Quantum Mechanics, the Stability of Matter, and Quantum

Ordinary matter is held together with electromagnetic forces, and the dynamical laws governing the constituents (electrons and nuclei) are those of quantum mechanics. These laws, found in the beginning of last century, were able to account for the fact that electrons do not fall into the nuclei and thus atoms are quite robust. It was only in 1967 that Dyson and Lenard were able to show that matter in bulk was also stable and that two stones had a volume twice that of one stone. Simple as this may sound, the conclusion is not at all obvious and hangs by a thread-- namely Pauli's 'exclusion principle' (which states that two electrons cannot be in the same state). In the ensuing 3 decades much was accomplished to clarify, simplify and extend this result. We now understand that matter can, indeed, be unstable when relativistic effects and magnetic fields are taken into account -- unless the electron's charge is small enough (which it is, fortunately). These delicate and non-intuitive conclusions will be summarized. The requisite mathematical apparatus needed for these results is itself interesting. Finally, we can now hope to begin an analysis of the half-century old question about the ultimate theory of ordinary matter, called quantum electrodynamics (QED). This is an experimentally successful theory, but one without a decent mathematical foundation. Some recent, preliminary steps to resolve the problems of QED will be presented.