Covariant Guiding Laws for Fields

After reviewing what is known about the passage from the classical Hamilton–Jacobi formulation of non-relativistic point-particle dynamics to the non-relativistic quantum dynamics of point particles whose motion is guided by a wave function that satisfies Schrödinger’s or Pauli’s equation, we study the analogous question for the Lorentz-covariant dynamics of fields on spacelike slices of spacetime. We establish a relationship, between the DeDonder–Weyl–Christodoulou formulation of covariant Hamilton–Jacobi equations for the classical field evolution, and the Lorentz-covariant Dirac-type wave equation proposed by Kanatchikov amended by a corresponding guiding equation for such fields. We show that Kanatchikov’s equation is well-posed and generally solvable, and we establish the correspondence between plane-wave solutions of Kanatchikov’s equation and solutions of the covariant Hamilton–Jacobi equations of DeDonder– Weyl–Christodoulou. Our proposed covariant guiding law for fields on spacelike slices yields the existence of an equivariant probability density for the resulting flow on generic field space. We show that it is local in the sense of Einstein’s special relativity, and we conclude by suggesting directions to be explored in future research.