Seven Sphere Quantization

Abstract

Co-oriented contact manifolds quite generally describe classical dynamical systems. Quantization is achieved by suitably associating a Schrödinger equation to every path in the contact manifold. We quantize the standard contact seven sphere by treating it as a homogeneous space of the quaternionic unitary group in order to construct a contact analog of Fedosov's formal connection on symplectic spinor bundles. We show that requiring convergence of the formal connection naturally filters the symplectic spinor bundle and yields an exact flat connection on each corresponding subbundle. A key ingredient is a generalization of the Holstein--Primakoff mechanism to the quaternionic unitary group. The passage from formal to bona fide quantization determines unitary irreducible representations of the quaternionic unitary group, whose dimensions tend to infinity as the formal deformation parameter approaches its classical limit. This appearance of finite-dimensional representations is not surprising since the contact seven sphere is closed and physically describes generalized positions, momenta and time variables.