Natural transformations associated with a locally compact group and universality of the global Terrell law

Abstract

Via the construction of a functor from $\mathsf{C}_{u}(H)$ to an auxiliary category we associate, with any triplet $(G,F,\rho)$, two natural transformations, $\mathfrak{m}_{\star}$ morphism of $\mathsf{Fct}(\mathsf{C}_{u}(H)^{op},\mathsf{Fct}(H,\mathsf{set}))$ and $\mathfrak{v}_{\natural}$ morphism of $\mathsf{Fct}(\mathsf{C}_{u}^{0}(H)^{op},\mathsf{Fct}(H,\mathsf{set}))$. $G$ and $F$ are locally compact groups, $\rho:F\to Aut(G)$ is a continuous morphism, $H$ is the external topological semidirect product of $G$ and $F$ relative to $\rho$, $\mathsf{C}_{u}^{0}(H)$ is a subcategory of $\mathsf{C}_{u}(H)$ a subcategory of the category of $C^{\ast}-$dynamical systems with symmetry group $H$ and equivariant morphisms. For $\mathfrak{A}$ in $\mathsf{C}_{u}^{0}(H)$ to assemble $\mathfrak{m}_{\star}^{\mathfrak{A}}$ we exploit the Connes characters generated by JLO cocycles $\Phi$ on the unitization of certain $C^{\ast}-$crossed products relative to $\mathfrak{A}$, while to construct $\mathfrak{v}_{\natural}^{\mathfrak{A}}$ we exert the states of the $C^{\ast}-$algebra underlying $\mathfrak{A}$ associated in a convenient manner with the $0-$dimensional components of the $\Phi$'s. Being $\mathsf{C}_{u}(H)$ the category of fissioning systems, we use $\mathfrak{m}_{\star}^{\mathfrak{A}}$ and $\mathfrak{v}_{\natural}^{\mathfrak{A}}$ to define the nucleon phases and the fragment states resulting next the fission processes of the fissioning system $\mathfrak{A}$ occur. We apply the naturality of $\mathfrak{m}_{\star}$ and $\mathfrak{v}_{\natural}$ to establish the universality of the global nucleon masses and the global Terrell law, stated as invariance of the light and heavy nucleon core masses and invariance of the prompt-neutron yield under controvariant action of $\mathsf{C}_{u}^{0}(H)$ and under action of $H$ over the field of fission processes.