We examine the mathematical solutions of the Dirac equation to predict the spontaneous electron-positron pair creation from the vacuum. The Dirac equation contains a position and time-dependent scalar potential to approximate the effect of an external force on the vacuum. We focus on forces that are localized in space as well as in time and find that the resulting creation process is also localized in time but delocalized in space. This illustrates that the Dirac equation can show nonlocal behavior as it predicts that particles can be created even in spatial regions where the force is zero. We also examine the spatial distribution of the created particles and show that for spatially extended force fields it is proportional to the square of the position dependence of the force. But when the force field is narrower than the Compton wavelength, the created electron density approaches a universal shape invariant form that is independent of the strength of the force for sufficiently weak field strength.
Cheng, T; Bowen, S P.; Gerry, C C.; and Grobe, Rainer, "Locality in the creation of electron-positron pairs" (2008). Faculty publications – Physics. Paper 19.