Lectures in Applied Mathematics, Proceedings of the Summer by I. E. Segal, George Whitelaw Mackey

By I. E. Segal, George Whitelaw Mackey

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Additional resources for Lectures in Applied Mathematics, Proceedings of the Summer Seminar, Boulder, Colorado, 1960, Vol 2: Mathematical Problems of Relativistic Physics

Example text

Cx(t) (c > 0, c 0 1) on Wiener space have highly pathological measure-theoretic properties. ), then the transformation described in Wiener space would be absolutely continuous. To sketch the connection briefly, the Brownian motion process x(t), say for 0 5 t < I, may be represented as a Fourier series xV() - qk q2xkt, qk *cqk, p---c- lpk k where the q, are independently distributed normal random variables of unit variance. It is evident that the transformations x(t) --* cx(t) and Now if the former transformation were qk --- cqk correspond.

The theoretical formulation of a relativistic elementary free particle species as an irreducible unitary representation of the Lorentz group seems on the whole physically justified. On the other hand, the conventional interactions between such particles cannot be readily expressed in terms of the corresponding unitary representations of the Lorentz group, in fact, the notion of a "local" interaction, such as have been conventionally assumed to be the only physical type, depends explicitly on the formulation of the vectors in the representation space as functions on space-time.

Under the action of the Lorentz group this space is far from irreducible, splitting as a direct integral, or continuous direct sum, of subspaces which derive essentially from an invariant quadratic expression in the generators of the 40 1. E. SEWAL Lorentz group, and related expressions, which must act as constants in an irreducible subspace, the expression of this requirement being the partial differential equations defining the elementary vector particles. e. the dual to space-time as a linear space.

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