Quantum cat maps with spin 1/2

We derive a semiclassical trace formula for quantized chaotic transformatio ns of the torus coupled to a 2-spinor precessing in a magnetic field. The t race formula is applied to semiclassical correlation densities of the quant um map, which, according to the conjecture of Bohigas, Giannoni and Schmit, are expected to converge to those of the circular symplectic ensemble (CSE ) of random matrices. In particular, we show that the diagonal approximatio n of the,spectral form factor for small arguments agrees with the CSE predi ction. The results are confirmed by numerical investigations.