1996/05/24 by Holger Bech Nielsen, H. B. Nielsen, Hans Henrik Rugh +5
Physics and Astronomy · #Chaotic Dynamics (nlin.CD) #Cosmology and Gravitation Theories #FOS: Physical sciences #High Energy Physics - Lattice (hep-lat) #High Energy Physics - Theory (hep-th) #Quantum chaos and dynamical systems #Theoretical and Computational Physics #chao-dyn #hep-lat #hep-th #nlin.CD
paper · pdf · doi:10.48550/arxiv.chao-dyn/9605013
16 pages, LaTeX, 3 Postscript figures
arxiv created 1996/05/24 · openalex publication_date 1996/05/24 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Without an ultraviolet cut-off, the time evolution of the classical Yang-Mills equations give rise to a never ending cascading of the modes towards the ultraviolet, and ergodic measures and dynamical averages, such as the spectrum of characteristic Lyapunov exponents (measures of temporal chaos) or spatial correlation functions, are ill defined. A lattice regularization (in space) provides an ultraviolet cut-off of the classical Yang-Mills theory, giving a possibility for the existence of ergodic measures and dynamical averages. We analyze in this investigation in particular the scaling behavior β= d log λ/ d log E of the principal Lyapunov exponent with the energy of the lattice system. A large body of recent literature claims a linear scaling relationship (β= 1) between the principal Lyapunov exponent and the average energy per lattice plaquette for the continuum limit of the lattice Yang-Mills equations. We question this result by providing rigorous upper bounds on the Lyapunov exponent for all energies, hence giving a non-positive exponent, β≤ 0, asymptotically for high energies, and we give plausible arguments for a scaling exponent close to β∼ 1/4 for low energies. We argue that the region of low energy is the region which comes closest to what could be termed a ``continuum limit'' for the classical lattice system.