Spin-Wave Scattering

Abstract

We consider an infinite array of ions, each fixed at a point of the lattice Z^v,v = 1,2 or 3. Each ion has no degrees of freedom except its spin, 1/2, which is described by a spinor, that is a unit vector in $C^2 ,\left( {\mathop {\dot 1}\limits_0 } \right)$ denoting spin up, ↑, and $\left( {\mathop 1\limits^0 } \right)$ spin down, ↓. A typical “product state” of all the ions can be pictured:

$$ ... \otimes \uparrow \otimes \nearrow \otimes \nwarrow \otimes \uparrow \otimes \nwarrow \otimes... $$

The “observables” associated with an ion j ε Z^v are the spins $J\mathop{j}\limits^{\alpha } = 1/2{{\sigma }^{\alpha }}$ regarded as acting on the j th factor in the product; here, α = 1,2 or 3 are the 3 spin (or isobaric spin) directions. For each ion j, its observables generate the 4 dimensional C* -algebra $a_j = B(c^2 )$ of 2 × 2 complex matrices. These are all independent observables, so the total system is described by $A = \mathop \otimes \limits_{j\varepsilon Z^v } a_j ,$a C* -algebra whose hermitian elements are called the quasi-local observables.