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The standard approximations of the Dyson-Schwinger equation lead to complex singularities of the fermion propagator. In three-dimensional QED one can show that this phenomenon might be related to confinement: a confining potential leads to masslike singularities at complex momenta, and thus to the absence of a mass singularity on the real timelike axis. The correct treatment of the vacuum polarization is essential for the confining nature of three-dimensional QED.
Pieter Maris (Wed,) studied this question.
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