The Arnol'd cat: failure of the correspondence principle

The classical Hamiltonian H=p2/2m+ε(q2/2)Σδ[s-(t/T)] has an integrable mapping of the plane, [qn+1, pn+1]=[qn+pn, qn+2pn], as its equations of motion. But then by introducing periodic boundary conditions via (mod 1) applied to both q and p variables, the equations of motion become the Arnol'd cat map, [qn+1, pn+1]=[qn+pn, qn+2pn], (mod 1), revealing it to be one of the simplest fully chaotic systems which can be derived from a Hamiltonian and analyzed. Consequently, we here quantize the Arnol'd cat and examine its quantum motion for signs of chaos using algorithmic complexity as the litmus. Our analysis reveals that the quantum cat is not chaotic in the deep quantum domain nor does it become chaotic in the classical limit as required by the correspondence principle. We therefore conclude that the correspondence principle, as defined herein, fails for the quantum Arnol'd cat. (orig.)