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.

  • $\begingroup$ To the close voters: there seems to be $1+\epsilon$ questions combined here, but they both stem from trying to understand what the displacement operator does. As for the homework style, it is tough because the underlying physics concept is the computation itself. I will try an edit $\endgroup$ Aug 15 at 15:39

1 Answer 1


The non-standard definition of displacement operators got me confused initially, but that just changes the phase factors.

All you need to do is rewrite an $x$-displacement as $$\exp(-i x\hat{p}/\hbar)=\int dx^\prime |x^\prime+x\rangle\langle x^\prime|$$ and verify that it acts appropriately on all position eigenstates. Then we can rewrite the displacement as $$\hat{D}(x,p)=\int dx^\prime |x^\prime+x\rangle\langle x^\prime| e^{i p\hat{x}/\hbar}=\int dx^\prime |x^\prime+x\rangle\langle x^\prime| e^{i px^\prime/\hbar}.$$ For the undisplaced kernel, we observe that the integral over $p^\prime$ now gives us a Dirac delta function, which simplifies everything as necessary: \begin{aligned} \frac{1}{2\pi\hbar}\int dx^\prime dp^\prime D(x^\prime ,p^\prime)e^{i p^\prime x^\prime/2\hbar}&=\frac{1}{2\pi\hbar}\int dx^\prime \int dp^\prime \int dx^{\prime\prime} |x^{\prime\prime}+x^\prime\rangle\langle x^{\prime\prime}| e^{i p^\prime x^{\prime\prime}/\hbar}e^{i p^\prime x^\prime/2\hbar}\\ &=\frac{1}{\hbar}\int dx^\prime \int dx^{\prime\prime} |x^{\prime\prime}+x^\prime\rangle\langle x^{\prime\prime}| \delta\left(\frac{x^{\prime\prime}}{\hbar}+\frac{x^{\prime}}{2\hbar}\right)\\ &=\int dx^\prime \int dx^{\prime\prime} |x^{\prime\prime}+x^\prime\rangle\langle x^{\prime\prime}| \delta\left(x^{\prime\prime}+\frac{x^{\prime}}{2}\right)\\ &= \int dx^\prime |-\frac{x^{\prime}}{2}+x^\prime\rangle\langle -\frac{x^{\prime}}{2}| . \end{aligned}

As for the second question, you should now be able to do this. You have the operator in the position basis, so you know how $\exp(i p\hat{x}/\hbar)$ acts in its eigenbasis and you know that $\exp(-i x\hat{p}/\hbar)$ shifts the eigenstates.

  • $\begingroup$ thank you, it helped a lot!!!! $\endgroup$ Aug 16 at 14:32
  • $\begingroup$ @WagnerCoelho you're very welcome $\endgroup$ Aug 16 at 14:38
  • $\begingroup$ Hi again, do you have any suggestions for bibliography that I can learn, from basics to advanced, about the stratonovich-weyl operator? would help a lot, I'm currently developing my master's thesis on the Wigner function (: $\endgroup$ Aug 29 at 1:11
  • $\begingroup$ @WagnerCoelho for x and p you just need Weyl - you can read our friend Cosmas Zachos's book hep.anl.gov/czachos/a.pdf for example. Could try reading Stratonovich's paper, or this recent thesis I just stumbled upon by googling proquest.com/docview/2088448582 $\endgroup$ Aug 29 at 1:54

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.