Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory object of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 31st 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexI badly need to polish the $n$Lab entries related to path integrals. Today a student asked me how the pull-push operation in [[string topology]] is a remnant of a quantum path integral. So a started writing now * [[path integral as a pull-push transformation]]. So far there is the description of the archetypical path integral for the quantum particle propagating on the line in terms of pull-tensor-push.

   I badly need to polish the $n$Lab entries related to path integrals. Today a student asked me how the pull-push operation in string topology is a remnant of a quantum path integral. So a started writing now

   • path integral as a pull-push transformation.

   So far there is the description of the archetypical path integral for the quantum particle propagating on the line in terms of pull-tensor-push.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeMay 31st 2011
   • PermaLink
   Author: zskoda
   Format: MarkdownItexThis is very interesting. The most rich geometric theory of quantization is so far developed for finite-dimensional mechanics. The central role are the cohomological classes related to Lagrangian geometry -- most notably the [[Maslov class]]. Cohomology is higher categorical subject, so can your approach predict the appearance of the Maslov class ? That would be so interesting.

   This is very interesting. The most rich geometric theory of quantization is so far developed for finite-dimensional mechanics. The central role are the cohomological classes related to Lagrangian geometry – most notably the Maslov class. Cohomology is higher categorical subject, so can your approach predict the appearance of the Maslov class ? That would be so interesting.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 31st 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexI haven't thought much about the Maslov index for a long while. Maybe I should. One question: in the entry you have a sentence > Lagrangean submanifold describes the phase of short-wave oscillations. I am not sure what this means.

   I haven’t thought much about the Maslov index for a long while. Maybe I should.

   One question: in the entry you have a sentence

   Lagrangean submanifold describes the phase of short-wave oscillations.

   I am not sure what this means.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeMay 31st 2011
   • (edited May 31st 2011)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI must have taken this phrase from somewhere. Well, it is roughly like the role of the real submanifold in the Fourier transform. My memory is that here one takes in the sense of the eikonal approximation, which is the splitting into short wave and long wave part and assigning the coordinates to each part. I should write some time an entry on eikonal. I think this is very important for us, as in the cases in which one has topological QFT, the quasiclassical approximations are exact (by localization), hence one is likely to see the connection best by looking at eikonal-like approaches.

   I must have taken this phrase from somewhere. Well, it is roughly like the role of the real submanifold in the Fourier transform. My memory is that here one takes in the sense of the eikonal approximation, which is the splitting into short wave and long wave part and assigning the coordinates to each part. I should write some time an entry on eikonal. I think this is very important for us, as in the cases in which one has topological QFT, the quasiclassical approximations are exact (by localization), hence one is likely to see the connection best by looking at eikonal-like approaches.