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Geometric Algebras in Physics: Eigenspinors and Dirac Theory by J. David Keselica A Dissertation Submitted to the Faculty of Graduate Studies through the Department of Physics in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy at the University of Windsor Windsor, Ontario, Canada 2008 Library and Bibliotheque et 1*1 Archives Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington Ottawa ON K1A0N4 Ottawa ON K1A0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-47108-1 Our file Notre reference ISBN: 978-0-494-47108-1 NOTICE: AVIS: The author has granted a non­ L'auteur a accorde une licence non exclusive exclusive license allowing Library permettant a la Bibliotheque et Archives and Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par I'lnternet, prefer, telecommunication or on the Internet, distribuer et vendre des theses partout dans loan, distribute and sell theses le monde, a des fins commerciales ou autres, worldwide, for commercial or non­ sur support microforme, papier, electronique commercial purposes, in microform, et/ou autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in et des droits moraux qui protege cette these. this thesis. Neither the thesis Ni la these ni des extraits substantiels de nor substantial extracts from it celle-ci ne doivent etre imprimes ou autrement may be printed or otherwise reproduits sans son autorisation. reproduced without the author's permission. In compliance with the Canadian Conformement a la loi canadienne Privacy Act some supporting sur la protection de la vie privee, forms may have been removed quelques formulaires secondaires from this thesis. ont ete enleves de cette these. While these forms may be included Bien que ces formulaires in the document page count, aient inclus dans la pagination, their removal does not represent il n'y aura aucun contenu manquant. any loss of content from the thesis. •*• Canada ©J. David Keselica, 2008 Windsor, Ontario, Canada Author's Declaration of Originality I hereby certify that I am the sole author of this thesis and that no part of this thesis has been published or submitted for publication. I certify that, to the best of my knowledge, my thesis does not infringe upon anyone's copyright nor violate any proprietary rights and that any ideas, techniques, quotations, or any other material from the work of other people included in my thesis, published or otherwise, are fully acknowledged in accordance with the standard referencing practices. Furthermore, to the extent that I have included copyrighted material that surpasses the bounds of fair dealing within the meaning of the Canada Copyright Act, I certify that I have obtained a written permission from the copyright owner(s) to include such material(s) in my thesis and have included copies of such copyright clearances to my appendix. I declare that this is a true copy of my thesis, including any final revisions, as approved by my thesis committee and the Graduate Studies office, and that this thesis has not been submitted for a higher degree to any other University or Institution. IV Abstract The foundations of quantum theory are closely tied to a formulation of clas­ sical relativistic physics. In Clifford's geometric algebra classical relativistic physics has a spinorial formulation that is closely related to the standard Dirac equation. The algebra of physical space, APS, gives clear insight into the quantum/classical interface. Here, APS is compared to other formula­ tions of relativistic quantum theory, especially the Dirac equation. These formulations are shown to be effectively equivalent to each other and to the standard theory, as demonstrated by establishing several isomorphisms. Dirac spinors are four-component complex entities, and so must be rep­ resented by objects containing 8 real degrees of freedom in the standard treatment (or 7 if a normalization constant is added). The relation 4 C -> Ch - Clfi3 cz C/Jj ~ H ® C. indicates that the 8-dimensional even subalgebra C7/~3 of the Space-time algebra, STA is isomorphic to APS CI3, which is isomorphic to complex quaternions HI © C. The complex quaternions should not be confused with the biquaternions, a name sometimes used for them. The biquaternions are more generally elements of the algebra H (B H. The algebras Cl\$ and C/3,1 are not isomorphic but their even sub-algebras are[l]. The Klein paradox is resolved in APS by considering Feynman's picture v of antiparticles as negative energy solutions traveling backward in time. It is also shown that the algebra of physical space can naturally describe an extended version of the De Broglie-Bohm approach to quantum theory. A relativistic causal account of a spin measurement in APS is given. The Stern- Gerlach magnet acts on the eigenspinor A field of a charged particle in a way that is analogous to the interaction of a birefringent medium acts on a beam of light. Then we introduce a covariant interpretation of complex algebra of physical space, CAPS, the complex extension of APS. This is done to solve a problem in that the space-time inversion, PT transformation, when P is parity inversion and T time reversal, although it is a proper transformation is not physical, yet, it has the form of a physical rotation in the traditional Dirac theory. The CAPS form of the PT transformation does not have the form of a physical rotation. A further problem is that the explicit form of the time inversion, T and charge conjugation, C transformations depends on the matrix representation chosen. A representation-independent form would be more fundamental. In CAPS, the complex extension of APS these problems are resolved. VI Acknowledgments I would like to thank my supervisor, Dr. W. E. Baylis, for his complete support and encouragement through all my PhD program. I would also like to thank Dr. J. Huschilt and my colleagues for their valuable suggestions and encouragement over the years. I also would like to give special thanks to my wife Annette, my son Alexander and my family for their love and support. Finally, I would like to thank the Physics Department for its support in providing a rich environment in which to work. VII Contents Author's Declaration of Originality iv Abstract v Acknowledgments vii 1 Introduction 1 2 Algebraic Background and the Dirac Equation 4 2.1 Clifford Algebras 5 2.2 Comparison of Different Formulations of the Dirac Equation . 9 2.3 Complex Quaternions 11 2.3.1 Quaternions 11 2.3.2 Complex Quaternions 15 2.3.3 Dirac Equation 17 2.4 Space-time Algebra 20 2.4.1 The Space/time Split 21 2.4.2 Dirac equation in STA 21 2.4.3 The Opposite Metric 25 2.5 The Algebra of Physical Space (APS) 30 2.5.1 Generating CI.3 31 viii 2.5.2 Complex Structure and Paravectors 35 2.5.3 Involutions of CI3 37 2.5.4 Dirac equation 40 2.5.5 Dirac Equation Relation to Standard Form 42 2.5.6 Equivalence of Dirac Theory to APS Version 44 3 Dirac Solutions in APS 45 3.1 Momentum Eigenstates 45 3.2 Standing Waves 47 3.3 Zitterbewegung 47 3.4 Klein Paradox 48 3.5 Basic Symmetry Transformations 54 4 De Broglie-Bohm Formulation 58 4.1 Pauli-Dirac equation 59 4.2 Stern-Gerlach Measurement 62 5 Complex Algebra of Physical Space 71 5.1 Conjugations 74 5.1.1 Metric 76 5.1.2 Matrix Representation of CAPS 76 5.1.3 Comparison of Space-Time Unit Vectors 77 5.2 CAPS Form of Dirac Equation 79 5.2.1 Relation to the Standard Form 81 5.3 Fundamental Symmetry Transformations 82 5.3.1 Momentum Eigenstates 85 5.4 Canonical Quantization •. 87 5.4.1 Annihilation and Creation Operators 90 IX 5.4.2 Number Operators 92 6 Conclusions 93 Bibliography 99 Vita Auctoris 103 x Chapter 1 Introduction Clifford (geometric) algebra is a powerful mathematical language for ex­ pressing physical ideas. It can unify diverse mathematical formalisms and provides geometric insights as well as physical ones. Clifford algebras art? associative algebras of vectors, where the vector product is generally non- commutative. Clifford algebras are extensions of vector spaces, complex numbers, quaternions and Grassmann algebras. William Kingdon Clifford (1845-1879) introduced them in part as an effort to unite the discovery of quaternions by William Rowan Hamilton (1805-1865) and the anticommut- ing variables of Herman Grassmann (1809-1879). Clifford algebras are well suited to modeling special relativity and quantum mechanics. The approach of recasting fundamental equations into Clifford algebras often reveals new insights that were previously obscured by the choice of the mathematical formalism used. In the case of the Dirac equation, Clifford algebras have been used to add to our understanding of the foundations of relativistie quantum theory. In particular, as we show below, it provides a useful tool for investigations of the quantum/classical (Q/C) interface[2][3]. Classical relativistie physics in Clifford's geometric algebra has a for- 1 mulation that is closely related to the standard quantum formalism. Tin1 present dissertation looks at how geometric approaches have been used to represent the Dirac equation. In chapter 2, we look at three different formu­ lations, first those based on complex quaternions, next the real even space- time algebra Cl^3, and then that based on the paravectors on the algebra of physical space(APS) CI3.

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