Emergence of quantum dynamics from chaos: The case of prequantum cat maps

Faure and Tsujii have recently proposed a novel quantization procedure, named natural quantization, for smooth symplectic Anosov diffeomorphisms. Their method starts with prequantization, which is also the first step of geometric quantization as proposed by Kostant-Souriau-Kirillov, and then relies on the Ruelle-Pollicott spectrum of the prequantum transfer operator, which they show to have a particular band structure. The appeal of this new quantization scheme resides in its naturalness: the quantum behavior appears dynamically in the classical correlation functions of the prequantum transfer operator. In this paper, we explicitly work out the case of cat maps on the $2n$-dimensional torus, showing in particular that the outcome is equivalent to that of the usual Weyl quantization. We also provide a concrete construction of all the prequantum cat maps.