~~ Physicists ~~ Some people like to define the "Fourier transform" on Minkowski space as $\hat f(\xi) = \int e^{i \eta(x,\xi)} f(x) dx$, where $\eta(x,\xi)$ is the Minkowski form. I'm used to thinking of the Fourier transform as a canonical isomorphism $L^2(K) \to L^2(\hat K)$ where $K$ is a locally compact abelian group and $\hat K$ is its Pontryagin dual. But this "Minkowski-Fourier transform" doesn't seem to arise in this way.

**Questions:**

Is there an abstract framework in which to understand this "Minkowski-Fourier transform"? For instance, is there a general theory of "Fourier transforms" on spaces equipped with a nondegenerate symmetric bilinear form? Is there a relationship between this "Minkowski-Fourier transform" and the representations of a suitable "Heisenberg group"?

Which properties of the usual Fourier transform on Euclidean space are shared by the "Minkowski-Fourier transform"? For instance, what is a precise statement of the Fourier inversion formula in this context?

Is there a good reference for the mathematical properties of the "Minkowski-Fourier transform"?

Perhaps it's worth adding that physicists seem to be a bit blase about using this "Minkowski-Fourier transform", and treat it as though it were an ordinary Fourier transform.

haveseen the Fourier transform written out explicitly with the signs from the Minkowski form, but it was in a more math-flavored source, so it seems they were just (somewhat perversely!) working with the dual of momentum space! $\endgroup$ – Tim Campion Apr 13 '20 at 21:56