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Spinnor

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Thanks for any help!

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- #1

Spinnor

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Thanks for any help!

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fzero

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found in Dan Freed's lectures on topological QFT.

Quantization, from a QFT perspective, typically adds structure on top of the classical geometry. I don't have any simple examples to try to illustrate with. A separate are a of study is geometricquantization, which is a mathematical formalism used to describe a quantization procedure starting with a classical geometry.

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Spinnor

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Thank you Fzero for your help!

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fzero

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A string can be modeled by a manifold structure as a curve in whatever space the string lives in.Can a vibrating string be though of as a curved manifold?

I'm not quite sure what fields you're referring to here. As I said, in general a field theory has a geometric structure associated with it that involves some fiber bundle. The fibers of the bundle are determined from the properties of the fields. Scalar fields lead to vector bundles, fermions lead to spin bundles, gauge fields lead to principal G-bundles, etc. The total space of a fiber bundle is often itself a manifold.If yes, why not more complicated fields.

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Spinnor

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A string can be modeled by a manifold structure as a curve in whatever space the string lives in.

I'm not quite sure what fields you're referring to here. As I said, in general a field theory has a geometric structure associated with it that involves some fiber bundle. The fibers of the bundle are determined from the properties of the fields. Scalar fields lead to vector bundles, fermions lead to spin bundles, gauge fields lead to principal G-bundles, etc. The total space of a fiber bundle is often itself a manifold.

Take the 2D spacetime Klein–Gordon equation, the base space is 2D spacetime and the fiber is the complex plane?

Then a plane wave solution looks like a wave on an infinite string (a string that satisfyies E^2 = P^2 + m^2) and a vibrating string can be thought of as a curved manifold?

Thank you for your help!

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