weinberg].

Once Weinberg’s formalism is expressed by means of the symplectic form in Eq. (\[eq:wein\_eqofm\]), it can be generalized very easily in order to obtain a non-Hamiltonian quantum algebra. To this end, one can substitute the antisymmetric matrix $\mbox{\boldmath$\cal B$}$ with another antisymmetric matrix $\mbox{\boldmath$\Omega$}=\mbox{\boldmath$\Omega$}[\mbox{\boldmath$\zeta$}]$, whose elements might be functionals of $\mbox{\boldmath$\zeta$}\equiv(\vert\Psi\rangle,\langle\Psi\vert)$ obeying the homogeneity condition in Eq. (\[eq:homogeneity\]). By means of $\mbox{\boldmath$\Omega$}$ a non-Hamiltonian bracket $\left\{ \ldots ,\ldots \right\}_{\mbox{\tiny\boldmath$\Omega$}}$ can be defined as $$\begin{aligned}
\left\{ a,b\right\}_{\mbox{\tiny\boldmath$\Omega$}}&=&
\sum_{\alpha=1}^2\frac{\partial a}{\partial\zeta{\alpha}}
{\Omega}_{\alpha\beta}[\zeta]\frac{\partial b}{\partial\zeta{\beta}}\;.
\label{eq:nhbracket}\end{aligned}$$ In general, the bracket in Eq. (\[eq:nhbracket\]) does no longer satisfy the Jacobi relation $${\cal J}=
\left\{ a,\left\{ b,c\right\}_{\mbox{\tiny\boldmath$\Omega$}}
\right\}_{\mbox{\tiny\boldmath$\Omega$}} 
+\left\{ c\left\{ a,b\right\}_{\mbox{\tiny\boldmath$\Omega$}}
\right\}_{\mbox{\tiny\boldmath$\Omega$}}
+\left\{ b,\left\{ c,a\right\}_{\mbox{\tiny\boldmath$\Omega$}}
\right\}_{\mbox{\tiny\boldmath$\Omega$}} 
\neq 0\;.\label{eq:njacobi}$$ Thus, non-Hamiltonian equations of motion can be written as $$\frac{\partial\mbox{\boldmath$\zeta$}}{\partial t}=\frac{i}{\hbar}
\left\{ {\cal H},\mbox{\boldmath$\zeta$}\right\}_{\mbox{\tiny\boldmath$\Omega$}}
\;.
\label{eq:wein_nheqofm}$$ In principle, the non-Hamiltonian theory, specified by Eqs. (\[eq:nhbracket\]), (\[eq:njacobi\]), and (\[eq:wein\_nheqofm\]), can be used to address the problem of non-linear correction to quantum mechanics, as it was done in Refs. [@weinberg]. In the present paper, it has been shown that such a non-Hamiltonian and non-linear version of quantum mechanics is already implied when one formulates qua