Suppressing the spurious states of the centre of mass

TL;DR

This paper proposes replacing the center of mass Hamiltonian with a new operator that commutes with all relevant constraints, effectively eliminating spurious states and uniformly shifting internal energy levels.

Contribution

It introduces a novel operator based on Dirac's quantization approach that suppresses spurious states in constrained systems.

Findings

The new Hamiltonian commutes with all algebra elements.

It is a multiple of the identity operator.

It shifts internal energy levels by a constant.

Abstract

Following Dirac's ideas concerning the quantization of constrained systems, we suggest to replace the free centre of mass Hamiltonian H_{CM} by another operator which commutes with all the elements of the algebra generated via the commutation relations by H_{CM} and the constraints which fix the centre of mass position. We show that the new Hamiltonian is a multiple of the identity operator and, as a result, its unique effect is to raise the internal energy levels by a constant amount.