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[[Eric Weinstein|Eric]] has suggested several alterations, listed below: | [[Eric Weinstein|Eric]] has suggested several alterations, listed below: | ||
* In (ii), “vector bundle | * In (ii), “vector bundle X” should be changed to principal G-bundle. | ||
* Also in (ii), “nonabelian gauge group G” should be changed to nonabelian structure group G. | * Also in (ii), “nonabelian gauge group G” should be changed to nonabelian structure group G. | ||
* In (iii), | * In (iii), \(R\) and \(\tilde{R}\) should be (complex) linear representations of G and so they are not equivalent. | ||
* He mentioned that some info was not required, and that the Higgs is remarkably absent. | * He mentioned that some info was not required, and that the Higgs is remarkably absent. | ||
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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. | ||
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If one wants to summarise our knowledge of physics in the briefest possible terms, there are three really fundamental observations: | If one wants to summarise our knowledge of physics in the briefest possible terms, there are three really fundamental observations: | ||
# [https://en.wikipedia.org/wiki/Spacetime Spacetime] is a [https://en.wikipedia.org/wiki/Pseudo-Riemannian_manifold pseudo-Riemannian manifold] | # [https://en.wikipedia.org/wiki/Spacetime Spacetime] is a [https://en.wikipedia.org/wiki/Pseudo-Riemannian_manifold pseudo-Riemannian manifold] $$M$$, endowed with a [[metric tensor]] and governed by [https://en.wikipedia.org/wiki/Geometry geometrical laws]. | ||
# Over | # Over $$M$$ is a [https://en.wikipedia.org/wiki/Principal_bundle principal bundle] $$P_{G}$$, with a [https://en.wikipedia.org/wiki/Non-abelian_group non-abelian structure group] $$G$$. | ||
# [https://en.wikipedia.org/wiki/Fermion Fermions] are sections of | # [https://en.wikipedia.org/wiki/Fermion Fermions] are sections of $$(\hat{S}_{+} \otimes V_{R}) \oplus (\hat{S}\_ \otimes V_{\bar{R}})$$. $$R$$ and $$\bar{R}$$ are not [https://en.wikipedia.org/wiki/Isomorphism isomorphic]; their failure to be isomorphic explains why the light fermions are light. | ||
# The masses of elementary particles are generated through the Higgs mechanism. | # The masses of elementary particles are generated through the Higgs mechanism. | ||