Constructive Quantum Field Theory

The construction of a relativistic quantum field is still an open problem for fields in spacetime dimension d ≥ 4. The conceptual difficulty that sometimes led to fear an incompatibility between nontrivial quantum systems and special relativity has however been solved in the case of dimension d = 2, 3 although, so far, has not influenced the corresponding debate on the foundations of quantum mechanics, still much alive. It began in the early 1960’s with Wightman’s work on the axioms and the attempts at understanding the mathematical aspects of renormalization theory and with Hepps’ renormalization theory for scalar fields. The breakthrough idea was, perhaps, Nelson’s realization that the problem could really be studied in Euclidean form. A solution in dimensions d = 2, 3 has been obtained in the 1960’s and 1970’s through a remarkable series of papers by Nelson, Glimm, Jaffe, Guerra. While the works of Nelson and Guerra relied on the “Euclidean approach” (see below) and on d = 2 the early works of Glimm and Jaffe dealt with d = 3 making use of the “Minkowskian approach” (based on second quantization) but making already use of a multiscale analysis technique. The latter received great impulsion and systematization by the adoption of Wilson’s views and methods on renormalization: in Physics terminology renormalization group methods; a point of view taken here following the Euclidean approach. The solution dealt initially with scalar fields but it has been subsequently considerably extended. The Euclidean approach studies quantum fields through the following problems