= Nonsinglet axial charge algebra of quark densities
{title2=$[Q_{5,a},S]=iP_a,\quad[Q_{5,a},P_b]=-i\delta_{ab}S$}
Let $S=\bar q q$, $P_a=i\bar q\gamma_5\tau_aq$ and $Q_{5,a}=\int q^\dagger\gamma_5\tau_aq/2\,d^3x$. The <canonical anticommutation relations> imply $[Q_{5,a},q]=-\gamma_5\tau_aq/2$ and $[Q_{5,a},\bar q]=-\bar q\gamma_5\tau_a/2$. The <Leibniz rule> for the <commutator>, together with $\{\tau_a,\tau_b\}=2\delta_{ab}$ and $\gamma_5^2=1$, gives the displayed transformations. These are equal-time operator relations; common nonzero <quark masses> remove conservation of the charges without changing their canonical action on the fields.
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