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If one wants to summarize our knowledge of physics in the briefest possible terms, there are three really fundamental observations: | If one wants to summarize our knowledge of physics in the briefest possible terms, there are three really fundamental observations: | ||
(i) Spacetime is a pseudo-Riemannian manifold | (i) Spacetime is a pseudo-Riemannian manifold $$M$$, endowed with a metric tensor and governed by geometrical laws. | ||
(ii) Over | (ii) Over $$M$$ is a vector bundle $$X$$ with a non-abelian gauge group $$G$$. | ||
(iii) Fermions are sections of | (iii) Fermions are sections of $$(\hat{S}_{+} \otimes V_{R}) \oplus (\hat{S}_{-} \otimes V_{\tilde{R}})$$. $$R$$ and $$\tilde{R}$$ are not isomorphic; their failure to be isomorphic explains why the light fermions are light and presumably has its origins in representation difference $$\Delta$$ in some underlying theory. | ||
All of this must be supplemented with the understanding that the geometrical laws obeyed by the metric tensor, the gauge fields, and the fermions are to be interpreted in quantum mechanical terms. | All of this must be supplemented with the understanding that the geometrical laws obeyed by the metric tensor, the gauge fields, and the fermions are to be interpreted in quantum mechanical terms. | ||