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April 20, 2007

Quantization and Cohomology (Week 21)

Posted by John Baez

This week in our course on Quantization and Cohomology we used Chen’s ‘smooth space’ technology to implement a new approach to Lagrangian mechanics, based on a smooth category equipped with an ‘action’ functor:

  • Week 21 (Apr. 17) - Any quotient of a smooth space becomes a smooth space. The category of smooth spaces has pushouts. The category of smooth spaces is cartesian closed. The path groupoid PXP X of a smooth space XX. The path groupoid is a smooth category. Smooth functors. Theorem: a smooth functor S:PXS: P X \to \mathbb{R} is the same as a 1-form on X.

    Supplementary reading:

    • John Baez and Urs Schreiber, Higher gauge theory II: 2-connections, draft version.
      Section 6.1: proof that for any Lie group GG, smooth functors S:PXGS: P X \to G are the same as Lie(G)Lie (G)-valued 1-forms on XX

Last week’s notes are here; next week’s notes are here.

In a bit more detail: we saw that any smooth space XX has a smooth groupoid of paths PXP X. The Lagrangian approach to classical mechanics involves a smooth groupoid CC where the objects are ‘configurations’ of our system and the morphisms are ‘processes’ or ‘paths’. The action should define a functor S:CS: C \to \mathbb{R}. So, it’s nice that in the special case when CC is a path groupoid, such functors turn out to be familiar entities! They’re just 1-forms on XX.

However, this isn’t quite general enough. What we really want is something that looks locally like a 1-form on XX, but not globally: a connection on a U(1)U(1) bundle over XX! This, after all, is what people use in geometric quantization — usually in the special case where XX is a symplectic manifold.

To get this answer, we’ll need to generalize from smooth functors to smooth anafunctors, as defined by Toby Bartels. A smooth anafunctor is something that’s locally isomorphic to a smooth functor!

Posted at April 20, 2007 7:39 PM UTC

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