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

The magnetic field of an infinitely long solenoid is immensely important for various reasons. For example, it is used to study the Aharanov-Bohm [1] effect. Similarity with this field also helps one to understand the physics of cosmic strings [2]. It is therefore important to study the quantum mechanics of a Dirac fermion in the field of an infinitely long solenoid, which is what we do in the present article. Several studies have been performed on Dirac particles in magnetic fields. The solution in a uniform field [3] is known for a long time. For a solenoidal background, the Dirac equation has been solved in 2+1 dimensions [4, 5, 6]. It shows that a self-adjoint extension of the Dirac Hamiltonian is necessary to obtain the solutions. In 3+1 dimensions, selfadjoint extension of the Dirac Hamiltonian has also been studied [7], but it involves a δ-sphere potential rather than solenoid interaction. Our aim is to solve the Dirac equation in 3+1 dimension in the background field of an infinite solenoid. This will enable us to obtain all four solutions of Dirac equation. We find that a self-adjoint extension is necessary for this case as well.