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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. | ||
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 above geometrical laws obeyed by the metric tensor, the gauge fields, and the fermions are to be interpreted in quantum mechanical terms. | ||
</blockquote> | </blockquote> | ||
</div> | </div> | ||
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# The masses of elementary particles are generated through the Higgs mechanism. | # The masses of elementary particles are generated through the Higgs mechanism. | ||
All of this must be supplemented with the understanding that the geometrical laws obeyed by the metric tensor, the [https://en.wikipedia.org/wiki/Introduction_to_gauge_theory gauge fields], and the fermions are to be interpreted in [https://en.wikipedia.org/wiki/Quantum_mechanics quantum mechanical] terms. | All of this must be supplemented with the understanding that the above geometrical laws obeyed by the metric tensor, the [https://en.wikipedia.org/wiki/Introduction_to_gauge_theory gauge fields], and the fermions are to be interpreted in [https://en.wikipedia.org/wiki/Quantum_mechanics quantum mechanical] terms. | ||
</blockquote> | </blockquote> | ||
</div> | </div> | ||