Not signed in (Sign In)

Not signed in

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

  • Sign in using OpenID

Site Tag Cloud

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 deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite 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 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 sheaves 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

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 14th 2011

    at geometric quantization of symplectic infinity-groupoids (which currently still redirects to symplectic infinity-groupoid) I am beginning to add some genuine substance. So far:

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeSep 14th 2011

    I am puzzled. The quantization goes from functions on symplectic manifold, Poisson manifold, symplectic stack, higher stack to the (noncommuting) operators of Hilbert space. If one were literally categorifying one would get operators on higher Hilbert space. But physics stays in the usual Hilbert space.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 14th 2011

    so far the entry only discusses pre-quantization.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeSep 14th 2011

    I am not discussing the entry. It is a question of the idea.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeSep 14th 2011
    • (edited Sep 14th 2011)

    Oh, now I understand what you are asking.

    The idea is this: where standard physics discusses nn-dimensional QFT in terms of ordinary Hilbert spaces (if it does) this is because it is not considering extended QFT. Because in codimension 1 we will always see a “1-space” of states. The higher categorical state spaces are assigned in higher codimension.

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeSep 14th 2011

    Say you consider the finite-dimensional higher symplectic groupoid/stack where finiteness is about the stack modification of finite-dimensional symplectic manifold. So it is a modification of finite-degree of freedom quantum mechanics. Are you implying that I should not consider the quantization of such examples without going to quantum field theory in higher dimension though the system is finite degrees of freedom, morally ? in the classical framework one gets to QFT only once one goes to infinite degrees of freedom. But going to infinite-dimensional manifold (say of paths, field configurations or so on) is not related to passing from a differentiable manifold to differentiable stack, is it ?

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 14th 2011

    Yes, this is not about “QFT as QM with infinitely many degrees of freedom” but about “QFT as QM on higher dimensional parameter spaces”.

    For instance Dijkgraaf-Witten theory is a 3-dimensional QFT with finitely many degrees of freedom. Its extended quantization is supposed to be a 3d-extended TQFT that assigns a 3-vector space to the point, a 2-vector space to the circle (some modular category) and vector spaces to surfaces.

    • CommentRowNumber8.
    • CommentAuthorzskoda
    • CommentTimeSep 14th 2011

    So infinity symplectic groupoid gives infinite-dimensional QFT ? If the higher stack is representable, the dimension of TQFT falls down ? And the universal TQFT for quantum computation fits into finite degree of freedom application ?

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeSep 14th 2011
    • (edited Sep 14th 2011)

    So infinity symplectic groupoid gives infinite-dimensional QFT ?

    A smooth \infty-groupoid equipped with a symplectic form of degree n+2n+2 corresponds to an (n+1)(n+1)-dimensional QFT.

    If the higher stack is representable, the dimension of TQFT falls down ?

    I am not sure what you have in mind here.

    And the universal TQFT for quantum computation fits into finite degree of freedom application ?

    What is the “universal TQFT for quantum computation”?

    • CommentRowNumber10.
    • CommentAuthorzskoda
    • CommentTimeSep 14th 2011
    • (edited Sep 14th 2011)

    Oh, I may be completely wrong.

    I thought that the CFT corresponding to the topological modular functors (where they look for a universal model for quantum computation) are some topological CFTs; here they also mention Chern-Simons theory in this context. Maybe you can understand better what is behind it:

    • Michael H. Freedman, Alexei Kitaev, Michael J. Larsen, Zhenghan Wang. Topological quantum computation, Bull. Amer. Math. Soc. 40 (2003), 31-38, web, pdf
    • M. Freedman, M. Larsen, and Z. Wang, A modular functor which is universal for quantum computation, Comm. Math. Phys. 227 (2002), no. 3, 605-622, pdf
    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeSep 14th 2011

    I have not been following the technical details of TQFTs as tools in quantum computation. That would take me a while to get up to speed there.

    • CommentRowNumber12.
    • CommentAuthorzskoda
    • CommentTimeSep 14th 2011

    OK, I saved the above references into the new entry topological quantum computation for future use.