Hamilton-Jacobi Treatment of The Scalar Field Coupled to Two Fermions

TL;DR

This paper applies the Hamilton-Jacobi method to analyze a scalar field coupled with two fermions via Yukawa interactions, deriving equations of motion and a path integral formulation without gauge fixing.

Contribution

It demonstrates the Hamilton-Jacobi approach as an effective alternative to Dirac's method for constrained systems with fermion-scalar couplings.

Findings

Equations of motion match Dirac's method results

Path integral obtained directly from phase space

No gauge fixing needed in the quantization process

Abstract

The constrained filed system, the scalar field coupled to two flavours of fermions through Yukawa couplings, is treated by using the Hamilton-Jacobi approach. The equations of motion are obtained as total differential equations in many variables. The equations of motion are in exact agreement with those equations obtained using Dirac's method. Due to the Hamilton-Jacobi quantization, the path integral of the scalar field coupled to two fermions is obtained directly as an integration over the canonical phase space without using any gauge fixing condition.