Lyapunov spectra in SU(2) lattice gauge theory.

We develop a method for calculating the Lyapunov characteristic exponents of lattice gauge theories. The complete Lyapunov spectrum of SU(2) gauge theory is obtained and Kolmogorov-Sinai entropy is calculated. Rapid convergence with lattice size is found. In Ref.1, we have studied chaos in lattice gauge systems by obtaining the largest Lyapunov exponents. The method used there, though straightforward, has two drawbacks. First the results have large error bars because the exponential divergences of trajectories have fluctuations, which results in an uncertainty in the determination of the exponential rate of divergence between trajectories in phase space. The second drawback of the method is that only the largest Lyapunov exponent can be obtained, but not the whole Lyapunov spectrum. There is a well developed method for calculating Lyapunov spectra of systems with many degrees of freedom, which is explained in Ref.2 and briefly outlined here. Given initially a point q(0) in the phase space and ν L vectors v i , i = 1, ..., ν L in the tangent space T q(0) , we can integrate the equations of motion in phase space and simultaneously the evolution equations for small perturbations in tangent space to obtain q(t) and v i (t) ∈ T q(t). At regular time intervals kτ , the Gram-Schmidt orthonormalization is applied to the tangent vectors v i. The scaling factors s i obtained by this procedure determine the Lyapunov exponents as follows, λ i = lim n→∞ n k=1 ln s k i τ , (1) where n is the number of iterations performed. The time needed to obtain the largest Lyapunov exponent depends on how fast the exponents converge with increasing n.