Q1: As we know, in classical mechanics(CM), according to Noether's theorem, there is always one conserved quantity corresponding to one particular symmetry. Now consider a classical system in a n dimensional general coordinates space described by the Lagrangian L(q,˙q), where q=(q1,...,qn) and ˙q=(˙q1,...,˙qn) are general coordinates and general velocities, respectively. If the system has SO(n) spatial rotation-symmetry, e.g.∀A∈SO(n),L(AqT,A˙qT)=L(q,˙q), then we can get n(n−1)2(number of the generators of group SO(n)) conserved angular momentum.
My question is as follows, now note that our spatial dimension is n, when n=3→ number of angular momentum(n(n−1)2)=spatial dimension(n)=3, otherwise, number of angular momentum≠spatial dimension, so why spatial dimension 3 is so special? Is there any deep reason for number 3 or it's just an accidental event? Or even does this phenomena have something to do with the fact that we "live" in a 3D world?
Q2: In quantum mechanics(QM), a Hermitian operator J=(Jx,Jy,Jz) is called an angular momentum iff [Jx,Jy]=iJz,[Jy,Jz]=iJx,[Jz,Jx]=iJy.
And my question is as follows, in QM, we can have 1-component momentum operator ˆp , or 2-component momentum operator(^px,^py), and so on. But Why we have never encountered a angular momentum with only two components J=(Jx,Jy)? Can we define a 2-component angular momentum? Like in the CM case, again the number 3 is special in QM case, why?
Thanks in advance.
By the way: More questions concerning the definition of rotation groups for angular momentum can be found here , who are interested in may have a look, thanks.
This post imported from StackExchange Physics at 2014-03-09 08:46 (UCT), posted by SE-user K-boy