In finite quantum many-body systems, such as nuclei, metalic clusters and dilute atomic gases in a trap, an approximation of rigid-rotor is useful. As far as rotation is concerned, anuglar momentum is the most important and natural choice for physical quantities in descriptions of these systems. Then, we have to think about the coordinates-space transformation between the Laboratory frame and the Intrinsic frames. (Let us denote Lab-frame and Intr-frame, hereafter.)
The transformation is an ordinary unitary (orthogonal) transformation, so that simply we tend to believe that physics does not change before and after the transformation (in particular, from a quantum mechanical viewpoint). However, Klein noticed more than 50 years ago that the commutatation relation of angular momentum operators are different in the Lab- and Intr-frames. This means that the transformation is not a canonical transformation.
Usually, the commutation relation is given as
[J_m, J_n] = i J_l \eps_{mnl}.
This is the case for the commutation rule in the Lab-frame. However, in the intrinsic frame, the commutation relation is given as
[J_m,J_n] = -i J_l \eps_{mnl}.
Namely, the sign is opposite in the right-hand side.
There are several proofs for the commutators, and they are roughly divided into two groups. One of them is to use differential operator representations, and the commutators are directly calculated. The other type is to use properties of tensors, in particular, with respect to the transformation in the Lab- or Intr-frame. Ring and Schuck, Eisenberg and Gleiner, and Iwanami's nuclear physics textbook employ the former approach, while Bohr and Mottelson the latter (although the detailed proofs are not written in the textbook of BM's....). My former supervisor in Tokyo said to me that the simplest proof he has ever seen is the one by Landau. It was true.... Only one math equation and three lines to explain briefly what has to be done in the proof. As usual, the equation can be derived "easily", according to Landau. I have to confess, at first, I could not understand why this equation can be derived "easily"... But, after having read Eisenberg and Gleiner, I realised that Landau's approach is exactly the same as the Germans', although they spent nearly three pages to demonstrate that "simple" equation.
Anyway, intuitively, this sign change in the intrinsic frame is very difficult to grasp. I heard many good and old professors who are not familar with nuclear structure physics but general nuclear physics say " I don't believe it" or "How come?" I myself tried once to prove this when I was in my PhD course in Tokyo, but I gave up proving the equation because I felt "the proof could eat up my precious time for numerical calculations and time to read recent publications in PRL and PRC in preparation for writing a paper....."
Last week, suddenly, I decided to come back to the old (personally) unsolved problem. The first several days, my calculations just repeated creations of the comments and complaints told by the old professors, that is, "How come?" Next several days, I noticed my physical interpretations about the transformation to the intrinsic frame was wrong and needed to be modified, but the calculations just ended up in a disastrous manner, simply gave nothing meaningful.... Then, finally, a last couple of days, I found my treatment of a rank-\lambda tensor was wrong, particularly, about the Wigner's function... If D(\Omega) is chosen, this is not a tensor at all. But the complex conjugate of D is a tensor!!! In addition, the direction of the transformation is reversed when we change an operator appraoch to a matrix-representation approach. Perphaps, this is the key point why the sign has to be reverted in the intrinsic frame for the angular momentum commutation relation.
Anyhow, it was very painful to complete the proof. I think Klein is great because he noticed the problem and solved correclty. I know this is because he did not have any prejuice, which is often very difficult not to do so. (Oscar Klein is the physicsts who derived the Klein-Gordon equation, Klein-Nishina formula and Kaluza-Klein theory!) Thinking from the first principle is sometimes very important, especially when you want to go beyond the currently established things and concepts. In this sense, I want to believe that my attempt this week is not simple a waste of time...
