Contact geometry and quantization (Subhobrata Chatterjee)

QG seminar
Speaker: Subhobrata Chatterjee
Title: "Contact geometry and quantization"

Dynamics and measurement are two key facets of any physical system. For dynamics, we need a notion of time, while for measurement we typically need conjugate pairs of phase-space variables. Contact geometry on an odd-dimensional manifold is a natural setting to describe dynamics on a time-covariant phase-space. Importantly, a contact manifold is characterized by a maximally non-integrable hyperplane field, allowing both an orientation of time as well as encoding a canonical Hilbert bundle of symplectic spinors.

We quantize a contact manifold by constructing a bundle of Hilbert spaces on it together with a flat connection. The “quantum” connection turns Schrödinger equation into a parallel transport problem: wavefunctions are parallel sections of the Hilbert bundle along paths on the manifold. While formal existence of such connections is a classic result due to Fedosov, constructing bona fide quantum connections for a general contact manifold is a non-trivial problem. Highly symmetric manifolds ought to be optimal for exact results. We show that it is possible to go beyond formality for the standard contact seven-sphere by realizing it as a coset space of the quaternionic unitary group U(2,H) modulo the closed subgroup U(1,H). We observe that convergence of the formal connection is reminiscent of the Holstein-Primakoff shortening effect of spin representations: the Hilbert space L^2(R^3) truncates to finite-dimensional representations of the quaternionic unitary group U(2,H) when the formal parameter is specialized to certain discrete values.