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Whoever performed that integration is using the following fact from Fourier analysis: The "delta function supported at the position 0" is the Fourier transform of the constant function 1. $\delta( …

In the first case, a gauge transformation changes S(A) by an integer. In the second case, by $2\pi$ times an integer. The second case is a useful normalization for physicists, who care about the beh …

The term "Yang-Mills theory" in the mass gap problem refers to a particular QFT. It is believed that this QFT (meaning its Hilbert space of states and its observable operators) should be defined in t …

This sounds to me like you're asking that your spacetime admit a family of Cauchy surfaces (modulo annoyances like having $f$ be closed and acausal). There's a theorem of Geroch which guarantees tha …

Take a look at Appendix A of the 2nd edition of Glimm & Jaffe's book. They give a rigorous construction of the measure you're after aka, the Ornstein-Uhlenbeck measure, which is the cylinder measure y …

Just to correct a small misconception: Physicists aren't claiming that an integral over the space of smooth maps from $WS$ to $X$ exists. The path integral measure for the bosonic string is defined …

I think a lot of the trouble people are having with this question comes from the phrase 'applications of quantum field theory'. Quantum field theory is a collection of ideas under active development …

I think the book you want is Frankel's The Geometry of Physics: An Introduction. It's accessible and well written, and largely aimed at the material you're talking about.

Let me add a few words of explanation to Aaron's comment. Perturbative string theory is (at least at the level of caricature) concerned with describing small corrections to classical gravitational ph …

What reasons are there for describing the harmonic oscillator as being so important in physics? The harmonic oscillator tends to show up when you're expanding a potential function around non-degenera …

That integral doesn't converge absolutely, so you'll have to choose some way of making sense of the sum. The one you've chosen above -- taking a limit of integrals on compact sets -- is not so easy t …

Sebastian points out correctly that $D$ and $d+d^{\dagger}$ are not the same objects. $D$ acts on spinors; $d + d^{\dagger}$ acts on differential forms. Physicists do sometimes make use of a trick …

What follows is a guess. I don't know enough about Symplectic Field Theory to be sure. In perturbative string theory, one describes physics in the target spacetime $X$ by summing over maps from Ri …

CQFT is very much still an open research subject. I don't think it is known what the best approach is. So all I can do is share my own opinion. (And a warning: I'm just an interested observer!) Fir …

Other people have said that the problem is that GR isn't renormalizable. I want to explain what that means in measure-theoretic terms. What I say won't be 100% rigorous, but it should get the general …