The main goal of this research project is to develop a description of the real -- time dynamics of field systems on a lattice which takes into account the essential quantum effects stemming from short wavelength excitations as well as the chaotic nonlinear dynamics of long wavelength time -- dependent fields.
In the recent past we have already developed a numerical simulation of the chaotic dynamical behavior of spatially extended field systems on a lattice [14]-[19]. Based on this we plan to derive an effective Hamiltonian for the long wavelength field components by integrating out components shorter than the lattice spacing. While we expect that the short wavelength action resembles the hard thermal loop generating part as it is known in hot perturbative field theory and the long wavelength contributions alone can be described by the classical action, we are especially interested here in obtaining a reliable description of the interaction between these modes of field excitation. This interaction is probably dissipative for the classical part and therefore may reduce chaos, but its mathematical form is presently unknown [34].
Related to this main subject of investigation also some further questions need to be answered.
What is the correspondence between exponentially growing fluctuations around a classically chaotic trajectory and the analytic structure of field correlations (poles, branch cuts) considered in a perturbative graphical expansion in field theory?
How to make the short wavelength -- long wavelength distinction not breaking gauge invariance and respecting scale invariance? In what extent can we use here renormalization group arguments?
Does the chaotic behavior of long wavelength (but high frequency) field components survive in the effective theory? If yes, is the characteristic time for developing chaos from an ordered state still in the sensible range to facilitate thermalization of quark -- gluon matter?