3. External Field Approximation: Dirichlet Boundary Conditions

This section uses the formalism introduced in Section 2 to discuss in some detail the behavior of a massless charged scalar field in 1 + 1 spacetime dimensions subject to a static external electric field . The field is restricted to the finite region , and satisfies the boundary conditions (2.9), (2.10) with . so that (Dirichlet boundary conditions). It is convenient to introduce the dimensionless variables

where is a gauge parameter, and is gauge invariant. Notice that in 1 + 1 dimensions, the electric charge has dimensions of mass. The field satisfies the Klein--Gordon equation (2.2a) with

The charge density associated with the field is obtained from (2.5c) and is given by

Here, as below, denotes a dimensionless charge density obtained by multiplication of the complete charge density by . For gauge dependent results, we choose . With this symmetrical choice, each positive eigenvalue has an associated negative eigenvalue . The charge densities of the corresponding modes are related by (so long as symmetrical boundary conditions are used).

When no electric field is applied , the modes of the field are given by with , where is an integer. The charge density of each mode is symmetrical about the point .

For non-zero electric field , the modes are formally given by

where , is a parabolic cylinder function (e.g., [8]), is a Whittaker function, and and are respectively regular Bessel functions and Neumann functions. The possible eigenvalues could in principle be found from (3.4) by solving the transcendental equation obtained from .

The form of the modes for small may be obtained from perturbation theory. To first order in ,

The total charge density associated with the nth positive and negative mode in this approximation is given by

In the absence of the external electric field the charge densities of the positive and negative modes are equal and opposite for all values of , and no net charge density exists: . Introduction of the electric field shifts the charge densities of the positive and negative modes oppositely . The positively charged modes are shifted in the direction of the electric field, as expected from classical considerations. The ``centre of mass'' for the charge density of the nth mode is given to first order in by

To this order, the total induced charge density is given by

This charge density leads, as expected, to screening of the applied external electric field.

Figure 3.1a shows the exact form of the lowest positive energy mode for (full curve) and (dashed curve). The direction of the electric field corresponds to a positive charge on the left-hand boundary and a negative charge on the right-hand boundary. The positively charged mode shown is thus shifted to the right in the presence of the electric field.

[ Figure 1 ] Modifications in low modes of a unit charge massless scalar field in an external electric field of strength directed from left to right. The field is taken to vanish at and (Dirichlet boundary conditions). A symmetrical gauge is chosen. (a) Charge density for the lowest energy positively charged mode. (b) Total charge density for the lowest energy positive and negatively charged modes with . Dashed line gives an approximate result obtained to first order in . (c) Charge density for the second positively charged mode.

[ Figure 2 ] Vacuum polarization for massless unit charge scalar field satisfying Dirichlet boundary conditions in a weak external electric field with strength . The contributions from positive and negative charge modes are included. In the case , the result obtained at first order in is shown (dashed curve).

Figure 3.1b gives the total charge density associated with the lowest-energy positive and negative charge modes. The full curve is the exact result; the dashed curve is the first-order form (3.6). The approximation remains comparatively accurate even at .

Figure 3.1c shows the distortion in the charge density of the positive charge mode in the presence of an external electric field.

The total induced charge density is in principle obtained by summing the contributions from each mode, according to Eq. (2.26). However, as suggested by the perturbative result (3.6), the numerical convergence of the resulting sum is not adequate. Nevertheless, consideration of the induced electric field

yields a suitably convergent sum. Figure 3.2 shows the vacuum polarization obtained with a sequence of partial mode sums, all for . In the result (Fig. 3.2a) for the lowest mode alone, the approximation is also given (dashed curve). The final vacuum polarization suggests that the ``vacuum'' behaves like a macroscopic dielectric medium with dielectric constant under the influence of a small applied electric field. Notice that the total vacuum polarization is dominated by the behaviour of the lowest mode.

Figure 3.3 shows the average vacuum polarization as a function of . The results show that the perturbative estimate remains accurate until .

[ Figure 3 ] Average vacuum polarization for massless unit charge scalar field satisfying Dirichlet boundary conditions, given as a function of the strength of the external electric field. labels the number of iterations in the self-consistent procedure used to account for back reaction effects. The dashed line gives the result obtained to first order in .

Equation (3.2) assumes an external electric field uniform throughout the region . As an alternative, one may take the field to be uniform only in the region , and to vanish elsewhere. In the limit , this corresponds to a potential jump with an infinite associated electric field. Figure 3.4 shows results for and various . The lowest mode is insensitive to short distance details of the electric field, and depends primarily on the total potential difference from to .

The results above are for small external electric fields . We now discuss the case of large . Throughout this section, we use the external field approximation. (2) For large , the back reaction effects thus ignored become important. Their consequences are considered in Section 5.

Figure 3.5 shows the energies and charge densities of the first few positive energy levels as a function of .

For , the charge density for the lowest positive energy, positive charge mode is positive everywhere. Above , a region of negative charge density develops. The integrated total charge nevertheless remains normalized to + 1. The Hamiltonian (2.7) suggests that for large , it may be favourable for a positively charged mode to develop a negative charge density in the region where is largest. The expression (3.3) shows that negative charge density appears in the region .

When , the energy levels are equally spaced with . Only the kinetic energy term in (2.7) contributes for . For small , the distortion of the charge density produces a small negative potential energy, which reduces the total energy of the mode. The effect is in a perturbation expansion. For large , the electric potential energy overwhelms the kinetic energy, and at , the total energy of the lowest mode drops vertically to zero. The region of negative charge density in the lowest mode expands as increases, and when reaches , it becomes as large as the region of positive charge density, so that the total charge of the mode vanishes. The formal aspects of this behaviour were discussed in Section 2. Each positive level in Fig. 3.5 is accompanied by a level with opposite and opposite total charge. Precisely at , the lowest positive and negative levels join into a single mode with zero energy and zero total charge (but non-zero charge density). As mentioned in Section 2, an additional ``associated eigenfunction'' exists at this value of , and the ground state is no longer unique [3,7]. At this point, the external field approximation therefore presumably ceases to yield correct physical results, even though it is formally possible to choose a vacuum state and perform canonical quantization [3,7]. The self-consistent approach described in Section 5 avoids these difficulties.

[ Figure 4 ] Results for massless unit charge scalar field satisfying Dirichlet boundary conditions with an external electric field of strength uniform in the region and zero elsewhere. (a) Charge density for lowest positive mode. (b) Total vacuum polarization.

[ Figure 5 ] Energies and charge densities for first few positive energy levels of a massless charged scalar field satisfying Dirichlet boundary conditions in an external electric field of strength . For some values of , the eigenvalues are complex. In such regions, Re is shown as a wavy curve. The vertical distance of the dashed curves indicates the value of Im. Cutting along the dashed lines and folding the upper half ellipses upwards and lower ones downwards yields a three dimensional plot with the Im axis out of the page.

Section 2 mentioned that in some cases, complex eigenvalues may occur. The first set of complex energy levels appear just above . The real part of their energy vanishes, while the imaginary part is non-zero for . The charge densities of the modes are shown in Fig. 3.5. As discussed in Sect. 2, the total charge obtained by integration of this charge density always vanishes.

Figure 3.5 shows that when , the pair of complex energy levels disappears, and a pair of positive and negative real energy levels appear. In the region , all eigenvalues are real. However, the lowest positive energy level has negative total charge and thus negative energy. The combination of this mode with its negative partner has zero charge and negative total energy . High occupation numbers in these modes lead to indefinitely negative energies and no definite vacuum state.

Figure 3.6a extends the energy levels shown in Figure 3.5 to higher values of . Curve segments with correspond to modes with positive charge and positive energy, and those with to negative modes [3,4]. At each point of vertical tangency, a zero norm mode exists. Between each adjacent pair of vertical tangents extend a pair of complex modes. When these complex modes are included, each energy level in Figure 3.6a forms a continuous curve as a function of . The curve develops an imaginary part whenever it crosses another curve. The wave functions for all modes associated with a particular continuous curve exhibit the same number of nodes in the interval . Notice that when , the fraction of possible values at which no complex modes exist tends to zero.

The existence of real modes even at high is essential in obtaining known results [6] for the continuum limit of a uniform electric field throughout space. This limit corresponds to and hence . As , the pattern of energy levels stabilizes, and the charge densities for real modes with take on the form illustrated in Figure 3.7. The modes have positive charge density in the region and negative charge density in the region . In the limit , the modes may be superposed to yield travelling waves carrying positive charge to the right and negative charge to the left, thereby corresponding to continuum modes, and allowing the interpretation of particle-antiparticle pair production.

[ Figure 6 ] Positive and negative energy levels for a massless charged scalar field satisfying Dirichlet boundary conditions with an applied external electric field of strength . (a) Electric field uniform throughout region ; (b) electric field uniform for , and zero elsewhere. Between each pair of vertical tangent points, a pair of complex energy eigenvalues exist. Curve segments with have positive charge and positive energy; others have negative charge and negative energy.

In finite systems, each mode is normalized if possible to unit total charge. In an infinite system, modes are normalized to give particle wave functions with unit charge density. Complex energy modes exhibit real exponential growth modulated by an oscillating factor in both time and space (see Eq. (3.4)). The oscillating factor allows modes to satisfy the necessary boundary conditions. However, away from the boundaries, the real exponential factor give rise to exponentially large charge densities. The normalized modes thus have exponentially small charge densities except in an exponentially small region. In the limit , the normalized complex energy modes have zero charge density almost everywhere, and are therefore of no physical relevance.

The energy levels shown in Figure 3.6a assume that a uniform electric field exists throughout the region . Figure 3.6b shows the energy levels obtained with a field uniform between and , and zero elsewhere. The results in Figs. 3.6a and 3.6b are qualitatively similar. The qualitative independence of the results on the width of the electric field reflect the similarity of particle production in a uniform electric field and a step potential.

[ Figure 7 ] Charge density for lowest positive energy level of scalar field with Dirichlet boundary conditions in an electric field of strength = 200. Separation into positive and negative ``particle'' regions is evident.

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