Can unquantized fields be considered smooth curved abstract manifolds?

[Spinnor]

Can unquantized fields be considered smooth curved abstract manifolds? Say free particle solutions of the Dirac equation or the Klein Gordon equation? Can quantized fields also be considered curved abstract manifolds?

Thanks for any help!

 

[fzero]

Classical fields can be thought of as sections of a fiber bundle. The bundle structure is that of a base identified with spacetime and fibers which take values in a vector space associated with the particular representation of the Lorentz and internal symmetry groups. This viewpoint is especially powerful when describing gauge theory in mathematical terms. A discussion, appropriate for physics grad students, can be found in Nakahara's Geometry, Topology and Physics. Another perspective, geared toward advanced math grad students, can be
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.

 

[Spinnor]

Can a vibrating string be though of as a curved manifold? If yes, why not more complicated fields.

Thank you Fzero for your help!

 

[fzero]

Can a vibrating string be though of as a curved manifold?

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

If yes, why not more complicated fields.

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.

 

[Spinnor]

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!