Table of contents

Presented in this volume are the Invited Lectures and the Contributed Papers of the Sixth International Workshop on Decoherence, Information, Complexity and Entropy – DICE 2012, held at Castello Pasquini, Castiglioncello (Tuscany), 17–21 September 2012.

These proceedings may document to the interested public and to the wider scientific community the stimulating exchange of ideas at the meeting. The number of participants has been steadily growing over the years, reflecting an increasing attraction, if not need, of such conference. Our very intention has always been to bring together leading researchers, advanced students, and renowned scholars from various areas, in order to stimulate new ideas and their exchange across the borders of specialization.

In this way, the series of meetings successfully continued from the beginning with DICE 2002¹, followed by DICE 2004², DICE 2006³, DICE 2008⁴, and DICE 2010⁵, Most recently, DICE 2012 brought together more than 120 participants representing more than 30 countries worldwide.

It has been a great honour and inspiration to have Professor Yakir Aharonov (Tel Aviv) with us, who presented the opening Keynote Lecture 'The two-vector quantum formalism'.

With the overarching theme 'Spacetime – Matter – Quantum Mechanics – from the Planck scale to emergent phenomena', the conference took place in the very pleasant and inspiring atmosphere of Castello Pasquini – in beautiful surroundings, overlooking a piece of Tuscany's coast. The 5-day program covered these major topics:

• Quantum Mechanics, Foundations and Quantum-Classical Border

• Quantum-Classical Hybrids and Many-Body Systems

• Spectral Geometry, Path Integrals and Experiments

• Quantum –/– Gravity –/– Spacetime

• Quantum Mechanics on all Scales?

A Roundtable Discussion under the theme 'Nuovi orizzonti nella ricerca scientifica. Ci troviamo di fronte ad una rivoluzione scientifica?' formed an integral part of the program. With participation of E Del Giudice (INFN & Università di Milano), F Guerra (Università 'La Sapienza', Roma) and G Vitiello (Università di Salerno), this event traditionally dedicated to the public drew a large audience involved in lively discussions until late.

The workshop was organized by L Diósi (Budapest), H-T Elze (Pisa, chair), L Fronzoni (Pisa), J J Halliwell (London), E Prati (Milano) and G Vitiello (Salerno), with most essential help from our conference secretaries L Fratino, N Lampo, I Pozzana, and A Sonnellini, all students from Pisa, and from our former secretaries M Pesce-Rollins and L Baldini.

Several institutions and sponsors supported the workshop and their representatives and, in particular, the citizens of Rosignano/Castiglioncello are deeply thanked for the generous help and kind hospitality:

• Comune di Rosignano – A Franchi (Sindaco di Rosignano), S Scarpellini (Segreteria sindaco), L Benini (Assessore ai lavori pubblici), M Pia (Assessore all' urbanistica)

• REA Rosignano Energia Ambiente s.p.a. – F Ghelardini (Presidente della REA), E Salvadori and C Peccianti (Segreteria)

• Associazione Armunia – A Nanni (Direttore), G Mannari (Programmazione), C Perna, F Bellini, M Nannerini, P Bruni and L Meucci (Tecnici). Special thanks go to G Mannari and her collaborators for advice and great help in all the practical matters that had to be dealt with, in order to run the meeting at Castello Pasquini smoothly

Funds made available by Università di Pisa, Domus Galilaeana (Pisa), Centro Interdisciplinare per lo Studio dei Sistemi Complessi – CISSC (Pisa), Dipartimento di Ingegneria Industriale (Università di Salerno), Istituto Italiano per gli Studi Filosofici – IISF (Napoli), Solvay Italia SA (Rosignano), Institute of Physics Publishing – IOP (Bristol), Springer Verlag (Heidelberg), and Hungarian Scientific Research Fund OTKA are gratefully acknowledged.

Last, but not least, special thanks are due to Laura Pesce (Vitrium Galleria, San Vincenzo) for the exposition of her artwork 'arte e scienza' at Castello Pasquini during the conference.

The papers submitted in the wake of the conference have been edited by L Diósi, H-T Elze, L Fronzoni, J J Halliwell, E Prati, G Vitiello and J Yearsley. The proceedings follow essentially the order of presentation during the conference, separating, however, invited lectures and contributed papers⁶. In the name of all participants, we would like to thank S Toms with her collaborators at IOP Publishing (Bristol) for friendly advice and most valuable immediate help during the editing process and, especially, for their continuing efforts to make the Journal of Physics: Conference Series available to all.

Budapest, Pisa, London, Milano, Salerno, Cambridge, April 2013 Lajos Diósi, Hans-Thomas Elze, Leone Fronzoni, Jonathan Halliwell, Enrico Prati, Giuseppe Vitiello and James Yearsley

¹Decoherence and Entropy in Complex Systems ed H-T Elze Lecture Notes in Physics633 (Berlin: Springer, 2004) ²Proceedings of the Second International Workshop on Decoherence, Information, Complexity and Entropy – DICE 2004 ed H-T Elze Braz. Journ. Phys.35 A & 2B (2005) pp 205–529; free access at: www.sbfisica.org.br/bjp³Proceedings of the Third International Workshop on Decoherence, Information, Complexity and Entropy – DICE 2006 eds H-T Elze, L Diósi and G Vitiello Journal of Physics: Conference Series67 (2007); free access at: www.iop.org/EJ/toc/1742-6596/67/1⁴Proceedings of the Fourth International Workshop on Decoherence, Information, Complexity and Entropy – DICE 2008> eds H-T Elze, L Diósi, L Fronzoni, J J Halliwell and G Vitiello Journal of Physics: Conference Series174 (2009); free access at: http://www.iop.org/EJ/toc/1742-6596/174/1⁵Proceedings of the Fifth International Workshop on Decoherence, Information, Complexity and Entropy – DICE 2010 eds H-T Elze, L Diósi, L Fronzoni, J J Halliwell, E Prati, G Vitiello and J Yearsley Journal of Physics: Conference Series306 (2011); free access at: http://iopscience.iop.org/1742-6596/306/1⁶ We regret that invited lectures by Y Aharonov, J Barbour, G Casati and X-G Wen could not be reproduced here, partly for copyright reasons

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.

Selected issues of the concept of spontaneous collapse are discussed, with the emphasis on the gravity-related model. We point out that without spontaneous collapses the Schrödinger cat states would macroscopically violate the standard conservation laws even in the presence of environmental noise. We prove that the collapse time of condensed matter c.o.m. superpositions is not sensitive to the natural uncertainty of the nuclear locations whereas we formulate the conjecture that superfluid. He may show an anomalous low rate of spontaneous collapse compared to common condensed matter.

Recently, it has been argued that quantum mechanics is complete, and that quantum states vectors are necessarily in one-to-one correspondence with the elements of reality, under the assumptions that quantum theory is correct and that measurement settings can be freely chosen. In this work, we argue that the adopted form of the free choice assumption is stronger than needed. In our perspective, there are hidden assumptions underlying these results, which limit their range of validity. We support our argument by a model for the bipartite two-level system, reproducing quantum mechanics, in which the free will assumption is respected, and different quantum states can be connected to the same state of reality.

The concept of open quantum walks (OQW), quantum walks exclusively driven by the interaction with the external environment, is reviewed. OQWs are formulated as discrete completely positive maps on graphs. The basic properties of OQWs are summarised and new examples of OQWs on and their simulation by means of quantum trajectories are presented.

Quantum entanglement in the motion of macroscopic solid bodies has implications both for quantum technologies and foundational studies of the boundary between the quantum and classical worlds. Entanglement is usually fragile in room-temperature solids, owing to strong interactions both internally and with the noisy environment. We generated motional entanglement between vibrational states of two spatially separated, millimeter-sized diamonds at room temperature. By measuring strong nonclassical correlations between Raman-scattered photons, we showed that the quantum state of the diamonds has positive concurrence with 98% probability. Our results show that entanglement can persist in the classical context of moving macroscopic solids in ambient conditions.

As shown in the famous EPR paper (Einstein, Podolsky e Rosen, 1935), Quantum Mechanics is non-local. The Bell theorem and the experiments by Aspect and many others, ruled out the possibility of explaining quantum correlations between entangled particles using local hidden variables models (except for implausible combinations of loopholes). Some authors (Bell, Eberhard, Bohm and Hiley) suggested that quantum correlations could be due to superluminal communications (tachyons) that propagate isotropically with velocity v_t > c in a preferred reference frame. For finite values of v_t, Quantum Mechanics and superluminal models lead to different predictions. Some years ago a Geneva group and our group did experiments on entangled photons to evidence possible discrepancies between experimental results and quantum predictions. Since no discrepancy was found, these experiments established only lower bounds for the possible tachyon velocities v_t. Here we propose an improved experiment that should lead us to explore a much larger range of possible tachyon velocities V_t for any possible direction of velocity of the tachyons preferred frame.

Hydrogen atom is studied as a quantum-classical hybrid system, where the proton is treated as a classical object while the electron is regarded as a quantum object. We use a well known mean-field approach to describe this hybrid hydrogen atom; the resulting dynamics for the electron and the proton is compared to their full quantum dynamics. The electron dynamics in the hybrid description is found to be only marginally different from its full quantum counterpart. The situation is very different for the proton: in the hybrid description, the proton behaves like a free particle; in the fully quantum description, the wave packet center of the proton orbits around the center of mass. Furthermore, we find that the failure to describe the proton dynamics properly can be regarded as a manifestation of the fact that there is no conservation of momentum in the mean-field hybrid approach. We expect that such a failure is a common feature for all existing approaches for quantum-classical hybrid systems of Born-Oppenheimer type.

A summary of a recently proposed description of quantum-classical hybrids is presented, which concerns quantum and classical degrees of freedom of a composite object that interact directly with each other. This is based on notions of classical Hamiltonian mechanics suitably extended to quantum mechanics.

The possibility to describe hybrid systems containing classical and quantum subsystems by means of conditional tomographic probability distributions (tomograms) is discussed. Tomographic Shannon and Rényi entropies associated with the tomograms are studied, and new tomographic uncertainty relations are obtained.

I consider the formulation of hybrid cosmological models that consists of a classical gravitational field interacting with a quantized massive scalar field in the formalism of ensembles on configuration space. This is a viable approach that provides an alternative to semiclassical gravity. I discuss a particular, highly nonclassical solution in two approximations, minisuperspace and spherically-symmetric midisuperspace. In both cases, the coupling of the quantum scalar field and classical gravitational field leads to a cosmological model which has a quantized radius of the universe.

Macroscopic quantum phenomena refer to quantum features in objects of 'large' sizes, systems with many components or degrees of freedom, organized in some ways where they can be identified as macroscopic objects. This emerging field is ushered in by several categories of definitive experiments in superconductivity, electromechanical systems, Bose-Einstein condensates and others. Yet this new field which is rich in open issues at the foundation of quantum and statistical physics remains little explored theoretically (with the important exception of the work of A J Leggett [1], while touched upon or implied by several groups of authors represented in this conference. Our attitude differs in that we believe in the full validity of quantum mechanics stretching from the testable micro to meso scales, with no need for the introduction of new laws of physics.) This talk summarizes our thoughts in attempting a systematic investigation into some key foundational issues of quantum macroscopic phenomena, with the goal of ultimately revealing or building a viable theoretical framework. Three major themes discussed in three intended essays are the large N expansion [2], the correlation hierarchy [3] and quantum entanglement [4]. We give a sketch of the first two themes and then discuss several key issues in the consideration of macro and quantum, namely, a) recognition that there exist many levels of structure in a composite body and only by judicious choice of an appropriate set of collective variables can one give the best description of the dynamics of a specific level of structure. Capturing the quantum features of a macroscopic object is greatly facilitated by the existence and functioning of these collective variables; b) quantum entanglement, an exclusively quantum feature [5], is known to persist to high temperatures [6] and large scales [7] under certain conditions, and may actually decrease with increased connectivity in a quantum network [8]. We use entanglement as a measure of quantumness here and pick out these somewhat counter-intuitive examples to show that there are blind spots worthy of our attention and issues which we need to analyze closer. Our purpose is to try to remove the stigma that quantum only pertains to micro, in order to make way for deeper probes into the conditions whereby quantum features of macroscopic systems manifest.

The quantum measurement problem as was formulated by von Neumann in 1933 can be solved by going beyond the operational quantum formalism. In our "prequantum model" quantum systems are symbolic representations of classical random fields. The Schrödinger's dynamics is a special form of the linear dynamics of classical fields. Measurements are described as interactions of classical fields with detectors. Discontinuity, the "collapse of the wave function", has the trivial origin: usage of threshold type detectors. The von Neumann projection postulate can be interpreted as the formal mathematical encoding of the absence of coincidence detection in measurement on a single quantum system, e.g., photon's polarization measurement. Our model, prequantum classical statistical field theory (PCSFT), in combination with measurements by threshold detectors satisfies the quantum restriction on coincidence detections: the second order coherence is less than one (opposite to all known semiclassical and classical feld models). The basic rule of quantum probability, the Born's rule, is derived from properties of prequantum random felds interacting with threshold type detectors. Comparison with De Broglie's views to quantum mechanics as theory of physical waves with singularities is presented.

In a new approach to explain double-slit interference "from the single particle perspective" via "systemic nonlocality", we answer the question of how a particle going through one slit can "know" about the state of the other slit. We show that this comes about by changed constraints on assumed classical sub-quantum currents, which we have recently employed [1] to derive probability distributions and Bohm-type trajectories in standard double-slit interference on the basis of a modern, 21st century classical physics. Despite claims in the literature that this scenario is to be described by a dynamical nonlocality that could best be understood in the framework of the Heisenberg picture [2], we show that an explanation can be cast within the framework of the intuitively appealing Schrödinger picture as well. We refer neither to potentials nor to a "quantum force" or some other dynamics, but show that a "systemic nonlocality" may be obtained as a phenomenon that emerges from an assumed sub-quantum kinematics, which is manipulated only by changing its constraints as determined by the changes of the apparatus. Consequences are discussed with respect to the prohibition of superluminal signaling by standard relativity theory.

We analyze the main aspects of the phenomenon of spontaneous replica symmetry breaking, introduced by Giorgio Parisi. We work in the frame of real replicas, by taking into account the simple case of the random energy model. In particular, we study the phase space diagram for systems of coupled replicas, and the connected phase transitions. Our considerations can be generalized to the more complicated models of mean field spin glasses and neural networks. We report also about a letter of Ettore Majorana, written in December 1937 to his uncle Dante, very interesting for its methodological content.

If a wave function does not describe microscopic reality then what does? Reformulating quantum mechanics in path-integral terms leads to a notion of "precluded event" and thence to the proposal that quantal reality differs from classical reality in the same way as a set of worldlines differs from a single worldline. One can then ask, for example, which sets of electron trajectories correspond to a Hydrogen atom in its ground state and how they differ from those of an excited state. We address the analogous questions for simple model that replaces the electron by a particle hopping (in discrete time) on a circular lattice.

We review the noncommutative spectral geometry, a gravitational model that combines noncommutative geometry with the spectral action principle, in an attempt to unify General Relativity and the Standard Model of electroweak and strong interactions. Despite the phenomenological successes of the model, the discrepancy between the predicted Higgs mass and the current experimental data indicate that one may have to go beyond the simple model considered at first. We review the current status of the phenomenological consequences and their implications. Since this model lives by construction at high energy scales, namely at the Grand Unified Theories scale, it provides a natural framework to investigate early universe cosmology. We briefly review some of its cosmological consequences.

We report the results obtained in the study of Alain Connes noncommutative spectral geometry construction focusing on its essential ingredient of the algebra doubling. We show that such a two-sheeted structure is related with the gauge structure of the theory, its dissipative character and carries in itself the seeds of quantization. From the algebraic point of view, the algebra doubling process has the same structure of the deformed Hops algebra structure which characterizes quantum field theory.

We explore the possibility to study the quantum dynamics of Dirac fermions in presence of a cosmic string by introducing a conical topological defect in gapped graphene in the presence of a Coulomb charge. When the Coulomb charge exceeds a certain critical strength, quantum instability sets in. Below the critical regime and for certain values of the system parameters, the allowed boundary conditions in gapped graphene cone can be classified in terms of a single real quantity. Observables such as local density of states, scattering phase shifts and the bound state spectra are dependent on the value of this real parameter, which has to be determined empirically. For a supercritical Coulomb charge, we analyze the system with a regularized potential as well as with a zigzag boundary condition and find the effect of the sample topology on the observable features of the system.

Path integrals appear to offer natural and intuitively appealing methods for defining quantum-mechanical amplitudes for questions involving spacetime regions. For example, the amplitude for entering a spatial region during a given time interval is typically defined by summing over all paths between given initial and final points but restricting them to pass through the region at any time. We argue that there is, however, under very general conditions, a significant complication in such constructions. This is the fact that the concrete implementation of the restrictions on paths over an interval of time corresponds, in an operator language, to sharp monitoring at every moment of time in the given time interval. Such processes suffer from the quantum Zeno effect – the continual monitoring of a quantum system in a Hilbert subspace prevents its state from leaving that subspace. As a consequence, path integral amplitudes defined in this seemingly obvious way have physically and intuitively unreasonable properties and in particular, no sensible classical limit. In this paper we describe this frequently-occurring but little-appreciated phenomenon in some detail, showing clearly the connection with the quantum Zeno effect. We then show that it may be avoided by implementing the restriction on paths in the path integral in a "softer" way. The resulting amplitudes then involve a new coarse graining parameter, which may be taken to be a timescale , describing the softening of the restrictions on the paths. We argue that the complications arising from the Zeno effect are then negligible as long as >> 1/E, where E is the energy scale of the incoming state. Our criticisms of path integral constructions largely apply to approaches to quantum theory such as the decoherent histories approach or quantum measure theory, which do not specifically involve measurements. We address some criticisms of our approach by Sokolovksi, concerning the relevance of our results to measurement-based models.

Free and weakly interacting particles perform approximately Gaussian random walks with collisions. They follow a second-quantized nonlinear Schrödinger equation, or relativistic versions of it. By contrast, the fields of strongly interacting particles extremize more involved effective actions obeying fractional wave equations with anomalous dimensions. Their particle orbits perform universal Lévy walks with heavy tails, in which rare events are much more frequent than in Gaussian random walks. Such rare events are observed in exceptionally strong windgusts, monster or rogue waves, earthquakes, and financial crashes. While earthquakes may destroy entire cities, the latter have the potential of devastating entire economies.

We explore leptogenesis induced by the propagation of neutrinos in non-trivial gravitational backgrounds that may characterise the Early Universe epochs of various theories, including string theory. The key point in all these models is that the background induces different populations of fermions as compared to antifermions, and hence CPT Violation (CPTV), already in thermal equilibrium. Depending on the model, then, such populations may freeze out at various conditions leading to leptogenesis and baryogenesis. Among the considered scenarios, is a stringy one, in which the CPTV is associated with a cosmological background with torsion provided by the Kalb-Ramond antisymmetric tensor field (axion) of the string gravitational multiplet. We also discuss CPTV models that go beyond the local effective lagrangian framework, such as a stochastic (Lorentz Violating) Finsler metric and D-particle foam, where the CPTV is due to populations of stochastically fluctuating point-like space-time defects that can be encountered in string/brane theory (D0 branes).

The OPERA neutrino detector at the Gran Sasso underground laboratory (LNGS) has been designed to perform the first detection of neutrino oscillations in direct appearance mode, through the study of the ν_μ → ν_τ channel. Tau leptons produced in the charged current interactions are identified by their decay topologies, using nuclear emulsions. After a brief description of the experimental setup, the first two tau candidates found in the runs 2008 to 2010 are reported. The OPERA experiment has measured the velocity of neutrinos sent from the CERN CNGS beam over a baseline of about 730 km. The measurement is based on data taken from 2009 to 2011. An arrival time of CNGS muon neutrinos with respect to the one computed, assuming the speed of light in vacuum, of (6.5±7.4(stat)^+3.4_−3.3(sys)) ns was measured.

Quantum properties of light and, in particular, quantum correlations have recently disclosed the possibility of realising protocols addressed to overcome limits of classical imaging, a field collectively christened quantum imaging. In particular quantum correlations between twin beams represent a fundamental resource for these studies. Here we present three experimental applications of these properties.

The advancements of nanofabrication, together with the control of implantation of individual atoms at nanometric precision, have opened the experimental study of phase transitions of electronic systems constituted by six electrons and less. Here I review the recent advancements made in the field of phase transitions at the few atoms scale, including the discretized version of the Anderson-Mott quantum phase transition and the melting of electrons arranged in a Wigner-like phase into a Fermi glass realized in a Si : X quantum device where X is a donor element. The rise of collective phenomena is directly probed by controlling the distance between few donor atoms. Four atoms are sufficient to observe emergent phenomena such as Hubbard bands instead of single particle behaviour.

We analyse particle creation and mode mixing for a quantum field in an accelerated cavity, assuming small accelerations but allowing arbitrary velocities, travel times and travel distances, and in particular including the regime of relativistic velocities. As an application, we identify a desktop experimental scenario where the mode mixing resonance frequency in linear sinusoidal motion or in uniform circular motion is significantly below the particle creation resonance frequencies of the Dynamical Casimir Effect, and arguably at the threshold of current technology. The mode mixing acts as a beamsplitter quantum gate, experimentally detectable not only via fluxes or particle numbers but also via entanglement generation.

We study possible observational effects from the Wheeler-DeWitt equation of quantum geometrodynamics. For this purpose, we perform a semiclassical expansion and derive quantum-gravitational correction terms that are inversely proportional to the Planck mass squared. We apply these results to cosmology and calculate the resulting modification of the CMB power spectrum. Although the correction terms are too small to be currently observable, they could provide the key for future tests of quantum gravity.

One interpretation of how the classical world emerges from quantum physics involves the build-up of certain robust entangled states between particles due to scattering events [1]. This is intriguing because it links classical behaviour with the uniquely quantum effect of entanglement and differs from other interpretations that say classicality arises when quantum correlations are lost or neglected in measurements. However, the problem with this new interpretation has been finding an experimental way of verifying it. Here we outline a straightforward scheme that enables just that and should, in principle, allow experiments to confirm the theory to any desired degree of accuracy.

Hamiltonian formulation of quantum dynamics and nonlinear constraints are used to derive dynamical equations of a hybrid classical-quantum system. Starting with a compound quantum system in the Hamiltonian formulation, conditions for classical behavior are imposed on one of its subsystems and the corresponding hybrid dynamical equations are derived. The dynamical equations for hybrid systems in pure and in mixed states indicate that the hybrid systems have properties that are not exhausted by those of quantum and classical systems.

Quantum Electrodynamics (QED) predicts the occurrence of a number of coherent dynamical phenomena in liquid water. In the present paper we focus our attention on the joint coherent oscillation of the almost free electrons produced by the coherent oscillation of the electron clouds of water molecules, which has been described in previous publications, and of the negative electric charges lying on the solid surfaces wet by water. This joint coherent oscillation gives rise to a number of phenomenological consequences which are found to exist in the physical reality and coincide with the layers of Exclusion Zone (EZ) water experimentally observed close to hydrophilic surfaces.

A paradigm model of modern atom optics is studied, strongly interacting ultracold bosons in an optical lattice. This many-body system can be artificially opened in a controlled manner by modern experimental techniques. We present results based on a non-hermitian effective Hamiltonian whose quantum spectrum is analyzed. The direct access to the spectrum of the metastable many-body system allows us to easily identify relatively stable quantum states, corresponding to previously predicted solitonic many-body structures.

The advent of the Hohenberg-Kohn theorem in 1964, its extension to finite-T, Kohn-Sham theory, and relativistic extensions provide the well-established formalism of density-functional theory (DFT). This theory enables the calculation of all static properties of quantum systems without the need for an n-body wavefunction ψ. DFT uses the one-body density distribution instead of ψ. The more recent time-dependent formulations of DFT attempt to describe the time evolution of quantum systems without using the time-dependent wavefunction. Although DFT has become the standard tool of condensed-matter computational quantum mechanics, its foundational implications have remained largely unexplored. While all systems require quantum mechanics (QM) at T=0, the pair-distribution functions (PDFs) of such quantum systems have been accurately mapped into classical models at effective finite-T, and using suitable non-local quantum potentials (e.g., to mimic Pauli exclusion effects). These approaches shed light on the quantum → hybrid → classical models, and provide a new way of looking at the existence of non- local correlations without appealing to Bell's theorem. They also provide insights regarding Bohmian mechanics. Furthermore, macroscopic systems even at 1 Kelvin have de Broglie wavelengths in the micro-femtometer range, thereby eliminating macroscopic cat states, and avoiding the need for ad hoc decoherence models.

Vibrational spectroscopy provides a powerful tool to understand the molecular structures. When applied to the liquid water, this technique reveals so many details which can also shed a light on the supramolecular arrangement of the most ubiquitous of the substances. In particular, the two fluid model of water, proposed several decades ago, founds experimental evidence. Moreover, some fundamental parameters calculated in the realm of the theory of Quantum ElectroDynamics applied to liquid water can be actually measured showing an excellent agreement with the theory. This allows to add a dynamical origin to the mixed cluster model of water well known by the biologists for fifty years and opens the way to the dawn of a real quantum biology.

A particle switching between two sites of a symmetric system in weak interaction with an environmental continuum of high density of states exhibits a telegraph-signal-like time development without the need of the Born-Bohr principle of reduction on eigenstates of the measuring equipment. The origin of the telegraph signal is a very weak local coupling of the particle to the continuum of gravitons which is connected with an enormous slow down of the particle motion. The proposed mechanism envisages entanglement to gravitons in high dimensional spacetime, which, in accord with Gauss law, gives rise to a local and much stronger gravitational law compared to classical gravitation. The physics behind the quantum jumps and the telegraph-signal-like time development is elucidated. The model might be useful for studying environmental decoherence effects on adsorbate localization and quantum diffusion on solid surface.

The emergent nature of quantum mechanics is shown to follow from a precise correspondence with the classical theory of irreversible thermodynamics. Specifically, the linear (or Gaussian) regime of the latter can be put in a 1-to-1 map with the semiclassical approximation to quantum mechanics. The very possibility of reinterpreting quantum mechanics as a thermodynamics proves that the former is an emergent phenomenon. That is, quantum mechanics is a coarse-grained description of some underlying degrees of freedom.

I outline some recent progress on the theory of topological preordered spaces, and comment on the role of these spaces for an approach to a quantum theory of spacetime.

We have demonstrated spatially-discontinuous quantum jumps of electrons at a distance as long as about 1cm. The effect occurs in a modified integer quantum Hall system consisted of a great number of extended Laughlin-Halperin-type states. Our observations directly contradict the no-aether Einstein's interpretation of special relativity together with the Minkowski's model of spacetime. However they are consistent with the aether-related Lorentz-Poincare's interpretation that allows absolute simultaneity. We thus strongly challenge the fundamental status of Lorentz invariance and hence break the basic argument against de Broglie-Bohm realistic quantum theory. We argue that both de Broglie-Bohm and Lorentz-Poincare theories are capable of providing a real synthesis of quantum and relativity theories. This synthesis is of such kind that quantum theory appears the most fundamental physical theory for which relativity is only a limiting case. In accordance with this hierarchy, quantum theory naturally resolves the problem of aether in Lorentz-Poincare's relativity. The role of aether could be played by a deeper Bohm-type undivided quantum pre-space, the relevance of which at any lengthscale directly follows from our observations.

The discovery of the accelerated expansion of the Universe has had a vast resonance on a number of physical disciplines. In recent years a few viable modified gravity models have been proposed, which naturally lead to a late-time de Sitter stage while reducing to General Relativity in the early Universe. We study two of these models during the contraction of a homogeneous cloud of pressureless dust. We show how the increasing energy/mass density may lead to a curvature singularity and derive the typical timescales for its development.

The conditional symmetries of the reduced Einstein-Hilbert action emerging from a static, spherically symmetric geometry are used as supplementary conditions on the wave function. Based on their integrability conditions, only one of the three existing symmetries can be consistently imposed, while the unique Casimir invariant, being the product of the remaining two symmetries, is calculated as the only possible second condition on the wave function. This quadratic integral of motion is identified with the reparametrization generator, as an implication of the uniqueness of the dynamical evolution, by fixing a suitable parametrization of the r-lapse function. In this parametrization, the determinant of the supermetric plays the role of the mesure. The combined Wheeler – DeWitt and linear conditional symmetry equations are analytically solved. The solutions obtained depend on the product of the two "scale factors".

In the standard QFT approach to the calculation of the cosmological constant contributions from the gravitational sector are usually neglected. Using a simple two-dimensional scenario, but in connection with higher dimensions, we discuss the possibility that quantum gravity can play an important role in the calculation of Λ, providing large quantum corrections and renormalizations. We reproduce the QFT result as a sub-case, while allowing a very large set of values.

This review presents noncommutative spacetimes as one of the approaches to Planck scale physics, with the main assumption that at this energy scale spacetime becomes quantized. Spacetime coordinates become noncommutative as observables in Quantum Mechanics. The basic elements of Drinfeld twist deformation theory are reminded. The Hopf algebra language provides natural framework for deformed relativistic symmetries which constitute Quantum Group of symmetry and noncommutative spacetime is in fact Hopf module algebra. The notion of realization for noncommutative coordinates in terms of differential operators is also presented.

This is a concise review of holographic superconductors and superfluids. We highlight some predictions of the holographic models and the emphasis is given to physical aspects rather than to the technical details, although some references to understand the latter are systematically provided. We include gapped systems in the discussion, motivated by the physics of high-temperature superconductivity. In order to do so we consider a compactified extra dimension (with radius R), or, alternatively, a dilatonic field. The first setup can also be used to model cylindrical superconductors; when these are probed by an axial magnetic field a universal property of holography emerges: while for large R (compared to the other scales in the problem) non-local operators are suppressed, leading to the so called Little-Parks periodicity, the opposite limit shows non-local effects, e.g. the uplifting of the Little-Parks periodicity. This difference corresponds in the gravity side to a Hawking-Page phase transition.

The De Donder-Weyl (DW) covariant Hamiltonian formulation of the Palatini first-order Lagrangian of vielbein (tetrad) gravity and its precanonical quantization are presented. No splitting into space and time is required in this formulation. Our recent generalization of Dirac brackets is used to treat the second class primary constraints appearing in the DW Hamiltonian formulation and to find the fundamental brackets. Quantization of the latter yields the representation of vielbeins as differential operators with respect to the spin connection coefficients and the Dirac-like precanonical Schrödinger equation on the space of spin connection coefficients and space time variables. The transition amplitudes on this space describe the quantum geometry of space-time. We also discuss the Hilbert space of the theory, the invariant measure on the spin connection coefficients, and point to the possible quantum singularity avoidance built in in the natural choice of the boundary conditions of the wave functions on the space of spin connection coefficients.

The model of a discrete pregeometry on a microscopic scale is an x-graph. This is a directed acyclic graph. An outdegree and an indegree of each vertex are not more than 2. The sets of vertices and edges of x-graph are particular cases of causal sets. The sequential growth of a graph is an addition of new vertices one by one. A simple stochastic algorithm of sequential growth of x-graph are considered. It is based on a random walk at the x-graph. The particles in this model must be self-organized repetitive structures. We introduce the method of search of such repetitive structures. It is based on a discrete Fourier transformation. An example of numerical simulation is introduced.

The paper proves that quantum mechanics is compatible with the constructive realism of modern philosophy of science. The proof is based on the observation that properties of quantum systems that are uniquely determined by their preparations can be assumed objective without the difficulties that are encountered by the same assumption about values of observables. The resulting realist interpretation of quantum mechanics is made rigorous by studying the space of quantum states—the convex set of state operators. Prepared states are classified according to their statistical structure into indecomposable and decomposable instead of pure and mixed. Simple objective properties are defined and showed to form a Boolean lattice.

Understanding quantum theory has been a subject of debate from its birth. Many different formulations and interpretations have been proposed. Here we examine a recent novel formulation, namely the coevents formulation. It is a histories formulation and has as starting point the Feynman path integral and the decoherence functional. The new ontology turns out to be that of a coarse-grained history. We start with a quantum measure defined on the space of histories, and the existence of zero covers rules out single-history as potential reality (the Kochen Specker theorem casted in histories form is a special case of a zero cover). We see that allowing coarse-grained histories as potential realities avoids the previous paradoxes, maintains deductive non-contextual logic (alas non-Boolean) and gives rise to a unique classical domain. Moreover, we can recover the probabilistic predictions of quantum theory with the use of the Cournot's principle. This formulation, being both a realist formulation and based on histories, is well suited conceptually for the purposes of quantum gravity and cosmology.

The Consistent Histories (CH) formalism attempts to construct a quantum framework which can be used without the need to introduce observers external to the studied system. The prime motivation in mind is the application of the formalism to the universe as a whole. In order to achieve this, CH maintains that a formulation of quantum mechanics should allow for the assignment of probabilities to alternative histories of a system. Therefore, it provides an observer-independent criterion to decide which sets of histories can be given probabilities and states rules to determine them. The framework establishes that each realm, that is, each set of histories to which probabilities can be assigned, provides a valid quantum-mechanical account of a system. Furthermore, the version of CH first presented in [1, 2] proposes an "evolutionary" explanation of our existence in the universe and of our preference for quasiclassical descriptions of nature. The present work critically evaluates claims to the effect that the formalism offered in [1, 2] solves many interpretational problems in quantum mechanics. In particular, it is pointed out that the interpretation of the proposed framework leaves vague two crucial points, namely, whether realms are fixed or chosen and the link between measurements and histories. The claim of this work is that by doing so, CH overlooks the main interpretational problems of quantum mechanics. Furthermore, we challenge the evolutionary explanation offered and we critically examine the proposed notion of a realm-dependent reality.

Assuming that quantum mechanics is obeyed exactly after averaging over hidden variables, and considering models that obey both the hypotheses of free will and locality, we establish the form of all possible hidden-variable models that reproduce the spin-singlet.

The purpose of this paper is to illustrate how cellular automaton can be used in order to arrive at discrete, but "classical", model leading to the emergent "quantum" field theory.

Wave-particle duality, together with the concept of elementary particles, was introduced by de Broglie in terms of intrinsically periodic phenomena. However, after nearly 90 years, the physical origin of such undulatory mechanics remains unrevealed. We propose a natural realization of the de Broglie periodic phenomenon in terms of harmonic vibrational modes associated to space-time periodicities. In this way we find that, similarly to a vibrating string or a particle in a box, the intrinsic recurrence imposed as a constraint to elementary particles represents a fully consistent quantization condition. The resulting cyclic dynamics formally match ordinary relativistic Quantum Mechanics in both the canonical and Feynman formulations. Interactions are introduced in a geometrodynamical way, similarly to general relativity, by simply considering that variations of kinematical state can be equivalently described in terms of modulations of space-time recurrences, as known from undulatory mechanics. We present this novel quantization prescription from an historical prospective.

The low temperature motion of hydrogen on solid metal surfaces displays some unexplained experimental features: in the quantum diffusion regime more than nine orders of magnitude difference between the diffusion rates on different metal surfaces have been measured, the lowest diffusion rates being established in the low temperature scanning tunnelling microscope. Furthermore telegraph-signal-like adsorption site change, rather than Rabi oscillations predicted by Schrödinger equation in 3+1 dimensions, is observed, signaling the breakdown of quantum mechanics in 3+1 dimensions. A theory is presented to resolve these problems, involving the entanglement of the adsorbate motion to gravitons in high-dimensional spacetime. Soft local massive gravonons, induced in the presence of the adsorbate, determine the time scale for surface diffusion. The γη-model is used for the evaluation of the soft gravonon modes. Weak and local entanglement of the adsorbate motion with a nearly degenerate graviton continuum of high density of states are the conditions for the telegraph-signal-like time development of adsorption site change. In contrast to the Copenhagen interpretation of quantum mechanics, this apparent "classical" behaviour of the adsorbate in 3+1 dimensional spacetime is the result of the solution of Schrödinger's time dependent equation in high-dimensional spacetime.

We study quantum behavior from a constructive "finite" point of view, since the introduction of continuum or other actual infinities into physics poses serious conceptual and technical difficulties without any need for these concepts in physics as an empirical science. Taking this approach, we can show that the quantum-mechanical problems can be formulated in the invariant subspaces of permutation representations of finite groups, while the quantum interferences occur as phenomena that are observable in these subspaces. The scalar products in the invariant subspaces (which are needed for formulating the Born rule – the main postulate of quantum mechanics that links mathematical description with experiment) are linear combinations of independent bilinear invariant forms of the permutation representation. A complete set of such forms for any permutation group can be easily calculated by a simple algorithm. Slightly more sophisticated algorithms are required for expressing quantum observables in terms of these forms.

A Newtonian mechanics model is essentially the model of a point body in an inertial reference frame. How to describe extended bodies in non-inertial (vibration) reference frames with the random initial conditions? One of the most generalized ways of descriptions (known as the higher derivatives formalism) consists in taking into account the infinite number of the higher temporal derivatives of the coordinates in the Lagrange function. Such formalism describing physical objects in the infinite dimensions space does not contradict to the quantum mechanics and infinite dimensions Hilbert space.

We study noncommutative deformation of manifolds by constructing star products. We start from a noncommutative ^d and discuss more genaral noncommutative manifolds. In general, star products can not be described in concrete expressions without some exceptions. In this article we introduce new examples of noncommutative manifolds with explicit star products. Karabegov's deformation quantization of P^N and H^N with separation of variables gives explicit calulable star products represented by gamma functions. Using the results of star products between inhomogeneous coordinates, we find creation and anihilation operators and obtain the Fock representation of the noncommutative P^N and H^N.

In this article, we review our recent works on fourth order Weyl gravity, and try to obtain same results for cosmological coefficients, using a different approach. Indeed we shall impose a de Sitter like condition on the original components of field equations. Afterwards using the reduced equations on a definite geometric background, we derive the same results as in [1], for same cosmological constraints.

We present a new theoretical evidence that a relativistically invariant quantum dynamics at large enough space-time scales can be derived from two inter-correlated genuinely non-relativistic stochastic processes that operate at different energy scales. This leads to Feynman amplitudes that are, in the Euclidean regime, identical to transition probability of a Brownian particle propagating through a granular space. Our observation implies a preferred frame and can have distinct experimental signatures. Ensuing implications for special and doubly-special relativity, quantum field theory, quantum gravity and cosmology are discussed.

We present an introduction to the backflow effect in quantum mechanics – the phenomenon in which a state consisting entirely of positive momenta may have negative current and the probability flows in the opposite direction to the momentum. We show that the effect is present even for simple states consisting of superpositions of gaussian wave packets, although the size of the effect is small. Inspired by the numerical results of Penz et al, we present a wave function whose current at any time may be computed analytically and which has periods of significant backflow, with a backwards flux equal to about 70 percent of the maximum possible backflow, a dimensionless number c_bm ≈ 0.04, discovered by Bracken and Melloy. This number has the unusual property of being independent of ℏ (and also of all other parameters of the model), despite corresponding to a quantum-mechanical effect, and we shed some light on this surprising property by considering the classical limit of backflow. We conclude by discussing a specific measurement model in which backflow may be identified in certain measurable probabilities.

A recently proposed step-by-step procedure, to merge the low-energy physics of the π-bonds electrons of graphene, and quantum field theory on curved spacetimes, is recalled. The last step there is the proposal of an experiment to test a Hawking-Unruh effect, emerging from the model, that manifests itself as an exact (within the model) prediction for the electronic local density of states, in the ideal case of the graphene membrane shaped as a Beltrami pseudosphere. A discussion about one particular attempt to experimentally test the model on molecular graphene is presented, and it is taken as an excuse to solve some basic issues that will help future experiments. In particular, it is stated that the effect should be visible on generic surfaces of constant negative Gaussian curvature, that are infinite in number.

The PVLAS collaboration is presently assembling a new apparatus to detect vacuum magnetic birefringence. This property is related to the structure of the QED vacuum and is predicted by the Euler-Heisenberg-Weisskopf effective Lagrangian. It can be detected by measuring the ellipticity acquired by a linearly polarised light beam propagating through a strong magnetic field. Here we report results of a scaled-down test setup and briefly describe the new PVLAS apparatus. This latter one is in construction and is based on a high-sensitivity ellipsometer with a high-finesse Fabry-Perot cavity (> 4×10⁵) and two 0.8 m long 2.5 T rotating permanent dipole magnets. Measurements with the test setup have improved by a factor 2 the previous upper bound on the parameter A_e, which determines the strength of the nonlinear terms in the QED Lagrangian: A_e^(PVLAS) < 3.3 × 10⁻²¹ T⁻² 95% c.l.

Based on Gaussian wave packet solutions of the time-dependent Schrödinger equation, a generalization of the conventional creation and annihilation operators and the corresponding coherent states can be obtained. This generalization includes systems where also the width of the coherent states is time-dependent as they occur for harmonic oscillators with time-dependent frequency or systems in contact with a dissipative environment. The key point is the replacement of the frequency ω₀ that occurs in the usual definition of the creation/annihilation operator by a complex time-dependent function that fulfils a nonlinear Riccati equation. This equation and its solutions depend on the system under consideration and on the (complex) initial conditions. Formal similarities also exist with supersymmetric quantum mechanics. The generalized creation and annihilation operators also allow to construct exact analytic solutions of the free motion Schrödinger equation in terms of Hermite polynomials with time-dependent variable.

A rigorous ab initio derivation of the quantum mechanics of a single particle with spin is presented starting by conformally Weyl-gauge-invariant principle. The particle is described as a relativistic top with six Euler's angles and quantum effects are introduced by assuming that the Weyl's curvature of the particle configuration space acts on it as an external scalar potential. Weyl's conformal covariance is made explicit in all steps of the theory. It is shown that metric of the configuration space accounts for non-quantum relativistic effects, while the affine connections account for quantum relativistic effects. In this way, classical and quantum features acquire well distinguished geometrical origin. A scalar wave function is also introduced to recover the connection with the standard quantum description based on Dirac's four-component spinors. Finally, the case of two entangled spins is considered in the nonrelativistic limit and it is found that the nonlocality rests on the entanglement of the spin internal orientational variables, playing the role of "hidden variables". The theory was carried out in the Minkowski space-time, but it can be easily extended to a space with nonzero Riemann curvature.

Bohmian mechanics also known as de Broglie-Bohm theory is the most popular alternative approach to quantum mechanics. Whereas the standard interpretation of quantum mechanics is based on the complementarity principle Bohmian mechanics assumes that both particle and wave are concrete physical objects. In 1993 Peter Holland has written an ardent account on the plausibility of the de Broglie-Bohm theory. He proved that it fully reproduces quantum mechanics if the initial particle distribution is consistent with a solution of the Schrödinger equation. Which may be the reasons that Bohmian mechanics has not yet found global acceptance? In this article it will be shown that predicted properties of atoms and molecules are in conflict with experimental findings. Moreover it will be demonstrated that repeatedly published ensembles of trajectories illustrating double slit diffraction processes do not agree with quantum mechanics. The credibility of a theory is undermined when recognizably wrong data presented frequently over years are finally not declared obsolete.

Notion about complexity of composite systems is applied to the problem of measurement of quantum states. An expression for the determination of the value of complexity of measurement is given. Its practical meaning is illustrated in the example how this value differs when measuring complexity of authorized and non-authorized receiving of information transferred by quantum channels in protocol of direct coding.

Unconditional security of information in direct coding protocol with information masking by uniformly distributed state of carrier is shown.

A fundamental requirement in quantum information processing and in many other areas of science is the capability of precisely controlling a quantum system by preparing a quantum state with the highest fidelity and/or in the fastest possible way. Here we present an experimental investigation of a two level system, characterized by a time-dependent Landau-Zener Hamiltonian, aiming to test general and optimal high-fidelity control protocols. The experiment is based on a Bose-Einstein condensate (BEC) loaded into an optical lattice, then accelerated, which provides a high degree of control over the experimental parameters. We implement generalized Landau-Zener sweeps, comparing them with the well-known linear Landau-Zener sweep. We drive the system from an initial state to a final state with fidelity close to unity in the shortest possible time (quantum brachistochrone), thus reaching the ultimate speed limit imposed by quantum mechanics. On the opposite extreme of the quantum control spectrum, the aim is not to minimize the total transition time but to maximize the adiabaticity during the time-evolution, the system being constrained to the adiabatic ground state at any time. We implement such transitionless superadiabatic protocols by an appropriate transformation of the Hamiltonian parameters. This transformation is general and independent of the physical system.

As a proof of principle, we show how a classical nonlinear Hamiltonian system can be driven resonantly over reasonably long times by appropriately shaped pulses. To keep the parameter space reasonably small, we limit ourselves to a driving force which consists of periodic pulses additionally modulated by a sinusoidal function. The main observables are the average increase of kinetic energy and of the action variable (of the non-driven system) with time. Applications of our scheme aim for driving high frequencies of a nonlinear system with a fixed modulation signal.

Based on accurate Lennard-Jones type interaction potentials, we derive a closed set of state equations for the description of warm atomic gases in the presence of ionization processes. The specific heat is predicted to exhibit peaks in correspondence to single and multiple ionizations. Such kinetic analogue in atomic gases of the Schottky anomaly in solids is enhanced at intermediate and low atomic densities. The case of adiabatic compression of noble gases is analyzed in detail and the implications on sonoluminescence are discussed.

We investigate the bulk properties of protoneutron stars in the framework of a relativistic mean field theory based on nonextensive statistical mechanics, originally proposed by C. Tsallis and characterized by power-law quantum distributions. We study the relevance of nonextensive statistical effects on the β-stable equation of state at fixed entropy per baryon, for nucleonic and hyperonic matter. We concentrate our analysis in the maximum heating and entropy per baryon s = 2 stage and T ≈ 40 ÷ 80 MeV. This is the phase, at high temperature and high baryon density, in which the presence of nonextensive effects may alter more sensibly the thermodynamical and mechanical properties of the protoneutron star. We show that nonextensive power-law effects could play a crucial role in the structure and in the evolution of the protoneutron stars also for small deviations from the standard Boltzmann-Gibbs statistics.

It is proposed here that due to rapid cooling at the end of the Planck-era a second order phase transition with Planck-size space-time-energy fluctuations may have led to a precipitation of Planck-dimension real objects with masses around m_P and hence a Schwarzschild radius ℜ = 2Gm_P/c². This equals twice their spatial dimension l_P; hence they are black holes. They could have survived through inflation and expansion because of their extremely small cross section for the fluctuations that are supposed to cause the hypothetical Hawking decay. In the case of our galaxy they could thus be the missing WIMPS of the dark mass halo, consisting of ~ 10⁵⁰ PPMBHs – in contrast to the ~ 10⁶⁸ baryons in the stars – both bound together by gravitation with negligible other interaction between the two. They would constitute a storage of gravitational energy three orders of magnitude larger than the energy stock of H → He fusion, an energy to be released somehow when they very slowly coalesce to form big black holes.

According to the formerly reported 4-D spherical model of the time and universe, any energy in the 3-dimensional space is a vibration of the intrinsic space energy. There is a special frame stationary to the space energy and the principle of relativity is no longer valid. Accordingly, abandonment of the Special Relativity and then introduction of a factor of acceleration for energy in the 3-D space are proposed.

By the study of physics at the Planck scale interesting and unusual peculiarities emerge, which make this sector an extremely fascinating field of theoretical investigation; in this paper it is given an overview concerning quantum foam, space rips, branes, duality, mirror simmetry, appeared with the study of the Planck scale physics.

We report on the recent results revealing the presence of geometric invariants in all the phenomena in which vacuum condensates appear and we show that Aharonov-Anandan phase can be used to provide the evidence of phenomena like Hawking and Unruh effects and to test some behavior of quantum field theory in curved space. A very precise quantum thermometer can be also built by using geometric invariants.

The phenomena of particle mixing and flavor oscillations in elementary particle physics are associated with multi-mode entanglement of single-particle states. We show that, in the framework of quantum field theory, these phenomena exhibit a fine structure of quantum correlations, as multi-mode multi-particle entanglement appears. Indeed, the presence of anti-particles adds further degrees of freedom, thus providing nontrivial contributions both to flavor entanglement and, more generally, to multi-partite entanglement. By using the global entanglement measure, based on the linear entropies associated with all the possible bipartitions, we analyze the entanglement in the multiparticle states of two-flavor neutrinos and anti-neutrinos. A direct comparison with the instance of the quantum mechanical Pontecorvo single-particle states is also performed.

This article resumes recent advances in the development of a formalism that allows the incorporation of a hypothetical spontaneous reduction or collapse of wave function of matter fields within the context of semiclassical gravity. The proposal is applied to the inflationary scenario for the emergence of the primordial inhomogeneities and anisotropies out of the initially homogeneous and isotropic quantum state, to which, the early stages of inflation are supposed to drive the universe. In previous works we have argued that a scheme of this kind is required if we want to satisfactorily account for the emergence of the seeds of cosmic structure.

It is shown in the framework of a harmonic system that the thermodynamical time arrow is induced by the environmental initial conditions in a manner similar to spontaneous symmetry breaking. The Closed Time Path formalism is introduced in classical mechanics to handle Green functions for initial condition problems by the action principle, in a systematic manner. The application of this scheme for quantum systems shows the common dynamical origin of the thermodynamical and the quantum time arrows. It is furthermore conjectured that the quantum-classical transition is strongly coupled.

We discuss the role of spin-rotation coupling and Thomas precession in the decay of hydrogen-like heavy ions that are injected in a storage ring. We find that the decay curve has a time modulation that vanishes if the magnetic field also disappears. The results can be applied to the recently reported modulation in the decay of the hydrogen-like ions ¹⁴⁰Pr⁵⁸⁺, ¹⁴²Pm⁶⁰⁺ and ¹²²I⁵²⁺, where rotation couples to the spins of electron and nucleus. Spin-rotation coupling and Thomas precession also generate oscillations in the decay of a muon bound to a nucleus rotating in a storage ring.