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Theory of Geometric Unity: Difference between revisions
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Geometric Unity is an attempt to produce a complete theory of fundamental physics through geometry. | |||
* A first video presentation of the theory is available on [https://www.youtube.com/watch?v=Z7rd04KzLcg Youtube] | * A first video presentation of the theory is available on [https://www.youtube.com/watch?v=Z7rd04KzLcg Youtube] | ||
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* Preliminary notes by the community on the talk are available in a [https://docs.google.com/document/d/1gPU_bJR5wBs7MCsNGCW5Y06Jh3SzarX5OnJHyLxQfDQ/edit Google Doc] | * Preliminary notes by the community on the talk are available in a [https://docs.google.com/document/d/1gPU_bJR5wBs7MCsNGCW5Y06Jh3SzarX5OnJHyLxQfDQ/edit Google Doc] | ||
* [https://www.youtube.com/watch?v=wf0_nMaQ6tA#t=2h16m27s Discussion on the Joe Rogan show] | * [https://www.youtube.com/watch?v=wf0_nMaQ6tA#t=2h16m27s Discussion on the Joe Rogan show] | ||
* [https://www.youtube.com/watch?v=N_aN8NnoeO0 PBS SpaceTime] | |||
<div style="float:right;padding:20px;">__TOC__</div> | |||
<blockquote style="width:500px">"The source code of the universe is overwhelmingly likely to determine a purely geometric operating system written in a uniform programming language." - Eric Weinstein </blockquote> | |||
== Project Ideas == | |||
* Unpack Eric's first talk by providing additional explanations for the concepts and problems introduced. One possible format would be to annotate it in a [https://genius.com/web-annotator Genius.com] format. | |||
* Organize voice/video chats to watch the talk together and stop every few minutes to discuss it. (Multiple calls would be needed to go through the whole talk.) | |||
== Key Ideas == | == Key Ideas == | ||
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{| class="wikitable" | {| class="wikitable" | ||
| '''1.''' The Arena (<math> Xg_{\mu\nu}</math>) | | '''1.''' The Arena (<math> Xg_{\mu\nu}</math>) | ||
| <math>R_{\mu\nu} - \frac{1}{2} Rg_{\mu\nu} + \Lambda g_{\mu\nu} = \left( \ | | <math>R_{\mu\nu} - \frac{1}{2} Rg_{\mu\nu} + \Lambda g_{\mu\nu} = \left( \dfrac{8 \pi G}{c^4} T_{\mu\nu}\right)</math> | ||
| the Einstein | | the Einstein field equations, which describe gravity in the theory of general relativity | ||
|- | |- | ||
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| '''3.''' Matter | | '''3.''' Matter | ||
Antisymmetric, therefore light | Antisymmetric, therefore light | ||
| <math>\ | | <math>(i \hbar \gamma^\mu \partial_\mu - m) \psi = 0</math> | ||
| the Dirac equation, | | the Dirac equation, the equation of motion describing matter particles, or fermions | ||
|} | |} | ||
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=== Problem Nr. 1: Einstein's Theory of General Relativity is not a proper Gauge Theory === | === Problem Nr. 1: Einstein's Theory of General Relativity is not a proper Gauge Theory === | ||
* From Einstein's general relativity, we take the Einstein projection of the curvature tensor of the Levi-Civita connection of the metric $$P_E(F_{\Delta^LC})$$ | |||
* From Yang-Mills-Maxwell-Anderson-Higgs theory of gauge fields, we take the adjoint exterior derivative coupled to a connection $$d^\star_A F_A$$ | |||
'''Idea:''' What if the $$F$$'s are the same in both contexts? | '''Idea:''' What if the $$F$$'s are the same in both contexts? | ||
Further, supposing these $$F$$'s are the same, then why apply two different operators? | |||
'''Thus the question becomes:''' Is there any opportunity to combine these two operators? | '''Thus the question becomes:''' Is there any opportunity to combine these two operators? | ||
A problem is that the hallmark of the Yang-Mills theory is the freedom to choose the data, the internal quantum numbers that give all the particles their personalities beyond the mass and the spin. We can allow the gauge group of symmetries to act on both sides of the equation, but the key problem is that: $$P_E(F_{\Delta^{LC} h}) \neq h^{-1} P_E(F_{\Delta^{LC} }) h $$. If we act on connections on the right and then take the Einstein projection, this is not equal to first taking the projection and then conjugating with the gauge action. The gauge rotation is only acting on one of the two factors. Yet the projection is making use of both of them. So there is a fundamental incompatibility in the claim that Einstein's theory is a gauge theory relies more on analogy than an exact mapping between the two theories. | A problem is that the hallmark of the Yang-Mills theory is the freedom to choose the data, the internal quantum numbers that give all the particles their personalities beyond the mass and the spin. We can allow the gauge group of symmetries to act on both sides of the equation, but the key problem is that: $$P_E(F_{\Delta^{LC} h}) \neq h^{-1} P_E(F_{\Delta^{LC} }) h $$. If we act on connections on the right and then take the Einstein projection, this is not equal to first taking the projection and then conjugating with the gauge action. The gauge rotation is only acting on one of the two factors. Yet the projection is making use of both of them. So there is a fundamental incompatibility in the claim that Einstein's theory is a gauge theory relies more on analogy than an exact mapping between the two theories. | ||
=== Problem Nr. 2: Spinors are sensitive to the metric === | === Problem Nr. 2: Spinors are sensitive to the metric === | ||
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How do we get the metric out from its responsibilities? It's been assigned far too many responsibilities. It is responsible for a volume form; for differential operators; it's responsible for measurement; it's responsible for being a dynamical field, part of the field content of the system." | How do we get the metric out from its responsibilities? It's been assigned far too many responsibilities. It is responsible for a volume form; for differential operators; it's responsible for measurement; it's responsible for being a dynamical field, part of the field content of the system." | ||
</blockquote> | </blockquote> | ||
<div class="toccolours mw-collapsible mw-collapsed" style="width:1000px; overflow:auto;"> | |||
<div style="font-weight:bold;line-height:1.6;">Comments</div> | |||
<div class="mw-collapsible-content"> | |||
'''Mark-Moon:''' Can anyone explicate Eric's point about spinor fields depending (in a bad way) on the metric in conventional theories, in a way that is no longer the case in GU? I feel like this is the original idea in GU that I'm closest to being able to understand, but I don't think I quite get it yet. | |||
'''Chain:''' Yeah I was wondering this as well, as far as I was aware you just need a spin structure, which only depends on the topology and atlas on the manifold and not on the choice of metric [https://math.stackexchange.com/questions/2836814/dependence-of-spinor-bundle-on-choice-of-metric]. Perhaps the point is that although each choice of metric yields an isomorphic spin structure, perhaps there is not a canonical isomorphism in the same way as in GU where the bundle of metrics Y (U in the talk) is isomorphic to the Chimeric bundle C, but the choice of isomorphism is given by a choice of connection on Y. Although I don't know why the chimeric bundle would come with a canonical choice of spin structure either, which seems to be Eric's claim | |||
to define spinors you would need a clifford bundle and hence a choice of metric on the chimeric bundle | |||
</div></div> | |||
=== Problem Nr. 3: The Higgs field introduces a lot of arbitrariness === | === Problem Nr. 3: The Higgs field introduces a lot of arbitrariness === | ||
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</blockquote> | </blockquote> | ||
== | == Frequently Asked Questions == | ||
What will this theory predict? | What will this theory predict? | ||