4. External Field Approximation: General Boundary Conditions

According to Eq. (2.10) the most general charge-conserving boundary conditions for a massless scalar field in 1 + 1 dimensions have the Robin form

Section 3 considered the special case , corresponding to Dirichlet boundary conditions.

We discuss first the implementation of the boundary conditions (4.1 ) in the absence of an external electric field . In this case, Eq. (2.2) is the equation of motion for a classical vibrating string with amplitude . The boundary conditions (4.1) specify the string to be attached at its two ends by elastic forces (). Equations (2.2) and (4.1) form a standard Sturm--Liouville system in this case. If one of the is negative, larger negative potential energy is achieved by increasing , and ``runaway'' solutions exist. The eigenvalues corresponding to these solutions are complex. (3) The complete set of eigenfunctions , including those with complex or vanishing , span the space of square-integrable functions on the interval .

Figure 4.1 shows the behaviour of the first few energy levels of the field as a function of the strength of the external electric field, for a selection of boundary conditions with . The results interpolate smoothly between the limiting cases of Neumann boundary conditions and Dirichlet boundary conditions . When decreasing shifts the energy levels as if the axis was shifted to move the point towards the origin. When , the zero norm eigenfunction which appeared at in the Dirichlet case appears at , as the eigenfunction . As expected, complex modes appear immediately above in the Neumann case.

[ Figure 4.1 ] Energy levels for a massless charged scalar field subject to a range of boundary conditions, as a function of the strength of an external applied electric field. parametrize general Robin boundary conditions. Dirichlet and Neumann boundary conditions are limiting cases.

The presence of complex modes at small for Neumann boundary conditions leads to very different results for vacuum polarization than in the Dirichlet case discussed in Section 3. To first order in , the induced charge density associated with the nth mode satisfying Neumann boundary conditions is

This result is essentially opposite to Eq. (3.6) obtained with Dirichlet boundary conditions. The induced charge density tends to increase (``antiscreen'') the electric field, rather than to decrease (``screen'') it, as in the Dirichlet case.

Figure 4.2 shows the total vacuum polarization obtained by summing the contributions of all modes with real eigenvalues for and Neumann boundary conditions. Modes with complex eigenvalues should also appear in the sum, but no unique prescription for their inclusion exists [3,7]. The result of Fig. 4.2 with these modes omitted clearly exhibits a positive average polarization, corresponding to an effective dielectric constant . Dielectric constants in the range usually imply instability [9,10]. As discussed in Section 5, however, inclusion of back reaction effects removes complex frequency modes and yields a stable system with dielectric constant .

[ Figure 4.2 ] Vacuum polarization obtained by summing modes of a massless charged scalar field subject to Neumann boundary conditions, omitting modes with complex eigenvalues. Antiscreening is evident.

Antiscreening is found at small whenever modes with real eigenvalues are summed, but a mode with complex eigenvalue is omitted. Boundary conditions for which no complex mode exists at sufficiently small always give rise to screening in this region.

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