A hint towards mass dimension one Flag-dipole spinors

Abstract

In this report we advance in exploring further details concerning the formal aspects of the construction of a Flag-dipole spinor. We report a (re-)definition of the dual structure which provide a Lorentz invariant and non-null norm, ensuring a local theory. With the new dual structure at hands, we look towards define relevant physical amounts, e.g., spin sums and quantum field operator. As we will see, the Flag-dipole and the Elko's theory are quite familar. In this vein, it is possible, via a matrix transformation, to write Flag-dipole spinors in terms of Elko spinor, evincing that both spinors are physically related and some physical amounts may be stated as equivalent.

Constraints on mapping the Lounesto’s classes

“…The distinction among the aforementioned spinors is present on the value of the bilinear form K. Such bilinear form, in general grounds, depends on the subtraction between the modulo of the phases, thus, if |α| 2 = |β| 2 it leads to a spinor to belong to class 5. On the other hand, for a spinor carrying |α| 2 = |β| 2 , it provides K = 0, leading, then, to belong to class 4, as it can easily been checked for the spinors in Refs [4,5,29].…”

On the generalized spinor classification: beyond the Lounesto’s classification

“…Furthermore, ( p) has to be idempotent, 2 ( p) = 1, ensuring an invertible mapping [22]. Thus, we have the following possibilities: h = h , for which ( p) = 1 and stands for the Dirac usual case, ( p) = 1 stands for the non-standard Dirac adjoint [21], and finally h = h leading to a more involved operator present in the mass-dimensionone theory [3] and also in flag-dipole theory [23]. Thus, the purpose of the present paper is to invoke a mathematical procedure for determining the spinorial dual structure based on the general form ∼ ψ= [ G ( p)ψ] † γ 0 , by analysing the bilinear forms and the related FPK identities.…”

“…A natural path to classify spinors resides on the Lounesto's classification. Such classification is built up taking into account the 16 bilinear forms, encompassing Dirac spinors, Flag-dipole spinors [23,33], Majorana spinors (neutrino), and Weyl spinors (massless neutrino) [9]. This specific classification is based on geometric FPK identities, given in (6) and (8), and displays exclusively six disjoint classes of spinors.…”

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

“…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.…”