Triviality of the 4D Ising Model
(Quantum) Field theory is a powerful tool in modern physics enabling one to perform many computations. For decades, mathematicians asked themselves whether fields involved in these theories can be rigorously defined, for instance in the famous case of Yang-Mills, and for a simpler example of phi4.
Since the work of Wightman in the sixties, mathematicians have lunched a program, entitled Constructive (quantum) Field Theory whose goal was to construct the phi4 theory rigorously. After successes by Glimm and Jaffe in dimensions 2 and 3, the program took a big hit with the works of Aizenman and Frohlich in 1982 proving that constructions of phi4 in dimension 5 (and more) based on the Ising model and the lattice phi4 model were necessary leading to a Gaussian theory, considered by physicists as a trivial theory. This being said, a certain mystery surrounded the case of dimension 4 (which is the most interesting physical dimension corresponding to 3 spatial and one time dimension), leaving open the question of whether the phi4 theory in 4D is Gaussian.
The goal of this talk is to present a recent result obtained with Michael Aizenman proving the equivalent statement of the famous Aizenman-Frohlich result in 4D. The talk will start with a historical and motivating part, before going to the description of the result and the probabilistic interpretation enabling one to understand it. No background is necessary.
Professor Hugo Duminil-Copin, U. Geneva / IHES