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P1:BinayaDash November1,2006 10:2 C7729 C7729˙C000 Geometric Algebra and Applications to Physics P1:BinayaDash November1,2006 10:2 C7729 C7729˙C000 P1:BinayaDash November1,2006 10:2 C7729 C7729˙C000 Geometric Algebra and Applications to Physics V S ENZO DE ABBATA B K D IDYUT UMAR ATTA P1:BinayaDash November1,2006 10:2 C7729 C7729˙C000 CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2007 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 1-58488-772-9 (Hardcover) International Standard Book Number-13: 978-1-58488-772-0 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. 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Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data De Sabbata, Venzo. Geometric algebra and applications to physics / Venzo de Sabbata and Bidyut Kumar Datta. p. cm. Includes bibliographical references and index. ISBN 1-58488-772-9 (alk. paper) 1. Geometry, Algebraic. 2. Mathematical physics. I. Datta, Bidyut Kumar. II. Title. QC20.7.A37D4 2006 530.15’1635--dc22 2006050868 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com P1:BinayaDash November1,2006 10:2 C7729 C7729˙C000 TheauthorswithPeterGabrielBergmann. Fromtheleft:Venzo,Peter,Datta. P1:BinayaDash November1,2006 10:2 C7729 C7729˙C000 P1:BinayaDash November1,2006 10:2 C7729 C7729˙C000 Preface Thisisatextbookongeometricalgebrawithapplicationstophysicsandserves also as an introduction to geometric algebra intended for research workers in physics who are interested in the study of this modern artefact. As it is extremelyusefulforallbranchesofphysicalscienceandveryimportantfor the new frontiers of physics, physicists are very much getting interested in thismodernmathematicalformalism. ThemathematicalfoundationofgeometricalgebraisbasedonHamilton’s andGrassmann’sworks.Cliffordthenunifiedtheirworksbyshowinghow Hamilton’s quaternion algebra could be included in Grassmann’s scheme throughtheintroductionofanewgeometricproduct.Theresultingalgebra is known as Clifford algebra (or geometric algebra) and was introduced to physicsbyHestenes.ItisacombinationofthealgebraicstructureofClifford algebra and the explicit geometric meaning of its mathematical elements at its foundation. Formally, it is Clifford algebra endowed with geometrical information of and physical interpretation to all mathematical elements of thealgebra. It is the largest possible associative algebra that integrates all algebraic systems (algebra of complex numbers, matrix algebra, quaternion algebra, etc.)intoacoherentmathematicallanguage.Itspotencyliesinthefactthatit canbeusedtodevelopallbranchesoftheoreticalphysicsenvisaginggeomet- rical meaning to all operations and physical interpretation to mathematical elements.Forinstance,thespinortheoryofrotationsandrotationaldynamics canbeformulatedinacoherentmannerwiththehelpofgeometricalgebra. Oneimportantfactistodeveloptheproblemofrotationsinrealspace-time intermsofspinors,whichareevenmultivectorsofspace-timealgebra.This fact is extremely important because it allows us to put tensors and spinors onthesamefooting:anecessarythingwhenwe,throughtorsion,introduce spininthegeneraltheoryofrelativity. Thislaterargumentseemstobeveryimportantwhenwewilltrytocon- sideraquantumtheoryforgravity.Moreover,theproblemofrotationsinreal space-time allows us to explain the neutron interferometer experiments in whichweknowthatafermiondoesnotreturntoitsinitialstatebyarotation of2π;infact,ittakesarotationof4π torestoreitsstateofinitialcondition. Geometric algebra provides the most powerful artefact for dealing with rotations and dilations. It generalizes the role of complex numbers in two dimensions, and quaternions in three dimensions, to a wider scheme for dealingwithrotationsinarbitrarydimensionsinasimpleandcomprehensive manner. P1:BinayaDash November1,2006 10:2 C7729 C7729˙C000 Thestrikingadvantageofanentirely“real”formalismoftheDiracequa- tion in space-time algebra (geometric algebra of “real” space-time) without using complex numbers is that the internal phase rotations and space-time rotationsareconsideredinasingleunifyingframecharacterizingtheminan identicalmanner. However,otherimportantphysicalinterpretationsarebasedongeometric algebra as we will show in this book. For instance, geometric algebra (GA) and electromagnetism, GA and polarization of electromagnetic waves, GA andtheDiracequationinspace-timealgebra,GAandquantumgravity,and also,GAinthecaseoftheMajorana–Weylequations,tomentiononlyafew. VenzodeSabbata BidyutKumarDatta P1:BinayaDash November1,2006 10:2 C7729 C7729˙C000 Introduction Therearemanycompetingviewsoftheevolutionofphysics.Someholdthe perspectivethatadvancesinitcomethroughgreatdiscoveriesthatsuddenly openvastnewfieldsofstudy.Othersseeaveryslow,continuousunfolding of knowledge, with each step along the path only painstakingly following its predecessor. Still others see great swings of the pendulum, with interest movingalmostcollectivelyfromtheoriginaledificeofclassicalphysicstothe 20thcenturydominanceofquantummechanics,andperhapsnowbackagain towardssomeintermediategroundheldbynonlineardynamicsandtheories of chaos. Superimposed on all of this, of course, is the overriding theme of unification,whichmostclearlymanifestsitselfinthequestforatheorythat fullyunifiesthebestdescriptionsofalltheknownforcesofnature. However,thereisstillanotherkindofevolutionofthoughtandunification of theory that has quietly yet effectively gone forward over the same scale of time, and it has been in the very mathematics itself used to describe the physicalattributesofnature.JustasNewtonandLeibnizintroducedcalculus in order to provide a centralized, rigorous framework for the development of mechanics, so have many others conceived of and applied ever-refined mathematical techniques to the needs of advancing physical science. One suchdevelopmentthatisonlynowbeginningtobetrulyappreciatedisthe adaptation by Clifford of Hamilton’s quaternions to Grassmann’s algebraic theory, which resulted in his creation of a geometric form of algebra. This powerfulapproachusestheconceptsofbivectorsandmultivectorstoprovide amuchsimplifiedmeansofexploringanddescribingawiderangeofphysical phenomena. Although several modern authors have done a great deal to introduce geometricalgebratothescientificcommunityatlarge,thereisstillroomfor effortsfocusedonbringingitmoreintothemainstreamofphysicspedagogy. ThefirststepsinthatdirectionwereoriginallytakenbyDavidHesteneswho wrote what have become classic books and papers on the subject. As the topic gets further incorporated into undergraduate and graduate curricula, theneedarisesfortheongoingdevelopmentoftextbooksforuseincovering thematerial.Amongtheauthorswhohaverecognizedthisneedandactedon itareVenzodeSabbataoftheUniversityofBolognaandBidyutKumarDatta of Tripura University in India, and the publication of their book Geometric AlgebraandItsApplicationstoPhysicsisthesatisfyingresult. The authors are well known for their research in general relativity. The roles of torsion and intrinsic spin in gravity have been recurring themes, especiallyintheworkofdeSabbata,andthesetopicshaveplayedacentral roleintheinterestingapproachesthathe,Datta,andothershavetakentothe