DIRAC SPINORS IN SOLENOIDAL FIELD AND SELF-ADJOINT EXTENSIONS OF ITS HAMILTONIAN

Abstract

We discuss Dirac equation and its solution in the presence of solenoid (infinitely long) field in (3+1) dimensions. Starting with a very restricted domain for the Hamiltonian, we show that a one-parameter family of self-adjoint extensions 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 to 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 spectrum, i.e. E ∈ (-M, M) for ν2[0, 1)! We also study the issue of relationship between scattering state and bound state in the region ν2 ∈ [0, 1) and recovered the BS solution and eigenvalue from the SS solution.