Implications of a deeper level explanation of the deBroglie-Bohm version of quantum mechanics
Gerhard Groessing, Siegfried Fussy, Johannes Mesa Pascasio, Herbert Schwabl
aa r X i v : . [ qu a n t - ph ] D ec Implications of a deeper level explanation of the deBroglie–Bohmversion of quantum mechanics
Gerhard Grössing, ∗ Siegfried Fussy, ∗ Johannes Mesa Pascasio, ∗ and Herbert Schwabl ∗ Austrian Institute for Nonlinear Studies, AkademiehofFriedrichstr. 10, 1010 Vienna, Austria (Dated: September 6, 2018)
Abstract
Elements of a “deeper level” explanation of the deBroglie–Bohm (dBB) version of quantum me-chanics are presented. Our explanation is based on an analogy of quantum wave-particle dual-ity with bouncing droplets in an oscillating medium, the latter being identified as the vacuum’szero-point field. A hydrodynamic analogy of a similar type has recently come under criticism byRichardson et al . [1], because despite striking similarities at a phenomenological level the governingequations related to the force on the particle are evidently different for the hydrodynamic and thequantum descriptions, respectively. However, said differences are not relevant if a radically differentuse of said analogy is being made, thereby essentially referring to emergent processes in our model.If the latter are taken into account, one can show that the forces on the particles are identical inboth the dBB and our model. In particular, this identity results from an exact matching of ouremergent velocity field with the Bohmian “guiding equation”. One thus arrives at an explanationinvolving a deeper, i.e. subquantum, level of the dBB version of quantum mechanics. We show inparticular how the classically-local approach of the usual hydrodynamical modeling can be overcomeand how, as a consequence, the configuration-space version of dBB theory for N particles can becompletely substituted by a “superclassical” emergent dynamics of N particles in real 3-dimensionalspace. Keywords: quantum mechanics, hydrodynamics, deBroglie–Bohm theory, guiding equation, configurationspace, zero-point field ∗ . INTRODUCTION The Schrödinger equation for
N > particles does not describe a wave function in ordi-nary 3-dimensional space, but instead in an abstract N -dimensional space. For quantumrealists, including Schrödinger and Einstein, for example, this has always been considered as“indigestible”. This holds even more so for a realist, causal approach to quantum phenomenasuch as the deBroglie–Bohm (dBB) version of quantum mechanics. David Bohm himself hasadmitted this, calling it a “serious problem”: “While our theory can be extended formally ina logically consistent way by introducing the concept of a wave in a N -dimensional space,it is evident that this procedure is not really acceptable in a physical theory, and should atleast be regarded as an artifice that one uses provisionally until one obtains a better theoryin which everything is expressed once more in ordinary -dimensional space.” [2] (For moredetailed accounts of this discussion already in the early years of quantum mechanics, see [3]and [4].)In the present paper, we shall refer to our attempt towards such a “better theory” interms of a deeper level, i.e. subquantum, approach to the dBB theory, and thus to quantumtheory in general. In fact, with our model we have in a series of papers already obtainedseveral essential elements of nonrelativistic quantum theory [5–8]. They derive from theassumption that a particle of energy E = ~ ω is actually an oscillator of angular frequency ω phase-locked with the zero-point oscillations of the surrounding environment, the lattercontaining both regular and fluctuating components and being constrained by the boundaryconditions of the experimental setup via the buildup and maintenance of standing waves.The particle in this approach is an off-equilibrium steady state oscillation maintained bya constant throughput of energy provided by the (“classical“) zero-point energy field. Wehave, for example, applied the model to the case of interference at a double slit, therebyobtaining the exact quantum mechanical probability density distributions on a screen behindthe double slit, the average trajectories (which because of the averaging are shown to beidentical to the Bohmian ones), and the involved probability density currents. Our wholemodel is constructed in close analogy to the bouncing/walking droplets above the surface ofa vibrated liquid in the experiments first performed by Yves Couder, Emmanuel Fort andco-workers [9–11], which in many respects can serve as a classical prototype guiding ourintuition for the modeling of quantum systems.2owever, there are also obvious differences between the mentioned physics of classicalbouncers/walkers on the one hand, and the hydrodynamic-like models for quantum systemslike our own model or the dBB one on the other hand. In a recent paper, Richardson et al . [1] have probed more thoroughly into the hydrodynamic analogy of dBB-type quantumwave-particle duality with that of the classical bouncing droplets. Apart from the obviousdifference in that Bohmian theory is distinctly nonlocal, whereas droplet-surface interactionsare rooted in classical hydrodynamics and thus in a manifestly local theory, Richardson et al .focus on the following observation: the evidently different nature of the Bohmian force upona quantum particle as compared to the force that a surface wave exerts upon a droplet. Infact, wherever the probability density in the dBB picture is close to zero, the quantum forcebecomes singular and will very quickly push any particle away from that area. Conversely,the hydrodynamic force directs the droplet into the trough of the wave! So, the probabilityof finding a droplet in the minima never reaches zero as it does for a quantum particle.The authors conclude that these discrepancies between the two models highlight “a majordifference between the hydrodynamic force and the quantum force”. [1]Although these authors generally recover in numerical hydrodynamic simulations theresults of the Paris group (later confirmed also by the group of John Bush at MIT [12]) onsingle-slit diffraction and double-slit interference, they also point out the (already known)striking contrast between the trajectory behaviors for the bouncing droplet systems anddBB-type quantum mechanics, respectively. Whereas the latter exhibits the well-knownno-crossing property, the trajectories of the former do to a large extent cross each other.So, again, the physics in the two models is apparently fundamentally different, despite somestriking similarities on a phenomenological level. As to the differences, one may very wellexpect that they will even become more severe when moving from one-particle to N -particlesystems.So, all in all, the paper by Richardson et al . [1] cautions against the assumption of too closea resemblance of bouncer/walker systems and the hydrodynamic-like modeling of quantumsystems like the dBB one, with their main argument being that the hydrodynamic force on adroplet strikingly contrasts with the quantum force on a particle in the dBB theory. However,we shall here argue against the possible conclusion that one has thus reached the limits ofapplicability of the hydrodynamic bouncer analogy for quantum modeling. On the contrary,as we have already pointed out in previous papers, it is a more detailed model inspired by3he bouncer/walker experiments that can show the fertility of said analogy. It enables us toshow that our model, being of the type of an “emergent quantum mechanics” [13, 14], canprovide a deeper-level explanation of the dBB version of quantum mechanics (Chapter 2).Moreover, as we shall also show, it turns out to provide an identity of an emergent forceon the bouncer in our hydrodynamic-like model with the quantum force in Bohmian theory(Chapter 3). Finally, in Chapter 4 we shall discuss the “price” to be paid in order to arriveat our explanation of dBB theory in that some kind of nonlocality, or a certain “systemicnonlocality”, has to be admitted in the model from the start. However, the simplicity andelegance of our derived formalism, combined with arguments about the reasonableness of acorresponding hydrodynamic-like modeling, will show that our approach may be a viableone w.r.t. understanding the emergence of quantum phenomena from the interactions andcontextualities provided by the combined levels of classical boundary conditions and thoseof a subquantum domain.
2. IDENTITY OF THE EMERGENT KINEMATICS OF N BOUNCERS IN REAL -DIMENSIONAL SPACE WITH THE CONFIGURATION-SPACE VERSION OFDEBROGLIE–BOHM THEORY FOR N PARTICLES
Consider one particle in an n -slit system. In quantum mechanics, as well as in ourquantum-like modeling via an emergent quantum mechanics approach, one can write downa formula for the total intensity distribution P which is very similar to the classical formula.For the general case of n slits, it holds with phase differences ϕ ij = ϕ i − ϕ j that P = n X i =1 P i + n X j = i +1 R i R j cos ϕ ij ! , (2.1)where the phase differences are defined over the whole domain of the experimental setup.Apart from the role of the relative phase with important implications for the discussions onnonlocality [8], there is one additional ingredient that distinguishes (2.1) from its classicalcounterpart, namely the “dispersion of the wavepacket”. As in our model the “particle” isactually a bouncer in a fluctuating wave-like environment, i.e. analogously to the bouncersof Couder and Fort’s group, one does have some (e.g. Gaussian) distribution, with its centerfollowing the Ehrenfest trajectory in the free case, but one also has a diffusion to the rightand to the left of the mean path which is just due to that stochastic bouncing. Thus the4otal velocity field of our bouncer in its fluctuating environment is given by the sum ofthe forward velocity v and the respective diffusive velocities u L and u R to the left and theright. As for any direction i the diffusion velocity u i = D ∇ i PP does not necessarily fall offwith the distance, one has long effective tails of the distributions which contribute to thenonlocal nature of the interference phenomena [8]. In sum, one has three, distinct velocity(or current) channels per slit in an n -slit system.We have previously shown [15, 16] how one can derive the Bohmian guidance formulafrom our bouncer/walker model. To recapitulate, we recall the basics of that derivation here.Introducing classical wave amplitudes R ( w i ) and generalized velocity field vectors w i , whichstand for either a forward velocity v i or a diffusive velocity u i in the direction transversal to v i , we account for the phase-dependent amplitude contributions of the total system’s wavefield projected on one channel’s amplitude R ( w i ) at the point ( x , t ) in the following way:We define a conditional probability density P ( w i ) as the local wave intensity P ( w i ) in onechannel (i.e. w i ) upon the condition that the totality of the superposing waves is given bythe “rest” of the n − channels (recalling that there are 3 velocity channels per slit). Theexpression for P ( w i ) represents what we have termed “relational causality”: any change inthe local intensity affects the total field, and vice versa , any change in the total field affectsthe local one. In an n -slit system, we thus obtain for the conditional probability densitiesand the corresponding currents, respectively, i.e. for each channel component i , P ( w i ) = R ( w i ) ˆw i · n X j =1 ˆw j R ( w j ) (2.2) J ( w i ) = w i P ( w i ) , i = 1 , . . . , n, (2.3)with cos ϕ i,j := ˆw i · ˆw j . (2.4)Consequently, the total intensity and current of our field read as P tot = n X i =1 P ( w i ) = n X i =1 ˆw i R ( w i ) ! (2.5) J tot = n X i =1 J ( w i ) = n X i =1 w i P ( w i ) , (2.6)5eading to the emergent total velocity v tot = J tot P tot = n X i =1 w i P ( w i ) n X i =1 P ( w i ) . (2.7)In [15, 16] we have shown with the example of n = 2 , i.e. a double slit system, thatEq. (2.7) can equivalently be written in the form v tot = R v + R v + R R ( v + v ) cos ϕ + R R ( u − u ) sin ϕR + R + 2 R R cos ϕ . (2.8)The trajectories or streamlines, respectively, are obtained according to ˙x = v tot in theusual way by integration. As first shown in [7], by re-inserting the expressions for convectiveand diffusive velocities, respectively, i.e. v i = ∇ S i m , u i = − ~ m ∇ R i R i , one immediately iden-tifies Eq. (2.8) with the Bohmian guidance formula. Naturally, employing the Madelungtransformation for each path j ( j = 1 or ), ψ j = R j e i S j / ~ , (2.9)and thus P j = R j = | ψ j | = ψ ∗ j ψ j , with ϕ = ( S − S ) / ~ , and recalling the usual trigono-metric identities such as cos ϕ = (cid:0) e i ϕ + e − i ϕ (cid:1) , one can rewrite the total average currentimmediately in the usual quantum mechanical form as J tot = P tot v tot = ( ψ + ψ ) ∗ ( ψ + ψ ) 12 (cid:20) m (cid:18) − i ~ ∇ ( ψ + ψ )( ψ + ψ ) (cid:19) + 1 m (cid:18) i ~ ∇ ( ψ + ψ ) ∗ ( ψ + ψ ) ∗ (cid:19)(cid:21) = − i ~ m [Ψ ∗ ∇ Ψ − Ψ ∇ Ψ ∗ ] = 1 m Re { Ψ ∗ ( − i ~ ∇ )Ψ } , (2.10)where P tot = | ψ + ψ | =: | Ψ | .Eq. (2.7) has been derived for one particle in an n -slit system. However, it is straightfor-ward to extend this derivation to the many-particle case. Due to the purely additive termsin the expressions for the total current and total probability density, respectively, also for N particles, the only difference now is that the currents’ nabla operators have to be appliedat all of the locations of the respective N particles, thus providing the quantum mechanicalformula J tot ( N ) = N X i =1 m i Re { Ψ ( t ) ∗ ( − i ~ ∇ i )Ψ ( t ) } , (2.11)6here Ψ ( t ) now is the total N -particle wave function, whereas the total velocity fields aregiven by v i ( t ) = ~ m i Im ∇ i Ψ ( t )Ψ ( t ) ∀ i = 1 , ..., N. (2.12)Note that this result is similar in spirit to that of Norsen et al. [3, 4] who with theintroduction of a conditional wave function ˜ ψ i , as opposed to the configuration-space wavefunction Ψ , rewrite the guidance formula, for each particle, in terms of the ˜ ψ i : d X i ( t )d t = ~ m i Im ∇ ΨΨ (cid:12)(cid:12)(cid:12)(cid:12) x = X ( t ) ≡ ~ m i Im ∇ ˜ ψ i ˜ ψ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = X i ( t ) , (2.13)where the X i denote the location of one specific particle and X ( t ) = { X ( t ) , ..., X N ( t ) } the actual configuration point. Thus, in this approach, each ˜ ψ i can be regarded as a wavepropagating in physical 3-dimensional space.In sum, with our introduction of a conditional probability P ( w i ) for channels w i , whichinclude subquantum velocity fields, we obtain the guidance formula also for N -particle sys-tems. Therefore, what looks like the necessity in the dBB theory to superpose wave func-tions in configuration space in order to provide an “indigestible” guiding wave, can equallybe obtained by superpositions of all relational amplitude configurations of waves in real3-dimensional space.
The central ingredient for this to be possible is to consider the emer-gence of the velocity field from the interplay of the totality of all of the system’s velocitychannels. We have termed the framework of our approach a “superclassical” one, becausein it are combined classical levels at vastly different scales, i.e. at the subquantum and themacroscopic levels, respectively. 7 . IDENTITY OF THE EMERGENT FORCE ON A PARTICLE MODELED BYA BOUNCER SYSTEM AND THE QUANTUM FORCE OF THE DEBROGLIE–BOHM THEORY
With the results of the foregoing Chapter, we can now return to and resolve the problemdiscussed in Chapter 1 of the apparent incompatibility between the Bohmian force upon aquantum particle and the force exerted on a bouncing droplet as formulated by Richardson et al . [1]. In fact, already a first look at the bouncer/walker model of our group provides aclear difference as compared to the hydrodynamical force studied by Richardson et al . For,whereas the latter investigate the effects of essentially a single bounce on the fluid surfaceand the acceleration of the bouncer as a consequence of this interaction, our bouncer/walkermodel for quantum particles involves a much more complex dynamical scenario: We considerthe effects of a huge number of bounces, i.e. typically of the order of / ω , like approximately bounces per second of an electron, which constitute effectively a “heating up” of thebouncer’s surrounding, i.e. the subquantum medium related to the zero-point energy field.Note that as soon as a microdynamics is assumed, the development of heat fluxes is alogical necessity if the microdynamics is constrained by some macroscopic boundaries likethat of a slit system, for example. As we have shown in some detail [17], the thermal fieldcreated by such a huge number of bounces in a slit system leads to an emergent averagebehavior of particle trajectories which is identified as anomalous, and more specifically asballistic, diffusion. As such, the particle trajectories exiting from, say, a Gaussian slit behaveexactly as if they were subject to a Bohmian quantum force. We were also able to showthat this applies also to n -slit systems, such that one arrives at a subquantum modeling ofthe emergent interference effects at n slits whose predicted average behavior is identical tothat provided by the dBB theory.It is then easily shown that the average force acting on a particle in our model is thesame as the Bohmian quantum force. For, due to the identity of our emerging velocity fieldwith the guidance formula, and because they essentially differ only via the notations dueto different forms of bookkeeping, their respective time derivatives must also be identical.Thus, from Eq. (2.7) one obtains the particle acceleration field (using a one-particle scenario8or simplicity) in an n -slit system as a tot ( t ) = d v tot d t = dd t n X i =1 w i P ( w i ) n X i =1 P ( w i ) = 1 n X i =1 P ( w i ) ! n X i =1 (cid:20) P ( w i ) d w i d t + w i d P ( w i )d t (cid:21) n X i =1 P ( w i ) ! (3.1) − n X i =1 w i P ( w i ) ! n X i =1 d P ( w i )d t ! . Note in particular that (3.1) typically becomes infinite for regions ( x , t ) where P tot = n X i =1 P ( w i ) → , in accordance with the Bohmian picture.From (3.1) we see that even the acceleration of one particle in an n -slit system is a highlycomplex affair, as it nonlocally depends on all other accelerations and temporal changes inthe probability densities across the whole experimental setup! In other words, this force istruly emergent, resulting from a huge amount of bouncer-medium interactions, both locallyand nonlocally. This is of course radically different from the scenario studied by Richardson et al . where the effect of only a single local bounce is compared with the quantum force.From our new perspective, it is then hardly a surprise that the comparison of the tworespective forces provides distinctive differences. However, as we just showed, with theemergent scenario proposed in our model, complete agreement with the Bohmian quantumforce is established.
4. CHOOSE YOUR POISON: HOW TO INTRODUCE NONLOCALITY IN AHYDRODYNAMIC-LIKE MODEL FOR QUANTUM SYSTEMS?
As already mentioned in the introduction of this paper, purely classical hydrodynami-cal models are manifestly local and thus inadequate tools to explain quantum mechanicalnonlocality. Although nonlocal correlations may also be obtainable within hydrodynamicalmodeling [18], there is no way to also account for dynamical nonlocality [19] in this manner.9o, as correctly observed by Richardson et al. [1], droplet-surface wave interaction scenariosare not enough to serve as a full-fledged analogy of the distinctly nonlocal dBB theory, forexample.The question thus arises how in our much more complex, but still “hydrodynamic-like”bouncer/walker model nonlocal, or nonlocal-like, effects can come about. To answer this,one needs to consider in more detail how the elements of our model are constructed, whichfinally provide an elegant formula, Eq. (2.7), identical with the guidance formula in a (forsimplicity: one-particle) system with n slits. (As shown above, the extension to N particlesis straightforward.) As we consider, without restriction of generality, the typical example ofGaussian slits, we introduce the Gaussians in the usual way, with σ related to the slit width,for the probability density distributions (which in our model coincide with “heat” distribu-tions due to the bouncers’ stirring up of the vacuum) just behind the slit. The importantfeature of these Gaussians is that we do not implement any cutoff for the distributions, butmaintain the long tails which actually then extend across the whole experimental setup,even if these are only very small and practically often negligible amplitudes in the regionsfar away from the slit proper. As the emerging probability density current is given by thedenominator of Eq. (2.8), we see that in fact the product R R may be negligibly small forregions where only a long tail of one Gaussian overlaps with another Gaussian, neverthelessthe last term in (2.8) can be very large despite the smallness of R or R . It is this latterpart which is responsible for the genuinely quantum-like nature of the average momentum,i.e. for its nonlocal nature. This is similar in the Bohmian picture, but here given a moredirect physical meaning in that this last term refers to a difference in diffusive currents asexplicitly formulated in the last term of (2.8). Because of the mixing of diffusion currentsfrom both channels, we call this decisive term in J tot = P tot v tot the “entangling current”.[20]Thus, one sees that formally one obtains genuine quantum mechanical nonlocality in ahydrodynamic-like model with one particular “unusual” characteristic: the extremely feeblebut long tails of (Gaussian or other) distribution functions for probability densities exitingfrom a slit extend nonlocally across the whole experimental setup. So, we have nonlocalityby explicitly putting it into our model. After all, if the world is nonlocal, it would not makemuch sense to attempt its reconstruction with purely local means. Still, so far we have juststated a formal reason for how nonlocality may come about. Somewhere in any theory, so10t seems, one has to “choose one’s poison” that would provide nonlocality in the end. Butwhat would be a truly “digestible” physical explanation? Here is where at present only somespeculative clues can be given.For one thing, strict nonlocality in the sense of absolute simultaneity of space-like sep-arated events can never be proven in any realistic experiment, because infinite precision isnot attainable. This means, however, that very short time lapses must be admitted in anyoperational scenario, with two basic options remaining: i) either there is a time lapse due tothe finitely fast “switching” of experimental arrangements in combination with instantaneousinformation transfer (but not signaling; see Walleczek and Grössing, [forthcoming]), or ii)the information transfer itself is not instantaneous, but happens at very high speeds v ≫ c .How, then, can the implementation of nonlocal or nonlocal-like processes with speeds v ≫ c be argued for in the context of a hydrodynamic-like bouncer/walker model? Webriefly mention two options here. Firstly, one can imagine that the “medium” we employin our model is characterized by oscillations of the zero-point energy throughout space, i.e.between any two or more macroscopic boundaries as given by experimental setups. Betweenthese boundaries standing wave configurations may emerge (similar to the Paris group’sexperiments, but now explicitly extending synchronously over nonlocal distances). Hereit might be helpful to remind ourselves that we deal with solutions of the diffusion (heatconduction) equation. At least (but perhaps only) formally, any change of the boundaryconditions is effective “instantaneously” across the whole setup. Alternatively, if the experi-mental setup is changed such that old boundary conditions are substituted by new ones, dueto the all-space-pervading zero-point energy oscillations, one “immediately” (i.e. after a veryshort time of the order t ∼ ω ) obtains a new standing wave configuration that now effectivelyimplies an almost instantaneous change of probability density distributions, or relative phasechanges, for example. The latter would then become “immediately” effective in that changedphase information is available across the whole domain of the extended probability densitydistribution. We have referred to this state of affairs as “systemic nonlocality” [8]. So, onemay speculate that it is something like “eigenvalues” of the universe’s network of zero-pointfluctuations that may be responsible for quantum mechanical nonlocality-eigenvalues which(almost?) instantaneously change whenever the boundary conditions are changed.A second option even more explicitly refers to the universe as a whole, or, more partic-ularly, to spacetime itself. If spacetime is an emergent phenomenon as some recent work11uggests [21], this would very likely have strong implications for the modeling and under-standing of quantum phenomena. Just as in our model of an emergent quantum mechanicswe consider quantum theory as a possible limiting case of a deeper level theory, present-dayrelativity and concepts of spacetime may be approximations of, and emergent from a super-classical, deeper level theory of gravity and/or spacetime. It is thus a potentially fruitfultask to bring both attempts together in the near future. ACKNOWLEDGMENTS
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