The observation of emergent particles in condensed matter physics such as phonons, holes, and recently magnetic monopoles [1] may lead one to speculate that real particles themselves could exist as emergent properties of a space-time lattice. There are many examples in the literature of the Dirac equation being supported by a lattice structure [2,3,4,5], related, for example, to the physics of ice [6] or of graphene [7]. In particular the case of the Dirac equation on the ice lattice is of potential practical interest as ice models can now be realised in a variety of new experimental systems, including spin ice materials [1] and artificial micromagnetic arrays [8]. We noticed that a Reichenbach causal net [9] has the same topology as an ice lattice and hence, if formulated in space-time, might support a version of relativistic quantum mechanics. In this paper we explore this idea at its simplest level. We propose a causal net for the free motion of a particle based on a relational concept of time as an ordered series of possible events that are linked by a principle of common cause. The causal net is 1+1 dimensional but applies along the direction of motion of the particle in three dimensions. It is reminiscent of the Feynman chequerboard [2] and the ice model [6] but differs from both in important respects. The causal net appears to support the 3+1 dimensional Dirac equation [10] for a free fermion and hence the Schrodinger equation in the low velocity limit. The causal net idea gives an intuitive picture of relativistic quantum mechanical motion and the origin of quantization although it is not a complete re-derivation of conventional quantum mechanics, partly because the space-time is discrete, and partly because it does not automatically imply a principle of superposition. If however we identify a causal net as representing a Dirac state vector then superposition of nets allows a development more consistent with conventional quantum mechanics. We therefore suggest that a more detailed study of effective quantum mechanics on causal nets, based on the transparent underlying concepts of causality and probability, might be a fruitful exercise.
The observation of emergent particles in condensed matter physics such as phonons, holes, and recently magnetic monopoles [1] may lead one to speculate that real particles themselves could exist as emergent properties of a space-time lattice. There are many examples in the literature of the Dirac equation being supported by a lattice structure [2,3,4,5], related, for example, to the physics of ice [6] or of graphene [7]. In particular the case of the Dirac equation on the ice lattice is of potential practical interest as ice models can now be realised in a variety of new experimental systems, including spin ice materials [1] and artificial micromagnetic arrays [8]. We noticed that a Reichenbach causal net [9] has the same topology as an ice lattice and hence, if formulated in space-time, might support a version of relativistic quantum mechanics. In this paper we explore this idea at its simplest level. We propose a causal net for the free motion of a particle based on a relational concept of time as an ordered series of possible events that are linked by a principle of common cause. The causal net is 1+1 dimensional but applies along the direction of motion of the particle in three dimensions. It is reminiscent of the Feynman chequerboard [2] and the ice model [6] but differs from both in important respects. The causal net appears to support the 3+1 dimensional Dirac equation [10] for a free fermion and hence the Schrodinger equation in the low velocity limit. The causal net idea gives an intuitive picture of relativistic quantum mechanical motion and the origin of quantization although it is not a complete re-derivation of conventional quantum mechanics, partly because the space-time is discrete, and partly because it does not automatically imply a principle of superposition. If however we identify a causal net as representing a Dirac state vector then superposition of nets allows a development more consistent with conventional quantum mechanics. We therefore suggest that a more detailed study of effective quantum mechanics on causal nets, based on the transparent underlying concepts of causality and probability, might be a fruitful exercise.
In this paper we discuss a causal network approach to describing relativistic quantum mechanics. Each vertex on the causal net represents a possible point event or particle observation. By constructing the simplest causal net based on Reichenbach-like conjunctive forks in proper time we can exactly derive the 1+1 dimension Dirac equation for a relativistic fermion and correctly model quantum mechanical statistics. Symmetries of the net provide various quantum mechanical effects such as quantum uncertainty and wavefunction, phase, spin, negative energy states and the effect of a potential. The causal net can be embedded in 3+1 dimensions and is consistent with the conventional Dirac equation. In the low velocity limit the causal net approximates to the Schrodinger equation and Pauli equation for an electromagnetic field. Extending to different momentum states the net is compatible with the Feynman path integral approach to quantum mechanics that allows calculation of well known quantum phenomena such as diffraction.
In this paper we discuss a causal network approach to describing relativistic quantum mechanics. Each vertex on the causal net represents a possible point event or particle observation. By constructing the simplest causal net based on Reichenbach-like conjunctive forks in proper time we can exactly derive the 1+1 dimension Dirac equation for a relativistic fermion and correctly model quantum mechanical statistics. Symmetries of the net provide various quantum mechanical effects such as quantum uncertainty and wavefunction, phase, spin, negative energy states and the effect of a potential. The causal net can be embedded in 3+1 dimensions and is consistent with the conventional Dirac equation. In the low velocity limit the causal net approximates to the Schrodinger equation and Pauli equation for an electromagnetic field. Extending to different momentum states the net is compatible with the Feynman path integral approach to quantum mechanics that allows calculation of well known quantum phenomena such as diffraction.
In this paper we discuss a causal network approach to describing relativistic quantum mechanics. Each vertex on the causal net represents a possible point event or particle observation. By constructing the simplest causal net based on Reichenbach-like conjunctive forks in proper time we can exactly derive the 1+1 dimension Dirac equation for a relativistic fermion and correctly model quantum mechanical statistics. Symmetries of the net provide various quantum mechanical effects such as quantum uncertainty and wavefunction, phase, spin, negative energy states and the effect of a potential. The causal net can be embedded in 3+1 dimensions and is consistent with the conventional Dirac equation. In the low velocity limit the causal net approximates to the Schrodinger equation and Pauli equation for an electromagnetic field. Extending to different momentum states the net is compatible with the Feynman path integral approach to quantum mechanics that allows calculation of well known quantum phenomena such as diffraction.
Abstract This paper is the third part of the third volume of the Back-Particle Theory. Building upon the dynamic intrinsic displacements, unified wave equation, and emergence of electromagnetic potentials established in Parts I and II of Volume 3, it systematically completes the geometric interpretation of quantum behavior, the emergent derivation of relativistic structure, and the geometric construction of the Dirac equation. Starting from the enormous degrees of freedom (∼ 10¹⁰⁴) of the back-particle array and its deterministic chaos, we demonstrate that quantum probability is not a fundamental postulate but an inevitable result of finite-information coarse-grained descriptions of the array’s chaotic dynamics, with the Born rule P = |ψ|² interpreted as a statistical expression of the excitation weights of specific vibration modes of the array. Through canonical quantization of the array displacement field, we rigorously derive the uncertainty principle Δx Δp ≥ ℏ_eff/2 and the zero-point fluctuation formula, and derive the decoherence rate Γ_dec ∼ (Θ_env/ℏ_eff)(Δr/d₀)² from the system-environment coupling model. Furthermore, starting from the homogeneous wave equation of the free array and requiring its form invariance under inertial frame transformations, we rigorously derive the Lorentz transformation, proving that the space-time structure of special relativity is a natural emergence of the array’s wave geometry. Based on the second-order classical dynamical equation of intrinsic displacements, we rigorously derive the Dirac equation through operator factorization by introducing the Clifford algebra γ^μ, where the spinor field Ψ is constructed from the intrinsic displacements (d_A, d_f) via symmetry mapping, the mass term ω₀/c originates from the array’s restoration frequency, the spin operator satisfies the SU(2) algebra, and spin 1/2 emerges naturally from the quantization of intrinsic angular momentum. All derivations strictly follow the principles of causal unidirectionality, field superposition, dimensional purity, and parameter emergence, with all parameters emerging in a closed manner from the basic set {A, F, f, f_ext, ρ₀, c} and the lattice geometric factors, achieving a complete logical closed loop from classical array dynamics to quantum mechanics. Keywords: Back-Particle; Emergence of Quantum Probability; Born Rule; Uncertainty Principle; Decoherence; Lorentz Transformation; Dirac Equation; Spinor Field; Spin 1/2; Clifford Algebra; Geometric Quantum Mechanics
The axiomatic foundations of quantum mechanics, while operationally successful, obscure the underlying physical reality. This work introduces a mechanical topological ontology intended to provide a causal basis for the theory. We propose a single postulate: that a fundamental fermion is the 0-dimensional boundary of a 1-dimensional, massive, string-like object topologically anchored to a dynamic vacuum field. From this postulate, we argue that the core structures of single-particle relativistic quantum mechanics can be understood as derivable theorems. First, we construct the object’s state space by identifying two independent, binary properties—a local, vibrational/topological duality (spin) and a global, orientational duality (matter/antimatter)—whose tensor product naturally yields the 4-component Dirac spinor. Second, we show that the requirement of relativistic consistency uniquely determines the algebraic operators (gamma matrices and their Clifford algebra) and the lowest-order dynamical law, leading to the Dirac equation as a theorem of consistency. Third, we demonstrate that the non-relativistic Schrödinger and Pauli equations emerge as its correct low-energy limits, where the quantum potential V (x) is identified with a local variation in the object’s rest mass. Finally, we argue that the foundational tenets of quantum measurement—the spin-1/2 property, the probabilistic Born rule, and the correlations of entanglement—can be reframed as necessary consequences of the object’s intrinsic topology and interaction mechanics. This framework thus offers a path toward demoting the axioms of quantum theory to derivable theorems, aiming to resolve its long-standing paradoxes by providing a clear and coherent physical picture.
A physical principle that has consequences for the topology of a relativistic space-time is the principle of Einstein-Weyl causality. We show that these consequences also have quantum mechanical manifestations. Borchers and Sen have rigorously investigated the mathematical implications of Einstein-Weyl causality, regarded as a partial order, for the underlying spaces. This partial order was axiomatized by them and shown to admit Q^2 as an ordered space. They subsequently proved that, given certain topological assumptions, these spaces satisfying Einstein-Weyl causality are densely imbedded in spaces that have locally the structure of differentiable manifolds. They were left, however, with important philosophical paradoxes regarding the nature of the physical real line E, e.g., whether E = R, the real line of mathematics. We can view their original result, without further assumptions, as an insight into a denumerable space-time. This would suggest an investigation into a constructible foundation for relativistic physics. A set theoretical foundation is proposed including an axiom of countable constructibility. We first show that this constructible theory actually contains polynomial functions which are locally homeomorphic with a dense, denumerable metric space R*. Eigenfunctions governing fields can then be effectively obtained by iteration. Postulating a Lagrangian for fields in a compactified space-time, we get a general field description of which the Schrodinger equation is a special case. From these results we can then also show that this denumerable space-time is relational (in the sense that space is not infinitesimally small if and only if it contains a quantized field) and that, since Q^2 is imbedded in R*^2, it directly fulfills the strict topological requirements for Einstein-Weyl causality. Therefore, the theory predicts that E = R*.