Recovering Unitarity of Lee Model in Bi-Orthogonal Basis

TL;DR

This paper introduces a bi-orthogonal basis approach to restore unitarity in the Lee model when the Hamiltonian becomes non-Hermitian due to strong coupling, resolving issues like negative norm and non-unitarity.

Contribution

It presents a systematic method using bi-orthogonal basis to define a non-trivial metric, restoring unitarity and Hermiticity in the Lee model beyond the critical coupling.

Findings

Successfully recovers unitarity in the Lee model

Resolves negative norm and probability issues

Provides a natural inner product with a non-trivial metric

Abstract

We study how to recover the unitarity of Lee model with the help of bi-orthogonal basis approach, when the physical coupling constant in renormalization exceeds its critical value, so that the Lee's Hamiltonian is non-Hermitian with respect to the conventional inner product. In a very natural fashion, our systematic approach based on bi-orthogonal basis leads to an elegant definition of inner product with a non-trivial metric, which can overcome all the previous problems in Lee model, such as non-Hermiticity of the Hamiltonian, the negative norm, the negative probability and the non-unitarity of the scattering matrix.