Computational approaches to examine the vacuum polarization density

Based on computational quantum field theory and solutions to the Dirac equation, we show how the vacuum’s polarization density induced by strong external field can be calculated based on five independent methods. We compare these methods for a spatially reduced dimensional system and show that some approaches rely on energy renormalizations, energy cutoffs or periodic boundaries. We also illustrate the spatial implications of the breakdown of the linear superposition principle for extremely high intensity fields. A main focus of the article is to discuss future challenges for each approach that might motivate new theoretical and computational studies.