# What is the analogue for symplectic structure in case of spin variables?

According to some (e.g. Haroche and Raimond in Exploring the quantum: atoms, cavities and photons), the quantum world consists (mainly) of spins and harmonic oscillators.

For harmonic oscillators (i.e. bosons), it is well known that they can be appropriately described in $$(x,p)$$ phase space, which satisfies a 'symplectic' structure (see e.g. Gaussian states in continuous variable quantum information). A system with coupled bosonic modes can be symplectically diagonalized into the eigenmodes.

My question is: is there a similar structure for spin states, living in $$(\sigma^x,\sigma^y,\sigma^z)$$ space (without resorting to a Holstein transformation or similar)? For simplicity, I'm mainly interested in the spin-1/2 case.

• This paper describes the phase space of classical particles with mass and spin: arxiv.org/abs/quant-ph/9601011, a construction of the phase space and the simplectic structure is provided. Oct 1, 2020 at 14:04

The phase space for spin is the two-sphere $$S^2$$ with the symplectic form being the area 2-form $$\omega= J \sin\theta d\theta\wedge d\phi.$$ Here $$\theta$$ and $$\phi$$ are the polar angles. Then, with $$S_x= J \sin\theta \cos\phi,\\ S_y= J \sin\theta \sin\phi,\\ S_z= J \cos\theta,$$ we have $$\{S_x,S_y\}= S_z$$ etc.
• @Wouter This the classical phase space for a single spin $J$. I don't see any room for quantum mixed states in this language. Oct 1, 2020 at 12:09
Angular momentum operators $$\hat{J}_a$$ satisfy an $$so(3)$$ Lie algebra $$[\hat{J}_a,\hat{J}_b]~=~i\hbar \epsilon_{abc} \hat{J}_c,\qquad a,b,c~\in~\{1,2,3\},\tag{C}$$ which at the classical level is a Poisson algebra $$\{J_a,J_b\}~=~ \epsilon_{abc} J_c,\qquad a,b,c~\in~\{1,2,3\}.\tag{P}$$ However, the Poisson structure (P) on $$\mathbb{R}^3$$ is not invertible/non-degenerate, so it is technically not a symplectic structure. But $$\mathbb{R}^3$$ equipped with (P) is a discrete union of symplectic leaves (namely concentric 2-spheres and the origin $$\{0\}$$).