L. Rafaj: Mechanization of a scalar field theory in 1+1 dimensions

Anotace:

Scalar field theory in (1+1)-dimensions is conceptually the simplest, non-linear relativistic field theory theory imaginable. Its dynamics should be therefore a well-understood territory. Yet, despite 40+ years of ongoing research, there are still mysteries without satisfactory resolutions and discoveries that have been made just recently. The tool that allowed the most recent progress is the so-called collective coordinate method (CCM). The CCM works by restricting the space of field configurations to a preselected subspace: the moduli (parameter) space of a given background ansatz that varies with time. In particular, CCM has been deployed with great success to capture essential features of kink-anti-kink collisions in various scalar field models. Given its specific nature, however, CCMs cannot be used for systematic and exhaustive exploration of the dynamics of scalar field theories. We present a universal tool that has the potential to see the entire range of dynamical phenomena, although in a simplified and coarse-grained setting. The idea is to replace a continuous field with a piece-wise linear function. In this way, the dynamics of infinitely many degrees of freedom turn into the dynamics of a finite (but arbitrarily large) number of "joints" connecting a sequence of straight segments carrying energy and momentum. We call this procedure a mechanization. Unlike the field theory, the dynamics of mechanized models can be explored gradually from the simplest "mech-fields". For instance, the $N=1$ mech-field is an analog of kink -- a "mech-kink" -- that behaves as a relativistic particle, despite an explicit breakdown of Lorentz symmetry. Further, we show that by adding more joints, new dynamic features appear. The $N=2$ mech-kink can shed a joint in a way resembling radiation loss or a quasi-normal mode decay. On the other hand, a topologically trivial $N=2$ mech-field, which we call "mech-oscillon", resembles a field theoretical oscillon in a way that it is a quasi-periodic solution that has a definite lifetime. In our talk, we present an exploration of the properties of mech-fields, including a detailed discussion of mech-kink-antikink scattering, lifetime of mech-oscillons, joint ejection, BPS mechanization and more. We also discuss the disadvantages and technical issues of the mechanization, that however can give us clues for future improvements. In particular, we will show how mechanization breaks integrability in free field theories and in the sine-Gordon model and what we can learn from it.