Difference between revisions of "A Portal Special Presentation- Geometric Unity: A First Look"
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A Portal Special Presentation- Geometric Unity: A First Look (view source)
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<p>[02:17:28] In Sector 1 of the Geometric Unity theory spacetime is replaced and recovered by the observerse contemplating itself. And so there are several sectors of GU and I wanted to go through at least four of them. In Einstein spacetime, we have not only four degrees of freedom, but also a spacetime metric representing rulers and protractors. | <p>[02:17:28] In Sector 1 of the Geometric Unity theory spacetime is replaced and recovered by the observerse contemplating itself. And so there are several sectors of GU and I wanted to go through at least four of them. In Einstein spacetime, we have not only four degrees of freedom, but also a spacetime metric representing rulers and protractors. | ||
<p>[02:17:56] If we're going to replace that. It's very tricky because it's almost impossible to think about what would be underneath Einstein’s theory. Now, there's a huge problem in the spinorial sector, which I don't why more people don't worry about. Which is that spinors aren't defined for representations of the double cover of GL four R the general linear group’s effective spin analog. And as such, if we imagine that we will one day quantize gravity, we will lose our definition, not of the electrons, but let's say of the medium in which the electrons operate. That is, there will be no spinorial bundle until we have an observation of a metric. So one thing we can do is to take a manifold $$X^d$$ as the starting point and see if we can create an entire universe from no other data. Not even with a metric. So since we don't choose a metric, what we instead do is to work over the space of all possible point-wise metrics. So not quite in the Feynman sense, but in the sense that | <p>[02:17:56] If we're going to replace that. It's very tricky because it's almost impossible to think about what would be underneath Einstein’s theory. Now, there's a huge problem in the spinorial sector, which I don't why more people don't worry about. Which is that spinors aren't defined for representations of the double cover of GL four R the general linear group’s effective spin analog. And as such, if we imagine that we will one day quantize gravity, we will lose our definition, not of the electrons, but let's say of the medium in which the electrons operate. That is, there will be no spinorial bundle until we have an observation of a metric. So one thing we can do is to take a manifold $$X^d$$ as the starting point and see if we can create an entire universe from no other data. Not even with a metric. So since we don't choose a metric, what we instead do is to work over the space of all possible point-wise metrics. So not quite in the Feynman sense, but in the sense that we will work over a bundle that is of a quite larger, quite a bit larger, dimension. | ||
<p>[02:19:10] So for example, if $$X$$ was a four dimensional, therefore d equals four, then $$Y$$, in this case, would be $$d^2$$, which would be 16 plus 3d, which would be 12 making 28 divided by two, which would be 14. So in other words, a four-dimensional proto-spacetime, not a spacetime, but a proto-spacetime with no metric would give rise to a 14-dimensional "observerse" portion called $$Y$$. Now, I believe that in the lecture in Oxford, I called that $$U$$, so I'm sorry for the confusion, but of course, this shifts around every time I take it out of the garage. And that's one of the problems with working on a theory in solitude for many years. So, we have two separate spaces and we have fields on the two spaces. | <p>[02:19:10] So for example, if $$X$$ was a four dimensional, therefore d equals four, then $$Y$$, in this case, would be $$d^2$$, which would be 16 plus 3d, which would be 12 making 28 divided by two, which would be 14. So in other words, a four-dimensional proto-spacetime, not a spacetime, but a proto-spacetime with no metric would give rise to a 14-dimensional "observerse" portion called $$Y$$. Now, I believe that in the lecture in Oxford, I called that $$U$$, so I'm sorry for the confusion, but of course, this shifts around every time I take it out of the garage. And that's one of the problems with working on a theory in solitude for many years. So, we have two separate spaces and we have fields on the two spaces. | ||
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<p>[02:27:46] We then get a bi connection. In other words, because we have two different ways of pushing a connection around. If we have a choice of a base connection, we can push the base connection around in two different ways according to the portion of it that is, uh, coming from the gauge transformations, uh, curly H or the affine translations coming from curly N. | <p>[02:27:46] We then get a bi connection. In other words, because we have two different ways of pushing a connection around. If we have a choice of a base connection, we can push the base connection around in two different ways according to the portion of it that is, uh, coming from the gauge transformations, uh, curly H or the affine translations coming from curly N. | ||
<p>[02:28:10] Yeah. We can call this map the bi connection, which gives us two separate connections for any point in the, uh, inhomogeneous gauge group. And we can notice that it can be viewed | <p>[02:28:10] Yeah. We can call this map the bi connection, which gives us two separate connections for any point in the, uh, inhomogeneous gauge group. And we can notice that it can be viewed as a section of a bundle over the base space to come. We find an interesting embedding of the gauge group that is not the standard one in the inhomogeneous gauge group. | ||
<p>[02:28:42] So our summary diagram looks something like this. | <p>[02:28:42] So our summary diagram looks something like this. Take a look at the Taus of $$A_0$$. We will find a homeomorphism of the gauge group into its inhomogeneous extension that isn't simply inclusion under the first factor. And I'm realizing that I have the wrong pi production. That should just be as simple $$\pi$$, projecting down, we have a map from the inhomogeneous gauge group, via the bi-connection to A cross A connections cross connections, and that that behaves well according to the difference operator $$\delta$$ that takes the difference of two connections and gives an honest add-value one-form. | ||
<p>[02:29:23] | <p>[02:29:23] The infinitesimal action of the gauge transformation of a gauge transformation, or at least an infinite testable one on a point inside of the group is given by a somewhat, almost familiar expression, which should remind us of how the first term in the gauge deformation complex for Self-Dual Yang-Mills actually gets started. | ||
<p>[02:29:47] And so by acting via this | <p>[02:29:47] And so by acting via this interesting embedding of the embedding of the gauge group inside its inhomogeneous extension, but the non-trivial one, we get something very close to the original first step of the two-step Deformation complex. Now in Sector 3, there are payoffs to the magic beans trade. | ||
<p>[02:30:14] The big issue here | <p>[02:30:14] The big issue here is that we've forgone the privilege of being able to choose and dial in our own field content, and we've decided to remain restricted to anything we can generate only from $$X^{d}$$, in this case $$X^{4}$$. So we generated $$Y^{14}$$ from $$X^{4}$$. And then we generated chimeric tangent bundles. On top of that, we built spinors off of the chimeric tangent bundle, and we have not made any other choices. | ||
<p>[02:30:42] So we're dealing with, I think it's $$U^{128}$$, $$U^{2^7}$$. That is our structure group and it's fixed by the choice of $$X^4$$ not anything. So what do we get? Well, as promised, there is a tilted homomomorphism which takes the gauge group into its inhomogeneous extension. It acts as inclusion on the first factor, but it uses the Levi-Civita connection to create a second sort of Maurer Cartan form. | <p>[02:30:42] So we're dealing with, I think it's $$U^{128}$$, $$U^{2^7}$$. That is our structure group and it's fixed by the choice of $$X^4$$ not anything. So what do we get? Well, as promised, there is a tilted homomomorphism which takes the gauge group into its inhomogeneous extension. It acts as inclusion on the first factor, but it uses the Levi-Civita connection to create a second sort of Maurer Cartan form. | ||
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<p>[02:31:17] I hope I remember the terminology right. It's been a long time. The map is injective because it's injective under the first factor, but it actually gives us a nontrivial embedding of the gauge group in its inhomogeneous extension, which makes the whole theory work. We then get to Shiab operators now, a Shiab operator a map from the group crossed the add-valued i forms. | <p>[02:31:17] I hope I remember the terminology right. It's been a long time. The map is injective because it's injective under the first factor, but it actually gives us a nontrivial embedding of the gauge group in its inhomogeneous extension, which makes the whole theory work. We then get to Shiab operators now, a Shiab operator a map from the group crossed the add-valued i forms. | ||
<p>[02:31:43] In this case, the particular Shiab operators we’re interested in is mapping i form is to d minus three, plus i forms. So, for example, uh, you would map a two form to a d minus three plus i. So if d, for example, were | <p>[02:31:43] In this case, the particular Shiab operators we’re interested in is mapping i form is to d minus three, plus i forms. So, for example, uh, you would map a two form to a d minus three plus i. So if d, for example, were 14, ..., and i was equal to two. Then 14 minus three is equal to 11 plus two is equal to 13. So that would be an add-valued 14 minus one form, which is exactly the right place for something to form a current. That is the differential of a Lagrangian on the space. | ||
<p>[02:32:38] Now the augmented torsion, the torsion is a very strange object. It's introduced sort of right at the beginning of learning a differential geometry, but it really doesn't get used very much. One of the reasons it doesn't get used, gauge theory is that it's not gauge invariant. It has a gauge and variant piece to it, but then a piece that spoils the gauge invariant. | <p>[02:32:38] Now the augmented torsion, the torsion is a very strange object. It's introduced sort of right at the beginning of learning a differential geometry, but it really doesn't get used very much. One of the reasons it doesn't get used, gauge theory is that it's not gauge invariant. It has a gauge and variant piece to it, but then a piece that spoils the gauge invariant. | ||
<p>[02:32:56] But we, because we have two connections, one of the ideas was to introduce two diseases and then to take a difference, and as long as the disease is the same in both | <p>[02:32:56] But we, because we have two connections, one of the ideas was to introduce two diseases and then to take a difference, and as long as the disease is the same in both, the difference will not have the disease because both diseases are included, but with a minus sign between them. So the augmented torsion is relatively well behaved relative to this particular slanted or tilted embedding of the gauge group in its inhomogeneous extension. | ||
<p>[02:33:28] Which this is very nice because now we actually have | <p>[02:33:28] Which this is very nice because now we actually have use for the torsion. We have an understanding of why it may never have figured, particularly into geometry, is that you need to have two connections rather than one to see the advantages of torsion at all. So here's an example of one Ship in a Bottle (Shiab) operator. | ||
<p>[02:33:47] I think this would be sort of analogous, if I'm not mistaken, to trying to take the Ricci curvature from the entire Riemann curvature, but if you think about what Einstein did, Einstein had to go further and reduce the Ricci curvature to the scalar | <p>[02:33:47] I think this would be sort of analogous, if I'm not mistaken, to trying to take the [[Ricci curvature]] from the entire [[Riemann curvature]], but if you think about what Einstein did, Einstein had to go further and reduce the [[Ricci curvature]] to the scalar curvature. And then, sort of dial the components of the traceless Ricci and the scalar curvature to get the right proportions. | ||
<p>[02:34:11] So | <p>[02:34:11] So, there are many Shiab operators and you have to be very careful about which one you want. And once you know exactly what it is you're trying to hit, you can choose the Shiab operator to be bespoke and get the contraction that you need. Now I've, I've made you guys sit through a lot. So I wanted to give you a sort of humorously, a feeling of positivity for the exhaustion. | ||
< | <blockquote> | ||
The years of anxious searching in the dark, with their intense longing, their alternations of confidence and exhaustion and the final emergence into the light—only those who have experienced it can understand it. -Albert Einstein | |||
</blockquote> | |||
<p>[02: | <p>[02:34:41] I've just always thought this was like the most sensitive and beautiful quote, and I wish it were one of his better known quotes, but I think it's so singular that it's hard to, it's hard to feel what it was that he was talking about because in fact, he sort of explains this in the last line. | ||
<p>[02:35: | <p>[02:35:10] So, given that you've been on a long journey, here is something of what Geometric Unity equations might look like. So, in the first place, you have the swerved curvature, the Shiab applied to the curvature tensor. That, in general, does not work out to be exact, so you can't have it as the differential of a Lagrangian. | ||
<p>[02:35: | <p>[02:35:31] And, in fact that we talked about swirls, swerves twirls eddies. There has to be a quadratic Eddy tensor. That I occasionally forget when I pull this thing out of mothballs and the two of those together make up what I call the total swervature. And, on the other side of that equation, you have the displaced torsion, which I've called the displasion. | ||
<p>[02: | <p>[02:35:54] And to get rid of the pesky sort of minus sign and Hodge star operator. This would be the replacement for the Einstein equation, not on $$X$$ where we would perceive it, but on $$Y$$ before being pulled back onto the manifold $$X$$. So a condensation of that would be very simple. In simplest terms, we would be saying that the swervature is equal to the displasion, and at least in this sector of the four main equations of theoretical physics. | ||
<p>[02:36: | <p>[02:36:25] This would be the replacement for the Einstein equations again on $$Y$$ before being pulled back to $$X$$. Next is the sketch of the fermionic field content. I'm not sure whether that should have been sector four, sector three, but it's going to be very brief. I showed some pictures during the lecture and I'm not going to go back through them, but I wanted to just give you an idea of where this mysterious third generation I think comes [from]. | ||
<p>[02: | <p>[02:36:55] So, if we review the three identities here, we see that if we have a space $$V$$, thought of like as a tangent bundle, and then you have spinors built on the tangent bundle. When you tensor product the tangent bundle with its own spinors, it breaks up into two pieces. One piece is the so-called Cartan product, which is sort of the some of the highest weights, and the other is a second copy of the spinors gotten through the Clifford contraction. | ||
<p>[02:37: | <p>[02:37:28] So, that's well known, but now what I think fewer people know, many people know that the spinors have a sort of an exponential property. That is, the spinors of a direct sum are the tensor product the spinors of the two summands of the direct sum. So that's a very nice sort of version of an exponential; an exponential would take a sum and turn it into a product. | ||
<p>[02:37:52] What happens when you're trying to think about a tangent space in $$Y$$ is being broken up into a tangent space along an immersed $$X$$. Together with its normal bundle. So imagine that $$X$$ and $$Y$$ are the tangent space to $$X$$ and a normal bundle. So the [[Rarita Schwinger]] piece -- that is, the spin 3/2 piece -- has a funny kind of almost exponential property. That is the [[Rarita Schwinger]] content of a direct sum of vector spaces is equal to the Rarita-Schiwnger. | |||
<p>[02:38:31] First, tensor producted with the ordinary spinors in the second direct sum with the ordinary spinors in the first tensor producted with the Ricci, with the, sorry, the Rarita Schwinger content of the second summand. but then there's this extra interesting term, which is the spinors on the first summand tensor producted with the spinors on the second summand. | <p>[02:38:31] First, tensor producted with the ordinary spinors in the second direct sum with the ordinary spinors in the first tensor producted with the Ricci, with the, sorry, the Rarita Schwinger content of the second summand. but then there's this extra interesting term, which is the spinors on the first summand tensor producted with the spinors on the second summand. | ||
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<p>[02:40:12] It's a daunting task to try. Address, uh, people about something you've been thinking about for a long time and have no idea whether it's even remotely correct. This is the Einsteinian replacement and it must be pulled back to X. That's the first thing. The yang mills, Maxwell piece comes from a direct square of the Einstein replaceme | <p>[02:40:12] It's a daunting task to try. Address, uh, people about something you've been thinking about for a long time and have no idea whether it's even remotely correct. This is the Einsteinian replacement and it must be pulled back to X. That's the first thing. The yang mills, Maxwell piece comes from a direct square of the Einstein replaceme | ||
<p>[02:40:29] That is, I don't believe that we're really looking for a unifying equation. I think we're looking for a unifying | <p>[02:40:29] That is, I don't believe that we're really looking for a unifying equation. I think we're looking for a unifying Dirac square. Dirac famously took the square root of the Klein-Gordon equation, and he gave the Dirac equation. And in fact, I believe that the Dirac equation and the Einstein equation are to be augmented and fit into the square root part of a Dirac square. | ||
<p>[02: | <p>[02:40:55] And, I believe that the [[Yang-Mills]] content and [[Higgs]] version of the [[Klein-Gordon]] equation would go in the square part of the Dirac square. So, two of these equations unified differently than to others. And the two pairs are unified in the content of a Dirac square. The Dirac piece will be done separately, elsewhere. | ||
<p>[02:41: | <p>[02:41:20] when we get around to it. And contains the Rarita-Schwinger field content, which is fundamental and new. There are only two generations in this model. I think people have accepted that there are three, but I don't believe that there are three. I think that there are two and that the third that unifies with other matter at higher energies. | ||
<p>[02: | <p>[02:41:39] The quartic Higgs piece comes from the Dirac Squaring of a quadratic. Remember, there's an eddy tensor, which is quadratic in the augmented torsion. The metric does multiple duties. Here it's the main field in this version of GU with the sort of strongest assumptions as field content on that is originally on $$X$$ where as most of the rest of the field content is on $$Y$$. | ||
<p>[02:42: | <p>[02:42:06] But it also acts as the observer pulling back the full content of $$Y$$ onto X to be interpreted as if it came from $$X$$ all along, generating the sort of illusion of internal quantum numbers. And I should say that the [[Pati-Salam]] theory, which is usually advertised as I think as SU(4) cross SU(2) cross SU(2) is really much more naturally Spin(6)xSpin(4), when the trace portion of the space of metrics is put in with the proper sign. | ||
<p>[02: | <p>[02:42:40] If you're trying to generate the sector that begins as X(1,3). Remember $$X^d$$, where d equals four is the generic situation. But you have all these different sectors. I believe that these sectors probably exist if this model's correct, but we are trapped in the (1,3) sector. | ||
<p>[02:43:00] So you have to figure out what the implications are for pushing that indefinite signature up into an indefinite signature on the $$Y$$ manifold. And there are signatures that make it look like the '[[Pati-Salam]] rather than Spin(10) [or] SU(5) line of thinking. | |||
====== Thanks & Final Thoughts ====== | ====== Thanks & Final Thoughts ====== | ||