Noneuclidean Tessellations and Their Relation to Regge Trajectories

B. H. Lavenda

Università degli Studi, Camerino, Italy.

DOI: 10.4236/jmp.2013.47128                

Abstract

The coefficients in the confluent hypergeometric equation specify the Regge trajectories and the degeneracy of the angular 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 conjugate 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

Conflicts of Interest

The authors declare no conflicts of interest.

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. Polkinghorne, “The Analytic S-Matrix,” Cambridge U.P., Cambridge, 1966, p. 12.
[3]	B. H. Lavenda, Journal of Modern Physics, Vol. 4, 9 p.
[4]	G. Veneziano, Nuovo Cimento, Vol. 57, 1968, pp. 190-197. doi:10.1007/BF02824451
[5]	N. F. Mott and H. S. W. Massey, “The Theory of Atomic Collisions,” 2nd Edition, Clarendon Press, Oxford, 1949, p. 52.
[6]	K. Gottfried, “Quantum Mechanics,” Vol. 1, Fundamentals, Benjamin, New York, 1966, p. 148.
[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. Choudhary, “New Relations between Analyticity, Regge Trajectories, Veneziano Amplitude, and Mobius Transformations,” arXiv: hep-th/0102019.
[12]	J. R. Forshaw and D. A. Ross, “Quantum Chromodynamics and the Pomeron,” Cambridge U.P., Cambridge, 1997, p. 16. doi:10.1017/CBO9780511524387
[13]	B. H. Lavenda, “Errors in the Bag Model of Strings, and Regge Trajectories Represent the Conservation of Angular Momentum in Hyperbolic Space,” arXiv:1112.4383.
[14]	J. Gray, “Linear Differential Equations and Group Theory from Riemann to Poincaré,” Birkhauser, Boston, 1986, p. 36.
[15]	L. R. Ford, “Automorphic Functions,” 2nd Edition, Chelsea Pub. Co., New York, 1929, p. 54.
[16]	D. Mumford, C. Series and D. Wright, “Indra’s Pearls: The Vision of Felix Klein,” Cambridge U.P., Cambridge, 2002, p. 171.
[17]	N. V. Efimov, “Higher Geometry,” Mir, Moscow, 1980, p. 413.
[18]	H. Busemann and P. J. Kelly, “Projective Geometry and Projective Metrics,” Academic Press, New York, 1953, p. 231.
[19]	J. M. Blatt and V. F. Weisskopf, “Theoretical Nuclear 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,” Benjamin, New York, 1964, p. 185.
[22]	A. R. Forysth, “Theory of Functions of a Complex Variable,” Vol. 2, 3rd Edition, Cambridge U.P., Cambridge, 1918, p. 685.
[23]	Z. Nehari, “Conformal Mapping,” McGraw-Hill, New York, 1952, p. 164.
[24]	S. C. Frautschi, “Regge Poles and S-Matrix Theory,” W. A. Benjamin, New York, 1963, p. 126.