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We study the question of diagonalizability of the Hamiltonian for theFaddeev-Reshetikhin (FR) model in the two particle sector. Although the twoparticle S-matrix element for the FR model, which may be relevant for thequantization of strings on $AdS_{5}\times S^{5}$, has been calculated recentlyusing field theoretic methods, we find that the Hamiltonian for the system inthis sector is not diagonalizable. We trace the difficulty to the fact that theinteraction term in the Hamiltonian violating Lorentz invariance leads todiscontinuity conditions (matching conditions) that cannot be satisfied. Wedetermine the most general quartic interaction Hamiltonian that can bediagonalized. This includes the bosonic Thirring model as well as the bosonicchiral Gross-Neveu model which we find share the same S-matrix. We explain thisby showing, through a Fierz transformation, that these two models are in factequivalent. In addition, we find a general quartic interaction Hamiltonian,violating Lorentz invariance, that can be diagonalized with the same twoparticle 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 thereality of the energy spectrum as well as the unitarity of the S-matrix.

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