Theoretical Physics

Not much by themselves, aparently. We try to reconstruct the scale factor a(t) of the universe from the SNe Ia data, i.e. the luminosity distance d L (z), using only the cosmological principle and the assumption that gravitation is governed by a metric theory. In our hence model-independent, or cosmographic study, we fit functions to d L (z) rather than a(t), since d L (z) is what is measured. We find that the acceleration history of the universe cannot be reliably determined in this approach due to the irregularity and parametrization-dependence of the results. However, adding the GRB data to the dataset cures most of the irregularities, at the cost of compromising the model-independent nature of the study slightly. Then we can determine the redshift of transition to cosmic acceleration as z t ∼ 0.50 ± 0.09 for a flat universe (larger for positive spatial curvature). If Einstein gravity (GR) is assumed, we find a redshift at which the density of the universe predicted from the d L (z) data is independent of curvature. We use this point to derive an upper limit on matter density, hence a lower limit on the density of dark energy. While these limits do not improve the generally accepted ones, they are derived only using the d L (z) data.

Not much by themselves, aparently. We try to reconstruct the scale factor a(t) of the universe from the SNe Ia data, i.e. the luminosity distance d L (z), using only the cosmological principle and the assumption that gravitation is governed by a metric theory. In our hence model-independent, or cosmographic study, we fit functions to d L (z) rather than a(t), since d L (z) is what is measured. We find that the acceleration history of the universe cannot be reliably determined in this approach due to the irregularity and parametrization-dependence of the results. However, adding the GRB data to the dataset cures most of the irregularities, at the cost of compromising the model-independent nature of the study slightly. Then we can determine the redshift of transition to cosmic acceleration as z t ∼ 0.50 ± 0.09 for a flat universe (larger for positive spatial curvature). If Einstein gravity (GR) is assumed, we find a redshift at which the density of the universe predicted from the d L (z) data is independent of curvature. We use this point to derive an upper limit on matter density, hence a lower limit on the density of dark energy. While these limits do not improve the generally accepted ones, they are derived only using the d L (z) data.

I develop a model of n-dimensional (n-D) complex Primordial Dynamic Space (briefly PDS) with peculiar characteristics, which is time-free (in the meaning indicated in the paper) and can be used as basic structure for a new theory of unification.

This paper reimagines thought as an evental intensity immanent to the cosmos, unbound by human agency or Platonic forms. Drawing on the philosophies of Gilles Deleuze, Alfred North Whitehead, and Gilbert Simondon, I propose that thinking emerges as a bifurcation within a field of intensities, rather than as a computational or reductive process. The concepts of Possest and Membrana frame thought as a "throw of the dice without number"---a rhythmic, twistorial splay that traverses physical, biological, and social domains. By integrating Roger Penrose's twistor theory with rigorous mathematical grounding, I argue that reality is a non-local, genealogical flux, dissolving the dualisms of mind and matter. Human cognition becomes a mere filament in a cosmic web of evental thought.

The aim of this paper is to constrain modified gravity with redshift space distortion observations and supernovae measurements. Compared with a standard ΛCDM analysis, we include three additional free parameters, namely the initial conditions of the matter perturbations, the overall perturbation normalization, and a scale-dependent modified gravity parameter modifying the Poisson equation, in an attempt to perform a more model-independent analysis. First, we constrain the Poisson parameter Y (also called G eff ) by using currently available f σ 8 data and the recent SN catalog JLA. We find that the inclusion of the additional free parameters makes the constraints significantly weaker than when fixing them to the standard cosmological value. Second, we forecast future constraints on Y by using the predicted growth-rate data for Euclid and SKA missions. Here again we point out the weakening of the constraints when the additional parameters are included. Finally, we adopt as modified gravity Poisson parameter the specific Horndeski form, and use scale-dependent forecasts to build an exclusion plot for the Yukawa potential akin to the ones realized in laboratory experiments, both for the Euclid and the SKA surveys.

We demonstrate that supergravity models containing the Standard Model, dilaton and modulus naturally lead to dynamical symmetry breaking with excellent phenomenology. We assume primordial supersymmetry breaking in the form of a constant contribution to the superpotential. String inspired relations link fundamental couplings to the dilaton vev. We specialize to a class of models inspired by the 4-D fermionic string. Non-renormalizable terms in the superpotential naturally produce the Higgs mixing parameter µ suitable for our mechanism. We discuss extensions and limitations of our approach.

Among the list of one-dimensional solvable Hamiltonians, we find the Hamiltonian with the Rosen–Morse II potential. The first objective is to analyse the scattering matrix corresponding to this potential. We show that it includes a series of poles corresponding to the types of redundant poles or anti-bound poles. In some cases, there are even bound states and this depends on the values of given parameters. Then, we perform different supersymmetric transformations on the original Hamiltonian using either the ground state (for those situations where there are bound states) wave functions, or other solutions that come from anti-bound states or redundant states. We study the properties of these transformations.

According to Einstein's Spacetime Theory, Gravitational Waves are "Spacetime Ripples". These ripples are distorted space which can cause oscillation to LIGO interferometer and such that interference signals can be detected. However, what exactly the spacetime is, how these ripples propagate and oscillate with LIGO interferometer remain unknown. In this paper, instead of "Spacetime Ripples", gravitational waves as the "Graviton Surges" in a combined graviton flux composed of two alternately blocking graviton fluxes emitted from a pair of rotating black holes (or mega stars) are proposed based on Graviton Radiation and Contact Interaction Theory. As a graviton flux meets a moving target object, dynamic graviton flux can be generated dependent on the speed of the target object and the relative direction between the moving target object and the static graviton flux observed at the parent object. Furthermore, according to Gravity Affected Wu's Spacetime Shrinkage Theory and Principle of Parallelism, Wu Unit Length (diameter) and Wu Unit Time (period) of Wu's Pairs (building blocks of the universe) in an object or event are dependent on the local gravitational field generated by either static graviton flux or dynamic graviton flux, as is the dimension, duration, velocity and acceleration of the object or event, as well as the wavelength and light speed of a photon. As a result, when Graviton Surges and Graviton Valleys in the gravitational waves hit LIGO Interferometer, different anisotropic light speeds can be produced in the two arms of LIGO interferometer subject to the dynamic gravitational field dependent on the effective mass of either one of the two black holes or two black holes caused by blocking in the rotation process between two black holes, and the direction between laser beams and gravitational waves (static graviton flux), such that interference can be generated and thus Graviton Surges and gravitational waves can be detected.

The speed of light has been repeatedly measured considering the time a photon spends travelling from an initial point to another and then coming back to the initial point, the so called two-way speed of light. Experiments aiming to measure the speed of light when it travels strictly an initial point to another, the so called one-way speed of light, have also been proposed. Here we will discuss the conceptual basics some of these experiments, proposing what seems to be a new feasible procedure to measure the one-way speed of light. According to Special Relativity it is expected that the measurable value will match the observed value for the two-way speed of light.

We find the spectrum P (ω)dω of the gravitational wave background produced in the early universe in string theory. We work in the framework of the recently discussed String Driven Cosmology, whose scale factors are computed with the low-energy effective string equations as well as selfconsistent solutions of General Relativity with a gas of strings as source. The scale factor evolution is described by an early string driven inflationary stage with an instantaneous transition to a radiation dominated stage and successive matter dominated stage. This is a string cosmology expanding evolution always running on positive proper cosmic time. The spectrum of the generated gravitons is computed in the framework of Quantum Field Theory and in the appropiated effective String Cosmology context. We study and show explicitly the effect of the dilaton field, characteristic to this kind of cosmologies. We compute the spectrum for the same evolution description with three differents approachs. Some features of gravitational wave spectra, as peaks and asymptotic behaviours, are found direct consequences of the dilaton involved in the string low energy effective action and not only of the scale factor evolution. We make use of a careful treatment of the scale factor evolution and involved transitions. This allows us to compute a full prediction on the power spectrum of gravitational waves without any free-parameters. A comparative analysis of different treatments, solutions and compatibility with observational bounds or detection perspectives is made.

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Recent results on classical and quantum strings in a variety of black hole and cosmological spacetimes, in various dimensions, are presented. The curved backgrounds under consideration include the 2 + 1 black hole anti de Sitter spacetime and its dual, the black string, the ordinary D ≥ 4 black holes with or without a cosmological constant, the de Sitter and anti de Sitter spacetimes and static Robertson-Walker spacetimes. Exact solutions to the string equations of motion and constraints, representing circular strings, stationary open strings and dynamical straight strings, are obtained in these backgrounds and their physical properties (length, energy, pressure) are described. The existence of multi-string solutions, describing finitely or infinitely many strings, is shown to be a general feature of spacetimes with a positive or negative cosmological constant. Generic approximative solutions are obtained using the string perturbation series approach, and the question of the stability of the solutions is addressed. Furthermore, using a canonical quantization procedure, we find the string mass spectrum in de Sitter and anti de Sitter spacetimes. New features as compared to the string spectrum in flat Minkowski spacetime appear, for instance the fine-structure effect at all levels beyond the graviton in both de Sitter and anti de Sitter spacetimes, and the non-existence of a Hagedorn temperature in anti de Sitter spacetime. We discuss the physical implications of these results. Finally, we consider the effect of spatial curvature on the string dynamics in Robertson-Walker spacetimes.

In the article, the effects of oil price changes on inflation and the exchange rate in Saudi Arabia, Iraq, Kuwait, and UAE, as well as in Azerbaijan, were econometrically assessed. The comparison between Azerbaijan and these Arab countries was made because oil revenues play an important role in the economy of these countries, but its effects are different. Saudi Arabian riyal, Iraqi dinar, Kuwaiti dinar, and UAE dirham depend on oil prices in the world market for the long run. In the short run, there is no such dependence. It is confirmed that Azerbaijani manat is dependent on oil revenues for the long run. Addiction is not felt in the short run. In the oil-rich Arab countries and Azerbaijan, the non-dependence of the national currency on the world market on oil prices is due to the state regulation of the national currency's exchange rate in these countries. In none of these countries is the national currency liberal, and its exchange rate is not determined by supply and demand in the market.

We consider the class of physical theories whose dynamics are given by natural equations, which are partial differential equations determined by a functor from the category of n manifolds, for some n, to the category of fiber bundles, satisfying certain further conditions. We show how the theory of natural equations clarifies several important foundational issues, including the status and meaning of minimal coupling, symmetries of theories, and background structure. We also state and prove a fundamental result about the initial value problem for natural equations.

It is shown that in classical general relativity, if space-time is nonempty at one time, it will be nonempty at all times provided that the energy momentum tensor of the matter satisfies a physically reasonable condition. The apparent contradiction with the quantum predictions for the creation and annihilation of matter particles by gravitons is discussed and is shown to arise from the lack of a good energy momentum operator for the matter in an unquantised curved space-time metric.

In the framework of generalized quantum theory using a W*-algebraic formalism, we introduce a completely symmetric coherence relation for states which is also applicable to nonpure states. Making use of lattice theoretic results the properties of this relation, especially its connection with incompatibility, are investigated. By means of algebraic decomposition theory the investigation is reduced to the case of factors where a complete classification of the coherence classes is given.

In this paper we discuss on recent attempts aimed at demonstrating that, contrary to well-known results, the second clock effect (SCE) does not take place in generalized Weyl spaces -spaces with arbitrary nonmetricity -denoted here as W4 spaces. These attempts include Weyl gauge theories of gravity, as well as the symmetric teleparallel theories (STTs). Our approach to this issue is based on the adoption of Weyl gauge symmetry (WGS) which is a manifest symmetry of the basic laws of Weyl geometry. We shall consistently adapt mathematical and geometrical quantities and concepts so that the resulting geometrical framework be gauge invariant. This issue is of special relevance for the fate of nonmetricity theories, including a class of the STTs which is being intensively applied in the cosmological framework. As we shall show, if realize that WGS is a manifest symmetry of generalized Weyl spaces W4, and identify physical vectors and tensors with corresponding hypothetical vectors and tensors living in W4, neither the Weyl gauge theories nor the nonmetricity theories are free of the SCE, unless Weyl integrable geometry (WIG) spaces are considered.

From a perspective of some 50 years or more, this paper reviews my recall of the early days of scalar-tensor alternatives to standard Einstein general relativistic theory of gravity. Of course, the story begins long before my involvement, going back to the proposals of Nordström in 1914, and that of Kaluza, Klein, et al., a few years later, so I include reviews of these seminal ideas and those that followed in the 1920's through the 1940's. This early work concerned the search for a Unified Field Theory, unifying gravity and Electromagnetism, using five dimensional manifolds. This formalism included not only the electromagnetic spacetime vector potential within the fivemetric, but also a spacetime scalar as the five-five metric component. Although this was at first regarded more as a nuisance, to be set to a constant, it turned out later that Fierz, Jordan, Einstein and Bergmann noticed that this scalar could be a field, possibly related to the Newtonian gravitational constant. Relatively little theoretical and experimental attention was given to these ideas until after the second world war when Bob Dicke, motivated by the ideas of Mach, Dirac, and others, suggested that this additional scalar, coupled only to the metric and matter, could provide a reasonable and viable alternative to standard Einstein theory. This is the point of my direct involvement with these topics. However, it was Dicke's prominence and expertise in experimental work, together with the blossoming of NASA's experimental tools, that caused the explosion of interest, experimental and theoretical, in this possible alternative to standard Einstein theory. This interest has waxed and waned over the last 50 years, and we summarize some of this work.

We propose a novel mutation mechanism for points and ordinary or palindromic sequences of DNA and RNA. We adopted non-Hermitian approaches based on quantum mechanics. Hermiticity is in the limelight of any physical structure with quantum character, like DNA, or RNA, as it creates quantum stability in that it yields real eigenvalues and orthonormal states. We show that, through the mutation mechanism we constructed based on non-Hermitian physics, the deterioration of the Hermitian character of the original DNA states, nucleotides, does not create a stability problem. We show that Weyl's perturbation theory helps us determine the stability of mutated DNA or RNA. We prove that mutations made in the laboratory with conventional nucleotides using non-Hermitian physics methods are not different from mutations that occur spontaneously in nature. This result may help to reveal the quantum nature of genetic diseases in the near future and may shape the molecular approaches.

This paper presents a new hypothesis that simultaneously addresses multiple unresolved problems in physics and cosmology: the origin of low initial entropy, the nature of dark matter and dark energy, the matter–antimatter asymmetry, early large-scale structures, the arrow of time, cosmic fine-tuning, and the apparent breakdown of unification.

By offering a unified explanation across these domains, the model aims to resolve the current cosmological crisis and connect phenomena that have long remained isolated. The proposal invites a fundamental rethinking of time, entropy, and the structure of the universe.

This chapter examines symmetries in the physical universe from the perspective of a geometry based on spheres of 4D space with a diameter equal to the Planck length. Through these spheres, a connection is proposed between the fundamental properties of particles and the fields they generate, highlighting that the properties of mass, energy, momentum and electric charge emerge as manifestations of the rotation and curvature of space.
The implications of these symmetries in the context of quantum mechanics and relativity are also reviewed, considering the possibility of unifying concepts in a deeper geometric framework. The exploration of these ideas could offer new perspectives on the nature of mass, energy and the fundamental interaction between particles.

It is possible to reconstruct a 'defragmented' Dirac equation in such a way as to generate automatic second quantization, supersymmetry without extra particles, finite energy of the free state without renormalization, propagators without an infrared divergence, string theory without strings, explicit state vectors for fermions, bosons and baryons, symmetry breaking between the weak, strong and electric interactions, quark confinement and automatic baryonic mass. Even the Dirac equation itself becomes unnecessary, as the entire information required to determine phase and amplitude becomes compactified within a single expression for the Dirac state.

Nilpotent quantum mechanics provides a powerful method of making efficient calculations. More importantly, however, it provides insights into a number of fundamental physical problems through its use of a dual vector space and its explicit construction of vacuum. Physical interpretation of the nilpotent formalism is discussed with respect to boson and baryon structures, the mass-gap problem, zitterbewgung, Berry phase, renormalization, and related issues.

A core level of basic information for physics is identified, based on an analysis of the characteristics of the parameters space, time, mass and charge. At this level, it is found that certain symmetries operate, which can be used to explain certain physical facts and even to derive new mathematical theorems. Applications are made to classical mechanics, electromagnetic theory and quantum mechanics.

It is possible to reconstruct a ‘defragmented ’ Dirac equation in such a way as to generate automatic second quantization, supersymmetry without extra particles, finite energy of the free state without renormalization, propagators without an infrared divergence, string theory without strings, explicit state vectors for fermions, bosons and baryons, symmetry breaking between the weak, strong and electric interactions, quark confinement and automatic baryonic mass. Even the Dirac equation itself becomes unnecessary, as the entire information required to determine phase and amplitude becomes compactified within a single expression for the Dirac state. 1 Defragmenting the Dirac equation To acquire a more profound understanding of the fermionic state, we need first to acquires a more profound understanding of the Dirac equation. The Dirac equation, as conventionally written, (γµ∂µ + im) ψ = 0, is the fundamental basis of particle physics, but it is inconvenient in many ways. The main pro...

The nilpotent version of the Dirac equation can be constructed on the basis of the algebra of a double vector space or complexified double quaternions. This algebra is isomorphic to the standard gamma matrix algebra, with 64 units which can be produced by just 5 generators. The H4 algebra used in the Berwald-Moor metric is a distinct subalgebra of this 64-part algebra. The creation of the 5 generators requires the rotation symmetry of one of the two component vector spaces to be preserved while the other is broken. It is convenient to identify the respective spaces as an observable real space and an unobservable ‘vacuum’ space, with corresponding physical properties. In combination the 5 generators produce a nilpotent structure which can be identified as a fermionic wavefunction or solution of the Dirac equation. The spinors required to generate the 4 components of the wavefunction can be derived from first principles and have exactly the same form as the four components of the Berw...

The Standard Model has three generations of fermions and antifermions, each with two states of isospin, and each of these has both a lepton and a quark in three possible colour states. In total there are 48 states. No known system exists for constructing these from first principles. Here, it is suggested that the number of degrees of freedom required is a consequence of the nilpotent complexified vector-quaternion Dirac algebra, which emerges from the representation of the fundamental parameters mass, time, charge and space as a Klein-4 group, and that these degrees of freedom lead to unique structural representations of each of the individual fermions.

Our notions of what is physically 'real' have long been based on the idea that the real is what is immediately apprehended, that is the local or observable, the physically tangible, though there has always been an alternative philosophical notion that the 'real' is some kind of ontological structure beyond immediate apprehension, and so inaccessible through physics. However, quantum mechanics, with its intrinsic nonlocal correlations, has seemingly left us with a dilemma by showing that fundamental physical theories cannot be both real and local. Reality cannot be reconstructed as a deterministic projection from physical observations. Many people think that the problem lies with quantum mechanics, but, in fact, it is more likely to be a result of unrealistic expectations. We have assumed that fundamental physics ought to be compatible with normal (macroscopic) experience. If, however, we go beyond our current high-level physical theories to the basic elements from which they are constructed, we see that a pattern emerges that gives us a very different and much more coherent understanding of what is meant by physical 'reality'.

Symmetry is everywhere in Nature and has a particularly significant role in physics at the most fundamental level. Here, we propose that the origin of all the fundamental symmetries is in a Klein-4 group structure that connects the fundamental parameters mass, time, charge and space. The algebras associated with these parameters emerge in a sequence which first generates real numbers, then complex numbers, quaternions and multivariate vectors. The combined algebra has a special significance in being identical to that of the Dirac equation of relativistic quantum mechanics. This is the equation that applies to the point-like fermion, the most fundamental physical state. Many other symmetries and the fundamental symmetry-breaking that occurs between the four physical interactions can be seen to emerge from this foundational symmetric structure.

The fermionic state is the foundation for the whole of physics. Physics is entirely concerned with fermions and their interactions, and nothing else. It is possible to derive a mathematical expression for the fermionic state, which is an operator only, not an equation, or wavefunction. This operator appears to contain within it all the information needed to construct fermion interactions and particle states. Extensions to particle representations using Finsler geometry could find this formalism a particularly accessible link.

A brief history is given of the factor 2, starting in the most elementary considerations of geometry and the kinematics of uniform acceleration, and moving to relativity, quantum mechanics and particle physics. The basic argument is that in all the significant cases in which the factor 2 or ½ occurs in fundamental physics, whether classical, quantum or relativistic, the same physical operation is taking place.

Vacuum can be defined with exact mathematical precision as the state which remains when a fermion, with all its special characteristics, is created out of absolutely nothing. The definition leads to a special form of relativistic quantum mechanics, which only requires the construction of a creation operator. This form of quantum mechanics is especially powerful for analytic calculation, at the same time as explaining, from first principles, many aspects of the Standard Model of particle physics. In particular, the characteristics of the weak, strong and electric interactions can be derived from the structure of the creation operator itself.

Based on a fundamental symmetry between space, time, mass and charge, a series of group structures of physical interest is generated, ranging from C2 to E8. The most significant result of this analysis is a version of the Dirac equation combining quaternions and multivariate vectors, which is already second quantized and intrinsically supersymmetric, and which automatically leads to a symmetry breaking, with the creation of specific particle structures and a mass-generating mechanism.

The nilpotent Dirac formalism has been shown, in previous publications, to generate new physical explanations for aspects of particle physics, with the additional possibility of calculating some of the parameters involved in the Standard Model. The applications so far obtained are summarised, with an outline of some more recent developments.

It is proposed that the Dirac equation, as normally interpreted, incorporates intrinsic redundancies whose removal necessarily leads to an enormous gain in calculating power and physical interpretation. Streamlined versions of the Dirac equation can be developed which remove the redundancies and singularities from many areas of quantum physics while giving quantum representations to specific particle states.

This is a very big, very deep, book about a proper foundation for mathematical physics. It is a wonderful book, with much information, some of which you may have encountered before in a fragmented way, but can now appreciate much better as part of a coherent whole, presented in this book. The book is about physics, but is not about any sort of model-dependent physical theory; for example, fluid dynamic aether theory, or quark-based elementary particle theory, or electromagnetic, or gravitational, or quantum, field theory. It does not buy into any sort of detail-dependent origin for physical laws, as in Big Bang theory. The book is about a process for the construction of a mathematical edifice that embraces much that we know to be needed in physics, and leaves little to be chosen arbitrarily among too many options. In fact, it leaves nothing important to be chosen arbitrarily. Contrast that to current string theory! The basic premise here is that "physics is the inherent creator of mathematics." (p. 47) I can"t really agree on that. Some of mathematics, such as the part discussed in this book, may be about physical reality, but a lot of it is about other things, such as probability, actuarial science, game theory, gambling, financial engineering, and other potentially destructive human activities! But that doesn"t matter; the book has inherent value. The book does not start with the usual concepts that have always been considered "fundamental" in mathematics, like counting by integers. Such counting actually presupposes quite a lot. On p. 83, Rowlands suggests that the premature adoption of the integers may lead to the system-incompleteness problems identified by Gödel. Interesting thought! I hope to find some response to it from A.A. Nassikas, [author of "Electromagnetic Space-Time-Ether", pp. 180-182, and of earlier NPA papers as well, and of the book Minimum Contradictions Everything, available at the 2009 NPA conference, and ever after from Hadronic Press]. There are also other NPA authors who may tell us something interesting in response to Rowland"s book. I look to Zbigniew Oziewicz, who invited me in on "Electric and Magnetic Fields According to Hermann Minkowski", p. 183-194. And I * Note by author. Correspondence between the author and reviewer has established that the symbolic convention employed on p. 26 is used only in describing the historical background, and is clearly distinguished from both modern conventions and the author"s own symbolism.

In order to address the central question at the heart of the concluding debate at the AAAI 07 Spring quantum interaction symposium, where it was proposed that other probabilistic and nondeterministic theories might replace quantum mechanics, even in relation to quantum physics, and in further explanation of our AAAI Spring 2007 paper 'How Intelligence Evolved?', we outline the derivation from first principles of what we call the universal rewrite system. For, once derived, this system has all the appearances of being nature's semantic computational machine order code, readily identifiable with a new version of quantum mechanics, and powerful predictive capabilities. These, fully detailed in the book Zero to Infinity, now published by World Scientific (Rowlands, 2007), especially in chapter 20, which is a joint contribution by the authors of this paper, are in total agreement with experiment so far, in particular in elementary particle physics, magnetic nuclear spectro...

A core level of basic information for physics is identified, based on an analysis of the characteristics of the parameters space, time, mass and charge. At this level, it is found that certain symmetries operate, which can be used to explain certain physical facts and even to derive new mathematical theorems. Applications are made to classical mechanics, electromagnetic theory and quantum mechanics.

It is shown that Schrödinger’s equation may be derived from three postulates. The first is a kind of statistical metamorphosis of classical mechanics, a set of two relations which are obtained from the canonical equations of particle mechanics by replacing all observables by statistical averages. The second is a local conservation law of probability. The third is energy conservation in the mean. The fact that Schrödinger’s equation may be derived from these premises, which are all purely statistical in character, is interpreted as an argument in favour of the statistical interpretation of quantum mechanics.

We present several ideas in direction of physical interpretation of qand f -oscillators as a nonlinear oscillators. First we show that an arbitrary one dimensional integrable system in action-angle variables can be naturally represented as a classical and quantum f -oscillator. As an example, the semi-relativistic oscillator as a descriptive of the Landau levels for relativistic electron in magnetic field is solved as an f -oscillator. By using dispersion relation for q-oscillator we solve the linear q-Schrödinger equation and corresponding nonlinear complex q-Burgers equation. The same dispersion allows us to construct integrable q-NLS model as a deformation of cubic NLS in terms of recursion operator of NLS hierarchy. Peculiar property of the model is to be completely integrable at any order of expansion in deformation parameter around q = 1. As another variation on the theme, we consider hydrodynamic flow in bounded domain. For the flow bounded by two concentric circles we formulate the two circle theorem and construct solution as the q-periodic flow by non-symmetric q-calculus. Then we generalize this theorem to the flow in the wedge domain bounded by two arcs. This two circular-wedge theorem determines images of the flow by extension of q-calculus to two bases: the real one, corresponding to circular arcs and the complex one, with q as a primitive root of unity. As an application, the vortex motion in annular domain as a nonlinear oscillator in the form of classical and quantum f-oscillator is studied. Extending idea of q-oscillator to two bases with the golden ratio, we describe Fibonacci numbers as a special type of q-numbers with matrix Binet formula. We derive the corresponding golden quantum oscillator, nonlinear coherent states and Fock-Bargman representation. The spectrum of it satisfies the triple relations, while the energy levels relative difference approaches asymptotically to the golden ratio and has no classical limit.

We propose here that certain observational features of granular matter in the infrared limit, exhibiting the phenomenon of jamming, arise from an underlying effective general relativistic description. The proposal stems from the assumption (which we justify on physical grounds) that grains in granular matter move freely in an effective curved Riemannian space. The termination of their trajectories at the onset of jamming is obtained from the focussing of a converging congruence of geodesics in such a space, as a solution of the Raychaudhuri equation for such congruences. This may happen irrespective of whether or not the curvature is sourced by external stresses (via an effective Einstein equation), although the properties of the resultant jammed state solution do differ in the two cases. A definite prediction of this geometrical approach is the negative role played by those trajectories which twist about each other, in reaching the jammed state. The local symmetries of granular interaction, translational and rotational invariance (corresponding to 'force balance' and 'torque balance' in standard force-based approaches to jamming) are inherent in the effective general relativity framework. A recently-proposed effective elasticity model of the jammed state, based on a tensorial variant of standard electrostatics (Vector Charge Theory), is seen to be entirely subsumed within the linearized version of the effective general relativistic description.

Inflationary spacetimes have been argued to be past geodesically incomplete in many situations. However, whether the geodesic incompleteness implies the existence of an initial spacetime curvature singularity or whether the spacetime may be extended (potentially into another phase of the universe) is generally unknown. Both questions have important physical implications. In this paper, we take a closer look at the geometrical structure of inflationary spacetimes and investigate these very questions. We first classify which past inflationary histories have a scalar curvature singularity and which might be extendible and/or non-singular in homogeneous and isotropic cosmology with flat spatial sections. Then, we derive rigorous extendibility criteria of various regularity classes for quasi-de Sitter spacetimes that evolve from infinite proper time in the past. Finally, we show that beyond homogeneity and isotropy, special continuous extensions respecting the Einstein field equations with a perfect fluid must have the equation of state of a de Sitter universe asymptotically. An interpretation of our results is that past-eternal inflationary scenarios are most likely physically singular, except in situations with very special initial conditions.

Please note, this is a draft preview!

Space, like time, may not be fundamental. By reformulating the Hamiltonian without reference to coordinates or geometry, and replacing it with recursive structure (graph Laplacian), the paper shows that quantum dynamics unfold identically. This invites a reconsideration of space's ontological role.