Quantum Mechanics from General Relativity An Approximation for a Theory of Inertia by Mendel Sachs Department of Physics and Astronomy, State University of New York at Buffalo, U. Reidel Publishing Company ta.t A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP " www.com Library of Congress Cataloging-in-Publication Data Sachs,Mendel. Quantum mechanics from general relativity.1'2 86-17902 ISBN 90-277-2247-1 Published by D. Reidel Publishing Company, P.
Box 17, 3300 AA Dordrecht, Holland. Sold and distributed in the U. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Assinippi Park, Norwell, MA 02061, U. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.
Box 322, 3300 AH Dordrecht, Holland. DEDICATED TO THE SIX MILLION MARTYRS · · · , A vraham, Chaje Taube, Rubin, Cili, .com Contents Preface x Chapter I / Fundamental Outlook 1 Chapter 2 / On the Comparison of the Quantum and Relativity Theories 7 2. Is the Quantum Jump Compatible with the Theory of Rela- tivity? 16 2. Is the Theory of Relativity Complete as a Theory of Matter? 19 2.
The Einstein-Podolsky-. Bohr's Reply to Einstein, Podolsky and Rosen 26 2. The Hidden Variable Approach 27 2. Bell's Inequalities and General Relativity 32 2.
The State Vector and Bell's Inequalities 33 Chapter 3 / Basis of a Matter Field Theory of Inertia - a Generalization of Quantum Mechanics 39 3. The General Mathematical Structure and Philosophical Implications 42 3. The Symmetry Group from Axiom 1 and Funda- mental Field Variables 42 3. The Generalized Mach Principle 46 3.
The Conservation of Interaction 47 3. Determinism 51 Chapter 4 / A Covariant Field Theory of Inertia 53 4. On the Origin of Inertial Mass 53 4. The Spinor Formalism in Special Relativity 56 V11 www.com Vll1 Contents 4.
The Spinor Variables in General Relativity 58 4. The Spinor Matter Field Equations in General Relativity 61 4. Matter and Antimatter 67 4. Proof of Force Symmetry of Matter and Antimatter 67 4.
On the Quantization of Electrical Charge from General Relativity 68 4. Conclusions 72 Chapter 5 / The Electromagnetic Interaction 74 5. On the Meaning of the Electromagnetic Field Equations 74 5. Generalization of the Elementary Interaction Formalism 76 5.
A Spinor Formulation of Electromagnetism 78 5. Invariants and Conservation Equations 80 5. The Interaction Lagrangian 83 5. The Electromagnetic Four-Potential 84 Chapter 6 / Quantum Mechanics from the Matter Field Equations and Derivation of the Pauli Exclusion Principle 88 6.
Approximations to Quantum Mechanics 89 6. The Free Field Limit 90 6. Coupling to an External Potential 92 6. The Pauli Exclusion Principle - a Derivation 93 6.
Sufficiency of the Three Conditions for Proof of the Pauli Principle 98 6. Fermi-Dirac Statistics from the Nonrelativistic Approximation for'll 99 6. The Hartree Approximation for the Matter Field Equations 101 6. Another Approximation for the Many-Electron Atom 103 6.
Scattering Cross Section 104 Chapter 7 / The Particle-Antiparticle Pair without Annihilation Creation 108 7. The Field Equations for the Particle-Antiparticle Pair 109 7. An Exact Bound State Solution for the Particle-Antiparticle Pair 112 7. The Energy and Momentum of the Bound Particle-Anti- particle in its Ground State 115 7.
The Free Particle Limit and Pair Creation 118 7. The Continuity of Energy Values 119 7. Rejection of the Photon Model in 'Pair Annihilation' 120 www.com Contents IX 7. Dynamical Properties of the Pair in the Ground State 121 7.
The Compton Effect 124 7. Blackbody Radiation - a Derivation of Plank's Law 125 7. The Anomalous Magnetic Moment of the Electron 130 Chapter 8 / The Electron-Proton System 134 8. Linearization of the Hydrogen Field Equations 135 8.
The Lamb Splitting 139 8. Deuterium and He+ 145 8. The Lifetimes of Atomic Excited States 146 8. Electron-Proton Scattering in a Vacuum 150 8.
Electron-Proton Scattering in a Background of Pairs 155 8. The Screening Effect of the Background Pairs on the e-p Interaction 156 8. The Generalized Electromagnetic Interaction 162 8. Summary 164 Chapter 9 / Elementary Particle Physics 167 9.
Polarization of the Pair Participation in the Neutron State 169 9. The Binding Energy of the Neutron 170 9. The Magnetic Moment of the Neutron 171 9. The Mass Ratio of Neutral to Charged Pions 174 9.
The Ratio of Neutral to Charged Pion Lifetimes 177 9. On the Possible Origin of CP-Violation in Neutral Kaon Decay 180 9. Neutral Kaon Decay 182 9. The Irreducible Spinor Matter Field Equations and CP-Violation 182 9.
The Generalized Electromagnetic Interaction 185 9. CP-Violation in Kaon Decay 186 9. Estimates of the Magnitude of CP-Violation m K~ Decay 188 9. On Time Reversal Noninvariance in Nuclear Forces - a Magnetic Resonance Experimental Test 190 9.
A Possible Source ofT-Violation in Nuclear Forces 192 9. Proton-Antiproton Collisions and the W±-Particle from General Relativity 193 9. Concluding Remarks 199 www.com x Contents Epilogue 200 Appendix A / Computation of the Lamb Splitting 207 Appendix B / Evaluation of the Scattering Correction Factor E( bq) 215 Bibliography 218 Index 222 www.com Preface This monograph is a sequel to my earlier work, General Relativity and Matter [1], which will be referred to henceforth as GRM. The monograph, GRM, focuses on the full set of implications of General Relativity Theory, as a fundamental theory of matter in all domains, from elementary particle physics to cosmology.
It is shown there to exhibit an explicit unification of the gravitational and electromagnetic fields of force with the inertial manifestations of matter, expressing the latter explicitly in terms of a covariant field theory within the structure of this general theory. This monograph will focus, primarily, on the special relativistic limit of the part of this general field theory of matter that deals with inertia, in the domain where quantum mechanics has been evoked in contemporary physics as a funda- mental explanation for the behavior of elementary matter. Many of the results presented in this book are based on earlier published works in the journals, which will be listed in the Bibliography. These results will be presented here in an expanded form, with more discussion on the motivation and explanation for the theoretical development of the subject than space would allow in normal journal articles, and they will be presented in one place where there would then be a more unified and coherent explication of the subject.
It goes without saying that Quantum Mechanics has been one of the outstanding successes of twentieth century physics - in its correctly predict- ing and representing many of the atomic, nuclear and elementary particle phenomena. From the point of view of the Philosophy of Science, it is indeed a necessary condition for any valid scientific theory to meet that it should accurately predict the empirical data relating to particular physical phenomena, if it is to claim to be a (scientifically) true explanation for these phenomena. Nevertheless, it is important to recognize that this requirement is not a sufficient condition to establish its scientific validity. For a valid theory in science must also be (1) logically and mathematically consistent, and (2) it should be successful in its full spectrum of potential predictions; that is to say, if some of its predictions should be verified and others not, the entire theory should then be subject to question.com Xli Preface In spite of the outstanding numerical successes of quantum mechanics in fitting the data of elementary matter experimentation, it has not been able to meet the criteria of consistency and completeness mentioned above, at least to this date.
As we will discuss in Chapter 2, the extension of nonrelativistic quantum mechanics to the relativistic domain, that is a necessary extension for the logical consistency of the theory, on its own terms, entails a breakdown of the essential logical and mathematical ingredients of the quantum theory, and indeed yields a mathematical formalism that has no solutions. Since the quantum theory, if generally true as a theory of elemen- tary matter, should apply equally to the relativistic region of elementary matter phenomena as to nonrelativistic phenomena, and since this has not been accomplished yet (for reasons that will be discussed in Chapter 2), in the form of a relativistically covariant 'quantum field theory' that would satisfy the requirements of both the quantum theory and the theory of relativity, simultaneously, it must be admitted by the objective scientist that the quantum theory has not yet established itself as a fundamental theory of elementary matter, even though it is an empirically correct description of atomic and elementary particle phenomena under particular experimental circums tances. In addition to the empirical successes of low energy (nonrelativistic) quantum mechanics, over the past 60 years of physics research, there has been a great deal of success in phenomenological approaches to high energy elementary particle physics, though always in the context of the quantum theory. These discoveries have entailed new kinds of 'hidden symmetries' in the expanded spaces to describe the probability functions of elementary particles [2].
Further, to classify their species, proposals are made about (a) new types of particles involved in the classification of strongly interacting particles, that make up those particles, though 'confined' to their domains (the 'quarks'), (b) a generalization of quantum electrodynamics to incorporate the quarks, called 'quantum chromodynamics' [3], (c) generalizations of the gauge symmetry so as to unify the phenomenological description of the weak and electromagnetic interactions [4], etc. If a new theory of matter is to replace the quantum theory, it must still yield the correct empirical data as predictions, much of it thus far represented brilliantly by the present day phenomenological schemes in high energy particle physics. These current researches in theoretical particle physics must then serve as a guide toward the form of an underlying theory of elementary matter, at least insofar as they represent the empirical data. The main aim of the research reported in this monograph is to present a fundamental theory of elementary matter, in terms of underlying principles, rather than taking a phenomenological approach.
It will be shown that such a theory, based fundamentally on the starting ideas of the theory of general relativity, as a theory of matter, does indeed lead to the formal structure of quantum mechanics - as a linear approximation for the part of this field www.com Preface XlII theory of matter that is associated with the phenomenon of inertia. In this way, seve!al of the features of matter in the microscopic domain are derived from first principles, rather than being imposed from the outset to fit the data. In Chapter 2, after comparing the underlying concepts of the quantum and relativity theories, there will be a discussion of the critique of Einstein, Podolsky and Rosen, on the Copenhagen view of quantum mechanics, and thence to Bohr's rejoinder. This will then lead to a brief outline of the program of hidden variable theories, and separated from the resolution of this monograph, which is toward the underlying basis of general relativity (which entails 'exposed variables' instead, in the fashion of the Einstein field theory).
Bell's inequalities will then be discussed in the context of their use as an asymptotic limit of the nonlinear field theory of matter implied by general relativity. In Chapters 3, 4, and 5, the logical and mathematical development of general relativity, as a theory of elementary matter, ·will be presented, including the new consequences as a result of incorporating the Mach principle. This leads to the new idea, expressed as the law of conservation of interaction (to replace the conservation of probability of the standard quantum view), and the derivation of the nonlinear inertial field equations will be demonstrated, whose linear limit is the formal structure of quantum mechanics, then to the full expression of the electromagnetic field equations that fully exploits the Mach principle in general relativity. It will be seen in Chapter 6 how this theory of inertia leads to the formal expression of quantum mechanics, as a low energy approximation.