Journal of Modern Dynamics (JMD)

Prequantum chaos: Resonances of the prequantum cat map

Pages: 255 - 285, Issue 2, April 2007      doi:10.3934/jmd.2007.1.255

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Frédéric Faure - Institut Fourier 100, rue des Maths, BP 74, 38402 St Martin d'Heres, France (email)

Abstract: Prequantum dynamics was introduced in the 70s by Kostant, Souriau and Kirillov as an intermediate between classical and quantum dynamics. In common with the classical dynamics, prequantum dynamics transports functions on phase space, but adds 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 $(2,\mathbb{Z})$ which generates a chaotic dynamical system on the torus. The dynamics is lifted to a prequantum fiber bundle. This gives a unitary prequantum (partially hyperbolic) map. We calculate its resonances and show that they are related to the quantum eigenvalues. A remarkable consequence is that quantum dynamics emerges from long-term behavior of prequantum dynamics. We present trace formulas, and discuss perspectives of this approach in the nonlinear case.

Keywords:  Prequantum dynamics, Quantum chaos, Linear hyperbolic map, Transfer operator, Resonances, Decay of correlations, Line bundle.
Mathematics Subject Classification:  Primary: 81Q50, 81S10; Secondary: 37D20, 37C30.

Received: June 2006;      Revised: December 2006;      Available Online: January 2007.