# zbMATH — the first resource for mathematics

Modular groups in Cantorian $$E^{(\infty)}$$ high-energy physics. (English) Zbl 1035.83503
Summary: This paper proposes that the geometry and topology of quantum spacetime is shadowed closely by the Möbius geometry of quasi-Fuschian and Kleinian groups and that is the cause behind the phenomena of high-energy particle physics.
In addition, on the large scale measurement of, for instance, the microwave background temperature, the universality of the Merger sponge provides an excellent limit set model for the Charlier–Zeldovich proposal of the fracticality of the universe today and the rather accurate estimate $$T_c=(\ln 20/\ln 3) = 2.726k$$.
In particular the paper shows the link between the fix points of the modular groups of the vacuum and the golden mean $$\phi = (1/(1+{\phi})) = (\sqrt 5 - 1)/2$$ of $$E^{(\infty)}$$ spacetime by analytical continuation of a Möbius transformation.

##### MSC:
 83C45 Quantization of the gravitational field 83F05 Cosmology
##### Keywords:
quantum spacetime; Möbius geometry
Full Text:
##### References:
 [1] Neumann, P.; Stoy, G.; Thompson, E., Groups and geometry, (1999), Oxford London [2] Henle, M., Modern geometries, (2001), Prentice-Hall Englewood Cliffs, NJ [3] El Naschie, M.S., Determining the temperature of the microwave background radiation from the topology and geometry of spacetime, Chaos, solitons & fractals, 14, 7, 1121-1126, (2002) · Zbl 1034.83503 [4] El Naschie, M.S., Wild topology, hyperbolic geometry and fusion algebra of high energy particle physics, Chaos, solitons & fractals, 13, 1935-1945, (2002) · Zbl 1024.81055 [5] El Naschie, M.S., On a class of general theories for high energy particle physics, Chaos, solitons & fractals, 14, 649-668, (2002) [6] El Naschie, M.S., On the exact mass spectrum of quarks, Chaos, solitons & fractals, 14, 369-376, (2002) [7] Thurston, W., Three-dimensional geometry and topology, (1997), Princeton Princeton, NJ [8] Ratcliffe, J.G., Foundation of hyperbolic manifolds, (1994), Springer Berlin [9] Tomuschitz, R., Tachyonic chaos and causality in the open universe, Chaos, solitons & fractals, 7, 5, 743-768, (1996) [10] Castro, C., Fractal strings as an alternative justification for el naschie’s Cantorian spacetime and the fine structure constant, Chaos, solitons & fractals, 14, 1341-1351, (2002) · Zbl 1033.81511 [11] El Naschie, M.S., Theoretical derivation and experiential confirmation of the topology of transfinite heterotic strings, Chaos, solitons & fractals, 12, 1167-1174, (2001) · Zbl 1022.81678 [12] El Naschie, M.S., Penrose universe and Cantorian spacetime as a model for non-commutative quantum geometry, Chaos, solitons & fractals, 9, 6, 931-933, (1998) · Zbl 0937.58006 [13] Gardener, M., Penrose tiles to trapadoor ciphers, (1989), W.H. Freeman New York [14] El Naschie, M.S., Superstrings knots and non-commutative geometry in E(∞) space, Int. J. theor. phys., 37, 12, (1998) · Zbl 0935.58005 [15] Kaku, M., Strings, conformal fields and M theory, (2002), Springer New York [16] Perkins, D.H., Introduction to high energy physics, (2000), Cambridge University Press Cambridge, UK [17] El Naschie, M.S., The exact mass of the electron via the transfinite way, Chaos, solitons & fractals, 14, 523-524, (2002) · Zbl 1011.81505 [18] El Naschie, M.S., Nash’s surface and the ξ(∞) space of quantum scale, Chaos, solitons & fractals, 14, 9, 1495-1497, (2002) · Zbl 1034.81541 [19] Nash, J., The embedding problem for Riemannian manifolds, Ann. math., 63, 20-63, (1956) · Zbl 0070.38603 [20] Marek-Crnjac, L., On mass spectrum of elementary particles of the standard model using el naschie’s Golden field theory, Chaos, solitons & fractals, 15, 611-618, (2003) · Zbl 1033.81521 [21] Matousek, J., Lectures on discrete geometry, (2002), Springer Berlin · Zbl 0999.52006 [22] Zeldovich, Y.; Starobinsky, A non-trivial topology universe and possibility of its quantum birth, Pisma astron. J. (SSSR), 10, 323-328, (1984) [23] Argyri, J.; Ciubotariu, C., On el naschie’s complex time and gravitation, Chaos, solitons & fractals, 8, 5, 743-751, (1997) · Zbl 0942.83005 [24] Ord, G., Classical particles and the Dirac equation with an electromagnetic force, Chaos, solitons & fractals, 8, 5, 727-741, (1997) · Zbl 0940.81003 [25] El Naschie, M.S., Nonlinear isometric bifurcation and shell buckling, ZAMM (zeitshrift fur angewanclete Mathematik und mechanik), 57, 293-296, (1977) · Zbl 0371.73050 [26] El Naschie, M.S., The Hausdorff dimensions of heterotic string fields are D(−)=26.18033989 and D(+)=10, Chaos, solitons & fractals, 12, 377-379, (2001) · Zbl 1066.81594 [27] Argyris, J.; Ciubotariu, C.; Matuttis, H.G., Fractal space, cosmic strings and spontaneous symmetry breaking, Chaos, solitons & fractals, 12, 1-48, (2001) · Zbl 1040.83520 [28] El Naschie, M.S., Time symmetry breaking, duality and Cantorian spacetime, Chaos, solitons & fractals, 7, 4, 499-518, (1996) · Zbl 1080.81514 [29] El Naschie, M.S., Derivation of the threshold and absolute temperature T0=273.16 K from the topology of quantum spacetime, Chaos, solitons & fractals, 14, 7, 1117-1121, (2002) · Zbl 1034.81540 [30] () [31] El Naschie MS. On John Nash’s crumpled surface. Chaos, Solitons & Fractals, in press · Zbl 1063.81603 [32] El Naschie, M.S., Correlation in E(∞), Chaos, solitons & fractals, 11, 1993-1994, (2000) · Zbl 1035.81566
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.