Space-Time
Algebra

David Hestenes

Second Edition

David Hestenes

Space-Time Algebra

Second Edition

Foreword by Anthony Lasenby

David Hestenes
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

Springer Cham Heidelberg New York Dordrecht London

© Springer International Publishing Switzerland 2015
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part
of the material is concerned, speci

ﬁcally the rights of translation, reprinting, reuse of illustrations,

recitation, broadcasting, reproduction on micro

ﬁlms or in any other physical way, and transmission

or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar
methodology now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publication does not imply, even in the absence of a speci

ﬁc statement, that such names are exempt from

the relevant protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authors or the editors give a warranty, express or implied, with respect to the material contained herein or
for any errors or omissions that may have been made.

Cover design: deblik, Berlin

Printed on acid-free paper

Springer International Publishing AG Switzerland is part of Springer Science+Business Media
(www.birkhauser-science.com)

Foreword

It is a pleasure and honour to write a Foreword for this new edition
of David Hestenes’ Space-Time Algebra. This small book started a
profound revolution in the development of mathematical physics, one
which has reached many working physicists already, and which stands
poised to bring about far-reaching change in the future.

At its heart is the use of Clifford algebra to unify otherwise dis-

parate mathematical languages, particularly those of spinors, quater-
nions, tensors and differential forms. It provides a unified approach
covering all these areas and thus leads to a very efficient ‘toolkit’ for
use 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,
in the form of the ’Geometric Algebra’, can be applied in many areas
of engineering, robotics and computer science, with no changes neces-
sary – it is the same underlying mathematics, and enables physicists
to understand topics in engineering, and engineers to understand top-
ics in physics (including aspects in frontier areas), in a way which no
other single mathematical system could hope to make possible.

As well as this, however, there is another aspect to Geometric

Algebra which is less tangible, and goes beyond questions of math-
ematical power and range. This is the remarkable insight it gives to
physical problems, and the way it constantly suggests new features of
the physics itself, not just the mathematics. Examples of this are pep-
pered throughout Space-Time Algebra, despite its short length, and
some of them are effectively still research topics for the future. As
an example, what is the real role of the unit imaginary in quantum

Foreword

mechanics? In Space-Time Algebra two possibilities were looked at,
and in intervening years David has settled on right multiplication by
the bivector iσ

= σ

as the correct answer between these two, thus

encoding rotation in the ‘internal’ xy plane as key to understanding
the role of complex numbers here. This has stimulated much of his
subsequent work in quantum mechanics, since it bears directly on the
zwitterbewegung interpretation of electron physics [1]. Even if we do
not follow all the way down this road, there are still profound ques-
tions to be understood about the role of the imaginary, such as the
generalization 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 had

already picked out generalizations of these internal transformations,
but still just using geometric entities in spacetime, as candidates for
describing 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 not
clear that we can yet claim a comparable spacetime ‘underpinning’
for QCD or the Standard Model itself, and it may well be that some
new feature, not yet understood, but perhaps still living within the
algebra 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 Manchester
University many years ago the idea that the Dirac matrices really rep-
resented vectors in 4d spacetime. He thought this was just ‘mad’, and
was vehemently against any contemplation of such heresy. For some-
one such as myself, however, coming from a background of cosmology
and astrophysics, this realization, which I gathered from this short
book and David’s subsequent papers, was a revelation, and showed
me that one could cut through pages of very unintuitive spin calcu-
lations in Dirac theory, which only experts in particle theory would
be comfortable with, and replace them with a few lines of intuitively
appealing calculations with rotors, of a type that for example an engi-
neer, also using Geometric Algebra, would be able to understand and
relate to immediately in the context of rigid body rotations.

A similar transformation and revelation also occurred for me with

respect to gravitational theory. Stimulated by Space-Time Algebra and

Foreword

vii

particularly further papers by David such as [6] and [7], then along
with 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 manipulations
with the same unified mathematical language as works for electromag-
netism and Dirac spinors, are great. This enabled us quickly to reach
interesting research topics in gravitational theory, such as the Dirac
equation around black holes, where we found solutions for electron
energies in a spherically symmetric gravitational potential analogous
to the Balmer series for electrons around nuclei, which had not been
obtained before [9].

Of course there are very clever people in all fields, and in both

relativistic quantum mechanics and gravity there have been those who
have been able to forge an intuitive understanding of the physics de-
spite the difficulties of e.g. spin sums and Dirac matrices on the one
hand, and tensor calculus index manipulations on the other. However,
the clarity and insight which Geometric Algebra brings to these areas
and many others is such that, for the rest of us who find the alternative
languages difficult, and for all those who are interested in spanning
different fields using the same language, the ideas first put forward in
’Space-time Algebra’ and the subsequent work by David, have been
pivotal, and we are all extremely grateful for his work and profound
insight.

Anthony Lasenby

Astrophysics Group, Cavendish Laboratory

and Kavli Institute for Cosmology

Cambridge, UK

References

[1] D. Hestenes, Zitterbewegung in Quantum Mechanics, Founda-

tions of Physics 40, 1 (2010).

[2] C. Doran, A. Lasenby, S. Gull, S. Somaroo and A. Challinor,

viii

Foreword

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 Electromagnetic

Interactions, Found. Phys. 12, 153 (1982).

[4] D. Hestenes, Gauge Gravity and Electroweak Theory. In: H. Klei-

nert, R. T. Jantzen and R. Ruffini (eds.), Proceedings of the
Eleventh Marcel Grossmann Meeting on General Relativity, p. 629
(World Scientific) (2008).

[5] C. Doran and A. Lasenby, Geometric Algebra for Physicists

(Cambridge University Press) (2003).

[6] D. Hestenes, Curvature Calculations with Spacetime Algebra, In-

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).

[8] A. Lasenby, C. Doran, and S. Gull, Gravity, gauge theories and

geometric 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 black
hole background, Phys. Rev. D. 72, 105014 (2005).

Preface after Fifty years

This book launched my career as a theoretical physicist fifty years
ago. I am most fortunate to have this opportunity for reflecting on its
influence and status today. Let me begin with the title Space-Time
Algebra. John Wheeler’s first comment on the manuscript about to
go to press was “Why don’t you call it Spacetime Algebra?” I have
followed his advice, and Spacetime Algebra (STA) is now the standard
term for the mathematical system that the book introduces.

I am pleased to report that STA is as relevant today as it was

when 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 from
someone who does not know too much. From many years of teaching
I know it is accessible to graduate physics students, but, because it
challenges 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 in

making claims for innovations in science or mathematics. Otherwise,
they will be too easily overlooked. Modestly presenting evidence and
arguing a case is seldom sufficient. Accordingly, with confidence that
comes 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 rela-

tivistic physics, including the Dirac equation, Maxwell’s equation
and General Relativity.

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

in space and spacetime respectively, with no necessary connection
to spin.

Preface after Fifty years

(3) STA reveals that the unit imaginary in quantum mechanics has

its origin in spacetime geometry.

(4) STA reduces the mathematical divide between classical, quantum

and relativistic physics, especially in the use of rotors for rota-
tional dynamics and gauge transformations.

Comments on these claims and their implications necessarily refer to
contents 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. It

implies that the physical significance of Dirac and Pauli algebras stems
entirely from the fact that they provide algebraic representation of
geometric structure. Their representations as matrices are irrelevant to
physics—perhaps inimical, because they introduce spurious complex
numbers without physical significance. The fact that they are spurious
is established by claim (3).

The crucial geometric relation between Dirac and Pauli algebras is

specified 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 i

= −1, but it should not to be confused with

the 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, then
the objects on the left side must be Dirac’s α-matrices, which, as is
seldom recognized, are 4 × 4 matrix representations of the 2 × 2 Pauli
matrices. That is a distinction without physical or geometric signifi-
cance, which only causes unnecessary complications and obscurity in
quantum 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-matrices
over the field of real numbers instead of the complex numbers in stan-
dard Dirac theory. Claim (3) declares that only the real numbers are
needed, 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. In
equation (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

Thus, the mere reformulation of the Dirac equation in terms of

STA automatically assigns a geometric meaning to the unit imaginary
in quantum mechanics! I was stunned by this revelation, and I set out
immediately 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 geometric
foundations for quantum mechanics ever since. The current state of
this 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 used
for an efficient coordinate-free treatment of Lorentz transformations
in Sections 16, 17 and 18. That provides the foundation for the Prin-
ciple of Local Relativity formulated in Section 23. It is an essential
gauge principle for incorporating Dirac spinors into General Relativ-
ity. That fact is demonstrated in a more general treatment of gauge
transformations in Section 24.

The most general gauge invariant derivative for a spinor field in

STA 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 very
suspicious of that tentative identification at the time. Later, when the
electroweak gauge group became well established, I reinterpreted the
D connection as electroweak [4]. That has the great virtue of grounding
electroweak interactions in the spacetime geometry of STA. The issue
is most thoroughly addressed in [5]. However, a definitive argument or
experimental test linking electroweak interactions to geometry in this
way remains to be found.

Finally, let me return to Claim (1) touting STA as a unified math-

ematical language for all spacetime physics. The whole book makes
the 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 required
taking them apart and thereby losing the advantage of invariant for-
mulation. To address that problem, soon after this book was published,

xii

Preface after Fifty years

I set forth on the huge task of reformulating standard mathematical
methods. That was a purely mathematical enterprise (but with one
eye on physics). I generalized STA to create a coordinate-free Geomet-
ric Algebra (GA) and Geometric Calculus (GC) for all dimensions and
signatures. There were already plenty of clues in STA on how to do it.
In particular, Section 22 showed how to define a vector derivatives and
integrals that generalize the concept of differential form. Another basic
task 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 de-

velopments of GA into a comprehensive mathematical system with a
vast range of applications that is still expanding today. These develop-
ments fall quite neatly into three Phases. Phase I covers the first two
decades, 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
of mathematician Roget Boudet, who promoted it enthusiastically in
France.

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

on Clifford Algebras organized by Roy Chisholm. That conference also
initiated 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 devote
time and energy to promoting the flow of scientific ideas. This essential
social service to science gets too little recognition.

The high point of Phase II was the publication of Gauge Theory

Gravity by Anthony Lasenby, Chris Doran and Steve Gull [10]. I see
this as a fundamental advance in spacetime physics and a capstone of
STA [11]. Phase II was capped with the comprehensive treatise [12]
by Doran and Lasenby.

Phase III was launched by my presentation of Conformal Geo-

metric Algebra (CGA) in July 1999 at a conference in Ixtapa, Mexico
[13]. After stimulating discussions with Hongbo Li, Alan Rockwood
and Leo Dorst, diverse applications of GA published during Phase II

Preface after Fifty years

xiii

had congealed quite quickly in my mind into a sharp formulation of
CGA. Response to my presentation was immediate and strong, initiat-
ing a steady stream of CGA applications to computer science [14] and
engineering (especially robotics). In physics, CGA greatly simplifies
the 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 de-

velopment of GA from STA to GC and CGA as reinvigorating the
mathematics of Hermann Grassmann and William Kingdon Clifford
with an infusion of twentieth century physics. The history of this de-
velopment and the present status of GA has been reviewed in [17].
I am sorry to say that few mathematicians are prepared to recog-
nize the central role of GA in their discipline, because it has become
increasingly insular and divorced from physics during the last century.

Personally, I dedicate my work with GA to the memory of my

mathematician father, Magnus Rudolph Hestenes, who was always
generous with his love but careful with his praise [9].

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 Electromagnetic

Interactions, Found. Phys. 12, 153–168 (1982).

[5] D. Hestenes, Gauge Gravity and Electroweak Theory. In: H. Klei-

nert, R.T. Jantzen and R. Ruffini (Eds.), Proceedings of the Ele-
venth Marcel Grossmann Meeting on General Relativity (World
Scientific: Singapore, 2008), pp. 629–647.

[6] D. Hestenes and G. Sobczyk, CLIFFORD ALGEBRA to GEO-

METRIC CALCULUS, A Unified Language for Mathematics and
Physics (Kluwer: Dordrecht/Boston, 1984).

xiv

Preface after Fifty years

[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 Quan-

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.

[10] A. Lasenby, C. Doran and S. Gull, Gravity, gauge theories and

geometric 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 frame-

work for computational geometry. In: E. Bayro-Corrochano and
G. Sobczyk (Eds), Advances in Geometric Algebra with Applica-
tions in Science and Engineering (Birkh¨

auser: Boston, 2001), pp.

1–14.

[14] L. Dorst, D. Fontijne and S. Mann, Geometric Algebra for Com-

puter Science (Morgan Kaufmann, San Francisco, 2007).

[15] D. Hestenes and J. Holt, The Crystallographic Space Groups in

Geometric 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’s
Work in Context (Birkh¨

auser: Berlin, 2011).

Preface

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 belief
that such is an essential step in the endeavor to simplify and unify
our theories of physical phenomena. The object has been to produce a
“space-time calculus” which is ready for physicists to use. Particular
attention has been given to the development of notation and theorems
which enhance the geometric meaning and algebraic efficiency of the
calculus; conventions have been chosen to differ as little as possible
from 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. The
reader will find that the “space-time algebra” introduces novelty of ex-
pression and interpretation into every topic. This naturally suggests
certain modifications of current physical theory; some are pointed out
in the text, but they are not pursued. The principle objective here has
been to formulate important physical ideas, not to modify or apply
them.

The mathematics in this book is relatively simple. Anyone who

knows what a vector space is should be able to understand the algebra
and geometry presented in chapter I. On the other hand, appreciation
of the physics involved in the remainder of the book will depend a great
deal 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 the
National Science Foundation—for which I am grateful.

I wish to make special mention of my debt to Marcel Riesz, whose

trenchant discussion of Clifford numbers and spinors (8, 9) provided
an initial stimulus and some of the key ideas for this work. I would

xvi

Preface

also like to express thanks to Professors V. Bargmann, J. A. Wheeler
and A. S. Wightman for suggesting some improvements in the final
manuscript and to Richard Shore and James Sock for correcting some
embarrassing errors.

David Hestenes

Palmer Laboratory

Princeton University

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

Geometric Algebra

Interpretation of Clifford Algebra . . . . . . . . . . . .

Definition of Clifford Algebra

. . . . . . . . . . . . . .

Inner and Outer Products . . . . . . . . . . . . . . . .

Structure of Clifford Algebra . . . . . . . . . . . . . . .

Reversion, Scalar Product . . . . . . . . . . . . . . . .

The Algebra of Space . . . . . . . . . . . . . . . . . . .

The Algebra of Space-Time

. . . . . . . . . . . . . . .

II Electrodynamics

Maxwell’s Equation . . . . . . . . . . . . . . . . . . . .

Stress-Energy Vectors . . . . . . . . . . . . . . . . . . .

Invariants . . . . . . . . . . . . . . . . . . . . . . . . .

Free Fields . . . . . . . . . . . . . . . . . . . . . . . . .

III Dirac Fields

Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . .

Dirac’s Equation

. . . . . . . . . . . . . . . . . . . . .

Conserved Currents . . . . . . . . . . . . . . . . . . . .

C, P , T

. . . . . . . . . . . . . . . . . . . . . . . . . .

IV Lorentz Transformations

Reflections and Rotations

. . . . . . . . . . . . . . . .

Coordinate Transformations . . . . . . . . . . . . . . .

Timelike Rotations . . . . . . . . . . . . . . . . . . . .

Scalar Product

. . . . . . . . . . . . . . . . . . . . . .

xvii

xviii

Contents

V Geometric Calculus

Differentiation . . . . . . . . . . . . . . . . . . . . . . .

Riemannian Space-Time . . . . . . . . . . . . . . . . .

Integration . . . . . . . . . . . . . . . . . . . . . . . . .

Global and Local Relativity . . . . . . . . . . . . . . .

Gauge Transformation and Spinor Derivatives . . . . .

Conclusion

Appendixes

Bases and Pseudoscalars . . . . . . . . . . . . . . . . .

Some Theorems . . . . . . . . . . . . . . . . . . . . . .

Composition of Spatial Rotations . . . . . . . . . . . .

Matrix Representation of the Pauli Algebra

. . . . . .

Bibliography

101

Introduction

A geometric algebra is an algebraic representation of geometric con-
cepts. Of particular importance to physicists is an algebra which can
efficiently describe the properties of objects in space-time and even
the properties of space-time itself, for it is only in terms of some such
algebra that a physical theory can be expressed and worked out. Many
different geometric algebras have been developed by mathematicians,
and from time to time some of these systems have been adapted to
meet the needs of physicists. Thus, during the latter part of the nine-
teenth century J. Willard Gibbs put together a vector algebra for three
dimensions based on ideas gleaned from Grassmann’s extensive alge-
bra and Hamilton’s quaternions [1, 2, 3]. By the time this system was
in relatively wide use, a new geometric algebra was needed for the
four-dimensional space-time continuum required by Einstein’s theory
of relativity. For the time being, this need was filled by tensor alge-
bra. But, it was not long before Pauli found it necessary to introduce
a new algebra to describe the electron spin. Subsequently, Dirac was
led to still another algebra which accommodates both spin and special
relativity. Each of these systems, vector and tensor algebra and the
algebras of Pauli and Dirac, is a geometric algebra with special ad-
vantages for the physicist. Because of this, each system is widely used
today. The development of a single geometric algebra which combines
advantages of all these systems is a principal object of this paper.

One of the oldest of the above-mentioned geometric algebras is

the one invented by Grassmann (∼1840) [4]. His approach went some-
thing like this: He represented the geometric statement that two points
determine 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

xix

Introduction

plane. In a similar fashion he represented geometric objects of higher
dimension. But more than that, he represented the symmetries and
relative orientations of these objects in the very rules of combination
of his algebra.

At about the same time, Hamilton invented his quaternion al-

gebra to represent the properties of rotations in three dimensions.
Evidently Grassmann and Hamilton each paid little attention to the
other’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
1930 that Clifford algebra had important applications in physics. At
that time Pauli and, subsequently, Dirac introduced matrix represen-
tations of Clifford algebra for physical but not for geometrical reasons

Clifford constructed his algebra as a direct product of quaternion

algebras. I rather think that Hamilton would have made a similar
construction earlier had he been troubled about the relation of his
quaternions to Grassmann’s algebra. But this is a formal algebraic
method. So is the now common procedure of arriving at the Pauli
and Dirac algebras from a study of representations of the Lorentz
group. Had Grassmann considered this problem, I think he would
have been led to Clifford algebra in quite a different way. He would
have insisted on beginning with algebraic objects directly representing
simple geometric objects and then introducing algebraic operations
with geometrical interpretations.

In this paper, ideas of Grassmann are used to motivate the con-

struction of Clifford algebra and to provide a geometric interpretation
of Clifford numbers. This is to be contrasted with other treatments of
Clifford algebra which are for the most part formal algebra. By insist-
ing on “Grassmann’s” geometric viewpoint, we are led to look upon
the Dirac algebra with new eyes. It appears simply as a vector algebra
for space-time. The familiar γ

are seen as four linearly independent

vectors, rather than as a single “4-vector” with matrices for compo-
nents. The Pauli algebra appears as a subalgebra of the Dirac algebra
and 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 no
geometrical reason for complex numbers to appear. Rather, the unit
pseudoscalar (denoted by γ

in matrix representations) plays the role

Introduction

of the unit imaginary i. All these features are foreign to the Dirac al-
gebra when seen from the usual viewpoint of matrix representations.
Yet, as will be demonstrated in the text, emphasis on the geometric
aspect 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.
In chapter I we review the properties of the Clifford algebra for a

finite dimensional vector space. We strive to develop notation and the-
orems which clarify the geometrical meaning of Clifford algebra and
make it an effective practical tool. Then we discuss the application of
our general viewpoint to space-time and the Dirac algebra. We deal
exclusively 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 clearly
between those properties of the Dirac algebra which are peculiar to
space-time and those which are not. A better reason is that Clifford
algebra has many important applications other than the one we work
out in this paper. Indeed, it is important wherever a vector space with
an inner product arises. For example, physicists will be interested in
the fact that quantum field operators form a Clifford algebra.

Many readers will be familiar with the matrix algebras of Pauli

and Dirac and will be anxious to see how these algebras can be viewed
differently. 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. To
get a full appreciation of the efficacy of the Dirac algebra as a vec-
tor algebra quite apart from any connection it has with spinors, the
discussion of the electromagnetic field in chapter II should be stud-
ied closely. It should particularly be noted how identification of the
Pauli algebra as a subalgebra of the Dirac algebra greatly facilitates
the 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

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

Introduction

spinors can be defined without reference to representations of the
Lorentz group, even without using matrices. The definition of a spinor
as an element of a minimal ideal in a Clifford algebra is due to Marcel
Riesz [8].

It can hardly be overemphasized that by our geometrical con-

struction we obtain the Dirac algebra without complex numbers. Nev-
ertheless, it is still possible to write down the Dirac equation. This
shows that complex numbers are superfluous in the Dirac theory. One
wonders if this circumstance has physical significance. To answer this
question, 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
in several different forms and raising some questions. The main pur-
pose of chapter III is to illustrate the most direct formulation of the
Dirac theory in the real Dirac algebra. We also indulge in speculating
a connection between isospin and the Dirac algebra.

Lorentz transformations are discussed in chapter IV. So many

treatments of Lorentz transformations have been given in the past
that publication of still another needs a thorough justification. The
approach presented here is unique in many details, but two features
deserve special mention. First, Lorentz transformations are introduced
as automorphisms of directions at a generic point of space-time. This
approach is so general that it applies to curved space-time, as can
be seen specifically when it is applied in section 23. Automorphisms
induced by coordinate transformations are discussed in detail as a
very special case. The discussion of coordinate transformations can be
further 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 into
the Dirac algebra when we specified its relation to the Pauli algebra.
The respective merits of representing Lorentz rotations in the Pauli
algebra 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

xxii

Introduction

considerable.

Since the Dirac algebra has physical significance, it is important

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

Having demonstrated that it provides an efficient description of

the 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 basis
for a versatile geometric calculus. This task is undertaken in chapter
V and proves to be relatively straightforward because of the geometric
meaning which has already been supplied to the Dirac algebra. The
main problem is to define a suitable differential operator . It ap-
pears that all the local properties of space-time can be construed as
properties of this remarkable operator. We express the gravitational
field equations in terms of , and we show how is related to two
different generalizations of Einstein’s special principle of relativity. We
discuss the application of and Clifford algebra to integration theory.
Finally, we illustrate a way to generalize to generate theories with
non-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 formu-
lated here. But it can be developed further. A tensor algebra can be
constructed over the entire Dirac algebra. This is most elegantly done
by emulating the treatment of dyadics by Gibbs [1, 2, 3]. In this way
equivalents 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 Dirac
algebra we are compelled to adopt a philosophy about the construc-
tion of physical theories which differs somewhat from the customary
one. Ordinarily the Lorentz group is taken to describe the primitive
geometrical 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 em-
bodies the local geometrical properties of space-time. Physical quan-
tities are then limited to objects which can be constructed from this

xxiii

Introduction

algebra.

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

it must arise because the second is more restricted than the first. This
can only occur if the assumptions about space-time which go into the
real Dirac algebra are more detailed than those which go into the
Lorentz group. That such may actually be the case is suggested by
the fact that the first philosophy permits physical theories which con-
tain complex numbers without direct geometrical meaning, whereas
the second does not. It should be possible to resolve this problem by
a study of the Dirac equation because it contains complex numbers
explicitly. But, we shall not attempt to do so here.

xxiv

Chapter I

Geometric Algebra

Interpretation of Clifford Algebra

To every n-dimensional vector space

with a scalar product there

corresponds a unique Clifford algebra

. In this section we give an

intuitive discussion of how

arises as an algebra of directions in

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

From two vectors a and b we can construct new geometric ob-

jects endowed with geometric properties which arise when a and b are
considered together. For one thing, we can form the projection of one
vector on the other. Since it is determined by the mutual properties of
a and b, we can express it as a product a · b. That a · b must be a num-
ber (scalar) and also symmetric in a and b is an algebraic expression
of the geometric concept of projection.

We can form another kind of product by noting that the vectors a

and b, if they are linearly independent, determine a parallelogram with
sides a and b. We can therefore construct from a and b a new vectorlike
entity with magnitude equal to the magnitude of the parallelogram and
direction 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 of
a∧b as an algebraic representation of the parallelogram since it retains
the information as to the area and plane of the parallelogram. However
a ∧ b does not determine the shape of the parallelogram and could

Ó Springer International Publishing Switzerland 2015
D. Hestenes, Space-Time Algebra,
DOI 10.1007/978-3-319-18413-5_1

Chapter I. Geometric Algebra

equally well be interpreted as a circular disc in the plane of a and b with
area equal to that of the parallelogram. In its shape the parallelogram
contains information about the vectors a and b individually, whereas
a ∧ b retains only properties which arise when a and b are considered
together—it has only magnitude and direction.

Actually, since we can distinguish two sides of the plane contain-

ing a and b, we can construct two different products which we write
as a ∧ b and b ∧ a. Intuitively speaking, if a ∧ b is represented by a
parallelogram 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 have
opposite orientation, just as do the vectors a and −a. We can express
this by an equation:

a ∧ b = −b ∧ a.

(1.1)

Thus the outer product of vectors is antisymmetric. The reader is
cautioned not to confuse a∧b with the familiar product a×b introduced
by Gibbs. For the vectors of

, a × b is the dual of a ∧ b and the same

interpretation can be given to both. However, the definition of a × b
depends on the dimensionality of the vector space in which a and b
are imbedded, whereas a ∧ b pertains only to the plane containing a
and b.

The reader is now directed to observe that since a · b and a ∧ b

have 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

ab = a · b + a ∧ b.

(1.2)

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

In our attempt to construct a geometrically meaningful multipli-

cation for vectors we were led to define other geometric objects which,
like vectors, can be characterized by magnitude and direction. This
suggests a generalization of the concept of vector. Every r-dimensional
subspace of

determines an oriented r-dimensional unit volume el-

ement which we call an r-direction. By associating a magnitude with
an r-direction we arrive at the concept of an r-vector. A few examples

1. Interpretation of Clifford Algebra

will help make this concept clear. The vectors of

are 1-vectors.

The unit vectors of

are 1-directions. Because they have magnitude

but no direction, we can interpret the scalars as 0-vectors. The outer
product 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

. Only two unit n-vectors can be

associated with

. They differ only in sign. They represent the two

possible orientations which can be given to the unit volume element
of

Let us return to our definition of vector product. In (1.2) we

introduced 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-vectors
and the 2-vectors are then linearly independent.

Now that we understand the product of two vectors, what is more

natural 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 of
vectors can be expressed as a sum of r-vectors of different degree. In
this way we generate the Clifford algebra

. An element of

will be called a Clifford number. Every r-vector of

is an element

, and, conversely, every Clifford number can be expressed as a

linear combination of r-directions. Thus, we can interpret

as an

algebra of directions in

, or, equivalently, an algebra of subspaces of

In the paper which introduced the algebra that goes by his name,

Clifford remarked on the dual role played by the real numbers. For
instance, the number 2 can be thought of as a magnitude, but it can
also be thought of as the operation of doubling, as in the expression
2 × 3 = 6. Clifford numbers exhibit the same dual role, but in a
more 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 one
Clifford number into another. This dual role enables us to discuss
directions and operations on directions without going outside of the
Clifford algebra. Combining our insights into the interpretation of the

Chapter I. Geometric Algebra

elements and operations in

we can characterize Clifford algebra

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

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 axiomatic
approach which does not make explicit mention of basis, because it
keeps the rules of combination clearly before us. In later sections we
again 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” which
already appear in the Clifford algebra over the reals. Our analysis will
suggest that the complex field arises in physics for geometric reasons.
In any case, it is a trivial matter to generalize our work to apply to
Clifford algebra over the complex field.

Let

be an n-dimensional vector space over the real numbers

with elements a, b, c, . . . called vectors. Represent multiplication of
vectors 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,

A + B = B + A

(commutative law),

(2.1)

(A + B) + C = A + (B + C)

(associative law).

(2.2)

Denote the resulting “polynomials” by capital letters also. Multipli-
cation of polynomials satisfies:

A(BC) = (AB)C

(associative law),

(2.3)

a0 = 0

(2.4)

for every a in

and 0 the null vector,

(A + B)C = AC + BC,

C(A + B) = CA + BA

(distributive law).

(2.5)

3. Inner and Outer Products

Assume scalars commute the vectors, i.e.,

αa = aα for scalar α and vector a.

(2.6)

Equations (2.3) and (2.6) imply that the usual rules for scalar mul-
tiplication hold also for vector polynomials. Scalars and vectors are
related by assuming:

For a, b in

, ab is a scalar

if and only if a and b are collinear.

(2.7)

We identify (|a

1
2

with the length of the vector a.

The algebra defined by the above rules is called the Clifford alge-

bra

of the vector space

Conversely,

is said to be the vector

space associated with

. The elements of

are called Clifford num-

bers, or simply c-numbers. We shall call AB the vector product of
c-numbers A and B.

From the mathematical point of view it is interesting to note that

there is a great deal of redundancy in our axioms. It is obvious, for
instance, that the rules of vector and scalar multiplication in

with

which we began are special cases of the operations on arbitrary c-
numbers which we defined later. A minimal set of axioms for Clifford
algebra over the reals is very nearly identical with the axioms for the
real numbers themselves.

Inner and Outer Products

Consider the quadratic form a

for a vector a. Observe that for vectors

a and b,

(a + b)

= a

+ ab + ba + b

ab + ba = (a + b)

− a

− b

which is a scalar. Thus a

has an associated symmetric bilinear form

which we denote by

a · b ≡

1
2

(ab + ba).

(3.1)

The special effects of various kinds of singular metric on

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.

Chapter I. Geometric Algebra

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

Now decompose the product of two vectors into symmetric and

anti-symmetric parts,

ab =

1
2

(ab + ba) +

1
2

(ab − ba).

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

a ∧ b ≡

1
2

(ab − ba).

(3.2)

So the product of 2 vectors can be written in the form

ab = a · b + a ∧ b.

(3.3)

We proceed to examine the outer product in some detail. Obvi-

ously a ∧ b = −b ∧ a and a ∧ a = 0. From (2.7) it is apparent that
a ∧ b = 0 means that a and b are collinear—a statement which is
complementary to the statement that a · b = 0 means that a and b are
perpendicular.

Geometrically, a ∧ b can be interpreted as an oriented area. b ∧ a

has opposite orientation. a ∧ b ∧ c can be interpretated as an oriented
volume, i.e., the parallelopiped with sides a, b, c. c ∧ b ∧ a is the volume
with opposite orientation. The outer product of r vectors,

= a

∧ a

∧ . . . ∧ a

(3.4)

is called a simple r-vector. It may be defined by recursion:

r+1

= a ∧ A

≡

1
2

(aA

+ (−1)

a) = a ∧ a

∧ . . . ∧ a

(3.5)

may be interpreted as an oriented r-dimensional volume. A partic-

ular simple r-vector may be thought of as representing the r-dimen-
sional subspace of

which contains the vectors of “its product” and

their linear combinations.

A linear combination of simple r-vectors (e.g., a

∧ a

+ a

∧ a

)

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 names
scalar, vector, bivector, trivector, . . . respectively. An n-vector in

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.

∧ a

∧ . . . ∧ a

) ∧ (b

∧ b

∧ . . . ∧ b

)

= (a

∧ . . . ∧ a

r−1

) ∧ (a

∧ b

∧ . . . ∧ b

(3.6)

This is clearly just the associative law for outer products.

From the associative law for outer products and (3.5) it follows

easily 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 at
will. Since a ∧ b = 0 if a and b are collinear, a r-vector is equal to
zero 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 have
defined the dimension of

as the degree of the largest r-vector in

the Clifford algebra of

The commutation rule for the outer product of an r-vector A

with an s-vector B

follows easily from antisymmetry and associativ-

ity,

∧ B

= (−1)

∧ A

(3.7)

We turn now to a generalization of the inner product which corre-

sponds to (3.5). The inner product has symmetry opposite to that of
the outer product. For a vector a and an r-vector A

= a

∧a

∧. . .∧a

write

a · A

≡

1
2

(aA

− (−1)

a).

(3.8)

The result is an (r − 1)-vector. The right side of (3.8) can be evaluated
to give the useful expansion

a · (a

∧ . . . ∧ a

)

k=1

(−1)

k+1

(a · a

∧ . . . ∧ a

k−1

∧ a

k+1

∧ . . . ∧ a

(3.9)

Chapter I. Geometric Algebra

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

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

to (3.6).

∧ . . . ∧ a

) · (b

∧ . . . ∧ b

)

= (a

∧ . . . ∧ a

r−1

) · [a

· (b

∧ . . . ∧ b

)].

(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.

∧ . . . ∧ a

) · (b

∧ . . . ∧ b

)

= (a

∧ . . . ∧ a

k−1

) · [(a

∧ . . . ∧ a

) · (b

∧ . . . ∧ b

)],

(3.11)

∧ . . . ∧ a

) · (b

∧ . . . ∧ b

)

= (a

∧ . . . ∧ a

) · (b

∧ . . . ∧ b

) b

r+1

∧ b

r+2

∧ . . . ∧ b

− (a

∧ . . . ∧ a

) · (b

∧ · · · ∧ b

r+1

∧ b

r+2

∧ · · · ∧ b

. . .

(3.12)

+ (−1)

s−r−1

∧ . . . ∧ a

) · (b

s−r

∧ . . . ∧ b

∧ b

∧ . . . ∧ b

s−r−1

(3.12) gives the explicit form of the multivector which is obtained from
the inner product of two simple multivectors.

By (3.8), a · (b

∧ . . . ∧ b

) = (−1)

s+1

∧ . . . ∧ b

) · a. So from

(3.10) we find that for any r-vector and s-vector with s = r,

· B

= (−1)

r(s+1)

· A

(3.13)

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]

= a

− (a · b)

b · b

b · a

a · b a · a

(3.14)

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)

a · u a · v

b · u

b · v

(3.15)

The sign measures the relative orientations of a ∧ b and u ∧ v.

Another example shows how (3.9) and (3.10) are related to the

expansion of a determinant by cofactors,

(a ∧ b ∧ c) · (u ∧ v ∧ w) = (a ∧ b) · [c · (u ∧ v ∧ w)]

= (a ∧ b) · [(c · u)v ∧ w − (c · v)u ∧ w + (c · w)u ∧ v]

= (c · u)

b · v

b · w

a · v a · w

− (c · v)

b · u

b · w

a · u a · w

+ (c · w)

b · u

b · v

a · u a · v

c · u

c · v

c · w

b · u

b · v

b · w

a · u a · v a · w

The generalization of this formula is evidently

∧ . . . ∧ a

) · (b

∧ . . . ∧ b

) =

· b

. . . a

· b

. . . a

· b

(3.17)

This is the projection of a

∧ . . . ∧ a

on b

∧ . . . ∧ b

We have shown how the Clifford algebra

generates a

vector algebra on

. The basic rules for vector multiplication are the

associative and anti-commutative rules for outer multiplication and
formulas (3.9) and (3.10) for inner multiplication. All other formulas
of vector analysis follow easily, as do all theorems on determinants.

In fact, (3.17) can be taken as the definition of determinant.

We have seen that our original rule of multiplication acts as a

kind of “neutral product” from which other rules of multiplication
can be obtained which have geometric significance. The inner and

Note, for instance, that the antisymmetry of rows (or columns) under interchange is obvious

from (3.17) by the antisymmetry of the outer product.

Chapter I. Geometric Algebra

outer products are not the only examples of such rules, but they are
the only ones to which we shall assign a special notation.

Our inner and outer products were invented independently of Clif-

ford algebra by Grassman [4] about 1844. Because of their geometric
significance they occur ubiquitously and have no doubt been rein-
vented many times The outer product is particularly important in
multiple integration theory. We will discuss this again in section 22.

Our “neutral algebra” was invented by Clifford [5] about 1878. He

appears to have arrived at it as a kind of union of Grassmann’s exterior
(outer) product with Hamilton’s quaternions. Grassmann’s original
ideas for algebraic representation of geometric notions are brought to
perfection by Clifford algebra, as is Hamilton’s idea that the generators
of rotations are the proper generalizations of pure imaginary numbers.
The latter idea is realized by the fact that the bivectors of

generate

the rotations in

; it will be amply illustrated in our treatment of

Lorentz transformations (chapter IV).

Structure of Clifford Algebra

In this section we discuss some important properties of

which for

the most part do not depend on the metric of

We learned in the last section that the product of a vector a with

an r-vector A

gives, in general, an (r−1)-vector plus an (r+1)-vector:

= a · A

+ a ∧ A

(4.1)

It follows that the result of any combination of multiplication and
addition of vectors can be expressed as a polynomial of multivectors.
In fact, since any vector can be expressed as a linear combination
of n-linearly independent vectors e

, e

, . . . , e

, any c-number can be

written as a polynomial of multi-vectors formed from the e

A = a + a

∧ e

+ . . . +

...i

∧ e

∧ . . . ∧ e

(4.2)

Here the a’s are scalar coefficients and summation is taken from 1 to
n over repeated indices.

4. Structure of Clifford Algebra

Let us pause to introduce some helpful notations. We will often

write

ij...k

≡ e

∧ e

∧ . . . ∧ e

(4.3)

Since e

ij...k

is antisymmetric under interchange of indices, it is usually

convenient to write the indices in natural order (i.e. i < j < . . . <
k). This eliminates redundancy in the sums of (4.2). As a further
abbreviation, we may represent a whole set of indices by a capital
letter. Let

J = (j

, . . . , j

)

(4.4)

where j

= k or 0. By deleting elements for which j

= 0, we may

write any one of the base multivectors in the form

= e

...j

= e

∧ e

∧ . . . ∧ e

(4.5)

For the case when all j

vanish, we take e

= 1. The expansion (4.2)

of any c-number can now be written simply as

A = a

(4.6)

Let

(r)

be the space of all r-vectors; thus,

(0)

is the scalar

field;

(1)

;

(n)

is the space of all pseudoscalars. The set of all

of degree r forms a basis for

(r)

. It is a simple matter of counting

to show that the dimension of the linear space

(r)

is given by the

binomial coefficient

The whole Clifford algebra

is itself a linear space, and the set

of all e

forms what we shall call a tensor basis for

. The dimension

is given by

dim

r=0

dim

(r)

r=0

= 2

(4.7)

It is a fact of great geometrical significance that

can be de-

composed into two subspaces composed respectively of even and odd
multivectors. We write

−

(4.8)

Chapter I. Geometric Algebra

where

is the space of all r-vectors for r even, and

is the space

of all r-vectors for r odd. The relation of

−

by multiplication

is expressed by the formulas

−

(4.9)

These formulas are implied by the more specific relations

(r)

(s)

⊂

k=0

(r+s−2k)

(4.10)

where m = min(r, s) .

It is clear from (4.9) that

always forms a subalgebra of

−

whereas

−

never does. We call

the even subalgebra of

itself a Clifford algebra, and by a proper identification of vectors we
may associate it with a vector space

n−1

. Let us express this by

n−1

(4.11)

This relation will be of great significance to us in section 7 when
we discuss the relation of the Pauli and Dirac algebras. There we will
discuss a geometrically significant rule which can be used to determine
the vectors of

n−1

from

. Here we will merely mention a curious

fact, namely, that the most important Clifford algebras are related
in a series by (4.11): The Pauli algebra is the even subalgebra of the
Dirac algebra. The quaternion algebra is the even subalgebra of the
Pauli algebra. The complex numbers compose the even subalgebra of
the quaternions, and, of course, the real numbers compose the even
subalgebra of the complex numbers.

The decomposition (4.8) of

into even and odd c-numbers is

as significant as the decomposition of complex numbers into real and
imaginary parts. In fact, for that Clifford algebra

which can be

identified with the complex numbers, the two decompositions are one
and the same. This leads us to an involution of

which may be

looked upon as a generalization of complex conjugation. We give the
name inversion

to the mapping which takes every r-vector A

into

Also called the main involution of

5. Reversion, Scalar Product

its conjugate A

∗
r

defined by

∗
r

= (−1)

(4.12)

Inversion distinguishes even and odd subspaces of

∗

= (

−

)

∗

−

(4.13)

Inversion may be thought of as an “invariant” kind of complex con-
jugation because it does not favor a particular tensor basis in the
algebra. It is worth mentioning that if n is odd inversion is an outer
automorphism of

, i.e. it cannot be induced by operations defined

(as can easily be verified for the complex numbers and for the

Pauli algebra). However, inversion is important enough to make it
worthwhile to extend the definition of

to include it.

In the one-dimensional Clifford algebra of the complex numbers

the vectors are also the pseudoscalars. We customarily represent the
unit positive pseudoscalar by the symbol i. A rotation in the complex
plane is a mapping of scalars into pseudoscalars by the exponential
operator e

iφ

. We call it a duality rotation. There is an analog of this

transformation in any non-singular Clifford algebra

. (A Clifford

algebra will be said to be non-singular if it contains pseudoscalars
with non-zero square.) Let us again use the symbol i to denote the
unit pseudoscalar in

. Again we will call the transformation

→ e

iφ

(4.14)

a duality rotation. In later sections we will see that duality rotations
in the Dirac algebra have physical significance.

Reversion, Scalar Product

To obtain the reverse A

†

of a c-number A, decompose A into a sum of

products of vectors and reverse the order of all products. We call this
operation reversion

or, alternatively, hermitian conjugation, because

it is equivalent to the operation by that name in a matrix representa-
tion of Clifford algebra which represents 1-vectors by hermitian matri-
ces. Since the inner product of vectors is symmetric, it does not matter

Also called the main antiautomorphism of

Chapter I. Geometric Algebra

whether the term “product” in the above definition is interpreted as
“vector product” or as “outer product”.

Let us define the scalar product (A, B) of two c-numbers A and

B as the scalar part of A

†

(A, B) ≡ (A

†

(5.1)

This is not to be confused with our “inner product” which might not
even have a scalar part.

We may say that A is orthogonal to B if (A, B) = 0. It is easy to

show that we can find a basis {e

} for

consisting of multivectors

which are orthogonal in this sense. If the metric of

is Euclidean the

can be chosen ortho-normal, i.e.

, e

) = δ

J K

(5.2)

The scalar product (5.1) determines a norm for each c-number A:

kAk = (A

†

(5.3)

This norm is positive definite if and only if the metric of

is Eu-

clidean, for only in the Euclidean case can we find an orthonormal
basis. Thus, using (5.2) and A = A

6= 0, we find

kAk =

)

> 0.

(5.4)

|kAk may be interpreted as the square of the length of a “vector” in
the vector space

. To see that this is a generalization of the square

of the length of a vector, note that for a vector a,

= (a

)

= (a

†

= kak.

(5.5)

Since (5.3) satisfies (5.4) only if the metric on

is Euclidean, any

singularity it may have in other cases will be exclusively due to a
singular metric on

. Therefore we conclude that (5.1) is a natural

generalization of the inner product on

We should mention that the scalar product (5.1) is symmetric,

(A, B) = (B, A).

(5.6)

5. Reversion, Scalar Product

This just follows from the fact that the scalar part of a product of
c-numbers is invariant under cyclic permutation of products, as can
be seen by using a tensor basis of orthogonal multivectors. Thus

(AB)

2
J

= (BA)

(5.7)

The problems of evaluating transition probabilities that occur in

quantum electrodynamics always involve finding the scalar part of
some c-numbers which may, for example, represent the scalar product
of c-numbers as in (5.1). This can always be reduced to evaluation of
the scalar part of a product of vectors. (When matrix representations
of the Clifford algebra are used, this is a problem in evaluating traces.)
We derive a fundamental formula for such calculations.

Let a, b

, b

, . . . , b

be vectors. We prove

a · (b

. . . b

) =

r=1

(−1)

r+1

(a · b

)(b

. . . b

r−1

r+1

. . . b

)

(5.8)

by adding up both sides of the equalities:

1
2

(ab

. . . b

+ b

. . . b

) = (a · b

. . . b

−

1
2

. . . b

+ b

. . . b

) = −(a · b

. . . b

. . .

(−1)

k+1 1

. . . b

k−1

+ b

. . . b

a) = (−1)

k+1

(a · b

. . . b

k−1

Multivectors of the same degree on both sides of (5.8) can be equated.
The largest multivector gives equation (3.9) so this is the promised
proof of that equation.

By (5.7),

· (a

. . . a

)]

1
2

[(a

. . . a

)

+ (a

. . . a

)

]

= (a

. . . a

)

So by (5.8)

. . . a

)

r=2

(−1)

· a

)(a

. . . a

r−1

r+1

. . . a

)

(5.9)

Chapter I. Geometric Algebra

. . . a

)

can be written entirely in terms of inner products of vectors

by successive applications of (5.9). This and other formulas can be
found in standard works on quantum field theory.

But (5.8) is the

most useful formula. Note, in passing, that (a

. . . a

)

= 0 if k is odd.

The Algebra of Space

The set of all vectors tangent to a generic point x in Euclidean 3-
space

forms a 3-dimensional vector space

(x) with an inner

product which expresses the metric on

. The Clifford algebra of

is so important we will give it a special name.

We call it the Pauli

algebra

P to agree with the name given its matrix representation. To

exploit the familiarity of physicists with the matrix representation of
P, we will begin our study of P with a basis in V

Select a set of orthonormal vectors σ

, σ

, tangent to the point

The vectors σ

satisfy the same multiplication rules as the familiar

Pauli matrices.

· σ

1
2

(σ

+ σ

) = δ

(6.1)

By taking all products of the σ

we generate a tensor basis for

P. P

is a linear space of dimension 2

= 8. The elements of the tensor basis

P can be given geometric interpretation. The bivector

= σ

∧ σ

1
2

(σ

− σ

)

(6.2)

can be interpreted as a unit element of area oriented in the jk-plane.
The unit element of volume is so important that we represent it by
the special symbol i,

i = σ

∧ σ

= σ

(6.3)

See, for example, page 437 of reference [7].

In the same way that we extend the concept of a vector to a vector field on a manifold, we

can extend the concept of a c-number to a c-number field. And just as it is often convenient to
suppress the distinction between vector and vector field, it is often convenient to suppress the
distinction between c-number and c-number field. In the rest of this paper we will frequently do
such things with little or no comment.

Throughout this paper we use boldface type only for points in Euclidean 3-space and 1-

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 section 7.

6. The Algebra of Space

We choose the sign of i to be positive when the σ

form a righthanded

set of vectors. To remind us of this, we call i the unit righthanded
pseudoscalar. The symbol i is appropriate because i commutes with
all elements of

P, and i

= −1. Of course, in this context, i also

has a geometrical meaning, and this enables us to give a geometrical
interpretation to complex conjugation. Let us give the name space
conjugation to the operation which reverses the direction of all vectors
at the point x.

It can be represented as an operation on the base

vectors:

→ −σ

(6.4)

This induces a change in orientation of the unit volume element:

i → −i.

(6.5)

We will see later that when applied to equations of physics space
conjugation changes particles into antiparticles and so can be given a
physical interpretation as well.

It win be convenient to call the elements of the Pauli algebra p-

numbers. Any p-number A can be written as a sum of scalar, vector,
bivector and pseudoscalar parts:

A = A

+ A

(6.6)

Or, in more explicit form,

A = α + iβ + a + ib

(6.7)

where α and β are scalars and a and b are vectors. This shows that
we can look upon a p-number as composed of a complex scalar plus a
complex vector. This algebraic interpretation is very convenient; how-
ever, it is important not to lose sight of the geometrical interpretation
of the various terms.

We will denote by A

∗

the element obtained from A by space con-

jugation. In terms of the decomposition (6.6),

∗

= A

− A

+ A

− A

(6.8)

Another invariant kind of conjugation is obtained by reversing the
order of all products of vectors in

P. We call this operation hermitian

Space conjugation is an example of the operation inversion introduced in section 4.

Chapter I. Geometric Algebra

conjugation, because it is equivalent to an operation with the same
name in the usual matrix representation of

P. The element obtained

from A by hermitian conjugation will be denoted by A

†

= A

+ A

− A

(6.9)

We can define an invariant kind of transpose by

A ≡ A

∗†

(6.10)

Obviously,

A = A

− A

+ A

(6.11)

As in (3.3), we can decompose the product of two vectors a and

b into symmetric and antisymmetric parts,

ab = a · b + a ∧ b.

(6.12)

The complete vector algebra developed in section 3 now applies to

It includes to the familiar vector algebra developed by Gibbs [2, 3].
This will become clear when we take account of some special properties
of

The product of iA of the unit pseudoscalar i with a p-number A

is called the dual of A. In the Pauli algebra the dual of a bivector is a
vector, and vice-versa. This special property of

P enables us to define

the familiar cross product of two vectors as follows,

a × b ≡ −ia ∧ b.

(6.13)

The sign of this definition has been chosen so that from (1.3) we obtain
σ

× σ

= σ

, which means that the σ

form a righthanded frame.

a × b is commonly called an “axial vector”. It is often said that the
behavior of a × b under “space reflection” differs from that of “true”

The reader familiar with quaternions will know that the Pauli algebra is equivalent to the al-

gebra of complex quaternions. He will recognize e

A as the quaternion conjugate of A. A discussion

in quaternion language which is similar to ours is given by G¨

ursey [15]. The quaternion viewpoint

suffers from two defects: It does not direct sufficient attention to geometric interpretation of

and it gives no hint as to the relation of

P to the Dirac algebra.

Any vector a in

P can be written as a linear combination of the σ

· : a = a

. It is a

common practice in the literature of physics to write a

= a · σ. This would be a flagrant

violation both of the notion of vector used in this paper and the definition of inner product (3.1),
for the a

are scalars and the σ

are vectors, so a

is not the inner product of two vectors

and should not be written a · σ.

6. The Algebra of Space

vectors such as a and b. Now all this depends critically on what is
meant by “space reflection”. Here we understand “space reflection” to
mean “reverse the direction of all vectors in space”. It is the operation
of space conjugation defined above. Vectors a and b change sign under
inversion, but bivector a ∧ b does not. We recorded in (6.5) that,
for consistency in multiplication, the orientation of the pseudoscalars
must change. Hence, according to our definition (6.13) a × b changes
sign under space conjugation and is in our sense a “true” vector. a∧b is
vectorlike except under inversion; it is appropriate to call it a pseudo-
vector because it is the dual of a vector, but the appellation “axial
vector” is not appropriate to a ∧ b because the “axial” property is an
accident of three dimensions.

By using the definition (6.13) for the cross product, all the usual

formulas of Gibbs’ vector analysis can be obtained easily from the
formulas of section 3. Before passing onto some examples, the reader
is cautioned to note that, although i commutes with all elements of
P, it does not commute with the inner and outer products. This can
be seen from the definitions (3.5) and (3.8).

Thus,

a × b = −i(a ∧ b) = −a · (ib) = (ib) · a.

(6.14)

Now observe the decomposition of the double cross product with the
help of (3.9):

a × (b × c)

= −i[a ∧ (−ib ∧ c)] = i

a · (b ∧ c) = −(a · b)c + (a · c)b.

(6.15)

Another example:

a ∧ b ∧ c = −i

(a ∧ b ∧ c) = −ia · (ib ∧ c) = −i[a · (b × c)]. (6.16)

So |a · (b × c)| is the magnitude of the oriented volume a ∧ b ∧ c. The
characteristic property of the Pauli algebra is that the outer product
of any three linearly independent vectors is a pseudoscalar. From this
it follows that the outer product of any four vectors in

P is zero:

a ∧ b ∧ c ∧ d = 0.

(6.17)

We have so far constructed a complete vector algebra for the

tangent vectors of Euclidean 3-space. The development of our vector

Chapter I. Geometric Algebra

calculus is completed by defining the following differential operator:

∇ ≡ σ

∂

∂x

(6.18)

where σ

is tangent vector for the coordinate x

. We call ∇ the gradi-

ent operator.

It has the multiplication properties of a vector super-

imposed on the Leibnitz rule, characteristic of a differential operator.
The gradient of a vector can be decomposed into two parts,

∇a = ∇ · a + ∇ ∧ a

= ∇ · a + i∇ × a.

(6.19)

We call ∇ · a the divergence of a; it is a scalar. We give the name
“curl of a” to ∇ ∧ a, rather than to ∇ × a as is customary, because
the outer product is more general than the cross product. Of course
∇ ∧ a is a bivector.

As a point of historical interest, we mention that in the latter part

of the nineteenth century there was a lively controversy as to whether
quaternion or vector methods were more suitable for the work of the-
oretical physicists.

The votaries of vectors were victorious. However

soon after the invention of quantum mechanics, quaternions reap-
peared in a new guise, for Clifford algebra (in matrix form) was found
necessary to represent the properties of particles with spin. Now the
root of the vector-quaternion controversy is to be found in the failure
of both sides to distinguish between what we have called vector and
bivector. It is amusing that vectors and quaternions are united in the
Pauli algebra, so that the whole basis for controversy is eliminated.
Neither side was wrong except in its opposition to the other.

The Algebra of Space-Time

The vector space of vectors tangent to a generic point x in Minkowski
space-time

determines a real Clifford algebra which we call the Dirac

The gradient operator is discussed in more detail in chapter V.

A short history of geometric algebra and the vector-quaternion controversy is given by Wills

[3]. Letters on the subject by Gibbs can be found in [1]. We mention that the significant comments
on notation made by Gibbs apply also to the calculus developed in this paper. These topics are
also wryly discussed by Heaviside [11].

The adjective “Minkowski” signifies the flat space-time of special relativity.

7. The Algebra of Space-Time

algebra. We begin our study of the Dirac algebra

D by selecting a set

of orthonormal vectors γ

, γ

tangent to the point x. The vectors

satisfy the same multiplication rules as the familiar Dirac matrices.

The γ

(k = 1, 2, 3) are spacelike vectors. The timelike vector γ

has

positive square.

= 1, γ

= −1, γ

· γ

= 0

(µ 6= ν).

(7.1)

These relations can be abbreviated,

· γ

1
2

(γ

+ γ

) = g

µν

(7.2)

By taking all products of the γ

we generate a tensor basis for

D. D is

a linear space of dimension 2

= 16. The elements of the tensor basis

D can be given geometric interpretation. The bivector

µν

= γ

∧ γ

1
2

(γ

− γ

)

(7.3)

can be interpreted as a unit element of area in the µν-plane. The
trivector (pseudovector)

µνσ

= γ

∧ γ

(7.4)

can be interpreted as an oriented unit 3-dimensional volume element.
The pseudoscalar

≡ γ

∧ γ

= γ

0123

(7.5)

can be interpreted as an oriented unit 4-dimensional volume element.

It will be convenient to call the elements of

D d-numbers. Any d-

number A can be written as a sum of scalar, vector, bivector, trivector,
and pseudoscalar parts:

A = A

+ A

(7.6)

We give the name space-time conjugation to the operation which

reverses the directions of all vectors in

D, and we denote by A the

element obtained from A by this operation.

Evidently,

A = A

− A

+ A

− A

+ A

(7.7)

This is another example of the inversion operation defined in section 4. We use e

A instead of

∗

for the space-time conjugate to distinguish it from the analogous space conjugate in

Chapter I. Geometric Algebra

If A = A, A will be called even. If A = −A, A will be called odd.
The other invariant kind of conjugation in

D reverses the order of all

vector products in

D and will be called reversion. We denote by

A the

element obtained from A by this operation. Evidently

A = A

+ A

− A

+ A

(7.8)

The Dirac algebra embodies the local geometric properties of

space-time, whereas the Pauli algebra represents the properties of
space. These two algebras must be related in a definite way, a way
which depends on the choice of a specific timelike direction. We fix
this relation by requiring that

P be a subalgebra of D. To see how

this partition depends on the selection of a timelike direction, let us
relate the basis σ

P to the basis γ

D. We choose σ

and γ

to refer to the same spacelike directions. Therefore, since they are el-
ements of the same algebra, σ

must be proportional to γ

. It is then

easy to see that, except for a sign, the only way we can satisfy (6.1)
and (7.1) is to have

= γ

(7.9)

From this it follows that

= −γ

(7.10)

and

i = γ

(7.11)

Thus we find that

P is the subalgebra of all even d-numbers.

The

scalars of

P are scalars of D. The pseudoscalars of P are pseu-

doscalars of

D. Only the partition of the bivectors of D into vectors

and bivectors of

P (as in (7.9) and (7.10)) depends on the singling

out of a particular timelike vector.

The uniqueness of our solution depends on the choice of metric we made in (7.1). If instead

we had chosen γ

= 1 and γ

= −1 we could entertain the solution σ

= γ

, which may

seem more natural, because then vectors in

P would be vectors in D. But this identification

is subject to two serious objections. First, “Dirac factorization” would be impossible with this
metric. By “Dirac factorization” we mean that the equation p

− m

= (p − m)(p + m) = 0

has a solution for a scalar m and timelike vector p. This is important in the study of the Dirac
equation. Second, we would have t = γ

= −γ

, so there would be more than one i—one

for each γ

. Consequently, the interpretation of complex numbers and the connection between

space and space-time would be more complicated than in the text.

It may be recalled that this point was discussed in section 4.

7. The Algebra of Space-Time

The Pauli algebra

P determined by a timelike vector γ

will be

called the Pauli algebra of γ

. We have just seen that the even d-

numbers are elements of the subalgebra

P. The odd d-numbers can

also be represented as elements of

P simply by multiplying them by

. Thus, a vector p in

D can be represented by a scalar p

and a

vector p in

P, for

pγ

= p

+ p

(7.12a)

where

≡ p · γ

, p = p ∧ γ

(7.12b)

We choose the sign in (7.12a) to agree with (7.9). We will follow this
convention throughout the rest of this paper.

We will hereafter always use the symbol i to represent the unit

righthanded pseudoscalar in

D. The symbol γ

may be retained to

represent the pseudo-scalar of an arbitrary basis, as in appendix A. i
will play the role of unit imaginary. Besides the property i

= −1, we

note that i commutes with the σ

but anticommutes with the γ

Now that we have established the connection between

P and D,

the operations of space conjugation and hermitian conjugation defined
for

P can be extended to D. The definitions

∗

≡ γ

Aγ

(7.13)

and

†

≡ γ

Aγ

= e

∗

(7.14)

apply to any element of

D and agree with the definitions given in

section 6. We will always remember that the symbols A

∗

and A

†

tac-

itly assume that a particular timelike direction has been singled out,
but we will usually not need to know which direction it is. Note, for
example, that i

∗

= −i holds for any choice of γ

The vector algebra of section 3 works in

D just as it did in P.

We remark again that no confusion should arise from our use of the
same symbols for inner and outer products in both

P and D because

we distinguish vectors in

P by boldface type.

To complete our development of a geometric calculus for space-

time, we define the gradient operator ,

≡ γ

∂

(7.15)

In chapter V a more careful definition of the gradient operator is given.

Chapter I. Geometric Algebra

If A is a differentiable d-number field, then

A = · A + ∧ A.

(7.16)

We call · A the divergence of A and ∧ A the curl of A. In agree-
ment with common parlance, we call

the d’Alembertian. Our use of

different symbols for and ∇ will preclude any confusion that might
follow from giving them the same name. Using (7.9) we find that

∇ = γ

∧

(7.17)

and

∂

= γ

· .

(7.18)

Thus

= ∂

+ ∇.

(7.19)

We make this one violation of the convention established by (7.12) in
order that our definition of ∇ agree with the usual one.

Chapter II

Electrodynamics

Maxwell’s Equation

The electromagnetic field can be written as a single bivector field F
in

D. Representing the charged current density by a vector J in D,

Maxwell’s equation can be written

F = J.

(8.1)

Using (7.15), we can separate (8.1) into vector and pseudovector parts:

· F = J,

(8.2a)

∧ F = 0.

(8.2b)

If these equations are expressed in terms of a tensor basis, it is found
that the coefficients show Maxwell’s equation in familiar tensor form.

Alternatively, we can re-express Maxwell’s equation as a set of

four equations in the Pauli algebra, by singling out a particular time-
like direction γ

. Using (7.12) we can write

F = E + iB

(8.3a)

where

E =

1
2

(F − F

∗

(8.3b)

iB =

1
2

(F + F

∗

(8.3c)

This form of Maxwell’s equation is due to Marcel Riesz [9].

Ó Springer International Publishing Switzerland 2015
D. Hestenes, Space-Time Algebra,
DOI 10.1007/978-3-319-18413-5_2

Chapter II. Electrodynamics

and we can write J in the form

J = (J γ

)γ

= (J · γ

+ J ∧ γ

)γ

≡ (ρ + J)γ

(8.4)

Now, multiplying (8.1) by γ

and recalling (7.19), we have

(∂

+ ∇)(E + iB) = ρ − J

(8.5)

∂

E + ∇E + i(∂

B + ∇B) = ρ − J.

Equating separately the scalar, vector, bivector and pseudoscalar parts
in

P, we get the four equations

∇ · E = ρ,

(8.6a)

∂

E + i∇ ∧ B = −J,

(8.6b)

i∂

B + ∇ ∧ E = 0,

(8.6c)

i∇ · B = 0.

(8.6d)

Using the definition (6.13) of cross product we easily convert (8.6) into
a form of Maxwell’s equation which is too familiar to require comment.

Equation (8.1) implies that J

is a conserved current, for

F = J = · J + ∧ J.

(8.7)

Since

is a scalar operator, the left side of (8.7) is a bivector, so

F = ∧ J

(8.8)

and

· J = 0 = ∂

ρ + ∇ · J.

(8.9)

We can, if we wish, express F as the gradient of a vector potential,

F = A = · A + ∧ A.

(8.10a)

But, since F is a bivector, (8.10a) says that

F = ∧ A

(8.10b)

and

· A = 0.

(8.10c)

Equation (8.5) is equivalent to the old quaternion form for Maxwell’s equation. I do not

know who first invented it. A convenient early reference is [12].

9. Stress-Energy Vectors

Substituting (8.10a) into (8.1), we arrive at the wave equation for A:

A = J.

(8.11)

The vector potential is not unique, for we can add to it the gradient
of any scalar χ for which

χ = 0. Thus, if

= A + χ,

(8.12)

then

= A +

χ = A.

If we wish, we can solve for E and B in terms of potentials. Let

Aγ

= A

+ A.

(8.13)

Then

F = E + iB = A = (γ

)(γ

= (∂

− ∇)(A

− A)

= −(∂

A + ∇A

) + ∇ ∧ A.

E = −∂

A − ∇A

, B = ∇ × A.

(8.14)

Stress-Energy Vectors

From the source-free Maxwell equation we can find four conserved
vector currents. From

F = 0,

F e

= 0

(9.1)

we get

F F + e

F e

F = ∂

F γ

F = 0.

(9.2)

We have used the symbol e

to mean differentiating to the left

instead of to the right. Define S

1
2

F γ

(9.3)

In (9.2) we have assumed inertial coordinates. Generalization to arbitrary coordinates is a

simple matter by the techniques of chapter V.

Chapter II. Electrodynamics

Since S

= e

= −S

, the S

are vectors. Let us call them the stress-

energy vectors of the electromagnetic field. S

gives the flux density

of electromagnetic stress-energy through the hypersurface orthogonal
to γ

. The components of the S

are

µν

= S

· γ

= (S

)

1
2

( e

F γ

)

(9.4)

µν

is the so-called energy-momentum tensor.

By writing F in terms

of a bivector basis, the righthand side of (9.4) can be reduced to fa-
miliar tensor form. But (9.4) is already more felicitous than the tensor
form. The following properties of S

µν

are easily verified directly from

(9.4):

µν

= S

νµ

= 0,

= 0.

(9.5)

We will find it more convenient to deal with the vectors S

rather

than the scalars S

µν

. To increase our familiarity with the S

, let us

re-express S

by writing F in the Pauli algebra of γ

= S

1
2

F γ

F = −

1
2

F γ

F = −

1
2

F F

∗

(9.6)

F F

∗

= (E + iB)(−E + iB) = −(E

+ B

) + iE ∧ B.

(9.7)

Hence

= [

1
2

+ B

) + E × B]γ

(9.8)

This shows that S

is the space-time generalization of the Poynting

vector.

We can write (9.2) in the form

∂

= 0.

(9.9)

Since the S

are vectors, (9.9) gives four conserved quantities. The

reader may show that (9.9) can also be written in the form

· S

= 0.

(9.10)

It is easy to find the modification of (9.9) produced by sources.

Since

1
2

( e

F J + J F ) = −(F J − J F ) = −F · J,

(9.11)

The form ( e

F γ

)

for the energy-momentum tensor of the electromagnetic field was first

given by Marcel Riesz [8].

10. Invariants

we get

∂

= J · F.

(9.12)

K = F · J is the space-time generalization of the Lorentz force. This
is easily seen by writing it as two equations in the Pauli algebra of γ

Kγ

+ K,

(9.13)

F · J γ

1
2

(F J − J F )γ

1
2

(F J γ

− Jγ

∗

)

1
2

[(E + iB)(ρ + J) − (ρ + J)(−E + iB)]

= ρE + J · E + J × B.

(9.14)

By separately equating the scalar and vector parts of (9.13) and (9.14),
we arrive at

= J · E,

(9.15)

K = ρE + J × B.

(9.16)

Invariants

The invariants of the electromagnetic field are given by F

The square

of a bivector has only scalar and pseudoscalar parts,

= F · F + F ∧ F.

(10.1)

To see the invariants in tensor form we must express F in terms of a
bivector basis: F =

1
2

µν

. Using (3.15), we find for the scalar part

F · F = −

1
2

µν

(10.2)

Using (A.30), we find for the pseudoscalar part

F ∧ F = −F

αβ

µν

αβµν

(10.3)

Alternatively, we may express the invariants in terms of E and B,

= (E + iB)

= E

− B

+ 2iE · B.

(10.4)

The decomposition F = E + iB depends on a particular timelike
vector. It would seem better to decompose F using invariants of the

Chapter II. Electrodynamics

field. We accomplish this here for the case F

6= 0. First we note that

F can be written

F = f e

iφ

(10.5)

where φ is a scalar and

f = e + ib

(10.6a)

with

e · b = 0.

(10.6b)

Now we observe that when F

6= 0 both φ and f

are determined by

, for

= f

2iφ

= (e

− b

2iφ

(10.7)

Here F

is represented as a scalar f

which is taken through an angle

2φ by a duality rotation to give the scalar and pseudoscalar parts of
F

Let us express the perhaps more significant invariants φ and f

in terms of E

− B

and E · B. The invariant of F

∗

= −E + iB is

∗

)

= (F

)

∗

= f

−2iφ

= E

− B

− 2iE · B.

(10.8)

Hence

∗

)

= f

= (E

− B

)

+ 4(E · B)

(10.9)

= ±[(E

− B

)

+ 4(E · B)

]

1
2

(10.10)

We can find another interesting expression for f

by noting that

∗

F = −(E

+ B

) + 2E × B.

(10.11)

Since E × B anticommutes with both E and B,

= F

∗

F F = (F

∗

F )(F

∗

F )

∗

= (E

+ B

) − 4(E × B)

= ±[(E

+ B

)

− 4(E × B)

]

1
2

(10.12)

The expression for φ is found to be

tan 2φ =

2E · B

− B

(10.13)

11. Free Fields

Evidently φ is determined by E

− B

and E · B only when F

6= 0

and then only to an additive multiple of π. It is natural to interpret φ
as the physical phase of the electromagnetic field. In the next section
we will do this for a free field. Other authors [13] have called the φ the
complexion of the Maxwell field. It seems that no one has pinned down
its physical manifestations. We note here that it does not contribute
to the stress-energy vectors, for, since i anticommutes with γ

1
2

F γ

F =

1
2

f γ

(10.14)

Free Fields

The reader has probably noticed that the source free Maxwell Equa-
tion has the same form as the Dirac equation for a free neutrino field,

F = 0.

(11.1)

It is therefore not surprising that we can describe the polarization
of a photon in the same way that we describe the polarization of a
neutrino. Just the same, it is worthwhile to briefly discuss photon
polarization, in order to see what special insights are provided by the
Dirac algebra. Consider the plane wave solutions of (11.1):

F (x) = f e

ik·x

(11.2)

Here x = x

is the position vector in Minkowski space-time, k =

is a constant vector and f is a constant bivector. Multiply (11.1)

by γ

to obtain an equation involving only p-numbers:

(∂

+ ∇)F = 0.

(11.3)

Let kγ

= k

+ k, and substitute (11.2) into (11.3) to get the equation

kf = k

(11.4)

We can multiply (11.4) by k

+ k to find k

= ±|k| (i.e. k

= 0). These

two solutions correspond to photons which are right or left circularly
polarized, or, in different words, photons with positive or negative
helicity. We can interpret the solution with k

= |k| as a particle,

and the solution with k

= −|k| as an antiparticle. This allows us to

Chapter II. Electrodynamics

identify the operation of space conjugation (defined in section 6) with
antiparticle conjugation, for, applying (7.12) to (11.4), we get

−kf

∗

= k

∗

(11.5)

In section 13 we will see that the same interpretation holds for the
Dirac equation with mass.

When the neutrino is described by the Dirac equation a side con-

dition is necessary to limit the number of solutions. The corresponding
condition on the electromagnetic field is

F = F .

(11.6)

This just follows from the fact that F is a bivector. The condition that
F be a bivector can be written

F = F = − e

F .

(11.7)

We can use the bivector property of f to get more detailed information
from (11.4). As in (8.3), we write

f = e + ib.

(11.8)

Equation (11.4) can be split into the two equations

e = ikb,

(11.9a)

b = −ike.

(11.9b)

It follows that

k = k

e × ˆ

(11.10a)

k · e = k · b = 0,

(11.10b)

e · b = 0,

(11.10c)

= b

(11.10d)

The last two conditions can be written f

= 0. This implies

= 0,

(11.11)

an invariant condition that F represents a field of circularly polarized
radiation.

11. Free Fields

Let us close this section with a geometrical interpretation of the

circularly polarized plane wave solutions (11.2). Operating on f, e

ik·x

is a duality rotation which rotates the vector e into a bivector and
rotates the bivector ib into a vector. Thus we get the picture of the
electric and magnetic vectors spinning about the momentum vector k
as the plane wave progresses, but this “spinning” is due to a duality
transformation rather than the usual kind of spatial rotation.

Chapter III

Dirac Fields

Spinors

Let

I be a subspace of an algebra A with the property that the sum

of elements in

I is also in I . I is called a two-sided ideal if it is

invariant under multiplication on both the left and the right by an
arbitrary element of

A . I is called a left (right) ideal if it is invariant

under multiplication from the left (right) only.

In both the Pauli and Dirac algebras the only two-sided ideals

are the zero element and the whole algebra. For brevity, we will apply
the appellation ideal only to left ideals. An ideal is called minimal
if it contains only itself and the zero element as ideals. An element
of a minimal left ideal

I will be called a left-spinor, or again for

brevity, simply a spinor. The minimal right ideal obtained from

I by

reversing products may be called the ideal conjugate to

I . An element

of a minimal right ideal may be called a conjugate spinor.
A definition of spinor in terms of transformations will be given in
section 19.

We will investigate the spinors of the Pauli algebra in detail. Any

p-number φ can be written in the form

φ = φ

+ φ

(12.1)

where the σ

are orthonormal vectors and φ

and the φ

are formally

“complex numbers”. We can take any unit vector, say σ

, and decom-

Ó Springer International Publishing Switzerland 2015
D. Hestenes, Space-Time Algebra,
DOI 10.1007/978-3-319-18413-5_3

Chapter III. Dirac Fields

pose φ as follows:

φ = φ[

1
2

(1 + σ

) +

1
2

(1 − σ

)] = φ

+ φ

−

(12.2)

The set of all φ

(φ

−

) is an ideal

(

−

≡ {φ

−

≡ {φ

−

(12.3)

and φ

−

are spinors.

and

−

are independent minimal left

ideals. These facts become obvious as we take the ideals apart. For a
basis in the ideal

we can choose u

and u

defined by

≡

√

(1 + σ

≡

√

(1 − σ

)σ

(12.4)

This basis satisfies the orthonormality and completeness relations:

†
a

)

= δ

(12.5)

†
a

= 2

(12.6)

(a, b = 1, 2) . The u

also satisfy

= u

= −u

= u

(12.7)

The conditions (12.5) and (12.7) determine the u

uniquely except for

a common phase factor. With the help of (12.7) we can write φ

the form

√

(φ

+ φ

√

(φ

+ iφ

(12.8)

The basis in

−

which corresponds to the u

≡

√

(1 + σ

)σ

≡

√

(1 − σ

(12.9)

The relations for the v

which correspond to (12.5), (12.6), (12.7) are

readily found. We also have the orthogonality relation

†

)

= (v

†

) = 0.

(12.10)

Now we can expand φ in terms of the “spinor basis” we have con-
structed for the Pauli algebra.

φ = φ

+ φ

1−

+ φ

2−

(12.11)

12. Spinors

where

√

(φ

+ φ

1−

√

(φ

− iφ

√

(φ

+ iφ

2−

√

(φ

− φ

(12.12)

The effect of space conjugation on the spinor basis is readily found
from the definitions. From (12.11) we get

∗

= φ

∗
2−

− φ

∗
2+

− φ

∗
1−

+ φ

(12.13)

By writing the coefficients of (12.8) in a column, we obtain a

representation of φ

as a column matrix Φ

(12.14)

Similarly, we can obtain a matrix representation Φ of φ by writing the
coefficients of (12.11) in an array.

Φ =

1−

2−

(12.15)

The correspondence between operations in the Pauli algebra and op-
erations in its matrix representation is given in appendix D.

We have chosen u

and v

to be “eigenvectors” of σ

operating

on both the left and the right. It is worth pointing out that this is
an unnecessarily restrictive choice of spinor basis. Multiplication of a
p-number on the left is completely independent of multiplication on
the right, and they may be considered as operations performed in in-
dependent spaces. Accordingly, we may choose a spinor basis which
“diagonalizes” σ

operating on the left and some other vector on the

right. Indeed, in the usual physical applications of spinors only the
matrix representation Φ

given by (12.14) and multiplication on the

left are used. In this case it does not matter what p-numbers are diag-
onalized on the right—it only matters that Φ

represents an element

of a minimal ideal. The base elements u

are not distinguishable from

the v

by left multiplication.

Chapter III. Dirac Fields

Our knowledge about spinors in the Pauli algebra will be helpful

in our study of spinors in the Dirac algebra. To make use of this
knowledge, we must distinguish some timelike vector γ

. Now any d-

number ψ can be written as the sum of an even part φ and an odd
part χγ

ψ = φ + χγ

(12.16)

φ and χ are p-numbers. As in the Pauli algebra, ψ can be written as
the sum of elements ψ

and ψ

−

of independent minimal ideals,

ψ = ψ

+ ψ

−

(12.17)

where

= ψ

1
2

(1 ± σ

) = φ

+ χ

∓

(12.18)

It is obvious that the set

(

−

) of all ψ

(ψ

−

) is a minimal ideal.

We can write ψ as a linear combination of orthonormal base ele-

ments w

a=1

(12.19)

The ψ

are “complex” coefficients and the w

are defined by

= u

= v

= −γ

= u

= v

= γ

(12.20)

and the u

and v

are defined by (12.4) and (12.9). A basis for

−

given simply by w

−

a=1

a−

(12.21)

It is now clear that the Dirac algebra can be written as a sum of

two independent minimal ideals, each of which is a four-dimensional
vector space over the “complex” numbers.

When the Dirac algebra is developed over the complex numbers instead of the reals there

are four independent minimal ideals.

13. Dirac’s Equation

Dirac’s Equation

In the last section, we observed that the usual “four-component spi-
nor” can be looked upon as an element of a minimal ideal in the Dirac
algebra. Consider a spinor ψ which is an eigenstate of σ

on the right,

ψσ

= ψ.

(13.1)

The Dirac equation for ψ interacting with an electromagnetic potential
A can be written

ψ = (m + eA)ψi.

(13.2)

The most important point to be noted here is that multiplication of
ψ by i on the right leaves (13.1) invariant and commutes with any
multiplication of ψ on the left.

Therefore i multiplying ψ on the right

plays the role of

√

−1 in the usual Dirac theory, whereas i multiplying

ψ on the left plays the role of γ

As is usual, (13.2) is invariant under gauge transformations.

ψ → ψe

−ieχ

(13.3a)

A → A + χ.

(13.3b)

To leave Maxwell’s Equation invariant, we must also have

χ = 0.

(13.3c)

We can write ψ in the form

ψ =

1
2

(1 + γ

)ψ

1
2

(1 − γ

)ψ

(13.4)

where ψ

, and ψ

are spinors in the Pauli algebra of γ

. For positive

energy solutions in the non-relativistic limit, ψ

is the socalled large

component and ψ

is the small component. Two coupled p-number

equations in ψ

and ψ

can be obtained by separately factoring out

1
2

(1 + γ

) and

1
2

(1 − γ

) from (13.2) on the left.

Rather than (13.4), it is perhaps more meaningful to separate ψ

into even and odd parts:

ψ = φ

+ χ

−

(13.5)

In (13.2), we cannot take ψ to be an eigenstate of γ

on the right because i anticommutes

with γ

and so right multiplication by i does not leave left ideals of γ

invariant.

Chapter III. Dirac Fields

Again φ

and χ

−

are spinors in the Pauli algebra of γ

= φ

−

= χ

−

(13.6)

We can get separate uncoupled equations for the even and odd parts
of ψ after properly “squaring” the operator in (13.2),

(

− e

+ m

)ψ + e(2A · + F )ψi = 0.

(13.7)

Here we have used F = ∧ A and · A = 0. We can separate (13.7)
into the two equations

(

+ 2ieA · − e

)φ

+ (ieF + m

)φ

= 0,

(13.8)

(

− 2ieA · − e

)χ

−

+ (−ieF + m

)χ

−

= 0.

(13.9)

Evidently the even and odd fields are coupled to the electromagnetic
field with opposite charge.

We can get a different form of the Dirac equation by separately

equating the even and odd parts of (13.2) and factoring out γ

. Using

the definitions (7.19) and (8.13), we get

(∂

+ ∇)φ

= e(A

− A)φ

i + mχ

∗
−

(13.10a)

(∂

+ ∇)χ

−

= −e(A

− A)χ

−

i − mφ

∗
+

(13.10b)

These can be recombined into. a single p-number equation:

(∂

+ ∇)Ψ = [e(A

− A)Ψ + mΨ

∗

]iσ

(13.11)

where

Ψ = φ

+ χ

−

(13.12)

Equation (13.11) was first obtained by G¨

ursey [19]. We can also write

it in the form

Ψ = (eAΨ + mΨ γ

)iσ

(13.13)

Here Ψ = Ψ ; we could equally well have arranged to have Ψ = −Ψ .

Equation (13.2) depends implicitly on a timelike bivector by

virtue of (13.1). The equivalent equation (13.13) has the merit that
this dependence is explicitly displayed. Evidently we will not have a

It has been shown that the results of quantum electrodynamics can be obtained by using

(13.8) rather than the Dirac equation (13.2). A review of this approach is given by Brown [14].

13. Dirac’s Equation

complete understanding of the Dirac equation until this circumstance
is interpreted physically.

The question arises as to why nature distinguishes one ideal from

another. Spinors in independent ideals are independent solutions of
the Dirac equation. Perhaps they should be identified with physically
independent fermion fields. Perhaps we should generalize the Dirac
equation by relaxing the condition that ψ be an element of a minimal
ideal. An arbitrary d-number ψ can be written

ψ = ψ

+ ψ

−

(13.14a)

where

= ψ

1
2

(1 ± σ

(13.14b)

Therefore the free particle equation

ψ = ψim

(13.15)

is equivalent to separate equations for two independent particles with
the same mass:

= ψ

im,

−

= ψ

−

im.

(13.16)

This circumstance at once brings to mind the nucleon. If we identify
ψ

with the proton and ψ

−

with the neutron, then the equation for

a nucleon interacting with the electromagnetic and pion fields can be
written

ψ = ψim + eAψ

1
2

(1 + σ

)i + gψπ.

(13.17)

Here g is the pion coupling constant, and π can of course be written

π = π

(13.18)

where the π

are scalars. It will be noted that ψ = −γ

ψi expresses the

pseudoscalar coupling of the π-nucleon interaction. Excluding the elec-
tromagnetic coupling, (13.17) is invariant under the constant gauge
transformation

ψ → ψe

1
2

(13.19a)

π → e

−

1
2

πe

1
2

(13.19b)

Chapter III. Dirac Fields

It is clear that (13.19) must be identified as an isospin transformation.

Following the procedure which led to (13.11), equation (13.17)

can be expressed as two equations of the G¨

ursey type. These equations

were found by G¨

ursey when he arrived at transformations equivalent

to (13.19) as a generalization of the Pauli group [16]. We will not here
attempt to generalize (13.17) further. Our discussion only suggests
that it may be possible to account for isospin within the Dirac alge-
bra and so give isospin a geometrical basis grounded in space-time.
To substantiate this suggestion it would be necessary to derive some
consequence peculiar to this representation. In any case, whether it
has any special consequences or not, the representation of the nucleon
field given here is conveniently succinct, because it is given entirely in
terms of the Dirac algebra and does not require any additional “isospin
algebra”.

Conserved Currents

In sections 9 and 10 we constructed invariants and conserved currents
from the electromagnetic field. Analogous constructions can be made
from a Dirac field. Let us consider a particular example. From equation
(13.2) we can construct the following equations:

ψψ = γ

ψ(m + eA)ψi,

(14.1a)

ψ e

ψγ

= i ˜

ψ(m + eA)ψγ

(14.1b)

The scalar part of the sum of these equations is

(γ

ψψ + ˜

ψ e

ψγ

)

= 0.

(14.2)

This is equivalent to

∂

(γ

ψγ

ψ)

= · T

= 0

(14.3)

where

= (ψγ

ψ)

(14.4)

The T

are evidently four independent conserved vector currents. How-

ever’ the condition (13.1) implies that T

and T

vanish identically and

15. C, P , T

that T

and T

differ only by a sign. T

is proportional to the usual

expression for the electromagnetic current vector J ,

∼ γ

· T

= (γ

ψγ

ψ)

= (γ

ψψ

†

)

= (ψ

†

ψ)

(14.5)

If the above procedure is applied to the nucleon equation (13.17),

it is found that T

is again conserved, but the other three vector cur-

rents, which can be identified as isospin currents, are not conserved.

C, P , T

In this section we use our space-time algebra to formulate antiparticle
conjugation, parity conjugation and time reversal. We also suggest
appropriate geometrical interpretations of these transformations. We
introduce them as transformations of equation (13.2) which either
leave the equation invariant or simply change the sign of the coupling
to the electromagnetic field. In our discussion we will always assume
that and A transform in the same way. This is natural if D, defined
by Dψ = ψ − eAψi, is viewed as a differential operator which is
covariant under the gauge transformation (13.3a).

We define antiparticle conjugation C by

C : ψ → ψ

= ψγ

(15.1)

The conjugate field ψ

satisfies the equation

= (m − eA)ψ

(15.2)

C interchanges even and odd parts of ψ and changes their relative
sign. A double application of (15.1) shows

= −ψ.

(15.3)

The usual space parity conjugation P is given by

P : ψ(x, t) → ψ

(x, t) = γ

ψ(−x, t).

(15.4)

This leaves equation (13.2) invariant. P interchanges even and odd
parts of ψ.

We define time reversal T by

T : ψ(x, t) → ψ

(x, t) = ψ

∗

(x, −t).

(15.5)

Chapter III. Dirac Fields

The time reversal field ψ

satisfies equation (15.2).

In our definitions of ψ

, ψ

and ψ

we could have included quite

general phase factors on the right, even bivector as well as scalar
phase factors. Our choice of phase factors makes possible a geometric
interpretation of CP conjugation,

CP : ψ(x, t) → ψ

(x, t) = φ

∗

(−x, t).

(15.6)

In chapter I we saw how to represent “surface” elements of any di-
mension by Clifford numbers. We can, for instance, represent a two-
dimensional oriented surface by the field of bivectors tangent to the
surface. The tangent bivector gives the orientation of the surface at
each point. (The magnitude of the bivector represents a density of
the surface.) We discussed, in chapter I, certain important transfor-
mations which change the orientation of “surface” elements, namely,
space conjugation and space-time conjugation. These transformations
can also be applied to physical fields represented by Clifford numbers,
and the corresponding geometrical interpretation can be carried over
as well. Applied to p-number fields in space, the transformation (15.6)
reverses the direction of all tangent vectors in space and reflects space
points through the origin. It literally turns finite volumes inside out.
This provides geometric meaning for CP conjugation as seen in the
inertial frame of γ

. The transformation (15.6) also changes the rel-

ative sign of even and odd d-numbers (space-time conjugation), but
this has no effect on the Pauli algebra of γ

. To summarize: From the

point of view of the Pauli algebra, (15.6) inverts volumes. But from
the point of view of the Dirac algebra, (15.6) reflects points in space
and reverses the time direction of tangent vectors.

The CP transformation (15.6) can be applied to the electromag-

netic as well as the Dirac field; it is just the antiparticle conjugation
which maps equation (11.4) into (11.5).

The Dirac equation is invariant under the product of C, P and T ,

CP T : ψ(x) → ψ

CP T

(x) = ψ(−x).

(15.7)

This transformation reflects all space-time points through the origin
and reverses the direction of all tangent vectors.

The operation C is more interesting when applied to the nucleon

equation (13.17). If we are to obtain the proper equation for antinu-

15. C, P , T

cleons, the transformation of the pion field must be

C : π → π

∗

= π

∗

= −π.

(15.8)

The reader will be left to satisfy himself that C is precisely the oper-
ation of G-conjugation first introduced by Lee and Yang [17].

Chapter IV

Lorentz Transformations

Reflections and Rotations

Let p be a vector tangent to a point x in space-time. The scalar p

a natural norm for p. We call p timelike if p

> 0, spacelike if p

< 0,

or lightlike if p

= 0.

The set

V (x) of all vectors tangent to the point x will be called

the tangent space at x. An automorphism of

V (x) which leaves the

square of every vector unchanged is called a Lorentz transformation.

The set of all Lorentz transformations is called the Lorentz group.

In the Dirac algebra, transformation of a d-number p into a d-

number p

can be written

p → p

= RpS.

(16.1)

This will be a Lorentz transformation if (16.1) takes any vector p into
a vector p

so that

)

= p

(16.2)

Substituting (16.1) into (16.2) we get

= RpSRpS.

(16.3)

An automorphism of a set

V is a one-to-one mapping of V onto itself.

At this point, it is worth mentioning that rotations and reflections in

can be handled

in the same way that we here handle Lorentz transformations in

D, with only minor

adjustments for metric and dimension.

We will suppress reference to the space-time point whenever it is consistent with clarity.

Ó Springer International Publishing Switzerland 2015
D. Hestenes, Space-Time Algebra,
DOI 10.1007/978-3-319-18413-5_4

Chapter IV. Lorentz Transformations

This will hold for every vector p if SR commutes with every p; this
can occur only if RS is a scalar;

therefore we write

S = ±R

−1

(16.4a)

where

−1

= R

−1

R = 1.

(16.4b)

We lose no generality by dealing only with the solution (16.4a) with
positive sign.

Only a 1-vector p in

D will satisfy the two conditions

p = p

(16.5)

and

p = −p.

(16.6)

Applying (16.6) to (16.1), we find that with proper normalization

−1

= ± e

(16.7)

Applying (16.6) to (16.1), we find that R must be either even or odd,
i.e.

R = R

(16.8)

R = −R.

(16.9)

The Lorentz transformation

p → p

= RpR

−1

(16.10)

will be called a rotation if (16.8) holds, and a reflection if (16.9) holds.

One of the most important kinds of reflection is the operation of

space conjugation defined by (7.13). This transformation reverses the
direction of every vector p

⊥

orthogonal to some unit timelike vector

while leaving unchanged every vector p

collinear with γ

. Thus,

space conjugation transforms any vector

p = p

+ p

⊥

(16.11)

The most general solution to (16.3) has RS = e

iϕ

, but such “phase factors” add nothing to

our discussion.

16. Reflections and Rotations

into

∗

= γ

pγ

= p

− p

⊥

(16.12)

Now observe that if R satisfies (16.9), then R

= γ

R satisfies (16.8).

It follows that any reflection of spacelike vectors can be produced by
a rotation followed by a space conjugation, or vice-versa.

Another basic transformation is the space-time conjugation (7.7).

This reverses the direction of every vector p,

p = ipi

−1

= −ipi = i

p = −p.

(16.13)

A time inversion, which reverses the direction of all vectors collinear
with γ

, is produced by the combination of (16.12) and (16.13),

∗

= −p

+ p

⊥

(16.14)

It follows that any reflection can be produced by some combination of
space conjugation, space-time conjugation and a rotation. Therefore
we may confine the rest of our discussion to rotations.

According to theorem 1 of appendix B, any d-number R which

satisfies R = R and e

RR = 1 can be written

R = ±e

−

1
2

(16.15)

where B is a bivector. Hence, for a Lorentz rotation, (16.10) can be
written

p → p

= e

−

1
2

(16.16)

We say that B generates the rotation (16.16).

A rotation can be characterized by the properties of the genera-

tor.

The square of any bivector B in the Dirac algebra has a scalar

part B · B and a pseudoscalar part B ∧ B,

= B · B + B ∧ B.

(16.17)

If B ∧ B = 0, B will be called simple. If B ∧ B 6= 0, B will be called
complex.

A simple bivector B will be called timelike, lightlike, spacelike

if B

> 0, B

= 0, B

< 0 respectively. The corresponding Lorentz

Relative to some bivector basis, B =

1
2

µν

. Since it takes six parameters B

µν

to deter-

mine B, the group of Lorentz rotations (16.16) is a six parameter group.

Chapter IV. Lorentz Transformations

rotations (16.16) can be distinguished by the same names. This agrees
with the analogous classification of vectors given in section 7.

A complex bivector B can always be written as the sum of two

orthogonal simple bivectors B

, and B

B = B

+ B

= B

(16.18)

If B

is timelike, then B

is spacelike. The Lorentz rotation generated

by B becomes

p → p

= e

−

1
2

)

1
2

)

= e

−

1
2

−

1
2

(16.19)

Therefore, any Lorentz rotation generated by a complex bivector can
be expressed as the commuting product of a timelike with a spacelike
Lorentz rotation. For this reason we can confine our study to Lorentz
rotations generated by simple bivectors.

Any d-number D can be expressed as some combination of sums

and products of vectors. If these vectors are subjected to the Lorentz
transformation (16.10), then

D → D

= RDR

−1

(16.20)

We say that (16.20) is induced by (16.10). A d-number which trans-
forms in this way will be called a rotor. The electromagnetic bivector is
an important example of a rotor. The reader will note that the trans-
formation (16.20) leaves invariant the scalar part of D

, so (D

)

is a

possible norm for D. However, in section 19 we suggest that ( e

DD)

is a more natural norm.

When some timelike direction is singled out as specially significant

there is a decomposition of a Lorentz rotation which is more convenient
than (16.19). Theorem 4 of appendix B enables us to write the Lorentz
rotation (16.16) in the form

p → p

= e

−

1
2

−

1
2

p e

1
2

(16.21)

where a and b are timelike bivectors proportional to some selected
timelike unit vector γ

. All Lorentz rotations which leave γ

invariant

are of the form

p → p

= e

−

1
2

p e

1
2

(16.22)

17. Coordinate Transformations

We call (16.22) a spatial rotation. A Lorentz rotation of the form

p → p

= e

−

1
2

p e

1
2

(16.23)

will be called a special timelike rotation. We can now describe (16.21)
qualitatively by the statement that every Lorentz rotation can be ex-
pressed as a special timelike rotation followed by a spatial rotation.

Coordinate Transformations

The Lorentz transformations which we discussed in section 16 are
rigid rotations and inversions of the tangent space

V (x) at a point

x of space-time. Of course, x can be a variable, so these transforma-
tions map vector fields into vector fields while preserving the length
of vectors at each point. In this section we will show how a subclass
of such transformations can be related to coordinate transformations
in Minkowski space-time. In section 23 we will discuss these problems
in the context of Riemannian space-time.

By taking the gradient of coordinate functions x

we form the

tangent vectors

= γ

(x) = x

(17.1)

Minkowski space-time is distinguished by the existence of coordinates
x

for which the corresponding tangent vectors γ

are constant and

orthonormal. Such a set of functions is commonly called an inertial
system (of coordinates). We will call the corresponding field of tangent
vectors an inertial frame. From one inertial system every other inertial
system can be obtained by a Poincar´

e transformation:

→ x

= a

µ
ν

+ a

(17.2)

The a

µ
ν

and the a

are constants. The coordinate transformation (17.2)

induces a Lorentz transformation of the tangent vectors, which gives,
by (17.1),

= x

∂x

= a

µ
ν

(17.3)

Also called an inhomogeneous Lorentz transformation.

Chapter IV. Lorentz Transformations

The inverse of this transformation gives

∂x

= b

ν
µ

(17.4)

Now, in a way we have already discussed, the timelike vector

field γ

of an inertial frame S uniquely determines a Pauli algebra at

every space-time point. This will be called the Pauli algebra of the
inertial system S, or, as before, the Pauli algebra of γ

. When we are

interested in the interpretation of physical processes as seen in some
inertial system S, it is useful to express the pertinent equations in
the Pauli algebra of S. We have already done this for some important
physical equations in parts II and III. To see how the Pauli algebra
of one inertial system is related to the Pauli algebra of another, we
express γ

above in terms of the Pauli algebra of γ

. From (17.3),

= a

0
0

+ a

0
k

(17.5)

To understand the physical meaning of the terms on the right of this
equation, let x

be the coordinates of a point at rest in the primed

inertial system. Then the velocity of this point in the unprimed inertial
system will be just the velocity of the primed system as observed in
the unprimed system. The components v

of this velocity are

∂x

(17.6)

Now

β ≡ a

0
0

= γ

· γ

= γ

· γ

= b

0
0

(17.7)

and

0
k

= γ

· γ

= −γ

· γ

= −b

k
0

= −βv

(17.8)

Therefore, using (7.9), we find

= β(1 + v).

(17.9)

By squaring this equation, we get an expression for β:

β = (1 − v

)

−

1
2

(17.10)

17. Coordinate Transformations

We emphasize that (17.9) holds for any Lorentz transformation. More
generally, for any timelike vector field p(x) we can write

p(x)γ

= |p(x)|

1 + v(x)

(1 − v

(x))

1
2

(17.11)

and interpret v(x) as the velocity of the vector p(x) at the point x as
observed in the inertial frame of γ

. As a physical example, consider

a particle with energy-momentum vector p and mass m,

pγ

= E + p = mβ(1 + v).

(17.12)

Hence, in the inertial frame of γ

the energy of the particle is

E = mβ

(17.13a)

and the momentum is

p = mβv = Ev.

(17.13b)

Using (16.10), we can write (17.3) in the form

= a

µ
ν

= R

−1

(17.14)

where R is a constant d-number field. A vector field p which represents
some property of a physical system will be independent of any inertial
system used for observational purposes. If p has components p

in the

unprimed system, then it has components p

0
µ

= b

σ
µ

in the primed

system,

p = p

= p

0
µ

(17.15)

If we say that the components p

transform covariantly under a change

of basis, then the base vectors γ

are said to transform contravariantly.

We can simulate a change of basis by applying the inverse of (17.14)
to the vector p:

p → p

= RpR

−1

= p

0
µ

(17.16)

To an observer in the {γ

} frame the vector p

appears the same

as the vector p appears to an observer in the {γ

} frame. Therefore,

Chapter IV. Lorentz Transformations

we can interpret (17.16) as a transformation from one inertial frame
to another which preserves a basis in the Dirac algebra.

It is sometimes of interest to know the a

µ
ν

as a function of R, or

vice-versa. To solve (17.14) for the a

µ
ν

is a simple matter.

µ
ν

= γ

· γ

= (γ

−1

(17.17)

To solve for R in terms of the a

µ
ν

is somewhat more difficult. We do it

here for the case of rotations. Define

A ≡ a

µ
ν

= a

µ
µ

+ a

µ
ν

∧ γ

(17.18)

From (17.14) we get

A = γ

= γ

Rγ

(17.19)

Since we are discussing rotations, R has the property R = R; this
implies that R can be written

R = R

+ R

(17.20)

where R

1
2

(R − e

R) is a bivector and R

1
2

(R + e

R) has only scalar

and pseudoscalar parts. Using (A.20), (A.21) and (A.25) of appendix
A, we find

Rγ

= γ

Rγ

= 4R

∗
I

(17.21)

Therefore (17.19) becomes

A = 4R

∗
I

(17.22)

Since R e

R = 1,

AA = 16(R

∗
I

)

(17.23)

Hence

R = ±

( e

AA)

1
2

(17.24)

When (17.18) is substituted this gives R explicitly as a function of
the a

µ
ν

. The indeterminate sign in (17.24) clearly occurs because R

appears twice in (17.14) whereas a

µ
ν

appears once. Formula (17.24)

is not of great practical value, because, as our discussion of Lorentz
transformations is designed to show, it is usually easier to construct
R than it is to construct a

µ
ν

from physical or geometrical data.

18. Timelike Rotations

Timelike Rotations

A timelike rotation is of special physical interest when it is interpreted
as a transformation from one inertial frame to another. In this section
we study timelike rotations from this point of view.

Recall from (16.23) that a timelike rotation relates a vector p to

a vector p by the formula

= Rp e

R = e

−

1
2

(18.1)

We learned in the last section that p and p

can be interpreted as the

same vector as seen in different inertial frames {γ

} and {γ

} respec-

tively. We also learned that, under this interpretation, the frames are
related by the inverse of (18.1); in particular,

= e

Rγ

R = e

1
2

−

1
2

(18.2)

For the purpose of physical interpretation, we must express R in

terms of the velocity v of the primed inertial system relative to the
unprimed system. First we remark that if (18.2) is to be interpreted
as a simple timelike rotation between inertial frames, then a must be
a vector in the Pauli algebra of γ

. In this case γ

anticommutes with

a, so

∗

= γ

Rγ

= e

(18.3)

From (18.2) and (17.9), we find

= γ

= e

−a

= β(1 − v).

(18.4)

In passing, we note that since

−a

= cosh a − sinh a,

(18.5)

we can write

β = (1 − v

)

−

1
2

= cosh a,

(18.6)

v = tanh a.

(18.7)

The half angle exponential can be written in the same form as (18.4).

R = e

−

1
2

1 − u

(1 − u

)

1
2

(18.8)

Chapter IV. Lorentz Transformations

To take the square root of (18.4) we must find u in terms of v. Thus





1 − u

(1 − u

)

1
2





= R

= β(1 − v).

(18.9)

The scalar part of this expression is

1 + u

1 − u

= β

(18.10)

from which it follows that

|u| =

β − 1

β + 1

1
2

|v|

1 + (1 − v

)

1
2

(18.11)

The ratio of vector to scalar part in (18.9) gives

v =

1 + u

(18.12)

Since v is collinear with u, (18.11) yields

u =

1 + (1 − v

)

1
2

βv

1 + β

(18.13)

At last we can express R as a function of v:

R =

1 + β − βv

[2(1 + β)]

1
2

= [

1
2

(β + 1)]

1
2

− ˆ

1
2

(β − 1)]

1
2

(18.14)

Physicists are frequently interested in a Lorentz transformation from
the laboratory system to the rest frame of a physical system charac-
terized by energy-momentum vector p and mass m. If E and p are the
energy and momentum of the system as observed in the lab., then,
according to (17.13),

R =

m + E − p

[2m(m + E)]

1
2

E + m

1
2

− ˆ

E − m

1
2

(18.15)

18. Timelike Rotations

Because of the complexity of R relative to R

(when expressed in phys-

ical variables), it is advantageous to express Lorentz transformations
in terms of R

whenever possible. We can always do this when spinors

are not involved.

Now let us reconsider the timelike rotation (18.1) of a vector p.

We express p and p

in the Pauli algebra of γ

by multiplying (18.1)

on the right by γ

, to get

0
0

+ p

= R(p

+ p)R.

(18.16)

Now p has a part p

collinear with v and a part p

⊥

orthogonal to v,

0
0

+ p

= R

+ p

) + p

⊥

= β(1 − v)(p

+ p

) + p

⊥

(18.17)

From the scalar and vector parts of this equation we get the usual
expression for the Lorentz transformation of a space-time vector:

0
0

= β(p

− v · p),

(18.18)

= β(p

− p

v) + p

⊥

(18.19)

Equation (18.19) can be written in a variety of different ways depend-
ing on the representation of p

and p

⊥

: for example,

= ˆ

v · pˆ

v = p − p

⊥

= p − ˆ

v × (p × ˆ

v),

(18.20a)

⊥

= ˆ

v · (ˆ

v ∧ p) = ˆ

v × (p × ˆ

v) = p − p

= p − ˆ

v · pˆ

(18.20b)

If p is the energy-momentum vector of some physical system, then

(18.18) shows how the energy of p as measured in one inertial system is
related to the energy measured in another inertial system moving with
velocity v relative to the first. Equation (18.19) is the corresponding
relation for momentum.

As we saw in (16.20), the Lorentz transformation of any rotor can

be accomplished in the same way as that of a vector. As an example,
we consider a timelike rotation of the electromagnetic bivector F :

F → F

= RF e

(18.21)

According to (8.13), we can write F = E + iB. By separating E and B
into parts collinear and parts orthogonal to v, we can express (18.21)
in terms of R

+ iB

= (E

+ iB

) + R

⊥

+ iB

⊥

Chapter IV. Lorentz Transformations

Using the expression (18.4) for R

, and noting that

⊥

= v ∧ E = iv × E

we can write (18.21) in the explicit form

+ iB

= E

+ β(E

⊥

+ v × B) + i[B

+ β(B

⊥

− v × E)]. (18.22)

The vector and bivector parts of this equation give familiar expressions
for E

and B

We will now study the composition of special timelike rotations,

because it occurs frequently in problems of physical interest. We will
be concerned with three different inertial systems. Let R

be the op-

erator which takes vectors in system 1 to system 2, and let R

take

system 1 to system 3. Let R

take system 2 to system 3.

According

to (18.4), we can write

2
i

= β

(1 − v

(18.23)

First we show that

R = R

= e

−

1
2

(18.24)

In words, (18.24) tells us that a Lorentz transformation from system
1 to system 2 followed by a transformation from system 2 to system 3
is equivalent to a transformation from system 1 to system 3 followed
by a spatial rotation.

According to (16.21), we can always write R in the form (18.24).

Still, we need to. verify that, given the interpretation of R

and R

, R

has the interpretation we have already assigned. Observe that

†

= R

2
2

= R

2
3

(1 − v

= β

(1 − v

(18.25)

Equation (18.25) is precisely of the form (18.16). It establishes an
equivalence of the velocity vector of system 1 as seen in system 2 with

It is difficult to find a system of labels for this problem which is both simple and suggestive.

The reader may find it useful to construct a simple triangle diagram to help keep the labels
straight.

19. Scalar Product

the velocity vector of system 1 as seen in system 3. It shows we have
the proper interpretation of R

. We can solve immediately for β

and

in terms of v

and v

= β

(1 + v

· v

(18.26)

+ v

+ (β

−1

− 1)ˆ

× (ˆ

× v

)

(1 + v

· v

)

(18.27)

From (18.25), it is easy to see that in (18.27) v

and v

can be inter-

changed if also the sign of v

is changed.

Our understanding of (18.24) will be complete when we have an

expression for b in terms of v

and v

. To this end, recall that, ac-

cording to (18.14), we can write the R

in the form

= N

(1 + β

− β

(18.28)

By substituting these expressions for R

into (18.24) and equating the

ratio of bivector to scalar part on each side of the resulting equation,
we find

tan

1
2

b =

× v

(1 + β

−1

)(1 + β

−1

) + v

· v

(18.29)

An expression for v

can also be found from (18.24), but, besides

being unduly complicated, it is superfluous, because we already have
(18.27).

Scalar Product

A scalar product (A, B) of d-numbers A and B is defined by

(A, B) ≡ (A e

(19.1)

From our discussion in section 5, where the notion of scalar product
was introduced, we know that the indefinite nature of (19.1) is entirely
due to the underlying Lorentz metric of space-time. This means that
(19.1) is a geometrically significant scalar product for the whole Dirac
algebra. It is interesting to examine the inner automorphisms of

which leave (19.1) invariant.

We will call them isometries of

An inner automorphism of an algebra

A is an automorphism which can be expressed in

terms of operations defined in

A .

Chapter IV. Lorentz Transformations

Any isometry which takes A into A

can be written

A → A

= RAS.

(19.2)

The scalar product of any two d-numbers is invariant under (19.2) if
and only if

R e

R = 1,

S e

S = 1,

(19.3a)

R e

R = −1,

S e

S = −1.

(19.3b)

If also S = R

−1

= ± e

R and R = ±R, then (19.2) becomes

A → A

= RAR

−1

(19.4)

According to (16.20) this is a Lorentz transformation of a rotor.

A rotor A can be written as the product of two d-numbers Ψ and

Φ,

A = Ψ e

Φ.

(19.5)

If we require that the Lorentz transformation on vectors (16.10) induce
the following transformations on Ψ and e

Φ,

Ψ → Ψ

= RΨ,

(19.6a)

Φ → e

= e

ΦR

−1

(19.6b)

then it is clear that Ψ e

Φ is a rotor. On the other hand, e

ΦΨ is invariant

under Lorentz transformations. D-numbers which transform accord-
ing to (19.6) are commonly called spinors. Since the transformation
(19.6) leaves minimal ideals invariant, this definition of spinor does
not conflict with section 12, where we defined a spinor as an element
of a minimal ideal. The two definitions are related by the fact that
(19.6) can be interpreted as a change of basis in a minimal ideal.

The set of all Lorentz transformations on a rotor is a group of

isometries of the Dirac algebra; so is the set of all spinor transforma-
tions. Closely related is the group of isometries which leave the even
and odd subspaces of the Dirac algebra separately invariant. We will
call it the group of complex Lorentz transformations. An element of the
subgroup for which R and S are even will be called a complex Lorentz

19. Scalar Product

rotation. Justification for the label “complex” comes from the fact
that these transformations are isometries among the odd d-numbers,
and any odd d-number k can be interpreted as a complex vector, as
is clear when it is written

k = a + ib,

(19.7)

where a and b are vectors. Furthermore, since it takes six parameters
to determine R and six more to determine S, the group of complex
Lorentz rotations is determined by twelve real parameters, or, equiv-
alently, by six “complex” parameters.

The complete group of complex Lorentz transformations can be

obtained by combining the continuous group of complex rotations with
the discrete operation of space conjugation. The complex rotations are
distinguished by leaving invariant both the scalar and pseudoscalar
parts of a product of d-numbers. Thus, the complex rotations leave
invariant the following norm for k:

k˜

k = (a + ib)(a + bi) = a

− b

+ 2ia · b.

(19.8)

Space conjugation complex conjugates (19.8).

∗

= (k˜

∗

= a

− b

− 2ia · b.

(19.9)

A canonical form for the d-numbers which produce the most gen-

eral isometry of

D is given in theorem 7 of appendix B. There it

is shown that the isometries continuously connected to unity are re-
lated to the group of rotations in a five-dimensional space with metric
(+ − − − −) .

We can entertain definitions other than (19.1) for a scalar product,

for instance, (AB)

, (AB)

or (A

†

. Any other definition of scalar

product must single out some space-time direction as special. The
considerations of section 14 give us good reason to choose

(φγ

ψ)

= (ψ

†

φ)

(19.10)

as the scalar product of Dirac fields φ and ψ. In section 24 we discuss
the possibility of a physical interpretation for the inner automorphisms
of the Dirac algebra which leave (19.10) invariant.

Chapter V

Geometric Calculus

Differentiation

To complete the space-time calculus developed in chapter I, we must
define the gradient operator for curved space-time. Evidently is
an invariant differential operator, but we will not attempt to define
it without reference to local coordinate systems. This approach sim-
plifies comparison with tensor analysis. However, once is defined,
manipulations can be carried out in a coordinate-independent manner.
To simplify our discussion, we will ignore all questions about differen-
tiability. Such questions can be answered in the same way as in tensor
analysis. We wish to emphasize the special algebraic features of our
geometric calculus.

A set of d-number fields on a region

R of space-time will be called

linearly independent if they are linearly independent at each point of
R. A set of four linearly independent 1-vector fields {γ

; i = 0, 1, 2, 3}

R will be called a frame field on R. The metric tensor g

of a

frame field is determined by the inner products of the γ

= γ

· γ

1
2

(γ

+ γ

(20.1)

It will be algebraically convenient to construct a frame field {γ

} of

vectors reciprocal to the γ

. The γ

are uniquely determined by the

condition

· γ

= δ

(20.2)

Ó Springer International Publishing Switzerland 2015
D. Hestenes, Space-Time Algebra,
DOI 10.1007/978-3-319-18413-5_5

Chapter V. Geometric Calculus

For further discussion of the reciprocal frame {γ

} we refer to appendix

Let {x

: µ = 0, 1, 2, 3} be a set of coordinate functions on

Through each point x of

R the µth coordinate function x

determines

a coordinate curve with tangent vector γ

. The set {γ

} of linearly in-

dependent vector fields so determined will be called a coordinate frame
field on

R. Throughout part V we will use Greek indices exclusively

for coordinate frame fields and Latin indices for more general frame
fields. It will also often be convenient to refer to a frame field simply
as a frame, and, since we will be dealing only with local properties of
space-time, to ignore the possibility that we may be able to cover only
a limited region of space-time by a single coordinate system.

Let {γ

} be a frame field. Corresponding to each vector field γ

we define a differential operator

by the following postulates:

(a)

maps scalars φ into scalars,

φ = ∂

φ.

(20.3)

If {γ

} is a coordinate frame, then ∂

is an ordinary partial deriva-

tive, i.e.

φ = ∂

φ =

∂φ

∂x

(20.3)

For other frame fields, ∂

can be expressed as a linear combination

of partial derivatives by postulate (e) below.

(b)

maps vectors into vectors.

In particular,

maps γ

into a

vector, which can, of course, be expressed as a linear combination
of the γ

. Therefore we can write

= −L

k
ij

(20.4)

This defines the so-called coefficients of connection L

k
ij

for the

frame {γ

(c)

obeys the Leibnitz rule. Thus, for any two d-number fields A

and B,

(AB) = (

A)B + A(

B).

(20.5)

Green [18] studies a

which maps a vector into an arbitrary d-number; however, he restricts

his treatment to a space-time with absolute parallelism. In section 23, we discuss a different
generalization.

20. Differentiation

(d) Differentiation is distributive with respect to addition,

(A + B) =

A +

(20.6)

(e) If a frame field {γ

} is related to a different frame field {e

} by

= h

i
m

(20.7a)

then the operators

corresponding to the γ

are related to the

operators

corresponding to the e

= h

i
m

(20.7b)

This postulate insures that differentiation has the same scale in all
frames.

It is important to remember that throughout this paper we use

the terms “scalar” and “vector” in the algebraic sense of section 2 and
not in the sense of tensor analysis. The derivative of a d-number is not
affected by the indices which label the d-number.

For sake of brevity, we call the operator

simply the derivative.

We now proceed to examine some of its properties. First note that the
derivative preserves the degree of any r-vector. This property will be
lost in section 24 when we modify the derivative to apply to spinors.

The coefficients of connection for a frame are not completely in-

dependent of the metric for the frame. To see this, apply the Leibnitz
rule to the metric tensor (20.1).

= (

) · γ

+ γ

· (

By (20.3),

= ∂

and by (20.4),

(

) · γ

= −g

m
ki

≡ −L

kij

So,

∂

= −L

kij

− L

kji

(20.8)

This shows that only

1
2

(4 − 1) = 24 components of L

kl
ij

are indepen-

dent.

Tensor analysis requires that the covariant derivative of the metric

tensor vanish in metric manifolds. (20.8) shows that this condition is

Chapter V. Geometric Calculus

equivalent to the Leibnitz rule in our language. Crudely speaking, the
Leibnitz rule says that when we differentiate a · b we differentiate a
and b but we do not differentiate the dot.

The coefficients of connection for {γ

} are completely determined

by those for {γ

) = (

)γ

+ g

By (20.4) and (20.8),

−L

ijk

= (−L

ijk

− L

ikj

)γ

+ g

= δ

= g

ikm

So,

= L

k
ij

(20.9)

We defined the minus sign in (20.4) in order to get a plus sign in
(20.9). We shall give reason soon for considering the γ

more basic

than the γ

Let us use our rules to differentiate an arbitrary vector field a.

a = a

a = (

)γ

+ a

(

a = (∂

+ a

k
ij

)γ

(20.10)

(20.10) shows us that when operating on vectors

is equivalent to

the “covariant derivative” of tensor analysis.

According to postulate (a), derivatives of scalars with respect to

coordinate frame always commute. For example,

= ∂

∂

= ∂

∂

(20.11)

But the derivatives of vectors in general do not commute,

µ
βσ

) = (

µ
βσ

)γ

+ L

µ
βσ

= (∂

µ
βσ

+ L

µ
βρ

ρ
ασ

)γ

[

]γ

= L

µ
αβσ

(20.12)

20. Differentiation

where

[

] ≡

−

(20.13)

µ
αβσ

= ∂

µ
βσ

− ∂

µ
ασ

+ L

µ
βρ

ρ
ασ

− L

µ
αρ

ρ
βσ

(20.14)

µ
αβσ

is the usual curvature tensor for a metric manifold with a linear

connection. Combining (20.12) and (20.11) we obtain

[

]a = a

µ
αβσ

(20.15)

From (20.15) it is clear that the vanishing of the curvature tensor is
a necessary and sufficient condition for the derivatives of a vector to
commute.

With respect to arbitrary frames, the derivatives of scalars do not

commute but they do form a Lie algebra with the commutation rule:

[∂

, ∂

] = Ω

∂

(20.16)

Thus, for arbitrary frames (20.15) becomes

[

]a = (a

m
ijk

+ Ω

∂

)γ

(20.17)

The derivatives

suffer from the mild defect of depending on

particular vector fields. We can remove this defect by defining a more
fundamental derivative by

≡ γ

(20.18)

as before, we call this vector differential operator the gradient. is
an invariant differential operator. It is independent of any particular
frame field Algebraically, it is characterized by superposition of vector
multiplication onto the Leibnitz rule for differentiating products. We
could have expressed these algebraic properties in “vector form” with-
out any reference to a frame field. Our procedure, however, achieves a
certain clarity by keeping the Leibnitz rule and the vector multiplica-
tion separate. Nevertheless, it is important to note that this separation
is not invariant, but depends on the selected frame field.

From we can obtain the directional derivative with respect to

any vector field a:

≡ a · .

(20.19)

Chapter V. Geometric Calculus

We get back the

by using (20.2) and (20.19):

= γ

· .

(20.20)

From (20.19) one finds that the gradient of the coordinate function is
the tangent vector γ

= γ

(20.21)

Because of this direct relation, we consider the frame {γ

} more fun-

damental than the frame {γ

}. Indeed, γ

depends not only on x

but on all the other coordinates as well. Likewise,

= γ

· = g

µν

(20.22)

is the directional derivative determined by the coordinate function x

whereas

depends on all the other coordinate functions as well.

Using our generally defined gradient operator and the vector al-

gebra of chapter I, we have a complete coordinate-independent vector
calculus. Except for some obvious modifications, our earlier formulas
now hold in curved space-time. For instance, the separation (7.16) of
the gradient into divergence and curl still holds, as does Maxwell’s
equation (8.1). As new examples, let us write down a couple of formu-
las that hold for the invariant vector calculus. If a and b are vectors,
then

(a · b) = (a · )b + (b · )a − a · ( ∧ b) − b · ( ∧ a).

(20.23)

This is a coordinate free form of (20.8). It is an expression of the
Leibnitz rule. The reader may recognize it as a generalization of a
familiar formula in the vector analysis of Gibbs. For an r-vector A

and an s-vector B

∧ A

∧ B

= ( ∧ A

) ∧ B

+ (−1)

∧ ∧ B

= (−1)

s(r+1)

∧ ∧ A

+ (−1)

∧ ∧ B

(20.24)

This can easily be proved by combining the Leibnitz rule with the
associative and anticommutative rules for outer products.

Riemannian Space-Time

The objective of this section is to express Einstein’s gravitational field
equations in terms of our space-time calculus. Therefore we direct

21. Riemannian Space-Time

our attention to Riemannian space-time. A Riemannian manifold is
distinguished by the simple requirement that the curl of the gradient
of every scalar vanish;

that is, for every scalar function φ

∧ φ = 0.

(21.1)

This implies that for a coordinate frame

∧ γ

= ∧ x

= 0.

(21.2)

But

∧ γ

1
2

µ
αβ

− L

µ
βα

)γ

αβ

µ
αβ

= L

µ
βα

(21.3)

By combining (21.3) with three copies of (20.8) with permuted indices,
we find

αβµ

1
2

(∂

αβ

− ∂

µβ

− ∂

µα

(21.4)

So the coefficients of connection are completely determined by the
metric tensor.

Let us now examine some properties of the curvature tensor. From

(20.12) we can get an equation containing only invariant derivatives

∧ γ

1
2

µ
αβσ

αβ

(21.5)

where R

µ
αβσ

is now the Riemann curvature tensor. We can analyze

(21.5) further in the following way:

= ( · + ∧ )γ

= ( · γ

+ ∧ γ

But because of (21.2),

∧ γ

= ( · γ

) − ( · )γ

(21.6)

The righthand side of (21.6) has only a vector part. Hence the trivector
part of (21.6) vanishes,

∧ ∧ γ

1
2

µ
αβσ

αβσ

= 0.

(21.7)

Remarks on the significance of this condition are made in section 22.

Chapter V. Geometric Calculus

This is equivalent to the well known identity

µ
αβσ

+ R

µ
βσα

+ R

µ
σαβ

= 0.

It is a simple matter to prove a more general version of (21.7), namely,
that for any differentiable d-number field A,

∧ ∧ A = 0.

(21.8)

To interpret the vector part of the right side of (21.5) we use equation
(3.9),

αβ

· γ

= γ

(γ

· γ

) − γ

(γ

· γ

µ
αβσ

αβ

= 2R

µ
αβσ

βσ

= 2R

µ
α

(21.9)

µ
α

is the Ricci tensor. Thus (21.5) and (21.6) give

≡ R

µ
α

= ( · γ

) − · γ

(21.10)

We see that the Ricci tensor involves only the gradient operator
whereas the total curvature tensor involves directional derivatives in
its definitions.

In space-time the free field gravitational equations are ordinarily

written R

µ
α

= 0. By (21.10) these equations become

· γ

= 0

(21.11)

if the γ

are subject to the subsidiary condition

· γ

= · x

= 0.

(21.12)

Coordinates which satisfy (21.12) will be called harmonic.

Equations (21.10) and (21.12) look quite like the wave equation

for the electromagnetic potential with Lorentz side condition. But the
appearance is deceiving. Observe that the operator · takes on its
simplest form in harmonic coordinates;

· = γ

· γ

+ ( · γ

)

(21.13)

Now it is true that when operating on a scalar

= ∂

∂

, the

d’Alembertian. Therefore (21.12) is the wave equation for the x

. But

when operating on vectors γ

in (21.11)

produces a nonlinear

21. Riemannian Space-Time

equation in the coefficients of connection, as can be seen by using
(20.10).

Because the gravitational field equations take on their simplest

form (21.11) for a harmonic frame, it is tempting to attribute some
special physical significance to harmonic coordinates. Fock [19] has
suggested that harmonic coordinates are the natural generalization
of the inertial coordinates of special relativity. We will return to this
suggestion in section 23. Here we will be content with the remark that
whether or not harmonic coordinates can be interpreted as inertial,
they retain a redoubtable significance because (21.12) is the simplest
coordinate condition involving only the gradient operator.

Einstein’s equations for the gravitational field in the presence of

matter are usually written

µ
ν

−

1
2

R = T

(21.14)

To complete our discussion, we must reformulate them in the geomet-
ric calculus. The scalar curvature is simply

R = R

= R

µν

(21.15)

From the matter field we can construct the stress-energy vectors T

, as we did for the electromagnetic field in section 9. The gravi-

tational field equations can now be written

γ(R

−

1
2

R) = γT

(21.16)

where γ = |γ

| =

√

− g is the magnitude of the pseudoscalar for the

frame {γ

The factor γ in (21.16) has been introduced so that the

right side satisfies the conservation law

(γT

) = 0,

(21.17)

and the left side satisfies the identity

[γ(R

−

1
2

R)] = 0.

(21.18)

The operator

differs somewhat from the covariant derivative of

tensor analysis, and the factor γ compensates for this differences in
(21.17) and (21.18).

The pseudoscalar of a frame is discussed in Appendix A.

Chapter V. Geometric Calculus

We can write (21.16) in a more suggestive form. By contracting

the tensors of (21.14) we find

T ≡ T

= −R.

(21.19)

Dropping the pseudoscalar factor γ from (21.15), the gravitational
field equations for a harmonic frame can be written

· γ

= 2T

− γ

(21.20)

Thus the gravitational field can be represented by four vector po-
tentials, each of which satisfies a “wave-like” equation with a source
determined by the stress-energy tensor of the matter field. In spite of
the apparent simplicity of (21.20), it is not clear that it can be solved
without being rewritten as an equation for the coefficients of connec-
tion of the harmonic frame, in which case the methods of solution are
the same as in tensor analysis.

Integration

Grassmann (or multilinear ) algebra is equivalent to Clifford algebra
using only the outer multiplication defined in section 3.

In modern

mathematical treatments of integration theory and differential geom-
etry, Grassmann algebra is widely used in the so-called algebra of
differential forms. Introductions to this subject which are designed
for physicists are given by Misner and Wheeler [13] and by Flanders
[20]. Here we wish to point out that our geometric calculus based
on Clifford algebra has all the advantages of the algebra of differen-
tial forms, and more. All the results given by Flanders can be easily
transformed into the language of Clifford algebra. We emphasize that
Clifford algebra is a superior language because, as we saw in chapter
I, it integrates inner and outer multiplication into a single operation;
this has the great advantage of making all multiplication associative.
By contrast, in treatments of differential forms, outer multiplication is
as simple as in Clifford algebra, but duality transformations are more
difficult, and the inner product is introduced as a combination of outer
multiplication and duality transformation; beyond these unnecessary

Or if you wish, the Grassmann algebra of a

is the Clifford algebra of a

for which the

inner products of all vectors in

are zero.

22. Integration

complications, the powerful associative law of Clifford algebra is never
found and exploited. The superiority of Clifford algebra is amply il-
lustrated by comparing our formulation of Maxwell’s equation with
that given in the above references.

To make our geometric calculus more useful, we show how Clifford

algebra enables us to express the fundamental theorem of integral
calculus in strikingly simple form. We call this theorem the boundary
theorem, and we formulate it first for flat space-time.

The Boundary Theorem. In Minkowski space-time, let Ω be a closed
(r + 1)-dimensional surface bounded by an r-dimensional surface Σ.
Represent the surface element of Ω by the (r + 1)-vector dω, and the
surface element of Σ by the r-vector dσ. If A is a differentiable d-
number field on Ω, then

Ω

dω · A =

dσA.

(22.1)

Actually, (22.1) holds for flat manifolds of any dimension as long

as the appropriate Clifford algebra is used. For Euclidean 3-space we
write

Ω

dω · ∇A =

dσA.

(22.2)

Of course for space-time 0 5 r 5 3, and for Euclidean 3-space 0 5
r 5 2.

Our formulation of the boundary theorem contains all special

cases, variously named as Stoke’s theorem, Green’s theorem, Gauss’s
theorem, the divergence theorem, etc.

The proof of the boundary theorem involves only the much stud-

ied analytical problems which arise in the usual treatment of Stoke’s
theorem, so we do not attempt it here. Our purpose is rather to demon-
strate the great algebraic convenience which accrues from the use of
Clifford algebra. However before we discuss applications, some remarks
on the boundary theorem for curved manifolds are in order. In flat
manifolds addition of vectors, hence addition of any c-numbers, at
different points can be uniquely defined. This means that the integral
of any c-number field has a definite meaning. But in curved mani-
folds only the sum of scalars and pseudoscalars at different points

Chapter V. Geometric Calculus

is well-defined. Therefore for curved manifolds only the scalar and
pseudoscalar parts of (22.1) are meaningful. For curved Riemannian
space-time (22.1) yields

Ω

(dω · A)

(dσA)

(22.3)

where the subscript I means invariant part = scalar part + pseu-
doscalar part. Let us examine (22.3) in more detail. Using equation
(3.10), we note that

(dω · A)

= (dω · ) · A

= dω · ( ∧ A

(22.4)

where A

is the r-vector part of A. Therefore the scalar part of (22.3)

can be written

Ω

dω · ( ∧ A

) =

dσ · A

(22.5a)

Similarly, the pseudoscalar part of (22.3) can be written

Ω

dω · ( ∧ B

) = i

dσ · B

(22.5b)

where B = Ai is the dual of A. With the help of (3.17), it is easy to
see that (22.5) is the same as the formulation of Stoke’s theorem given
in references [17] and [24].

We may take equation (22.3) as the definition of the boundary op-

erator . For a manifold with a general linear connection the bound-
ary operator so defined will not be equivalent with the gradient op-
erator defined in section 20. However, by virtue of equation (21.1), it
can easily be shown that in Riemannian geometry the gradient and
the boundary operator may be taken as one and the same. This is
the beautiful property that sets Riemannian geometry apart. It also
explains why the basic differential operator should be a 1-vector,
for as boundary operator, mediates between manifolds differing by
one dimension.

Now return to flat space-time and equation (22.1). If r = 3 in

space-time dω is pseudoscalar so dω ∧ = 0. Therefore for this special
case we can write (22.1) as

Ω

dωA =

dσA.

(22.6)

22. Integration

This circumstance enables us to give a definition of A without ex-
plicit reference to coordinates:

A = lim

dω→0

dω

dσA.

(22.7)

The limiting procedure, clearly involves the shrinking of the volume
Ω to the point where A is defined.

As further illustration, let us see how the integral theorems for

Euclidean 3-space are special cases of (22.2).

For r = 0, Ω is a curve with tangent 1-vector dω = ds and end

points x

and x

. Σ is the 0-dimensional surface consisting of the

points x

and x

. Integration over Σ consists merely in multiplication

by the 0-vectors 1 or −1 depending respectively on whether the curve
Ω is arriving at or leaving Σ. Thus (22.2) takes on the familiar form

ds · ∇A = A(x

) − A(x

(22.8)

For r = 1, Ω is a surface with tangent 2-vector dω = −indS

where n is the unit normal to Ω and dS = |dω| is the magnitude of
dω. The surface Ω is bounded by a curve Σ with tangent 1-vector
dσ = ds. For this case (22.2) becomes

Ω

dSn × ∇A =

dsA.

(22.9)

Using inner and outer products we can separate (22.9) into two parts.
For example if A is a vector a, the scalar and bivector parts of (22.9)
are, after multiplying the bivector part by −i,

Ω

dSn × ∇ · a =

ds · a,

(22.10)

Ω

dS(n × ∇) × a =

ds × a.

(22.11)

If A is a scalar φ, (22.9) has only one part:

Ω

dSn × ∇φ =

dsφ.

(22.12)

Chapter V. Geometric Calculus

Formulas for bivector or pseudoscalar A are similar to the above, be-
cause in Euclidean 3-space they can be expressed as duals of a vector
or scalar.

For r = 2, Ω is a volume with volume element dω = idV and

surface Σ with area element dσ = indS, where n is the unit outer
normal to Ω. For this case we can write (22.2) in the form

Ω

dV ∇A =

dS nA.

(22.13)

If A is a vector a, (22.13) is equivalent to the two equations

Ω

dV ∇ · a =

dS n · a,

(22.14)

Ω

dV ∇ × a =

dS n × a.

(22.15)

If A is a scalar φ, (22.13) becomes

Ω

dV ∇φ =

dS nφ.

(22.16)

Global and Local Relativity

In chapter IV we introduced the Lorentz transformation as an isometry
of the tangent space

V (x) at a point x of space-time. We found that

it could be written in the form

V (x) → V

(x) = R(x)

V (x)R

−1

(x).

(23.1)

Now in the theory of special relativity Lorentz transformations are in-
terpreted as coordinate transformations among inertial systems, and
they have a priori nothing to do with (23.1), although in section 17
we related them to isometries with R(x) constant. This brings us to
a very important question: How is the notion of Lorentz invariance,
which has proved so pregnant in special relativity, properly general-
ized to nonuniform space-time? Is the physical significance of a Lorentz
transformation to be found in its interpretation as a coordinate trans-
formation? Or as a change in direction of tangent vectors? Perhaps

23. Global and Local Relativity

each interpretation relates to a different physical fact, and perhaps
they are connected in some physically significant way.

Einstein [21] generalized the idea of coordinate transformation.

In the special theory of relativity, Einstein had introduced covariance
under Lorentz transformations as a statement of the equivalence of
observations made in different inertial systems. In the general theory
of relativity, he extended this to the principle that the equations of
physics must be covariant under any coordinate transformation. How-
ever, although of heuristic value, the principle of general covariance
proved to be without physical content, as Einstein himself admitted,
for any equation can be cast in covariant form.

Evidently the success

of his theory of gravitation has some basis other than this principle,
and the physical significance of Lorentz transformations is to be found
somewhere else.

Fock [19] rejects Einstein’s idea that in nonuniform space-time no

class of coordinate systems can have objective physical significance. He
generalizes the definition of inertial coordinates by identifying them
with harmonic coordinates in Riemannian space-time. The Riemann
and the harmonic conditions can be combined in a natural way into
a single definition of inertial systems: Coordinate functions x

will be

called inertial if they everywhere satisfy the wave equation:

= 0.

(23.2)

The geometry of space-time can now be largely determined by a pos-
tulate which we call the Principle of Global Relativity (PGR): Inertial
coordinates (systems) exist in space-time.

Let us mention some of the physical implications of PGR. First,

note that the bivector part of (23.2) is equivalent to (21.2), and we
saw that this implies that space-time is Riemannian. Second, observe
that PGR leads to a relativity group which characterizes space-time;
namely, the group of all coordinate transformations which leaves the
metric of inertial systems invariant. This is a subgroup of the group
of all transformations which leave (23.2) invariant. In flat space-time,
for instance, the larger group permits arbitrary scale transformations
for each coordinate in addition to the usual Lorentz transformation.

Indeed, equation (8.1) is an invariant formulation of Maxwell’s equation—it makes no ref-

erence whatever to coordinates. What can general covariance mean here except that coordinate
systems are irrelevant?

Chapter V. Geometric Calculus

We can eliminate these scale transformations by suitable boundary
conditions. Therefore, we can alternatively define the relativity group
as a group of transformations, subject to certain boundary conditions,
which leaves (23.2) invariant.

Perhaps it is not necessary to construe PGR as asserting that all

of space-time can be covered by a single coordinate system. We must
be able to cover certain subregions of space-time with inertial coordi-
nates, but we leave the exact nature of such regions to be determined
by some future physical principle. The structure of the relativity group
is not solely determined by PGR but depends on boundary conditions
for space-time, and so involves a fundamental question of cosmology.
This leads to the problem of determining the relativity group for var-
ious models of space-time. If space-time is flat, the relativity group is
just the Poincar´

e group, and our theory reduces to the special theory

of relativity. Fock [19] has further argued that for any Riemannian
manifold which is uniform at infinity harmonic coordinates are unique
to within a Poincar´

e transformation. Beyond this, the problem of de-

termining “suitable boundary conditions” is wide open.

The original special theory of relativity suffers from the lack of a

clear definition of inertial coordinates. Einstein tacitly assumed that
space-time is flat by assuming that rigid rods exist in each inertial
system and can be used to lay out coordinates. Definition (23.2) is de-
signed to remedy this defect in a way which unites both of Einstein’s
great theories. It is fitting that this definition involves a generaliza-
tion of the wave equation, for it was Einstein’s brilliant analysis of
electrodynamics which first led to the formulation of the relativity
principle.

PGR leads to a uniqueness in interpretation of physical phenom-

ena which is absent in the viewpoint of general relativity. Assuming
suitable boundary conditions in space-time, a straight line is uniquely
defined as a straight line in an inertial system, and the relativity group
can be looked upon as the group of congruences of all straight lines in
space-time. Acceleration is the same in all inertial systems, and this
gives absolute significance to gravitational forces and the stress-energy
tensor. Unfortunately, these so called advantages of PGR can be con-
strued as mere conveniences, for no decisive physical consequence of
identifying harmonic and inertial coordinates has yet been discovered.

23. Global and Local Relativity

In our formulation of global relativity we assumed that the nature

of the gradient operator was understood. Thus, the d’Alembertian in
(23.2) makes implicit use of the metric of space-time; it incorporates
Einstein’s original assumption of the constancy of the speed of light.
The gradient operator is, moreover, a “local” operator. We can for-
mulate this property in terms of a “relativity principle” which has
nothing to do with coordinates. We will show that it agrees with the
postulates for differentiation given in section 20 and so could be used
to construct an alternative definition for the gradient operator. The
insight we gain leads us to the more general notion of differentiation
discussed in the next section.

Let us examine what may be called the Principle of Local Rela-

tivity (PLR): The gradient operator is covariant under local Lorentz
transformations.

The transformation (23.1) will be called a local Lorentz trans-

formation at x if R(x) is differentiable. Supposing that is not yet
defined, let us consider how to give meaning to the word “differen-
tiable”. We cover a neighborhood of the point x with coordinates x

and their tangent vectors γ

. We take the partial derivatives of the γ

with respect to x

to be zero by definition. Now, by expressing R(x) in

terms of the tensor basis γ

determined by the γ

(defined in section

4), we have

∂

R(x) = ∂

) = (∂

)γ

(23.3)

i.e., the partial derivative operates only on the scalar coefficients. The
main point to be made is that differentiability implies that a local
Lorentz transformation at x entails a local Lorentz transformation at
neighboring points of x.

A d-number which transforms according to (23.1) will be called

a local rotor. Let A be a local rotor. The principle of local relativity
says that the local Lorentz transformation

A → A

= RAR

−1

(23.4)

induces

A → (A)

= R(A)R

−1

(23.5)

In order to see that this can be identified with the gradient

operator defined in section 20, we cover a neighborhood of the point

Chapter V. Geometric Calculus

x under consideration by a frame field of orthonormal vectors γ

which we call a vierbein. We can, of course, express the γ

as a linear

combination of the vectors γ

of a coordinate frame,

= h

k
µ

= h

µ
k

i
µ

ν
i

= δ

i
µ

µ
j

= δ

(23.6)

Just as we fixed the partial derivatives ∂

by requiring ∂

= 0,

we can define operators ∂

corresponding to the γ

by the condition

∂

= 0. The ∂

can be expressed in terms of the ∂

∂

= h

µ
k

∂

(23.7)

and differentiation carried out in the usual way, but it must be re-
membered that only the components of a d-number taken relative to
the γ

are to be differentiated.

Now, in a well-known way [22, 23], we can construct a covariant

derivative

which is associated with the γ

. We limit our discussion

to local Lorentz rotations. First we introduce a d-number field C

for

which (23.4) induces the transformation

→ C

= RC

R + R∂

(23.8)

By differentiating R e

R = 1, we find that

R∂

R = −(∂

R) e

(23.9)

which shows that R∂

R is a bivector. Therefore, (23.8) can be satisfied

if C

is chosen to be a bivector field. The operation of

on a local

rotor A can now be written explicitly.

A = ∂

A + [C

, A].

(23.10)

It is easily verified that (23.4) induces

A → (

= R(

A) e

(23.11)

Since also

→ Rγ

(23.12)

= γ

satisfies (23.5).

23. Global and Local Relativity

We can verify that

is the covariant derivative of section 20 by

applying it to the γ

= [C

, γ

] = 2C

· γ

(23.13)

Now, we can write

1
4

imn

, γ

(23.14)

so, using (3.9),

· γ

1
4

imn

· γ

1
2

= C

(23.15)

This agrees with (20.9). Of course, the coefficients of connection for a
vierbein satisfy

ijk

= −C

ikj

(23.16)

which agrees with (20.8), since for a vierbein ∂

= 0. The coefficients

of connection for a coordinate frame field can be found from those for
a vierbein by substitution,

ν
i

) = (

ν
i

)γ

+ h

ν
i

k
µ

(

)

= (∂

ν
j

+ h

ν
i

k
µ

j
σ

But

= L

ν
µσ

So,

ν
µσ

= A

ν
µσ

+ C

µσ

(23.17a)

where

ν
µσ

≡ h

j
σ

(∂

ν
j

µσ

≡ h

ν
i

k
µ

j
σ

(23.17b)

This completes our demonstration that the gradient operator of sec-
tion 20 is covariant under local rotations.

The covariance of the gradient operator expresses the local char-

acter of differentiation. Differentiation involves comparison only of
d-numbers at neighboring points. A local rotation can change the rel-
ative directions of tangent vectors at points separated by a finite dis-
tance, but it leaves the relative directions of vectors at infinitesimally

Chapter V. Geometric Calculus

separated points unchanged. For this reason the algebraic properties
of the gradient operator are preserved by local rotations.

The connection C

in (23.10) includes the gravitational force on

the rotor A. The separation of the right side of (23.10) into two parts
is not invariant, but depends on the choice of a frame. A unique sep-
aration of the directional derivative into a part which is due to the
gravitational field and a part which is not can only be accomplished by
some global postulate such as PGR. Just the same, since the gradient
operator includes the gravitational force, PLR can be interpreted as
an expression of the local character of the gravitational interaction.

Gauge Transformation and Spinor Derivatives

In the last section, we saw that the gradient operator could be de-
scribed, at least in part, by the condition of covariance under local
Lorentz transformations. Now, in section 19 we found isometries of
the Dirac algebra which are more general than Lorentz transforma-
tions. This leads us immediately to a more general kind of derivative.

Let us define a gauge transformation as any differentiable isome-

try of the Dirac algebra at a generic point x of space-time. The term
“differentiability” is to be understood in the same sense as in the last
section, and again differentiability implies that a gauge transforma-
tion at x entails a gauge transformation at neighboring points of x.
The set of all gauge transformations forms what we call a gauge group.
The structure of the gauge group is determined by the scalar prod-
uct it leaves invariant. Appropriate definitions of scalar product were
discussed in section 19. Any gauge transformation has the form

ψ → ψ

= RψS.

(24.1)

To illustrate the general method, we will study the gauge group which
leaves invariant the norm

(ψγ

ψ)

= (ψ

†

ψ)

(24.2)

where γ

is a field of timelike unit vectors. In order that (24.1) leave

(24.2) invariant, we must have

R e

R = ±1

(24.3)

24. Gauge Transformation and Spinor Derivatives

and

†

= ±1,

(24.4)

where both (24.3) and (24.4) must have the same sign. Our purpose
will be served if we allow only the positive signs in (24.3) and (24.4),
and, further, if we restrict our discussion to gauge transformations
continuously connected to the identity.

We consider the following generalization of the principle of local

relativity: The gradient operator is covariant under gauge transforma-
tions. This means that on the derivative

ψ of the d-number field

ψ with respect to the vierbein {γ

}, the gauge transformation (24.1)

induces the transformation

ψ → (

ψ)

= R(

ψ)S.

(24.5)

The covariance of

ψ can be assured by requiring that it have the

form

ψ = (∂

+ C

)ψ + ψD

(24.6)

where ∂

is defined as in section 23, and (24.1) induces the following

transformations on C

and D

→ C

= RC

R + R∂

(24.7)

→ D

= S

†

S + (∂

†

)S.

(24.8)

and D

must be proportional to the generators of R and S.

Define the gradient operator as before:

≡ γ

(24.9)

Now the γ

were introduced as vectors, and they must remain so after

a gauge transformation. Therefore if the gradient is to be covariant,
i.e., if (24.1) induces the transformation

ψ → (ψ)

= R(ψ)S,

(24.10)

then the induced transformation

→ γ

= Rγ

(24.11)

must be a local Lorentz rotation, that is, the γ

must be local rotors,

and the C

in (24.6) must be the same as the C

defined in section 23.

Chapter V. Geometric Calculus

We can now write

ψ = (∂ + C)ψ + γ

ψD

(24.12)

where

∂ ≡ γ

∂

(24.13)

and

C ≡ γ

(24.14)

It is very important to remember that the separation of ∂ and C de-
pends on the selected vierbein {γ

}, so the comments we made about

(23.6) apply here. Also note that the connection C can be expressed
in terms of the connection for a coordinate frame field by (23.13) and
(23.16),

C =

1
4

αβ

= (L

σ
αβ

− A

σ
αβ

)γ

(24.15)

The term Cψ in (24.12) is the usual expression for the interaction

of the gravitational field with a spinor field. The term γ

ψD

must be

interpreted as the coupling of ψ to nongravitational fields D

. From

theorem 5 of appendix C, we find that D

is proportional to the gen-

erators of the group of rotations in a 5-dimensional Euclidean space.
Therefore D

represents 10 independent vector fields interacting with

ψ. We can write

= ia

+ γ

+ ic

) + id

(24.16)

where, as usual, the boldface letters represent vectors in the Pauli
algebra of γ

. If, as in section 13, we interpret ψ as a nucleon field, then

certain physical interpretations are possible for the terms on the right
of (24.16). The first term can be identified with the ρ meson and the
last term with the ω meson.

It is not clear how the second and third

terms might be associated with any known vector mesons; certainly
no identification can be made until the factor γ

is interpreted.

At this point we terminate the discussion before it leads too far

along speculative lines. Equation (24.12) stands only as an example
of interactions invariant under local isometries of the Dirac algebra.

Cf. reference [24].

24. Gauge Transformation and Spinor Derivatives

At the very least, it has the attractive feature that both gravitational
and electromagnetic interactions are generated in the same way—by
covariance under gauge transformations. In as much as the interaction
terms arise naturally from the Dirac algebra, they may be said to have
a geometric origin.

Conclusion

We have constructed a single algebraic system which combines many of
the advantages of matrix algebra, Grassmann algebra and vector and
tensor analysis with some peculiar advantages of its own. The simple
form of equations and the ease of manipulation provided by our space-
time algebra have been amply illustrated. Much of this simplicity is
due to specific assumptions about space-time which we incorporated
into the algebra.

All the elements and operations in our space-time algebra can be

interpreted, either directly or indirectly, as representing geometrical
features of space-time. This is the great epistemological virtue of our
system. We have seen how it forces us to identify the unit pseudoscalar
in both space and space-time as the complex imaginary i. This, in turn,
connects the indefinite property of the space-time metric to the equa-
tion i

= −1. Further, it implies that space-time is something more

intricate than three dimensions of space plus one dimension of time,
as illustrated by our conclusion that a vector in space corresponds to
a timelike bivector in space-time. Space-time is related to space as an
algebra is related to one of its subalgebras.

Since our space-time algebra is more than a mathematical sys-

tem, but something of a physical theory as well, it is natural to ask
for its experimental consequences. Unfortunately, such consequences
can be obtained only in the context of a more detailed physical the-
ory. Just the same the striking successes of the Dirac electron theory
furnish strong evidence for the fundamental role of the Dirac alge-
bra. Although spin was discovered before the Dirac algebra, the Dirac
algebra provides a theoretical justification for the existence of spin.
It should not be too surprising if the Dirac algebra also provides a
justification for isospin after the manner suggested in section 13. The

Ó Springer International Publishing Switzerland 2015
D. Hestenes, Space-Time Algebra,
DOI 10.1007/978-3-319-18413-5

Conclusion

ultimate justification of our space-time calculus as a “theory” awaits
a physical prediction which depends on our identification of i as a
pseudoscalar.

Appendixes

Bases and Pseudoscalars

In this appendix we give some miscellaneous conventions and formulas
which are useful for work with arbitrary bases in

, together with

some special formulas in the Dirac and Pauli algebras.

A set of linearly independent vectors {e

: i = 1, 2, . . . , n} will be

called a frame in

. The metric tensor g

is defined by

= e

· e

1
2

+ e

(A.1)

A vector a can be written as a linear combination of the e

: a = a

The inner product of vectors a and b is

a · b = a

· e

) = a

= a

(A.2)

where

= a

(A.3)

Consider the simple r-vector e

∧. . .∧e

. The number of transpo-

sition necessary to obtain e

∧. . .∧e

is 1+2+3 . . .+(r−1) =

1
2

r(r−1).

It follows that the reverse (defined in section 5) of any r-vector A

†
r

= (−1)

1
2

r(r−1)

(A.4)

The pseudoscalar of a frame {e

} is an n-vector e defined by

e = e

∧ e

∧ . . . ∧ e

(A.5)

The determinant g of the metric tensor is related to the pseudoscalar
e by (3.17):

†

e = e

†

· e = (e

∧ . . . ∧ e

) · (e

∧ . . . ∧ e

)

= det g

= g.

(A.6)

Ó Springer International Publishing Switzerland 2015
D. Hestenes, Space-Time Algebra,
DOI 10.1007/978-3-319-18413-5

Appendixes

By (A.4),

= (−1)

1
2

n(n−1)+s

|g|,

(A.7)

where the signature s is the number of vectors in

with negative

square.

In order to simplify algebraic manipulation with base vectors, it

is convenient to define another set of base vectors e

related to the e

by the conditions

· e

= δ

(A.8)

where δ

is a Kronecker delta. The raised index on e

is strictly a

mnemonic device to help us remember the relations (A.8) between
the two bases, and is not meant to indicate that the e

are in any

sense vectors in a “space” different from that containing the e

Each e

can be written as a linear combination of e

= g

(A.9)

from which it follows immediately that

= δ

(A.10)

and

= e

· e

1
2

+ e

(A.11)

We can construct the e

explicitly in terms of the e

in the follow-

ing way: Define the pseudovector E

= (−1)

j+1

∧ . . . ∧ ˆe

∧ . . . ∧ e

(A.12)

where the circumflex means omit e

. Observe that

∧ E

= δ

(A.13)

Also define the reciprocal pseudoscalar e

−1

= (−1)

(A.14)

Now we can define e

≡ E

−1

(A.15)

A. Bases and Pseudoscalars

The frame {e

} is often called dual to the frame {e

} because of the

duality operation used in constructing it. With (A.15) we can verify
(A.8):

· e

= (e

∧ E

−1

= δ

−1

= δ

We can also write

−1

= (−1)

∧ . . . ∧ e

∧ e

(A.16)

We now restrict ourselves to the Dirac algebra. The pseudoscalar

for the frame {γ

: µ = 0, 1, 2, 3} will be denoted by γ

≡ γ

0123

= γ

∧ γ

(A.17)

Similarly, the pseudoscalar for the dual frame is

≡ γ

0123

(A.18)

All of the following formulas can be proved with the aid of formulas
in section 3,

= 1,

(γ

)

(A.19)

= 4,

(A.20)

= −4γ

(A.21)

For any vectors a, b, c,

aγ

= −2a,

(A.22)

abγ

= 4a · b,

(A.23)

abcγ

= −2cba.

(A.24)

The last two formulas admit the following special cases:

a ∧ bγ

= 0,

(A.25)

a ∧ b ∧ cγ

= 2a ∧ b ∧ c.

(A.26)

The completely antisymmetric unit ε

µναβ

can be defined by

µναβ

≡ γ

µναβ

(A.27)

Appendixes

Similarly,

µναβ

= γ

µναβ

(A.28)

It follows that

0123

= 1,

(A.29)

µναβ

= γ

µναβ

(A.30)

µναβ

= γ

µνα

(A.31)

µναβ

αβ

= 2γ

µν

(A.32)

µναβ

ναβ

= 6γ

(A.33)

The magnitude of γ

can be expressed in terms of the metric tensor,

g ≡ (γ

)

= g

0µ

1ν

2α

3β

µναβ

= g

0µ

1ν

2α

3β

µναβ

= det g

µν

(A.34)

Thus

= ±i

√

− g.

(A.35)

The reader is invited to find formulas analogous to those above

in the Pauli algebra. Bases in

D and P can be related by

≡ γ

(A.36)

The pseudoscalar in space for the frame {σ

} is

ijk

= γ

ijk

(A.37)

where

ijk

≡ ε

ijk0

(A.38)

Of course, for a righthanded orthonormal frame,

ijk

= iε

ijk

(A.39)

Some Theorems

Theorem 1. If R = R and R e

R = 1, then

R = ±e

(B.1)

where B is a bivector. B can be chosen so that the sign in (B.1) is
positive except in the special case when B

= 0 and R = −e

B. Some Theorems

Proof. Since R = R, R is the sum of a scalar S, a bivector T and a
pseudo-scalar P ,

R = S + T + P,

R = S − T + P.

If T

= 0, we can express T as a linear combination of orthogonal

simple bivectors T

and T

T = S

+ S

· T

= 0,

5 0.

The condition R e

R = 1 gives us two equations:

= S

− S

+ P

= 1,

= SP.

We can adjust the magnitude of T

so T

= P , then S

= S.

This enables us to factor R,

R = (S

+ T

)(S

+ T

So the condition R e

R = 1 becomes

− T

)(S

− T

) = 1.

We can take S

− T

= 1, so, since T

< 0,

+ T

= e

= cos B

+ sin B

Similarly, since T

> 0,

+ T

= e

= cosh B

+ sinh B

Since T

· T

= 0, T

commutes with T

, and B

commutes with B

Therefore

R = e

= e

(B.2)

This equation shows slightly more than (B.1), because B is expressed
as the sum of orthogonal simple bivectors.

To complete our proof, we must treat the special case T

= 0.

The condition e

RR = 1 gives

+ P

= 1,

SP = 0.

Appendixes

Since P

5 0, we must have P = 0 and S = 1. Consider the case

S = +1. Then, since T

= 0,

R = 1 + T =

∞

n=0

= e

(B.3)

This is in the form (B.1). If S = −1, then R = e

Theorem 2. If U e

U = 1 and U = U = U

∗

, then

U = e

(B.4)

where b is a simple timelike bivector.

Proof. From theorem 1 we have U = ±e

, so

∗

= γ

U γ

= ±e

Bγ

= ±e

∗

= U

implies B = B

∗

= γ

Bγ

. Therefore

· B =

1
2

(γ

B − Bγ

) = 0,

which means that B is orthogonal to γ

and so is simple and spacelike.

We can write B as the dual of a simple timelike bivector b: B = ib.
Since −e

= e

Bπ

= e

Bπ+B

= e

, where B = |B| ˆ

B and ˆ

= −1,

we can always choose B so U = e

Theorem 3. If H e

H = 1 and H = H = H

†

, then

H = ±e

(B.5)

where a is a simple timelike bivector.

Proof. From theorem 1 we have H = ±e

, so

†

= γ

Hγ

= ±e

−γ

Bγ

= H

implies B = −B

∗

= −γ

Bγ

. Therefore

∧ B =

1
2

(γ

B + Bγ

) = 0,

which means that B is proportional to γ

and so is simple and timelike.

Hence we can write B = a.

B. Some Theorems

Theorem 4. If R e

R = 1 and R = R, then a timelike unit vector γ

uniquely determines the decomposition

R = HU,

(B.6a)

where

U = U

∗

= e

(B.6b)

and

H = H

†

= ±e

(B.6c)

Proof. Note that A ≡ RR

†

satisfies A e

A = 1 and A = A = A

†

, so by

theorem 3 we can write

†

= e

Define H by

H = (RR

†

)

1
2

= ±e

Note that R

†

∗

= 1, so R = RR

†

∗

= H(HR

∗

). Therefore define u

U = HR

∗

We verify that

U e

U = HR

∗

†

H = H e

H = 1

and

∗

= H

∗

R = H

∗

HHR

∗

= HR

∗

= U.

So, by theorem 2, we can write U in the form (B.6b).

Theorem 5. If SS

†

= 1, then

S = e

(B.7)

S = ve

(B.8)

where v is a unit vector and B is a bivector for the five-dimensional
Euclidean space

spanned by vectors e

, defined by

= σ

= γ

(B.9)

= e

1234

= iγ

Appendixes

where {γ

: µ = 0, 1, 2, 3} is a righthanded orthonormal basis of vectors

in the Dirac algebra. Relative to the basis (B.9),

B = B

mn 1

, e

], v = v

(B.10)

(m, n = 1, 2, 3, 4, 5).

The Clifford algebra of a 4-dimensional Euclidean space

isomorphic to the Clifford algebra of a 4-dimensional Minkowski space
(we call the latter the Dirac algebra). The two algebras differ only
in geometric interpretation, i.e. in what elements are called vectors,
bivectors, etc. An isomorphism between the two algebras is given by
(B.9). If we take e

, e

as an orthonormal basis for

, then e

can be identified as the unit pseudoscalar.

We can identify the transformation

u → u

= SuS

†

(B.11)

as a rotation of a vector u in

if S is given by (B.7) and as a

reflection if S is given by (B.8). The reflections can be distinguished
from rotations in

by the method used in section 7. We can prove

that for rotations S has the form (B.7) in the same way that we proved
theorem 1 of this appendix. In fact, the proof here is somewhat simpler
because the scalar part of B

is negative definite. We also find, just

as in theorem 1, that B can be written as the sum of two orthogonal
simple bivectors. This represents the fact that the rotation group in
five dimensions is a group of rank 2.

Theorem 6. If a and b are timelike vectors and a · b > 0, then, for any
d-number A,

(aAb e

= 0,

(B.12)

and equality holds only if A vanishes.

Proof. We lose no generality by taking a and b to be unit vectors.
Write a = γ

. Since, as we saw in section 18, b can be obtained from

by a time-like rotation, we can write b = Rγ

R. Now define a new

d-number B = AR, so

(aAb e

= (γ

Bγ

B) = (BB

†

)

(B.13)

C. Composition of Spatial Rotations

By continuing our discussion following theorem 5, we identify (B.13)
as a Euclidean norm, and we recall that we proved this was positive
definite in section 5.

Theorem 7. If S e

S = 1, then

S = ±e

B+A

(B.14)

S = (V + P )e

B+A

(B.15)

where B is a bivector, A is a pseudovector, V is a vector, P is a
pseudoscalar and (V + P )

= 1.

If u is an element of the 5-dimensional linear space

consisting

of the vectors and pseudoscalars, we can identify the transformation

u → u

= Su e

(B.16)

as a rotation if S is given by (B.14) and as a reflection if S is given
by (B.15). Theorem 7 can be proved in the same way as theorem 5,
with due account taken of the metric (+ − − − −) on

Composition of Spatial Rotations

Recall from section 16 that a spatial rotation of a vector p can be
written

p → p

= e

−

1
2

(C.1)

This is a righthanded rotation through an angle b = |b| about the vec-
tor b. The exponential can be written as a sum of scalar and bivector
parts,

1
2

= cos

1
2

b + i sin

1
2

(C.2)

The series expansion for the exponential clearly shows

cos

1
2

b = cos

1
2

(C.3a)

sin

1
2

b = ˆ

b sin

1
2

(C.3b)

Appendixes

where ˆ

b is the unit vector in the direction of b.

The composition of rotations in Euclidean 3-space is reduced by

equation (C.1) to a problem in composition of exponentials, which is
fairly simple. Suppose the rotation (C.1) is followed by the rotation

→ p

= e

−

1
2

(C.4)

Then the rotation of p into p

is given by

p → p

= e

−

1
2

= e

−

1
2

−

1
2

(C.5)

From

1
2

= e

1
2

(C.6)

we can find c in terms of b and a. The scalar part of (C.6) is

cos

1
2

c = cos

1
2

a cos

1
2

b − (sin

1
2

a) · (sin

1
2

b).

(C.7)

From the bivector part of (C.6) we get

sin

1
2

c = cos

1
2

a sin

1
2

b + cos

1
2

b sin

1
2

a + (sin

1
2

a) × (sin

1
2

b).

(C.8)

Or, in more explicit form, (C.7) and (C.8) become

cos

1
2

c = cos

1
2

a cos

1
2

b − ˆ

a · ˆ

b sin

1
2

a sin

1
2

(C.7

)

c sin

1
2

c = ˆ

b cos

1
2

a sin

1
2

b + ˆ

a cos

1
2

b sin

1
2

a + ˆ

a × ˆ

b sin

1
2

a sin

1
2

(C.8

)

The ratio of (C.8) to (C.7) gives the law of tangents

tan

1
2

c =

tan

1
2

a + tan

1
2

b + (tan

1
2

a) × (tan

1
2

1 − (tan

1
2

a) · (tan

1
2

(C.9)

When a and b are collinear these formulas reduce to the familiar
trigonometric formulas for the addition of angles in a plane. From these
the full angle formulas can be obtained, but the half-angle formulas
are simpler and adequate in most problems.

D. Matrix Representation of the Pauli Algebra

Matrix Representation of the Pauli Algebra

In this appendix we give the correspondence between operations in
the Pauli algebra and operations in its matrix representation.

As we saw in section 12, a p-number φ can be represented by a

two by two matrix Φ over the field of the complex numbers,

φ = φ

+ φ

= φ

+ φ

(D.1)

Φ =

(D.2)

The matrix elements of Φ can be looked upon as matrix elements of
φ,

= (u

†
b

φu

)

= (v

†

φv

)

(D.3)

where the subscript I means invariant part, i.e. scalar plus pseu-
doscalar part. Using (12.7) and (12.5) we can find the matrix rep-
resentations of the base vectors σ

, σ

−i

, σ

−1

. (D.4)

Let Φ

be the transpose of Φ, let Φ

be the complex conjugate of Φ,

and let Φ† be the hermitian conjugate Φ. The following correspon-
dences are easily established:

∗

= φ

∗
0

− φ

∗
i

∼σ

(D.5)

†

= φ

∗
0

+ φ

∗
i

∼Φ

†

(D.6)

φ = φ

− φ

∼σ

(D.7)

φ e

φ = e

φφ

∼ det Φ,

(D.8)

= φ

∼

1
2

T rΦ.

(D.9)

For the two-component spinors we have the correspondences:

∼

(D.10)

∼

(D.11)

100

Appendixes

The operations of complex conjugation and transpose while of inter-
est to matrix algebra are only of minor interest to Clifford algebra, for
these operations are defined and meaningful only relative to a particu-
lar basis in the algebra. On the other hand, the operations of inversion
and hermitian conjugation are independent of basis; for this reason
they may be considered to be more fundamental than the operations
of complex conjugation and transpose.

Bibliography

[1] J. Gibbs, The Scientific Papers of J. Willard Gibbs, Dover, New

York (1961).

[2] E. B. Wilson, Vector Analysis, Dover, New York (1961).

[3] A. Wills, Vector Analysis, Dover, New York (1958).

[4] H. Grassmann, Die Ausdehnungslehre von 1862.

[5] W. K. Clifford, Amer. Jour. of Math. 1, 350 (1878).

[6] E. R. Caianiello, Nuovo Cimento 14 (Supp.), 177 (1959).

[7] H. Jauch and F. Rohrlich, The Theory of Photons and Electrons,

Addison-Wesley, Cambridge (1955).

[8] M. Riesz, Dixi´

eme Congr´

es des Mathematicians Scandinaves,

Copenhague, 1946, p. 123.

[9] M. Riesz, Clifford Numbers and Spinors, Lecture series No. 38,

The Institute for Fluid Dynamics and Applied Mathematics, Uni-
versity of Maryland (1958).

[10] F. G¨

ursey, Rev. Fac. Sci., Instanbul A20, 149 (1956).

[11] O. Heaviside, Electromagnetic Theory, Dover, New York (1950).

[12] L. Silberstein, Theory of Relativity, Macmillan, London (1914).

[13] C. Misner and J. Wheeler, Ann. Phys. 2, 525 (1957).

[14] L. M. Brown, Lectures in Theoretical Physics, Interscience, N.Y.,

IV, 324 (1962).

[15] F. G¨

ursey, Rev. Fac. Sci. Univ. Istanbul A21, 33 (1956).

[16] F. G¨

ursey, Nuovo Cimento 7, 411 (1958).

Ó Springer International Publishing Switzerland 2015
D. Hestenes, Space-Time Algebra,
DOI 10.1007/978-3-319-18413-5

101

102

Bibliography

[17] T. D. Lee and C. N. Yang, Nuovo Cimento 3, 749 (1956).

[18] H. Green, Nuclear Phys. 7, 373 (1958).

[19] V. Fock, Space, Time and Gravitation, Pergamon Press, New

York (1958).

[20] H. Flanders, Differential Forms, New York, Academic Press

(1963).

[21] A. Einstein and others, The Special Principle of Relativity, Dover,

New York (1923), p. 115.

[22] C. Yang and R. Mills, Phys. Rev. 96, 191 (1956).

[23] R. Utiyama, Phys. Rev. 101, 1597 (1956).

[24] J. J. Sakurai, Annals of Physics 11, 1 (1960).

Document Outline

Cover
Space-Time Algebra, Second Edition
Foreword
- References
Preface after Fifty years
- References
Preface
Contents
Introduction
Chapter 1 Geometric Algebra
Chapter 2 Electrodynamics
Chapter 3 Dirac Fields
Chapter 4 Lorentz Transformations
Chapter 5 Geometric Calculus
Conclusion
Appendixes
Bibliography

Wyszukiwarka

Podobne podstrony:
65 Czasoprzestrzeń Space Time Continuum Jay Friedman Jul 4 2015
Jose Wudka Space Time, Relativity and Cosmology Ch5 The Clouds Gather
Bearden GRAVITATIONAL AND EM ENERGY FROM CURVED SPACE TIME
Space time representation and analytics
Shan A Model of Wavefunction Collapse in Discrete Space Time
Stephen Hawking Space & Time Warps
(eBook) Charles Tart Space, Time, and Mind
Demidov A S Generalized Functions in Mathematical Physics Main Ideas and Concepts (Nova Science Pub
Laudal O A Geometry of time spaces Non commutative algebraic geometry, applied to quantum theory (WS
Rucker Master of Space and Time
Isaac Asimov Of Time and Space and Other Things
To Love Through Space and Time Tinnean
Vicci Quaternions and rotations in 3d space Algebra and its Geometric Interpretation (2001) [share
FIDE Trainers Surveys 2015 02 26 Reynaldo Vera Time Trouble
62 Nie spiesz się, ODETCHNIJ Take your time and BREATHE Jay Friedman Kirk Lundbeck Mar 22 2015
Hestenes D Reforming the math language of physics (geometric algebra)(Oersted medal lecture, 2002)(4
Asimov, Isaac Of Time and Space and Other Things(1)
Quest Through Space and Time Clark Darlton

więcej podobnych podstron

Hestenes D Space time algebra (2ed , Birkhauser, 2015)(ISBN 9783319184128)(O)(122s) MP

Document Outline