A Portal Special Presentation- Geometric Unity: A First Look: Difference between revisions

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].

[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.

[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.

[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.

[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.

[02:38:53] Now recalling that. Um, when I started my career, we did not know that neutrinos were massive. And I figured that they probably had to be massive because I desperately wanted a 16 dimensional space of internal quantum numbers, not 15, because my, my ideas only work if the space of internal quantum numbers as a dimension to to the n.

[02:39:14] And, one of my favorite equations at the time was 15 equals two to the four. Not literally true, but almost true. And thankfully in the late 1990s, the case for 16 particles in a generation was strengthened when neutrinos were found to have mass. But that remaining term in the southeast corner, the spinors on $$X$$ tensor spinors on $$Y$$ looks like the term above it in line 2.15.

[02:39:46] And that, in fact, is the third generation of matter, in my opinion. That is, it is not a true generation. It is broken off and would unify very differently if we were able to heat the universe to the proper temperature. So, starting to sum up, this is not the full theory. I'm just presenting this in part to dip my toe back into the water.

[02:40:12] It's a daunting task to try to address 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 Dirac Square of the Einstein replacement.

[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.

[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.

[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.

[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 $$Y$$.

[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)xSU(2)xSU(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.

[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.

 

====== Thanks & Final Thoughts ======