There are many research projects devoted to chaos. We propose here to investigate chaos of a very specific substance: relativistic (quantum) fields, especially gauge fields, which form the foundation of our modern description of the world of elementary particles and their interactions.
The development and the success of gauge theories have their deep roots in quantum mechanics, where the principle of gauge invariance was first discovered. Although the full scope of this principle became apparent when Yang and Mills [1] introduced the first nonabelian gauge field theory in 1954, gauge theories begun their complete triumph as fundamental description of all interactions only around 1970 [2].
The study of chaotic behavior of gauge fields has been started in the 1980-s when analytically solvable nonlinear equations were investigated. This Yang-Mills and Yang-Mills-Higgs mechanics reduced the wealth of field theory to a few degrees of freedom and explored dynamical instabilities through analytic considerations as well as numerical demonstrations [3,4,5]. In the time dependent, spherically symmetric case the well known analytic solution, the Wu-Yang monopole [6], was shown to be unstable by applying the method of Fermi, Pasta and Ulam [7]-[10]. Generally a stabilizing effect of dynamically generated masses was established [4].
The advance of numerical methods in field theory, in particular
its lattice formulation [11,12,13], provided a basis of numerical
studies of a large scale field system. With the availability
of high-performance computational facilities systems with
(N=2,...30) degrees of freedom became feasible for the
numerical investigation at the beginning of 1990-s.
Using the Hamiltonian lattice gauge theory formalism
classical real -- time simulations of U(1), SU(2) and SU(3)
gauge field systems has been carried out [14]-[17].
In these studies the spectrum of Lyapunov exponents
(connected to the entropy growth by coarse graining the
fields) and scaling with system size and total energy
were calculated. Also the instability of nonabelian
plasma waves has been demonstrated by real -- time
lattice calculations [18].
The validity of the classical approximation was investigated in 1993 using nonabelian Gaussian wave packets on the lattice [19]. An explanation of a numerical coincidence between the leading Lyapunov exponent and the gluon damping rate obtained in hot perturbative QCD has been given recently (1995) [20]. While all of these studies were motivated by the desire to gain a better understanding of the physics at high energy densities characteristic for accelerator experiments and presumably the early universe, there are also speculations that the chaotic behavior of elementary fields plays a role in important properties of the ground state (vacuum) as well [21,22]. A celebrated example is the confinement of quarks and other color charges into hadrons.
The study of field theoretical chaos has arrived at a state when the classical -- quantum correspondence and the dynamical interplay between short wavelength (more quantum) and long wavelength (more classical) field components in hot elementary particle media have to be studied. We shall elaborate on this point in our research plan, but before that we review other research activities related to this project. They are important for offering a stimulating working environment as well as for being potential beneficiaries of our research results.