A note on the Hamiltonian of the real scalar field

TL;DR

This paper examines the ambiguity in defining the Hamiltonian for a real scalar field, emphasizing the importance of consistency with energy density and equations of motion, and proposes a refined Hamiltonian that aligns with non-relativistic limits.

Contribution

The authors identify the need for adding surface terms to the standard Hamiltonian to ensure consistency with energy density and equations of motion, providing a more accurate formulation for the scalar field.

Findings

Standard Hamiltonian may not reproduce equations of motion correctly.

Modified Hamiltonian aligns with non-relativistic energy density.

Explicit example with kink solution illustrates the improved formulation.

Abstract

We address the question of ambiguity in defining a Hamiltonian for a scalar field. We point out that the Hamiltonian for a real Klein-Gordon scalar field must be consistent with the energy density obtained from the Schrodinger equation in the non-relativistic regime. To achieve this we had to add some surface terms (total divergencies) to the standard Hamiltonian, which in general will not change the equations of motion, but will redefine energy. As an additional requirement, a Hamiltonian must be able to reproduce the equations of motion directly from Hamilton's equations defined by the principle of the least action. We find that the standard Hamiltonian does not always do so and that the proposed Hamiltonian provides a good non-relativistic limit. This is a hint that something as simple as the Hamiltonian of the real Klein-Gordon scalar field has to be treated carefully. We had illustrated our discussion with an explicit example of the kink solution.