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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: | ||
(i) Spacetime is a pseudo-Riemannian manifold $$M$$, endowed with a metric tensor and governed by geometrical laws. | (i) Spacetime is a pseudo-Riemannian manifold $$M$$, endowed with a metric tensor and governed by geometrical laws. | ||
(ii) Over M is a vector bundle $$X$$ with a nonabelian gauge group | (ii) Over M is a vector bundle $$X$$ with a nonabelian gauge group $$G$$. | ||
(iii) Fermions are sections of $$(\hat{S}{+} \otimes V{R}) \oplus (\hat{S}_ \otimes V_{\bar{R}})$$. $$R$$ and $$\bar{R}$$ 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. | (iii) Fermions are sections of $$(\hat{S}{+} \otimes V{R}) \oplus (\hat{S}_ \otimes V_{\bar{R}})$$. $$R$$ and $$\bar{R}$$ 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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# [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]. | # [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 $$M$$ is a [https://en.wikipedia.org/wiki/Principal_bundle principal bundle] | # 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 $$(\hat{S}_{+} \otimes V_{R}) \oplus (\hat{S}\_ \otimes V_{\bar{R}})$$. $$R$$ and $$\bar{R}$$ are not [https://en.wikipedia.org/wiki/Isomorphism isomorphic]; their failure to be isomorphic explains why the light fermions are light. | # [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. | ||
# Add something about Higgs | # Add something about Higgs | ||