Recently during my studies, I came across an alternative construction of the Wigner function. This construction starts from the notion of the Stratonovich-Weyl operator kernel. I saw this construction in a paper called 'Quantum phase-space representation for curved configuration spaces'. And the development is as follows:
The author defines the Stratonovich-Weyl operator kernel as:
$$\hat{\Delta}(x,p) = \hat{D}(x,p)\hat{\Delta}(0,0)\hat{D}^{\dagger}(x,p)$$
where $\hat{D}$ is the already known displacement operator, given by:
$$\hat{D}(x,p) = e^{\frac{-ix\hat{p}}{\hslash}}e^{\frac{ip\hat{x}}{\hslash}}$$
with the undisplaced kernel:
$$\hat{\Delta}(0,0) =\frac{1}{2\pi\hslash}\int dp' \int dx'\hat{D}(x',p')e^{\frac{ix'p'}{2\hslash}} = \int dx' |x'/2\rangle\langle -x'/2|$$
I couldn't understand how he got to the last equality. I think he inserted the completeness relation to get the last equality. However, I performed the calculations but could not get the correct result. What does it mean to take the Fourier transform of a displacement operator? Is there a standard way to manipulate displacement operators, such as to express them in bra-ket notation?
Answering these questions will hopefully help me understand how to perform other calculations that I cannot currently do involving the Stratonovich-Weyl operator kernel, such as proving that $$\hat{\Delta}(x,p) = \int dx'e^{\frac{px'}{\hslash}}|x+x'/2\rangle\langle x-x'/2|.$$
I tried to get to that relationship, but I couldn't either.