Quantum mechanics for multivalued Hamiltonians

When the Lagrangian is not quadratic in the velocities, the situation may arise that the expression for the velocities in terms of the momenta is multivalued. As a consequence, the classical motion is unpredictable since at any time one can jump from one branch of the Hamiltonian to another. Yet, the quantum theory turns out to be perfectly smooth, with wave functions which are regular functions of time. We show that the path integral automatically picks up a unique combination of the branch Hamiltonians, which is a natural generalization of the Brouwer degree of the Legendre map.