 Physicists  Some people like to define the "Fourier transform" on [Minkowski space](https://en.wikipedia.org/wiki/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:**


1. 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"?
2. 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?
3. 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.



Fourier transform on Minkowski space
Well you seem to have worked it out but I wrote most of this before your comment happened: I claim there isn't any material difference between your "Minkowski space Fourier transform" and the usual Fourier transform on ${\mathbb R}^n$: in fact write $$ \hat f(\xi)\equiv \int e^{i\eta(x,\xi)} f(x) dx $$ for *any* non-degenerate bilinear form $\eta$. Then there exists another such form $\eta^{-1}$ so $\eta(x,\eta^{-1}\zeta)= \langle x,\zeta\rangle$ where $\zeta \in ({\mathbb R}^n)^\star$. Clearly $$ \hat f(\eta^{-1} \zeta)=({\mathcal F}f)(\zeta)\,,$$ where $\mathcal F$ is the usual --- ``Euclidean'' --- Fourier transform.


In physics texts this $\zeta$ variable is the down-index momentum ($k\_\mu$ in e.g. Peskin and Schroeder) while $\xi$ is the up-index momentum ($k^\mu$ in e.g. Peskin and Schroeder). Derivatives play perfectly nice with up/down index notation, which allows one to be blase about whether the Fourier transform involves the up- or down-index $k$.


Mathematically speaking you're taking the Pontryagin dual of Minkowski space seen as a group of translations, which is exactly the same as that of the corresponding Euclidean space. More abstractly the conserved charge in the sense of Noether's theorem corresponding to a translation is the down-index momentum rather than the up-index one.


