Quantization of linear maps on a torus-fresnel diffraction by a periodic grating

Quantization on a phase space q, p in the form of a torus (or periodized plane) with dimensions Δq, Δp requires the Planck's constant take one of the values h = ΔqΔp/N, where N is an integer. Corresponding to a linear classical map T of points q, p is a unitary operator U mapping quantum states that are periodic in q and p; the construction of U involves techniques from number theory. U has eigenvalues exp(iα). The ‘eigenangles’ α must be multiples of $\text{2π}\text{n(N)}$, where n(N) is the lowest common multiple of the lengths of the classical ‘cycles’ mapped under T by those rational points in q, p which are multiples of $\text{Δq}\text{N}$ and $\text{Δp}\text{N}$ (i.e. n(N) is the ‘period of T mod N′), at least for odd N. If T is hyperbolic, n is a very erratic function of N, and the classical limit N → ∞ is very different from the ‘Bohr-Sommerfeld’ behaviour for parabolic maps. The degeneracy structure of the eigenangle spectrum is related to the distribution of cycle lengths. Computation of the quantal Wigner function shows that eigenstates of U do not correspond to individual cycles.