Armand Siegel

  • Differential Space, Quantum Systems, and Prediction

    Norbert Wiener, Armand Siegel, Bayard Rankin, and William Ted Martin

    Permeated with the spirit of the late Norbert Wiener, and reflecting his approach to mathematical abstraction by way of physical reality, these integrated chapters were in fact prepared under his aegis. Both constructional and postulational methods are emphasized in elaborating the subject matter; both historical and formal developments in the subject are reviewed; both mathematical and physical insights are called into play. In this way, the close mathematical association and physical connection of a number of seemingly disparate topics become evident: the theory and construction of the Brownian motion process, integration in differential space, prediction of single series stationary processes, the development of quantum concepts on the basis of differential space, multiple prediction using the successive projection technique, the Fourier-Hermite development of nonlinear functionals, the postulational basis for measure-theoretic probability theory.

    Included is a verification, by means of differential space theory, of the conjecture that ensembles of completely described systems obeying a postulated dynamics can be so constructed as to have the same statistical properties as those expressed by a given quantum=mechanical wave function.

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