Converting Classical Theories to Quantum Theories by Solutions of the Hamilton-Jacobi Equation

TL;DR

This paper proposes a novel method to transform classical theories into quantum theories using solutions of the Hamilton-Jacobi equation and lattice theory tools, successfully applied to gauge and fermion fields.

Contribution

It introduces a new approach for quantization of classical theories by leveraging Hamilton-Jacobi solutions and lattice techniques, providing explicit results for gauge and fermion fields.

Findings

Derived a first order Lagrangian with mass for a topologically massive gauge theory.

Produced a massive fermion term by adding a surface term to the Lagrangian.

Demonstrated the feasibility of the approach for gauge and fermion fields.

Abstract

By employing special solutions of the Hamilton-Jacobi equation and tools from lattice theories, we suggest an approach to convert classical theories to quantum theories for mechanics and field theories. Some nontrivial results are obtained for a gauge field and a fermion field. For a topologically massive gauge theory, we can obtain a first order Lagrangian with mass term. For the fermion field, in order to make our approach feasible, we supplement the conventional Lagrangian with a surface term. This surface term can also produce the massive term for the fermion.