Dirac spinors in solenoidal field and self adjoint extensions of its Hamiltonian

TL;DR

This paper analyzes the Dirac equation in a solenoidal magnetic field, demonstrating the necessity of self-adjoint extensions for proper evolution, and explores bound and scattering states, highlighting the impact of scaling anomalies on energy bounds.

Contribution

It introduces a one-parameter family of self-adjoint extensions for the Dirac Hamiltonian in a solenoid field and investigates their effects on bound and scattering states, revealing the role of scaling anomalies.

Findings

Bound states exist within the energy range (-M, M) due to scaling anomaly.

Self-adjoint extensions are essential for correct Dirac evolution in this system.

Relationship between scattering and bound states is established for certain parameter ranges.

Abstract

We discuss Dirac equation (DE) and its solution in presence of solenoid (infinitely long) field in (3+1) dimensions. Starting with a very restricted domain for the Hamiltonian, we show that a 1-parameter family of self adjoint extensions (SAE) are necessary to make sure the correct evolution of the Dirac spinors. Within the extended domain bound state (BS) and scattering state (SS) solutions are obtained. We argue that the existence of bound state in such system is basically due the breaking of classical scaling symmetry by the quantization procedure. A remarkable effect of the scaling anomaly is that it puts an open bound on both sides of the Dirac sea, i.e., E\in(-M,M) for \nu^2[0,1)! We also study the issue of relationship between scattering state and bound state in the region \nu^2 \in[0,1) and recovered the bound state solution and eigenvalue from the scattering state solution.