Covariant guiding laws for fields

After reviewing the passage from 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 our proposed 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 ofDeDonder–Weyl–Christodoulou. We propose a covariant guiding law for the temporal evolution of fields defined on constant time slices of spacetime, and show that it yields, at each spacetime point, the existence of a finite measure on the space of field values at that point that is equivariant with respect to the flow induced by the solution of Kanatchikov’s equation that is guiding the actual field, so long as it is a plane-wave solution. We show that our guiding law is local in the sense of Einstein’s special relativity, and therefore it cannot be used to analyze Bell-type experiments. We conclude by suggesting directions to be explored in future research.

Keywords

Field quantization, DeDonder-Weyl, Hamilton-Jacobi equations, action principle, Lorentz covariance

2010 Mathematics Subject Classification

35Q41, 70H20, 70S05, 81Q65, 81S99, 81T99

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Dedicated to Demetrios Christodoulou on the occasion of his 70th birthday

Received 18 October 2021

Received revised 2 February 2023

Accepted 12 February 2023

Published 18 July 2024