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RESEARCH PRODUCT

On Riemann’s Ideas on Space and Schwinger’s Treatment of Low-Energy Pion-Nucleon Physics

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We begin with a demonstration of how great an influence Riemann’s habilitation essay had on the development of field theory. His ideas about the origin of physical space and the importance of a metric field were clearly outlined as early as 1854, and praised highly by the old C.F. Gauss, who died 1 year later. There is but one formula in Riemann’s article. This formula and its relevance will be explained at the beginning of the present chapter. The basic principle which is omnipresent in Riemann’s entire work is to understand the physical behavior of nature from its smallness. Hence partial differential equations stand at the beginning of any field theory. In our case, it is not the metric field, but the pion field, the nucleon field, etc. and their interactions that are studied. The action principle and its infinitesimal variation (δ) to generate field equations are our basic ingredients. This will be exemplified in detail for the interacting pion-nucleon system. We follow mostly Schwinger’s so-called “source theory”, which deals with the infinitesimally small—in the spirit of Riemann—and we occasionally compare it with S. Weinberg’s current-algebra approach.