Editing the Graph: Difference between revisions

Revision as of 19:15, 2 November 2020

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:

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]\displaystyle{ \ R }[/math] and [math]\displaystyle{ \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.

Original Graph

The Graph is a paragraph from Edward Witten's paper Physics and Geometry, at the bottom of page 20:

The Original Graph

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 $$M$$, endowed with a metric tensor and governed by geometrical laws.
(ii) Over $$M$$ is a vector bundle $$X$$ with a non-abelian gauge group $$G$$.
(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.

Edited Graphs

This is a modified version of the paragraph:

Edited Graph Version 1

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.

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 gauge fields, and the fermions are to be interpreted in quantum mechanical terms.

Further Resources

Eric Weinstein tweeted about the paragraph here.

@@ Line 7: / Line 7: @@
 * He mentioned that some info was not required, and that the Higgs is remarkably absent.
-This is a modified version of the paragraph:
+== Original Graph ==
+The Graph is a paragraph from Edward Witten's paper [https://cds.cern.ch/record/181783/files/cer-000093203.pdf Physics and Geometry], at the bottom of page 20:
 <div class="toccolours mw-collapsible mw-collapsed" style="width: 600px; overflow:auto;">
-<div style="font-weight:bold;line-height:1.6;">Edited Graph Version 1</div>
+<div style="font-weight:bold;line-height:1.6;">The Original Graph</div>
 <div class="mw-collapsible-content">
 <blockquote style="max-width:575px; min-width:300px; font-size: 125%; background: #f3f3ff; border-color: #ddd;">
-If one wants to summarise 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 $$M$$, endowed with a metric tensor and governed by geometrical laws.
+(ii) Over $$M$$ is a vector bundle $$X$$ with a non-abelian gauge group $$G$$.
-# [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].
+(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.
-# 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.
-# 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 geometrical laws obeyed by the metric tensor, the gauge fields, and the fermions are to be interpreted in quantum mechanical terms.
 </blockquote>
 </div>
 </div>
+== Edited Graphs ==
-The Graph is a paragraph from Edward Witten's paper [https://cds.cern.ch/record/181783/files/cer-000093203.pdf Physics and Geometry], at the bottom of page 20:
+This is a modified version of the paragraph:
 <div class="toccolours mw-collapsible mw-collapsed" style="width: 600px; overflow:auto;">
-<div style="font-weight:bold;line-height:1.6;">The Original Graph</div>
+<div style="font-weight:bold;line-height:1.6;">Edited Graph Version 1</div>
 <div class="mw-collapsible-content">
 <blockquote style="max-width:575px; min-width:300px; font-size: 125%; background: #f3f3ff; border-color: #ddd;">
-If one wants to summarize 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.
+# [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] $$P_{G}$$, 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.
+# The masses of elementary particles are generated through the Higgs mechanism.
-(ii) Over $$M$$ is a vector bundle $$X$$ with a non-abelian gauge group $$G$$.
+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.
-(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.
 </blockquote>
 </div>