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On the Structure of Geometries with Spinor ON THE STRUCTURE OF GEOMETRIES WITH SPINOR-TYPE CONNEXION M. c. Cullinan A dissertation presented in partial fulfilment of requirements for the degree of Doctor of Philosphy universitj of New South Wal~s C March 1975 .... .,,..-,·-,; ..,. ., I am indebted to Professor G. Szekeres for his constant help and guidance during the writing of this thesis. I would also like to thank Ms. Helen Cook for her careful and expert typing, and Dr. H.A. Cohen for many stimulating conversations. In addition it gives me great pleasure to acknowledge the continued support of my family and friends during the preparation of this thesis; without them this research could not have been completed. Abstract The aim of this thesis is to present a synthesis, using standard techniques of algebra and of differential geometry, of some of the principal ideas and structures of the classical field­ theoretic description of spinor fields in curved space-time. A geometric model for spinor fields in curved space-time is described which is based upon properties of so-called tangent Clifford algebras. The tangent Clifford algebra to a space-time manifold at a certain point is the quotient space of the algebra of covariant tensors at the point by a certain two-sided ideal, and is uniquely defined once the metric structure of the space-time manifold is given. Minimal left ideals of each tangent Clifford algebra are identified with spaces of four component spinors at the point. This model therefore features a direct method for synthesizing spinor fields from vector and tensor fields on a space­ time manifold. Since a spanning set for each tangent Clifford algebra necessarily involves basis elements of four minimal left ideals, one is led to associate four four-dimensional spin-spaces with each point of a space-time manifold. The possibility of making transformations between these four spin spaces at each point offers a simple geometrical model for the class of unitary symmetry theories which relate to symmetries over a quartet of four dimensional spin spaces. Fields of higher spin are also readily accommodated in this model; as with the four-component spinor objects mentioned earlier, geometrical concomitants of (spinor-} fields of higher spin are constructed entirely from vector and tensor fields of various kinds. iii IV A theory of connexion is proposed for the spinor objects described above; this theory uses only standard methods of classical differential geometry, together with a certain amount of algebra, and leads, amongst other things, to the realization of a rather explicit relationship between the operation of taking the covariant differential of a tensor quantity on the one hand, and that of taking a 'Dirac'-type derivative (Q~( ) ,~> of associated spinor quantities on the other. This theory is more general than existing models for spinor geometry as it embraces not only the usual theory of spinor connexion, but also a geometrical model for a certain class of generalized (Yang-Mills type) gauge fields in curved space-time. Of particular interest in this context are those space-time manifolds which do not admit a so-called spinor structure, for on these manifolds a geometry of spinors will generally involve connexion quantities which are direct geometrical concomitants of gauge fields such as the isotopic gauge field of Yang and Mills. At each stage in the development of this model, explicit calculations are given which serve to relate the various results to those of standard physical theory. In particular, a description of the so-called generalized Dirac matrices which embodies both algebraic and differential properties is obtained from the general structure of the model. Like the other spinor objects with which this model deals, these matrices and their properties derive entirely from space-time properties of vector and tensor fields. CONTENTS Abstract iii Introduction vi CHAPTER 1 The Algebra of Tangent Clifford Numbers 1 §1.1 The space-time manifold 2 §1.2 Tensors and tangent Clifford numbers 8 §1. 3 On minimal ideals of e; 22 §1.4 Frame transformations 35 CHAPTER 2 The Geometry of Tangent Clifford Numbers 53 §2,1 Linear connexion and covariant tensors 54 §2,2 Connexion and T,C, numbers; holonomic 62 c.onsiderations §2,3 Connexion and T.C. numbers (continued); non-holonomic considerations 70 §2.4 Connexion and T,C, numbers (continued); spinor connexion 80 §2,5 On invariant distributions of minimal left ideals 88 CHAPTER 3 Physical Applications 108 §3,1 One-ideal distributions and four-spinor fields 109 §3,2 Multiple-ideal distributions; geometry and intemal symmetry 116 CHAPTER 4 Summary and Conclusions 128 Appendix Matrix representations of the r=ab etc. 132 Bibliography 137 Introduction The literature of mathematical physics testifies to the existence of a close relationship between the development of our understanding of general relativity on the one hand and that of modern differential geometry on the other. The closeness of this relationship has stimulated research in both fields and has led, amongst other things, to our presently having a remarkably clear appreciation of the geometric foundations and content of relativity theory and of the theory of vector and tensor fields in curved space-time. To date we have no understanding of the geometrical foundations of spinor field theory, nor of its geometrical content, that can compare with current geometrical models for vector and tensor field theory in terms of either depth or clarity. In fact our understanding of the genesis and geometrical roots of spinor field theory has progressed very little since Dirac's original treatment of the quantum mechanics of the hydrogen atom and the well-known early work of INFELD and VAN DER WAERDEN [13]. Although the reasons for this are not entirely clear, one important contributing factor is surely that a full and satisfactory account of the geometrical foundations of spinor analysis has yet to be given; the geometrical framework necessary for a restructuring of the foundations of spinor field theory has quite simply not been available. What would be the likely benefits, if any, accruing from a clearer understanding of the geometrical structures underlying spinor field theory, and what would one be entitled to expect of a definitive account of the geometry of spinor fields? An examination of the literature of orthodox spinor geometry, ranging from the by now classic paper of Infeld and Van der Waerden to more recent accounts such as that due to LUEHR and ROSENBAUM [16), reveals that the classical theory of spinor fields in curved-space time is predominantly composed of an inter-relation of ad hoe assumptions and constructions about which it is known that they are (generally) consistent and that they seem to "work", but about which, it appears to this author at least, little else is known at a systematic level. As a minimal requirement it would therefore be a desirable feature of any sensible geometrical model for spinor fields in curved space-time that it be able to provide a unifying perspective for at least some of the diverse assumptions and ideas to which we refer above, and a means of relating them to orthodox constructions of classical differential geometry. At a more specific level it would be of considerable interest to know answers to a number of questions of a geometrical kind, questions for which solutions would seem to lie largely beyond the scope of orthodox ad hoe and axiomatic treatments of spinor theory. Typical of these questions are the following: (i) What is the qeometrical/topoloqical content of classical spinor field theory? Can a knowledqe of solutions of Di l"ac' s equation, for example, tell us anything about the underlying geometry and topology of space-time? How are solutions of Dirac' s equation ( the equation of Rari ta and Schwinger, equations for zero rest-mass fields) related to the harmonic fields of classical geometry and to topoloqical invariants of the space-time manifold? (ii) What can a knowledge of space-time geometry and topology tell us about the existence and properties of spinor fields? For example, could a knowledge of space-time geometry and topoloqy provide selection rules for solutions of wave Viii equations representing fields of particular spin? charge? iso-spin? And ultimately, (iii) What is the geometric content of Lagrangian formulations of field theory? Could the manipulation of field Lagrangians as a means of obtaining self-consistent field equations be replaced by an equivalent set of manifestly geometrical postulates? It is towards obtaining a basis for a possible clarification of some of these issues that this thesis is ultimately directed. The model which we will describe and which will hopefully form this basis consists of a more or less systematic exploration of one central idea: that the key to a satisfactory geometrical understanding of the behaviour and properties of fields of spinor-type objects and of those issues raised above is contained in the notion of a suitably constructed local or - as we have called it - tangent, Clifford algebra. The idea of basing an account of the properties of spinor objects in curved space-time on the concept of a localized Clifford algebra structure is due to M.R!ESZ ([21]~ see also [20]). In his model Ri.esz identifies a "local Clifford algebra" with each point of space-time, and minimal left ideals of each local Clifford algebra with spaces of four component spinors at the point. By a "local Clifford algebra" at a point P of a space-time manifold Jt one is evidently to understand an algebra generated by the actual tangent vector basis elements in the tangent plane to..,llg at :f>•J if in a suitable coordinate system around P these tangent vector basis elements -+ are denoted (eA)P (A=~· 4) then each local Clifford algebra in Riesz -+ model is to be thought.of as being generated by the (e\)p• the latter being subject to a constraint ix 'IEITLER ( [27] - [29]) has explored some flat space-time applications of an idea of this kind, and in the first mentioned of these papers refers to the earlier use of a similar idea by workers such as Sommerfeld and Sauter.
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