Gone

Reconstruction of Schr¨odinger Dynamics from C3Phase Geometry Quantum Mechanics on C3: The Unique Anti-Hermitian Axis and the Two-Channel Schr¨odinger Equation Bora Akta¸s This monograph presents an ongoing research program. Future editions will refine the structure and expand the mathematical and physical developments introduced here. December 2, 2025 2 Preface Quantum mechanics can be consistently formulated on the cyclic algebra C3by replacing the complex imaginary unit with the unique anti-Hermitian direction j+j2, thereby producing a two-channel (visible–hidden) generalization of Schr¨odinger evolution. This conceptual shift extends the traditional complex formalism (based on a single imaginary axis) into a phase geometry with richer internal structure, one that naturally decomposes into Hermitian and anti-Hermitian components and provides a setting for multi-carrier wave dynamics. The purpose of this book is not to assert a final, fully polished theory, but rather to consolidate a rapidly growing body of technical results into a coherent reference framework. Over the course of this research program, many partial derivations, operator identities, geometric arguments, and algebraic constructions have accumulated across various manuscripts, preprints, and notebooks. The present volume gathers these pieces into a single, structured document, providing a foundation from which more optimized and systematically organized versions will emerge. The reader should therefore understand this monograph as a developmental edition. Several chapters contain material that will later be reorganized or expanded; some proofs are included in preliminary form; and a number of topics—such as the C3-based trigonometric system, the multi-carrier interpretation of the wavefunction, and the visible– hidden phase projection formalism—are presented in their earliest complete versions. Future editions will refine the conceptual hierarchy, introduce additional examples, and develop connections to relativistic extensions, operator algebras on Cn, and geometric interpretations of phase curvature. Nevertheless, the central mathematical structure presented here remains stable: the C3algebra possesses a single anti-Hermitian direction that generalizes the role of the complex imaginary unit, enabling a reconstruction of Schr¨odinger dynamics and providing a natural two-channel framework for quantum evolution. This monograph documents that structure, records the derivations that lead to it, and establishes a reference point for future developments. I am grateful to the mathematical physics community for the ongoing discussions and 3 4 the many inspirations that shaped this work. I hope that this volume serves as a useful foundation for researchers interested in alternative algebraic formulations of quantum mechanics, multi-layered phase geometries, and the analytic generalization of quantum dynamics beyond the complex field. Bora Akta¸s Independent Researcher December 2, 2025 Contents Preface 3 1 Introduction 11 1.1 Extending Quantum Mechanics Beyond the Complex Field . . . . . . . . 11 1.2 The Algebraic Structure of the C3Phase Space . . . . . . . . . . . . . . 11 1.3 Visible–Hidden Phase Decomposition . . . . . . . . . . . . . . . . . . . . 12 1.4 Overview of the C3Schr¨odinger Reconstruction . . . . . . . . . . . . . . 12 2 Algebraic Foundations of C315 2.1 Definition and Multiplication Structure . . . . . . . . . . . . . . . . . . . 15 2.2 ConjugationStructure ............................ 15 2.3 Hermitian and Anti-Hermitian Subspaces . . . . . . . . . . . . . . . . . . 16 2.4 Minimal Polynomials and Spectral Properties . . . . . . . . . . . . . . . 17 2.5 The C3Norm: Derivation and Full Structure . . . . . . . . . . . . . . . . 17 2.6 Exponential and Power Identities for the Visible and Hidden Directions . 19 2.6.1 Powers of the Visible Direction vvis ................. 19 2.6.2 Powers of the Hidden Direction vhid ................. 19 2.6.3 Fundamental Identity Linking the Two Directions . . . . . . . . . 20 2.7 Hermitian Operator Bases, Spectral Projections, and C3-Geometric Norm Morphisms................................... 21 2.7.1 Introduction.............................. 21 2.7.2 Hermitian Operator Bases . . . . . . . . . . . . . . . . . . . . . . 21 2.7.3 Spectral Projections and Visible–Hidden Decomposition . . . . . 22 2.7.4 C3-Geometric Norm Morphisms . . . . . . . . . . . . . . . . . . . 22 2.7.5 Summary ............................... 23 2.8 Geometric Derivation of the Power Sequence in C3............. 23 2.8.1 Phase Geometry and Distinguished Directions . . . . . . . . . . . 24 2.8.2 Multiplication by xas a Linear Transformation . . . . . . . . . . 24 2.8.3 Eigen-Decomposition of xn...................... 25 2.8.4 Conversion to the {1, x}Basis.................... 25 2.8.5 Final Geometric Interpretation . . . . . . . . . . . . . . . . . . . 26 2.9 Norm Dynamics and Visible–Hidden Energy Distribution in C3...... 26 5 6CONTENTS 2.9.1 General Power Structure in C3.................... 26 2.9.2 Visible and Hidden Phase Directions . . . . . . . . . . . . . . . . 27 2.9.3 Norm of xnand Energy Distribution . . . . . . . . . . . . . . . . 27 2.9.4 Classical Limit and Dominant Mode . . . . . . . . . . . . . . . . 27 2.10 Conjugation, Norm, and the Forced Decomposition into Visible and HiddenPhaseDirections............................. 28 2.10.1 Conjugation eigenstructure on the j–plane............. 29 2.10.2 The C3norm eliminates the hidden direction . . . . . . . . . . . . 30 2.10.3 Uniqueness of the visible/hidden decomposition . . . . . . . . . . 30 2.11 Visible and Hidden Phase Directions in C3: A Canonical Decomposition Imposed by the Conjugation Structure . . . . . . . . . . . . . . . . . . . 31 2.11.1 Conjugation Eigenstructure on the Phase Plane . . . . . . . . . . 31 2.11.2 Norm Structure and the Annihilation of the Hidden Direction . . 32 2.11.3 Uniqueness of the Hidden Direction . . . . . . . . . . . . . . . . . 33 2.11.4 Physical Interpretation (Optional) . . . . . . . . . . . . . . . . . . 33 2.12 From C2to C3: Suppression of Imaginary Components and the Emergence ofaHiddenPhaseAxis............................ 33 2.12.1 Generalization to C3......................... 34 2.12.2 Eigenstructure of the Conjugation Operator . . . . . . . . . . . . 35 2.12.3 Interpretation: Visible and Hidden Phase Directions . . . . . . . . 35 2.13 Visible and Hidden Phase Subspaces in General Cn: Conjugation Spectra and Norm–Projection Structure . . . . . . . . . . . . . . . . . . . . . . . 35 2.13.1 Conjugation as an Involutive Anti-Automorphism . . . . . . . . . 36 2.13.2 Phase Subspace and Conjugation Eigenvectors . . . . . . . . . . . 36 2.13.3 Dimensions of the Visible and Hidden Subspaces . . . . . . . . . . 37 2.13.4 Norm Map and the Annihilation of Hidden Phases . . . . . . . . . 37 2.13.5 Interpretation and the Generalization of the Complex Case . . . . 38 2.14 Hidden Phase Channel and Its Dynamical Role in the Algebra Cn. . . . 38 2.14.1 Conjugation-Induced Decomposition of the Phase Space . . . . . 38 2.14.2 Norm and the Collapse of the Hidden Phase Channel . . . . . . . 39 2.14.3 Dynamical Role of the Hidden Phase Channel . . . . . . . . . . . 40 2.14.4 Specialization to C3.......................... 40 2.15 Projection Operators onto Visible and Hidden Phase Channels in Cn. . 41 2.15.1 Conjugation Structure and Phase-Space Decomposition . . . . . . 41 2.15.2 Definition of Projections onto Visible and Hidden Subspaces . . . 42 2.15.3 Interaction with the Norm . . . . . . . . . . . . . . . . . . . . . . 42 2.15.4 Dynamical Role of the Hidden Channel . . . . . . . . . . . . . . . 43 2.15.5 Specialization to C3.......................... 43 2.16 Generalized Born Rule in the Cyclic Algebra C3.............. 44 2.17 Double-Slit Interference in C3........................ 45 CONTENTS 7 2.17.1Interpretation............................. 45 2.18 Visible Phase Difference and Generalized Trigonometric Structure in C3. 46 2.18.1 Conjugation and the visible phase direction . . . . . . . . . . . . 46 2.18.2 Powers of the visible phase generator and the minimal polynomial 47 2.18.3 Generalized trigonometric functions cos3and sin3......... 47 2.18.4 C3double-slit interference and the cos3carrier . . . . . . . . . . . 48 2.19 Hidden-Phase Differences and Carrier Precession in the Cyclic Algebra C350 2.19.1 Wave decomposition and hidden-phase parametrization . . . . . . 51 2.19.2 Structure of the hidden-axis exponential . . . . . . . . . . . . . . 51 2.19.3 Hidden-phase interference kernel . . . . . . . . . . . . . . . . . . . 52 2.19.4 Carrier precession and the rotation of the visible mode . . . . . . 53 2.20 Norm Growth of the Powers of x=j−j2and Minimal Polynomial . . . 53 2.20.1 Euclidean Norm Behaviour of xn.................. 54 2.20.2 Minimal and Characteristic Polynomials of x............ 55 2.21 Power Structure of the Hidden Axis Element s=j+j2in the Algebra C356 2.21.1 Direct Computations for the First Powers . . . . . . . . . . . . . 56 2.21.2 Recurrence Relation and General Closed Form . . . . . . . . . . . 56 2.21.3 Geometric and Physical Interpretation . . . . . . . . . . . . . . . 57 2.22 Effective Two–Dimensional Phase Dynamics Generated by s=j+j2. . 57 2.22.1 Projection Structure and Phase–Channel Energy . . . . . . . . . . 58 2.23 Trigonometric Representation of j,j2, and Their Symmetric/Antisymmetric Combinations ................................. 59 2.23.1 Complex-plane embedding compatible with j3=−1........ 59 2.23.2 Trigonometric form . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.23.3 Symmetric and antisymmetric phase combinations . . . . . . . . . 60 2.23.4 Reconstruction of jand j2from the phase axes . . . . . . . . . . 60 2.23.5 Geometric interpretation . . . . . . . . . . . . . . . . . . . . . . . 61 2.24 Norm Decomposition, Projection, and Interference in the C3Framework . 62 2.24.1 The C3Algebra and Conjugation . . . . . . . . . . . . . . . . . . 62 2.24.2 Explicit Expansion of the C3Norm ................. 62 2.24.3 Projection to C2and the Visible Norm . . . . . . . . . . . . . . . 63 2.24.4 Two-Slit C3–BornRule........................ 63 2.24.5 Relation to the C3–Schr¨odinger Dynamics . . . . . . . . . . . . . 64 2.25 Spherical Symmetry and Phase–Channel Energetics in the Cubic Algebra C3....................................... 65 2.25.1 Visible and Hidden Phase Channels . . . . . . . . . . . . . . . . . 65 2.25.2 Spherical Symmetry of the Scalar Kernel . . . . . . . . . . . . . . 66 2.25.3 Phase–Cone Anisotropy . . . . . . . . . . . . . . . . . . . . . . . 66 2.25.4 Full Norm in Phase–Channel Coordinates . . . . . . . . . . . . . 66 2.25.5 Geometric Summary . . . . . . . . . . . . . . . . . . . . . . . . . 67 8CONTENTS 2.26 Decomposition of a Single-Particle C3Wavefunction into Visible and HiddenPhaseAxes................................ 67 2.26.1 Phase–Axis Decomposition . . . . . . . . . . . . . . . . . . . . . . 68 2.26.2 Norm in the (1, u, v)Basis...................... 68 2.26.3 Visible (Complex) Projection of the Norm . . . . . . . . . . . . . 68 2.26.4 Mechanism by Which the Hidden Axis Contributes to the Visible Norm ................................. 69 2.27 Angular Parametrization of the C3Phase Components . . . . . . . . . . 70 2.27.1 Spherical Phase Coordinates (R, α, β)................ 70 2.27.2 Visible Norm Nvis as a Function of (R, α, β) ............ 71 2.28 Embedding into a Single-Particle Double-Slit Scenario . . . . . . . . . . . 72 2.28.1 Two Paths, One Particle: Path-Dependent Phase Geometry . . . 72 2.28.2 Effective Visible Intensity and Phase Geometry . . . . . . . . . . 72 2.29 Angular Parameterization of Internal C3Phase Geometry and Its Impact on Single-Particle Double-Slit Interference . . . . . . . . . . . . . . . . . 74 2.29.1 Decomposition of the C3Wavefunction . . . . . . . . . . . . . . . 74 2.29.2 Spherical Phase Parameterization . . . . . . . . . . . . . . . . . . 74 2.29.3 Two-Path Single-Particle Superposition . . . . . . . . . . . . . . . 75 2.29.4 General C3Double-Slit Intensity . . . . . . . . . . . . . . . . . . . 75 2.29.5 Interpretation of the Geometry Factor . . . . . . . . . . . . . . . 76 2.30 C3–Born Rule in Visible/Hidden Phase Parametrization and the X↔Y CouplingMatrix ............................... 76 2.30.1 Norm and Visible Projection . . . . . . . . . . . . . . . . . . . . . 77 2.30.2 Visible/Hidden Phase Parametrization and Trigonometric Form of the C3–BornRule........................... 77 2.30.3 Internal C3–Schr¨odinger Dynamics and the X↔YCoupling Matrix 79 2.31 Extremal Behavior of the C3Geometry Factor G.............. 81 2.31.1 Case I: Full visibility enhancement (G=1)............. 81 2.31.2 Case II: Complete suppression of interference (G= 0) . . . . . . . 81 2.31.3 Case III: Partial visibility (0 <G<1) ............... 82 2.31.4 Summary of extremal regimes . . . . . . . . . . . . . . . . . . . . 82 2.32 Single-Particle Double-Slit Interference in the C3Framework . . . . . . . 83 2.32.1 Classical C2Reference: Two Gaussian Wavepackets . . . . . . . . 83 2.32.2 Embedding into C3: Visible/Hidden Phase Structure . . . . . . . 84 2.32.3 C3–Born Rule for the Double-Slit Geometry . . . . . . . . . . . . 85 2.32.4 Internal C3Dynamics and X↔YCoupling in the Double-Slit Setup 86 2.33 Angular Dynamics of the Internal C3Phase under the Schr¨odinger Evolution 86 2.33.1 From Cartesian (S, X, Y ) to angular (R, α, β) coordinates . . . . . 87 2.33.2 General evolution equations for R,αand β............. 87 2.33.3 Interpretation in terms of visible and hidden currents . . . . . . . 89 CONTENTS 9 2.34 Time-Dependent C3Interference Visibility and Internal Phase Precession 89 2.34.1 Angular dynamics from the C3–Schr¨odinger evolution . . . . . . . 90 2.34.2 Time evolution of the geometry factor G(t, x) ........... 91 2.34.3 Relation to hidden current and experimental signatures . . . . . . 92 3 Phase Geometry on C393 3.1 Visible Directions: 1, j−j2......................... 93 3.2 The Unique Anti-Hermitian Axis j+j2................... 93 3.3 Norm Geometry and Probability Projections . . . . . . . . . . . . . . . . 93 3.4 Analogy with Classical Complex Geometry . . . . . . . . . . . . . . . . . 93 4 Reconstruction of Schr¨odinger Dynamics 95 4.1 The Role of the Imaginary Unit in C2.................... 95 4.2 Generalization to the C3Anti-Hermitian Axis . . . . . . . . . . . . . . . 95 4.3 The C3Schr¨odinger Equation . . . . . . . . . . . . . . . . . . . . . . . . 95 4.4 Hermiticity, Unitarity, and Probability Conservation . . . . . . . . . . . . 95 4.5 Projection to the Visible Channel . . . . . . . . . . . . . . . . . . . . . . 95 5 Two-Channel Quantum Dynamics 97 5.1 Wavefunction Decomposition Ψ = ψvis +χhid(j+j2) ........... 97 5.2 Coupled Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . 97 5.3 Phase Winding and Interference . . . . . . . . . . . . . . . . . . . . . . . 97 5.4 Measurement and the Visible Projection . . . . . . . . . . . . . . . . . . 97 6 Physical Implications 99 6.1 Generalized Interference Patterns . . . . . . . . . . . . . . . . . . . . . . 99 6.2 Multi-Carrier Interpretation of Wavefunctions . . . . . . . . . . . . . . . 99 6.3 Classicalization and Phase Gradient Criteria . . . . . . . . . . . . . . . . 99 6.4 Connections to Relativistic Extensions . . . . . . . . . . . . . . . . . . . 99 A Multiplication and Conjugation Tables for C3101 B Hermitian Operator Bases 103 C Generalized Trigonometric Functions on C3105 D Comparison with C2and Higher CnAlgebras 107 16 CHAPTER 2. ALGEBRAIC FOUNDATIONS OF C3 extended linearly across C3. For an arbitrary element z=a+bj +cj2, its conjugate is z∗=a−cj −bj2. This involution satisfies: 1. involutive property: (z∗)∗=z, 2. anti-automorphism: (xy)∗=y∗x∗, 3. real-line invariance: a∗=afor a∈R. The chosen conjugation is the essential algebraic ingredient that permits a Hermitian/antiHermitian decomposition analogous to the real/imaginary split of the complex field. 2.3 Hermitian and Anti-Hermitian Subspaces Under the involution ∗, the algebra C3decomposes into a direct sum: C3=CHerm 3⊕Canti 3. The Hermitian (self-conjugate) subspace is CHerm 3={z∈C3:z∗=z}= spanR{1, j −j2}. The anti-Hermitian (skew-conjugate) subspace is Canti 3={z∈C3:z∗=−z}= spanR{j+j2}. A number of essential properties follow: •The Hermitian subspace is two-dimensional, consisting of the real axis and a “visible” phase direction (j−j2). •The anti-Hermitian subspace is one-dimensional, generated solely by (j+j2), which plays the exact analogue of the imaginary unit in the complex field. •This decomposition provides the algebraic basis for the visible–hidden phase structure developed in later chapters. In particular, the existence of a unique anti-Hermitian axis distinguishes C3from all other Cnwith n>3, and enables a single-channel generalization of the imaginary unit needed to reconstruct Schr¨odinger dynamics. 2.4. MINIMAL POLYNOMIALS AND SPECTRAL PROPERTIES 17 2.4 Minimal Polynomials and Spectral Properties The generator jsatisfies the irreducible cubic relation: j3+ 1 = 0. This is the minimal polynomial of j, since no lower-degree polynomial with real coefficients annihilates j. When C3is embedded into Cfor spectral analysis (via any faithful representation), the eigenvalues of jare the cube roots of −1: Spec(j) = eiπ/3, eiπ, ei5π/3. The conjugation induces a Z2-grading: (j−j2)∗= +(j−j2),(j+j2)∗=−(j+j2), so the Hermitian component has “parity” +1, while the anti-Hermitian direction has parity −1. This grading plays a central role in the reconstruction of quantum mechanics: •Hermitian operators (observables) must lie in CHerm 3, •the generator of unitary time evolution must lie in Canti 3, •and wavefunctions naturally decompose into visible and hidden channels. The interplay between these components is explored in detail in the subsequent chapters, where the Schr¨odinger equation, probability structure, unitary evolution, and generalized interference phenomena are developed on the C3phase geometry. 2.5 The C3Norm: Derivation and Full Structure In the algebra C3equipped with the involution (a+bj +cj2)∗=a−cj −bj2, the natural quadratic norm is defined by N(z) = z∗z, z ∈C3. Let z=a+bj +cj2. 18 CHAPTER 2. ALGEBRAIC FOUNDATIONS OF C3 Then z∗=a−cj −bj2. We compute the product explicitly: N(z) = (a−cj −bj2)(a+bj +cj2), and expand term by term using the multiplication rules j2=j2, j3=−1, j4=−j, j5=−j2. Carrying out the full expansion: N(z) = a2+abj +acj2−acj −bcj2−c2j3−abj2−b2j3−bcj4. Substituting j3=−1 and j4=−j, we obtain: N(z) = a2+ (ab −ac −bc +bc)j+ (ac −ab −bc +bc)j2+c2+b2. The cross terms simplify to j-component = (b−c)(a+c), j2-component = (c−b)(a+b). Therefore, the full norm takes the canonical form: N(z) = (a2+b2+c2)+(b−c)(a+c)j+ (c−b)(a+b)j2. This shows that: •The scalar part of N(z) is always non-negative. •The jand j2components indicate that the C3norm retains a visible phase component that does not collapse to a purely real number. •Only the projection onto the Hermitian subspace ΠHerm(N(z)) = (a2+b2+c2) + (b−c)(a+c)(j−j2) represents the measurable probability structure. This norm is not multiplicative but respects the grading induced by the Hermitian/antiHermitian decomposition. Its structure underlies the two-channel phase geometry developed in later chapters. 2.6. EXPONENTIAL AND POWER IDENTITIES FOR THE VISIBLE AND HIDDEN DIRECTIONS19 2.6 Exponential and Power Identities for the Visible and Hidden Directions Two distinguished phase directions appear naturally in C3: vvis := j−j2, vhid := j+j2. These vectors generate, respectively, the Hermitian and anti-Hermitian subspaces of C3. Their powers obey highly structured identities that play the role of fundamental trigonometric relations in the generalized phase theory. 2.6.1 Powers of the Visible Direction vvis Let vvis =j−j2. Compute its powers: v2 vis = (j−j2)2=j2−2j3+j4=j2+ 2 + (−j) = 2 + (j2−j). Since j2−j=−(j−j2), we obtain the identity: v2 vis = 2 −vvis. Similarly, the cubic power gives: v3 vis =vvis(2 −vvis)=2vvis −(2 −vvis) = 3vvis −2. Thus, v3 vis = 3 vvis −2. These two relations form a closed recurrence, showing that the sequence {vn vis}satisfies a linear recurrence of order two: vn vis = 3vn−1 vis −2vn−2 vis . This is the C3analogue of the trigonometric recurrences for cos(nθ) and sin(nθ). 2.6.2 Powers of the Hidden Direction vhid Let vhid =j+j2. 20 CHAPTER 2. ALGEBRAIC FOUNDATIONS OF C3 Compute its square: v2 hid = (j+j2)2=j2+ 2j3+j4=j2−2 + (−j) = −2 + (j2−j)=−2−vvis. Thus: v2 hid =−2−vvis. Cubic power: v3 hid =vhid(−2−vvis) = −2vhid −vhidvvis. Since vhidvvis = (j+j2)(j−j2)=j2−j3+j3−j4=j2+1+1+j=2+vhid, we obtain: v3 hid =−2vhid −(2 + vhid) = −3vhid −2. Thus: v3 hid =−3vhid −2. This recurrence: vn hid =−3vn−1 hid −2vn−2 hid , is the hidden-sector analogue of the visible-sector recurrence. 2.6.3 Fundamental Identity Linking the Two Directions A key structural relation is: vvis vhid = (j−j2)(j+j2) = 2 + vhid. vvis vhid =2+vhid This identity expresses the geometric coupling between the visible and hidden directions and will form the basis for the two-channel phase dynamics used in the reconstruction of the Schr¨odinger equation. 2.7. HERMITIAN OPERATOR BASES, SPECTRAL PROJECTIONS, AND C3-GEOMETRIC NORM MORPHISMS21 2.7 Hermitian Operator Bases, Spectral Projections, and C3-Geometric Norm Morphisms 2.7.1 Introduction The involution ∗on C3induces a natural Z2-grading that splits the algebra into a Hermitian (visible) and an anti-Hermitian (hidden) subspace. This structure is fundamental for the reconstruction of Schr¨odinger dynamics on C3, as it determines which components of an operator are physically observable, which components generate time evolution, and how probability is encoded through the norm. This section develops three central tools: 1. Hermitian operator bases for observables, 2. spectral projection operators associated with the involution, 3. the geometric morphism that sends the full C3-norm into the visible sector. Together these structures form the algebraic backbone of all physical interpretations in subsequent chapters. 2.7.2 Hermitian Operator Bases The involution on the generators, 1∗= 1,(j−j2)∗= +(j−j2),(j+j2)∗=−(j+j2), immediately identifies the Hermitian subspace: CHerm 3= spanR{1, j −j2}. This two-dimensional space contains all physically observable operators. Any Hermitian operator Hcan therefore be written uniquely as H=Hre 1+Hvis (j−j2), where Hre, Hvis ∈R. Key features: •The direction (j−j2) generalizes the role of the imaginary unit iin complex quantum mechanics: it represents a measurable phase gradient. •The scalar part encodes classical observables. 22 CHAPTER 2. ALGEBRAIC FOUNDATIONS OF C3 •Anti-Hermitian components cannot appear in observable quantities, but they will play a role in time evolution. Thus the Hermitian operator basis provides the necessary framework for a generalized quantum mechanical observable algebra on C3. 2.7.3 Spectral Projections and Visible–Hidden Decomposition The involution ∗acts as a reflection, with eigenvalues +1 on CHerm 3,−1onCanti 3= span{j+j2}. This allows the natural definition of projection operators: Pvis(z) = 1 2(z+z∗), Phid(z) = 1 2(z−z∗). For z=a+bj +cj2we obtain: Pvis(z)=a+ (b−c)(j−j2), Phid(z)=(b+c)(j+j2). These formulas exhibit the essential two-channel structure: •Pvis preserves the Hermitian (observable) components, •Phid extracts the anti-Hermitian (internal phase) component. The decomposition z=Pvis(z)+Phid(z) is orthogonal with respect to the involution and provides the algebraic basis for visible–hidden phase separation in the dynamics developed in Chapter 4. 2.7.4 C3-Geometric Norm Morphisms The full C3-norm is defined by N(z) = z∗z. For z=a+bj +cj2, the computation yields N(z) = (a2+b2+c2)+(b−c)(a+c)j+ (c−b)(a+b)j2. 2.8. GEOMETRIC DERIVATION OF THE POWER SEQUENCE IN C323 Unlike the complex case, N(z) is not purely real; it contains visible-phase components. To obtain a physically meaningful probability quantity, the norm must be projected onto the Hermitian subspace: Nphys(z) = Pvis(N(z)). This defines a morphism: Nphys :C3−→ CHerm 3. Properties: 1. Positivity: The scalar part a2+b2+c2ensures non-negativity. 2. Visibility preservation: Only the visible phase (j−j2) survives projection; the hidden axis (j+j2) is always eliminated. 3. Physical consistency: Nphys provides the correct probability interpretation for wavefunctions in the C3framework. 4. Channel separation: The projection implements the fundamental measurement postulate: measurements access only the visible channel. Thus, the norm is not merely a quadratic form but a geometric transformation relating the full three-dimensional phase space to the two-dimensional observable sector. 2.7.5 Summary This section establishes: •the complete Hermitian operator basis on C3, •the spectral projections associated with the involution, •the geometric morphism sending the full C3-norm into the physically observable Hermitian subspace. These constructions form the structural foundation for the generalized Schr¨odinger equation on C3and its two-channel (visible–hidden) interpretation in the chapters that follow. 2.8 Geometric Derivation of the Power Sequence in C3 Let C3= spanR{1, j, j2}be the real cyclic phase algebra generated by an element j satisfying j3=−1. In this section we show that the power sequence of the special element x:= j−j2 24 CHAPTER 2. ALGEBRAIC FOUNDATIONS OF C3 is completely determined by the geometric decomposition of C3into two distinguished phase directions: a “hidden” axis and a mixed real–visible axis. We demonstrate that multiplication by xacts as a linear transformation with two eigenvalues, 1 and −2, and we derive from this the closed form of the coefficients in xn=An+Bnx. 2.8.1 Phase Geometry and Distinguished Directions The algebra C3admits a natural geometric interpretation: the basis elements {1, j, j2} correspond to three vertices of an equilateral phase triangle, and linear combinations describe phase configurations in a two-dimensional subspace of R3with one real and two rotational components. A key structural feature is the existence of two special phase directions: h:= j+j2,(hidden axis),(2.1) u:= 1 −j+j2,(mixed real–visible axis).(2.2) These directions are not arbitrary; they appear naturally when analyzing how multiplication by xacts on the algebra. 2.8.2 Multiplication by xas a Linear Transformation We compute the action of x=j−j2on the two special directions. Hidden axis. A direct calculation gives xh = (j−j2)(j+j2)=j(j+j2)−j2(j+j2) = j2−1+1+j=h. Thus the hidden axis is an eigen-direction: xh = 1 ·h. Mixed real–visible axis. Similarly, xu = (j−j2)(1 −j+j2) = −2(1 −j+j2) = −2u, so xu =−2·u. Eigenstructure. Hence multiplication by xhas two eigenvalues and eigenvectors: λ1= 1, h =j+j2;λ2=−2, u = 1 −j+j2. 2.8. GEOMETRIC DERIVATION OF THE POWER SEQUENCE IN C325 Every power xntherefore decomposes along these two geometric directions. 2.8.3 Eigen-Decomposition of xn Since {h, u}span the same real subspace as {1, x}, each power xnadmits a decomposition xn=αnu+βnh, and multiplication by xyields xn+1 =x(αnu+βnh) = αn(−2u)+βn(1h) = αn+1u+βn+1h. Thus αn+1 =−2αn, βn+1 =βn. Solving gives αn=C1(−2)n−1, βn=C2. To determine C1and C2, we express the initial vector xin the eigenbasis: x=j−j2=1 3u−2 3h. Thus C1=1 3, C2=−2 3. Therefore, xn=1 3(−2)n−1u−2 3h. 2.8.4 Conversion to the {1, x}Basis Since u= 1 −x, h =x−1+2j2 but more simply in the (1, j, j2) basis, u= 1 −j+j2, h =j+j2, we write xnin the standard form xn=An+Bnx. Matching coefficients of 1, j, and j2yields the closed-form expressions: An=2+(−2)n 3, Bn=1−(−2)n 3. 32 CHAPTER 2. ALGEBRAIC FOUNDATIONS OF C3 Thus xis an eigenvector of the conjugation map iff x∗=λx (λ=±1). Equating coefficients gives the system −β=λα, −α=λβ. There are exactly two solutions: (i) λ= +1 (self-conjugate direction): β=−α, hence v:= j−j2, v∗=v. (ii) λ=−1 (anti-self-conjugate direction): β=α, hence h:= j+j2, h∗=−h. Thus the phase plane decomposes into two conjugation eigenspaces: F= span{v}⊕span{h}, v =j−j2, h =j+j2. 2.11.2 Norm Structure and the Annihilation of the Hidden Direction Let z=a+bj +cj2. Using the above conjugation we obtain z∗=a−cj −bj2. A direct computation yields z∗z= (a2+b2+c2)+(ab −ac +bc) (j−j2). Two structural properties follow immediately: 1. The phase part of z∗zlies entirely in span{j−j2}: Phase(z∗z)∈span{v}. 2.12. FROM C2TO C3: SUPPRESSION OF IMAGINARY COMPONENTS AND THE EMERGENCE OF A HIDDEN PHASE AXIS33 2. No component proportional to h=j+j2ever appears: (z∗z)h= 0 for all z∈C3. Hence the Hermitian form N(z)=z∗zprojects the entire phase plane Fonto the one-dimensional subspace span{v}. The orthogonal complement (in the conjugationeigenvector sense), namely span{h}, is always annihilated. Thus, N(z) = z∗zis sensitive only to v=j−j2and completely insensitive to h=j+j2. 2.11.3 Uniqueness of the Hidden Direction Because Fhas real dimension 2, and N(z) restricts to a rank-1 form on F, the kernel of this restriction must be one-dimensional. As shown above, this kernel is exactly the −1 eigenspace of conjugation: ker(N|F) = span{h}. Therefore: The hidden direction is algebraically forced to be h=j+j2. No other linear combination of jand j2satisfies h∗=−hwhile simultaneously being orthogonal to vunder z∗z. Hence the visible/hidden phase decomposition is not a matter of definition but a structural consequence of the conjugation on C3. 2.11.4 Physical Interpretation (Optional) In a quantum-mechanical interpretation, the surviving direction v=j−j2represents the measurable (“visible”) phase channel, as it survives in z∗z, probability density, current, and all Hermitian observables. The direction h=j+j2, being annihilated by the Hermitian form, corresponds to an unobservable (“hidden”) phase mode. However, the physical naming is secondary; the algebraic decomposition itself is canonical and unavoidable. 2.12 From C2to C3: Suppression of Imaginary Components and the Emergence of a Hidden Phase Axis A fundamental property of the complex field C2= spanR{1, i}is that the norm N(z) = z∗z= (a−ib)(a+ib) (2.5) 34 CHAPTER 2. ALGEBRAIC FOUNDATIONS OF C3 removes the imaginary component entirely, yielding the purely real expression N(z) = a2+b2.(2.6) This phenomenon is usually regarded as a trivial consequence of i∗=−i, but it has a deeper structural meaning: the imaginary axis span{i}is the anti-self-adjoint eigenspace of the conjugation operator, and therefore cannot contribute to any Hermitian quadratic form such as a norm. In this sense, the “imaginary direction” is systematically erased at the level of measurable quantities. 2.12.1 Generalization to C3 The cyclic algebra C3= spanR{1, j, j2}, j3=−1, equipped with the conjugation structure 1∗= 1, j∗=−j2,(j2)∗=−j, (2.7) admits a direct analogue of the C2decomposition into self-adjoint and anti-self-adjoint parts. For a general element z=a+bj +cj2, the conjugate is z∗=a−cj −bj2,(2.8) and the associated norm takes the form N(z) = z∗z= (a2+b2+c2)+(ab −ac +bc) (j−j2).(2.9) A key structural feature of (2.9) is that the norm never produces a component along the direction j+j2. In other words, although the “phase plane” F= span{j, j2} is two-dimensional, the norm projects it onto the one-dimensional subspace Fvis = span{j−j2}, leaving its orthogonal complement Fhid = span{j+j2} entirely invisible to N(z). 2.13. VISIBLE AND HIDDEN PHASE SUBSPACES IN GENERAL Cn: CONJUGATION SPECTRA AND NORM–PROJECTION STRUCTURE35 2.12.2 Eigenstructure of the Conjugation Operator To understand this phenomenon, observe that conjugation restricted to the phase plane acts as a linear operator with two eigenvectors: (j−j2)∗= +(j−j2),(2.10) (j+j2)∗=−(j+j2).(2.11) Thus, j−j2is self-adjoint, j +j2is anti-self-adjoint. Since a Hermitian quantity such as z∗zcan only depend on the self-adjoint eigenspace, the anti-self-adjoint axis span{j+j2}is necessarily eliminated from all norm expressions. This is the exact structural analogue of the familiar C2fact that i∗=−i=⇒span{i}contributes nothing to z∗z. 2.12.3 Interpretation: Visible and Hidden Phase Directions The decomposition F= span{j−j2} |{z } visible phase ⊕span{j+j2} | {z } hidden phase is therefore not a matter of convention nor a physical assumption, but a direct consequence of the conjugation structure and of the Hermitian nature of the norm. The visible axis persists in all measurable quantities, while the hidden axis is systematically suppressed, exactly mirroring the suppression of the imaginary axis in C2. This observation establishes C3as the natural three-mode generalization of the complex numbers, with “imaginary suppression” replaced not by elimination of the entire phase plane, but by the projection of a two-dimensional phase space onto its unique self-adjoint component. 2.13 Visible and Hidden Phase Subspaces in General Cn: Conjugation Spectra and Norm–Projection Structure In this section we develop the general decomposition of the cyclic phase algebra Cn= spanR{1, j, j2, . . . , jn−1}, jn=−1, into visible and hidden phase subspaces. The construction generalizes the familiar real/imaginary split of the complex numbers (C2) and the visible/hidden phase splitting previously ob- 36 CHAPTER 2. ALGEBRAIC FOUNDATIONS OF C3 tained in C3. Our goal is to determine, for arbitrary n, (i) which phase directions survive under the norm map N(z) = z∗z, (ii) which directions are systematically annihilated, and (iii) the precise dimensions of the surviving (“visible”) and suppressed (“hidden”) phase degrees of freedom. 2.13.1 Conjugation as an Involutive Anti-Automorphism Following the C3construction, we introduce the conjugation operation (jk)∗=   1, k = 0, −jn−k,1≤k≤n−1. This map satisfies the involutive anti-automorphism properties (zw)∗=w∗z∗,(z∗)∗=z, and therefore plays the role of the Hermitian adjoint in the generalized algebra. The real axis is invariant under ∗, while the remaining (n−1) generators mix pairwise as jk←→ −jn−k. 2.13.2 Phase Subspace and Conjugation Eigenvectors Define the phase subspace Fn:= spanR{j, j2, . . . , jn−1},dim Fn=n−1. The conjugation operator acts linearly on Fnand admits a complete set of eigenvectors. For every 1 ≤k < n/2 we introduce the symmetric and antisymmetric phase combinations v(+) k:= jk−jn−k, v(−) k:= jk+jn−k. A direct calculation shows v(+) k∗= + v(+) k,v(−) k∗=−v(−) k. Thus v(+) kspans a +1 eigenvector (self-conjugate), while v(−) kspans a −1 eigenvector (anti–self-conjugate). If nis even, the midpoint generator jn/2satisfies jn/2∗=−jn/2 and therefore contributes an additional −1 eigenvector. 2.13. VISIBLE AND HIDDEN PHASE SUBSPACES IN GENERAL Cn: CONJUGATION SPECTRA AND NORM–PROJECTION STRUCTURE37 Hence the phase space decomposes orthogonally as Fn=Vvis ⊕Vhid, where Vvis = span{v(+) k: 1 ≤k < n/2}, Vhid = span{v(−) k: 1 ≤k < n/2}⊕   span{jn/2}, n even, 0, n odd. 2.13.3 Dimensions of the Visible and Hidden Subspaces The above construction immediately gives the subspace dimensions: dim Vvis =n−1 2,dim Vhid =n−1 2. Thus approximately half of the phase degrees of freedom are self-conjugate and half are anti–self-conjugate. For nodd the split is exactly symmetric, while for neven the hidden sector contains one additional dimension corresponding to jn/2. 2.13.4 Norm Map and the Annihilation of Hidden Phases We define the norm for arbitrary z∈Cnas N(z) := z∗z. Because conjugation is an involution, one always has N(z)∗= (z∗z)∗=z∗(z∗)∗=z∗z=N(z). Hence N(z)isself–conjugate. Consequently, N(z) may contain components only along the +1 eigenspace of the conjugation operator. It follows immediately that N(z)∈R·1⊕Vvis, and the projection onto the hidden sector satisfies Πhid N(z) = 0. In other words, the entire hidden phase subspace Vhid is systematically erased by the quadratic norm map. No component of N(z) can lie in any −1 eigenspace of conjugation, for such components would contradict the requirement N(z)∗=N(z). 38 CHAPTER 2. ALGEBRAIC FOUNDATIONS OF C3 2.13.5 Interpretation and the Generalization of the Complex Case For n= 2 the algebra C2reduces to the ordinary complex numbers. The phase space has dimension 1 and is entirely anti–self-conjugate: F2=Vhid, Vvis = 0. Thus the norm N(z) = |z|2eliminates the entire imaginary axis. For n= 3 the phase space has dimension 2, which splits into Vvis = span{j−j2}, Vhid = span{j+j2}. The norm therefore retains only the “visible” direction j−j2and annihilates the “hidden” direction j+j2, exactly as in the previously analyzed C3case. For general n, the norm retains precisely the ⌊(n−1)/2⌋visible directions and annihilates the remaining ⌈(n−1)/2⌉hidden directions. This structure constitutes the unique and universal generalization of the real/imaginary split of the complex plane to higher cyclic phase algebras. 2.14 Hidden Phase Channel and Its Dynamical Role in the Algebra Cn 2.14.1 Conjugation-Induced Decomposition of the Phase Space Let the cyclic algebra Cnbe defined as Cn= spanR{1, j, j2, . . . , jn−1}, jn=−1, equipped with the involutive conjugation (jk)∗=   1, k = 0, −jn−k,1≤k≤n−1. The phase subspace is the real linear hull Fn:= spanR{j, j2, . . . , jn−1}. The conjugation operator acts as an involutory linear map on Fnand splits it into two 2.14. HIDDEN PHASE CHANNEL AND ITS DYNAMICAL ROLE IN THE ALGEBRA Cn39 invariant subspaces: (jk)∗=−jn−k=⇒ Fn=Vvis ⊕Vhid, where Vvis := span{jk−jn−k: 1 ≤k < n/2}, Vhid := span{jk+jn−k:1≤k < n/2} ⊕    span{jn/2}, n even, 0, n odd. These satisfy v∈Vvis =⇒v∗=v, h∈Vhid =⇒h∗=−h, i.e. Vvis is the +1 eigenspace (self-conjugate) and Vhid is the −1 eigenspace (anti-selfconjugate) of the conjugation. Dimensions are dim Vvis =jn−1 2k,dim Vhid =ln−1 2m. 2.14.2 Norm and the Collapse of the Hidden Phase Channel The natural norm is defined by N(z) := z∗z, z ∈Cn. Using involutivity, N(z)∗= (z∗z)∗=z∗(z∗)∗=z∗z=N(z), so the norm is always self-conjugate. It follows immediately that N(z)∈R·1⊕Vvis, and therefore Πhid(N(z)) = 0,∀z∈Cn, where Πhid is the projection onto Vhid. Thus all components of the wavefunction lying in the hidden subspace Vhid are completely invisible to the norm and therefore to any probability functional. In particular, for the physically relevant probability density P(z) := Πvis(N(z)), one always has P(z)≥0 and P(z) = 0 iff z= 0, establishing the interpretation of the norm as a valid probability measure. 40 CHAPTER 2. ALGEBRAIC FOUNDATIONS OF C3 2.14.3 Dynamical Role of the Hidden Phase Channel Although Vhid is invisible to the norm, it plays an essential role in the dynamics. Any linear evolution equation of Schr¨odinger type, ∂tψ=D ψ, contains an anti-self-adjoint part Dasatisfying D∗ a=−Da. Such components necessarily lie in the hidden phase subspace: Da∈Vhid. This is the direct generalization of the fact that in the complex case C2, the imaginary unit iis anti-self-adjoint and generates phase motion while remaining invisible to the norm |ψ|2. Hence: •The visible subspace Vvis determines measurable interference patterns. •The hidden subspace Vhid governs pure phase evolution, internal rotation, and the anti-Hermitian sector of the generator of time evolution. •Measurement projects ψonto R⊕Vvis, eliminating hidden components. 2.14.4 Specialization to C3 For n= 3, the decomposition reduces to Vvis = span{j−j2}, Vhid = span{j+j2}, and any norm satisfies N(a+bj +cj2) = (a2+b2+c2)+(ab −ac +bc) (j−j2), with no contribution along the hidden direction j+j2. The hidden direction enters exclusively through the anti-Hermitian component of the evolution generator, providing the higher-dimensional analogue of the imaginary unit in standard complex quantum mechanics. 2.15. PROJECTION OPERATORS ONTO VISIBLE AND HIDDEN PHASE CHANNELS IN Cn41 2.15 Projection Operators onto Visible and Hidden Phase Channels in Cn 2.15.1 Conjugation Structure and Phase-Space Decomposition The cyclic algebra Cn= spanR{1, j, j2, . . . , jn−1}, jn=−1, is equipped with the conjugation (jk)∗=   1, k = 0, −jn−k,1≤k≤n−1. The phase subspace, Fn:= spanR{j, j2, . . . , jn−1}, is invariant under this conjugation. For each 1 ≤k < n/2, define the symmetric and anti-symmetric combinations v(+) k:= jk−jn−k, v(−) k:= jk+jn−k. These satisfy (v(+) k)∗=v(+) k,(v(−) k)∗=−v(−) k. If nis even, the midpoint generator jn/2also satisfies (jn/2)∗=−jn/2, and therefore belongs to the anti-self-conjugate sector. Thus the phase subspace decomposes as the direct sum Fn=Vvis ⊕Vhid, where Vvis := span{v(+) k:1≤k < n/2}, Vhid := span{v(−) k: 1 ≤k < n/2} ⊕    span{jn/2}, n even, 0, n odd. The visible subspace Vvis is the +1 eigenspace of the conjugation, and the hidden subspace Vhid is its −1 eigenspace. 48 CHAPTER 2. ALGEBRAIC FOUNDATIONS OF C3 Using the quadratic relation (2.33), v2 vis = 2 −vvis, this becomes α′(t)+β′(t)vvis =α(t)vvis +β(t)2−vvis= 2β(t) + α(t)−β(t)vvis.(2.40) Equating coefficients of 1 and vvis yields the coupled system α′(t) = 2 β(t), β′(t) = α(t)−β(t),(2.41) with initial conditions determined from e0·vvis = 1: α(0) = 1, β(0) = 0.(2.42) Solving (2.41) gives α(t) = 2et+e−2t 3, β(t) = et−e−2t 3.(2.43) Substituting into (2.37), we obtain the closed form etvvis =2et+e−2t 3+et−e−2t 3vvis.(2.44) This motivates the definition of generalized C3–trigonometric functions cos3(t) := 2et+e−2t 3,sin3(t) := et−e−2t 3,(2.45) so that etvvis = cos3(t) + sin3(t)vvis.(2.46) In direct analogy with the real and imaginary parts of eit in the complex case, cos3 and sin3encode the scalar and visible-phase components of the exponential along the self-conjugate direction vvis. 2.18.4 C3double-slit interference and the cos3carrier We now consider a double-slit configuration in which the C3-valued wavefunction is the superposition of two contributions, Ψ(x) = Ψ1(x)+Ψ2(x),(2.47) with Ψk(x) = Rk(x)eΦk(x), k = 1,2,(2.48) 2.18. VISIBLE PHASE DIFFERENCE AND GENERALIZED TRIGONOMETRIC STRUCTURE IN C349 where Rk(x)≥0 are real amplitudes and Φk(x) = θvis,k(x)vvis,(2.49) i.e. we isolate for the moment only the visible phase component (we will ignore the hidden phase vhid in this subsection). The C3-norm of Ψ is defined by N(Ψ)(x) := Ψ(x)∗Ψ(x),(2.50) and takes values in C3. A direct expansion gives N(Ψ) = Ψ∗ 1Ψ1+ Ψ∗ 2Ψ2+ Ψ∗ 1Ψ2+ Ψ∗ 2Ψ1.(2.51) The first two terms are the individual intensities, Ψ∗ kΨk=N(Ψk)∈C3,MN(Ψk)=R2 k,(2.52) where Mdenotes the measurement functional (defined below) that extracts a real probability density. The interference term is Iint(x) := Ψ∗ 1(x)Ψ2(x)+Ψ∗ 2(x)Ψ1(x).(2.53) Since v∗ vis =vvis and C3is commutative, we have Ψ∗ 1Ψ2=R1R2e−θvis,1vvis eθvis,2vvis =R1R2e(θvis,2−θvis,1)vvis .(2.54) Denoting the visible phase difference by ∆θvis(x):=θvis,2(x)−θvis,1(x),(2.55) and using (2.46), we obtain Ψ∗ 1Ψ2=R1(x)R2(x)cos3(∆θvis(x)) + sin3(∆θvis(x)) vvis.(2.56) The Hermitian part of Ψ∗ 1Ψ2(with respect to the conjugation (2.28)) is obtained by taking the C3-real component in the subalgebra R⊕Rvvis: ℜC3(Ψ∗ 1Ψ2) = R1R2cos3(∆θvis) + sin3(∆θvis)vvis.(2.57) 50 CHAPTER 2. ALGEBRAIC FOUNDATIONS OF C3 The full interference contribution is then Iint(x) = Ψ∗ 1Ψ2+ Ψ∗ 2Ψ1= 2 ℜC3Ψ∗ 1Ψ2= 2R1R2cos3(∆θvis) + sin3(∆θvis)vvis.(2.58) To obtain a real probability density, we define the Born-type measurement functional M:C3→R,Ma+b vvis:= a, (2.59) which projects C3onto its scalar part. The C3-Born rule is then PC3(x) = MN(Ψ)(x),(2.60) so that the observed intensity pattern becomes PC3(x) = MN(Ψ1)+MN(Ψ2)+MIint(x) =R1(x)2+R2(x)2+ 2R1(x)R2(x) cos3∆θvis(x). (2.61) Equation (2.61) shows that the usual complex double-slit interference pattern PC2(x)=R1(x)2+R2(x)2+ 2R1(x)R2(x) cos ∆θ(x),(2.62) is generalized in C3by the replacement cos ∆θ−→ cos3∆θvis,(2.63) where ∆θvis is the phase difference along the self-conjugate visible direction vvis and cos3is the generalized trigonometric function defined in (2.45). The structure of the interference term is thus preserved in form, but the carrier is deformed by the internal C3phase geometry. 2.19 Hidden-Phase Differences and Carrier Precession in the Cyclic Algebra C3 In the cyclic algebra C3= spanR{1, j, j2}, j3=−1, j∗=−j2, the internal phase space splits into two orthogonal directions with distinct conjugation parity. Writing vvis := j−j2, vhid := j+j2, 2.19. HIDDEN-PHASE DIFFERENCES AND CARRIER PRECESSION IN THE CYCLIC ALGEBRA C351 we obtain v∗ vis =vvis, v∗ hid =−vhid, so that the visible axis is self-adjoint while the hidden axis is anti-self-adjoint. This section analyzes in full mathematical detail how hidden-phase differences affect interference in a two-path configuration, and why they do not appear directly in the Born probability but nevertheless induce an internal rotation (“carrier precession”) of the observable interference carrier. 2.19.1 Wave decomposition and hidden-phase parametrization Consider two contributions to a C3-valued wavefunction, Ψk(x) = Rk(x)eΦk(x), k = 1,2, where Rk≥0, and the phase is decomposed as Φk(x) = θvis,k(x)vvis +θhid,k(x)vhid. In this section we focus exclusively on the hidden component and set θvis,1=θvis,2= 0. Thus the phases reduce to Φk(x) = θhid,k(x)vhid. The total wave is Ψ(x) = Ψ1(x)+Ψ2(x), and the interference contribution to the C3-norm N(Ψ) = Ψ∗Ψ is governed by the cross terms Ψ∗ 1Ψ2and Ψ∗ 2Ψ1. 2.19.2 Structure of the hidden-axis exponential Since vhid is anti-self-adjoint, (vhid)∗=−vhid, its exponential satisfies the adjoint relation e(θvhid)∗=eθvhid ∗=e−θvhid . 52 CHAPTER 2. ALGEBRAIC FOUNDATIONS OF C3 Unlike the visible axis, the hidden axis is not closed under multiplication: v2 hid = (j+j2)2=j2+ 2jj2+j4=j2+ 2j3+j= (j+j2)−2, which implies the quadratic identity v2 hid =vhid −2.(2.64) This relation shows that eθvhid expands into the full three-dimensional subspace span{1, vvis, vhid}, because the square of vhid generates a component along vvis =j−j2. Hence the hidden-axis exponential admits a decomposition eθvhid =A(θ)·1 + B(θ)vvis +C(θ)vhid, for real analytic functions A(θ), B(θ), and C(θ). 2.19.3 Hidden-phase interference kernel The interference cross term is Ψ∗ 1Ψ2=R1R2e−θhid,1vhid eθhid,2vhid =R1R2eΣθhid vhid , where the hidden-phase sum is Σθhid := θhid,2−θhid,1. Using the above three-term expansion, Ψ∗ 1Ψ2=R1R2A(Σ) ·1 + B(Σ) vvis +C(Σ) vhid.(2.65) The adjoint cross term is obtained via (Ψ∗ 1Ψ2)∗=R1R2A(Σ) ·1 + B(Σ) vvis −C(Σ) vhid, since v∗ vis =vvis and v∗ hid =−vhid. Summing both contributions yields the self-conjugate interference kernel Ψ∗ 1Ψ2+ Ψ∗ 2Ψ1= 2R1R2A(Σ) ·1 + B(Σ) vvis.(2.66) The hidden component C(Σ)vhid cancels identically. 2.20. NORM GROWTH OF THE POWERS OF x=j−j2AND MINIMAL POLYNOMIAL53 2.19.4 Carrier precession and the rotation of the visible mode Equation (2.66) shows that hidden-phase contributions survive only through the pair of real functions (A(Σ), B(Σ)) multiplying the carrier basis {1, vvis}. The pair can be written in polar form, A(Σ) = Chid(Σ) cosφhid(Σ), B(Σ) = Chid(Σ) sinφhid(Σ), where Chid(Σ) = pA(Σ)2+B(Σ)2, and the precession angle is φhid(Σ) = arctan 2 B(Σ), A(Σ). Substituting this into (2.66) yields the final form Ψ∗ 1Ψ2+ Ψ∗ 2Ψ1= 2R1R2Chid(Σ)cosφhid(Σ) + sinφhid(Σ) vvis.(2.67) The expression inside parentheses is precisely the visible carrier K(Σ) := cosφhid(Σ) + sinφhid(Σ) vvis, which has been rotated by the hidden-phase angle φhid(Σ) in the visible subspace. This internal rotation—which depends solely on the hidden-phase sum Σθhid—constitutes the carrier precession of the C3interference pattern. Although the hidden phase does not contribute directly to the Born probability, its effect survives as a rotation and amplitude modulation of the visible-mode interference carrier. 2.20 Norm Growth of the Powers of x=j−j2and Minimal Polynomial We work in the real cyclic phase algebra C3= spanR{1, j, j2}, j3=−1,(2.68) and consider the distinguished element x:= j−j2.(2.69) Every power of xcan be written in the form xn=An+Bnx, n ≥1,(2.70) 54 CHAPTER 2. ALGEBRAIC FOUNDATIONS OF C3 for suitable real coefficients An, Bn. Using the algebraic relation x2= (j−j2)2= 2 −x, (2.71) one obtains the linear recurrence xn+1 =x xn=x(An+Bnx)=Anx+Bnx2=Anx+Bn(2 −x),(2.72) hence xn+1 = 2Bn+ (An−Bn)x. (2.73) Therefore the coefficient vector vn:= (An, Bn)Tsatisfies vn+1 =Mvn, M := 0 2 1−1!, v1= 0 1!.(2.74) Diagonalising M(with eigenvalues 1 and −2) yields the closed form An=2+(−2)n 3, Bn=1−(−2)n 3, n ≥1.(2.75) In the basis {1, j, j2}this implies xn=An+Bn(j−j2) = An·1 + Bn·j+ (−Bn)·j2,(2.76) so the coefficient triple of xnis (An, Bn,−Bn). 2.20.1 Euclidean Norm Behaviour of xn We now endow C3with the standard Euclidean norm induced from its identification with R3: z=a+bj +cj27−→ N(z):=a2+b2+c2.(2.77) For xnwe have an=An,bn=Bn,cn=−Bn, hence N(xn)=A2 n+B2 n+ (−Bn)2=A2 n+ 2B2 n.(2.78) Substituting the closed forms for Anand Bnyields N(xn) = 2+(−2)n 32 + 2 1−(−2)n 32 .(2.79) A straightforward simplification gives the remarkably simple formula N(xn) = 4n+ 2 3(n≥1).(2.80) 2.20. NORM GROWTH OF THE POWERS OF x=j−j2AND MINIMAL POLYNOMIAL55 In particular, the norm of xnexhibits exponential growth with base 4: N(xn)∼4n 3as n→ ∞.(2.81) This growth is directly tied to the spectral data of the multiplication operator by x: the eigenvalue −2 contributes the dominant mode |−2|2n= 4n, while the eigenvalue 1 contributes only a bounded component to the norm. 2.20.2 Minimal and Characteristic Polynomials of x The defining quadratic relation x2= 2 −x(2.82) can be rewritten as x2+x−2 = 0.(2.83) Hence the monic polynomial of least degree that annihilates xis mX(λ) = λ2+λ−2 = (λ−1)(λ+ 2).(2.84) Therefore the minimal polynomial of xhas distinct real roots λ= 1 and λ=−2. If we consider xas a left multiplication operator Lx:C3→C3, Lx(z)=xz, (2.85) then in the basis {1, j, j2}the corresponding 3 ×3 real matrix has eigenvalues λ1=−2, λ2= 1, λ3= 1,(2.86) so the characteristic polynomial of Lxis χX(λ) = (λ+ 2)(λ−1)2.(2.87) The minimal polynomial mXis then the square-free factor of the characteristic polynomial, namely mX(λ) = (λ+ 2)(λ−1).(2.88) The two spectral values 1 and −2 are precisely those that govern the decomposition of xninto a sum of a bounded “unit-eigenmode” and a rapidly growing “−2-eigenmode”, which is reflected in the norm growth law N(xn) = (4n+ 2)/3. 56 CHAPTER 2. ALGEBRAIC FOUNDATIONS OF C3 2.21 Power Structure of the Hidden Axis Element s=j+j2in the Algebra C3 Let C3= spanR{1, j, j2}denote the cyclic phase algebra with the defining relation j3= −1. One of the central objects in the hidden–phase analysis is the element s:= j+j2, which spans the anti-Hermitian “hidden” axis according to the conjugation rule j∗= −j2, (j2)∗=−j. In this section we derive a closed recurrence relation for the powers sn, provide explicit expressions for the first several values, and show that the sequence (sn)n≥1collapses to a two-dimensional invariant subspace that controls the internal phase dynamics. 2.21.1 Direct Computations for the First Powers Using only the relation j3=−1, a straightforward computation yields s2= (j+j2)2=−2−j+j2,(2.89) s3= (j+j2)3=−3(j+j2) = −3s, (2.90) s4= (j+j2)4=−3s2= 6 + 3j−3j2,(2.91) s5= 9(j+j2)=9s, (2.92) s6= 9(−2−j+j2).(2.93) These identities already reveal a striking pattern: odd powers are proportional to s, whereas even powers are proportional to s2. 2.21.2 Recurrence Relation and General Closed Form From the identity s3=−3s, multiplication by sn−3implies sn=−3sn−2, n ≥3.(2.94) Thus the entire sequence is generated by a second-order linear recurrence with constant coefficient −3. Iterating (2.102) yields the explicit closed forms s2k+1 = (−3)ks, k ≥0,(2.95) s2k= (−3)k−1s2, k ≥1.(2.96) 2.22. EFFECTIVE TWO–DIMENSIONAL PHASE DYNAMICS GENERATED BY s=j+j257 Hence all powers of slie in the two-dimensional real subspace spanR{s, s2}= spanR{j+j2,−2−j+j2}. This shows that the hidden axis does not generate the full algebra under power iteration; instead it collapses dynamically onto a minimal invariant cone spanned by {s, s2}. 2.21.3 Geometric and Physical Interpretation The element s=j+j2represents the purely hidden (anti-Hermitian) direction in C3, orthogonal to the visible Hermitian axis span{1, j−j2}. The recurrence relation s3=−3s implies that repeated internal phase evolution along this axis produces only two effective modes: a “hidden carrier” mode sand a “compressed curvature” mode s2. The sign alternation and the factor (−3)kappearing in the powers encode the curvature-weighted precession associated with the hidden phase channel. Thus, in the generalized Schr¨odinger dynamics, the action of the time-axis operator on the hidden sector reduces to oscillation between two coupled internal modes, rather than a full three-dimensional cyclic evolution. This two-mode closure is one of the key structural reasons why the hidden phase influences the observable sector only through higher-order phase gradients, even though it remains dynamically active in the full C3amplitude. 2.22 Effective Two–Dimensional Phase Dynamics Generated by s=j+j2 Let C3= spanR{1, j, j2}with j3=−1. Define the element s:= j+j2, which we interpret as the generator of the hidden phase axis. A direct computation shows that its first powers satisfy the identities s2=−2−j+j2,(2.97) s3=−3s, (2.98) s4=−3s2,(2.99) s5= 9s, (2.100) s6= 9s2.(2.101) From these relations we obtain the universal recurrence law sn=−3sn−2, n ≥3.(2.102) 64 CHAPTER 2. ALGEBRAIC FOUNDATIONS OF C3 2.24.5 Relation to the C3–Schr¨odinger Dynamics Let the wavefunction be Ψ(x, t) = a(x, t)+jb(x, t)+j2c(x, t). Define A(x, t) = a2+b2+c2, B(x, t)=ab +bc −ac, so that N(x, t)=A(x, t) + B(x, t)(j−j2). AC3–Hermitian Schr¨odinger-type equation Kℏ∂tΨ = ˆ HΨ, with Kproportional to j+j2, leads to the continuity equation ∂tN=−∇·J. Separating the visible and hidden components gives ∂tA+B+∇·Jvis = 0, ∂tB+∇·Jhid = Ω(x, t)A(x, t), where Ω is an effective precession frequency generated by the action of Kin the Hamiltonian. Thus: •Agoverns the visible probability density, •Bencodes the hidden phase-tension, •Jvis is the classical probability current, and •Jhid is a carrier-precession current associated with the hidden (j+j2) axis. In this way the triplet (A, B, ∆ϕeff) closes the triangle: norm ←→ Born rule ←→ phase gradient. 2.25. SPHERICAL SYMMETRY AND PHASE–CHANNEL ENERGETICS IN THE CUBIC ALGEBRA C365 2.25 Spherical Symmetry and Phase–Channel Energetics in the Cubic Algebra C3 Let z=a+jb+j2c, j3=−1, a, b, c ∈R, denote an arbitrary element of the cubic algebra C3. We adopt the algebraic involution 1∗= 1, j∗=−j2,(j2)∗=−j, under which the quadratic form N(z) := z z∗ splits into a purely scalar part and a directional phase component: N(z) = S(z) + F(z)j−j2.(2.117) The coefficients in (2.117) are explicitly S(z) = a2+b2+c2,(2.118) F(z) = ab +bc −ac. (2.119) The scalar part S(z) is rotationally invariant in R3, while F(z) measures an anisotropic phase imbalance along the distinguished phase-direction j−j2. 2.25.1 Visible and Hidden Phase Channels To reveal the intrinsic two-phase geometry of C3, we introduce the orthonormal change of variables u0:= a, uvis := b−c √2, uhid := b+c √2.(2.120) This separates the original coordinates into: •u0: the central (classical) amplitude channel, •uvis: the visible phase–contrast channel, •uhid: the hidden symmetric phase channel. The inverse transformation is a=u0, b =uvis +uhid √2, c =uhid −uvis √2.(2.121) 66 CHAPTER 2. ALGEBRAIC FOUNDATIONS OF C3 2.25.2 Spherical Symmetry of the Scalar Kernel Substituting (2.120) into (2.118), we obtain S(z) = u2 0+u2 vis +u2 hid.(2.122) Thus the scalar part of the norm is a perfect Euclidean radius in three dimensions: S(z) = R2⇐⇒ (u0, uvis, uhid)∈S2 R. It defines a spherically symmetric energy kernel, invariant under all rotations in the (u0, uvis, uhid)-space. 2.25.3 Phase–Cone Anisotropy Using (2.120)–(2.121) in (2.119), the phase-interference term becomes F(z) = √2u0uvis +1 2u2 hid −1 2u2 vis.(2.123) Equation (2.123) exhibits three physically distinct contributions: 1. Central–visible coupling √2u0uvis, encoding a direct coupling between classical amplitude and visible phase contrast. 2. Hidden phase pressure +1 2u2 hid, providing a positive, isotropic contribution to phase-tension. 3. Visible phase saturation −1 2u2 vis, representing a suppression effect induced by strong visible-phase contrast. 2.25.4 Full Norm in Phase–Channel Coordinates Combining (2.122) and (2.123) with (2.117), we obtain: N(z) = u2 0+u2 vis +u2 hid | {z } spherical energy kernel +√2u0uvis +1 2u2 hid −1 2u2 vis | {z } phase–cone anisotropy j−j2.(2.124) 2.26. DECOMPOSITION OF A SINGLE-PARTICLE C3WAVEFUNCTION INTO VISIBLE AND HIDDEN PHASE AXES67 The norm N(z) therefore consists of: •a rotationally invariant energy sphere, and •a directional phase–cone whose orientation and curvature are controlled by the visible and hidden phase-channel amplitudes. 2.25.5 Geometric Summary •The quantity u2 0+u2 vis +u2 hid measures total amplitude and is independent of phase distribution. •The function F(z) generates a conical deformation of the energy sphere, determined by the imbalance between visible and hidden channels. •Hidden phase modes (uhid)inflate the phase cone, while visible modes (uvis)narrow it. •Measurement-accessible physics aligns with the visible channel uvis, while latent phase dynamics align with the hidden channel uhid. This decomposition is intrinsic to the two-dimensional internal phase geometry of the cubic algebra C3. It provides a natural framework for interpreting amplitude, phase imbalance, and measurement back-action in multi-carrier wave or quantum systems. 2.26 Decomposition of a Single-Particle C3Wavefunction into Visible and Hidden Phase Axes In the C3algebra, a single particle is represented not by a single complex amplitude (as in C2), but by a triplet of phase-carriers. These do not correspond to multiple waves or multiple particles. A single physical wavefunction is instead expanded in a threedimensional phase basis: Ψ = a+j b +j2c, where the components a, b, c represent projections of the same physical state on three internal phase directions separated by 120◦. The quantity Ψ will now be re-expressed in the orthogonal phase basis u:= j−j2, v := j+j2, corresponding respectively to the visible and hidden phase axes. The inverse relations follow immediately: j=u+v 2, j2=v−u 2. 68 CHAPTER 2. ALGEBRAIC FOUNDATIONS OF C3 2.26.1 Phase–Axis Decomposition Substituting these into the original form of Ψ yields: Ψ = a+bu+v 2+cv−u 2 =a+b−c 2u+b+c 2v. Thus the single-particle wavefunction decomposes as Ψ = S+X u +Y v, S := a, X := b−c 2, Y := b+c 2. The physical interpretation is immediate: •Sis the scalar (phase–neutral) component. •Xis the projection onto the visible phase axis u=j−j2. •Yis the projection onto the hidden phase axis v=j+j2. All three components together form a single physical wave; the C3representation does not introduce multiple particles or multiple independent waves. 2.26.2 Norm in the (1, u, v)Basis The C3norm of Ψ in the original (a, b, c) components is N(Ψ) = A+B(j−j2), A =a2+b2+c2, B =ab +bc −ac. Expressing Aand Bin terms of (S, X, Y ) gives A=S2+ 2X2+ 2Y2, B=−X2+ 2SX +Y2. Hence the full norm becomes N(Ψ) = S2+ 2X2+ 2Y2+−X2+ 2SX +Y2u. 2.26.3 Visible (Complex) Projection of the Norm The passage to the physical, measurable C2complex plane corresponds to the projection uC= (j−j2)C= 1, vC= (j+j2)C=i√3. 2.26. DECOMPOSITION OF A SINGLE-PARTICLE C3WAVEFUNCTION INTO VISIBLE AND HIDDEN PHASE AXES69 Since the norm contains no v–component, the hidden axis does not appear directly in the complex projection. The visible norm is therefore Nvis := A+B. Substituting the expressions for Aand Bfrom above: Nvis =S2+ 2X2+ 2Y2+−X2+ 2SX +Y2 = (S+X)2+ 3Y2. Thus the measurable intensity of the single-particle C3wavefunction is Nvis = (S+X)2+ 3Y2. 2.26.4 Mechanism by Which the Hidden Axis Contributes to the Visible Norm The decomposition Ψ = S+X u +Y v shows that the hidden component Ylies entirely along the axis v=j+j2, which has no direct projection into the complex plane. Nevertheless, the hidden amplitude contributes to the observable norm through the term 3Y2, which is strictly positive and modifies the total visible intensity. This occurs because the C3–Schr¨odinger dynamics couples the visible and hidden components (X, Y ) through the internal phase geometry, so that any evolution of the hidden phase modifies Y(t) and hence feeds back into the visible norm Nvis. In short: The hidden axis does not appear as a direction in the norm, but its amplitude Ycontributes through 3Y2. Thus the hidden phase influences measurable quantum behaviour not through its orientation (which is unobservable), but through its dynamical coupling to the visible component and through its quadratic contribution to the intensity. 70 CHAPTER 2. ALGEBRAIC FOUNDATIONS OF C3 2.27 Angular Parametrization of the C3Phase Components We start from the decomposition of a single-particle C3wavefunction in the phase basis Ψ = S+X u +Y v, u := j−j2, v := j+j2, where S=a, X =b−c 2, Y =b+c 2 encode, respectively, the scalar, visible-phase and hidden-phase projections of the same physical state. 2.27.1 Spherical Phase Coordinates (R, α, β) To separate the overall amplitude from the internal phase geometry, we introduce a spherical parametrization in the (S, X, Y ) space: S=Rcos α, X =Rsin αcos β, Y =Rsin αsin β, with R≥0, α ∈[0, π], β ∈[0,2π). The interpretation of these angles is: •Ris the overall amplitude scale of the single-particle wavefunction. •αcontrols the splitting between the scalar component and the two phase components: α→0⇒Sdominates (almost purely scalar), α→π 2⇒purely phase-type components (X, Y ). •βcontrols the ratio between the visible and hidden phase axes: β→0⇒Y≈0, X dominates (purely visible phase), β→π 2⇒X≈0, Y dominates (purely hidden phase). Thus αmeasures the scalar vs. phase balance, while βmeasures the visible vs. hidden phase balance inside the phase subspace. 2.27. ANGULAR PARAMETRIZATION OF THE C3PHASE COMPONENTS 71 2.27.2 Visible Norm Nvis as a Function of (R, α, β) In the (S, X, Y ) basis, the visible norm—i.e. the norm projected to the C2complex plane—was found to be Nvis = (S+X)2+ 3Y2. Substituting the angular parametrization, S=Rcos α, X =Rsin αcos β, Y =Rsin αsin β, we obtain Nvis =Rcos α+Rsin αcos β2+ 3Rsin αsin β2 =R2hcos α+ sin αcos β2+ 3 sin2αsin2βi. Thus the visible norm factorizes as Nvis(R, α, β)=R2F(α, β), with F(α, β) := cos α+ sin αcos β2+ 3 sin2αsin2β. The function F(α, β) plays the role of an effective phase–geometry factor: it encodes how the internal C3phase structure redistributes the single-particle amplitude between •the purely visible part (S+X)2, and •the hidden-induced contribution 3Y2. Special cases: •Purely visible phase subspace (β= 0): Y= 0, X =Rsin α⇒F(α, 0) = (cos α+ sin α)2. •Purely hidden phase subspace (β=π 2): X= 0, Y =Rsin α⇒Fα, π 2= cos2α+ 3 sin2α. In both cases the hidden axis does not appear as a direction in the norm, but its amplitude Ygives a strictly positive contribution through the 3Y2term. 72 CHAPTER 2. ALGEBRAIC FOUNDATIONS OF C3 2.28 Embedding into a Single-Particle Double-Slit Scenario We now sketch how this angular parametrization naturally fits into a single-particle double-slit configuration. The key point is conceptual: the C3structure describes one particle, and each path of the double-slit experiment carries its own internal phase geometry (R, α, β), but the superposition still represents a single particle. 2.28.1 Two Paths, One Particle: Path-Dependent Phase Geometry Consider a single particle whose wavefunction splits into two paths after the slits. For each path k= 1,2, we associate a C3-wave with parameters Ψk=Sk+Xku+Ykv, k = 1,2, with Sk=Rkcos αk, Xk=Rksin αkcos βk, Yk=Rksin αksin βk. In addition, each path carries a visible phase ϕk(x) (the usual path-dependent phase in standard C2quantum mechanics), so that upon projection to the complex plane, their contributions acquire relative phase factors eiϕk(x). The visible phase difference is ∆ϕvis(x) := ϕ2(x)−ϕ1(x). 2.28.2 Effective Visible Intensity and Phase Geometry For a given path k, the visible norm is N(k) vis (x)=R2 k(x)F(αk, βk), so that we may identify the effective single-path intensity as Ik(x):=N(k) vis (x) = R2 k(x)F(αk, βk). The total visible intensity on the screen for the superposed state Ψtot = Ψ1+ Ψ2then has the schematic form IC3(x)=I1(x)+I2(x)+2pI1(x)I2(x)G(α1, β1;α2, β2) cos ∆ϕvis(x). Here the factor G(α1, β1;α2, β2) 2.28. EMBEDDING INTO A SINGLE-PARTICLE DOUBLE-SLIT SCENARIO 73 encodes the effect of the internal C3phase geometry on the interference contrast. In the simplest symmetric case, α1=α2=α, β1=β2=β, the two paths share the same internal phase structure, so I1(x)=R2 1(x)F(α, β), I2(x)=R2 2(x)F(α, β), and one finds G= 1, leading to I(symmetric) C3(x) = F(α, β)hR2 1(x)+R2 2(x)+2R1(x)R2(x) cos ∆ϕvis(x)i. In this symmetric configuration, the internal C3geometry appears as a global multiplicative factor F(α, β) which renormalizes the overall intensity profile while preserving the usual cos ∆ϕvis interference pattern. More generally, when α1=α2or β1=β2, the function Gdeviates from unity and the interference contrast is modified in a nontrivial way by the mismatch of visible/hidden phase ratios on the two paths. In that sense: •The visible phase difference ∆ϕvis controls the usual oscillatory pattern along the screen. •The hidden phase ratio, parametrized by (αk, βk), controls how strongly each path contributes to the visible intensity and how much of the total amplitude is stored along the hidden axis v. Thus the pair of angles (α, β) define a “phase-geometry point” in the internal C3 space, and together with the visible phase difference ∆ϕvis(x) they provide a complete trigonometric diagram for the C3-Born rule in the double-slit configuration of a single particle. 80 CHAPTER 2. ALGEBRAIC FOUNDATIONS OF C3 Back to (S, X, Y )Variables: Explicit X↔YCoupling The inverse transformation is S=U+W 2, X =U−W 2, Y =√3 3V. Using the rotation equations for Uand Vand the fact that ∂tW= 0, we obtain ∂tS=−Ω 2V, ∂tX=−Ω 2V, ∂tY=√3 Ω 3U. Substituting U=S+Xand V=√3Y, we find ∂tS=−√3 Ω 2Y, ∂tX=−√3 Ω 2Y, ∂tY=√3 Ω 3(S+X). In matrix form, ∂t    S X Y    = Ω         0 0 −√3 2 0 0 −√3 2 √3 3 √3 30             S X Y    . This is an explicit realization of an internal C3–Schr¨odinger dynamics in which: •The visible and hidden amplitudes mix through the off–diagonal entries coupling Xand Y, as well as Sand Y. •The visible norm Nvis = (S+X)2+ 3Y2=U2+V2 is strictly conserved by construction: a direct computation yields ∂tNvis = 0. This coupling matrix thus represents a minimal precession of the single-particle wavefunction in the internal (u, v) phase space: a rotation between the visible amplitude (S+X) and the hidden amplitude Y, while preserving the C3–Born norm. 2.31. EXTREMAL BEHAVIOR OF THE C3GEOMETRY FACTOR G81 2.31 Extremal Behavior of the C3Geometry Factor G For two internal C3phase states characterized by (α1, β1) and (α2, β2), the interference visibility in the double-slit configuration is governed by the geometry factor G(α1, β1;α2, β2) = (cos α1+ sin α1cos β1) (cos α2+ sin α2cos β2) pF(α1, β1)F(α2, β2), where F(α, β) = (cos α+ sin αcos β)2+ 3 sin2αsin2β. The quantity Gsatisfies 0≤ G ≤ 1, and the limit cases correspond to physically meaningful configurations of the internal phase geometry. We now characterize these extremal regimes. 2.31.1 Case I: Full visibility enhancement (G= 1) The geometry factor attains its maximum value when the two paths share identical internal phase geometry: (α1, β1)=(α2, β2). In this case, G= 1, and the C3double-slit intensity reduces to the standard quantum mechanical expression, I(x) = I1(x)+I2(x)+2pI1(x)I2(x) cos ∆ϕvis(x). Thus, full interference contrast is preserved. This regime corresponds to perfect alignment of the two C3internal-phase states within the visible axis of the algebra. 2.31.2 Case II: Complete suppression of interference (G= 0) A qualitatively new phenomenon emerges in the C3formalism: interference can vanish even when both paths carry finite visible intensity. This occurs when one path lies fully along the visible phase axis (β= 0), while the other lies fully along the hidden axis (β=π 2): β1= 0, β2=π 2. 82 CHAPTER 2. ALGEBRAIC FOUNDATIONS OF C3 In this configuration, the visible projections Vk=Sk+Xk=Rkcos αk+ sin αkcos βk satisfy V1= 0, V2≈R2cos α2, but the visible axes of the two paths have no compatible overlap. Consequently, G= 0. Although the hidden component Ykcontributes positively to the total visible norm via the term 3Y2 k, it does not contribute to the interfering phase-carrying direction. Thus, interference is entirely suppressed, even though I1and I2remain non-zero. Such complete suppression has no analogue in standard C2quantum mechanics and arises purely from the multi-axis internal structure of C3. 2.31.3 Case III: Partial visibility (0<G<1) The generic situation corresponds to internal phase geometries that differ but still retain non-zero visible projections: (α1, β1)= (α2, β2), V1, V2= 0. In this case, 0<G<1, and interference is partially suppressed. The degree of suppression is governed by the mismatch between the visible projections of the two paths, and thus by the relative distribution of phase amplitude between the visible (u) and hidden (v) axes encoded by βk. This regime has no counterpart in the C2theory, where interference visibility depends solely on the phase difference ∆ϕvis. In C3quantum mechanics, the internal phase geometry introduces an additional visibility-control channel through the angles (α, β), representing intrinsic ”phase-geometric” decoherence. 2.31.4 Summary of extremal regimes Regime Condition Value of G Full visibility (α1, β1)=(α2, β2)G= 1 Complete suppression (β1, β2) = (0,π 2)G= 0 Partial visibility general mismatch 0 <G<1 2.32. SINGLE-PARTICLE DOUBLE-SLIT INTERFERENCE IN THE C3FRAMEWORK83 These limits demonstrate that the geometry factor Gis a quantitative measure of internal phase alignment between the two C3states. The hidden phase axis vcontributes to the visible norm but cannot carry the interfering phase. As a result, misalignment along the (u, v) decomposition produces systematic reductions in interference contrast, a phenomenon unique to C3internal geometry. 2.32 Single-Particle Double-Slit Interference in the C3Framework In this section we embed the visible/hidden phase parametrization of the C3theory into a standard single-particle double-slit geometry. The key point is that the C3structure does not describe multiple particles or multiple independent waves: it is a refined, multicarrier representation of a single wavefunction. Each path of the double-slit experiment is still part of a single C3state. 2.32.1 Classical C2Reference: Two Gaussian Wavepackets We consider a one-dimensional detection coordinate xalong the screen and two slits separated by a distance d, at a distance Lfrom the screen. For a monochromatic source with wavenumber k, the optical path difference in the Fraunhofer limit is ∆ℓ(x)≃d Lx, ∆ϕvis(x):=k∆ℓ(x)≃kd Lx. Let the two partial wavepackets on the screen be modeled by Gaussians ψ1(x)=A0exp−(x+d/2)2 4σ2eiϕ1(x), ψ2(x) = A0exp−(x−d/2)2 4σ2eiϕ2(x), with equal amplitudes and a visible phase difference ∆ϕvis(x) := ϕ2(x)−ϕ1(x). In the symmetric case (equal intensities, same envelope), we can write ψ1(x) = R0(x)eiϕ1(x), ψ2(x) = R0(x)eiϕ2(x), so that R2 0(x)=|ψ1(x)|2=|ψ2(x)|2. 84 CHAPTER 2. ALGEBRAIC FOUNDATIONS OF C3 The usual C2total wavefunction is ψC2(x)=ψ1(x)+ψ2(x), and the classical Born intensity is IC2(x)=|ψ1(x)+ψ2(x)|2 =|ψ1|2+|ψ2|2+ 2 ℜψ∗ 1ψ2 = 2R2 0(x)1 + cos ∆ϕvis(x). We identify the envelope intensity I0(x) := 2R2 0(x) and write IC2(x) = I0(x)1 + cos ∆ϕvis(x). 2.32.2 Embedding into C3: Visible/Hidden Phase Structure In the C3description, the same single-particle state is represented by a multi-carrier wavefunction Ψ(x) = S(x)+X(x)u+Y(x)v, u := j−j2, v := j+j2. The three real functions S, X, Y are not independent waves; they are internal phase components of the same physical wavefunction. The visible C3norm for a given (S, X, Y ) is Nvis(x) = (S+X)2+ 3Y2. We now parametrize (S, X, Y ) by an overall amplitude R(x) and two angles (α(x), β(x)): S=Rcos α, X =Rsin αcos β, Y =R √3sin αsin β. Inserting into Nvis yields Nvis(x) R2(x)= 1 + sin(2α(x)) cos β(x), so that Nvis(x) = R2(x)1 + sin(2α(x)) cos β(x). 2.32. SINGLE-PARTICLE DOUBLE-SLIT INTERFERENCE IN THE C3FRAMEWORK85 2.32.3 C3–Born Rule for the Double-Slit Geometry To connect this with the classical double-slit pattern, we identify the overall amplitude with the classical envelope: R2(x) := I0(x) = 2R2 0(x). The classical C2interference term is IC2(x)−I0(x) = I0(x) cos ∆ϕvis(x). In the C3parametrization, the deviation from the envelope is Nvis(x)−R2(x) = R2(x) sin(2α(x)) cos β(x). We therefore identify the effective interference factor as sin(2α(x)) cos β(x)←→ VC3(x) cos ∆ϕvis(x), where VC3(x) is the C3–modified visibility. A particularly transparent choice is to set sin(2α(x)) = cos ∆ϕvis(x),cos β(x) = cos ϕhid(x), where ϕhid(x) is an internal (hidden) C3phase, so that sin(2α(x)) cos β(x) = cos ∆ϕvis(x) cos ϕhid(x). With this choice, the C3–Born intensity becomes IC3(x) := Nvis(x)=I0(x)h1 + cos ∆ϕvis(x) cos ϕhid(x)i. We recover the classical pattern for ϕhid(x) = 0: IC3(x)ϕhid=0 =I0(x)1 + cos ∆ϕvis(x)=IC2(x). For nonzero hidden phase, the classical interference contrast is modulated by cos ϕhid(x), which is an internal C3degree of freedom. The hidden phase does not appear as an angle in the complex plane; instead, it renormalizes the effective visibility of the fringes: VC3(x) = cos ϕhid(x). 86 CHAPTER 2. ALGEBRAIC FOUNDATIONS OF C3 2.32.4 Internal C3Dynamics and X↔YCoupling in the DoubleSlit Setup The internal C3–Schr¨odinger dynamics couples the visible and hidden amplitudes. In the (S, X, Y ) basis, a minimal norm-preserving evolution can be written as ∂t    S X Y    = Ω         0 0 −√3 2 0 0 −√3 2 √3 3 √3 30             S X Y    , with Ω an internal precession frequency. The corresponding visible norm Nvis(x, t) = (S+X)2+ 3Y2 is conserved by construction: ∂tNvis(x, t) = 0. However, the distribution of intensity between (S+X) and Ychanges in time. In the double-slit geometry, this leads to a slow modulation of the effective hidden phase ϕhid(x, t), and therefore of the fringe visibility VC3(x, t) = cos ϕhid(x, t). Thus, the internal C3dynamics provides a concrete mechanism by which an unobservable hidden phase in the (u, v) internal space feeds back into the visible interference pattern as a timeand space-dependent modulation of the fringe contrast. 2.33 Angular Dynamics of the Internal C3Phase under the Schr¨odinger Evolution We consider a single-particle C3wavefunction written in the phase basis Ψ(t, x) = S(t, x)+X(t, x)u+Y(t, x)v, u := j−j2, v := j+j2, and evolving according to a C3-Schr¨odinger-type equation v ∂tΨ(t, x) = HΨ(t, x), H := −ℏ2 2m∂2 x+V(x). 2.33. ANGULAR DYNAMICS OF THE INTERNAL C3PHASE UNDER THE SCHR ¨ ODINGER EVOLUTION87 Projecting this equation onto the basis {1, u, v}of the internal phase space determines a system of three coupled evolution equations for the real fields S(t, x), X(t, x) and Y(t, x): ∂tS=FS[S, X, Y ;H], ∂tX=FX[S, X, Y ;H], ∂tY=FY[S, X, Y ;H], where FS, FX, FYare linear functionals of (S, X, Y ) built from Hand the C3multiplication rules. Their explicit form depends on the chosen C3representation and conjugation, but the angular dynamics presented below is independent of this choice and follows purely from the definition of (R, α, β). 2.33.1 From Cartesian (S, X, Y )to angular (R, α, β)coordinates We introduce the spherical parametrization of the internal phase vector S=Rcos α, X =Rsin αcos β, Y =Rsin αsin β, with R≥0, α∈[0, π] and β∈[0,2π). The quantity R(t, x) represents the total internal amplitude, α(t, x) controls the scalar vs. phase balance, and β(t, x) controls the visible vs. hidden phase ratio. It is useful to introduce ρ:= √X2+Y2, so that R2=S2+ρ2,tan α=ρ S,tan β=Y X. 2.33.2 General evolution equations for R,αand β We first express the time derivatives of (S, X, Y ) in terms of ˙ R:= ∂tR, ˙α:= ∂tαand ˙ β:= ∂tβ: ∂tS=˙ Rcos α−R˙αsin α, ∂tX=˙ Rsin αcos β+R˙αcos αcos β−R˙ βsin αsin β, ∂tY=˙ Rsin αsin β+R˙αcos αsin β+R˙ βsin αcos β. Conversely, one can solve for ( ˙ R, ˙α, ˙ β) in terms of (∂tS, ∂tX, ∂tY) in a representationfree way. Amplitude evolution. From R2=S2+X2+Y2, we obtain 2R˙ R= 2S ∂tS+ 2X ∂tX+ 2Y ∂tY, 88 CHAPTER 2. ALGEBRAIC FOUNDATIONS OF C3 and therefore ˙ R=S ∂tS+X ∂tX+Y ∂tY R. Substituting ∂tS=FS,∂tX=FX,∂tY=FYfrom the C3-Schr¨odinger projection yields ˙ R(t, x) = S FS+X FX+Y FY R, which controls the evolution of the total internal amplitude and hence the visible norm Nvis =R2F(α, β). Scalar–phase mixing angle α.Define ρ=√X2+Y2, so that tan α=ρ/S. Differentiating, ˙α=1 1 + tan2α∂tρ S= cos2αS˙ρ−ρ ∂tS S2. Using cos2α=S2/R2, we obtain ˙α=S˙ρ−ρ ∂tS R2. Since ˙ρ=X ∂tX+Y ∂tY ρ, we finally get ˙α=S(X ∂tX+Y ∂tY)/ρ −ρ ∂tS R2. Substituting again ∂tS=FS,∂tX=FX,∂tY=FYyields the explicit scalar–phase mixing dynamics: ˙α(t, x) = SX FX+Y FY/ρ −ρ FS R2. Visible–hidden mixing angle β(precession). The angle βis defined by tan β= Y/X. Differentiating, ˙ β=1 1 + tan2β∂tY X=X ∂tY−Y ∂tX X2+Y2. Thus, ˙ β=X ∂tY−Y ∂tX X2+Y2. Inserting the C3-Schr¨odinger projections ∂tX=FX,∂tY=FYgives ˙ β(t, x) = X FY−Y FX X2+Y2. This equation shows explicitly that the visible–hidden phase angle βevolves whenever 2.34. TIME-DEPENDENT C3INTERFERENCE VISIBILITY AND INTERNAL PHASE PRECESSION89 the C3dynamics couples Xand Y(i.e. whenever FXand FYcontain off-diagonal terms mixing the visible and hidden axes). In particular, non-zero X FY−Y FXcorresponds to an internal precession of the phase vector in the (X, Y ) plane, which has no analogue in the purely complex (C2) theory. 2.33.3 Interpretation in terms of visible and hidden currents The above angular evolution equations can be related to generalized probability currents. Defining Jvis ∼S∇X−X∇S, Jhid ∼X∇Y−Y∇X, one finds that •˙ Ris controlled by the divergence of a total current built from (Jvis, Jhid), •˙αmeasures the exchange between scalar and phase-type amplitudes, and •˙ βis directly governed by the “hidden” current Jhid and encodes the precession of the internal phase vector around the hidden axis. In this way, the C3internal geometry naturally decomposes the Schr¨odinger dynamics into an amplitude evolution R(t, x), a scalar–phase mixing angle α(t, x), and a visible–hidden precession angle β(t, x), providing a geometric interpretation of how hidden phases feed into observable interference patterns. 2.34 Time-Dependent C3Interference Visibility and Internal Phase Precession In the C3double-slit configuration, a single particle is described by the superposition of two internal-phase states, Ψtot(t, x) = Ψ1(t, x)+Ψ2(t, x),Ψk=Sk+Xku+Ykv, each associated with a path k= 1,2. For each path we use the angular parameterization Sk=Rkcos αk, Xk=Rksin αkcos βk, Yk=Rksin αksin βk, where Rk(t, x) is the internal amplitude, αk(t, x) controls the scalar vs. phase splitting, and βk(t, x) the visible vs. hidden phase splitting. The visible intensity on the screen is IC3(t, x) = I1(t, x)+I2(t, x)+2pI1(t, x)I2(t, x)G(t, x) cos ∆ϕvis(t, x), with Ik(t, x) = R2 k(t, x)F(αk(t, x), βk(t, x)), 96 CHAPTER 4. RECONSTRUCTION OF SCHR ¨ ODINGER DYNAMICS Chapter 5 Two-Channel Quantum Dynamics 5.1 Wavefunction Decomposition Ψ=ψvis +χhid(j+j2) 5.2 Coupled Evolution Equations 5.3 Phase Winding and Interference 5.4 Measurement and the Visible Projection 97 98 CHAPTER 5. TWO-CHANNEL QUANTUM DYNAMICS Chapter 6 Physical Implications 6.1 Generalized Interference Patterns 6.2 Multi-Carrier Interpretation of Wavefunctions 6.3 Classicalization and Phase Gradient Criteria 6.4 Connections to Relativistic Extensions 99 100 CHAPTER 6. PHYSICAL IMPLICATIONS Appendix A Multiplication and Conjugation Tables for C3 101 102 APPENDIX A. MULTIPLICATION AND CONJUGATION TABLES FOR C3 Appendix B Hermitian Operator Bases 103 104 APPENDIX B. HERMITIAN OPERATOR BASES Appendix C Generalized Trigonometric Functions on C3 105