Quantum field theory is a vast theory in physics which has many deep connections to various fields of mathematics. Here we are interested in analytic aspects of quantum field theory typically represented by Wightman axioms.

In classical field theory, a field is some kind of function on a spacetime. In quantum mechanics, numbers are replaced with operators, so we consider operator-valued functions on a spacetime, but it turns out that we have to deal with something like a δ-function, so we consider operator-valued distributions on a spacetime. Such an operator-valued distribution on a spacetime is called a quantum field. From this viewpoint, one quantum field theory consists of a spacetime, its symmetry group and a family of operator-valued distributions on the spacetime. The Wightman axioms give a mathematical axiomatization of such objects.

We have another approach to quantum field theory based on operator algebras, which is called algebraic quantum field theory. We now explain this idea. The operatorvalued distributions are technically difficult to handle, since distributions are more difficult than functions and we have to deal with unbounded operators. In algebraic quantum field theory, we deal with only bounded linear operators. These operators appears as “observables”, so observables are more emphasized than states in this approach.