Sixfold Discrete Symmetry of Fermion Fields as Explanation for Dark Matter

We address the question, what is dark matter? The method used is the Pauli algebra form of the Dirac equation, equivalent to the standard one but allowing to use the multiplicative structure of the algebra. In this form discrete symmetries of fermion fields are the same as the automorphisms of the Pauli group. We construct the prototype Dirac equation in the Clifford algebra and then use its six representations by complex two by two matrices to construct the six symmetric versions, indexed by permutations of three letters. The solutions of symmetric equations form the six sectors of fermion fields. It is shown that the sectors are genuinely distinct, by proving that any fermion field belonging to two different sectors must have mass zero. Also shown is the lack of electromagnetic interaction between one sector and another, since each sector has its own matrix coupling the fermion field to the electromagnetic field. The key tool used is the mass inversion symmetry, introduced in []. The sixfold symmetry predicts the ratio of dark to ordinary matter of 5:1 which is close to the observed ratio of 5.2:1. However this symmetry is constructed only for interactions between fermion fields and the electromagnetic field, not yet taking into account the weak and strong interactions. So this article is an indication that maybe the complete answer can be found if the sixfold symmetry extends to these interactions.