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KFKI-1989-09/B Z. PERJÉS UNITARY SPINOR METHODS IN GENERAL RELATIVITY Hungarian Academy of Sciences CENTRAL RESEARCH INSTITUTE FOR PHYSICS BUDAPEST KFKI-1989-09/B PREPRINT UNITARY SPINOR METHODS IN GENERAL RELATIVITY Z. PERJES Central Research Institute for Physics H 1526 Budapest 114, P.OB.49, Hungary Mathematical Relativity miniconference, Australian National University. 1988. HU ISSN 0368 5330 Z. Perjés: Unitary spinor methods in general relativity KFKI 1989 09/B ABSTRACT A survey is given of the structure and applications of spinor fields in three dimensional (pseudo ) Riemannian manifolds. A systematic treatment, independent of the metric signature, is possible since there exists a fairly general structure, to be associated with unitary splnnors, which encompasses all but the reality properties. The discussion begins with the algebraic and analytic properties of unitary splnors, the Ricci identities and curvature spinor, followed by the spinor adjungation as space reflection, and 'he SUÍ2) and SU(1.1) spin coefficients with some applications. The rapidly increasing range of applications includes space times with Killing symmetries, the initial value formulation, positivity theorems on gravitational energy and topotogically massive gauge theories. 3. Перьеш: Унитарные спиномые методы в общей теории относительности. KFKI-1989-09/B АННОТАЦИЯ Рассматривается структура спинорных полей и их применение в трехмерных (псевдо-) римановских множествах. Независимо от сигнатуры метрики имеется воз­ можность систематического, независимого приближения, поскольку существует об­ щая структура, охватывающая, кроме действительности, все другие свойства. Об­ суждение начинается с рассмотрения алгебраических и аналитических свойств уни­ тарных спиноров, затем дается ознакомление с тождественностью Риччи и спинором кривизны, обратимостью спиноров, как пространственное отражение, а также со спиновыми коэффициентами SU(2) и SU(1,1). Количество применений быстро растет, в их числе: пространства-времена, допускающие симметрию Киллинга, задача с на­ чальными условиями, положительность гравитационной энергии и калибровочные по­ ля с топологической массой. I Perjós Z.: Unitér spinormodszerek az általános relativitáselméletben. KFKI 1989 09/B KIVONAT Áttekintjük a splnormezők szerkezetét és alkalmazásait háromdimenziós (pszeudo ) Ríemann sokaságokban A metrika szignatúrájától független rendszeres megközelítés lehetséges, mert létezik a valósságon kívül minden más tulajdonságot felölelő általános struktúra. A tárgyalást a spinorok algebrai és analitikus tulajdonságaival kezdjük, majd a Ricci azonosságokat, a görbületi spinort, spinorok adjungáltját, mint térbeli tükrőzöttet, az SU(2) ós SU(1,1) splnegyütthatókat ismertetjük Az alkalmazások száma gyorsan nő, közöttük: a Killing szimmetriát megengedő téridők, a kezdőértékprobléma, a gravitációs energia pozitivitása ós a mértékterek topológiai tömeggel 1. A LITTLE HISTORY One of the earliest successes of spinor techniques in general relativity is Witten's [l] version of the Petrov classification which was later perfected by Penrose [2] and complemented by the spin coefficient techniques of Newman and Penrose [3]. In the following years, the SL(2, C) spinors have gradually been accepted as a useful mathe­ matical working tool in relativity, even though they never ranked to the straightforward physical utility which spinors in particle physics enjoy. The prevailing view as to why spinors sail so well in curved space-time is that they are seen closely related to null and causal structures. Other insights of more physical nature have only recently been provided by the supersymmetric theories [4]. The applications of the spinors of unitary and pseudo-unitary subgroups of the Lorentz group escaped the attention of relativists until as late as 1970. Then, in an at­ tempt to establish a spinor approach to stationary space-times, the present author has worked out an SU(2) spinor formalism [5]. These technique* have been used for obtain­ ing exact solutions of the gravitational field equations [6]. A corresponding formalism for the non-compact group SU(1,1) has also been developed [7]. It hau been 2 shown possible to formulát» the stationary axisymmetric vacuum gravitational equations such that the complete description is carried by a set of SU(1,1) spin coefficients, and with no curvature quantities appearing. In 1980, Sommers [8] introduced the notion of 'space spinors' in connection with SU(2) spinors relative to a space-like foliation of the space-time. This approach has been taken up by Sen [9] for quantizing the spin-3/2 massless field in a curved background. Recently, the pace of applications has increased, due partly to the SU(2) spinorial nature of many supergravity models [4]. But SU(2) spinors enter Witten's expression for the gravitational energy [10], as well as the theory of topologically massive gauge fields [11]. A new spate of works followed Ashtekar's discovery of new canonical variables for the quantization of the gravitational degrees of freedom [12]. His configuration space is coordinatized essentially by the soldering forms of SU(2) spinors. Unitary spinors have been introduced on manifolds of arbitrary non-null congruences [13]. The ensuing description of space-time contains both the spinors in hypersurfaces and in the manifolds of Killing trajectories as special cases. It has already been noted by Barut [14] that SU(2) and SU(1,1) spinors can be treated in a unified formalism in which one does not specify the signature of the 3- metric. The signature-independent properties, characterizing what will be called uni­ tary spinors, embrace all spinorial relations but those involving adjunction. In the present note we shall show that there exists a remarkably elaborate structure shared by both genres of spinor fields in (pseudo-) Riemannian З-врасев. Adjoining of spinors is relegated to Sec. 5. Applications in general relativity will be discussed at the end of •ach section. s '{"* t )~% 4É* 3 2. UNITARY SPINORS The algebra of unitary spinors encompasses those properties of the SU(2) and SU(1,1) spinors which do not depend on the signature of the associated metric. 2.0 DEFINITION [15] Spin apace is a pair (E,c) where Б is a two-dimensional vector space over the field of complex (or real) numbers and с a symplectic structure on E. Note that с provides an ismorphism (2.1) £ : E -» E between E and the dual space E*. We shall use an index notation such that £д € E is an element of the vector space E and £A € E* an element of the dual space. Capital Roman spinor indices A, B,... may take the values 0 and 1. Then (2.1) can be written A д £ -» £в = £ «лв where {еАв) = ( « ft )• Note that cAB = -€вл- The inverse map employe the spinor tAB satisfying (2.2) елв*св=6* where 6£ is the identity map on E. 2.1 DEFINITION A spinor nfd0- of p upstairs and q downstairs indices is a (p, q) tensor over E. 2.2 PROPOSITION The linear map L: E -» E preserving the spinor eAB has the determinant 1. The group of the maps L is SL(2,C). The proof is all too easy. 2.3 DEFINITION The spinor (2.3) QABCD — *л[с*о)в> 4 called the metric, yields a bilinear map V ® V —» С where V is the vector space (2.4) V = {Ив № = 0}. The inclusion of V in E ® E* defines the map a : E ® E* -»V rendering to each v* e E ® E* its trace-free part. This map is sometimes called a soldering form and has the index structure o\ B where (2.5) о\в=<ГВА. Lower case Roman indices refer concisely to the vector space V. 2.4 DEFINITION Unitary spin space is the triplet (E,«, o) where a : E®E" -• V is the map satisfying i A i C i A i C ii (2.6) °c ° B +<>c <>B =9 6E Í ADCD andíT =<BO OD9 - The vector space V is oriented because Trvx (v3, v3] defines canonically a three-form e(v1,V2,v3) on V [16]. In the index notation we have tijk = е^ыу/д where eijk is the skew numerical Levi-Civita symbol and g = det[gik]. (Thus с<>* is imaginary when the signature of the metric is negative.) A 2.5 COROLLARY [5] The soldering forms oB satisfy the Lie product rule J (2.7) O~ÍACO% ~°ÍAO^ÍB — v2ieiiho AB. Here the right-hand side changes sign under reversal of the orientation of V. 2.6 PROPOSITION Let F<» = F{ih] be a bivector. Then the spinor FABCD = F**aABacD has the decomposition (2.8) FABOD — ЖвоФло -гСлсФво)- Proof. In three dimensions, the tensor Fik can be written equivalently ik (2.9) Fi = y/TfitiikF . 5 The (axial) vector Fi has the spinor components Флв = алвЪ- в в Using the identity <^ <т*в = -Í*, we have £ = -о? ФАв- Substitution of (2.9) in флв yields Флв=у/Що*лви,шР**. This contains the terms on the right hand side of Eq. (2.7), hence С (2.10) Флв=-%РАО в- Now adding the terms %(FADCB - FADCB) to FABCD, it can be written identically as FABCD = FA\B\C\D\ + F\A\D\C)B • For an arbitrary spinor 17 with a pair of skew indices R we have ibis -riBA = €ABnR . Thus FABCD = \(*BDFARC * + eAcFRD*B). Inserting here (2.10), the decomposition (2.8) follows. 2.7 DIGRESSION TO SL(2,C) The algebraic properties of unitary spinors can be derived [5] from the 5L(2, C) spinor algebra in the presence of a preferred non-null four-vector a. The soldering forms a of 5L(2, C) satisfy the defining relation [3] (211) a»AC'°vB +аиВС'°цА =9**еЛВ where Greek indices /1,1/,... range through the values 0, 1, 2 and 3 and primed spinor indices refer to the complex conjugate representation. A hat will signify that the entity refers to the 4-space-time whenever such an emphasis is necessary. Choose coordinates adapted to the non-null vector a such that the metric $M„ is independent of x° = t.
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    The Construction of Spinors in Geometric Algebra Matthew R. Francis∗ and Arthur Kosowsky† Dept. of Physics and Astronomy, Rutgers University 136 Frelinghuysen Road, Piscataway, NJ 08854 (Dated: February 4, 2008) The relationship between spinors and Clifford (or geometric) algebra has long been studied, but little consistency may be found between the various approaches. However, when spinors are defined to be elements of the even subalgebra of some real geometric algebra, the gap between algebraic, geometric, and physical methods is closed. Spinors are developed in any number of dimensions from a discussion of spin groups, followed by the specific cases of U(1), SU(2), and SL(2, C) spinors. The physical observables in Schr¨odinger-Pauli theory and Dirac theory are found, and the relationship between Dirac, Lorentz, Weyl, and Majorana spinors is made explicit. The use of a real geometric algebra, as opposed to one defined over the complex numbers, provides a simpler construction and advantages of conceptual and theoretical clarity not available in other approaches. I. INTRODUCTION Spinors are used in a wide range of fields, from the quantum physics of fermions and general relativity, to fairly abstract areas of algebra and geometry. Independent of the particular application, the defining characteristic of spinors is their behavior under rotations: for a given angle θ that a vector or tensorial object rotates, a spinor rotates by θ/2, and hence takes two full rotations to return to its original configuration. The spin groups, which are universal coverings of the rotation groups, govern this behavior, and are frequently defined in the language of geometric (Clifford) algebras [1, 2].
  • On Certain Identities Involving Basic Spinors and Curvature Spinors

    On Certain Identities Involving Basic Spinors and Curvature Spinors

    ON CERTAIN IDENTITIES INVOLVING BASIC SPINORS AND CURVATURE SPINORS H. A. BUCHDAHL (received 22 May 1961) 1. Introduction The spinor analysis of Infeld and van der Waerden [1] is particularly well suited to the transcription of given flat space wave equations into forms which'constitute possible generalizations appropriate to Riemann spaces [2]. The basic elements of this calculus are (i) the skew-symmetric spinor y^, (ii) the hermitian tensor-spinor a*** (generalized Pauli matrices), and (iii) M the curvature spinor P ykl. When one deals with wave equations in Riemann spaces F4 one is apt to be confronted with expressions of somewhat be- wildering appearance in so far as they may involve products of a large number of <r-symbols many of the indices of which may be paired in all sorts of ways either with each other or with the indices of the components of the curvature spinors. Such expressions are generally capable of great simplifi- cation, but how the latter may be achieved is often far from obvious. It is the purpose of this paper to present a number of useful relations between basic tensors and spinors, commonly known relations being taken for gran- ted [3], [4], [5]. That some of these new relations appear as more or less trivial consequences of elementary identities is largely the result of a diligent search for their simplest derivation, once they had been obtained in more roundabout ways. Certain relations take a particularly simple form when the Vt is an Einstein space, and some necessary and sufficient conditions relating to Ein- stein spaces and to spaces of constant Riemannian curvature are considered in the last section.