Paper XVII - Temporal Holonomy and Chronology Protection

DOI: 10.5281/zenodo.18009216 Temporal Holonomy and Chronology Protection Paper XVII of the Ordered-Dynamics Reconstruction Program Paul Cooneya aIndependent Researcher, Innisfil, Ontario, Canada E-mail: paul.co[email protected]to.ca Contents 1 Introduction 1 2 Evolution as a Path Functor 1 3 Temporal Holonomy and Obstruction 2 4 Operational Time and Integrability 2 5 Discrete Interpretation and Ordered Graphs 3 6 Chronology Protection and Operational Causality 3 7 Conclusion 3 losed timelike curves are traditionally treated as sources of causal paradox. We reformulate the problem at a more primitive level appropriate to the ordered-dynamics framework: the existence of a globally single-valued evolution rule. We show that any theory assigning evolution operators to timelike paths with composition under concatenation generically develops temporal holonomy in the presence of closed timelike loops. Nontrivial temporal holonomy obstructs the existence of a single-valued state assignment and therefore obstructs quantum mechanics itself. Chronology protection is thus derived as a structural consistency condition on ordered dynamics, rather than as a dynamical response to paradoxes or boundary conditions. Any universe admitting quantum states must enforce trivial temporal holonomy and hence acyclicity of its underlying interaction structure. 1 Introduction General Relativity admits exact solutions containing closed timelike curves, most famously the G¨odel universe. Such solutions raise a fundamental question: are closed timelike curves physically admissible, or do they signal a breakdown of predictability? Most discussions focus on paradox resolution. In contrast, the Ordered-Dynamics Reconstruction Program places a more basic requirement on physical theories: the existence of a globally single-valued evolution rule compatible with record-based observables. In earlier papers, space, influence cones, quantum state spaces, and operational time were reconstructed from bounded information flow. In this paper we show that these structures are dynamically viable only if evolution is globally single-valued. This requirement alone excludes nontrivial temporal holonomy and therefore enforces chronology protection. Chronology protection is not introduced to avoid paradoxes. It is required for quantum dynamics to exist at all. 2 Evolution as a Path Functor Let Mbe a time-oriented Lorentzian manifold or, more generally, an ordered interaction structure admitting timelike paths. – 1 – Let Tim(M) be the category whose objects are events and whose morphisms are futuredirected timelike curves, with composition given by concatenation. Let Sbe a state space and Aut(S) the group of invertible state transformations. Definition 1 (Evolution assignment).An evolution assignment is a map U:Tim(M)→Aut(S) satisfying: 1. U(γ2◦γ1)=U(γ2)U(γ1), 2. U(γ) = for trivial curves, 3. U(γ) is generated locally along γ. Remark 1.Only concatenation and single-valuedness are used below. No quantum axioms are assumed at this stage. 3 Temporal Holonomy and Obstruction Let Ωpdenote the set of closed timelike curves based at an event p. Definition 2 (Temporal holonomy group). p:= {U(γ):γ∈Ωp}. [Holonomy obstruction] If p={} at any event p, then no globally single-valued state assignment exists. Proof. If U(γ)= for some closed loop γ, then transporting a state around γproduces two distinct states at the same event p, violating single-valuedness. Globally consistent evolution requires p={} for all p. 4 Operational Time and Integrability Definition 3 (Operational time 1-form).A 1-form ωis operational if ω(v)>0 for all future-directed timelike vectors v. Definition 4 (Clock-driven evolution).Evolution is clock-driven if U(γ) = exp−GZγ ω for some generator G. Definition 5 (Temporal integrability).The operational time form ωis integrable if ω=d˜ t for some scalar operational time ˜ t. If ωis integrable, all closed timelike loops have zero accumulated operational time. Remark 2 (Relation to influence delay).In the interaction-graph formulation (Papers XII– XIII), the operational time 1-form corresponds to accumulated influence delay. Nonintegrability of ωcorresponds to path-dependent delay and hence temporal holonomy. Remark 3 (Gauge analogy).Gauge holonomy obstructs global trivialization of a bundle. Temporal holonomy obstructs global trivialization of evolution. Unlike gauge holonomy, temporal holonomy is operationally forbidden. – 2 – 5 Discrete Interpretation and Ordered Graphs In the ordered interaction graph, a G¨odel-type configuration corresponds to a directed cycle v1→v2→···→v1. Proposition 3 implies that such cycles cannot support a Hilbert space of states unless the loop evolution satisfies Uloop =. Generic dynamics therefore enforce acyclicity of the interaction graph. This condition is structural, not dynamical. 6 Chronology Protection and Operational Causality Remark 4.Acyclicity of the interaction graph is equivalent to trivial temporal holonomy. As shown in subsequent papers, acyclicity is precisely the condition required for conditional probabilities to be single-valued and for operational no-signaling to hold. Temporal holonomy would render marginal probabilities multi-valued and destroy relativistic causality at the operational level. 7 Conclusion Closed timelike curves are not fundamentally paradoxical. They are structurally incompatible with single-valued evolution. Temporal holonomy obstructs the existence of a consistent evolution functor and therefore obstructs quantum mechanics itself. Chronology protection is thus not a conjectural dynamical effect but a structural requirement: any universe admitting quantum states must enforce trivial temporal holonomy and hence acyclicity of its underlying interaction structure. This result completes the global consistency constraints required by ordered dynamics. – 3 –