Quantum Solitons in any Dimension: Derrick's Theorem v. AQFT

TL;DR

This paper challenges Derrick's theorem in classical field theories by demonstrating that algebraic quantum field theory (AQFT) allows for stable solitons in any dimension, with implications for scalar fields and Yang-Mills theory.

Contribution

It introduces a proof using relative entropy showing no obstruction to soliton construction in AQFT, contrasting classical limitations.

Findings

Derrick's theorem does not apply in AQFT framework.

Stable solitons can exist in scalar fields and Yang-Mills theory within AQFT.

Implications for topologically trivial sectors in quantum field theories.

Abstract

A powerful tool for studying the behavior of classical field theories is Derrick's theorem: one may rule out the existence of localized inhomogeneous stable field configurations (solitons) by inspecting the Hamiltonian and making scaling arguments. For example, the theorem can be used to rule out compact domain wall configurations for the classic $\phi^4$ theory in $3+1$ dimensions and greater. We argue no such obstruction to constructing solitons exists in the framework of algebraic/axiomatic quantum field theory (AQFT), and that states like the example given lie in topologically trivial superselection sectors of the Hilbert space. A proof is presented making use of the relative entropy, and the implications are explored for a few common models of scalar fields and the pure Yang-Mills theory.