この論文は国立情報学研究所の電子図書館事業により電子化されました。研究会報告There is a class of polygonal billiards which can be mapped into classical flows on the surface of a g-handled sphere. In particular the π/3-rhombus billiard flow is equivalent to a flow on the surface of a two-handled sphere. The quantisation of this system is as yet an unsolved problem

Crehan, P.

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Title Quantisation on the Two-Handled SphereAuthor(s)Crehan, P.Citation物性研究 (1996), 67(3): 296-302Issue Date1996-12-20URL http://hdl.handle.net/2433/95964RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityQuantisation on the Two-Handled SphereP. CrehanDept. of Mathematics, Faculty of Science, University of KyotoPAT@KUSM.KYOTO- U.A C.JPAbstractThere is a class of polygonal billiards which can be mapped into clas-sical flows on the surface of a g-handled sphere. In particular the 11"/3-rhombus billiard flow is equivalent to a flow on the surface of a two-handledsphere. The quantisation of this system is as yet an unsolved problem.1 IntroductionFlows in phase-space of compact two-dimensional fully integrable Hamiltoniansystems have a second conservation law and are confined to the surface of two-dimensional tori. Generic systems are not confined in this way. Their attractorsmay have fractal dimension greater than two, the orbits may fill up parts ofthe three dimensional energy surface and in extreme cases they may even beergodic. Many working in chaos-theory have found it interesting to quantisegeneric Hamiltonian systems and to consider the asymptotic ,properties of theirspectra. These systems are not solvable as a rule and such work has to be donenumerically. Generally the asymptotic energy gaps of fully integrable systemshave a Poisson distribution, whereas chaotic systems with time reversal in-variance are modelled by the Gaussian-Orthogonal-Ensemble. In between fullychaotic-ergodic systems and the fully integrable regular systems lie the mixedsystems whose level-spacings are sometimes described by the Berry-Robnic dis-tribution. These are not mathematically exact statements. They are reasonableapproximations born out by numerical experiment, and the reader is referred 'to[1] for more information, as well as to [2] and [3] for general background on thesetopics. The question as to whether or not quantum chaos exists is a circularone, so anyone who insists on asking it is referred to [4] to find out why it does,as well as to [5] to find out why it does not!The overall picture is that there are nice regular fully integrable systemswhich we can solve explicitly, whose classical motions lie on tori, and whosequantum level statistics are Poisson. Examples of these are provided by theintegrable billiards. At the other end of the scale are chaotic systems which areeasy to understand classically because they are ergodic, whose quantum me-chanics can be solved if we are lucky, and whose level spacings have statisticswhich are GOE. An important example of which is the Sinai billiard [6]. In be-tween lie systems which are very complicated, which we can not solve explicitlyand which are qualitatively hard to understand. An interesting family of suchin-between systems is provided by a class of billiard flows equivalent to flows on- 296-g-handled spheres. These have an extra conservation law, so naievely we mightexpect to analytically compute the energy levels and wavefunctions of the quan-tum problem. Although classically they are very like fully integrable systems,their quantum properties are very like those of classically chaotic systems. Evenin the simplest case that of the 7r13-rhombus billiard, a full analytic solutionto the quantum problem has not yet been found. The difficulty stems from thenon-trivial topology of the invariant manifolds in phase-space.The purpose of my talk is simply to try to provide some classical intuitionwhich might bear on the problems of quantisation on a manifold with non-trivialtopology, as well as an appreciation of the non-trivial nature of the task.2 Classical flows on the g-handled sphereFirst we take a look at some general features of classical flows on g-handledspheres. Most of what I will say can be found in [7]. Many elegant theoremsrelate the topology of these surfaces to properties of functions, vector fields ormetrics with which they may be endowed. In particular if M is a compactorientable two dimensional manifold we haveX(M) - V-E+F (1)-2-2g (2)- Lix(Po) (3)0- A-B+C (4)- 2~LKdA. (5)If V is the number of vertices, E the number of edges and'F the number of facesin a simplicial decomposition of M, equation (1) defines the Euler characteristicX(M). The Euler formula (2) relates this to the genus of M. These are relatedto the properties of vector fields on M through the Hopf-index theorem (3). Inthis formula X is a vector field with a singular point at each POt whose indexis denoted ix(po). The index is a topological quantity which essentially countsthe number of seperatrices going in and going out of a small circle around thesingular point Po. For example it can be calculated according to1- 2- { da271'" 1stwhere 8 1 is a small circle around POt which does not contain any other singularpoint and da is the angle between the tangent to the circle and the vectorX. In the case of non-degenerate singular points, the index of a node and a- 297-saddle point is +1 or -1 respectively. If the vector field is the gradient of asmooth function f(x, y) on M with non-degenerate critical points, then theMorse-relation is given by (4) where A is the number of maxima of f(x,y), Bthe number of saddle points and C the number of minima. The last relation(5) is the Gauss-Bonnet formula and this holds whenever M is provided with aRiemannian structure whose curvature is K and the volume element dA.In the case of the torus g = 1 and it is not hard to avoid dealing withcurvature or singularities in the classical flow. As handles are added howeverg > 2, every flow must have some singular point and every metric must havesome curvature. In this sense the g-handled spheres for g > 1 can not supportfree-particles in the normal sense of the word, not even classically.3 Polygonal Billiard FlowsConsider a polygon in 2-dimensional Euclidean space which contains a billiardmoving freely and colliding elastically with each of the sides. This providesan example of geodesic flow on a manifold with boundary. Simple as theyseem such systems are very challenging mathematically and there remain manynatural yet unsolved problems. It is not yet known if every polygonal billiardposesses a periodic orbit. On the other hand it is possible to say that if thereexists a peridodic orbit, it can be classified as odd or even depending on thenumber of segments of which it is made, and that the odd ones occur in isolationwhereas the even ones occur in bands. There are many interesting known facts,conjectures as well as unsolved problems relating to the existence of periodicorbits, their distribution, and the transitivity, minimality or ergodicity of theflows. Due to considerations of space all I can do is refer the reader to an articleby Gutkin [8], and references therein.Now let us consider the so-called rational billiards - billiards in p-sided poly-gons whose contained-angles are all of the form mi1r/ni for i = 1, ... ,p, whereeach pair of integers ni, and mi is relatively prime. Later we use N to denotethe LCM of the ni. The configuration space of this system is the polygon itselfand since the total momentum is conserved, the reduced momentum-space is acircle. The momentum remains constant except when the particle collides witha wall. For this reason the evolution of momentum is described by a finite set ofmaps on the circle, one for each side of the polygon. With respect to any set ofaxes if 4> describes the direction of momentum of a billiard which collides witha side whose perpendicular has direction Bi , then the direction of momentum istransformed by the circle map 4> -+ 2B i - 4>. We choose to ignore what happenswhen a billiard collides with a ve~tex. The momentum state at anytime is there-fore of the form 4> = ±4> + 2 2::~i kiBi moc}. 21r where ki is a positive integer.General billiard flows fill up the circle, but in the case of rational billiards theset of possible directions of momentum is finite.The phase-space accessible to a rational billiard is covered by a finite number- 298-of copies of the original polygon, one copy for each of these momentum direc-tions. In general the number of copies required is 2N, and inequivalent flowson the same polygon can be labelled by the angle a E [0, 7r/N]. All angles inthe interior of this interval correspond to what are known as generic flows, andfor irrational a they are spatially dense. The flows corresponding to the endpoints are termed exceptional flows. In their case only N copies of the originalpolygon is required.In the case of generic flows, the 2N copies of the original polygon are con-nected along their edges, and they join up to form a compact, 2-dimensionalorientable invariant-manifold M a without boundary whose genus is given byi=pN Lmi-11+-2 .n·i=l aM a is therefore a g(Ma)-handled sphere. When a given flow is transfered fromthe polygon to the g-handled sphere, it is found that where there used to bevertices there are now saddle points. Each vertex Vi gives rise to N /ni multi-saddle points of index 1-mi, saddle points with mi incoming and mi outgoingseparatrices.The invariant manifolds of the exceptional flows cover the polygon N times.They have a boundary, they mayor may not be orientable and they are oftenreferred to in the 'literature as invariant knots.The M a are all of dimension two in the four dimensional phase-space. Thismeans that there exists a second conservation law and for this reason thesesystems are called pseudo-integrable or quasi-integrable. Special cases of thes<=:are the almost-integrable billiards. They are more integrable than the pseudo-integrable billiards [8]. Finally there are the integrable billiards. These arisewhen the vertices of the polygon are such that mi = 1 for all i = 1, ... ,po Insuch cases 9 = 1 and the invariant manifolds are tori. There are only a smallnumber of integrable billiards, these are the rectangles and the triangles withvertex-angles {7r/3,7r/3,7r/3}, {1r/2,7r/4,7r/4}, and {7r/2,7r/3,7r/6}.The simplest case of a rational billiard with 9 = 2 is the 7r/3-rhombus billiard.In this case the polygon has four sides with opposite angles 7r/3 and 27r/3. Thecircle maps corresponding to collisions with the sides are¢J -+ 41r/3 - ¢J.Under the action of these transformations, any initial direction of momentum ¢Jwill become either one of±¢J, ±¢J + 27r/3, ±</> + 47r /3.These values degenerate for the exceptional flows </> = °or 27r/3, into 0, 27r /3and 47r/3.. The generic orbit therefore has length six and the invariant manifoldcorresponding to a generic flow covers the rhombus six times.- 299-To construct these manifolds from six copies of the rhombus, lay down thefirst copy labelling all of the vertices and all of the sides. Reflecting the rhombusin one of the sides uses up one of the remaining five and induces a labelling ofits sides and vertices. The original rhombus has become a polygon made of twoglued rhombi all of whose vertices and edges are labelled. Now choose one ofthese rhombi, and one of its exterior edges, and reflect the chosen rhombus inthis edge. This uses up another one of the copies and induces a labelling of itsvertices and edges. Continue in this way until all six copies are used up. Thelabelling on the exterior edges and vertices tells you how to glue these togetherinto a closed compact orientable two-dimensional surface.There is no unique way of doing this, but the resulting surface is unique -in this case a two-handled sphere. Furthermore the total angle around the eachobtuse vertex will add up to 47r testifying to the presence of negative curvatureconcentrated at that point. If you go to even more trouble and draw the flowlines on each of the copies of the rhombus, you will find saddle points in placeof these vertices. Each with two ingoing and two outgoing separatrices.Although a little tedious, it is ~ell worth the trouble to verify all of this atleast in the case of the 7r13 rhombus. Further hints and contructions can befound in [8] or [11]. The picture which should emerge is that the flow in phase-space of a free particle moving inside the 1("13-rhombus and colliding elasticallywith its walls, is equivalent to a flow on the surface of a two-handled sphere.This flow has a pair of ordinary saddle points. It resembles an integrable systemin the sense that the motion is confined to a two dimensional ~urface in the fourdimensional phase-space. It can also be shown that in such systems neighbouringtrajectories do not diverge exponentially as they do in chaotic systems, and thatthe Kolmogorov entropy is zero. The only hint of irregularity is the higher genusof the invariant surface and the consequent presence of saddle points which causebifurcations in the flow.4 QuantisationThis simply means solving the Schrodinger equation on the polygon8~x~(x,y)+8;y~(x,y) +E ~(x,y) = 0subject to the vanishing of ~(x, y) on the boundary. Richens and Berry haveconsidered this problem for the 7r13-rhombus [11]. They note that it is easy tofind half of the solutions, those which vanish along the short diagonal. Thesesolutions correspond to energy levels deduced from a WKB argument. Theypoint out that the WKB method is highly inadequate for higher genus surfaces.Analytic solutions have still not been found for the other eigenfunctions of the7r13-rhombus, although they have been studied numerically by Biswas and Jain[9]. In the absense of analytic solutions, and in view of the failure of the WKBmethod even to provide useful asymptotic information, other strategies have- 300-been tried. These are based on periodic-orbit methods. Mathematically theconnection between periodic orbits of a classical dynamical system and eigen-values of the Laplacian on a Riemann surface is established by Selberg's traceformula, a readable account of which has been provided by Mc Keane [12]. Theimportance of periodic orbits for understanding both classical and quantum dy-namics can easily be appreciated from an article by Cvitanovic [13]. A moregeneral introduction to this subject can be found in [2] and [3]. With a viewto applying these techniques to the case of the 1r13-rhombus, its periodic orbitshave been studied by Jain and Parab [14] and Parab and Jain [15).Apart from the question of finding analytic solutions, numerical work on thefull set of eigenvalues has shown that even for the 1r13-rhombus the asymptoticdistribution of energy level spacings closely fits the GOE [9]. As one moves upthrough higher genus this fit quickly improves [10]. Richens and Berry havealso considered a 'square torus billiard' [11]. In this case the flow in phase-spaceis equivalent to a flow on a 5-handled sphere which has 4 saddle-points, eachwith 3 ingoing and" 3 outgoing seperatrices. Numerically they found that theenergy level spacings exhibit the phenomenon of level-crossing avoidance. Boththis and GOE statistics of the level spacings are properties asscoiated with thequantum mechanics of classically chaotic systems.5 ConclusionTo conclude - there exists a class of flows on the g-handled spheres which arerelated to the rational billiards. Classically they are very like regular fully inte-grable systems, whereas quantum mechanically their properties are those asso-ciated with the classical chaos. In each case it is possible to consider solving theSchrodinger equation on the g-handled sphere, however an easier although notidentical problem is to solve it on the polygon from which the g-handled sphereis constructed. In general the WKB method is of little use computing the energylevels. The simplest non-trivial case is the 1r13-rhombus. This corresponds to aclass of flows on a two-handled sphere. In this case the WKB method providesonly half of the eigenvalues. The corresponding wavefunctions are analytic. Asyet the full quantum problem remains unsolved.References[1] B. Eckhardt: Phys. Rep. 163. p.205-297 (1988): Quantum Mechanics ofClassically Non-Integrable Systems.[2] Martin C. Gutzwiller: Chaos in Classical and Quantum Mechanics. Inter-disciplinary Applied Mathematics Vol.l. Springer Verlag 1990.- 301-[3] Ozorio De Almeida: Hamiltonian Systems - Chaos and Quantisation. Cam-bridge Monographs in Mathematical Physics 1988.[4] F. Steiner: Desy 94-013 Jan. 1994 - chao-dyn/9402001: Quantum Chaos:[5] J. Ford & G. Mantica: Am. J. Phys. 60 (12) p.1086-1098. Dec.1992: DoesQuantum Mechanics Obey the Correspondence Principle? Is it Complete?[6] M. V. Berry: Ann. Phys. (N.Y.) 131 p163-216 (1981): Quantizing a Clas-sically Ergodic System - Sinai's Billiard and the KKR Method.[7] C. Godbillon: Dynamical Systems on Surfaces - Springer Verlag 1983[8] E. Gutkin: Physica 19D (1986) p.311-333: Billiards in Polygons.[9] D. Biswas & S. R. Jain: Phys. Rev. A Vo1.42 NO.6 p.3170-3185 Sept. 1990:Quantum Description of a Pseudointegrable system - the 7r /3-RhombusBilliard.[10] Taksu Cheon & Thomas D. Cohen: Phys. Rev. Let. Vol.62 No.24 p.2769-2772 June 1989: Quantum Level Statistics of Pseudointegrable Billiards.[11] P. J. Richens & M. V. Berry: Physica 2D (1981) p.495-512: ·PseudintegrableSystems in Classical and Quantum Mechanics.[12] H. P. McKeane: Comm. Pure and Applied Mathematics, Vol. XXV p.225-246 (1972): Selberg's Trace Formula as Applied to a Compact RiemannSurface.[13] P. Cvitanovic: Physica D 51 (1991) p.138-151: Periodic Orbits as the Skele-ton of Classical and Quantum Chaos.[14] S. R. Jain & H. D. Parab: J. Phys. A: Math & Gen. 25 (1992) p.6669-6683:Periodic Orbits of an Almost Integrable System.[15] H. D. Parab & S. R. Jain: Phys. Rev. E. Vol.47 I p.R776-9 1993: Law ofProliferation of Periodic Orbits in Pseudointegrable Billiards.- 302-

Quantisation on the Two-Handled Sphere

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Quantisation on the Two-Handled Sphere

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