Towards a unified network theory of space, time and matter: A Conceptual and Philosophical Exploration

1 Towards a Unified Dynamic Network Theory of Space, Time and Matter: A Conceptual and Philosophical Exploration Rudi Van Nieuwenhove Independent Researcher, Dessel, Belgium (previously working at the Belgian Nuclear Research Centre, SCKCEN, Belgium and at the Institute for Energy research IFE in Norway) E-mail: [email protected] ORCID ID : 0000-0003-1265-8718 Abstract This article proposes a deterministic foundation for quantum mechanics and spacetime based on a Unified Dynamic Network Theory (UDNT), in which nodes and links form the sole constituents of physical reality. Space, matter, and time are not independent entities but emergent aspects of the same underlying network dynamics. Nodes undergo continuous join–split cycles, producing the intrinsic oscillatory behavior normally attributed to zeropoint fluctuations. Distance is defined by the typical number of network transitions between two regions rather than by pre-existing geometric separation. Phases attached to the network links enable multiple microscopic paths to coexist, providing a deterministic mechanism for superposition, interference, and wave-like behavior. Spin is incorporated through a minimal two-component internal structure of the link phases. Matter arises as a stable pattern of enhanced connectivity that retains its identity under the network’s evolution. In this way, space and matter are unified: particles are excitations of the same relational structure that constitutes the geometry around them. Such excitations naturally modify the surrounding network, suggesting a path toward a geometric description of gravitation. Quantum entanglement is reinterpreted as the presence of direct nonlocal connections that impose correlations without transmitting signals through space. In this framework, quantum randomness is not fundamental but emerges from the complexity of the network’s deterministic evolution. The UDNT approach thus offers a coherent relational picture in which spacetime, matter, and quantum phenomena jointly arise from a single, evolving network substrate. 2 Keywords: Space-matter unification, Emergent geometry, Discrete spacetime, Network theory, Quantum foundations 1. Introduction String theory, despite its mathematical elegance and promise of unifying gravity with quantum mechanics, remains fundamentally background-dependent. It assumes a fixed spacetime geometry upon which strings propagate, rather than allowing spacetime itself to emerge dynamically from the theory. This reliance on a pre-defined geometric backdrop stands in contrast with general relativity, which is fully background-independent. Furthermore, string theory lacks predictive power due to the enormous 'landscape' of possible vacua (∼10500), making it difficult to identify a unique low-energy limit corresponding to our universe. It also fails to offer direct experimental evidence after decades of development (Smolin, 2006; Witten, 1996; Ellis & Silk, 2014). Loop Quantum Gravity (LQG) (Rovelli, 2004) presents a bold attempt to quantize general relativity in a background-independent manner by replacing the smooth geometry of spacetime with discrete quantum states of geometry, so-called spin networks. While this is conceptually appealing, LQG also faces unresolved issues in dynamics, notably with the Hamiltonian constraint and the semiclassical limit (Nicolai & Zamaklar, 2005; Dittrich & Thiemann, 2009). Other approaches include causal dynamical triangulations (Ambjørn & Loll, 2005), group field theory (Oriti, 2009), and asymptotic safety (Niedermaier & Reuter, 2006). While mathematically rich, none yet provide a complete account of low-energy physics or matter couplings. Quantum Graphity (Konopka & Smolin, 2008; Caravelli & Markopoulou, 2011) tries to derive geometry from a pre-geometric network but struggles with node creation and embedding matter. One guiding theme in physics is unification: Maxwell’s unification of electricity and magnetism, Einstein’s unification of space and time, the electroweak theory (Weinberg, 1967; Salam, 1968). Yet geometry and matter remain distinct in general relativity. A deeper theory may need to unify both as emergent aspects of one substrate. In this work we propose the Unified Dynamic Network Theory (UDNT), a deterministic local-rule-based model where both space and matter emerge from node-link dynamics. Matter can emerge from this network as a self-sustained dynamic structure of the network. Apparent randomness arises from deterministic complexity, not indeterminacy. Recent work supports this line: tensor-network holography shows spacetime emerging from entanglement (Swingle, 2012; Sahay & Cotler, 2025); deterministic cellular-automation approaches (’t Hooft, 2016) pursue similar goals; and information-theoretic gravity emphasizes spacetime as emergent from quantum information (Van Raamsdonk, 2010; Cao & Carroll, 2018). These reinforce the plausibility of deterministic microdynamics yielding the quantum world. In most random tensor-network approaches, geometry 3 emerges from entanglement, while matter typically appears as excitations or defects of the underlying network. For instance, in (Sahay & Cotler, 2025) localized “hologron” excitations in a multiscale entanglement renormalization ansatz (MERA) tensor network are interpreted as matter-like disturbances within an emergent AdS-like bulk geometry. This resembles the hologron excitations of MERA, which are part of the network itself but interpreted as disturbances of an underlying background geometry. In contrast, in the present model particles are conceived as self-sustained configurations of the node–link fabric, co-generating geometry rather than emerging as excitations upon it. Whereas tensornetwork approaches encode local structure in high-rank tensors with thousands of parameters, the present model assigns only simple link relations to each node, shifting the burden of complexity from local objects to the global dynamics of the network. 2. Network description The network consists of nodes and links. The nodes carry complex amplitudes and the links complex phases (see later). It is assumed that the links have no direction. The network evolves through deterministic rules governing node splitting and merging: A node may split into two, provided local connectivity rules are respected (no node may fall below three links). For the case of a non-expanding space, the average number of nodes must remain constant. This means that for every node which splits, there must be a corresponding pair of nodes which merge. The continuous splitting and merging of nodes correspond to the zero-point fluctuations of the vacuum. Further, we require that links are not allowed to “snap”, reflecting the conservation of information. As mentioned before, In UDNT, particles are self-sustained dynamical structures of the network. 2. 1 Compatibility with Special Relativity It is important that the dynamical network of space is compatible with Special Relativity. This imposes the axioms described next. i) A node may only update its state from states of nodes to which it is directly linked. (Locality: No instantaneous influence between non-neighbor nodes). ii) Influences may propagate at most one link per update cycle (or per fixed small number of cycles). This defines the maximum signal speed — the analogue of the speed of light. iii) On large scales (coarse-grained over many nodes), the statistical distribution of links is uniform in every spatial direction. So, there is no built-in preferred rest direction. (Statistical homogeneity and isotropy). iv) Any observer whose internal dynamics propagate at a finite fraction of the maximum 4 influence speed cannot detect an absolute rest frame using measurements internal to the network. If these constraints hold, then: • Lorentz symmetry appears in the coarse-grained long-range behavior of the network. • Length contraction and time dilation follow naturally as different slicings through the same causal structure. • Minkowski spacetime is not fundamental. Instead, it is an emergent geometry of the network. Lorentz invariance is emergent from causal constraints on the network evolution. 2.2 Superposition A key requirement of the network is that it can reproduce the phenomenon of superposition. This means that a single system can evolve along multiple compatible configurations simultaneously, with complex phases assigned to each configuration. So, superposition arises from the assignment of phases to links. A particle moves by many allowed merge–split paths. Each path contributes a complex amplitude: 𝒜path =∏𝑈𝑖𝑗 links (𝑖,𝑗) The quantity 𝑈𝑖𝑗 is the phase-propagation operator associated with the link between node 𝑖 and node 𝑗 . In other words, for any link (𝑖,𝑗) in the network, 𝑈𝑖𝑗 is a unitary operator that updates the particle’s internal state when the excitation moves from node 𝑖to node 𝑗. If the system is spinless, 𝑈𝑖𝑗 =𝑒𝑖𝜙𝑖𝑗 (complex phase factor).If the system carries spin1/2, 𝑈𝑖𝑗 = 𝑒𝑖𝜙𝑖𝑗 𝑅𝑖𝑗, where 𝑅𝑖𝑗 ∈SU(2)is a 2×2 rotation acting on the spinor. So, Uij is both a phase shifter and a spin rotator. The final state is the sum over all allowed paths: Ψfinal =∑𝒜path paths A particle’s wave-like behavior emerges from how these phases accumulate along alternative paths through the network. Superposition reflects the coexistence of all pathdependent phase contributions, while interference arises from their relative phases. Thus, 5 even with undirected links, the network fully supports quantum superposition, interference, and spin, without requiring a built-in direction structure. These phases evolve deterministically through the laws governing merge–split cycles. When a particle (a self-sustained excitation) propagates through the network, it follows many micro-paths simultaneously, not because the excitation physically splits, but because its internal consistency condition couples it to the phases of all neighboring links. What appears in standard quantum mechanics as a “superposition of paths” is, in the UDNT, a single excitation interacting with multiple coherent phase assignments on the surrounding links. The particle does not choose one path; it samples all of them through deterministic phase evolution. Interference arises when the excitation reaches a site where different link paths contribute phases that add or cancel. Superposition, in this model, is neither mysterious nor fundamental. It emerges because the particle samples many deterministic phase relationships simultaneously, and because its internal stability conditions depend on the full neighborhood of link phases. Spin emerges not from literal rotation of a structure, but from the transformation properties of the twocomponent phase under merge–split updates. Thus, the UDNT reproduces key features of quantum behavior such as interference, superposition and spin through deterministic, local rules applied to a richly phased network. 2.4 Discrete Conservation Rules A dynamical network intended to reproduce known physics must incorporate some analogue of local conservation laws. In the continuum, conservation of charge, probability, or energy–momentum follows from symmetry principles and is mathematically expressed by continuity equations. Their discrete counterpart in a network reads Δ𝑄(𝑣)+∑𝐽ℓ ℓ→𝑣 −∑𝐽ℓ ℓ\from𝑣=0 where 𝑄(𝑣)denotes a quantity associated with a node 𝑣, and 𝐽ℓ is a flux or transfer along a link ℓ. The expression states that any change in the quantity stored at a node must be balanced by fluxes along neighboring links. Without such a constraint, the network’s evolution would be too unconstrained to yield stable excitations or well-defined propagation of information. This discrete continuity principle is therefore essential to ensure that the effective large-scale behavior of the network resembles the structure of physical field theories. 6 2.5 Emergent Smoothness and Coarse-Grain Stability To recover familiar dynamical laws, such as Schrödinger, Dirac, or wave equations, a discrete network must support stable long-wavelength excitations and admit an effective coarse-grained continuum limit. This requires that microscopic irregularities do not grow under the update rules, and that the network is statistically homogeneous and isotropic at scales much larger than the fundamental link length. In practice, this imposes constraints on the network dynamics: local update rules must not introduce ultraviolet instabilities, and perturbations should disperse in an approximately linear fashion at coarse scales. These requirements guarantee that smooth wave-like behavior emerges naturally from the underlying discrete structure, allowing the continuum limit to approximate a differentiable manifold with well-defined propagation speeds. 2.6 Gauge-like Redundancies and Phase Freedom Quantum theory fundamentally relies on the existence of phase degrees of freedom, which manifest through interference and the relative phases of wave amplitudes. To reproduce these effects, the network must possess an internal redundancy analogous to gauge freedom. A minimal requirement is that node amplitudes 𝜓𝑣may be rephased locally, 𝜓𝑣 → 𝑒𝑖𝜃(𝑣)𝜓𝑣 with compensating transformations in the phases assigned to links. Such transformations leave all observable predictions unchanged but are essential for ensuring the correct structure of interference, the existence of Berry-phase effects, and the potential emergence of electromagnetic-like interactions as collective phenomena of the network. Without this gauge-like redundancy, the network would lack the correct mathematical degrees of freedom to represent quantum fields or to support the rich phenomenology of phasecoherent dynamics. 2.7 Causality Constraints and the No-Signalling Condition Although the network may support nonlocal links, which are required to account for quantum entanglement, it must still enforce the causal structure compatible with special relativity. In particular, nonlocal connections cannot be used to transmit signals or controllable information faster than light. This constraint may be expressed probabilistically as 𝑃(𝐴 ∣𝐵) =𝑃(𝐴), for events 𝐴and 𝐵associated with spacelikeseparated network updates. While joint correlations between 𝐴 and 𝐵 may exist, local outcome statistics must remain unaffected by operations performed at distant nodes. This condition preserves Lorentz invariance at the emergent level and mirrors the structure of quantum nonlocality, where entanglement generates strong correlations but does not permit superluminal communication. Thus, the network must allow nonlocal correlations while simultaneously ensuring that its update rules respect operational causality. 7 2.8 Entanglement and Nonlocal Links To reproduce quantum entanglement within the network framework, the structure must accommodate correlations that cannot be mediated by ordinary geometric adjacency alone. This can be achieved by introducing nonlocal links, connections between nodes that are separated by many intermediate nodes in the geometric network, but that are directly connected at the level of the underlying dynamical structure. Each such link may be assigned an internal complex weight 𝜓𝑖𝑗 =∣𝜓𝑖𝑗 ∣𝑒𝑖𝜙𝑖𝑗 , representing the strength and relative phase of the correlation between nodes 𝑖 and 𝑗. These complex weights do not correspond to propagating signals; instead, they encode constraints on the joint probability distributions of measurement outcomes. During a measurement, the network locally resolves the amplitudes associated with the entanglement links connected to the measured node, and the correlations implied by the set {𝜓𝑖𝑗} determine the allowed collapse outcomes across distant regions. Because the update rules governing geometric links limit the propagation of classical influence to a finite number of hops per cycle, relativistic causality is preserved even in the presence of nonlocal correlations. In this way, the network supports both a causal, Lorentz-compatible geometry and a layer of nonlocal structure that accounts for quantum entanglement. The familiar interference phenomena of quantum mechanics arise when multiple nonlocal link patterns contribute amplitudes whose phases 𝜙𝑖𝑗 combine constructively or destructively, suggesting that entanglement is encoded in the global pattern of complex-valued connectivity rather than in any superluminal transmission of information. 2.9 Proposing Candidate Microscopic Rules Identifying the microscopic rules governing the underlying space-network requires a systematic approach grounded in known physical principles. Several broad classes of rules offer a promising starting point. 2.9.1 Local Growth and Rewiring Rules A natural assumption is that the network evolves through local update rules governing the creation, deletion, or rewiring of links based solely on the immediate neighborhood of each node. Let 𝐴(𝑡) denote the adjacency matrix at discrete time 𝑡. Local dynamics can be expressed as 𝐴(𝑡+1) = 𝐹[𝐴(𝑡)] where 𝐹 affects only nodes within a finite graph radius. Exploring families of such rules (deterministic, stochastic, or threshold-based) makes it possible to search for combinations whose coarse-grained behavior reproduces smooth geometry and relativistic propagation. 2.9.2 Connectivity and Conservation Constraints To avoid uncontrolled growth or collapse of the network, one may impose constraints resembling conservation laws. For example, the number of active links attached to each node may fluctuate around an equilibrium degree, 8 ∑𝐴𝑖𝑗(𝑡) 𝑗≈𝑘0 ensuring that the emergent geometry remains approximately homogeneous. Such constraints also help stabilize wave-like excitations and support a meaningful continuum limit. 2.9.3 Complex Link Weights and Superposition Quantum behavior suggests that links may carry complex amplitudes rather than binary values. In this case the network evolves through update rules of the form 𝐴(𝑡+1)=𝑈[𝐴(𝑡)] 𝐴(𝑡), where 𝑈[𝐴(𝑡)]acts as a local, unitary-like operator. This allows superposition and interference to emerge naturally from spreading complex amplitudes on the graph. The Schrödinger equation can then appear as a continuum approximation of discrete, phaseweighted propagation on the network. 2.9.4 Nonlocal Links and Entanglement Structure To accommodate entanglement, the network may include a second layer of nonlocal links that do not represent spatial adjacency but encode quantum correlations. These links obey separate update rules that determine when they form, how they evolve, and under what conditions they decohere or collapse. Their presence allows distant regions of the local graph to share correlated amplitudes without transmitting signals, thereby reproducing quantum nonlocality. 2.9.5 Compatibility with Emergent Lorentz Symmetry Perhaps the strongest constraint is the requirement that long-wavelength excitations propagate according to relativistic symmetries. This eliminates update rules that create preferred frames or anisotropies. Candidate rules should approximate a discrete wave equation, 𝜙(𝑡+1)−2𝜙(𝑡)+𝜙(𝑡−1)= Δgraph𝜙(𝑡), where the graph Laplacian Δgraph becomes rotationally symmetric at large scales. Only a restricted subset of microscopic rules flows toward Lorentz-invariant behavior under coarse-graining. 2.9.6 Geometry from Connectivity Finally, curvature may arise from deviations from an equilibrium connectivity pattern. A simple curvature surrogate is 𝑅𝑖∼ 𝑘𝑖−𝑘0, where 𝑘𝑖 is the degree of node 𝑖. Update rules 9 that redistribute such curvature can mimic geometric relaxation and may provide a route to gravitational behavior in the continuum limit. Together, these classes offer a structured space of candidate microscopic rules. Their viability can be tested through coarse-graining analysis, numerical simulation, and comparison with the symmetries and dynamical laws observed in nature. 2.9.7 Outlook and Future Directions Taken together, these proposed classes of microscopic rules offer a coherent path toward a fundamental network-based description of space, matter, and quantum phenomena. The strategy is not to guess the exact rules from the outset, but to progressively narrow the space of possibilities by requiring that the emergent large-scale behavior reproduces the robust symmetries and dynamical structures of physics: Lorentz invariance, linear quantum evolution, entanglement correlations, and the stability of particle-like excitations. In this sense, the familiar laws of physics serve as fixed points toward which viable network dynamics must flow under coarse-graining. The next steps involve exploring specific rule sets through analytical approximations and numerical simulations, seeking those that naturally give rise to smooth geometry, wave propagation, and nonlocal quantum correlations. Any predicted deviations, such as slight modifications to dispersion relations or subtle structures in entangled correlations, could eventually provide empirical tests of the model. By combining theoretical filtering with potential observational signatures, it may become possible to progressively converge on the underlying dynamical rules governing the network structure of space itself. 3. Time as emergent network activity In UDNT, time is not introduced as an external parameter or imposed background structure. Instead, it emerges from the intrinsic dynamics of the underlying node-link network. Specifically, time is associated with the fluctuating behavior of local network configurations, where links may transiently shift, and nodes may split and merge in reversible processes. In regions of the network that are spatially stable, i.e. not undergoing net expansion or large-scale structural change, such activity does not result in permanent alterations to geometry or connectivity. However, these fluctuations still represent physical processes, and their cumulative count serves as a natural candidate for measuring the passage of time. Just as spatial distance is defined by the number of links connecting two nodes, temporal duration is defined by the number of local network transitions (e.g., oscillatory split-merge cycles) that occur within a given region. This approach offers a relational and quantized concept of time, grounded in the network’s internal evolution rather than imposed from outside. 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