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

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 $Ad{S}_5\times S^5$, 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.

arXiv comment32 pages, 1 figure; references added, version published in JHEP

References in corpus (42)

• Making Sense of Non-Hermitian Hamiltonians
• Complex Extension of Quantum Mechanics
• PT-Symmetric Quantum Mechanics
• Hidden Symmetries of the AdS_5 x S^5 Superstring
• Type IIB superstring action in AdS_5 x S^5 background
• Bethe Ansatz for Quantum Strings
• Introduction to PT-Symmetric Quantum Theory
• Classical/quantum integrability in AdS/CFT
• The Factorized S-Matrix of CFT/AdS
• The AdS_5xS^5 superstring worldsheet S-matrix and crossing symmetry
• The Dilatation Operator of N=4 Super Yang-Mills Theory and Integrability
• The su(2|3) Dynamic Spin Chain
• The Algebraic Curve of Classical Superstrings on AdS_5xS^5
• Supersymmetry and the spontaneous breakdown of PT symmetry
• The Zamolodchikov-Faddeev Algebra for AdS_5 x S^5 Superstring
• Quantum corrections to the string Bethe ansatz
• Integrable Hamiltonian for Classical Strings on AdS_5 x S^5
• On AdS_5 x S^5 String S-matrix
• Integrability in QCD and beyond
• Algebraic Curve for the SO(6) sector of AdS/CFT
• Worldsheet Scattering in AdS_5 x S^5
• Classical/quantum integrability in non-compact sector of AdS/CFT
• Uniform Light-Cone Gauge for Strings in AdS_5 x S^5: Solving su(1|1) Sector
• On Integrability of Classical SuperStrings in AdS_5 x S^5
• The AdS5xS5 Superstring in Light-Cone Gauge and its Bethe Equations
• The Algebraic Curve of 1-loop Planar N=4 SYM
• On the Hopf algebra structure of the AdS/CFT S-matrix
• On the Dynamics of Finite-Gap Solutions in Classical String Theory
• A Universality Test of the Quantum String Bethe Ansatz
• Bethe Ansatz for a Quantum Supercoset Sigma Model
• Bethe Ansatz in Stringy Sigma Models
• New Integrable System of 2dim Fermions from Strings on AdS_5 x S^5
• The Algebra of Transition Matrices for the AdS_5 x S^5 Superstring
• Strings as Multi-Particle States of Quantum Sigma-Models
• The algebra of flat currents for the string on AdS_5 x S^5 in the light-cone gauge
• Asymptotic Bethe Ansatz from String Sigma Model on S^3 x R
• Reduced sigma-model on AdS_5 x S^5: one-loop scattering amplitudes
• On the Scattering Phase for AdS_5 x S^5 Strings
• Asymptotic Bethe Ansatz S-matrix and Landau-Lifshitz type effective 2-d actions
• Sigma model from SU(1,1|2) spin chain
• Quantum integrability of bosonic Massive Thirring model in continuum
• A Lax Equation for the Non-Linear Sigma Model

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• On the equivalence theorem for integrable systems
• Higher charges and regularized quantum trace identities in su(1,1) Landau-Lifshitz model
• Exceptional solutions to the eight-vertex model and integrability of anisotropic extensions of massive fermionic models