(no subject)

OK, I think I finally figured out how to show that the algebra of local observables in a free field theory is a von Neumann algebra of Type III. Or at least, why it is not Type I. Everything boils down to checking that a certain integral operator on the space of functions in a bounded region is not trace-class, which in turn is due to short-distance singularities in the correlators of fields across the boundary. As a by product, I learned some representation theory for canonical commutation relations (CCR) due to Araki.