Positivity of the Invariant Kernel Underlying Quantum Theory of the Coulomb Field

TL;DR

This paper proves the positivity of an invariant kernel fundamental to quantum Coulomb field theory, using classical theorems, independent of the original theoretical framework.

Contribution

It provides a novel, independent proof of kernel positivity based on classical mathematical theorems, enhancing the theoretical foundation of quantum Coulomb field models.

Findings

Established positivity of the invariant kernel

Applied Schoenberg's and Bochner's theorems in a new context

Strengthened mathematical basis for quantum Coulomb field theory

Abstract

We present a proof of positivity of an invariant kernel, which is of basic importance for the Staruszkiewicz theory of the quantum Coulomb field. Presented proof of positivity is independent of the Staruszkiewicz theory and is based on the classical Schoenberg's theorem for conditionally negative definite functions, as well as on the generalized Bochner's theorem.