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If one wants to summarize our knowledge of physics in the briefest possible terms, there are four really fundamental observations: | If one wants to summarize our knowledge of physics in the briefest possible terms, there are four really fundamental observations: | ||
1. Space-time is a pseudo-Riemannian manifold | 1. Space-time is a pseudo-Riemannian manifold <math>M</math>, endowed with a metric tensor and governed by geometrical laws | ||
2. Over | 2. Over <math>M</math> is a principal G-bundle <math>P_{G}</math> with a nonabelian structure group <math>G</math>. | ||
3. Fermions are sections of | 3. Fermions are sections of <math>(\hat{S}_+ \otimes V_R) \oplus (\hat{S}_- \otimes V_{\tilde{R}})</math>. <math>R</math> and <math>\tilde{R}</math> are complex linear representations of <math>G</math> and thus are not isomorphic. Their failure to be isomorphic explains why the light fermions are light. | ||
4. Yukawa couplings between the fermion field and the Higgs field endow fermions with the property of mass. Massive bosons also acquire their mass through this Higgs mechanism. | 4. Yukawa couplings between the fermion field and the Higgs field endow fermions with the property of mass. Massive bosons also acquire their mass through this Higgs mechanism. | ||