This article presents a simplified version of the author's previous work. We first construct a causal growth process (CGP). We then form path Hilbert spaces using paths of varying lengths in the CGP. A sequence of positive operators on these Hilbert spaces that satisfy certain normalization and consistency conditions is called a quantum sequential growth process (QSGP). The operators of a QSGP are employed to define natural decoherence functionals and quantum measures. These quantum measures are extended to a single quantum measure defined on a suitable collection of subsets of a space of all paths. Continuing our general formalism, we define curvature operators and a discrete analogue of Einstein's field equations on the Hilbert space of causal sets. We next present a method for constructing a QSGP using an amplitude process (AP). We then consider a specific AP that employs a discrete analogue of a quantum action. Finally, we consider the special case in which the QSGP is classical. It is pointed out that this formalism not only gives a discrete version of general relativity, there is also emerging a discrete analogue of quantum field theory. We therefore have discrete versions of these two theories within one unifying framework.
This article presents a simplified version of the author's previous work. We first construct a causal growth process (CGP). We then form path Hilbert spaces using paths of varying lengths in the CGP. A sequence of positive operators on these Hilbert spaces that satisfy certain normalization and consistency conditions is called a quantum sequential growth process (QSGP). The operators of a QSGP are employed to define natural decoherence functionals and quantum measures. These quantum measures are extended to a single quantum measure defined on a suitable collection of subsets of a space of all paths. Continuing our general formalism, we define curvature operators and a discrete analogue of Einstein's field equations on the Hilbert space of causal sets. We next present a method for constructing a QSGP using an amplitude process (AP). We then consider a specific AP that employs a discrete analogue of a quantum action. Finally, we consider the special case in which the QSGP is classical. It is pointed out that this formalism not only gives a discrete version of general relativity, there is also emerging a discrete analogue of quantum field theory. We therefore have discrete versions of these two theories within one unifying framework.
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.
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.
Abstract This work develops a process-based framework for quantum gravity in which physical space is rigorously described as the topology of a dynamically evolving graph of discrete quantum processes called P-tacts. A P-tact is defined as the primitive unit of quantum evolution, unifying local superposition and entangling operations into a single indivisible process step. Within this formalism, spacetime is not assumed as a pre-existing background but emerges from the connectivity structure of entanglement relations between P-tacts. The present study synthesizes and extends a coherent research program published throughout 2026. The foundational definition of the P-tact as the primitive unit of quantum theory was introduced in: Pavlov, M. (2026). P Tact (П-такт) as the Primitive Unit of Quantum Theory. Zenodo.https://doi.org/10.5281/zenodo.18516711 The emergence of the quantum switch as a process structure and the reconstruction of indefinite causal order were developed in: Pavlov, M. (2026). Quantum Switch as an Emergent Process Structure from Discrete Process Tacts (П-такт).https://doi.org/10.5281/zenodo.18498181 Pavlov, M. (2026). Indefinite Causal Order as Superposition of Process Graphs.https://doi.org/10.5281/zenodo.18519010 Pavlov, M. (2026). From Indefinite Causal Order to Definite Process Order: A Topological Reconstruction of the Quantum Switch via Discrete P-Tacts (П-такт).https://doi.org/10.5281/zenodo.18526959 Pavlov, M. (2026). Quantum Switch as a Minimal P Tact (П-такт) Cycle.https://doi.org/10.5281/zenodo.18607385 The connection between P-tacts, teleportation, and non-classical causal structures was presented in: Pavlov, M. (2026). Quantum Switch and Teleportation.https://doi.org/10.5281/zenodo.18509173 The emergence of time, causality, gravity, and three-dimensional space from discrete P-tact dynamics was developed in: Pavlov, M. (2026). Emergence of Causality, Gravity, Time, and Three-Dimensional Space from Discrete P-Tact (П-такт) Dynamics.https://doi.org/10.5281/zenodo.18504244 Pavlov, M. (2026). Emergence of Time from P Tact Entanglement (Происхождение времени).https://doi.org/10.5281/zenodo.18526015 Pavlov, M. (2026). Causality and Arrow of Time.https://doi.org/10.5281/zenodo.18549931 The Everett interpretation was explicitly formalized within the P-tact graph framework in: Pavlov, M. (2026). Everett Interpretation Formalized via P-Tact (П-такт) Path Graphs.https://doi.org/10.5281/zenodo.18524745 The present work integrates these results into a unified topological model in which space is identified with graph connectivity, curvature corresponds to dynamic restructuring of entanglement relations, and gravitational phenomena emerge as large-scale patterns of connectivity evolution. In this framework, geometry is not fundamental but process-derived. The theory is fully compatible with Everettian quantum mechanics and eliminates the need for wavefunction collapse. Branching structures correspond to graph bifurcations in the P-tact network, providing a structural representation of many-world dynamics. Gravity is thus interpreted not as a fundamental interaction but as an emergent property of quantum process topology. This approach proposes a formally consistent, unitary, and experimentally testable pathway toward quantum gravity grounded entirely in discrete quantum processes. Potential experimental implementations include quantum simulators and BQM platforms capable of modeling evolving entanglement graphs.
An explanation for quantum mechanics is given in terms of a classical theory (general relativity) for the first time. Specifically, it is shown that certain structures in classical general relativity can give rise to the non-classical logic normally associated with quantum mechanics. An artificial classical model of quantum logic is constructed to show how the Hilbert space structure of quantum mechanics is a natural way to describe a measurement-dependent stochastic process. A 4-geon model of an elementary particle is proposed which is asymptotically flat, particle-like and has a non-trivial causal structure. The usual Cauchy data are no longer sufficient to determine a unique evolution; the measurement apparatus itself can impose further non-redundant boundary conditions. When measurements of an object provide additional non-redundant boundary conditions, the associated propositions would fail to satisfy the distributive law of classical physics. Using the 4-geon model, an orthomodular lattice of propositions, characteristic of quantum mechanics, is formally constructed within the framework of classical general relativity. The model described provides a classical gravitational basis for quantum mechanics, obviating the need for quantum gravity. The equations of quantum mechanics are unmodified, but quantum behaviour is not universal; classical particles and waves could exist and there is no graviton.
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.
Description This work presents a process-based quantum framework in which causality, gravity, time, and three-dimensional space emerge from discrete quantum processes (P-tacts). A P-tact is defined as a minimal unit of quantum evolution combining a local superposition operation and an entangling unitary transformation. The global dynamics of a system is described as a graph of P-tacts, where causal order, branching, and relational structure arise intrinsically, without introducing an external time parameter. The framework demonstrates that causal branching appears first, followed by emergent quantum gravity as an effect of entanglement density and network connectivity. From this structure, relational time emerges as a discrete sequence of P-tacts, and only at later stages does three-dimensional space arise from the global topology of the process graph. Space, time, and gravity are thus shown to be mutually emergent properties of a single discrete quantum process network. This work directly builds upon the previously published concept of the quantum switch as an emergent process structure from discrete P-tacts (see Ref. [1]) and extends it toward a unified description of spacetime emergence. The formalism provides a concrete realization of Everettian branching, making the many-worlds interpretation operational and graph-theoretic. The proposed framework connects quantum information theory, indefinite causal order, quantum gravity, and emergent spacetime, offering both conceptual clarity and experimental relevance. It is applicable to quantum switches, multi-qubit entanglement, quantum networks, and quantum simulations, and provides a testable route toward understanding spacetime as a derived structure rather than a fundamental background. Reference to previous foundational work: DOI 10.5281/zenodo.18498181.