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====Acknowledgments====
====Acknowledgments====
''[https://youtu.be/Z7rd04KzLcg?t=1909 00:31:49]''<br>
''[https://youtu.be/Z7rd04KzLcg?t=1909 00:31:49]''<br>
What I hope to show you is a lecture that was the first of three versions of this lecture that were delivered in Oxford over the course of a week. And, one of the things that has held me back is that I have a great number of people who I have to thank, for effectively being my Underground Railroad when my career got into serious trouble, to make sure—who made sure that I always had an opportunity to fight another day. And some of the most important of those people, one of whom occurs on this video, is [https://en.wikipedia.org/wiki/Marcus_du_Sautoy Marcus du Sautoy]. And Marcus, I just wanted to say thank you for your bravery, your courage, your friendship, and your encouragement. I know I've been absolutely impossible to you. I've made you wait for this, and I just want to say how much I love you.
What I hope to show you is a lecture that was the first of three versions of this lecture that were delivered in Oxford over the course of a week. And, one of the things that has held me back is that I have a great number of people who I have to thank, for effectively being my Underground Railroad when my career got into serious trouble, to make sure—who made sure that I always had an opportunity to fight another day. And some of the most important of those people, one of whom occurs on this video, is [https://en.wikipedia.org/wiki/Marcus_du_Sautoy Marcus du Sautoy]. And Marcus, I just wanted to say thank you for your bravery, your courage, your friendship, and your encouragement. I know I've been absolutely impossible to you. I've made you wait for this, and I just want to say how much I love you.
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''[https://youtu.be/Z7rd04KzLcg?t=3911 01:05:11]''<br>
''[https://youtu.be/Z7rd04KzLcg?t=3911 01:05:11]''<br>
So, if here is physical reality, standard physics is over here, we're going to start with the sandbox, and all we're going to put in it is \(X^4\). And we're going to set ourselves a strait-jacketed task of seeing how close we can come to dragging out a model that looks like the natural world that follows this trajectory. While it may appear that that is not a particularly smart thing to do, I would like to think that we could agree that it is quite possible that if that were to be the case, we might say that this is what Einstein meant by a creator, which was his anthropomorphic concept for necessity and elegance and design, having no choice in the making of the world.
So, if here is physical reality, standard physics is over here, we're going to start with the sandbox, and all we're going to put in it is \(X^4\). And we're going to set ourselves a strait-jacketed task of seeing how close we can come to dragging out a model that looks like the natural world that follows this trajectory. While it may appear that that is not a particularly smart thing to do, I would like to think that we could agree that it is quite possible that if that were to be the case, we might say that this is what Einstein meant by a creator, which was his anthropomorphic concept for necessity and elegance and design, having no choice in the making of the world.


====== Four flavors of GU with a focus on the endogenous version ======
====== Four flavors of GU with a focus on the endogenous version ======
[[File:GU Presentation Flavors Diagram.png|thumb|right]]


[01:06:19] So with that, let us begin to think about what we mean today by Geometric Unity. GU comes in four flavors, but I'm only getting one shot to do this, so I'm going to do the most exciting of them to me.  
''[https://youtu.be/Z7rd04KzLcg?t=3979 01:06:19]''<br>
So with that, let us begin to think about what we mean today by Geometric Unity. GU comes in four flavors, but I'm only getting one shot to do this, so I'm going to do the most exciting of them to me.


[01:06:42] There's a completely exogenous flavor.  
''[https://youtu.be/Z7rd04KzLcg?t=4002 01:06:42]''<br>
There's a completely exogenous flavor. And what I'm gonna do is I'm going to take the concept of observation and I'm going to break the world into two pieces: a place where we do our observation and a place where most of the activity takes place. And I'm going to try and do this without loss of generality. So in this case, we have \(X^4\) and it can map into some other space, and we are going to call this an '''observerse'''. The idea of an observerse is a bit like a stadium: you have a playing field and you have stands. They aren't distinct entities, they're coupled. And so fundamentally, we're going to replace one space with two. Exogenous model simply means that \(U\) is unrestricted, although larger than \(X^4\), so any manifold of four dimensions or higher that is capable of admitting \(X^4\) as an immersion.


What I'm gonna do is I'm going to take the concept of observation and I'm going to break the world into two pieces. A place where we do our observation and a place where most of the activity takes place. And I'm going to try and do this without loss of generality. So in this case, we have $$X^4$$ and it can map into some other space, and we are going to call this an observerse.
''[https://youtu.be/Z7rd04KzLcg?t=4061 01:07:41]''<br>
The next model we have is the bundle theoretic, in which case, \(U\) sits over \(X\) as a fiber bundle.


[01:07:15] The idea of an observerse is a bit like a stadium. You have a playing field and you have stands. They aren't distinct entities, they're coupled. And so, fundamentally, we're going to replace one space with two. Exogenous model simply means that $$U$$ is unrestricted, although larger than $$X^4$$, so any manifold of four dimensions or higher that is capable of admitting $$X^4$$ as an immersion.
''[https://youtu.be/Z7rd04KzLcg?t=4078 01:07:58]''<br>
The most exciting, which is the one we'll deal with today, is the endogenous model, where \(X^4\) actually grows the space \(U\) where the activity takes place. So we talk about extra dimensions, but these are, in some sense, not extra dimensions. They are implicit dimensions within \(X^4\).


[01:07:41] The next model we have is the bundle-theoretic, in which case, $$U$$ sits over $$X$$, as a fiber bundle.
''[https://youtu.be/Z7rd04KzLcg?t=4097 01:08:17]''<br>
And last, to proceed without loss of generality, we have the tautological model. In that case, \(X^4 = U\), and the immersion is the identity, and without loss of generality, we simply play our games on one space. Okay?


[01:07:58] The most exciting, which is the one we'll deal with today, is the endogenous model where $$X^4$$ actually grows the space $$U$$, where the activity takes place. So, we talked about extra dimensions, but these are, in some sense, not extra dimensions. They are implicit dimensions within $$X^4$$.  
=====Rules for Constructing GU=====
''[https://youtu.be/Z7rd04KzLcg?t=4116 01:08:36]''<br>
Now, we need rules. The rules are... sorry, feedback. No choice of fundamental metric. So we imagine that Einstein was presented with a fork in the road, and it's always disturbing not to follow Einstein's path, but we're in fact going to turn Einstein's game on its head and see if we can get anywhere with that, right? It's also a possibility that because Einstein's theory is so perfect, that if there's anything wrong with it, it's very hard to unthink it because everything is built on top of it.


And last, to proceed without loss of generality, we have the tautological model. In that case, $$X^4$$ equals $$U$$. And the immersion is the identity. And without loss of generality, we simply play our games on one space.  
''[https://youtu.be/Z7rd04KzLcg?t=4153 01:09:13]''<br>
So let's make no choice of fundamental metric, and in fact, let's go more ambitious, and let's say we're going to reverse the logic of Einstein. In Einstein, the metric is fundamental, but the Levi-Civita connection from which we deduce the curvature is emergent, right? So the fundamental theorem of Riemannian geometry is that every metric causes a connection to emerge, and then the curvature is built on the connection. We turn this around. We imagine we're looking for a connection, and we wish it to build a metric, because connections are amenable to quantization in a way that metrics are not.


Okay? Now we need rules. The rules  a choice of fundamental metric. So we imagine that Einstein was presented with a fork in the road, and it's always disturbing not to follow Einstein's path, but we're in fact going to turn Einstein's game on its head and see if we can get anywhere with that. Right. It's also a possibility that because Einstein's theory is so perfect that if there's anything wrong with it, it's very hard to unthink it because everything is built on top of it. So let's make no choice of fundamental metric.
''[https://youtu.be/Z7rd04KzLcg?t=4187 01:09:47]''<br>
The next point is that we always want to have a plan to return to finite dimensions without losing sight of the quantum.


[01:09:16] And in fact, let's go more ambitious. And let's say we're going to reverse the logic of Einstein. The metric is fundamental, but the Levi-Civita connection from which we deduced the curvature is emergent, right? So the fundamental theorem of Riemannian geometry is that every connection causes every metric causes a connection to emerge.
''[https://youtu.be/Z7rd04KzLcg?t=4196 01:09:56]''<br>
And lastly, we want to liberate matter from its dependence on the metric for its very existence. So we now need to build fermions onto our four-dimensional manifold in some way without ever choosing a metric, if we're even going to have any hope of playing a game involving matter, starting from this perspective of no information other than the most bare information. Let's get started.


[01:09:36] And then the curvature is built on the connection. We turn this around. We imagine we're looking for a connection and we wish it to build a metric, because connections are amenable to quantization in a way that metrics are not. The next point is that we always want to have a plan to return to finite dimensions without losing sight of the quantum.
=====Constructing Endogenous GU: Choosing All Metrics=====
''[https://youtu.be/Z7rd04KzLcg?t=4224 01:10:24]''<br>
We take \(X^4\). We need metrics, we have none. We're not allowed to choose one, so we do the standard trick: we choose them all.


[01:09:56] And lastly, we want to liberate matter from its dependence on the metric for its very existence. So we now need to build [[fermions]] onto our four-dimensional manifold in some way without ever choosing a metric, if we're even going to have any hope of playing a game involving matter, starting from this perspective of no information other than the most bare information.


[01:10:23] Let's get started. We take $$X^4$$, we need metrics. We have none. We're not allowed to choose one. So we do the standard trick. We choose them all.
<div style="text-align: center; margin-left: auto; margin-right: auto;">$$U^{14} = \text{met}(X^4)$$</div>


====== Choosing All Metrics ======


[01:10:36] So we allow $$U^{14}$$ to equal the space of metrics on $$X^4$$ pointwise. Therefore, if we propagate on top of this, let me call this the projection operator. If we propagate on $$U^{14}$$ we are, in some sense, following a Feynman-like idea of propagating over the space of all metrics, but not at a field level, at a pointwise tensorial level.
''[https://youtu.be/Z7rd04KzLcg?t=4236 01:10:36]''<br>
So we allow \(U^{14}\) to equal the space of metrics on \(X^4\) pointwise. Therefore, if we propagate on top of this—let me call this (\(\pi\)) the projection operator. If we propagate on \(U^{14}\), we are, in some sense, following a Feynman-like idea of propagating over the space of all metrics, but not at a field level, at a pointwise tensorial level.


[01:11:03] Is there a metric on $$U^{14}$$? Well, we both want one and don't want one. If we had a metric from the space of all metrics, we could define fermions, but we would also lock out any ability to do dynamics. We want some choice over what this metric is, but we don't want full choice, because we want enough to be able to define the matter fields to begin with.
''[https://youtu.be/Z7rd04KzLcg?t=4263 01:11:03]''<br>
Is there a metric on \(U^{14}\)? Well, we both want one and don't want one. If we had a metric from the space of all metrics, we could define fermions, but we would also lock out any ability to do dynamics. We want some choice over what this metric is, but we don't want full choice, because we want enough to be able to define the matter fields to begin with.


[01:11:22] It turns out if this is $$X^4$$ and this is this particular endogenous choice of $$U^{14}$$, we have a 10-dimensional metric along the fibers. So we have a $$g^{10}_{\mu\nu}$$.  
''[https://youtu.be/Z7rd04KzLcg?t=4282 01:11:22]''<br>
It turns out that if this is \(X^4\), and this is this particular endogenous choice of \(U^{14}\), we have a 10-dimensional metric along the fibers. So we have a \(g^{10}_{\mu \nu}\). Further, for every point in the fibers, we get a metric downstairs on the base space. So if we pull back the cotangent bundle, we get a metric \(g^4_{\mu \nu}\) on \(\pi*\) of the cotangent bundle of \(X\).


Further, for every point in the fibers, we get a metric downstairs on the base space. So if we pull back the cotangent bundle, we get a metric $$g^4_{\mu\nu}$$ on $$\Pi^*$$ of the cotangent bundle of $$X$$.
=====Chimeric Bundle=====
''[https://youtu.be/Z7rd04KzLcg?t=4337 01:12:17]''<br>
We now define the '''chimeric bundle''', right? And the chimeric bundle is this direct sum of the vertical tangent bundle along the fibers with the pullback, which we're going to call the horizontal bundle, from the base space. So the chimeric bundle is going to be the vertical tangent space of 10 dimensions to \(U\), direct sum the cotangent space, which we're going to call horizontal, to \(U\). And the great thing about the chimeric bundle is that it has an a priori metric. It's got a metric on the 4, a metric on the 10, and we can always decide that the two of them are naturally perpendicular to each other.


[01:12:17] We now define the chimeric bundle, right? And the chimeric bundle is this direct sum of the vertical tangent bundle along the fibers with the pullback, which we're going to call the horizontal bundle from the base space.


So the chimeric bundle is going to be $$T^{10}_V(U) \oplus T^{4}_H(U)$$. And the great thing about the chimeric bundle is that it has an a priori metric. It's got a metric on the four, a metric on the 10, and we can always decide that the two of them are naturally perpendicular to each other.
<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ T_V^{10}(U) \oplus T_H^{4*}(U) $$</div>


[01:13:00] Furthermore, it is almost canonically isomorphic to the tangent bundle or the cotangent bundle, because we either have 4 out of 14, or 10 out of 14 dimensions, on the nose. So the question is, what are we missing? And the answer is that we're missing exactly the data of a connection. So this bundle, chimeric $$C$$. We have $$C$$ equal to the tangent bundle of $$U$$ up to a choice of a connection $$\theta$$.


[01:13:32] And this is exactly what we wanted. We have a situation where we have some field on the manifold $$X$$ in the form of a connection which is amenable, more friendly to quantization, which is now determining a metric turning around the Levi-Civita game. And the only problem is that we've had to buy ourselves into a different space than the one we thought we wanted to work on.
''[https://youtu.be/Z7rd04KzLcg?t=4380 01:13:00]''<br>
Furthermore, it is almost canonically isomorphic to the tangent bundle or the cotangent bundle, because we either have 4 out of 14, or 10 out of 14 dimensions, on the nose. So the question is, what are we missing? And the answer is that we're missing exactly the data of a connection. So this bundle, chimeric \(C\)—we have \(C\) is equal to the tangent bundle of \(U\) up to a choice of a connection \(\theta\).


[01:13:55] But now as $$\theta$$ changes, the fermions are defined on the chimeric bundle, and it's the isomorphism from the chimeric bundle to the tangent bundle of the space $$U$$, which is variant, which means that the fermions no longer depend on the metric. They no longer depend on the $$\theta$$ connection. They are there if things go quantum mechanical, and we've achieved our objective of putting the matter fields and the spin-one fields on something of the same footing.


[01:14:27] But, and I want to emphasize this one thing.
<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ CU \overset{\theta}{=} TU $$</div>


Most of us, we think a lot about final theories and and about unification, but until you actually start daring to try to do it, you don't realize what the process of it feels like.


[01:14:54] And, I try to imagine conducting your life where you have no children and let's say no philanthropic urges. And what you want to do is you want to use all of your money for yourself. And die penniless, right? Like a perfect finish. Assuming that that's what you wanted to do, it would be pretty nerve-wracking at the end, right? How many days left do I have? How many dollars left do I have? This is the process of unification.
''[https://youtu.be/Z7rd04KzLcg?t=4412 01:13:32]''<br>
And this is exactly what we wanted. We have a situation where we have some field on the manifold \(X\) in the form of a connection, which is amenable, more friendly to quantization, which is now determining a metric, turning around the Levi-Civita game. And the only problem is that we've had to buy ourselves into a different space than the one we thought we wanted to work on.


In physics, you start giving away all of your most valuable possessions, and you don't know whether you've given them away too early, whether you husband them too long. And so in this process, what we've just done is we've started to paint ourselves into a corner. And we got something we wanted, but we've given away freedom.
''[https://youtu.be/Z7rd04KzLcg?t=4435 01:13:55]''<br>
But now, as \(\theta\) changes, the fermions are defined on the chimeric bundle, and it's the isomorphism from the chimeric bundle to the tangent bundle of the space \(U\), which is variant—which means that the fermions no longer depend on the metric. They no longer depend on the \(\theta\) connection. They are there if things go quantum mechanical, and we've achieved our objective of putting the matter fields and the spin-one fields on something of the same footing.


We're now dealing with a 14-dimensional world.
=====Observerse Conclusion=====
''[https://youtu.be/Z7rd04KzLcg?t=4467 01:14:27]''<br>
But, and I want to emphasize this: one thing most of us—we think a lot about final theories and about unification, but until you actually start daring to try to do it, you don't realize what the process of it feels like. And, try to imagine conducting your life where you have no children, let's say, and no philanthropic urges, and what you want to do is you want to use all of your money for yourself and die penniless, right? Like a perfect finish. Assuming that that's what you wanted to do, it would be pretty nerve-wracking at the end, right? How many days left do I have? How many dollars left do I have? This is the process of unification in physics. You start giving away all of your most valuable possessions, and you don't know whether you've given them away too early, whether you husbanded them too long, and so in this process, what we've just done is we've started to paint ourselves into a corner. We got something we wanted, but we've given away freedom. We're now dealing with a 14-dimensional world.


[01:15:39] Well, let me just sum this up by saying: between fundamental and emergent, standard model and GR (general relativity). Let's do GR (general relativity). Fundamental is the metric; emergent is the connection. Here in GU (Geometric Unity), it is the connection that's fundamental and the metric that's emergent.
[[File:GU Presentation Fund-Emerg Diagram.png|thumb|right]]


[01:16:23] And the next unit of GU. So this is sort of the first unit of GU. Are there any quick questions having to do with confusion or may I proceed to the next unit?
''[https://youtu.be/Z7rd04KzLcg?t=4539 01:15:39]''<br>
Well, let me just sum this up by saying: between fundamental and emergent, standard model and GR... Let's do GR. Fundamental is the metric, emergent is the connection. Here in GU, it is the connection that's fundamental and the metric that's emergent.


==== GU II: From Magic Beans to Unified Field Content ====
==== GU II: From Magic Beans to Unified Field Content ====
====== Magic Beans trade ======
====== Magic Beans trade ======
[01:16:23] And the next unit of GU. So this is sort of the first unit of GU. Are there any quick questions having to do with confusion or may I proceed to the next unit?
[01:16:36] Okay. The next unit of GU is the unified field content. What does it mean for our fields to become unified? There are, in fact, only at this moment, two fields that know about $$X$$. $$\theta$$, which is the connection that we've just talked about, and a section, $$\sigma$$ that takes us back so that we can communicate back and forth between $$U$$ and $$X$$. We now need field content that only knows about $$U$$, which now has a metric depending on $$\theta$$.
[01:16:36] Okay. The next unit of GU is the unified field content. What does it mean for our fields to become unified? There are, in fact, only at this moment, two fields that know about $$X$$. $$\theta$$, which is the connection that we've just talked about, and a section, $$\sigma$$ that takes us back so that we can communicate back and forth between $$U$$ and $$X$$. We now need field content that only knows about $$U$$, which now has a metric depending on $$\theta$$.


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