6. Discussion and Extensions

Sections 3 and 4 described the behaviour of a charged scalar field with an applied electric field in the ``external field approximation.'' Section 5 gave results including back reaction effects by a self-consistent semiclassical procedure. Complex frequency modes indicating instabilities were found for large electric field strengths in the external field approximation. These instabilities never occurred when back reaction effects were included. The magnitude of vacuum polarization in the self-consistent limit was always found to reduce the effective electric field until no instabilities appeared. In some cases, the vacuum polarization was sufficient to reverse the electric field around .

The self-consistent semiclassical procedure of Section 5 takes a quantized field and a classical electromagnetic field determined from the expectation value of . Results depend not only on (as in the external field approximation), but also on the charge of a single quantum. Whereas the external field approximation fails for large , the semiclassical approximation is accurate for large so long as is small. The field is quantized in the semiclassical approximation by decomposition into energy eigenstates in the required external electric field. In the vacuum state, each mode of the field has zero occupation number: non-zero occupation numbers lead to higher energies. States with non-zero occupation numbers should, however, contribute at a finite temperature. Finite temperature typically serves only to enhance the contribution of the lowest mode, and has no qualitative effect on vacuum polarization.

The results in Sections 3, 4 and 5 were all given for massless scalar particles. Results for massive particles are qualitatively similar: the presence of the mass typically reduces the vacuum polarization for a given external electric field. There is no adequate definition of a charged vector field in 1 + 1 dimensions with an external electric field. Spinor fields may be defined, but yield no vacuum polarization in the external field approximation. Extension of the results given here to more space dimensions is in principle possible. Scalar electrodynamics remains formally superrenormalizable in 2 + 1 dimensions, so that calculations analogous to those of this paper may in principle be performed without including renormalization. In 2 + 1 dimensions, non-trivial charged vector or Yang-Mills fields exist, and calculations of suitable gauge invariant quantities using the methods of this paper should explicitly exhibit antiscreening and asymptotic freedom.

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