Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics 9781441947468, 9781475723151, 0792328906, 9780792328902

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Division Algebras: Octonions, Quatemions, Complex Numbers and the Algebraic Design of Physics

Mathematics and Its Applications

Managing Editor:

M. HAZEWINKEL Centre for Mathematics anti Computer Science. Amsterdam. The NetherlantIs

Volume290

Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics by

Geoffrey M. Dixon BrandeisUniversity, U.S.A.

Springer-Science+Business Media, B.\(

Library of Congress Cataloging-in-Publication Data Dlxon, Geoffrey M. Division algebras : octonlons, quaternions, co~plex numbers, and the algebralc design of physlcs I by Geoffrey M. Dixon. p. cm. -- (Mathematlcs and Its appllcations ; v. 290) Includes Index. 1. Algebra. 2. Mathematical physlcs. I. Tltle. II. Serles: Mathematlcs and Its applicatlons (Kluwer Academic Publlshers) ; v. 290. OC20.7.A4D59 1994 512' .57--dc20 94-13948

ISBN 978-1-4419-4746-8 ISBN 978-1-4757-2315-1 (eBook) DOI 10.1007/978-1-4757-2315-1

Printed on acid-jree paper

This printing is a digital duplication of the original edition. All Rights Reserved ©, 1994 Springer Science+Business Media Dordrecht. Second Printing 2002. Originally published by Kluwer Academic Publishers in 1994 Softcover reprint of the hardcover 1st edition 1994 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without wriUen permission from the copyright owner.

Table of Contents I U nderpinnings

1

1.1' The Argument Screed I Screed 11 Screed 111 1.2 Clifford Aigebras Clifford Algebra R3,1 Pauli Algebra Clifford Algebra Rl,3 Clifford Algebra Rl,9 1.3 Conjugations and Spinors Nilpotent Clifford Aigebras Symplectic Nilpotent Clifford Aigebras 1.4 Aigebraic Fundamentals of the Standard Model

1 1 3 7 11 16 18 19 20 22 24 25 29

11 Division Aigebras Alone 2.1 Mostly Octonions 2.2 Adjoint Aigebras 2.3 Clifford Aigebras, Spinors 2.4 Resolving the Identity of 0 L 2.5 Lie Aigebras, Lie Groups, from 0 L 2.6 From Galois Fields to Division Aigebras: An Insight

31 31 35 40 43 46 49

111 Tensor Aigebras 3.1 Tensoring Two: Clifford Aigebras and Spinors 3.2 Tensoring Two: Spinor Inner Product Link to Internal Symmetry

59 59 61 64

vi

Table oE Contents

3.3 Tensoring Three: Clifford Algebras and Spinors 3.4 Tensoring Three: Spinor Inner Produet Resolving the Identity of T The Traee of X 3.5 Derivation of the Standard Symmetry 3.6 SU(2) x SU(3) Multiplets, and U(I) U(I) Charges

66 68 68 70 73 78 81

IV Connecting to Physics 4.1 Connecting to Geometry Dimensional Reduction 4.2 Connecting to Particles 4.3 Parity Nonconservation Righthanded Dirae Operator Lefthanded Dirae Operator Full Parity Violating Dirac Operator 4.4 Gauge Fields 4.5 Weak Mixing 4.6 Gauging SU(3)

83 83 85 90 94 94 95 96 97 100 105

V Spontaneous Symmetry Breaking 5.1 Sealar Fields 5.2 Scalar Lagrangians 5.3 Fermions and Scalars

109 109 111 115

VI 10 Dimensions 6.1 Fermion Lagrangian Matter / Antimatter Mixing 6.2 More SU(3) 6.3 Freedom from Matter-Antimatter Mixing 6.4 (1,9)-Scalar Lagrangian 6.5 Charge Conjugation on TL(2) 6.6 Charge Conjugation on T 2 The Meaning of Majorana 6.7 10 Other Dimensions The Clifford Algebra

117 117 120 122 124 126 128 130 132 133 137

Table oi Contents

VII Doorways 7.1 Moufang and Other Identities Two Identities The Moufang Identities 7.2 Spheres and Lie Algebras

S3 S7 Sphere Fibrations 7.3 Triality Triality Representations of 80(8) The Tri in Triality Freudenthal's Principle of Triality 7.4 LG 2 and Tri LG2 Triality Triplet 7.5 LG2 Triplets and the X-Produet LG: General Solution LG: and the X-Adjoint Algebra OLX

VIII Corridors 8.1 Magie Square 8.2 The Ten MSKK, 8.3 SpinorKKI Outer Produets C Outer Produet Q Outer Produets o Outer Produets 8.4 LF4 ~ MSRO 8.5 Jf and F4 8.6 More Magie Square Appendix Appendix Appendix Appendix

i. OL Aetions: Produet Rule ea e a+1 = e a+5 ii. OR Aetions: Produet Rule ea ea+l = ea+5 iii. 0 L Aetions: Product Rule e a e a+l = ea+3 iv. OR Actions: Product Rule ea ea+1 = ea+3

Bi bliography Index

vii 141 141 145 148 150 151 153 159 160 164 165 169 170 171 175 180 187 191 191 192 198 198 200 202 203 208 213 217 221 225 229 233 235

Preface I don't know who Gigerenzer is, but he wrote something very clever that I saw quoted in a popular glossy magazine: "Evolution has tuned the way we think to frequencies of co-occurances, as with the hunter who remembers the area where he has had the most success killing game." This sanguine thought explains my obsession with the division algebras. Every effort I have ever made to connect them to physics - to the design of reality - has succeeded, with my expectations often surpassed. Doubtless this strong statement is colored by a selective memory, but the kind of game I sought, and still seek, seems to frowst about this particular watering hole in droves. I settled down there some years ago and have never feIt like Ieaving. This book is about the beasts I selected for attention (if you will, to render this metaphor politically correct, let's say I was a nature photographer), and the kind of tools I had to develop to get the kind of shots Iwanted (the tools that I found there were for my taste overly abstract and theoretical). Half of thisbook is about these tools, and some applications thereof that should demonstrate their power. The rest is devoted to a demonstration of the intimate connection between the mathematics of the division algebras and the Standard Model of quarks and leptons with U(l) x SU(2) x SU(3) gauge fields, and the connection of this model to lO-dimensional spacetime implied by the mathematics. If you understand what I have written to this point, then there is a good chance you have the background to understand this book. As to agreeing with its philosophical premises, you probably won't. I've never yet found two mathematicians or physicists who agree on the connection between mathematics and physics, and its depth. However, such agreement is not required to appreciate the material presented. ix

x

Preface.

I owe thanks to several people: to my wife, Suzanne Young of Harvard, whose work in archaeometry compelled me in some weird way to write the book; to my thesis advisor, Hugh Pendleton of Brandeis, for his advice, interest, and forbearance; and to Rafal Ablamowicz, Martin Cederwall, Pertti Lounesto, Corinne Manogue, lan Porteous, and Tony Smith, all of whom at my instigation converged on Göteborg, Sweden, this last January (1994) to talk about division algebras, Clifford algebras, and physics. Each played a crucial role at this meeting, and the breadth and depth of their insights has greatly enriched this monograph.

Geoffrey Dixon, Waltham, Massachusetts, 21 February 1994.

1. Underpinnings. This is a book of ideas. Most of them arise out of my fascination with the division algebras and their tantalizing potential for a deep connection to physies. Part of the book is a development of ideas that compellingly demonstrate that there is a connection. The rest is mathematies, applications of techniques originally developed for physies, but here applied to some of the special algebraie and geometrie notions, like triality, associated with the division algebras. Inevitably many of these mathematieal ideas will be shown to be intimately linked to the design of our physieal reality.

1.1. The Argument. Shortly after I began this monograph, in an attempt to crystallise the mathematieal ideas that had motivated, and continue to motivate, this work, I wrote aseries of essays, whieh 1 called Screeds, on Truth in physies, and how the mathematieal ideas presented here conform to my not ion of what that Truth should look like. The first two of these I call Gedanken Fantasies: SCREED I As you crack your eyes one morning your reason is assaulted by astrange sight. Over your bed, humming quietly, there floats a monitor, an ethereal, otherworldly screen on which is written a curious message: "I am the Screen of ultimate Truth. I am bulging with information and ask not hing better than to be allowed to impart it." In an attempt to establish the bonafides of the thing, or failing that, to drive it away that you may be about your business, you mumble a question. "What is the proof of Fermat's last theorem?" The screen fliekers slightly, goes blank, and then there, in its righthand 1

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CHAPTER 1

margin, under the heading "A Proof', is a proof. So Fermat was wrong, you think. It does fit in the margin of a page. [Of course, subsequent mathematical his tory has rendered this piece of the screed outdated, but although now proven, or perhaps only nearly proven, there is still no proof that will fit in the margin of a page.] Excitement builds as you realize that you, a theoretical physicist, have before you the last word in labor-saving devices. What should you ask it? The screen flickers again, ;:tnd a new message appears: "You have but one hour more." No doubt it has many calls on its time. What should you ask? Should you ask how the uni verse came into being? But what if this question has no answer in the context of ultimate Truth? Perhaps it were better to ask why, than how. What abotit your own work on modelling physical reality? Do you suppose it could tell you if you are at least on the right path? Do you want to know? Then a sobering thought occurs: there is no reason to believe you will understand the answers you are given. Do you believe sincerely that we have already at our disposal nearly an of the mathematical and physical concepts required to ans wer ultimate questions, and that, given time and some talented juggling of those concepts, a best fit to Truth may be found? What of the principles underlying the best fit to physical reality? What of the principles underlying the existence of physical reality? Do you believe in ultimate truths? Do you believe that evolution has given rise in the human cranium to an instrument capable of understanding? Not wishing to disturb the person snoring softly at your side, you lean carefully forward and whisper a question. Wh at question?

The thought that the human brain, even taken collectively and over the entire past and future of the species, may ultimately prove inadequate to the task, occured to me as I wrote. Once I had argued the contrary view. But why should Nature produce a creature capable of stripping Nature bare? One needs first, perhaps, to believe that Nature has a plan, and that we fit prominently into it. And ultimately selfreferentiality may preclude any creature arising out of Nature from fuHy understanding the Nature whence it sterns. But long before we encounter problems of that sort, we shall doubtless become bogged down by arbitrary and prejudiced human nature; it i8

UNDERPINNINGS

3

our lower brain that drives us to strive, and which will finally keep us from succeeding. The second Gedanken Fantasy follows from the first: seREED 11

To airy not hing give a local habitation and a name. Uruk lay back in bed and stared at the ceiling. The Monitor of ultimate Truth had departed, and he little doubted he would not see it again. He had asked it but one question: why does the universe exist? The monitor had responded with a faint crackling, what sounded like a weary sigh, and then the screen had filled with symbols and words for which Uruk had no context. Ignoring his pleas to cease, it used up his hour with notions too abstruse for Uruk to comprehend even in part. At the end of an hour it stopped and a message appeared: "1 am of course not done, but my time is up. 1 am sorry if you feel cheated by what you saw, for 1 know you can not have understood even a small fraction of the answer. But I have been strangely remiss if I have left you with the impression that the answer is not deep. Let me leave you with a thought. Those theorists accounted by you as great were humans of intelligence, but the magnitude of their accomplishments sterns more from the paths they chose than it does from their mental powers. When all the world is mired even a single individual who finds a right path can accomplish seeming wonders. But right paths are harder to find than identify. If you find one you will experience a sudden lack of resistance, new realms of thought and application open rapidly and compellingly before you. To find one you must be guided by a right philosophy, by right principles, and luck. Without these, though all the greatest minds on earth should commit their energies to the search, no path from the mire will be found. Look about you and see if it is not so. Be prepared to unlearn when faced with discovery, to relearn in a new light." Uruk watched the ceiling - it wasn't doing anything - and thought. Had he harbored any notion that the ans wer to his question might be religious, he was now disabused. In no conventional sense could the monitor's answer be so construed. He had been looking at ... what? Mathematics? Perhaps, but not as he knew it. Metamathematics? But of what else could the answer have consisted? Perhaps he had imagined he might be told the universe existed because of quantum bubbles, or something of that sort, but that would

4

CHAPTER 1

beg the question. Whence quantum? Whenee bubble? Whenee any of the form with which he was familiar? The answer had to He in mathematics, but not as done by mathematicians. The universe of Uruk's experience is very specifie. Conventional mathematics is full of generalizations, mostly of ideas drawn from the universe. The attention of physicists is focussed on the extant, with a hand in mathematics. Mathematicians keep a hand in physics, but gloved, lest their fingernails get dirty. One studies the tail, the other the trunk, of an elephant. Neither is even much aware of the vast body twixt the two ends that unifies them into a single science. In part, thought Uruk, it is the science of mathematical specialness. Is it an accident that of an the general notions of our mathematics, those we fin~ that help us best to deseribe our universe are so often in so me sense seleet? Could this be a guiding principle? Uruk shut his eyes and listened to the softly sussurating snoring of the sleeper sleeping soundly at his side. If custom reconciles us to everything, he thought, is custom binding us to the mire? Look about you said the monitor, and see if it is not so.

In another unfinished and more aeerbic Screed, the Screen of ultimate Truth had more to say:

"The form of reality, that you as theorists are slowly beginning to understand, was before you were. The Truth that underlies it was before you were, and was before the invention of your mathematical formalisms. When you are gone, and a time will come when all of you are gone, the Truth will continue to exist, without the benefit of your formalisms, your definitions, theorems, your rigor. "Do you believe that were I to explainas mueh of what I know as you could comprehend that you would recognize it, that you would say, oh yes, this is but an extension of what we have already done, and though the mathematics is broader, the principles deeper, I am not surprised? Do you think you have asked even a fraction of the questions you need to ask?"

The Screeds were intended to be read by others (which they were), the thought being that by posing such questions as those above, the readers' preeonceptions might be softened, and their minds might be opened to what I subsequently had to say about the division algebras and physics. But we are

UNDERPINNINGS

5

most easily persuaded by ideas we are on the verge of discovering ourselves, and I have discovered that among physicists there is nothing remotely like a philosophical consensus. Still, a summation of my beliefs, or faith, if you will, just might prove a useful preamble. 27 million years ago there was a reality governed by the same mathematical/physical Truth that governs ours. The Truth was and iso It does not change. What we call mathematics is founded on a formal system. The system is a tool adapted to the human brain that it might see into this Truth a little, to give Truth a comprehensible shape. The tool, the system, is an invention. Gödel's famous undecidability theorem, for example, says nothing about Truth. It pertains to the system, and to its shortcomings, An eye may not see all of itself. Truth is 'not an invention. With our mathematics we may discover parts of it, but in being undiscovered, even undiscoverable, Truth functions. 27 million years ago there was matter composed of quarks, and the quarks were subject to an SU(3) symmetry. In a very real sense SU(3) existed and functioned during all that vast his tory of the uni verse prior to our formulation of the notion of Lie group. More significantly, the symmetry of the quarks was not, and is not, SU(7), SO(39), Sp(12), or G2 • It was and is SU(3). There are layers of Truth underneath this fact, which, in my opinion, force it to be so. They force observable space-time to have a (1,3)-pseudometric, and force isospin SU(2) to break. They force reality to take the fundamental form it has taken. In showing that the alteration of a given feature, or collection of features, of our physical reality would lead to an inconsistent theory, we have not necessarily uncovered another layer of Truth. We should expect reality to be consistent, and any alteration of reality that leads to inconsistency to be impossible. Consistency, after all, is a concept that connects our mathematical system to reality. It is acheck, both of the system, and of our application of it. I personally would be surprised, confronted with the Screen of ultimate Truth, if in telling me what it knows, it needed our mathematical axioms and physical principles, and that consistency was other than a consequence of something far more profound. If one is to believe the literature most physicists think of mathematics as a collection of methods. Mathematics is accounted secondary to primary reality. It is laid on. Differential geometry applied to gravitation, algebraic topology [o,pplied to string theory, group representation theory applied to GUTs, are all mathematical methods. Again judging from the literature, mathematicians are no doser to think-

CHAPTER 1

6

ing that deep beneath their scienee, and physical s~ienee, there is a Truth to which both are profoundly tied. Mathematicians spend mueh time either in a selfreferential study of the system, or in using the system to generalize not ions spawned by reality. So, for example, geometers do not study merely 3-spaee or 4-space, but n-spaee. Curiously they diseovered that the most rem ar kable of all such geometries has dimension = 4. This fascinating result leads one to wonder if this is related to the fact that our reality's geometry has 4 apparent dimensions. Conjeeture of this sort - stemming from the observation of mathematical seleetness in many aspeets of the design of reality as we pereeive it - arises out of the attraetive notion that this design may not be arbitrary. It is an admission of faith that there is an undiscovered prereality Truth governing the features of this design. While we may not be, perhaps never will or can be, in a position to formulate comprehensibly such a Truth, never-the-Iess faith in its existence can have a practical influenee on research. The association of truth to beauty is such an influence, but this association is evidently one-way. Not all that is considered beautiful is true. Attesting to this is the pie thora of often incompatible theoretical models each ofwhose adherents considers their baby beautiful. Adding to beauty the not ion of selectness results in a more usefully stringent guiding principle. It is this eombination of qualities that so excites those theorists interested in applying the division algebras to physics. These algebras are eertainly select. There are only four real normed division algebras: the reals, R (dim=l); the complexes, C (dim=2); the quaternions, Q (dim=4) (often denoted H, for Hamilton, their discoverer); and the oetonions, (dim=8) (sometimes called the Cayley algebra). And there the sequence ends. There are at least two ways to extend the sequence (the so-called Cayley-Dickson prescription is one, and I outline a more natural prescription in ehapter 2), but none can result in more division algebras. There are only four. As it turns out, in being division algebras they are also extremely generative. The Lie algebras so( n; K) of nxn traceless, antihermitian matrices over the division algebra K, are just the sequenees so( n) (K =R), su( n) (K =C), and sp(n) (K = Q). (0 is nonassociative, so it can not generate a Lie algebra in this way.) The Lie algebras of the five exceptional groups, G 2 , F4 , Ek, k=6,7,8, are all associated with the oetonions (see ehapters 2, 7 and 8). The three parallelizable spheres, Si, i=1,3,7, arise from C, Q, and 0, and the sphere fibrations, Sk ~ S2k +l ~ Sk+ 1 , for k=O,1,3,7, are intimately

°

UNDERPINNINGS

7

°

linked to R, C, Q, and (see chapter 7). The list goes on. In physics most attention has been paid to their link to quantum mechanics, this interest principally spawned by the work of von Neumann and Jordan. Quantum meehanics must rest on a division algebra (C is the usual choiee). In addition, the algebraie underpinnings of modern particle/field theory ean be shown to arise from the division algebras. The Pauli algebra is isomorphie to C 0 Q (whieh is just the complexification of Q, or the quaternionization of C). Via exponentiation the imaginary element of C generates U(l); those of Q generate SU(2); and the automorphism group of 0, G 2 , has an SU(3) subgroup that is the stability group of a fixed imaginary direction. U(1) X SU(2) X SU(3) is, of course, the famous standard symmetry of quark and lepton theory. This brings us to the final Screed, whieh will serve as a segue to wh at follows: SCREEDIII Lest too frequent repetition should blunt the foree of the rant, herewith the deed. The real normed division algebras are the reals, complexes, quaternions, and octonions (R, C, Q, 0). There are no others. For this reason there are no other classes of classieal Lie groups than orthogonal, unitary, symplectie, and exceptional, for they arise out of the division algebras. The Clifford algebras of the real pseudo-orthogonal geometries are all isomorphie to matrix algebras over R, C, and Q (if we use the left-adjoint algebras, RL = R, CL = C, QL = Q, OL = R(8), then the octonions can be included as weH). Many (most?) of the special structures of analysis and topology are associated with the division algebras. The parallelizable spheres, for example, SI, S3, S7, arise direetly from exponentiation of the hypercomplex elements of C, Q, and 0, and they are parallelizable because C, Q, and 0 are division algebras. In mathematics the division algebras are fundamental and generative. Accepting that there is a subtle conjoining of mathematies and physies at depths beyond our present understanding, it would be impossibly strange did not the division algebras surface on the physies end of the pool equally fundamental and generative. Given that R®C0Q = RL0CL0QL (tensor product ) [Note: R is unnecessary in these tensor products over the reals. It was added only for emphasis.] is just the Pauli algebra in another guise; that RL ® CL 0 QL 0 OL (isomorphie to C(16)) is to the increasingly important

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(1,9)-geometry what the Pauli algebra is to the (1,3)-geometry, and that the individual groups of the standard symmetry, U(1), SU(2), SU(3), have direct links to C, Q, and 0, it would require an effort of will to deny it is so. But an even stronger case has been made. Let T = R ® C ® Q ® 0, TL RL ® CL ® QL ® 0L. T is the object or spinor space of TL. T is 64dimensional, so the T inner product has an SO(64) symmetry. This is not gaugable, but the mechanism of the inner product has an U(1) x SU(2) x SU(3) symmetry that iso With respect to this symmetry the elements of T transform precisely, down to the quantum numbers, as the direct sum of a family and antifamily of quark and lepton Pauli spinors. Moreover, this can happen in two ways, one precisely like left-chiral matter, right-chiral antirnatter, and the other like right-chiral matter, left-chiral antirnatter. So the mathematics accounts for the observed violation of parity. Moreover, spontaneous symmetry breaking can break U(1) x SU(2), while SU(3) is necessarily exact. This is a mathematical result. Moreover, a phase arises from the mathematics that weak mixes the quarks, not the leptons, mathematics again in agreement with observation. The spinors naturally link to (1,9)-space-time, in that TL is the "Pauli" algebra of this geometry. With respect to this geometry leptons and quarks, even matter and antirnatter, are indistinguishable. With respect to the derived U(1) x SU(2) x SU(3) symmetry the spinors ofTL (ie., T itself) split into matter and antirnatter, and into the individual quark-lepton, antiquarkantilepton parts. The (anti)matter projector of the algebra reduces (1,9) to (1,3) (one for matter, one for antirnatter). All of the algebraic features of the standard model are derived from T and its adjoints (internal SU(2) is actually associated with QR, the right adjoint algebra of Qj the right adjoint algebra of is the same as the left), including quarks, leptons, the standard symmetry, parity nonconservation, symmetry breaking, quantum numbers, family structure, family replication, and weak mixing. These are associated via the identificatioll of the component spinors as of Pauli type with the appropriate (1,3)-space-time. And going beyond the experimentally confirmed, T and TL give rise to a link to a (1,9)-spacetime, and that link is precisely determined. To dismiss the exact correspondence of T-maths to observed reality as accidental or insignificant is wrong. Theories of which the standard model is a desired part but that are not based on T-maths are wrong. Extended theories, like string theories, that require a (1,9)space-time, but link it to the standard model in a way different from that derived in T-maths, are wrong.

=

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They are wrong in the same way theories of the electron not based upon the Pauli and Dirac algebras and their spinors are wrong. Uruk lay back with a sigh. End of Screeds.

The words at the end of this Screed were written with generous strength, but I wished to avoid the mistake of the fellow who was so afraid of making a bad impression, he made no impression at all. In particular I wish now to avoid leaving the reader with the idea that I consider the division algebras to be a mathematical method, interchangeable with other methods, like the theories of matrix Clifford algebras and associated spinors, and Lie groups and associated multiplets and representations, by means of which the algebraic features of the standard model may be easily represented. Such a representation is sewn together by arbitrariness. There are infinitely many Clifford algebras and Lie groups, and having picked from these, there are infinitely many multiplets and representations. And when one has chosen a phenomenologically suitable collection, one has shot one's bolt. Laying on the methods of quantum field theory can give rise, of course, to many testable results, but in a broader sense the theory is a elosed context with few or no pointers to deeper Truth. Even worse, like many scientific paradigms, this one is an ideological cage. It is filled with an infinite number of blocks with which to play, and amidst these blocks have been discovered some golden nuggets, but the rush that inevitably resulted from this discovery stranded a generation amidst boulders of basalt. The division algebras not only represent the standard model, they come very elose to deriving it. All the multiplets and algebra representations, even the allowable partiele transitions, are fixed by the mathematics. What I shall present in this monograph is a development of this mathematics. The algebraic features of the standard model drop out en route. This is not an accidentj this is a layer of Truth, and a doorway to deeper layers. Finally, the physical consequences of the mathematics do not stop with the standard model. As I shall demonstrate, this theory beautifully connects the standard model to (1,9)-space-time. But waiting in the wings are all the little explored connections to those special topological and analytical structures associated with the division algebras. This is a seed waiting to germinatej a vast cavern waiting to be explored. Indeed, I do not wish to give the impression that I believe the division algebras to be a universal cure

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for all that aHs uso Our symbolic mathematics is a tool, the microscope with which we attempt to see wh at we can not directly touch, and from which we are in all likelihood precluded from touching. But the Truth of Reality gives every indication of being accreted about the most select and beautiful of mathematical ideas, and the division algebras are just apart of the web of these. This monograph begins in ehapter 2 with a development of general division algebra theory with the focus on the octonions. The eyclic multiplieation tables I employ are not new, but they are also not eommon in physical applications. They make ealculations eonsiderably easier than the eonventional multiplieation tables, whieh rely on the Cayley-Diekson proeess, which I will not cover. In section 2.6 I present a mathematical motivation for the approach which I hope will be viewed as compelling. Heavily stressed throughout this monograph is the notion that with respeet to the left, right, or two-sided adjoint actions of a division algebra on itself, the adjoint algebras behave like Clifford algebras, and the division algebras themselves, which are the objeets of the adjoint actions, behave like spinor spaees. This idea is not new, but I am unaware of any treatment of the matter that makes the relation as obvious as that developed here. In chapter 3 tensor products of the division algebras and their adjoints are considered, and inner products on the tensored division algebras (the spinor spaees) are developed. In the ease of the tensor product of C, Q, and 0, there is associated with this inner produet a symmetry from which the standard symmetry of elementary particle theory is drawn. In chapters 4 and 5 the mathematies developed in ehapters 2 and 3 is applied to physics in the obvious way, as outlined already in this section. This includes identifying the quark and lepton fields, developing gauge fields from the internal symmetry, weak mixing, and the identification of the Higgs fields, whose action on the fermion fields is heavily restricted by the mathematies. These are used to spontaneously break the symmetry. In chapter 6 it is shown that this entire theory can be seen to result from a theory of fermions based on a lO-dimensional space. With respect to this lO-dimensional space quarks and leptons, even matter and antirnatter, are subsumed into one big hyperfield in which all are indistinguishable. Only by restricting the dependence on the extra six dimensions of the various identifiable quark and lepton bits of the hyperfield (these extra dimensions carry SU(3) eh arges ) does the theory dimensionally reduce to a more familiar form. The further eonsequenees of this lO-dimensional theory are in the

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developmental stage. At the end of chapter 6 I take a brief look at the link of the division algebras to string theory, especially focussing on applications of the octonions to string theory on 10-dimensional spacetime. In chapter 7 some of the mathematical ideas used in the construction of the physical application of chapters 3 through 6 are applied to other problems, in particular to the parallelizable spheres, the Moufang identities, and 80(8) triality. Chapter 8 continues the furt her application of the mathematics, beginning with an unconventional development ofthe Lie algebras ofthe FreudenthalTits Magic Square. The commutation relations of one of these algebras, the exceptional F4 , are explicitly developed as an example. Several appendices are inc1uded, the use of which can make algebraic calculations easier. In the remainder of this chapter some background material is developed, beginning with a review of Clifford algebra theory.

1.2. Clifford Aigebras. To any m ::; n linearly independent vectors in an n-dimensional Euclidean space there corresponds an antisymmetric m-tensor (m-form) representing the oriented m-volume of the m-parallelopiped determined by the vectors. This m-form can be constructed via an antisymmetrie "outer" product of the vectors. Of equal importance is the symmetrie "inner" product of vectors from which a scalar norm can be constructed. Clifford algebras are associative algebras with identity that embody these outer and inner products in an elegant algebraic formalism [1]. Via the Dirac algebra, the foundation of the Dirac equation, and also the complexified Clifford algebra of (1,3 )-spacetime (with pseudo-orthogonal metric signature {+ - - - }), the importance of Clifford algebras to physics has been cemented and solidified. The (mtiversal) Clifford algebra of (p,q)-spacetime (RP,q), denoted Rp,q, is generated by n = p + q linearly independent elements, T GO Q = 1, ... , n, satisfying

(1.1)

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where TJCiß = 0, a

f:. ß,

TJCiCi = 1, 1 :::; a :::; p, TJCiCi = -1, p

+ 1 :::; a :::; p + q

(E = identity). Therefore, if x = Ea xai Ci and y = Ea yai Ci are algebraic representations of RP,q vectors in Rp,q, then (1.2)

where ECiß XCiTJCißyß is the Rp,q inner product ofthe vectors with components {XCi} and {yCi}. The elements i Ci are a basis for the 1-vector subset of Rp,q (when p=l the values of the index a will be taken from {O, 1, ... , n - I}). The 2-vector subset is spanned by the (

~)

=

2!(:~2)! distinct (1.3)

If x and y are defined as above, then ~(xy-yx)

~ ECiß(xCiyßiCiiß - yßxCiißi a ) ECiß xCiyßi Ciß ECiß ~(XCiyß - yCixß)iCiß

(1.4)

where !(xCiyß - yCiXß) in the (p = n, q = 0) or (p = 0, q = n) case are the components of the oriented 2-form representing the area (2-volume) of the parallelogram determined by the two vectors. Likewise, a basis for the space of general m- vectors consists of the ( ;

)

distinct elements of the form (1.5) where al, a2, ... , a m , are distinct (in which case this element is antisymmetric in the indices). A "universal" Clifford algebra satisfies (1.6)

UNDERPINNINGS

13

This element is called the pseudoscalar of the Clifford algebra. Unless otherwise specified the term Clifford algebra will mean universal Clifford algebra. Only certain (p,q)-spacetimes admit non universal Clifford algebras. Some of these will appear in chapter 8. Universal Clifford algebras are 2 n _ dimensional, while the nonuniversal are 2n - 1 -dimensional. The subspace of 2-ve.ctors of Rp,q is easily shown to elose under commutation, and in this case it is isomorphie to the Lie algebra so(p, q). In particular, the set of all U = exp«(JaßYaß) form a Lie group under multiplication. If x

= xcvy a , then the action (1. 7)

results in an SO(p, q) rotation on th~ components {XCV} of x, and this rotation leaves the bilinear form 1.2 invariant. This same element U, if acting on the spinor space of Rp,q (a column vector space upon which a matrix representation of Rp,q naturally acts; the concept is further developed on the following pages) is an element of Spin(p, q), which defines what this means. Spin(p, q) is a double covering of SO(p,q). There is a well- known table of isomorphisms between the Clifford algebras Rp,q and matrix algebras over R, C, or Q. 1'11 present the table in a slightly unconventional way, splitting it into two tables, one for p + q = n even, and one for p + q = n odd.

14

CHAPTER 1 Clifford Algebras: n even n=

0

2

4

6

10

8

p-q

R(32)

10

R(I6)

R(32)

Q(4)

Q(8)

Q(I6)

Q(2)

Q(4)

Q(8)

Q(16)

R(2)

R(4)

R(8)

R(I6)' R(32)

R(2)

R(4)

R(8)

R(I6)

R(32)

Q

Q(2)

Q(4)

Q(8)

Q(16)

Q(2)

Q(4)

Q(8)

Q(16)

R(8)

R(i6)

R(32)

R(i6)

R(32)

8 6

4 2 0 -2 -4 -6

-8 -10

R

Q(16)

There are some obvious patterns. In particular note the

... RRQQRRQQ ... pattern of the columns, and along the constant p and constant q diagonals in the n even table.

UNDERPINNINGS

15

Clifford Algebras: n odd n=

1

3

5

11

9

7

p-q C(32)

11

2R(16)

2R(32)

C(8)

C(16)

C(32)

2Q(2)

2Q(4)

2Q(8)

2Q(16)

C(2)

C(4)

C(8)

C(16)

C(32)

9 7

5 3 1

2R

2R(2)

2R(4)

2R(8)

2R(16)

2R(32)

-1

C

C(2)

C(4)

C(8)

C(16)

C(32)

2Q

2Q(2)

2Q(4)

2Q(8)

2Q(16)

C(4)

C(8)

C(16)

C(32)

2R(8)

2R(16)

2R(32)

C(16)

C(32)

-3 -5 -7

-9

2Q(16)

-11

The pattern across the rows is clear in both tables. In the second table, for example, 2R(2) is the algebra of matrices of the form

[~ ~ l, where X, Y E R(2), and 0 is the 2 X 2 zero matrix.

(1.8)

CHAPTER 1

16

CLIFFORD ALGEBRA

R3,1

Of particular interest to physicists, and to the application to be developed in chapters 3 to 6, are the spacetimes R 1,3 and R 3 ,1, and R 1,9 and R 9 ,1. Consider first the Clifford algebra

R 3 ,1

~

R(4).

(1.9)

We can represent the generators by the matrices

[1. . ] .

1

-1

. . -1

[: -1 -1

[,,1,]

l' _ . '

1 .'

1 . 1

2 -

1T, [~1 -1 ; 1

(1.10)

=

In this representation .

l' 1234

=

. [ 1

.

-1

.

.]

1 .

'

(1.11)

-1

The elements of R 3 ,1 can be viewed as endomorphisms of the real vector space of 4 X 1 matrices, which are in this context called the spinors of the Clifford algebra. Nature, however, seems unimpressed with this scheme, prefering instead the complexification of the Clifford algebra and its spinors. In particular, the Dirac algebra, upon which the physlcS of fermions is based, is the complexification of R 3 ,1, and its spinors are 4 x lover C (Dirac spinors), instead of 4 x 1 over R (Majorana spinors). The addition of the complex unit i allows the construction of the following more interesting and physically more relevant representation of the generators of R3,1, which we

UNDERPINNINGS

17

now denote by the more conventional lcx , a

11

~

[;

1 1

1~ 12

= 1,2,3,4:

[; -i ;

13 ~ [i -I ~I], 1. ~ [-I -I 1

~1 1

J

(1.12)

where now all the 10 are 2 X 2 block off-diagonal, and the laß, which generate the Lorentz group SO(3, 1), are block diagonal, as is the pseudoscalar

(1.13)

Prom this we define

IS = h1234

=

[ ~. i -·1 -:.1].

(1.14)

Cis anticommutes with 10' a = 1,2,3,4, and it is not an element of R 3 ,t, the real Clifford algebra. Hence the set {'Yl, 12, 13, 14, IS} can be viewed as a basis for the 1-vector subspace of R 4 ,1 ~ C( 4).) Nature employs IS in the

following way. Let 'P

'Pt

~ ~(I

h)'P

~ ~ [

] be a Dirac spinor, and define

~ T], [

and 'P,

~ ~(1-1S)'P ~ i: ]. [

( 1.15)

These are the so-called left and right (chiral) Weyl spinors of which the Dirac ± IS) are the chiral projection operators. spinor tp is composed, and the

!(1

CHAPTER 1

18

Note that because the generators of the Lorentz group are block diagonal, Lorentz rotations map the spaces of left/right Weyl spinors to themselves. Note also that without the complex unit i, is not constructable. Weyl spinors, instead of Dirac spinors, are the indecomposible bits upon whieh fermion physies is based. For example, the left and right chiral parts of the electron's Dirac spinor interact differently with the gauge fields arising from the electroweak gauge symmetry, U(l) x SU(2). In partieular, SU(2) gauge fields arise as though the lefthanded electron and its associated lefthanded neutrino are an SU(2) doublet, while the righthanded electron interacts as though a singlet. Its associated righthanded neutrino is often assumed not to exist at all. So although mathematieally the block off-diagonal representation of the ,-matriees in 1.12 is not singled out, a representation of this sort is singled out by the physics. Because the left and right chiral projections of spinor fields corresponding to real fermionic particles interact differently with some gauge bosons, nature requires the complexification of the R 3 ,1 Clifford algebra, for without the complex unit i, the chiral projectors ~(1 + /5) can not be constructed.

,5

PAULI ALGEBRA

More interestingly, we see that the Dirac algebra is not so much 4 x 4 over C as.it is 2 X 2 over P=Pauli algebra; that P more properly constitutes the parts of wruch the Dirac algebra is composed. The Pauli algebra (1.16) and its basis of generating I-vectors is conventionally represented by (1.17) Let (1.18) then " J = [ aj 0

aoj

,] = 1,2, 3,'4 = [0 -ao

].

a0o ] .

(1.19)

The linear subspace ofP with basis {ao, iaj,j = 1,2,3} is a subalgebra of P isomorphie to the quaternion algebra Q (ao -+ 1, iaj -+ -qj,j

= 1,2,3.)

19

UNDERPINNINGS Hence,

(1.20)

CLIFFORD ALGEBRA R 1 ,3 The Clifford algebra R 1 ,3 basis by

o

qj

~

Q(2), and we may represent the 1-vector

l',)=1, 2,3,

~4=

'V'

[10 -10 1'

(1.21 )

so that (1.22) This algebra can be viewed as an algebra of endomorphisms on the space of 2 X 1 matrices over Q, which is an unconventional spinor space for R 1,3, being neither real nor complex. It is interesting in that the algebra of left actions of Q(2) on this spinor space is incomplete, and is only completed by induding right actions of Q on the space of column spinors. This algebra of right actions commutes with the larger algebra of left actions, and is in that sense "internaI" with respect to the algebra of left actions, which, in representing the Clifford algebra of the relevant Minkowski geometry, are considered "external" . In that the set of unit quaternions is the Lie group SU(2), which in the context just outlined could be viewed as an internal symmetry, it is curious that this representation of R 1 ,3 has never been empIoyed (although something similar will be empIoyed in this monograph). In complexifying R 1 ,3 we are again able to define a chiral projection operator. In particular redefine in this new context, 10 =

[01 01]

"j =

[0

iqj

-iqj

0

1,).= 1,2,3,

(1.23)

where now we use the index 0 to denote the time direction. Given this basis, (1.24)

CHAPTER 1

20 and from this we define /5

. = [10 = Z/0123

01.

(1.25)

-1

As before, /5 together with the /P.' J.l = 0,1,2,3, form a 1-vector basis for a larger Clifford algebra, in this case R 2 ,3. This is an unconventional representation still, involving as it does the quaternions. The more conventional representation is /0

= [ 0"00 0'0] 0 ,/j = [0 -O'j

O'j 0

l'

3

,} = 1,2, .

(1.26)

This has the conventionally perceived advantage that its spinor space is the space of 4 x 1 complex column matriees, while the spinor space of the representation 1.23 is the space of 2 x 1 column matrices over C®Q. That the representation 1.23 has an associated inherent intern al symmetry U(1) x 8U(2) is largely unrecognized, but a detailed study ofthis will follow in subsequent chapters. CLIFFORD ALGEBRA R 1 ,9

Both R 1 ,9 and Rg,l are isomorphie to R(32), and they can be similarly analysed. We will develop a representation of R 1 ,9. Let

(=

[~ ~], a = [~ ~1 1'

ß=

[~ ~ 1,w = [~1 ~ 1

(1.27)

be a basis for R(2). Define, for example, in R(4), (1.28)

the tensor product of ß and a, where the first matrix, ß, determines the block structure, and the second, a, what goes in the blocks. Therefore, (1.29)

UNDERPINNINGS

21

This easily generalizes to tensor products of arbitrary length, and in this manner we can construct anyelement of any R(2 k ). In particular, define in R(32), 10 = ß.f.f.f.f

11

= W.a.f.f.f

12 = w·ß·ß·ß.ß 13

= w·ß·a.a.a

14 = w·ß·f..a.ß 15

(1.30)

= w·ß·ß·f.a

16 = w·ß·a·ß·f 17

= w.w.w.a.ß

18 = w.w.ß·w.a. 19 = w.w.a·ß.w where for the sake of simplicity I've replaced the symbol ® with a dot. These 10 elements of R(32) satisfy 1 "2(1k 11 + 111k) = "lklf32, (1.31) where {"lkl} = diag( + - - - - - - - --), and matrix. Furthermore, the pseudoscalar

101 1 ••. 1 9

= ±a.f.f.f.f = ± [

f32

f16 0

is the 32

0 -f.16

1

x 32 identity (1.32)

(I leave it to the reader to work out the sign). Clearly then the 1k, k=0, ... ,9, are I-vector generators of the Clifford algebra R 1 ,9' (Note: Rg,l is easily constructable by changing the lead ß in 10 to w, and the lead w's in 1k, k=I, ... ,9, to ß's.) Note that in this case the chiral projectors

1(

"2

f.f.f.f.f ± a.f.f..f.f.)

=

[f 0] 016 0

and

(1.33)

22

CHAPTER 1

are elements of the Clifford algebra R 1 ,9. Therefore it is not necessary to complexify the Clifford algebra and its spinors to project Weyl spinors. So R 1 ,9 spinors can be simultaneously Majorana and Weyl. In the present representation a Majorana spinor would be a 32 X 1 real column, and a MajoranajWeyl spinor would be all zero in the first or last 16 positions.

1.3. Conjugations and Spinors. There is a pair of natural (product reversing) conjugations on any Clifford algebra [1] determined by their action on the 1-vector generators:

(1.34) and

(1.35) In both cases, because these maps are antiautomorphic,

(1.36) In general,

= +x = -x XII = -x XII = +x

1 - vector 2 - vector 3 - vector 4 - vector

XII

XV

XII

XV

5 - vector 6 - vector 7 - vector 8 - vector

XII

XV

XII

xV

9 - vector

XII

XV

10 - vector

XII

XV

11 - vector 12 - vector

= +x = -x XII = -x XII

= +x

= +x = -x XII = -x XII = +x

XV XV

XV XV

XV XV

=-x =-x = +x = +x =-x =-x = +x = +x

(1.37)

=-x =-x = +x = +x

In certain cases these conjugations correspond to ordinary Hermitian

UNDERPINNINGS

23

conjugation, denotedx t:

x"

= xt :

xE Rn,a,

n

= 1,2,3, ... ;

Xv

= xt .

xE R a,n,

n = 1,2,3, ....

( 1.38)

In all other cases the conjugations are more difficult to define. In the representations 1.19 of R 3 ,1, and 1.23,26 of R 1 ,3, these conjugations are defined the same for each. Given x, y, u, v E C(2) (for the 1.19 and 1.26 cases), or (equivalently) x,y,u,v E C0Q (for the 1.23 case), we define

(1.39) and

(1.40)

These conjugations are generalizable to within a sign or phase to the spinors of the complexifications of R 3 ,1, and R 1 ,3. In particular, let

and

w= [ ~~

1E c4,

a Dirac spinor for the Dirac algebra, C 0 R 1 ,3, as represented in 1.26. Then we define

(1.41) and

WV = [

~:

r

= [#

-1/Jt], (WV)V

= -Wo

(1.42)

The former I shall call the spinor conjugate associated with the Clifford algebra. Note that in both cases,

(1.43)

CHAPTER 1

24

(1.44)

Also note that in the Dirac algebra case lJIA is usually denoted

q, ,and (1.45)

NILPOTENT CLIFFORD ALGEBRAS The reader is warned that what follows is an unconventional approach to Clifford algebra theory and to the best of my knowledge it has not found its way into the mainstream literat ure on physieal model building [2][3][4]. It will also play no role in what folIows, although the attentive reader will see how it might. Define (1.46) This is wh at I refer to as the nilpotent Clifford algebra of the pseudoorthogonal space RP,q. Note that unlike the ordinary Clifford algebra case, where Rp,q and Rq,p are seldom isomorphie, Np,q is always isomorphie to Nq,p' For example, (1.47) N 1 ,3 ~ N 3 ,l ~ C(4), the Dirac algebra for (1,3)-spacetime. Let (1.48) be a 1-vector basis for Rp+1,q, where n = p + q. The Y':VI a assumed to satisfy 1.1, while

= 1, ... , n are

y2 = 1, YY cx + Y cx Y = O.

(1.49)

(Note that the collection of 1-vectors 1.48, given the commutator product, generates the conformal Lie algebra so(p + 1, q + 1).) Define (1.50) whieh are orthogonal projection operators (Jl±Jl T = 0), and define (1.51 )

UNDERPINNINGS

25

Because of 1.49 these elements are nilpotent. In fact, for all

T;T~

(x,

ß E {I, ... , n},

= P-± TO/JL± Tß = JL±JL~ TO/ Tß = O.

(In the (1,3)-case, p-± are the chiral projectors, so nilpotent Clifford algebras have chirality buHt in.) Also, if (x, ß, v E {I, ... , n}, then

T;T;T~+l~l;T;

=

JL±(TO/lvTß+TßTvTO/) 2T;1Jvß

+ 2T~1JvO/ -

21~1JO/ß'

We shall write these results in the following ways: (1.52) and

(1.53)

SYMPLECTIC NILPOTENT CLIFFORD ALGEBRA The equations 1.52 and 1.53 are sufficient to completely determine Np,q' They are presented in what must seem a peculiar way to allow for a generalization to the symplectic (or alternating) case, as opposed to the pseudoorthogonal. In particular, replace T; by i = 1,2, so n = 2, and replace 1JO/ß by Eij, such that E12 = -E21 = 1; EU = E22 = O.

S;,

In this case the equations 1.52,53 reduce to (1.54) and

st S; st + st S; st = 2EijSt + 2EkjSt, (1.55)

s; sj Sk + Sk sj Si- = 2S; Ejk + 2SkEj;.

CHAPTER 1

26

Therefore,

st s:; st = 2St; SI Si SI = -2S1 ;

(1.56)

st s:; Si + si s:; st = 2Si; SI Si s:; + s:; si SI = -2S:;;

Using the equations 1.54,56 one can show that the algebra generated by the is isomorphie to the 18-dimensional matrix algebra 2R(3). We can represent the generators as

Sr

S 2+ --

S1+ --

.l [j

G.l

. . 1

. .

.

. . -1

..

[J.. '

.

. . .

-1

w·· . . .

1 .

1 1 .

(1.57)

SI =

[J.. .

. .

-1

1 .

U·1 . . -1

..

w·· . ..

1 1 .

'

U·..

.

..

1 1

.

In the pseudo-orthogonal case, the elements la = lt + l~ are ordinary Clifford algebra I-vectors, and the set of all commutators

1

laß

= "2[la, lßl

UNDERPINNINGS

27

form a basis for so(p, q), which generates the invariance group ofthe bilinear form associated with 17Ciß. In the symplectie case we define

ISi = st+S;,

i=

1,2,1

(1.58)

and we replace commutators with anticommutators:

(no entry, or a dot entry, signifies a zero component). These matriees constitute a basis for sp(2), whieh generates the invariance group of the alternating bilinear form associated with (ij. Together with the Si, i = 1,2, and given the appropriate mix of commutator and anticommutator products, these five elements constitute a basis for the Lie superalgebra osp(1, 2) [5]. Clearly this approach to the idea of geometrie algebra is rieher than the conventional one. To this point, however, little has been done to apply these symplectie nilpotent Clifford algebras, nor is much known about their general mathematical properties. They are presented here just as an idea.

28

CHAPTER 1

A final word about Clifford algebras in general: they are, like Lie groups, profligate. There are too many ofthem, an infinite number ofboth Lie groups and Clifford algebras that are physieally irrelevant, not apart of the design of reality. This is and always was the problem with GUT theories based on a unifying large Lie group, and it is and always will be the problem with unification theories based on large Clifford algebras .. In both cases it is the principal of the educated guess that leads to the choiee of unifying algebraic object. This is unsatisfactory. Nature can not be so aruitrarily ugly. The first three division algebras, R, C, and Q, are clearly intimately linked to Clifford algebras in general, and it will be demonstrated in chapter 2 that the octonions, despite their nonassociativity, give rise to Clifford algebras as weH (the algebra of left adjoint actions of 0 on 0 is isomorphie to the associative algebra R(8), so it could be incorporated into the Clifford algebra tables listed in this section (for that matter, in the isomorphisms listed that involve Q it should be specified that it is Q's left adjoint algebra that is being used, for it is this that acts on the column spinor space)). As will be pointed out in chapter 6 the sequence of division algebras R, C, Q, and o is linked to the sequence of Clifford algebras of spacetimes of dimension 3,4,6, and 10, critieal dimensions for classieal string theory. This monograph, however, is not about string theory, and it is not this link of the division algebras to the critieal dimensions of string theory that gave birth to my interest in these algebras. Instead it was the isomorphism P ~ C ® Q, linking the Pauli algebra of 4-dimensional spacetime to the tensor product of the first two hypercomplex division algebras, and the fact that the use of C ® Q instead of P in building the Dirac algebra leads automatieally to the consideration of a U(l) x SU(2) internat symmetry. Tensoring C ® Q with 0 connects to 10-dimensional spacetime in the same way, and leads to the inclusion of an SU(3) internal symmetry and the reduction of 10dimensional spacetime to 4-dimensional [6]. In my opinion these are the cleanest and clearest links of the division algebras to the design of physieal reality. This is not to say that the string theory links [7][8], or, for that matter, the quantum mechanieallinks via Jordan algebras [9], are wrong or irrelevant, just that their ultimate relevance has not been demonstrated. It is my personal hope, in that these different approaches to application are not inherently incompatible, that eventually they may combine, and their union be more universally unifying than any is individuaHy.

UNDERPINNINGS

29

1.4. Aigebraic Fundamentals ofthe Standard Model. Our observable spacetime has 1 time and 3 space dimensions. However, instead of housing physical fermion fields in the spinor spaces of either the Clifford algebra R 1 ,3 or R 3 ,b Nature requires a complexification ofthe Clifford algebra, C ® R 1,3 ~ C ® R3,1 ~ C(4), and She exploits Dirac spinor fields involving 4 complex components. However, these are not the fundamental spinors of physics, for it is observed that the two Weyl spinors of which a Dirac spinor is composed behave differently in the real world. Each fundamental fermion (leptons and quarks) is representable by a pair of complex 2 component Weyl spinor fields (left and right Chiral projections). [Righthanded neutrino fields are thought by some not to exist in Nature, as they evidently can have no interactions save for gravitational. However, no evidence for their absence exists.] The internal gauge symmetry from which vector boson gauge fields arise has been discovered to be U(1)

X

SU(2) x SU(3),

the so-called standard symmetry. Lepton and quark fields interact with each other via these 1 + 3 +8 = 12 gauge fields. The nature of each fermion field is determined by its U(1) X SU(2) X SU(3) charges, which in turn determine how the fermions will interact with the gauge bosons of the theory. The standard symmetry is put into the theory by hand, as are the various leptoquark charges. Lepton and quark fields are observed to fall into either SU(2) singlets or SU(2) doublets. Lefthanded Weyl spinor fields fall into doublets; righthanded Weyl spinor fields fall into singlets. Hence SU(2) is said to be chiral; it violates parjty. Leptons fall into SU(3) singlets, and quarks into SU(3) triplets. This js independent of handedness, so SU(3) js said to be nonchiral; it conserves parity.

30

CHAPTER 1

The leptoquark multiplet structure associated with SU(2) and SU(3) is put into the theory by hand, as is the chiral nature of SU(2). An exact symmetry will mix the fields of a multiplet while preserving masses. Not all of the groups of the standard symmetry are observed to be exact. SU(3) is exact (hence its gauge fields, the gluons, are massless), as is a U(1) C U(1) X SU(2) giving rise to the massless photon, the gauge field of electromagnetism. The rest of U(1) X SU(2) is broken, and the corresponding gauge fields (W±, ZO) are massive. Scalar fields are introduced that are designed to break most of U(1) X SU(2), but leave a U(1) x SU(3) symmetry exact. This process, called spontaneous symmetry breaking, accounts for the mass differences observed between the charge (-1) and charge (0) leptons, and between the charge (-1/3) and (2/3) quarks. Because SU(3) is exact, the three "colors" of quarks in an SU(3) triplet have the same mass. Finally, this pattern of fermion multiplets, which accounts at lowest level for the electron and its neutrino, the up-quark and down-quark (3 colors eachj a total of 2(1 + 3) = 8 fermion fields called a family), is replicated at least twice more. Each of these additional families is distinguished from the first, and from each other, only in the masses of its fields. For example, each of the masses of the particles in the second family, the muon and its neutrino, the strange-quark and charm-quark, is more massive than the corresponding fields of the electron family. The explication of these algebraic features via the mathematics of the division algebras is a central goal of the first half of this monograph.

2. Division Algebras Alone. 2.1. Mostly Octonions. The octonion algebra, 0, is generally developed as an extension of the quaternion algebra, Q. Let % i=1,2,3, be a conventional basis for the hypercomplex quaternions. These elements associate, anticommute, and satisfy = -1. Together with the cyclic multiplication rule

q;

(2.1) i=1, ... ,3, indices modulo 3, from 1 to 3, this completely determines the multiplication table for Q.

Relabel these quaternion units ei, i=1,2,3, and introduce a new unit, e7, anticommuting with each of the ei, which satisfies e~ = -1. Define three more units: (2.2) Let 0 be the real algebra (with identity) generated from the ea a=1, ... ,7, such that {ql ~ ea , q2 ~ eb, q3 ~ ec } defines an injection of Q into 0 for

(a,b,c)= (1,2,3),(1,7,4),(2, 7,5),(3, 7,6),(1,6,5),(2,4,6),(3,5,4). Therefore, for example,

31

32

CHAPTER 2

So unlike the complexes and quaternions, the octonions are nonassociative. In general, if ea, eb, and ec generate a quaternion subalgebra of 0, then they associate, and if not, then they do not. Like C and Q, however, 0 is a division algebra, and it is normed. In particular, if x = x O+ xae a, (sum a=1, ... ,7), and

(an antiau tomorphism), then (2.3)

defines the square of the norm of x (so

This octonion multiplication is not, how,ever, the most natural (see section 2.6), and it will not be employed here. Again let ea , a = 1, ... , 7, represent the hypercomplex units of 0, but now adopt the cyclic multiplication rule (2.4) a=1, ... ,7, all indices modulo 7, from 1 to 7 . In particular,

define injections of Q into 0 for a=1, ... ,7. I am accustomed to using the symbol eo to represent unity, and I bot her to remember that although 7 = 0 mod 7, e7 ::I eo, and in the multiplication rule 2.4 the indices range from 1 to 7, and the index 0 is not subject to the rule. (In [10] 00 is used as the index for unity, and this has advantages, which I find intermittently persuasive.) [Note: Another cyclic multiplication rule for 0, dual to that above in a way outlined in section 2.6, is (2.5)

This one is more commonly used than 2.4. Indeed, of the two cyclic multiplication rules 2.4 and 2.5 I may be the only one who uses 2.4. None of the mathematical or theoretical development of this monograph depends on the

DIVISION ALGEBRAS ALONE

33

choice, and both are pretty in the same way. Using 2.5 the three octonion units {eI, e2, e4} (aH subscripts powers of 2) associate: el(e2e4) = (ele2)e4 = -1.

In this case they generate a Q subalgebra. Using the rule 2.4 these same units do not associate: el(e2e4)

= -(ele2)e4 = e7.

In this case they generate the whole of O. I thought that a nice property, and on that questionable basis I made my choice.] Both the octonion multiplication rules 2.4 and 2.5 have some very nice properties. For example, 1 if

eaeb

= ec,

then e(2a)e(2b) = e(2c) ·1

(2.6)

2.6 in combination with 2.4 immediately implies

=

eaea+2 ea+3, eaea+4 = ea+6

(2.7)

(so or eaea+b = [b 3 mod 7]ea-2b4, b = 1, ... ,6,

where b3 out front provides the sign of the product (modulo 7, 13 = 23 = 43 = 1, and 33 = 53 = 63 = -1 )). Also, 2(7)=7 mod 7, so 2.6 and 2.4 imply

(2.8) These modulo 7 periodicity properties are reflected in the fuH multiplication table:

1 el e2 e3 e4 es e6 e7 e6 e7 -e2 -es el -1 e4 -e3 e7 es -e4 e2 -e6 -1 el -e3 el e6 -es e2 e3 -e4 -e7 -1 e7 -e6 e4 e3 -es -eI -1 e2 e4 -e6 -e2 -1 e3 es -e7 el e6 e2 -eI es -e7 -e3 -1 e4 es e3 -e2 e6 -eI -e4 -1 e7

(2.9)

CHAPTER 2

34

The naturalness of this 2.9: 1 1 1 1 0= 1 1 1 1

table is reflected in the matrix of signs derived from 1 -1 -1 -1 1 -1 1 1

1 1 -1 -1 -1 1 -1 1

1 1 1 -1 -1 -1 1 -1

1 -1 1 1 -1 -1 -1 1

1 1 -1 1 1 -1 -1 -1

1 1 -1 -1 1 -1 -1 1 1 -1 1 1 -1 1 -1 -1

(2.10)

[This is what is called a normalized Hadamard matrix of order 8 [10]. It is normalized because the first row and column are an1 's, and it is a Hadamard matrix in containing only l's and -1 's, and in satisfying oot = 81, where ot is the transpose of 0, and I is the 8x8 identity matrix.] Note that if a :f 0, b :f 0, then the components Oa,b = Oa+l,b+l' indices from 1 to 7, modulo 7 (first row and column of 0 are given the index 0). Let Oa be the ath row of 0, a=0,1, ... ,7, and define the product (2.11)

Oa. Ob = Oa,bOe, where the components of Oe are Oe,d

= Oa,dOb,d,

for each d=O,1, ... ,7, and Oa,b gives a sign to the product. For example,

= +[1 . 1, (-1) . ( -1), 1 . ( -1),1 . 1, ( -1) . 1,1 . ( -1), ( -1) . 1, ( -1) . ( -1)]

= +[1,1, -1, 1, -1, -1, -1, 1] =

06, where the plus sign out front arises from the component 0 1 ,2 = +1. The resulting multiplication table of the 0~8 is exactly the same as 2.9, giving rise to the obvious isomorphism ea ~ Oa, a = 0, 1, ... ,7. The quaternion algebra arises in exact1y the same way from the Hadamard matrix of order 4: 1 1 -1 11 -1 -1 -1

1.

(2.12)

DIVISION ALGEBRAS ALONE

35

In this case, while index cycling (modulo 3) is an automorphism, index doubling is a (product reversing) antiautomorphism. Likewise the complexes arise from the order 2 Hadamard matrix C=

[~ _ ~ ].

(!d3)

Index cycling and doubling, both modulo 1, have no effect here.

2.2. Adjoint algebras. The octonion algebra is nonassociative and so is not representable as a matrix algebra. The adjoint algebras of left and right actions of 0 on itself are associative. For example, let Ul, ••• , U n , x be elements of o. Consider the left adjoint map (2.14) The nesting of parentheses forces the products to occur in a certain order, hence this algebra of left-actions is trivially associative, and it is representable by a matrix algebra. One such representation can be derived immediately from the multiplication table 2.9. For example, the actions

36

CHAPTER 2

(only nonzero entries are indicated). Note that eLt [] x

= el ( x 0 eo

+ x I et + x 2 e2 + x 3 e3 + x 4e4 + x Ses + x 6 e6 + x 7e7)

xo

-1

xl

1

x2 x3 x4

-1 -1

1 -1

XS

x6 x7

1 1

which gives rise to the first of the matrix identifications in 2.16 (please note that we can identify these adjoint elements with matrices because of the associativity of OL, the algebra of left actions of 0 on itselfj this is not possible for the nonassociative 0). Note that e2e6 = e3e4 = eSe7 == e}, but because of the nonassociativity of 0, for example, (2.17)

DIVISION ALGEBRAS ALONE

37

in general. In particular, -1

1 -1

1 -1

1 1 -1 -1

1 1 -1

eL34 --+

1 1 -1

-1 -1

1 1 1

eL57 --+

(2.18)

-1 -1 -1

1

These matrices highlight a peculiar trait of the octonions. They satisfy

IeRl = (-eLl + eL26 + eL34 + eL57 )/2·1

(2.19)

This is because for all x in 0

Likewise, so

(2.20)

38

CHAPTER 2

In general, any product from the right can be reproduced as the sum of products from the left, and visa versa (see chapter 7). [Note: Because of the cyclic nature of the octonion product, together with the index doubling automorphism, the rules 2.19 and 2.20 are quite general. For example, via doubling 2.20 implies

Cycling the indiees by -1, we get in turn, eR34 = (eLl

+ eL26 -

eL34

+ eL57)/2

(the terms have been arranged in the order they appear in 2.20).] This property provides a great contrast with the quaternion algebra. If x and b are in Q, then there is no general method of expressing the right

product xb of b on x as the sum of left products. The left and right adjoint algebras of Q on Q are distinct and commute with one another (assured by the associativity of Q). Also assured by the associativity of Q, the left and right adjoint algebras are each isomorphic to Q itself. In the octonion case this is clearly not true. The left and right adjoint algebras (0 Land 0 R) are the same algebra, and this algebra is larger than o itself. In fact, it is isomorphie to R(8). It is not difficult to prove that eLa ... bc ... d

= -eLa ... cb ...d

(2.21)

if b =f:. c, all indices from 1 to 7. So, for example,

In addition, eLab ...pp .•• c

=

-eLab ... c

(2.22)

(cancellation of like indices). Together with eL7654321

=

1L

(2.23)

(the left adjoint identity, also denoted eLo), 2.22 and 2.23 imply that a complete basis for the left=right adjoint algebra of 0 consists of elements of the form (2.24)

DIVISION ALGEBRAS ALONE

39

This yields 1+7+21+35=64 as the dimension of the adjoint algebra of 0, also the dimension of R(8). (2.25) The 8-dimensional 0 itself is the object, or spinor, space of the adjoint algebra. Since the left(right) adjoint algebra of 0 is isomorphie to R(8), and 0 is 8-dimensional, we call 0 left( right) complete. Again there is a contrast to the other hypercomplex algebras. Q is 4-dimensional, but QL = QR = Q (QL and QR are isomorphie, but distinct). Therefore, since neither is isomorphie to R(4), Q is neither left nor right complete. However, QA, the combined left-right adjoint algebra, with a 16 dimensional basis consisting of the actions (2.26) x - X, qiX, xqj, qixqj, i,j E {l, 2, 3}, is isomorphie to R(4). Hence we may call Q two-sided complete.

(2.27)

(I shall occasionally let the context distinguish qLj from qRj, and denote both qj.) The case for Cis even worse. In being both commutative and associative C is complete in neither of the above senses. (2.28) and all are identieal. To generate a full R(2) algebra of actions for the 2dimensional (over the reals) ob ject algebra C, one has to resort to actions arising outside the algebra. In particular, the four actions . * , zx . x - x, x * , zx

(2.29)

form a basis for a complete R(2) algebra of actions on C (x* = the complex conjugate of x: note that the last three actions in 2.30 anticommute). All of this brings up an interesting point. Because C is a subalgebra of Q is a subalgebra of 0, quite often one finds in applications of the division

CHAPTER 2

40

algebras to physies that they are applied singly. Why, after-all, in applying o should one separately include Q and/or C? But the properties that define an algebra or sub algebra are anthropogenie. It is not recorded that there were any higher deities hanging about at the time who could speak with absolute authority on the relevance of these properties to the Truth underlying the design of reality. These adjoint completeness properties set the algebras apart. In the next section, in which the adjoint algebras are identified as Clifford algebras, and the division algebras upon which they act as spinor spaces, these distinctions become more pronounced. And in section 2.6 it will be seen that the fact that "C is a sub algebra of Q is a subalgebra of 0" is not as fundamental as it might seem, and in one light it can even be viewed as accidental.

2.3. Clifford Algebras, Spinors. The left adjoint algebras are

whieh imply the Clifford algebra isomorphisms

(Rp,q the Clifford algebra associated with (p( +), q( - ))-space-time). In particular, multivector bases for 0 L ~ R o,6 may be represented as below:

Clil lord Algebra OL

scalar vector 2-vector 3-vector 4-vector 5-vector 6-vector

~

R o,6

lL eLp eLpq eLpqr

(2.30)

eLpq7 eLp7 eL7,

p, q, rE {I, 2, 3, 4, 5, 6} (note that each of these multivector bases is invariant

under index doubling). The spinor space of Ro,6 ~ R(8) is the space of real 8-spinors, but if represented as in 2.30, the spinor space is just 0 itself, the 8-dimensional

DIVISION ALGEBRAS ALONE

41

object space of OL. The basis {1, ea , a = 1, ... , 7} ofO is also an orthonormal basis for the spinor space, orthonormal with respect to the inner product (2.31 ) where x t is the conventiomiJ. Hermitian transpose antiautomorphism of 0, changing the signs of the ea , a = 1, ... , 7. In general, on C, Q, and 0, (X O

+ Li xaea)t = x O - Li xae a, (2.32)

(ZO + zli)t = (zO + zli)* = zO - zli, and 2.31 defines the inner product on each. Product reversing Hermitian transposition can be extended to 0 L in the obvious way:

(2.33) t eLabc

-eLcba

eLabc,

(a,b,c distinct, from 1 to 7). These actions are consistent with the matrix representations 2.16 and 2.18. Each of those matrices is antisymmetric. It is worth pointing out that even though, for example, eL235[1] = et, we have eL235 =F eLt. and the matrix associated with eL235 (in the representation o( 2.16 and 2.18) is 1 1 -1 -1 -1 -1 -1 -1

which is Ijymmetric, consistent with 4235 = eL235' With the definition 2.33 in hand one can easily prove that for all x and y in 0, and A in OL,

< x, A[y] >=< At[x], Y > .

(2.34)

42

CHAPTER 2

Certain subsets of 0L elose under eommutation and so beeome Lie algebras. The 2-veetor subspaee of the Clifford algebra 2.30 is an example (see also seetion 5):

I

{eLpq

:p,q::J 7}

-t

8u(4) ~ 80(6) ~ 8pin(6)

I

(2.35)

(see seetion 2.5 and ehapters 7 and 8 for a more eomplete development). The ease for the quaternions, Q, is in some respects more eomplieated, beeause Q is less algebraieally complete than O. Multivector bases of QL, identified with R O,2, are (r=1,2):

Clil lord Algebra QL sealar veetor 2-vector

~

R O,2

1 (2.36)

The spinor spaee of QL, namely Q itself, seems larger than it need be, and indeed the elements {qRi, i = 1,2, 3} eommute with the Clifford algebra 2.36, and so ean be considered 'internal' actions relative to the 'external' or geometrie actions 2.36. This extra set of internal actions generates an internal SU(2), whieh will play an important role in what follows. That is, the span of the set {qRi, i = 1,2, 3} eloses under commutation, and

I{qRi, i = 1,2, 3}

-t

8u(2)·1

(2.37)

The inner produet of Q is defined as it was for O. If Ais an element of QL, then 2.34 is valid in this context as well. In addition, if x - t xU is an internal SU(2) action (from QR), then

< xU,yU >=< xuut,y >=< xUU-1,y >=< x,y >.

(2.38)

As pointed out in 2.27, only the 1-sided adjoint algebras of Q are algebraically incomplete. QA ~ R( 4), whieh is the clifford algebra R 3 ,1. Its spinor spaee, which we expect to be 4-dimensional, is also Q, which is 4dimensional. The multivectors of QA, identified with R 3 ,1l are representable as follows (i E {1,2,3}):

DIVISION ALGEBRAS ALONE

43

Clil lord Algebra QA

scalar vector 2-vector 3-vector 4-vector

~

R 3 ,1

1 qLlqRi, qL2 qRi, qL3qRi

(2.39)

qL2qRi, qLl QL3·

With respect to QA, the 4-dimensional object (spinar) space Q has no internal degrees of freedom.

2.4. Resolving the Identity of 0

L.

An algebraic idempotent, A, is by definition a nonzero element satisfying: (2.40) A is nontrivial if A

and

.:p 1, the identity.

In this case

A( 1 - A) = A - A 2 = A - A = 0

(2.41 )

(1 - A)2 = 1- 2A + A 2 = 1 - 2A + A = 1- A.

(2.42)

So by 2.42 1 - A is also an idempotent, and by 2.41, A and 1 - Aare orthogonal. Because of 2.41, nontrivial idempotents are divisors of zero, hence the identity is the sole idempotent of any division algebra. This applies to C, Q, their left and right adjoint algebras, and to 0, but not to OL = 0R ~ R(8), which is not a division algebra. In particular, if a, b, cE {1, ... , 7} are distinct, then (2.43) is an idempotent (note: eLabc

= eLabcabc = 1L, by 2.21 and 2.22).

1L - A

=

So is

1 2(lL - eLabc).

Certain elements of 0 L are diagonal in the adjoint representation. A basis for these consists of the identity, 1L, together with the eLabc satisfying (2.44)

CHAPTER 2

44

(ie., ql - e a , q2 - e c, q3 - eb defines an injection of Q into 0, in which case, ea(ebec) = (eaeb)e c = eaebec = qlq3q2 = 1). In particular, define (2.45) (indices from 1 to 7, modulo 7), and let 10 be the identity. Their adjoint representations are

10 = 1L - diag( + + + + + + ++),

lt =

eL476 -

diag(+ - - - + - ++),

12

= eL517 - diag(+ + - - - + -+),

13

=

14

= eL732 - diag(+ -+ + - - -+),

15

= eL143 - diag(+ + - + + - --),

ls = 17

eL621 -

eL254 -

diag( + + + - - - +-),

diag(+ - + - + + --),

= eL365 - diag(+ - - + - + +-).

(2.46)

(Thus, for example,

eL517[e a]

= -ea,a = 2,3,4,6.)

Being diagonal, the la clearly commute. They also satisfy (2.47) E {I, ... , 7} (had eaea+I = e a+3 been chosen as the multiplication for 0, then 2.47 would change to lala+I = la+S, so these choices are in this manner dual to each other (see section 2.6)). Likewise, if laIb = le, then

a

1(2a)I(2b)

Notice that (2·7)

= 1(2c)'

= 14 = 7 mod 7, and

(2.48)

DIVISION ALGEBRAS ALONE

45

(only h is invariant with respect to this index doubling operation). These elements satisfy some other fascinating properties. For example, for all x in 0, (2.49) (This result is not obvious, but it will be more easily provable after the tools of chapter 7 are developed, which will happen, not coincidentally, in chapter 7.) Moreover, X ---4 la[x] (2.50) is an octonion automorphism (see the next section), so (2.51 )

for all x, y in O. Finally, the identity of 0 L can be elegantly resolved into orthogonal primitive idempotents using the I a • A primitive idempotent can not be expressed as the sum of two other idempotents. So, for example, 1 4(1

+ h + 12 + 14 ) and

1 4(1

+11 -

h - 14 )

are both idempotents, and so is their sum, !(1 + It) (note: h/2 = 14 ), The sum is therefore not a primitive idempotent. As it turns out, neither are the initial two. Let i ac = ±1 be the cth sign on the diagonal of the adjoint representation of the la in 2.46, c=0,1, ... ,7. Define (2.52) So, for example,

These satisfy PaPb = OabPb

(2.53)

(no sum; a, bE {O, 1, ... , 7}), and 7

Epa = 10 . a=O

(2.54)

46

CHAPTER 2

And they are primitive idempotents (in the adjoint representation Pa is the 8x8 real diagonal matrix with 1 in the (a, a) position, all other components zero). Hence they resolve the identity of 0L. Finally, if x = xo +L:~=I xae a, then Po[x] = xo, Pa[x] = xae a (no sum), a E {l, ... , 7}.

(2.55)

So, for example, (Po

+ PI + P2 + P6)[x] = xO + xIel + x 2e2 + x6e6'

That is, (PO+P1+P2+P6 ) is an idempotent projecting fr um 0 a subalgebra isomorphie to Q: Likewise, for example, and of course,

PolO]

~

R.

2.5. Lie Algebras, Lie Groups, from 0

L.

The automorphism group of 0 is the 14-dimensional exceptional Lie group G2 • A basis for the Lie algebra of G2 (denoted, LG2 ) is simply represented in OL: (2.56) {eLab - eLcd : eaeb = eced} -+ LG2 1

I

(this result will be proven in chapter 7). For example, e1e5 = e2e3(= e7), therefore eL15 - eL23 is an element of LG 2 • Therefore, for all realO,

G(O) = (2.57)

=

!(1 + h) + !(1- h)cosO + !(eLI5 - eL23)sinO

is an element of G 2 • Let () =

G(7I")

71":

1

= 2(1 + It) -

1

2(1- h)

=h

(see 2.46), thus confirming that 11 (hence all the I a ) is an (sign changing) automorphism.

47

DIVISION ALGEBRAS ALONE In 0 R the basis is much the same: {eRab - eRcd : eaeb

= eced} -+ LG 2,

and in fact it is not difficult to prove that eLab - eLcd

= -(eRab -

eRcd)

when eaeb = eced. It is easily verified that G(O) defined in 2.57 leaves 1, e4, e6, and e7 invariant. In general, eLab - eLcd E LG2 will generate G2 actions that leave 1 and eJ invariant for f ::J a, b, c, d. The stability group of any fixed octonion direction is the 8-dimensional SU(3) C G2 • Therefore, for example, a basis for the Lie algebra of the stability group of e7 is: {eLpq - eLrs E G 2 : p, q, T, S ::J 7}

-+

su(3).

(2.58)

This su(3) is the intersection of LG2 with spin(6), defined in 2.35, and it will appear as color-su(3) in coming chapters. Note that the index doubling isomorphism of 0 is an element of the SU(3) stability group of e7, since e2x7 = e14 = e7. This can be explicitly represented as

= [!(1 +12 )

+ !(eL24 -

eL63)][!(1 + lt) + !(eL12 - eL35)),

and index quadrupalling as exp[~(eL35 - eL12)]exp[~(eL56 - eL4t))

Bases for some other subsets of 0L that dose under commutation, and their representation types, are listed below: {eLab}

-+

so(7), spinor, (2.59)

48

CHAPTER 2

These three examples become more obvious when we tensor OL with R(2). A basis for R(2) consists of the four matriees

€=[~ ~],O=[~ ~1]' ß=

[~ ~] ,w = [~1 ~].

(2.60)

OL(2) ~ R(8)®R(2) ~ R(16), whieh is the Clifford algebra R s,o • Bases for

the vector and 2-vector parts of R s,o are vector 2-vector

ß, eLaW,. eLao, eLab€.

(2.61)

The subspace of 2-vectors closes under commutation and is isomorphie to so(8). The commutator of a 2-vector with a vector is a vector, this giving rise to the vector representation of 80(8). Let XL = xo + L:i xaeLal and if A is in OL, let At be the conjugation defined in 2.33. Then

The operation of eLaXL L:i xaea ) E 0)

+ xLeLa

on the identity of 0 yields (let x = (Xo

+

(2.62) Likewise [[ eLObC

0 l, [0xl

eLbc

XL 0

II = [ (same)t 0

(eLbcxL

+ xLeLbc) 0

1'

and (2.63)

= eLbe[X]- eRa[X], where ea = eb( ee1) = ebec' This justifies the claim that the third of the 80(8) representations in 2.59 is vector.

DIVISION ALGEBRAS ALONE

49

The other two are simpler. The spinor space of OL(2) is 0 2 , 2x1 matrices over o. Let X and Y be in O. The 2-vector (80(8)) actions from 2.61 on 0 2 take the forms [ e La

o

0 -eLa

] [ X ] = [ e La [X] ], Y

-eLa[Y]

(2.64) [ eLbc

o

0 eLbc

1[ YX ] = [ eLbc[X] 1' eLbc[Y]

thereby justifying the two spinor representations in 2.59. Thus the two spinor representations and the vector representation all aet on 8-dimensional o. These three 80(8) representations are also linked by what is known as 80(8) triality. This coneept will be developed more eompletely in section 7.3.

2.6. From Galois Fields to Division Aigebras: An Insight. The real numbers are the paradigm for mathematical field theory. There is addition (and subtraction), an additive identity, 0, and every element X has an additive inverse, -x. There is multiplieation (and division), a multiplieative identity, 1, andevery element x " 0 has a multiplicative inverse, X-I. Multiplication by zero gives zero, and for all x " 0 and Y =f 0, we also have xy " 0 (no divisors of zero). Finally, xy = yx (eommutative), and x(yz) = (xy)z (associative). R is an infinite field, but there also exist finite fields. For any prime p there exist (unique up to isomorphism) fields of order pk for all k = 1,2,3, ... , denoted GF(pk) (G for Galois, their ill-fated founder, F for field). For no other positive integers are there fields of that order. The pk elements of GF(pk) are easily written: {0,1,h,h 2 , •.• ,hpk - 2 }, where his the multiplicative generator. That is, the multiplication of G F(pk) is eyelic and for all x =f 0 in G F(pk), (2.65) hpk_I • ( le.,

1) =.

All that remains then is to eonstruct an addition table for G F(pk) eonsistent with its being a field. This problem ean be redueed to finding what is ealled a Galois sequenee for GF(pk), which eonsists of pk -1 elements of Zp

50

CHAPTER 2

(the integers modulo p). Its furt her properties can be best illustrated byan example. (Mathematicians have a more elaborate development in terms of polynomials and quotient modules; the elements of a Galois sequence appear in that context as coefficients of a polynomial. ) [0 1 1 2 0 2 2 1] is a Galois sequence for GF(3 2 := 9). We identify it with hO == 1, the multiplicative identity of GF(9), and we'H identify its kth cyclic permutation with h k • That is, h 1 == [1 0 1 1 2 0 2 2], h 2 == [2 1 0 1 1 2 0 2], h 3 == [2 2 1 0 1 1 2 0], h 4 = [0 2 2 1 0 1 1 2], h 5 == [2 0 2 2 1 0

1 1],

h 6 == [1 2 0 2 2 1 0 1], h7 == [1 1 2 0 2 2 1 0], h8

= [0

1 1 2 0 2 2 1],

(2.66)

where h8 = hO := 1 gets us back to where we started from (any cyclic permutation of the initial sequence would have been a valid starting point). Notice that the first k := 2 elements of each sequence are unique, and can be used as labels for the elements (we are using instead the exponents). And notice that by adjoining to this colJection the zero sequence, 0=[00 00 0000J,wehaveasetof p k:=3 2 :=9vectors(sequences), each pk - 1 := 32 - 1 == 8-dimensional over Zp := Z3, and that the set is closed with respect to Z3 vector addition. Let +p represent addition modulo p. Then for example, h2

+3 h4 := [2

1 0 1 1 2 0 2]

+3 [0

:= [2 0 2 2 1 0 1 1]

2 2 1 0 1 1 2

= h5 •

1

DIVISION ALGEBRAS ALONE

51

A fuH addition table for GF(9) resulting from this sequence is listed below: 0

h1 h5 h8 h4 h6

h2 h8 h6 h1 h5 h7

h3 h4 h1 h7 h2 h6 h8

h1 h2 h3 h4 h5 0 h6 h3 0 h7 h2 h4 0 h8 h7 h3 h5 (recall that h8

= 1).

h4 h6 h5 h2 h8 h3 h7 h1

h5 h6 0 h3 h7 0 h6 h8 h3 h7 h1 h4 h4 h2 h8 h3 h2 h1

0

h7 h8 h2 h7 h4 h3 0 h5 h1 0 h8 h2 h5 h1 h3 h6 h4 h4

(2.67)

Note that

hk +3 hk = hkH and

hk +3 hk +3 hk

= hk +3 hkH ::: o.

Also,

hk +3 hk +1

= hk+ 7 •

Because for any x and y in any G F( 3m ),

(X +3 Y)3

= X 3 +3 Y3,

(2.68)

cubing the last equation above results in

hk +3 hk+3

= hk+ 5

(exponents are taken modulo 8 from 1 to 8, and although strictly speaking the exponents k cube to 3k, because 3 and 8 are relatively prime we are aHowed to replace 3k by k in constructing new addition rules), and cubing this leads back to hk +3 hk +1 = hk +7. There is also,

hk +3 hkH = hk+3 , which cubed yields, and finaHy

52

CHAPTER 2

which cubed yields, Of more interest to us here are the fields G F(2 n ), n ular, a Galois sequence for GF(2 1 ) is [ 1 ],

= 1,2,3.

In partic-

for GF(2 2 ) is [ 0 1 1 ], for GF(2 3 ) is [ 0 0 1 0 1 1 1 ]. In this last case we define

=[ 1 e2 = [ 1 e3 = [ 1 e4 = [ 0 e5 = [ 1 e6 = [ 0 e7 = [ 0 e1

0 0 1 0 1 1 ], 1 0 0 1 0 1 ], 1 1 0 0 1

o ],

1 1 1 0 0 1 ], 0 1 1 1 0 1 0 1 1 1

o ], o ],

0 1 0 1 1 1 ].

(2.69)

Addition in this case can also be completely described by cyclic equations in the ea • To begin with, (2.70) (every element is its own additive inverse, and tion). Also,

+2

is the binary vector addi-

(2.71) Since in this case (p = 2) ( X +2

2

2

2

Y) = X +2 Y ,

(2.72)

squaring the above addition rule leads to a new rule, (2.73) and squaring this leads to

(2.74)

DIVISION ALGEBRAS ALONE

53

(exponents are taken modulo 7 from 1 to 7). The link of GF(8) to the octonions should now be obvious. The matrix of signs in 2.10, used to construct an octonion multiplication, could have been replaced by the following matrix of elements of Z2 (ie., O's and l's):

0'=

0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 1 1 0 1 0 1 0 0 1 0

0 0 0 1 1 0 0 1 0 0 1 0

0

1 1 0

1

(2.75)

0

1 1 0 1 1 1

Note that (_l)o~b = Oab (see 2.10), so if we define

O~ * O~ = (-l)O~b[O~

+2 O~],

(2.76)

then we have on ce again ereated an octonion product, where this time the rows of 0' are identified with the basis of the octonions. Note! We have used GF(8) addition to ereate an octonion multiplieation. The first row of 0' is the multiplicative identity of 0, and we must ereate a new 0 to play the role of the additive identity of O. With respeet to 0 addition, the rows of 0' are now treated as linearly independent, a basis for a real algebra. Relabel the rows of 0' as ea,a = 0,1, ... , 7. So the exponents of GF(8) in 2.69 have beeome the subseripts of the oetonions. Beeause the oetonion product (now denoted just eaeb) is derived direet1y from the GF(8) addition, the exponent rules 2.71,73,74 are valid for the octonion pr~duct, the rules now applied to subseripts (see 2.4,7). In addition, the index doubling automorphism for the octonions (2.6) is now seen to follow from 2.72. [Note: The sum rules 2.71,73,74 for GF(8) correspond to 2.4,7, but in general we ean only make such eorrespondenees up to a sign. For example, while it is true in GF(8) that ea +2 ea+5 = eaH , in 0 we have eaea+5 =-eaH' Index doubling is also tricky, and in Q it works out slightly differently.] [Also not\:!: In GP(8), e7 = eO = 1. The reason that it was listed as e7 in 2.69 is to make the eorrespondenee e7 ---7 e7 of GF(8) to O. Therefore, sinee eO == e7, we have eO ---7 e7, too! That is, we use eo to denote the oetonion identity, 1, and it has no correspondence to any power of e1 E GF(8).

CHAPTER 2

54

In [10] the identity is given the suhscript 00, so we could write eoo = 1. This notation has distinct advantages, hut also for our purposes disadvantages. Pd apologize for the confusion of all this were it not that this section is only intended to highlight the naturalness of the chosen 0 product, and this material will not playapart in what follows.] [Finally note: the transpose of 0' also results in a valid GF(8) addition and 0 multiplication. In this case, however, ea ea+1 = -e a+3 in O. Except for the sign change, this is the dual multiplication mentioned in section 2.4. If we replace 2.76 by

we generate the 0 multiplication rule, eaea+1 = -e a+5; and if we use the transpose of 0', the rule eaea+1 = ea+3. So strictly speaking there are 4 cyclic octonionic multiplication rules, 2 for the quaternions, and 1 for the complexes.] Having made the correspondence between GF(8) addition and 0 multiplication, one is naturally led to consider the role of G F( 8) multiplication in O. Since in G F( 8), ea eb = ea+b, this operation on the indices of 0 is just a cyclic shift (of the index a for a = 1, ... ,7; eo is left unaltered). Let S he the 0 automorphism that shifts the indices of eb, b 1, ... , 7 hy 1. So shifts the 0 indices by a, and S7 = SO is the identity map. Let

3 as weH. For example, let 9 1 = [0 0 0 1 0 0 1 1 0 1 0 1 1 1 1], 15-dimensional over Z2. This is a Galois sequence for GF(16), and it can be used to construct a new 16-dimensional algebra, extending the sequence, R, C, Q, 0 (this is distinct from the Cayley-Diekson prescription, whieh is founded on the inclusion property, and in fact 0 is not a subalgebra of this new 16-dimensional algebra, whieh is noncommutative, nonassociative, and nonalternative). One final path down whieh I have no intention of travelling far: we should be able to construct algebras in like manner from any GF(pn), for any prime p. For example, take the h k , k = 1, ... ,8, in GF(9) listed in 2.66, and map

CHAPTER 2

56

them to hk, k = 1, ... ,8, part of a basis for a new algebra. Map the zero sequence to 1, completing the basis. Form the stacked sequences in 2.66 into a matrix, H (8 x 8). If hi +3 hi = hk in GF(9), then define (2.81)

If j - i = 4 mod 8, then replace hk by 1. At this point I'm just spewing out ideas without a dear not ion of their interest or vi ability, so 1'11 shift directions a bit in hopes of bringing order out of chaos. It would seem in light of the material presented to this point in this

section that the division algebras are four out of an infinite co11ection of possible algebras constructable in like manner. Worse, it is a co11ection, not a sequence. Highlighting this is the fact that the first rows of Q' and 0' (ignoring the intitial O's) had to be [1 0 1] and [1 0 0 1 0 1 1] for Q and 0 with the multiplication rules we are adopting to result. Completely different algebras result from most of the other cydic permutations of these sequences. We could just as weH have begun with the dual sequences [1 1 0] and [1 1 1 0 1 0 0] (ending with the same element, but reverse the order of the sequence). These sequences also give rise to Q and 0, and they are Galois sequences for GF(4) and GF(8). They are in addition quadratic residue codes of lengths 3 and 7 over GF(2) (see [10]). For example, the quadratic residues modulo 7 are 02 = 72 = 0, 12 = 62 = 1, 22 = 52 = 4, 3 2 = 4 2 = 2, so confusingly renumbering the positions of the sequence [1 1 1 0 1 0 0] from 0 to 6, we see that the l's appear in the 0,1,2, and 4 positions, which are determined by the quadratic residues. Likewise, modulo 3, 02 = 32 = 0, 12 = 22 = 1, and the l's of [1 1 0] appear in the 0 and 1 positions. The quadratic residue code oflength lover GF(2) is [ 1 ], also the Galois sequence of GF2, and associated with C. There are no other examples of quadratic residue codes over GF(2) that' correspond to Galois sequences. To even have a chance we must have a code of length 2k - 1, and 2k - 1 must be prime. So 15 is out. The quadratic residue codeof length 31 is [1110110111100010101110000100100];

(2.82)

and a Galois sequence, equal to 2.82 in the first seven positions, is [1110110001111100110100100001010].

(2.83)

DIVISION ALGEBRAS ALONE

57

Let U be the 31x31 matrix formed of the first of these sequenees and all its cyclic permutations, and let V be the 31x31 matrix formed from the second. The first has the nice property shared by all quadratic residue codes over GF(2) that (2.84) In the 22 - 1 = 3 and 23 - 1 = 7 eases this gives rise to the noneommutativity among the imaginary basis elements (# 1) of Q and 0, which together with (2.85)

ensures that Q and 0 are division algebras (replaee U by the appropriate 3x3 and 7x7 matrices). Unfortunately the rows of U are not closed under Z2 addition. Those of V are: (2.86)

for some c, with a # b. Without this property the matrix will not give rise to an algebra in the manner outlined (of course, an initial eolumn and row of O's has to be tacked on). Requiring of our generating sequences that they be both Galois and quadratic residue is a heavy restriction, and the division algehras are the only algehras that result. They onee again form a sequence, hut it is definitely finite.

3. Tensor Algebras. 3.1. Tensoring Two: Clifford Aigebras and Spinors. Continuing the policy of viewing adjoint division algebras as Clifford algebras, and the division algebras themselves as spinor spaces, define

whieh is isomorphie to C(2) and is identified with the Clifford algebra R 3 ,o. Multivector bases for PL ~ R 3 ,o are listed below 0=1,2,3):

Clil lord Algebra PL scalar vector 2-vector 3-vector

~

R 3 ,o

1 iqLj qLj

(3.1)

t.

As usual the subspace of 2-vectors closes under commutation. In this case it is isomorphie to

Iso(3) ~ su(2) ~ spin(3).1 Let j

j

and let 'IjJ E C ® Q, the spinor space of P L. Then the action X _ UXU- 1

is an SO(3) rotation of the Euclidean vector {xj}, and the action

59

60

CHAPTER 3

is and SU(2) = Spin(3) rotation of the spinor

t/J.

The object space of C(2) is the space of 2xl complex Pauli spinors. The object (spinor) space of PL is P = C ® Q, which is 4-dimensional over the complexes, hence it has twice the dimension of the space of Pauli spinors. With the help of orthogonal Hermitian idempotents, P can be decomposed into the sum of two Pauli spinor spaces. Let x = L:~ x j qj, x j real, such that x2 = -1. Define (3.2) A± = !(1 ± ix),1

I

These satisfy

= orsAr,

(3.3)

+ A_ = 1.

(3.4)

ArAs and A+

As they are primitive idempotents, by 3.3 and 3.4 they constitute aresolution of the identity of P. The subspaces P A± are stable under the action of P L. Each is 2-dimensional over C and each transforms as a Pauli spinor with respect to PL. In particular, if we set .... 2 X=xq2

+ xq3 3

(3.5)

(which is done for the purposes of chapter 4), then {qt,X,qlX} form an orthonormal basis for the hypercomplex part of Q. Since (3.6)

any element

Z

of PA± will, with this choice for

x, reduce to the form (3.7)

ZO, zl

E C. Therefore, for example, the action

TENSOR ALGEBRAS

61

is equivalent to the action

which is the action of a Pauli matrix on a Pauli spinor. All of the elements of {1L, iL, qLl, iqLt. XL = x 2qL2 + x 3qL3' iXL, qLIXL, iqLIXL} can be be similarly identified with elements of C(2). Note that the subspaces {PA±} transform as an su(2) doublet with respect to the right action of the qj. For example,

Xe 8x = XA+e 8x + XA_e 8x = XA+e- i8 So the action X - Xx action

+ XA_e i8 •

(3.8)

= X A+( -i) + X A_(i) can be written as the matrix (3.9)

which is recognizably su(2). Note that the action 3.8 is an element of QR, and that QR commutes with PL = R 3 •o, which is the "external" geometrie (Clifford) algebra. Any action on the spinor space which is not an element of the Clifford algebra, and commutes with the action of the Clifford algebra, is considered "internal" relative to that algebra. The set of elements of QR of unit length is isomorphie to SU(2) (the action 3.8 arises from this group), hence this SU(2) is inherently internal.

3.2. Tensoring Two: Spinor Inner Product. Let A and B be elements of P. Each decomposes with respect to A± into a pair of Pauli spinors:

(3.10) Consider the expression

(3.11)

62

CHAPTER 3

where the < At B >± are complex. This follows from the fact that all X E P may be expressed as

where X m E C, m = 0,1,2,3. Since

and

therefore

In fact, < AtB >± is just the conventional complex inner product of the Pauli spinor AA± with the Pauli spinor BA±. We define the real inner product of A and B to be (3.12)

< A,B > is just the real part of AtB (or BtA). To see this, let X E P, and define the four components of X with respect to the A± by AsX Ar (r,8 = ±). In particular, the diagonal components are (3.13) where the X± are complex (so< AtB >± are the diagonal components of AtB). Define the trace of X with respect to the A± by

tr(X) = X+ Let X % i

-+

+ X_.

(3.14)

X- be the antiautomorphism on P that reverses the signs of not the sign of i (so X- = xt*, Hermitian conjugation

= 1,2,3, but

TENSOR ALGEBRAS

63

followed by complex conjugation). Therefore, A± = AT' Then

X++X_

+ A_) + X_(A+ + A_)

=

X+(A+

=

X+A+

=

A+XA+

+ (A+XA+)- + (A_XA_)- + A_XA_

=

A+X A+

+ A_X- A_ + A+X- A+ + A_X A_

+ (X+A+t + (X_A_t + X_A_

=

A+(X + X-)A+

+ A_(X + X-)L

=

(X

+ X-)A+ + (X + X-)A_

=

(X

+ X-)(A+ + A_)

=

X+X-,

where X + X- is clearly complex, so A±(X + X-)A± = (X Therefore tr(X) = X+ + X_ = X + X-

(3.15)

+ X-)A±.

is just twice the complex part of X. Define Mo(X) = X, the identity map, an automorphic involution (note, * is complex conjugation on P, not on C(2». The inner product may now be constructively expressed:

I< A, B >= i {[2:rs Mr«AAs)t(BAs»] + [same]t} I (r = 0,1,8 = ±). This follows from the fact that M1 (A±) = AT' so that, for example,

(3.16)

64

CHAPTER 3 Another, simpler, expression for

< A, B > is the following:

The expression 3.16 has the advantage that it highlights the connection between the Pauli spinor parts of A and B. In seetion 4 this form will be generalized to the more elaborate case of C ® Q ® O. In both cases it facilitates the ultimate connection to physics, to be made in chapter 4, and the derivation of important symmetries. LINK TO INTERNAL SYMMETRY In particular, if in 3.16 we replace A± by I\;± satisfying (3.17) then the result is the same. This is so because by 3.17 the components of the identity of P with respect to I\;± are the same as the components with respect to >'± (see 3.13), namely, Crs. The symmetry of 3.17 is

IU(1)

X

U(1)

X

SU(2).\

Note first that

= If xt X

= 0, X

E P, then X

= O.

(3.18)

Therefore,

That is,

!I\;± E P>'±.!

(3.19)

Therefore I\;± may be expressed in the form

(3.20)

TENSOR ALGEBRAS

65

u± and V± complex (i in the form 3.5, see 3.7). From 3.17 we obtain 4K± = (u±u± + V±V±)A± = A±, (3.21 ) implying (3.22) The symmetry of the equations 3.22 is U+(1) X U_(1) x SU(2). The SU(2) arises from the left action of the qj. Relative to this action u+ and v+ transform as the components of an SU(2) doublet, as do u_ and v_ (see 3.7). U±(1) arises from the action of iA± from the right. In particular note that

=

A_

+ A+ + A+[iO] + A+[!(iO)2] + ... A_

+ A+[1 + iO + !(iO)2 + ...) A_

+ A+[eili ).

Therefore, for example,

K+e iAtO

= K+(A_ + A+eill ) = K+e iO ,

so iA± from the right acts nontrivially only on K±. We may re-express all this in a more conventional way using matrices, which is done to help the reader get a handle on the mathematics before we get to the more complicated tripie tensor product algebra of the next section. T.lte A± can be represented as follows:

so that

66

CHAPTER 3

= ei9 [00 [ 0o U_] v_ In this case the left action of a 2 accounts for

X

-~+]. u+

2 representation of U(1)

~+ = [~: ~], and the map MI

X

SU(2) on A+

takes this to

[~ ~~+].

Finally the U(1) phase ei9 completes the action of [U(1) Schematically

A+

[U(1)

X

SU(2)]

A_ [U(l)

X

SU(2)]

xU_(1)

-+~_

X

SU(2)]

X

U(1).

(3.23)

We may now re-express 3.16 in the general form (3.24)

3.3. Tensoring Three: Clifford Aigebras and Spinors. Let and

As PL is viewed as the Pauli algebra of (1,3)-space-time, from which the Dirac algebra for that space is built, so too will we view TL as the "Pauli" algebra of (1,9)-space-time, from wh ich the "Dirac" algebra of that space will be built. In particular note that since P L ~ C(2), and 0 L ~ R(8), then

ITL ~ C(16), t and so we may identify it with the Clifford algebra

Ro,9'

TENSOR ALGEBRAS

67

The object space of RO.9 ~ C(16) is the space of 16 x 1 complex spinors. As was the case for P and PL, the P idempotents A± defined in 3.1 resolve the elements X of T (whieh is 32-dimensional over C) into X A+ and X A_, a pair of 16-dimensional (over C) R O•9 ~ TL spinors. And again, the SU(2) action 3.5 on X in T commutes with TL and is therefore internal. The example action developed in 3.6-8 go es through without change. It should seem surprising that T relative to TL has the same internal structure as P relative to P L. In both cases the extra internal structure arises from QR, which commutes with QL, hence also with the external geometrie algebras PL and TL. No internal depth arises from the C because i E TL and is identified below as a basis for the 9-vector subspace of R O•9 (when we expand to Dirac algebras, i will regain its independent, nongeometrie status). Also, although 0 is noncommutative, never-the-Iess OL = OR (see chapter 2). Hence the parts of TR, or PR, distinct from and commuting with TL, or PL, are all associated with QR, and so the internal structures of T and P are the same.

The chosen multivector bases for TL {I, ... , 6}:

~

R O•9 are (j E {I, 2, 3}, p, q, r E

Clil lord Algebra TL ~ R O•9

scalar vector 2-vector 3-vector 4-vector 5-vector 6-vector 7-vector 8-vector 9-vector

1 ieL7qj,eLp qj, eLpq, iqjeLp7 ieL7, iqjeLpq7, qjeLp, eLpqr qjeLpq, eLpq7, ieLp7' iqjeLpqr iqjeLpq, ieLpq7, eLp7, qjeLpqr eL7, qjeLpq7, iqjeLp, ieLpqr iqj, ieLpq, qjeLp7 eL7qj, ieLp

(3.25)

t.

As usual the space of 2-vectors c10ses under commutation, and is in this case isomorphie to

80(9)

~

8pin(9).

The sub set {qj} is a basis for

so(3)

~

su(2)

~

spin(3),

CHAPTER 3

68 and the subset {e Lpq} a basis for

80(6)

~

8u(4)

~

8pin(6).

As has been pointed out, the intersection of this 80(6) with LG2 is 8u(3) (see 2.58). This 8u(3) generates the color symmetry in what follows, hence color SU(3)is intern al only relative to RI,3, not to RI,9, and in this latter case SU(3) rotations are space rotations (see chapter 6).

3.4. Tensoring Three: Spinor Inner Product. If A and B are elements ofT, then A>.± and B>.± are four Ro.9 spinors. Put into the righthand side of 3.16, the result of this algebraically more complicated case is no Ion ger real, but is linear in the identity, and the 21 elements, qje a , j=1,2,3, a=1, ... ,7. No automorphism or antiautomorphism will simultaneously change the signs of all these 21 elements, so the introduction of another conjugation (beyond Mt and Hermitian conjugation) into 3.16 will not reduce the righthand side to areal number. However, the expression 3.16 for the inner product is constructed from aresolution of the identity of P, and a group of automorphisms permuting the idempotents of the resolution. The same technique may be successfully applied to T, although some care is required, for T is not only nonassociative (inherited from 0), but, as will be shown below, also nonalternative. RESOLVING THE IDENTITY OF T Not surprisingly the structure of T = C 0 Q 0 0 is more complicated than that of P. T admits at least three inequivalent resolutions of its identity into 4 orthogonal primitive idempotents. For example, let

ao =

H1 +qtet + q2 e2 + q3e6), (3.26)

(recall that {et, e2, e6} are a quaternion tripie of 0 ). These satisfy (3.27)

69

TENSOR ALGEBRAS Note, however, that

This curious result (O'o( O'Oe7) =1= (O'ö)e7) demonstrates that T is not an alternative algebra (0 by itself is). Furthermore it demonstrates that while the O'm may be idempotents, they are not projection operators. This suggests we restrict the definition of aresolution of the identity to consist of orthogonal primitive projection operators. (In fact this is what is generally required of an idempotent, but so rarely does the issue of nonalternativity arise that it is seldom necessary explicitly to specify this.) In particular, ß m , m = 0,1,2,3, will be considered to resolve the identity of T if for all X in T,

(3.28) In particular, define ßo

= (1 + ix)(l + ie7)/4 = AOP+,

ß2

= (1 + iy)(l- ie7)/4 = A2P-,

(3.29)

ß3 = (1- iy)(1- ie7)/4 = A3P-,

-2 -2 1 x=y=-,

and P± = (1 ± ie7 )/2, which alone resolve the identity of C ® o. The x into yflexibility in the idempotents ß m is allowed because p+p_ = o ensures that ßo and ßl are orthogonal to ß2 and ß3. If we set x y, then an additional degree of freedom can arise in the octonion. For example,

=

CHAPTER 3

70 the four idempotents

ßo = (1

ßl

+ ix)(l + ie7)/4,

= (1- ix)(1 + ies)/4,

ß2 = (1

+ ix)(1 -

(3.30)

ie7 )/4,

ß3 = (1 - ix)(l - ies)/4,

also resolve the identity of T, and they satisfy 3.28. But thp ß m also satisfy (3.31)

which the ßm do not. So components of X in T with respect to the ß m , namely ßmX ß m are consistently definahle in this case, hut in neither the Qm nor ßm cases. THE TRACE OF X

For all X in T, ßmX ß m = Xmß m ,

(3.32)

where X m , m = 0,1,2,3, are complex. Again we define

tr(X) = EXm . m

[ It is easy to prove 3.32. For p = 1, ... ,6,

(parentheses unnecessary). Also,

Any element X of T may he written as

X = XO +

7

L Xaie a, a=l

XO, X a E C ® Q, a = 1, ... ,7.

(3.33)

TENSOR ALGEBRAS

71

Therefore, for example,

We have already seen that for elements Y E C ® Q,

Yo E C. Therefore,

X o E C. (This confusing use of indices is now at an end.) Let Mm , m = 0,1,2,3, be a group of automorphisms on T defined by

x, h[X*], (3.34)

(h could have been replaced with Ip,p = 3,5,6), where X~X"

is complex conjugation, and

changes the signs of the quaternion units qn r = 1,3. h is defined in chapter 2 (if p = 3,5,6,7, then I p (e7) = -e7, and in particular I 7 (e p ) = -e p , p

= 1,2,4,7).

We may assume without a loss in generality that

72

CHAPTER 3

Therefore,

Mm('ö'o)

= 'ö'm

(3.35)

(in addition, M l M 2 = M 3 , M~ = Mo for all m). The actions of the automorphisms Mm, and the M m followed by Hermitian conjugation, on T are given below (r = 1,2, P = 1,2,4,7, q = 3,5,6):

Mo MI 1

qr q2 ep eq ~qr

iq2 ~ep

ie q qrep q2 ep qreq q2 eq iqrep iq2 ep iqreq iq2eq

M 2 M3 Mt0 Mt1 Mt2

+ + + + + + + + + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + + + + + + + + +

Mt3

+ + + + + + + + + + + + + + + + + + + + + + + + + + + +

(3.36)

+ + + + + +

Therefore,

X o + Xl + X 2 + X3

=L

Am[X + Ml(Xt) + M2(X)

+ M3(Xt)]A m.

(3.37)

m

The term in the brackets of 3.37 is 4 times those parts of X linear in 1, i, q2ep, and iq2ep, p = 3,5,6. But AmepA m = 0, p = 3,5,6 (because p±epp± = P±P'fep = 0), therefore I: Xm is just four times the complex part of X. Using 3.37 we can constnict an inner product of A and B in T, which is just the real part of At B. We employ an expression far the inner product identical in form ta 3.16:

< A, B > = =

~(tr(AtB

+ BtA))

~{[I:mn Mm((AAn)t(B'ö'n))]

+ [same]t}.

(3.38)

TENSOR ALGEBRAS

73

3.5. Derivation of the Standard Symmetry. As was done in 3.24 relative to 3.16, the expression 3.38 for < A, B > can be generalized by replacing ß n in 3.38 by the algebraically more complicated r n, n = 0,1,2,3: (3.39) We require that with respect to the r n the components of the identity be the same as those obtained from ß n . That is,

(3.40) (see 3.17). In addition we need a consistency requirement, that the components of any X in T, with respect to the r n, be free of nonassociativity problems. That is, (r~x)r m

= r~(Xr m), n, mE {O, 1,2, 3}.

(3.41)

From 3.40 we obtain so

ßm(r~r n)ß m = (rnßm)t(rnßm) =

omnßn,

(making use of the fact that p±(AB)p± = (p±A)(Bp±), and if A E T, and AtA = 0, then A = O. Therefore,

ßt = ß n ).

But

So That is, (3.42) For the sake of simplicity, and without loss of generality, we now put the 2-dimensional degree offreedom in and yin ß m (see 3.29) in the (Q2,Q3)plane. That is, set

x

74

CHAPTER 3

Y- = -q2 X- q2

= X 2 q2 -

X3

Q3·

So {Ql, X, ql X} and {ql! y, qd} are two orthonormal bases for the hypercomplex parts of Q. Therefore, since

x(l

± ix)/2 =

=fi(l

± ix)/2,

y(l ± iy)/2 = =fi(l ± iy)/2, (3.43) and e3 = -e2er, es = -eIer, and e6 be written in the reduced form

= -e4er, the elements f n

of T~n can

(3.44) where the u, v coefficients are complex. This can be further simplified by imposing the associativity condition 3.41 at this point. Suppose ~ n f; 0 and/or +v n f; O. For example, let f o = el~O = eIAOp+. Consider in this case the (0,0) component of e2 (note: (p+el)e2 = p-(ele2)):

(f6 e 2)f o

=

-[(AoP+el)e2]( elAoP+)

=

-[ Aop_( el e2)]( elAoP+)

=

-(p- e6)( elP+ )Ao

=

-p+(e6(elP+))Ao

=

-p+((e6edp-)Ao

=

-AoP+e2P-

=

-AOe2'

TENSOR ALGEBRAS

75

It's bad enough that this diagonal component is nonzero (it should pick out the complex (C) part of e2, which of course is zero), but in addition,

r6(e2r o) =

-e2 ß O

#

-ßOe2

= (r~e2)rO.

A voidance of these problems is possible only if U n +--

°

= 0, +-V n = 0, n = ,1,2,3,

that is, if (3.45) n=0,1,2,3, U n and V n complex. In this case, because P±P=F = 0, it follows automatically that r~r m = if n=O,l (or 2,3), and m=2,3 (or 0,1). T~erefore the only nontrivial equations left in 3.40 are

°

r6rl = 0, (3.46) and their Hermitian conjugates. But because of 3.45, the symmetry of

rtnr m = ßm,m = 0,1, r~rl = 0, is the same as the symmetry of 3.17, that is,

The same is true of the equations r~rm = ßm,m = 2,3,

r~r3 =

0,

the

U2 (1)

X

U3 (1)

X

SU_(2)

symmetry of this case being independent of that of the (O,l)-case. So the total symmetry to this point is (3.47)

('+' associated with m=0,1, '-' with m=2,3).

CHAPTER 3

76

The SU(2)'s arise from the qj,j = 1,2,3, and the U(I)'s from the complex unit i, or the octonion unit e7. In particular, if a± = ai±qj, ai± real, and ()m real, m=0,I,2,3, then

Li

(3.48) performs the rotation 3.47. For example,

exp( a- p_)r n (3.49)

The symmetry 3.47 of 3.40 and 3.41 is not complete. The SU(3) stability subgroup of G 2 leaving e7 fixed (see 2.57) also clearly maintains the validity of 3.40 and 3.41, for the 6. n and r n are invariant under this SU(3). (Since SU(3) is a subgroup of the automorphism group G2 , it may be extended to the spinors A and B as wen.) So each of the 6. m is subject to a symmetry

IU(I) x SU(2) x SU(3) I

(3.50)

(the SU(3)'s may be considered independent for m=0,I,2,3, but as all the

r m 's are invariant, this is irrelevant).

In 3.48 one method of generating the SU(2) actions was presented. There is another way that will prove more useful in applications. First redefine

IA± = MI ± i(X 2q2 + x3ie7q3»

=

~(I ± ix) I

(3.51)

(x 2 = -1), so that A+P+ = AOP+ = 6. 0 , A_P+ = AIP+ = 6. 1, A+P_ = A2P- = 6. 2 , A_P_ = A3P- = 6. 3

(3.52)

(note: ie7P± = ±p±). Define (3.53)

TENSOR ALGEBRAS

77

(note that the real span of the set {ie7Ql, Q2, ie7Q3} closes under commutation and is isomorphie to su(2). Let er- = er 1 ql

Then

eo,ßo

e&ßl

=

+ er 2 q2 + er 3 q3.

eäßo

Mo(e&ßo),

eäßl

M1(e&ßo),

(3.54)

. (3.55) eo,ß2

e-q2äq2ß2

M 2(e&ßo),

eo,ß3

e-q2äq2ß3

M 3(e&ßo)

(note that Mn(ii) = ii). That is, ii generates an SU(2) that is carried intact by the Mn from ßo to the other ß n. (There is an additional SU(2) degree of freedom on ß2 and ß3 (see 3.47 and 3.48). We can ac count for this by an SU(2) action like 3.49 that acts nontrivially only on ß2 and ß3') . For each n=O,1,2,3, ß n and qlß n form an SU(2) doublet with respect to eo,. The diagonal SU(2) generator is x. In particular,

=

(see 3.52). The standard symmetry is then identified with a single copy of U(1) x SU(2) x SU(3) that is associated with ßo and r o, and carried from there by the Mn to the remaining ß n and r n' The other symmetry degrees offreedom that arise on the ßj, j = 1,2.,3, then arise separately. In particular, the symmetry 3.47, together withSU(3), can be rewritten (3.56) where [U(.t) x SU(2) x SU(3)] is shared by all the r n (carried along by the maps Mn), Uj(l),j = 1,2,3, acts only on rj, and SU23 (2) acts only on r 2

CHAPTER 3

78

and r 3 • Note that the SU_(2) in 3.47 is a mix of SU23 (2) and the SU(2) in brackets. Schematically,

Ll o [U(l)

X

SU(2)

X

SU(3)]

--t

ro

--t

r1

Ll 1 [U(l) X SU(2) X SU(3)]

xU1(l)

Ll 2 [U(l)

X

SU(2)

X

SU(3)]

xU2(1)

XSU23 (2)

--t

r2

Ll 3 [U(l)

X

SU(2)

X

SU(3)]

xU3(1)

XSU23(2)

--t

r3 •

[ Curiously U(l)4 X SU(2)2 X SU(3) also arises in [11], where the division algebras are applied to the problem of particle masses. It is not yet known if there is a connection. ]

3.6. SU(2) Let X = X O +

X

SU(3) Multiplets, and U(l).

Ei Xae a be an element of O.

(3.57) The p+Zo part of 3.57 transforms as a singlet with respect to SU(3). The rest of P+X transforms as a triplet. This may be seen most explicitly by representing a basis for su(3) in 0 L adjointly as 3x3 matrices acting on the column [

~: ]. Tm, I. done bclow

TENSOR ALGEBRAS

!(eL34 - eL26) -

!(eL61 - eL4s) -

[:

i

[

.

J~(eL63

- eL24)

~

~ ] , HeL56 - ew) ~

[;

~(eL52 - eLl3) ~ [ ; !(eL46 - eL23) -

79

: ]

i

[

i ],

-1

~1

,~(eL3' - eL12) ~ [ ~1

:.] -~

[

,~(eL23 - ew) ~ [~

1 1

-i

(3.58)

:]. :]

For example,

!(eL34 - eL26)(p+€1)

= !p+[e3(e4 et} -

e2(e6el)]

= 0,

!(eL34 - eL26)(p+e2)

= !p+[e3(e4 e2) -

e2(e6 e2)]

= !P+( - 2e6) = ip+e4,

!(eL34 - eL26)(p+e4)

= !p+[e3(e4e4) -

e2(e6e4)] = !P+( -2e3) = ip+e2.

Assigning p+X the multiplet structure 1$3

(singlet plus triplet ) means p_ X transforms as

(antisinglet plus antitriplet). At this stage, however, the P+X and p_X multiplets are merely conjugates of each other, hence not independent. Now let X E C ® 0, so xa E C,a = 0,1, ... ,7, in 3.57. In this case P+X and p_X have independent multiplets (ie., they are no longer conjugates of each other). Therefore, under

ISU(3) : C ® 0 -

1 $ 3 $ 1 $ 3.1

(3.59)

80

CHAPTER 3

Therefore, T = (C ® 0) ® Q has the same multiplet structure as C ® 0, but times four. Explicitly,

(3.60)

Suppose in addition that X transforms with respect to SU(2) as r~. That is, (3.61) So X transforms like SU(2) doublets (see 3.9 and 3.56). [Note: of course we could as weH have let X transform with respect to SU(2) as r n , so that X --t i" X, but for two reasons 3.61 is preferable. Firstly, with that action the product xr n will be SU(2) invariant; and secondly, if SU(2) acts on the spinor X from the right, then it is internal with respect to the action of the geometrie algebra R 1 ,3 from the left. This aspect is developed in chapter 4.J Thus SU(3) divides T into 8 spinor bits (see 3.59), and SU(2) divides each of these in two, making 16 bits in al1. Each of these 16 SU(2) X SU(3) bits is 64/16=4-dimensional over the re als , 2-dimensional over C. In fact they are Weyl spinors., Consider for example the SU(3) singlet, SU(2) doublet,

Each ofthese subspaces is SU(3) invariant, and they are mixed by the action of e- ci from the right. They are also stable under the action of P L, the Pauli algebra, and with respect to that algebra they transform as Pauli spinor spaces. We conclude, then, that with respect to SU(2) X SU(3), T transforms exactly like the direct sum of a family of lefthanded leptons and quarks, and an antifamily of righthanded antileptons and antiquarks.

TENSOR ALGEBRAS

81

U(l) CHARGES As was done in 3.55 for SU(2), we may carry a U(l) charge from f o to the other fj, j=1,2,3, via the Mj. That is,

Mo(eillf o )

eillfo

fo

-t

f1

-t

f2

-t

ei of 2

f3

-t

e- i of 3

e-iof 1 =

M1(eiOf o )

M 1(e- e7 0f0) ,

M 2 (e i Of o) =

M 2(e- e7 0f 0 ) ,

M3(e i Or o ) =

M3(Ce70fo).

(3.62) =

In 3.55, f 1 and f 3 transform under a conjugate SU(2) representation relative to f o and f 2 (see 3.56). The same is true of the U(l) in 3.62. Therefore they combine to form U(l) x SU(2) - t U(2). (3.63) The remaining three U(l) degrees of freedom can be performed separately on rj, j=1,2,3. One final note: the U(l) transformation (3.64) (the factor 1/6 sets a conventional scale) acts triviallyon the f m , but on the other elements of T combines with SU(3) to form U(3). Note that if XE T, then

X

-t

e-,per!6 Xe,per!6 = p+Xp+

+ p_Xp_ + ei ,p/3 p+Xp_ + e- i ,p/3 p_Xp+.

(3.65) So p+X P+ - t 1 and p_X p_ - t i are invariant', and p+X p_ - t 3 and p_X P+ - t 3 transform as U(l) conjugates. So the total symmetry carried by the M m from f o to the other f m, including the trivial U(l) 3.64, is

U(2) x U(3),

(3.66)

and the quantum numbers arising from this group on the various Pauli (Weyl) spinor components are precisely those of a lefthanded lepto quark family, and righthanded antifamily. Obviously to assurne this is coincidental requires a faith and vested interest in some other scheme. To assurne it is not coincidental requires a faith and vested interest in this scheme. On the persuasiveness of this latter faith does half of this book rest. As to the other half, while strictly mathematical, it will be seen to bolster the first half.

4. Connecting to Physics. 4.1. Connecting to Geometry. The parity violation of Dirac spinor fields occurs to the level of the Pauli (or Weyl) subspinor fields. So in physies the fundamental fermion fields are Pauli spinors instead of Dirac spinors. Dirac spinors consist of a pair of Pauli spinors; the Dirac algebra is the Pauli algebra tensored with R(2). This extra R(2) is required not only of this physieal context, but also of mathematieal contexts, like using OL(2) to represent RS,Q and triality operations (see chapter 7), and in the construction of the groups of the magie square (see chapter 8). This paradigm, that the Clifford algebra of spacetime starts with a Pauli algebra for just space, and that this when tensored with R(2) provides room for time, will also be used below in the construction of a Clifford algebra for (1,9)-spacetime. Define in R(2),

ß=

[~ ~ 1,w = [~1 ~].

(4.1)

These four matriees form a basis for R(2), and a, ß, and w mutually anticommute. P L(2) ~ C( 4), the Dirac algebra, whieh is the complexification of the Clifford algebra R 1,3' Therefore a basis for R 1 ,3 can be represented in terms of the elements i and qLj of P L, and the matriees in 4.1.

84

CHAPTER 4

MultivectoT bases fOT R 1 ,3 are (j = 1,2,3): scalar vector (4.2) 3-vector 4-vector

ia,

and we define 1'0 =

ß, 1'j =

iqjW, 1'5 = -i')'01'11'2'Y3 = a.

(4.3)

The spinor space of P L(2) is p2, 2x1 matrices over P. An element A of p 2 can be decomposed into AAr, T = 0,1, and each of these is a 4-dimensional (complex) Dirac spinor. Left multiplication by the chiral projectors ~(f±1'5) project a pair of Weyl spinors from each Dirac spinor.

TL(2) ~ C(32), which is the complexification of R 1 ,9. So TL ~ R 9 ,o is, so-to-speak, the "Pauli" algebra to the (1,9)-"Dirac" algebra TL(2). [Note: The generalization I am making is an obvious one. CL ® QL becomes CL ® QL ® OL, This is a mathematically motivated generalization, for it will be remembered that the principal reason the Clifford algebra of (1,3)-spacetime requires complexification is a physical one, for without complexification we can not construct a chiral projector, which is phenomenologically required. In the Clifford algebra for (1,9)-spacetime, however, no such complexification is required, for chiral projectors can be constructed in the absence of the complex unit i. Hence, mathematics, as opposed to physics, is leading us in what folIows. The results will speak for themselves.]

CONNECTING TO PHYSICS

M ultivector bases for R t ,9 are (j

85

= 1,2,3; p, q, r = 1, ... , 6) :

scalar

€

10-vector

a.

(4.4) Of course, only half of TL(2) appears in 4.4 - the rest arises from multi plication by i. Note that with the aid of the pseudovector (lO-vector basis) we can in this case construct chiral projectors without complexification. But we are already complexified, and without the indusion of the algebra C in T we could not have constructed a basis for the Clifford algebra R t ,9' In this case complexification is mathematically required. DIMENSIONAL REDUCTION

The object space of TL(2) is T 2 , 2xl matrices over T. Each A in T 2 can be decomposed into a pair of complex (1,9)-Dirac spinors, AAr, r = 0,1. T 2 can also be decomposed into SU(2) X SU(3) family plus antifamily bits using the projectors P± and ß m (see chapter 3, section 5). In particular,

86

CHAPTER 4

Family (SU(2) doublets)

{p+ T2ßo, P+ T 2ßd SU(3)

-t

1,

{p+T2ß2, p+ T2ß 3} SU(3)

-t

3,

{p_ T2ß2, p_ T2ß3} SU(3)

-t

i,

{p_ T2ßo, p_ T 2ßd SU(3)

-t

3.

Antifamily

(4.5)

Therefore, P+ T 2 is the family half of T 2, and p_ T 2 the antifamily half. Corresponding to the family projection on the spinor space,

is the projection

ITL(2)

-t

PL+ TL(2)pL+

I

(4.6)

on the Clifford algebra, where

PL+ =

~(1 + ieL1)'

So

(PL+TL(2)PL+)[p+T 2] = p+T 2. PL+ TL(2)pL+ is a subalgebra of TL(2) isomorphie to C(16). While C(16) is isomorphie to a Clifford algebra, PL+ TL(2)pL+ should not be thought of in this way. Rather, from the Clifford algebra R 1 ,9 C TL(2) is projected the Clifford algebra R 1 •3 crossed with the 15 generators of 80(6) ~ 8pin(6) ~ 8u(4), and this Lie algebra contains the generators for color SU(3), so they survive the projection, while the extra 6 space dimensions do not. The (1,9)-multivectors surviving the projection 4.6 are 0=1,2,3; p,q = 1, ... ,6):

CONNECTING TO PHYSICS

87

Dimensionally Reduced Rt,9

scalar veetor 2-veetor

IfPL+ I

IßPL+, iqjWPL+ I IqjfPL+, iqjapL+, I eLpqfpL+,

4-vector

IiWPL+, qjßPL+, I eLpqßPL+, iqjeLpqWpL+ IiapL+, I iqjeLpqapL+, qjeLpqfPL+, ieLpqfpL+

5-veetor

qjeLpqßPL+, ieLpqßPL+, ieLpqwPL+, qjeLpqWPL+

6-veetor

ÜPL+, iqjeLpqfpL+, qjeLpqapL+, ieLpqapL+

7-veetor

ißPL+, qjWPL+, eLpqwPL+, iqjeLpqßpL+

8-veetor

qjapL+, iqjfPL+, eLpqapL+

9-veetor

WPL+, iqjßPL+

lO-veetor

apL+

3-vector

(4.7)

For example, the reduction of the I-veetors of 4.4 occurs as follows:

leaving the veetor basis in 4.7, which is a basis for the vector part ofR1 ,3PL+ (see 4.2). The rest of this isomorphie eopy of R 1,3PL+ is indicated by boxes in 4.7.

CHAPTER4

88

The ten (1,9)-vectors reduce to the four (1,3)-vectors (pL+ tacked on), but higher multivectors do not similarly reduce simply from (1,9) to (1,3). Under commutation the 2-vectors of R 1 ,9 form the Lie algebra 80(1,9). The projection 4.6 reduces this to 80(1,3) x 80(6). In all cases, however, the sub algebra 4.7 has a nontrivial effect only on the matter half P+ T 2 of T 2 • The antirnatter projection (4.8) has a similar result, the only change being that in 4.6 each PL+ becomes PL-. The subalgebra PL- TL(2)PL- is also isomorphie to C(16), and its object space is the antirnatter subspace p_ T2 of T 2 • PL±T L(2)PL± account for only half of TL(2). The remaining elements mix the matter and antirnatter subspaces of T 2 • More on this in chapter 6. On specific elements of R 1 ,9 the reductions 4.6 and 4.8 are illuminating in their distinctions. For example, the (1,9)-Dirac operator,

reduces via 4.6 to the {1,3)-matter-Dirac operator

(the signs in 4.9 were arranged so that the signs in 4.10 would be conventional), and via 4.8 to the {1,3)-antimatter-Dirac operator

. fjl,3- = PL- fjl,9PL- = (ß80" iqjw{)3)PL_. 3

L.J

(4.11)

1

Likewise, if

then ( 4.12) This associates left( right )handed matter with right(1eft )handed antirnatter , and of course that is the observed associatioll.

CONNECTING TO PHYSICS

89

Perhaps a word of explication is needed here. The matrices {'o

= ß, {'j = iqjw,j = 1,2,3,

via the maps

[~ ~ 1' iql - [~ ~ 1' iq2 - [~ -~ 1' iq3 - [-~ ~ 1'

1 = qo -

transform to

{'o -

-~o ~i 1, . . which are Dirac {'-matrices as commonly represented. The context with w'hieh we are dealing precludes the use of these conventional{'-matrices. We must play the hand we are dealt, and that is TL(2) ~ C ® R I ,9. Let

io

= ß,

'::0

= -eL7qjW,j = 1,2,3, ih = -ieLhw, h = 4, ... , 9,

be the R I ,9 generators (l-vector basis) listed in 4.4 (signs specified to accomodate 4.9, leading to 4.10). The (1,3)-part of this basis consists of the il-',11 = 0,1,2,3, but at this point these operators are not matter / antimatter specific - the same set is valid for both. In making the projections

io - PL±ioPL± = {'OPL±,

ij -

PL±ijPL± = ±{'jPL±,j = 1,2,3,

we achieve the more conventional{'l-" with the antimatter {"s space reflected relative to the matter {"~so In addition, the matter/antimatter {"s are a8sociated with the matter/antimatter projectors, PL±. They do not appear without these projectors. And finally,

ih - PL±ihPL± = 0, h = 4, ... ,9. The matter/antimatter projectors kill the extra six dimensions. We shall look at these issues in greater detail in chapter 6.

90

CHAPTER4

4.2. Connecting to Particles. Let \{I be an element of T 2 transforming with respect to U(2) (any would do). That is,

rtn

x U(3) like r~

( 4.13)

(the factor 1/2 sets a conventional scale), so \{IrQ

-U(2)

'lirQ

(invariance)

(4.14)

(recall that Ö

= a1ierql + a 2q2 + a3ierq3 ~ ÖP± =

(±a1ql

+ a 2q2 ± a 3q3)p±,

hence it is natural to associate the U(1) generator (-er) with Ö, since -erP± = ±ip±). It will be understood henceforth that by the symbol \{I we mean the canonical T 2 spinor such that \{IrQ \{I~Q (modulo U(3)). However, even though this will be understood, the former notation will be used to avoid confusion with the ~Q in the decomposition \{I = E~ \{I~m.

=

With respect to U(3) the situation is different, for all the r mare U(3) invariant. The action of U(1) C U(3) on \{I is the inverse of 3.64:

( 4.15)

With respect to SU(3) \{I transforms as 1 EB the respective terms in 4.15). In particular,

I EB 3 EB 3 (corresponding to ( 4.16)

where W'l and the 'li a , a = r,g,b,each consist ofan SU(2) doublet ofDirac spinors. The subscripts r= red, g= green, b= blue, are conventional SU(3) 'color' labels. We assign them conventional SU(3) color charges:

CONNECTING TO PHYSICS Let

91

Ja be the quantum number of SU(2).

It is associated with the action

13w,>,± = W'>'±ii/2 = ±~W'>'±. Y2, the hypercharge of U(l) C U(2), is associated with the action,

Y2 Wp±

= Wp±(-ie7/2) = +~WP±,

and Y3 , the hypercharge of U(l) C U(3), with the action

1 Y3p+Wp± = (-ie7/6)p+Wp± + p+Wp±(ie7/6) = 0 or - 3P+WP-. Below is a table of values for 13 , Y2 , Y3 for the SU(3) singlet and triplet parts of W, together with the charge Q = 13 + Y2 + Y3 .

Ja

Y2 1 -2

Y3 0

Q 0

-2

0

-1

+~

+12

-'3

1

+12

-'3

p+Wp+'>'+

= WIßO

+~

p+Wp+.>._

= WIßl

-2

p+\lfp_.>.+

= \lf3ß2

p+\lfp_.>._

= W3ß3

-2

1

1

1

1

( 4.17)

+3 1

-'3

The obvious particle assignments are:

P+ \lf P+'>'+ p+Wp+.>._ p+Wp_.>.+

-+

P+ \lf p-'>'-

-+

-+ -+

neutrino, electron, up-quark, down-quark.

(4.18)

The antifamily charges are

Y2

Y3

p_\lfp_.>._

= Wiß3

13 1 -2

+~

0

Q 0

p_Wp_,>,+

= \lfiß2

+~ +~

0

+1

p_\lfp+.>._

= Waßl

-2

-'2

p- Wp+.>.+

= Waßo +12

-'2

1

2

1

+13

1

+13 +l

-'3

( 4.19)

92

CHAPTER4

In fact, however, due to parity nonconservation the quantum numbers 4.17, and the particle assignments 4.18, are not entirely correct. Only the lefthanded Weyl spinor parts of the associated Dirac spinors have these quantum numbers. The righthanded spinors are SU(2) singlets (h = 0), and have different values of Y2. Spinors with the correct righthanded quantum numbers arise as follows. The inner product of q, with any other spinor in T 2 , a la equation 3.39, associates q, with the r m' New spinors, q,r m, arise in this way. The action of SU(2) on q, was defined so as to cancel the action on r m: ( 4.20) So each of the eight spinors P±q,r m is SU(2) chargeless. The Y2 charges for the r mare carried by the automorphisms M m from r o to rj, j=1,2,3 (see 3.62). Therefore, if

then

The U (3) charges of q, r mare the same as those for q, ~m above. SU (3) is a subgroup of the automorphism group of 0, so its action on P± q,r m is the same as that on p±q,~m, as the only term in either not SU(3) invariant is q,. Likewise, the Y3 charges are the same since e7 commutes with r m, and if e7 ~m = ±id m , then e7 r m = ±irm too.

In particular, it should be emphasized that the fact that parity nonconservation occurs with respect to SU(2), but not SU(3), is being demonstrated to arise from the fact that the r m are not SU(2) invariant, but are SU(3) invariant.

CONNECTING TO PHYSICS

93

Quantum numbers for the p+ wr mare listed below:

p+ wrt = p+wr2 =

Y2 0

Y3 0

Q

0 0

-1

0

-1

+1

t -3

+~

h

. p+ wf o = Wl f O

WIrt War 2

0

0 (4.21)

o For example, under the action of the U(1) of Y2 ,

P+ wr 2

_

P+ We(}er!2 e i(}!2f 2

=

P+ wr 2 et

'()

•

The quantum numbers 4.21 give rise to the same particle assignments as 4.17, but this time they are those of righthanded leptons and quarks. The question therefore arises, how can we mathematically associate the quantum numbers 4.17 with

I~(f. +1's)P+ Wd m =

lefthanded family,

I

(4.22)

and the quantum numbers 4.21 with

I~ (€ - 1'5 )p+ wfm =

righthanded family?

(4.23)

The antifamily charges corresponding to 4.21 are

p_wr3 = wi f 3

p_ wf t = Wsf t p- wfo =

h

Y2

Y3

Q

0

0

0

0

o

+1

0 -1

wsro 0 0

+~

-~

+~

+1

(4.24)

CHAPTER4

94

In this case we'd want to associate the charges 4.19 to

!(

f -

'Ys)p_1lJ ß m = righthanded antifamily,

(4.25)

and those in 4.24 to

I!( + 'Ys)p_llJr f

m

= lefthanded antifamily.

(4.26)

4.3. Parity N onconservation. RIGHTHANDED DIRAC OPERATOR.

Define

1

L± = '2( f 1

+ 'YS)PL±,

R± = '2(f-'YS)PL±.

(4.27)

Define (4.28) where 0'0 = 1, and O'j = iqj, j=1,2,3. Let llJ in T 2 be a space-time field (for nOWj later we'll consider llJ a (l,9)-space-time field). Consider the inner product

where (4.30) and (4.31)

CONNECTING TO PHYSICS

95

is the spinor eonjugate. In 4.29 eonsider the term (4.32) The derivative in 4.32 'sees' both WR and r n, so it sees the U(2) x U(3) aetions on eaeh. That is, in making this internal symmetry loeal, gauge fields arise eonsistent with the eh arges 4.21 associated with righthanded leptoquark fields. Note that SU(3) gauge fields survive beeause the r n are invariant, and so have no effect on the gauge fields arising from the action of SU(3) on q,. LEFTHANDED DIRAC OPERATOR A trick is used to manifest the eorreet gauge fields for lefthanded leptons and quarks. Define (4.33) where CfÖ = Cfo, Cf; = -Cfj, j=1,2,3, and

is differentiation from the right. That is, it is assumed that since 0 00 [ Cf;;

1PL+ = M

1(

[00 Cf,.,. 0

1PL+),t

( 4.34)

where A -+ M 1(A)t is an antiautomorphism (product reversing), that this reverses the direction of the action of the derivative (ie., under the action of this antiautomorphism,

Take the inner produet

< q"

~L+ q, >= ~([L Mm((q,r n)t\(~L+ q,rn))] + [same]t}, mn

( 4.35)

96

CHAPTER4

( 4.36)

-

In 4.36 the derivative ()J.' sees only 3

Ilf L =

I: Ilf LD. m ,

o hence only the U(2) x U(3) actions on this field. Therefore gauge fields associated with the charges 4.17 arise in this case, those of lefthanded leptons and quarks. SU(3) gauge fields are the same as the righthanded case. FULL PARITY VIOLATING DIRAC OPERATOR The operator (4.37) differs from "1,3+ (our earlier (1,3)-matter-Dirac operator defined in 4.10) only in the direction of the derivative in the lefthanded part, so it is also a valid (1,3)-matter-Dirac operator. Its advantage, of course, is that its use in the spinor Lagrangian term

l.co =< 1lf,"+1lf >= 1{[Emn Mm((wrn)"(,,+wrn))] + [same]t} I

(4.38)

results after gauging in the ohserved lepto-quark gauge field interactions. To this point the only unconventional result is the appearance of a chargeless righthanded neutrino. And, of course, the structures heing manifested are intrinsic to the mathematics. Because of the appearance of p+ in .co is only a matter Lagrangian, and it gives rise to only matter to matter transitions. That is,

"+'

.co =< Ilf, ~+ Ilf >=< p+ Ilf, ~+p+ Ilf > . The p_1lf parts of W fall out in taking the inner product, hecause in general

< p±A,B >=< A,p±B >=< p±A,p±B > (p± could he replaced by any Hermitian associative projection operator). An

antirnatter version of .co is easily constructed along the lines outlined ahove, hut in the next few sections attention will he restricted to matter and .co.

CONNECTING TO PHYSICS

97

4.4. Gauge Fjelds. The core of the Lagrangian Co consists of the terms

where Define ( 4.40) These are alternate SU(2) generators, and T3 = x is diagonal with respect to the projection operators A± = !(1 ± ix). Consider the n=O term of the lefthanded part of 4.39. U(2) gauge fields arise as follows (p+ q,"r o is U(3) invariant ):

(q," LrO)t( a; p+ Wd~gAJ.ljTj - !g' BJ.l e7 ]rO)

+ ßl)[!gAlLjTj + !g'iBJ.ljr o)

=

(q,"LrO)t(a;p+q,"L(ßo

=

(q,"LrO)t(a;p+q,"LßO[!gAIL3x

+ !g'iBIL]ro)

+(q,"LrO)t(a;p+WLßl[!g(AJ.llql ~

+ AJ.l2q1X )]rO)

(NLrO)t(a; Nd!ßoi(g'BIL - gAJ.l3)j r O)

(ß2,ß3 terms drop 01 (ßO(TI

+ T2)r O = 0)

(ßt(T3

+ i)r o = 0)

(ßoxro = -iro)

+(NLrO)t(a; Ed!ßlqlg(AJ.ll - iAJ.l2)]ro) (4.41) (summation over repeated indices), where

is a lefthanded neutrino field, and

is a lefthanded charged lepton field. The last step in 4.41 is not a strict equality hecause (q,"LrO)t = «p+ + p_)q,"LrO)t, hut in completing the inner product 4.38 the (p_ q," LrO)t term drops out. 9 and g' are conventional gauge coupling constants for SU(2) and U(l).

98

CHAPTER 4 In like manner, setting n=1 in the lefthanded part of 4.39, we obtain

(IJILfdt(a;p+lJld!gAi'jTj - !g'Bi'e7]fd ~ (ELfdt(a; Ed!ß1i(g' Bi'

+gAi'3)]ft)

+(ELr1)t(a; Nd!ßOqlg(Ai'l

+ iAi'2)]f 1).

(4.42)

Another word in explication is needed here. The last lines of 4.41 are those parts of the first line that survive the inner product. The gauge field terms, and the indecomposable (with respect to U(2) X U(3)) Pauli spinor bits of IJI, are all easily identifiable with physical particles. Combinations of these algebraic pieces of the gauged Lagrangian that survive the inner product correspond to viable transitions; those that do not are not viable. They will not occur. No actual cross-sections or probabilities are being calculated here. As yet we may only read from the mathematics a binary resu1t: yes, this transition is allowed; no, that one is not. The algebraic gauge fields are indicated by square brackets in the last terms of 4.41 and 4.42, and from these we can figure their Y2, 13 (and Q = Y2 + h) charges (U(3) invariant). Only two of these charges are nonzero. . The first is:

where we have exploited the fact that 1'_ A

-2" zxu lql

1._ + ulql2"ZX = uIQ1· A

A

So h = 1 for this gauge field, and Q

= h + Y2 = 1 + 0 = 1,

and we make the conventional identification (4.44) So the last term in 4.41 represents the transition (4.45)

CONNECTING TO PHYSICS

99

Likewise, (4.46)

(J3

= -1, Y2 = 0) and the last term in 4.42 corresponds to the transition (4.47)

It will be noted that the issue of second quantization has not yet arisen. When algebraic fields identifiable with particular particles (by their symmetry transformation properties) appear in combination in a Lagrangian, and the combination survives the inner product, then an allowable transition results. Part of the U(2) X U(3) gauging of Lo, denote it Lg (ie., Lg is distinct from Lo and contains all the gauge field-fermion interaction terms), is 1

< EL, 0';; Nd .J2ßOQ19WiL-j >1 0, which in having a non zero real part implies 4.47 is an allowable transition. On the other hand, 1

< EL, 0';; Ed .J2ßOQ19 WiL-j >= 0, so EL - EL

+ W-

is not allowed.

Two other conventional definitions are made:

Z iL =

g' B/J _ 9 AiL3 (9 2 + 9 12 )1/2

::'-':---::,.,....,...,c::-

(4.48)

and (4.49) so

(4.50) (with 99' e - ---:...:..--..,-= - (g2 + g/2)1/2 the electromagnetic coupling constant). These arise in the terms

(4.51)

CHAPTER4

100 from 4.41 (neutral current), and

from 4.42 (electromagnetic and neutral currents). The righthanded term in 4.39 also gauges in the conventional manner. However, this term contains the following: (4.53)

(NR = p+WRrO). NR is a righthanded neutrino. Since it is U(2) x U(3} chargeless, it does not appear in Cg • The presence of a righthanded neutrino in this theory is a prediction, along with all that it would imply in astrophysics, in particular regarding the problem of dark matter. Its only inter action may be gravitational.

4.5. Weak Mixing. The symmetry of equation 3.40 is larger than U(2) x U(3). There are extra U(1)'s and an extra SU(2) (see 3.47). Let ( 4.54)

U1 E U(1), U2 E SU(2). The remaining rj,j = 1,2,3, differ from Mj(r o} by the extra U(1} and SU(2} phases. Let

and let P

= exp(ßp_).

Therefore,

pr m = pr m

r m,

m = 0,1,

= eßr m , m = 2,3

(see 3.48). Since P has a nontrivial effect only on r 2 and r 3 , it can be used to account for the additional SU(2) freedom in the r m • We may express the

101

CONNECTING TO PHYSICS

rm

as follows:

(4.55)

In the square brackets are incorporated the additional symmetry bits, but outside there appears a revamped

Since

pu2 P- 1 r m

=

u2 r m ,

m = 0,1,

the SU(2) gauge fields of the previous section are unaltered by this change. This is true even if we alter the U (2) action on the matter spinor field p+ W: p+1I1

-+U(2)

p+II1(PU;lp-l)U1-

1

+p+ Wß2(eßU;le-ß)Ul-l

+ P+ II1ß3(eßU;le-ß)Utl.

( 4.56) That is, the neutral and charged lepton fields, p+ 111 ßo and p+ Wßl, are unaltered. Not so the quark fields, p+ 111 ß2 and p+ 111 ß3. It is necessary in wh at follows to specify that the Bj, j = 1,2,3, and the components of 13, are to be treated as constant phases. Only the U(2) x U(3) previously defined will give rise to gauge fields. For the moment the justification for this a8sumption will rest on its consequences. Making use of 4.55,56, the lefthanded n=2 part of the Co terms 4.39 transform into {(WL(PUi 1 p-l )U1- 1 U1 (PU2 P-l )[ei92 Pß2])tX

(0'; P+ WL(PU;l p- 1 )Ut 1 {FUl(PU2P- 1 )[e i92 Pß2])}

(4.57)

102

CHAPTER4

Including U(l) C U(3), but ignoring SU(3), this gauges as follows:

(p+'I1LPr 2)t(u;;p+'I1LP[!gAlLiTj - ~g'Bl'e7]r2) ~ (p+'I1LPr2)t(U;;P+'I1L(~2

(reverting back to the symbol set

+ ~3)P[!gAlLiTj -

(4.58) ~g'BlLe7]r2)

r 2 instead of the ~2 appearing in 4.57).

Now

ß real, so that

Also,

for the canonical

r2

(=

~2)'

In this case 4.58 reduces to

(p+ '11 Le- iß / 2r 2)t( u;; p+ '11 L~2e-iß/2[ !gAl'jTj - ~gl Bl' e7]r 2) +(p+ '11 Le- iß / 2r 2)t( u;; p+ '11 L~3e+iß/2[!gAl'iTj -

ig' Bl'e7]r2) (4.59)

P+'I1~2 is a Q = ~ quark, P+'I1~3 a Q =

U

= up-quark, C = charm quark, and

-! quark.

Let

CONNECTING TO PHYSICS

103

D = down-quark, S = strange quark. Then 4.59 can be written

+«UL + iCL)r 2)t(u;(DL + iSL)eiß[~qlgw+/Llr2) =

((UL)r 2)t(u;(UL)[i( -~eA/L +

392_9'2! 6(g2+g l2)

z/L]r 2)

( 4.60)

+(ULr 2)t(u;(DLCOS({3) - SLsin({3))[hq19W+/Ljr2) +(CLr 2)t(U;(SLCOS({3) + DLsin(ß))[~qlgW+/Ljr2) +i{(ULr 2)t(U;(SLCOS({3) + DLsin(ß))[~qlgW+/Ljr2) -(CLr 2)t(u;(DLCOS(ß) - SLsin(ß))[~qlgW+Jl]r2)}. By replacing p+ W, = (electron family)

P+W'

+ i(muon family)

= (electron family) -

i(muon family),

the terms in 4.60 delimited i{ ... } change sign, so in adding the Wand w' versions tagether, these terms drop out. What remains are the U-to-U and C-to-C electromagnetic and neutral currents, plus charged currents with

104

CHAPTER 4

families mixed. For example,

(ULf 2)t((1;;(DLCOS(ß) - SLsin(ß))[~qlgW+lLlf2)

=

(ULf 2 )t( (1;;(D L)[~qlgW+lLlf2)cos(ß)

-( ULf 2)t( (1;; (SL)[ ~qlgW+lLlf 2)sin(ß) indicates a transition of DL via W+ to UL, proportional to cos(ß), and a transition of SL via W+ to UL, proportional to sin(ß), this latter transition intergenerational. The angle ß is just the Cabbibo angle, and the mixing is just conventional weak mixing. Leptons do not similarly mix because the extra SU(2) phase does not appear in those cases (see 4.56, where p+"\[Ißo and p+"\[I ßl are lepton terms). As to righthanded fields, because they are free of SU(2) charged currents, they are free of mixing as weIl. Certainly this is not a derivation of the Cabbibo angle. It nicely explains why leptons are immune to weak mixing, but it doesn't explain the exact SU(2) phase needed for this purpose. Nor does it explain why there should be another family, nor force the inclusion of a third. When I first began work with the tensored division algebras I assumed that the occurance in T of a family plus antifamily was an indication that eventually there would be found to be an even number of families. Experimental evidence seems to indicate there are only three. For the time being, therefore, I consider the generation structure of this model an open question, with the ideas presented in this section an indication of how things might work out for the first two generations.

CONNECTING TO PHYSICS

105

4.6. Gauging 8U(3). A set of eight generators for SU(3) are:

Al

= ~(eL26 -

eL34),

A2

= ~(eL45 -

eL6t),

A3

= ~(eL56 -

eL41),

A4

= !(eL13 - eL52), (4.61)

A5

= ~(eL63 - eL24),

A6

= ~(eL35 - eL12),

A7

= ~(eL23 - eL15),

A8

= H-eL23 -

eL15

+ 2eL46)

(we'lliet the context distinguish the A-projection operators from these generators, which will only be employed in this section; this notation for SU(3) generators is conventional, and originated with Gell-Mann). The last two ~re diagonal. We see this by operating the Hermitian generators ~A7 and ~'\8 on p+er, r = 1,2,4 (since

and we have identified the p+e n r For example,

= 1,2,4, directions

as red-green-blue).

CHAPTER4

106 and

=

-![p+el]'

In like manner we determine a complete table of SU(3) charges for p+er. r 1,2,4:

p7[p+el]

.

1 = -'2[p+el]

and

~A8[p+el]

= -~[p+el]'

~A7[p+e2]

= +![p+e2]

and

~A8[p+e2]

= -~[p+e2]'

= 0[p+e3]

and

~A8[p+e3]

= +Mp+e3]'

~A7[p+e3]

=

(4.62)

These quantum numbers justify the red-green-blue labels above, and the use of the Gell-Mann symbols Aa , a = 1, ... ,8, to signify the generators. The antimatter eh arges of p_1T! are opposite those above (beeause ie7P± = ±p±). SU(3) gauge fields arise from the (1,3)-matter Lagrangian 4.38 as follows: ( 4.63) where the G/Ja are the gluon fields (I' = 0,1,2,3, a = 1, ... ,8). Onee again we ean obtain from those parts of CSU (3) that ean eontribute a nonzero part to the result of the inner produet 4.63 a list of possible transitions. For example, eonsider the simpler looking term 1

= -2'(e4P+)(p+(e6(e3e2) -

e2(e4e2)))

= (e4P+ )(p+e4) = e4P+p+e4 = e4P+e4 = p_(e4)2 = -pI

=

1.

-2' + 2ze7 ( 4.64)

107

CONNECTING TO PHYSICS

(the nonassociativity of 0 plays no role after the second line), which has a nonzero real part. The occurance of this expression in SU (3)' together with its contributing a non zero real value to the inner product, tells us that the green quark, p+e2, can interact with "Yp.Gp.5 A5 to produce a blue quark, p+e4· In addition,

.c

1 =-"2(e2P+)(p+(e6(e3e4)e2(e4e4»)

= = = =

-(e2P+)(p+e2) -e2P+p+e2 -e2P+e2 -p_(e2)2

= P1 1. = "2 - 2U7 ' which again has a nonzero real part, so with the help of "Yp.Gp.5 A5, a blue quark, p+e4, can trans form to a green quark, p+e2. However,

= (etP+)(p+e4) = (( etP+ )p- )e4 = (et (P+P- »e4

=0 (nonassociativity is responsible for P+ p_ in the second line), so the green quark, p+e2, can not interact with "Yp.Gp.5 A5 to produce a red quark, p+et. On the other hand, although --j.

= = = =

(etp- )(p+e4) ((etp-)p-)e4 (etp-)e4 (p+et)e4 = p-(et e4) = -P- e3 ~ 0,

108

CHAPTER4

it has no real part, hence drops out of the inner product, so the green quark, p+e2, can not interact with 'Y/JG/J5AS to produce an antired antiquark, P-el, either. Note that the space index J.L in 4.63 appears in 'Y/J' not just (T/J' and that Wis the fullleft-right spinor hyperfield. That is, parity is conserved in this case, which is a direct consequence of the SU(3) invariance of the r m, in contrast to SU(2). In the next chapter we shall see that SU(3) is also necessarily exact. Also note that index doubling takes el -+ e2 -+ e4 -+ el, and makes corresponding changes in the Aa • Index quadrupalling takes el -+ e4 -+ e2 -+ e}, with yet another set of Aa • Doubling again gets us back where we started. One mlght conjecture that these three representations are somehow linked to there being three generations, but this is admittedly numerological groping, and it will not be pursued here. In fact, please forget I even mentioned it. There are more likely candidate explanations for family replication.

5. Spontaneous Symmetry Breaking. 5.1. Scalar Fields. Of the positive attributes of model building with T, rigidity ranks high on the list. The ability to mirror the structural features of our physieal reality with flexible mathematies bespeaks more the talents of the model builder than any fundamental connection of the mathematics to reality. Ta lay claim to such a connection a mathematieal idea must, in my opinion, satisfy at least the three qualities that so distinguish T: selectness; inflexibility; and consistency with those aspects of physieal reality it is intended to explicate. This chapter should highlight these qualities. Two kinds of fields have already been defined on this developing context: the fermion hyperfield, '1; and the gauge fields BI-', AI-'i, and GiJ a • The former resides in T 2 , the spinor/object space for the external, or geometrie algebra TL(2), and also for the internal algebra PR. The latter fields reside in, or are associated with, these adjoint algebras. The former are in same sense primary, the latter derived or secondary. The inner product, < ... , ... >, is defined for primary fields, elements of T or T2, and in what follows we shall be considering scalar Lagrangians of the form

where is a scalar field. Because our is not a gauge field, and because we shall be required to take its inner product, it is defined here to be a primary field. This, as it turns out, is a nontrivial constraint. We may define a field to be scalar with respect to a given geometry if its algebraic action on a spinor field commutes with the action of the geometrie (Clifford) algebra of the geometry on the same spinor. For example, let be

109

CHAPTER 5

110

a scalar with respect to R 1 •9 s:; T L(2). In this case we may assurne 4> E P, and that its algebraic connection to 111 takes the form,

(5.2) Note that as OL = OR s:; R 1•9 ,

4> must be free ofhypercomplex octonions, and if dependent on hypercomplex quaternions it must connect to 111 from the right to commute with QL s:; R 1 •9 • A second possibility is to consider scalars with respect to the subalgebras PL±TL(2)PL± (see 4.7), the matter and antirnatter projections from TL(2). Multiplication from the left by e7 on 111 commutes with these algebras, so the dimension of the scalar space is doubled, and we may define a new action on 111 by, (5.3)

where 4>± E P. [Note that although OL = OR, eL7 =f eR7, and in fact eR7 does not commute with PL±TL(2)PL±, even though eL7 does. For example,

hut

Hence the commutator [PL+eL12 , eR7]

=f 0,

so eR7 does not commute with PL+ TL(2)pL+.] This scalar has the advantage that it distinguishes matter from antirnatter, but it has the problem that in giving masses to leptons (via spontaneous

SPONTANEOUS SYMMETRY BREAKING

111

symmetry breaking), it is unable to do the sam.e for the quarks, which are not diagonal with respect to the resulting mass operator. Consequently we shall loosen the definition one final step (the last possible) and take our scalars with respect to the matter and antirnatter Dirac algebras, R 1 ,3PL± (R 1 ,3PL+ is the boxed subalgebra in 4.7). These algebras commute with eR7 as weH as eL7, which again doubles the dimension of our algebraic scalar space. In this case we define the connection to \l1 by

IP+ \l1 P+4Jo + P+ \l1 P-4JI + P- \l1 P+4J2 + P- \l1 P-4J3, I

(5.4)

where 4Jm E P, m=O,1,2,3. Recall that we have made the following identifications: P+ \l1 P+ = lepton doublet; P+ \l1 p_ = quark doublet, color triplet; p_ \l1 p_ and p_ \l1 P+ respective antirnatter hyperfields. Each of the cPm is an element of P, hence SU(3) singlets, so they mix the doublet parts of these hyperfields, but not the SU(3) triplet parts. Consequently spontaneous symmetry breaking can occur only to the SU(2) level. SU(3) is necessarily exact. As was the case with parity nonconservation (see section 4.3), the exactness of SU(3), and the breaking of SU(2), follow from the mathematics.

5.2. Scalar Lagrangians. Let cP E P be one of the scalar fields defined in 5.4. Define

l.coP = E~=o < ovcP,E)v cP > _j.t2 < cP, cP > -A < cP, cP >2 .1

(5.5)

This is very nearly as conventional a scalar Lagrangian as it looks, even though cP in its present form is not an SU(2) doublet, or even a pair of doublets. If eä / 2 E SU(2) (see 3.55 and 4.13), it is natural to define the SU(2) transformation of 4J by:

(5.6) (so cP would be U(l) invariant». cP therefore transforms like the direct sum of an SU(2) singlet and triplet.

CHAPTER 5

112

However,

and under SU(2), (5.8) That is, .+>._ = >._>.+ = 0, >.~ = >.±. Also, ql>'± = >'=Fql. Decompose

1> = >'+1>>'+ + A+1>>'- + >'-1>>'+ + >'-1>A-

where

1>++,1>+-,1>-+,1>-- E C. Therefore, for n=0,2,

[-~g'iBIl - ~9Alliri]1>r n

=

[... ](1)++A+

+ 1>-+Qt>.+)rn

+{~WIl- - i(eAIl

+9z2 ZIl )Qd1>-+rn , (5.13)

where the n=1,3,

>.± have been absorbed into r n in the last step. Likewise, for

[+!g'iBIl- !gAJti ri ]1>r n

=

[... ](A+Ql1>+-

=

{i(eAIl

+ >'-1>-_)r n

+9z2ZIl)Ql + ~WIl+}1>+_rn

+{ -~ WIl- Qt - igztZIl}1> __ r n' (5.14) As usual, when J.l2 in 5.5 is taken negative, 1> takes on a nonzero vacuum expectation value (vev) [12], in this case somewhere on the 7-sphere

< 1>,1> >= constant

(5.15)

CHAPTER 5

114

(which is the equation of a 7-sphere). Because AI' is identified with the massless photon field, the vacuum expectation values of

= < p_(iI1~ + iI1~el + iI1:'e2 + iI1~e4)'

p_{iw(el(ß4 + i8S)'li~

+ ... ) > .

(6.15) Since, for example, < P_el,p_el >f:. 0, the last line above indicates a transition . W~ = lepton --+ antired-quark = 'li ~. (6.16)

122

CHAPTER6

In fact, the last line in 6.15 gives rise to all possible transitions from fields with SU(3) color white, red, green, and blue, to antifields of all anticolors save the one initiating the transition. That is, the red-quark may go to antiwhite (antilepton), antigreen, or antiblue, but not antired. In any case, baryon and lepton numbers are not conserved in these transitions. It is also curious that these transitions are not mediated by any gauge fieldsj it is the operator fJo 6 itself that provides the necessary compensating charges. Reca11 that as c~lor SU(3) = Spin(6) n G2 , the color symmetry SU(3) is a subgroup of the (1,9)-Lorentz group, and the operator fJo,6 is not an SU(3) singlet. An objection might be made to this on the grounds that color SU(3) is an internal symmetry, and therefore ought to commute with the spacetime Clifford algebra. However, there is absolutely no evidence that SU(3) must commute with any Clifford algebra larger than R 1 ,3, that of our observable spacetime. Indeed, there is also no evidence for the extra six dimensions of Rl,9. That there are an extra six dimensions, and that they carry color SU(3) charges, is a prediction of model building on T. To better understand these charges, and to figure out some conditions sufficient to rid us of the last two terms of 6.10, we must first look more closely at SU(3).

6.2. More 8U(3). The action of SU(3) on p+X, where X E 0 (or C ® 0 or T), was developed in section 3.6. For convenience 1'11 reproduce the principal results here. In particular, SU(3) is generated by elements of the form: eLpq - eLrs, where epe q = ere s , and p, q, T, S i= 7. These generators leave 1 and e7 invariant, hence also W~. Their action on W±, m= 1,2,4, can be represented as a C( 3) action on the eolumn [

:f ].

The corresponding matriees are:

10 DIMENSIONS

!( €L34 !(€L61 -

123

[: [;

€L26) -

eL4S) -

: ] ,

: ] ,

~(eL63 - eL24) ~ [ ~(eL56 - eL',) ~

~(eL5' - eL13) ~ [ ; :1' ~(eL35 - ew)

~(eL46 - eL23) ~ [ :

i

:.] , -z

-1

i ],

~l

1 ~ ~1 J ~ [: J [

(6.17)

1

[

~(eL23 - ew)

-i

Th. eo"esponding matrie.. ",ting on th. eolumn [

:~ ] are the cornplex

conjugates of those in 6.12. SO(6), hence also SU(3), acts on RO,6. The action of SU(3) may be determined by commuting the generators listed in 6.17 with the element 6

X

= EXP+3iW€Lp 1

of R 1,9 ~ TL(2). Since these generators commute with iw, we may look only at their action on the eLp. For example, the commutator

[~( €L34 -

6

€L26),

EX

P+ 3 eLp]

= X g eL2

-

X 7€L3

1

Therefore, under the action 6.18,

[

:.

-i

+ X 6 €L4 -

X S€L6. (6.18)

124

CHAPTER 6

Likewise the commutator

[~(eL56 -

6

eL4t) , EXP+3 eLp ] =

-X 7eLl

+X 4eL4 -

X geL5 + X 8eL6. (6.20)

1

-+

+ ~X: ] . [ -X7X +0 iX9iX8 ]_-[..1... .-1] [ XX:X7 +zX + iX9 4 -

X4 + iX 8 [ Therefore X5 + iX 6

(6.21)

]

transforms with respect to SU(3) under a represenX7+iX9 tation the inverse O[f ~~t~ni~~l]e 6.17, which acts on P+ Wp_, a quark-matter

X5 - iX 6 transforms under a representation the inX 7 - iX9 verse of that on p_ WP+, an antiquark-antimatter spinor.

spinor. Likewise,

6.3. Freedom from Matter-Antimatter Mixing. If fis a suitably smooth function of a complex variable, z

= x + iy, then

(8x

+ i8y)f(x + iy) = O.

(6.22)

(8x

-

i8y)f(x - iy) = O.

(6.23)

Likewise, The parts of .c 1 ,9 (see 6.5) that give rise to matter-antimatter mixing are those in which the differential operator tJo 6 appears. In 6.9 it was shown that tJ06(P+W) can be rewritten using the ~peratorR 8 4 ± i88 , 8 5 ± i86, and 8 7 ± i89• Combining 6.14 with 6.22 and 6.23, it is apparent that if

10 DIMENSIONS

125

where I-" = 0,1,2,3, then (6.25) identically (for the corresponding bits of p_1)} we would require the complex conjugate dependencies). The grou p U(3) acts transitivelyon S5, the 5-sphere determined by 9

}:)xP)2 = r2• 4

Consequently, since I)}~ is supposed to be U(3) invariant, we should expect that its only dependence on x P , P = 4, ... ,9, be radial (a function of r). But this is not possible given the functional dependence in 6.24, for no nontrivial function of the complex tripie

(x 4 + ix 8,x 5 + ix 6,x7 + ix 9) = (zt, z2,z4) can be dependent only on

r

= (zl zh + z2 z 2.. + z4 Z4*)L

Arguing along these phenomenologicallines we might then expect I)}~ to be independent entirely of x P, p = 4, ... , 9. Likewise we might expect I)}~, identified as the red quark, to be independent of x 5 , x 6 , x 7 , x 9 , which transform nontrivially with respect to the SU(2) subgroup of SU(3) that leaves red (ie., et, e5, and xl +ix 5 ) invariant. Carrying on like this leads to reduced dependencies:

where , ., indicates independence of that coordinate. Because of the results of section 6.2 it is clear that the Ro,6 dependencies in 6.24 have complicated SU(3) transformation properties. The dependenci.. 6.26 are phenomenologically preferable. Let

w\'''''

~

[n l'

and

CHAPTER 6

126

z=

[

~: ~ ~~: ]. Then if U E SU(3), we have the transformation

X 7 + iX 9

(6.27) where even after such mixing, for example, we expect (Uq,~ark)l (first component) to be independent of the second two components of U- 1 Z. Finally, the dependencies 6.26 (or 6.24) reduce Cl ,9 to

IC l ,9 =< P+ 1l1, 9Jl,3P+ ll1 > + < p_ll1, 9Jl,3P- ~ >, I

(6.28)

which is where we want to be. Although I am not astring theorist, it is interesting that in both string theory and the context developed here the space Rl,9 should playaprominent role. Morevover, in string theory the extra six dimensions are balled up into a small complex 3-manifold, inorder to ren der them unobservable, and in the T context, inorder to be rid of matter-antimatter mixing, the extra six dimensions should also appear as a complex tripie (although it is unclear if there is a unique compactification arising from this context, much less if there is an associated SU(3) holonomy group).

6.4. (1,9)-Scalar Lagrangian. Redefine

IC", = L~=O < oa4J, oa4J > _J.L2 < 4J,4J > -A < 4J,4J >2 .1

(6.29)

This is the obvious generalization of 5.1 from R l ,3 to R l ,9. That is, now it is assumed (6.30) 4J = 4J(x a),a = 0,1, ... ,9, rat her than merely

4J( xlJ.), J.L = 0, 1,2,3. Euler's equations, (6.31)

applied to 6.24, imply 9

-2J.L2i - 4Ai < 4J,4J >= 2 ~)OaOa)4Ji. o

10 DIMENSIONS

127

Therefore 3

9

o

4

'2){}jJ{}jJ )(Pi + '2) {}p{}P) + < x,go[y], z > + < x, y,g2[Z] >= 0·1

(7.122)

CHAPTER 7

170

By 7.96 the second term can be expressed

= «go[y])t,xz>

inner product

= < Togo[Y] , xz > = < TogoTo[ytJ, xz >

(7.123)

< yt, (TogoTo)t[xz] >

= < yt, -TogoTo[xz] > . Together with 7.96, this implies we may rewrite 7.122

< yt,(gl[X])Z > + < yt,-TogoTo[xz] > + < yt,X(g2[Z)) >= 0, which implies (7.124)

wh ich is an expression of Freudenthal's principle of triality [1]. We see from 7.123 that 80(8) almost acts as a derivation on the octonions (and of course, elements of LG2, a Lie subalgebra of 80(8), do act as derivations, since G 2 is the automorphism group of 0).

7.4. LG2 and Tri. In order to prove that some 9 in the adjoint algebra of 0 is an element of LG 2 we need merely prove it acts as a derivation on O. That is, that

g[xz]

= (g[x))z + x(g[z]).

(7.125)

Our claim is that LG 2 is spanned by elements of the form 9

= eLab -

eLcd :

a, b, c, d E {1, ... , 7} distinct, g[lJ

Notice that in the particular case a = 2, b = 3, c Tri{eL23 - eL46}

-eR23

= O.

(7.126)

= 4, d = 6,

+ eR46 (7.127)

=

eL23 - eL46·

171

DOORWAYS Also,

(7.128)

As this same result can be duplicated for any gof the form 7.126, it should now be dear in general that for 9 of this form

Tri{g} = g, and

T og7'o = g. Combined with 7.124 this implies that

TogTo[xz]

g[xz] (Tri{g }[x])z + x(Tri 2 {g }[z]) (g[x])z + x(g[ z]).

Therefore gis a derivation of 0, and

Since the set of such 9 is 14-dimensional, as is LG 2 , the span of all such 9 is in fact LG 2 , as daimed.

LG2 TRIALITY TRIP LET The copy of LG 2 spanned by elements of the form 7.126 generates the automorphism group of O. Each element of this Lie algebra is invariant with respect to the action of the triality map Tri (and with respect to the map 9 ~ TogTo). In this sense this copy of LG2 is a triality singlet. There is also a triality triplet of copies of LG 2 which is constructed below. The LG 2 defined by 7.126 can be seen to arise in the following way from the Clifford algebra RO,6 ~ 0 L. This can be represented in two natural ways

CHAPTER 7

172 (p,q,rE {1, ... ,6}):

scalar 1L 1L vector eLp eLp7 2-vector eLpq eLpq 3-vector eLpqr eLpqr (7.129) 4-vector eLpq7 eLpq7 5-vector eLp7 eLp 6-vector eL7 eL7 Recall that under the commutator product the 1- and 2-vectors of Ra,n generate so(n + 1). Therefore, both the sets {eLp, eLqr : p, q, rE {I, ... , 6}}

(7.130)

and {eLp7, eLqr : p, q, rE {I, ... , 6}} = {eLab : a, bE {I, ... , 7}}

(7.131)

form bases for so(7). And in both cases the Lie subalgebras which kill the identity are co pies of LG 2 • In the case of 7.131 this yields the LG 2 defined in 7.126. In the case 7.130 it yields a copy of LG 2 spanned by elements of the form 9

= eLpq -

eLrs : p, q, r, sE {I, ... , 6} distinct, 9[1]

= 0,

(7.132)

which generate su(3), together with the 6-dimensional subspace of elements ofthe form (one for each p = 1, ... ,6)

9

= 2eLp ~ eLqr -

eLst : p, q, r, s, tE {I, ... , 6}, 9[1]

= o.

(7.133)

That yields 8 + 6 = 14 elements all together. The 8-dimensional su(3) sub algebra is shared by this copy of LG 2 and that defined in 7.126, so it is invariant under the action of Tri. The remaining 6-dimensional subspace is not shared, and is not Tri invariant. In particular, consider the action of the triality map on the element 2eLl - eL26 - eL34 of this subspace. Tri{2eLl - eL26 - eL34}

=

2eRl

+ eR26 + eR34

DOORWAYS

173

Therefore Tri maps the 6-dimensional space of elements 7.133 to the 6dimensional space of elements of the form Ig

= 2eLp1+ e Lqr +eLst:

p,q,r,s,tE {1, ... ,6}, g[e11

= 0·1

(7.134)

These elements, together with the su(3) in 7.132, elose under the commutator product to form another copy of LG 2 • In this case, however, the G 2 generated by these elements leaves e1 invariant, instead of the identity. Carrying on in this manner, we note that

=

-2eR51 - eR26 - eR34

= -( eLl +eL26 +eL34 -

eL51)

(7.135)

=

-2eLl - eL26 - eL34·

Therefore Tri maps the 6-dimensional space of elements 7.134 to the 6dimensional space of elements of the form 9

= -2eLp -

eLqr - eLst : p, q, r, s, tE

{I, ... , 6},

g[e1]

= 0,

(7.136)

which again with the su(3) in 7.132 elose under the commutator product to form another copy of LG 2 • Finally,

Tri 3 {2eLl

- eL26 - eL34}

= Tri{ -2eLl - eL26 - eL34}

+ eR26 + eR34

=

-2eRl

=

-( -eLl

=

2eLl - eL26 - eL34,

+ eL26 + eL34 + eL51)

CHAPTER 7

174

which is where we started from. Therefore, the three LG 2 's consisting ofthe su(3) 7.132 together with the 6-dimensional spaces 7.133, or 7.134, or 7.136, form a triplet with respect to Tri. The first of these, like the LG 2 defined in 7.126, generate copies of G 2 that leave the identity invariant, hence act nontriviallyon the 7-dimensional subspace of 0 spanned by {e a , a = 1, ... , 7}. The last two generate G 2 '8 that leave e7 E 0 invariant, hence act nontriviallyon the 7-dimensional 8ubspace ofO spanned by {l,e p ,p= 1, ... ,6}. It is interesting how often the (1,3)-motif arises in the context of the division algebras, but no speculation on the meaning of this example is offered at this point. As noted earlier, if 9 takes the form 7.126, then

TogTo = g.

(7.137)

Likewise, if 9 takes the form 7.132, then, for example,

=

-( -eLl

+ eL26 + eL34 + eL57)

That is, elements 9 of the form 7.132 also satisfy 7.137. On the other hand,

-!( +eLl + eL26 -

eL34

-2eLl - eL26 - eL34·

+ eL57)

175

DOORWAYS Therefore,

So elements ofthe form 7.134 and 7.136 are interchanged by the map, 9 ~ T 0 9T o. Therefore, for example, if 90 is an element ofthe first LG 2 (7.132,133),91 the second (7.132,134), and 92 the third (7.132,136), then by 7.124 (Freudenthal's principle of triality), for all x, z E 0,

since 90 satisfies 7.137. In the next section we'll give some meaning to this, and other, LG 2 triality triplets.

7.5. LG2 Triplets and the X-Product. Some preliminary results are needed. Let 9 E 80(8), spanned by the set {eLa, eLab: a, bE {I, ... , 7}}. Note that

Tri{ToTri{ eLa}To}

and

Tri{ToTri{ eLab}To}

=

Tri{ToeRaTO}

=

Tri{ -eLa}

=

-eRa

=

TOeLaTO'

=

Tri{To( -eRab)To}

=

Tri{eLab}

=

-eRab

=

TOeLabTO·

CHAPTER 7

176

That is, in general,

ITri{ToTri{g}To} = TogTo, I (7.138)

where the second equality follows immediately from the first. These results, together with Tri 3 = identity, imply

ITri{TogTo} = ToTri {g}To, I 2

(7.139)

As noted in section 7.2, if A, B E 0, and X is a unit octonion (that is, XE S7 j xxt = XtX = 1), then

IA °X B = (AX)(XtB)

I

(7.140)

defines a modified octonion product. Let 9 E 80(8), and note that To(TogTo)To = g. This, together with 7.138,139, and multiple applications of 7.124, implies

g[A 0x B]

=

g[(AX)(XtB)]

=

(Tri{TogTo}[AX])(XtB)

=

(ToTri 2 {g}To[AXJ)(XtB)

=

(Tri 3 {g}[A]X + ATri4 {g}[X])(XtB) +(AX)(Tri 2 {g}[Xt]B

=

(g[A]) 0x B

+ (AX)(Tri 2{TogTo}[XtB]) + (AX)(ToTri{g}To[XtB])

+ XtTri 3 {g}[BJ)

+ A 0x (g[B])

+(ATri{g}[X])(XtB) + (AX)(Tri 2 {g}[Xt]B). (7.141)

177

DOORWAYS

Gf

Let be the automorphism group ofthe product Aox B, and let LGf be its Lie algebra. Then by 7.141,

IgE LGf iff (ATri{g}[X])(XtB) + (AX)(Tri {g}[Xt]B) = 0 I 2

(7.142)

for all A, B E O. Consider the special case of 1

X

= EXae a ==> X t = -X ==> X

E

S6,

(7.143)

1

and in particu1ar, and without a 10ss of genera1ity, set X = e1.

(7.144)

g E eL1eR1LG 2 eR1eL1.

(7.145)

Let That is, there exists an h E LG 2 such that (7.146) Note that (7.147)

LG2 can be divided into two parts: the 8-dimensional su(3) spanned by {h = eLpq - eLrs : p, q, r, sE {I, ... , 6} distinct , h[1] == O}j

(7.148)

and the six dimensional subspace spanned by

{h=2eLp1-eLqr-eLst: p,q,r,s,tE {1, ... ,6}distinct, h[1]=O}. (7.149) If h 'E su(3), then Tri{h}

= Tri 2 {h} = h, and h[e1] = O.

In addition,

h E su(3) ==> eL1eR1heR1eL1 = h.

(7.150)

Therefore, 7.142 is trivial1y satisfied as

Tri{g}[e1]

= h[e1] = 0 = h[-e1] = Tri 2 {g}[et].

(7.151)

CHAPTER 7

178

Now suppose his an element of the span of the set 7.149. In particular, and again without a loss of generality, set (7.152) With the help of multiple applications of the equations

(7.153)

which are also valid with the subscripts Rand L interchanged, we derive 9

=

eR7 e L7(2eL57 - eL26 - eL34)eL7e R7

=

eR7(2eL57 eR7(2eRl

+ eL26 + eL34)eR7

+ eR26 + eR34)eR7

(7.154)

= 2eRl - eR26 - eR34 =

-2eLl

+ eL26 + eL34·

Therefore,

Tri{g}

=

-2eRI - eR26 - eR34 -2eL57 - eL26 - eL34

= and

Tri 2 {g}

(7.155)

-eL7 heL7, 2eR57

+ eR26 + eR34

=

TogTo

=

-eR7heR7,

(7.156)

DOORWAYS since TohTo

179

= h for all hE LG 2. This implies (7.157)

0, and

= eR7h[1]

(7.158)

o. Therefore,

9 E LG~7. Finally, since -eL7su(3)eL7 = -eR7su(3)eR7 = su(3), every element of the 14-dimensional eL7eR7LG2eR7eL7 is in LG~7. In summary, on dimensional grounds we conelude that (7.159)

(7.160)

(7.161)

This LG 2-triality triplet is the same as the one developed in the previous section. However, we can now generalize this result to all of S6. That is, (7.162)

CHAPTER 7

180

LG?f GENERAL SOLUTION Nowassume 7

X = XO + EXae a; XO

7

i' 0,

EXae a i' O.

1

1

In particular, and again without a 10ss of generality, we may set

X = XO + X7e7; XO =I- 0, X 7 =I-

o.

(7.163)

Let 9 E LG?f C 80(8), and let Z = Tri{g }[X).

(7.164)

Since any infinitesimal 80(8) rotation of X E S7 is orthogonal to X (see section 7.2),

< Z,X >= !(ZtX + XtZ) = !(ZX t + XZ t ) = O. 2

2

(7.165)

Therefore, Z takes the form (7.166)

9 E LG?f implies

= (g[AJX)(XtB) + (AX)(Xtg[BJ) Setting A = B = 1 implies g[IJ = g[l] + g[I],

g[(AX)(XtB)J for all A, BE O.

so

Ig E LG?f

==?

g[l] =

0·1

(7.167)

(7.168)

g[IJ = 0 implies 9 is an element of the vector 80(7) spanned by {eLa - eLbc: (eLa - eLbc)[IJ =O}.

(7.169)

DOORWAYS

181

Consider the 80(7) element eL7 - eL23, which satisfies

Tri{eL7 - eL23}

=

eR7 + eR23

=

eL15

+ eL46·

Therefore, the elements of this Tri{ 80(7)} are linear in the 0 L bilinears eLab, which satisfy

So 1gE

LG;

=}

Tri 2{g} = ToTri{g}To·1

(7.170)

By 7.139 this implies

= Tri 2{TogTo}.

Tri 2 {g}

Applying Tri to both si des implies

Ig E LGf

Again setting Z

=}

9 = To9To·1

(7.171)

= Tri{g }[X], we find Tri 2{g}[ xt] = ToToTri 2{g} To[X] =

ToTri{g}[X] (7.172)

= To[Z]

= zt. Therefore, 7.142 may be written in the form (7.173)

Given 7.166 this may be written

H(A, B)

=

m(Ae 7X)(XtB) - m(AX)(e7XtB) +n(AU)(XtB) - n(AX)(U B)

=

O.

(7.174)

CHAPTER 7

182 So

= m(Ae7X) - m(AX)e7 + n(AU) - n(AX)(UX) = O. But m(Ae7X) - m(AX)e7 = 0, since X = X O + X7e7, so H(A,X) = n(AU) - n(AX)(UX) = n(AU) - n(AX)(XtU) = O. (7.175) H(A,X)

Since U is orthogonal to X and xt, the only way 7.175 can hold for all A E 0 is if n = O. Therefore,

But

H(elXt,Xe2) so finally, m

= m(ele7)e2 -

mel(e7e2)

= -2me4 = 0,

= 0 too, and in general

I9 E LG~ iff Tri{g }[X] = Tri 2{g }[xt] = 0, I

(7.176)

so each of the terms in 7.142 is separately zero (given 7.142, the reverse implication is trivial). Now we are ready to derive more central results. Application of the results above, together with 7.124, imply

g[Axt]

=

TogTo[AXtj

=

(Tri{g }[A])Xt + A(Tri 2{g }[xt])

=

(Tri{g }[A])Xt.

In addition,

g[AXt]

since g[1]

= O.

=

g[((AXt)X)(Xt(1))]

=

((g[AXt]X)Xt + ((AXt)X)(Xtg[1])

Therefore,

I9 E LG~ ~ Tri{g} = XR9Xk'\

(7.177)

183

DOORWAYS Together with 7.170,171, this implies

Ig E LG~ ==> Tri {g} = Xlg X L·1 2

(7.178)

These results are entirely consistent with 7.159,160,161, but they do not actually solve for LG~ in this general case the way that 7.159 does in the case X = e7. Note that with X = XO + X 7e7, and identifying

su(3)

= {g

E LG z : g[e7)

= O}

(7.179)

(as usual, the Lie algebra of the stability group of e7), then

g[(AX)(XtB»)

=

(g[A)X)(XtB)

= (g[A)X)(XtB)

+ (Ag[X))(XtB) + (AX)(g[Xt)B) + (AX)(Xtg[BJ) + (AX)(Xtg[B)), (7.180)

since

g[X)

= XOg[1) + X 7g[e7) = O.

(7.181)

That is,

Isu(3) C LGf,1

(7.182)

accounting for 8 of the 14 dimensions of LG~. The other 6 dimensions of LG z (not LGf) are found in

span{h

= 2eLp7 -

eLqr - eLst : p, q, r, s, tE {1, ... , 6} distinct ,h[l]

= O}. (7.183)

My claim is (and I will spare you the tedious details of its derivation)

span{g

= 2XLeLp7 -

Xh(eLqr

+ eLst) : 2eLp7 - eLqr - eLst E LG z/su(3)}. (7.184)

184

CHAPTER 7

We investigate this claim with the element (the different forms of which are derived using 7.153 and its cyclic variations)

9

=

2XLeL57 - Xh.(eL26

+ eL34)

=

2XLeL57 - Xk(eRl

+ eR57)

XO(2eL57 - eL26 - eL34) +X 7(2eL5 - eL63 - eL24)

=

XO( -2eR57 + eR26 + eR34)

(7.185)

+X 7( -2eR5 + eR63 + eR24)

h

=

- 2X eR57 + XL(eLl

+ eL57)

=

!XL(3eL57 + eLt} - !Xk(3eR57 + eRl).

The last equality above makes it clear that 9 = TogTo.

(7.186)

From this equality we also derive g[1] =

=

!XL(3eL57 + eLt}[1]- !Xt(3eR57 + eRt}[1]

Xd2e l] -

Xk[2e l] (7.187)

=

0,

since X ep = epXt, p = 1, ... ,6. This verifies 7.171 and 7.168.

185

DOORWAYS Using the third equality in 7.185 we determine (again using 7.153)

=

Tri{g}

XO( -2eR57 + eR26 + eR34)

+ eR63 + eR24)

+X 7(2eR5

=

-2(XOeR57 + X 7eR7eR57) + XO(eLl

=

-2XReR57 + XL(eLl

=

-2eR57 Xk

+ eL57) + X 7(eL5 + eL71)

+ eL57)

+ (eLl + eL57)X!. (7.188)

Therefore, Tri{g}[X]

=

-2eR57Xk[X] + (eLl

=

-2eR57[1] + (eLl

=

0,

+ eL57)Xl[X]

+ eL57)[1]

which verifies half of 7.176. As to the other half, using 7.153 rewrite Tri{g}

=

XO(2eL57 - eL26 - eL34) +X 7(2eL71

so that

+ eL63 + eL24),

XO( -2eR57 + eR26 + eR34)

Tri 2{g}

+X 7( ~2eR71 - eR63 - eR24)

=

XO(2eL57 - eL26 - eL34) (7.189) +X 7(-2eL5 - eL63 - eL24)

=

ToTri{g}To

=

2eL57X L - (eR!

+ eR57)XR.

CHAPTER 7

186 Therefore,

Tri 2 {g}To[X] ToTri{g }[X]

O. Finally, in satisfying all of 7.176, this proves gE

La:,

and proves 7.184in general. I leave it to the reader to prove 7.177 and 7.178. Note, for example, that 7.177 may be rewritten

and that

(eLl

+ eL57).

Finally note that if X O = 1, and X 7 = 0, then by 7.185

And if X O = 0, and X 7

where which verifies that

= 1, then

DOORWAYS

187

LG~ AND THE X-ADJOINT ALGEBRA 0LX

Let 0 X be the X -product modified octonions, and let eax, a = 1, ... , 7, be a basis for the pure hypercomplex subspace of Ox, arranged so that (7.190) For simplicity again set (7.191)

In this case if we set emX

=

Xe m ,

m = 1,2,3,5, (7.192)

= ek, k = 4,6,7,

ekX

then 7.188 is satisfied. To see this let J + = ~ (1

+ 1t) = Po + P4 + P6 + P7 , (7.193)

L

= ~ (1 -

11)

= Pt + P2 + P3 + Ps,

an A E 0, 7.192 is equivalent to lAx = J+[A] + X L[A] = J+[A] + J_[A]xt.1

and note that for

(7.194)

Then 7.190 is equivalent to

lAx °x Bx = (AB)x for

I

(7.195)

an A, B E O. It is easily proven that

J+[AB]

= J+[A]J+[B] + J_[A]L[B], (7.196)

J_[AB] = J+[A]J_[B]

+ L[A]J+[B],

for all A, BE O. Also,

(J+[A]X)(XtJ+[B)) = J+[A]J+[B], XL[A]X

= J_[A] = XtJ_[A]Xt.

(7.197)

CHAPTER 7

188 By 7.74

(J+[A]X)J_[B] = (J+[A]X)(Xt(X L[B)))

=

(J+[A](X L[B]X»X t (7.198)

= (J+ [A]L [B])Xt

X(J+[AlJ_[B]),

=

since J+[A]L[B] is linear in em , m = 1,2,3,5. Taking the Hermitian conjugate of 7.74, and interchanging the indices a and b, implies (7.199) Therefore,

L[A](xtJ+[B)) = ((J_[A]Xt)X)(XtJ+[B)) =

X((XtL[A]Xt)J+[B))

=

X(L[A]J+[B]).

(7.200)

Combining 7.196-200 implies

Ax 0x Bx

=

((J+[A] + L[A]Xt)X)(Xt(J+[B] + X L[B)))

=

(J+[A]X + J_[A])(XtJ+[B]

=

(J+[A]J+[B]

=

J+[AB]

+ X(J+[A]L[B] + L[A]J+[B])

=

J+[AB]

+ X L[AB]

=

(AB)x,

+ L[B])

+ J_[A]L[B]) + ((J+[A]X)J_[B] + L[A](XtJ+[B]

(7.201) thereby proving 7.195. Let OLX be the associative left adjoint algebra of Ox. Let A,B E 0, and Ax, Bx E Ox their images via the isomorphism 7.194. Then we define

IALX[Bx] = Ax 0X Bx·1

(7.202)

189

DOORWAYS Therefore, for k

= 4,6,7, eLkx[Bx]

=

(ekX)(XtBx)

=

(ek(Bx X))X t

(7.203)

But if k = 7, then (7.204) Also, for m = 1,2,3,5,

eLmx[BxJ

=

(emxX)(XtBx)

=

(X emX)(XtBx)

=

em(XtBx)

(7.205)

eLm X l[Bx]

=

XLeLm[Bx].

Because of the ea ~ eax isomorphism it is now clear that (7.206) Consider the specific case X = er. The equations 7.203-205 imply in this case,

(7.207)

Let SU X (3) C

Gf

be the stability group of erx = er. A basis for its

190

CHAPTER 7

8-dimensional Lie algebra consists of €L2X€L6X - €L3X€L4X

=

€L24 - €L63

€L6X€LIX - €L4X€L5X

=

€L41 - €L56 E

€L4X€LIX - €L5X€L6X

=

-€L61

€L5X€L2X - €LIX€L3X

=

€L52 - €L13 E

€UX€L3X - €L2X€L4X

=

-€L34

€LIX€L2X - €L3X€L5X

=

€L12 - €L35

E su(3),

€L2X€L3X - €L4X€L6X

=

€L23 - €L46

E su(3),

€L2X€L3X - €LIX€L5X

=

€L23 - €L15

E su(3).

E su(3),

su(3),

+ €L45 E su(3), su(3),

+ €L26 E su(3),

(7.208)

That is, su e7 (3) = su(3), as expected (su(3) = su1 (3)). As to the rest of LG~7 , consider the element (7.209) which is what you get from 7.185 if you set XO = 0, and X 7 = 1. Similar results hold for the other elements of LG~7 / su(3), which demonstrates the consistency of the 0 LX approach with what went before. Using SU(3) we can construct a nicer looking isomorphism from 0 to o x. Redefine (where now X E S7 is arbitrary), (7.210) (note that Xl/3 is not uniquely defined.) In this case, (7.211) which gives S7 an almost group structure (what I would call a sliding group structurej sliding on the X -product). This also makes it c1ear that if X = ±1 (that is, Xl/3 is a 6th root of unity), then the X -product is ordinary octonion multiplication, and A - t X 1 / 3 AX- 1 / 3 is a G 2 action (see also [7],[8]). Much more could be said, and I dare say it eventually will be.

8. Corridors. 8.1. Magie Square. The Freudenthal-Tits Magie Square [15] is a 4 X 4 array of groups associated with the Jordan algebras of 3 X 3 Hermitian matriees over R, C, Q, or 0, complexified by (tensored with) R, C, Q, or O. That is, each group is associated with a pair of division algebras. The mathematical context upon whieh the model building of chapters 3 through 6 rests relied heavily on treating the (tensored) division algebras as spinor spaces of their left adjoint algebras (identified as Clifford algebras), of tensoring those adjoint algebras with R(2) (doubling the size of the spinor space), and of treating the parts of the right adjoint algebras that commute with the left as internal. These exact same methods will be employed here to generate bases for the Lie algebras of the groups of aversion of the magie square. Each will be derived from a tensor product of two division algebras, and as the order in whieh they are taken is immaterial, only ten distinct Lie algebras will result. The foundation upon whieh the method rests is R(2). In R(2) define again

€=[~ ~],a=[~ _~], ß=

[~ ~] ,w = [_~ ~].

(8.1)

Let K ® K' be the tensor product of two of the four division algebras. Let and denote bases for the pure hypercomplex parts of K and K' (ie., i, qj, or ea ). In KL ® K' L(2) the elements

Ck

cI

w, CLka, c'rAß 191

(8.2)

CHAPTER 8

192

anticommute (and associate) and form the basis for the 1-vector generators of a (possibly nonuniversal, meaning the product of these 1-vectors may equal the identity) Clifford algebra with negative definite Euclidean metric. Under commutation they generate the 2-vectors

(8.3) Together the elements 7.2 and 7.3 form a basis for a representation of the Lie algebra so( dimK + dimK'). I'll denote this ExternaIKK" and call it the external subalgebra. To this collection we now add the spinors of KL ® K' L(2), namely the elements of (K ® K')2, without yet specifying a commutator product on this linear space. I'll denote this SpinorKKI. Finally we add an the elements of KR ® K' R whose action on the spinor space is distinct from and commutes with (ie., is internal with respect to) the action of so( dimK + dimK') on the spinor space, and which together generate a compact subgroup. I'll denote this InternaIKK" and call it the internal subalgebra. Note that because of the nonassociativity of 0, only OR will be excluded by these conditions (CR and QR both commute with ExternaIKK" whatever the algebras K and K'). That OR can be excIuded is in a sense responsible for the existence of the exceptional Lie groups F4 , Es, E7 , and Es. The total resulting linear space will be denoted M S K K', M S for magie square.

8.2. The Ten MSKK'. R®R R has no complex part, so the external subalgebra is spanned by w (1vector basis for Ro,d.

ExternalRR

= u(1) = 80(2)

(8.4)

(there is no InternaiRR). SpinorRR is 2-dimensional, so MSRR has a total of 3 dimensions and we make the identification:

M SRR = 80(3) = su(2).

(8.5)

CORRIDORS

193

R®C ExternalRc is spanned by wand iß (l-vector basis for Ro,2), and ia (2-vector). (8.6) ExternalRc = su(2) = so(3). SpinorRc is 4-dimensional, and

= u(l) = so(2)

Interna/Rc

is spanned by iR (ie., its action on [ : ] E SpinorRc is iR [ : i [ :

(8.7)

1= [ : ] i =

l). That's 8 elements altogether, and we make the identification: M SRC

= su(3).

(8.8)

R®Q ExternalRQ is spanned by wand (2-vectors).

qLjß

(l-vector basis for R O,4)' and

qLja, qLj€

ExternalRQ = sp(2) = so(5).

(8.9)

SpinorRQ is 8-dimensional, and InternaiRQ is spanned by qRj the identification:

.

= sp(l) = so(3)

(8.10)

That's 10+8+3 = 21 elements altogether, and we make

M SRQ

= sp(3).

(8.11)

CHAPTER8

194 R®O

ExternalRo is spanned by wand eLaß (l-vector basis for Ro,s), and eLaa, eLab( (2-vectors). ExternalRO

= 80(9).

(8.12)

SpinorRO is 16-dimensional, hut because OL = OR, InternalRo is empty. (The inclusion of OR causes 80(9) to expand to 80(16), leading to a quite different result.) That's 36+16 = 52 elements altogether, and we make the identification: (8.13) MSRO = LF4 ,

where LF4 is the Lie algebra of F4.

Let i and i' he distinct imaginary units that commute with each other. ExternalRc is spanned by w, ia, i'ß (l-vector basis for nonuniversal Ro,3), and iß, i'a, ii'w (2-vector). Externa1cc = 80(4) = 8u(2)

X

8u(2).

(8.14)

Spinorcc is 8-dimensional, and Internalcc = u(l) X u(l)

(8.15)

is spanned by iR and iR. That's 6+8+2 =16 elements altogether, and we make the identification: M Scc = 8u(3)

X

8u(3).

(8.16)

CORRIDORS

195

C®Q ExternalcQ is spanned by w, ia:, qLjß (l-vector basis for R O,5), and iß, qLja:, iqLjW, qLj€ (2-vectors). ExternalcQ = 80(6).

(8.17)

SpinorcQ is 16-dimensional, and InternalcQ = u{l) X su(2)

is spanned by i and qRj . That's 15+16+4 we make the identification:

(8.18)

= 35 elements altogether, and

M SCQ = 8u(6).

(8.19)

Externalco is spanned by w, ia:, eLaß (l-vector basis for Ro,9), and iß, eLaa:, ieLaW, eLab€ (2-vectors).

(8.20)

Externalco = 80(10). Spinorco is 32-dimensional, and again because OL spanned only by iR' Internalco = u(l) = 80(2).

That 's 45+32+ 1

= OR,

InternalRo is

(8.21)

= 78 elements altogether, and we make the identification: MSco = LE6 •

(8.22)

CHAPTER8

196 Q®Q

Let qj and qj be distinct and mutually commuting bases for the hypercomplex quaternions. ExternalQQ is spanned by w, qLjCX, qtjß (l-vector basis for nonuniversal R O,T ), and qLjß, ql,jCX, qLjqhw, qLjf, ql,jf (2-vectors).

= 80(8).

(8.23)

= 8u(2) X 8u(2)

(8.24)

ExternalQQ SpinorQQ is 32-dimensional, and InternalQQ

is spanned by qRj and qRj . That's 28+32+6 = 66 elements altogether, and we make the identification: M SQQ = 80(12).

(8.25)

Q®O ExternalQo is spanned by w, qLjCX, eLaß (l-vector basis for nonuniversal R o,n ), and qLjß, eLaQ!, qLjeLaW, qLjf, eLabf (2-vectors). ExternalQo = 80(12). SpinorQo is 64-dimensional, and again because OL spanned only by qRj. InternalQo

= 8u(2) = 80(3).

(8.26)

= OR, InternalRo

is

(8.27)

That's 66+64+3 = 133 elements altogether, and we make the identification: (8.28)

CORRIDORS

197

000 Let eLa and eLb be distinct and mutually commuting bases for the hypercomplex octonions. Externaloo is spanned by w, eLaa, eLaß (l-vector basis for nonuniversal R O,l5), and eLaß, eLaa, eLaeLbw, eLabf, eLabf (2-vectors). Externaloo

= 80(16).

(8.29)

Spinoroo is 128-dimensional, and again because OL = OR, InternalRo is empty. That's 120+128 = 248 elements altogether, and we make the identification: (8.30) MSOO = LEB.

Part of the significance of these constructions is the manner in which they parallel the construction of a physical model in the earlier chapters. In particular, that model was based on TL(2) 0 PR acting on T 2 , with TL(2) identified as 'external', and PR 'internal'. Some important distinctions can be made, however. TL(2) can be viewed as either the complexification of R 1 ,9, or as the Clifford algebra R l,lO • For example, the elements (8.31) are a 1-vector basis for R1,l0 ~ TL(2) (see the Q ® 0 case abovej this is only a slight modification of the 1-vector basis for RO,ll used there). For that matter, the set (8.32) is another 1-vector basis for R 1,9 • Why is the 1-vector basis given in 4.4 preferable to this? Perhaps it isn't, but there is a difference. The former basis is buHt up from a "Pauli" algebra underlying the (1,9)-Dirac algebra (see 6.1), just as the (1,3)-Dirac algebra is buHt up from the conventional Pauli algebra. The basis 8.31 is not so constructed. It should be possible to generate a basis for the Lie algebra of E7 starting from the 1-vector basis for Ro,lt. (8.33) (a slight modification of the I-vector basis for Rl,9 in 4.4, where the basis above is founded on the "Pauli" algebra 6.1). To this add the basis of

CHAPTER8

198

corresponding 2-vectors, completing the external subalgebra, 80(12). The internal space should be the same (ie., we are ignoring the presence of the complex unit i in 8.33), 8u(2). The spinor space should be half (a Majorana doublet) of T 2 , where the other half is obtained by' multiplication by i, and each half is closed under the action of 80(12) X su(2)). This construction of LE7 , however, is not valid for all of the other algebras of the magie square. Questions arise from these ideas, for whieh there are presently no answers. Do these constructions signify a connection of the T-model to some GUT (for example, the E6 GUT)? Or is the attractiveness of GUTs accidental, arising from algebraie similarities to the T-model? In either case, how may we naturally determine the global structure of the varivus manifold bits of the T-model?

8.3. SpinorJ{J{1 Outer Products. In the next section we shall define commutator products for the M SRO case

(LF4 ). As a first step in that direction we define the outer product of spinors. Three cases are developed here: C, Q, and O. Only the last is relevant. This is because in this section I am treating 2-dimensional C and 4-dimensional Q as spinor spaces for R(2) and R(4), but CL ::j:. R(2), and QL ::j:. R(4) (the Q spinor outer product below is valid for QA ~ R( 4) (see 2.28), in whieh case 4-dimensional Q is the complete spinor space). On the other hand, 8-dimensional 0 is the full spinor space for OL, because OL ~ R(8). The first two cases establish the method. C OUTER PRODUCTS

Consider the example of CL acting on the spinor space C. C is 2dimensional over Rand as areal column spinor space its elements can be represented by (8.34) This leads naturally to the representations

1L

-t

1 .

[ 01 01 '

ZL

-t

1

[01 -10 .

CORRIDORS

199

We expect the outer product of 1 with its transpose to be representable as (8.36)

which is not a linear combination of the matrices 8.35. We need to expand the set by

1* [10 -101' L --+

.*

zL --+

[01 10 1'

(8.37)

the former just complex conjugation, the latter complex conjugation followed by iL (see 2.30). Therefore, Re = 11

~(1L + 11),

>< 11 =

(8.38)

which is the projector of the real part of C. That is 1 "2(1 L

+ 1L)[x] = Re(x)

(8.39)

far all x E C. Any element x of C can be mapped to an element XL of CL by appending the subscript 'L'. We may now define a general outer product of x and y in C: (8.40) Ix >< yl = xLReyL where yt = y* in this case. For example,

(8.41 ) so we expect 11 >< Using 8.41,

11 >< and

as expected.

il =

il to map i (= li »

1LRe( -iL) =

to 1 (= 11

», and to map 1 to zero.

-~(1L + 1iJiL = ~(i'i -

~(i'i -

iL)[i] =

~(i'i -

iL)[l]

~(i(-i) -

= ~(i(l) -

i(i» = 1, i(1»

= 0,

iL),

(8.42)

CHAPTER 8

200

Equation 8.42 has exact counterparts when we replace C by Q or O. The problem is to define RQ and Ra, which project the real parts of those algebras (Ra = Po; see sections 2.4 and 7.1).

Q OUTER PRODUCTS In the case of QL and QR acting on the spinor the basis for Q by

1-

[

~ l'

q, -

[

~ 1'

q2

-+

sp~ce

Q, if we represent

[n [n

(8.43)

%-

then

qLl -

[~

-1 0 0 0 0 0 0 1

qu-

qRl -

[

~

_~]

[~

0 0

1 0

0

-1 0 0

!1

-1

0 0 0 0 0 0 -1

qru -

,QL2 -

,qm -

[

~

0 0 0 1 -1 0 0 0

[~

0 -1 0 0 0 0 -1 0

-~o1'

n

0

(8.44)

[

~

0 -1 0 0 0 0 1 0

-~o ]' 0

-~o]' 0

CORRIDORS

201

so that

-9L!9R1

~ [~

i], ~ ~

0 0 1 0 0 -1 0 0 -1

0 -1 [ 0 0 0

-QL,qR2

0 0 1 0 -1

i], (8.45)

-Q~qm~ [~

0]

0 o 0 -1 0 -1 0 o 1 0

.

Therefore, RQ

and for all

X

1

= 4(1L -

3

~qLjqRj)

-

;=1

0 0 0 [ 01 0 0 0 0 0 0 0

and Y in Q we define

,

Ix >< Yl = xLRQYL'

~

~ ~

n.

t

(8.46)

(8.4 7)

As an example,

Iq, >< q,1

QLlRQ( -qL>J

[

0 0 0 0

0 1 0 0

This should map q2 - q1, and everything else to zero.

n

= -qL1RQ[q2] = -qLdRe(q2)] = 0, -qL1 RQqL2[q1] = -qL1RQ[-q3] = -qLdRe( -q3)] = 0, -qL1RQqL2[q2] = -qL1RQ[-1] = -qL1[Re( -1)] = -qLd-1] = qll -qL1RQqL2[q3] = -qL1RQ[q1] = -qL1[Re(q1)] = O. -qL1RQqL2[1]

202

CHAPTER 8

o OUTER PRODUCTS In the case of the octonions,

(see sections 2.4 and 7.1), and for all x,y E 0,

(8.49) Therefore, for example, (8.50)

And the vector 80(8) generators in 7.102 can be written, (8.51)

For a, bE {O, ... , 7}, this yields all 28 of the vector 80(8) generators on the last line of 7.102. Note that if X E T, then

Ro[X] E C® Q, RQRo[X] E C,

(8.52)

RcRQRo[X] E R. So the inner product

which is SO(64). invariant. This expression for the inner product, however, hides the U(1) X SU(2) X SU(3) symmetry that is manifested by constructing the inner product using projection operators from T, the spinor space, instead of TA.

203

CORRIDORS

8.4. LF4

~

M S RO.

A basis for ExternalRo

~

80(9) consists of the 36 elements (8.53)

Denote these elements Jj, f = 1, ... , 36 (the order is immaterial). SpinorRo is just 0 2 ,2 xl matrices over O. Together ExternalRo and SpinorRo define a representation of LF4 (InternalRo is empty). Let A, B E ExternalRo, and X, Y E SpinorRo. The commutator

[A,B] = AB - BA,

(8.54)

as expected. External/ Spinor commutators are also simply defined:

[A, X]

= -[X, A] = A[X].

(8.55)

It only remains to define Spinor / Spinor commutators. Following [Baaklini],

we define

1

[X,Y] =

36

4 L < X,J1[YJ > J1

(8.56)

1=1

(note: the spinor conjugation used in the definition of the inner product is in this case ordinary Hermitian transposition). We can gain some insight into 8.57 by considering a few examples. Start with

(8.57) For the inner product on the righthand side of 8.58 to be nonzero, we must have

This occurs for J1 = [[

eLlO, eL26f, eL34f, eL57f.

e~ ] , [~]J

Therefore

=

HeL10+eL26f+eL34E+eL57f)

=

HeLl

+ eL26 + eL34 + eL57)oal

(8.58)

CHAPTER8

204 where

1 - a) a e = -(€ 2

= [ 00

The coefficient of aE!), using table i.1 in appendix i, is just the outer product

= lel >< 11- 11 >< ell = eLlPO- PoeL = eLlPO+ POeLl·

(8.59)

The coefficient of a e is recognized to be (see 2.20) (8.60)

Likewise, it is simply determined using table i.6 in appendix i, that 1

"4( -eL6a + eL74€ - eLl2€ + eL3S€) =

1

"4( -eL6 + eL74 - eLl2 + eL3S)a 1 +:t( eL6 + eL74 -

eLl2

E!)

+ eL3S)a e .

(8.61)

In this case the coefficient of aE!) is

= lel >< e21-l e2>< ell = eLlPoe12 - eL2 PoeL = eRlPOeh2 - eR2Poehl'

(8.62)

and the coefficient of a e is (8.63)

where

CORRIDORS

205

The pattern is dear. In general,

= (xRPoYk- YRPoxh)a E9 1 t t e +4(XRYR - YRxR)a ,

(8.64)

Note: From the results of section 7.3 we see that the coefficient of a E9 is vector 80(8), and of ae is spinor 80(8). That this should be so is related to the fact that F4 is the automorphism group of the Jordan algebra of 3 x 3 Hermitian matrices over 0, which is also intimately linked to the notion of triality. Indeed, note that by 7.113 and 7.119, under the triality map Tri, if a, bE {I, ... , 7},

Tri{ !eRoeha}

=

Tri{ -!eRa}

=

eLaPO + POeLa

=

eLa Poelo - eLoPoeL,

and

Tri{ -!eRaeRb}

Tri{ !eRaehb}

Tri{~eRab}

=

-eLaPOeLb + eLbPOeLa

=

eLa Poe

lb - eLbPoeL·

Therefore, in general, (8.65) So

(8.66) That is, the coefficient of a E9 in 8.64 is minus the triality rotation of the coefficient of a e .

206

CHAPTER 8 In like manner we determine

Now consider the commutator

[ ~ l,[~l]=~t.< [e~ ],Jj[~l >Jj.

(8.68)

In this case the inner product on the righthand side of 8.68 is nünzero only if

This üccurs in the single case

Jj = eLIß.

Hence, (8.69)

Similarly,

< [ only if

Jj = eLIß.

e~ ] ,Jj [ ~6 >~ 0 ]

Since eLl[e6] = -e2, (8.70)

Therefore, using the identities of section 7.1, if a, bE {O, ... , 7} are distinct, then

[[ ~ 1,[ ~]l

=

HeRa(2Po-

~)ekb -

eRb(2Po -

~)ekalß

Finally, für all a E {O, ... , 7}, (8.72)

CORRIDORS

207

Define

(8.73) Combining these results, we see in general that

[[ ~ ] , [ ~]J = =

HXR(2Po -

~)Yh- YR(2Po - !)xtJß + ~[xRYh + YRxtJW

ß[XR(2Po - !)Yh- YR(2Po -

!)xkJ + l[xRYh + YRxknß

+H[XR(2Po - !)Yh- YR(2Po =

{!(xRPoYh- YRPoxk)

!)xkJ - ~[xRYh + YRxkn,

+ hRxt}ßEll

+H(XRPOYh- YRPoxk) - ~xRYh}ße

=

~(xyt)LßEll

- ~(yxt)Lße.

(8.74) Fiddling with all these commutators leads to a unified expression of a general LF4 spinor commutator:

[u]l -

x] ' v [[ Y

-

{!2 [ YRP XRPO ] [ Pt0 utR o

E

PtovRt J - (same)t}

+ {same}T

+~ [ ~~R ] [-vh ut 1- (same)t, (8.75) where AT is the transpose of the matrix A. Taking advantage of the triality map, we may rewrite 8.75,

(8.76)

CHAPTER 8

208

8.5.

J§> and F4 •

Let Jp be the set of all 3 X 3 Hermitian matrices over the octonions (a 27-dimensional real vector space). This set is closed under the symmetrie Jordan product 1 (8.77) X 0 Y = "2(XY +Y X), for all X, Y E Jp, and in addition,

X

0

(Y

X 2 ) = (X 0 Y)

0

0

X 2•

(8.78)

These conditions make Jp a Jordan algebra [9], a dass of algebras initially studied for their connection to quantum theory. Jp is the exceptional Jordan algebra, the only one not representable (given the product 8.77) by a set of associative matrices (another in a growing list of exceptional mathematical objects associated with 0). The automorphism group of Jp is F4 , and in this section we will construct a representation of its Lie algebra, LF4 , that will look quite different from that developed in the previous section. Let X = [ :t

Y

;

;t ] E

Jp,

(8.79)

xt c

a,b,c ER, and x,y,z E O. Clearly xt = X. Let Y be another element of Jp. Since XoY !(XY+YX)

=

H(X

F4 consists of those maps M on

+ y)2 _ X2 _ y 2),

Jp satisfying (8.80)

for all X E Jp. Therefore, LF4 consists of those maps mon

Jp

Im[X2] = (m[X])X + X(m[X])·1

satisfying (8.81)

The most obvious of such maps arise from the 80(3) subalgebra of LF4 arising from commutation of X in 8.79 with the 80(3) basis

[

-~o 0~ ~], [~1 ~0 -~], [ ~ ~ ~]. 0 0 0 -1 0

(8.82)

CORRIDORS

209

If u is a 3 x 3 real matrix linear in the matrices 8.82, and U = exp( u), then the SO(3) action of the form

x

-jo

UXU t = UXU- 1

(8.83)

clearly maps Jp Jp and is an Jp automorphism. The matrices 8.82 account for 3 of the 52 dimensions of LF4 • The remaining 49 arise from octonion actions and can be most easily constructed using the triality map Tri defined in section 7.3. Recall that the set -jo

{eLa, eLbc :

a, b, cE {l, ... , 7}}

(8.84)

forms a basis for a representation of 80(8), and that if 90 is linear in the elements 8.84, then (8.85) form a triality tripie of 80(8) actions on 0 satisfying Freudenthal's principle of triality, (8.86) the indices taken modulo 3, from 0 to 2. Now define the following action on X E

m[X]

=

[0

JP:

g,[ z] (g,[y])t 90[X]

... 0 ......

1

0

(8.87)

g,[z] [ , 0 , o =

...

...

T09'TO[yt]l 90[x] , 0

where for the rest of this section, ... in a matrix will indicate that what belongs in that position is the Hermitian conjugate of the term across the

CHAPTER 8

210 principal diagonal. Then

(m[X])X + X(m[X])

~ [ (g'[Z])zt~: (gI [y])t y g,[z]b ~ ~~y])txt

---j - -(8.88)

+[

z(g,[z])t ~ yt(91[Y]) a9'[z]:+- ~t(go[X])t

---j --

-

The one diagonal and one off-diagonal terms shown in 8.88 will prove suffident to finish its construction. The rest of the terms are analysed similarly. The diagonal term is

=

2 < g2[Z]), Z > +2< gt[y], Y >

= 0, since, for example, 92[Z], an infinitesimal 80(8) rotation of z, is perpendicular to z. The off-diagonal term is

g2[z](a + b)

+ (xgl[Y] + go[x]y)t = g2[z](a + b) + (TOg2To[xy])t

=

g2[z](a + b) + ToTog2TO[xy]

= g2[z](a + b) + g2[yt xt]

= Therefore,

g2[z(a

(8.89)

+ b) + ytxt].

(m[X])X + X(m[X])

= [

.~.

...

g2[z(a + b) + ytxt]

o

- - - .j o

(8.90)

CORRIDORS

211

The square of the matrix X is

X' = [ .' + z~t

+ yt y

z(a H)+ ytxt

~~~1

(8.91)

Therefore

g2[z(a +~) + ytxt]

===1

... =

0

(8.92)

(m[X])X + X(m[X]).

That is, the actions m defined in 8.87, the collection ofwhich is 28-dimensional (80(8», are elements of LF4. Together with the the 80(3) actions defined above, this accounts for 3 + 28 = 31 of the 52-dimensional LF4 • The remaining 21 dimensions of LF4 arise from 80(3) commutators with 80(8). Let S be the first ofthe matrices 8.82. Then the following commutator establishes the pattern for all [80(3),80(8)] commutators:

m[S, X] - [S,m[X]]

(8.93)

__ [ -2R:o:9: 2[Z] g2[b - a] (-gO + ToYtTo)[x] 2POg2[z] (-gO + T~YITo)[yt]

1,

since x + x t = 2Po[x]. Note that (8.94)

212

CHAPTER 8

So if we give go the general form 7

7

1 '" a go = 2 L... goeLa

+ 21 '" L... gobe eLbe,

1

(8.95)

1

g8 and gge real, then 7

- go +TOgITo = - LgoeLa.

(8.96)

I

Furthermore,

g2 = Tri 2{go} =

Li gö(eLaPO + POeLa)

+ EiggC(eLbPOeLe Therefore, since POeLaPO = 0, a

(8.97)

eLePOeLb)..

= 1, ... ,7, and pJ = Po, 7

(8.98)

Pog2 = LgoPOeLa' I

Also, if r is real, then PoeLa[r] = 0, and Po[r] = r. This implies 7

(8.99)

g2[b - a] = EgoeLa[b - al· I

Finally, using 8.96,98,99, we may rewrite 8.93

m[S, X] - [s, m[X]]

= [

-2 Li g8 eLa[Z] ...

...

Ei g8 eLa[b -

a] 2 Li g8 eLa[zl ...

Ei g8 eLa[X] 1 Li g8 eLa[yt l .

(8.100)

0

That is, the commutator of the so(3) action ([S, Xl) with the so(8) action (m[X]) is the 7-dimensional action 8.100. Carrying on in this way, commuting the other two so( 3) basis actions (associated with the other two generators in 8.82), yields similar results. Together with 8.100 that yields a 21-dimensional subspace of LF4 generators. Since 3 + 28 + 21

= 52,

that completes. this new basis for LF4 • Notiee that S, together with the 28 generators 8.87, and the subsequent 7 generators 8.100 resulting from commutation, form a 36-dimensional subalgebra of LF4 isomorphie to so(9) ~ ExternalRo,

CORRIDORS

213

8.6. More Magie Square.

Jl'

The Lie algebra of the automorphism group of (Jordan algebra of 3 X 3 real Hermitian matrices) is just so(3), with basis given in 8.82; that of Jf (3 X 3 complex Hermitian matrices) is su(3), with five additional generators

[

oi o

0] , -i00] 0 , riO 0 i 0 0 0 0 0 -2i (8.101)

° i]

0 i 0] [00 0 0 , [ 0 0O 0] O i. [ iOO, 000 iOO OiO The space of matrices C ® satisfying

Jp is the set of all 3 X 3 matrices over C ® 0 x'" = X,

(8.102)

where X'" is transposition together with octonion conjugation. That is, if X* is simple complex conjugation, no transposition, then X'"

= X*t,

(8.103)

where now (8.104)

a,b,c E C, and x,y,z E C ® O. If U E SU(3) is generated from the matrices in 8.82,101, then the set C® is invariant under the action

Jp

X

-+

UXU'"

= UXU T ,

(8.105)

where UT is the transpose of U (U T = U*t). Of course, an action ofthe form 8.104 is not an automorphism of C ® given the product 8.77. The infinitesimal (Lie algebra) version ofthe action 8.105 is commutation for the matrices 8.82, and anticommutation for the matrices in 8.101. Let R be the first ofthe matrices in 8.101, T the third (both nonzero in the upper left 2 X 2 block). Then the anticommutator

Jp

2ia {R,X}= [

?

zy

o iy'" -2ib -ix -ix'" 0

1,

(8.106)

214

CHAPTER 8

and 0

0

m{R,X} = [ .~. .~.

T091TO[iytPll

9o[~ixl

(note that if y E C ® 0, then To[Y]

= ytP).

(8.107)

= {R,m[X]} That is,

m{R,X} - {R,m[X]}

= 0;

(8.108)

the commutator of the action 8.106 arising from the diagonal su(3) generator R, and the action 8.87, which is also in a sense diagonal (in that while x, y, z may be changed, they are not shifted from their matrix component slots), is zero. On the other hand

m{T, X} - {T, m[X]} 0 92[i(b + a)) T091TO[iX)l [ ... 0 90 [iytP]

...

- [

...

0

o i(92[Z) + (92[Z])tP) i9o[X] 0 i(92[Z) + (92[Z])tP) iT091~O[YtP)

= [

1

(8.109)

2iPo92[Z] i~2[b + a) ... 2zPo92[Z] ... ...

A quick comparison of this commutator with the commutator 8.93 should be enough to convince the reader that the COITlmutator above, like 8.93, gives rise to 7 additional generators. The same will be true of the other two off-diagonal matrices in 8.101. And there the Lie algebra closes, with 28 elements from 8.87, 8 from 8.82,101, and then from the 6 off-diagonal elements of su(3), 6 X 7 42 additional elements, for

=

28+ 8 + 42 = 78 all together. That is, once again from a combination of C and 0 we have arrived at LE6 •

CORRIDORS

215

Again note that the 28 elements 8.87, together with the generators R, S, T (the 8u(2) subalgebra of 8u(3) nonzero in the upper left 2 X 2 matrix block), and their commutators with 80(8) (8.87) (which give rise to 14 additional elements, 7 for the off-diagonal S, 7 for T), yields a sub algebra of LE6 of order 28 + 3 + 14 = 45 isomorphie to

80(10)

~

Externalco.

LE7 arises in the same way if we replace C by Q. That is, we extend 80(3), the automorphism Lie algebra of Jp., to 8p(3), the automorphism Lie algebra of J~ (3 X 3 Hermitian matrices over Q), by augmenting the 380(3) generators 8.82 with the 18 additional generators qj [

o o

0

0 0 (8.110)

If U E Sp(3) is generated from a linear combination ofthe matrices 8.82,110, maps this set to itself. Again, this then the action 8.105 on X E Q 0 translates to an anticommutator Lie algebra action on X arising from the extra generators 8.110. Of these, 9 are diagonal, 9 off-diagonal, and as was the case with 8u(3), commutators of the form 8.93,105 give rise to non zero results only for the off-diagonal generators of 8p(3), and in each case the resultant space of new generators is 7-dimensional. This closes the algebra on the

J:p

28 + 21 + 84 = 133 elements of a new representation of LE7 • Getting LEB from 0 ® Jp is slightly trickier. In this case there are two distinct copies of 0 commuting with each other (denote them 0 1 and O 2 ), and while X E 0 1 ® is still of the form 8.103, where a, b, C E Ob and x, y, z E 0 1 0 O 2 , the map x - t xl/> conjugates only the O 2 part of x.

JP2

CHAPTER8

216

We begin again with the 28 80(8) generators 8.87, and the 3 80(3) generators 8.82. As before we expand the set of 80(3) generators. In the C ® Jp case we expanded 80(3) to 8u(3), the Lie algebra of the automorphism of Jfj in the Q ® Jp case we expanded 80(3) to 8p(3), the Lie algebra of the case we expand 80(3) to LF4, the Lie automorphism of J~j in the 0 1 ® algebra of the automorphism of That gives us 28 elements from 80(8), and 52 elements from LF4 (which contains another distinct 80(8)). Of the 52 generators of this new LF4 , 28 are diagonal in the sense that the maps 8.87 are diagonal, and 24 are off-diagonal. Commutators of the 28 diagonal generators (the 80(8) of 0d with the 80(8) of O 2 yield nothing new, but each of the 24 off-diagonal generators gives rise to a 7-dimensional space of new generators. That yields,

J?2 J?l.

28 + 52 + 168 = 248

generators a11 together, and the set closes he re on LEs . (The interested readers may carry out the details for themselves, as weH as the identification of the 80( 16) ~ E xternaloo of L Es in this case.)

Appendix i. 0 L Actions: Product Rule ea e a +l == e a +5. Given the octonion multiplication, eaea+l = ea+s,

the resulting multiplication table is 1 el

el -1

e2 e3 e4

-e6 -e4 e3 -e7 e2

es e6 e7

es

e2 e6

-1

e3 e4 e7

e4 -e3

-1 -es -eI

es el -1

-e6 e5 -e2

-e2 -e7 e6

-e7 e4

-eI e3

es e7 -e4 e6 e2

e6 -e2

el -es

e7 -e5 -e3 e2 -e6

-1

e7 e3

-e3

-1

e4

-eI

-e4

-1

(LO)

el

o is the object space of the 64-dimensional OL.

In the tables listed below are the actions of the basis elements of 0 L on the basis of O.

As an example of how to use these tables, consider the bold-faced elements in table i.O. These indicate that

where the sign is drawn from the slot in the relevant row and column. As another example, from table i.1 we see that eL34[eS] = -e7,

where es is on the left side, determining the row, e7 is on the right at the end of the same row, and eL34 is on the top, determining the column. A

217

218

APPENDIX 1.

minus sign is in the appropriate row and column, which is the sign of the action. The La table is associated with ea , where eo = 1. Note that the adjoint elements along the top of table LO are the diagonal elements of OL, denoted I a in 2.46. These were used to resolve the identity of 0L in section 2.4.

1 1 el e2

ea e4

es e6 e7

1 el e2 e3

eL476

+ + + + + + + + + + + +

eL732

eLl43

eL2S4

eL36S

+ + +

+

+ +

+

+

+ +

+

e7

+

+ + +

eLS7

eLS46

eL23S

eL427

eL367

+

+ +

+

+ + + + +

+ +

+ +

+ + +

+

+

+ + +

+

+ +

eL2

eL37

eL4S

eL61

eL657

+

+ +

+ +

+

+

+ + +

+ +

+ +

+ + +

+ + +

+

es e6

eL346

eL531

eL471

+

+ + +

+ + + +

+ + +

e6

(i.2)

e3

+

+

el 1 e4

+ + +

+

+ + + +

(i.1)

e7

+ +

ea e4

eL34

es e6

+ +

1 el e2

+

e2 e3

+ +

+

eL26

e7

e4

+ +

+

e6

1 el

eL621

+ +

eLI

e4

es

eLS17

e7 (:2

es

e2 e6

1 e7

es e4

+ +

el e3

(i.3)

219

1

eL3

eL41

eL56

eL72

eL761

eL457

eL642

eL512

+

+

+ + +

+ +

+ +

+

+

+ + + +

+

+ + + + +

e1

+

e2 e3 e4 e5 e6 e7

1 e1 e2

+ + +

+ +

+

+ +

eL52

eL67

eL13

eL172

eL561

eL753

eL623

+

+

+

+

+ +

+

+ +

+ + +

+ + + + + +

+ +

e7

1

+ +

+

+

+ + + +

+

+ +

eL71

eL24

eL213

eL672

eL164

eL734

+

+ +

+

+ + +

+

+

+ +

+ + + +

+ +

+

+

+ + + +

+

+ + +

(iA)

e1 e6 e5 e2

e4 e3 e5 e1

(i.5)

1 e2 e7

+ +

+

1

e6

eL63

e4

e7

+

+

e3 e5 e6

+

+

eL5 e1

e2

+

+ +

e4 e6

+ + +

eL4

e3

e5

+ +

e3 e4 e7

+ + + + +

e5 e7 e4

e6 e2

1 e3 e1

(i.6)

APPENDIX 1.

220

1 el e2 e3

eL6

eL74

eL12

eL35

eL324

eL713

eL275

eL145

+ +

+

+ +

+

+ + +

+

+ + + +

+

e6 e2 el

+ + + + +

e5 e7

+

+

e4

+

e5

+ +

+ + +

e6 e7

1 el e2 e3 e4 e5 e6 e7

+ + +

+

+ +

+

+

eL7

eL15

eL23

eL46

eL435

eL124

eL316

eL256

+ + +

+ +

+

+

+ + + +

+

+

+ +

+

+ +

+ + +

+ + +

+ + +

+

+

+

+ +

+ + + +

(i.7)

e3

1 e4

e7 e5 e3 e2 e6 el e4

1

(i.8)

Appendix ii. 0 R Actions: Product Rule ea e a+l = e a +5. For the sake of completeness, the actions of the basis of 0R on 0 are also given.

1 1 el e2 e3 e4 e5 e6 e7

1 el

eR476

eR517

eR621

eR732

eR143

eR254

eR365

+ + + + + + + + + + + +

+ +

+ + +

+

+ +

+

+

eR!

eR26

eR34

eR57

eR564

eR253

eR472

eR376

+

+

+

+

+

+ +

+ +

+ + + + + + + +

+ + +

+ +

e2 e3 e4

+

e5 e6 e7

+ +

+ + +

+ +

+ +

+

+ +

+ + +

+

+ +

+ +

+

1 el e2

+

e3

(ii.O)

e4

+ +

e5 e6 e7

el 1 e6

+ +

e4 e3 e7

+

e2

es

(ii.1 )

APPENDIX 11.

222

1 e1

eR2

eR37

eR4S

eR61

eR675

eR364

eRS13

eR417

+ +

+

+

+ +

+ + +

+ + + + + + + +

+

+

e2

+

e6 1

e2 e3

e4 es

+

1 el e2 e3

e2 e3

eR41

eR56

eR72

eR716

eR475

eR624

eR521

+ +

+

+

+

+ +

+

+ +

+

+

+ + + + + + + +

+

+

+ +

+ +

+ +

+

e7

+ +

eRS2

eR67

eR13

eR127

eRS16

eR73S

eR632

+

+ + +

+ +

+

+ +

+ + + + + + + +

+

+

es e6

+

+

+ +

+ +

+ +

e3 e4

eR4

+

e4 e3

+

+ +

(ii.2)

es el

+ +

e4

e7

+ +

+ + +

e7

1 el

e7

eR3

es

+

+ +

+

e4

e6

+ +

+ +

+ +

e6

e7

+

+

1 el e6 es e2

(H.3)

e4 e3

+ + +

+

es el 1 e2

+ +

e7 e6

(H.4)

223

1

el e2 e3 e4 es e6 e7

1 el e2

eRS

eR63

eR71

eR24

eR231

eR627

eRl46

eR743

+ +

+

+ + +

+

+

+ + + + + + + +

+

+ +

+ +

+

es

+ +

+

+ +

eR342

eR731

eR2S7

eRIS4

+

+ +

+

+ +

+

+ + + + + + + +

+

+ + +

+ +

+

+ + +

+ +

+ +

+ +

+

+

+ +

+

CR23

CR46

eR4S3

CR142

eR361

CR26S

+ +

+ + +

+

+ + + + + + + +

+ +

+

+ +

+ +

+ +

+ +

+ +

el

e6 e2

el (ii.6)

1 e4

+

+

(ii.5 )

e3

CRIS

e4

e7 e4 e6 1 e3

e7

+

+

es

es

eR7

C2

e6

+

eR35

el

C7

+

eRl2

e7

es

+ +

eR74

e6

e3

+ +

+

eR6

e3 e4

+ +

es

e7

es

+ +

e3 e2 e6

+

el e4

+

+

1

(ii. 7)

Appendix iii. 0 L Actions: Product Rule ea e a +l == ea +3. Given the octonion Illultiplication, eaea+l

= e a+3,

the resulting multiplicatioll table is 1

el

el

-1

ez

-e4

e3

-e7

CI

ez

e3

e4

e5

e6

e7

e4

e7

-1

e5

-ez

e6

-e5

-e3

el

-e3

e7

-e5

-e6

-1

e6

e2

-e4

e2

el

-Cl

-e6

-1

e7

C3

-e5

e5

-e6

e3

-e2

-e7

-1

el

e4

e6

e5

-e7

e4

-e3

-eI

-1

e2

e7

e3

e6

-eI

e5

-e4

-e2

-1

(iii.O)

The resulting adjoint actions are

1 el

e2 e3 e4 e5 e6 e7

1

eL325

eL436

eL547

eL651

eL762

eL173

CL214

+ +

+

+

+

+

+

+

+

+

+ +

+

+ + + + +

+ + +

+ + +

+ + +

+

+ +

+ +

1 el e2 e3

+

e4 e5

+ +

e6

+

e7

(iiU)

226

APPENDIX III

1 el e2 e3

eLl

eL37

eL56

eL24

eL453

eL764

eL572

eL632

+

+

+

+

+ + +

+ + + + + + + +

+ +

+ +

+ +

e4 e5

+

e6

+ + +

e7

1 el

+ + +

+

+

+ +

e3 e4

eL35

eL564

eL175

eL613

eL743

+ +

+

+ + + + + + + +

+ + +

+

+

+

+ + +

+ + +

+ +

es e6 e7

+

+

el

+ +

eL675

eL216

eL724

eL154

+ +

+ + +

+

+ + + + + + + +

+

+ +

+ +

+ +

+ + +

(iii.3)

e3

eL46

+

1 e5

+

e3

+ +

+

eL71

+

e2 e4

eL52

e2

e6 e5 e3

+ +

eL3

e4

+

eL67

+ +

(iii,2)

e2

+ +

+

e7

1 el

e7

eL41

e5 e6

+

+

+ +

1 e4

eL2

e2

el

+ +

e7 e6

e3 e7 e5

+

1 e6

+

e2 e4

+

el

(iii.4 )

227

1 el

eL4

eU3

eL12

eLS7

eL716

eL327

eL13S

eL26S

+ +

+

+ +

+

+

+

+ + +

+

+ + + + + + + +

e2

+

e3

+

+ +

e4

es e6

+ +

eL74

eL23

eL61

eL127

eL431

eL246

eL376

+

+ +

+ + +

+

+ +

+ + + + + + + +

+

+

+

+

e4

es e7

1 el

+ +

+ + +

eL72

eL231

eLS42

eL3S7

eL417

+ +

+ + +

+

+

+

+ + + + + + + +

+ +

+

+

+

+

+ +

e6

+

+

+ +

e3 e2

(iii.6)

e7

eL34

+

es e6

eL1S

es e7

es

eL6

+ +

(iiL5 )

1 e3

+

e4

+

+ +

+

+ +

e2 e3

+

+ +

el

e7

eLS

e3

e6

+

+

el e2

+ +

e2

e6

+ +

e7

1

+

+ +

e4

+

1 el e4

e6

es e7

+ +

+ + +

e4 e3

el 1 e2

(Hi.7)

APPENDIX III

228

1 el e2

eL7

eL26

eL4S

eL13

eL342

eL6S3

eL461

eLS21

+ + +

+

+

+ +

+ +

+ + + + + + + +

+

+

e3 e4

es e6 e7

+

+ + +

+ + +

+ + +

+

e7 e3

+ + +

e6

+ +

el es e4 e2

+

1

(iii.8)

Appendix iv. 0 R Actions: Product Rule ea e a +l == e a +3. Finally

1 el e2 e3 e4 e5 e6 e7

1

1

eR325

eR436

eR547

eR651

eR762

eR173

eR214

+ + + + + + + +

+

+

+

+ +

+

+ +

+ + +

+ + +

+ + +

+ + +

+

e7

eR24

eR453

eR764

eR572

eR632

+

+

+

+ +

+ +

+ +

+ + +

+ + + + +

+ + +

+ +

+

+ + + + 229

(iv.O)

e4 e6

eR56

+ + +

e2

e5

+ +

+

e5

e7

+

eR37

e3

el e3

+

e2

es-

+

eR!

et

e4

+ +

+

1

+ + +

el

1 e4 e7 e2 e6 e5

+

e3

(iv.1 )

APPENDIX IV

230

1

el

eR2

eR41

eR67

eR3S

eRS64

eR17S

eR613

eR743

+ +

+ +

+

+

+ + +

+

+

+ + + + + +

e2

+

e3 e4

es

+

e6 e7

1

el e2

+

+

eR3

eRS2

+ + +

+ +

+ + +

+ + +

+

e5 e6

+

e7

eR4

1

el e2 e3

+ + +

+ + + +

e6 e7

+

+ +

eR216

eR724

eR154

+ +

+ + + +

+

+ +

+ .+ + + + +

+ +

eR12

+ +

+ + +

+

eRS7

eR716

+ +

+ +

+ +

+ +

+

+ + + + + + -

+ +

eR327

eR135

eR265

+

+ + +

+

+

+ + +

1

es el

(iv.2)

e3

e6

eR675

+

e4

e7

+

e4 e5

+ + +

eR46

+

eR63

+

+ +

eR71

e3 e4

+

e2

e3 e7

es 1

(iv.3)

e6 e2 e4

el

e4 e2

el

+ + + + +

e6

1 e7 e3

es

(iv.4 )

231

1 e1

eRS

eR74

eR23

eR61

eR127

eR431

eR246

eR376

+ +

+

+

+ + +

+

+

+ + + +

+ +

e2 e3 e4

+ +

+ +

+

es

+

e6 e7

1

es

+

+

eR72

eR231

eRS42

eR357

eR417

+

+

+ +

+ + + +

+ +

+

+

+ + +

+ + +

+ +

+

e7

+ +

+ + +

eR26

eR4S

eR13

eR342

eR6S3

eR461

eRS21

+

+ +

+ + +

+

+

+

+ +

+ + + +

+ + +

+ +

+

+ + +

+ + + + +

+ + + +

+

1 e1 e4

e6

es e7

(iv.6)

e3

eR7

e4

e6

+

(iv.5)

e7

e4

+

e2

es

+ + + +

+ + +

+

e7

e3

+ + + +

eR34

e6

1 el

+ +

e2

eR1S

e3 e4

+ +

+

e6 e3

eR6 e1 e2

+ + + +

es

el

1 e2

e7 e3 e6

el es e4

+ +

e2

1

(iv.7)

Bibliography [1] I.R. Porteous, Topological Geometry, (Cambridge, 2nd Ed., 1981). R. Ablamowicz, P. Lounesto, J. Maks: Report on the Second "Workshop on Clifford Aigebras and Their Applications in Mathematical Physics," Found. Phys. 21 (1989) 201. [2] G.M. Dixon, Aigebraic Unification, Phys. Rev. D 28 (1983) 833. [3] G.M. Dixon, Generalized Clifford Aigebras: Orthogonal and Symplectic Cases, Lett. Math. Phys. 5 (1981) 411. [4] G.M. Dixon, Fermionic Clifford Aigebras and Supersymmetry, Clifford Algebras and their Application in Mathematical Physics, (D. Reidel Publishing Co., 1986) 393.

[5] V.G. Kac, A Sketch of Lie Superalgebra Theory, Comm. Math. Phys. 53 (1977) 31. [6] G.M. Dixon, Derivation of the Standard Model, Il Nuovo Cimento 105B

(1990) 349. [7] C.A. Manogue, A. Sudbery, General Solutions of Covariant Superstring Equations of Motion, Phys. Rev. D 40 (1989) 4073. C.A. Manogue, J. Schray, Finite Lorentz Transformations, Automorphisms, and Division Aigebras, J. Math. Phys. 34 (1993) 3746. [8] I. Bengtsson, M. Cederwall, Particles, Twistors and Division Aigebras, Nuc. Phys. B302 (1988) 81. /\

M. Cederwall, C.R. Preitschopf, S1 and S1, hep-th-9309030. 233

BIBLIOGRAPHY

234

[9J M. Günaydin, G. Sierra, P.K. Townsend, Exeeptional Supergravity Theories and the Magie Square, Phys. Lett. 133B (1983) 72. [10] J.H. Conway, N.J .A. Sloane, Sphere Paclcings, Lattices and Groups, (Springer-Verlag, 2nd Ed., 1991). [l1J F.D. Smith, Calculation of 130 GeV Mass for T-quark, hep-ph-9301210. F.D. Smith, SU(3) x SU(2) x U(l), Higgs, and Gravity from Spin(O, 8) Clifford Algebra Cl(O, 8), hep-th-9402003. [12J E.s. Abers, B.W. Lee, Phys. Rep. 9C (1973) 1. [13J P. Candelas, G.T. Horowitz, A. Strominger, E. Witten, Superstring Phenomenology, Symposium on Anomolies, Geometry, Topology, (World Scientifie Publishing Co. Pte. Ltd., 1985) 377. [14J A. Sudbery, Division Algebras, (Pseudo)orthogonal Groups and Spinors, J. Phys. A 17 (1984) 939. [15J M. Günaydin, G. Sierra, P.K. Townsend,The Geometry of N=2 Maxwell-Einstein Supergravity and Jordan Algebras, Nue. Phys. B242 (1984) 244.

Index Adjoint Division Algebras, 35,39 Charge Conjugation, 128 Chirality, 29 Chiral Projeetor, 17,21 CL, CR, 39 Clifford Algebra, 11,40,59,137 - universal, 12,192 - nonuniversal, 13,192 - tables of, 14,15 Dimensional Reduetion, 85 Dirae Operator, 88,94,117 - parity violating, 96,117 Division Algebra, 32 Family (Lepto-quark), 30,79,90 G 2 ,46 Galois Field, 49 Galois Sequenee,49 Gauge Fields, 97,105 Hadamard Matrix, 34 Idempotent, 43,60,69 - primitive, 43,60,69 Internal Symmetry, 64,73 Jordan Algebra, 208 Lepton, 29,90 LE6 , 195,214 LE7, 196,215 LEs,197,216 LF4 , 194,203,208 LG 2 ,46,170 - triality triplet LG~, 176,179,187

Lie Superalgebra, 27 Magie Square, 191,213 Matter / antimatter Mixing, 120,12 Moufang Identities, 148 Multiplets, 78 Nilpotent Clifford Algebra, 24 - symplectic, 25 Oetonion, 31 OL,36,39 OLX, 187 OR,37,39 Po,45,141 Parity Noneonservation, 29,92,94 Pauli Algebra, 18,60 Periodicity (Index), 32,33,53 Pseudosealar, 13,17 QL, QR, 39 Quadratie Residue Code, 56 Quark, 29,90 Quaternion, 31 R O,2,42 R 3 ,o, 18,59 R 1 ,3, 19,84 R 3,11 16,43 R O,6,40 R s ,o,48 Ro,g, 67 Rt,g, 20,85,137 Resolution ofthe Identity, 43,60,68 Sealar Field, 109,126 Special Linear Groups, 135

236

Spheres (parallelizable), 150 Sphere Fibrations, 159 Spin Group, 13 Spinor, 13,40,59 - conjugations, 22 - Dirac, 16,132 - inner product, 61,68 .. Majorana, 16,132 - Majorana-Weyl, 22,138 - outer product, 198 - Pauli, 16,61 - Weyl, 17 Spontaneo.us Symmetry Breaking, 30,109 Stability Group, 47 Standard Model, 29 Standard Symmetry, 29,73 SU(3),47,105,111,122,177

To, 142 Tangent Space, 151 Tensored Division Algebra, 59,66 Torsion, 158 Triality, 160,209 - Freudenthal's principle, 169 - Tri, 165,167,169,209 - trilinear form, 161,162 Twistor, 133 Weak Mixing, 100 X -Product, 155,175

INDEX

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