Non-orientable foam and an effective planck mass for point-like fermions

Abstract

A path-integral formalism is used to describe the propagation of fermions on spacetime foam in which virtual non-orientable handles are present. Because 2-component spinors cannot be defined on a non-orientable background, non-orientability is incompatible with chiral symmetry. Every fermion must have a partner (mirror fermion) with opposite chirality, that couples in the same way to the gauge bosons. Fermions appear to acquire an effective mass of the order of the Planck mass from propagation through non-orientable factors of the topologyy (a mechanism first suggested by Zel'dovich). A lower limit on the intrinsic size of leptons and quarks is apparently required to avoid excessively large masses. If one includes in the path integral a sum over all spinor structures, the effective mass arising from this mechanism is suppressed, but mirror fermions are still required. If one replaces the usual definition of parity by CP and assumes that only CP-reversing handles occur, and if, in addition, one sums over spinor structures, one can avoid both mirror fermions and a Planck-size effective mass for point fermions. However, in the context of a Kaluza-Klein theory, the restriction to CP-reversing handles is unnatural: both CP- and P-reversing handles can occur as vacuum fluctuations.