MORE ON THE SUBTRACTION ALGORITHM 1 Damiano Anselmi

We go on in the program of investigating the removal of divergences of a generical quantum gauge field theory, in the context of the Batalin-Vilkovisky formalism. We extend to open gauge-algebræ a recently formulated algorithm, based on redefinitions δλ of the parameters λ of the classical Lagrangian and canonical transformations. The key point is to generalize a well-known conjecture on the form of the divergent terms to the case of open gauge-algebræ. We also show that it is possible to reach a complete control on the effects of the subtraction algorithm on the space Mgf of the gauge-fixing parameters. We develop a differential calculus on Mgf providing an intuitive geometrical description of the fact the on shell physical amplitudes cannot depend on Mgf . A principal fiber bundle E → Mgf with a connection ω1 is defined, such that the canonical transformations are gauge transformations for ω1. A geometrical description of the effect of the subtraction algorithm on the space Mph of the physical parameters λ is also proposed. At the end, the full subtraction algorithm can be described as a series of diffeomorphisms on Mph, orthogonal to Mgf (under which the action transforms as a scalar), and gauge transformations on E . In this geometrical context, a suitable concept of predictivity is formulated. Finally, we give some examples of (unphysical) toy models that satisfy this requirement, though being neither power counting renormalizable, nor finite. Partially supported by EEC, Science Project SC1∗-CT92-0789.