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      ELKO Spinor Fields: Lagrangians for Gravity derived from Supergravity

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          Abstract

          Dual-helicity eigenspinors of the charge conjugation operator (ELKO spinor fields) belong -- together with Majorana spinor fields -- to a wider class of spinor fields, the so-called flagpole spinor fields, corresponding to the class-(5), according to Lounesto spinor field classification based on the relations and values taken by their associated bilinear covariants. There exists only six such disjoint classes: the first three corresponding to Dirac spinor fields, and the other three respectively corresponding to flagpole, flag-dipole and Weyl spinor fields. Using the mapping from ELKO spinor fields to the three classes Dirac spinor fields, it is shown that the Einstein-Hilbert, the Einstein-Palatini, and the Holst actions can be derived from the Quadratic Spinor Lagrangian (QSL), as the prime Lagrangian for supergravity. The Holst action is related to the Ashtekar's quantum gravity formulation. To each one of these classes, there corresponds a unique kind of action for a covariant gravity theory. Furthermore we consider the necessary and sufficient conditions to map Dirac spinor fields (DSFs) to ELKO, in order to naturally extend the Standard Model to spinor fields possessing mass dimension one. As ELKO is a prime candidate to describe dark matter and can be obtained from the DSFs, via a mapping explicitly constructed that does not preserve spinor field classes, we prove that in particular the Einstein-Hilbert, Einstein-Palatini, and Holst actions can be derived from the QSL, as a fundamental Lagrangian for supergravity, via ELKO spinor fields. The geometric meaning of the mass dimension-transmuting operator - leading ELKO Lagrangian into the Dirac Lagrangian - is also pointed out, together with its relationship to the instanton Hopf fibration.

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          Most cited references 24

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            Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action

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            Barbero recently suggested a modification of Ashtekar's choice of canonical variables for general relativity. Although leading to a more complicated Hamiltonian constraint this modified version, in which the configuration variable still is a connection, has the advantage of being real. In this article we derive Barbero's Hamiltonian formulation from an action, which can be considered as a generalization of the ordinary Hilbert-Palatini action.
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                Author and article information

                Journal
                07 January 2009
                10.1142/S0219887809003618
                0901.0883

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                Custom metadata
                15A66, 81Q05
                Int.J.Geom.Meth.Mod.Phys.6:461-477, 2009
                11 pages, RevTeX, accepted for publication in Int.J.Geom.Meth.Mod.Phys. (2009)
                math-ph gr-qc hep-th math.MP

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