Given a nice function $f\colon\mathbb{R}\to\mathbb{S}^1\subset\mathbb{C}$ and an unbounded self-adjoint operator $P$ acting on the Hilbert space $L^2(\mathbb{R})$, one can define a unitary operator $f(P)$ via continuous functional calculus. Moreover, we require that $f$ is represented by a Fourier transformation $$f(x)=\int_{-\infty}^{+\infty}\psi(y)e^{-2\pi ixy}dy,$$ does it make sense to have $$f(P)=\int_{-\infty}^{+\infty}\psi(y)e^{-2\pi iyP}dy\;?$$ Note that here $e^{-2\pi iyP}$ is also a well-defined unitary operator.
I came up with this question when reading a paper https://link.springer.com/article/10.1007/s002200100412 by Faddeev-Kashaev-Volkov. There they use such arguments to show the pentagon relation of quantum dilogarithm (in the appendix).