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On a Relation Between Twistors and Killing Spinors Cumhuriyet Science Journal CSJ e-ISSN: 2587-246X Cumhuriyet Sci. J., Vol.39-4(2018) 954-969 ISSN: 2587-2680 On a Relation Between Twistors and Killing Spinors Özgür AÇIK Ankara University, Faculty of Sciences, Department of Physics, Ankara, TURKEY Received: 19.06.2018; Accepted: 13.11.2018 http://dx.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. Twistörler ve Killing Spinörleri Arasındaki Bir Ilişki Üzerine Özet. 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. Anahtar Kelimeler: Clifford cebirleri ve spinörler, Twistörler, Killing-Yano formları, Killing spinörleri. 1. INTRODUCTION with the quantum electrodynamical pair annihilation process, Killing spinors correspond to Killing spinors are fruitful objects that give rise to the electron-positron pair and twistors correspond many differential equations that appear in to the pure radiation field. The mathematics mathematical physics. Up to now bosonic literature include the classification of pseudo- equations were constructed from the real bilinear Riemannian manifolds admitting twistors [6,7] and covariants of Killing spinors which include Duffin- the classification of pseudo-Riemannian manifolds Kemmer-Petiau and Maxwell-like equations in admitting Killing spinors [8]; so our method will gravitational fields [1], but relation to possible form a bridge between these classifications. fermionic field equations were absent. Connectedly the construction of twistors from Killing spinors is We prefer to use coordinate-free differential to be seen by us as a first step in this direction, that geometry for the sake of notational elegance and may be tied [2] to the generation of higher spin brevity and also for clarity in direct geometrical or fields by known methods such as spin physical interpretations. The usage of raising/lowering of non-gravitational fields in componentwise tensor calculus is more common in curved spacetimes [3-5]. We show that, in analogy physics community; closing this gap could only be * Corresponding author. Email address: [email protected] http://dergipark.gov.tr/csj ©2016 Faculty of Science, Sivas Cumhuriyet University Açık / Cumhuriyet Sci. J., Vol.39-4 (2018) 954-969 955 possible by transforming one language to the other. 2. BUILDING UP CONFORMAL KILLING- We tried to do this translation at least for a set of YANO FORMS equations that we feel important and any reader can use this example as a dictionary in between; this is A. A Different Perspective for Defining Killing- done in Appendix C. Pre-metric notions are Yano Forms handled through the usage of Cartan calculus on manifolds [9, 10], the participation of a metric The important roles played by KY forms and CKY tensor field into the set of spacetime structures [37] forms are various. They define the hidden forces us to use the full power of Clifford bundle symmetries of the ambient spacetime generalising formulation of physics and geometry. This latter to higher degree forms the ones induced by Killing choice is precious for admitting a local equivalence and Conformal Killing vector fields, and they between spinor fields and the sections of minimal respectively can be used for setting up first order left ideal bundle of the Clifford bundle. A good symmetry operators for massive and massless account of this tradition can be found in [11, 12]. Dirac equations in curved spacetimes [14, 15]. Throughout the text we assume a Riemannian These symmetry operators reduce to the Lie spacetime [38] i.e. a smooth manifold endowed derivative on spinor fields for the lowest possible with an indefinite metric 푔 and its Levi-Civita degree, so if carefully worked out they may define Lie multi-flows of spinor fields in a classical connection ∇. The organisation of the paper is as spacetime. Achieving this shall also extend the follows. In section II an explicit method for concept of Lie multi-flows to form fields and multi- constructing conformal Killing-Yano (CKY) forms vector fields. As a consequence, the analysis of from Killing-Yano (KY) forms and closed higher rank infinitesimal symmetries will heavily conformal Killing-Yano (CCKY) forms is given; simplify [16, 17]. On the other side, via Noether's before that a new form for the definitive equation theorem, they induce conserved quantities for KY forms is built up. Then the first order associated to geodesics of (point) particles [18] and symmetry operators for massive and massless more generally to world-immersions of p-branes Dirac equations are reconsidered and are related in [19-21]; where there is much work to do for the an elegant way in the view of the above setup. An later case [22]. Also the definition of some charges intuitional question, which leads to the core of this for higher dimensional black holes rely on the paper, is asked at the end of section II. The positive existence of these symmetry generating totally answer of this main question is given in terms of a anti-symmetric tensor fields [23]. proposition and its proof. The analysis is bifurcated into fermionic and bosonic sectors both of which Now we want to redefine KY forms from a contain geometric identities that are derived in the different perspective that is more general. The corresponding subsections; the fermionic sector is usual definition of a KY-form of degree p is the detailed by considerations on charge conjugation, solution of time reversal, helicity and their relation to Killing reversal due to the preceding main physical 1 ∇푋휔 = i푋d휔. (1) interpretation. After the conclusion of our paper at 푝+1 section IV, two appendices are given at the end. Exterior derivative 푑 and co-derivative d† ≔ Appendix A is based on the calculation of the ∗−1 d ∗ 휂 can be given by the Riemannian relations adjoints of the symmetry operators associated to 푎 † d = 푒 ∧ ∇푋 , d = −푖푋푎 ∇푋 andi푋 is the internal KY and CCKY forms and ends up with a comment 푎 푎 contraction with respect to the vector field 푋. If 푇 on symmetry algebra. Appendix B contains the is a mixed tensor field of covariant degree r and derivation of the coordinate expressions of our contravariant degree s, its covariant differential ∇T primitive set of equations which is important for which has one contravariant degree more than 푇 is the understanding of general reader who are used defined to the notation of tensor components. as 956 Açık / Cumhuriyet Sci. J., Vol.39-4 (2018) 954-969 1 푠 1 푠 (∇T)(X, Y1, … , Y푟, 훽 , … , 훽 ) = (∇푋푇)( Y1, … , Y푟, 훽 , … , 훽 ) (2) 푗 in terms of vector fields X, Y푖 and co-vector fields 훽 . Also remembering the identity (i푋휙)( Y1, … , Y푝−1) = 푝휙(푋, Y1, … , Y푝−1) (3) for any p-form 휙, we can define KY-forms in a different manner. If we write (1) as i d휔 (∇ 휔)(Y , … , Y ) = ( 푋 ) (Y , … , Y ) 푋 1 푝 푝 + 1 1 푝 then by using (2) and (3) for the left and right hand sides respectively we obtain (∇휔)(푋, Y1, … , Y푝) = (d휔)(푋, Y1, … , Y푝) (4) which is valid for any X, Y푖 ∈ Γ푇푀. As a result the new defining equation for a KY form is (∇ − d)휔 = 0 (5) and is not necessarily homogeneous. An integrability condition equivalent to Poincaré's lemma is ∇2휔 = 0. Remembering the defnition of the alternating idempotent map; i.e. 퐴푙푡: Γ푇푟(푀)⟶ Γ푇Λ푟(푀) 퐺 ↣ 퐴푙푡(퐺) such that 1 퐴푙푡(퐺)(Y , … , Y ) = ∑ 푠푖푔푛() 퐺(Y , … , Y ) 1 푟 푟! (1) (푟) ∈푆(푟) it is possible to write (5) as ((1 − 퐴푙푡)∇)휔 = 0. B. Adding Closed Conformal Killing-Yano forms to Killing-Yano forms In this section our main objects will be CKY forms and we will give a trivial method for their construction. CKY form equation is 1 1 ∇ = i d − 푋̃ ∧ d† (6) 푋 푝+1 푋 푛−푝+1 where an inhomogeneous generalisation was given in [1], 푋̃ is the 푔-dual of the vector field 푋. We want to emphasize the duality between the space of Killing-Yano forms and the space of CCKY forms, that is if 휔̂ is a CCKY p-form then its Hodge dual ∗ 휔̂ is a KY (n - p)-form and vice-versa. Although the usual definition of a CCKY p-form 휔̂ is given by 1 ∇ 휔̂ = − 푋̃ ∧ d†휔̂ (7) 푋 푛−푝+1 it is trivial from (6) that this equation can also be written as (∇† − d†)휔̂ = 0 (8) Açık / Cumhuriyet Sci. J., Vol.39-4 (2018) 954-969 957 where ∇†=∗−1 ∇ ∗ 휂 . 휂 is the main automorphism of the tensor algebra.
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