History of quantum field theory
The third thread in the development of quantum field theory was the need to handle the statistics of many-particle systems consistently and with ease. In 1927, Pascual Jordan tried to extend the canonical quantization of fields to the many-body wave functions of identical particles using a formalism which is known as statistical transformation theory; this procedure is now sometimes called second quantization. In 1928, Jordan and Eugene Wigner found that the quantum field describing electrons, or other fermions, had to be expanded using anti-commuting creation and annihilation operators due to the Pauli exclusion principle (see Jordan–Wigner transformation). This thread of development was incorporated into many-body theory and strongly influenced condensed matter physics and nuclear physics.
Constructive quantum field theory
The traditional basis of constructive quantum field theory is the set of Wightman axioms. Osterwalder and Schrader showed that there is an equivalent problem in mathematical probability theory. The examples with "d" < 4 satisfy the Wightman axioms as well as the Osterwalder-Schrader axioms. They also fall in the related framework introduced by Haag and Kastler, called algebraic quantum field theory. There is a firm belief in the physics community that the gauge theory of Yang and Mills can lead to a tractable theory, but new ideas and new methods will be required to actually establish this, and this could take many years.