What is the Weyl transform of narrow Gaussians and/or the Dirac delta?

Question

Consider the family of Gaussians in $q$, $p$ with decreasing widths $σ$ $$Φ_σ(q,p) = \frac{2}{π σ^2} e^{-\frac{2}{σ^2}(q^2+p^2)}$$ or in complex plane coordinates $$\tilde Φ_σ(α) = \frac{1}{π σ^2} e^{-\frac{|α|^2}{σ^2} }.$$ While the corresponding characteristic function $$\chi_σ(β) = e^{-σ^2 β^2}$$ is well-defined for each $σ \in \mathbb{R}_ +$, there seems to be a lower bound $σ^2 \ge 1/2$ for the Weyl transform to produce an operator, namely the thermal state $ρ_\text{th}$ with $\bar n = σ^2 - 1/2$, the value of $σ^2 = 1/2$ corresponding to the pure state $|0⟩⟨0|$.

Is the Weyl transform of Gaussians narrower than the Wigner function of vacuum ill-defined, or is it just some operator which happens not to satisfy some of the conditions on density matrices? If this does not work, what exactly breaks? If it does, certainly a limit $σ → 0_+$ is possible in some respect; in such case, what would the Weyl transform of $Φ_0(q,p) = δ(q)δ(p)$ be?

Surely I could simply write $$A_σ = \frac1π \int_{\mathbb{C}} χ_σ(β) D(-β) \mathrm{d}^2β,$$ which in the latter case would give $$A_σ = \frac1π \int_{\mathbb{C}} D(β) \mathrm{d}^2β,$$ but I can't seem to identify what the action of such operator would excplicitly be, or even if the integral does not diverge.

Answer 1

The short answer is that not all Gaussians are Wigner functions, and you should not think of them as such. But you may certainly Weyl-transform them!

I'm staying out of optical phase space, to which you may translate, and nondimensionalizing $\hbar=1$ for simplicity. It is then straightforward to see, in plain QM, what operator your normalized phase space Gaussian transforms to, $$ {1\over 2\pi} \int dx dp dy ~~\bigl \vert ~x+\frac{ y}{2} \left \rangle e^{iyp}~ Φ_σ (x,p) ~\right \langle x- \frac{ y}{2}~ \bigr |~~~\tag{142}. $$

Taking the narrow/vanishing $\sigma\to 0$ limit, the integrals are even easier to do: they collapse to $$ {1\over \pi} \int\!\! dx~\vert \!-\!x\rangle \langle x| ~~~ \tag{151}, $$ the celebrated parity operator !

This, of course, cannot be a bona-fide density matrix summarizing a normalizable wave function. A fundamental requirement for a ($\hbar=1$) phase-space function to be representing (the density matrix of) a physical wave function as a Wigner function is $$ |Φ_σ|\leq {1\over \pi}. $$

• There are no spiky Wigner functions.