Discrete Spacetime Research Papers - Academia.edu (original) (raw)

We present a novel derivation of both the Minkowski metric and Lorentz transformations from the consistent quantification of a causally ordered set of events with respect to an embedded observer. Unlike past derivations, which have relied... more

We present a novel derivation of both the Minkowski metric and Lorentz transformations from the consistent quantification of a causally ordered set of events with respect to an embedded observer. Unlike past derivations, which have relied on assumptions such as the existence of a 4-dimensional manifold, symmetries of space-time, or the constant speed of light, we demonstrate that these now familiar mathematics can be derived as the unique means to consistently quantify a network of events. This suggests that space-time need not be physical, but instead the mathematics of space and time emerges as the unique way in which an observer can consistently quantify events and their relationships to one another. The result is a potential foundation for emergent space-time.

Abstract: In questi ultimi anni si sono sviluppate diverse teorie di quantizzazione primordiale sulla scala di Planck. Il loro obiettivo è quello di fornire un modello di vuoto in grado di sostenere la ricerca oltre il Modello Standard.... more

Abstract: In questi ultimi anni si sono sviluppate diverse teorie di quantizzazione primordiale sulla scala di Planck. Il loro obiettivo è quello di fornire un modello di vuoto in grado di sostenere la ricerca oltre il Modello Standard. Benché il loro obiettivo sia dunque assai più ambizioso e punti in
direzione della fisica delle particelle, una conseguenza necessaria notevole è una lettura emergente
della Meccanica Quantistica. Sono possibili diverse ipotesi sulle celle elementari. In questa sede ci soffermeremo principalmente sugli aspetti concettuali delle teorie di G. ‘t Hooft e F. Winterberg, con un attenzione speciale per l’emergere delle correlazioni non–locali.

Modern physics is based on two paradigms that emerged from the Second Scientific Revolution - relativity and the quantum. Since their inception these two theoretical giants have advanced science and society far beyond anything that their... more

Modern physics is based on two paradigms that emerged from the Second Scientific Revolution - relativity and the quantum. Since their inception these two theoretical giants have advanced science and society far beyond anything that their originators could have even dreamed up, but in a sense they seem to be progressing in different directions and are regarded mutually incompatible. Yet they come to a point in history where the next step in their own advancement would be to unify into a single overwhelming physical paradigm. The concept of unification is well known in relativity theory with many attempts dating from just a few short years after Einstein first developed relativity. But the hyperspace unification that seems to have originated with Kaluza in 1921 and was modified by Klein a few years later has become the best known attempt. This book has two parts. The first, Hyperspace, deals with the original Kaluza theory in all of its aspects and incarnations up until the 1960s. After the 1960s the idea of unification was adopted by quantum theoreticians based on their belief that the quantum is more fundamental than relative space-time. However, since the 1980s Klein’s adaptation of Kaluza’s theory has been used to unify physics under the quantum banner as has the purely quantum theory Standard Model. Yet both of these quantum attempts have fundamental problems that can only be solved by returning to Kaluza’s original theory as modified by Einstein and Bergmann in the 1930s. So the second part of the book, Continuum, tells the rest of the story and offers a new theory, a five-dimensional single field theory, the merges the best of both paradigms together. By merging the two paradigms instead of replacing one by the other, single field theory preserves all of the many successes of both theories. This is the book that every physicist, physical scientist and non-professional science advocate should read and every theoretical physicist and physics students must read.

Attempt is made to shed more Light on the concept of conjugatemirrored space-time dimensions by being able to better visualize them, as these dimensions are stealth to one another and one represents the ethereal aspect of the other. This... more

Attempt is made to shed more Light on the concept of conjugatemirrored space-time dimensions by being able to better visualize them, as these dimensions are stealth to one another and one represents the ethereal aspect of the other. This was inspired by a question posed by a soul-seeker and a spiritual scholar who had a problem comprehending or visualizing them.

A fundamental asymmetry currently exists between the Einstein-Minkowski definition of a single unified spacetime and the separation of space and time variables required by wave theory. The source of this asymmetry is traced back to... more

A fundamental asymmetry currently exists between the Einstein-Minkowski definition of a single unified spacetime and the separation of space and time variables required by wave theory. The source of this asymmetry is traced back to Einstein's definition of time dilation which, by following the 'world line' of a discrete material particle, not only contravenes Heisenberg's Uncertainty Principle but also the more general condition that time-frequency measurement must be carried out at a single rest point in space relative to each inertial system. Redefining space-time accordingly, the space and time axes for "moving" systems, expressed in the coordinates of the "stationary" system, become identical to the phase and group velocities of spherical standing waves. By recognising that Einstein's "array" of synchronised clocks, and the Michelson-Morely experiment upon which it was based, has all of the salient features of equal and opposite standing waves, the Lorentz Transformation Equations can then be directly deduced as the wave arguments of these standing wave motions when transformed to other systems of coordinates. Only under this definition can the principle of relativity be upheld. Assigning the proper frame to the observer has the effect of inverting time dilation while leaving the mathematical structure essentially in tact; the standard time-like, space-like and light-like interpretations of the Minkowski interval, for example, find their most natural expression depending upon whether we are measuring frequency, wavelength, or the wavefront of an electromagnetic wave. Inasmuch as energy-momentum depend upon frequency-wavelength, then the equivalence of mass and energy still holds under this interpretation. However, this is no mere trivial change in our `point of view'. Time is now proportional to and transformed together with length, which was always a fundamental condition of wave theory. Since the array of clocks have been synchronised via light waves of universal speed c according to Einstein's second principle, then this leads to a satisfactory definition of time "in general". Reviewing some of the experimental evidence or arguments that are usually cited in defence of time dilation - the Transverse Doppler Effect, the Twins Paradox and acceleration - it is shown that they can and must be reinterpreted in terms of time contraction if the equations are to remain consistent. The empirical evidence obtained by Ives-Stillwel, Hafele-Keating et al. merely proved the existence of a Transverse Doppler Effect, which is here identified with time contraction; the "stationary" clocks, now defined as objective wave motions, have been contracted with respect to clocks in relative motion and not dilated inversely. Since the empirical predictions are precisely the same as the conventional interpretation - it is the "stationary" clock's that are running fast - then it becomes a clear case of affirming the consequent. Hence, all of this implies that our sense of space and time, and perhaps even our biological apparatus used in sensory perception, evolved from our experience of observable wave phenomena and not post hoc from the rods and clocks of our own making. As a final consideration, if we generalise this Minkowski metric for curved space-time, then the motion of quantum wave-particles under the influence of gravity should follow as a matter of course.

This work logically and scientifically explores the possibility that a universe comprising of matter, just, existing, moveing, and interacting, may be enough to mislead us into wrongly assuming that a 'temporal past', 'future', and thus... more

This work logically and scientifically explores the possibility that a universe comprising of matter, just, existing, moveing, and interacting, may be enough to mislead us into wrongly assuming that a 'temporal past', 'future', and thus thing called 'time' exist.
In other words, how we -may- be wrong to assume the existence of unobservable, intangible, 'time', from the outset.

In this paper, I introduce a particular discrete spacetime that should be seriously considered as part of physics because it allows to explain the characteristics of the motion properly, contrary to what happens with the continuous... more

In this paper, I introduce a particular discrete spacetime that should
be seriously considered as part of physics because it allows to explain the characteristics of the motion properly, contrary to what happens with the continuous spacetime of the common conception.

Causal set theory and the theory of linear structures (which has recently been developed by Tim Maudlin as an alternative to standard topology) share some of their main motivations. In view of that, I raise and answer the question how... more

Causal set theory and the theory of linear structures (which has recently been developed by Tim Maudlin as an alternative to standard topology) share some of their main motivations. In view of that, I raise and answer the question how these two theories are related to each other and to standard topology. I show that causal set theory can be embedded into Maudlin's more general framework and I characterise what Maudlin's topological concepts boil down to when applied to discrete linear structures that correspond to causal sets. Moreover, I show that all topological aspects of causal sets that can be described in Maudlin's theory can also be described in the framework of standard topology. Finally, I discuss why these results are relevant for evaluating Maudlin's theory. The value of this theory depends crucially on whether it is true that (a) its conceptual framework is as expressive as that of standard topology when it comes to describing well-known continuous as well as discrete models of spacetime and (b) it is even more expressive or fruitful when it comes to analysing topological aspects of discrete structures that are intended as models of spacetime. On one hand, my theorems support (a). The theory is rich enough to incorporate causal set theory and its definitions of topological notions yield a plausible outcome in the case of causal sets. On the other hand, the results undermine (b). Standard topology, too, has the conceptual resources to capture those topological aspects of causal sets that are analysable within Maudlin's framework. This fact poses a challenge for the proponents of Maudlin's theory to prove it fruitful.

This work proposes a mathematical model about how a reaction is created in the human brain in response to a particular incoming Information/Event using quantum mechanics and more precisely path integrals theory. The set of action... more

This work proposes a mathematical model about how a reaction is created in the human brain in response to a particular incoming Information/Event using quantum mechanics and more precisely path integrals theory. The set of action potentials created in a particular neuron N2 is a result of temporal and spatial summation of the signals coming from different neighboring neurons Nx with different dendrite-paths. Each dendrite-path of N2 is assumed to be determined by its respective synapse with its neurotransmitters and assumed to have an action S due to the neurotransmitter types (for example: excitatory or inhibitory). An external incoming signal information being initially modulated by receptor neurons (in eyes, ears...) travels through the neighboring neurons that are linked to the excited receptor neurons. A potential reaction responses are subsequently created thanks to a final deformed signal in the motor neurons by all the correlated neural paths. The total deformation at each neuron is created by different incoming dendrite-paths and their structures (inhibitory or excitatory neurotransmitters and their type), and of course the existence or not of the signal and its frequency coming from each path. Using path Integrals theory, we compute the probability of existence of the signal-Information or the potential reaction to the incoming information at each neuron. In this paper we also compute how much the signal-Information has been distorted between two neighboring linked neural points including if it arrives or not to the neighboring neurons. We propose an Information entropy similar to Shannon one and we demonstrate that this entropy is equivalent to timespace curvature in the Brain.

The central motivating idea behind the development of this work is the concept of prespace, a hypothetical structure that is postulated by some physicists to underlie the fabric of space or space-time. I consider how such a structure... more

The central motivating idea behind the development of this work is the concept of prespace, a hypothetical structure that is postulated by some physicists to underlie the fabric of space or space-time. I consider how such a structure could relate to space and space-time, and the rest of reality as we know it, and the implications of the existence of this structure for quantum theory. Understanding how this structure could relate to space and to the rest of reality requires, I believe, that we consider how space itself relates to reality, and how other so-called "spaces" used in physics relate to reality. In chapter 2, I compare space and space-time to other spaces used in physics, such as configuration space, phase space and Hilbert space. I support what is known as the "property view" of space, opposing both the traditional views of space and space-time, substantivalism and relationism. I argue that all these spaces are property spaces. After examining the relationships of these spaces to causality, I argue that configuration space has, due to its role in quantum mechanics, a special status in the microscopic world similar to the status of position space in the macroscopic world. In chapter 3, prespace itself is considered. One way of approaching this structure is through the comparison of the prespace structure with a computational system, in particular to a cellular automaton, in which space or space-time and all other physical quantities are broken down into discrete units. I suggest that one way open for a prespace metaphysics can be found if physics is made fully discrete in this way. I suggest as a heuristic principle that the physical laws of our world are such that the computational cost of implementing those laws on an arbitrary computational system is minimized, adapting a heuristic principle of this type proposed by Feynman. In chapter 4, some of the ideas of the previous chapters are applied in an examination of the physics and metaphysics of quantum theory. I first discuss the "measurement problem" of quantum mechanics: this problem and its proposed solution are the primary subjects of chapter 4. It turns out that considering how quantum theory could be made fully discrete leads naturally to a suggestion of how standard linear quantum mechanics could be modified to give rise to a solution to the measurement problem. The computational heuristic principle reinforces the same solution. I call the modified quantum mechanics Critical Complexity Quantum Mechanics (CCQM). I compare CCQM with some of the other proposed solutions to the measurement problem, in particular the spontaneous localization model of Ghirardi, Rimini and Weber. Finally, in chapters 5 and 6, I argue that the measure of complexity of quantum mechanical states I introduce in CCQM also provides a new definition of entropy for quantum mechanics, and suggests a solution to the problem of providing an objective foundation for statistical mechanics, thermodynamics, and the arrow of time.

It is generally believed that physical laws, reflecting an inherent order in the universe, are ordained by nature. However, in modern physics the observer plays a central role raising questions about how an observer-centric physics can... more

It is generally believed that physical laws, reflecting an inherent order in the universe, are ordained by nature. However, in modern physics the observer plays a central role raising questions about how an observer-centric physics can result in laws apparently worthy of a universal nature-centric physics. Over the last decade, we have found that the consistent apt quantification of algebraic and order-theoretic structures results in calculi that possess constraint equations taking the form of what are often considered to be physical laws. I review recent derivations of the formal relations among relevant variables central to special relativity, probability theory and quantum mechanics in this context by considering a problem where two observers form consistent descriptions of and make optimal inferences about a free particle that simply influences them. I show that this approach to describing such a particle based only on available information leads to the mathematics of relativistic quantum mechanics as well as a description of a free particle that reproduces many of the basic properties of a fermion. The result is an approach to foundational physics where laws derive from both consistent descriptions and optimal information-based inferences made by embedded observers.

Zatrikean progeometry is a model based on the assumption that the space is discrete. The discrete space is called abacus and looks like a chess board. Αny set of restrictions imposed on the motion of particles on the abacus is called... more

Zatrikean progeometry is a model based on the assumption that the space is discrete. The discrete space is called abacus and looks like a chess board. Αny set of restrictions imposed on the motion of particles on the abacus is called premetrics. Several premetrics are considered with emphasis on those induced by rotation. The concept of the entropy of a curve leads to the ability of interrelating pregeometry and thermodynamics. Furthermore, the maximum entropy theorem assigns the maximum probability to the thermodynamic geodesic. We apply zatrikean pregeometry to the study of perihelion shift, photon deflection, radar echo delay, double pulsars, polytropic stars, black holes and cosmological scenarios with variation of the mass of the universe, of the physical constants, or of the zatrikean constant of dilation.

Over the years, a number of measures have been defined for the purpose of determining the number of independent dimensions contained in a space. The most common dimensionality measures are the topological dimensionality and various kinds... more

Over the years, a number of measures have been defined for the purpose of determining the number of independent dimensions contained in a space. The most common dimensionality measures are the topological dimensionality and various kinds of fractal dimensionalities. While each of these dimensionality measures is useful in its own right, none of them accurately quantifies the effective number of independent directions passing through locations contained in a local region of a space. This article introduces a new dimensionality measure, called the connectivity dimensionality field, which is the true measure for the effective number of independent directions passing through locations in a space. In contrast to the fractal dimensionality, the connectivity dimensionality field is a topological property because its value at each material location is invariant to deformations of the space preserving connectivity. The connectivity dimensionality field is a fundamental concept that applies to many different kinds of discrete spaces, continuous spaces, and discrete-continuous dual spaces. A discrete space is a space in which positions cannot be varied differentially, and a continuous space is a space in which positions can be varied differentially. A discrete-continuous dual space has complementary discrete and continuous representations, and a process called discrete-continuous dual matching relates the discrete and continuous representations to each other. This article formally defines two basic types of discrete-continuous duality: (a) asymptotic and (b) strict. A rigorous method is provided for computing the connectivity dimensionality field in edge-vertex graphs, continuous spaces, and discrete-continuous dual spaces. Many examples are given to illustrate the key concepts. Single points, unbranched lines, and periodic lattices are examples of discrete-continuous dual spaces in which the connectivity dimensionality field is a constant nonnegative integer. In other types of discrete-continuous dual spaces, the connectivity dimensionality field contains inherent uncertainty. For the first time, a comprehensive theory is derived that predicts the inherent uncertainty associated with the connectivity dimensionality field in discrete-continuous dual spaces. The study of discrete-continuous dual spaces with variable connectivity dimensionality fields transcends variable-based mathematics.

A new model of space in the form of a Planck scale, deformable, discrete lattice is proposed. The local deformations of the lattice, which are limited in number and precisely defined in structure, are identified with the standard model's... more

A new model of space in the form of a Planck scale, deformable, discrete lattice is proposed. The local deformations of the lattice, which are limited in number and precisely defined in structure, are identified with the standard model's elementary particles. The lattice is proposed as the ground from which quantum field theory's vacuum state and relativity's spacetime both arise simultaneously as emergent theoretical frameworks. We refer to the framework that encompasses these ideas as synergetic lattice field theory: synergetic because of its roots in Buckminster Fuller's synergetic geometry and lattice field theory because of its description of an all-pervading, discrete field within which all the observed phenomena of the universe arise. We refer to this universal lattice as the synergetic field.

This paper proposes a set of relatively new conjectures and hypotheses in modern physics, mainly concept of subquantum movement (SQM), the finite “elasticity” of (charged/neutral) spacetime hypothesis (FESTH) (a unifying concept which may... more

This paper proposes a set of relatively new conjectures and hypotheses in modern physics, mainly concept of subquantum movement (SQM), the finite “elasticity” of (charged/neutral) spacetime hypothesis (FESTH) (a unifying concept which may bring under the “same umbrella” both Einstein’s General relativity [EGR] and Quantum Field Theory [QFT]), the self-repulsiveness of electromagnetic charge (SR-EMC) and the gravitational significance (GS) of the fine structure constant (GS-FSC): each conjecture (or hypothesis) in part is based on at least one observation and generates some interesting predictions. This paper continues (from alternative angles of view!) the work of other past articles/preprints of the same author.
#DONATIONS. Anyone can donate for dr. Dragoi’s independent research and original music at: https://www.paypal.com/donate/?hosted_button_id=AQYGGDVDR7KH2

Wu's Spacetime is a four dimensional system based on Wu's Unit Length l yy and Wu's Unit Time t yy which are related to each other by Wu's Spacetime Theory t yy = γl yy 3/2. Einstein's Spacetime is a special Wu's Spacetime based on earth.... more

Wu's Spacetime is a four dimensional system based on Wu's Unit Length l yy and Wu's Unit Time t yy which are related to each other by Wu's Spacetime Theory t yy = γl yy 3/2. Einstein's Spacetime is a special Wu's Spacetime based on earth. According to Yangton and Yington Theory, Wu's Unit Length l yy on a massive star is much bigger than l yy0 on earth. Because a ∞ C-4 ∞ l yy 2 , the Amount of Normal Unit Acceleration "a" measured on the star is much bigger than "a 0 "measured on earth. In other words, for a massive star, Wu's Spacetime Field Equation measured on the star has much deeper slope (curvature) than that of Einstein's Field Equation measured on earth. Furthermore, because of the large Wu’s Unit Length lyy caused by the huge gravitational force, a hollow structure in the center of a black hole is expected. Also because of the Photon Inertia Transformation and the large acceleration in the center of a black hole based on Wu’s Spacetime Field Equations, it is predicted that photon can be trapped inside the event horizon of a black hole.

Editor: Ignazio Licata ( ISEM, Institute for Scientific Methodology, Palermo and School of Advanced International Studies on Applied Theoretical and Non Linear Methodologies of Physics, Bari. G. 't Hooft Foreword Authors: B. Hiley,... more

Editor: Ignazio Licata ( ISEM, Institute for
Scientific Methodology, Palermo and School of Advanced
International Studies on Applied Theoretical and Non Linear Methodologies of
Physics, Bari.
G. 't Hooft Foreword
Authors: B. Hiley, Maurice de Gosson, Geoffrey F. Chew, D. Dolce, George Jaroszkiewicz,Rui Vilela Mendes,R. Zapatrin,Cecilia Flori, Reiner Hedrich, P. Jarvis, J. Munkhammar, H. Kleinert,T. Elze,Manfred Requardt,Erasmo Recami,G. Chapline,M. Consoli, G. Vitiello,R. Kastner, W. M. Stuckey, L. Chiatti, S. Vacaru, M. Pavsic, Imperial College Press.

We analyze the possible implications of spacetime discreteness for the special and general relativity and quantum theory. It is argued that the existence of a minimum size of spacetime may explain the invariance of the speed of light in... more

We analyze the possible implications of spacetime discreteness for the special and general relativity and quantum theory. It is argued that the existence of a minimum size of spacetime may explain the invariance of the speed of light in special relativity and Einstein’s equivalence principle in general relativity. Moreover, the discreteness of spacetime may also result in the collapse of the wave function in quantum mechanics, which may provide a possible solution to the quantum measurement problem. These interesting results might have some important implications for a complete theory of quantum gravity.

This paper assumes the existence of a fabric of space that is locally Euclidean with a preferred coordinate system. These assumptions are shown to produce the special relativity transformations for two bodies in collinear motion. Of... more

This paper assumes the existence of a fabric of space that is locally Euclidean with a preferred coordinate system. These assumptions are shown to produce the special relativity transformations for two bodies in collinear motion. Of primary importance is the insight gained into the transformations of special relativity. There is an observational error factor and an actual component factor for each transformation. Purpose This papers allows one to view the fabric of space as being a structure that supports a preferred coordinate system. Special relativity is shown to be consistent with this view. The inability to detect when an observer is stationary in the structure of space does not prevent a stationary structure from existing. Special relativity allows an egocentric view for all observers. This can mislead one into believing special relativity precludes a stationary structure of space. It is important to realize that a stationary structure may be considered when one contemplates the fabric of space. The Michelson-Morley (M-M) experiment is considered by some to support rejection of an ether based preferred coordinate system consistent with Euclidean geometry. It seems that the MM null results must occur if the apparatus used for the experiment if the physical apparatus is held together electromagnetically and as such undergoes the same changes as the light waves being examined.

This paper proposes an extended Special relativity (eSR) containing a set of universal equivalence principles (UEPs), offering an alternative interpretation of the universal physical constants and predicting a "digital"/quantized... more

This paper proposes an extended Special relativity (eSR) containing a set of universal equivalence principles (UEPs), offering an alternative interpretation of the universal physical constants and predicting a "digital"/quantized spacetime, together with the possible existence of superluminal gravitons and a set of maximum speeds (in perfect vacuum) for each type of elementary particle.
Keywords: extended Special relativity (eSR), universal equivalence principles (UEPs); universal physical constants; “digital”/quantized spacetime; superluminal gravitons; set of maximum speeds (in perfect vacuum)
#DONATIONS. Anyone can donate for dr. Dragoi’s independent research and original music at: https://www.paypal.com/donate/?hosted_button_id=AQYGGDVDR7KH2

This set of notes act as development on, and supplement to the paper, 'The Emergence of Massive Equilibrium States from Fully Connected Stochastic Substitution Systems'. Topics covered in these notes develop on topics already introduced... more

This set of notes act as development on, and supplement to the paper, 'The Emergence of Massive Equilibrium States from Fully Connected Stochastic Substitution Systems'. Topics covered in these notes develop on topics already introduced in the paper, add extra detail, strengthen arguments and propose additional extensions to the structure of the system described in the paper.

We present a novel derivation of both the Minkowski metric and Lorentz transformations from the consistent quantification of a causally-ordered set of events with respect to an embedded observer. Unlike past derivations, which have relied... more

We present a novel derivation of both the Minkowski metric and Lorentz transformations from the consistent quantification of a causally-ordered set of events with respect to an embedded observer. Unlike past derivations, which have relied on assumptions such as the existence of a 4-dimensional manifold, symmetries of space-time, or the constant speed of light, we demonstrate that these now familiar mathematics can be derived as the unique means to consistently quantify a network of events. This suggests that space-time need not be physical, but instead the mathematics of space and time emerges as the unique way in which an observer can consistently quantify events and their relationships to one another. The result is a potential foundation for emergent space-time.

This article investigates the properties and emergent behaviour of a new kind of discrete substitution system. The micro-states of these systems are modelled as complete weighted graphs over N, the weights of which are stored in a state... more

This article investigates the properties and emergent behaviour of a new kind of discrete substitution system. The micro-states of these systems are modelled as complete weighted graphs over N, the weights of which are stored in a state matrix S t = {s i j } t , and evolve via a set of constituent specific, independent probabilistic substitution rules. The state is then embedded in R n by treating flat space geometrical violations as internal curvature within the system that may be minimized via a process of stress minimization. This paper gives arguments for a definition of energy within the system, observes an emergent intrinsic inertia affecting vertex clusters, and shows the emergence of particle like massive equilibrium states. An retentive effect due to clustering is also observed within this system.

We attempt a connection between thermodynamics and zatrikean pregeometry, i.e., a chess-like pregeometry. In zatrikean pregeometry space is represented by the abacus, a discrete chessboard-like structure consisting of a sufficiently large... more

We attempt a connection between thermodynamics and zatrikean pregeometry, i.e., a chess-like pregeometry. In zatrikean pregeometry space is represented by the abacus, a discrete chessboard-like structure consisting of a sufficiently large number of plaquettes called geobits. The particles move on the abacus from one geobit to the next following certain rules that resemble the game of chess. The sets of rules imposed on the motions of particles on the abacus are called premetrics. There is a variety of paths (called subabaces) leading from one geobit to another, and there is a class consisting of subabaces with the minimum number of geobits. These are called alyssoids (respectively, class of alyssoids) for the particular premetric, while those alyssoids with minimum length are called geodesics (respectively, class of geodesics) for the particular premetric. The so-called zatrikean geodesic was originally defined in G93 (Section 2) as the geodesic most closely following the line segment joining the two geobits. It is also called algorithmic geodesic since it is drawn with the assistance of four simple algorithms. This is a rectifiable curve; and a connection between rectifiable curves and thermodynamics is already available (DuPain, Kamae and Mendes-France 1986). Consequently, the so-called thermodynamic geodesic is defined as the particular member of the class of geodesics with maximum entropy. Since it does not necessarily correspond to the algorithmic geodesic, a new algorithm is devised that draws the geodesic with maximum entropy. Furthermore, the probability of each member of the class of geodesics can be determined as the difference of its entropy from the entropy of the thermodynamic geodesic.

The purpose of this essay is to propose a perspective about the nature of space in the context of quantum mechanics. We will argue that it could be heuristically advantageous to consider the space of quantum entities as being discrete. We... more

The purpose of this essay is to propose a perspective about the nature of space in the context of quantum mechanics. We will argue that it could be heuristically advantageous to consider the space of quantum entities as being discrete. We show that the hypothesis of discrete (well call it this way as opposed to the continuum hypothesis) does not involves any substantial change in quantum mechanics and may offer a conceptual framework for the understanding of the success of certain mathematical tools actually used in the application of the theory.