Correlation Functions of Integrable Flows

The first chapter of this thesis is devoted to the detailed explanation of the method just mentioned of computing the correlation functions for integrable models. It contains the starting sum rule formula for a correlation function in terms of form factors, and their definition in terms of the Zamolodchikov-Faddeev algebra, together with a short survey of factorized scattering properties. General recursive equations for the form factors are obtained from the analytic properties of the scattering amplitudes. A description of the principal features in the approach of form factors of two privileged operators then will follow. They are the trace of the stressenergy tensor, which drives the renormalization group flow, and the elementary interpolating field, which plays the role of the lagrangian field creating the particles. These properties are then shown for the very easy example of the free massive boson. The second chapter describes the implementation of this method for lagrangian massive integrable theories. The form factors of the most significant fields of the sinh-Gordon and the Bullough-Dodd models are computed. The ultraviolet behaviour of the theories is discussed in terms of the c-theorem. Very interesting exact non-perturbative features of these theories are described. The two quantum equations of motion are established, and are identified at a particular value of the coupling constants. It is then shown that a line of ultraviolet fixed points can be described in terms of a persistent marginal perturbation of these models, that is within the same quantum dynamics. The form factor program described in the first chapter and carried on in the second chapter for massive flows shows a flexibility which allows us a natural generalization to the massless case. It is mainly based on disentangling the massless particles into right and left movers. The massless form factor program is proposed in the third chapter, where it is also worked out for the massless flow from the tricritical Ising model to the Ising model. The form factors of the principal operators flowing from a Kac-table to the other are obtained. The expected infrared divergency problem emerges only in the last step of this massless program, that is computing the correlation functions. However, due to the special feature of the infrared theory, which is free, the correlation functions of the operators in the energy sector of the Kac-table are not plagued by the occurence of infrared divergencies. So they can be computed without the introduction of a volume cut-off and with the same efficiency of the massive case. The computed correlation functions confirm the conjectured identification of the operator content. In the fourth chapter we face the problem of integrable theories for inhomogeneous systems. The general scattering theory of integrable particles on lines of integrable defects is settled. It is treated by means of a very natural generalization of the Zamolodchikov-Faddeev algebra. It is shown how the only massive non-diagonal bulk theories which admit non trivial reflectiontransmission amplitudes are the free fermionic (Ising) and bosonic ones. The detailed description of these two theories is developed. Some physical features are exhibited for the case of many defect lines. The correlation functions for the energy and the magnetization of Ising model with one defect are computed by taking advantage of the complete knowledge of their form factors in the bulk and of the scattering amplitudes on the defect. Our results are in complete agreement with the results obtained in the lattice approach.