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By American Mathematical Society, János Kollár, Robert Lazarsfeld

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Additional resources for Algebraic Geometry Santa Cruz 1995, Part 2: Summer Research Institute on Algebraic Geometry, July 9-29, 1995, University of California, Santa Cruz

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Here lies the deeper connection of non-Iocal fields and the stochastic space-time which we shall discuss in latter chapter. 18) is implicit in the assumption that the probability depends on these variables. Therefore, it is natural to consider the variable as the intern al variable. A stochastic space-time then be constructed with the line element defined by e e. 19) Here, the metric tensor G 1'11 is function of both x and This is popularly known as Finsler metric [Asanov 1985]. We shall show in latter chapter that Finsler metric is inherently probabilistic in nature.

59) 2 4>(i, t), T denoting the correlation time. It is assumed with V(i, t) that T is much sm aller than the relevant quantum time of the system. For example, a typical colloid grain is of the order of 10- 5 cm. Let us take L = 10-4 cm, then T = 3 X 10- 15 s. The corresponding quantum time is = which becomes (for proton) ~ 10-9 s. Hence our above assumption mf, STATISTICAL GEOMETRY, MICROPHYSICS AND COSMOLOGY 41 holds. We can calculate the function 9 considering the correlation function for 4>(i, t).

4) Within this framework, it is possible to establish the following uncertainty principle for position and momentum observables q and p respectively. 5) In quantum mechanics, the wave function contains more information rather than the probability density. The wave function contains the phase which is very important in describing the interference phenomena. But here, in the frame of stochastic space-time we are dealing directly with P(z, t). So, it appears to be problematic to explain the interference phenomena within this framework.

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