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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.

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    $\begingroup$ 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. $\endgroup$
    – Quillo
    Oct 1, 2020 at 14:04

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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.

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  • $\begingroup$ thanks, but if the two-sphere is the phase space, what happens for a mixed state, which is located at the inside of the Bloch Sphere? It seems to me that this is also a relevant question in the case of coupled spins with two-mode squeezing? $\endgroup$
    – Wouter
    Oct 1, 2020 at 11:53
  • $\begingroup$ @Wouter This the classical phase space for a single spin $J$. I don't see any room for quantum mixed states in this language. $\endgroup$
    – mike stone
    Oct 1, 2020 at 12:09
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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 $\{0\}$).

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  • $\begingroup$ Any idea if this would allow for the use of symplectic integrators when considering coupled equations for the different J-components? $\endgroup$
    – Wouter
    Oct 8, 2020 at 15:01

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