Abstract
It is suggested that the world is locally projectively flat rather than Euclidean. From this postulate it is shown that an (N+1)-particle system has the global geometry of the symmetric spaceSO(4,N+1)/SO(4)×SO(N+1). A complex representation also exists, with structureSU(2,N+1)/S[U(2)×U(N+1)]. Several aspects of these geometrics are developed. Physical states are taken to be eigenfunctions of the Laplace-Beltrami operators. The theory may provide a rational basis for comprehending the groupsSO(4, 2),SU(2)×U(1),SU(3), etc., of current interest.
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References
F. H. Colson,The Week (Cambridge Univ. Press, London, 1926).
F. Hoyle and J. V. Narlikar,Proc. R. Soc. A282, 191 (1964).
P. A. M. Dirac,Ann. Math. 37, 429 (1936).
G. Mack and A. Salam,Ann. Phys. (N.Y.) 53, 174 (1969), and references contained therein.
A. O. Barut and H. Kleinert,Phys. Rev. 160, 1149 (1967).
H. Kleinert,Springer Tracts in Modern Physics, Vol. 49 (1969), p. 90.
R. Wilson and R. F. Peierls,Ann. Phys. (N.Y.) 81, 15 (1973).
O. Schreier and E. Sperner,Projective Geometry of N Dimensions, C. A. Rogers, transl. (Chelsea, New York, 1961).
W. V. D. Hodge and D. Pedoe,Methods of Algebraic Geometry (Cambridge Univ. Press, London, 1946).
L. K. Hua and B. A. Rozenfel'd,Acta Math. Sinica 8, 132 (1958) [English transl.,Chinese Math. 8, 726 (1966)].
L. K. Hua,Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, Trans. Math. Monographs, Vol. 6 (Am. Math. Soc., Providence, Rhode Island, 1963).
S. Helgason,Differential Geometry and Symmetric Spaces (Academic Press, New York, 1962).
E. Calabi,Ann. Math. 58, 1 (1953); E. Calabi and E. Vesentini,Ann. Math. 71, 472 (1960).
L. K. Hua,Am. J. Math. 66, 470 (1944).
C. L. Siegel,Symplectic Geometry (Academic Press, New York, 1964); reprinted fromAmer. J. Math. 65, 1 (1943).
S. I. Goldberg,Curvature and Homology (Academic Press, New York, 1962), pp. 158–168.
S. Kobayashi and K. Nomizu,Foundations of Differential Geometry, Vols. I and II (Interscience, New York, 1963).
M. Bander and C. Itzykson,Rev. Mod. Phys. 38, 330 (1966).
H. Bateman,Proc. Lond. Math. Soc. 8, 223 (1910).
I. A. Malkin and V. I. Man'ko,Sov. Phys.—JETP Lett. 2, 146 (1965).
K. C. Tripathy, R. Gupta, and J. D. Anand,J. Math. Phys. 16, 1139 (1975).
R. Penrose and M. A. H. MacCallum,Phys. Reports 6, 241 (1972).
S. Weinberg,Rev. Mod. Phys. 46, 255 (1974).
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Supported in part by the National Science Foundation.
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Eichinger, B.E. Projective spacetime. Found Phys 7, 673–703 (1977). https://doi.org/10.1007/BF00708589
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DOI: https://doi.org/10.1007/BF00708589