On a Relation Between Twistors and Killing Spinors

Özgür Açık

doi:10.17776/csj.434630

TR EN

On a Relation Between Twistors and Killing Spinors

Abstract

Inspiring from the consequence of constructing conformal Killing-Yano forms out of Killing-Yano forms and closed conformal Killing-Yano forms, this work includes a method for building up twistors from Killing spinors which can be analogously interpreted as the quantum electrodynamical pair annihilation process in background gravitational fields. The former consequence is easily verified if one introduces a new defining differential equation for (possibly inhomogeneous) Killing-Yano forms which is free from auxiliary vector fields, as is done in this text. From this point of view a neat relation between the symmetry operators of massive and massless Dirac equation is also introduced. Some other physical interpretations are also included.

Keywords

Clifford Algebras and Spinors,Twistors,Killing-Yano forms,Killing spinors

References

[1]. Açık Ö. And Ertem Ü.,Higher degree Dirac currents of twistor and Killing spinors in supergravity theories, Class. Quantum Grav. 32, (2015) 175007; Açık Ö. and Ertem Ü., "Generating dynamical bosons from kinematical fermions", CQG+, (19 August 2015).
[2]. Açık Ö., Field equations from Killing spinors, J. Math. Phys. 59, (2018) 023501.
[3]. Penrose R. and Rindler W., Spinors and Space-time, Vol.2, Cambridge Univ. Press, 1987.
[4]. Benn I. M. and Kress J., Differential forms relating twistors to Dirac fields, in: Differential Geometry and its Applications, Proceedings of the 10th International Conference DGA 2007, World Scientific Publishing, Singapore, 2008, pp. 573.
[5]. Charlton P., The Geometry of Pure Spinors with Applications, PhD thesis, University of Newcastle 1997.
[6]. Baum H., Leitner F., The twistor equation in Lorentzian spin geometry, Math. Z. 247 795812.14 (2004).
[7]. Lischewski A., Towards a Classification of pseudo-Riemannian Geometries Admitting Twistor Spinors, arXiv:1303.7246v2.
[8]. Kath I., Killing spinors on pseudo-Riemannian manifolds, Habilit., Humboldt-Universitat zu Berlin (1999).

[9]. Tucker R. W., Extended Particles and Exterior Calculus, Rutherford Laboratory, Chilton-Didcot-Oxon, OX11 0QX, RL-76-022 (1976).
[10]. Burton D. A., A primer on exterior differential calculus, Theoret. Appl. Mech., Vol. 30, No. 2, 85-162, Belgrade (2003).
[11]. Benn I. M. and Tucker R. W., An Introduction to Spinors and Geometry with Applications in Physics, IOP Publishing Ltd, Bristol, 1987.
[12]. Tucker R. W., A Clifford calculus for physical field theories, in J. S. R. Chisholm and A. K. Common (Eds.), Clifford Algebras and Their Applications in Mathematical Physics, Dordrecht: D. Reidel Publishing Company, 1986.
[13]. Toretti R., Relativity and Geometry, Dover Publications, Inc. New York, 1983.
[14]. Benn I. M. and Kress J., First-order Dirac symmetry operators", Class. Quantum Grav. 21 (2004) 427.
[15]. Acik O., Ertem U., Onder M. and Vercin A., First-order symmetries of the Dirac equation in a curved background: a unified dynamical symmetry condition, Class. Quantum Grav. 26 (2009) 075001.
[16]. Ertem Ü., Symmetry operators of Killing spinors and superalgebras in AdS5, J. Math. Phys. 57, (2016) 042502.
[17]. Ertem Ü., Twistor spinors and extended conformal superalgebras, arXiv:1605.03361.
[18]. Hughston L. P., Penrose R., Sommers P. and Walker M., On a quadratic first integral for the charged particle orbits in the charged Kerr solution, Commun. Math. Phys. 27 (1972) 303.
[19]. Kastor D. and Traschen J., Conserved gravitational charges from Yano tensors, J. High Energy Phys. JHEP08 (2004) 045.
[20]. Acik O., Ertem U., Onder M. and Vercin A., Basic gravitational currents and Killing-Yano forms, Gen. Relativ. Gravit. 42 (2010) 2543.
[21]. Açık Ö. and Ertem Ü., Hidden symmetries and Lie algebra structures from geometric and supergravity Killing spinors, Class. Quantum Grav. 33, (2016) 165002.
[22]. Açık Ö., New developments in Killing spinor programme and more motivations for physics, arXiv:1611.04424v2.
[23]. Krtous P., Kubiznak D., Page D. N. and Frolov V. P., KillingYano tensors, rank-2 Killing tensors, and conserved quantities in higher dimensions, J. High Energy Phys. JHEP02 (2007) 004.
[24]. Trautmann A., Complex structures in physics, arXiv:math-ph/9809022v1.
[25]. Cariglia M., Krtous P. and Kubiznak D., Commuting symmetry operators of the Dirac equation, Killing-Yano and Schouten-Nijenhuis brackets, Phys. Rev. D 84, (2011) 024004.
[26]. Geroch R. and Traschen J., Strings and other distributional sources in general relativity, Phys. Rev. D 36, (1987) 1017.
[27]. Stachel J., Thickening the string I, The string perfect dust, Phys. Rev. D 21, (1980) 2171.
[28]. Stachel J., Thickening the string II, The null-string dust, Phys. Rev. D 21, (1980) 2182.
[29]. Tucker R. W., Motion of Membranes in Spacetime, Conference on Mathematical Relativity, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra AUS, (1989) 238-243.
[30]. Hartley D., Tucker R. W., Tuckey P. A. and Dray T., Tensor Distributions on Signature-changing Space-times", Gen. Relativ. Gravit., 32, 3 (2000).
[31]. Dirac P. A. M., An extensible model of the electron, Proc. Roy. Soc. of London A, 268, (1962) 57-67.
[32]. Önder M. and Tucker R. W., Membrane Interactions and Total Mean Curvature, Phys. Lett. B 202(4), (1988) 501-504.
[33]. Fujii Y. and Yamagishi K., Killing spinors on spheres and hyperbolic manifolds, J. Math. Phys. 27 (4) (1986).
[34]. Fulling S. A., Aspects of Quantum Field Theory in Curved Space-Time, London Mathematical Society Student Texts 17 (1989).
[35]. Açık Ö., On a relation between twistors and Killing spinors II, (in preparation).
[36]. Açık Ö., Spin 1/2 and spin 3/2 field solutions in plane wave spacetimes, Gen. Relativ. Gravit., 50 33 (2018).
[37]. Of course in almost all geometric theories of gravity metric is the fundamental field, especially in General Relativity.
[38]. Although we have used the term pseudo-Riemannain in the abstract and in the introduction, we go against this usage with the same lines of thought as R. Toretti [13], and instead we prefer to use Riemannian spacetime generally and especially Lorentzian spacetime for manifolds endowed with one time-like direction in a local g-frame.
[39]. We use the term Yano form to indicate a KY form or a CCKY form, then the term dual Yano form indicates a CCKY form or a KY form respectively.
[40]. Confusion with the Riemann curvature R tensor should be avoided.
[41]. The properties associated to these projectors are given at the Appendix A of [1].

Details

Primary Language

English

Subjects

-

Journal Section

Research Article

Authors

Özgür Açık ^*
Türkiye

Publication Date

December 24, 2018

Submission Date

June 19, 2018

Acceptance Date

November 13, 2018

Published in Issue

Year 2018 Volume: 39 Number: 4

DOI

https://doi.org/10.17776/csj.434630

IZ

https://izlik.org/JA28LK68PU

Cite

RIS / Bibtex

APA

Açık, Ö. (2018). On a Relation Between Twistors and Killing Spinors. Cumhuriyet Science Journal, 39(4), 954-969. https://doi.org/10.17776/csj.434630

On a Relation Between Twistors and Killing Spinors

Twistörler ve Killing Spinörleri Arasındaki Bir Ilişki Üzerine

Öz

Anahtar Kelimeler

On a Relation Between Twistors and Killing Spinors

Abstract

Keywords

References

Details

Primary Language

Subjects

Journal Section

Authors

Publication Date

Submission Date

Acceptance Date

Published in Issue

DOI

IZ

Cite