= Skyrme baryon density
{c}
{title2=$B=\int\mathcal B\,d^3x$}
The local <Skyrme baryon density> is $\mathcal B(x)=-\epsilon_{ijk}\operatorname{tr}(L_iL_jL_k)/(24\pi^2)$, with $L_i=U^\dagger\partial_iU$ and $B=\int\mathcal B\,d^3x$. It is invariant under global <isorotations> and transforms as a scalar under proper spatial rotations, so its contours reveal the geometric shape of a <Skyrmion>. For the <rational map approximation for Skyrmions>, $\mathcal B=-f'\sin^2f\,J_R/(2\pi^2r^2)$, where $J_R$ is the <angular Jacobian of a rational map>. Integrating $J_R$ over the sphere gives $4\pi\deg R$, recovering the integer total charge. A decreasing profile and holomorphic angular map give a nonnegative density, but a general field may have regions of negative density; positivity is not a general topological theorem. The density is different from the <energy density>.
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