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We construct a mass dimension one fermionic field associated with flag-dipolespinors. These spinors are related to Elko (flag-pole spinors) by aone-parameter matrix transformation $\mathcal{Z}(z)$ where $z$ is a complexnumber. The theory is non-local and non-covariant. While it is possible toobtain a Lorentz-invariant theory via $\tau$-deformation, we choose to studythe effects of non-locality and non-covariance. Our motivation for doing so isexplained. We show that a fermionic field with $|z|\neq1$ and $|z|=1$ arephysically equivalent. But for fermionic fields with more than one value of$z$, their interactions are $z$-dependent thus introducing an additionalfermionic degeneracy that is absent in the Lorentz-invariant theory. We studythe fermionic self-interaction and the local $U(1)$ interaction. In theprocess, we obtained non-local contributions for fermionic self-interactionthat have previously been neglected. For the local $U(1)$ theory, theinteractions contain time derivatives that renders the interacting densitynon-commutative at space-like separation. We show that this problem can beresolved by working in the temporal gauge. This issue is also discussed in thecontext of gravity.

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