A Portal Special Presentation- Geometric Unity: A First Look: Difference between revisions
A Portal Special Presentation- Geometric Unity: A First Look (edit)
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<p>[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 $$\theta$$. | <p>[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 $$\theta$$. | ||
<p>[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 | <p>[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. | ||
<p>[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. | <p>[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. | ||