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Articles General Physics Properties of the Vacuum. 2. Electrodynamic (1983)
Properties of the Vacuum. 2. Electrodynamic (1983) |
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Sections 3 and 4 considered the effect of a fixed classical external electric field on a charged quantized 
field. However, the field charge density induced by the external field itself generates an electric field, which may in turn modify the vacuum polarization. Assuming that the electric field may still be treated as classical, this ``back reaction'' may be included exactly. In this section, we describe the necessary procedure, and discuss results for large external electric fields including back reaction effects.
The complete result for vacuum polarization including back reaction effects is obtained by solving the coupled equations (see Section 2)
where 
is the original external potential. In the second form of Eq. (5.1c), the factor 
is the Coulomb Green function in 1 + 1 dimensions.
Equations (5.1) take a quantized field, but assume a classical A field depending only on the expectation value of the charge operator (2.25). This approximation should be accurate for arbitrarily large 
, so long as the charge 
on a single quantum is sufficiently small.
In terms of Feynman diagrams, the external field approximation used in Sections 3 and 4 sums all diagrams with a single loop and an arbitrary number of external field vertices. Equation (5.1) includes also diagrams with many loops in a tree connected by Coulomb photons. It does not account for diagrams involving photons within a single loop.
The integro-differential equations (5.1) yield a ``self-consistent'' solution for vacuum polarization. The equations may often be solved by a simple iterative procedure. At the first step in the procedure the vacuum polarization obtained with the original external electric field is calculated as in the external field approximation. The next step consists in calculating the correction to the electric field resulting from the vacuum polarization. This modified field is then used to calculate a new vacuum polarization. The iteration continues until a self-consistent vacuum polarization is obtained. Labelling successive steps in this procedure by the parameter 
, the necessary sequence of equations is obtained by replacing 
, 
and 
by 
and 
respectively in Eqs. (5.1), and replacing 
by 
in (5.1a) and (5.1b) and by 
in (5.1c).
At small 
, the vacuum polarization obtained in the external field approximation is small. So long as is small the resulting modification to the electric field is small, and back reaction effects are negligible. With Dirichlet boundary conditions at 
, inclusion of back reaction effects changes the final vacuum polarization by 
. As approaches 
however, the charge density associated with the lowest mode increases, and the total vacuum polarization becomes larger. Figure 5.1 shows the total vacuum polarization at successive steps in the iterative procedure for a field with Dirichlet boundary conditions and 
close to 
. 
is the original zero vacuum polarization. 
is the result in the external field approximation. The result for 
is the final self-consistent one. When 
, the vacuum polarization depends not only on , but also on the charge . Notice that in the case shown, the iterative procedure converges rapidly.
close to . Back reaction effects are included by an iterative procedure. The steps in the procedure are labelled by . corresponds to the original zero polarization. gives the external field approximation result. In the limit , the vacuum polarization tends to the exact self-consistent form obtained from Eq. (5.1).When increases above , the external field approximation yields complex energy eigenvalues, and is no longer appropriate, as discussed in Sections 3 and 4. Equations (5.1) may nevertheless still be solved iteratively if a suitable starting form for 
is chosen. Figure 5.2 shows the vacuum polarization just above as a function of for two choices of . The procedure is seen to converge to the same final self-consistent vacuum polarization in the two cases.
When approaches the energy 
of the lowest mode in the external field approximation goes to zero. Figures 5.2c and 5.2d show, however, the sequence of values for obtained in the iterative procedure. The two starting chosen yield the same final self-consistent non-zero value for in the limit .
. Steps in the iterative procedure used to include back reaction effects are labelled by . (a) and (b) show results for vacuum polarization at various stages in the iterative procedure. (c) and (d) give the behaviour of the lowest energy eigenvalue as a function of . Two choices are made for the starting potential in the iterative procedure. (The original external electric field cannot be used because of the complex eigenvalues it yields when 
The starting polarizations are labelled . In (a) and (c) the potential 
was used; in (b) and (d) the potential from vacuum polarization at 
. The results from the two starting potentials are seen to converge to the same final self consistent form.
of an external electric field. Steps in the iterative procedure used to account for back reaction effects are labelled by . gives the external field approximation result. The curve marked 
is the second step in an iteration starting from zero vacuum polarization. 
gives the final self-consistent result.Back reaction effects generally reduce the effective electric field. They thus tend to raise , and prevent the appearance of complex eigenvalues. Figure 5.3 shows the behaviour of 
as a function of with Dirichlet boundary conditions and 
. The self-consistent result extends continuously above . No complex eigenvalues appear. The absence of complex eigenvalues allows straightforward quantization of field modes. For large , the charge density of the lowest mode far exceeds that of the higher modes. In the remainder of this section, we shall usually ignore contributions from higher modes. Figure 5.4a shows the vacuum polarization obtained in the self-consistent limit with Dirichlet boundary conditions and . Figure 3.3 shows the average polarization as a function of . The vacuum polarization increases rapidly at large , always reducing the effective electric field sufficiently to avoid complex eigenvalues. The 
for different shown in Fig. 5.4a differ essentially only in overall scale. Large screening of the external electric field occurs around 
, but boundary conditions on the field enforce zero vacuum polarization for 
.
induced by an external electric field of strength . (a) Dirichlet boundary conditions; . (b) 
; Dirichlet boundary conditions. (c) 
; Robin boundary conditions with 
. 
corresponds to Dirichlet and 
to Neumann boundary conditions.) Only the lowest energy mode is included: small contributions from higher modes are dropped.The behaviour of a physical system with a large external electric field depends on the mechanism by which the field is introduced. In typical situations, an external electric potential is increased as a function of time (for example, by separating ``capacitor'' plates carrying fixed electric charges). Transients resulting from a rapid increase in the electric field would mix normal modes and require a complete solution of the time-dependent Klein--Gordon equation (2.2). However, if the rate of change of the electric field is small compared to the frequency of the low modes of the field, then time-independent normal mode analysis provides an adequate approximation. As the external electric field is increased, the vacuum polarization achieves its self-consistent form at every . Suitable choices for in Eq. (5.1) are obtained by solving the equations for one value of , and using the resulting vacuum polarization in the starting potential for a slightly higher value of . Numerical instabilities in practice require rather small steps to be used in this procedure.
for a massless charge scalar field in an external electric field of strength . labels steps in the iterative procedure used to account for back reaction effects. corresponds to the external field approximation. Final self-consistent results are obtained in the limit . (a) for Robin boundary conditions with 
as a function of . (b) as a function of for Dirichlet boundary conditions with 
. (c) for 
as a function of the parameter 
specifying Robin boundary conditions. corresponds to Neumann boundary conditions.Figure 4.1 shows that with Robin boundary conditions 
, the critical in the external field approximation decreases with decreasing 
. The behaviour of as a function of for 
is shown in Fig. 5.5a. Once again, the self-consistent remains non-zero above 
and no complex eigenvalues appear. Different self-consistent results are obtained for different charges . Smaller charge implies smaller back reaction; the value of thus tends to be smaller for 
than for , but it remains non-zero.
Figure 5.5b shows the dependence of on in the presence of back reaction effects. Figure 5.4b shows the corresponding vacuum polarization. For 
, results become unreliable as the semiclassical approximation fails.
Figures 5.4c and 5.5c show the dependence of the vacuum polarization and first energy level on the parameter specifying Robin boundary conditions. remains non-zero in the self-consistent limit for all 
. As approaches zero, corresponding to Neumann boundary conditions, however, tends to zero. The vacuum polarization increases as decreases. When 
reaches zero. As discussed in Section 4, Neumann boundary conditions lie on the edge of the Robin parameter region in which free field modes have complex eigenvalues. The Neumann case is thus at the boundary between stable and unstable systems in the self consistent limit. For 
, the lowest energy level in the external field approximation 
shown in Fig. 5.5c has a complex energy eigenvalue. In the self-consistent limit no complex eigenvalues occur. Figure 4.2 showed the vacuum polarization obtained with Neumann boundary conditions, ignoring complex energy eigenvalues, and exhibited antiscreening. Figure 5.6a shows vacuum polarization resulting from the lowest real frequency mode for Robin boundary conditions with 
in the external field approximation. Antiscreening is again apparent. Figure 5.6b shows the vacuum polarization found in the self-consistent limit. No complex energy eigenvalues exist, and the vacuum polarization gives screening. It appears that in all cases for which the external field approximation implies complex energy eigenvalues and antiscreening of real modes, inclusion of back reaction effects by the self consistent method removes the complex modes, and restores screening.
For some choices of parameters, the self-consistent vacuum polarization shown in Fig. 5.4 screens a significant part of the original external electric field. However, as mentioned above the shape of the vacuum polarization as a function of z remains roughly unchanged. In some cases, the vacuum polarization may become so large that around , it reverses the total electric field.
Equation (2.24) shows that in the external field approximation, all modes of the field have zero occupation number in the vacuum state. The vacuum state is generally the state of lowest energy: when back reaction effects are included, the lowest energy state might involve non-zero occupation numbers (cf. [4]). Nevertheless, in the self-consistent limit it appears however that non-zero occupation numbers always increase the total energy obtained from Eq. (2.6). The vacuum state therefore remains as the unique state with zero occupation number for each mode.
in an external electric field of strength 
. (a) Result in the external field approximation (ignoring complex frequency modes). (b) Result including back reaction through self-consistent procedure.
