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SCHRÖDINGER, DIRAC, AND THE PRIMITIVE ARCHITECTURE OF DUALITY

Cláudio Vicente da Silva

Published Sep 14, 2026• 0 citations
Publisher

Abstract

Description This work investigates the mathematical and structural relationship between Schrödinger wave mechanics, Dirac’s relativistic quantum theory, charge conjugation, duality, opposition, equilibrium, and the concept of transduction developed within the framework of Primitive Architecture. The analysis begins with the wave function and the Schrödinger equation and proceeds to the relativistic structure introduced by Dirac. Particular attention is given to the pair of relativistic energy branches E±(p)=±p2c2+m2c4,E_{\pm}(p)=\pm\sqrt{p^2c^2+m^2c^4}, and to the mathematical structure of opposition represented by the transformation d↦−d.d\mapsto-d. The work formalizes this opposition as an involution, C2=I,C^2=I, and establishes the centered representation X=x0+d,X′=x0−d,X=x_0+d,\qquad X'=x_0-d, with x0=X+X′2,d=X−X′2.x_0=\frac{X+X'}{2}, \qquad d=\frac{X-X'}{2}. Within the normalized architectural representation, the structural center is x0=12,x_0=\frac12, with the corresponding involution C(x)=1−x.C(x)=1-x. The work distinguishes the abstract mathematical structure of duality from its specific physical realizations. In particular, it distinguishes the transformation of the sign of energy in the Dirac spectrum from charge conjugation and does not identify these operations as physically identical. Instead, it investigates their common formal structure as transformations between complementary configurations. The analysis also examines the Clifford-algebraic structure of the Dirac formalism through the decomposition AB=12{A,B}+12[A,B],AB= \frac12\{A,B\} + \frac12[A,B], where the anticommutator represents the symmetric component and the commutator represents the antisymmetric component. The corresponding Dirac relation {γμ,γν}=2ημνI\{\gamma^\mu,\gamma^\nu\} = 2\eta^{\mu\nu}I is considered in relation to the architectural distinction between equilibrium and opposition. The work then develops the concept of Relational Conjugation as a structural operation satisfying C(X)=X′,C(X′)=X,C2=I,C(X)=X', \qquad C(X')=X, \qquad C^2=I, together with the preservation of an appropriate invariant, I(C(X))=I(X).I(C(X))=I(X). This produces the closed relation X→C(X)→C2(X)=X.X\rightarrow C(X)\rightarrow C^2(X)=X. The same structure is examined in relation to Spiral Geometry, rs(t,θ)=A(t)esθ,s∈{−1,+1},r_s(t,\theta)=A(t)e^{s\theta}, \qquad s\in\{-1,+1\}, for which rsr−s=A(t)2.r_s r_{-s}=A(t)^2. The work therefore establishes a formal correspondence between opposition, centered duality, conjugation, invariance, transduction, and closure while maintaining the distinction between established physical theory and the architectural layer proposed in this research. A mathematical-computational auditing framework is also formulated. For a configuration XiX_i, its conjugate Xi′=C(Xi)X_i'=C(X_i), and an invariant II, the framework evaluates involution, invariant preservation, centered symmetry, and a coherence metric: Φi=w1∣C2(Xi)−Xi∣+w2∣I(Xi′)−I(Xi)∣+w3∣Xi+Xi′−2X0∣.\Phi_i = w_1|C^2(X_i)-X_i| + w_2|I(X_i')-I(X_i)| + w_3|X_i+X_i'-2X_0|. Exact structural compatibility corresponds to Φi=0.\Phi_i=0. The resulting framework is organized according to the sequence: Relation → Opposition → Equilibrium → Transformation → Invariant → Closure. Within the broader Primitive Architecture research program, this sequence is connected to the more general processes of movement, latency, transduction, form, memory, resonance, and cycle. The contribution of this work is not the creation of the Schrödinger equation, Dirac equation, Clifford algebra, relativistic energy-momentum relation, or the established physical theory of particles and antiparticles. These are treated as established mathematical and physical foundations. The proposed contribution consists in organizing selected structures from these theories within the conceptual and mathematical framework of Primitive Architecture, particularly through involution, center-deviation decomposition, relational conjugation, invariant preservation, transductive transformation, and structural closure. The work therefore presents Primitive Architecture as a formal organizational framework for examining duality and conjugation across different mathematical representations while preserving the distinction between mathematical construction, physical formalism, computational auditing, and theoretical interpretation.

DOI
10.5281/zenodo.22758575

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Description This work investigates the mathematical and structural relationship between Schrödinger wave mechanics, Dirac’s relativistic quantum theory, charge conjugation, duality, opposition, equilibrium, and the concept of transduction developed within the framework of Primitive Architecture. The analysis begins with the wave function and the Schrödinger equation and proceeds to the relativistic structure introduced by Dirac. Particular attention is given to the pair of relativistic energy branches E±(p)=±p2c2+m2c4,E_{\pm}(p)=\pm\sqrt{p^2c^2+m^2c^4}, and to the mathematical structure of opposition represented by the transformation d↦−d.d\mapsto-d. The work formalizes this opposition as an involution, C2=I,C^2=I, and establishes the centered representation X=x0+d,X′=x0−d,X=x_0+d,\qquad X'=x_0-d, with x0=X+X′2,d=X−X′2.x_0=\frac{X+X'}{2}, \qquad d=\frac{X-X'}{2}. Within the normalized architectural representation, the structural center is x0=12,x_0=\frac12, with the corresponding involution C(x)=1−x.C(x)=1-x. The work distinguishes the abstract mathematical structure of duality from its specific physical realizations. In particular, it distinguishes the transformation of the sign of energy in the Dirac spectrum from charge conjugation and does not identify these operations as physically identical. Instead, it investigates their common formal structure as transformations between complementary configurations. The analysis also examines the Clifford-algebraic structure of the Dirac formalism through the decomposition AB=12{A,B}+12[A,B],AB= \frac12{A,B} + \frac12[A,B], where the anticommutator represents the symmetric component and the commutator represents the antisymmetric component. The corresponding Dirac relation {γμ,γν}=2ημνI{\gamma^\mu,\gamma^\nu} = 2\eta^{\mu\nu}I is considered in relation to the architectural distinction between equilibrium and opposition. The work then develops the concept of Relational Conjugation as a structural operation satisfying C(X)=X′,C(X′)=X,C2=I,C(X)=X', \qquad C(X')=X, \qquad C^2=I, together with the preservation of an appropriate invariant, I(C(X))=I(X).I(C(X))=I(X). This produces the closed relation X→C(X)→C2(X)=X.X\rightarrow C(X)\rightarrow C^2(X)=X. The same structure is examined in relation to Spiral Geometry, rs(t,θ)=A(t)esθ,s∈{−1,+1},r_s(t,\theta)=A(t)e^{s\theta}, \qquad s\in{-1,+1}, for which rsr−s=A(t)2.r_s r_{-s}=A(t)^2. The work therefore establishes a formal correspondence between opposition, centered duality, conjugation, invariance, transduction, and closure while maintaining the distinction between established physical theory and the architectural layer proposed in this research. A mathematical-computational auditing framework is also formulated. For a configuration XiX_i, its conjugate Xi′=C(Xi)X_i'=C(X_i), and an invariant II, the framework evaluates involution, invariant preservation, centered symmetry, and a coherence metric: Φi=w1∣C2(Xi)−Xi∣+w2∣I(Xi′)−I(Xi)∣+w3∣Xi+Xi′−2X0∣.\Phi_i = w_1|C^2(X_i)-X_i| + w_2|I(X_i')-I(X_i)| + w_3|X_i+X_i'-2X_0|. Exact structural compatibility corresponds to Φi=0.\Phi_i=0. The resulting framework is organized according to the sequence: Relation → Opposition → Equilibrium → Transformation → Invariant → Closure. Within the broader Primitive Architecture research program, this sequence is connected to the more general processes of movement, latency, transduction, form, memory, resonance, and cycle. The contribution of this work is not the creation of the Schrödinger equation, Dirac equation, Clifford algebra, relativistic energy-momentum relation, or the established physical theory of particles and antiparticles. These are treated as established mathematical and physical foundations. The proposed contribution consists in organizing selected structures from these theories within the conceptual and mathematical framework of Primitive Architecture, particularly through involution, center-deviation decomposition, relational conjugation, invariant preservation, transductive transformation, and structural closure. The work therefore presents Primitive Architecture as a formal organizational framework for examining duality and conjugation across different mathematical representations while preserving the distinction between mathematical construction, physical formalism, computational auditing, and theoretical interpretation.
Modern physics is formulated through several mathematical frameworks that are individually successful but structurally distinct. Quantum mechanics is expressed through Hilbert spaces and self-adjoint operators, relativistic quantum theory through representations of spacetime symmetries and Clifford algebras, gauge theories through connections and curvature, quantum field theory through propagators and Green functions, statistical mechanics through state multiplicities and probability measures, and general relativity through Lorentzian geometry. The present work asks whether some of these structures can be reconstructed as representations of a more primitive mathematical architecture rather than independently postulated. An architecture is introduced in terms of four structural sectors: composable transport, interaction, holonomy, and recursive closure. Physical theories are then interpreted as representation classes of this architecture subject to explicitly stated additional conditions. Continuous norm-preserving transport in a complex Hilbert representation yields Schrödinger evolution. Lorentz-covariant spacetime transport gives the relativistic mass-shell relation, while scalar and first-order representations lead respectively to Klein–Gordon and Dirac structures. The central methodological requirement is that established equations must arise from primitive rules plus explicit representation conditions rather than being inserted retrospectively.
This work resolves a foundational tension in Dirac's formulation of relativistic quantum mechanics: momentum appears in two distinct but undifferentiated roles—as a dispersion-locked parameter generating unitary evolution in the relativistic wave equation, and as a free phase variable subject to symplectic flow in the constraint formalism. We demonstrate that this duality reflects an intrinsic Lorentzian structure in energy-momentum space, characterized by the interval dΣ² = dE²/c² − dp². This geometry divides naturally into three sectors: symmetric (dispersion-locked), null (massless, E/|p| = c), and antisymmetric (free-momentum) regimes, each governed by different geometric constraints. Dispersion emerges as the global integral of an exact action one-form, while the (E,p) symplectic structure arises locally as the exterior derivative of the action's inexact form θ_E = E dp. Energy quantization appears as a topological stability condition requiring closed spectral loops to satisfy ∮ E dp = nh, defining "proto-eigenstates" whose Fourier encodings yield standard Hilbert-space eigenstates. We recover both the Bohr spectrum and Dirac fine-structure energies from spectral phase closure, revealing α = e²/(ℏc) as a geometric action-normalization ratio between Maxwell U(1) curvature and Lorentz spinor action—not a coupling parameter from QED perturbation theory. The framework synthesizes unitary flow and symplectic uncertainty into a single bilinear Lorentzian structure, showing quantum states as objective in their symmetric unitary existence but path-dependent in their antisymmetric observed manifestation. This geometric reconstruction bridges Dirac's wave mechanics and constraint formalism through native (E,p) phase space, offering a foundation for quantum theory that treats energy-momentum as primary coordinates rather than conjugate variables derived from spacetime position.
We derive the Dirac equation from three postulates concerning the internal complex structure of time: (P1) physical time comprises two orthogonal components, past-fixed (pf) and future-pending (fp); (P2) pf ⊥ fp; and (P3) |T|² = pf² + fp² = 1. These postulates entail that temporal evolution is a first-order phase rotation e^{iθ} (yielding the Schrödinger equation) and that spatial extension of the conservation law gives E² = m²c⁴ + c²p² (yielding the Minkowski metric). The first-order requirement and the causality constraint (both derived from the same postulates) jointly exclude the square-root Hamiltonian on grounds of non-locality, leaving matrix linearisation as the unique path. The resulting Clifford algebra {γ^μ, γ^ν} = 2g^{μν} encodes the Minkowski metric already derived from the postulates, completing a self-consistent chain from temporal ontology to the Dirac equation. The four-component spinor is interpreted as the direct product of the pf–fp doublet (2 components) and the two directions of phase rotation (particle/antiparticle), providing a physical origin for spin-1/2 and its 4π periodicity (Möbius structure). Charge conjugation C, parity P, and time reversal T are reformulated as phase-rotation reversal, spatial phase-gradient reversal, and phase-angle reversal, respectively; CPT invariance is proved as a direct consequence of |T|² = 1. Connections to prior quantum-circuit experiments — in which entanglement was found to be a necessary condition for the linear accumulation of measurement-induced irreversibility — are discussed in the context of the Dirac spinor’s inter-sector coupling. complex time; Dirac equation; Clifford algebra; spinor structure; CPT invariance; Minkowski metric; quantum–relativity unification; temporal ontology; measurement irreversibility
This paper presents a geometric reinterpretation of the Dirac equation within the Dimensional Evolution (DE) framework. Without modifying the mathematical structure or empirical predictions of relativistic quantum mechanics, the Dirac equation is reformulated as a constraint on relativistic identity formation rather than as a purely kinematic wave equation. In this interpretation, the Dirac spinor encodes internal resonance structure required for relativistic coherence, while gamma matrices act as directional collapse operators mediating between internal geometry and spacetime constraints. The intrinsic doubling of solutions is explained as a structural duality between verse-domain localization and inverse-domain harmonic support, providing a physical account of antimatter without violating CPT symmetry or energy positivity. Spin, Zitterbewegung, and relativistic measurement are reinterpreted as consequences of resonance geometry rather than unexplained algebraic features. Measurement and localization are treated as local geometric stabilization processes consistent with Lorentz covariance, extending the collapse-geometry account developed in DE-SP6 (Mott) to relativistic systems. This work is interpretive and foundational in scope. It preserves full compatibility with quantum field theory and established relativistic quantum mechanics while clarifying the physical meaning of the Dirac equation’s most enigmatic structures. --- Project website: https://de-theory.com This work is part of the Dimensional Evolution research program. Additional materials, updates, and contact information will be made available via the project website as the work expands.
We extend the framework of Unified Quantum Mechanics (UQM) to the relativistic domain by formulating the Dirac equation entirely in terms of real-valued fields. In this approach, the complex imaginary unit iii is systematically replaced by the temporal Hilbert transform operator Ht\mathcal{H}_{t}Ht​, which satisfies the anti-involution property Ht2=−I\mathcal{H}_{t}^{2}=-\mathcal{I}Ht2​=−I and acts as a dimensionless isometric operator. The resulting theory is expressed as a coupled system of real-valued equations for a primary field ψ\psiψ and its Hilbert-transformed partner ϕ=Ht[ψ]\phi=\mathcal{H}_{t}[\psi]ϕ=Ht​[ψ], whose evolution is governed exclusively by local differential operators. We demonstrate that all essential algebraic and symmetry structures of relativistic quantum mechanics are preserved within this real-valued formulation. In particular, the full Clifford algebra of the Dirac matrices, Lorentz covariance, and the relativistic energy-momentum relation are recovered exactly. A rigorous isomorphism with the conventional complex Dirac equation is established via the analytic signal representation ψa=ψ−iϕ\psi_{a}=\psi-i\phiψa​=ψ−iϕ, under which the standard U(1)U(1)U(1) gauge symmetry emerges naturally as a global SO(2)SO(2)SO(2) rotation in the real field space. The resulting formulation is therefore operationally and predictively equivalent to the standard complex theory. The conserved scalar density ρ=ψ†ψ+ϕ†ϕ=∣ψa∣2\rho=\psi^{\dagger}\psi+\phi^{\dagger}\phi=|\psi_{a}|^{2}ρ=ψ†ψ+ϕ†ϕ=∣ψa​∣2 is shown to arise as the squared envelope (intensity) of a real physical field and admits a direct interpretation as a local field intensity with well-defined physical dimensions. Probability conservation is recovered through the continuity equation, while normalization in standard quantum mechanics is reinterpreted as a rescaling by the conserved total field. Stability of matter, expressed as energy positivity, is shown to restrict the physical solution space to analytic signals, enforcing a Hilbert-pair structure and eliminating independent complex degrees of freedom. This work demonstrates that the complex structure of relativistic quantum mechanics is not fundamental but emerges from the analytically constrained dynamics of real-valued fields, providing a unified and physically transparent foundation for the Dirac theory.
Quantum computation has traditionally been formulated by postulating abstract information carriers and subsequently identifying physical systems that realize them. We adopt the opposite viewpoint and ask what computational structure is supplied by a fundamental relativistic quantum system itself. Focusing on the simplest nontrivial massive spin carrier, spin-1/2, we show that its relativistic description supplies a native four-dimensional information carrier through the Dirac equation. The resulting Dirac information carrier possesses an intrinsic positive- and negative-energy decomposition that induces a physics-constrained computational structure comprising sector-preserving and sector-coupling quantum logic. Restricting the relativistic computational structure to the positive-energy sector and taking the nonrelativistic regime recovers the familiar Pauli qubit description. We distinguish mathematical unitary transformations from physically admissible gates and show how charge-conjugation structure, charge superselection, and associated reference-frame resources can constrain quantum logic beyond the sector-preserving structure. Finally, under appropriate physical assumptions, we establish the conditions under which this structure supports full single-carrier controllability. Our work establishes the Dirac information carrier as the massive spin-1/2 instance of a broader programme in which computational structures are derived from the relativistic representation carried by the underlying physical system.