Classical and quantum complex Hamiltonian curl forces

Source abstract

Abstract A class of Newtonian forces, determining the acceleration F ( x , y ) of particles in the plane, is F =(Re F ( z ), Im F ( z )), where z is the complex variable x + i y . Curl F is non-zero, so these forces are nonconservative. These complex curl forces correspond to completely integrable Hamiltonians that are anisotropic in the momenta, separable in z and z * but not in x and y if the curl is nonzero. The Hamiltonians can be quantised, leading to unfamiliar wavefunctions, even for the (non-curl) isotropic harmonic oscillator. The formalism provides an alternative interpretation of the analytic continuation of one-dimensional real Hamiltonian particle dynamics, where trajectories are known to exhibit intricate structure (though not chaos), and is a Hermitian alternative to non-Hermitian quantisation.