Abstract
Pinor and spinor fields are sections of the subbundles whose fibers are the representation spaces of the Clifford algebra of the forms, equipped with the Graf product. In this context, pinors and spinors are here considered and the geometric generalized Fierz identities provide the necessary framework to derive and construct new spinor classes on the space of smooth sections of the exterior bundle, endowed with the Graf product, for prominent specific signatures, whose applications are discussed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Vaz Jr. and R. da Rocha, An Introduction to Clifford Algebras and Spinors, Oxford University Press, Oxford, U.K., (2016).
L. Bonora, F.F. Ruffino and R. Savelli, Revisiting pinors, spinors and orientability, arXiv:0907.4334 [INSPIRE].
P. Lounesto, Clifford Algebras and Spinors, Cambridge University Press, (2002).
L.S. Randriamihamison, Identites de Fierz et formes bilineaires dans les espaces spinoriels, J. Geom. Phys. 10 (1992) 19.
C.I. Lazaroiu, E.M. Babalic and I.A. Coman, The geometric algebra of Fierz identities in arbitrary dimensions and signatures, JHEP 09 (2013) 156 [arXiv:1304.4403] [INSPIRE].
L. Fabbri, Least-order torsion-gravity for Dirac fields and their non-linearity terms, Gen. Rel. Grav. 47 (2015) 1837 [arXiv:1405.5129] [INSPIRE].
L. Fabbri, A generally-relativistic gauge classification of the Dirac fields, Int. J. Geom. Meth. Mod. Phys. 13 (2016) 1650078 [arXiv:1603.02554] [INSPIRE].
L. Fabbri, S. Vignolo and S. Carloni, Renormalizability of the Dirac equation in torsion gravity with nonminimal coupling, Phys. Rev. D 90 (2014) 024012 [arXiv:1404.5784] [INSPIRE].
S. Vignolo, L. Fabbri and R. Cianci, Dirac spinors in Bianchi-I f(R)-cosmology with torsion, J. Math. Phys. 52 (2011) 112502 [arXiv:1106.0414] [INSPIRE].
J.M. Hoff da Silva and R. da Rocha, Unfolding Physics from the Algebraic Classification of Spinor Fields, Phys. Lett. B 718 (2013) 1519 [arXiv:1212.2406] [INSPIRE].
R.J. Bueno Rogerio, J.M. Hoff da Silva, M. Dias and S.H. Pereira, Effective lagrangian for a mass dimension one fermionic field in curved spacetime, JHEP 02 (2018) 145 [arXiv:1709.08707] [INSPIRE].
J.P. Crawford, On the algebra of Dirac bispinor densities: factorization and inversion theorems, J. Math. Phys. 26 (1985) 1439 [INSPIRE].
R.T. Cavalcanti, Classification of Singular Spinor Fields and Other Mass Dimension One Fermions, Int. J. Mod. Phys. D 23 (2014) 1444002 [arXiv:1408.0720] [INSPIRE].
L. Bonora, K.P.S. de Brito and R. da Rocha, Spinor Fields Classification in Arbitrary Dimensions and New Classes of Spinor Fields on 7-Manifolds, JHEP 02 (2015) 069 [arXiv:1411.1590] [INSPIRE].
L. Bonora and R. da Rocha, New Spinor Fields on Lorentzian 7-Manifolds, JHEP 01 (2016) 133 [arXiv:1508.01357] [INSPIRE].
K.P.S. de Brito and R. da Rocha, New fermions in the bulk, J. Phys. A 49 (2016) 415403 [arXiv:1609.06495] [INSPIRE].
J. Fröhlich and P.A. Marchetti, Quantum Field Theories of Vortices and Anyons, Commun. Math. Phys. 121 (1989) 177 [INSPIRE].
G. Grignani, M. Plyushchay and P. Sodano, A pseudoclassical model for P, T invariant planar fermions, Nucl. Phys. B 464 (1996) 189 [hep-th/9511072] [INSPIRE].
S.P. Gavrilov, D.M. Gitman and N. Yokomizo, Dirac fermions in strong electric field and quantum transport in graphene, Phys. Rev. D 86 (2012) 125022 [arXiv:1207.1749] [INSPIRE].
J. Gonzalez and J. Herrero, Graphene wormholes: A condensed matter illustration of Dirac fermions in curved space, Nucl. Phys. B 825 (2010) 426 [arXiv:0909.3057] [INSPIRE].
C. Dutreix, M. Guigou, D. Chevallier and C. Bena, Majorana Fermions in Graphene and Graphene-Like Materials, Eur. Phys. J. B 87 (2014) 296 [arXiv:1309.1143] [INSPIRE].
W.M. Mendes, G. Alencar and R.R. Landim, Spinors Fields in Co-dimension One Braneworlds, JHEP 02 (2018) 018 [arXiv:1712.02590] [INSPIRE].
C.-I. Lazaroiu, E.-M. Babalic and I.-A. Coman, Geometric algebra techniques in flux compactifications, Adv. High Energy Phys. 2016 (2016) 7292534 [arXiv:1212.6766] [INSPIRE].
R. Lopes and R. da Rocha, The Graf product: a Clifford structure framework on the exterior bundle, Adv. Appl. Clifford Alg. 28 (2018) 57 [arXiv:1712.02737].
T. Houri, D. Kubiznák, C. Warnick and Y. Yasui, Symmetries of the Dirac Operator with Skew-Symmetric Torsion, Class. Quant. Grav. 27 (2010) 185019 [arXiv:1002.3616] [INSPIRE].
C.A. Linhares and J.A. Mignaco, SU(4) for the Dirac equation, Phys. Lett. B 153 (1985) 82 [INSPIRE].
W. Graf, Differential forms as spinors, Annales de l’I.H.P. Physique théorique 29 (1978) 85 https://eudml.org/doc/75997.
M. Cariglia, P. Krtous and D. Kubiznák, Commuting symmetry operators of the Dirac equation, Killing-Yano and Schouten-Nijenhuis brackets, Phys. Rev. D 84 (2011) 024004 [arXiv:1102.4501] [INSPIRE].
A. Van Proeyen, Tools for supersymmetry, Ann. U. Craiova Phys. 9 (1999) 1 [hep-th/9910030] [INSPIRE].
S. Okubo, Real representations of finite Clifford algebras. 1. Classification, J. Math. Phys. 32 (1991) 1657 [INSPIRE].
S. Okubo, Representation of Clifford algebras and its applications, Math. Jap. 41 (1995) 59 [hep-th/9408165] [INSPIRE].
D.V. Alekseevsky, V. Cortes, C. Devchand and A. Van Proeyen, Polyvector superPoincaré algebras, Commun. Math. Phys. 253 (2004) 385 [hep-th/0311107] [INSPIRE].
H.L. Carrion, M. Rojas and F. Toppan, Quaternionic and octonionic spinors: A classification, JHEP 04 (2003) 040 [hep-th/0302113] [INSPIRE].
R. Abłamowicz, I. Gonçalves and R. da Rocha, Bilinear Covariants and Spinor Fields Duality in Quantum Clifford Algebras, J. Math. Phys. 55 (2014) 103501 [arXiv:1409.4550] [INSPIRE].
I. Bengtsson and M. Cederwall, Particles, Twistors and the Division Algebras, Nucl. Phys. B 302 (1988) 81 [INSPIRE].
L. Bonora, J.M. Hoff da Silva and R. da Rocha, Opening the Pandora’s box of quantum spinor fields, Eur. Phys. J. C 78 (2018) 157 [arXiv:1711.00544] [INSPIRE].
T. Fleury, On the Pure Spinor Heterotic Superstring b Ghost, JHEP 03 (2016) 200 [arXiv:1512.00807] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1802.06413
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Lopes, R., da Rocha, R. New spinor classes on the Graf-Clifford algebra. J. High Energ. Phys. 2018, 84 (2018). https://doi.org/10.1007/JHEP08(2018)084
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2018)084