Paper V — Angular Information Locality and Spin

Angular Information Locality and Spin Paper V of the Ordered-Dynamics Reconstruction Program Paul Cooney DOI: 10.5281/zenodo.17925682 Abstract Papers I–IV of the Ordered-Dynamics Reconstruction Program established the kinematical structure of quantum theory, bounded transport dynamics, relativistic compatibility, and the rigidity of interaction structure. The present paper addresses intrinsic degrees of freedom associated with spatial orientation. We introduce Angular Information Locality (AIL), a principle bounding the operational distinguishability of orientation under finite rotations. We show that bounded angular information forces compact rotation structure and restricts admissible representations to projective representations of SO(3). Minimal angular capacity selects spin-1 2as the elementary carrier of orientation. Requiring relativistic stability and positive-definite probability currents then forces linear factorization of relativistic transport, yielding the Dirac equation. Finally, we show that the topology of the rotation group links angular representations to exchange statistics, establishing the spin–statistics connection without additional postulates. Spin emerges not as an internal label, but as the minimal quantum degree of freedom required to encode spatial orientation in a boundedinformation universe. Contents 1 Introduction 2 2 Orientation as an Operational Resource 2 2.1 Operational meaning of orientation ............... 2 2.2 Bounded angular information .................. 3 3 Compactness and Rotation Representations 3 3.1 Compactness from bounded information ............ 3 3.2 Projective representations and SU(2) .............. 3 4 Spin as Minimal Angular Information 4 4.1 Informational minimality .................... 4 1 5 Relativistic Stability and Linear Transport 4 5.1 Square root of transport ..................... 4 6 Topology of Rotation and Statistics 5 6.1 Rotation–exchange connection .................. 5 6.2 Spin–statistics ........................... 5 7 Summary and Outlook 5 1 Introduction Quantum mechanics contains intrinsic degrees of freedom not reducible to spatial motion. Chief among these is spin. In standard treatments, spin is introduced either algebraically (as representations of SU(2)) or phenomenologically (to fit experimental data). Neither approach explains why spin exists, nor why nature selects particular spin values. Within the Ordered-Dynamics Reconstruction Program, previous papers derived: •Quantum kinematics from ordered reversible dynamics (Paper I), •Free transport dynamics from bounded information growth (Paper II), •Relativistic compatibility and mass-shell constraints (Paper III), •Interaction structure from operational locality (Paper IV). The present paper completes the single-particle sector by answering: Why must physical systems carry intrinsic angular degrees of freedom, and why is spin-1 2fundamental? Our answer is operational. Orientation itself is a form of information. Once one requires that orientation be distinguishable, stable under transport, and bounded in informational capacity, spin becomes inevitable. 2 Orientation as an Operational Resource 2.1 Operational meaning of orientation Orientation is not an abstract geometric notion; it is defined operationally by comparisons between physical systems. To say that a system has an orientation means that different rotations lead to distinguishable experimental outcomes. 2 Definition 2.1 (Operational orientation).A system possesses operational orientation if there exists a set of rotations {R(θ)}such that the states {R(θ)|ψ⟩} are mutually distinguishable by some measurement. Orientation therefore encodes information. The question is how much. 2.2 Bounded angular information Just as spatial distinguishability is bounded by Operational Information Locality (OIL), angular distinguishability must be bounded as well. Axiom 2.1 (Angular Information Locality (AIL)).There exists a finite upper bound on the distinguishability of system states under finite angular rotations. Infinitesimal rotations cannot generate arbitrarily large distinguishability. AIL is the angular analogue of OIL. It rules out systems whose internal state space would allow unbounded orientation resolution at arbitrarily small angular scales. 3 Compactness and Rotation Representations 3.1 Compactness from bounded information Theorem 3.1 (Compact rotation group).AIL forces the operational rotation group to be compact. Proof. If the rotation group were noncompact, repeated rotations could generate an infinite family of mutually distinguishable states within finite angular intervals, violating AIL. Compactness ensures finite informational capacity under rotation. The physically relevant compact rotation group in three dimensions is SO(3). However, quantum states need only furnish projective representations. 3.2 Projective representations and SU(2) Theorem 3.2. All irreducible operationally admissible representations of spatial rotations lift to unitary representations of SU(2). This introduces half-integer spin naturally, without postulating it. 3 4 Spin as Minimal Angular Information 4.1 Informational minimality We now sharpen the key conceptual result. Definition 4.1 (Angular capacity).The angular capacity of a system is the logarithm of the number of perfectly distinguishable orientation states it can encode. •Spin-0 systems encode zero angular information. •Spin-1 2systems encode one qubit of orientation. •Spin-1 systems encode redundant angular information for a single entity. Theorem 4.1 (Minimal angular carrier).Spin-1 2is the minimal nontrivial representation consistent with AIL. Proof. The smallest nontrivial irreducible projective representation of SO(3) has dimension two. Any higher-dimensional representation encodes additional angular information not required for minimal orientation discrimination, violating informational efficiency. Remark 4.1. Spin-1 2is the qubit of orientation: the smallest possible quantum system capable of encoding directional information. 5 Relativistic Stability and Linear Transport Paper III showed that relativistic compatibility imposes a quadratic massshell constraint: E2=p2+m2. However, quadratic dynamics alone do not guarantee positive-definite probability densities. 5.1 Square root of transport Theorem 5.1 (Linear factorization of relativistic transport).Requiring a conserved, positive-definite probability current forces linear factorization of the relativistic dispersion relation, yielding the Dirac equation: (iγµ∂µ−m)ψ= 0. 4 Sketch. A conserved current jµwith j0≥0 cannot be constructed from second-order time evolution without introducing negative-norm components. Linearizing the generator of evolution requires matrices γµsatisfying the Clifford algebra, whose minimal faithful representation is four-dimensional and acts on spinors. Remark 5.1. The Dirac operator is the square root of transport. Just as Paper IV decomposed evolution into transport and interaction, here transport itself is decomposed into a linear structure required for probabilistic stability. 6 Topology of Rotation and Statistics 6.1 Rotation–exchange connection The rotation group SO(3) is not simply connected. A 2πrotation is not contractible to the identity. Theorem 6.1 (Rotation–exchange equivalence).For indistinguishable particles, exchanging two particles is topologically equivalent to a 2πrotation. Proof. The configuration space of two identical particles modulo exchange has fundamental group Z2, matching that of SO(3). The exchange path lifts to a nontrivial loop corresponding to a 2πrotation. 6.2 Spin–statistics Theorem 6.2 (Spin–statistics connection).Half-integer spin implies fermionic statistics; integer spin implies bosonic statistics. Remark 6.1. Statistics are not an independent postulate. They are fixed by the topology of rotation combined with indistinguishability. 7 Summary and Outlook We have shown that: •Orientation is an informational resource, •Bounded angular information forces compact rotation structure, •Minimal angular capacity selects spin-1 2, •Relativistic stability yields the Dirac equation, •Rotation topology fixes particle statistics. Spin is therefore not an arbitrary quantum number. It is the minimal way nature encodes orientation in a bounded-information universe. 5 Next papers Paper VI will extend these ideas to many-body systems and entanglement, while Paper VII will complete the transition to quantum field theory. 6