Abstract
In this paper, we explore the field propagator with a structure that is general enough to encompas both the case of newly-defined mass-dimension 1 fermions and spin-1/2 bosons. The method we employ is to define a map between spinors of different Lounesto classes, and then write the propagator in terms of the corresponding dual structures.
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Notes
It is important to remark, if one take into account a more specific framework, like the adjoints defined in [1, 9], thus, we may set the following relations
$$\begin{aligned} {\mathop {\lambda }\limits ^{\lnot }} = [\Xi _{\lambda }\lambda ]^{\dag }\eta _0{\mathcal {O}}_{\lambda } \quad \text{ and }\quad {\mathop {\psi }\limits ^{\lnot }} = [\Xi _{\psi }\psi ]^{\dag }\eta _0{\mathcal {O}}_{\psi }, \end{aligned}$$(31)in which the \({\mathcal {O}}\) operator holds important physical information regarding the spin-sums. Thus, a more involved amplitude of propagation may emerge as
$$\begin{aligned} {\mathcal {A}}_{{\textbf{M}},\Xi }(x-x^{\prime })= & {} [{\textbf{M}}{\mathcal {A}}(x-x^{\prime }){\mathcal {O}}^{-1}_{\lambda }\eta _0\Xi _{\lambda }^{\dag }{\textbf{M}}^{\dag }\Xi _{\psi }^{\dag }\eta _0{\mathcal {O}}_{\psi }]_{_{particle}}\nonumber \\{} & {} \pm [{\textbf{M}}{\mathcal {A}}(x-x^{\prime }){\mathcal {O}}^{-1}_{\lambda }\eta _0\Xi _{\lambda }^{\dag }{\textbf{M}}^{\dag }\Xi _{\psi }^{\dag }\eta _0{\mathcal {O}}_{\psi }]_{_{{anti}{\text {-}}{particle}}}. \end{aligned}$$(32)However, for the purposes of this work, we will not investigate this case further.
Where the sub-index Ah-7 means Ahluwalia class-7 spinors.
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Appendix A: Some Results Based on New Dual Definitions
Appendix A: Some Results Based on New Dual Definitions
Quite recently, some new definitions of dual structure opened windows to a new interpretation of how spin-half bosons emerge (from Dirac spinors) and also how they evade the spin-statistic theorem [5]. As claimed, the new dual structure furnishes a local and Lorentz-invariant theory and also provide a positive-definite Hamiltonian. Such aforementioned features are carried by a new dual structure, which reads
in which \(s=1\) stands for fermionic field and \(s=-1\) stands for bosonic field. Bearing in mind the dual definition in (A1), the spin sums (8) and (9) are now replaced by the following set
With these new results at hands, Eq.(34) is written, accordingly [5, page3], as
According to the type of field, the statistic will be dictated by the s parameter. If the field is to be fermionic, \(s=1\), forcing the anticommutative relations among the creator and annihilator operators. If the field is bosonic, \(s=-1\), and commutative relations must be taken into account. For more details, we refer to [5].
Now, following the very same steps as in Sect.(3.1), and bearing in mind the correct relation among the creator/annihilator operators as well as the value of s, the above amplitude of propagation yields
as expected for Dirac fermions and also for spin-half Therefore, we show the similarity of both results presented in [3] and [5].
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Rogerio, R.J.B., Fabbri, L. Propagators Beyond The Standard Model. Adv. Appl. Clifford Algebras 33, 39 (2023). https://doi.org/10.1007/s00006-023-01287-7
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DOI: https://doi.org/10.1007/s00006-023-01287-7