Algebraic Properties of BRST Coupled Doublets

We characterize the dependence on doublets of the cohomology of an arbitrary nilpotent differential s (including BRST differentials and classical linearized Slavnov-Taylor (ST) operators) in terms of the cohomology of the doublets-independent component of s. All cohomologies are computed in the space of local integrated formal power series. We drop the usual assumption that the counting operator for the doublets commutes with s (decoupled doublets) and discuss the general case where the counting operator does not commute with s (coupled doublets). The results are purely algebraic and do not rely on power-counting arguments.Comment: Some explanations enlarged, references adde

An Approach to the Equivalence Theorem by the Slavnov-Taylor Identities

We discuss the Equivalence Theorem (ET) in the BRST formalism. The existence of a local inverse of the field transformation (at least as a formal power expansion) suggests a formulation of the ET, which allows a nilpotent BRST symmetry. This strategy cannot be implemented at the quantum level if the inverse is non-local. In this case we propose an alternative formulation of the ET, where, by using Faddeev-Popov fields, this difficulty is circumvented. We study the quantum deformation of the associated ST identity, which turns out to be anomaly free, and show that a selected set of Green functions, which in some cases can be identified with the physical observables of the model, does not depend on the choice of the transformation of the fields. In general the transformation of the fields yields a non-renormalizable theory. When the equivalence is established between a renormalizable and a non-renormalizable theory, the ET provides a way to give a meaning to the last one by using the resulting ST identity. In this case the Quantum Action Principle cannot be of any help in the discussion of the ET. We assume and discuss the validity of a Quasi Classical Action Principle, which turns out to be sufficient for the present work. As an example we study the renormalizability and unitarity of massive QED in Proca's gauge by starting from a linear Lorentz-covariant gauge.Comment: 26 page

Endowing the Nonlinear Sigma Model with a Flat Connection Structure: a Way to Renormalization

We discuss the quantized theory of a pure-gauge non-abelian vector field (flat connection) as it would appear in a mass term a` la Stueckelberg. However the paper is limited to the case where only the flat connection is present (no field strength term). The perturbative solution is constructed by using only the functional equations and by expanding in the number of loops. In particular we do not use a perturbative approach based on the path integral or on a canonical quantization. It is shown that there is no solution with trivial S-matrix. Then the model is embedded in a nonlinear sigma model. The solution is constructed by exploiting a natural hierarchy in the functional equations given by the number of insertions of the flat connection and of the constrained component of the sigma field. The amplitudes with the sigma field are simply derived from those of the flat connection and of the constraint component. Unitarity is enforced by hand by using Feynman rules. We demonstrate the remarkable fact that in generic dimensions the naive Feynman rules yield amplitudes that satisfy the functional equations. This allows a dimensional renormalization of the theory in D=4 by recursive subtractions of the poles in the Laurent expansion. Thus one gets a finite theory depending only on two parameters. The novelty of the paper is the use of the functional equation associated to the local left multiplication introduced by Faddeev and Slavnov, here improved by adding the external source coupled to the constrained component. It gives a powerful tool to renormalize the nonlinear sigma model.Comment: 42 pages, 7 figures, Latex; improved presentation of the subtraction procedur