Arthur Jaffe

"One should mention right at the start that one still does not understand whether quantum mechanics and special relativity are compatible at a fundamental level in our Minkowski four-space world. One generally assumes that this means finding a complete Yang-Mills gauge theory or the interaction of gauge fields with fermionic matter fields, the simplest form being quantum chromodynamics (QCD). Associated with this picture is the belief that the fundamental vector meson excitations are massive (as opposed to photons, which arise in the limiting case of an abelian gauge symmetry. The proof of the existence of a “mass gap” appears a necessary integral part of solving the entire puzzle. This question remains one of the deepest open issues in theoretical physics, as well as in mathematics. Basically the question remains: can one give a mathematical foundation to the theory of fields in four-dimensions? In other words, can do quantum mechanics and special relativity lie on the same footing as the classical physics of Newton, Maxwell, Einstein, or Schrödinger—all of which fits into a mathematical framework that we describe as the language of physics. This glaring gap in our fundamental knowledge even dwarfs questions of whether there are other more complicated and sophisticated approaches to physics—those that incorporate gravity, strings, or branes—for understanding their fundamental significance lies far in the future. In fact, one believes that stringy proposals, if they can be fully implemented, have limiting cases that appear as relativistic quantum fields, just as relativistic quantum fields describe non-relativistic quantum theory and classical physics in various limiting cases."