Soutenance d'Alex Simon

Quantum field theories are physical models that provide a relevant and highly accurate description of many quantum systems when relativistic effects begin to come into play. Correlation functions, which are of major physical interest from being directly measured experimentally, can be calculated perturbatively using formal expansions, at least in the case of theories with weak interactions. However, in a generic theory at full interaction strength, an explicit, non-perturbative calculation of these functions remains out of reach to this day. In this thesis, we focus on a particular class of 1+1-dimensional quantum field theories that are integrable. For these theories, there exists an infinite number of conserved charges and a symmetry algebra, thereby allowing, at least in principle, access to explicit formulas for all quantities of interest in the model, in particular the correlation functions. To obtain these expressions, the works of Karowski-Weisz, Kirillov-Smirnov, and Zamolodchikov-Zamolodchikov led to the development of the integrable bootstrap approach. This method, applicable to a 1+1-dimensional integrable quantum field theory, has its roots in the historical scattering matrix (S-matrix) approach proposed by Heisenberg and Wheeler. It paves the way for the explicit calculation of the model’s correlation functions from the explicit expression of the S-matrix. The results obtained to date concern exclusively 2-point correlation functions. In this thesis, we obtain various types of representations in terms of series of multiple integrals for the multipoint correlation functions of the Sinh-Gordon model, which is the simplest non-trivial integrable model in 1+1-dimensional quantum field theory. Under the assumption that these series of multiple integrals converge, we have shown that they define tempered distributions satisfying Wightman’s axioms. This result thus connects the integrable bootstrap approach to the standard framework for the rigorous construction of a quantum field theory. This thesis also explores the implementation of the integrable bootstrap for certain correlators of the Sine-Gordon model. This model exhibits a much more complex structure due to the presence of bound states, known as breathers, and additional degrees of freedom, known as solitons and antisolitons. For this model, we were able to calculate, using the off-shell Bethe ansatz method, the explicit expressions for the form factors associated with the “field exponential” operator and its charge-non-conserving variants. Our results are consistent with various conjectures previously obtained by Lukyanov in the context of angular quantization. We note here that the form factors constitute the building blocks for calculating correlation functions.