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

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.