Fermions and bosons interacting with arbitrarily strong external fields

The question, "What happens to the electron orbitals as the charge of the nucleus is increased without bounds?" has inspired much of the interest in the description of particles bound strongly by external fields. Interest in this problem and in the related Klein paradox extends back nearly to the beginnings of relativistic quantum mechanics. However, the correct interpretation of the theory for overcritical potentials, where the parts of the complete set of single particle solutions associated with particles and antiparticles are no longer distinct, was given only recently. The understanding of the spectrum of the Dirac and Klein-Gordon equations is essential in order to obtain an appropriate physical description with quantum field theory. The strong binding by more than twice the rest mass of the particles in overcritical external potentials leads to qualitatively new effects. In the case of fermions we find spontaneous positron emission accompanied by creation of a charged lowest energy state, i.e. a charged vacuum. The number of positrons produced spontaneously is limited by the Pauli exclusion principle. For bosons we find that depending on the character of the external potential, either neutral or charged Bose condensates develop. While the questions associated with the meson fields seem academic at the moment, the effects attributed to the fermion field stand a good chance of being tested in an experiment in the near future. It is expected that in heavy ion collisions such as uranium on uranium near the Coulomb barrier overcritical electromagnetic fields will be created.