Abstract
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 {mathcal {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|ne 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.
Highlights
In the Standard Model (SM), the Dirac and Weyl spinors and their quantum fields have both played important roles in describing the dynamics of the fermions as well as elucidating the structure of the model
We find that the flagdipole spinors in the helicity basis which give rise to a physical fermionic field, is related to Elko given in Ref. [10] by a matrix transformation1
It is natural to ask whether the remaining spinors have any applications in particle physics
Summary
In the Standard Model (SM), the Dirac and Weyl spinors and their quantum fields have both played important roles in describing the dynamics of the fermions as well as elucidating the structure of the model. There remains many outstanding questions which led to the common consensus that the model is incomplete. In the chosen basis given by Eq (2), the Dirac spinors that are associated with the fermionic field belong to the 2nd class. Lounesto identified the 1st-3rd classes to be the Dirac spinors. The 4th and 5th classes are known as the flag-dipole and flag-pole spinors respectively. The 6th class is the Weyl spinors. From the Lounesto classification, a natural question arises –What are the quantum field theories associated with the
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