Non-unitarity in quantum affine Toda theory and perturbed conformal field theory

Takács, Gábor; Watts, Gérard

doi:10.1016/s0550-3213(99)00100-5

Cited by 19 publications

(7 citation statements)

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“…The fascinating possibility was raised [6,7] that it could nevertheless describe a unitary field theory in a sufficiently strong quantum regime. This possibility was ruled out in the interesting paper [8], and we confirm and extend their observations here.…”

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confidence: 90%

Thermodynamics of the complex su(3) Toda theory

Saleur¹,

Wehefritz-Kaufmann²

2000

Physics Letters B

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We present the first computation of the thermodynamic properties of the complex su(3) Toda theory. This is possible thanks to a new string hypothesis, which involves bound states that are non self-conjugate solutions of the Bethe equations. Our method provides equivalently the solution of the su(3) generalization of the XXZ chain. In the repulsive regime, we confirm that the scattering theory proposed over the past few years -made only of solitons with non diagonal S matrices -is complete. But we show that unitarity does not follow, contrary to early claims, eigenvalues of the monodromy matrix not being pure phases. In the attractive regime, we find that the proposed minimal solution of the bootstrap equations is actually far from being complete. We discuss some simple values of the couplings, where, instead of the few conjectured breathers, a very complex structure (involving E6, or two E8) of bound states is necessary to close the bootstrap. Formal developments, as well as practical applications, would largely benefit from an extension of these results to the case of other Lie algebras, in particular su(n). The situation here is somewhat embarrassing, however. Although the pillars of the su(2) case -the XXZ chain and the associated sine-Gordon model -have been under control for a long time, even the simplest su(3) case is only very partially understood.One of the difficulties here -and, from the field theory point of view, one of the most interesting issues at stakehas to do with unitarity. Indeed, the simplest integrable generalizations of the sine-Gordon model are the complex 1 affine su(n) Toda theories defined by the Lagrangian:1 Real Toda theories involve entirely different issues. For a recent review see [5].where λ > 0, α 1 , . . . , α n−1 form the root system of the classical lie algebra a n−1 , α 0 = − n−1 j=1 α j is the negative of the longest root. The conformal weights of the perturbing operator in (1) are ∆ =∆ = β 2 4π . In the following, we shall parameterize ∆ = t−1 t . The theory described by (1) is obviously non unitary at the classical level. The fascinating possibility was raised [6,7] that it could nevertheless describe a unitary field theory in a sufficiently strong quantum regime. This possibility was ruled out in the interesting paper [8], and we confirm and extend their observations here.From a practical point of view, unitarity is not such a key issue. In fact, the most interesting applications of complex Toda theories are potentially found in disordered systems of statistical mechanics, where Toda theories based on superalgebras naturally seem to appear [9,10], leading most likely to even stronger violations of unitarity. More crucial then are the questions of completeness of the bootstrap, the physical meaning of the bound states, and the calculation of physical quantities.The main progress in the study of complex Toda theories have been based on non perturbative S matrix analysis, following the pioneering work of [6]. One of the difficulties in this approach for su(n) is the appeara...

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confidence: 90%

Thermodynamics of the complex su(3) Toda theory

Saleur¹,

Wehefritz-Kaufmann²

2000

Physics Letters B

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“…In contrast, in the quantum group approach the role of unitarity is played by another condition called 'R-matrix unitarity' (RU). In fact, Takács and Watts have recently highlighted that some of the resulting S-matrices are not unitary, which does not prevent them describing physically relevant (non-unitary) models [13]. We will show that Hermitian analyticity ensures that RU is equivalent to physical unitarity without any extra requirements.…”

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confidence: 81%

“…This shows that the amplitudes with a trivial matrix structure will be Real analytic, but Hermitian analyticity will not be satisfied unless the S-matrix exhibits additional symmetry properties, like parity invariance if (42) holds. All this results in the non-unitarity of these S-matrices reported in [13].…”

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confidence: 89%

“…which is nothing else than the condition of 'R-matrix unitarity' (RU) that arises naturally in the quantum group approach to factorised S-matrices [9,10,11,12,13]. Actually, to be precise, RU is the analytic continuation of (24) to the complex θ-plane.…”

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confidence: 99%

“…However, they found rather difficult to check it directly and proposed to investigate instead if the two-and three-particle S-matrices have pure phase eigenvalues, which is a necessary condition. Following this method, they have singled out a number of S-matrix theories where such changes of basis should exist [13,21]. Concerning this, we have obtained a sufficient condition for the existence of a basis where Hermitian analyticity is satisfied; we call it 'weak Hermitian analyticity', and it is summarized by eqs.…”

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confidence: 99%

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Hermitian analyticity versus Real analyticity in two-dimensional factorised S-matrix theories

Miramontes

1999

Physics Letters B

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The constraints implied by analyticity in two-dimensional factorised S-matrix theories are reviewed. Whenever the theory is not time-reversal invariant, it is argued that the familiar condition of 'Real analyticity' for the S-matrix amplitudes has to be superseded by a different one known as 'Hermitian analyticity'. Examples are provided of integrable quantum field theories whose (diagonal) two-particle Smatrix amplitudes are Hermitian analytic but not Real analytic. It is also shown that Hermitian analyticity is consistent with the bootstrap equations and that it ensures the equivalence between the notion of unitarity in the quantum group approach to factorised S-matrices and the genuine unitarity of the S-matrix.

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The Faddeev–Jackiw Approach and the Conformal Affine sl(2) Toda Model Coupled to the Matter Field

Blas¹,

Pimentel²

2000

Annals of Physics

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The conformal affine sl(2) Toda model coupled to the matter field is treated as a constrained system in the context of Faddeev Jackiw and the (constrained) symplectic schemes. We recover from this theory either the sine-Gordon or the massive Thirring model, through a process of Hamiltonian reduction, considering the equivalence of the Noether and topological currrents as a constraint and gauge fixing the conformal symmetry. Academic Press

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Non-unitarity in quantum affine Toda theory and perturbed conformal field theory

Cited by 19 publications

References 50 publications

Thermodynamics of the complex su(3) Toda theory

Thermodynamics of the complex su(3) Toda theory

Hermitian analyticity versus Real analyticity in two-dimensional factorised S-matrix theories

The Faddeev–Jackiw Approach and the Conformal Affine sl(2) Toda Model Coupled to the Matter Field

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