Projective Geometry as the Mother Tongue of Quantum Mechanics: From Brody & Hughston's $\mathbb{CP}^n$ to the Fano Orbit Decomposition

Abstract

PENDING QUALITY AUDIT (2026-08). The file is restricted while this record is reviewed as part of a systematic audit of the author's corpus. It has not yet been assessed. Metadata and DOI remain public, and access can be requested. Two programmes have placed projective geometry at the centre of quantum mechanics, arriving from opposite directions. The Brody–Hughston geometric quantum mechanics programme identifies the space of pure states with the complex projective space $\mathbb{CP}^n$, equipped with the Fubini-Study symplectic form. Our companion programme identifies the 3-qubit discrete phase space with the finite projective space $PG(5,2)$ over $\mathrm{GF}(2)$, equipped with a binary symplectic form. We show that both are specialisations of a single abstract structure: a projective space over a field $\mathbb{F}$ carrying a non-degenerate alternating bilinear form and a central extension of its symplectic group. The central result concerns the Wigner functor $W$, sending each quantum system to its Wigner function. Numerical experiment shows that $W$ is covariant under the real symplectic subgroup of the Clifford group (CNOT-type gates) but fails for phase gates (CS, S). The obstruction is the metaplectic character $\chi: \mathrm{Sp}(2n, \mathrm{GF}(2)) \to U(1)$ satisfying a group 2-cocycle condition. The precise unification is: Brody–Hughston's prequantum $U(1)$ line bundle on $\mathbb{CP}^n$ (the Maslov bundle, required for geometric quantisation) and the metaplectic character $\chi$ on $PG(2n-1,2)$ (required for discrete Wigner covariance) are the same construction over different base fields — a central extension of the symplectic group by a circle. Three additional connections are established. (i) The Fano orbit decomposition of $PG(5,2)$ — seven orbits of eight points, one per Fano line — is a purely discrete phenomenon with no continuous analogue in $\mathbb{CP}^n$; magic valence is intrinsically a $\mathrm{GF}(2)$ structure invisible to characteristic-zero geometry. (ii) The 6j symbols governing both programmes — $SU(2)$ spin networks in the Brody–Hughston setting, $\mathbb{Z}_2^3$ Fano symbols in ours — are evaluations of the same abstract Racah coefficient for different groups; the Origami ISA is the spin-$\frac{1}{2}$ discrete limit of Penrose's spin-network geometry; Racah (1942), Mac Lane (1963), Penrose (1971), and the present work all evaluate the same Pentagon identity for different groups. (iii) The finite Klein correspondence ($PG(5,2)$ points $\leftrightarrow$ $PG(3,2)$ lines) suggests a discrete twistor interpretation of the magic-valence structure, directly connecting to Hughston's twistor programme. The unifying category $\mathbf{MetaProjSymp}(\mathbb{F})$ is defined, the Wigner functor is shown to be natural on it, and the bridge theorem is proved: Kibble's geometrisation of quantum mechanics, the Brody–Hughston programme, and the discrete Fano programme are all full subcategories of the same abstract construction, distinguished only by the choice of field.