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Journal of Modern Physics, 2013, 4, 950-962 http://dx.doi.org/10.4236/jmp.2013.47128 Published Online July 2013 (http://www.scirp.org/journal/jmp) Noneuclidean Tessellations and Their Relation to Regge Trajectories B. H. Lavenda Università degli Studi, Camerino, Italy Email: bernard.lavenda@unicam.it Received January 14, 2013; revised February 17, 2013; accepted March 14, 2013 Copyright © 2013 B. H. Lavenda. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT The coefficients in the confluent hypergeometric equation specify the Regge trajectories and the degeneracy of the an- gular momentum states. Bound states are associated with real angular momenta while resonances are characterized by complex angular momenta. With a centrifugal potential, the half-plane is tessellated by crescents. The addition of an electrostatic potential converts it into a hydrogen atom, and the crescents into triangles which may have complex con- jugate angles; the angle through which a rotation takes place is accompanied by a stretching. Rather than studying the properties of the wave functions themselves, we study their symmetry groups. A complex angle indicates that the group contains loxodromic elements. Since the domain of such groups is not the disc, hyperbolic plane geometry cannot be used. Rather, the theory of the isometric circle is adapted since it treats all groups symmetrically. The pairing of circles and their inverses is likened to pairing particles with their antiparticles which then go on to produce nested circles, or a proliferation of particles. A corollary to Laguerre’s theorem, which states that the euclidean angle is represented by a pure imaginary projective invariant, represents the imaginary angle in the form of a real projective invariant. Keywords: Tessellations; Reggie Trajectories 1. Introduction Poincaré discovered his conformal models of hyperbolic geometry in an attempt to understand whether there were solutions to the hypergeometric equation of higher peri- odicities than the then known circular and elliptic func- tions (of genus 0 and 1, respectively). The names, Fuch- sian and Kleinian functions, which he coined, belong to a certain class of automorphic functions that live on tiles that tessellate the half-plane or disc, depending on which model is chosen. The indicial equation is a solution to a differential equ- ation in the neighborhood of a singular point. An equa- tion of second-order can have at most three branch points, which are conveniently taken to be 0, 1, and ∞. The use of a matrix to describe how an algebraic function is branched had been introduced by Hermite, but it was Riemann who first considered products of such matrices. Frobenius showed that the hypergeometric equation is completely determined by its exponents at its singular point. If the only effect of analytically continuing two solutions around a singular point is to multipy them by a constant, then the differences in the exponents must all be integers, without the solution containing a logarithmic term. In other words, the matrices are rotations about each of the singular points where performing a complete circuit multiplies the solution by a constant factor. We will generalize these rotation matrices through real an- gles to complex ones, and in so doing elliptic transforma- tions will become loxodromic ones. The monodromy matrices of Riemann are generators of a group, and are either homothetic (magnification) or rotation. These are related to the hyperbolic and elliptic geometries, respectively, both of which preserve the unit circle. According to the Riemann mapping theorem, any arbitrary region bounded by a closed curve can be map- ped in a one-to-one fashion onto the interior of a unit disc by an analytic function. Riemann arrived at his theorem as an intuitive conjecture in 1852, and it is hardly com- prehensible that it took almost fifty years to prove it. According to Schwarz, the only biunique analytic map- pings of the interior of the unit disc onto itself is a linear, fractional (Möbius) transform of the hyperbolic or ellip- tic type; that is, magnification or rotation. The relevant equation for quantum theory is the con- fluent hypergeometric equation, where two the branch points of the hypergeometric equation merge at ∞ to be- C opyright © 2013 SciRes. JMP B. H. LAVENDA 951 come an essential singularity. The origin is the regular singularity. The two parameters of the equation deter- mine the Regge trajectories, and the degeneracy of the angular momentum states. The Regge trajectories express the angular momentum in terms of the energy, and if the potential is real, or the energy is greater than the potential energy, the angular momentum will be real and discrete. The parameter that determines the trajectories discrimi- nates between quantized motion of bound states and un- stable resonance states. In the former case it is negative, and identified as the radial quantum number, whereas in the latter case it can be associated with a complex angle. In the case of the nonrelativistic coulomb interaction, the Bohr formula for the energy levels of the hydrogen atom result when the parameter is set equal to a negative inte- ger, which is also the index of the Laguerre polynomials that represent the radial component of the wave function. All this can be obtained without the usual procedure of expanding the radial wave function in a series and ter- minating it at a certain point to obtain a quantum condi- tion. Rather, if the energy becomes positive, the potential complex, or the potential energy greater than the total energy, quantization does not occur, and the parameter represents an angular point whose homologue is branch point. The new point is the generalization of the angular point to complex values. This is not unlike the generali- zation of the scattering amplitude to make it a function of the angular momentum. In order to make the scattering amplitude a function of the angular momentum, it had to be made complex and continuous so that one could take advantage of Poincaré’s theorem which says that if a parameter in a differential equation, the angular momen- tum, or the wavenumber, appears only in the analytic function in some domain of the parameter, and if some other domain a solution of the equation is defined by a boundary condition that is independent of the parameter, then this solution is analytic in the parameter in the do- main formed from the intersection of the two domains. In other words, Poincaré had shown that, under suitable conditions, the smooth solution to the differential equa- tion could be made an analytic function of the parameters of that equation by allowing them to become complex and continuous, instead of real and discrete. The reason for naming the trajectories after Regge [1] was that he brought Poincaré’s theorem to the attention of high energy physicists. Instead of confining his atten- tion to integer angular momenta, Regge transformed the scattering amplitude so that it became a function of the, continuous, angular momenta. In order to do so, he had to allow it to become complex. Regge’s idea was not new, it had already been used by Poincaré himself, and Nicholson in 1910, to describe the bending of electro- magnetic waves by a sphere. Sommerfeld and Watson used it to describe the propagation of radio waves on the surface of the earth, and their scattering from various potentials. It has become known as the Sommerfeld- Watson representation. The new, unphysical, regions provided proving grounds for speculative high energy physics. The passage from a real to a complex parameter is not nearly as radical as that from a discrete to a continuous one, or from a posi- tive to a negative one. How does one define negative angular momenta? The angular momentum is represented in the equation for the energy as a centrifugal repulsion. At a constant attractive potential, the only way the en- ergy could be made more negative, thereby allowing more bound states to be formed, is to convert a centrifu- gal repulsion into a centripetal attraction by allowing the angular momentum to become negative. The limit occurs where the two indicial solutions to the Schrödinger equa- tion with a centrifugal term coincide. One is called the regular solution because it goes to zero at the origin, while the other is the irregular solution because it blows up there. Conventionally, the latter solution is rejected because any admixture of the two would not lead to a unique solution. However this is incorrect because both indicial exponents determine how the ratio of the solu- tions, which is an automorphic function, transform. An automorphic function is a periodic function under the group of linear (fractional) substitutions. When Po- incaré came on the scene in 1880 the only two periodic functions that were known were the trigonometrical and elliptical functions. By cutting and pasting edges of the fundamental region together one could get solid figures with different amount of holes, or genus. Trigonometric functions had no holes, elliptic functions, one hole be- longing to a torus, and Poincaré wondered if there were automorphic functions with a greater number of holes. Any given point of the fundamental region would be transformed into the same point in an adjacent funda- mental region by the linear fractional transformation. It would not connect points in the same fundamental region, for, otherwise, it would not be “fundamental”. For instance, if the angular momentum is negative and in the interval 1 0, 2 , the plane would be tessellated by crescents formed from the intersection of nonconcen- tric circles whose angle would be the degeneracy of states. It is precisely in the unphysical region that the greater than unity cosine has become a hyperbolic cosine with a complex angle. The crescent of the plane with a given angle will be successively transformed by the fractional linear transformation ultimately returning to itself. Thus, the entire plane is divided into portions equal in number of the periodic order of the substitution. When two particle collide there is a scattering angle, , whose cosine resides between and 11 in the Copyright © 2013 SciRes. JMP B. H. LAVENDA 952 physical region. However, by allowing cos to go to either plus or minus infinity, enables one to consider in- finite momentum transfer. Such large momentum transfer occur over extremely small distances. Thus, large cos is adapted to the study of strongly scattered waves that can bind and resonate. Poles can be expanded in a bilinear series of the pro- ducts of Legendre functions of the first and second kinds. Only when the pole is within a given ellipse will the series converge. The ellipse is determined by the trace of the monodromy matrix which is a hyperbolic cosine with a complex argument. The hyperbolic cosine of the real component is the semi-major axis of the ellipse, while the imaginary component is the eccentric angle of the ellipse. In the case of coulomb scattering the real com- ponent is the ratio of the charges to the velocity of the incoming particle so that the ellipse will be larger the smaller the velocity of the incoming particle. This is referred to as the classical region. As the velocity in- creases we are transformed into the relativistic region with a decrease in the size of the ellipse. What is a complex angle? In optics, complex angles arise when a refractive wave does not penetrate into the second medium, but rather, propagates parallel to the surface. The system is then said to suffer total internal reflection. There is no energy flow across the surface, and at the angle of incidence there is total reflection. For angles of incidence greater than the critical value, the angle of reflection becomes complex with a pure imagi- nary cosine meaning the wave is attenuated exponentially beyond the interface. The three poles of the second order differential equ- ation are associated with the thresholds of particle crea- tion in high energy physics [2]. Their residues are given by the angles of a triangle which tessellate the complex plane. The angles themselves may be complex like the argument of the hyperbolic cosine above. In order to tessellate the plane it must reproduce itself by rotating about any given axis. When the angles become complex, it will not only rotate the sides of the triangle but will also deform them. The fact that any two angles are complex conjugates, related to source and sink, will render the sum of the angles real but, may not be equal to . For Regge trajectories these angles represent the com- plex angular momentum. Below threshold the imaginary component of the angle momentum vanishes for there is no state to decay into so the resonance width is zero. Regge gave an interesting interpretation to the imaginary component of the angular momentum. Just as the longer the time the smaller the uncertainty in energy, so that long time uncertainty is related to small resonance widths, imaginary angular momentum is related to change in angle through which the particle orbits during the course of a resonance. For extremely long resonances, the angle of orbit is large until it becomes permanent in a bound state. Triangle functions which tessellate the complex plane are described by automorphic functions which represent solid figures. The automorphic functions are the inverses of the quotients of the two independent solutions to a second order differential equation. These quotients can be moved around the complex plane by linear fractional transformations which delineate the fundamental regions. Schwarz showed how these automorphic functions map one complex plane onto another, just like the hyperbolic cosine maps ellipses in one plane onto circles in another. The angles are either the interior, or exterior, angles of the triangle which is the fundamental region. In the case of large, but real, cos , the automorphic function is none other than the expression for the Legendre function of the second kind at a large value of its order, the angular momentum [3]. It tessellates the surface of a sphere with triangles whose bases lie on the equator of the sphere, each angle being radians, and one vertex at the north pole whose angle is proportional to the diffe- rence between the order of the Legendre function and the angular momentum of the Regge trajectory. The solid is a double pyramid, or a dihedron, whose triangles have sums greater than radians, and, therefore, belong to elliptic geometry. In fact, this automorphic function has been proposed as a partial wave scattering amplitude. Another proposal was made by Veneziano [4] who showed that the Euler beta integral satisfies the duality principle of high energy physical where the scattering amplitude remains the same under the exchange of total energy and momentum transfer. Experimentally, this is achieved by replacing the particle with its anti-particle. It so happens that the beta integral is the automorphic function of Schwarz for tri- angle tessellations. The angles are the Regge trajectories which become complex above threshold. In fact, a beta integral with real arguments could not represent a com- plex scattering amplitude for it would be physically measureable being the distance between the angles in the triangle. The Veneziano model, which has served as the impetus of string theories, was found to be wanting in the hard sphere limit because it did not reflect the granular, or parton-like, behavior observed in deep inelastic scat- tering experiments. What is the use of automorphic functions in high en- ergy particle physics? First, it provides restrictions on the nature of the complex Regge trajectories and on the na- ture of the potentials. The potentials must be real for bound states, complex, or imaginary, for resonances. Second, the possibility of their being a complementarity between continuous groups in quantized systems and discrete groups with a continuous range of non-quantized Copyright © 2013 SciRes. JMP B. H. LAVENDA 953 parameters for bound states and resonances. Third, the spectrum of resonances that lie along a Regge trajectory is likened to the nesting and proliferation of circles when the number of generators is increased. 2. Nonrelativistic Coulomb Interaction The confluent hypergeometric equation arises from the confluence of two singularities in Riemann’s hyper- geometric equation leaving the regular and irregular sin- gularities at 0 and ∞. It can therefore describe an infi- nite-range potential like the coulomb potential for which it is given by1 2 2 dd 10 d d ca rrr r 21,c 1,ai . (1) The parameters for the coulomb interaction are (2) and (3) where is the total angular momentum, 22 Z Ze E is the coulomb parameter which is negative if the charges Z e and Z e E 1cm an n 0E are opposite, and is the energy of the incoming particle in units . The parameter a determines the nature of the tra- jectories. If , the system has bound states where is the radial quantum number, and the total energy, . This specifies the Kummer function as a Laguerre polynomial of index . Specifying the Regge trajectory (3) avoids the introduction of a series ex- pansion in the Schrödinger equation, and imposing a cut- off. When n 0 , the solution to Equation (1) reduces to a product of an exponential function and a hyperbolic Bessel function. The confluent hypergeometric Equation, (1), can be easily converted into the Schrödinger equation, 22 22 d11 41 0, 4 drr ir (4) by the substitution 11d 2 ecr r 21 0r 1, . , where Alternatively, if we didn’t know Equation (1), we could transform (4) into it by inverting the substitution. The indicial equation as is A rBr (5) where the constant A is conventionally set equal to zero in order for not to diverge at the origin. How- ever, at 1 2 r both terms give the same dependence upon . It is precisely at 1 2 B where the regular, , and irregular, A , solutions in Equation (5), coincide. There is no reason to constrain the angular momentum to positive integral or semi-integral values since we are considering “elementary” and “composite” particles, which may be stable or unstable. We may look for a solution to Equation (1) in the form of a Laplace transform [6], 2 1 ed, r r (6) for the (normalized) wavenumber, . Introducing it into (1) results in 2 1 2 1 2 1 d 1ed d 1e d1d, d r r ca ca after an integration by parts has been performed. If the limits 1 and 2 can be chosen so as to make the integrated part to vanish, then will satisfy d10. d ca 1 11, ca a C C 1 11ed, ca ar rC The solution to this first order equation is (7) where is a constant of integration. In view of the Laplace transform, (6), we find (8) where, if the contour is not closed the integrand in (8) is required to have the same value at the endpoints. In contrast to the original confluent hypergeometric Equation (1), which has branch points at 0 and , we have added an additional branch point at 1 by con- sidering the wave number. This branch point may be thought of as placing a bound, 1This is formerly identical with Equation (26) on page 52 in Ref. [5]. However, the definition of 2,pE p on the momentum by the square root of twice the total energy, like the maximum momentum of a Fermi gas of elementary particles at absolute zero [8]. there is real. The same expression can be found in Ref. [6]. But then the quantization condition an is complex [6, Equation (27) p. 156]. Since E 2 is imaginary for bounded states, and not the velocity , the quantization condition is real. Rather, in Ref. [7] the condition is taken as the condi- tion of the poles of the -matrix, giving the position of the v an S n Re gg e p ole. Copyright © 2013 SciRes. JMP B. H. LAVENDA 954 The difference between (2) and (3) determines the second angle as 1,i a y y ca (9) which is the complex conjugate of (3). We will appreci- ate that attractive and repulsive coulomb potentials al- ways appear as complex conjugates, or, equivalently, for every source there is a sink. If 1 and 2 are any two particular solutions to the hypergeometric equation, then the Wronskian is given by [9] 1 1, cab rr 2 12 21 dd dd c yy yyC rr where C is a constant. Dividing both sides by 2 y , it be- comes the derivative of the ratio 12 of the two particular solutions. Now, any other two solutions, say and can be expressed as linear combinations of and , viz., wyy 1 y2 y 1 y2 y 112 yyy 212 yyy so that the quotient of the new solutions, 12 is related to the quotient of the old, , by a linear fractional transformation wyy w . w ww (10) When the quotient of the solutions is inverted, we get a function automorphic with respect to a certain group of the linear fractional transformations. For the hyper- geometric equation, the group of automorphisms are tri- angular tessellations of the unit disc. Our interest will be focused on the momentum space at in (6). Since the hypergeometric equation with the coefficient , 0r 0b 2 2 11 d 22 dd 0, 1d YY aa (11) has one solution, 1 11 d, a t t 20 ,, a Yac Ct (12) which is an incomplete beta function, ,;Baa , if the constant of integration is set equal to unity. The second solution 12 ,YwY 1 w (13) is given by an automorphic function . Dividing the Wronskian, 1 11, ca 2 2 Y 12 21 dd dd a YY YY C 1 1 2 2 1, a a wC Y by , it becomes the derivative of the ratio, viz., which has the Schwarzian derivative, 2 22 22 3 ,: 2 11 11 , 1 221 ww www aa aa (14) where the prime now stands for the derivative with respect to . With the transformation, ln1 ln 1 e, ca YY 0,YIY (11) can be converted into (15) 2,Iw where , with the Schwarzian derivative given by (14). The Schwarzian derivative has a long and glorius history dating back to Lagrange’s investigations on stereographic projection used in map making [10]. The third angle can be read off from the Schwarzian, (14), and is 1,c (16) which is the original angle of the crescent, having branch points at 0 and . This is due to the centrifugal poten- tial in the Schrödinger equation. The coulomb potential introduces the complex conjugate angles, and aa, which make the interaction independent of whether it is attractive or repulsive since they appear symmetrically. If we adhere to the triangle representation, the requirement that the angles be less than limits to the closed interval 1,0 2 . In this interval, centrifugal repulsion 2 1r becomes centrifugal “attraction”. For 1 2, the sum of the angles of the triangle, 1 2i , 1 2i , and 0 is . [11] The analytic function ,;Baa given by (12) for maps the upper half-plane 1C 0 a onto the interior of a half- strip formed by the two base angles and a corre- sponding to the points 01 and in the - plane. The distance between the two vertices in the - plane is B 11 1 0 11 1d22 cosh a a ttt ii (17) Copyright © 2013 SciRes. JMP B. H. LAVENDA 955 In general, the amplitude will be given by the com- plete Beta function 11 . 21 ii ,;1=Baa (18) otentials. In the attractive case, there is an infinite number of Reggie poles determin poles in the numerator of (18) [7], i.e., The beta function is symmetric with respect to attract- tive and repulsive p ed by the 2 of the numerator 8) are equasite so alyticity of thde. In the tial, with Eation (18) tells us there there pole esp n t po 1. 2 ZZ en E (19) These are bound states with 0E. In the case of resonances, 0E, the imaginary parts of (1 an poten wi l and oppo e amplitu 0, Equ to as to preserve the case of a repulsive ll again be an infinity of Regge s. However, these will not corrod to bound states or resonances be- cause the Regge poles are restricted to the left half of the plane. This information is contained in the amplitude, and it is equivalentwo S-matrix elements, one being the inverse of the other [7]. Parenthetically, we would like to point out that the Regge pole behavior for an infinite-range potential is quite different than that for short range ones. A Regge trajectory for a short-range tential would have the form 1,E EE (20) where the prime stands for differentiation. The intercept of 1 is called for by the form of the angular momen- tum term 1 in the Schrödinger equation. The fact that the cross-sections do not vanish asym the energy, but increase slowly with it is at r, a gula ptotically with tributed to an exchange of a Reggeon whose intercept is +1 [12]. How- eve positive intercept cannot be interpreted as origina- ting in the anr momentum. This is substantiated by the fact that only negative intercepts give rise to the con- servation of angular momentum in hyperbolic space [13]. In Equation (20) there would be no violent jump in the trajectories as the energy passes through zero. In the case 0E the poles are complex, but there is no violation of analyticity since they are complex conjugates. In the asymptotic case of large angular momentum the ampli- tude (18) will be modulated by oscillations, viz., 2 1 ,;1 e. i i Baa i The Schwarz-Christoffel transform remains valid even when one of the vertices of the triangle coincides with the point at infinity. The lengths of the sides from either vertex to the vertex at infinity are infinite, so that 11 22 0 ,;1 d ii i Baattt (21) 3. Generators from Monodromy Relations If 2 0 lim ,I 0 1d 1 tt ttt will map the upper half-plane 0 onto the interior of the “half-strip” shown in Figure 1. (22) the indicial equation for the branch point 0 is nn 10. The two unequal roots, 1 n and 2 n, are the exponents e integrals of the equation 1 1 n Y of th so that ratio of the solutions is 2 2, n Y 12 12 nn wYY quadratic form is 2 14 ,a we have (23) Since the discriminant of the 2 2 11 4 a I in view of (22). en as 1i w The ratio of the two solutions, (23), can be writt (24) ing to Riemann, the exponents of the indicial equation, Accord 1,2 11, 2 na (25) Figure 1. The half-strip obtained by the conformal mapping Equation (21). Copyright © 2013 SciRes. JMP B. H. LAVENDA 956 “completely determine the periodicity of the function” [14], in reference to the solutions of the hypergeometric equa- tion. Equation (25) shows that the difference in roots of the indicial equation, or exponents as they are commonly referred to, is the angle at the singular point. The mono- dromy relations imply that the roots to the indicial equ- ations are rational numbers, which, in turn imply elliptic generators. Here, they are generalized to complex num- bers so as to allow for loxodromic generators, and take as a definition of monodromy as the invariance of auto- morphic functions, or functions inverse to the quotint of two independent solutions of a differential equation, e under a certain group of transformations. A circuit around in the positive direction that returns 1 Y as 1 2 1 ein Y , while it returns 2 Y as 2 2 2 ein Y , is performed by 1 2 2 2 e0 . 0e in in The monodromy theorem asserts that any global analy- tic function can be continued along all curves in a simply connected region that determines a single-valued analytic function on every sheet of the branch points, one sheet per branch point. The monodromy matrix about , 2 2 e0 , 0e i i (26) has unittermind trace denant, a 2cos2, so that it is an elliptic transformatio Rather, around 0 , the monodromy matrix, Tr n. e0 , 0e ii ii (27) also has unit determinant, but trace, Tr 2cos ,i (28) so that it is a loxodromic generator around 1 , where , 0e ii (29) ose . This is also true e0 ii wh trace is cos .i (30) The product of any two matrices, or thei yield the third, or its inverse. Stated slightly differently, a points in a positive (anti-clockwise) direction will give a third branch point in the negative (clo atrices are abelian since they have the same fixed points, 0 and Tr 2 r inverses, will circuit around two of the branch circuit around the ckwise) direction, since is the unit matrix. The m . and are loxodromic be- cause their traces, (28) and (30), are com Loxodromic transformations do not leave the disc in- sfer the inside of the disc to the outside The generators, (27), (29), and (26 points 0 and ∞ in common. Conjugation with the Cayley m plex and 2. variant: they tran of its inverse. ), have the fixed apping, , zi zzi (31) carries these fixed points to 1 and 1, respectively, while se, 1 its inver , takes them to i and i , re- sp ca 1 cosh si sinh cosh ii ii i s to ww carries points in the interior of w I to the exterior of w ectively. For instance, the conjugate generator of (27), 1 coshsinh , sinh cosh ii ii (32) rries the fixed points 0 and ∞ to −1 and 1, respectively, while those of the inverse conjugation, nh , i (33) carry the fixed point i and i, respectively. The loxodromic transformation, (32), pairs the iso- metric circle, I, with its inverse, I. That is, I. The fixed points of (32), 1 and 1, lie i w n w I and I, respectively. With 0 , (32) is pure stretching, pushing points from 1, the source, to 1, the sink. The cyclic group consists of one generator, and apply- ing it n times gives coshsinh , sinh cosh nnn nn (34) for 0 . Whereas the isometric circle of (32) is sinhcosh 1, z (35) the isometric circle of (34) is sinhcosh 1.nz n (36) he isometric circle, (35), has its center at T coth , and radius 11sinRh . The isomec circle, (36tri), on the other hand, has its center at coth n , and radius 1sinh n Rn . Since sinh sinhn , it follows that 1n RR. And since this is true for any 1n, the isometric circles will be neste another, becoming ever smaller until the limit point is hed. xposed d inside one reac The loodromic generator, (33), can be decom Copyright © 2013 SciRes. JMP B. H. LAVENDA 957 into a product, sinh cosh coshsinh cos i ii i i sinhcoshsin cosi sin cosh sinhii of hyp with the same fi nts. The isometric circles of erbolic, , and elliptic, , transformations, xed poi s inverse, 1 , and it sinhcosh 1,iz have their centers on the imaginary axis at 0cothzi , and radius 1sinh . Part of the ental on for the grted by [15]. In other words, it does not detic transformation Since the isometric circle is defined by plane exterior to these two circles is the fundam oup genera pend on the ellip regi . 21, sinh coshiz and that of its inverse by 2 1nh 2 si cosh sinh cosh iz iz , , whatever is inside the isometric circle of sinhcosh 1,iz i.e., 1 is outside of its inverse, because 11 , and vice versa. Let and be the generators whic ometric circles, w I with its inverse, h pair off the is w I, and v I , respectandwith v Iively. The matrices will al to one anot be externher provided cosh , and 2 2. pairs circles coth The matrix, (32), has fixed p oints 1 with centers, radii, , and the same sinh , on the real will be located in the circle on the negative axis, while the attracting fixed point +1, which is a sink, will be located axis. The fixed point 1 in the circle on the positive axis, since 1 ta 1 2 nh for whatever value happenbe. rtherm the iso s to Fuore, let be associated with- metric circle z I. If Iternal to one an- ot en v her th and w I are ex z I isn w I [15, p. 53, Thm 12]. For sup- pose that the circles are not tangent to oneother, th if p a point outside of, and not on, , the generator i an en is v I will carry the point p into, or on, v I, say p with a decrease in length, or at least no change in length. Since p is w I, will transforoutside of m it with a decreaseine in length. Consequently, the combd opera- tion, , will transform p with a decrease in length, implying the p is outside of z . And since every point on or outside of v I is also outside of z I, the latter sting in a neith their v I must be inside the former. Each time we add a generator, we get a ne sting of circles w proliferation [16, p. 170]. The isometric circle will contain three nested circles, z I, and z Ifor the generator , and another for . This is shown in Figure 2. Ech of the other three discs will also have three sted discs. Increasing th generator by e, so that there are now three generators, or “letters”, there will be three nested discs in each thormer discs, and so on. Thus, there would be no limit of an elementary particle, but, rher, particles withn particles within particles and so on. There may result inhigh energy collisions additional particles to those of the compound particle disintegrating into its compon a nee on ofe f at i ent parts because there m lativ tio ometric ci geneand one for its in- ve ay be sufficient energy that can be converted into mat- ter before disintegrating again into other forms of mat- ter. Moreover, the reistic phenomenon of pair crea- n may be related to pairs of isrcles, one for the rator of the transformation, rse. 4. The Coulomb Phase Shift as a Projective Invariant The absolute conic for elliptic geometry is the null conic. It is defined in projective coordinates by an equation with real coefficients, but it is composed exclusively of imagi- nary points. The secant through the points 1 k and 2 k join the conic at i and i. The cross ratio is 2If the inequality becomes an equality, it is treaded by the example Ref. [16] which is the condition that the four circles are tangent to one an- other. The trace of the commutator is –2 indicating that the two fixed p oints have coalesced into one at the point where the circles touch. Both groups are Fuchsian since their limit points are either on a line or on a circle. igure 2. The nesting of circles in the isometric circle and its inverse. F Copyright © 2013 SciRes. JMP B. H. LAVENDA 958 12 12 21 2 ,;, 1tan e, 1tan i kk ii kiik kk kiikkk i i 1221 1221 1 1 ikk ikk (37) tan where tan 21 , and tan ii k , for er a real conic with imagi- 1 ik and 2 ik , so that their join cuts the ab- solute at and 1. The cross ratio is now 1, 2i. Equivalently, we can consid nary points, 1 12 211221 2 ,;1,1 11 1 1tan e, 1tan i ikik ikikk kikk i i which is the complex conjugate of (37). Thus, the eucli- dean angle, 12 1221 11 1ikikk kikk (38) Figure 3. The real conic, tangents, pole P, and polar. 1 1 0 0 tan , j jj (42) which is none other than a generalized Breit-Wigner expression [20] in the neighborhood of a resonance where the angular momentum, j, stands in the for resonance energy, and 2 1 2 , ; 22 kki iikik ii (39) ca ar he (3any valued quantities. In order to associatent 12 kk with the logarithm of the cross ratio, a pure imaginary absolute constant must be chosen [17]. Conjugacy with respect to a polarity general pendicularity with respect to an inner product thus allow- ing euclidean geometry to be defined from affine geo- metry by singling out a polarity. [18] The imaginary t poi1 and 1hown in Figure 3. The coulomase shift, 12 11 ln,;,ln1,1, n be expressed in terms of a cross ratio, and, hence, is a projective inviant. T logarithms of the cross ratios, (37) and 8), are pure imaginary and m is the width. It was Laguerre’s great achievement to define eucli- dean geometry from affine geometry by singling out a polarity. The projective invariant (42) is a projective in- variant in that it expresses an euclidean angle directly as the logarithm of the cross ratio [18]. It seems odd that the same name, Laguerre, should be associated with both the orthogonal polynomials when a is a negative integer in the an eometry from affine geometry through a projective invariant. 5. Relativistic Coulomb Interaction ts e a segm izes per- points, 1 ik and 2 ik , lie on the polar whose pole, P, is determined by the point of contact of two tangent lines to the real conic ants , as s b ph , which determines pure- ly electrostatic, or Rutherford, scattering is also a pro- jective invariant. It is defined by [19] 0 1 2 2 0 ! ee. ! ji i j iji iji (40) Transposing and taking the logarithm of both sides give 0 0 1 1 0 1ln 2 tanh . j j j j ji iji i j (41) The coulomb phase shift ius givy coulomb interaction and the derivation of euclide g The relativistic generalization of the nonrelativistic cou- lomb interaction is given in this section. By specifying the coefficienin the confluent hypergeometric equation, we can obtain Dirac’s expression for the energy of the hydrogen atom from the Klein-Gordon equation instead of from the Dirac equation. The coefficients in the confluent hypergeometric equa- tion for the relativistic coulomb interaction are 2 222 11 22 aiEEm (43) 2 2,c 2 11 22 (44) s then b Copyright © 2013 SciRes. JMP B. H. LAVENDA 959 where Z , is the fine structure constant, and we have reinstated the mass m. If 0a and Em, we have scattering, and the second angle will be 2 222 11 . 22 caiEEm (45) The two base angles, corresponding to the branch points at 0 and 1 in the -plane, are, again, complex conjugates, the + and – signs, before the energy term (43) and (45), correspond, respectively, to an attractive and repulsive coulomb potential. The third angle is 2 1 12 2 c to 2, (46) and in order it be , the second inequality in for 22 11, 4 (47) has to be satisfied. The other inequality is the con that (46) is real. The upper bound converts a repu centrifugal wal quantum number. This avoids the necessity of looking for a series solution to the radial wave equation, and imposing a cut-off on the condition that a be equal to a negative integer implies a- Equation (48) gives the exact energy particle bound by a coulomb potential dition lsive force into an attractive force, as can be seen in the Klein-Gordon equation, (50), below. Alternatively, for 0a, and Em, the Regge tra- jectories for the bound states are given by 1,an n (48) here n is the princip series. The that the Kummer function becomes a generalized L guerre polynomial. levels of a Dirac , 1 2 n Em 2 2 1 11 22 1 222 nn 2 (49) This is rather surprising since the confluent hyper- geometric equation with coefficients, (43) and (44), is completely equivalent to the Klein-Gordon equation 2 22 2 22 1 d1 0. d EmE r rr (50) 4 It is commonly believed that (50) describes a spinless particle in a coulomb field, and is, therefore, not capable of describing the hydrogen atom since electrons have spin 1 2 [21]. Now the indicial equation for (50) about the origin has exponents 2 2 1,2 11 , 22 n which reduce to 1 n (51) and 21n in the non- relativistic limit [cf. Equation (5)]. This would lead us to consider a solution to the indicial equation with only the former exponent. However, with 0, the square root can become complex for 1 2 , and the solution would diverge. A further complication is that the exponents, (51), become complex for 1 2 and 0. With a complex exponents, the solutions near the origin would at [21] contends that the value of Z oscille. Bethe that w avelength so as to invalidate the solution ould be required to make the exponents complex would correspond to atoms whose radii are several times the Compton w ln e, ir r (52) for small r . However true thit would still make the energy levels, (49), complex, again making it unacceptable. We now address this in some detail. In the nonrelativistic limit, the of the two solu- is may be, ratio tions is r , where . The conformal mapping between the z a planes is [22] 21 nd r 1 2cot eip r 1 2cot , i zi (53) ep r which upon solving for r becomes 1 ln2cot ee . rip zi zi (54) The circles intersect at i and i, and p is the dis- tance from the smaller circle to the origin as shown in Figure 4. These correspond to the branch points 0r . Now, 1 cot si 1 cothpi ip and the and r right-hand de of (54) is the cross ratio, 2 11 co cot 11 e i pz ii ii (55) 11 2 cothcoth ln eeip iz rpizi pizi t, 11 cot cot 11 pz ii ii Copyright © 2013 SciRes. JMP B. H. LAVENDA 960 Figure 4. Crescent formed from intersecting circles making an angle λπ. e pwher cot and cotz . Taking the logarithm of both sides of (55) gives 2ln,ir (56) where the cross ratio is the distance between the points cos ,s cos, sin, 0 with respect to the circnity, and 1, ,0i in omplane called circular points at infinity because they lie on the complexification of every real circle. Both points satisfy the homogeneous equation, 2 233 0.xEx (57) By specifying the line at infinity, 30x, the circular points then satisfy, 22 12 0,xx (58) in, 0 and ular points at infi plex projective 1, ,0i . They arethe c 22 12 13 22AxBxCx xDx if 1 A B. The involution, (58), is elliptic since it has maginary ciircular points for its fixed elements. Equation fines a pai 12 0xix. (58) der of imlaaginary pnes, In the relativistic case, where 1 Z the in 2dices, (51), become 1,2 . 2 1 ni (59) The ratio of the two solutions is 2i r , which is the crescent problem, but with an imaginary angle. Rotate the crescent by 2, and the circles intersect at 1 and 1, which lie on the disc. Again specifying the line at infinity, x30, the c7), reces to 22 0,xx onic, (5du 12 for 1 (60) A B . Equation (60) is a hy involution, since it has 1 as perbolic its fixed elements, and defines a pair , 0xx. eted, here p of real planes With the cr 12 scent rotapip, w , the ss ratio ocrof the 4 points, ,1,0p , ,1,0z, 1,1,0, and 1,1,0 is real and is given by 2 11cot1c 1pz e 11 2 cothcoth 2ln 1 ee pz ir 1pz1 1 11 11 1111 coth 1 11 oth pz pz coth 1 where cothp (61) and cothz . Hence, ln .ir (62) The substitution is hyperbolic s hich lie ince it has real fixed points w on the disc in terms of the homogeneous coordinates osh ,sinh,0 and cosh ,sinh ,0 . A compariso (56) and (62) shows that whereas in the fo c n e the lo the a many-valued quantity, i the logarithm o es e, i r rmer the angle is real whilgarithm ofcross ratio is pure imaginary, and n the latter, the angle is imaginary whilef the cross ratio is real. Thus, in both cas (63) where the condition, determines the single- valued principal value of ln r. Both (55) and (61) ex- press angles in the form of a projective invariant the logarithm of both sides of the first equality in (55) yields . Taking 11 1lncotcot . 2ri zp (64) ducing (63taking thf both sides yield Intro), and e cotangent o 11cotcot1 cot 2cotcot , zp pz cot (65) which, upon equating arguments, becomes 2. (66) Likewise, taking the logarithm (61) gives 1 h of the first equality in 1 lncotcoth .irz p (67) Introducing (63) and taking the hyperbolic cotangent of both sides result in 1 cothcoth1 coth coth coth coth , zp zp or (68) upon equating the arguments give Copyright © 2013 SciRes. JMP B. H. LAVENDA 961 . (69) Whereas the Möbius transformation in (65), , 1 zp zpz (70) an imaginary circle, the Möbius transform in (68), has imaginary fixed points, and is related to , zp z (71) points and is the most general analytic function that maps the unit circle onto itself i 1pz has real fixed . [23] That is, for ez , 1z since e iiii zpp e e1e1.pp pz Consequently, the small r solution to the Klein-Gor equation, (52), not allow r to be interpre real, radial coordinate. This is in contrast to the usual interpretation wher don ted as a does eby the factor 1 22 1 2 du intro- ces a fixed branch cut along the real axis from 1 2 to 1 2 Wh. en 1 takes the cal state 0, and there is no consistent ant in (55) is pure imaginary, meaning that we are dealing substitution, while the angle is jective invariant in (61) is real, meaning that hyperbolic inally, all r solution to the non- relativistic Schrödinger equation and show that if r is real, 2 this cut over- physi solution. [24] In summary, the projective invari with an elliptic real, whereas the pro- substitutions map the real circle into itself, while the angle pure imaginary. Fwe return to the sm must be clex. If we agrotate the cre ce omp ain s- nt in Figure 4 so that the vertices are at 1 and 1 , while keeping p real, we have 11 12 cothcoth 2cot 1 ee 2 e. 1 z ip ip z r i z (72) as a Equation (72) shows that in order to interpret r real, radial coordinate, the angle must be complex, except iit at lim confo n the lim. In thit the crescent s pa degenerates into a half circle, and (72) becomes real. This is in contrast with rmal analysis which holds that r is complex and the angle is real and positive. The corresponding Möbius transform, , 1 zip zipz (73) has fixed points at 1 and 1 , which are the vertices of the crescent. In the limit as p, (73) becomes an inversion, 1zz . This, again, mps circles into circles for which straight lines are regarded as circles that ass through the point of infinity. In other words, this transformation associates points in the interior of a uit circle with poin exterior to it. S a p n ts o that it would appear that classical quantum mechanics emerges when confor- mality disappears, and the independent coordi comes real as well as the coefficient in the geometric equation, (1). nate be- hyper- Equation (73) can be written as 11 , 11 zz zz K where the multiplier of the transform, 1i ip 2 shows that the transformation e, 1 Kip is elliptic, i.e., 1K . The angle is constrained to the interval 02 , and is determined by theeight ohe center of the smaller circle of the crescent from the origin of the r plane. The transformation (72) will therefore map the upper half of the z plane onto the interior of the crescent in the r plane with angles hf t . REFERENCES [1] T. Regge, Nuovo Cimento, Vol. 14, 1959, pp. 951-976. [2] R. J. Eden, P. V. Landshoff, D. I. Olive and J. C. Polk- inghorne, “The Analytic S-Matrix,” Cambridge U.P., Cam- bridge, 1966, p. 12. [3] , Vol. 4, 9 p. [4] G. Veneziano, Nuovo Cimento, Vol. 57, 1968, pp. 190- 197. doi:10.1007/BF02824451 B. H. Lavenda, Journal of Modern Physics e Theory of Atomic Press, Oxford, 1949, p. 52. [6] K. Gottfried, “Quantum Mechanics,” Vol. 1, Fundamen- tals, Benjamin, New York, 1966, p. 148. [5] N. F. Mott and H. S. W. Massey, “Th Collisions,” 2nd Edition, Clarendon [7] V. Singh, Physical Review, Vol. 127, 1962, pp. 632-636. doi:10.1103/PhysRev.127.632 [8] L. D. Landau and E. M. Lifshitz, “Statistical Physics,” 2nd Edition, Pergamon, Oxford, 1959, p. 152. [9] A. R. Forsyth, “A Treatise on Differential Equations,” 6th Edition, Macmillan, London, 1956, p. 228. [10] V. Ovsienko and S. Tabachnikov, Notices AMS, Vol. 56, 2009, pp. 34-36. [11] A. R. Choudhtween Analyticity, Regge Trajector Möbius 017/CBO9780511524387 ary, “New Relations be ies, Veneziano Amplitude, and Transformations,” arXiv: hep-th/0102019. [12] J. R. Forshaw and D. A. Ross, “Quantum Chromody- namics and the Pomeron,” Cambridge U.P., Cambridge, 1997, p. 16. doi:10.1 d [13] B. H. Lavenda, “Errors in the Bag Model of Strings, an Copyright © 2013 SciRes. JMP B. H. LAVENDA Copyright © 2013 SciRes. JMP 962 ace,” arXiv:1112.4383. mbridge U.P., Cambridge, , 1980, p. eometry and . Blatt and V. F. Weisskopf, “Theoretical Nuclear Regge Trajectories Represent the Conservation of Angu- lar Momentum in Hyperbolic Sp [14] J. Gray, “Linear Differential Equations and Group Theory from Riemann to Poincaré,” Birkhäuser, Boston, 1986, p. 36. [15] L. R. Ford, “Automorphic Functions,” 2nd Edition, Chel- sea Pub. Co., New York, 1929, p. 54. [16] D. Mumford, C. Series and D. Wright, “Indra’s Pearls: The Vision of Felix Klein,” Ca 2002, p. 171. [17] N. V. Efimov, “Higher Geometry,” Mir, Moscow 413. [18] H. Busemann and P. J. Kelly, “Projective G Projective Metrics,” Academic Press, New York, 1953, p. 231. [19] J. M Physics,” Springer, New York, 1979, p. 330. doi:10.1007/978-1-4612-9959-2 [20] R. Omnès and M. Froissart, “Mandelstam Theory and Regge Poles,” Benjamin, New York, 1963, p. 27. [21] H. A. Bethe, “Intermediate Quantum Mechanics,” Ben- Forysth, “Theory of Functions of a Complex Vari- ehari, “Conformal Mapping,” McGraw-Hill, New heory,” W. 126. jamin, New York, 1964, p. 185. [22] A. R. able,” Vol. 2, 3rd Edition, Cambridge U.P., Cambridge, 1918, p. 685. [23] Z. N York, 1952, p. 164. [24] S. C. Frautschi, “Regge Poles and S-Matrix T A. Benjamin, New York, 1963, p. |