An Introduction to Clifford Algebras and Spinors

257 Pages·2016·1.492 MB·English

by  Jayme Vaz, Roldao da Rocha

AN INTRODUCTION TO CLIFFORD
ALGEBRAS AND SPINORS

An Introduction
to Clifford Algebras and Spinors
Jayme Vaz, Jr.
IMECC,UniversidadeEstadualdeCampinas,Campinas,SP,Brazil
Roldão da Rocha, Jr.
CMCC–UniversidadeFederaldoABC,SantoAndré,SP,Brazil
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Preface
In 1878 William Kingdom Clifford published an article in the American Journal
of Mathematics entitled ‘Applications of Grassmann’s Extensive Algebra’ (Clifford,
1878). Clifford presented a new mathematical structure which he called ‘Geometric
Algebra’, but which is now usually known as Clifford Algebra. Some years earlier,
in 1873, Clifford had published in Proceedings of the London Mathematical Society
an article called ‘Preliminary Sketch of Biquaternions’, where he presented the first
rudiments of his ideas. These later evolved in 1876 in two unpublished manuscripts –
called ‘Further note on so-called biquaternions’ and ‘On the classification of geomet-
ric algebra’, respectively (published posthumously as a single article entitled ‘On the
Classification of Geometric Algebras’ (Clifford, 1882)) – culminating in the ideas pre-
sentedinthe1978article.Briefly,whatCliffordaccomplishedwasthesynthesisoftwo
apparentlydissociatedmathematicalstructures:thequaternionsofSirWilliamRowan
Hamilton, and the algebra of extensions (Ausdehnungslehre) of Hermann Grassmann.
WecantracetheoriginoftheCliffordalgebrastotheeffortstogeometricallyrepresent
a complex number.
TheideaofunifyinggeometricandalgebraicoperationswasfirstadvocatedbyGot-
tfriedLeibnizin1679,inalettertoChristiaanHuygens(thisletterwaslaterpublished
in 1833). This idea was manifested for complex numbers in terms of the Argand, or
Argand–Gauss plane, which has been given the name tribute to the results published
onthistopicin1806byJean-RobertArgandandthenin1831byCarlFriedrichGauss.
Meanwhile, the first to succeed in representing the complex numbers in a plane was
the Norwegian Caspar Wessel, who presented his results in 1797 to the Royal Danish
AcademyofSciencesandLetters,inhiswork‘Om Directionens analytiske Betegning’.
Thisworkwaspublishedin1799butremainedinobscurityuntilitwastranslatedinto
French in 1897. The success of the geometrical representation of complex numbers led
Hamilton to search for a generalisation of this approach to three-dimensional space.
Aftersomeyearsoffailedattempts,Hamiltonsucceededin1843withthediscoveryof
quaternions.1 Afterwards, Hamilton (and his followers) dedicated years to developing
the theory and applications of quaternions. Meanwhile, in 1844 Grassmann published
his outstanding work, Die lineale Ausdehnungslehre, (Grassmann, 1844, 1894). In this
work – influenced by ideas from his father, Justus Grassmann – he introduced an
algebraic system based on a geometric product that, because of its innovative and
abstract character, proved to be difficult to understand at the time. In an attempt
to facilitate the knowledge of his work, Grassmann presented in 1862 a new version
of Die lineale Ausdehnungslehre, which he called Die Ausdehnungslehre: Vollstandig
und in strenger Form bearbeitet (Grassmann, 1862). Nevertheless, Grassmann’s work
1The present-day formalism of vector algebra was extracted from the quaternion product of two
vectors,byJosiahWillardGibbsin1901.
viii Preface
remained obscure for but a few like Clifford. Interestingly, in 1844, after the publi-
cation in 1833 of Leibniz’s 1679 letter to Huygens, the Jablonowskischen Gesellschaft
der Wissenschaft offered an award to those who developed Leibniz’s ideas. The prize
was granted to Grassmann in 1846 for taking Leibniz’s ideas into the realm of the
algebraic system presented in Die lineale Ausdehnungslehre and was later published
as a separate work (Grassmann, 1847). Yet, this award was not enough to draw the
attention of the majority of the scientific community to Grassmann’s work.
MoreinformationontheworkofHamilton,Grassmannandotherimportantnames
in the history of the vector analysis can be found in, for example, the book by Crown
(1994). Clifford’s major achievement in 1878 was essentially introducing a quaternion
framework into Grassmann’s algebra of extensions, thus obtaining a system naturally
adaptedtotheorthogonalgeometryofanarbitraryspace:inthissystem,quaternions
represent merely a very particular case of a Clifford algebra. Furthermore, one of the
many (at least ten) products defined by Grassmann is the so-called ‘stereometrisches
Produkt’,whichimplementsquaternionmultiplicationforthreegenerators,andisthus
itselfaCliffordproduct.Unfortunately,Cliffordwasunabletocontinuehisworkashe
diedinthefollowingyear,attheageof33.However,itwasnotlongbeforetheClifford
algebras were reinvented. In 1880, Lipschitz studied representations of rotations by
complexnumbersandquaternions,andtheirgeneralisationstohighdimensionsledto
the Clifford geometric algebra and to the group Spin.
Many scientists have contributed to the development of the Clifford algebras. One
ofthemostimportantiscertainlyE´lieCartan.Aswellasmakingothercontributions,
he described Clifford algebras as algebras of matrices and found the periodicity 8 of
thesealgebras(theperiodicitytheorem).Cartanintroducedin1913theconceptofthe
spinor, and in 1938 the concept of the pure spinor. Although the term ‘spinor’ had
been coined by Paul Ehrenfest in the 1920s, the intrinsic concept of a spinor is much
older than that, as a spinor is a special linear structure, which had been studied (and
had been used in civil engineering for calculating statics) long before Ehrenfest used
it in quantum theory.
It was through the concept of the spinor that Clifford algebras had a decisive
presenceinscience.ItwasprominentlyWolfgangPauliwhointroducedtheconceptof
spintophysics,firsttoexplain,forexample,theZeemaneffect,andlatertoformulate
theexclusionprinciple(hisNobel-prizewinningwork).PaulDiracshowedin1928that
the main equation in relativistic quantum mechanics is written in terms of a Clifford
algebraandthattheelectronisdescribedbyaspinor,orrather,byaspinorfield(Dirac,
1928). Cornelius Lanczos rewrote Dirac’s equation in terms of quaternions (Gsponer
andHurni,1994)in1929,andGustaveJuvetandFritzSauterin1930replacedcolumn
spinors by square matrix spinors, in minimal left ideals, where only the first column
containednon-zeroelements(Lounesto,2001a).MarcelRieszin1947wasthefirstone
toconsiderspinorsaselementsofaminimalleftidealinaCliffordalgebra.Thereafter,
Feza Gu¨rsey (1956, 1958) expressed the Dirac equation in terms of 2×2 quaternionic
matrices.AfterKustaanheimopresentedthespinorregularisationofKeplermotionin
1964, proposing the so-called Kustaanheimo–Stiefel transformation, David Hestenes
(1966, 1967) reformulated the Dirac theory in this context, providing new aspects
and insights. Since then, a variety of other applications of Clifford algebras have been
Preface ix
found, not only in physical sciences, but also in engineering and computation (Doran
and Lasenby, 2003; Baylis, 1996; Dorst, Fontijne, and Mann, 2007; Perwass, 2008).
Thisbookisdividedintosixchapters.Inthefirstone,areviewofthemainconcepts
and results from linear algebra is presented, as these are essential in order to develop
thesubsequentideasinthisbook.Then,tensoralgebraisintroduced,asitisrequired
for the definitions of exterior algebras and Clifford algebras. The second chapter is
dedicated to the presentation of exterior algebras and Grassmann algebras. Although
someauthorsusethesetermsassynonyms,thatwillnotbethecasehere.Theexterior
algebra is understood in this book as a structure devoid of a metric, whereas Grass-
mannalgebrasarestructuresconstructedonavectorspaceendowedwithametric.In
this chapter, some fundamental concepts, such as the exterior product, the contrac-
tion, and the quasi-Hodge isomorphism, and the Hodge isomorphism are introduced.
Clifford algebras are then defined in the third chapter, together with their properties.
Three different definitions of real Clifford algebras are introduced, emphasising the
numberofexcellentfeaturespossessedbyCliffordalgebras.Inthefourthchapter,the-
orems about the structure of Clifford algebras are presented; these theorems are later
used for the classification and representation of the Clifford algebras in terms of ma-
trix algebras. Moreover, we introduce procedures for explicitly building these matrix
representations, as these are important when it comes to computations in physics and
otherapplications.The groupsassociatedwiththeCliffordalgebras aretheobjectsof
discussion in the fifth chapter. The groups Pin and Spin are comprehensively exam-
ined.Inaddition,theLiealgebrasassociatedwiththosegroupsarediscussedandsome
of their applications provided. In particular, conformal transformations and twistors
are introduced. The sixth chapter is dedicated to the study of spinors. Three different
definitions of spinors are presented, as well as the properties associated with each def-
inition. In addition, the relations between the different definitions are examined. The
so-called pure spinors are also introduced and the triality principle and the Penrose
flagpole are discussed. A detailed study of the so-called Weyl spinors, which form the
basisofthePenroseandRindlerformalism(PenroseandRindler,1984),concludesthis
chapter. Finally, in an appendix, the Lorentz transformations and the classical coun-
terparts of dotted/undotted Weyl spinors, the Penrose flagpole and supersymmetry
algebra are presented, in the context of the Van der Waerden framework.
This book is based on lecture notes written by Jayme Vaz Jr for two courses on
Clifford algebras in 1999 and 2005 at the Institute of Mathematics, Statistics and Sci-
entific Computation (IMECC) at the University of Campinas (Unicamp). From 2008
to 2015 Rold˜ao da Rocha Jr used and improved the lecture notes, in particular for
the courses ‘Clifford Algebras and Spinors’, ‘Spinors in Hilbert Spaces’, and ‘Quan-
tumFieldTheory’forgraduateprogramsattheCenterofMathematics,Computation
and Cognition and at the Center of Natural and Human Sciences at ABC Federal
University, Santo Andr´e, Brazil, in the process of completing and ameliorating the
manuscript, as well as adding topics. Those who are familiar with the Clifford alge-
bras and their applications certainly will realise that an important topic, the Dirac
operators, which play a significant role in many areas of modern mathematics and
physics, is missing. The reason for its absence is that, although a discussion of the
Dirac operators would have been welcome in the course, it was beyond the scope of