In this chapter, we quantize (preferentially) the sinh–Gordon model in the Lagrangian framework. We consider this theory as the perturbation of the free massive boson (Klein–Gordon model), which is quantized first. The potential is treated perturbatively; it is supposed to be weak in the sense that the particle spectrum of the free model is not changed. This assumption is valid for the sinh–Gordon theory, where the only particle already exists in the Klein–Gordon model. In the sine–Gordon case, however, additionally to the breather-type solution, which is the analog of the sinh–Gordon particle, there are non-perturbative particles like the soliton and the anti-soliton. Nevertheless, our approach is based on general field theoretical investigations and the conceptual consequences are valid for any theory of quantum particles, even for the quantum counterparts of the soliton and anti-soliton.

We start the chapter by introducing the quantum analog of the classical time shift, the scattering phase.We show how they are related in the semi-classical limit, which makes a bridge between the classical and quantum descriptions [214]. Then we turn to the quantization of the sinh–Gordon theory in the perturbative scheme: the free Klein–Gordon part is quantized first and then the interaction is taken into account in the interaction picture. We introduce the notion of asymptotic states and their scattering S-matrix. A reduction formula links the S-matrix to the correlation functions. This makes it possible to derive the crossing symmetry of the S-matrix and to investigate its analytical structure [215].

In classical mechanics, a dynamical system of interacting bodies with 2N-dimensional phase space is said to be integrable if there exist N conserved functions (charges) whose Poisson brackets vanish. For an integrable system in quantum field theory (QFT) there exists an infinite set of commuting conserved charges. The existence of the conserved charges allows us to solve the physical system exactly and in this way to describe the modeled phenomena without any approximation. Although the integrability is restricted to low dimensions, the exact solution often provides general information about the physical phenomena. At present, we know precisely how to generate systematically integrable models and how to solve them, explicitly or implicitly in the form of integral equations.

Integrable models cover many domains of quantum mechanics and statistical physics:

• Non-relativistic one-dimensional (1D) continuum Fermi and Bose quantum gases with specific types of singular and short-range interactions.

• 1D lattice and continuum quantum models of condensed-matter physics, like the Heisenberg model of interacting quantum spins, the Hubbard model of hopping electrons with one-site interactions between electrons of opposite spins, the Kondo model describing the interaction of a conduction band with a localized spin impurity, microscopic models of superconductors, etc.

• Relativistic models of QFT in a (1+1)-dimensional spacetime like the sine-Gordon model and its fermionic analog, the Thirring model, and so on.

• Two-dimensional (2D) lattice and continuum classical models in thermal equilibrium like the lattice Ising model of interacting nearest-neighbor ±1 spins, the six- and eight-vertex models, the continuum Coulomb gas of ±1 charges interacting by a logarithmic potential, etc.