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Causal set theory offers a simple and elegant picture of discrete physics. But the vast majority of causal sets look nothing at all like continuum spacetimes, and must be excluded in some way to obtain a realistic theory. I describe recent results showing that almost all non-manifoldlike causal sets are, in fact, very strongly suppressed in the gravitational path integral. This does not quite demonstrate the emergence of a continuum—we do not yet understand the remaining unsuppressed causal sets well enough—but it is a significant step in that direction. [ABSTRACT FROM AUTHOR]

We investigate a model of becoming—classical sequential growth (CSG)—that has been proposed within the framework of causal sets (causets), with the latter defined as order types of certain partial orderings. To investigate how causets grow, we introduce special sequences of causets, which we call “csg-paths”. We prove a number of results concerning relations between csg-paths and causets. These results paint a highly non-trivial picture of csg-paths. There are uncountably many csg-paths, all of them sharing the same beginning, after which they branch. Every infinite csg-path achieves in the limit an infinite causet, and vice versa, every infinite causet is achieved in the limit by an infinite csg-path. However, coalescing csg-paths, i.e., ones that achieve the same causet even after forking off at some point, are ubiquitous. [ABSTRACT FROM AUTHOR]

We propose an ansatz for encoding the physics of nonlocal spacetime defects in the Green's functions for a scalar field theory defined on a causal set. This allows us to numerically study the effects of nonlocal spacetime defects on the discrete Feynman propagator of the theory defined on the causal set in 1+1 dimensions, and to compare to the defect-free limit. The latter approaches the expected continuum result, on average, when the number of points becomes large. When defects are present, two points with the same invariant spacetime interval can have different propagation amplitudes, depending on whether the propagation is between two ordinary spacetime points, two defects, or a defect and an ordinary point. We show that a coarse-grained description that is only sensitive to the average effect of the defects can be interpreted as a defect-induced mass and wave-function renormalization of the scalar theory. [ABSTRACT FROM AUTHOR]

In the Standard Model of particle physics, the mass of the Higgs particle can be linked to the scale at which the Standard Model breaks down due to a Landau pole/triviality problem: for a Higgs mass somewhat higher than the measured value, the Standard Model breaks down before the Planck scale. We take a first step towards investigating this relation in the context of causal set quantum gravity. We use a scalar-field propagator that carries the imprints of spacetime discreteness in a modified ultraviolet behavior that depends on a nonlocality scale. We investigate whether the modification can shift the scale of the Landau pole in a scalar field theory with quartic interaction. We discover that the modifications speed up the onset of the Landau pole considerably, so that the scale of new physics occurs roughly at the nonlocality scale. Our results call into question, whether a separation between the nonlocality scale and the discreteness scale, which is postulated within causal set quantum gravity, and which has been argued to give rise to phenomenological consequences, is in fact achievable. Methodologically, our paper is the first to apply continuum functional Renormalization Group techniques in the context of a causal-set inspired setting. [ABSTRACT FROM AUTHOR]

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 ℵ 0 , 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. [ABSTRACT FROM AUTHOR]

Causal set theory is a discrete model of spacetime that retains a notion of causal structure. We understand how to construct causal sets that approximate a given spacetime, but most causal sets are not at all manifold-like, and must be dynamically excluded if something like our Universe is to emerge from the theory. Here we show that the most common of these 'bad' causal sets, the Kleitman–Rothschild orders, are strongly suppressed in the gravitational path integral, and we provide evidence that a large class of other 'bad' causal sets are similarly suppressed. It thus becomes plausible that continuum behavior could emerge naturally from causal set quantum theory. [ABSTRACT FROM AUTHOR]

While it is possible to build causal sets that approximate spacetime manifolds, most causal sets are not at all manifold-like. We show that a Lorentzian path integral with the Einstein-Hilbert action has a phase in which one large class of non-manifold-like causal sets is strongly suppressed, and suggest a direction for generalization to other classes. While we cannot yet show our argument holds for all non-manifold-like sets, our results make it plausible that the path integral might lead to emergent manifold-like behavior with no need for further conditions.

Results from a number of different approaches to quantum gravity suggest that the effective dimension of spacetime may drop to d = 2 at small scales. I show that two different dimensional estimators in causal set theory display the same behavior, and argue that a third, the spectral dimension, may exhibit a related phenomenon of 'asymptotic silence.'

The true nature of space and time has been a topic of natural philosophy, passed down since the presocratic era. In modern times reflection has particularly been inspired by the physical theories of Newton and Einstein and, more recently, by the quest for a theory of quantum gravity. In this paper we want to specify the idea that material systems and their spatio-temporal distances emerge from quantum-events. We will show a mechanism, by which quantum-events induce a metric field between material systems, which is governed by Einstein's equation including a cosmological constant. [ABSTRACT FROM AUTHOR]

This paper begins the study of the relation between causality and quantum mechanics, taking advantage of the groupoidal description of quantum mechanical systems inspired by Schwinger’s picture of quantum mechanics. After identifying causal structures on groupoids with a particular class of subcategories, called causal categories accordingly, it will be shown that causal structures can be recovered from a particular class of non-selfadjoint class of algebras, known as triangular operator algebras, contained in the von Neumann algebra of the groupoid of the quantum system. As a consequence of this, Sorkin’s incidence theorem will be proved and some illustrative examples will be discussed.

This article presents a sequential growth model for the Universe that acts like a quantum computer. The basic constituents of the model are a special type of causal set (causet) called a c-causet. A c-causet is defined to be a causet that has a unique labeling. We characterize c-causets as those causets that form a multipartite graph or equivalently those causets whose elements are comparable whenever their heights are different. We show that a c-causet has precisely two c-causet offspring. It follows that there are 2n c-causets of cardinality n + 1. This enables us to classify c-causets of cardinality n + 1 in terms of n-bits. We then quantize the model by introducing a quantum sequential growth process. This is accomplished by replacing the n-bits by n-qubits and defining transition amplitudes for the growth transitions. We mainly consider two types of processes, called stationary and completely stationary. We show that for stationary processes, the probability operators are tensor products of positive rank-one qubit operators. Moreover, the converse of this result holds. Simplifications occur for completely stationary processes. We close with examples of precluded events.

Causal sets (or causets) are a particular class of partially ordered sets, which are proposed as basic models of discrete space-time, specially in the field of quantum gravity. In this context, we show the existence of temporal foliations for any causal set, or more generally, for a causal space. Moreover, we show that (order-preserving) automorphisms of a large class of infinite causal sets fall into two classes 1) Automorphisms of spacelike hypersurfaces in some given foliation (i.e. spacelike automorphisms), or 2) Translations in time. More generally, we show that for any automorphism Φ of a generic causal set C, there exists a partition of C into finitely many subcausets, on each of which (1) or (2) above hold. These subcausets can be assumed connected if, in addition, there are enough distinct orbits under Φ. [ABSTRACT FROM AUTHOR]

A proposal is made for a fundamental theory, in which the history of the universe is constituted of diverse views of itself. Views are attributes of events, and the theory’s only be-ables; they comprise information about energy and momentum transferred to an event from its causal past. A dynamics is proposed for a universe constituted of views of events, which combines the energetic causal set dynamics with a potential energy based on a measure of the distinctiveness of the views, called the variety (Smolin in Found Phys 46(6):736–758, 2016). As in the real ensemble formulation of quantum mechanics (Barbour and Smolin in Variety, complexity and cosmology, arXiv: hep-th/9203041), quantum pure states are associated to ensembles of similar events; the quantum potential of Bohm then arises from the variety. [ABSTRACT FROM AUTHOR]

Path summation offers a flexible general approach to quantum theory, including quantum gravity. In the latter setting, summation is performed over a space of evolutionary pathways in a history configuration space. Discrete causal histories called acyclic directed sets offer certain advantages over similar models appearing in the literature, such as causal sets. Path summation defined in terms of these histories enables derivation of discrete Schrödinger-type equations describing quantum spacetime dynamics for any suitable choice of algebraic quantities associated with each evolutionary pathway. These quantities, called phases, collectively define a phase map from the space of evolutionary pathways to a target object, such as the unit circle S1 ⊂ C, or an analogue such as S3 or S7. This paper explores the problem of identifying suitable phase maps for discrete quantum gravity, focusing on a class of S1-valued maps defined in terms of "structural increments" of histories, called terminal states. Invariants such as state automorphism groups determine multiplicities of states, and induce families of natural entropy functions. A phase map defined in terms of such a function is called an entropic phase map. The associated dynamical law may be viewed as an abstract combination of Schrödinger's equation and the second law of thermodynamics. [ABSTRACT FROM AUTHOR]

The causal set approach to the problem of quantum gravity is based on the hypothesis that spacetime is fundamentally discrete. Spacetime discreteness opens the door to novel types of dynamical law for cosmology and the Classical Sequential Growth (CSG) models of Rideout and Sorkin form an interesting class of such laws. It has been shown that a renormalisation of the dynamical parameters of a CSG model occurs whenever the universe undergoes a Big Crunch–Big Bang bounce. In this paper we propose a way to model the creation of a new universe after the singularity of a black hole. We show that renormalisation of dynamical parameters occurs in a CSG model after such a creation event. We speculate that this could realise aspects of Smolin's Cosmological Natural Selection proposal. [ABSTRACT FROM AUTHOR]

It will be shown in this article that a pluralism of causality is needed. Not only, as might be expected, for such different domains as natural sciences and humanities, but even within the domain of physics different causal relations are necessary. This will be illustrated with ex- amples from Classical Mechanics and Special Relativity, Thermodynamics and Quantum Mechanics. In these domains causal relations differ in their properties. A model for five types of causal relations is provided that is based on a decidable 6-valued logic. [ABSTRACT FROM AUTHOR]

Path summation offers a flexible general approach to quantum theory, including quantum gravity. In the latter setting, summation is performed over a space of evolutionary pathways in a history configuration space. Discrete causal histories called acyclic directed sets offer certain advantages over similar models appearing in the literature, such as causal sets. Path summation defined in terms of these histories enables derivation of discrete Schrödinger-type equations describing quantum spacetime dynamics for any suitable choice of algebraic quantities associated with each evolutionary pathway. These quantities, called phases, collectively define a phase map from the space of evolutionary pathways to a target object, such as the unit circle S 1 ⊂ C , or an analogue such as S 3 or S 7 . This paper explores the problem of identifying suitable phase maps for discrete quantum gravity, focusing on a class of S 1 -valued maps defined in terms of “structural increments” of histories, called terminal states. Invariants such as state automorphism groups determine multiplicities of states, and induce families of natural entropy functions. A phase map defined in terms of such a function is called an entropic phase map. The associated dynamical law may be viewed as an abstract combination of Schrödinger’s equation and the second law of thermodynamics.

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. [ABSTRACT FROM AUTHOR]

This article presents a sequential growth model for the Universe that acts like a quantum computer. The basic constituents of the model are a special type of causal set (causet) called a c-causet. A c-causet is defined to be a causet that has a unique labeling. We characterize c-causets as those causets that form a multipartite graph or equivalently those causets whose elements are comparable whenever their heights are different. We show that a c-causet has precisely two c-causet offspring. It follows that there are 2n c-causets of cardinality n + 1. This enables us to classify c-causets of cardinality n + 1 in terms of n-bits. We then quantize the model by introducing a quantum sequential growth process. This is accomplished by replacing the n-bits by n-qubits and defining transition amplitudes for the growth transitions. We mainly consider two types of processes, called stationary and completely stationary. We show that for stationary processes, the probability operators are tensor products of positive rank-one qubit operators. Moreover, the converse of this result holds. Simplifications occur for completely stationary processes. We close with examples of precluded events. [ABSTRACT FROM AUTHOR]

This paper is based on the causal set approach to discrete quantum gravity. We first describe a classical sequential growth process (CSGP) in which the universe grows one element at a time in discrete steps. At each step the process has the form of a causal set (causet) and the 'completed' universe is given by a path through a discretely growing chain of causets. We then quantize the CSGP by forming a Hilbert space H on the set of paths. The quantum dynamics is governed by a sequence of positive operators ρ on H that satisfy normalization and consistency conditions. The pair ( H,{ ρ}) is called a quantum sequential growth process (QSGP). We next discuss a concrete realization of a QSGP in terms of a natural quantum action. This gives an amplitude process related to the 'sum over histories' approach to quantum mechanics. Finally, we briefly discuss a discrete form of Einstein's field equation and speculate how this may be employed to compare the present framework with classical general relativity theory. [ABSTRACT FROM AUTHOR]

We first discuss a framework for discrete quantum processes (DQP). It is shown that the set of q-probability operators is convex and its set of extreme elements is found. The property of consistency for a DQP is studied and the quadratic algebra of suitable sets is introduced. A classical sequential growth process is “quantized” to obtain a model for discrete quantum gravity called a quantum sequential growth process (QSGP). Two methods for constructing concrete examples of QSGP are provided. [Copyright &y& Elsevier]

This paper concerns random sprinklings of points into Minkowski spacetime (Poisson processes). It proves that there exists no equivariant measurable map from sprinklings to spacetime directions (even locally). Therefore, if a discrete structure is associated to a sprinkling in an intrinsic manner, then the structure will not pick out a preferred frame, locally or globally. This implies that the discreteness of a sprinkled causal set will not give rise to "Lorentz breaking" effects like modified dispersion relations. Another consequence is that there is no way to associate a finite-valency graph to a sprinkling consistently with Lorentz invariance. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we put forward the definition of particular subsets on a unital -algebra, that we call isocones, and which reduce in the commutative case to the set of continuous non-decreasing functions with real values for a partial order relation defined on the spectrum of the algebra, which satisfies a compatibility condition with the topology (complete separateness). We prove that this space/algebra correspondence is a dual equivalence of categories, which generalizes the Gelfand–Naimark duality. Thus we can expect that general isocones could serve to define a notion of “noncommutative ordered spaces”. We also explore some basic algebraic constructions involving isocones, and classify those which are defined in . [Copyright &y& Elsevier]