Tetrads and their intrinsic spin structures

Abstract

In Minkowski spacetime an observer's moving frame may be associated with a tetrad: four orthogonal directions, one timelike and three spacelike. Locally, a physical measurement is always made in relation to some tetrad, and relativistic spin measurements in particular raise subtleties because spin and momentum cannot be cleanly separated under Lorentz boosts. This paper shows that every tetrad is endowed with an intrinsic spin structure. Each spatial direction may be generated from the timelike direction by a linear operator, and the requirement that these generate an orthogonal tetrad for every timelike direction is sufficient, on its own, to endow them with the Lie algebra $\mathfrak{so}(1,3)$, the Pauli algebra $\mathrm{Cl}_{3,0}(\mathbb{R})$ and the Lie algebra $\mathfrak{su}(2)$ simultaneously. These generators serve three roles at once: they build the spatial legs of every tetrad, they generate the Lorentz boosts between tetrads, and they endow each tetrad with an intrinsic spin structure. The eigenvectors of the spin generators form a null-ray tetrad which is simultaneously a Newman--Penrose-type frame and the set of Dirac rest spinors, while the invertibility of boosts between tetrads yields the Dirac factorisation of the mass-shell condition. Minimal coupling of gauge and spinor fields is forced rather than imposed, the potential being absorbed by the covariant derivative because the path dependence of the phase prevents it from being incorporated by the spinor phase. Wigner rotations, relativistic spin correlations, quantum reference frames and the Coulomb problem are examined within this tetrad framework. This framework is equivalent to the Hilbert-space formalism of relativistic quantum mechanics wherever the two have been compared, and so offers a geometric formulation of that formalism rather than an alternative to it.