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. There will still be a cas- structure) in inhomogeneous Yang-Mills fields one cading of modes towards the ultraviolet, i.e. to- has in some way to discretize the space-time con- wards the lattice cut-off, and this ultraviolet cas- tinuum on which the Yang-Mills fields are defined. cade will still dominate the dynamical evolution This can be achieved in a way which breaks gauge of smooth initial field configurations. However, in invariance (see e.g.14) or in a gauge invariant way a lattice formulation of the Yang-Mills fields on which is called lattice gauge theory15. a large but finite lattice, the phase space is com- pact for any given energy and thus the system can lution of the wave function is of little importance to us since the number of degrees of freedom involved here are so ''This is an obstacle which, in our opinion, has received large that the time it will take to evolve quasi-periodically insufficient attention in the various studies attempting at through the states will be substantially larger than the life- discussing and modeling chaotic properties of spatially in- time of the Universe. homogeneous classical Yang-Mills fields. reach an equilibrium state among the modes (a a Hamiltonian is of course ambiguous in the sense 'thermodynamic equilibrium'). The lattice regu- that extra terms of order 0(a) may be added to larization of the theory opens up for the definition the Hamiltonian. Alluding to some sort of "univer- of dynamic and thermodynamic properties, which sality" (especially in the limit a —*• 0), we expect are not defined in the classical Yang-Mills field that the precise choice of Hamiltonian is not so theory without regularization. It could e.g. be important in what follows. A Hamiltonian which ergodic (modulo constraints) with respect to the is often employed to generate the time evolution of Liouville measure, in which case it makes sense the orbit X(i) is the Kogut-Susskind Hamiltonian, to talk about its micro-canonical distribution and which can be written in the following wayd : approximating this by looking at 'typical' classi- cal trajectories. The fundamental assumption of thermodynamics asserts that on average the two approaches give the same result if we have a large i £ \tT(PiP}) + i £(1 - i • (4) system, and we may then naturally introduce cor- relation functions and possibly a correlation length Here the last sum is over elementary plaquettes £ (measured in lattice units) of the system. bounded by 4 links, and Ua denotes the path- One hopes to define a continuum theory if, by ordered product of the 4 gauge elements along the judicious choice of the parameters in the system, boundary of the plaquette d. The last term, the one obtains a physical correlation length in the potential term, is automatically bounded and, for limit when the lattice constant goes to zero, i.e. a given finite total energy, the same is the case for the first term, the kinetic term. Thus the phase S(a,E(a),...)xa — t £ 0 , as a -> 0. (2) space corresponding to a given energy-surface is In equation (2) the correlation length £ in the lat- compact. tice system is a function of lattice model parame- As is standard we shall discuss the temporal ters such as lattice spacing a, average energy den- correlations (temporal chaos) of the lattice gauge sity E(a), etc. Condition (2) implies that the cor- system in terms of its spectrum of Lyapunov ex- relation length diverges when measured in lattice ponents. The compactness of the phase space im- units and only if this is the case do we expect the plies 4 that the spectrum of Lyapunov exponents lattice system to lose its memory of the underlying (which we overall will assume to be well defined lattice structure. quantities for the lattice system) is independent of the choice of norm on the space of field con- We shall restrict attention to the lattice gauge 3>4 theory in 3+1 dimensions based on the gauge figurations. In fact, it follows from the scaling group 577(2). We consider a finite size 3 di- properties of the equations of motion generated by mensional hypercubic lattice having N3 points the lattice Hamiltonian (4), that in order to study where nearest neighbor points are separated by the dependence of the maximal Lyapunov expo- a distance a > 0. The phase space is a fibered nent with the energy density of the system, it is space where the tangent manifold of the Lie-group sufficient to consider the equations of motion for a SU(2) is assigned to each of the links, i € A, con- fixed value of the lattice constant a, e.g. a = 1, as necting nearest neighbor lattice points: To each a function of energy density (oc energy/plaquette) i € A is associated a link variable U% G SU(2) as and then rescale the results back afterwards. well as its canonical momentum6 P{ 6 TuiSU(2). We have several different forms of lattice arti- A point in the entire phase space will be denoted facts in the lattice simulation of real-time dynami- cal behavior of the continuum classical Yang-Mills X = {Ut, Pi},-6A € M = JIT SU{2) . (3) fields: A (1) Lattice artifacts due to the compactness A Hamiltonian is constructed so as to correspond of the group. The magnetic term (the second to the continuum, classical Yang-Mills Hamilto- term) in the Kogut-Susskind Hamiltonian (4) is nian in the limit a —* 0. The construction of such dFor a derivation we refer to15 and3'16 from which we cIn principle Pi is a cotangent vector, but the Lie al- adapt our notation. For simplicity we omit the coupling gebra inner product gives a natural identification of the constant factor 2/g"2 which anyway is arbitrary in a classical cotangent space with the tangent space. theory. uniformly bounded, 0 < 1 - \TrUo < 2, due to in the region of low energy per plaquette.* the SU(2) compactification. Thinking in terms of Gauge fields exist, however, as quantum the- statistical mechanics for our classical lattice sys- ories in Nature and the interesting definition - as tem, we expect that after some time the typical concerns applications in physics - of a "continuum field configuration has equally much energy in all limit" of the classical lattice gauge theory, is thus modes of vibration - independent of the frequency" to identify regions in the parameter space for the - and the total amplitude of the classical field, and classical lattice gauge theory which probe the be- the energy per lattice plaquette, is thus small for a havior of the time evolution of semi-classical initial fixed low energy. For low energy, when the average configurations (with many quanta) of the quantum energy per plaquette is small, the lattice artifacts theory for a shorter or longer interval of time or in due to the compactness of the gauge group are an equilibrium situation. (See also e.g. discussion thus negligible. in Ref.22). Since the classical lattice gauge the- (2) For small energy per plaquette, we thus ory does not contain a relationship like E = hu expect that the dominant form for lattice artifacts (implying a damping of the high frequency modes is due to the fact that an appreciable amount of relative to the soft modes), we must expect that the activity (for example the energy) is in the field the simulation of quantum gauge theory will be modes with wavelengths comparable to the lattice distorted by this fact (even if implemented with constant a. This short wavelength activity at lat- effective lattice Hamiltonians as e.g. devised by tice cut-off scales is unavoidable in the limit of Ref22). long time simulation of an initially smooth field We shall in the following restrict attention to configuration (relative to the lattice spacing), or real-time simulations of the classical lattice gauge already after a short time if we initially have an theory which have reached an equilibrium situa- irregular field configuration. tion on the lattice. Such simulations are hoped In which way can the classical lattice gauge 3 to yield insights into the dynamical behavior of theory approach a continuum limit ? This is a (long wavelength modes in) the high temperature difficult question to which we shall only be able to Q.C.D. fields in situations where an equilibrium provide a very partial answer here. situation has been reached. For the study of the time evolution of Yang- Before we attempt an analysis of aspects of Mills fields which initially are far from an equi- chaos (in time and space) of the lattice gauge the- librium situation, the (classical) field modes will ory let us first note that it is far from obvious how exhibit a never ending dynamical cascade towards to establish contact between an effective lattice the ultraviolet and after a certain transient time, temperature of the classical lattice gauge theory the cut-off provided by the spatial lattice will pre- (which has reached an equilibrium situation due vent the lattice gauge theory from simulating this to the presence of the lattice cutoff A = I/a) and cascade. It is therefore immediately clear that the physical temperature T of the Q.C.D. theory the lattice regularized, classical fields will not ap- which is a quantum theory and which can reach an proach a "continuum limit" in the sense of simulat- equilibrium situation due to a cut off of quantum ing the dynamical behavior of the classical contin- uum fields in the t —* oo limit. For the simulation urations which are 'good' approximations to continuum of classical continuum gauge fields far from equi- configurations. 9 librium, we conclude that we will have the best I.e. we are here imagining a situation where the spa- tial correlations in the monitored field variables are so large "continuum limit" if we simulate, for a short pe- that they lose memory of the underlying lattice structure riod of time, an initial smooth ansata^ for the fields (including the lattice spacing a). In the extreme opposite limit, one could imagine situations with randomly fluctu- eNote, this situation is very different in the quantum ating fields on the scale of the lattice constant, i.e. with case. Planck's constant ft introduces a relation E = ftw be- (almost) no spatial correlations from link to link. If the tween the energy of a mode of vibration and its frequency, field variables fluctuate independently of each other (inde- implying that a mode with a high frequency also has a high pendent of their neighbors), one could imagine the model energy. With a given available finite total energy, modes to be invariant (with respect to the monitoring of many with high frequencies will therefore be suppressed. Quan- variables) under changes of the lattice spacing, a. Thus, tum mechanically, we thus have that at low energy only it appears that lattice cut-off independence of numerical excitations of the longest wavelengths appear. results can not be a sufficient criterion for the results to •'By "smooth" configurations we mean lattice config- report "continuumphysics". mechanical origin (i.e. ultimately due to the exis- cancellation of a factor a implies that Amar(a) = tence of a Planck constant ft). Some discrepancy const x Eo(a) and thus there is a continuum limit in the literature 20-25>26 illustrates that this is not a —* 0, either of both sides simultaneously or of an easy question. none of them. In the particular case where the In the classical lattice gauge theory we have energy per mode (a energy per plaquette) is in- an "effective lattice temperature" Ti = 1//3 where terpreted to be a fixed temperature T (cf. equa- /? has been determined by looking at the probabil- tion (5) above), one deduces that the maximal ity distribution (Gibbs distribution) of the kinetic Lyapunov exponent has a continuum limit in real energy (or the magnetic energy in; a plaquette), time, proportional to the temperature of the gauge p(Ek) oc exp(—0Ek) (after equilibration). Cf. e.g. field. Ref. 3 (p. 206-207) and Ref. 26. Note, also, the suggested relationship3'16'24 be- In Ref.3'18'26 it is asserted that the physical tween the maximal Lyapunov exponent A of the temperature T (of the quantum Q.C.D. fields in an gauge fields on the lattice and the "gluon damp- equilibrium situation) and the average energy Ea ing rate" 7(0) for a thermal gluon at rest, arrived per plaquette in the classical lattice gauge study at in re-summed perturbation theory in finite tem- in equilibrium are related by perature quantum field theory, see also Refs.25'26. For the SU(2) gauge theory, this suggested rela- 2 tion reads £Q«|(n -l)T (for SU(n)). (5) 2 2 'It is not clear to us how serious one should take A = 2T(0) = 2x -^-<7 T~ 0.34g T (7) 24?r this relationship (one objection being that a scale It is also our understanding that 7(0) is a quan- has not been fixed in the continuum limit of the 17 classical theory) but we shall leave a discussion of tity of semiclassical origin. A relation like (7) this issue aside in the following. is nevertheless remarkable in chaos theory since it suggests that a complicated dynamical quan- Let us now attempt an understanding of tity like a temporal Lyapunov exponent (which is chaotic aspects of the lattice gauge theory - i.e. usually only possible to extract after a consider- the temporal chaos (as monitored by the spectrum able numerical effort) is analytically calculable by of Lyapunov exponents) and its relationship with summing up some diagrams in finite temperature spatial chaos (as monitored by the spatial corre- quantum field theory. lations, e.g. a spatial correlation length) in the In Ref.4 we argued (in view of the numerical dynamical system. 3 16 18 16 17 18 evidence presented in e.g. > i ) that the appar- A sequence of articles ' ' and a recent book 3 ent linear scaling relation (6) is a transient phe- by Biro, Matinyan and Muller present numerical nomena residing in a region extending at most results for the classical 5(7(2) lattice gauge model a decade between two scaling regions, namely and provide evidence that the maximal Lyapunov for small energies where the Lyapunov exponent exponent is a monotonically increasing continuous scales with an exponent which could be close to function of the scale free energy/plaquette with ib ~ 1/4, and a high energy region where the scal- the value zero at zero energy. ing exponent is at most zero: Ref.3'16'18 reports a particularly interesting in- terpretation of numerical results for the dynamics dlog A = k 1/4 for E- •0 (8) on the lattice, namely that there is a linear scaling d\ogE 0 for E- • 00 relation between the scale free maximal Lyapunov = {< exponent, \max{a = 1) and the average energy per The proposed scaling relation k ~ 1/4 was, for plaquette Ea(a = 1). The possible physical rele- reasons we shall give below, suggested to hold in vance of this result follows from the observation16 the limit E —*• 0 from general scaling arguments that when we rescale back to a variable lattice 8 of the continuum classical Yang-Mills equations spacing a we note that the observed relationship in accordance with simulations on homogeneous 16 is in fact a graph of a\max(a) as a function of models and consistent with the figures in and a Ea{a). Thus being linear, 18. However, a more recent numerical analysis5 ar- gues rather convincingly that the data points pre- 16 18 3 a Xmax = const x a Ea(a) (6) sented in e.g. > ' were subject to "finite time" aitifacts, and that long time simulations with a ity to reach contact with "continuum physics" - correct procedure for extracting the principal Lya- not in the sense (as we have seen) that the lattice punov exponent could support that Jb = 1 even in system simulates the original continuum classical the limit as E —• 0. ; gauge fields (over a long time span), but in the Before we discuss the limit of low energies sense that the lattice gauge theory - considered as (E —* 0) let us note that regarding the limit for a statistical system in its own right - will develop high energies per plaquette a rigorous result7 (also spatial correlations in the fields (as monitored by reported in 4'6) shows that the 5(7(2) scale free a correlation length f, say) which may approach a (a = 1) lattice Hamiltonian in d spatial dimen- formal "continuum limit", sions has an upper bound for the maximal Lya- punov exponent £-•00 as E-+0 (10) There is a well known analogy19 between a quan- (9) tum field theory in the Euclidean formulation with which for d = 3 becomes \B = 4.03... This result a compactified (periodic) imaginary time axis 0 < is arrived at by constructing an appropriate norm T < 0 and finite temperature statistical mechan- on the phase space and showing that the time ics of the quantum field theory at a temperature derivative of this norm can be bounded by a con- T = 1/0 which is inversely proportional to the stant times the norm itself, hence giving us an up- above-mentioned time extension. per bound as to how exponentially fast the norm Calculations in finite temperature Euclidean can grow in time. The upper bound (9) shows that quantum field theory suggest that a characteris- a linear scaling region, i.e. TJ = d\og\/dlogE = 1, tic correlation length of static magnetic fields in cannot extend further than around E ~ 10 on the thermal quantum gauge theory is of the order 2 -1 the figure 1. Beyond that point the maximal Lya- f ~ (• oo limit of equation (8). configurations on the lattice will have characteris- tic correlation Jengths of the order It should be noted that the upper bound (9) is independent of the lattice size and the energy (but O 1 1 O , scales with I/a). There is quite a simple intuitive £ ~ \g 1) ~ \—g tia) —»• oo as iba —*• U . explanation for the saturation of the maximal Lya- (11) punov exponent in the regime for high energy per In systems with space-time chaos (and local prop- plaquette of the 517(2) lattice gauge model, since agation of disturbances with a fixed speed ~ c) the potential (magnetic) term in the Hamiltonian one often has a relationship between a correlation (4) is uniformly bounded, 0 < 1 - \TrUa < 2, due length (coherence length) £ and the maximal Lya- to the 5(7(2) compactification. For high energies, punov exponent A (see also e.g. discussions in almost all the energy is thus put in the (integrable) Ref.23) of the form \ ~ c/A. This suggests, via kinetic energy term in the lattice Hamiltonian (4), equation (11), and the spectrum of Lyapunov exponents will sat- urate as the energy per plaquette increases for the A ~ c/f ~ g2 E (12) model. (The chaos generated by the non-linear a potential energy term does not increase, only the i.e. a Lyapunov exponent which scales linearly energy in the kinetic energy (the electric fields) with the average energy per plaquette Ea- Clearly, increases). it deserves further investigation to establish a pre- As we shall argue (a discussion which is rather cise relationship between Amar and the spatial cor- suppressed in16'17'18'3), it is however in the oppo- relation length £ (in lattice units). However, we site limit, i.e. in the limit where the average energy emphasize that it is in the limit as the energy per per plaquette goes to zero, i.e. E —* 0, that the plaquette goes to zero (and where the lattice Lya- equilibrium lattice gauge theory has the possibil- punov exponent Amar goes to zero) that we expect the spatial correlation length £ (in lattice units) with a4. This indicates 4 that if we perform a will diverge. The finite size of the lattice makes it measurement of the maximal Lyapunov exponent difficult to analyze the behavior of the gauge fields ^max over a time short enough for the solutions to 1 4 and the principal Lyapunov exponent on the lat- stay smooth, then Xmas scales with E ^ . Such a tice in the limit E —• 0. scaling was also observed for "smooth" configura- Let us reiterate the arguments which led us to tions by Miiller and Trayanov p. 3389 in Ref.16. the conclusion (in Ref.4) that the scaling exponent These scaling arguments do not carry over to in- of the Lyapunov exponent likely will be closer to ~ finite time averages since solutions on the lattice 1/4 in the limit E —•• 0 than the scaling exponent tend to be irregular. ~ 1 observed in the intermediate energy region In any case we expect that the lattice gauge 1/2
8 if the 1/4 classical-scaling of the Lyapunov expo- Foundation) from University of California Santa nent emphasized above renormalizes to an expo- Barbara and discussions with Hans-Thomas Elze nent which equals unity. Even if this did happen are also acknowledged. it is not clear that such a behavior should be inde- pendent of the regularization procedure employed, since Lyapunov exponents are measures of local in- References stabilities, i.e. short range rather than long range structures. 1. H.B. Nielsen, "Dual Strings - Section 6. As regards the numerical evidence for the re- Catastrophe Theory Programme", in I.M. lationship (7) between the maximal Lyapunov ex- Barbour and A.T. Davies (eds.), Fundamen- ponent and the "gluon damping rate" 24 we note tals of Quark Models, Scottish Univ. Sum- that in the hierarchy of scales g2T < gT < T mer School in Phys. (1976) pp. 528-543. See (this separation of scales is assumed in hot per- also (e.g.) reprints and discussion in CD. turbative gauge theory) the "gluon damping rate" Froggatt and H.B. Nielsen, Origin of Sym- is connected to the decay of the "soft" modes metries. World Scientific. 1991. ~ g2T. In order for the lattice gauge theory to 2. D. Hobill et al. (eds.), Deterministic Chaos probe decays of "soft" gauge modes, this requires in General Relativity, NATO ASI Series B; the existence of some "soft" modes on the lat- Physics Vol. 332. Plenum Press. New York. tice (in a background of "hard" modes). Thus • 1994. we must monitor the lattice gauge system in a re- 3. T.S. Biro, S.G. Matinyan and B. Miiller, gion where there are fields (in equilibrium with Chaos and Gauge Field Theory. World Sci- "hard" modes A ~ I/a) which have spatial corre- entific. 1994. lations substantially larger than the lattice unit 4. H.B. Nielsen, H.H. Rugh and S.E. Rugh, a. As we have seen this is not the case in the "Chaos and Scaling in Classical Non-Abelian regime where the lattice gauge theory was studied Gauge Fields", Preprint LA-UR-96-1577, (N = 20, g tts 2,E ~ 1 - 4) in Ref. 3-16-18 pro- chao-dyn/9605013 (May 1996). viding the numerical support for equation (7). It 5. B. Miiller, "Study of Chaos and Scaling in appears, that the numerical support of the relation Classical SU(2) Gauge Theory", Preprint (7) is somewhat of an accident if the relation (7) is DUKE-TH-96-118, chao-dyn/9607001 (July to be interpreted as a "continuum result". Studies 1996). of the lattice gauge theory have to be conducted 6. H.B. Nielsen, H.H. Rugh and S.E. Rugh, (in for much smaller energies per plaquette which are, preparation). however, also difficult, since finite size effects then 7. H. H. Rugh, "Uniform Bounds on Lyapunov will become of appreciable size.J Exponents in Lattice Gauge Theories", (in preparation). Acknowledgements 8. S.E. Rugh, Aspects of Chaos in the Funda- mental Interactions. Part I. Non-Abelian We thank Alexander Krasnitz, Salman Habib and Gauge Fields. (Part of) Licentiate Thesis, Emil Mottola for several discussions on the subject The Niels Bohr Institute. September 1994. and we thank Berndt Miiller for sending us numer- Available upon request. To appear (in a sec- ical data from Ref.16 and illuminating discussions ond revised edition) on the e-print server. following the submission of our initial manuscript 9. M. Axenides, A. Johansen, H.B. Nielsen and Ref.4. S.E.R. would like to thank the U.S. Depart- O. Tornkvist, Nucl.Phys. B 474, 3 (1996). ment of Energy and Wojciech H. Zurek at the Los 10. Cf. (e.g.) Sec. "Thermalization - Entropy", Alamos National Laboratory for support. Support pp. 165 - 250 in J. Letessier et al. (eds.), from Grant No. PHY94-07194 (National Science Hot Hadronic Matter - Theory and Exper- iment. NATO ASI Series B: Physics Vol. 3 Finite size effects are of course - in the context of our 346. Plenum Press. New York. 1995. discussion - a "good sign" which witnesses that the lattice 11. E. Fermi, J. Pasta and S. Ulam. Studies of monitors field configurations which have correlations of the order of the lattice size ~ Na, i.e. much beyond a single nonlinear problems. Report LA-1940, Los lattice unit. Alamos Scientific Laboratory, 1955. 12. S.G. Matinyan, G.K. Savvidy and N.G. Ter-Arutyuyan-Savvidy, Sov.Phys. JETP 53, 421 (1981); G.K. Savvidy, Nucl.Phys. B 246, 302 (1984). S.G. Matinyan, SovJ.Part.Nuci 16, 226 (1985). 13. B.V. Chirikov and D.L. Shepelyanskii, JETP Leu. 34, 163 (1981); Sov. J. Nucl. Phys. 36, 908 (1982); E.S. Nikolaevskii and L.N. Shchur, Sov. Phys.JETP 58, 1 (1983). 14. M. Wellner, Phys. Rev. Lett. 68, 1811 (1992); Phys. Rev. E 50, 780 (1994). 15. K. Wilson, Phys.Rev. D XX (1974). J. Kogut and L. Susskind, Phys. Rev. D 11, 395 (1975). 16. B. Müller and A. Trayanov, Phys. Rev. Lett. 68, 3387 (1992). 17. C. Gong, Phys. Lett. B 298, 257 (1993); Phys. Rev. D 49, 2642 (1994). 18. T.S. Birö, C. Gong, B. Muller and A. Trayanov, Int. J. Mod. Phys. C 5, 113 (1994). 19. See e.g. Chapt. 10 in R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals. McGraw-Hill, Inc. (1965); Chapt. 17-18 in H.J. Rothe, Lattice Gauge Theories. World Scientific (1992). 20. J.Ambjørn, T.Askgaard, H.Porter and M.E.Shaposhnikov, Nucl. Phys. B353, 346 (1991); J. Ambjørn and A. Krasnitz, Phys. Lett. B 362, 97 (1995). 21. K. Kanaya, "Finite Temperature QCD on the Lattice", Nucl.Phys. B (Proc.Suppl.) 47, 144 (1996). 22. D. Bödeker, L. McLerran and A. Smilga, Phys. Rev. D 52, 4675 (1995). 23. T. Bohr, "Chaos and Turbulence", in Ap- plications of Statistical Mechanics and Field Theory to Condensed Matter, edited by A.R. Bishop et al. Plenum Press. New York (1990); W. van de Water and T. Bohr, Chaos 3, 747 (1993). 24. T.S. Birö, C Gong and B. Müller, Phys.Rev. D 52, 1260 (1995). 25. T.S. Birö and M.H. Thoma, Phys.Rev. D 54, 3465 (1996). 26. U. Heinz, C.R. Hu, S. Leupold, S.G. Matinyan and B. Müller, "Thermalization and Lyapunov exponents in the Yang-Mills- Higgs Theory", Preprint DUKE-TH-96-129, hep-th/9608181 (August 1996).
10