Clifford_Report 2 wilson

A Glance Through the Gate Cli®ord's Geometrical Algebra Andrew Jowan Wilson 11th February 2005 Dr. Andrew Baker ..for geometry,you know,is the gateof science, and the gateis so low and smallthat one canonly enter it as a little child. –WilliamKingdom Clifford Preface Clifford’s algebras are a rich subject, born in the later half of the nineteenth century, they have been lying dormant for many years. Recently they have received much interest from both physi- cists and computer scientists; this in turn has generated new interest in their mathematics. Pre- sented here is an introduction to this diverse subject. It is aimed at a final year undergraduate or first year postgraduate interested in discovering the basic notions and ideas concerned with this areaof mathematics. Weassume,forthepurposeofthisreport,thatthereaderiscomfortablewithbasicvectorspace, matrix and ring theory. In particular with; basic set theory, basis and dimension of vector spaces, divisors of zero, direct products/sums of rings and vector spaces, homomorphisms, surjections, injections,bijections,thekernelandimageofhomomorphisms,ideals,generatingsets,theorthog- onalgroup and special orthogonal group. Perhapsitisworthyofnotethatinmanyplacesthisreportisnotasgeneralascouldbethecase. Preferencethroughouthasbeengiventopresentingtheunderlyingconceptsasclearlyaspossible. Thisapproachwillhopefullyimparttothereaderanintuitivefeelforalgebras. Asanappendixwe haveincluded a more general definition of a Cliffordalgebra for the interestedreader. Section 1 presents a basic introduction to algebras. We assume that the reader is familiar with somebasicexamplesofvectorspacesandringsandusetheseasourstartingpoint. Asisoftenthe casewith newmathematics, the ratio of definitions to theorems is rather high in this section. Section 2 gives some of the reasons for the importance of matrix algebras in studying algebras in general,including a method for finding the regularrepresentationsof anygivenalgebra. Section 3 contains the main exposition of this report, here we acquaint the reader with Clifford algebras and guide them through the basic definitions. The section continues with a discovery of the regular representations of these algebras and examines a fascinating theorem describing the ‘periodicity’of Cliffordalgebras. Section 4 is included to give the reader an idea of which directions further study could lead. Alsoofferedhere are some applications for this theory to other sciences. I would especially like to thank Andy Baker for his guidance in helping me to understand the materialin this report. i Contents Preface i Introduction 1 1. Algebras 2 2. Matrix Algebras 6 3. Cli®ord Algebras 9 3.1. Grade and Dimension 10 3.2. Some Remarks 12 3.3. Matrix Representations of Cli®ord Algebras 12 3.4. Periodicity of 8 14 4. Further Theory and Applications of Cli®ord Algebras 16 4.1. Division Algebras 16 4.2. Spinors 16 4.3. Embedding C`ninC`n+1 17 4.4. More Applications in Physics and Computing 19 5. Appendix 20 References 21 Cover Illustration: Dave Chisholm’s depiction of William Clifford performing his ‘corkscrew’. It appears that Clifford was not only theoretically butalso athletically expertat rotations! Figure 7: an illustration by Roger Penrose from his book [Pnrse 2004]. Introduction Confusingly, mathematicians use the word ‘algebra’ in two distinct ways. On the one hand al- gebra is a broad branch on the mathematical tree of knowledge; on the other, it is the name given to a very precise structure. It is often the case in mathematics that where subject areas overlap, newfertileandfascinatingareasarisefromtheirintersection. Thisisthecasewithalgebras,where vectorsspaces and rings interact togetherin intriguing ways. In this report we shall take the approach of expanding a vector space in such a way as to allow multiplicationof vectors. What shall we require of this product? It would be advantageous to have it satisfy the same axioms as the multiplication for real numbers, that is distributivity, associativity and commutivity. How about the complex numbers? Vectors in R2can be represented by complex numbers, whose additionandmultiplicationsatisfiesthesameaxiomsastherealnumbers. Canthisbeextendedfor dimensionsgreater than two? Itturnsoutthatthereisnowaytodeveloptheseaxiomsforlargerdimensions. In1843,consid- ering a similar question was William Hamilton. Whilst walking along the Royal Canal in Dublin with his wife, he had a flash of insight. In an infamous act he defaced Brougham bridge with the quaternionsdefining equations i2=j2=k2=¡1 succeeding in generalising the concept of the complex numbers to four dimensions. He achieved thisfeat by dropping the requirement of commutivity. Thirty years later, another William, William Kingdom Clifford desired a consistent framework for extending these observations to higher dimensions. This is what we study in detail here: Clif- ford’salgebras . OnthepathtoinvestigatingClifford’salgebrasweshallencounterrichtheory,includingmatrix algebras and their importance in understanding algebras in general. Finally, we conclude the re- port by pointing the reader to future paths for study and research. In addition to this, we unearth applications for Clifford algebras – revealing , amongst other things, their significance in modern physics. 1 1. Algebras Informally, an algebra is a extension of the structure of a vector space. In addition to the vector spaceaxioms, we define a multiplication between vectorsin such a wayas to form a ring. LetFbe a field throughout. More formally then, we havethe following. De¯nition 1. Analgebra AoverFis a set on which three operations are defined; addition, mul- tiplicationand multiplication by scalars. These must satisfy the followingconditions. (i)Ais a ring under addition and multiplication. (ii)Ais anF-vectorspace under addition and multiplication by scalars. (iii)8a;b2Aandλ2F (λa)b=a(λb) =λ(ab). Condition(iii)ensuresthatscalarmultiplicationcommutesamongthevectorsinasensibleway. Since an algebra Ais both a vector space and a ring, we use definitions and terminology estab- lished in connection with these structures to describe A. For example, we may refer to elements ofAas vectors; we may consider bases for A; we may also consider what it means for Ato have divisorsof zero and so on. It is in this spirit we makethe followingdefinition. De¯nition 2. An algebra Ais said to be of dimension nwhen it is ndimensional as an F-vector space. When Ais an infinite dimensional F-vector space, we say that it is of infinite dimension as an algebra. Examples 3. (i) As a first example, the familiar complex numbers Ccan be thought of as an R-algebra of dimension 2, with basis f1;ig. Let’sworkthrough this in detail. Anyelements α;β2Ccan be written as α=a11+a2iβ=b11+b2i witha1;a2;b1;b22R. Multiplication is then, αβ= (a11+a2i)(b11+b2i) =a1b11+(a1b2+a2b1)i+a2b2i2 butof course i2=¡1, so we have, αβ= (a1b1¡a2b2)1+(a1b2+a2b1)i. The multiplication of vectors in an algebra can be described completely in terms of multiplica- tion of basis elements. Indeed, every algebra has a unique multiplication table, with respect to a particularbasis. Belowin Table 1we see the multiplication table for C,with this basis. 2 1i 11i ii¡1 Table 1. Multiplication Tablefor C (ii) Next, the quaternions H, of which we will see a lot more later, are a 4-dim R-algebra, with standardbasis f1;i;j;kg, whose multiplication table is shownin Table 2. 1i j k 11i j k ii¡1k¡j jj¡k¡1i kk j ¡i¡1 Table 2. Multiplication Tablefor H Notice that i2=j2=k2=¡1,ij=¡ji=k. (iii) What about Has aC-algebra? On inspection, we see that His both a ring and a C-vector space. Unfortunately, the scalars (i.e. elements of C) do not commute evenly among the vectors. Forexample,take λ=i;a=jandb=1in condition (iii) of the definition to get (ij)1=j(i1) =i(j1)i.e.ij=ji which,on checking Table 2,is clearly false. (iv)Mn(F), then£nmatrices overFare anndimensionalF-algebra, these are very important examplesof algebras. Forthis reason we devote Section2 to studying their basic properties. (v) LetGbe a group. Then we may take the group elements as a basis of an F-vector space and form what is called the group algebra overF, sometimes denoted FG. The group multiplication tablethenbecomesamultiplicationtableforthealgebra. Althoughwewon’tgointoanymorede- tailhereonthissubject,itisworthnotingthatthesealgebrasareveryimportantinstudyinggroups. Note 4. As with finite groups, to show an isomorphism between algebras it is enough to show thattheir multiplication tables are the same. We conclude this introduction to algebras by making some definitions that will be useful in the followingsections. Theseshouldappearfamiliartothereader;theyaredirectanaloguesofsimilar definitionsmade in group, ring or module theory. 3 Forthe remainder of this section, let A,B,Cbealgebras overthe same field F. De¯nition 5. Wesaythata Bisasub-algebra ofA,ifBisanalgebraunderthesameoperations asAandB½A. Example 6. For example,Cis a sub-algebra of H(asR-algebras). To prove this is it enough to show that the multiplication table of Cis contained in the multiplication table for H, which can be easily checked in Tables 1and2. In fact, the multiplication table for Cis contained more than once. Thatistosay,thereismorethanonesubalgebraof Hthatisisomorphicto C,asanalgebra. De¯nition 7. Thedirectproduct ofBandCisdefinedtobetheset B£C=f(b;c):b2B;c2Cg satisfying,for all b;b02B;c;c02Candλ2F. (i)(b;c)+(b0;c0) = (b+b0;c+c0) (ii)(b;c)(b0;c0) = (bb0;cc0) (iii) λ(b;c) = (λb;λc) Note that these conditions imply the following.¡ λ(b;c)¢ (b0;c0) = (b;c)¡ λ(b0;c0)¢ =λ¡ (b;c)(b0;c0)¢ Clearlythisdefinitioncanbeextended,byinduction,todirectproductsofanynumberofalgebras. De¯nition 8. A mapϕ:B!Cis analgebra homomorphism if it is both a ring and a vector spacehomomorphism. That is for all b;b02Bandλ2F,ϕsatisfies the following. (i) ϕ(b+b0) =ϕ(b)+ϕ(b0) (ii) ϕ(bb0) =ϕ(b)ϕ(b0) (iii)ϕ(λb) =λϕ(b) We say that ϕis amonomorphism (respectively epimorphism ) when it is injective (respectively surjective). When ϕis both a monomorphism and an epimorphism we say that it is an isomor- phism, writing B»=C. De¯nition 9. Letϕ:B!Cbe an algebra homomorphism. Then we define the kernelofϕ, writtenkerϕ,to be the set kerϕ=fb2B:ϕ(b) =0g: Wealso define the imageofϕ,written im ϕ,to be the set imϕ=fc2C:c=ϕ(b);for some b2Bg: We should perhaps note that, by construction, any algebra homomorphism is simultaneously a vector space and ring homomorphism, for which we already have definitions of the image and kernel. It may be easily checkedhoweverthat these definitionscoincide. 4 Theorem 10. (1stIsomorphism Theorem for Algebras) Letϕ:B!Cbe an algebra homomorphism. Then kerϕis an ideal of B,imϕis a subalgebra of Cand B=kerϕ»=imϕ. Thisisanimportanttheorem,essentiallycommentingonthepossiblefactorisationsofanygiven homomorphism. Perhaps it is more readily absorbed in a pictorial form, displayed below. Bϕ // π ²²C B=kerϕeϕ //imϕι OO Hereπ:B!B=kerϕistheprojectionmapsending b7!b+kerϕ,eϕistheisomorphismofthe theorem and ι:imϕ!Cis the inclusion map c7!c. The theorem can then be stated by saying thatthe abovediagram commutes, that is ϕ=ι±eϕ±π. In other algebraic structures, groups, modules and rings for instance, analogues of the first iso- morphismtheoremareapowerfultoolinidentifyingisomorphismsbetweenspaces. Thispoweris increased in algebras since we can bring to bear on problems dimension arguments. For example, withϕas above, if we know that dim(imϕ) =dimCthen as im ϕis a subalgebra Cwe may con- cludethat im ϕ=C. Exercises. Letϕ:B!Cbe an algebra homomorphism. (i)Showthat the direct product of twoalgebras is also an algebra. (ii)Showthat kerϕis an ideal of Band imϕisa sub-algebra of C. (iii)Provethat ϕisinjective ()kerϕ=0: (iv)Provealso that ϕissurjective ()imϕ=C: (v)Findthe isomorphism of Theorem 10 . 5 2. Matrix Algebras We noted in the previous section that matrix algebras are very important in algebra theory. Withouthesitation then, we present thereason for this statement. Theorem 11. Everyalgebra AoverFisisomorphictoasubalgebraofthematrixalgebra Mn(F), forsomen. Proof(Abian;[Abn 1971]) We will prove the theorem for the case n=3, which illustrates the proof in general. Thus, we assume that Ais a3dimensional algebra over Fand that fI;A;Bgis a basis of AwhereIis the identityof A. Let the multiplication table of Awith respect to the basis fI;A;Bgbe givenby I A B I1I+0A+0B0I+1A+0B0I+0A+1B A0I+1A+0B aI +bA+cB gI +hA+kB B0I+0A+1B qI +rA+tB uI +vA+wB Clearly, we may represent Iby(1;0;0),Aby(0;1;0)andBby(0;0;1)and rewrite this table as follows. I A B (1;0;0)(1;0;0) (0;1;0) (0;0;1) (0;1;0)(0;1;0) (a;b;c) (g;h;k) (0;0;1)(0;0;1) (q;r;t) (u;v;w) From the rewrittentable we see that (1;0;0):I= (1;0;0);(0;1;0):I= (0;1;0);(0;0;1):I= (0;0;1): The aboveequalities suggest substituting forIthematrix0 @1 0 0 0 1 0 0 0 11 A: Again,from this table we see that (1;0;0):A= (0;1;0);(0;1;0):A= (a;b;c);(0;0;1):A= (q;r;t): The aboveequalities suggest substituting forAthe matrix0 @0 1 0 a b c q r t1 A: Finally,we see that (1;0;0):B= (0;0;1);(0;1;0):B= (g;h;k);(0;0;1):B= (u;v;w): 6 The aboveequalities suggest substituting forBthematrix0 @0 0 1 g h k u v w1 A: Motivatedbythesesubstitutions,weconsiderthemapping ϕfromAintoM3(F)whereforevery elementS2AwithS=s1I+s2A+s1Bwedefine ϕ(S) =s1ϕ(I)+s2ϕ(A)+s1ϕ(B) where ϕ(I) =0 @1 0 0 0 1 0 0 0 11 Aϕ(A) =0 @0 1 0 a b c q r t1 Aϕ(B) =0 @0 0 1 g h k u v w1 A: We claim that ϕis an algebra homomorphism and leave it as an exercise to check the three conditionsprescribed in De¯nition 8 hold. Finally,weseethematrices ϕ(I);ϕ(A)andϕ(B)arelinearlyindependent. Therefore kerϕ=0. Bythefirstisomorphismtheorem( Theorem 10 ),wehavethat ϕisanisomorphismfrom Aontoa subalgebraof M3(F), as desired. ¤ This process of representing an algebra over Fby a subalgebra of the matrix algebra is some- timesreferred to as regularrepresentation . Example 12. For a concrete example of Theorem 11 we shall endeavor to discover the regular representationof the algebra C. First we rewrite Table 1as 1i (1;0)(1;0) (0;1) (0;1)(0;1) (¡1;0) whichsuggests the mappings 17!µ 1 0 0 1¶ i7!µ 0 1 ¡1 0¶ and a quickcheckµ 0 1 ¡1 0¶2 =¡µ 1 0 0 1¶ showsthat we are on the right track. Finallywe see that, C»=(µ α β ¡β α¶ :α;β2R) : 7 Exercise. Find the regularrepresentation for H, the quaternions. Another reason for matrix algebras importance is the following. Proposition 13. LetAbean algebrawith BandMn(F)as subalgebrassuchthat B\Mn(F) =fλ:1A:λ2Fg. Supposethat BandMn(F)commute element-wise and that BandMn(F)generate A. Then A»=Mn(B). This is, for those readers who are familiar, ostensibly a fact from tensor products (i.e. B­ Mn(F)»=Mn(B)). Indeed this is the correct way to think of this result. However, since this is not theplacetointroducesuchtheorywewillgiveanilluminatingexampleinplaceofaformalproof. Example 14. Considerthematrixalgebra M2(C). Thishassubalgebras CandM2(R),withbases(µ 1 0 0 1¶ ;µ i0 0i¶) (µ 1 0 0 0¶ ;µ 0 1 0 0¶ ;µ 0 0 1 0¶ ;µ 0 0 0 1¶) ; respectively. It is easy to see that elements of these bases commute with each other and that C\M2(R) =(µ λ0 0λ¶ :λ2R) . Further,theygenerate an algebra with basis(µ 1 0 0 0¶ ;µ 0 1 0 0¶ ;µ 0 0 1 0¶ ;µ 0 0 0 1¶ ;µ i0 0 0¶ ;µ 0i 0 0¶ ;µ 0 0 i0¶ ;µ 0 0 0i¶) i.e. with basis(µ 1+i0 0 0¶ ;µ 0 1+i 0 0¶ ;µ 0 0 1+i0¶ ;µ 0 0 0 1+i¶) Whichcan be seen to be a basis for M2(C). 8 3. Cli®ord Algebras With our introduction to algebras finished, what follows is the main exposition of this report. Starting with the basic definition below, we shall expand to look at the regular representations of Clifford algebras. Culminating this section with a marvellous theorem ( Theorem 22 ) on the ‘peri- odicity’of Cliffordalgebras. De¯nition 15. Given a real vector space V, theClifford algebra C`Vis the associative algebra freelygenerated by Vsatisfying x2=¡jxj2 for allx2V. Werestrict our explorationof Cliffordalgebras here to considering only the case where V=Rnand jxj=·n ∑ i=1x2 i¸1=2 wherex= (x1;x2;:::;xn), we denote this family of Clifford algebras C`n. A more general defini- tion is included in the appendix as De¯nition 24 . For a broader approach to Clifford algebras see [Artn1957], [Bkr 2002], [Lousto 1995] and [Prts 1995]. Example 16. Let’sget our hands dirty with some calculations. Take an orthonormal basis of R2,fe1;e2g. We wish to form all finite products of these basis elements, subject to the condition given in the definition. Any vector in R2can be written as λe1+µe2forsome λ;µ2R. Using the definition abovewe require that (λe1+µe2)2=¡(λ2+µ2), i.e.λ2e2 1+µ2e2 2+λµ(e1e2+e2e1) = ¡λ2¡µ2 Hence, equating coefficients,we havethe relations e2 1=e2 2=¡1ande1e2+e2e1=0. Thusanyfurtherproductsofbasiselementsislinearlydependantontheelements f1;e1;e2;e1e2g (where1denotesthe empty product). Therefore, this set forms a basis for C`2. Observe that the subset of this basis fe1;e2gof products of odd length generates R2as a vector space. Alsothattheproductsofevenlengthareclosedundermultiplicationshowingthatthespace spannedby f1;e1e2gisa subalgebra of C`2. The full multiplication table for C`2isshownbelowin Table 3below. 9 1e1e2e1e2 11e1e2e1e2 e1e1¡1e1e2¡e2 e2e2¡e1e2¡1e1 e1e2e1e2e2¡e1¡1 Table 3. Multiplication Tablefor C`2 Notice the similarity between this multiplication table and Table 2for the quaternions. As we remarkedin Note 4,this is sufficientto showthe following. Proposition 17. C`2»=H As a corollary to this, we note that the even subalgebra mentioned above, spanned by f1;e1e2g isisomorphic to the complexnumbers C. Generalisingtheprocessfollowedintheexample,bybuilding C`nfromanorthonormalbasisof Rn, we recoverthe subsequent proposition. Proposition 18. For1·i·n, letfeigbe an orthonormal basis for Rn. Then in C`nwe have the followingrelations. e2 i=¡1eiej+ejei=0 for1·i;j·nwithi6=j. Remarks. In our definition of Clifford algebras, De¯nition 15 , we made two assertions that are notatalltransparent. Firstly,wesaidthataCliffordalgebrais freelygeneratedbythevectorspace; secondly,that this algebra is unique (up to isomorphism). The concern is that we have made choices that could apparently lead to the construction of distinct Clifford algebras. Namely, we choose a basis of the vector space V, then we multiply elements of this basis together to get the basis for an algebra. The miraculous part is if we choose a different basis of Vand form the Clifford algebra over Vwith respect to this new basis, we get an isomorphic Clifford algebra! For more details on this consult [Bkr 2002] or [Chvy 1997]. An algebrawith this property is often called an universal algebra. 3.1.Grade and Dimension. De¯nition 19. We define the gradeof a basis element of C`nto be the length of the product of thatelement. 10 Forexample,the basis of C`2consists of: 1element of grade zero 1 scalar 2elements of grade one e1,e2vectors 1element of grade two e1e2bi-vector In general, the basis of C`nconsistsof: 1elementof grade zero 1 scalar nelementsof grade one e1,:::,en vectors ¡n 2¢ elementsof grade two e1e2,:::,en-1enbi-vectors ¡n 3¢ elementsof grade three e1e2e3,:::,en-2en-1entri-vectors ...... 1elementof grade n e 1e2¢¢¢en n-vector The grade structure of Cliffordalgebras followsthe pattern of Pascal’striangle, seen below. n 2n 0 1 1 1 1 1 2 2 1 2 1 4 3 1 3 3 1 8 4 1 4 6 4 1 16 5 1 5 10 10 5 1 32 6 1 6 15 20 15 6 1 64 7 1 7 21 35 35 21 7 1 128 ........................... Fromvectorspaces,weknowthatitsdimensionisthenumberofelementsinitsbasis. Thusthe dimension of an algebra is the number of elements in the basis. The above remarks on the grade structure of Clifford algebras give us the following proposition. Indeed, this also follows quite naturallyfrom the factthat we are constructing our algebras freely. Proposition 20. dimC`n=2n. 11 3.2.Some Remarks. (i)WehavethroughoutbeenconstructingexamplesofCliffordalgebrasfromthevectorspace Rnwith the standard inner product. Reflecting on the containment 1½R½R2½ ¢¢¢ ½Rn¡1½Rn½ ¢¢¢ itfollowsthat C`0½C`1½C`2½ ¢¢¢ ½C`n¡1½C`n½ ¢¢¢ (ii)Notealso that in C`3, (1+e1e2e3)(1¡e1e2e3) =0; with(1+e1e2e3);(1¡e1e2e3)6=0. ThusC`3has a divisorof zero. These tworemarks bring to light the following. Proposition 21. Forn>2,C`nhasdivisorsof zero. 3.3.MatrixRepresentationsofCliffordAlgebras. Inthespiritoftheprevioussection (Section 2) on matrix algebras, we show here isomorphisms between the first nine Clifford algebras and standard matrix algebras. As we will see in Section3.4, something quite wonderful happens after then. We saw in Example16 thatC`2»=Hand we hinted in the introduction that there exist Clifford algebrasisomorphic to RandC. A method due to Alan Wiederhold gives us a uniform way of obtaining the representations of C`0toC`4shownbelowin table 4. nC`n 0R 1C 2H 3H£H 4M2(H) Table 4. Matrix Representations of C`0toC`4 12 From our knowledge of matrix theory, we know that any element αof a fieldFcan be repre- sented with the elementµ α0 0α¶ 2M2(F). Further, we can represent an element (α1;α2)of F£Fwith the elementµ α10 0α2¶ 2M2(F). Using this knowledge,combined with the observations R½C½H½H£H½M2(H) C`0½C`1½C`2½ ¢¢¢ ½C`n¡1½C`n½ ¢¢¢ weconsider the followingassignments. 1 e1 e2e3e4µ 1 0 0 1¶ µ i0 0i¶ µ j0 0j¶ µ k0 0¡k¶µ 0k k0¶_OO )44iiiiiiiii v ¾¾66666666666 H ¤¤©©©©©©©©©©©¸jjUUUUUUUUU wherei;jandkare defined as on Brougham bridge! That is i2=j2=k2=¡1,ij=¡ji=k. The dedicated reader may check that this indeed givesthe isomorphisms in Table 4. Next we would like to find a representation for C`5. The assignment of the basis vectors 1;e1;:::;e5ofR5shown below induces the required isomorphism C`5!M4(C). Here, as usual, i2=¡1. 13 1 e1 e2 e3e4e50 BB@1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 11 CCA0 BB@i0 0 0 0¡i0 0 0 0i0 0 0 0 ¡i1 CCA 0 BB@0¡1 0 0 1 0 0 0 0 0 0 ¡1 0 0 1 01 CCA0 BB@0i0 0 i0 0 0 0 0 0 ¡i 0 0 ¡i01 CCA0 BB@0 0 0 ¡1 0 0 ¡1 0 0 1 0 0 1 0 0 01 CCA0 BB@0 0 0 ¡i 0 0 ¡i0 0¡i0 0 ¡i0 0 01 CCA _OO °ffMMMMMMMMM 1 xxqqqqqqqqq _ ²²° &&MMMMMMMMM188qqqqqqqqq By direct calculation, using the method described in the proof of Theorem 11 for finding a reg- ularrepresentation, we discoverthe representations shownbelowin Table 5. n C`n 6 M8(R) 7M8(R)£M8(R) 8 M16(R) Table 5. Matrix Representations of C`6toC`8 3.4.Periodicity of 8. The result that was alluded to at the start of the last section is the topic of this section. Informally, it characterises all of the real Clifford algebras (with the standard inner product)in terms of the first eight, which we havesummerised belowin Table 6. 14 n C`n 0R 1C 2H 3H£H 4 M2(H) 5 M4(C) 6 M8(R) 7M8(R)£M8(R) Table 6. Matrix Representations of C`0toC`7 Theorem 22. (Cartan,1908) C`n+8»=M16(C`n) Proof Take an orthonormal basis fe1;e2;:::;en;en+1;:::;en+8gofRn+8and sete0 i=eien+1¢¢¢en+8fori= 1;2;:::;n. Then the subset fe0 1;e0 2;:::;e0 ngofC`n+8generates a subalgebra isomorphic to C`n. The subalgebra generated by en+1;:::;en+8is isomorphic to C`8»=M16(R). These two subalgebras com- mute with each other element-wise and generate all of C`n+8. Finally, Proposition 13 inSection 2 givesus the result. ¤ Example 23. Toilluminatetheabovetheorem,weshallworkthroughtheproofforthecase n=1. First, takean orthonormal basis of R9,fe1;e2;:::;e9g. Wesete0=e1e2¢¢¢e9and observethat (e0)2= (e1e2¢¢¢e9)(e1e2¢¢¢e9) =:::= (e1e2¢¢¢e9)(e9e8¢¢¢e1) =:::=¡1. Hencee0=e1e2¢¢¢e92C`9generates the subalgebra C»=C`1. Next,itisclearthat fe2;e3;:::;e9gisabasisforR8andsothesubalgebrageneratedbyitis C`8, by construction. To show that these two algebras commute element-wise we are required to show that e0ei= eie0i.e. (e1e2e3¢¢¢e9)ei=ei(e1e2e3¢¢¢e9) (2·i·9), which follows from the properties of Proposition 18 . Itremainstoshowthat fe0;e2;e3;:::;e9ggeneratesallof C`9,thisisachievedbynoticingthat e1 canbe recoveredfrom e0on post-multiplication by the element ¡e9e8¢¢¢e2. Putting this together we have C`9»=M16(C`1)»=M16(C). 15 4. Further Theory and Applications of Cli®ord Algebras We conclude this report by looking at the directions in which further study of Clifford algebras maylead, both in developingadditional theory and exploringapplications to other sciences. 4.1.Division Algebras. The alert reader may have noticed that although in our introduction to Clifford algebras we promised to generalise the complex and quaternion algebras into higher di- mensionswe havehad to drop one of their nice properties: division. Roughly speaking (and perhaps dangerously close to sounding patronising), a division algebra isanalgebrawheredivisionispossible. Wehavedefinedouralgebrastobeassociative. Ifwefur- ther assume that they are of finite dimension, then being a division algebra is equivalent to having no divisors of zero. Alternatively, every non-zero element has a multiplicative inverse, that is to say,everynon-zero element is a unit. There is nothing to stop us however, from defining this concept for a non-associative algebra. We say that a non-associative algebra is a (non-associative) division algebra if the operations of leftand right multiplication by anynon-zero elementare invertible. Remarkably, it turns out that there exist, up to isomorphism, only four real division algebras. Three of these are associative, they are the familiar real, complex and quaternion algebras R;C andH, respectively. The octonions are our fourth division algebra. They are eight dimensional overRandnon-associative,although theydo satisfy a weakercondition, called alternative . Albert, in his book [Albt 1939], states that the study of linear algebras ‘reached its zenith when thesolutionwasfoundfortheproblemofdeterminingallrationaldivisionalgebras’. Thestandard construction of these four algebras is named after the mathematicians Cayley and Dickson. For a detailedaccountofthisandtherelationtoClifford’salgebraswedirectthereaderto[Bz 2001],an excellentpaper entitled ‘The Octonions’. 4.2.Spinors. Let us look now at the concept of a Spinor(pronounced spin-or). The mathematics of these objects is central to the understanding of the quantum physics of basic particles – like protons, neutrons and electrons. Indeed, as Penrose states in [Pnrse 2004], ‘ordinary solid matter couldnot existwithout its consequences’. Essentially, a spinor is an object that, on completion of a rotation through an angle of 2π, turns intoitsnegative. Thismayseemcounterintuitivetooureverydayexperience,absurdeven. Inplace of a formaldefinition, we will describe spinors by analogy. 16 Picture a book lying on a table in front of you. We want to keep track of the rotations of the book, so let’s open it and place a belt in between the pages. Next, fix the buckle end of the belt (under another pile of books, say). This set-up is shown in Figure 7 (a). Now, rotating the book through2πputs a twist in the belt, which cannot be undone without further rotation (as shown in (b)). Butsomethingcurioushappensifwerotatethebookthroughanother 2π,thetwistinthebelt canbe undone by looping the belt overthe book (shownin (c)). Figure 7. Spinorial Book Thus, the belt keepstrack of the parity of the number of 2πrotations. That is to say, if the book is rotated through an even number of 2πrotations the twist in the belt can be removed without further rotations of the book, whereas an odd number of 2πrotations leaves an inevitable twist in thebelt. This holds true for anycombinationof rotations through anyaxes. Spinors are connected with Clifford algebras in a very fundamental way, which we will illumi- nateat the end of the followingsection. 4.3.Embedding C`ninC`n+1.As we noted in Section 3.2 (i), there is a canonical embedding of C`ninC`n+1givenby the containment Rn½Rn+1. A less obviousembedding can be found by considering Rn!C`n+1x7!xen+1. It turnsout that this can be extendedto an algebra isomorphism C`n!C`+ n+1ei7!eien+1 17 (1<i<n+1), whereC`+nis the subalgebra of C`ngenerated by a subset of the basis for C`n consisting entirely of products of even length. For example, C`+ 2has basis f1;e1e2gand similarly, C`+ 3hasbasis f1;e1e2;e1e3;e2e3g. Returningtotheconceptofspinors,weshallmakeacoupleofdefinitionstoaidourexplanation of theconnections between these and Cliffordalgebras. A closely related concept is that of pinors. Let Pin nbe the group of pinors sitting inside C`n, consistingof all the products of unit vectorsin Rn. Consider now the following homomorphism between Pin nand the orthogonal group, On. For a unit vector x2Rn, we map both §vto the element of Onrepresenting the reflection in the hyper- plane perpendicular to v. Since every element of Onis composed by reflections, it is easily seen thatthis homomorphism is surjective. Moreover, Pin nis a double cover for On, that is to say every element of the orthogonal group is represented by two opposite pinors. Another way of expressing this is to say that the kernel of this mapping consists of just two elements, namely §1. The concept of double covers should be familiar to the reader – you need look only as far as the hands on your wrist-watch! As each hand positioncorresponds to twopositions of the sun. Now, let Spin nbe the group consisting of all the products of an even number of unit vectors in Rn. This group, a subgroup of Pin n,is the group of spinors as described in Section 4.2 . An element of Onis also an element of SOn, the special orthogonal group, precisely when it is the product of an evennumber of reflections. Thus, just as Pin nis a double coverof On, Spinnis a doublecoverof SOn. As we know, every rotation in Rncan be represented by an element of SOn. The trouble is that elements of SOndon’t so much as represent rotations, but represent the final result of such a rotation–thesenseoftherotationhasbeenlost. ThisiswhereSpin nentersandtheveilshrouding its importance is lifted, as with one element of this group we can truly represent a rotation, its magnitudeand sense. Intriguingly, the epimorphism above from Pin ntoOn, also tells us that pinors in ndimensions arespinors in n+1dimensions! 18 4.4.More Applications in Physics and Computing. Next, a brief glance at some other appli- cations of Clifford algebras. In 1928, the Dirac electron equation provided a turning point for physics. Dirac himself, unaware of Clifford and Hamilton’s earlier work, was driven to reinvent partsofCliffordalgebrainanattempttounderstandthephysicalspinofthefundamentalparticles of nature. Toconcludethereportletustakeabrieflookatsomerecentdevelopmentsincomputing. Here, Clifford algebras are presently being groomed for use in modelling geometries. These are in turn used in complex computer graphics. The current methods for modelling geometries, according to Dorst[Drst 2001], are fragmented at best. :::in every application a bit of linear algebra, a bit of differential geometry, a bit ofvectorcalculus, each sensible used, but ad hocintheir connections ::: He goes on to say that this approach leads to unnatural splits in the program – rather than a di- vision of tasks matching the nature of the problem. The solution he offers is a single ‘Clifford’ toolbox. Allowing calculations to be performed in a single framework, free from coordinates. Claimingalso,thatgeneralisationofprogramstohigherdimensionsbecomesintuitiveunderthese constructions. Altogether, it seems as if he and his colleagues are trying to start a Clifford algebra revolutionin computing! 19 5. Appendix For the interested reader, here is the more general definition for a Clifford algebra that was promisedin the preface. De¯nition 24. Givenaninnerproductspace (V;(¢j¢))andassociatedquadraticform Q(x) = (xjx), thereis a Cliffordalgebra C`V(Q)whichcontains Vasa subspace and satisfies x2=¡Q(x)1 for allx2V. 20 References [Abn 1971] A. Abian, LinearAssociative Algebras , PergamonPress, 1971 [Albt1939] A. A. Albert, Structureof Algebras ,AMS, 1939 [Artn1957] E.Artin,GeometricAlgebra , Interscience Publications, 1957 [Bz WWW] J. C. Baez, http://math.ucr.edu/home/baez , Universityof California Riverside,2005 [Bz 2001] J.C. Baez, The Octonions ,AMS, 2001 [Bkr2002] A. Baker, MatrixGroups , Springer,2002 [Bn&Tkr1987] I.M.BennandR.W.Tucker, AnIntroductiontoSpinorsandGeometrywithApplicationsinPhysics , IOP Publishing Ltd., 1987 [Brner1963] H. Boerner, Representationsof Groups ,North-Holland Publishing Company,1963 [Crtn1966] E. Cartan, TheTheory of Spinors , MIT Press, 1966 [Chvy 1997] C.Chevalley, The AlgebraicTheory of Spinorsand CliffordAlgebras ,Springer,1997 [Chvy 1956] C.Chevalley, Fundamental Concepts of Algebra ,Academic Press, 1956 [Crts1979] M.L. Curtis, MatrixGroups ,Springer-Verlag,1979 [Dksn1930] L.E. Dickson, Linear Algebras , Cambridge UniversityPress, 1930 [Drst1999] L.Dorst, Geometric(Clifford)Algebra:apracticaltoolforefficientgeometricalrepresentation ,Dept.of Computer Science, Universityof Amsterdam, 1999 [Drst2001] L. Dorst, Honing geometric algebra for its use in the computer sciences , Dept. of Computer Science, Universityof Amsterdam, 19?? [Hmltn1853] Hamilton, Lecturesin Quaternions ,MacMillan and Co., 1853 [Hrvy1990] F.R. Harvey, Spinorsand Calibrations , Academic Press Inc., 1990 [Jly1905] C.J. Joly, Manualof Quaternions , MacMillian and Co., 1905 [Klnd&Tt1873] Kellandand Tait, Introductionto Quaternions ,MacMillan and Co., 1873 [Kprs1999] J. B. Kuipers, Quaternionsand Rotation Sequences ,Princeton UniversityPress, 1999 [Lousto1995] P.Lounesto, CliffordAlgebrasand Spinors , Cambridge UniverstiyPress, 1995 [Pnrse2004] R. Penrose, TheRoad to Reality ,Jonathan Cape, 2004 [Prce 1968] R. S. Pierce, Introductionto the Theory of AbstractAlgebras ,Holt, Reinhart and Winston,1968 [Pymn&Rbnson1994] Plymen and Robinson, Spinorsin Hilbert Space , Cambridge UniversityPress, 1994 [Prts1995] I. R. Porteous, CliffordAlgebrasand the Classical Groups , Cambridge UniversityPress, 1995 [SmthWWW] T.Smith, http://www.innerx.net/personal/tsmith/clfpq.html [Tt 1867] Tait,Quaternions , Clarendon Press Series, 1867 [WfrmWWW] E. W.Weisstein, http://mathworld.wolfram.com ,2005 [DnkrWWW] J.S. Denker, http://www.av8n.com/physics/complex-clifford.htm ,2002 [clfrdWWW] W.M. Pezzaglia Jr., http://www.clifford.org ,Santa Clara University,2003 [FlrWWW] J. Fletcher, http://www.ceac.aston.ac.uk/clifford ,Aston University,2004 DEPARTMENT OF MATHEMATICS , UNIVERSITY OF GLASGOW, UNIVERSITY GARDENS, GLASGOW G12 8QW E-mailaddress: [email protected] 21