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Algebraic and Dirac–Hestenes Spinors and Spinor Fields Waldyr A Algebraic and Dirac–Hestenes spinors and spinor fields Waldyr A. Rodrigues Jr. Citation: Journal of Mathematical Physics 45, 2908 (2004); doi: 10.1063/1.1757037 View online: http://dx.doi.org/10.1063/1.1757037 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/45/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Bohr-Sommerfeld quantization condition for Dirac states derived from an Ermakov-type invariant J. Math. Phys. 54, 052301 (2013); 10.1063/1.4803030 From Dirac spinor fields to eigenspinoren des ladungskonjugationsoperators J. Math. Phys. 48, 123517 (2007); 10.1063/1.2825840 Symmetry Breaking And The Nilpotent Dirac Equation AIP Conf. Proc. 718, 102 (2004); 10.1063/1.1787315 The bundles of algebraic and Dirac–Hestenes spinor fields J. Math. Phys. 45, 2945 (2004); 10.1063/1.1757038 The Dirac Equation in Classical Statistical Mechanics AIP Conf. Proc. 646, 249 (2002); 10.1063/1.1524581 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 143.106.108.134 On: Wed, 03 Jun 2015 16:56:21 JOURNAL OF MATHEMATICAL PHYSICS VOLUME 45, NUMBER 7 JULY 2004 Algebraic and Dirac–Hestenes spinors and spinor fields Waldyr A. Rodrigues, Jr.a) Institute of Mathematics, Statistics and Scientific Computation, IMECC–UNICAMP CP 6065, 13083-970 Campinas, SP, Brazil ͑Received 25 November 2003; accepted 5 April 2004; published online 14 June 2004͒ Almost all presentations of Dirac theory in first or second quantization in physics ͑and mathematics͒ textbooks make use of covariant Dirac spinor fields. An excep- tion is the presentation of that theory ͑first quantization͒ offered originally by Hestenes and now used by many authors. There, a new concept of spinor field ͑as a sum of nonhomogeneous even multivectors fields͒ is used. However, a careful analysis ͑detailed below͒ shows that the original Hestenes definition cannot be correct since it conflicts with the meaning of the Fierz identities. In this paper we start a program dedicated to the examination of the mathematical and physical basis for a comprehensive definition of the objects used by Hestenes. In order to do that we give a preliminary definition of algebraic spinor fields ͑ASF͒ and Dirac– Hestenes spinor fields ͑DHSF͒ on Minkowski space–time as some equivalence classes of pairs (⌶ ,␺⌶ ), where ⌶ is a spinorial frame field and ␺⌶ is an u u u u appropriate sum of multivectors fields ͑to be specified below͒. The necessity of our definitions are shown by a careful analysis of possible formulations of Dirac theory and the meaning of the set of Fierz identities associated with the bilinear covariants ͑on Minkowski space–time͒ made with ASF or DHSF. We believe that the present paper clarifies some misunderstandings ͑past and recent͒ appearing on the literature of the subject. It will be followed by a sequel paper where definitive definitions of ASF and DHSF are given as appropriate sections of a vector bundle called the left spin-Clifford bundle. The bundle formulation is essential in order to be possible to produce a coherent theory for the covariant derivatives of these fields on arbitrary Riemann–Cartan space–times. The present paper contains also Appendixes A–E which exhibits a truly useful collection of results concerning the theory of Clifford algebras ͑including many tricks of the trade͒ necessary for the intelligibility of the text. © 2004 American Institute of Physics. ͓DOI: 10.1063/1.1757037͔ I. INTRODUCTION Physicists usually make first contact with Dirac spinors and Dirac spinor fields when they study relativistic quantum theory. At that stage they are supposed to have had contact with a good introduction to relativity theory and know the importance of the Lorentz and Poincare´ groups. So, they are told that Dirac spinors are elements of a complex four-dimensional space C4, which are the carrier space of a particular representation of the Lorentz group. They are told that when you do Lorentz transformations Dirac spinors behave in a certain way, which is different from the way vectors and tensors behave under the same transformation. Dirac matrices are introduced as certain matrices on C͑4͒ satisfying certain anticommutation rules and it is said that they close a particular Clifford algebra, known as Dirac algebra. The next step is to introduce Dirac wave functions. These are mappings, ⌿:M!C4, from Minkowski space–time M ͑at that stage often introduced as an affine space͒ to the space C4, which must have the structure of a Hilbert space. After that, Dirac equation, which is a first order partial differential equation is introduced for ⌿(x). Physics come into play by interpreting ⌿(x) as the quantum wave function of the electron. Problems with a͒Electronic mail: [email protected] or [email protected] 0022-2488/2004/45(7)/2908/37/$22.002908 © 2004 American Institute of Physics This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 143.106.108.134 On: Wed, 03 Jun 2015 16:56:21 J. Math. Phys., Vol. 45, No. 7, July 2004 Algebraic and Dirac–Hestenes spinors 2909 this theory are discussed and it is pointed out that the difficulties can only be solved in relativistic quantum theory, where the Dirac spinor field, gains a new status. It is no more simply a mapping ⌿:M!C4, but a more complicated object ͓it becomes an operator valued distribution in a given Hilbert space ͑see, e.g., Ref. 162 for a correct characterization of these objects͔͒ whose expecta- tion values on certain one particle states can be represented by objects like ⌿. From a pragmatic point of view, only this knowledge is more than satisfactory. However, that approach, we believe, is not a satisfactory one to any scientist with an enquiring mind, in particular to one that is worried with the foundations of quantum theory. For such person the first questions which certainly occur are what is the geometrical meaning of the Dirac spinor wave function? From where did this concept come from? Pure mathematicians, who study the theory of Clifford algebras, e.g., using Chevalley’s clas- sical books,38,39 learn that spinors are elements of certain minimal ideals ͑do not worry if you did not know the meaning of this concept, it is not a difficult one and is introduced in Appendix B͒ in Clifford algebras. In particular Dirac spinors are the elements of a minimal ideal in a particular Clifford algebra, the Dirac algebra. Of course, the relation of that approach ͑algebraic spinors͒, with the one learned by physicists ͑covariant spinors͒ is known ͑see, e.g., Refs. 14, 67, and 68͒, but is not well known by the great majority of physicists, even for many which specialize in general relativity and more advanced theories, like string and M-theory. Now, the fact is that the algebraic spinor concept ͑algebraic spinor fields on Minkowski space–time will be studied in details in what follows, and in Ref. 126 where the concept is introduced using fiber bundle theory on general Lorentzian manifolds͒͑as it is the case of the covariant spinor concept͒ fail to reveal the true geometrical meaning of spinor in general and Dirac spinors in particular. In 1966, Hestenes81 introduced a new definition of spinor field, that he called later operator spinor field. Objects in this class which in this paper, will be called Dirac–Hestenes spinor fields, ␺ M! 0 0 have been introduced by Hestenes as mappings : R1,3, where R1,3 is the even subalgebra of ͓ R1,3, a particular Clifford algebra, technically known as the space–time algebra. R1,3 is not the original Dirac algebra, which is the Clifford algebra R4,1, but is closely related to it, indeed R1,3 is the even subalgebra of the Dirac algebra ͑see the Appendix B for details͒.͔ Hestenes in a series of remarkable papers80,82–85,75 applied his new concept of spinor to the study of Dirac theory. He introduced an equation, now known as the Dirac–Hestenes equation, which does not contain ͑explicitly͒ imaginary numbers and obtained a very clever interpretation of that theory through the study of the geometrical meaning of the so-called bilinear covariants, which are the observables of the theory. He further developed an interpretation of quantum theory from his formalism,88,89 that he called the Zitterbewegung interpretation. Also, he showed how his approach suggests a geo- metrical link between electromagnetism and the weak interactions, different from the original one of the standard model.87 Hestenes papers and his book with Sobczyk86 have been the inspiration for a series of inter- national conferences on ‘‘Clifford Algebras and their Applications in Mathematical Physics’’ which in 2002 has had its sixth edition. A consultation of the table of contents of the last two conferences1,145,2 certainly will show that Clifford algebras and their applications generated a wider interest among many physicists, mathematicians, and even in engineering and computer sciences. ͑In what follows we quote some of the principal papers that we have had opportunity to study. We apologize to any author who thinks that his work is a worthy one concerning the subject and is not quoted in the present paper.͒ Physicists used Clifford algebras concepts and Hestenes methods, in many different applications. As some examples, we quote some developments in relativistic quantum theory as, e.g., Refs. 36, 37, 45, 46, 48–52, 56, 58, 74, and 70. The papers by De Leo and collaborators exhibit a close relationship between Hestenes methods and quaternionic quantum mechanics, as developed, e.g., by Adler,4 a subject that is finding a renewed interest.
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