Prequantum chaos: Resonances of the prequantum cat map

Abstract

Prequantum dynamics has been introduced in the 70' by Kostant-Souriau-Kirillov as an intermediate between classical and quantum dynamics. In common with the classical dynamics, prequantum dynamics transports functions on phase space, but add some phases which are important in quantum interference effects. In the case of hyperbolic dynamical systems, it is believed that the study of the prequantum dynamics will give a better understanding of the quantum interference effects for large time, and of their statistical properties. We consider a linear hyperbolic map M in SL(2,Z) which generates a chaotic dynamics on the torus. This dynamics is lifted on a prequantum fiber bundle. This gives a unitary prequantum (partially hyperbolic) map. We calculate its resonances and show that they are related with the quantum eigenvalues. A remarkable consequence is that quantum dynamics emerges from long time behaviour of prequantum dynamics. We present trace formulas, and discuss perspectives of this approach in the non linear case.