Examples: Explicitly Soluble Two-Dimensional Models

Abstract

The two-dimensional free scalar massless field ø(x) is of interest for several reasons. Because of the infrared singularities of the Wightman functions, it goes beyond the scope of the Wightman formalism (and requires a modification of the Wightman formalism by allowing an indefinite metric in the non-physical representation, and by replacing the ordinary vacuum vector by a generalized one in the physical representation). The model enables us to illustrate the importance of non-local quantities, since there are non-local quantities among the physically interesting ones; here by non-local quantities we mean those not generated by the algebra of local quantities (for example, the operators of creation of states with non-zero “topological” charge). Finally, this model lies at the basis of a number of other explicitly soluble models.