
 [No title] 
  Furthermore, as already could be understood from (\ref{sp}), the fluctuation operators $F_\lambda(A),F_\lambda(B),\ldots$ satisfy the canonical commutation relations: \begin{equation}\label{ccr} \mathrm{[} F_\lambda(A), F_\lambda(B)\mathrm{]} = \omega_{\beta\lambda}(\mathrm{[} A,B \mathrm{]}) \mathbf{1} \end{equation} \ie the fluctuation operators are quantum canonical variables (compare $\mathrm{[} q,p \mathrm{]}= \mathrm{i}\hbar$) with quantisation parameter $ \omega_{\beta\lambda}(\mathrm{[} A,B \mathrm{]})$ (instead of $\mathrm{i}\hbar$) and the state $\left(\tilde{\Omega}_\lambda,. 
  We know from section \ref{ssb} that the fluctuation operators are canonical variables, satisfying the canonical commutation relations. 
  Hence, the four pairs of canonical fluctuation operators are all independent, proving the twofold degeneracy of the plasmon frequency $\xi_+$ and $\xi_$. 
 www.ma.utexas.edu /mp_arc/html/papers/00124 (3601 words) 
