space time algebra second edition pdf

It may be helpful to refer to STA as the “Real Dirac Algebra”,because it is isomorphic to the algebra generated by Dirac γ-matricesover the field of real numbers instead of the complex n

Space-Time Algebra

David Hestenes

Second Edition

David Hestenes

Space-Time Algebra Second Edition

Foreword by Anthony Lasenby

Department of Physics

Arizona State University

Tempe, AZ

USA

Originally published by Gordon and Breach Science Publishers, New York, 1966

ISBN 978-3-319-18412-8 ISBN 978-3-319-18413-5 (eBook)

DOI 10.1007/978-3-319-18413-5

Library of Congress Control Number: 2015937947

Mathematics Subject Classi fication (2010): 53-01, 83-01, 53C27, 81R25, 53B30, 83C60

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It is a pleasure and honour to write a Foreword for this new edition

of David Hestenes’ Space-Time Algebra This small book started aprofound revolution in the development of mathematical physics, onewhich has reached many working physicists already, and which standspoised to bring about far-reaching change in the future

At its heart is the use of Clifford algebra to unify otherwise parate mathematical languages, particularly those of spinors, quater-nions, tensors and differential forms It provides a unified approachcovering all these areas and thus leads to a very efficient ‘toolkit’ foruse in physical problems including quantum mechanics, classical me-chanics, electromagnetism and relativity (both special and general) –only one mathematical system needs to be learned and understood,and one can use it at levels which extend right through to current re-search topics in each of these areas Moreover, these same techniques,

dis-in the form of the ’Geometric Algebra’, can be applied dis-in many areas

of engineering, robotics and computer science, with no changes sary – it is the same underlying mathematics, and enables physicists

neces-to understand neces-topics in engineering, and engineers neces-to understand neces-ics in physics (including aspects in frontier areas), in a way which noother single mathematical system could hope to make possible

top-As well as this, however, there is another aspect to GeometricAlgebra which is less tangible, and goes beyond questions of math-ematical power and range This is the remarkable insight it gives tophysical problems, and the way it constantly suggests new features ofthe physics itself, not just the mathematics Examples of this are pep-pered throughout Space-Time Algebra, despite its short length, andsome of them are effectively still research topics for the future As

an example, what is the real role of the unit imaginary in quantum

mechanics? In Space-Time Algebra two possibilities were looked at,and in intervening years David has settled on right multiplication bythe bivector iσ3 = σ1σ2 as the correct answer between these two, thusencoding rotation in the ‘internal’ xy plane as key to understandingthe role of complex numbers here This has stimulated much of hissubsequent work in quantum mechanics, since it bears directly on thezwitterbewegung interpretation of electron physics [1] Even if we donot follow all the way down this road, there are still profound ques-tions to be understood about the role of the imaginary, such as thegeneralization to multiparticle systems, and how a single correlated ‘i’

is picked out when there is more than one particle [2]

As another related example, in Space-Time Algebra David hadalready picked out generalizations of these internal transformations,but still just using geometric entities in spacetime, as candidates fordescribing the then-known particle interactions Over the years since,

it has become clear that this does seem to work for electroweak theory,and provides a good representation for it [3,4,5] However, it is notclear that we can yet claim a comparable spacetime ‘underpinning’for QCD or the Standard Model itself, and it may well be that somenew feature, not yet understood, but perhaps still living within thealgebra of spacetime, needs to be bought in to accomplish this.The ideas in Space-Time Algebra have not met universal acclaim

I well remember discussing with a particle physicist at ManchesterUniversity many years ago the idea that the Dirac matrices really rep-resented vectors in 4d spacetime He thought this was just ‘mad’, andwas vehemently against any contemplation of such heresy For some-one such as myself, however, coming from a background of cosmologyand astrophysics, this realization, which I gathered from this shortbook and David’s subsequent papers, was a revelation, and showed

me that one could cut through pages of very unintuitive spin lations in Dirac theory, which only experts in particle theory would

calcu-be comfortable with, and replace them with a few lines of intuitivelyappealing calculations with rotors, of a type that for example an engi-neer, also using Geometric Algebra, would be able to understand andrelate to immediately in the context of rigid body rotations

A similar transformation and revelation also occurred for me withrespect to gravitational theory Stimulated by Space-Time Algebra and

Foreword vii

particularly further papers by David such as [6] and [7], then alongwith Chris Doran and Stephen Gull, we realized that general relativ-ity, another area traditionally thought very difficult mathematically,could be replaced by a coordinate-free gauge theory on flat spacetime,

of a type similar to electroweak and other Yang-Mills theories [8].The conceptual advantages of being able to deal with gauge fields in

a flat space, and to be able to replace tensor calculus manipulationswith the same unified mathematical language as works for electromag-netism and Dirac spinors, are great This enabled us quickly to reachinteresting research topics in gravitational theory, such as the Diracequation around black holes, where we found solutions for electronenergies in a spherically symmetric gravitational potential analogous

to the Balmer series for electrons around nuclei, which had not beenobtained before [9]

Of course there are very clever people in all fields, and in bothrelativistic quantum mechanics and gravity there have been those whohave been able to forge an intuitive understanding of the physics de-spite the difficulties of e.g spin sums and Dirac matrices on the onehand, and tensor calculus index manipulations on the other However,the clarity and insight which Geometric Algebra brings to these areasand many others is such that, for the rest of us who find the alternativelanguages difficult, and for all those who are interested in spanningdifferent fields using the same language, the ideas first put forward in

’Space-time Algebra’ and the subsequent work by David, have beenpivotal, and we are all extremely grateful for his work and profoundinsight

Anthony LasenbyAstrophysics Group, Cavendish Laboratory

and Kavli Institute for Cosmology

Spacetime Algebra and Electron Physics In: P.W Hawkes (ed.),Advances in Imaging and Electron Physics, Vol 95, p 271 (Aca-demic Press) (1996).

[3] D Hestenes, Space-Time Structure of Weak and ElectromagneticInteractions, Found Phys 12, 153 (1982)

[4] D Hestenes, Gauge Gravity and Electroweak Theory In: H nert, R T Jantzen and R Ruffini (eds.), Proceedings of theEleventh Marcel Grossmann Meeting on General Relativity, p 629(World Scientific) (2008)

Klei-[5] C Doran and A Lasenby, Geometric Algebra for Physicists(Cambridge University Press) (2003)

[6] D Hestenes, Curvature Calculations with Spacetime Algebra, ternational Journal of Theoretical Physics 25, 581 (1986).[7] D Hestenes, Spinor Approach to Gravitational Motion and Pre-cession, International Journal of Theoretical Physics 25, 589(1986)

In-[8] A Lasenby, C Doran, and S Gull, Gravity, gauge theories andgeometric algebra, Phil Trans R Soc Lond A 356, 487 (1998).[9] A Lasenby, C Doran, J Pritchard, A Caceres and S Dolan,Bound states and decay times of fermions in a Schwarzschild blackhole background, Phys Rev D 72, 105014 (2005)

Preface after Fifty years

This book launched my career as a theoretical physicist fifty yearsago I am most fortunate to have this opportunity for reflecting on itsinfluence and status today Let me begin with the title Space-TimeAlgebra John Wheeler’s first comment on the manuscript about to

go to press was “Why don’t you call it Spacetime Algebra?” I havefollowed his advice, and Spacetime Algebra (STA) is now the standardterm for the mathematical system that the book introduces

I am pleased to report that STA is as relevant today as it waswhen first published I regard nothing in the book as out of date or

in need of revision Indeed, it may still be the best quick introduction

to the subject It retains that first blush of compact explanation fromsomeone who does not know too much From many years of teaching

I know it is accessible to graduate physics students, but, because itchallenges standard mathematical formalisms, it can present difficul-ties even to experienced physicists

One lesson I learned in my career is to be bold and explicit inmaking claims for innovations in science or mathematics Otherwise,they will be too easily overlooked Modestly presenting evidence andarguing a case is seldom sufficient Accordingly, with confidence thatcomes from decades of hindsight, I make the following Claims for STA

as formulated in this book:

(1) STA enables a unified, coordinate-free formulation for all of tivistic physics, including the Dirac equation, Maxwell’s equationand General Relativity

rela-(2) Pauli and Dirac matrices are represented in STA as basis vectors

in space and spacetime respectively, with no necessary connection

to spin

(3) STA reveals that the unit imaginary in quantum mechanics hasits origin in spacetime geometry.

(4) STA reduces the mathematical divide between classical, quantumand relativistic physics, especially in the use of rotors for rota-tional dynamics and gauge transformations

Comments on these claims and their implications necessarily refer tocontents of the book and ensuing publications, so the reader may wish

to reserve them for later reference when questions arise

Claim (2) expresses the crucial secret to the power of STA Itimplies that the physical significance of Dirac and Pauli algebras stemsentirely from the fact that they provide algebraic representation ofgeometric structure Their representations as matrices are irrelevant tophysics—perhaps inimical, because they introduce spurious complexnumbers without physical significance The fact that they are spurious

is established by claim (3)

The crucial geometric relation between Dirac and Pauli algebras isspecified by equations (7.9) to (7.11) in the text It is so important that

I dubbed it space-time split in later publications Note that the symbol

i is used to denote the pseudoscalar in (6.3) and (7.9) That symbol

is appropriate because i2= −1, but it should not to be confused withthe imaginary unit in the matrix algebra

Readers familiar with Dirac matrices will note that if the gammas

on the right side of (7.9) are interpreted as Dirac’s γ-matrices, thenthe objects on the left side must be Dirac’s α-matrices, which, as isseldom recognized, are 4 × 4 matrix representations of the 2 × 2 Paulimatrices That is a distinction without physical or geometric signifi-cance, which only causes unnecessary complications and obscurity inquantum mechanics STA eliminates it completely

It may be helpful to refer to STA as the “Real Dirac Algebra”,because it is isomorphic to the algebra generated by Dirac γ-matricesover the field of real numbers instead of the complex numbers in stan-dard Dirac theory Claim (3) declares that only the real numbers areneeded, so standard Dirac theory has a superfluous degree of freedom.That claim is backed up in Section 13 of the text where Dirac’s equa-tion is given two different but equivalent formulations within STA Inequation (13.1) the role of unit imaginary is played by the pseudoscalar

i, while, in equation (13.13) it is played by a spacelike bivector

Preface after Fifty years xi

Thus, the mere reformulation of the Dirac equation in terms ofSTA automatically assigns a geometric meaning to the unit imaginary

in quantum mechanics! I was stunned by this revelation, and I set outimmediately to ascertain what its physical implications might be Be-fore the ink was dry as the STA book went to press, I had established

in [1] that (13.13) was the most significant form for the Dirac equation,with the bivector unit imaginary related to spin in an intriguing way.This insight has been a guiding light for my research into geometricfoundations for quantum mechanics ever since The current state ofthis so-called “Real Dirac Theory” is reviewed in [2, 3]

Concerning Claim (4): Rotors are mathematically defined by(16.7) and (16.8), but I failed to mention there what has since be-come a standard name for this important concept Rotors are usedfor an efficient coordinate-free treatment of Lorentz transformations

in Sections 16, 17 and 18 That provides the foundation for the ciple of Local Relativity formulated in Section 23 It is an essentialgauge principle for incorporating Dirac spinors into General Relativ-ity That fact is demonstrated in a more general treatment of gaugetransformations in Section 24

Prin-The most general gauge invariant derivative for a spinor field inSTA is given by equations (24.6) and (24.12) The “C connection”

is the coupling to the gravitational field, while “D connection” in(24.16) was tentatively identified with strong interactions I was verysuspicious of that tentative identification at the time Later, when theelectroweak gauge group became well established, I reinterpreted the

D connection as electroweak [4] That has the great virtue of groundingelectroweak interactions in the spacetime geometry of STA The issue

is most thoroughly addressed in [5] However, a definitive argument orexperimental test linking electroweak interactions to geometry in thisway remains to be found

Finally, let me return to Claim (1) touting STA as a unified ematical language for all spacetime physics The whole book makesthe case for that Claim However, though STA provides coordinate-free formulations for the most fundamental equations of physics, solv-ing those equations with standard coordinate-based methods requiredtaking them apart and thereby losing the advantage of invariant for-mulation To address that problem, soon after this book was published,

math-I set forth on the huge task of reformulating standard mathematicalmethods That was a purely mathematical enterprise (but with oneeye on physics) I generalized STA to create a coordinate-free Geomet-ric Algebra (GA) and Geometric Calculus (GC) for all dimensions andsignatures There were already plenty of clues in STA on how to do it.

In particular, Section 22 showed how to define a vector derivatives andintegrals that generalize the concept of differential form Another basictask was to reformulate linear algebra to enable coordinate-free cal-culations with GA The outcome of this initiative, after two decades,

is the book [6] Also during this period I demonstrated the efficiency

of GA in introductory physics and Classical Mechanics, as presented

in my book [7]

By fulfilling the four Claims just discussed, STA initiated velopments of GA into a comprehensive mathematical system with avast range of applications that is still expanding today These develop-ments fall quite neatly into three Phases Phase I covers the first twodecades, when I worked alone with assistance of my students, prin-cipally Garret Sobczyk, Richard Gurtler and Robert Hecht-Nielsen.This work attracted little notice in the literature, with the exception

de-of mathematician Roget Boudet, who promoted it enthusiastically inFrance

Phase I was capped by my two talks [8, 9] at a NATO conference

on Clifford Algebras organized by Roy Chisholm That conference alsoinitiated Phase II and a steady stream of similar conferences still flow-ing today Let me take this opportunity to applaud the contribution

of Chisholm and the other conference organizers who selflessly devotetime and energy to promoting the flow of scientific ideas This essentialsocial service to science gets too little recognition

The high point of Phase II was the publication of Gauge TheoryGravity by Anthony Lasenby, Chris Doran and Steve Gull [10] I seethis as a fundamental advance in spacetime physics and a capstone ofSTA [11] Phase II was capped with the comprehensive treatise [12]

by Doran and Lasenby

Phase III was launched by my presentation of Conformal metric Algebra (CGA) in July 1999 at a conference in Ixtapa, Mexico[13] After stimulating discussions with Hongbo Li, Alan Rockwoodand Leo Dorst, diverse applications of GA published during Phase II

Geo-Preface after Fifty years xiii

had congealed quite quickly in my mind into a sharp formulation ofCGA Response to my presentation was immediate and strong, initiat-ing a steady stream of CGA applications to computer science [14] andengineering (especially robotics) In physics, CGA greatly simplifiesthe treatment of the crystallographic space groups [15], and applica-tions are facilitated by the powerful Space Group Visualizer created

by Echard Hitzer and Christian Perwass [16]

I count myself as much mathematician as physicist, and I see velopment of GA from STA to GC and CGA as reinvigorating themathematics of Hermann Grassmann and William Kingdon Cliffordwith an infusion of twentieth century physics The history of this de-velopment and the present status of GA has been reviewed in [17]

de-I am sorry to say that few mathematicians are prepared to nize the central role of GA in their discipline, because it has becomeincreasingly insular and divorced from physics during the last century.Personally, I dedicate my work with GA to the memory of mymathematician father, Magnus Rudolph Hestenes, who was alwaysgenerous with his love but careful with his praise [9]

recog-References

[1] D Hestenes, Real Spinor Fields, J Math Phys 8, 798–808 (1967).[2] D Hestenes, Oersted Medal Lecture 2002: Reforming the Math-ematical Language of Physics Am J Phys 71, 104–121 (2003).[3] D Hestenes, Spacetime Physics with Geometric Algebra, Am J.Phys 71, 691–714 (2003)

[4] D Hestenes, Space-Time Structure of Weak and ElectromagneticInteractions, Found Phys 12, 153–168 (1982)

[5] D Hestenes, Gauge Gravity and Electroweak Theory In: H nert, R.T Jantzen and R Ruffini (Eds.), Proceedings of the Ele-venth Marcel Grossmann Meeting on General Relativity (WorldScientific: Singapore, 2008), pp 629–647

Klei-[6] D Hestenes and G Sobczyk, CLIFFORD ALGEBRA to METRIC CALCULUS, A Unified Language for Mathematics andPhysics (Kluwer: Dordrecht/Boston, 1984)

GEO-[7] D Hestenes, New Foundations for Classical Mechanics (Kluwer:Dordrecht/Boston, 1986).

[8] D Hestenes, A Unified Language for Mathematics and Physics.In: J.S.R Chisholm and A.K Common (Eds.), Clifford Alge-bras and their Applications in Mathematical Physics (Reidel: Dor-drecht/Boston, 1986), pp 1–23

[9] D Hestenes, Clifford Algebra and the Interpretation of tum Mechanics In: J.S.R Chisholm and A.K Common (Eds.),Clifford Algebras and their Applications in Mathematical Physics(Reidel: Dordrecht/Boston, 1986), pp 321–346

Quan-[10] A Lasenby, C Doran and S Gull, Gravity, gauge theories andgeometric algebra, Phil Trans R Lond A 356, 487–582 (1998).[11] D Hestenes, Gauge Theory Gravity with Geometric Calculus,Foundations of Physics 36, 903–970 (2005)

[12] C Doran and A Lasenby, Geometric Algebra for Physicists(Cambridge: Cambridge University Press, 2003)

[13] D Hestenes, Old Wine in New Bottles: A new algebraic work for computational geometry In: E Bayro-Corrochano and

frame-G Sobczyk (Eds), Advances in Geometric Algebra with tions in Science and Engineering (Birkh¨auser: Boston, 2001), pp.1–14

Applica-[14] L Dorst, D Fontijne and S Mann, Geometric Algebra for puter Science (Morgan Kaufmann, San Francisco, 2007)

Com-[15] D Hestenes and J Holt, The Crystallographic Space Groups inGeometric Algebra, Journal of Mathematical Physics 48, 023514(2007)

[16] E Hitzer and C Perwas: http://www.spacegroup.info

[17] D Hestenes, Grassmann’s Legacy In: H.-J Petsche, A Lewis,

J Liesen and S Russ (Eds.), From Past to Future: Grassmann’sWork in Context (Birkh¨auser: Berlin, 2011)

This book is an attempt to simplify and clarify the language we use

to express ideas about space and time It was motivated by the beliefthat such is an essential step in the endeavor to simplify and unifyour theories of physical phenomena The object has been to produce a

“space-time calculus” which is ready for physicists to use Particularattention has been given to the development of notation and theoremswhich enhance the geometric meaning and algebraic efficiency of thecalculus; conventions have been chosen to differ as little as possiblefrom those in formalisms with which physicists are already familiar.Physical concepts and equations which have an intimate connec-tion with our notions of space-time are formulated and discussed Thereader will find that the “space-time algebra” introduces novelty of ex-pression and interpretation into every topic This naturally suggestscertain modifications of current physical theory; some are pointed out

in the text, but they are not pursued The principle objective here hasbeen to formulate important physical ideas, not to modify or applythem

The mathematics in this book is relatively simple Anyone whoknows what a vector space is should be able to understand the algebraand geometry presented in chapter I On the other hand, appreciation

of the physics involved in the remainder of the book will depend a greatdeal on the reader’s prior acquaintance with the topics discussed.This work was completed in 1964 while I was at the University

of California at Los Angeles It was supported by a grant from theNational Science Foundation—for which I am grateful

I wish to make special mention of my debt to Marcel Riesz, whosetrenchant discussion of Clifford numbers and spinors (8, 9) provided

an initial stimulus and some of the key ideas for this work I would

also like to express thanks to Professors V Bargmann, J A Wheelerand A S Wightman for suggesting some improvements in the finalmanuscript and to Richard Shore and James Sock for correcting someembarrassing errors.

David HestenesPalmer LaboratoryPrinceton University

Introduction xix

I Geometric Algebra 1 1 Interpretation of Clifford Algebra 1

2 Definition of Clifford Algebra 4

3 Inner and Outer Products

4 Structure of Clifford Algebra 10

5 Reversion, Scalar Product 13

6 The Algebra of Space 16

7 The Algebra of Space-Time 20

II Electrodynamics 25 8 Maxwell’s Equation 25

9 Stress-Energy Vectors 27

10 Invariants 29

11 Free Fields 31

III Dirac Fields 35 12 Spinors 35

13 Dirac’s Equation 39

14 Conserved Currents 42

15 C, P , T 43

IV Lorentz Transformations 47 16 Reflections and Rotations 47

17 Coordinate Transformations 51

18 Timelike Rotations 55

19 Scalar Product 59

5

V Geometric Calculus 63

20 Differentiation 63

21 Riemannian Space-Time 68

22 Integration 72

23 Global and Local Relativity 76

24 Gauge Transformation and Spinor Derivatives 82

Conclusion 87 Appendixes 89 A Bases and Pseudoscalars 89

B Some Theorems 92

C Composition of Spatial Rotations 97

D Matrix Representation of the Pauli Algebra 99

A geometric algebra is an algebraic representation of geometric cepts Of particular importance to physicists is an algebra which canefficiently describe the properties of objects in space-time and eventhe properties of space-time itself, for it is only in terms of some suchalgebra that a physical theory can be expressed and worked out Manydifferent geometric algebras have been developed by mathematicians,and from time to time some of these systems have been adapted tomeet the needs of physicists Thus, during the latter part of the nine-teenth century J Willard Gibbs put together a vector algebra for threedimensions based on ideas gleaned from Grassmann’s extensive alge-bra and Hamilton’s quaternions [1, 2, 3] By the time this system was

con-in relatively wide use, a new geometric algebra was needed for thefour-dimensional space-time continuum required by Einstein’s theory

of relativity For the time being, this need was filled by tensor bra But, it was not long before Pauli found it necessary to introduce

alge-a new alge-algebralge-a to describe the electron spin Subsequently, Diralge-ac walge-asled to still another algebra which accommodates both spin and specialrelativity Each of these systems, vector and tensor algebra and thealgebras of Pauli and Dirac, is a geometric algebra with special ad-vantages for the physicist Because of this, each system is widely usedtoday The development of a single geometric algebra which combinesadvantages of all these systems is a principal object of this paper.One of the oldest of the above-mentioned geometric algebras isthe one invented by Grassmann (∼1840) [4] His approach went some-thing like this: He represented the geometric statement that two pointsdetermine a straight line by the product of two algebraic entities rep-resenting points to form a new algebraic entity representing a line

He represented a plane by the product of two lines determining the

plane In a similar fashion he represented geometric objects of higherdimension But more than that, he represented the symmetries andrelative orientations of these objects in the very rules of combination

of his algebra

At about the same time, Hamilton invented his quaternion gebra to represent the properties of rotations in three dimensions.Evidently Grassmann and Hamilton each paid little attention to theother’s work It was not until 1878 that Clifford [5] united these sys-tems into a single geometric algebra, and it was not until just before

al-1930 that Clifford algebra had important applications in physics Atthat time Pauli and, subsequently, Dirac introduced matrix represen-tations of Clifford algebra for physical but not for geometrical reasonsClifford constructed his algebra as a direct product of quaternionalgebras I rather think that Hamilton would have made a similarconstruction earlier had he been troubled about the relation of hisquaternions to Grassmann’s algebra But this is a formal algebraicmethod So is the now common procedure of arriving at the Pauliand Dirac algebras from a study of representations of the Lorentzgroup Had Grassmann considered this problem, I think he wouldhave been led to Clifford algebra in quite a different way He wouldhave insisted on beginning with algebraic objects directly representingsimple geometric objects and then introducing algebraic operationswith geometrical interpretations

In this paper, ideas of Grassmann are used to motivate the struction of Clifford algebra and to provide a geometric interpretation

con-of Clifford numbers This is to be contrasted with other treatments con-ofClifford algebra which are for the most part formal algebra By insist-ing on “Grassmann’s” geometric viewpoint, we are led to look uponthe Dirac algebra with new eyes It appears simply as a vector algebrafor space-time The familiar γµ are seen as four linearly independentvectors, rather than as a single “4-vector” with matrices for compo-nents The Pauli algebra appears as a subalgebra of the Dirac algebraand is simply the vector algebra for the 3-space of some inertial frame.The vector algebra of Gibbs is seen not as a separate algebraic system,but as a subalgebra of the Pauli algebra Most important, there is nogeometrical reason for complex numbers to appear Rather, the unitpseudoscalar (denoted by γ5 in matrix representations) plays the role

of the unit imaginary i All these features are foreign to the Dirac gebra when seen from the usual viewpoint of matrix representations.Yet, as will be demonstrated in the text, emphasis on the geometricaspect broadens the range of applicability of the Dirac algebra, sim-plifies manipulations and imbues algebraic expressions with meaning.The main body of this paper consists of five chapters

al-In chapter I we review the properties of the Clifford algebra for afinite dimensional vector space We strive to develop notation and the-orems which clarify the geometrical meaning of Clifford algebra andmake it an effective practical tool Then we discuss the application ofour general viewpoint to space-time and the Dirac algebra We dealexclusively with the Dirac algebra in the rest of the paper Neverthe-less, there are good reasons for first giving a more general discussion

of Clifford algebra For one thing, it enables us to distinguish clearlybetween those properties of the Dirac algebra which are peculiar tospace-time and those which are not A better reason is that Cliffordalgebra has many important applications other than the one we workout in this paper Indeed, it is important wherever a vector space with

an inner product arises For example, physicists will be interested inthe fact that quantum field operators form a Clifford algebra.1

Many readers will be familiar with the matrix algebras of Pauliand Dirac and will be anxious to see how these algebras can be vieweddifferently They can take a short cut, first reading the intuitive dis-cussion of section 1 and then proceeding directly to sections 6 and 7.The intervening sections can be referred to as the occasion arises Toget a full appreciation of the efficacy of the Dirac algebra as a vec-tor algebra quite apart from any connection it has with spinors, thediscussion of the electromagnetic field in chapter II should be stud-ied closely It should particularly be noted how identification of thePauli algebra as a subalgebra of the Dirac algebra greatly facilitatesthe transformation from 4-dimensional to 3-dimensional descriptions.Spinors are discussed in chapter II before any mention is made

of Lorentz transformations to emphasize the little-known fact that

1 See reference [6] An account of the Clifford algebra of fermion field operators is given in any text on quantum field theory; see, for instance, reference [7], especially sections (3–10) There is

no clear physical connection between the Clifford algebra of space-time discussed in this book and the algebra of quantum field operators, although one might suspect that there should be because of the well-known theorem relating “spin and statistics”.

xxi

spinors can be defined without reference to representations of theLorentz group, even without using matrices The definition of a spinor

as an element of a minimal ideal in a Clifford algebra is due to MarcelRiesz [8]

It can hardly be overemphasized that by our geometrical struction we obtain the Dirac algebra without complex numbers Nev-ertheless, it is still possible to write down the Dirac equation Thisshows that complex numbers are superfluous in the Dirac theory Onewonders if this circumstance has physical significance To answer thisquestion, a fuller analysis of the Dirac theory will be carried out else-where In section 13 we will be content with writing the Dirac equation

con-in several different forms and raiscon-ing some questions The macon-in pose of chapter III is to illustrate the most direct formulation of theDirac theory in the real Dirac algebra We also indulge in speculating

pur-a connection between isospin pur-and the Dirpur-ac pur-algebrpur-a

Lorentz transformations are discussed in chapter IV So manytreatments of Lorentz transformations have been given in the pastthat publication of still another needs a thorough justification Theapproach presented here is unique in many details, but two featuresdeserve special mention First, Lorentz transformations are introduced

as automorphisms of directions at a generic point of space-time Thisapproach is so general that it applies to curved space-time, as can

be seen specifically when it is applied in section 23 Automorphismsinduced by coordinate transformations are discussed in detail as avery special case The discussion of coordinate transformations can befurther compressed if points in space-time are represented by vectors,but such a procedure risks an insidious confusion of transformations

of points in space-time with transformations of tangent vectors.The most distinctive feature in our approach to Lorentz transfor-mations arises from information about space-time which was built intothe Dirac algebra when we specified its relation to the Pauli algebra.The respective merits of representing Lorentz rotations in the Paulialgebra and in the Dirac algebra have been discussed on many occa-sions We are able to combine all the advantages of both approaches,because both approaches become one when the Pauli algebra is iden-tified as the even subalgebra of the Dirac algebra The resultant gain

in perspicuity and ease in manipulation of Lorentz transformations is

considerable

Since the Dirac algebra has physical significance, it is important

to be thoroughly acquainted with its special properties For this reasonsome space is devoted in section 19 to a discussion of isometries of theDirac algebra The technique of using isometries to generate “possible”physical interactions is illustrated in section 24

Having demonstrated that it provides an efficient description ofthe electro-magnetic field as well as the Dirac equation, the Dirac alge-bra must be adapted to curved space-time if it is to provide the basisfor a versatile geometric calculus This task is undertaken in chapter

V and proves to be relatively straightforward because of the geometricmeaning which has already been supplied to the Dirac algebra Themain problem is to define a suitable differential operator  It ap-pears that all the local properties of space-time can be construed asproperties of this remarkable operator We express the gravitationalfield equations in terms of , and we show how  is related to twodifferent generalizations of Einstein’s special principle of relativity Wediscuss the application of and Clifford algebra to integration theory.Finally, we illustrate a way to generalize  to generate theories withnon-gravitational as well as gravitational interactions

The space-time calculus is sufficiently well-developed to be used

as a practical tool in the study of the basic equations of physics lated here But it can be developed further A tensor algebra can beconstructed over the entire Dirac algebra This is most elegantly done

formu-by emulating the treatment of dyadics formu-by Gibbs [1, 2, 3] In this wayequivalents of the higher rank tensors and spinors can be constructed.Above and beyond algebraic details, it is important to realize that

by insisting on a direct and universal geometric meaning of the Diracalgebra we are compelled to adopt a philosophy about the construc-tion of physical theories which differs somewhat from the customaryone Ordinarily the Lorentz group is taken to describe the primitivegeometrical properties of space-time These properties are then imbed-ded in a physical theory by requiring that all mathematical quantities

in the theory be covariant under Lorentz transformations In contrast,

we construct the Dirac algebra as an “algebra of directions” which bodies the local geometrical properties of space-time Physical quan-tities are then limited to objects which can be constructed from this

em-xxiii

If there is more than a formal difference between these approaches,

it must arise because the second is more restricted than the first Thiscan only occur if the assumptions about space-time which go into thereal Dirac algebra are more detailed than those which go into theLorentz group That such may actually be the case is suggested bythe fact that the first philosophy permits physical theories which con-tain complex numbers without direct geometrical meaning, whereasthe second does not It should be possible to resolve this problem by

a study of the Dirac equation because it contains complex numbersexplicitly But, we shall not attempt to do so here

Chapter I

Geometric Algebra

1 Interpretation of Clifford Algebra

To every n-dimensional vector space Vn with a scalar product therecorresponds a unique Clifford algebra Cn In this section we give anintuitive discussion of howCn arises as an algebra of directions inVn

In the next section we proceed with a formal algebraic definition of

Cn

From two vectors a and b we can construct new geometric jects endowed with geometric properties which arise when a and b areconsidered together For one thing, we can form the projection of onevector on the other Since it is determined by the mutual properties of

ob-a ob-and b, we cob-an express it ob-as ob-a product ob-a · b Thob-at ob-a · b must be ob-a ber (scalar) and also symmetric in a and b is an algebraic expression

num-of the geometric concept num-of projection

We can form another kind of product by noting that the vectors aand b, if they are linearly independent, determine a parallelogram withsides a and b We can therefore construct from a and b a new vectorlikeentity with magnitude equal to the magnitude of the parallelogram anddirection determined by the plane containing the parallelogram Since

it is completely determined by a and b, we can write it as a product:a∧b We call a∧b the outer product of a and b Loosely, we can think ofa∧b as an algebraic representation of the parallelogram since it retainsthe information as to the area and plane of the parallelogram However

a ∧ b does not determine the shape of the parallelogram and could

equally well be interpreted as a circular disc in the plane of a and b witharea equal to that of the parallelogram In its shape the parallelogramcontains information about the vectors a and b individually, whereas

a ∧ b retains only properties which arise when a and b are consideredtogether—it has only magnitude and direction

Actually, since we can distinguish two sides of the plane ing a and b, we can construct two different products which we write

contain-as a ∧ b and b ∧ a Intuitively speaking, if a ∧ b is represented by aparallelogram facing up on the plane, then b ∧ a can be represented by

a similar parallelogram facing down in the plane a ∧ b and b ∧ a haveopposite orientation, just as do the vectors a and −a We can expressthis by an equation:

The reader is now directed to observe that since a · b and a ∧ bhave opposite symmetry we can form a new kind of product ab with

a · b and a ∧ b as its symmetric and antisymmetric parts We write

Let us call this the vector product of a and b It is the fundamentalkind of multiplication in Clifford algebra In the subtle way we havedescribed, it unites the two geometrically significant kinds of multi-plication of vectors

In our attempt to construct a geometrically meaningful cation for vectors we were led to define other geometric objects which,like vectors, can be characterized by magnitude and direction Thissuggests a generalization of the concept of vector Every r-dimensionalsubspace of Vn determines an oriented r-dimensional unit volume el-ement which we call an r-direction By associating a magnitude with

multipli-an r-direction we arrive at the concept of multipli-an r-vector A few examples

1 Interpretation of Clifford Algebra

will help make this concept clear The vectors of Vn are 1-vectors.The unit vectors ofVn are 1-directions Because they have magnitudebut no direction, we can interpret the scalars as 0-vectors The outerproduct a ∧ b is a 2-vector If it has unit magnitude it is a 2-direction.Let us call r the degree of an r-vector Evidently n is the largest de-gree possible for an r-vector of Vn Only two unit n-vectors can beassociated with Vn They differ only in sign They represent the twopossible orientations which can be given to the unit volume element

of Vn

Let us return to our definition of vector product In (1.2) weintroduced the sum of a 0-vector and a 2-vector In view of the inter-pretation of a · b and a ∧ b as “vectors”, it is natural that they should

be added according to the usual rules of vector addition The 0-vectorsand the 2-vectors are then linearly independent

Now that we understand the product of two vectors, what is morenatural than to study the product of three? We find that it is consis-tent with our geometrical interpretation of (1.2) to allow multiplica-tion which is associative and distributive with respect to addition As

we will show in the text, it follows that the product of any number ofvectors can be expressed as a sum of r-vectors of different degree Inthis way we generate the Clifford algebra Cn of Vn An element of Cn

will be called a Clifford number Every r-vector of Vn is an element

of Cn, and, conversely, every Clifford number can be expressed as alinear combination of r-directions Thus, we can interpret Cn as analgebra of directions inVn, or, equivalently, an algebra of subspaces of

Vn

In the paper which introduced the algebra that goes by his name,Clifford remarked on the dual role played by the real numbers Forinstance, the number 2 can be thought of as a magnitude, but it canalso be thought of as the operation of doubling, as in the expression

2 × 3 = 6 Clifford numbers exhibit the same dual role, but in amore significant way since directions are involved An r-vector can

be thought of as an r-direction with a magnitude, but it can also

be thought of as an operator which by multiplication changes oneClifford number into another This dual role enables us to discussdirections and operations on directions without going outside of theClifford algebra Combining our insights into the interpretation of the

3

elements and operations in Cn we can characterize Clifford algebra

as a generalization of the real numbers which unites the geometricalnotion of direction with the notion of magnitude

2 Definition of Clifford Algebra

To exploit our familiarity with vectors, we introduce Clifford algebra as

a multiplication defined on a vector space We begin with an axiomaticapproach which does not make explicit mention of basis, because itkeeps the rules of combination clearly before us In later sections weagain emphasize geometrical interpretation

We develop Clifford algebra over the field of the real numbers.Development over the complex field gives only a spurious kind of gen-erality since it adds nothing of geometric significance Indeed, it tends

to obscure the fascinating variety of “hypercomplex numbers” whichalready appear in the Clifford algebra over the reals Our analysis willsuggest that the complex field arises in physics for geometric reasons

In any case, it is a trivial matter to generalize our work to apply toClifford algebra over the complex field

Let Vn be an n-dimensional vector space over the real numberswith elements a, b, c, called vectors Represent multiplication ofvectors by juxtaposition, and denote the product of an indetermi-nate number of vectors by capital letters A, B, C, Assume these

“monomials” may be “added” according to the usual rules,

3 Inner and Outer Products

Assume scalars commute the vectors, i.e.,

Equations (2.3) and (2.6) imply that the usual rules for scalar tiplication hold also for vector polynomials Scalars and vectors arerelated by assuming:

mul-For a, b inVn, ab is a scalar

We identify (|a2|)12 with the length of the vector a

The algebra defined by the above rules is called the Clifford bra Cn of the vector spaceVn.1 Conversely,Vn is said to be the vectorspace associated withCn The elements ofCn are called Clifford num-bers, or simply c-numbers We shall call AB the vector product ofc-numbers A and B

alge-From the mathematical point of view it is interesting to note thatthere is a great deal of redundancy in our axioms It is obvious, forinstance, that the rules of vector and scalar multiplication in Vn withwhich we began are special cases of the operations on arbitrary c-numbers which we defined later A minimal set of axioms for Cliffordalgebra over the reals is very nearly identical with the axioms for thereal numbers themselves

3 Inner and Outer Products

Consider the quadratic form a2for a vector a Observe that for vectors

a and b,

(a + b)2 = a2+ ab + ba + b2,

ab + ba = (a + b)2− a2− b2,which is a scalar Thus a2 has an associated symmetric bilinear formwhich we denote by

1 The special effects of various kinds of singular metric on V n are discussed by Riesz [9] Most

of our work is indifferent to these features, so we will take them into account only as the occasion arises.

5

Call it the inner product of a and b Obviously a · a = a2.

Now decompose the product of two vectors into symmetric andanti-symmetric parts,

ab = 12(ab + ba) + 12(ab − ba)

Anticipating a special significance for the antisymmetric part, call itthe outer product of a and b and write

Geometrically, a ∧ b can be interpreted as an oriented area b ∧ ahas opposite orientation a ∧ b ∧ c can be interpretated as an orientedvolume, i.e., the parallelopiped with sides a, b, c c ∧ b ∧ a is the volumewith opposite orientation The outer product of r vectors,

partic-A linear combination of simple r-vectors (e.g., a1∧ a2+ a3∧ a4)will be called simply an r-vector As an alternate terminology to 0-vector, 1-vector, 2-vector, 3-vector, we shall often use the namesscalar, vector, bivector, trivector, respectively An n-vector in Cn

will be called a pseudoscalar, an (n − 1)-vector a pseudovector, etc

We shall call r the degree of an r-vector The geometric term

“dimension’ is perhaps more appropriate, but we shall be using it to

3 Inner and Outer Products

signify the number of linearly independent vectors in a vector space

By multivector we shall understand an r-vector of unspecified degree.Generalizing (3.5), we define the outer product of a simple r-vector with a simple s-vector

(a1∧ a2∧ ∧ ar) ∧ (b1∧ b2∧ ∧ bs)

= (a1∧ ∧ ar−1) ∧ (ar ∧ b1∧ ∧ bs) (3.6)This is clearly just the associative law for outer products

From the associative law for outer products and (3.5) it followseasily that the outer products of r vectors is antisymmetric with re-spect to interchange of any two vectors Therefore, with proper atten-tion to sign, we may permute the vectors “in” a simple r-vector atwill Since a ∧ b = 0 if a and b are collinear, a r-vector is equal tozero if it “contains” two linearly dependent vectors This fact gives us

a simple test for linear dependence: r vectors are linearly dependent

if and only if their associated r-vector is zero Thus, we could havedefined the dimension of Vn as the degree of the largest r-vector inthe Clifford algebra of Vn

with an s-vector Bs follows easily from antisymmetry and ity,

We turn now to a generalization of the inner product which sponds to (3.5) The inner product has symmetry opposite to that ofthe outer product For a vector a and an r-vector Ar = a1∧a2∧ .∧ar

corre-write

a · Ar ≡ 1

The result is an (r − 1)-vector The right side of (3.8) can be evaluated

to give the useful expansion

This expansion is most easily obtained from a more general formulawhich we prove later.

We define the inner product of simple r- and s-vectors in analogy

to (3.6)

(a1∧ ∧ ar) · (b1∧ ∧ bs)

= (a1∧ ∧ ar−1) · [ar· (b1∧ ∧ bs)] (3.10)

By (3.8), this is well defined for r 5 s The expansion (3.9) shows that

if one of the a’s is orthogonal to all the b’s, the inner product (3.10)

is zero Successive applications of (3.10) and (3.9) shows that (3.10)

is an |r − s|-vector Two other forms of (3.10) are often convenient.(a1∧ ∧ ar) · (b1∧ ∧ bs)

By (3.8), a · (b1∧ ∧ bs) = (−1)s+1(b1∧ ∧ bs) · a So from(3.10) we find that for any r-vector and s-vector with s = r,

This corresponds to (3.7) for outer products

To get some understanding of the geometric significance of (3.10)

we look at a couple of simple cases By (3.10) and (3.9)

(a ∧ b) · (b ∧ a) = a · [b · (b ∧ a)] = a · [(b · b)a − (b · a)b]

= a2b2− (a · b)2=

b · b b · a

a · b a · a

3 Inner and Outer Products

This is the square of the area of a ∧ b The projection of a ∧ b on u ∧ v

is expanded the same way as (3.14),

(b ∧ a) · (u ∧ v) = (a · u)(b · v) − (a · v)(b · u)

=

... important Clifford algebras are related

in a series by (4.11): The Pauli algebra is the even subalgebra of theDirac algebra The quaternion algebra is the even subalgebra of thePauli algebra The... only for points in Euclidean 3 -space and vectors in the Pauli algebra Some such convention as this is necessary to distinguish vectors in space from vectors in space- time, as will become clear in... is odd

6 The Algebra of Space< /h3>

The set of all vectors tangent to a generic point x in Euclidean spaceE3forms a 3-dimensional vector spaceV3=V3(x)