Taulalai

Sharan solving Moyal's equation in Wigner quasiprobability distribution

Well, Sharan does evolve any phase space observable, and, naturally, the Wigner function—recall the flipped signs, as the WF is in the Schroedinger rep! So, his evaluation of the *-exponential U of the hamiltonian, (1.2)—(1.3) through path integral trajectories, determines the WF evolution by the obvious P(x,p,t)=U^-1 * P(x,p,t=0) * U, as detailed in any self-respecting book on phase-space quantum mechanics. (The double *-product is only 4 integrals, since 4 out of the needed 8 in phase space collapse to δ -functions, added onto the infinite ones of the path integral.) The majority of papers taking that route since, like, e.g., J Samson (2000) JPhysA33, simply emulate and elaborate on Sharan. The strategy of Marinov is the same.

Motion is a canonical transformation.

Actually, Moyal's equation was solved, in principle, by Moyal himself, in his 1949 paper, his eqns (61.) → (7.4), through his phase-space propagator K ("temporal transformation function"), (9.1)→ (10.2) . He then proceeds to evaluate it for the oscillator in his 1949 Manchester paper with Bartlett, eqns (1.2)—(1.3), which yields the time-evolved WF, mutatis mutandis, both in the U and the equivalent K convolution language! This is the reason I was steering you into the (admittedly flawed and challenging) quantum characteristics article. In practice, if you follow actual results in practical applications, the majority of time-evolved WFs are obtained through more direct techniques like these, and it is misleading to let the novice walk away from the Wigner quasiprobability distribution article with the impression that he/she has to descend to the level of quantum characteristics if one wishes to monitor the evolution of a WF, or that any serious results were obtained that way. I have been looking for years for an instance where quantum characteristics could yield a nontrivial novel answer less easily accessible to standard techniques. I disapproved quantum characteristics taking up peremptory residence in the WQPDF article, but packing in more emphasis on them, not less, is unhelpful to the odd grad student who wishes to get an honest view of WFs. A large number of applied mathematicians are devoting themselves to boundary layer theory in perturbation theory solving the Moyal equation. What about them? etc... That's why I think a section by you on quantum charcteristics might get something going... Cuzkatzimhut (talk) 01:03, 30 September 2012 (UTC)

Sharan has a double copy of phase space and, as you have correctly pointed out, a 4-integral over the path. Marinov made two integrations and left with one copy of the phase space and a 2-integral. He got a path integral representation for Moyal’s “temporal transformation function”. I think the difference between Sharan and Marinov is substantial. It is so substantial that after Marinov it is no longer clear what is more fundamental, the path integral for the amplitude or the Wigner function.

Matrix quantum mechanics is not very powerful method also. Let's hope that there will be some applications in the future. Taulalai (talk) 15:12, 2 October 2012 (UTC)

Hi; I wonder if you had something useful to tell Trompedo in Talk:Wigner quasiprobability distribution#Recent additions in the Evolution equation for Wigner function section or, more appropriately, on his talk page, User talk:Trompedo. He's oddly championing Leaf's (library research) review as some sort of a grandfather of path integration in phase space. It is not clear to me he is bothering to read Moyal's paper and his phase space integrals there.... Chris Howard is proposing to consolidate evolution questions in a separate article——my take is to over-write/overhaul/rectify the hapless Method of quantum characteristics article, possibly leaving its original stuff in a late section thereof. Cuzkatzimhut (talk) 15:03, 16 November 2012 (UTC)