Unveiling mapping structures of spinor duals

Abstract

Following the program of investigation of alternative spinor duals potentially applicable to fermions beyond the standard model, we demonstrate explicitly the existence of several well defined spinor duals. Going further we define a mapping structure among them and the conditions under which sets of such dual maps do form a group. We also study the covariance of bilinear quantities constructed with the several possible duals, the invariant eigenspaces of those group elements and its connections with spinors classification, as well as dual maps defined as elements of group algebras. Elko is an acronym of the German term "Eigenspinoren des Ladungskonjugationsoperators", which means "eigenspinor of the charge conjugation operator".

Polar form of spinor fields from regular to singular: the flag-dipoles

“…This adjoint, or dual, can be proven to be uniquely defined, up to a re-naming of all the bi-linear spinor quantities, if we assume that the dualization be done universally. However, one might drop this assumption and require that dualization be performed in a momentum-dependent way [5].…”

“…This development is general. Now is time to perform the classification of spinor fields, and for that we will closely follow the Lounesto classification, based on the bi-linear spinor fields [1][2][3][4][5][6][7][8][9][10]. Since all the bi-linear spinors are tensors, such a classification, based on vanishing these tensors, is manifestly generally covariant.…”

“…However, this leaves the door open for an altogether different type of spinors having both scalar and pseudo-scalar bi-linear quantities vanishing identically, called singular [1,2]. Singular spinor fields, or flag-dipole spinor fields, may be unusual but they still have many important things to tell [3][4][5][6][7][8][9][10]. In fact, these spinors can be further split into subclasses, obtained when the axial-vector bi-linear quantity, that is the spin, is zero, and in this case they are called flagpole spinors, or when the antisymmetric tensor bi-linear quantity, that is the momentum, is zero, and in this case they are called dipole spinors.…”

Flag-dipole spinors: On the dual structure derivation and $${\mathcal {C}}$$, $${\mathcal {P}}$$ and $${\mathcal {T}}$$ symmetries

“…Moreover, Γ(p µ ) has to be idempotent ensuring an invertible mapping. From (12) we are able to have the following two possibilities: h = h ′ , for which Γ(p µ ) = ½, has it is the case for the Dirac spinors, or h = h ′ leading to a more involved operator [6,24,[29][30][31].…”

“…Interesting enough, the contrasting dual structure carry an involved operator, which is responsible to ensure a Lorentz invariant and non-null norm besides carrying much of the physical information encoded on flag-dipole spinors. In such a way, given emergent operator just playing an important role when one deal with phenomenological applications [21,22], cosmological applications and mathematical physics analysis [6,[23][24][25]. Therefore, we highlight its matrix form in addition to exploring some of its main characteristics.…”