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arXiv:gr-qc/0109053 · International Journal of Theoretical Physics

Discrete Quantum Causal Dynamics

Richard F. Blute, Ivan T. Ivanov, Prakash Panangaden

Published Sep 1, 2003• 37 citations
arXivHTMLPDF
DOI
10.1023/a:1027335119549

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Jean-François Rigollet

Discrete-to-Continuum Limit of Causal Chains in the Kähler Reconstruction Previous paper (Causal Reconstruction: Elementary Succession and the Planck Energy) proposed that the Schrödinger equation should arise as the continuum limit of discrete causal chains. Article 18 establishes this result under explicit regularity assumptions, completing the bridge between elementary causal succession and continuous quantum dynamics. Purpose The reconstruction programme contains two distinct transitions: Discrete causal chain → continuum reconstructive action → Madelung–Schrödinger dynamics. The second step was already established in Article 5. The purpose of this paper is to prove the first step: that discrete causal chains converge to the continuous reconstructive action in a mathematically controlled way. Main Idea The reconstructive cost between two states is interpreted as a discrete Lagrangian in the sense of variational integrator theory. A causal chain therefore becomes a discrete variational system whose dynamics is governed by discrete Euler–Lagrange equations. The key observation is that the discrete action SN = Σ C(ρk,ρk+1) plays exactly the role of a discrete Lagrangian approximation of the reconstructive action Srec. Main Results Discrete Lagrangian formulation. The reconstructive cost is shown to define a discrete Lagrangian under suitable regularity conditions. Discrete Euler–Lagrange equations. Critical causal chains satisfy discrete Euler–Lagrange equations expressing momentum continuity across each causal node. Γ-convergence. The discrete action SN Γ-converges to the continuous reconstructive action Srec as the elementary action quantum tends to zero while total duration remains fixed. Convergence of solutions. Solutions of the discrete causal-chain equations converge to solutions of the continuous Madelung equations. Preservation of Kähler structure. The discrete symplectic structure converges to the canonical Kähler symplectic form of the reconstruction phase space. Recovery of Schrödinger dynamics. Combining the present result with Article 5 yields a complete chain from discrete causal succession to the Schrödinger equation. Regularity Analysis A significant contribution of the paper is the analysis of the Fisher information functional in alternative coordinates. In the density variable ρ, Fisher information becomes singular as ρ approaches zero. In the amplitude variable u = √ρ, Fisher information becomes exactly a Dirichlet energy: I(ρ) = 4||∇u||². In the wavefunction variable ψ = √ρ exp(iS/ℏ), the reconstructive action becomes smooth on the natural Hilbert space H¹(M,C). This shows that the apparent singularities of Fisher information are coordinate artefacts rather than genuine obstructions. The Three Operative Conditions The continuum limit ultimately depends on three explicit conditions: C1 — Tightness: minimising sequences remain precompact. C2 — Small-step regime: elementary causal transitions remain inside the injectivity radius. C3 — Non-degeneracy: the second variation of the reconstructive action remains positive, excluding conjugate points and ensuring local uniqueness. Condition C2 becomes automatic in the continuum limit. Conditions C1 and C3 remain the principal analytical assumptions. Conceptual Significance The paper transforms the causal-chain picture from a heuristic interpretation into a variational approximation theorem. Discrete causal succession is no longer merely compatible with quantum dynamics; it converges to it. In this view, a quantum trajectory is not a fundamental continuous object. It is the continuum limit of a chain of elementary reconstructive transitions, each carrying one action quantum. Conclusion This paper completes the mathematical bridge between the causal layer and the quantum layer of the reconstruction programme. Under explicit regularity assumptions, discrete causal chains converge to the reconstructive action, the reconstructive action yields the Madelung equations, and the Madelung formulation is equivalent to Schrödinger dynamics. The resulting picture is a fully variational route from elementary causal succession to quantum mechanics.

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In this paper, we consider fermionic systems in discrete space-time evolving with a strict notion of causality, meaning they evolve unitarily and with a bounded propagation speed. First, we show that the evolution of these systems has a natural decomposition into a product of local unitaries, which also holds if we include bosons. Next, we show that causal evolution of fermions in discrete space-time can also be viewed as the causal evolution of a lattice of qubits, meaning these systems can be viewed as quantum cellular automata. Following this, we discuss some examples of causal fermionic models in discrete space-time that become interesting physical systems in the continuum limit: Dirac fermions in one and three spatial dimensions, Dirac fields, and briefly the Thirring model. Finally, we show that the dynamics of causal fermions in discrete space-time can be efficiently simulated on a quantum computer.

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Discrete-to-Continuum Limit of Causal Chains in the Kähler Reconstruction Previous paper (Causal Reconstruction: Elementary Succession and the Planck Energy) proposed that the Schrödinger equation should arise as the continuum limit of discrete causal chains. Article 18 establishes this result under explicit regularity assumptions, completing the bridge between elementary causal succession and continuous quantum dynamics. Purpose The reconstruction programme contains two distinct transitions: Discrete causal chain → continuum reconstructive action → Madelung–Schrödinger dynamics. The second step was already established in Article 5. The purpose of this paper is to prove the first step: that discrete causal chains converge to the continuous reconstructive action in a mathematically controlled way. Main Idea The reconstructive cost between two states is interpreted as a discrete Lagrangian in the sense of variational integrator theory. A causal chain therefore becomes a discrete variational system whose dynamics is governed by discrete Euler–Lagrange equations. The key observation is that the discrete action SN = Σ C(ρk,ρk+1) plays exactly the role of a discrete Lagrangian approximation of the reconstructive action Srec. Main Results Discrete Lagrangian formulation. The reconstructive cost is shown to define a discrete Lagrangian under suitable regularity conditions. Discrete Euler–Lagrange equations. Critical causal chains satisfy discrete Euler–Lagrange equations expressing momentum continuity across each causal node. Γ-convergence. The discrete action SN Γ-converges to the continuous reconstructive action Srec as the elementary action quantum tends to zero while total duration remains fixed. Convergence of solutions. Solutions of the discrete causal-chain equations converge to solutions of the continuous Madelung equations. Preservation of Kähler structure. The discrete symplectic structure converges to the canonical Kähler symplectic form of the reconstruction phase space. Recovery of Schrödinger dynamics. Combining the present result with Article 5 yields a complete chain from discrete causal succession to the Schrödinger equation. Regularity Analysis A significant contribution of the paper is the analysis of the Fisher information functional in alternative coordinates. In the density variable ρ, Fisher information becomes singular as ρ approaches zero. In the amplitude variable u = √ρ, Fisher information becomes exactly a Dirichlet energy: I(ρ) = 4||∇u||². In the wavefunction variable ψ = √ρ exp(iS/ℏ), the reconstructive action becomes smooth on the natural Hilbert space H¹(M,C). This shows that the apparent singularities of Fisher information are coordinate artefacts rather than genuine obstructions. The Three Operative Conditions The continuum limit ultimately depends on three explicit conditions: C1 — Tightness: minimising sequences remain precompact. C2 — Small-step regime: elementary causal transitions remain inside the injectivity radius. C3 — Non-degeneracy: the second variation of the reconstructive action remains positive, excluding conjugate points and ensuring local uniqueness. Condition C2 becomes automatic in the continuum limit. Conditions C1 and C3 remain the principal analytical assumptions. Conceptual Significance The paper transforms the causal-chain picture from a heuristic interpretation into a variational approximation theorem. Discrete causal succession is no longer merely compatible with quantum dynamics; it converges to it. In this view, a quantum trajectory is not a fundamental continuous object. It is the continuum limit of a chain of elementary reconstructive transitions, each carrying one action quantum. Conclusion This paper completes the mathematical bridge between the causal layer and the quantum layer of the reconstruction programme. Under explicit regularity assumptions, discrete causal chains converge to the reconstructive action, the reconstructive action yields the Madelung equations, and the Madelung formulation is equivalent to Schrödinger dynamics. The resulting picture is a fully variational route from elementary causal succession to quantum mechanics.
This work develops a graph-theoretic and process-based foundation for indefinite causal order in quantum theory, demonstrating that causal indefiniteness is not an ad hoc or exotic phenomenon but an emergent structural property of quantum dynamics when the primitive unit of evolution is taken to be the П-такт (P-Tact) — a discrete quantum process composed of local unitary superposition and entangling unitary interaction. Building upon a sequence of previously published foundational works, this paper formulates indefinite causal order as a superposition of quantum process graphs, where each graph encodes a definite causal structure, and the quantum state corresponds to a coherent superposition over such graphs. Within this framework, causal order is not fundamental but arises only in coarse-grained or classical limits. We show that: Indefinite causal order naturally emerges from the superposition of process graphs, without invoking external control systems or higher-order operations. The quantum switch is reinterpreted as a special finite realization of a more general graph-superposition principle. Time and causality emerge as effective parameters in the infinite-graph (continuum) limit of discrete П-такт dynamics. The graph-theoretic formulation is fully compatible with — and structurally supports — the Everettian (Many-Worlds) interpretation, where branching corresponds to graph bifurcations rather than wavefunction collapse. Classical spacetime causality appears as a macroscopic ordering imposed on an underlying non-causal quantum substrate. This approach unifies quantum information, indefinite causal order, emergent temporality, and Everettian branching within a single formal framework based on discrete process units and their graph composition. The results close several open conceptual gaps in the interpretation of quantum switches, process matrices, and non-classical causal structures, offering a consistent ontology grounded in quantum dynamics itself. This paper directly builds upon and extends the following foundational works: Pavlov, M. (2026). Quantum Switch as an Emergent Process Structure from Discrete Process Tacts (П-такт). Zenodo.https://doi.org/10.5281/zenodo.18498181 Pavlov, M. (2026). Quantum Switch and Teleportation. Zenodo.https://doi.org/10.5281/zenodo.18509173 Pavlov, M. (2026). Emergence of Causality, Gravity, Time, and Three-Dimensional Space from Discrete P-Tact (П-такт) Dynamics. Zenodo.https://doi.org/10.5281/zenodo.18504244 Pavlov, M. (2026). P-Tact (П-такт) as the Primitive Unit of Quantum Theory. Zenodo.https://doi.org/10.5281/zenodo.18516711 Together, these works establish a coherent program aimed at demonstrating the universality of quantum mechanics, the absence of fundamental collapse, and the emergence of spacetime and causality from discrete quantum process dynamics.
In this paper, we consider fermionic systems in discrete space-time evolving with a strict notion of causality, meaning they evolve unitarily and with a bounded propagation speed. First, we show that the evolution of these systems has a natural decomposition into a product of local unitaries, which also holds if we include bosons. Next, we show that causal evolution of fermions in discrete space-time can also be viewed as the causal evolution of a lattice of qubits, meaning these systems can be viewed as quantum cellular automata. Following this, we discuss some examples of causal fermionic models in discrete space-time that become interesting physical systems in the continuum limit: Dirac fermions in one and three spatial dimensions, Dirac fields, and briefly the Thirring model. Finally, we show that the dynamics of causal fermions in discrete space-time can be efficiently simulated on a quantum computer.