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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:
an Idea-section
and a section Prequantum circle (n+1)-bundle with the general abstract definition and the beginning of some examples
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.
so far the entry only discusses pre-quantization.
I am not discussing the entry. It is a question of the idea.
Oh, now I understand what you are asking.
The idea is this: where standard physics discusses $n$-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.
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 ?
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.
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 ?
So infinity symplectic groupoid gives infinite-dimensional QFT ?
A smooth $\infty$-groupoid equipped with a symplectic form of degree $n+2$ corresponds to an $(n+1)$-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”?
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:
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.
OK, I saved the above references into the new entry topological quantum computation for future use.
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