On the Inequivalence of Renormalization and Self-Adjoint Extensions for Quantum Singular Interactions

TL;DR

This paper compares self-adjoint extensions and renormalization in quantum singular interactions, showing they are not equivalent for long-range conformal interactions, with renormalization selecting preferred extensions and regulating Hamiltonians.

Contribution

It demonstrates the non-equivalence of renormalization and self-adjoint extensions in certain quantum singular interactions, providing a unified S-matrix framework.

Findings

Renormalization acts as a selector of preferred extensions.

Renormalization regulates unbounded Hamiltonians.

For long-range conformal interactions, the two methods differ.

Abstract

A unified S-matrix framework of quantum singular interactions is presented for the comparison of self-adjoint extensions and physical renormalization. For the long-range conformal interaction the two methods are not equivalent, with renormalization acting as selector of a preferred extension and regulator of the unbounded Hamiltonian.