Fermionic degeneracy and non-local contributions in flag-dipole spinors and mass dimension one fermions

We construct a mass dimension one fermionic field associated with flag-dipole spinors. These spinors are related to Elko (flag-pole spinors) by a one-parameter matrix transformation Z(z) where z is a complex number. The theory is non-local and non-covariant. While it is possible to obtain a Lorentz-invariant theory via tau -deformation, we choose to study the effects of non-locality and non-covariance. Our motivation for doing so is explained. We show that a fermionic field with |z|not equal 1 and |z|=1 are physically equivalent. But for fermionic fields with more than one value of z, their interactions are z-dependent thus introducing an additional fermionic degeneracy that is absent in the Lorentz-invariant theory. We study the fermionic self-interaction and the local U(1) interaction. In the process, we obtained non-local contributions for fermionic self-interaction that have previously been neglected. For the local U(1) theory, the interactions contain time derivatives that renders the interacting density non-commutative at space-like separation. We show that this problem can be resolved by working in the temporal gauge. This issue is also discussed in the context of gravity.

Automated summary

The research focuses on a non-local and non-covariant theory of fermionic fields associated with flag-pole spinors, utilizing a matrix transformation and varying a specific parameter z. It is found that fermionic fields are physically equivalent at |z| not equal to 1 and |z| = 1, but exhibit additional fermionic degeneracy for multiple values of z. The study explores fermionic self-interaction and local U(1) interaction, revealing non-local contributions and non-commutative interaction density at space-like separation.