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I am learning about the Wigner-Weyl transformations to move a $c$-number Lindblad operator $A(x,p)$ back into operator form. As far as I know, to move back and forth normally requires a four variable integral transform. However, a colleague has pointed out that there is a simpler method to achieve the transform from operator to c-number, the so-called Bopp operators:

$$\hat{x}=x+\frac{i\hbar}{2}\frac{\partial}{\partial p},\qquad \hat{p}=p-\frac{i\hbar}{2}\frac{\partial}{\partial x}.$$

Using this substitution is a quick method to the Weyl symbol (for example, with $\hat{x}\hat{p}$ straight substitution leads to the correct WW form).

I have two questions:

  • If I would like to go from phase-space to operator form, is it as simple as substituting $x=\hat{x}-\frac{i\hbar}{2}\frac{\partial}{\partial p}$?

  • The example given is written as $\hat{x}\hat{p}\hat{1}$, so the derivatives operate on the identity operator and give $0$. However, in the c-number function $A(x,p)$ there is no unity operator. When the derivative operates to the left, what does it operate on?

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Indeed, the Bopp shift is a clumsy Lagrange translation operator transcription of the celebrated * product, a 4-variable integral transform, cf. eqns (12-15) in Ref. 1.

There is an infinity of phase-space functions corresponding to differently ordered operators, as their ps and qs may be ordered in different ways with the intercalated * s enforcing noncommutativity; or, equivalently, their Bopp shifts acting in differently ordered sequences. They thus all coincide to vanishing order in $\hbar$, but differ in their hbar-dependence.

Identity operators in Hilbert space map to constants in phase space and vice versa.

The Wigner transform of the operator you wrote down, x̂ p̂ 1̂, is, hence, directly, xp+iħ/2. Note the p-derivative term is idle, so only the leading, conventional, term 1 survives as a contribution of the identity.

References:

  1. Thomas L. Curtright, David B. Fairlie, & Cosmas K. Zachos, A Concise Treatise on Quantum Mechanics in Phase Space, World Scientific, 2014. The PDF file is available here.
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