Bosonization and even Grassmann Variables

We test a new approach to bosonization in relativistic field theories and many-body systems, based on the use of fermionic composites as integration variables in the Berezin integral defining the partition function of the system. The method appears promising since at zeroth order it correctly describes the propagators of the composites, which can be evaluated in a number of significant cases. Still to be established 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. In this simple case the action of the composites can indeed be obtained: whether this result can be generalized it remains hewever to be seen. Yet an interesting property of the wave operators appearing in the free actions of bilinear composites already emerges from the simple problem we have treated: here the wave operators do not describe any time evolution, even though they generate the right propagators. This outcome relates to the basic properties of the integrals over the even elements of a Grassmann algebra where the composites live, which entails that the propagators are no longer the inverse of the wave operators.