The paper considers the possibility of constructing ultralocal theories, whose Hamiltonians contain no gradient terms and are therefore diagonal in position space, entirely in terms of currents with an equal time current algebra replacing the canonical commutation relations. It is shown that the free current theory can be defined in terms of a certain representation of the current algebra related to the group, S L(2, R). This representation is then constructed by using certain results of Araki and in the process a new infinitely divisible state on the universal covering group of SL(2, R) is displayed. An ultralocal free theory can also be constructed for the canonical fields, and its relation to the free current theory is shown to involve a certain renormalization procedure reminiscent of the thermodynamic limit.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics