\hat{b}^{il}_{-}
( u - \frac{1}{2} (k+l) \hbar )
- \sum_{l=i+2}^{N} \hat{b}^{i+1,l}_{-}
( u- \frac{1}{2} (k+l-1) \hbar) \right): \nonumber\\
& & ~~~~- :\mbox{exp} \left( (b+c)^{i,i+1}
( u + \frac{1}{2} (k+i) \hbar ) \right)  \nonumber\\
& & ~~~~~~~~\times \mbox{exp} \left( \hat{a}^i_+ (u)
+ \sum_{l=i+1}^N \hat{b}^{il}_{+}
( u + \frac{1}{2} (k+l) \hbar )
- \sum_{l=i+2}^{N} \hat{b}^{i+1,l}_{-}
( u+ \frac{1}{2} (k+l-1) \hbar) \right): \nonumber\\
& &~~~~- \sum_{m=i+2}^N :\mbox{exp} \left( (b+c)^{im}
( u + \frac{1}{2} (k+m-1) \hbar ) \right) \nonumber\\
& & ~~~~~~~~\times \left[
\mbox{exp} \left( \hat{b}^{i+1,m}_+ ( u + \frac{1}{2} (k+m-1) \hbar )
-(b+c)^{i+1,m} ( u + \frac{1}{2} (k+m) \hbar ) \right) \right. \nonumber\\
& &~~~~~~~~~~~~ - \left. \mbox{exp} \left( \hat{b}^{i+1,m}_-
( u + \frac{1}{2} (k+m-1) \hbar )
-(b+c)^{i+1,m} ( u + \frac{1}{2} (k+m-2) \hbar ) \right) \right] \nonumber\\
& &~~~~~~~~\times \left.
\mbox{exp} \left( \sum_{l=m}^{N} \left[
\hat{b}^{il}_+ ( u + \frac{1}{2} (k+l) \hbar )
-\hat{b}^{i+1,l}_+ ( u + \frac{1}{2} (k+l-1) \hbar ) \right]
\right) : \right\}~. \label{en}\end{aligned}$$

The following proposition is the main result of this paper:

\[prop1\] The fields $H^{\pm}_i(u)$, $E^{\pm}_i(u)$ defined in equations (\[hpn\]), (\[ep\]) and (\[en\]) are well-defined on the Fock space ${\cal F}(l_a,l_b,l_c)$ and satisfy equations (\[y1\]-\[y4\]) with $c=k$ and

$$\begin{aligned}
& & E^{\pm}_i(u) E^{\pm}_j(v) \simeq
E^{\pm}_j(v) E^{\pm}_i(u) \sim reg. ~~\mbox{for}~B_{ij}=0,\\
& & (u-v \mp B_{ij} \hbar) E^{\pm}_i(u) E^{\pm}_j(v) \simeq
(u-v \pm B_{ij} \hbar) E^{\pm}_j(v) E^{\pm}_i(u)
\sim reg. ~~\mbox{for}~B_{ij} \neq 0,\\
& & E^{+}_i(u) E^{-}_j(v) - E^{-}_j(v) E^{+}_i(u) \\
& &~~~~ \sim reg. + \frac{1}{\hbar} \left(
\delta( u_- - v_+) H^+_i(v_+) - \delta( u_+ - v_-) H^-_i(v_-) \right),\end{aligned}$$

where $reg.$ means some regular expressions and $\simeq$ and $\sim$ imply “equals up to” such expressions.

[*Proof*]{}: The proposition follow by straightforward but tedious calculations. Actuall