The Complex Structure of Quantum Mechanics from a Real Spinor in Signature (3,3)

Abstract

Quantum mechanics is written over the complex numbers, and the question why is rarely pressed. That a canonical answer exists is already known. Moretti and Oppio prove that a relativistic system in a real Hilbert space with non-negative squared mass admits a Poincaré-invariant complex structure, unique up to sign, commuting with the whole algebra of observables. What no result supplies is an identification: their theorem establishes that the object exists and does not exhibit it.

This paper exhibits it, and their uniqueness is what turns an exhibition into an identification. Let a real spinor of Cl(3,3) be viewed from a four-dimensional Lorentzian subspace. The complement is a plane, and its bivector is a real operator squaring to −1 that commutes with the entire Lorentzian Clifford action. Under it the real eight-component spinor becomes the complex four-component Dirac spinor, with the Dirac algebra recovered exactly. Centrality follows from orthogonality in two lines rather than by postulate.

One property does all the work: definiteness of the complement. It supplies the complex structure; it fixes the complement at two dimensions, across all signatures and all total dimensions; and it forces the four-dimensional shadow to be Lorentzian. The equality of the space and time counts is an output rather than an assumption: a Lorentzian shadow with a definite complement leaves two temporal directions over, giving three of each.

What the paper claims

The paper proposes that the imaginary unit of quantum mechanics is the bivector of the two directions a frame cannot show, and that quantum mechanics is complex because any frame shows only four of the six dimensions. This is offered as an explanation. It is not an empirical test, and the paper explains why an argument of this kind cannot be one. It does not change the formalism of quantum mechanics or any of its predictions.

Verification

Every algebraic claim in the paper is checked by constructing the real matrices explicitly in numpy, and the code is included with the manuscript so that anyone can re-run it. These checks catch sign errors, wrong dimensions and algebraic slips. They do not check whether the argument itself is sound, since the scripts follow the same reading of it, and no one outside the project has verified it yet. The appendix states this.

Status

Manuscript, September 2026, not yet submitted. A companion paper, What the Complex Structure Forbids, calculates three restrictions that follow from the same geometry. Expert review is being sought before submission; a preprint link will appear here once it is posted.