[Paper Review] Recovering Unitarity of Lee Model in Bi-Orthogonal Basis

Tao Shi, C. P. Sun|ArXiv.org|May 12, 2009

Quantum Mechanics and Non-Hermitian Physics2 references4 citations

TL;DR

This paper proposes a bi-orthogonal basis approach to restore unitarity in the Lee model when the coupling constant exceeds its critical value, rendering the Hamiltonian non-Hermitian under the conventional inner product. By introducing a non-trivial metric operator η, the method defines a new inner product that ensures the Hamiltonian becomes Hermitian, all states have positive norms, and the S-matrix is unitary, thereby resolving issues like ghost states and negative probabilities in the strong-coupling regime.

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.

Motivation & Objective

• To address the breakdown of unitarity in the Lee model when the physical coupling constant exceeds its critical value.
• To resolve the emergence of ghost states with negative norms and non-unitary scattering matrices in the strong-coupling regime.
• To construct a consistent quantum theory with a non-Hermitian Hamiltonian by defining a new inner product via a bi-orthogonal basis.
• To ensure the Hamiltonian becomes Hermitian and the S-matrix unitary under the new inner product, even when the coupling is strong.
• To provide a physically acceptable framework for the Lee model applicable to realistic systems like coupled resonator arrays.

Proposed method

• Utilizes a bi-orthogonal basis formed by two complete sets of eigenstates: {|Eₙ⟩} for the Hamiltonian H and {|Dₙ⟩} for its adjoint H†.
• Defines a non-trivial metric operator η via |Dₙ⟩ = η|Eₙ⟩, which induces a new inner product (φ, ψ) = ⟨φ|η|ψ⟩.
• Explicitly calculates the metric operator η for both the quantum mechanical (one-boson-mode) and quantum field theory versions of the Lee model.
• Demonstrates that under the new inner product, the Hamiltonian becomes Hermitian and all eigenstates have positive norms.
• Shows that the S-matrix is unitary in this new Hilbert space, restoring physical consistency.
• Establishes that the new metric ensures orthogonality of different eigenstates and avoids artificial pole removal.

Experimental results

Research questions

• RQ1 How can unitarity be restored in the Lee model when the coupling constant exceeds the critical value, leading to a non-Hermitian Hamiltonian?
• RQ2 What is the role of a non-trivial metric in defining a physically consistent inner product for non-Hermitian Hamiltonians?
• RQ3 Can the bi-orthogonal basis approach systematically resolve issues like ghost states, negative norms, and non-unitary S-matrices in the Lee model?
• RQ4 How does the new inner product ensure that the Hamiltonian is Hermitian and the evolution unitary?
• RQ5 Is the metric operator derived via this method distinct from those in previous approaches, such as the CPT-based metric?

Key findings

• The bi-orthogonal basis approach successfully defines a non-trivial metric operator η that restores Hermiticity of the Lee model Hamiltonian in the strong-coupling regime (g > g_c).
• All eigenstates, including the physical V-state and scattering states, acquire positive norms under the new inner product, eliminating the ghost state problem.
• The S-matrix becomes unitary in the Hilbert space equipped with the new inner product, resolving the issue of non-unitary scattering.
• The metric operator η is explicitly calculated for both the QM and QFT versions of the Lee model, providing a concrete and systematic framework.
• The new metric ensures orthogonality of different eigenstates and is distinct from the CPT-based metric used in previous works.
• The method allows the Lee model to be physically acceptable and applicable to realistic systems, such as coupled resonator arrays, even for arbitrary coupling strength.