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 ${\mathfrak{v}}_{♮}$ 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 ${\mathfrak{v}}_{♮}^{\mathfrak{A}}$ 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 ${\mathfrak{v}}_{♮}^{\mathfrak{A}}$ 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 ${\mathfrak{v}}_{♮}$ 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.

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