Induced Vacuum Energy Density of Quantum Charged Scalar Matter in the Background of an Impenetrable Magnetic Tube with the Neumann Boundary Condition

Abstract

We consider the vacuum polarization of a charged scalar matter field outside the tube with magnetic flux inside. The tube is impenetrable for quantum matter, and the perfectly rigid (Neumann) boundary condition is imposed at its surface. We write expressions for the induced vacuum energy density for the case of a space with arbitrary dimension and for an arbitrary value of the magnetic flux. We do the numerical computation for the case of a half-integer flux value in the London flux units and the (2 + 1)-dimensional space-time. We show that the induced vacuum energy of the charged scalar matter field is induced, if the Compton wavelength of the matter field exceeds the transverse size of the tube considerably. We show that the vacuum energy is periodic in the value of the magnetic flux of the tube, providing a quantumfield-theoretical manifestation of the Aharonov–Bohm effect. The dependencies of the induced vacuum energy upon the distance from the center of the tube for different values of its thickness are obtained. The results are compared to those obtained earlier in the case of the perfectly reflecting (Dirichlet) boundary condition. It is shown that the value of the induced vacuum energy density in the case of the Neumann boundary condition is greater than in the case of the Dirichlet boundary condition.