Is space-time discrete or continuous? ? An empirical question

Eric A. B. Cole

George Jaroszkiewicz

Introduction In this chapter we apply the concepts presented in Chapter 22 to a topic that has been of much theoretical interest in recent decades: causal set theory (CST). It is interesting and useful to review the status of CST relative to other foundational topics in mathematical physics that have been and remain of prime significance to empirical physics, namely, special relativity (SR), general relativity (GR), and quantum mechanics (QM). Those great frameworks are built on foundations that presuppose the continuity of space and time. Despite this, it is an empirical fact that we are surrounded by discreteness in the form of atoms, so it is natural to see how that discreteness fits in with the continuity of space and time. We have attempted in this book to show that the application of discrete principles to QM can go far beyond the basic Planck quantum of action. We have gone as far as discretizing empirical time, in the form of the stage concept. However, as far as relativity in either of its manifestations is concerned, the case for any form of discreteness has not yet been established empirically, although great efforts are being made in that direction in the program known as quantum gravity . We have particularly in mind the discretization approaches to GR of Regge (Regge, 1961) and Penrose (Penrose, 1971). Regge's approach is a classically based discretization of the Riemannian manifold concept, so has at best a generalized propositional classification (GPC) of two. Penrose's spin network paradigm started as a quantum description of spacetime and has led to much work in the area of spin networks in the hope of unifying GR and QM. However, workers in spin networks and related branches of quantum gravity have focued their attention on the mathematical complexities and hardly any on the role of observers and their apparatus, leading to a GPC classification of quantum gravity as one, that is, as mathematical physics and not physics. Claims made that such programs represent empirical physics have not been empirically substantiated to date; quantum gravity at this time remains empirically vacuous, despite the best efforts of many theorists over many decades.

Alexey L. Krugly

Joseph A. Smiga, Marco Radaelli, Felix C. Binder +1 more

We study the problem of parameter estimation in time series stemming from general stochastic processes, where the outcomes may exhibit arbitrary temporal correlations. In particular, we address the question of how much Fisher information is lost if the stochastic process is compressed into a single histogram, known as the empirical distribution. As we show, the answer is nontrivial due to the correlations between outcomes. We derive practical formulas for the resulting Fisher information for various scenarios, from generic stationary processes to discrete-time Markov chains to continuous-time classical master equations. The results are illustrated with several examples.

M. Garc�a-Sucre

Jordan Gabriel Farrell

Why does nature organize itself into discrete, persistent structures rather than dissolving into continuous, transient fluctuations? While modern physics successfully describes how particles behave, it largely presupposes their existence. In this work, we address the more fundamental question of why particle-like structures exist at all.Building on Time-Scalar Field Theory (TSFT) and its associated domain-of-validity constraint, we show that discrete particles emerge as a consequence of survival-conditioned evolution. Forward propagation in time is not guaranteed: configurations that approach the Noneness boundary—where temporal density vanishes—encounter divergent costs and effectively terminate. As a result, only a sparse set of configurations that minimize temporal leakage while preserving functional coherence can persist. We formalize this selection principle as a constrained variational problem and demonstrate that its discrete minimizers—which we term rivet modes—naturally exhibit particle-like stability and quantization without postulation. We further show that the same mechanism appears across scales, from quantum eigenstates and composite quasiparticles to biological survival strategies and macroscopic system evolution. Recent experimental observations of anomalously enhanced exciton transport near Mott insulating transitions are discussed as empirical manifestations of survival-conditioned path selection. Together, these results suggest that particles are not fundamental primitives, but the minimal survivable answers to the question of continued existence.