Table of contents

These proceedings comprise the plenary lectures and poster contributions of the 'Heinz von Foerster Conference 2011' on Emergent Quantum Mechanics (EmerQuM11), which was held at the University of Vienna, 11–13 November 2011.

With the 5th International Heinz von Foerster Conference convened at the occasion of von Foerster's 100th birthday, the organizers opted for a twin conference to take place at the Large and Small Ceremonial Halls of the University's main building, respectively. The overall topic was chosen as 'Self-Organization and Emergence', a topic to which von Foerster was an early contributor. While the first conference ('Self-Organization and Emergence in Nature and Society') addressed a more general audience, the second one ('Emergent Quantum Mechanics') was intended as a specialist meeting with a contemporary topic that could both serve as an illustration of von Foerster's intellectual heritage and, more generally, point towards future directions in physics. We thus intended to bring together many of those physicists who are interested in or are working on attempts to understand quantum mechanics as emerging from a suitable classical (or, more generally, deeper level) physics.

EmerQuM11 was organized by the Austrian Institute for Nonlinear Studies (AINS), with essential support from the Wiener Institute for Social Science Documentation and Methodology (WISDOM), the Department of Contemporary History at the University of Vienna, and the Heinz von Foerster-Gesellschaft. There were a number of individuals who contributed to the smooth course of our meeting and whom I would like to sincerely thank: Christian Bischof, Thomas Elze, Marianne Ertl, Gertrud Hafner, Werner Korn, Angelika Krawanja, Florian Krug and his team, Sonja Lang, Albert Müller, Ilse Müller, Irene Müller, Karl Müller, Armin Reautschnig, Marion Schirrmacher, Anton Staudinger, Roman Zlabinger, and, last but not least, my AINS colleagues Siegfried Fussy, Herbert Schwabl and Johannes Mesa Pascasio, the latter in particular for his invaluable technical help with these proceedings.

Funds made available by the Federal Ministry of Science and Research (BMWF), the City of Vienna MA7 Science Funding, the Faculty of Historical and Cultural Studies, Blaha Office Furniture, and Padma AG Zurich are gratefully acknowledged.

As for the nature of the search for a 'deeper level' foundation of quantum mechanics, a first difficulty already arises with respect to the question: Where do we start? One may look for quite different points of departure, such as an encompassing theory of quantum gravity. Or one may find arguments for the necessity to base one's approach at least on a relativistic formulation of the problem. Or one may discard searching for general principles for the time being, and develop an explicit physical model first. And so on. In fact, this is actually what is happening today in different research programs for emergent quantum mechanics, a fact which is also reflected in the rich variety of approaches presented at our meeting. This may be considered a very welcome situation, reminding us of Heinz von Foerster's dictum: 'Act always so as to increase the number of choices.' However, some may view this variety also as a drawback: There is not (yet?) a single, definite alternative theory that would challenge orthodox positions, for example, by providing different experimental predictions. However, the prevailing orthodoxy has shown throughout the 20th century to the present day, that a too restrictive attitude towards theoretical alternatives can lead to almost a standstill in coping with the serious shortcomings and contradictions of present-day physics. Many of us remember famous quantum physicists repeating in an almost mantra-like fashion that quantum theory, or experimental evidence, 'excludes hidden variables as a possibility', along with a reference to some or other newly found 'impossibility proof'. Yet we also recall John Bell's famous counter-statement: 'What is proved by impossibility proofs is lack of imagination.' In this sense, therefore, our task is to make use of the variety of the different approaches offered, for it is in scrutinizing each of them that a chance for further progress and understanding may emerge.

The papers collected in these proceedings essentially follow the order of the plenary talks during the conference program, with the addition of the poster presentations. Prior to the contributions to EmerQuM11, the very first paper of these proceedings presents the opening lecture by Yves Couder who addressed both twin conferences with his talk on wave-particle duality in a classical system. (Although he was not able to participate personally, the contribution of Robert Carroll, a member of the academic advisory board, is included here. Regrettably, the lectures by John Bush, Marek Czachor, Mark Everitt, Felix Finster, and Lee Smolin could not be included, partly for copyright reasons.) Finally, I would like to thank Sarah Toms and Graham Douglas and their team at IOP Publishing (Bristol) for their friendly advice and help during the preparation of these proceedings.

Vienna, April 2012

Gerhard Grössing

All papers published in this volume of Journal of Physics: Conference Series have been peer reviewed through processes administered by the proceedings Editors. Reviews were conducted by expert referees to the professional and scientific standards expected of a proceedings journal published by IOP Publishing.

Several recent experiments were devoted to walkers, structures that associate a droplet bouncing on a vibrated liquid with the surface waves it excites. They reveal that a form of wave-particle duality exists in this classical system with the emergence of quantum-like behaviours. Here we revisit the single particle diffraction experiment and show the coexistence of two waves. The measured probability distributions are ruled by the diffraction of a quantumlike probability wave. But the observation of a single walker reveals that the droplet is driven by a pilot wave of different spatial structure that determines its trajectory in real space. The existence of two waves of these types had been proposed by de Broglie in his "double solution" model of quantum mechanics. A difference with the latter is that the pilot-wave is, in our experiment, endowed with a "path memory". When intrusive measurements are performed, this memory effect induces transient chaotic individual trajectories that generate the resulting statistical behaviour.

I review the proposal made in my 2004 book [1], that quantum theory is an emergent theory arising from a deeper level of dynamics. The dynamics at this deeper level is taken to be an extension of classical dynamics to non-commuting matrix variables, with cyclic permutation inside a trace used as the basic calculational tool. With plausible assumptions, quantum theory is shown to emerge as the statistical thermodynamics of this underlying theory, with the canonical commutation-anticommutation relations derived from a generalized equipartition theorem. Brownian motion corrections to this thermodynamics are argued to lead to state vector reduction and to the probabilistic interpretation of quantum theory, making contact with phenomenological proposals [2, 3] for stochastic modifications to Schrödinger dynamics.

In this essay I begin with a sketch of three senses of emergence and a suggestive view on the emergence of time and the direction of time. After trying to identify what issues representative philosophers view as important in their inquires into emergence phenomena in physics I make several observations pertaining to the concepts, methodology and mechanisms required to understand emergence and describe a platform for its investigation. I then identify some key physical issues which I feel need be better appreciated by the philosophers in this pursuit. I end with some comments on the issue of coarse-graining and persistence structures.

Four questions are discussed which may be addressed to any proposal of a quantum-classical hybrid theory which concerns the direct coupling of classical and quantum mechanical degrees of freedom. In particular, we consider the formulation of hybrid dynamics presented recently in Ref. [1]. This linear theory differs from the nonlinear ensemble theory of Hall and Reginatto, but shares with it to fulfil all consistency requirements discussed in the literature, while earlier attempts failed. - Here, we additionally ask: Does the theory allow for superposition, separable, and entangled states originating in the quantum mechanical sector? Does it allow for "Free Will", as introduced, in this context, by Diósi [2]? Is it local? Does it provide hints for the emergence of quantum mechanics from dynamics beneath?

A hypothetical equation of motion is proposed for Kerr-Newman particles. Obtained by analytic continuation of the Lorentz-Dirac equation into complex space-time, the goal is to study the implications of this theory for emergent quantum mechanics. A new class of "runaway" solutions is found which bear similarity to the quantum phenomenon of Zitterbewegung. Other solutions incorporating external forces are also presented. The electromagnetic fields generated by these motions are studied. It is found that the retarded times are multi-sheeted functions of the field points. As a consequence, the solutions are not unique. A cloaking mechanism is found which inhibits electromagnetic radiation by combining the fields from several Riemann sheets for the retarded time. These results reinforce to some extent, but also pose additional conceptual questions for the idea that Kerr-Newman solutions provide insight into elementary particles and into emergent quantum mechanics.

The Pauli exclusion principle (PEP) and, more generally, the spin-statistics connection, is at the very basis of our understanding of matter. The PEP spurs, presently, a lively debate on its possible limits, deeply rooted in the very foundations of Quantum Field Theory. Therefore, it is extremely important to test the limits of its validity. Quon theory provides a suitable mathematical framework of possible violation of PEP, where the q violation parameter translates into a probability of violating PEP. Experimentally, setting a bound on PEP violation means confining the q-parameter to a value very close to either 1 (for bosons) or -1 (for fermions). The VIP (Violation of the Pauli exclusion principle) experiment established a limit on the probability that PEP is violated by electrons, using the method of searching for PEP forbidden atomic transitions in copper. We describe the experimental method, the obtained results, both in terms of the q-parameter and as probability of PEP violation, we briefly discuss the results and present future plans to go beyond the actual limit by upgrading the experimental technique using vetoed new spectroscopic fast Silicon Drift Detectors. We mention as well the possibility of using a similar experimental technique to search for eventual X-rays generated as a signature of the spontaneous collapse of the wave function, predicted by continuous spontaneous localization type theories.

The modern concept of spacetime usually emerges from the consideration of moving clocks on the assumption that world-lines are continuous. In this paper we start with the assumption that natural clocks are digital and that events are discrete. By taking different continuum limits we show that the phase of non-relativistic quantum mechanics and the odd metric of spacetime both emerge from the consideration of discrete clocks in relative motion. From this perspective, the continuum limit that manifests itself in 'spacetime' is an infinite mass limit. The continuum limit that gives rise to the Schrödinger equation retains a finite mass as a beat frequency superimposed on the 'Zitterbewegung' at the Compton frequency. We illustrate this in a simple model in which a Poisson process drives a relativistic clock that gives rise to a Feynman path integral, where the phase is a manifestation of the twin paradox. The example shows that the non-Euclidean character of spacetime and the wave-particle duality of quantum mechanics share a common origin. They both emerge from the necessity that clocks age at rates that are path dependent.

Double slit interference is explained with the aid of what we call "21st century classical physics". We model a particle as an oscillator ("bouncer") in a thermal context, which is given by some assumed "zero-point" field of the vacuum. In this way, the quantum is understood as an emergent system, i.e., a steady-state system maintained by a constant throughput of (vacuum) energy. To account for the particle's thermal environment, we introduce a "path excitation field", which derives from the thermodynamics of the zero-point vacuum and which represents all possible paths a particle can take via thermal path fluctuations. The intensity distribution on a screen behind a double slit is calculated, as well as the corresponding trajectories and the probability density current. Further, particular features of the relative phase are shown to be responsible for nonlocal effects not only in ordinary quantum theory, but also in our classical approach.

In this paper we discuss a causal network approach to describing relativistic quantum mechanics. Each vertex on the causal net represents a possible point event or particle observation. By constructing the simplest causal net based on Reichenbach-like conjunctive forks in proper time we can exactly derive the 1+1 dimension Dirac equation for a relativistic fermion and correctly model quantum mechanical statistics. Symmetries of the net provide various quantum mechanical effects such as quantum uncertainty and wavefunction, phase, spin, negative energy states and the effect of a potential. The causal net can be embedded in 3+1 dimensions and is consistent with the conventional Dirac equation. In the low velocity limit the causal net approximates to the Schrodinger equation and Pauli equation for an electromagnetic field. Extending to different momentum states the net is compatible with the Feynman path integral approach to quantum mechanics that allows calculation of well known quantum phenomena such as diffraction.

Some relations of the quantum potential to Weyl geometry are indicated with applications to the Friedmann equations for a toy quantum cosmology. Osmotic velocity and pressure are briefly discussed in terms of quantum mechanics and superfluids with connections to gravity.

Stochastic mechanics is an interpretation of nonrelativistic quantum mechanics in which the trajectories of the configuration, described as a Markov stochastic process, are regarded as physically real. The natural stochastic generalization of classical variational principles leads to a derivation of the Schrödinger equation. A brief review of the successes and failures of the theory is given, with references.

We review major appearances of the functional expression ±Δρ^1/2/ρ^1/2 in the theory of diffusion-type processes and in quantum mechanically supported dynamical scenarios. Attention is paid to various manifestations of "pressure" terms and their meaning.

Quantization is derived as an emergent phenomenon, resulting from the permanent interaction between matter and radiation field. The starting point for the derivation is the existence of the (continuous) random zero-point electromagnetic radiation field (zpf) of mean energy ℏω/2 per normal mode. A thermodynamic and statistical analysis leads unequivocally (and without quantum assumptions) to the Planck distribution law for the complete field in equilibrium. The problem of the quantization of matter is then approached from the same perspective: A detailed study of the dynamics of a particle embedded in the zpf shows that when the entire system eventually reaches a situation of energy balance thanks to the combined effect of diffusion and dissipation, the particle has acquired its characteristic quantum properties. To obtain the quantum-mechanical description it has been necessary to do a partial averaging and take the radiationless approximation. Consideration of the neglected radiative terms allows to establish contact with nonrelativistic quantum electrodynamics and derive the correct formulas for the first-order radiative corrections. Quantum mechanics emerges therefore as a partial, approximate and time-asymptotic description of a phenomenon that in its original (pre-quantum) description is entirely local and causal.

In this paper we calculate various transition probability amplitudes, TPAs, known as 'weak values' for the Schrödinger and Pauli particles. It is shown that these values are related to the Bohm momentum, the Bohm energy and the quantum potential in each case. The results for the Schrödinger particle are obtained in three ways, the standard approach, the Clifford algebra approach of Hiley and Callaghan, and the Moyal approach. To obtain the results for the Pauli particle, we combine the Clifford and Moyal algebras into one structure. The consequences of these results are discussed.

We show that the paradigmatic quantum triad "Schrödinger equation–Uncertainty principle–Heisenberg group" emerges mathematically from classical mechanics. In the case of the Schrödinger equation, this is done by extending the metaplectic representation of linear Hamiltonian flows to arbitrary flows; for the Heisenberg group this follows from a careful analysis of the notion of phase of a Lagrangian manifold, and for the uncertainty principle it suffices to use tools from multivariate statistics together with the theory of John's minimum volume ellipsoid. Thus, the mathematical structure needed to make quantum mechanics emerge already exists in classical mechanics.

Bohmian mechanics allows us to understand quantum systems in the light of other quantum traits than the well-known ones (coherence, diffraction, interference, tunnelling, discreteness, entanglement, etc.). Here the discussion focusses precisely on two of these interesting aspects, which arise when quantum mechanics is thought within this theoretical framework: the non-crossing property, which allows for distinguishability without erasing interference patterns, and the possibility to define quantum probability tubes, along which the probability remains constant all the way. Furthermore, taking into account this hydrodynamic-like description as a link, it is also shown how this knowledge (concepts and ideas) can be straightforwardly transferred to other fields of physics (for example, the transmission of light along waveguides).

The notion of non-equilibrium, in the sense of a particle distribution other than ρ = |ψ|², is imported into Nelson's stochastic mechanics, and described in terms of effective wavefunctions obeying non-linear equations. These techniques are applied to the discussion of non-locality in non-linear Schrodinger equations.

We argue that to solve the foundational problems of quantum theory one has to first understand what it means to quantize a classical system. We then propose a quantization method based on replacement of deterministic c-numbers by stochastically-parametrized c-numbers. Unlike canonical quantization, the method is free from operator ordering ambiguity and the resulting quantum system has a straightforward interpretation as statistical modification of an ensemble of classical trajectories. We then develop measurement without wave function collapse à la pilot-wave theory and point out new testable predictions.

Neutron interferometry provides a powerful tool to investigate particle and wave features in quantum physics. Single particle interference phenomena can be observed with neutrons and the entanglement of degrees of freedom, i.e., contextuality can be verified and used in further experiments. Entanglement of two photons, or atoms, is analogous to a double slit diffraction of a single photon, neutron or atom. Neutrons are proper tools for testing quantum mechanics because they are massive, they couple to electromagnetic fields due to their magnetic moment, they are subject to all basic interactions, and they are sensitive to topological effects, as well. The 4π-symmetry of spinor wave functions, the spin-superposition law and many topological phenomena can be made visible, thus showing interesting intrinsic features of quantum physics. Related experiments will be discussed. Deterministic and stochastic partial absorption experiments can be described by Bell-type inequalities. Neutron interferometry experiments based on post-selection methods renewed the discussion about quantum non-locality and the quantum measuring process. It has been shown that interference phenomena can be revived even when the overall interference pattern has lost its contrast. This indicates a persisting coupling in phase space even in cases of spatially separated Schrödinger cat-like situations. These states are extremely fragile and sensitive against any kind of fluctuations and other decoherence processes. More complete quantum experiments also show that a complete retrieval of quantum states behind an interaction volume becomes impossible in principle, but where and when a collapse of the wave-field occurs depends on the level of experiment.

Phase and amplitude of the complex wave function are not independent of each other, but coupled, which becomes obvious looking at Madelung's hydrodynamic formulation of quantum mechanics. In the time-independent case, this leads to a kind of conservation law that allows for the reformulation of the linear Schrödinger equation in terms of a nonlinear Ermakov equation which is equivalent to a complex Riccati equation where the quadratic term in this equation explains the origin of the phase-amplitude coupling. A similar conservation law and corresponding nonlinear equations can also be found in the time-dependent case. The gain from the nonlinear formulations emerges when open systems with dissipation and irreversibility are considered. Describing this kind of systems by an effective nonlinear Schrödinger equation leads to a modification of the above-mentioned equations with new qualitative effects like Hopf bifurcations.

In our bouncer-walker model a quantum is a nonequilibrium steady-state maintained by a permanent throughput of energy. Specifically, we consider a "particle" as a bouncer whose oscillations are phase-locked with those of the energy-momentum reservoir of the zero-point field (ZPF), and we combine this with the random-walk model of the walker, again driven by the ZPF. Starting with this classical toy model of the bouncer-walker we were able to derive fundamental elements of quantum theory [1]. Here this toy model is revisited with special emphasis on the mechanism of emergence. Especially the derivation of the total energy ℏω_o and the coupling to the ZPF are clarified. For this we make use of a sub-quantum equipartition theorem. It can further be shown that the couplings of both bouncer and walker to the ZPF are identical. Then we follow this path in accordance with Ref. [2], expanding the view from the particle in its rest frame to a particle in motion. The basic features of ballistic diffusion are derived, especially the diffusion constant D, thus providing a missing link between the different approaches of our previous works [1, 2].

We discuss a model for charged particles and their fields where mass is field energy only, charge appears as a topological quantum number, photons are Goldstone bosons of spontaneous symmetry breaking, spin and chirality are described as topological charge of solitons interacting by Coulomb and Lorentz forces. The model is U(l) gauge invariant. We discuss open problems and add some speculations. The model is as simple as possible, with three degrees of freedom only describing the rotations of Dreibeins (triads) in space-time and dynamics denned by a Lorentz covariant Lagrangian.

In a recent paper we introduced a model of extended electrons, which is fully compatible with quantum mechanics in the formulation of Schrödinger. However, it contradicts the current interpretation of electrons as point-particles. Here, we show by a statistical analysis of high-resolution scanning tunneling microscopy (STM) experiments, that the interpretation of electrons as point particles and, consequently, the interpretation of the density of electron charge as a statistical quantity will lead to a conflict with the Heisenberg uncertainty principle. Given the precision in these experiments we find that the uncertainty principle would be violated by close to two orders of magnitude, if this interpretation were correct. We are thus forced to conclude that the density of electron charge is a physically real, i.e. in principle precisely measurable quantity, as derived in a recent paper. Experimental evidence to the contrary, in particular high-energy scattering experiments, is briefly discussed. The finding is expected to have wide implications in condensed matter physics, chemistry, and biology, scientific disciplines which are based on the properties and interactions of electrons.

Although quantum mechanics is generally considered to be fundamentally incompatible with classical logic, it is argued here that the gap is not as great as it seems. Any classical, discrete, time reversible system can be naturally described using a quantum Hubert space, operators, and a Schrödinger equation. The quantum states generated this way resemble the ones in the real world so much that one wonders why this could not be used to interpret all of quantum mechanics this way. Indeed, such an interpretation leads to the most natural explanation as to why a wave function appears to "collapse" when a measurement is made, and why probabilities obey the Born rule. Because it is real quantum mechanics that we generate, Bell's inequalities should not be an obstacle.

We present a physical interpretation of the doubling of the algebra, which is the basic ingredient of the noncommutative spectral geometry, developed by Connes and collaborators as an approach to unification. We discuss its connection to dissipation and to the gauge structure of the theory. We then argue, following 't Hooft's conjecture, that noncommutative spectral geometry classical construction carries implicit in its feature of the doubling of the algebra the seeds of quantization.

We show how a Brownian motion on a short scale can originate a relativistic motion on scales larger than the particle's Compton wavelength. Thus, Lorentz symmetry appears to be not a primitive concept, but rather it statistically emerges when a coarse graining average over distances of order, or longer than the Compton wavelength, is taken. We also present the generalizations needed to accommodate in our scheme the doubly special relativistic dynamics. In this way, a previously unsuspected, common stochastic origin of the two frameworks is revealed for the first time. Issues such as generalized commutation relations are also discussed.

We propose a holographic correspondence between the action integral I describing the mechanics of a finite number of degrees of freedom in the bulk, and the entropy S of the boundary (a holographic screen) enclosing that same volume. The action integral must be measured in units of (i times) Planck's constant, while the entropy must be measured in units of Boltzmann's constant. In this way we are led to an intriguing relation between the second law of thermodynamics and the uncertainty principle of quantum mechanics.

Von Neumann's statistical theory of quantum measurement interprets the instantaneous quantum state and derives instantaneous classical variables. In reality, quantum states and classical variables coexist and can influence each other in a time-continuous way. This has been motivating investigations since a long time in quite different fields from quantum cosmology to optics as well as in foundations. Different theories (mean-field, Bohm, decoherence, dynamical collapse, continuous measurement, hybrid dynamics, etc.) emerged for what I call 'coexistence of classical continuum with quantum'. I apply to these theories a sort of 'free will' test to distinguish 'tangible' classical variables useful for causal control from useless ones.

Until recently, wave-particle duality has been thought of as quantum principle without a counterpart in classical physics. This belief was challenged after (i) finding that average dynamics of a classical particle in a strong inhomogeneous oscillating field resembles that of a quantum object and (ii) experimental discovery of "walkers" - macroscopic droplets that bounce on a vertically vibrating bath of the same fluid and can self-propel via interaction with the surface waves they generate. This paper exposes a new family of objects that can display both particle and wave features all together while strictly obeying laws of the Newtonian mechanics. In contrast to the previously known duality examples in classical physics, oscillating field or constant inflow of energy are not required for their existence. These objects behave deterministically provided that all their degrees of freedom are known to an observer. If, however, some degrees of freedom are unknown, an observer can describe such objects only probabilistically and they manifest weird features similar to that of quantum particles. We show new classical counterparts of such quantum phenomena as particle interference, tunneling, above-barrier reflection, trapping on top of a barrier, and spontaneous emission of radiation. In the light of these findings, we hypothesize that quantum mechanics may emerge as approximation from a more profound theory on a deeper level.

We present a simple classical (random) signal model reproducing Born's rule. The crucial point of our approach is that the presence of detector's threshold and calibration procedure have to be treated not as simply experimental technicalities, but as the basic counterparts of the theoretical model. We call this approach threshold signal detection model (TSD). The experiment on coincidence detection which was done by Grangier in 1986 [22] played a crucial role in rejection of (semi-)classical field models in favour of quantum mechanics (QM): impossibility to resolve the wave-particle duality in favour of a purely wave model. QM predicts that the relative probability of coincidence detection, the coefficient g⁽²⁾ (0), is zero (for one photon states), but in (semi-)classical models g⁽²⁾(0) ≥ 1. In TSD the coefficient g⁽²⁾(0) decreases as 1/ε²_d, where ε_d > 0 is the detection threshold. Hence, by increasing this threshold an experimenter can make the coefficient g⁽²⁾ (0) essentially less than 1. The TSD-prediction can be tested experimentally in new Grangier type experiments presenting a detailed monitoring of dependence of the coefficient g⁽²⁾(0) on the detection threshold. Structurally our model has some similarity with the prequantum model of Grossing et al. Subquantum stochasticity is composed of the two counterparts: a stationary process in the space of internal degrees of freedom and the random walk type motion describing the temporal dynamics.

An Ising-type classical statistical ensemble can describe the quantum physics of fermions if one chooses a particular law for the time evolution of the probability distribution. It accounts for the time evolution of a quantum field theory for Dirac particles in an external electromagnetic field. This yields in the non-relativistic one-particle limit the Schrödinger equation for a quantum particle in a potential. Interference or tunneling arise from classical probabilities.

The observable parameters of the electron indicate unambiguously that its gravitational background should be the Kerr-Newman solution without horizons. This background is not flat and has a non-trivial topology created by the Kerr singular ring. This ring was identified with a closed gravitational string. We discuss the relation of this string to the closed heterotic string of the low energy string theory and show that travelling waves along the KN string give rise to the Dirac theory of electron. Gravitational strings form a bridge between gravity and quantum theory, indicating a new way to consistent Quantum Gravity. We explain the pointlike experimental exhibition of the electron and argue that the predicted closed string may be observed by the novel experimental method of the "nonforward" Compton scattering.

Based on the Chetaev theorem on stable dynamical trajectories in the presence of dissipative forces, we obtain the generalized condition for stability of relativistic classical Hamiltonian systems (with an invariant evolution parameter) in the form of the Stückelberg equation. As is known, this equation is the basis of a competing paradigm known as parametrized relativistic quantum mechanics (pRQM).

It is shown that the energy of dissipative forces, which generate the Chetaev generalized condition of stability, coincides exactly with Bohmian relativistic quantum potential. Within the framework of Bohmian RQM supplemented by the generalized Chetaev theorem and on the basis of the principle of least action for dissipative forces, we show that the squared amplitude of a wave function in the Stückelberg equation is equivalent to the probability density function for the number of particle trajectories, relative to which the velocity and the position of the particle are not hidden parameters.

The conditions for reasonableness of trajectory interpretation of pRQM are discussed. Based on analysis of a general formalism for vacuum-flavor mixing of neutrino within the context of the standard and pRQM models we show that the corresponding expressions for the probability of transition from one neutrino flavour to another differ appreciably, but they are experimentally testable: the estimations of absolute value for neutrino mass based on modern experimental data for solar and atmospheric neutrinos show that the pRQM results have a preference. It is noted that the selection criterion of mass solutions relies on proximity between the average size of condensed neutrino clouds, which is described by the Muraki formula (29th ICRC, 2005) and depends on the neutrino mass, and the average size of typical observed void structure (dark matter + hydrogen gas), which plays the role of characteristic dimension of large-scale structure of the Universe.

This essay is a discussion of the concept of eigenform, due to Heinz von Foerster, and its relationship with discrete physics and quantum mechanics. We interpret the square root of minus one as a simple oscillatory process - a clock, and as an eigenform. By taking a generalization of this identification of i as a clock and eigenform, we show how quantum mechanics emerges from discrete physics.

Three problems stand in the way of deriving classical theories from quantum mechanics: those of realist interpretation, of classical properties and of quantum measurement. Recently, we have identified some tacit assumptions that lie at the roots of these problems. Thus, a realist interpretation is hindered by the assumption that the only properties of quantum systems are values of observables. If one simply postulates the properties to be objective that are uniquely defined by preparation then all difficulties disappear. As for classical properties, the wrong assumption is that there are arbitrarily sharp classical trajectories. It turns out that fuzzy classical trajectories can be obtained from quantum mechanics by taking the limit of high entropy. Finally, standard quantum mechanics implies that any registration on a quantum system is disturbed by all quantum systems of the same kind existing somewhere in the universe. If one works out systematically how quantum mechanics must be corrected so that there is no such disturbance, one finds a new interpretation of von Neumann's "first kind of dynamics", and so a new way to a solution of the quantum measurement problem. The present paper gives a very short review of this work.

With general arguments it is motivated that we are still in search of a hidden-variables theory or sub-quantum mechanics. Aspects of a solvable model for quantum measurements are discussed, and a solution of the quantum measurement problem within ordinary quantum theory. The related statistical interpretation of quantum mechanics is advocated. Soliton approaches for elementary particles are proposed and some classical looking aspects of the relativistic hydrogen atom are mentioned. The integration of these solitons with stochastic electrodynamics is sketched.

Is it possible to offer a "no drama" quantum theory? Something as simple (in principle) as classical electrodynamics - a theory described by a system of partial differential equations in 3+1 dimensions, but reproducing unitary evolution of a quantum field theory in the configuration space? The following results suggest an affirmative answer:

1. The scalar field can be algebraically eliminated from scalar electrodynamics; the resulting equations describe independent evolution of the electromagnetic field.

2. After introduction of a complex 4-potential (producing the same electromagnetic field as the standard real 4-potential), the spinor field can be algebraically eliminated from spinor electrodynamics; the resulting equations describe independent evolution of the electromagnetic field.

3. The resulting theories for the electromagnetic field can be embedded into quantum field theories.

Another fundamental result: in a general case, the Dirac equation is equivalent to a 4th order partial differential equations for just one component, which can be made real by a gauge transform.

Issues related to the Bell theorem are discussed.

In this letter we generalize the Dirac equation in the context of the Magueijo-Smolin (MS) [1] approach to the Doubly Special Relativity. This recent theory introduced by Amelino-Camelia [2] is based on the addition to the speed of light of a second relativistic invariant κ related to the Planck energy that implies a noncommutative space-time structure, the κ–Minkowski space-time. In a recent paper [3] we deduced the "Dirac equation" in momentum space in MS base (see also [4]). Interesting studies with this equation have also been conducted, in particular in [5] where it was applied to a calculation of the hydrogen atom spectrum, or in [6] where the discreteness of the "generalized uncertainty principle" was used. Here we will establish the Dirac equation in position space.

Surprisingly, the natural looking random walk leading to Brownian motion occurs to be often biased in a very subtle way: emphasizing some possibilities by only approximating maximal uncertainty principle. A new philosophy of stochastic modelling has been recently introduced, in which we use the only maximizing entropy choice of transition probabilities instead. Local behaviour of both approaches is similar, but they usually lead to completely different global situations. In contrast to Brownian motion leading to nearly uniform stationary density, this recent approach turns out in agreement with having strong localization properties, thermodynamical predictions of quantum mechanics, like thermalizing to dynamical equilibrium state of probability density as the quantum ground state: squares of coordinates of the lowest energy eigenvector of the Bose-Hubbard Hamiltonian for single particle in discrete case, or of the standard Schrödinger operator while including potential and making infinitesimal limit.

The Schrödinger equation for the particle wave function introduced via its action was derived from Newton equation for a point-like particle moving under effect of both potential force and fluctuation-dissipative environment, provided considering only stable motion of the particle. The model considered assumes existence of wave function as a physical field rather than just a mathematical abstraction.

Based on a proposed classical explanation, the quantum mechanical "decay of the wave packet" is shown to simply result from sub-quantum diffusion with a specific diffusivity varying in time due to a particle's changing thermal environment. The exact quantum mechanical intensity distribution, as well as the corresponding trajectory distribution and the velocity field of a Gaussian wave packet are therewith computed. We utilize no quantum mechanics, but only familiar simulation techniques for diffusion, e.g., finite differences or coupled map lattices (CML).