The covering S²↓ℝℙ² as the locus of quantum mechanics: A rereading, and four calculations

Quantum theory reads several of its quantities on the loops of configuration space rather than at its points: the inequivalent quantizations of a system are classified by the characters of the fundamental group (Schulman 1968; Laidlaw and DeWitt 1971), and the phase factors on loops give a complete description of a gauge field (Wu and Yang 1975). We propose a framework in which reading on loops is the rule. It has three elements: the site ℝℙ², realized in Euclidean space by the cross-cap; its covering S²↓ℝℙ², the only connected non-trivial covering of the site, hence its universal covering; and the reading, with values in a sheaf on the site. The obstruction of the site is the class a = w₁(ℝℙ²). Four known calculations are reread in this framework, and each reads the same class a: the Schrödinger equation on ℝℙ² and on SO(3) (sectors split by the parity of ℓ, or of 2j); the Dirac equation on ℝℙ² (a single type of Pin structure, in two copies, and an asymmetric spectrum); the two degrees of the reading (the holonomy −1 and the torsion class of H²(ℝℙ²; ℤ) are one element); and the 4π periodicity of spin ½ (the monodromy of SU(2)↓SO(3)). The thesis (Th) is that this framework offers a common perspective on scattered approaches. Numerical checks are supplied as scripts. This record contains the same paper in two versions: English (primary) and French (original). The French text prevails in case of discrepancy. — Ce dépôt contient le même texte en deux versions : anglaise (principale) et française (originale) ; le texte français fait foi.