On a Relation Between Twistors and Killing Spinors

Özgür Açık

doi:10.17776/csj.434630

Research Article

Twistörler ve Killing Spinörleri Arasındaki Bir Ilişki Üzerine

Year 2018, Volume: 39 Issue: 4, 954 - 969, 24.12.2018

Özgür Açık

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

Abstract

Killing-Yano formları ve kapalı konformal
Killing-Yano formlarından, konformal Killing-Yano formları oluşturabilme sonucundan
esinlenmek suretiyle, bu çalışma Killing spinörlerinden twistörler inşa etmek
için bir yöntem içermektedir ki bu benzeşik olarak arka-zemin kütle-çekimsel
alanlarda kuantum elektrodinamiksel çift yokoluşu olarak yorumlanabilmektedir.
Eğer bu makalede yapıldığı gibi (homojen olmaması muhtemel) Killing-Yano
formları için yardımcı vektör alanlarından muhaf yeni tanımlayıcı bir
diferensiyel denklem tanıtılırsa ilk sonuç kolyaca doğrulanabilir. Bu bakış
açısıyla kütleli ve kütlesiz Dirac denkleminin simetri işlemcileri arasında
muntazam bir ilişki tanıtılabilmektedir. Bazı diğer fiziksel yorumlar da
içerilmektedir.

Keywords

Clifford cebirleri ve spinörler, Twistörler, Killing-Yano formları, Killing spinörleri

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].

On a Relation Between Twistors and Killing Spinors

Year 2018, Volume: 39 Issue: 4, 954 - 969, 24.12.2018

Özgür Açık

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

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].

There are 41 citations in total.

Details

Primary Language	English
Journal Section	Natural Sciences
Authors	Özgür Açık
Publication Date	December 24, 2018
Submission Date	June 19, 2018
Acceptance Date	November 13, 2018
Published in Issue	Year 2018Volume: 39 Issue: 4

Cite

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

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