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We're doing okay.
We're doing okay.


==== GU III: Physics ====
==== Swervature and Swerve with Shiab Operators ====


[01:43:06] Okay. We're not doing physics yet. We're just building tools. We've built ourselves a little bit of freedom. We have some reprieves. We've still got some very big debts to pay back for this magic beans trade. We're in the wrong dimension. We don't have good field content. We're stuck on this one spinor. We've built ourselves an projection operators. We've picked up some symmetric, nonlinear $$\sigma$$ field.
''[https://youtu.be/Z7rd04KzLcg?t=6186 01:43:06]''<br>
Okay. We're not doing physics yet. We're just building tools. We've built ourselves a little bit of freedom, we have some reprieves, we've still got some very big debts to pay back for this magic beans trade. We're in the wrong dimension. We don't have good field content. We're stuck on this one spinor. We've built ourselves some projection operators. We've picked up some symmetric, nonlinear \(\sigma\) field.


[01:43:33] What can we write down in terms of equations of motion. Let's start with Einstein's concept. If we do Shiab of the curvature tensor of the gauge potential hit with an operator defined by the $$\epsilon$$-sigma field plus the star operator acting on the augmented torsion of the pair. This contains all of the information when $$\pi$$ is zero in Einstein's tensor.
''[https://youtu.be/Z7rd04KzLcg?t=6212 01:43:32]''<br>
What can we write down in terms of equations of motion? Let's start with Einstein's concept. If we do shiab of the curvature tensor of a gauge potential, hit with an operator defined by the epsilon-sigma field, plus the star operator acting on the augmented torsion of the pair, this contains all of the information, when \(\pi\) is zero, in Einstein's tensor.


[01:44:16] In other words, there is a spinorial analog of the Weyl curvature, a spinorial analog of the traceless Ricci curvature, and a spinorial analog of the scalar curvature. This operator should shear off the analog of the Weyl curvature -- just the way the Einstein's projection shears off the Weyl curvature when you're looking at the tangent bundle. And this term, which is now gauge invariant may be considered as containing a piece that looks like $$\Lambda$$ times $$G_{\mu \nu}$$ or a cosmological constant.


[01:44:51] And this piece here can be made to contain a piece that looks like Einstein's tensor. And so this looks very much like the vacuum field equations. Okay, well we have to add in something else. I'll be a little bit vague cause I'm still giving myself some freedom as we write this up.
<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \bigcirc\kern-0.940em\circ\kern-0.58em·\text{}\kern0.3em_\epsilon F_\pi +[\bigcirc\kern-0.940em\circ\kern-0.58em·\text{}\kern0.3em_\epsilon T_\omega, T_{\epsilon, \pi}] + *T_{\epsilon, \pi} = 0 $$</div>


[01:45:25] But we are going to define whatever tensor we need. This is gauge invariant. This is gauge invariant. And this is gauge invariant with respect to the tilted gauge group. These two tensors together should be exact. And this tensor on its own should be exact.


[01:45:53] We're going to call the exact tensor, the **swervature**.
''[https://youtu.be/Z7rd04KzLcg?t=6256 01:44:16]''<br>
In other words, there is a spinorial analog of the Weyl curvature, a spinorial analog of the traceless Ricci curvature, and a spinorial analog of the scalar curvature. This operator should shear off the analog of the Weyl curvature just the way the Einstein's projection shears off the Weyl curvature when you're looking at the tangent bundle. And this term, which is now gauge invariant, may be considered as containing a piece that looks like \(\Lambda g_{\mu \nu}\), or a cosmological constant, and this piece here can be made to contain a piece that looks like Einstein's tensor, and so this looks very much like the vacuum field equations. But we have to add in something else. I'll be a little bit vague 'cause I'm still giving myself some freedom as we write this up.


[01:45:59] The particular Shiab operator we call the **swerve**. So that's 'swerve-curvature' plus the adjustment needed for exactness and another gauge invariant term which is not usually gauge invariant.
''[https://youtu.be/Z7rd04KzLcg?t=6325 01:45:25]''<br>
But we are going to define whatever tensor we need, for this term, for these terms here, this is gauge invariant, this is gauge invariant, and this is gauge invariant, with respect to the tilted gauge group. These two tensors together should be exact, and this tensor on its own should be exact.


[01:46:17] That is pretty cool. If that works, we've now taken the Einstein equation and we've put it not on the space of metrics, but we've put a generalization and an analog on the space of gauge potentials, much more amenable to quantization with much more algebraic structure and symmetry in the form of the inhomogeneous gauge group and its homogeneous vector bundle, some of which may be supersymmetric.
''[https://youtu.be/Z7rd04KzLcg?t=6353 01:45:53]''<br>
We're going to call the exact tensor the '''swervature'''. So the particular shiab operator we call the '''swerve''', so that it's ''swerve-curvature'' plus the adjustment needed for exactness, and another gauge invariant term which is not usually gauge invariant.
 
''[https://youtu.be/Z7rd04KzLcg?t=6377 01:46:17]''<br>
So that's pretty cool. If that works, we've now taken the Einstein equation and we've put it not on the space of metrics, but we've put a generalization and an analog on the space of gauge potentials, much more amenable to quantization, with much more algebraic structure and symmetry in the form of the inhomogeneous gauge group and its homogeneous vector bundle, some of which may be supersymmetric.
 
==== Somatic Complex and Two More Operators ====


[01:46:44] Now, the question is: "We've integrated so tightly with the matter field" -- we have to ask ourselves the question -- "can we see unification here?"
[01:46:44] Now, the question is: "We've integrated so tightly with the matter field" -- we have to ask ourselves the question -- "can we see unification here?"
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