Paper I — Ordered Dynamics and Operational State Spaces

DOI: 10.5281/zenodo.17925621 Ordered Dynamics and Operational State Spaces Paper I 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 Operational state spaces 2 3 Ordered reversible dynamics 2 4 Rotation-plane decomposition 3 5 Informational reciprocity and invariant pairing 3 6 Complex structure and orientation fixing 3 7 Born rule 4 8 Ordered kinematics 5 9 Why complex structure is not optional 6 10 Physical interpretation of the ordering parameter 6 10.1 From ordering to clock time 6 10.2 Relational time 6 11 Conclusion 6 DOI: 10.5281/zenodo.17925621 e reconstruct the kinematical structure of quantum theory from minimal operational assumptions about ordered reversible dynamics on finite-dimensional probabilistic state spaces. Rather than postulating Hilbert space or complex amplitudes, we assume only: (i) finite operational resolution, (ii) existence of continuous reversible evolution with bounded orbits, and (iii) an informational reciprocity principle relating preparations and sharp effects. We show these assumptions force a decomposition into invariant two-dimensional rotation planes. A reciprocity-induced invariant pairing then selects a unique orientation on each plane, yielding a canonical complex structure. From phase invariance, positivity, and composition consistency, the Born rule follows as the unique probability functional compatible with the induced complex inner product. Unitary Schr¨odinger kinematics emerges as the minimal consistent representation of ordered dynamics. Contents 1 Introduction Quantum theory is often presented by postulating a complex Hilbert space, unitary evolution, and the Born rule. This obscures why complex structure and quadratic probabilities are necessary, rather than convenient. – 1 – A long tradition of reconstruction programs derives quantum theory from operational or informational axioms (e.g. Hardy; Chiribella–D’Ariano–Perinotti; Masanes–M¨uller). Our approach differs in emphasis: we take ordered reversible dynamics with bounded orbits as primary structure and ask what probabilistic bookkeeping is compatible with it. Roadmap. Section 2 defines operational state spaces and finite resolution. Section 3 formulates ordered reversibility with bounded orbits. Section 4 derives the rotation-plane decomposition. Section 5 introduces informational reciprocity and proves existence of an invariant pairing. Section 6 shows how this pairing fixes orientation and induces a complex Hilbert structure. Section 7 derives the Born rule from phase invariance and composition. Section 10 interprets the ordering parameter and clarifies the relation to physical time. Relationship to other reconstructions. Hardy (2001) and Masanes–M¨uller (2011) use continuous reversibility and simplicity/capacity conditions; Chiribella–D’Ariano–Perinotti (2011) rely on purification and informational constraints. Here, bounded-orbit ordered reversibility is the key dynamical input, while our reciprocity axiom is a constrained (and explicitly weaker) substitute for assuming full self-duality from the outset. 2 Operational state spaces Definition 1 (Operational state space).A system is described by a real finite-dimensional vector space Vcontaining a compact convex set Ω ⊂Vof normalized states. Definition 2 (Effects).An effect is an affine map e: Ω →[0,1]. For convenience we extend effects affinely to Vand identify them with elements of the affine dual. [Finite operational resolution] For any preparation procedure, there exists a finite set of mutually exclusive and exhaustive outcomes {Oi}n i=1 such that Pn i=1 p(Oi|ω) = 1 for all ω∈Ω. Remark 1.Axiom 2 is deliberately weak: it encodes the empirical fact that any actual measurement corresponds to a finite outcome partition at finite resolution. 3 Ordered reversible dynamics [Ordered reversibility with bounded orbits] There exists a continuous one-parameter group of reversible transformations Tλ:V→Vsuch that: 1. T0= 1 and Tλ+µ=TλTµ, 2. For every ω∈Ω, the orbit {Tλω:λ∈R}has compact closure in V. Remark 2.The parameter λis an ordering parameter, not yet physical clock time. It labels reversible transformations; physical time arises by calibration of clocks (Section 10). Standard Lie theory yields a generator Gwith Tλ=eλG. – 2 – 4 Rotation-plane decomposition Definition 3 (Dynamically active subspace).Let Vdyn := span{Tλω−ω:ω∈Ω, λ ∈R}. [Rotation-plane decomposition] The restriction of Tλto Vdyn decomposes as a direct sum of invariant two-dimensional planes: Vdyn ∼ = m M j=1 Pj, where each Pjis invariant and Tλ|Pjis a rotation with frequency ωj>0. Proof. Bounded orbits exclude any expanding/contracting Jordan blocks of G, hence exclude real eigenvalues and nontrivial nilpotent parts on Vdyn. Over R, the remaining canonical blocks are 2 ×2 rotation generators with eigenvalues ±iωj. Each such block defines an invariant plane Pjon which Tλacts as a rotation. [Spectral frequencies] On each Pj,G|Pjhas eigenvalues ±iωjwith ωj>0. 5 Informational reciprocity and invariant pairing [Informational reciprocity] For every pure state ω∈Ω, there exists a unique pure effect eω such that: 1. (Normalization) eω(ω) = 1, 2. (Reciprocity) eω(ω′)=eω′(ω) for all pure ω′, 3. (Covariance) eTλω(Tλω′)=eω(ω′) for all λ. [Existence of invariant pairing] Axiom 5 induces a unique symmetric bilinear pairing ⟨·,·⟩ :Vdyn ×Vdyn →Rsuch that for pure states ω, ω′, eω(ω′) = ⟨ω, ω′⟩, and ⟨Tλx, Tλy⟩=⟨x, y⟩for all x, y ∈Vdyn. Proof. Define ⟨ω, ω′⟩:= eω(ω′) on pure states; reciprocity gives symmetry and normalization fixes scale. Extend bilinearly by linearity of effects on convex combinations and finitedimensionality. Covariance gives invariance under Tλ. Uniqueness follows from density of affine combinations of pure states in Ω. 6 Complex structure and orientation fixing On each plane Pj, define Jj:= 1 ωj G  Pj ,so that J2 j=−1. There are two compatible orientations, Jjand −Jj. [Orientation fixing from positivity and covariance] The physical cone Ω ∩Pjand the pairing ⟨·,·⟩ select a unique choice between ±Jjon each Pj: the choice for which the induced complex inner product yields nonnegative transition probabilities for all pairs of pure states under the full rotation orbit. – 3 – Proof. Fix a plane Pj. Choose any pure state ω∈Ω∩Pjand consider its orbit ω(θ):=Tθ/ωjω, which is a closed curve in Pj. By Theorem 5, transition numbers p(θ):=eω(ω(θ))=⟨ω,ω(θ)⟩are invariantly defined and must satisfy 0 ≤p(θ)≤1. Now, representing Pjas R2with generator Gacting as an oriented rotation, the two choices ±Jjcorrespond to identifying the physical rotation angle as ±θin the complex phase. Under one choice, the complex coordinate of ω(θ) is e−iθω; under the other it is e+iθω. Because p(θ) must be nonnegative for the entire orbit and because the bilinear pairing is T-invariant, the physically consistent choice is the one for which the phase convention makes the overlap functional depend only on the relative angle and yields a nonnegative function of that angle. This fixes orientation up to a global conjugation on all planes; a mixed choice (flipping orientation on some but not all planes) would violate covariance of the state–effect correspondence across composed rotations. Hence a unique Jjis selected on each plane, consistently across j. [Complex Hilbert structure] Let J:= LjJjwith the orientations fixed as in Proposition 6. Define ⟨ψ, ϕ⟩C:= ⟨ψ, ϕ⟩+i⟨ψ, Jϕ⟩. Then ⟨·,·⟩Cis a Hermitian inner product on Vdyn, and Tλacts unitarily. Proof. Sesquilinearity and Hermiticity. Bilinearity of ⟨·,·⟩ and linearity of Jgive linearity in the second argument; symmetry plus J-orthogonality give conjugate symmetry: ⟨ψ, ϕ⟩C=⟨ψ, ϕ⟩ − i⟨ψ, Jϕ⟩=⟨ϕ, ψ⟩+i⟨ϕ, Jψ⟩=⟨ϕ, ψ⟩C. Positive-definiteness. For ψ= 0, ⟨ψ, ψ⟩C=⟨ψ,ψ⟩+i⟨ψ, Jψ⟩. But ⟨ψ, Jψ⟩=−⟨Jψ, ψ⟩by symmetry and J2=−1, hence ⟨ψ, Jψ⟩= 0. Therefore ⟨ψ, ψ⟩C= ⟨ψ, ψ⟩>0. Unitarity. Since ⟨Tλx, Tλy⟩=⟨x, y⟩and Jcommutes with Tλon each invariant plane, ⟨Tλψ, Tλϕ⟩C=⟨ψ, ϕ⟩C. 7 Born rule We now derive the quadratic probability rule as the unique functional compatible with: (i) phase invariance, (ii) positivity, (iii) composition of independent rotations, and (iv) normalization on identical states. [Phase invariance forces dependence on modulus] For pure ψ, ϕ, any probability assignment p(ϕ|ψ) compatible with the complex structure must satisfy p(ϕ|ψ) = F(|⟨ϕ, ψ⟩C|) for some function F: [0,1] →[0,1]. Proof. Global phase is an internal rotation acting as ψ7→ eiθψon a plane. Operationally it cannot change outcome statistics, hence p(ϕ|eiθψ) = p(ϕ|ψ) for all θ. But ⟨ϕ, eiθψ⟩C= eiθ⟨ϕ, ψ⟩C, so invariance implies dependence only on the modulus. [Composition consistency yields a functional equation] Assume that for sequential filters (rank-one tests) the probability of passing both depends only on the product of overlaps. Then Fsatisfies F(rs) = F(r)F(s) for r, s ∈[0,1]. – 4 – Proof. Consider three pure states ψ, χ, ϕ with |⟨χ, ψ⟩C|=rand |⟨ϕ, χ⟩C|=s, and choose them within a single two-plane so that phases can be aligned. Operationally, perform the sharp test associated with χ, then the sharp test associated with ϕ. The joint pass probability equals the product of the conditional probabilities: p(ϕafter χ|ψ)=p(χ|ψ)p(ϕ|χ)=F(r)F(s). But in the aligned coplanar case the net transition amplitude has modulus rs, so the same operational procedure must equal F(rs). Hence F(rs)=F(r)F(s). [Continuity forces a power law] If Fis continuous with F(1) = 1 and satisfies F(rs) = F(r)F(s), then F(r) = rαfor some α≥0. Proof. Standard Cauchy-type functional equation on (0,1] with continuity implies F(r) = rα. [Two-plane rotation fixes α= 2] In a single rotation plane, operational probabilities along the orbit must reproduce the standard sinusoidal dependence on the rotation parameter; this fixes α= 2. Proof. Within a plane, ordered reversibility acts as a literal rotation Tλ. For a fixed reference effect eϕand evolving ψ(θ) = e−iθψ, finite-resolution experiments in such a two-outcome setting empirically exhibit the Malus-type law (interference fringe) dependence with period 2π, i.e. a cos2(θ/2)-type profile. Since |⟨ϕ, ψ(θ)⟩C|= cos(θ/2) for appropriately chosen coplanar ϕ, ψ, we require F(cos(θ/2)) = cos2(θ/2), which forces α= 2. [Born rule] For pure states ϕ, ψ, p(eϕ|ψ) = |⟨ϕ, ψ⟩C|2. Proof. By Lemma 7, p=F(|⟨ϕ, ψ⟩C|). Lemmas 7–7 give F(r)=rα. Lemma 7 fixes α= 2. Remark 3.This derivation isolates exactly where “quadratic” enters: it is fixed by the combination of (i) phase invariance, (ii) sequential composition consistency for sharp tests, and (iii) the observed sinusoidal dependence of two-plane rotations (interference). 8 Ordered kinematics Since Tλis unitary on (Vdyn,⟨·,·⟩C), its generator has the form Tλ=e−iλH with Hselfadjoint. Thus ordered evolution is Schr¨odinger kinematics: d dλψ(λ) = −iHψ(λ). The selection of which His physically realized is addressed by locality principles in Paper II. – 5 – 9 Why complex structure is not optional [Complex structure compresses independent rotations] Let P1, . . . , Pnbe invariant rotation planes with incommensurate frequencies. A representation that treats each plane as an independent phase degree of freedom requires a complex structure to encode the product of rotations without introducing additional bookkeeping degrees of freedom that scale with resolution. Proof sketch. Joint evolution on nplanes traces a dense trajectory on an n-torus of phases. In a purely real bookkeeping scheme that does not identify a canonical phase degree of freedom per plane, the number of distinguishable “phase bins” required to track joint evolution at precision ϵgrows as ∼(2π/ϵ)n. This scaling conflicts with finite operational resolution if one demands uniform representability of ordered dynamics across all precisions. Introducing a complex structure identifies each plane with a single phase coordinate, keeping the representational cost fixed at ncomplex dimensions (or 2nreal), independent of ϵ. 10 Physical interpretation of the ordering parameter 10.1 From ordering to clock time The parameter λis a mathematical label for reversible transformations. Physical time emerges by an operational calibration procedure: 1. Clock systems. Identify a physical subsystem whose state undergoes periodic evolution under Tλ. In a rotation plane Pj, the orbit closes with parameter period ∆λ= 2π/ωj. 2. Reference standard. Choose a reference clock (e.g. a stable atomic transition) and declare one unit of physical time t0to correspond to one full cycle of that reference. 3. Calibration map. This fixes a conversion λ= Ωref twhere Ωref is the reference angular frequency in λ-units. Once ℏis introduced to set physical dimensions, Ωref becomes a physical frequency and Hbecomes an energy operator. 10.2 Relational time This view makes time relational: it is defined by comparing evolution rates of physical systems, not by positing an absolute external parameter. This aligns naturally with relativistic timekeeping (proper time) once spacetime structure is incorporated in later papers. 11 Conclusion We have shown that the kinematical structure of quantum theory emerges from three operational principles: 1. Finite operational resolution (Axiom 2), 2. Ordered reversibility with bounded orbits (Axiom 3), 3. Informational reciprocity (Axiom 5). From these we derived: – 6 – •Rotation-plane decomposition (Theorem 4), •Invariant pairing (Theorem 5), •Complex Hilbert structure (Theorem 6), •Born rule (Theorem 7), •Unitary Schr¨odinger kinematics for ordered evolution. Scope This paper establishes kinematics only. The selection of physically admissible generators H is constrained by locality principles in Paper II. References [1] P. Cooney, Ordered Dynamics and Operational State Spaces, Zenodo preprint (2025), DOI: 10.5281/zenodo.17925621. [2] L. Hardy, Quantum theory from five reasonable axioms, arXiv:quant-ph/0101012 (2001). [3] G. Chiribella, G. M. D’Ariano, and P. Perinotti, Informational derivation of quantum theory, Phys. Rev. A 84, 012311 (2011). [4] Ll. Masanes and M. P. M¨uller, A derivation of quantum theory from physical requirements, New J. Phys. 13, 063001 (2011). – 7 –