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

TL;DR

This paper demonstrates that Majorana mass terms for Weyl fermions are only possible in three (modulo eight) spatial dimensions, linking space dimension, time-reversal symmetry, and neutrino properties.

Contribution

It establishes a dimension-specific condition for Majorana mass acquisition by Weyl fermions, connecting algebraic properties with physical implications.

Findings

Majorana masses only in 3 (mod 8) dimensions

Connection between time-reversal symmetry and space dimension

Implications for neutrino mass and lepton number violation

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.