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Shortly after posting my question, I realized that the answer to the original question is that the Hessian is not necessarily positive-definite. In particular, consider picking a different antideriva …

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 …

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 …

Note that the word "conformal field theory" (indeed, "field theory" in general) has many meanings, depending on the area of mathematics of the user, and in general the relationships between the differ …

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,\ …

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 …

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 …

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 …

Witten, I think, deserves much of the credit for getting mathematicians interested in the path integral, with his paper Quantum field theory and the Jones polynomial. In particular, path integrals ar …

Charles already posted Takhtajan's book, which is my first choice --- it's geared towards early graduate students. A more elementary book, geared towards math undergrads, is by Fadeev (who is, incide …

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 …

Many definitions I've seen of "star quantization of a Poisson manifold" include the request that each coefficient in $\hbar$ of $a\star b$ be a differential operator in $a,b$. Such a star quantizatio …

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 …

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 …

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 …