Paper XV - Emergence of Complex Quantum Structure from Ordered Dynamics

DOI: 10.5281/zenodo.18009175 Emergence of Complex Quantum Structure from Ordered Dynamics Paper XV of the Ordered-Dynamics Reconstruction Program Paul Cooneya aIndependent Researcher, Innisfil, Ontario, Canada E-mail: paul.co[email protected]to.ca Abstract. We derive the complex Hilbert-space structure of quantum theory from informationtheoretic principles within the ordered-dynamics framework. Building on Papers XII–XIV, where space, gravity, and gauge structure emerge from bounded influence propagation and local record redundancy, we show that quantum superposition and linear dynamics are forced by fluctuating influence. We then demonstrate that consistent encoding of operational phase requires the state space to decompose into invariant two-dimensional rotation planes. This structure uniquely induces a complex field, excluding real and quaternionic alternatives. Complex Hilbert space thus emerges not as a mathematical convenience but as the minimal structure supporting reversible, continuous dynamics with stable interference under finite influence speed. Contents 1 Introduction 1 2 Fluctuating Influence and Linearity 2 2.1 Influence fluctuations 2 2.2 Necessity of linear state space 2 3 Operational Phase 2 3.1 Relative ordering as phase 2 3.2 Why phase requires rotations 2 4 Invariant Rotation Planes 3 5 Emergence of the Complex Field 3 5.1 Complex structure from rotation 3 5.2 Exclusion of real and quaternionic alternatives 3 6 Unitary Dynamics and Probabilities 3 7 Relation to Other Papers 4 8 Conclusion 4 1 Introduction Quantum theory is distinguished from classical probability theory by superposition, interference, and complex amplitudes. While numerous reconstructions derive aspects of quantum structure from operational axioms, the appearance of complex numbers is often assumed rather than explained. Why should nature use the complex field Crather than real Ror quaternionic Halternatives? Existing derivations typically appeal to abstract symmetry principles or mathematical convenience. In this paper we derive complex quantum structure from the same information-theoretic foundations that produced space (Paper XII), gravity (Paper XIII), and gauge structure (Paper XIV). No Hilbert space is assumed. Instead, we show that complex structure is forced by: •finite influence speed, •reversible dynamics, •fluctuating influence propagation, •stable record encoding. The result is a principled explanation of why quantum mechanics is complex at the kinematical level. – 1 – 2 Fluctuating Influence and Linearity 2.1 Influence fluctuations In classical regimes, influence propagation can be treated deterministically. At sufficiently small scales, bounded influence capacity implies stochastic ordering of influence events. Different microscopic orderings may produce the same macroscopic records. These fluctuations are not epistemic ignorance but operationally unavoidable: finite influence speed prevents total ordering of all microscopic processes. 2.2 Necessity of linear state space Let ρ1and ρ2be two distinguishable preparation procedures. Fluctuating influence ordering implies that convex combinations ρ=pρ1+(1−p)ρ2(2.1) must themselves be valid states, representing mixtures of influence histories. More strongly, reversible coarse-graining of fluctuating influence requires closure under linear superposition. Nonlinear state spaces generically amplify microscopic ordering noise, destroying record stability. Finite influence fluctuations force the state space to be linear. Remark 1.This establishes linearity and superposition but does not yet determine the scalar field. That is the task of the next sections. 3 Operational Phase 3.1 Relative ordering as phase Distinct influence orderings that yield identical records may differ by a relative temporal offset invisible to classical observation. Such offsets cannot be represented as probabilities; they are relational. This relational degree of freedom is operational phase. Phase has three defining properties: •it is unobservable in isolation, •it is observable through interference, •it is conserved under reversible dynamics. Any consistent theory must encode phase without introducing new observable degrees of freedom. 3.2 Why phase requires rotations A scalar parameter cannot encode phase invariantly. Phase must be represented by a structure admitting continuous, norm-preserving transformations. This forces phase to act as a rotation. – 2 – 4 Invariant Rotation Planes [Invariant rotation planes] Any finite-dimensional real linear state space supporting continuous, reversible, bounded dynamics decomposes into orthogonal invariant two-dimensional subspaces on which dynamics act as rotations. Sketch. Bounded reversible dynamics generate a compact one-parameter group. By the real spectral theorem, generators decompose into blocks that are either trivial or 2 ×2 antisymmetric matrices. Each nontrivial block generates rotations in a two-dimensional plane. Each such plane supports transformations of the form x y7→ cos θ−sin θ sin θcos θx y. 5 Emergence of the Complex Field 5.1 Complex structure from rotation Each invariant rotation plane admits a canonical linear operator Jsatisfying J2=−I, defined by a π/2 rotation. This operator equips the plane with the algebraic structure of C. Superpositions across independent planes assemble into a complex vector space. 5.2 Exclusion of real and quaternionic alternatives Why not R?Real vector spaces cannot encode continuous phase. Interference effects are unstable under composition. Why not H?Quaternionic structure introduces noncommuting phases. This destroys path-independence of influence composition and violates record consistency under finite influence speed. The complex field Cis the unique scalar field compatible with reversible, continuous phase encoding under bounded influence. 6 Unitary Dynamics and Probabilities Reversible dynamics preserving interference must preserve the inner product. This forces evolution to be unitary on the emergent complex Hilbert space. Measurement probabilities arise from record statistics. Consistency under composition and coarse-graining uniquely selects the quadratic Born rule. A full treatment of records, irreversibility, and measurement is deferred to later papers in the series. – 3 – 7 Relation to Other Papers The foundational reconstruction proceeds as follows: •Paper XII: Space and locality from bounded influence, •Paper XIII: Gravity from inhomogeneous influence delay, •Paper XIV: Gauge structure from local redundancy, •Paper XV: Quantum kinematics from ordered influence. Subsequent papers address records, measurement, irreversibility, and phenomenological applications. 8 Conclusion We have shown that complex quantum structure emerges necessarily from informationtheoretic constraints on influence propagation. Fluctuating influence forces linearity; operational phase forces rotation; invariant rotation planes force complex structure. Real and quaternionic alternatives are excluded by reversibility and record consistency. Quantum theory thus emerges as the unique stable kinematical framework compatible with ordered dynamics under finite influence speed. References [1] P. Cooney, Emergent Spatial Locality from Bounded Influence, Paper XII of the Ordered-Dynamics Reconstruction Program (2025). [2] P. Cooney, Gravity as Inhomogeneous Influence Propagation, Paper XIII of the Ordered-Dynamics Reconstruction Program (2025). [3] P. Cooney, Emergent Gauge Structure from Local Redundancy, Paper XIV of the Ordered-Dynamics Reconstruction Program (2025). [4] L. Hardy, Quantum Theory From Five Reasonable Axioms, arXiv:quant-ph/0101012. [5] G. Chiribella, G. M. D’Ariano, and P. Perinotti, Phys. Rev. A 84, 012311 (2011). [6] M. P. Soler, Commun. Math. Phys. 4, 165 (1967). – 4 –