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Theory of Geometric Unity: Difference between revisions
→Problem Nr. 2: Spinors are sensitive to the metric
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How do we get the metric out from its responsibilities? It's been assigned far too many responsibilities. It is responsible for a volume form; for differential operators; it's responsible for measurement; it's responsible for being a dynamical field, part of the field content of the system." | How do we get the metric out from its responsibilities? It's been assigned far too many responsibilities. It is responsible for a volume form; for differential operators; it's responsible for measurement; it's responsible for being a dynamical field, part of the field content of the system." | ||
</blockquote> | </blockquote> | ||
'''Comments''' | |||
'''Mark-Moon:''' Can anyone explicate Eric's point about spinor fields depending (in a bad way) on the metric in conventional theories, in a way that is no longer the case in GU? I feel like this is the original idea in GU that I'm closest to being able to understand, but I don't think I quite get it yet. | |||
'''Chain:''' Yeah I was wondering this as well, as far as I was aware you just need a spin structure, which only depends on the topology and atlas on the manifold and not on the choice of metric [https://math.stackexchange.com/questions/2836814/dependence-of-spinor-bundle-on-choice-of-metric]. Perhaps the point is that although each choice of metric yields an isomorphic spin structure, perhaps there is not a canonical isomorphism in the same way as in GU where the bundle of metrics Y (U in the talk) is isomorphic to the Chimeric bundle C, but the choice of isomorphism is given by a choice of connection on Y. Although I don't know why the chimeric bundle would come with a canonical choice of spin structure either, which seems to be Eric's claim | |||
to define spinors you would need a clifford bundle and hence a choice of metric on the chimeric bundle | |||
=== Problem Nr. 3: The Higgs field introduces a lot of arbitrariness === | === Problem Nr. 3: The Higgs field introduces a lot of arbitrariness === | ||