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84927826499
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Theor
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articles by M. Taketani and S. Sakata, in
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See articles by M. Taketani and S. Sakata, in Suppl. Progr. Theor. Phys. 50, (1971).
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(1971)
Phys.
, vol.50
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2
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84927826498
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A good recent example is the theory of hadrons, with the three stages being respectively the observation of resonances, the discovery of quarks, and the formulation of QCD.
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3
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84927826497
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G. F. B. Riemann, in Über die Hypothesen, welche der Geometrie zugrunde liegen, edited by H. Weyl (Springer-Verlag, Berlin, 1919).
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7
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84927826496
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There have also been proposals in which a partial ordering has been used as fundamental structure, but with a directly topological meaning, rather than a causal one. See, e.g., R. D. Sorkin, in General Relativity and Gravitation, Vol. 1, edited by B. Bertotti, F. de Felice, and A. Pascolini (Consiglio Nazionale delle Ricerche, Roma, 1983), and to be published.
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8
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84927826495
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G. 't Hooft, in Recent Developments in Gravitation, Cargèse 1978, edited by M. Levy and S. Deser (Plenum, New York, 1979); J. Myrheim, CERN Report No. TH-2538, 1978 (unpublished).
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10
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84927826494
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L. Bombelli, P. Jain, J. Lee, D. Meyer, and R. D. Sorkin, to be published.
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11
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84927826493
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The notion of ``uniformly distributed'' can be slightly tricky to define, but, roughly, we mean that every Alexandroff neighborhood J- ( p ) INTER J+ ( p ) in the manifold contains a number of points of C equal to its volume, within the Poisson-type fluctuations which could be expected from a random ``sprinkling'' of points.
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12
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84927826492
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Notice that condition (2) on the density would not help us to determine a unique approximate metric if we did not also have condition (3) on the characteristic length lambda: Given any manifold with the right causal structure, i.e., conformal metric, we could always arrange the density to be unity by setting the conformal factor appropriately; but in doing so, we would in general introduce an unreasonably large curvature, or other small characteristic lengths. However, it seems plausible that conditions (1) and (2) alone determine the continuum geometry ``up to arbitrary variations on small scales, and small variations on arbitrary scales'' (where small scale means size unity or smaller).
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