Abstract
We use a vector parameter description of the Lorentz groups in ℝ2,1 and ℝ3,1 to obtain an exact expression for the Thomas factor as a geometric phase. The effect of phase accumulation in Thomas-Wigner precession phenomena is seen as a manifestation of the hyperbolic solid angle theorem. On the infinitesimal level, our description involves affine connections on the noncompact Hopf fibrations U(1) → SU(1, 1) → Δ and SU(2) → PSL(2,ℂ) → H 3. The associated gauge field is a restriction of the familiar Yang-Mills anti-instanton. We also consider the dual compact case, and we discuss generalizations to arbitrary dimensions and applications in various branches of theoretical physics.
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