The S-matrix of the Faddeev-Reshetikhin Model, Diagonalizability and PT Symmetry

TL;DR

This paper investigates the diagonalizability of the Hamiltonian in the Faddeev-Reshetikhin model, revealing non-diagonalizability issues due to Lorentz invariance violation, and introduces PT-symmetric Hamiltonians that preserve the S-matrix's unitarity and reality.

Contribution

It identifies the non-diagonalizability problem in the FR model's Hamiltonian and proposes PT-symmetric, non-Hermitian Hamiltonians that maintain the same S-matrix and are diagonalizable.

Findings

The FR model's Hamiltonian is not diagonalizable in the two-particle sector.

Certain PT-symmetric, non-Hermitian Hamiltonians can be diagonalized and share the same S-matrix.

Wave functions in these models are PT symmetric, ensuring real spectra and unitarity.

Abstract

We study the question of diagonalizability of the Hamiltonian for the Faddeev-Reshetikhin (FR) model in the two particle sector. Although the two particle S-matrix element for the FR model, which may be relevant for the quantization of strings on $AdS_{5}\times S^{5}$, has been calculated recently using field theoretic methods, we find that the Hamiltonian for the system in this sector is not diagonalizable. We trace the difficulty to the fact that the interaction term in the Hamiltonian violating Lorentz invariance leads to discontinuity conditions (matching conditions) that cannot be satisfied. We determine the most general quartic interaction Hamiltonian that can be diagonalized. This includes the bosonic Thirring model as well as the bosonic chiral Gross-Neveu model which we find share the same S-matrix. We explain this by showing, through a Fierz transformation, that these two models are in fact equivalent. In addition, we find a general quartic interaction Hamiltonian, violating Lorentz invariance, that can be diagonalized with the same two particle S-matrix element as calculated by Klose and Zarembo for the FR model. This family of generalized interaction Hamiltonians is not Hermitian, but is $PT$ symmetric. We show that the wave functions for this system are also $PT$ symmetric. Thus, the theory is in a $PT$ unbroken phase which guarantees the reality of the energy spectrum as well as the unitarity of the S-matrix.