Field Fluctuations and Casimir Energy of 1D-Fermions

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.

Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian

“…The eigenfunctions depend on the angles (ψ, β 0 , β 1 ) only. These angles are those defined in ( 24) and ( 25); note the difference with the angles defined in (22). If we take the limits β 0 → 0 and β 1 → 0, we recover the eigenfunctions for the textbook extension.…”

“…This parameterization is quite similar to (22), where we have interchanged the expressions for m 0 and m 2 . In terms of the new angular variables, an expression for the energy levels as functions of s is given by…”

“…Taking into account that Equation (13) gives the values of s, and hence the energy levels, in terms of the triplet of parameters (ψ, m 0 , m 1 ), then, according to (22), s will depend on the angular variables (ψ, θ 0 , θ 1 ) only. In general, we cannot solve (13) to find s(ψ, θ 0 , θ 1 ) explicitly.…”

“…In the sequel we are going to deal with the cases of Table I that can be treated in some way, either graphically or analytically. For m 2 = m 3 = 0, we have in ( 21) that m 2 0 + m 2 1 = 1, that is, we can take θ 1 = 0 in (22) and therefore m 0 = cos θ 0 and m 1 = sin θ 0 . Then, let us go back to (10), so as to see that Tr(N (s)) = 0.…”

“…Although the idea of self-adjoint extensions of symmetric (or Hermitian) operators on (infinite dimensional) Hilbert spaces is not yet very popular among physicists, it is, however, possible to find recent papers on the topic [16][17][18][19][20][21][22][23][24].…”