The quantum theory of fields is a formalism providing a universal language for a large part of contemporary physics. Originally successfully developed to model fundamental interactions of elementary particles, the quantum theory of fields has then found applications in condensed matter, statistical physics, and cosmology, in each case leading to profound and novel results that considerably improved our understanding of Nature. Not a single experiment has ever falsified a prediction of QFT. As an aside, the formalism also has deep connections with pure mathematics, leading to surprising and ground-breaking discoveries.
In a quantum field theory (QFT), the dynamics of a given microscopical system is described in terms of correlations among fluctuations of a medium that fills space and time, the quantum field. It is convenient to organize quantum fields according to the strength of their interactions, from weakly-interacting to strongly-interacting ones. Nature provides numerous realizations of both.
Weakly-interacting QFTs can be analyzed using perturbation theory, and the predictions of weakly interacting QFTs are among the most striking successes of modern theoretical physics. Strongly interacting QFTs are intrinsically non-perturbative, meaning that their dynamics differs significantly from that of a free field, and hence perturbation theory does not apply. Non-perturbative regimes are among the most challenging ones: phase transitions, the physics of confined quarks within protons, and the dynamics near the horizons of black holes are all examples of non-perturbative phenomena. Despite their importance, we lack tools to characterize non-perturbative quantum fields: any progress in this direction is a ground-breaking achievement with far-reaching interdisciplinary consequences in many areas of physics and mathematics.
A hallmark of non-perturbative QFTs is the presence of solitons in the spectrum of the quantum field excitations. Solitons are very special ripples of a given field, lumps of energy held together by the strength of field interactions that cannot be described perturbatively and are often protected by topological quantum numbers, meaning that their features are robust against continuous deformations of the parameters of the theory at hand. An important feature of quantum fields is given by the fact that their behavior changes as a function of energy. Such changes are captured by the so-called renormalization flow. Along a renormalization flow from a given phase to another, relativistic QFTs often develop solitonic sectors which are typically massive and decouple at low energies. In the context of various toy models obtained by constraining the quantum field imposing it has extra symmetries that allows to have a better control over its dynamics, one can see that it is possible to reconstruct some properties of the original highly energetic, ultraviolet phase exactly from the knowledge of the spectrum of decoupled solitons along the renormalization flow, including the values of certain observables. This effect is the memory of solitons. The main objective of the memory of solitons project is to develop this effect into a series of tools to constrain the non-perturbative regimes of various quantum fields and compute their properites. More concretely we will determine the full spectrum of excitations of given quantum fields in non-perturbative regimes (objective I: spectral problem), and we will establish dictionaries between the properties of said excitations and the properties of the non-perturbative quantum field of interest, thus providing a different pathway towards extracting the values of several observables of interest (objective II: reconstruction of observables).
