Talk:Moyal bracket

cosine bracket

The cosine bracket was actually intriduced by Groenewold, ref 3, in his 1946 thesis paper, eqn (4.28). So was the sine bracket, by the way, but historical misuse, starting from the UK, where Moyal's work was more accessible then, is by now entrenched. Cuzkatzimhut (talk) 15:33, 1 January 2013 (UTC)

Glad to see your edits (I was guessing you would pick up on mine).

Indeed, in this book (Cosmas Zachos, David Fairlie und Thomas Curtwright: Quantum Mechanics in Phase Space: An Overview with Selected Papers (World Scientific Series in 20th Century Physics), World Scientific, April 2006) there is Groenewold's paper, and the cosine bracket in introduced in eq. (4.28). Furthermore from that book: "G. Baker (1958)^Bak58 (Baker's thesis paper) [...] appreciates the significance of the anticommutator of the ★-product as well, thus shifting the emphasis to the ★-product itself, over and above its commutator." (In Curtright, T.; Fairlie, D.; Zachos, C. (1998). "Features of time-independent Wigner functions". Physical Review D. 58 (2). arXiv:hep-th/9711183. Bibcode:1998PhRvD..58b5002C. doi:10.1103/PhysRevD.58.025002., arXiv:hep-th/9711183v3 only Baker is cited, not Groenewold.)

Also, do you have any reference on who actually first pointed out that the cosine bracket leads to the HJE in the classical limit? It could be useful information for the article.

Cheers --Chris Howard (talk) 18:21, 2 January 2013 (UTC)

Thanks, pleased by your concordance. Baker did focus quite adroitly on the cosine bracket, e.g. in his all-important eqn (3), which he uses to beautiful effect throughout his paper−−and he is aware of Groenewold, of course. The citation to Baker for the cosine bracket may be residual usage at that time, since the bulk of the mechanics of the * product often used the Bayen et al papers as the point of reference, and these authors barely acknowledged the monumental work of Groenewold. I suspect the naming Moyal brackets is irreversible by now, but Baker's brackets might be qualified... but who am I to judge? As for the Hamilton-Jacobi equation, the classics I know are the ultra-rich Voros papers: A Voros, Ann Inst H Poincare ́ 24 (1976) 31-90; ibid 26 (1977) 343-403; but they are hard work to fully parse out, and I'd be most reluctant to do the history research to claim absolute priority for them and establish who did what first. Apologies for my sloth.... Cuzkatzimhut (talk) 20:06, 2 January 2013 (UTC)

Thanks too for your comments. For easier reference I list the direct links for the references you cited (André Voros: Semi-classical approximations, Annales de l'institut Henri Poincaré, Reihe A, Physique théorique, Band 24, 1976, S. 31-90, numdam; André Voros: Asymptotic $\hbar$-expansions of stationary quantum states, Annales de l'institut Henri Poincaré, A, Band 26, 1977, S. 343-403, numdam). Wouldn't be up to me either at this point to sort out all details of who did what first, nor to assess what's in the publications of Voros or not (yet what is striking at first glance it that Voros speaks of a semiclassical approximation whereas the phase space approach turns out to be equivalent to the standard formalism of quantum mechanics...).

I had come across the cosine bracket from the work of Hiley, who gives the cosine bracket an interesting generalization. In "Phase space descriptions of quantum phenomena" [1] Hiley goes further than just stating that the cosine bracket gives rise to an equation reduces to the HJE in the classical limit. He shows that the sine and cosine brackets have their analogue in a more general, purely algebraic description of quantum mechanics (Bohm cites G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley-Interscience, 1972, as well as O. Bratteli, DW Robertson, Operator Algebras and Quantum Statistical Mechanics. I, Springer, New York, Heidelberg, Berlin, 1979). In particular, Hiley shows that the equation that involves the cosine bracket generalizes to an algebraic "phase equation" (eq. (22) of Brown & Hiley arXiv:quant-ph/0005026, cited as eq. (32) in "Phase space descriptions of quantum phenomena" - according to Brown & Hiley was not mentioned earlier except that "something similar has been used" by Cl. George, F. Henin, F. Mayne and Ilya Prigogine in 1978). This algebraic description in turn gives rise to multiple descriptions, including the phase space description and the Bohm trajectory description. As proposal, I now add a brief indication of all this into the current article:

The sine and cosine bracket also stand in relation to equations of a purely algebraic description of quantum mechanics.

(with a link into the relevant section of the bio article on Basil Hiley, and two references). --Chris Howard (talk) 12:17, 6 January 2013 (UTC)

Epstein, S. (1953). "The Causal Interpretation of Quantum Mechanics". Physical Review. 91 (4): 985. Bibcode:1953PhRv...91..985E. doi:10.1103/PhysRev.91.985. Cuzkatzimhut (talk) 16:14, 28 February 2013 (UTC)