Talk:Method of quantum characteristics

Deleted reference
The following reference was deleted today: Apparently, to me, in excessive overprotective zeal. While I have not met the authors, I and several respected colleagues have internalized the rather natural lessons therein, and the diffidence ("must wait") in the comment box of said deletion is unwarranted, in my opinion. Nevertheless, here it is, for access by the motivated editor and, of course, posterity of good sense. Cuzkatzimhut (talk) 23:00, 15 November 2018 (UTC)

This interesting paper is about trajectories, which appear in the argument of the Wigner function. In Eq. (19) we see the dot-composition of two functions W(q(x,t)) in the sense of mathematical analysis. The wiki article discusses star-product trajectories that also appear in the argument of the Wigner function. The resultant object is, however, the star-composition of two functions W(*q(x,t)). Readers would be grateful if somebody wrote a section on why the dot- and star-product trajectories as well as the de Broglie Bohm trajectories are pairwise different. It is clear that objects related to the dot-composition of functions do not belong to the Moyal dynamics. I am not surprised therefore by the inconsistencies that arise when replacing W(*q(x,t)) -> W(q(x,t)). Mr Renovator (talk) 13:49, 20 November 2018 (UTC)

Average of operator
The average value of the operator Ĝ can be expressed in terms of the Wigner function in two ways. The first method uses the completeness property of the basis B(ξ) with respect to the density matrix ρ and the operator Ĝ:

\langle \hat{G} \rangle = \mathrm{Tr}[\hat{\rho}\hat{G}]=\int \frac{d^{2n}\xi }{(2\pi \hbar )^{n}}\mathrm{Tr}[\hat{B}(\xi )\hat{\rho}]\mathrm{Tr}[\hat{B}(\xi )\hat{G}]=\int \frac{d^{2n}\xi }{(2\pi \hbar )^{n}}W(\xi )g(\xi )\mathrm{,}$$ where
 * $$ W(\xi ) = \mathrm{Tr}[\hat{B}(\xi )\hat{\rho}],$$
 * $$ g(\xi ) = \mathrm{Tr}[\hat{B}(\xi )\hat{G}].$$

From other hand, the completeness property with respect to the product of the operators ρ and Ĝ means
 * $$\langle \hat{G} \rangle = \mathrm{Tr}[\hat{\rho}\hat{G}]=\int \frac{d^{2n}\xi }{(2\pi \hbar )^{n}}\mathrm{Tr}[\hat{B}(\xi )]\mathrm{Tr}[\hat{B}(\xi )\hat{\rho}\hat{G}]=\int \frac{d^{2n}\xi }{(2\pi \hbar )^{n}}W(\xi )\star g(\xi )\mathrm{,}$$

where conditions are taken into account
 * $$ \mathrm{Tr}[\hat{B}(\xi )] =1,$$
 * $$ \mathrm{Tr}[\hat{B}(\xi )\hat{\rho}\hat{G}] = W(\xi )\star g(\xi ).$$

We thus conclude
 * $$ \int \frac{d^{2n}\xi }{(2\pi \hbar )^{n}}W(\xi )g(\xi )=\int \frac{d^{2n}\xi }{(2\pi \hbar )^{n}}W(\xi )\star g(\xi ).$$

For a family of Wigner functions tending to zero at infinity together with all their derivatives, integration in parts proves this equality in an independent way (cf. Lemma 2 of Ref. [5] in the Wigner–Weyl transform), entering, oddly enough, into contradiction with the footnote 19 on page 71 of the textbook "Lectures on Quantum Mechanics for Mathematics Students" (Student Mathematical Library) L. D. Faddeev and O. A. Yakubovskii (American Mathematical Society, 2009). When calculating the average, star-product can also be omitted in the Glauber–Sudarshan P representation and the Husimi Q representation. -- Edehdu (talk) 06:19, 6 May 2022 (UTC)


 * The last sentence is flat wrong and misleading, as discussed in the main article, Talk:Wigner_quasiprobability_distribution. Cuzkatzimhut  The canonical demonstration is in Lee, Hai-Woong, "Theory and application of the quantum phase-space distribution functions." Physics Reports 259 (1995): 147-211.   Make sure you appreciate the essential configuration of optical phase space, and how star products are defined there: with extraordinary difficulty. (talk) 13:54, 21 May 2022 (UTC)


 * Many thanks for the reference. The density matrix is ordered reversely with regard to other operators in the Glauber-Sudarshan P and Husimi Q representations, allowing the average values (and only the average values) to be calculated in terms of the ordinary product in accordance with the so-called optical equivalence theorem. The same is indicated in Eq. (4.9) of H.-W. Lee's review paper. -- Edehdu (talk) 10:42, 24 November 2022 (UTC)