Fermion mass gap in the loop representation of quantum gravity

Abstract

An essential step towards the identification of a fermion mass generation mechanism at Planck scale is to analyse massive fermions in a given quantum gravity framework. In this letter the two mass terms entering the Hamiltonian constraint for the Einstein-Majorana system are studied in the loop representation of quantum gravity and fermions. One resembles a bare mass gap because it is not zero for states with zero (fermion) kinetic energy as opposite to the other that is interpreted as `dressing' the mass. The former contribution originates from (at least) triple intersections of the loop states acted on whilst the latter is traced back to every couple of coinciding end points, where fermions sit. Thus, fermion mass terms get encoded in the combinatorics of loop states. At last the possibility is discussed of relating fermion masses to the topology of space.