Cut-Off Kinks

TL;DR

This paper investigates how regularization by energy cutoff affects the one-kink sector Hamiltonian in quantum field theory, revealing it is not regularized by energy cutoff but constrained by mode coefficients, ensuring correct kink mass calculation.

Contribution

It demonstrates that the one-kink sector Hamiltonian's regularization differs from the vacuum sector, with mode coefficients constrained, and establishes conditions for unitary equivalence to ensure correct physical results.

Findings

Normal modes are present at all energies in the kink Hamiltonian.

Mode coefficients are constrained to become small above the cutoff.

Regularization by energy cutoff does not directly regularize the kink Hamiltonian.

Abstract

We answer the question: If a vacuum sector Hamiltonian is regularized by an energy cutoff, how is the one-kink sector Hamiltonian regularized? We find that it is not regularized by an energy cutoff, indeed normal modes of all energies are present in the kink Hamiltonian, but rather the decomposition of the field into normal mode operators yields coefficients which lie on a constrained surface that forces them to become small for energies above the cutoff. This explains the old observation that an energy cutoff of the kink Hamiltonian leads to an incorrect one-loop kink mass. To arrive at our conclusion, we impose that the regularized kink sector Hamiltonian is unitarily equivalent to the regularized vacuum sector Hamiltonian. This condition implies that the two regularized Hamiltonians have the same spectrum and so guarantees that the kink Hamiltonian yields the correct kink mass.