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. (shrink)

Recent work in quantum gravity has prompted a re-evaluation of the fundamental nature of spacetime. Spacetime is potentially emergent from non-spatiotemporal entities posited by a theory of quantum gravity. Recent efforts have sought to interpret the relationship between spacetime and the fundamental entities through a mereological framework. These frameworks propose that spacetime can be conceived as either having non-spatiotemporal entities as its constituents or being a constituent part of a non-spatiotemporal structure. I present a roadmap for (...) those interested in exploring the role of composition in understanding the emergence of spacetime. I establish a taxonomy based on four crucial parameters that should be considered when constructing potential mereological models. Subsequently, I connect these models to a spectrum of perspectives found in current literature, with the aim of pinpointing areas that require further exploration. Finally, I identify three potential challenges facing mereological models of spacetime emergence, rooted in issues of mereological harmony, the distinction between continuous and discrete spacetime, and the implications of quantum superposition. (shrink)

Quantum gravity is understood as a theory that, in some sense, unifies general relativity (GR) and quantum theory, and is supposed to replace GR at extremely small distances (high-energies). It may be that quantum gravity represents the breakdown of spacetime geometry described by GR. The relationship between quantum gravity and spacetime has been deemed ``emergence'', and the aim of this thesis is to investigate and explicate this relation. After finding traditional philosophical accounts of emergence to be inappropriate, I (...) develop a new conception of emergence by considering physical case studies including condensed matter physics, hydrodynamics, critical phenomena and quantum field theory understood as effective field theory. This new conception of emergence is unconcerned with the ideas of reduction and derivation (i.e. it holds that we may have emergence with reduction or without it). Instead, a low-energy theory (or model) is understood as emergent from a high-energy theory if it is novel and autonomous compared to the high-energy theory, and the low-energy physics is dependent in a particular, minimal sense on the high-energy physics (this dependence is revealed by the techniques of effective field theory and the renormalisation group). While novelty is construed in a broad sense, the autonomy comes essentially from the underdetermination of the high-energy theory by the low-energy theory, which reflects the minimal way in which the emergent, low-energy theory depends on the high-energy one. It results from the scaling behaviour of the theories and the limiting relations between them, and is demonstrated by the renormalisation group and effective field theory techniques, the idea of universality, and the phenomenon of symmetry-breaking. These ideas are important in exploring the relationship between quantum gravity and GR, where GR is understood as an effective, low-energy theory of quantum gravity. Without experimental data or a theory of quantum gravity, we rely on principles and techniques from other areas of physics to guide the way. As well as considering the idea of emergence appropriate to treating GR as an effective field theory, I investigate the emergence of spacetime (and other aspects of GR) in several concrete approaches to quantum gravity, including examples of the condensed matter approaches, the ``discrete approaches'' (causal set theory, causal dynamical triangulations, quantum causal histories and quantum graphity) and loop quantum gravity. (shrink)

This paper presents a realistic, stochastic, and local model that reproduces nonrelativistic quantum mechanics results without using its mathematical formulation. The proposed model only uses integer-valued quantities and operations on probabilities, in particular assuming a discrete spacetime under the form of a Euclidean lattice. Individual particle trajectories are described as random walks. Transition probabilities are simple functions of a few quantities that are either randomly associated to the particles during their preparation, or stored in the lattice nodes they visit (...) during the walk. QM predictions are retrieved as probability distributions of similarly-prepared ensembles of particles. The scenarios considered to assess the model comprise of free particle, constant external force, harmonic oscillator, particle in a box, the Delta potential, particle on a ring, particle on a sphere and include quantization of energy levels and angular momentum, as well as momentum entanglement. (shrink)

We give an analysis over a variation of causal sets where the light cone of an event is represented by finitely branching trees with respect to any given arbitrary dynamics. We argue through basic topological properties of Cantor space that under certain assumptions about the universe, spacetime structure and causation, given any event x, the number of all possible future worldlines of x within the many-worlds interpretation is uncountable. However, if all worldlines extending the event x are ‘eventually deterministic’, (...) then the cardinality of the set of future worldlines with respect to x is exactly ℵ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\aleph _0$$\end{document}, i.e., countably infinite. We also observe that if there are countably many future worldlines with respect to x, then at least one of them must be necessarily ‘decidable’ in the sense that there is an algorithm which determines whether or not any given event belongs to that worldline. We then show that if there are only finitely many worldlines in the future of an event x, then they are all decidable. We finally point out the fact that there can be only countably many terminating worldlines. (shrink)

This book discusses the notion that quantum gravity may represent the "breakdown" of spacetime at extremely high energy scales. If spacetime does not exist at the fundamental level, then it has to be considered "emergent", in other words an effective structure, valid at low energy scales. The author develops a conception of emergence appropriate to effective theories in physics, and shows how it applies (or could apply) in various approaches to quantum gravity, including condensed matter approaches, discrete approaches, (...) and loop quantum gravity. (shrink)

We give a new argument supporting a gravitational role in quantum collapse. It is demonstrated that the discreteness of space-time, which results from the proper combination of quantum theory and general relativity, may inevitably result in the dynamical collapse of thewave function. Moreover, the minimum size of discrete space-time yields a plausible collapse criterion consistent with experiments. By assuming that the source to collapse the wave function is the inherent random motion of particles described by the wave function, we further (...) propose a concrete model of wavefunction collapse in the discrete space-time. It is shown that the model is consistent with the existing experiments and macroscopic experiences. (shrink)

This paper discusses the discrete symmetries of off-shell electromagnetism, the Stueckelberg–Schrodinger relativistic quantum theory and its associated 5D local gauge theory. Seeking a dynamical description of particle/antiparticle interactions, Stueckelberg developed a covariant mechanics with a monotonically increasing Poincaré-invariant parameter. In Stueckelberg’s framework, worldlines are traced out through the parameterized evolution of spacetime events, which may advance or retreat with respect to the laboratory clock, depending on the sign of the energy, so that negative energy trajectories appear as antiparticles when (...) the observer describes the evolution using the laboratory clock. The associated gauge theory describes local interactions between events (correlated by the invariant parameter) mediated by five off-shell gauge fields. These gauge fields are shown to transform tensorially under under space and time reflections—unlike the standard Maxwell fields—and the interacting quantum theory therefore remains manifestly Lorentz covariant. Charge conjugation symmetry in the quantum theory is achieved by simultaneous reflection of the sense of evolution and the fifth scalar field. Applying this procedure to the classical gauge theory leads to a purely classical manifestation of charge conjugation, placing the CPT symmetries on the same footing in the classical and quantum domains. In the resulting picture, interactions do not distinguish between particle and antiparticle trajectories—charge conjugation merely describes the interpretation of observed negative energy trajectories according to the laboratory clock. (shrink)

Since the emergence of computing as a mode of investigation in the sciences, computational approaches have revolutionised many fields of inquiry. Recently in philosophy, the question has begun rendering bit by bit—could computation be considered a deeper fundamental building block to all of reality? This paper proposes a continuum computing construct, predicated on a set of core computational principles: computability, discretisation, stability and optimisation. The construct is applied to the set of most fundamental physical laws, in the form of non-relativistic (...) conservation equations which underpin our governing physics. The discretisation approach divides all of space into a mesh of discrete computational cells, and evolution of data in those cells must obey pre-defined stability conditions from established continuum computing theory, namely, the Courant–Friedrichs–Lewy condition. Evolving cell-state data in a manner which logically optimises computational efficiency, combined with the defined stability condition, the construct derives a central coupling of space, time, and the fastest speed of information propagation. This coupling formed at the lowest level by the computing construct, naturally and inherently produces aspects of special relativity and general relativity at the macroscale. This paper therefore proposes a new explanation of why the nature of space and time may be fused, and explores simple emergent congruities with relativity. The theory invites us to reverse the philosophical premise of the original Simulation Argument. This argument currently leads us to consider: how might all of the observed physics of our reality be reproduced computationally in a simulation of our existence? This work asks instead: could foundations of the physics of our universe—namely spacetime coupling—emerge inherently, and necessarily, from fundamental principles of computation? (shrink)

The physical, mathematical, and philosophical foundations of the quantum theory of free Bose fields in fixed general relativistic spacetimes are examined. It is argued that the theory is logically and mathematically consistent whereas semiclassical prescriptions for incorporating the back-reaction of the quantum field on the geometry lead to inconsistencies. Still, the relations and heuristic value of the semiclassical approach to canonical and covariant schemes of quantum gravity-plus-matter are assessed. Both conventional and rigorous formulations of the theory and of its principal (...) predictions, cosmological particle creation and horizon radiation, are expounded and compared. Special attention is devoted to spacetime properties needed for the existence or uniqueness of the relevant theoretical elements , renormalization of the stress tensor). The emergence of unitarily inequivalent representations in a single dynamical context is used as motivation for the introduction of the abstract $\rm C\sp{\*}$-algebraic axiomatic formalism. The operationalist and conventionalist claims of the original abstract algebraic program are criticized in favor of its tempered outgrowth, local quantum physics. The interpretation of the theory as a wave mechanics of classical field configurations, deriving from the Schrodinger representations of the abstract algebra, is discussed and is found superior, at least on the level of analogy, to particle or harmonic oscillator interpretations. Further, it is argued that the various detector results and the Fulling nonuniqueness problem do not undermine the particle concept in the ways commonly claimed. In particular, arguments are offered against the attribution of particle status to the Rindler quanta, against the physical realizability of the Rindler vacuum, and against the more general notion of observer-dependence as to the definition of 'particle' or 'vacuum'. However, the question of the ontological status of particles is raised in terms of the consistency of quantum field theory with non-reductive realism about particles, the latter being conceived as entities exhibiting attributes of discreteness and localizability. Two arguments against non-reductive realism about particles, one from axiomatic algebraic local quantum theory in Minkowski spacetime and one from quantum field theory in curved spacetime, are developed. (shrink)

The causal set programme towards a quantum theory of gravity is situated vis-à-vis the long-standing debate between eternalism (block theory) and past-presentism or possibilism (growing block theory) in the philosophy of time. It is argued that despite 'appearances' and declarations to the contrary, the programme does not side with growing block theorists when it comes to harboring a robust notion of Becoming - at least, not more than familiar relativistic theories on continuous spacetime manifolds. The problem stems mainly from (...) the postulate of discrete general covariance - a requirement imposed upon the only fully worked out kind of dynamics for causal sets to date, a dynamics of a classical stochastic process. (shrink)

The main project of this dissertation is to analyze various temporal conceptions of modality for discrete n-dimensional spacetime. The first chapter contains an introduction to the problem and known results. Chapter 2 consists of a study of logics which are analogues of the so-called 'logic of today and tomorrow' and 'logic of tomorrow' investigated by Segerberg and others. We consider the analogues of these successor logics for 2-dimensional integral spacetime. We provide axiomatizations in monomodal and multimodal languages and (...) prove completeness theorems. We also establish that the irreflexive successor logic in the "standard" modal language is not finitely axiomatizable. ;Chapter 3 investigates the axiomatization problem for 2-dimensional integral spacetime frames with Robb's irreflexive 'after' relation. We provide an axiomatization in a multimodal language and prove that this axiomatization is complete. We use this result to prove the system in the "standard" modal language is decidable. ;In Chapter 4 we take up the problem of what the correct Diodorean modal logic is for 2-dimensional integral spacetime. We show that the corresponding Diodorean modal logic is not S4.2, nor indeed any known modal system. We then present several candidate axioms and prove their independence in the context of S4.2. We also study some of the sublogics of the Diodorean modal logic for 2-dimensional discrete spacetime . ;The final chapter contains some general results. First we prove that for every n ≥ 2 the n-dimensional frame determines a unique Diodorean modal logic. We also consider temporal logics in the language with F and P for two kinds of irreflexive spacetime frames. We prove that for every n ≥ 2 the n-dimensional frame with Robb's 'after' relation determines a unique temporal logic. We also prove a generalized completeness theorem for n-dimensional irreflexive successor logics. Finally, in an appendix, we provide a list of selected open problems and indicate directions for future research in this area. (shrink)

A potentially new interpretation of quantum mechanics posits the state of the universe as a consistent set of facts that are instantiated in the correlations among entangled objects. A fact (or event) occurs exactly when the number or density of future possibilities decreases, and a quantum superposition exists if and only if the facts of the universe are consistent with the superposition. The interpretation sheds light on both in-principle and real-world predictability of the universe.

This thesis is an attempt to reconstruct the conceptual foundations of quantum mechanics. First, we argue that the wave function in quantum mechanics is a description of random discontinuous motion of particles, and the modulus square of the wave function gives the probability density of the particles being in certain locations in space. Next, we show that the linear non-relativistic evolution of the wave function of an isolated system obeys the free Schrödinger equation due to the requirements of spacetime (...) translation invariance and relativistic invariance. Thirdly, we argue that the random discontinuous motion of particles may lead to a stochastic, nonlinear collapse evolution of the wave function. A discrete model of energy-conserved wavefunction collapse is proposed and shown to be consistent with existing experiments and our macroscopic experience. In addition, we also give a critical analysis of the de Broglie-Bohm theory, the many-worlds interpretation and other dynamical collapse theories, and briefly discuss the issue of unifying quantum mechanics and special relativity. (shrink)

It has been argued within some philosophy of quantum gravity circles that endorsing Lewisian modal metaphysics is incompatible with endorsing the fundamental physical ontology of any quantum gravity theory. Speaking concisely, the unsolvable tension would be between Lewis' metaphysical commitment to the fundamentality of space and time, and the physical lesson of quantum gravity about the disappearance of space and time from the fundamental structure of the world. In this essay I argue against the idea that the tension is unsolvable. (...) The analysis does not apply to quantum gravity in general, but only to the specific perspective delivered by quantum string theory. In the first two sections I describe what I think is the general formal payoff of Lewisian possible worlds semantics, and more importantly the general metaphysical payoff generated by some Lewisian metaphysical applications of that formal framework. Lewisian possible worlds semantics is not among the topics of this essay, still a concise presentation of its basic features is required for a more accurate description of the type of modal metaphysics Lewis delivers by applying that formal machinery. The moral of the story in the first two sections is that an attempt of preserving crucial features of Lewisian modal metaphysics, although endorsing the quantum string theory lesson on spacetime non fundamentality, would be not only possible, but also philosophically worthy in a general sense. Then, section three is divided in three parts. In the first part I summarize some formal tools of the methodology used elsewhere - (Vistarini, Routledge, 2019, chapter 6)- to argue for string theory background independence. More precisely, I briefly re-describe some crucial features of a topological-vector bundle structure there defined on a "space" associated to the theory. Such structure is there shown to carry physical and dynamical information about quantum strings systems supporting the thesis of background independence of quantum strings laws (string theory background independence is not a topic of this essay, and the construction of the full formal structure as well as the full argumentative line can be found in the work cited above). The rationale behind the choice of including these formal tools is explained in the second and third parts of section three: that very same topological-fiber bundle structure associated to string theory, if read differently, is shown to also carry structural properties of the Lewisian metaphysical scheme. More precisely, in the second part of section three I attempt to show that such topological-fiber bundle structure owns a topology which qualifies as a plausible accessibility structure extending the Lewisian similarity accessibility. To avoid confusion, I will denote the extended similarity order with S*, or S*-similarity. The latter has some interesting properties. It is quantitatively describable, and even more interestingly its spectrum of variation is a countable one. Then, insofar Lewisian similarity order can be regained from the S*-similarity, one might say that the latter is a metaphysical accessibility relation underlying the former because "more fundamental". As we will see, the explanation of what "more fundamental" means in this metaphysical context mainly involves the idea that the formal nature of S*-similarity, in particular its property of having a countable spectrum of variation, calibrates a metaphysical similarity that reflects the discreteness of the fundamental quantum nature of reality. The extended modal metaphysics with accessibility structure S*-similarity is informed by quantum string physics. Within the extended scheme, the Lewisian claim "things could have been different in countless ways" (Lewis 1973) is an approximate one because only permitted by ordinary language. As soon as one refines the accessibility structure by using the formal language of quantum string theory, it appears things could have been different in countable ways. But the Lewisian variety of alternative states of affairs can be regained from the underlying countable variety via a specific interpretation of the Humean supervenience thesis, one that changes the metaphysical status of the thesis' content and restricts the thesis' domain of application. Finally, in the third part of section three I attempt to show that the same topological-fiber bundle structure used to extend Lewisian similarity has a bundle component producing the formal scheme for a tentative extension of Lewisian nomological accessibility through the lens of string dualities. This will prove to be a non trivial task, and the Lewisian accessibility structure will be regained from the extended one only partially. Finally, the essay contains two short appendices summarizing two argumentative lines outside the main topic of the essay. They briefly describe different ways in which the non fundamentality of spatiotemporal structures can be found in quantum string theory. These sections can be skipped. Nevertheless, they might contribute to understand more robustly the general view endorsed by this essay, i.e. that there isn't any unsolvable tension between Lewis' commitment to the fundamentality of spatiotemporal relations and quantum gravity's lesson about their non fundamentality. -/- . (shrink)

In the present paper the concept of a covering is presented and developed. The relationship between cover schemes, frames (complete Heyting algebras), Kripke models, and frame-valued set theory is discussed. Finally cover schemes and framevalued set theory are applied in the context of Markopoulou’s account of discrete spacetime as sets “evolving” over a causal set. We observe that Markopoulou’s proposal may be effectively realized by working within an appropriate frame-valued model of set theory. We go on to show that, (...) within this framework, cover schemes may be used to force certain conditions to prevail in the associated models: for example, rendering the universe timeless, obliterating a given event or forcing it to become the universe’s “beginning”. (shrink)

The search for a quantum theory of the gravitational field is one of the great open problems in theoretical physics. This book presents a self-contained discussion of the concepts, methods and applications that can be expected in such a theory. The two main approaches to its construction - the direct quantisation of Einstein's general theory of relativity and string theory - are covered. Whereas the first attempts to construct a viable theory for the gravitational field alone, string theory assumes that (...) a quantum theory of gravity will be achieved only through a unification of all the interactions. However, both employ the general method of quantisation of constrained systems, which is described together with illustrative examples relevant for quantum gravity. There is a detailed presentation of the main approaches employed in quantum general relativity: path-integral quantisation, the background-field method and canonical quantum gravity in the metric, connection and loop formulations. The discussion of string theory centres around its quantum-gravitational aspects and the comparison with quantum general relativity. Physical applications discussed at length include the quantisation of black holes, quantum cosmology, the indications of a discrete structure of spacetime, and the origin of irreversibility. This book will be of interest to researchers and students working in relativity and gravitation, cosmology, quantum field theory and related topics. It will also be of interest to mathematicians and philosophers of science. (shrink)

Since Einstein’s failure to define a Grand Unified Theory, physicists have pursued a comprehensive theory explaining nature, a Theory of Everything. But because General Relativity, Quantum Field Theory, and Cosmology have little in common, defining one theory is an imposing task, having eluded the best scientists for ninety years. So are we close to defining a Theory of Everything? This analysis, after defining requirements, identifies four possible options for a Theory of Everything. Quotes from prominent physicists express divergent views on (...) each alternative.The leading proposal for a Theory of Everything is String Theory, which has possible issues. It requires supersymmetry, has extra compacted dimensions, and is background-dependent. A second less comprehensive theory is Loop Quantum Gravity, which emphases background-independence based on discrete quantum space. Explaining how theories define relationships between spacetime, quantum space, quantum time, and quantum gravity, positions the role of background-independence in proposed theories. If supersymmetry and extra dimensions are discovered, String Theory or a modified String Theory integrated with Loop Quantum Gravity, may be the option. Or, possibly a radically new theory might be developed. Or, as the last option considers, the answer to the question — Does a Theory of Everything exist? — may be no; if so, nature will always be a mystery. (shrink)

Since Einstein’s failure to define a Grand Unified Theory, physicists have pursued a comprehensive theory explaining nature, a Theory of Everything. But because General Relativity, Quantum Field Theory, and Cosmology have little in common, defining one theory is an imposing task, having eluded the best scientists for ninety years. So are we close to defining a Theory of Everything? This analysis, after defining requirements, identifies four possible options for a Theory of Everything. Quotes from prominent physicists express divergent views on (...) each alternative.The leading proposal for a Theory of Everything is String Theory, which has possible issues. It requires supersymmetry, has extra compacted dimensions, and is background-dependent. A second less comprehensive theory is Loop Quantum Gravity, which emphases background-independence based on discrete quantum space. Explaining how theories define relationships between spacetime, quantum space, quantum time, and quantum gravity, positions the role of background-independence in proposed theories. If supersymmetry and extra dimensions are discovered, String Theory or a modified String Theory integrated with Loop Quantum Gravity, may be the option. Or, possibly a radically new theory might be developed. Or, as the last option considers, the answer to the question — Does a Theory of Everything exist? — may be no; if so, nature will always be a mystery. (shrink)

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 the 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. (shrink)

Perceptual completion fills the gap for discrete perception to become continuous. Similarly, dynamic perceptual completion provides an experience of dynamic continuity. Our recent discovery of the ‘happening’ element of DPC completes the total experience for dynamism in the flow of time. However, a phenomenological explanation for these experiences is essential. The Snapshot Hypotheses especially the Dynamic Snapshot View provides the most comprehensive explanation. From that understanding the ‘two times’ problem can be addressed. The static time of spacetime cosmologies has (...) been irreconcilable with the dynamic FOT. Dismissing the FOT as an illusion is unsatisfactory. Therefore, we provide four hypotheses for the TTP.1) Since cosmological static time demands that all events are discrete, DPC elements for dynamism should likewise be expected to be discrete and accounted for by a snapshot phenomenology such as the DSV. 2) If temporality can be demonstrated to be similar to apparent motion by being a snapshot phenomenon and not demanding temporal extension it would confirm the DSV and permit reconciliation with static time. 3) If the ‘present moment’ is subjective as static time theories suggest, it should be possible experimentally for an observer to choose his own ‘present’ by moving to various points in the past with the aid of virtual reality. 4) If dynamism e.g. motion can be precluded without significant information loss or violating physics principles it is a cognitive add-on, thereby contradicting non-static time theories which suggest that time is ‘real.’ We confirm those hypotheses. (shrink)

Causal set theory and the theory of linear structures 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 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 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 the one hand, my theorems support : 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 : 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. (shrink)

A new variational method, the principle of least radix economy, is formulated. The mathematical and physical relevance of the radix economy, also called digit capacity, is established, showing how physical laws can be derived from this concept in a unified way. The principle reinterprets and generalizes the principle of least action yielding two classes of physical solutions: least action paths and quantum wavefunctions. A new physical foundation of the Hilbert space of quantum mechanics is then accomplished and it is used (...) to derive the Schrödinger and Dirac equations and the breaking of the commutativity of spacetime geometry. The formulation provides an explanation of how determinism and random statistical behavior coexist in spacetime and a framework is developed that allows dynamical processes to be formulated in terms of chains of digits. These methods lead to a new foundation for Lorentz transformations and special relativity. The Parker-Rhodes combinatorial hierarchy is encompassed within our approach and this leads to an estimate of the interaction strength of the electromagnetic and gravitational forces that agrees with the experimental values to an error of less than one thousandth. Finally, it is shown how the principle of least-radix economy naturally gives rise to Boltzmann’s principle of classical statistical thermodynamics. A new expression for a general nonequilibrium entropy is proposed satisfying the Second Law of Thermodynamics. (shrink)

Stationary spacetimes containing a black hole have several properties akin to those of atoms. For instance, such spacetimes have only three classical degrees of freedom, or observables, which may be taken to be the mass, the angular momentum, and the electric charge of the hole. There are several arguments supporting a proposal originally made by Bekenstein that quantization of these classical degrees of freedom gives an equal spacing for the horizon area spectrum of black holes. We review some of these (...) arguments and introduce a specific Hamiltonian quantum theory of black holes. Our Hamiltonian quantum theory gives, among other things, a discrete spectrum for the classical observables, and it produces an area spectrum which is closely related to Bekenstein's proposal. We also present a foamlike model of horizons of spacetime. In our model spacetime horizon consists of microscopic Schwarzschild black holes. Applying our Hamiltonian approach to this model we find that the entropy of any horizon is one quarter of its area. (shrink)

We review connections between the metric of spacetime and the quantum fluctuations of fields. We start with the finding that the spacetime metric can be expressed entirely in terms of the 2-point correlator of the fluctuations of quantum fields. We then discuss the open question whether the knowledge of only the spectra of the quantum fluctuations of fields also suffices to determine the spacetime metric. This question is of interest because spectra are geometric invariants and their quantization (...) would, therefore, have the benefit of not requiring the modding out of diffeomorphisms. Further, we discuss the fact that spacetime at the Planck scale need not necessarily be either discrete or continuous. Instead, results from information theory show that spacetime may be simultaneously discrete and continuous in the same way that information can. Finally, we review the recent finding that a covariant natural ultraviolet cutoff at the Planck scale implies a signature in the cosmic microwave background that may become observable. (shrink)

The meaning of the wave function and its evolution are investigated. First, we argue that the wave function in quantum mechanics is a description of random discontinuous motion of particles, and the modulus square of the wave function gives the probability density of the particles being in certain locations in space. Next, we show that the linear non-relativistic evolution of the wave function of an isolated system obeys the free Schrödinger equation due to the requirements of spacetime translation invariance (...) and relativistic invariance. Thirdly, we argue that the random discontinuous motion of particles may lead to a stochastic, nonlinear collapse evolution of the wave function. A discrete model of energy-conserved wavefunction collapse is proposed and shown consistent with existing experiments and our macroscopic experience. Besides, we also give a critical analysis of the de Broglie-Bohm theory, the many-worlds interpretation and other dynamical collapse theories, and briefly discuss the issues of unifying quantum mechanics and relativity. (shrink)

By forming the intersections of the parity and time reversal equivalence classes of physical entities that are represented by differential forms and differential form densities, a number of subsets of discrete symmetry classes for electromagnetic systems can be generated. Only one of these subsets is consistent with elementary thermodynamic arguments for dissipative systems and at the same time yields the notion that both charge and mass are spacetime scalars. This subset is not in correspondence with the two self-consistent presentations (...) that now are implied in the literature. (shrink)

A fiber bundle constructed over spacetime is used as the basic underlying framework for a differential geometric description of extended hadrons. The bundle has a Cartan connection and possesses the de Sitter groupSO(4, 1) as structural group, operating as a group of motion in a locally defined space of constant curvature (the fiber) characterized by a radius of curvatureR≈10−13 cm related to the strong interactions. A hadronic matter field ω(x, ζ) is defined on the bundle space, withx the (...) class='Hi'>spacetime coordinate and ζ varying in the local fiber. The components of a generalized tensor current ℑ μab (M) (x) are introduced, involving a bilinear expression in the fields ω(x, ζ) and ωΔ(x, ζ) integrated over the local fiber at the pointx. This hadronic matter current is considered as a source current for the underlying fiber geometry by coupling it in a gauge-invariant manner to the curvature expressions derived from the bundle connection coefficients, which are associated here with strong interaction effects, i.e., play the role of meson fields induced in the geometry. Studying discrete symmetry transformations between the 16 differently soldered Cartan bundles, a generalized matter-antimatter conjugation Ĉ is established which leaves the basic current-curvature equations Ĉ-invariant. The discrete symmetry transformation Ĉ turns out to be the direct product of an ordinary charge conjugation for the Dirac point-spinor part of ω(x, ζ) and an internal $\hat P\hat T$ transformation applied globally on the bundle to the fiber (i.e., de Sitter) part of ω(x, ζ). (shrink)

Consider the following pair of theses: all fundamental physical objects are spatiotemporal and all non-fundamental physical objects are ultimately composed of fundamental objects. Work on the physics of quantum gravity suggests that spacetime is a non-fundamental, emergent phenomenon and thus that thesis is false. The fundamentals are non-spatiotemporal in nature. This paper will argue against on the grounds that non-fundamental spatiotemporal objects cannot be composed of fundamental non-spatiotemporal objects. So, assuming that spacetime is emergent, new metaphysical resources are (...) needed to explain the relationship between the fundamental objects posited by quantum gravity and spacetime. (shrink)

This contributed volume is the result of a July 2010 workshop at the University of Wuppertal Interdisciplinary Centre for Science and Technology Studies which brought together world-wide experts from physics, philosophy and history, in order to address a set of questions first posed in the 1950s: How do we compare spacetime theories? How do we judge, objectively, which is the “best” theory? Is there even a unique answer to this question? -/- The goal of the workshop, and of this (...) book, is to contribute to the development of a meta-theory of spacetime theories. Such a meta-theory would reveal insights about specific spacetime theories by distilling their essential similarities and differences, deliver a framework for a class of theories that could be helpful as a blueprint to build other meta-theories, and provide a higher-level viewpoint for judging which theory most accurately describes nature. But rather than drawing a map in broad strokes, the focus is on particularly rich regions in the “space of spacetime theories.” -/- This work will be of interest to physicists, as well as philosophers and historians of science working with or interested in General Relativity and/or Space, Time and Gravitation more generally. (shrink)

How does the modern scientific conception of time constrain the project of assigning the mind its proper place in nature? On the scientific conception, it makes no sense to speak of the duration of a pain, or the simultaneity of sensations occurring in different parts of the brain. Such considerations led Henri Poincaré, one of the founders of the modern conception, to conclude that consciousness does not exist in spacetime, but serves as the basic material out of which we (...) must create the physical world. The central claim of Sensorama is that Poincaré was substantially correct. The best way to reconcile the scientific conception of time with the evidence of introspection is through a phenomenalist metaphysic according to which consciousness exists in neither time nor space, but serves as a basis for the logical construction of spacetime and its contents. (shrink)

Leibniz argues against Descartes’s conception of material substance based on considerations of unity. I examine a key premise of Leibniz’s argument, what I call the Plurality Thesis—the claim that matter (i.e. extension alone) is a plurality of parts. More specifically, I engage an objection to the Plurality Thesis stemming from what I call Material Monism—the claim that the physical world is a single material substance. I argue that Leibniz can productively engage this objection based on his view that matter is (...) discrete. The discreteness of matter provides two aspects of support for the Plurality Thesis. First, it indicates that the parts of matter do not share boundaries and are, therefore, independent in an important sense. Second, it indicates that the parts of matter are determinate and are, therefore, ontologically prior to the wholes they compose. (shrink)

This paper argues that all of the standard theories about the divisions of space and time can benefit from, and may need to rely on, parsimony considerations. More specifically, whether spacetime is discrete, gunky or pointy, there are wildly unparsimonious rivals to standard accounts that need to be resisted by proponents of those accounts, and only parsimony considerations offer a natural way of doing that resisting. Furthermore, quantitative parsimony considerations appear to be needed in many of these cases.

The remarkable connections between gravity and thermodynamics seem to imply that gravity is not fundamental but emergent, and in particular, as Verlinde suggested, gravity is probably an entropic force. In this paper, we will argue that the idea of gravity as an entropic force is debatable. It is shown that there is no convincing analogy between gravity and entropic force in Verlinde’s example. Neither holographic screen nor test particle satisfies all requirements for the existence of entropic force in a thermodynamics (...) system. As a result, there is no entropic force in the gravity system. Furthermore, we show that the entropy increase of the screen is not caused by its statistical tendency to increase entropy as required by the existence of entropic force, but in fact caused by gravity. Therefore, Verlinde’s argument for the entropic origin of gravity is problematic. In addition, we argue that the existence of a minimum size of spacetime, together with the Heisenberg uncertainty principle in quantum theory, may imply the fundamental existence of gravity as a geometric property of spacetime. This provides a further support for the conclusion that gravity is not an entropic force. (shrink)

The free Schrödinger equation is shown to be a consequence of spacetime homogeneity in the non-relativistic domain. This may help understand the origin of the wave equations in quantum theory.

I present a discussion of some issues in the ontology of spacetime. After a characterisation of the controversies among relationists, substantivalists, eternalists, and presentists, I offer a new argument for rejecting presentism, the doctrine that only present objects exist. Then, I outline and defend a form of spacetime realism that I call event substantivalism. I propose an ontological theory for the emergence of spacetime from more basic entities. Finally, I argue that a relational theory of pre-geometric entities (...) can give rise to substantival spacetime in such a way that relationism and substantivalism are not necessarily opposed positions, but rather complementary. In an appendix I give axiomatic formulations of my ontological views. (shrink)

We develop a new version of the causal theory of spacetime. Whereas traditional versions of the theory seek to identify spatiotemporal relations with causal relations, the version we develop takes causal relations to be the grounds for spatiotemporal relations. Causation is thus distinct from, and more basic than, spacetime. We argue that this non-identity theory, suitably developed, avoids the challenges facing the traditional identity theory.

I examine the debate between substantivalists and relationalists about the ontological character of spacetime and conclude it is not well posed. I argue that the hole argument does not bear on the debate, because it provides no clear criterion to distinguish the positions. I propose two such precise criteria and construct separate arguments based on each to yield contrary conclusions, one supportive of something like relationalism and the other of something like substantivalism. The lesson is that one must fix (...) an investigative context in order to make such criteria precise, but different investigative contexts yield inconsistent results. I examine questions of existence about spacetime structures other than the spacetime manifold itself to argue that it is more fruitful to focus on pragmatic issues of physicality, a notion that lends itself to several different explications, all of philosophical interest, none privileged a priori over any of the others. I conclude by suggesting an extension of the lessons of my arguments to the broader debate between realists and instrumentalists. 1 Introduction2 The Hole Argument3 Limits of Spacetimes4 Pointless Constructions5 The Debate between Substantivalists and Relationalists6 Existence and Physicality: An Embarassment of Spacetime Structures7 Valedictory Remarks on Realism and Instrumentalism, and the Structure of Our Knowledge of Physics. (shrink)

This article tells the story of Ed Fredkin, a pilot, programmer, engineer, hardware designer and entrepreneur, whose work inside and outside academia has influenced major developments in computer science and in the foundations of theoretical physics for the past fifty years.

The nature of space and time is one of the most fascinating and fundamental areas of the philosophy of physics. This study aims to provide a complete account of current debates in the application of spacetime to string theory. String theory has been an important discipline within physics for many years but is only now being applied to the problems faced by philosophers of science. This emerging area of physics is discussed in relation to a number of theories including (...) general relativity, T-duality and moduli space, and set in the context of current and future research. (shrink)

In this paper, we argue that what is often discussed under the umbrella of `spacetime emergence' in the philosophy of quantum gravity in fact consists of a plethora of distinct and even highly different problems. We therefore advocate to cast such debates more specifically in terms of the emergent spatiotemporal aspects, as is already done in the physics literature. We first show how ambiguous the notion of spacetime is already in general relativity. We then argue against three ways (...) to reject our call for specificity: the many distinct philosophical problems relating to the emergence of the various spatiotemporal aspects do not suggest any singular, overarching, or exceptional problem of spacetime emergence. We also discuss how the objections are reflected by five conceptions of spacetime. Next, we observe that different spatiotemporal aspects are emergent in the different quantum gravity approaches, whereby investigating emergence in quantum gravity collectively is problematic. Finally, we illustrate how philosophical inquiries that employ notions of spacetime emergence are aided by conducting the investigation at the level of specific spatiotemporal aspects. (shrink)

I examine the debate between substantivalists and relationalists about the ontological character of spacetime and conclude it is not well posed. I argue that the hole argument does not bear on the debate, because it provides no clear criterion to distinguish the positions. I propose two such precise criteria and construct separate arguments based on each to yield contrary conclusions, one supportive of something like relationalism and the other of something like substantivalism. The lesson is that one must fix (...) an investigative context in order to make such criteria precise, but different investigative contexts yield inconsistent results. I examine questions of existence about spacetime structures other than the spacetime manifold itself to argue that it is more fruitful to focus on pragmatic issues of physicality, a notion that lends itself to several different explications, all of philosophical interest, none privileged a priori over any of the others. I conclude by suggesting an extension of the lessons of my arguments to the broader debate between realists and instrumentalists. 1 Introduction2 The Hole Argument3 Limits of Spacetimes4 Pointless Constructions5 The Debate between Substantivalists and Relationalists6 Existence and Physicality: An Embarassment of Spacetime Structures7 Valedictory Remarks on Realism and Instrumentalism, and the Structure of Our Knowledge of Physics. (shrink)

Work in quantum gravity suggests that spacetime is not fundamental. Rather, spacetime emerges from an underlying, non-spatiotemporal reality. After clarifying the type of emergence at issue, I argue that standard conceptions of emergence available in metaphysics won’t work for the emergence of spacetime. I go on to consider spacetime functionalism as a way to make sense of spacetime emergence. I argue that a functionalist approach to spacetime modelled on mental state functionalism is not a (...) viable alternative to the standard conception of emergence in metaphysics. I go on to consider an alternative: ‘partial’ functionalism, whereby certain aspects of spacetime are functionalised, rather than spacetime as a whole. (shrink)

The answer to some of the longstanding issues in the 20th century theoretical physics, such as those of the incompatibility between general relativity and quantum mechanics, the broken symmetries of the electroweak force acting at the subatomic scale and the missing mass of Higgs particle, and also those of the cosmic singularity and the black matter and energy, appear to be closely related to the problem of the quantum texture of space-time and the fluctuations of its underlying geometry. Each region (...) of space landscape seem to be filled with spacetime weaved and knotted networks, for example, spacetime has immaterial curvature and structures, such as topological singularities, and obeys the laws of quantum physics. Thus, it is filled with potentialparticles, pairs of virtual matter and anti-matter units, and potential properties at the quantum scale. For example, quantum entities (like fields and particles) have both wave (i.e., continuous) and particle (i.e., discrete) properties and behaviors. At the quantum level (precisely, the Planck scale) of space-time such properties and behaviors could emerge from some underlying (dynamic) phase space related to some field theory. Accordingly, these properties and behaviors leave their signature on objects and phenomena in the real Universe. In this paper we consider some conceptual issues of this question. (shrink)

This paper examines two cosmological models of quantum gravity to investigate the foundational and conceptual issues arising from quantum treatments of the big bang. While the classical singularity is erased, the quantum evolution that replaces it may not correspond to classical spacetime: it may instead be a non-spatiotemporal region, which somehow transitions to a spatiotemporal state. The different kinds of transition involved are partially characterized, the concept of a physical transition without time is investigated, and the problem of empirical (...) incoherence for regions without spacetime is discussed. (shrink)

I focus on the stochastic gravity program, a program that conceptualizes spacetime as the hydrodynamic limit of the correlation hierarchy of an underlying quantum theory, that is, a theory of the microscopic theory of gravity. This approach is relatively obscure, and so I begin by outlining the stochastic gravity program in enough detail to make clear the basic sense in which, on this approach, spacetime emerges from more fundamental physical structures. The theory, insofar as it is a univocal (...) theory, is quite clear in its basic features, and so issues of philosophical interpretation can be readily isolated.The most obvious reason to investigate the theory as a model for the emergence of spacetime structure is how close it is to the stage at which the behavior that we recognize as spacetime actually emerges from the micro gravitational system. Approaches that begin with fully quantum gravity treat a system that is conceptually quite far removed from the stage at which emergence is relevant. The stochastic approach however begins by identifying the point at which spacetime emerges as a phenomena of interest.I begin with an analysis of the emergence question generally and ask how best we should understand it, especially from the point of view of thinking of spacetime as emergent. A nice feature of the stochastic program is how clear the question of emergence is on this approach. In part this is because of its similarity by design to the kinetic theory of gases and solid state physics. And so many of the analyses of the emergence of macroscopic variables in the thermodynamic limit can be repurposed to understand how an apparently continuous metrical space emerges from the behavior of a non-spatial system.A serious interpretive problem looms however. The problem is that there is no clear connection between features of the kinetic theory of gravity, as a quantum theory, and any final theory of gravity. In the third part of the paper I will argue that as far as questions of emergence are concerned, we need not begin with a final, underlying theory, and I attempt to identify general issues connected to the emergence of spacetime that can be addressed in isolation from our certainty about that final theory. I will argue that this is a common way in which we treat our other, after all, provisional theories. We begin with the theories we have and ask about their implications without assuming that they are final theories, and yet also without explicitly downplaying the significance of the results we derive. Moreover I will attempt to show that, whatever character a final theory of micro gravity has, spacetime as an emergent structure in that theory is likely to be similar in important respects to the way it manifests in the stochastic gravity program. Briefly this is precisely because of the metaphysical neutrality of the kinetic theory. I will expand, in this section, on the nature of the emergence of the spacetime structure in the context of the stochastic gravity program and explain how the emergence is tied not to the particular model of interactions appealed to, but rather to the generic features of quantum fields with correlated fluctuations at all orders. (shrink)

Endurantism, the view that material objects are wholly present at each moment of their careers, is under threat from supersubstantivalism, the view that material objects are identical to spacetime regions. I discuss three compromise positions. They are alike in that they all take material objects to be composed of spacetime points or regions without being identical to any such point or region. They differ in whether they permit multilocation and in whether they generate cases of mereologically coincident entities.

The purpose of the study is to reveal the relationship between the discreteness of historical consciousness and the processes of spiritual and intellectual development of the individual; to offer his vision of what the discreteness of historical consciousness is and to determine the degree of its influence on the process of spiritual and intellectual formation of the individual. The article focuses on the fact that the discreteness of historical consciousness has a destructive effect on public consciousness and its forms. The (...) degree of discreteness of historical consciousness and the level of spiritual and intellectual development of the individual are interdependent and have a mutual influence on each other. As a result of the conducted research, it was revealed that the discrete historical consciousness is a fragmented "torn" historical consciousness, which poses a threat to the spiritual and intellectual formation of the individual. (shrink)

Condensed matter approaches to quantum gravity suggest that spacetime emerges in the low-energy sector of a fundamental condensate. This essay investigates what could be meant by this claim. In particular, I offer an account of low-energy emergence that is appropriate to effective field theories in general, and consider the extent to which it underwrites claims about the emergence of spacetime in effective field theories of condensed matter systems of the type that are relevant to quantum gravity.