# Angular Momentum Addition in Phase Space QM

In my very limited understanding of geometric quantization, we quantize spin by choosing as our phase space $S^2$ with a suitably normalized area form as the symplectic form. Depending on the normalization we get out a finite-dimensional Hilbert space corresponding to some spin $j$ particle.

My question is if/how angular momentum addition may be understood in this picture. In the Hilbert-space picture, we can write things like $1/2\otimes 1/2=0\oplus 1$; i.e., we may decompose tensor products of reps of $SU(2)$ into irreps. If we naively try translating such an equation into phase space, I assume tensor products of Hilbert spaces correspond to Cartesian products of phase spaces, but I have no idea how direct sums would be interpreted (disjoint unions?). Does thinking carefully about quantizing products of spheres give a nice geometric interpretation of angular momentum addition in phase space?

Plain vanilla phase-space QM in a 2d phase space replicates the standard ladder algebraic features of quantized angular momentum in Hilbert space, with very parallel algebraic tricks, cf Exercise 0.8 of p.33 of that book. A long moment's thought should convince you then that Hilbert space addition of angular momentum maps isomorphically to a phase space equivalent, on tensor-product phase spaces, one-to-one corresponding to the Hilbert spaces. The Hilbert space comultiplication representing Kronecker multiplication of SU(2), $$\Delta(J_a)\equiv J_a \otimes 1\!\!\!1 + 1\!\!\!1 \otimes J_a$$ will satisfy the same Lie Algebra as the left-and-right $J_a$s, and the Casimir $\Delta(J_a)\Delta(J_a)$ will be reducible to a direct sum of the Casimirs of the Clebsched reduced reps, so, then the respective eigenvalues times the respective identities in the block subspaces of each irrep.
• Thank you so much for your answer! If I understand correctly, you are saying that on a 6-d phase space (I wasn't quite sure on how to do it with just one position coordinate) with coordinates $(\vec{x},\vec{p})$, if we define $\vec{L}=\vec{x}\times\vec{p}$, the $L_i$'s will commute under the moyal bracket identically to the way the corresponding operators do in QM, so on the product phase space with coordinates $(\vec{x}_1,\vec{p}_1,\vec{x}_2,\vec{p}_2)$ the usual highest-weight decomposition procedure works the exact same way. (1/3) – commutatertot Apr 9 '16 at 22:17
• I was hoping there could be more geometrical picture. For instance, the adjoint rep for spin-1/2 is $0\oplus 1$, so my understanding is we may visualize hermitian operators over the spin-1/2 hilbert space (including density matrices) as functions over $S^2$, to first order in spherical harmonics, and endowed with the "fuzzy sphere" noncommutative product. Now $1/2\otimes 1/2=0\oplus 1$, and the adjoint rep of $0\oplus 1$ breaks up as a direct sum of operators from 0 to 0, from 0 to 1, from 1 to 0, and from 1 to 1. (2/3) – commutatertot Apr 9 '16 at 22:17
• I wanted a way to visualize the "fuzzy functions" on $S^2\times S^2$ breaking up as a direct sum in a similar way, hopefully with a nice picture that would work for any addition-of-angular-momentum situation, but that may not be doable. In general, I was curious how phase spaces like $S^2\times S^2$ would be quantized. I apologize for confusing the terms "phase space QM" and "geometric QM," which are probably both different than what this is. Also, if I should edit my question with these details (I can also try to write more clearly and give better examples), let me know. (3/3) – commutatertot Apr 9 '16 at 22:17