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Spinors in Relativity

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Spinors in Relativity Spinors in Relativity Jörg Frauendiener December 13, 2018 Department of Mathematics and Statistics University of Otago Since this script is available online I would appreciate any feedback from whoever reads it. Please send comments, suggestions, corrections to [email protected]. Contents 1 Introduction and Literature 4 2 The geometry of Minkowski space-time 8 2.1 Affine space . 8 2.2 Metric space . 10 2.3 Basic notions . 11 3 Spin transformations 15 3.1 Representation of the sphere . 15 3.2 Null directions and the celestial sphere . 17 3.3 Lorentz and spin transformations . 18 3.4 Lorentz transformations: a closer look . 21 4 Spinorial objects and spin vectors 23 4.1 Null flags . 23 4.2 Spin vectors . 27 4.3 Some global considerations . 28 5 Spinor algebra 30 5.1 Foundations of tensors . 30 5.2 Spinors . 35 5.2.1 # spinors . 36 5.2.2 Complex conjugate spinors . 37 5.2.3 Spin frames . 39 6 4-vectors as spinors 40 6.1 Spin frames and their tetrads . 40 6.2 Van-der-Waerden symbols . 42 6.3 Clifford algebra, Dirac spinors, Weyl spinors . 46 6.4 Null vectors, null flags, etc. 50 3 1 Introduction and Literature Einstein’s theory of special relativity is of fundamental importance in today’s mod- ern Physics. It is not a theory of interaction, i.e., describing a fundamental force such as for instance electrodynamics. Rather it is a kinematical theory which describes the framework within which Physics takes place. It takes the place of classical Newtonian Physics which was shaped by Galilei’s principle of relativity. This principle postulates that all reference frames which are related by Galilei transformations are equivalent for the description of physical phenomena. This stipulation distinguishes a class of pre- ferred reference frames, called inertial frames. It was expected that all laws of Nature “look the same” with respect to each of these inertial frames, i.e., that they are invariant under Galilei transformations. However, after the realisation that Maxwell’s equations of electrodynamics are not Galilei invariant, it was Einstein to dissolved the ensuing dilemma (parts of Physics being Galilei invariant, others not) by postulating a different principle of relativity. According to his principle of relativity not the Galilei transfor- mations map inertial frames to inertial frames but it is Poincaré transformations, also known as inhomogeneous Lorentz transformations. The space-time continuum of relativistic (quantum-) mechanics is the Minkowski space- time, an affine space which is equipped with a bilinear form of signature −2. The invari- ance group of this space is the Poincaré group, the group of inhomogeneous Lorentz transformations. The Lorentz group itself is the subgroup of the Poincaré group which consists of transformations which keep one selected point fixed. It can be regarded as the invariance group of the Minkowski vector space of position vectors based on the selected point. The importance of the Poincaré group is highlighted by the fact that elementary particles are classified in relativistic quantum mechanics according to its irreducible representations. The fact that the Poincaré group is intimately connected to the global structure of Minkowski space (which can even be defined as a homogeneous space on which the Poincaré group acts) already suggests that there might be difficulties in the transition to Einstein’s theory of gravity. Now the space-time is no longer an affine space but a more general (curved) manifold whose curvature is determined by the matter contents in the space-time. The only commonality with Minkowski space is the fact that at every point there is attached a Minkowski vector space which also carries a bilinear form of signature −2. Accordingly, the importance of the Poincaré group evaporates because only in the rarest cases will there exists a global invariance group for the space-time which in addition will not agree with the Poincaré group. Furthermore, in this general case the notion of an elementary particle gets blurred. Together with other difficulties 4 this is a manifestation of the current incompatibility of quantum theory and general rel- ativity. However, the Lorentz group does survive in the transition from flat to curved space. It remains the invariance group of the Minkowski vector space and, hence, only acts at each point individually. It is the truly fundamental group of relativistic physics and its representations should be of physical relevance in all aspects of Physics. The representations of the Lorentz group are treated at length in [4]. The best known representation of the Lorentz group is its defining representation in the 4-dimensional Minkowski vector space. It is the basis for the usual tensorial repre- sentations and the mathematical process leads naturally to the development of tensor algebra. The Lorentz group is a semi-simple Lie group whose finite dimensional rep- resentations are fully reducible. Hence, it is enough to find and classify the irreducible representations. Among those there are two distinguished ones: the two 2-dimensional representations are fundamental representations, i.e., all representations of the Lorentz group can be obtained as appropriate tensor products of only those two. The vectors of these representation spaces are the primed and unprimed spin-vectors and the ten- sor algebra built upon the fundamental representations is called spinor algebra. This means that spin-vectors can be used to construct all other tensors. The 2-dimensionality of the spin-vectors makes calculating with them quite easy. This is a peculiarity of the four dimensions of the world we live in. In higher dimensions one can find analogous spinor representations related to the corresponding invariance groups. However, their dimension grows exponentially with the dimension of the space-time so that already for space-time dimension 8 the corresponding spinor rep- resentation is also 8-dimensional. Only for space-time dimensions 3 and 4 can we ex- pect to gain simplifications in the study of space-time geometry using the calculus of spinors. The theory of spinors was already developed and applied at the beginning of the last century. The general notion of a spinor was introduced by Cartan [1, 2] while van der Waerden [11] developed the calculus of two-component spinors. Dirac discovered the fundamental importance of spinors for relativistic physics when he found the equation the electron. With the work of Penrose [7] the calculus of spinors was firmly intro- duced into General Relativity. This formalism as well as the related Newman-Penrose (NP) formalism was very successfully applied in many different areas of gravitational theory, in particular in the treatment of asymptotically flat space-times, the analysis of gravitational waves and last but not least in the proof of the positive-energy-theorem of General Relativity, one of the most fundamental results in that area. The two volumes by Penrose and Rindler [8] contain an impressive introduction into the theory of spinors and many application thereof. They also explain the relationship between spin-vectors and space-time geometry. In the book by Stewart [10] one can find the treatment of asymptotically flat space-times using the calculus of two-component spinors. Spinors are also discussed in the book by Naber [6]. 5 Another good source for the mutual relations between spinors, relativity and quantum theory is the book by Sexl and Urbantke [9]. It discusses in particular the group theoret- ical aspects. Finally, it is worthwhile to point out a classic which does not deal explicitly with spinors but presents a very good treatment of the foundations of General Relativ- ity: the book by Hermann Weyl [12] is warmly recommended. It appeared first in 1918 and has been published now in its 7th edition. The following course will provide an introduction into the calculus of two-component spinor-vectors. The aim of the course is to reduce the timidity towards and overcome the hesitation to apply the formalism which may seem daunting at the first encounter. The first chapters deal with the fundamental properties of Minkowski geometry. How- ever, the point of view is a bit different from other approaches. We regard the geometry of light rays as fundamental. The “celestial sphere” of past directed light rays is intro- duced and with its aid the null flags are defined. These are the geometric imprints of spin-vectors. Then we adopt a more abstract point of view and introduce the spinor algebra. After a chapter on the main tools of spinor calculus (principal null directions, dualisation, symmetry operations) the results are transformed to manifolds. The last three chapters are devoted to applications, the classification of the Weyl tensor, the NP formalism, and finally a discussion of some spinor equations. Except for some parts of the last chapter the whole material is standard and can be found in the cited litera- ture. 6 Bibliography [1] É. CARTAN, Les groupes projectifs qui ne laissent invariante aucune multiplicité plane, Bull. Soc. Math. France, 41 (1913), pp. 53–96. [2] , The Theory of Spinors, Hermann, Paris, 1966. [3] T. FRIEDRICH, Dirac-Operatoren in der Riemannschen Geometrie, Verlag Vieweg, Braunschweig, 1997. [4] I. M. GELFAND, R. A. MINLOS, UND Z. Y. SHAPIRO, Representations of the Rotation and Lorentz Group and Their Applications, Pergamon Press, Oxford, 1963. [5] H. B. Lawson und M.-L. Michelsohn, Spin Geometry, Princeton University Press, 1989. [6] G. L. NABER, The Geometry of Minkowski Spacetime, Springer-Verlag, Berlin, 1992. [7] R. PENROSE, A spinor approach to general relativity, Ann. Phys., 10 (1960), pp. 171– 201. [8] R. PENROSE UND W. RINDLER, Spinors and Spacetime, vol. 1,2, Cambridge Univer- sity Press, 1984, 1986. [9] R. SEXL UND H. URBANTKE, Relativität, Gruppen, Teilchen, Springer-Verlag, Wien, 3.
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