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Though the original Graph aptly describes our physical knowledge, there are some minor alterations and additions to be made for it to wholly capture the current state of physics. | |||
Eric Weinstein suggested several alterations, listed below: | |||
Β | * In (ii), βvector bundle Xβ should be changed to principal G-bundle. | ||
* In (ii), βvector bundle | |||
* Also in (ii), βnonabelian gauge group Gβ should be changed to nonabelian structure group G. | * Also in (ii), βnonabelian gauge group Gβ should be changed to nonabelian structure group G. | ||
* In (iii), <math>R</math> and <math>\tilde | * In (iii), <math>\ R</math> and <math>\tilde R</math> 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 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 `M`, endowed with a metric tensor and governed be geometrical laws | ||
2. Over | 2. Over `M` is a principal bundle with a nonabelian structure group `G`. | ||
3. Fermions are sections of | 3. Fermions are sections of `(\hat{S}_+ \otimes V_R) \oplus (\hat{S}_- \otimes V_{\tilde{R}})`. `R` and `\tilde{R}` are complex linear representations of `G` 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 | 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. | ||
</blockquote> | </blockquote> | ||
</div> | </div> | ||
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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. | ||
All of this must be supplemented with the understanding that the | 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. | ||
</blockquote> | </blockquote> | ||
</div> | </div> | ||
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[[Category:Graph, Wall, Tome]] | [[Category:Graph, Wall, Tome]] | ||
[[Category:Projects]] | [[Category:Projects]] | ||