General quantization of canonical maps on a two-torus

Canonical maps on a two-torus in phase space are quantized under most general conditions. Recent results by Keating et al (1999 Nonlinearity 12 579) are thus fully extended in two directions: (a) The translational component of a general canonical map is included in the quantization, (b) All values of Planck's constant, consistent with the toral boundary conditions (BCs), are considered; generically, these values are rational numbers whose numerator must satisfy a number-theoretical condition. Besides the condition on Planck's constant, the quantization is possible only for particular, 'allowed' BCs on the torus. The general equation determining these BCs is derived. Allowed BCs may not exist in some cases; representative examples are the irrational skew translations and Kronecker maps. Exact versions of Egorov's theorem are shown to hold under some conditions. Composition and representation properties of the quantization scheme are studied.