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)
Revision as of 15:34, 10 April 2020
, 15:34, 10 April 2020→Choosing All Metrics
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===== Choosing All Metrics ===== | ===== Choosing All Metrics ===== | ||
<p>[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. | <p>[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. | At a pointwise tensorial level. | ||
<p>[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. | <p>[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. | ||
<p>[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 $$G^{10}_{\mu\nu}$$. | <p>[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 $$G^{10}_{\mu\nu}$$. | ||
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 $$X$$. | ||