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

TL;DR

This paper constructs natural transformations linked to locally compact groups to demonstrate the universality of the global Terrell law and nucleon mass invariance in nuclear fission processes.

Contribution

It introduces a functorial framework associating natural transformations with fission systems, establishing invariance properties of the Terrell law.

Findings

Proves invariance of nucleon core masses under group actions.

Establishes universality of the global Terrell law.

Defines natural transformations using Connes characters and states.

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.