An Alternative Dimensional Reduction Prescription

We propose an alternative dimensional reduction prescription which in respect with Green functions corresponds to drop the extra spatial coordinate. From this, we construct the dimensionally reduced Lagrangians both for scalars and fermions, discussing bosonization and supersymmetry in the particular 2-dimensional case. We argue that our proposal is in some situations more physical in the sense that it mantains the form of the interactions between particles thus preserving the dynamics correspondig to the higher dimensional space. CONICET, Argentina Investigador CICBA, Argentina Dimensional reduction is a well-honoured procedure to build theories in a given number of space-time dimensions starting from higher dimensional theories. Initiated with the Kaluza-Klein proposal for unifying electromagnetic and gravitational forces, there are many fields where it can be fruitfully exploited. Let us mention for example its wide application in Supersymmetry and Supergravity for constructing extended supersymmetric theories, studying spontaneous breaking of supersymmetry, etc [1]. Also in condensed matter problems where charged particles are constrained to move on a plane, the connection between the models in the higher (3 + 1) and lower (2 + 1) dimensional spaces is of relevance [2]-[3]. The dimensional reduction procedure, as it is usually applied, consists in dropping out extra coordinates at the Lagrangian level, this amounting to mantain unchanged the differential (kinetic energy) operator appearing in the Lagrangian. As a result, the Green function in momentum space preserves its form but in configuration space drastically changes. For example, if in the higher dimensional space the kinetic energy operator is a D’Alembertian, the same operator will appear in the lower dimensional theory. Evidently, the D’Alembertian Green function is different for different number of spacetime dimensions. This, in a sense to be clarified below, implies that the dynamics of the interacting particles described by the dimensionally reduced Lagrangian is changed in configuration space. The statement above can be clarified with an example which is relevant for the study of the Quantum Hall effect : electrodynamics of planar systems (see for example [2]). If one decides that electrons compelled to move in a plane are to be described by a (2 + 1) gauge theory coupled to matter then, the resulting Coulomb potential is logarithmic instead of the 1/r potential to which electrons are actually subject even if they move on a plane. In passing from 3 to 2 spatial dimensions dynamics has changed. The opposite attitude for implementing a dimensional reduction can also be thought of. Indeed, one can preserve the form of the Green function in configuration space dropping its extra space coordinate dependence. In the case of electrons in the plane, their interaction will then still be 1/r (with r depending only on planar coordinates). In this way one keeps the dynamics of the higher dimensional space. As we shall see below, for the case of planar electrons the Coulomb potential restricted to the plane, 1/ √