Seminar

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The Sinh-Gordon 1+1 dimensional quantum field theory is the simplest example of a non-trivial interacting quantum integrable model, namely one that is not equivalent to free fermions. As such, it exhibits many new features that are absent in free fermion equivalent models, while displaying for a moderately low level of technical intricacy. This lecture aims at providing a rudimentary introduction to this model which will culminate in a closed expression for its simplest correlation functions. This will already be enough so as to highlight the deep structural differences between the correlators in interacting models and those of free fermion equivalent models which are typically expressed in terms of Fredholm determinants of integrable integral operators. After reviewing the motivations for being interested in this kind of model, I will briefly sketch the requirements one needs to impose on any viable model of quantum field theory and then construct the quantum fields of the Sinh-Gordon model. This will be achieved by solving a system of scalar Riemann–Hilbert problems in n variables, n = 1, 2 . . . , known as Smirnov’s bootstrap program. No prior knowledge of Riemann–Hilbert techniques will be necessary to follow the steps of the construction. Then, I will show how these pieces of information allow one to compute the two-point correlation functions in terms of series of multiple integrals which significantly differ from the Fredholm series for a Fredholm determinant of an integral operator. I will conclude by listing some open problems pertaining to the analysis of correlation functions in interacting quantum integrable models, such as universality.

Correlation Functions of the Sinh-Gordon Quantum Field Theory in 1+1 Dimensions