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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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

The stationary phase approximation is strictly easier for fermionic manifolds than for bosonic ones. Indeed, suppose $M = \mathbb{R}^{0|n}$ is a purely odd supermanifold, with (odd) coordinates $x_1,\ …

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 …

In addition to DBM's (totally correct) answer above, I realized that there's probably a much simpler answer. If I'm wrong, hopefully someone will set me right. Let $\mathcal O$ be an open neighborho …

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 …

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. …

Quantization is not a functor.

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 …