Classical Chaos of SU(2) Gauge Fields

Dynamical chaos has been known for some time to exist at the classical level in non-abelian gauge fields, which describe the fundamental forces of nature. Recently, it has been shown by computer simulations that SU(2) gauge theory exhibits quantum chaos. In particular, simulations have shown that the quantum gauge theory satisfies the Eigenstate Thermalization Hypothesis which describes thermalization of an isolated system. While gauge fields are known to be chaotic at the classical level, the specific systems for which quantum calculations are done have not been studied in the classical limit. If the relation between classical and quantum chaos were known in these systems, extrapolations to much larger systems more relevant to the universe would be possible. In order to understand how the properties of chaos differ between the classical and quantum limits, it is necessary to solve the classical gauge field equations on the same system and compare the results with those from the quantum calculations. We use Mathematica to solve the differential equations that describe the classical dynamics of SU(2) gauge fields on a lattice. We examine how the Lyapunov exponent depends on various parameters, such as the energy and the coupling constant of the gauge field. We find that, in the classical limit, the Lyapunov exponent grows with the energy. This finding will be compared with further calculations in the quantum theory to determine how the properties of chaos differ in the classical and quantum limits.