limit representsthe WFM) suppressing them on a distance much shorter than the theory-defined quantumcoherence length c = /(2 m k )½ so that the quantum mechanics is achieved when c goesto infinity with respect to physical scale of the problem (as for the deterministic limit of null noiseamplitude = 0). The investigation shows that the non-local quantum interaction (that in the QHA originates by theQP) can extend itself beyond the quantum coherence length but, in the stochastic case, with arange of interaction that may be finite (maintaining the quantum non-local interactions up to adistance (the “quantum non - locality length” L ) whose order of magnitude can be evaluated by anintegral formula . The analysis shows that the so-defined L depends by the strength of theparticles interaction and by the fluctuation amplitude. The model shows that in linear systems L can be infinite (even if c is finite) so that fluctuationsare not sufficient (from the general point of view) to break the non-local interaction of thequantum mechanics.On the contrary, for non-linear interactions, the noise may produce quantum non-locality breakingwhen the force of the QP decreases and becomes vanishing at large distance (beyond L ) beingnegligible with respect to the fluctuations. The SPDE of motion exhibits various limiting dynamics, depending on the two characteristiclengths c and L . For c as well as L are negligibly small with respect to the physical length of the problem, while the deterministicclassical mechanics is realized only for = 0.The SQHA model furnishes a non-contradictory logical pathway from the quantum to the classicalbehavior. The quantum mechanics is deterministic while the classical one is achieved when onlarge distance fluctuations overcome the QP interaction (that builds up the quantum eigenstatesand their superposition of states). In the frame of the SQHA it is possible to achieve a unified understanding of quantum andclassical mechanics.The open quantum mechanics, the meso-scale quantum dynamics and the irreversible quantumphenomena are the fields where the SQHA model can be fruitful utilized and tested.Finally, it must be underlined that the SQHA model sees the standard quantum mechanics as a deterministic “mechanics” satisfying the long waited philosoph ical need of such a theory.
Appendix A
Wave-particle equivalence in the deterministic limit of the SPDE of motion Here we derive the condition to which the noise must obey in the motion equation (17) in order towarrant the wave-particle equivalence in the deterministic quantum mechanics limit. In the QHAmodel the wave particle equivalence is warranted by the separate variable solution of type )Sp( q(q,t)t),(q,p n (A.1) where the momentum p equals the gradient of the action in the quantum limit S qu that in a plane wave represents the wave vector multiplied by the Plank’s constant. By introducing the separate variable solution (p,t)(q,t)t),(q,p ~ n in equation (17) we obtainthe equation ),t,p,q(ququtt N))xx((~))xx(~(~~ HH nnnn (A.2) that is satisfied by system of equations ))xx(~(~ qut H (A.3) ),t,p,q(qut N~))xx(( H nn (A.4) If we impose that ),t,q(nN~ ),t,p,q( (A.5)(A.4) is a SPDE function of q, t and ; while (A.3) is a deterministic PDE that can bestraightforwardly solved with the general initial condition )pp(~ )t()(p,t (A.6) that leads to )Sp(~ q(p,t) (A.7)where stqutt )(qu)q(tt )(qu)q( SS*)IVVmpp(dt)VVmpp(dtS nn (A.8)with tt )(qu)q(qu )VVmpp(dtS n (A.9) and hence to )Sp(),t,q(nN q),t,p,q( (A.10) Moreover, providing, as shown in appendix D , that *Ilim (A.11) in the quantum deterministic limit, condition (A.7) gives )Sp(~lim quq(p,t) (A.12) and hence that quq Splim (A.13)namely, the wave-particle equivalence. Appendix B
Convergence of the SQHA SPDE to the quantum deterministic limit
The request that *Ilim)VV(lim )(ququ (B.1) the root mean square of the quantum potential is vanishing in order to have theconvergence to the deterministic limit for tending to zero is based upon the physicalrequirement that the energy of the system has to be finite also in the fluctuating state.Actually, a priori this is only a necessary condition while, to completely warrant thequantum deterministic limit for = 0, in principle, we have to warrant (?). Given that thequantum force is the derivative of the quantum potential, a small quantum potentialfluctuation may lead to a great quantum force inputs *I*F q (that could deeplychange the evolution of the system from the deterministic one). The same possibilityapplies also to the term *)F( q . Therefore, in principle, in addition to the condition / *Ilim (B.2) we progressively must impose that / *)I(lim q (B.3) / *)F(lim q (B.4) As shown in the section (2.1.2), in the limit of going to zero, condition (B.2) iswarranted by the constraint applied to the coefficients a , a , a , and a in (54). As far asit concerns (B.3) and (B.4), since the derivative operation increases by one order the -exponent in (51-53), they will bring additional constraints on the coefficients a n (with j close to zero the termswith n This hierarchy shows that the energy plays a primary role also in the realization of thequantum deterministic limit.
Appendix C
The discretized SPDE of motion
Practically, the introduction of condition (25) can be implemented by operating on the discreteversion of the SPDE (19) whose variable reads )t,p,q( pdqdY nn mi im , (C.1) where the indexes i and m are actually vectors of integers i = (i , i , …., i ,…..) m = (m , m , …., m ,……) that define the discrete point, as in the following ,...)p,...,p,p,...,q,...,q,q()p,q(x mmmiii miim (C.2) and where i and m are the hyper-cells sides and , respectively, of the phase-spacearound the discrete point x im =(q i , p m ) that read ]q,q[ ,...),..,,( iii iii i (C.3) ]p,p[ ,...),..,,( mmm mmm m (C.4) The discretization procedure gives rise to an infinite system of discrete stochastic Langevinequations for the discrete field Y i , m (t) . After standard manipulations [57] for Hamiltonians of type(2), the equation (19) for the discrete variable Y im Y i m reads imimsmhkjisjkhiskmikim NYP]DmYp[YQpdtdY ququ m nnn2 (C.5) where }{)m(Q )( )( ikkIiik , (C.6) where I ( ) = (0,….0, 1( - th place), 0, ….,0) and where i qHp , (C.7) n m Y n jmj (C.8) )t,p,q( pdqdY nn mi im (C.9) being the solution of the deterministic PDE and where }{P )( )( smImssm , (C.10) n )()( )()()( ]ln[YD )()()( )()()()(qu nnn nnn hkjihkIIijIi hkjhIikIijIIiisisjkh (C.11) )Sp(pdqdN q),t,q( nn mi im (C.12) and where *)F(pdqdY t),(q,p nnqu mi im (C.13) represents the correction to the discretized quantum force coming from the fluctuations thatcritically depends by the correlation distance of the field fluctuations. The discrete SPDE of motion close to the deterministic limit
In the discrete form, the small noise system of approximated equation (33-35) can be written as aninfinite system Stochastic differential equations (SDE) smhkjisjkhiskmikim
P]DmYp[YQpdtdY qu m nnn2 (C.14) imimsmhkjisjkhiskmikim NYP]DmYp[YQpdtdY ququ m nnn2 (C.15) imimsmhkjisjkhiskmikim NYP]DmYp[YQpdtdY ququ m k2kkk nnn2 (C.16) where )*F(pdqdY kt),(q,pk nnqu mi im (C.17)and )t,p,q(k pdqdY nn mi im (C.18) In the case of converging to zero, the small noise series expansion ...Y...YYY uk1k0kk imimimim (C.19) can be used to solve the system of equation (C.14-C.16) . In this case, the first order solution ...Y...YY...Y...YYY u1u10 imimimimimimim (C.20) where it has been introduced the information that the zero order solution im Y is given by thedeterministic one im Y , can be used to solve (C.15) leading to imimim dYdYdY . (C.21) Moreover, being im qu Y , (C.22) the SDE (C.15) [57] at the first order reads dtNdtYKdY )Y( imimim im (C.23) where }Y }P]DmYp[YQp{{K qu m)Y( nnn2 im smhkjisjkhiskmik im (C.24)that with the initial conditions descending from (30) that reads )t( Y im , leads to dtNdY imim (C.25)and thence to the first order WFM fluctuations [57] dt)}Sp(pdqd{dY dYdYdY q),t,q( nn mi im imimim (C.26) Moreover, being n m Y n jmj (C.27) in the small noise approximation it follows that nnn iii (C.28) dt})Sp(pdqd{d ddd n m q),t,q( nn n nnn mii iii (C.29) that passing to the continuous limit (i.e., )Sp(pd)Sp( )Sp(pd qn m qn m q nn m m (C.30) gives ),t,q((q))q((q)(q)(q) dtddtddtddtddtd nnnnn (C.31) Appendix D
Quantum potential fluctuations in the small noise limit
Given that }VVVV{lim)VV(lim )(qu)(quququ)(ququ (D.1)and that }VV{limV,Vlim ququququ (D.2) it clearly appears that the QP divergence generated by the shape of the WFM fluctuations in bothterms (D.1,D.2) are brought by the quadratic mean qu Vlim that is sensitive toamplitude of the QP fluctuations.On the contrary, the linear mean terms: )(ququ VV n and qu V are expected to be quiteinsensitive to fluctuations since in the small noise approximation the probability transitionfunction is Gaussian and fluctuations are practically symmetric respect the change of sign.Therefore, both ququ V,Vlim and )VV(lim )(ququ do not diverge (asa function of the shape of the WFM fluctuations) if qu Vlim does not, and vice versa.Thence, in order to warrant (D.1) we impose that ququ V,Vlim does not diverge inorder to derive the correlation length of the WFM fluctuations.Given the quantum potential }((){m(V qqqqqu n)nnn)n2 (D.3) the divergence of ququ V,Vlim due to the shape of WFM fluctuations can be evaluatedby writing down the discrete expression of the derivative terms as follows }{q ),q,......,q( )t,q()t,q( nq )) (i1(i nnlimn nnn
10 31 (D.4) nnnn ][lim )i()i( q(q(qq (D.5) and ][lim )i()i()i( q(q(q(qq nn2nn , (D.6) by deriving )i( q( n from (C.31) (once (74*) by using (C.27) and the correlation function (40)we obtain t)(G)(g,dt,dt, dt,dt,, t )t,q(t )t,q()t,q()t,q( t )t,q(t )t,q()t,q()t,q()t,q()t,q( ) (i (D.7) that for normalized states (localized particles) for which it holds , (D.8) since n is a deterministic solution, finally reads t)(G)(g, )t,q()t,q( (D.9) where the relations t)(G)(gdt,dt, tt t )t,q(tt t )t,q()t,q()t,q( ) (i (D.10) dtnnnn t ),t,q(t)(q,t)(q,t)(q,t)(q, (D.11)have been used. Therefore, we can write )},(),( ),(),({ })()( ))(({qq )t,q()t,q()t,q()t,q( )t,q()t,q()t,q()t,q( )t,q()t,q()t,q()t,q( )t,q()t,q()t,q()t,q( )))) )))) )))) )))) nnnn nnnnlim nnnn nnnnlimn,n
20 20 (D.12)
Moreover, since for uncorrelated noises on different spatial components we have ),(),( )t,q()t,q()t,q()t,q( nnnn (D.13)it follows that )](G[g )},(),( ),(),({ )},(),( ),(),({qq )i( )t,q()t,q()t,q()t,q( )t,q()t,q()t,q()t,q( )t,q()t,q()t,q()t,q( )t,q()t,q()t,q()t,q( )))) )))) )))) ))))
120 20 20 (D.14) where )i()i()i( ggg (D.15) t)(gt)(g,),(g )t,q()t,q()i( )) (i(i (D.16) )i( )t,q()t,q( g ),()(G )) (i1(i nn (D.17) Given that the WFM fluctuations at lowest order are approximately Gaussian, all higher momentsare functions of the moments of second order. By using the identity holding for a Gaussianrandom variables )( ],[ ,,, (D.18) we obtain nnnn2nnn,n ]qqq,q[ qqqq (D.19) Since the terms q n reads qdt)(q dtqdqq dtqqq dtqq )dtq t )t,q()t,q( t )t,q()t,q(t )t,q( t )t,q(t )t,q(t )t,q( nlimn limnn nn(nn (D.20) we obtain nnnn2nnn,n ]qqq,q[ qqqq (D.21) Given that the terms q n (that are independent by the fluctuations) are ineffective for thedivergence of the root mean square QP fluctuations, we can discharge it to obtain ]q,qAq,q[ qqqq nn2nn2nnn,n (D.22) with qqA nn (D.23) Therefore, at the smallest order in , we obtain )](G[gA })](G[g{ )i( )i(qqqq
120 22140 (D.24) where )(Gt)(g ),(g ),()(G )t,q()t,q()i( )t,q()t,q( )) (i1(i (D.25) Moreover, by using the identity t)(gt)(g,g )i( (D.26) finally (D.24) reads )](G)[(gtA })](G[)(gt{ qqqq
20 22240 (D.27)
Furthermore, writing down the second partial derivative in the discrete form as }{ )t,q()t,q()t,q(qq ))) (i1(i2(i nnnlimn (D.28) applying the condition of uncorrelated WFM fluctuations on different spatial components, thevariance of (D.28) reads } }{ } }-{ )t,q()t,q()t,q( )t,q()t,q()t,q( )t,q()t,q()t,q( )t,q()t,q()t,q(qqqq ))) ))) ))) )))
22 2240 22 240 (i1(i2(i (i1(i2(i (i1(i2(i (i1(i2(i nnn nn{nlim nnn n2nn{limnn, (D.29) that unwinding the terms inside (D.29), leads to (i1(i1(i2(i(i2(i (i(i1(i1(i2(i2(i ]g g)(G)(G[g }{ )i( )i()i( )t,q()t,q()t,q()t,q()t,q()t,q( )t,q()t,q()t,q()t,q()t,q()t,q(qqqq )))))) )))))) (D.30) where )i()i()i()i( gggg (D.31) )i( )t,q()t,q( g ),()(G )) (i2(i nn32 (D.32) (10.19)Moreover, by using the identities t)(gt)(gt)(gt)(g gggg )i()i()i()i(
032 0040 24limlim (D.33) )(Gt)(g ),(g ),()(G )t,q()t,q()i( )t,q()t,q( ))
20 nnnn3lim2lim (D.34) it follows that )](G)(G)[(gt qqqq (D.35) Furthermore, being that nn2nn{ n2nn nn2nn{ n2nn{lim nnn, (i(i1(i2(i (i1(i2(i (i(i1(i2(i (i1(i2(i
22 2240 }}-- }- }- )t,q()t,q()t,q()t,q( )t,q()t,q()t,q( )t,q()t,q()t,q()t,q( )t,q()t,q()t,q( qqqq )))) ))) )))) ))) (D.36) by grouping the terms and by using the property that the third moments of a Gaussian randomvariable are null, it follows that } )t,q(),t,q()t,q( )t,q(),t,q()t,q()t,q(),t,q( )t,q(),t,q()t,q(),t,q()t,q(),t,q( )t,q(),t,q()t,q(),t,q( qqqq ))) ))))) )))))) )))) (D.38) Therefore, in force of the above calculations, finally, the quantum potential variance reads ][)m(lim V,Vlim qqqqqqqqququ nnn,n-dnn,d2 (D.39) that by (D.27,D.35) leads to })](G[tA)(g))](G[(t)(g[ )](G)(G[t)(g{lim)m(V,Vlim ququ (D.40) where the (weighted) mean particle densities d , d , given by (??) are finite positive values (e.g.for n constant over all the space d = d = n )Moreover, given that c a)](G[lim (D.41) c)( a]G[lim (D.42) c)()( a]GG[lim (D.43)it follows that, at smallest order in k , ququ V,V goes like )k()(g*I*,IlimV,Vlim cququ (D.44)Moreover, given that *Ilim it also follows that )k()(g*Ilim*I*,Ilim c . Furthermore, given also that c *I*I )k()(g*Ilim*)I(lim cc and that )k()k()()k()k(lim )k()()k()k(lim *I*I)(*I*Ilim *I*I)(*I*Ilim *I*I)*I(*)I(lim )*I*I(lim*)F(lim q,q qcqcc qccqcc qccqcc qqqqqqqq qqqqq it follows that )Sp()Sp( )Sp(*)F(*)F(Nlim q),t,q(q),t,q( q),t,q(qq since )k( ),t,q( , and finally that )Sp(Nlim q),t,q( (D.46) that supports the approximated expression (37) from which it follows that *F . Appendix E If we consider a system (e.g., an ideal gas) at equilibrium in a fluctuating vacuumenvironment in a container of side L we have two way to calculate the system energyfluctuations. One is the standard one given by the statistical physics where the energyfluctuations are function of the temperature T, the second one is given by using the SQHAapproach that will furnish an expression function of the amplitude of the vacuum noise.Since the result must be same because independent by the way it is calculated, a connectionbetween the temperature of a gas at equilibrium in a vacuum environment and the amplitude of its fluctuations can be established.To this end, let’s start by calculating the energy fluctuations amplitude in the SQHAnotations pd...,pd...E,E nn HH (E.1)By using the identity )Sp( q(q,t)t),(q,p n (introducing (111) in (A.1) at zeroorder for small ) we obtain pd)p δ(p ...,pd)p δ(p ...E,E nqunqu HnHn (E.2) )tt(,n )tt(,,E,E
H)(3 HH HnHn ququ ququ (E.3)where the system volume has been assumed unitary and where n ququnqu mpppd)p δ(p ... qu . (E.4)is brought out the variance operation since is not function of the space being the quantumpotential of a pure sine or cosine wave function constant.Moreover, by introducing the notation k,, (E.5)where by (61) c , (E.6)we obtain H)(3 //// )k(,nVE,EVE ( E .7)where mp}mpp)n{( / n ququ
23 11 (E.8)and where, conceptually, = (t-t0) is the interval of time along which the vacuumfluctuations are observed. Since for a particle of an ideal gas the time for a free path can be at maximum the interval of time between two collisions with the vessel walls (e.g., a cube ofside L) it follows that the maximum time of fluctuations observation reads H2 // )m( Lmp Lm . (E.9)Moreover, by equating the energy fluctuations of n independent point mass particles of theideal gas at equilibrium [65], given by the formula kT)n(kT)C(E // v (E.10)with (E.7), it follows that kT)n()k(,nLE //// (E.11)where V is the volume of the system.The value of H depends by the state of the system and for T close to the absolute zero it canbe easily calculated.
Even if does not coincide with the thermodynamic temperature T, going toward theabsolute null temperature by steps of thermodynamic equilibrium, correspondingly, mustdecrease to zero since the systems fluctuations must vanish both as a function of as wellas a function of T. Therefore, in this case we expect that T lim (E.12)We suppose the gas sufficiently rarefied so that N/V is very small and the Bose-Einstaincondensation temperature TBE is smaller than that one of the probing ideal gas temperature.The value of (p)0T Hlim depends by the state of the system. When the limit of
0 isachieved at thermodynamic equilibrium and all the particles are in the fundamental state, atT=0 in a vessel of side L, it follows that Lm , (E.14)that Lm (E.15)and that kT)n()k(,LmLE //// (E.16).Moreover, given that by (E.6), as can be checked at the end, it holds k, c , (E.17)the energy fluctuations are linear both as a function of as well as of T . Therefore for anideal gas (E.12) coherently reads Tlim T (E.18)and hence, we can redefine by a proportionality constant to have Tlim T (E.19)Thence, if we measure in a scale such as = T , we obtain lim82 /// ,LmV)k( (E.20)that by re-writing (61) as c , (E.21)leads to mk L c . (E.22)Moreover, given that the term L has the dimension of a mobility, it follows that the term LL , (E.23)(E.24)has the dimension of a number and, hence, it follows that // )mk())(L( c . (E.25) Finally, in order to define the numerical value of )L( we need an additional information.Since the length c defines the maximum of quantum state delocalization, it is linked to theindetermination principle. It is matter of fact that smaller is the value of c larger are is the energyof vacuum fluctuations (and hence the connected momentum variance).Moreover, given that on distance shorter that c for any system it holds the wave-particleequivalence and hence we cannot perturb a part of it without disturbing all the system (non localbehavior), to perform a statistical measurement the system and the measuring apparatus must befar apart a distance larger that c . This fact influences both the time of measurement as well as itsprecision. Therefore the numerical value of )L( is determined by the experimental value ofthe physical uncertainty. This can be ascertained as follow: given that the time for traveling the distance c is c cc ,for performing a statistical measurement between quantum uncorrelated systems we need a time c t . Moreover, given the relativistic energy of a particle of mass m in presence of vacuumenergy fluctuations )( E such as kE )( , for the classical case ( kmc )we obtain
22 22222222
22 2 // // )kmc()Emc( ))mc(E)mc(())mc()Emc((E )( )()( from which, by imposing the uncertainty relation that reads hc )kmc(EtE cc / ,it follows that / )()L( .For that goes to zero (i.e., standard quantum mechanics) the measuring time goes to infinity andthe energy fluctuation E goes to zero (we have a perfect overall quantum system with exactlydefined energy levels). This makes clear that in a perfect quantum universe the measuring processis endless (i.e., not possible) confirming that the classical behavior is needed to the definition ofthe quantum mechanics based on the measuring process.The minimum uncertainty principle comes by the fact that below the length c the locality is lost(I cannot divide such a system into parts in order to improve my precision) united to the fact that Icannot collect information (ultimately to make a measure) in a time shorter than that one needed tothe interactions and information to travel such distance. By using (in the limit of small ) the convention that equals the temperature of an ideal gasat equilibrium (E.19) and by imposing that the physical uncertainty principle isverified, c remains defined by Formula (E.??). Moreover, by measuring the temperature of anideal gas at equilibrium in a fluctuating vacuum environment, we detect the value of of thetheory. Appendix F
Large-scale quantum potential
If we write the WFM in the form )]q(fexp[Mlim||lim )q(|q||q| / n , ( F .1)with the sufficiently general condition )q(flim |q| ( F .2) it follows that for a finite L condition (??) that implies that // qqq|q|quq|q| )m(limVlim ( F .3) leads to )]q(fexp[M)]q(fexp[Mlim )q(qq)q(q|q| . ( F .4) The condition (above) implies that the following differential equalities must contemporarily beverified ))q(f(lim qqq|q| ( F .5) ))q(f)q(f(lim qqq|q| ( F .6) ))q(M(Mlim qq)q(q|q| ( F .7) ))q(M)q(f(Mlim qq)q(q|q| ( F .8) The system of equations (F .5 , F .8 ) has no simple and immediate solutions in three dimensionalspace. For sake of simplicity, here we give the solution for the mono-dimensional case, since formany practical cases (e.g., the interaction of a couple of particles, as in a real gas or in a chain ofneighbors interacting atoms) this is still of great interest. In this case (F .5 , F .8 ) reads ])dq )q(df[(dqdlim |q| , ( F .9) )]dq )q(fd[(dqdlim |q| , ( F .10) )]dqdM)(dqdf(M[dqdlim )q()q()q(|q| , ( F .11) )]dqMd(M[dqdlim )q()q(|q| . ( F .12) If we approximate )q(f for large |q| by a polynomial expression of maximum degree h such as )q(h|q| P)q(flim ( F .13)we obtain h ( F .14)If we are interested in bounded or localized states (owing the property ||lim |q| ) , thatrequires |)q(f|lim |q| and hence that | h)q(h|q| q||P||)q(f|lim , ( F .15)it necessary follows that h . ( F .16)If we want to comprehend the case ttancons||lim |q| , the equality must added tocondition ( F .16.a), to obtain h ( F .17)As far as it concerns (14.4 -5), if we approximates )q( M for large |q| by a polynomial expressionsuch as n )q(pnnm)q(|q| ]iAexp[aqMlim ( F .18) where )q(pn A is a polynomial expression of maximum order p , conditions (F .9 , F .12 ) are contemporarily satisfied by m and p qAlim )q(pn|q| ( F .19) (as for L-J type potentials where the proportionality constant reads (2mE) ½ ) we have p = 1 and,hence, (14.1) is warranted just by the condition h ( F .20) Nomenclature References
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