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Chaos, Scaling and Existence of a Continuum Limit in Classical Non-Abelian Lattice Gauge Theory

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Chaos, Scaling and Existence of a Continuum Limit in Classical Non-Abelian Lattice Gauge Theory LA-UR- 9 6 " 4 43"6" Los Alamos National Laboratory is operated by the University of Caiifomia for the United States Department of Energy under contract W-7405-ENG-36 TITLE: CHAOS, SCALING AND EXISTENCE OF A CONTINUUM Q&TJ LIMIT IN CLASSICAL NON-ABELIAN LATTICE GAUGE THEORY AUTHOR(S): Holger Nielsen, Niels Bohr Institute Hans Rugh, Universty of Warwick Svend Rugh, T-6 SUBMITTED TO: 28TH INTERNATIONAL CONFERENCE ON HIGH ENERGY PHYSICS, WARSAW POLAND, JULY 1996 By acceptance of this article, the publisher recognizes that the U.S. Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or to allow others to do so, for U.S. Government purposes. The Los Alamos National Laboratory requests that the publisher identify this article as work performed under the auspices of the U.S. Department of Energy. Los Alamos National Laboratory L< Los Alamos, New Mexico 87545 FORM NO. 836 R4 ST. NO. 2629 5/81 DISTRIBUTION OF THIS DOCUMENT IS UNLIMITED DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or use- fulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any spe- cific commercial product, process, or service by trade name, trademark, manufac- turer, or otherwise does not necessarily constitute or imply its endorsement, recom- mendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. CHAOS, SCALING AND EXISTENCE OF A CONTINUUM LIMIT IN CLASSICAL NON-ABELIAN LATTICE GAUGE THEORY Holger Bech Nielsen The Niels Bohr Institute, Blegdamsvej 17, 2100 Ktbenhavn 0, Denmark Hans Henrik Rugh Department of Mathematics, University of Warwick, Coventry, CV4 7AL, England Svend Erik Rugh Theoretical Division, T-6, MS B 288, University of California, Los Alamos National Laboratory, Los Alamos, New Mexico, NM 87545, U.S.A. We discuss space-time chaos and scaling properties for classical non-Abelian gauge fields discretized on a spatial lattice. We emphasize that there is a "no go" for simulating the original continuum classical gauge fields over a long time span since there is a never ending dynamical cascading towards the ultraviolet. We note that the temporal chaotic properties of the original continuum gauge fields and the lattice gauge system have entirely different scaling properties thereby emphasizing that they are entirely different dynamical systems which have only very little in common. Considered as a statistical system in its own right the lattice gauge system in a situation where it has reached equilibrium comes closest to what could be termed a "continuum limit" in the limit of very small energies (weak non-linearities). We discuss the lattice system both in the limit for small energies and in the limit of high energies where we show that there is a saturation of the temporal chaos as a pure lattice artifact. Our discussion focuses not only on the temporal correlations but to a large extent also on the spatial correlations in the lattice system. We argue that various conclusions of physics have been based on monitoring the non-Abelian lattice system in regimes where the fields are correlated over few lattice units only. This is further evidenced by comparison with results for Abelian lattice gauge theory. How the real time simulations of the classical lattice gauge theory may reach contact with the real time evolution of (semi-classical aspects of) the quantum gauge theory (e.g. Q.C.D.) is left as an important question to be further examined. There are some indications - and it would be lattice implementation of space-time chaos of clas- a beautiful principle if it was true - that we have sical Yang-Mills fields, but let us first note some the "approximative laws and regularities" which few physical motivations for studying dynamical we know, because they are infrared stable against chaos in classical Yang-Mills fields: modifications of "short distance physics" in the ul- (1) Despite the word "chaos" at first suggests traviolet1. These "approximative laws", which at something structureless it is really a study of ways present accessible scales to the best of our knowl- in which "structure" may be generated (from var- edge comprise a sector consisting of the Stan- ious initial configurations) during the evolution of dard Model of quantum Yang-Mills fields based the governing equations of motion. It is of in- on the group S{Ui x U3) ~ U(1)®SU(2)®SU(S) terest - but so far only little is known - to know and a sector consisting of the gravitational inter- how structures form and evolve (e.g. the formation actions, may (thus) be robust and stable in the and evolution of embedded topological structures) sense of Ref.1, but they are full of unstable and during the time evolution of the Yang-Mills equa- chaotic solutions! Indeed, both the gravitational tions. In the semi-classical regimes of the Stan- field (the Einstein equations) and the Yang-Mills dard Model, the quantum solutions build around fields exhibit dynamical chaos 2>3 for generic solu- the classical solutions (and it is not unlikely that tions (solutions without too much symmetry) in quantum fluctuations on top of classical chaos only regions where (semi)classical treatments are justi- enhance chaos"). fied and non-linearities of the interactions are non- negligible. In the present contribution we shall aThe remark that the full quantum theory for a system 4>5>6 with bounded configuration space (and a discrete spectrum report on some observations concerning the of quantum states) has no chaos due to quasiperiodic evo- (2) For example, the possibility of baryon We shall refer to4>6 for more details in the dis- non-conservation (via the famous ABJ-anomaly) cussion which will follow. The study of chaos in within the electroweak theory has achieved quite toy-models where the gauge fields are spatially ho- some attention recently, and it is related to a de- mogeneous was initiated by Sergei Matinyan and tailed understanding of the dynamics of the elec- George Savvidy 12. An important new qualita- troweak fields (as the Universe cools down). It tive dynamical feature comes into play, however, is quite natural to speculate about a relationship when one considers the spatially inhomogeneous between dynamical chaos and an activity of for- classical Yang-Mills equations: There is a never mation and destruction of embedded topologically ending cascading of the dynamical degrees of free- interesting field configurations (such] a relationship dom towards the ultraviolet, generated by the time is, for example, well known for the' complex Lan- evolution of the Yang-Mills equations! (In spite dau Ginzburg equation, see also e.g. discussion in of this "ultraviolet catastrophe", the solutions are Ref. 8 which contemplates the relevance of a con- well behaved in the sense that there are no "finite cept of topological turbulence of the gauge fields). time blow up of singularities"). The non-linear We note that field configurations for which the rate self-coupling terms which open up the possibility of production of baryon number for a chaotic behavior in the classical evolution lead in the non-homogeneous case to the infinite z cascade of energy from the long wavelength modes B~ f d xTr{FF') (1) JncR3 towards the ultraviolet. (Note, there is a priori no concept of temperature and in the case of Abelian vanishes will span a surface of co-dimension one. (electromagnetic) fields this cascading would not Thus, in fact, most field configurations (in the hot show up dynamically, unless one couples the fields electroweak plasma) will contribute to the right to charged particles). This tendency of the mode hand side of equation (1). For a particular class frequencies cascading towards the ultraviolet will of field configurations which contributes to the 9 completely dominate the qualitative behavior of baryon non-conservation see also Ref. the classical Yang-Mills equations, and the "ultra- (3) Fast equilibration processes which take violet catastrophe" has for some time been em- place in heavy-ion collisions10 are very likely con- phasized by us (cf. e.g. discussion in Ref.8) as nected to non-linear chaotic dynamics; an idea a major obstacle to simulate the classical contin- which in principle dates back e.g. to Fermi, Pasta uum Yang-Mills fields in a numerical experiment and Ulam, Ref. ". over a long time span. * There is no mechanism, As a further motivation of the study of clas- within the classical equations, which prevents this sical Yang-Mills chaos, we should also note that never ending cascading of the modes towards the not many non-perturbative tools are available to ultraviolet. Nature needs H, the Planck constant, study the time evolution of quantum Yang-Mills as an ultraviolet regulator. Indeed, both Abelian fields, so the study of classical Yang-Mills fields is and non-Abelian gauge fields are implemented as a natural starting point for semi-classical under- quantum theories in Nature. standing of the dynamics. Here we shall discuss the possibility of using In order to facilitate a numerical study of a lattice cutoff in a purely classical treatment to spatio-temporal chaos (formation of space-time regularize the equations.
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