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Edward Witten (original) | Edward Witten (original) | ||
If one wants to | If one wants to summarize 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] : $$M$$, 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/ | # Over $$M$$ is a [https://en.wikipedia.org/wiki/Vector_bundle vector bundle] : $$X$$, 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. | ||
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 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. | ||
Revision as of 04:40, 5 February 2020
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This is the version of "the paragraph" by Edward Witten posted by Eric via Twitter.
Edward Witten (original)
If one wants to summarize our knowledge of physics in the briefest possible terms, there are three really fundamental observations:
- Spacetime is a pseudo-Riemannian manifold : $$M$$, endowed with a metric tensor and governed by geometrical laws.
- Over $$M$$ is a vector bundle : $$X$$, with a non-abelian structure group : $$G$$.
- 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.
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.
Eric Weinstein suggested several alterations:
- 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.
- 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 higgs is remarkably absent.
Eric Weinstein (update)
If one wants to summarise our knowledge of physics in the briefest possible terms, there are three really fundamental observations:
- Spacetime is a pseudo-Riemannian manifold : $$M$$, endowed with a metric tensor and governed by geometrical laws.
- Over $$M$$ is a principal bundle : $$P_{G}$$, with a non-abelian structure group : $$G$$.
- 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.
- Add something about Higgs
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.