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Geometric Origin of Complex Wavefunctions: From the Spacetime Algebra Cl(1,3) to the Schrödinger Equation

Marek Hubka

Published Mar 23, 2026• 0 citations
Publisher

Abstract

The Schrödinger equation need not be postulated. This paper demonstrates that the imaginary unit $i$ in quantum mechanics and the Schrödinger equation itself are both derived consequences of the Dirac equation when formulated in the real Clifford algebra $\mathrm{Cl}(1,3)$ - the Spacetime Algebra introduced by Hestenes (1966). No new physics is proposed; the result collects and organises existing mathematical results into a single self-contained derivation chain: $$\mathrm{Cl}(1,3) \;\to\; \text{Hestenes-Dirac} \;\to\; \gamma_0\text{-split} \;\to\; \text{Pauli} \;\to\; \text{Schroedinger}$$ The role of $i$ is played by the spin-plane bivector $\gamma_{21} = \gamma_2\gamma_1$, a real geometric object satisfying $\gamma_{21}^{\;2} = -1$ as a consequence of the Minkowski metric. Complex-valued wavefunctions are thus projection artifacts: the residue of real four-dimensional geometry viewed from a three-dimensional observer frame. Key Results - The imaginary unit $i$ in the Schrödinger equation is identified as the spin-plane bivector $\gamma_{21}$ - a directed plane in spacetime, not an abstract algebraic postulate.- The Schrödinger equation is derived as the non-relativistic limit of the Hestenes--Dirac equation via the $\gamma_0$-split (observer frame choice), with $i$ inherited from the spacetime bivector structure.- Two of the five standard axioms of quantum mechanics (complex Hilbert space and the Schrödinger equation) are demoted from postulates to derived theorems. The measurement postulate and tensor product rule are retained.- Full compatibility with the Renou et al. (2021) no-go theorem and the Chen et al. (2022) experimental confirmation: the formulation is isomorphic to complex quantum mechanics, not a real restriction of it. $\mathrm{Cl}(1,3)$ contains the required complex structure via $\gamma_{21}$.- Convergence with Penrose's twistor programme: both approaches conclude that complex structure in quantum mechanics and Lorentzian spacetime geometry have a common origin. Document Contents - complex_wavefunctions.pdf - Main paper (13 pages). Self-contained derivation from $\mathrm{Cl}(1,3)$ to Schrödinger, with compatibility analysis (Renou/Chen/Penrose), axiom restructuring, and pedagogical appendix.- complex_wavefunctions.tex - LaTeX source. Standalone, no external dependencies. Scope and Originality This paper presents no new mathematical results. The identification $i \leftrightarrow \gamma_{21}$ is due to Hestenes (1967); the non-relativistic reduction in geometric algebra is given in Doran & Lasenby (2003). The contribution is expository: collecting the derivation chain into a single path with the geometric identity of $i$ tracked at each step, and situating the result in the context of the Renou et al. no-go theorem and its experimental confirmation.

DOI
10.5281/zenodo.19162378

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Complex numbers are the mathematical foundation of quantum mechanics, yet a fundamental question has remained unasked: where does the imaginary unit i come from? The standard formulation treats it as a primitive mathematical object without tracing its physical or geometric origin. This paper proves that i is not a mysterious attribute peculiar to quantum mechanics, but an algebraic necessity of spatial geometry. Starting from the Oh point group of a simple cubic lattice, the rigorous construction of the generators of the antisymmetric T1g subspace demonstrates that the commutation relation [J^a, J^b] = iε^{abc}J^c necessarily contains i, which is the algebraic signature of the su(2) structure constants arising from the non-commutativity of rotations about the three spatial axes. Furthermore, within the Clifford algebra Cl(3,0), the square of a bivector equals -1, providing a direct geometric counterpart for i. Recent experimental tests have confirmed the physical reality of complex numbers in quantum theory with extremely high confidence—this paper establishes the geometric necessity of i: it is not an introduced mathematical tool, but an intrinsic feature of the algebraic structure of any vector degree of freedom in three-dimensional space.
Standard quantum mechanics relies fundamentally on complex Hilbert spaces, utilizing the imaginary unit i to govern phase evolution. While empirically flawless, this algebraic abstraction obscures the potential underlying physical geometry of quantum states. Building upon the Spacetime Algebra formalism established by Hestenes, this paper presents a fully real-geometric holographic formulation of non-relativistic quantum mechanics. We map the state evolution from a unitary, higher-dimensional Clifford algebra bulk (Cℓ6,1) to a physical 3D observable boundary (Cℓ3,1) via a specific Holographic Projection Operator (Pholo). By replacing the imaginary unit with a geometric bivector rotor and treating fundamental states as 1D topological spinors, we successfully recover the time-evolution of the Schrödinger equation. We subsequently apply this formalism to two standard models: the 1D Infinite Square Well and the Quantum Harmonic Oscillator. In both cases, the geometric formulation exactly recovers the rigorously validated quantized energy spectra while offering a strictly topological, deterministic mechanism for ladder operators and baseline zero-point energy.
While the complex formalism of quantum mechanics has been extraordinarily successful, the fundamental geometric origin of its core elements—the imaginary unit, operator non-commutativity, quantum phase, and the intrinsic geometry with quantum spin—has remained unresolved for more than a century. Here we resolve these foundational questions by deriving an exact and physically equivalent formulation of the Schrödinger equation as a Real Geometric Flow—a moving-frame evolution on a 2𝑁-dimensional real state space. In this restored framework, the global imaginary unit 𝑖 fully decomposes into a set of local skew-symmetric generators that drive planar rotations in real space, revealing that the traditional complex formalism is a compressed representation of an intrinsic rotational geometry.Within the real geometric flow, energy eigenvalues appear as the Curvatures of the state trajectory, energy gaps correspond to Torsions that couple internal planes—naturally generating the geometry of quantum spin—and quantum phase emerges naturally as the Holonomy of the real moving frame. This formulation preserves the full empirical content of quantum mechanics while exposing the geometric mechanisms hidden beneath its notation, providing the first coherent explanation of curvature, torsion, non-commutativity, spin behavior, and quantum phase within a single framework.By demonstrating that the algebraic postulates of quantum mechanics arise from a deeper geometric order—one that stands alongside curvature-based theories such as General Relativity—this work establishes a unified differential-geometric foundation for quantum dynamics.It opens new directions for quantum control, quantum information, and fundamental physics.
The complex field structure of quantum mechanics is traditionally treated as an irreducible postulate. We derive this structure from the physical requirement of the stability of matter. We show that enforcing a lower-bounded energy spectrum (E≥0)(E\ge0)(E≥0) constrains the temporal dependence of physical states to the positive-frequency sector of tempered distributions. The resulting generalized solution space of the Schrödinger equation is given by the completed tensor product Pgen≃S+′(Rt) ⊗^ L2(Rx3),\mathcal P_{\mathrm{gen}} \simeq \mathcal S'_+(\mathbb R_t)\,\widehat\otimes\,L^2(\mathbb R^3_{\mathbf x}),Pgen​≃S+′​(Rt​)⊗​L2(Rx3​), with the Hardy space H+2(Rt)H^2_+(\mathbb R_t)H+2​(Rt​) emerging as the Hardy-class subspace of finite analytic norm. This topological factorization separates temporal analyticity-which enforces causality and energy positivity-from spatial probabilistic localization. We prove a unified no-go theorem demonstrating that treating the real and imaginary components of quantum wavefunctions as independent degrees of freedom leads to an energy spectrum unbounded from below and is therefore incompatible with the stability of matter. As a consequence, the imaginary component of the wavefunction is not an independent degree of freedom but is uniquely determined by the real component via the temporal Hilbert transform, ψI=Ht[ψR]\psi_I = \mathcal{H}_t[\psi_R]ψI​=Ht​[ψR​]. The formulation thus admits a description in terms of strictly real-valued fields, with the complex unit arising as an operator representation of time-domain analyticity.
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 proves that quantum mechanics emerges as a mathematical necessity when a deterministic classical system possesses compact Kähler dimensions inaccessible to observation. The proof rests on three independent results from established mathematics: Spectral discreteness: The spectral theorem for elliptic operators on compact manifolds guarantees a discrete eigenspectrum, producing superposition and complex phase structure. The uncertainty principle: The Gromov non-squeezing theorem establishes an irreducible lower bound on observable uncertainties, with Planck's constant identified as the symplectic capacity of the compact space. The Born rule: The unique Kähler-compatible L² inner product determines the probability rule P = |ψ|² with no alternatives. Together, these results demonstrate that the complete axiomatic structure of quantum mechanics — complex Hilbert space, superposition, interference, the uncertainty principle, the Born rule, and entanglement — is a necessary consequence of dimensional projection rather than a set of independent postulates. The paper further shows that: The complex numbers in quantum mechanics originate from the complex structure (J) of the compact manifold, providing a geometric explanation for recent experiments ruling out real-valued quantum mechanics (Renou et al., Nature 2021; Chen et al., PRL 2022). Unitary time evolution arises because the complex structure converts what would otherwise be dissipative dynamics into probability-preserving oscillation. The number field of quantum mechanics (real, complex, or quaternionic) is determined by the topology of the compact dimensions (Riemannian, Kähler, or hyperkähler respectively), making laboratory quantum experiments probes of extra-dimensional geometry. CPT symmetry corresponds to orientation-invariance of the compact manifold, connecting discrete symmetry tests to the topology of the extra dimensions. The theorem requires no physical assumptions beyond the existence of inaccessible compact Kähler dimensions. It applies to any number of dimensions, any compact Kähler geometry, and any deterministic dynamics. The paper concludes that quantum mechanics is not a law of nature but a law of perspective — the second such law in physics, following relativity. Where relativity revealed that the separation of spacetime into space and time is observer-dependent, this work shows that the separation of dynamics into deterministic and probabilistic is equally observer-dependent. Originally submitted to Foundations of Physics on March 16, 2026