Superluminality and a curious phenomenon in the relativistic quantum Hamilton-Jacobi equation

A basic problem in the relativistic quantum Hamilton-Jacobi theory is to understand whether it may admit superluminal solutions. Here we consider the averaging of the speed on a period of the oscillating term which is similar to Dirac's averaging of the oscillating part of the free electron's speed. Such an averaging solves the problem of getting the ℏ → 0 limit of the speed of the free particle, and leads to solutions that, depending on the integration constants, may be superluminal.