Majorana mass, time reversal symmetry, and the dimension of space

Abstract

The Weyl fermions with a well defined chirality are known to demand that the dimension of space which they inhabit must be odd. It is shown here, however, that not all odd dimensional spaces are equally good hosts: in particular, an arbitrary number of chiral Weyl fermions can acquire a Majorana mass only in three (modulo eight) dimensions. The argument utilizes a) the precise analogy that exists between the Majorana mass term and the Cooper pairing of time-reversed Weyl fermions, and b) the conditions on the requisite time-reversal operator, which are implied by the Clifford algebra. The theorem connects the observed odd number of neutrino flavors, the time reversal symmetry, and the dimension of our space, and strengthens the argument for the possible violation of the lepton number conservation law.

Comments: seven pages, slightly revised and restructured version; new section on connection to reality properties of spinor representations; typos corrected, new appendix on Majorana-Weyl fermions in D=1 (modulo eight); close to published version