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

Question

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.

Qmechanic · Accepted Answer

Angular momentum operators $hat{J}_a$ satisfy an $so(3)$ Lie algebra
$$ [hat{J}_a,hat{J}_b]~=~ihbar epsilon_{abc} hat{J}_c,qquad a,b,c~in~{1,2,3},tag{C}$$
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 ${0}$).

mike stone · Answer

The phase space for spin is the two-sphere $S^2$ with the symplectic form being the area 2-form
$$
omega= J sintheta dthetawedge dphi.
$$
Here $theta$ and $phi$ are the polar angles.  Then, with
$$
S_x= J sintheta cosphi,
S_y= J sintheta sinphi,
S_z= J costheta,
$$
we have ${S_x,S_y}= S_z$ etc.

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

2 Answers

Add your own answers!

Ask a Question