The electron goes forward and backward in time and interacts with itself. (In quantum field theory you can replace a positron with an electron going backwards in time). Regrettably the theory does not work, since it would predict the same number of electrons and positrons in the universe.
Feynman later proposed this interpretation of the positron as an electron moving backward in time in his 1949 paper "The Theory of Positrons".[2] Yoichiro Nambu later applied it to all production and annihilation of particle-antiparticle pairs, stating that "the eventual creation and annihilation of pairs that may occur now and then is no creation or annihilation, but only a change of direction of moving particles, from past to future, or from future to past."[3]
Physics is sufficiently weird that I can't tell if this is just spitballing or an actual theory I haven't heard of before.
Edit: holy crap, these people think nucleons are made of muons that are made of electrons and positrons: http://wlsprojects.com/structure-inside-proton.html . Solves the matter vs antimatter problem, I guess.
