Bosonization and even Grassmann variables

We test a new approach to bosonization in relativistic field theories and many-body systems, whose purpose is to set up a perturbative scheme where the unperturbed action is the free action of the composites. The method is of practical relevance since the free propagators of the composites can be evaluated in a number of interesting cases. This is achieved by performing a generalized change of variables in the Berezin integral which defines the partition function of the system, whereby one assumes the composites as new integration variables. Still to be established, however, is a general procedure for deriving the free action of the composites starting from the one of the constituents. To shed light on this problem and to explore further features of the method we study a simplified version of the BCS model, whose spectrum consists of the excitations of the composite field associated to a Cooper pair. We are able to obtain the free action of this field, which displays a peculiar feature which we conjecture to characterize all the actions of quadratic fermionic composites, namely it does not contain a time derivative. Nevertheless it yields the right propagator, because, due to the properties of the integral over even elements of a Grassmann algebra, the propagator turns out not to be the inverse of the wave operator.