Sub-Quantum Thermodynamics as a Basis of Emergent Quantum Mechanics

Abstract

This review presents results obtained from our group's approach to model quantum mechanics with the aid of nonequilibrium thermodynamics. As has been shown, the exact Schrödinger equation can be derived by assuming that a particle of energy E equal to the reduced Planck constant times its angular frequency is actually a dissipative system maintained in a nonequilibrium steady state by a constant throughput of energy (heat flow). Here, also other typical quantum mechanical features are discussed and shown to be completely understandable within our approach, i.e., on the basis of the assumed sub-quantum thermodynamics. In particular, Planck's relation for the energy of a particle, the Heisenberg uncertainty relations, the quantum mechanical superposition principle and Born's rule, or the "dispersion of the Gaussian wave packet", a.o., are all explained on the basis of purely classical physics.

Keywords: nonequilibrium thermodynamics; dissipative systems; diffusion wave fields; quantum mechanics; Schrödinger equation; uncertainty relations; superposition principle; Born's rule.

PACS codes: 03.65.-w, 03.65.Ta, 05.40.-a, 05.70.Ln

1. Introduction

Considering a theory as emergent if it "contains or reduces to another theory in a significant manner or if its laws are tied to those of another theory via mathematical connections", it is proposed that quantum mechanics is such a theory. More precisely, it is proposed that quantum theory emerges from a deeper, more exact theory on a sub-quantum level. In our approach, one assumes that the latter can be described with the aid of nonequilibrium thermodynamics. We ask ourselves how quantum theory would have evolved, had the "tool" of modern nonequilibrium thermodynamics existed, say, a century ago. As has recently been shown, one can derive the exact Schrödinger equation with said tool, where the relation between energy E and frequency, respectively, is used as the only empirical input, with the additional option that even the appearance of Planck's constant may have its origin in classical physics.

In the present review, we shall more generally summarize the results of our works relating to the derivation from purely classical physics of the following quantum mechanical features:

• Planck's relation for the energy of a particle,
• the Schrödinger equation for conservative and non-conservative systems,
• the Heisenberg uncertainty relations,
• the quantum mechanical superposition principle,
• Born's rule, and
• the quantum mechanical "decay of a Gaussian wave packet".

Moreover, the energy spectrum of a quantum mechanical harmonical oscillator is derived classically, as well as that of a "particle in a box", the latter thereby providing both a resolution of an objection by Einstein, and a clarification with respect to the differences between the de Broglie-Bohm interpretation and the present approach, respectively.

Further, it will be proven that free quantum motion exactly equals sub-quantum anomalous (i.e., "ballistic") diffusion, and, via computer simulations with coupled map lattices, it will be shown how to calculate averaged (Bohmian) trajectories purely from a real-valued classical model. This is illustrated with the cases of the dispersion of a Gaussian wave packet, both for free quantum motion and for motion in a linear (e.g., gravitational) potential. The results are shown to be in excellent agreement with analytical expressions as they are obtained both via our approach, and also via the Bohmian theory. However, in the context of the explanation of Gaussian wave packet dispersion, quantitative statements on the trajectories' characteristic behavior are presented, which cannot be formulated in any other existing model for quantum systems. Finally, an outlook is provided on some of the possible next steps of our thus presented research program.

As is well known, the main features of quantum mechanics, like the Schrödinger equation, for example, have only been postulated, but never derived from some basic principles. (Cf. Murray Gell-Mann: "Quantum mechanics is not a theory, but rather a framework within which we believe any correct theory must fit.") Even in causal interpretations of the quantum mechanical formalism, such as the de Broglie-Bohm theory, the quantum mechanical wave function, or the solution of the Schrödinger equation, respectively, is taken as input to the theory (sometimes even as a "real" ontological field), without further explanation of why this should have to be so. Still, the Bohmian approach has brought some essential insight into the nature of quantum systems, particularly by exploiting the physics of the "guiding equation" (in what is called "Bohmian mechanics") or, respectively, by providing a detailed analysis of the "quantum potential". The latter was shown, in the context of the Hamilton-Jacobi theory, to represent the only difference to the dynamics of classical systems.

However, in 1965, Edward Nelson suggested a derivation of the Schrödinger equation from classical, Newtonian mechanics via the introduction of a new differential calculus. Thus it was possible to show, e.g., that the quantum potential can be understood as resulting from an underlying stochastic mechanics, thereby referring to a hypothesized sub-quantum level. However, ambiguities within said calculus, e.g., as to the formula for the mean acceleration, as well as an apparent impossibility to cope with quantum mechanical nonlocality (which had become rather firmly established in the meantime) has led to a temporary decline of interest in stochastic mechanics. Still, it is legitimate to enquire also today whether the stochastic mechanics envisioned is not just one part of a necessarily larger picture, with the other part or parts of it yet to be established.

Considering the history of quantum mechanics, for example, with its many differences in emphasizing particle and wave aspects of quantum systems, one must concede that in general the particle framework was the dominant one throughout the twentieth century. (Cf., as a representative example, Richard Feynman: "It is very important to know that light behaves like particles, especially for those of you who have gone to school, where you were probably told about light behaving like waves. I'm telling you the way it does behave — like particles.") However, a purely particle-centered approach may not be enough, as the quantum phenomena to be explained may just be more complex than to be reducible to a one-level point-particle mechanics only. In other words, it is possible that by the attempts to reduce quantum dynamics to simple point-by-point interactions, the phenomenon to be discussed would remain without reach, because it is too complex to be described on just one (i.e., an assumed "basic") level. In still other words, a quantum system may be an emergent phenomenon, where a stochastic point-mechanics on just one level of description would still be a necessary ingredient for its description, but not the only relevant one. So, there may exist two or more relevant levels (e.g., on different time and/or spatial scales), where only the combination, or interactions, of them would result in the possibility to completely describe quantum systems. The latter may thus be more complex than it is assumed in any one-level stochastic mechanics model. In fact, recent results from classical physics suggest that this more complex scenario is even highly probable, since the said new results exhibit phenomena which previously were considered to be possible exclusively as quantum phenomena.

One is here reminded of Feynman's famous discussion of the double slit, and his introductory remark: "We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way and has in it the heart of quantum mechanics. In reality, it contains the only mystery." However, the above-mentioned recent classical physics experiments not only disprove Feynman's statement with respect to the double slit, but prove that a whole set of "quantum" features can be shown to occur in completely classical ones, among them being the Heisenberg uncertainty principle, indeterministic behaviour of a particle despite a deterministic evolution of its statistical ensemble over many runs, nonlocal interaction, tunnelling, and, of course, a combination of all these. I am referring to the beautiful series of experiments performed by the group of Yves Couder using small liquid drops that can be kept bouncing on the surface of a bath of the same fluid for an unlimited time when the substrate oscillates vertically. These "bouncers" can become coupled to the surface waves they generate and thus become "walkers" moving at constant velocity on the liquid surface. A "walker" is defined by a lock-in phenomenon so that the drop falls systematically on the forward front of the wave generated by its previous bouncings. It is thus a "symbiotic" dynamical phenomenon consisting of the moving droplet dressed with the Faraday wave packet it emits. Couder and Fort report on single-particle diffraction and interference of walkers. They show "how this wavelike behaviour of particle trajectories can result from the feedback of a remote sensing of the surrounding world by the waves they emit". Of course, the "walkers" of Couder's group, despite showing so many features they have in common with quantum systems, cannot be employed one-to-one as a model for the latter, with the most obvious difference being that quantum systems are not restricted to two-dimensional surfaces. However, along with the understanding of how the Schrödinger equation can be derived via nonequilibrium thermodynamics, also the mutual relationship of particle and wave behaviour has become clearer. Just as in the experiments with walkers, there exists an average orthogonality also for particle trajectories and wave fronts in the quantum case. This is going to be of central importance for our modelling of quantum mechanics with the aid of an assumed sub-quantum thermodynamics.

In the remainder of this introduction, a first sketch shall be given of how the said modelling can be carried out. At first, and foremost, we observe that the so-called "vacuum" unambiguously turned out during the twentieth century to be permeated by what is generally called the "zero-point energy", or "zero-point fluctuations", respectively, i.e., by a residual field in any accessible spacetime volume, even as the temperature T goes toward zero. Also, any particle of nature has turned out to be characterized by a fundamental angular frequency (i.e., in its rest frame) such that its total energy E is described by Planck's relation, E equals the reduced Planck constant times that angular frequency.

So, if there were only one particle in the world, the totality of such a universe would consist of an oscillator (i.e., our "particle" characterized by its angular frequency) and its environment, i.e., the all-pervasive field of the zero-point fluctuations. Therefore, to start with, we shall investigate possible models for the dynamics between the said oscillator and the said vacuum fluctuations. From a thermodynamical viewpoint, then, it is clear that one will in general have to employ a nonequilibrium scenario. For, any oscillating system in nature is the result of a dissipative process, so that the mentioned frequency can be understood as belonging to the properties of driven, off-equilibrium steady-state systems maintained by a permanent throughput of (kinetic) energy. Of course, for our single oscillator, the said energy must come from the zero-point energy, or, more precisely, the energy throughput for the maintenance of that frequency is determined, respectively, by the absorption from and the dissipation into the zero-point energy field of the particle's environment.

In this context, it is helpful to return again to the above-mentioned experiments of Couder's group. To guide our imagination, the following analogy may be considered. As the "bouncer" in the said experiments both oscillates due to being driven by the surrounding waves and causes the emission of radially symmetrical waves into the environment, with a self-sustained phase-locking mechanism guaranteeing coherent oscillations of bouncer and surrounding medium, respectively, one may in a first approximation also consider the frequency of a quantum as representing a similarly emergent "symbiotic" process. In other words, we consider a quantum "particle" as a dissipative phase-locked steady state, where an amount of zero-point energy of the wave-like environment is absorbed by the particle, and then, during a characteristic relaxation time given by the inverse of that frequency, dissipated into the environment again. In the simplest scenario of a universe with only one particle in it, this dissipation will occur radially-symmetrically, thus creating a (thermal) wave, or rather, maintaining the zero-point energy's wave-like structure through the phase-locking.

In what follows it will be shown that if one considers such a corresponding free quantum particle along a single path, its description cannot be distinguished from that of a classical particle. That is, in this case the quantum potential will vanish identically. The said vanishing quantum potential is then proven to be exactly equivalent to a classical heat (diffusion) equation, the solutions of which are given as radially symmetric thermal diffusion wave fields. In other words, a non-vanishing quantum potential is thus generally an expression of a particle's surrounding diffusion wave field scenario when that radial symmetry is broken by "something else" in the thus established more complex universe, i.e., in the cases when the particle is not free (or not facing a single, possible path, respectively). The gradient of the quantum potential will then be described as a completely "thermalized" fluctuating force field, where the origin of the latter is exactly identical to the zero-point fluctuation field.

So, our modelling approach will consist of two basic steps. In a first step, we just consider the simple "one particle in the universe" scenario, with the particle as the origin of thermal waves in synchrony with the surrounding medium. (Note that these thermal waves themselves could be considered as emerging from millions of millions of individual sub-quantum Brownian motions, but we are here interested only in the emergent waves and their possible interactions with others.)

In a second step, we consider the particle's environment to be more realistic, i.e., the simple regular zero-point energy oscillations will then have to be substituted by oscillations within constraints, as, e.g., given by an experimental setup in which our particle is embedded. Then, even a single particle may not be considered as being free in general. For example, even the description of a source of quantum particles represents constraints on an otherwise unconstrained zero-point energy environment. Representing an initial particle distribution in some experimental setup by a Gaussian, for example, already implies that the heat of the zero-point field will be "squeezed" (i.e., within the limits of a "Gaussian slit", for example). This squeezing, then, is equivalent to a non-vanishing fluctuating force on the particle. In other words, the first step in our modelling procedure basically refers to an oscillator and a radially symmetric diffusion wave field. The second step, however, will have to employ a stochastic element in order to account for the interactions of many different (though phase-locked) diffusion waves, thus referring directly to the zero-point fluctuations of the embedding environment and the effective Brownian-type "jumps" of our bouncer within the given geometric (and "vacuum compressing") constraints of the setup.

In the following chapters, we shall employ this two-steps strategy twice. To begin with, we shall concentrate on the question of the appearance of Planck's constant in a classical context, i.e., in a simple "driven harmonic oscillator" scenario, and then move to include a stochastic level, thus referring also to a fluctuating environment. Later, when concretely modelling the quantum mechanical dispersion of a Gaussian wave packet with classical means, we shall again start with the simple scenario of undisturbed diffusion waves, only to be modified in a second step to include the more realistic stochasticity of the processes involved. We shall then see that the model exactly reproduces the quantum mechanical results.

2. The "Walking Bouncer": A Classical Explanation of Quantization

2.1. Introduction

The Schrödinger equation was derived in the context of modelling quantum systems via nonequilibrium thermodynamics, i.e., by the requirement that the dissipation function, or the time-averaged work over the system of interest, vanish identically. The "system of interest" is a "particle" in terms of a harmonic oscillator embedded in a thermal environment of non-zero average temperature (i.e., of the "vacuum"). In more recent papers, we have illustrated the "particle" more concretely by using the concept of a "bouncer" (or "walker", respectively) gleaned from the beautiful experiments by Couder's group. Thus we assume that the thermal environment is oscillating itself, with the kinetic energy of these latter oscillations providing the energy necessary for the "particle" to maintain a constant energy, i.e., to remain in a nonequilibrium steady state. Regarding the respective ("zero-point") oscillations of the vacuum, we simply assume the particle oscillator to be embedded in an environment comprising a corresponding energy bath.

(Sections 2.2 and 2.3, which treat the classical oscillator driven by its environment's energy bath — the "bouncer" — and the Brownian motion of a particle — the "walker" — are omitted for length; the complete text is at the source.)

2.4. The "Walking Bouncer": Derivation of Planck's relation

Let us summarize what we have achieved so far. We have for both systems, i.e., oscillator and particle in Brownian-type motion (or "bouncer" and "walker", respectively), obtained a net work-energy flow into each system, respectively, in order to compensate for the respective energy losses due to friction. There is a continuous flow from the bath to the oscillator, and vice versa. Moreover, and most importantly, during that flow, for long enough times, the friction of the bouncer can be assumed to be exactly identical with the friction of the walker. For this reason we directly compare the results for the work done on each.

Now, one generally has that the total energy of a sinusoidal oscillator exactly equals twice its average kinetic energy. Moreover, despite having a nonequilibrium framework of our system, the fact that we deal with a steady state means that our oscillator is in local thermal equilibrium with its environment. As the average kinetic energy of the latter is always given by one half of the Boltzmann constant times the temperature, one obtains for the total energy exactly the usual quantum mechanical expression: the total energy equals the reduced Planck constant times the oscillator's angular frequency.

Equation (2.4.13) provides an "entropic" view of a harmonic oscillator in its thermal bath. First, the total energy of a simple harmonic oscillator is given as one half of the reduced Planck constant times its frequency. Now, the average kinetic energy of a harmonic oscillator is given by half of its total energy — which, because of the local equilibrium, is both the average kinetic energy of the bath and that of the "bouncer" particle. As the latter during one oscillation varies between zero and its maximum, one has the following entropic scenario. When it is minimal, the tendency towards maximal entropy will provide an entropic force equivalent to the absorption of a heat quantity of one quarter of the reduced Planck constant times the frequency. Similarly, when it is maximal, the same tendency will now enforce that the same heat quantity is given off again to the "thermostat" of the thermal bath. In sum, then, the total energy throughput along a full circle will equal the reduced Planck constant times the frequency. In other words, Planck's formula does not refer to a classical "object" oscillating with that frequency, but rather to a process of a "fleeting constancy": due to entropic requirements, the energy exchange between bouncer and heat bath will constantly consist of absorbing and emitting heat quantities such that in sum the "total particle energy" emerges.

(Section 2.5, deriving the energy spectrum of the harmonic oscillator from classical physics, is omitted for length; the complete text is at the source.)

3. Derivation of the Exact Schrödinger Equation from Classical Physics

3.1. Introduction

Based on the results of the foregoing chapter, it already follows from classical physics that to each particle of nature one associates an energy equal to the reduced Planck constant times its angular frequency. As it is well known that oscillations in general are the result of dissipative processes, the frequencies can be understood within the framework of nonequilibrium thermodynamics, or, more precisely, as properties of off-equilibrium steady-state systems maintained by a permanent throughput of energy from the environment.

So, we deal here with a "hidden" thermodynamics, out of which the known features of quantum theory should emerge. (This says, among other things, that we do not occupy ourselves here with the usual quantum versions of thermodynamics, out of which classical thermodynamics is assumed to emerge, since we intend to deal with a level "below" that of quantum theory, to begin with.)

Of course, there is a priori no guarantee that nonequilibrium thermodynamics is in fact operative on the level of a hypothetical sub-quantum "medium", but, as will be shown here, the straightforwardness and simplicity of how the exact central features of quantum theory emerge from this ansatz will speak for themselves. Moreover, one can even reverse the doubter's questions and ask for compelling reasons, once one does assume the existence of some sub-quantum domain with real physics going on in it, why this medium should not obey the known laws of, say, statistical mechanics. For, one also has to bear in mind, a number of physical systems exhibit very similar, if not identical, behaviours at vastly different length scales. For example, the laws of hydrodynamics are successfully applied even to the largest structures in the known universe, as well as on scales of kilometres, or centimetres, or even in the collective behaviour of quantum systems. In short, although there is no a priori guarantee of success, there is also no principle that could prevent us from applying present-day thermodynamics to the sub-quantum regime.

What is proposed here can also be considered as a gedanken experiment: what if our knowledge of classical physics (including wave mechanics and nonequilibrium thermodynamics) of today had been available 100 years ago? The answer is as follows: One could have thus, without any further assumptions or any ad hoc choices of constants, derived the exact Schrödinger equation, both for conservative and non-conservative systems, using only universal properties of oscillators and nonequilibrium thermostatting. It is particularly the latter feature which is rather appealing, since the use of universality properties guarantees model independence. That is, it will turn out unnecessary to have much knowledge about the detailed sub-quantum mechanisms, as the universal properties of the systems in question will be shown to suffice to obtain the results looked for. Moreover, the approach to be presented here not only re-produces the Schrödinger equation, but also puts forward some new results, such as the sub-quantum fluctuation theorem, which can thus help shed light on problems not properly understood today within the known quantum formalism.

In section 2 of this chapter, a short review is given of some results from nonequilibrium thermodynamics, which are particularly useful for our purposes. Section 3 then presents the application of the corresponding sub-quantum modelling of conservative systems, thus providing a straightforward derivation of the Schrödinger equation from modern classical physics. It is claimed that this represents the only exact derivation of the Schrödinger equation from classical physics in the literature. In section 4, then, the scheme is extended to include the Schrödinger equation for integrable non-conservative systems. Finally, the more encompassing scope of the present approach is presented, culminating in a formulation and discussion of the "vacuum fluctuation theorem", with particular emphasis being put on possible applications for a better understanding of quantum mechanical nonlocality.

(Sections 3.2, 3.3 and 3.4 — the results from nonequilibrium thermodynamics, the basic assumptions, the derivation itself, and the extension to integrable non-conservative systems and the vacuum fluctuation theorem — are omitted for length; the complete text is at the source.)

4. Derivation of the Heisenberg Uncertainty Relations

We have seen that the velocity fluctuation must be added to the classical velocity to obtain the total velocity of the "particle immersed in the zero-point field". So, if for the time being we assume that our knowledge of the particle's momentum is given to one part by the classical momentum, we can consider the latter to be "smeared" by the presence of this "osmotic" velocity term such that the uncertainty in the particle's momentum is then given by the average root-mean-square momentum fluctuation. Now we recall that a classical measure of minimal position uncertainty is given by the "Fisher length". Comparing the two immediately provides an "exact uncertainty relation" which has been proposed by Hall and Reginatto.

(The remainder of Chapter 4 is omitted for length; the complete text is at the source.)

5. Thermodynamic Origin of the Quantum Potential

5.1. The Case of a Vanishing Quantum Potential: Equivalence with the Classical Heat Equation

The energetic scenario of a steady-state oscillator in nonequilibrium thermodynamics is given by a throughput of heat, i.e., a kinetic energy at the sub-quantum level providing, first, the necessary energy to maintain a constant oscillation frequency, and second, some excess kinetic energy resulting in a fluctuating momentum contribution to the momentum of the particle. From a perspective out of everyday life, one can compare this to the situation of some small convex half-sphere, say, lying on a flat vibrating membrane. Due to resonance, the half-sphere will oscillate with the same frequency as the membrane, but if the energy of the membrane's vibration is higher than that required for the half-sphere to co-oscillate, the latter will start to perform an irregular motion, thus reflecting minute irregularities in the membrane (or the half-sphere itself) such as to amplify them in a momentum fluctuation. However, there is one more element in the energy scenario that is important. In our everyday life example, it is the friction between the half-sphere and the membrane, which causes the half-sphere to dissipate heat energy into its environment.

Very similarly, the steady-state resonator representing a "particle" in a thermodynamic environment will not only receive kinetic energy from it, but, in order to balance the stochastic influence of the buffeting momentum fluctuations, it will also dissipate heat into the environment. In fact, the "Vacuum Fluctuation Theorem" proposes, as all fluctuation theorems, that the larger the energy fluctuation of the oscillating "system of interest" is, the higher is the probability that heat will be dissipated into the environment rather than be absorbed.

(The remainder of Chapter 5, including section 5.2 on the case of a non-vanishing quantum potential, is omitted for length; the complete text is at the source.)

6. Diffusion Waves in Sub-Quantum Thermodynamics: Resolution of Einstein's "Particle-in-a-box" Objection and Explanation of Planck's Quantization Assumption

6.1. Einstein's Objection

In 1953, Albert Einstein summarized his arguments against the claim of the completeness of quantum theory, and also his criticism of Bohm's interpretation, by referring to the one-dimensional quantum mechanical problem of a particle of mass m being trapped between two totally-reflecting walls of a box of length L. With quantum mechanics being a universal theory, Einstein argued, it should in principle also apply to macroscopic objects. Thus, the solution of the respective quantum mechanical problem should, at least approximately, approach the classical one when passing to the "macroscopic limit", like, e.g., when having a sphere of mass m and diameter one millimetre entrapped in a box of length one metre. As is well known, in the latter case there should be a classical to-and-fro uniform motion between the walls, which is something that not all quantum predictions do converge to.

(The remainder of Chapter 6, including section 6.2 on the thermodynamic meaning of the quantum potential and section 6.3 on the resolution of Einstein's objection and the vacuum fluctuation theorem, is omitted for length; the complete text is at the source.)

7. The Superposition Principle and Born's Rule from Classical Physics

7.1. The "Translation Scheme"

The "translation" between the language of classical physics employed so far in this review on one hand, and that of traditional quantum theory on the other, can easily be established. As was argued in Chapter 3, a main condition for being able to derive the Schrödinger equation is an average orthogonality condition holding between momenta and fluctuations of them, respectively. Later we shall see that this also corresponds to an orthogonality between reversible physics (i.e., as represented by the classical velocity of the center of a Gaussian wave packet, for example) and irreversible diffusion due to a "heated" environment (i.e., as represented by a velocity fluctuation).

(The remainder of Chapter 7, including the treatment of two alternative paths, of two consecutive paths for one particle and the anti-correlated two-particle system, and section 7.2 towards a classical theory of the collapse of quantum mechanical superposition, is omitted for length; the complete text is at the source.)

8. Free Quantum Motion Identified as Sub-quantum Ballistic Diffusion

8.1. Co-existence of Reversible Schrödinger Dynamics and Irreversible Diffusion

In recent years, G. N. Ord has provided a lattice random walk model which in the continuum approximation produces the Schrödinger equation as a projection from an ensemble of random walks. To our knowledge, this is the first application in the literature of the strategy to "leave microscopic irreversibility untouched (keeping the random walk completely intact) and simply look carefully for reversible features which are independent of the intrinsic irreversibility of the full system." In other words, "the fact that the projection is orthogonal to that responsible for diffusion allows the reversible dynamics of Schrödinger's equation to coexist with the irreversible behaviour of particle densities (i.e. diffusion)."

Historically, it had already been Schrödinger himself who pointed out the close resemblance of his time-dependent equation with the classical diffusion equation. This formal analogy, with the equations differing only in that Schrödinger's uses an "imaginary diffusion constant" (i.e., instead of a real-valued one), has been extensively discussed by R. Fürth. In his treatise, much space is devoted to a discussion of the behaviour of Gaussian wave packets, both in classical diffusion and in quantum theory. It is there where one can see very clearly the many similarities, but also the subtle differences between both types of evolutions. Therefore, we shall also in the present chapter discuss Gaussian wave packets, to begin with, and we shall see how Ord's strategy will provide a fresh look at the whole topic.

Moreover, along with the understanding of how the Schrödinger equation can be derived via nonequilibrium thermodynamics, also the mutual relationship of particle and wave behaviour has become clearer. Just as in the experiments with bouncers or walkers, there exists an average orthogonality also for particle trajectories and wave fronts in the quantum case. In fact, it lies at the heart of the reasons for the emergence of quantum from sub-quantum behaviour in general, and of the superposition principle in particular.

(Sections 8.2, 8.3 and 8.4 — the dispersion of a free Gaussian wave packet with particle trajectories and velocities from purely classical physics, the addition of a linear potential, and the ballistic-diffusion conclusions — are omitted for length; the complete text is at the source.)

9. Conclusions and Outlook

In this review, an extensive discussion was presented of various aspects of a suggested sub-quantum thermodynamics as a basis for emergent quantum theory. On this basis, it has been explicitly shown how the following quantum mechanical features can be derived from purely classical physics: Planck's relation for the energy of a particle, the Schrödinger equation for conservative and non-conservative systems, the Heisenberg uncertainty relations, the quantum mechanical superposition principle, Born's rule, and the quantum mechanical "decay of a Gaussian wave packet". Moreover, also the energy spectrum of a quantum mechanical harmonic oscillator has been derived classically, as well as that of a "particle in a box".

Further, it has been proven that free quantum motion exactly equals sub-quantum anomalous (i.e., "ballistic") diffusion, and, via computer simulations with coupled map lattices, it has been shown how to calculate averaged (Bohmian) trajectories purely from a real-valued classical model. This has been illustrated with the cases of the dispersion of a Gaussian wave packet, both for free quantum motion and for motion in a linear (e.g., gravitational) potential. It has been shown that the results are in excellent agreement with analytical expressions as they are obtained both via our approach, and also via the Bohmian theory. However, in the context of the explanation of Gaussian wave packet dispersion, quantitative statements on the trajectories' characteristic behaviour were presented, which cannot be formulated in any other existing model for quantum systems.

Concerning the computer simulations, much more should be possible, and we are only just beginning to exploit this simple and practical tool to arrive at classical simulations of "quantum processes". Of course, as a next step, relative phases will have to be implemented, so as to be able to simulate truly wave-related phenomena such as interference at a double slit, and the like. Moreover, the simulation of interactions with potentials will constitute a major challenge, as well as many-particle processes, or the extension to higher-dimensional scenarios. In principle, however, there is one area where the simulation might prove to have a big advantage, i.e., in situations of high complexity, where the usual analytic tools of ordinary quantum mechanics would be insufficient. In sum, there is a great potential for novelty with our classical simulation approach.

Finally, one must also mention the challenges as given by quantum mechanical nonlocality and the model's possible relativistic extensions, respectively. The fact that with diffusion wave fields spatial coherence can be created out of random ensembles of diffusive energy is per se already highly interesting. Moreover, as has been pointed out, in this context one needs to mention that the equations of the type used here yield "the physical artefact of infinite speed of field propagation, though with vanishingly small amplitude, at remote locations away from the source. (…) Because propagation is instantaneous, the equations yield no travelling waves, no wave-fronts, and no phase velocity. Rather, the entire domain 'breathes' in phase with the oscillating source. In the world of diffusion waves, there are only spatially correlated phase lags controlled by the diffusion length." Naturally, if any phenomena from classical physics should be helpful at all in this regard, these features make diffusion waves particularly amenable for modelling quantum mechanical nonlocality.

One can thus imagine the following scenario for, e.g., an experiment in neutron interferometry. With a prepared neutron source in a reactor, one immediately has a thermal field in the "vacuum" that nonlocally links the neutron oven, the apparatus (including, e.g., a Mach-Zehnder interferometer), and the detectors. The (typical) Gaussians used to describe the initial quantum mechanical particle distributions thus also contribute in their totality to the form of the heat distribution in the overall system, no matter which particle actually is on its way through the interferometer. In this way, all "potential" paths are implicitly present throughout the experiment (i.e., under constant boundary conditions) in that the corresponding thermal field is spread out no matter where the particle actually is. That this can be assumed is, of course, solely due to the fact of the infinite propagation of diffusion wave fields. Moreover, eventual relativistic formulations of the physics of diffusion wave fields would thus become of primary importance.

At last, let me mention that this review is in no way intended to provide a closed chapter of, say, emergent quantum mechanics. On the contrary, as with research in general, a comprehensive view of ongoing research activity is, particularly when devoted to such foundational issues, almost a self-contradictory project per se, because the process goes on as we write or read. One can say that work in the domain of basic research is always work in progress.

Acknowledgements

I sincerely thank my long-time friends and collaborators at AINS, Siegfried Fussy, Johannes Mesa Pascasio, and Herbert Schwabl, for their invaluable contributions and support, and, of course, also for the lots of fun we had, and still have, by being involved in this exciting field of basic research.

(The 50-item reference list is omitted for length; the complete text is at the source.)