Space-Time Algebra

122 Pages·2015·0.87 MB·English

by  David Hestenes

David Hestenes
Space -Time
Algebra
Second Edition

David Hestenes
Space-Time Algebra
Second Edition
Foreword by Anthony Lasenby
DavidHestenes
Department ofPhysics
Arizona State University
Tempe,AZ
USA
Originally published byGordonandBreach Science Publishers,New York,1966
ISBN978-3-319-18412-8 ISBN978-3-319-18413-5 (eBook)
DOI 10.1007/978-3-319-18413-5
LibraryofCongressControlNumber:2015937947
MathematicsSubjectClassification(2010):53-01,83-01,53C27,81R25,53B30,83C60
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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 Cli(cid:11)ord algebra to unify otherwise dis-
parate mathematical languages, particularly those of spinors, quater-
nions, tensors and di(cid:11)erential forms. It provides a uni(cid:12)ed approach
covering all these areas and thus leads to a very e(cid:14)cient ‘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 e(cid:11)ectively still research topics for the future. As
an example, what is the real role of the unit imaginary in quantum
v
vi Foreword
mechanics? In Space-Time Algebra two possibilities were looked at,
and in intervening years David has settled on right multiplication by
the bivector i(cid:27) = (cid:27) (cid:27) as the correct answer between these two, thus
3 1 2
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,
ithasbecomeclearthatthisdoesseemtoworkforelectroweaktheory,
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.
Asimilartransformationandrevelationalsooccurredformewith
respecttogravitationaltheory.StimulatedbySpace-TimeAlgebra 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 di(cid:14)cult mathematically,
could be replaced by a coordinate-free gauge theory on (cid:13)at spacetime,
of a type similar to electroweak and other Yang-Mills theories [8].
The conceptual advantages of being able to deal with gauge (cid:12)elds in
a (cid:13)at space, and to be able to replace tensor calculus manipulations
withthesameuni(cid:12)edmathematicallanguageasworksforelectromag-
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 (cid:12)elds, and in both
relativisticquantummechanicsandgravitytherehavebeenthosewho
have been able to forge an intuitive understanding of the physics de-
spite the di(cid:14)culties 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
andmanyothersissuchthat,fortherestofuswho(cid:12)ndthealternative
languages di(cid:14)cult, and for all those who are interested in spanning
di(cid:11)erent (cid:12)elds using the same language, the ideas (cid:12)rst 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. Ru(cid:14)ni (eds.), Proceedings of the
Eleventh Marcel Grossmann Meeting on General Relativity,p.629
(World Scienti(cid:12)c) (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,
BoundstatesanddecaytimesoffermionsinaSchwarzschildblack
hole background, Phys. Rev. D. 72, 105014 (2005).
Preface after Fifty years
This book launched my career as a theoretical physicist (cid:12)fty years
ago. I am most fortunate to have this opportunity for re(cid:13)ecting on its
in(cid:13)uence and status today. Let me begin with the title Space-Time
Algebra. John Wheeler’s (cid:12)rst 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 (cid:12)rst 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 (cid:12)rst 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 di(cid:14)cul-
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 su(cid:14)cient. Accordingly, with con(cid:12)dence that
comes from decades of hindsight, I make the following Claims for STA
as formulated in this book:
(1) STA enables a uni(cid:12)ed, 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
inspaceandspacetimerespectively,withnonecessaryconnection
to spin.
ix
x 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
impliesthatthephysicalsigni(cid:12)canceofDiracandPaulialgebrasstems
entirely from the fact that they provide algebraic representation of
geometricstructure.Theirrepresentationsasmatricesareirrelevantto
physics|perhaps inimical, because they introduce spurious complex
numberswithoutphysicalsigni(cid:12)cance.Thefactthattheyarespurious
is established by claim (3).
ThecrucialgeometricrelationbetweenDiracandPaulialgebrasis
speci(cid:12)edbyequations(7.9)to(7.11)inthetext.Itissoimportantthat
Idubbeditspace-timesplitinlaterpublications.Notethatthesymbol
i is used to denote the pseudoscalar in (6.3) and (7.9). That symbol
is appropriate because i2 = (cid:0)1, but it should not to be confused with
the imaginary unit in the matrix algebra.
ReadersfamiliarwithDiracmatriceswillnotethatifthegammas
on the right side of (7.9) are interpreted as Dirac’s (cid:13)-matrices, then
the objects on the left side must be Dirac’s (cid:11)-matrices, which, as is
seldom recognized, are 4(cid:2)4 matrix representations of the 2(cid:2)2 Pauli
matrices. That is a distinction without physical or geometric signi(cid:12)-
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 (cid:13)-matrices
over the (cid:12)eld 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 super(cid:13)uous degree of freedom.
That claim is backed up in Section 13 of the text where Dirac’s equa-
tion is given two di(cid:11)erent but equivalent formulations within STA. In
equation(13.1)theroleofunitimaginaryisplayedbythepseudoscalar
i, while, in equation (13.13) it is played by a spacelike bivector.