Editing User:Aardvark/GU

The '''Theory of Geometric Unity''' is an attempt by Eric Weinstein to produce a unified field theory by recovering the different, seemingly incompatible geometries of fundamental physics from a general structure with minimal assumptions. This structure is a mapping from a manifold <math>X^4</math> to a manifold <math>Y</math> called the [[Observerse|observerse]], which replaces Einstein's spacetime. For the latest updates on the theory, visit '''https://geometricunity.org/'''.

This page currently summarizes information surrounding the theory.

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== Reframing the Central Question of Physics ==
Riemannian and Ehresmannian. Each simplest in their category. Don't quantize gravity, unite the geometries.

== Concepts ==
=== Observerse ===
The '''observerse''' is the central mathematical object in the [[Theory of Geometric Unity]]. It is a mapping from a four-dimensional manifold <math>X^4</math> to a manifold <math>Y</math>, which replaces Einstein's spacetime. There are four different constructions of the observerse: exogenous, bundle-theoretic, endogenous, and tautological. Each generates a possible Geometric Unity theory.

Read more at [[Observerse|observerse]].
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== Motivations ==
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=== Beauty ===
Dirac on beauty.

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=== Source Code and the End of Physics ===
In his answer to the last [https://www.edge.org/ Edge.org] annual question "What is the Last Question?", Eric [https://www.edge.org/response-detail/27761 responded:]
<blockquote style="font-size: 125%; background: #f3f3ff; border-color: #ddd;">
Does something unprecedented happen when we finally learn our own source code?
</blockquote>
The source code, or a theory of everything, would mean the end of theoretical discovery at the most fundamental level of physics; all new discovery would take place at levels of more complexity.

==== Twin Nuclei Problem of Cell and Atom ====
Geometric Unity is hoped to solve the [[Twin Nuclei Problem]] by finding a means of breaking the Einsteinian speed limit, the speed of light, so that human life can spread across the cosmos. Eric says that does not necessarily imply faster-than-light travel or other concepts in that vein, and that he does not know whether this hope will bear fruit. A theory of everything, though, would provide definite means of exploring possible ways around Einstein's constraint.

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=== New April Fools' Day Tradition ===
The release of updates to Geometric Unity on (or around) April 1st is to establish a new April Fools' Day tradition. The tradition was conceived in recognition that many ideas may never be expressed due to social pressures and the risk to livelihood and reputation. To address that, it is proposed that on one day a year, people should be allowed to put forward competent, partial ideas that would normally tarnish one's reputation or destroy one's career. Then let the "fools" develop their ideas for a year, and report back, with the choice of abandoning the idea without reputational cost or continuing their pursuit. 

Eric first outlined this idea in two Twitter threads in the Spring of 2018, and discussed it further at the beginning of [[A Portal Special Presentation- Geometric Unity: A First Look]]. During the video's preface, in relation to Geometric Unity, he said, "[If] there is a fool, it is certainly me, because I have sat on this theory for almost 40 years."

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== History ==
Eric Weinstein first began developing the theory in the mid to late 1980s while he was a PhD student at Harvard University.

=== 2013 First Presentation ===
[[File:Oxford Lecture Cover Image.jpg|thumb|right]]
Geometric Unity was [[A Portal Special Presentation- Geometric Unity: A First Look|first presented]] in three Simonyi Special Lectures delivered over the course of a week at the University of Oxford. The lectures were organized by [https://twitter.com/MarcusduSautoy Marcus du Sautoy], the Simonyi Professor for the Public Understanding of Science. The lectures provided a broad overview of the mathematical structures in the theory’s endogenous version, discussed where current effective theories are recovered in those structures, and showed general predictions based on those structures.

=== 2016 Private Talk at FQXi ===
Three years after the first presentation at Oxford, Eric gave a private talk on Geometric Unity at [https://fqxi.org/ FQXi.]

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=== 2020 Video Release of First Presentation ===
[[A Portal Special Presentation- Geometric Unity: A First Look]] was uploaded on April 2nd, 2020 as a general introduction to Geometric Unity. The video is divided into three sections: a preface, the lecture proper, and a supplementary PowerPoint presentation. The preface provides context based on historical and contemporary events and introduces the concept of a theory of everything. The lecture shown is a recording of the first lecture given at the University of Oxford. The PowerPoint reviews the lecture’s major concepts in a more up-to-date notation.

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=== 2021 Manuscript and Presentation ===
In April, Eric will be presenting Geometric Unity on the Joe Rogan Experience and releasing a manuscript.

== Eric Discussing Geometric Unity ==
Appearances where Eric discusses Geometric Unity are listed below.

{| class="wikitable sortable"
|-
! Outlet !! Title !! Link !! Date
|-
| Joe Rogan Experience || [[Joe Rogan Experience 1453 - Eric Weinstein (YouTube Content)|Joe Rogan Experience #1453 - Eric Weinstein]] || [https://www.youtube.com/watch?v=wf0_nMaQ6tA Watch] || 2020-04-03
|-
| Lex Fridman Podcast || [[Eric Weinstein: Geometric Unity and the Call for New Ideas, Leaders & Institutions (YouTube Content)|Eric Weinstein: Geometric Unity and the Call for New Ideas, Leaders & Institutions]]|| [https://www.youtube.com/watch?v=rIAZJNe7YtE Watch] || 2020-04-13
|-
| Into the Impossible (Dr. Brian Keating) || [[Eric Weinstein: Theories of Everything, Geometric Unity & Science’s Paths. Ep 49 (YouTube Content)|Eric Weinstein: Theories of Everything, Geometric Unity, Mathematical Reality]] || [https://www.youtube.com/watch?v=YjsPb3kBGnk Watch] || 2020-05-19
|-
| DarkHorse Podcast (Bret Weinstein) || [[Bret Weinstein and Eric Weinstein: Fundamental Truth and How to Think About it (YouTube Content)|Bret Weinstein and Eric Weinstein: Fundamental Truth and How to Think About it]] || [https://www.youtube.com/watch?v=XjOg-OP_69Y Watch] || 2020-06-26
|-
| PBS Space Time<br>Into the Impossible (Dr. Brian Keating) || [[Theory of Everything Controversies: Livestream (YouTube Content)|Theory of Everything Controversies: Livestream]] || [https://www.youtube.com/watch?v=N_aN8NnoeO0 Watch] || 2020-08-12
|-
| Into the Impossible (Dr. Brian Keating) || [[Stephen Wolfram & Eric Weinstein: The Nature of Mathematical Reality (YouTube Content)|Stephen Wolfram & Eric Weinstein: The Nature of Mathematical Reality]] || [https://www.youtube.com/watch?v=OI0AZ4Y4Ip4 Watch] || 2020-08-12
|-
| Into the Impossible (Dr. Brian Keating) || [[Eric Weinstein: Imposter Syndrome, Donald Trump, & the Future of Theoretical Physics (YouTube Content)|Eric Weinstein: Imposter Syndrome, Donald Trump, & the Future of Theoretical Physics]] || [https://www.youtube.com/watch?v=DvPvFDF9dAE Watch] || 2020-11-20
|-
| Into the Impossible (Dr. Brian Keating) || [[Eric Weinstein: Ask Me Anything! (YouTube Content)|Eric Weinstein: Ask Me Anything!]] || [https://www.youtube.com/watch?v=jqgEOCMu7B4 Watch] || 2020-12-04
|-
| Into the Impossible (Dr. Brian Keating) || [[Max Tegmark & Eric Weinstein in Conversation (YouTube Content)| Max Tegmark & Eric Weinstein in Conversation]] || [https://www.youtube.com/watch?v=qRmbQP6ho4c Watch] || 2020-12-31
|-
| Into the Impossible (Dr. Brian Keating) || [[Eric Weinstein & Garrett Lisi: Theories (& Experiments) of Everything LIVE! (YouTube Content)|Eric Weinstein & Garrett Lisi: Theories (& Experiments) of Everything LIVE!]] || [https://www.youtube.com/watch?v=QCKCQNFsJUw Watch] || 2021-01-21
|-
| Joe Rogan Experience || Joe Rogan Experience <nowiki>#Placeholder</nowiki> - Eric Weinstein || Placeholder || 2021-04-<nowiki>##Placeholder</nowiki>
|-
|}

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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]

<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>

== Key Ideas ==

=== Starting point: three observations by Edward Witten === 

[[File:GU_triangle.png|500px|right]]


{| class="wikitable"
| '''1.''' The Arena (<math> X, g_{\mu\nu}</math>)
| <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 field equations, which describe gravity in the theory of general relativity

|-
| '''2.''' <math>G</math> (non abelian)
<math> SU(3) \times SU(2) \times U(1)</math>
| <math>d_A^*F_A=J(\psi)</math>
| the Yang-Mills equation, which governs all other force fields in Yang-Mill-Maxwell theory
|-
| '''3.''' Matter
Antisymmetric, therefore light
| <math>(i \hbar \gamma^\mu \partial_\mu - m) \psi = 0</math>
| the Dirac equation, the equation of motion describing matter particles, or fermions
|}

'''Key guiding question:''' what are the compatibilities and incompatibilities between these puzzle pieces on the geometric level before the theory is created quantum mechanical.

=== 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 <math>\nabla</math> of the metric <math>P_E(F_{\nabla})</math>
* From Yang-Mills-Maxwell-Anderson-Higgs theory of gauge fields, we take the adjoint exterior derivative coupled to a connection <math>d^\star_A F_A</math>

'''Idea:''' What if the <math>F</math>'s are the same in both contexts?

Further, supposing these <math>F</math>'s are the same, then why apply two different 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: <math>P_E(F_{\nabla h}) \neq  h^{-1} P_E(F_{\nabla}) h </math>. 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 ===

'''Observation:''' Gauge fields do not depend on the existence of a metric. One-forms are defined whether or not a metric is present. But for spinors (fermion fields) this is not the case. 

<blockquote>
"So if we're going to take the spin-2 <math>G_{\mu\nu}</math> field to be quantum mechanical, if it blinks out and does whatever the quantum does between observations. In the case of the photon, it is saying that the waves may blink out, but the ocean need not blink out. In the case of the Dirac theory, it is the ocean, the medium, in which the waves live that becomes uncertain itself. So even if you're comfortable with the quantum, to me, this becomes a bridge too far. So the question is: "How do we liberate the definition?"
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>

<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 ===

<blockquote>"The Dirac field, Einstein's field, and the connection fields are all geometrically well-motivated but we push a lot of the artificiality that we do not understand into the potential for the scalar field that gives everything its mass. We tend to treat it as something of a mysterious fudge factor. So the question is, ''if we have a Higgs field: "why is it here and why is it geometric?''""</blockquote>


=== Proposed Solution ===


<blockquote>
[[File:Geometric unity puzzle pieces.png|500px|right]]

[[File:GU proposal2.png|right|500px|right]]

We may have to generalize all three vertices before we can make progress. That's daunting because in each case, it would appear that we can make an argument that the three vertices are already the simplest possible theories that could live at these vertices.

* We know, for example, the Dirac operator is the most fundamental of all the elliptic operators and Euclidean signature generating all of the Atiyah-Singer theory.
* We know that Einstein's theory describes, in some sense, a unique spin two massless field capable of communicating gravity, which can be arrived at from field-theoretic rather than geometric consideration. 
* In the Yang-Mills case, it can also be argued that the Yang-Mills theory is the simplest theory that we can write down. In the Yang-Mills case, we have no substructure, and so we're doing the most simple-minded thing we can do by taking the norm-squared of the curvature and saying whatever the field strength is, let's measure that size. 

So if each one of these is simplest possible, doesn't Occam’s razor tell us that if we wish to remain in geometric field theory, that we've already reached bottom? 

I would say that there are other possibilities that while each of these may be simplest in its category, they are not simplest in their interaction.

For example, we know that Dirac famously took the square root of the Klein-Gordon equation to achieve the Dirac equation. He actually took two square roots, one of the differential operator, and another of the algebra on which it acts. But could we not do the same thing by re-interpreting what we saw in Donaldson theory and Chern-Simons theory and finding that there are first-order equations that imply second-order equations that are nonlinear in the curvature?

So, let's imagine the following: we replaced the standard model with a true second-order theory. We imagine the general relativity is replaced by a true first-order theory. And then we find that the true second-order theory admits of a square root and can be linked with the true first order theory.

''This would be a program for some kind of unification of Dirac's type, but in the force sector. The question is, "does this really make any sense? Are there any possibilities to do any such thing?"''



</blockquote>


<blockquote >
[[File:GU vision.png|right|400px|right]]

Let's talk about what the Geometric Unity (GU) proposal is. First of all, we observe that we have a division into intrinsic theories and auxiliary theory and between physics and mathematics. And intrinsic physical theory would be general relativity. An auxiliary physical theory would be the Yang-Mills theory, with the freedom to choose internal quantum numbers. 

At the mathematical level, an intrinsic theory would be, the older semi-Riemannian geometry. The study of manifolds with length and angle. Auxiliary geometry is what we're going to call fiber bundle theory or modern gauge theory. 

''Geometric Unity is the search for some way to break down the walls between these four boxes.''

What we'd like to come up with is some theory that is intrinsic, but allows us to play some of the games that exist in other boxes. How can we fit? How can we try to have our cake and eat it too? And use all the full suite of techniques that are available to us?

Our perspective is that the quantum that may be the comparatively easy part and that the unification of the geometry, which has not occurred, may be what we're being asked to do. 
</blockquote>

== Frequently Asked Questions ==


What will this theory predict?

When will Eric release the next part?

Why hasn't Eric gone through the normal scientific route? Arxiv.org? Academic journals?
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