Spin-1/2 Wave Equations in Relativistic Quantum Mechanics Donald Stephen Staudte June, 1993 A thesis submitted for the degree of Doctor of Philosophy of The Australian National University. S tatem en t The (1972) idea to construct a spin-1/2 analogue of the spin-0 Feshbach- Villars equation belongs to Dr. B. A. Robson and equation l.VII.l was his (1972) attempt at the construction of such an equation. He (1991) suggested to me that I attempt to obtain the energy spectrum of the hydrogen atom using equation l.VII.l. The (1990) idea to study the representation theory of the Lorentz group belongs to Dr. J. M. C. F. Govaerts. However, all of the material given in chapters 2, 3, 4, 5 and 6 is solely my own original work, except where explicitly acknowledged otherwise. I developed this material in the period from July 1992 to June 1993. My development of this material has benefited from discussions with a number of people who are mentioned in the acknowledgements. Acknowledgements I would like to thank my supervisor Dr. B. A. Robson for introducing me to his problem of developing a spin-1/2 wave equation with indefinite metric in rela­ tivistic quantum mechanics, and for his patience and initial guidance to enable me to understand his studies of the subject. I thank him also in his role as Head of Department for making it possible for me to undertake a PhD at the Australian National University and for arranging the financial support. I would also like to thank Professor C. Jarlskog and Rickard Lindkvist for ar­ ranging a most fruitful and rewarding six month visit to Stockholms Universitet, during the period March-September 1992. It was during this period that I derived the first major results of this thesis. I also wish to thank Dr. J. M. C. F. Govaerts for his invaluable insights, inspira­ tion and constant encouragement throughout the course of this thesis. I wish to thank Dr. A. M. Pendrill for arranging a research position for me in Göteborg, beginning August 1993. This has allowed me to concentrate on my thesis over the last few months, without having to find future employment as well. In my development of the material given in chapters 2, 3, 4, 5 and 6, I have ben­ efited from invaluable discussions with and comments from the following people: Professors K. I. Golden, C. Jarlskog, A. Klein, B. Laurent, I. Lindgren, B. H. J. McKellar, P. Minnhagen, J. Sucher, J. A. Wheeler; Drs F. Bastianelli, J. M. C. F. Govaerts, C. J. Griffin, H. Hansson, H. H. Lee, U. Lindgren, E. Lindroth, H. Nordstrom, A. M. Pendrill, R. G. Staudte, B. Sund- borg; and P. A. Blackmore, J. P. Costella, J. A. Daicic, D. J. Daniel, K. Halliwell, S. Lane, K. Laughlin and S. A. Moten. I wish to thank Drs M. Andrews, J. M. C. F. Govaerts and B. A. Robson for their assistance during the proofreading stage of this thesis. I wish to thank my fellow students at the ANU; Perry and Kim, Kenneth and Kutluyil; Damian, Derek, Kang-Mei, Ken and Simon; Kit and the posture and flexibility group; Blair and all my other friends at the ANU; also those at Stock­ holm; Bo, Erik, Fiorenzo, Hakan and Jan; for all the fun times. I wish to thank the ACT Academy of Sport, Perry Blackmore, Steve Craig, Rob Jessop, Rickard Lindkvist and all my other friends who have contributed to these rewarding years of my life. I wish to gratefully acknowledge Sally Moten for making my life wonderful. Finally I wish to thank my family for their encouragement and support. Abstract This thesis is concerned with the development of a new spin-1/2 wave equa­ tion in relativistic quantum mechanics. This equation is the spin-1/2 analogue of the spin-0 Feshbach-Villars equation. The thesis begins in chapter 1 with a review of the subject of relativistic wave equations and its place in the development of modern physics. Chapter 2 sees the derivation of the new equation (hereafter referred to as the FV1/2 equation). This derivation is presented at three levels of understanding, and is combined with an analysis of the equation using elements of the underlying mathematics of the theories of special relativity and quantum mechanics. The analysis justifies the methods of derivation of the equation, and its suitability as a wave equation in relativistic quantum mechanics. It is shown that the FV1/2 equation is different to the Dirac equation and other equations in the literature. One of the striking features of the FV1/2 equation is that it contains a less restrictive dynamics than the conventional Dirac equation. Chapter 3 examines the solution space and proves some results using the discrete symmetries C, P and T, which enable a subset of the solution space to be used for physical applications. It is possible to use a subset of solutions which contain the usual Dirac negative energy states, or it is possible to use a subset which contains only positive energy states, but includes states of opposite signs of charge. The second subset corresponds to the physical properties of particles and antiparticles and suggests that the FV1/2 equation could be useful in atomic physics. Chapter 4 represents a start on the development of the FV1/2 equation for use in atomic physics. It contains a study of hydrogenic atoms. The energy spectra and wavefunctions for hydrogenic atoms are derived using the FV1/2 equation. The method of solution is of comparable difficulty to that using the Dirac equation. The spectra are found to be identical to that of the Dirac equation, but the wavefunctions differ. A calculation of the expectation value of the Coulomb energy is performed and the results obtained differ from the Dirac equation results by 6 x 10-3% for Z — 1, and 40% for Z = 70. This suggests that physically measureable quantities should be calculated and compared using the FV1/2 and Dirac equations. A calculation is presented to show the energy spectra and wavefunctions obtained are consistent with the literature. The results obtained in this chapter (again) suggest that the FV1/2 equation could be useful for calculations in atomic physics. Chapter 5 discusses important further work which should be undertaken if the FV1/2 equation is to become useful to physics. The development in the previous chapters concentrated on the FV1/2 equation in relativistic quantum mechanics. It should be also considered as a quantised field equation. The two electron problem and the calculation of physical quantities are crucial tests of the FV1/2 equation’s applicability in atomic physics. The mathematical analysis in chapter 2 can be extended. Suggestions for gaining a deeper mathematical understanding of the less restrictive dynamics of the FV1/2 equation are given. Finally some conclusions on what has been achieved in the thesis are pre­ sented in chapter 6. Table of Contents Title page Statement of originality of research Acknowledgements Abstract Table of Contents Preface Notation Chapter 1. Introduction 1.1. The historical development of relativistic wave equations............................... 1 1 .II. A brief overview of R Q M ......................................................................................... 4 1 .III. Operators and expectation values in RQM ...................................................... 7 l.IV . Free particle solutions in R Q M ..............................................................................8 l.V. The non-relativistic lim it....................................................................................... 10 l.V I. Historical development, continued...................................................................... 11 l.VII. Initial motivations for a spin-1/2 analogue of the FVO eq u atio n ............................................................................................................ 13 1. VIII. Outline of this th esis........................................................................................... 17 Chapter 2. Derivations and analysis of the FV1/2 equation 2.1. Introduction................................................................................................................... 21 2.II. A first derivation of the FV1/2 equation .......................................................... 25 2.III. The KG1/2 and Dirac equations in spinor notation .....................................27 2.IV. Comparison of the solutions of the KG 1/2 and Dirac equations...................................................................................... 31 2. V. A second derivation of the FV1/2 equation................................................... 32 2.VI.. The indefinite metric formalism in RQM ........................................................ 36 2.VII. A third derivation of the FV1/2 equation...................................................... 49 2.VIII. Various linearisation procedures applied to the second order equations.......................................................................................... 53 2.IX. Operators and expectation values for the FV1/2 equation ....................... 59 2.X. A study of the FV1/2 equation using the representation theory of the Lorentz group ........................................................ 64 2. XL The eight solutions................................................................................. 65 Chapter 3. The use of the FV1/2 equation in physical problems 3.1. Introduction..........................................................................................................69 3.11. The number of solutions..................................................................................70