Inequivalent quantization in the field of a ferromagnetic wire

TL;DR

This paper explores how inequivalent quantization methods can lead to bound states of neutral atoms around a ferromagnetic wire, revealing a scale anomaly and re-normalization in the quantum regime.

Contribution

It introduces a novel application of self-adjoint extensions to induce bound states in a ferromagnetic wire system, highlighting the role of inequivalent quantization.

Findings

Bound states exist for coupling constants in [0,1).

Quantization breaks classical scale symmetry, leading to a scaling anomaly.

Strong coupling region supports bound states and re-normalization.

Abstract

We argue that it is possible to bind neutral atom (NA) to the ferromagnetic wire (FW) by inequivalent quantization of the Hamiltonian. We follow the well known von Neumann's method of self-adjoint extensions (SAE) to get this inequivalent quantization, which is characterized by a parameter \Sigma\in\mathbb{R}({mod}2\pi). There exists a single bound state for the coupling constant \eta^2\in[0,1). Although this bound state should not occur due to the existence of classical scale symmetry in the problem. But since quantization procedure breaks this classical symmetry, bound state comes out as a scale in the problem leading to scaling anomaly. We also discuss the strong coupling region \eta^2< 0, which supports bound state making the problem re-normalizable.