Abstract
We investigate the self-adjoint extensions of the Dirac operator of a massive one-dimensional field of mass m confined in a finite filament of length L. We compute the spectrum of vacuum fluctuations of the Dirac field under the most general dispersionless boundary conditions. We identify its edge states in the mass gap within a set of values of the boundary parameters, and compute the Casimir energy of the discrete normal modes. Two limit cases are considered, namely, that of light fermions with m L ≪ 1 , and that of heavy fermions for which m L ≫ 1 . It is found that both positive and negative energies are obtained for different sets of values of the boundary parameters. As a consequence of our calculation we demonstrate that the sign of the quantum vacuum energy is not fixed for exchange-symmetric plates (parity-invariant configurations), unlike for electromagnetic and scalar fields.
Highlights
Over the last two decades, the application of the theory of self-adjoint extensions of elliptic operators to problems of quantum physics has motivated an intensive research activity
The quantum vacuum energy for Dirac fields has been largely studied previously, for the case that the self-adjoint extension of the Dirac operator that represents the interaction of the quantum field with the boundary is given by the M.I.T. bag boundary condition in its most general form
We have studied the self-adjoint extensions of the one-dimensional Dirac operator for massive fermions confined in a filament of length L, as well as in its dual geometry
Summary
Over the last two decades, the application of the theory of self-adjoint extensions of elliptic operators to problems of quantum physics has motivated an intensive research activity. In the seminal paper of Asorey, Marmo and Ibort [1], the authors have reformulated the classical theory of self-adjoint extensions pioneered by Von Neumann They have studied the properties of the Laplace-Beltrami operator over a compact manifold with boundary, in terms of physically meaningful boundary data. [3,4,5,6,7] to develop the theory of scalar quantum fields confined in domains with boundaries Those works have applied the theory of self-adjoint extensions to the computation of the so-called Casimir energy of scalar fields [8,9]. The quantum vacuum energy for Dirac fields has been largely studied previously, for the case that the self-adjoint extension of the Dirac operator that represents the interaction of the quantum field with the boundary is given by the M.I.T. bag boundary condition in its most general form
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