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As I'm sure you'll see from the many answers you'll get, there are lots of notions of "quantization". Here's another perspective. Recall the primary motivation of, say, algebraic geometry: a geometr …

I can't comment on the Lion hypotheses. I'm pretty sure the SHLT is nothing more than the fact that: A linear endormophism of a $k$-dimensional vector space factors through a $(k-1)$-dimensional …

Yes, of course, there is much research on mathematical rigor in quantum field theory. Of course, I don't know what "reasonable", "essentially different", and "realistic" mean to you, but I would say …

Let's take for granted the Gaussian integration formula, which holds for both bosonic and fermionic integrals, if they are properly interpreted: Theoreom (Gauss, Wick): Let $X$ be a vector space with …

It may take a bit of extraction, but positive answers to both of your questions follow from my results joint with Gaiotto in Condensations in higher categories (arXiv:1905.09566). In that paper we bui …

tl;dr: The reason for operator-valued distributions is because the physically meaningful "measurements" in QFT are things that preserve locality and that can be measured at any location. In quantum m …

Incidentally, I more or less disagree that symplectic geometry captures what I would consider "classical mechanics". The reason is that in all the examples that I think deserve to be called "classica …

To understand the possible spaces of boundary conditions for a TQFT, it is helpful to start in highest dimension. Suppose you have a $(d+1)$-dimensional nonanomalous TQFT $\mathcal Q$. (The anomalous …

My understanding is that Segal invented his formalism (which was then adapted by Atiyah) by thinking about the same thing Wightman was thinking about: formalising the theory of local operators. In hin …

As far as I can tell, a double group is a double cover of a group. Specifically, if $G \subset \operatorname{SO}(n)$ is a group acting by rotations of $n$-dimensional space, its double group is the li …

Some good references are the papers by Dan Freed and the book The geometry and physics of knots by Michael Atiyah. But by far the best answer to your question is in Witten's paper "Quantum field theo …

The question of what "natural transformation of QFTs" should be is a somewhat subtle one. The issue is most apparent if you work with TQFTs, but it doesn't completely go away if you work with dynamica …

Suppose you are given a super Hilbert space $\mathcal{H} = \mathcal{H}_0 \oplus \mathcal{H}_1$, with bosonic and fermionic subspaces $\mathcal{H}_0$ and $\mathcal{H}_1$ respectively. Define a new supe …

Quantization is not a functor.

As to the bulk of your question, which I take to be a reference request for mathematical accounts of quantum mechanics, I am partial to the book Quantum Mechanics for Mathematicians by L. Takhtajan. …