On the 'Millenium' mass problem in quantum Yang-Mills Theory: an explanation to mathematicians

Ludwig D. Faddeev

The quantum Yang-Mills theory is the only realistic example of a quantum field theory on four dimensional space-time which has some chance to be defined in a mathematically correct way. In its classical formulation it is scale invariant and so does not contain any dimensional parameters. However it is believed that its quantum version has massive excitations. The Clay Millenium Problem consists in proving that statement. This requires to give a mathematical description of the Yang-Mills system and to investigate its spectrum.

 

In my talk I shall present the exposition of the problem and give some hints how the classical scale invariance can be broken by quantisation. The reason consists in the appearance of divergences in the quantum version, the regularisation of which requires the use of a dimensional parameter. I shall explain in some detail how this phenomenon, called "dimensional transmutation," appears in the process of renormalisation in the functional integral formalism. Since the latter object does not yet have a rigorous mathematical definition, my exposition will be only heuristic. However I hope that it will attract the interest of a variety of mathematically minded colleagues to the problem in question. I shall also present arguments showing why it was natural to include it into the list of the seven Clay Millenium Problems.