CliffordGeometricAlgebra

“wherealgebrameetsgeometry(andviceverse)”

SeminarnotesMarch-21,2016

RobertForchheimer

SomepreparaGons:•  Complexnumbersare2-dimensionalenGGes:

•  a+jbwheretheaxesunitvectorsare1andj

•  butitisalsoanAlgebrawiththegeneraGngelements1andjandthetwooperaGons“+”and“*”.

•  MulGplicaGonrulesfortheunitvectors:1*1=1,1*j=j*1=j,j*j=-1.

•  MulGplyinganytwocomplexnumbersgivesanewcomplexnumber(thesetisclosedundermulGplicaGon).

SomepreparaGons–2Considera3DCartesianvectorspace:•  Orthogonalunitvectorse1,e2,e3.

•  v=a1e1+a2e2+a3e3isageneralvector

•  u=b1e1+b2e2+b3e3isanothergeneralvector

FormalmulGplicaGongives:•  v*u=a1b1e1e1+a2b2e2e2+a3b3e3e3+

a1b2e1e2+a2b1e2e1+a2b3e2e3+a3b2e3e2+a1b3e1e3+a3b1e3e1

UsingthefollowingmulGplicaGonrules(scalarmulGplicaGon):(ei)2=1,eiej=0,i≠ j.gives:v*u=a1b1+a2b2+a3b3=v•u(ascalar-outsidethe3Dvectorspace!)

UsinginsteadthefollowingmulGplicaGonrules:(ei)2=0,e1e2=-e2e1=e3;e2e3=-e3e2=e1;e3e1=-e1e3=e2AndinserGngin:v*u=a1b1e1e1+a2b2e2e2+a3b3e3e3+

a1b2e1e2+a2b1e2e1+a2b3e2e3+a3b2e3e2+a1b3e1e3+a3b1e3e1gives:v*u=(a2b3-a3b2)e1-(a1b3-a3b1)e2+(a1b2-a2b1)e3=v×u(crossproduct)

Theresultwillbeanewvectorintheoriginal3Dspace.Thesetofvectorsisclosedunderthecrossproduct.

w=v×u

3DCartesianvectorstogetherwiththescalarproductandvectorproductformthebasisofVectorAnalysiswhichhasmanyapplicaGonsinmathemaGcsandphysics.Butbewareofthefollowing...,letsassumethataphysicalenGtyisgivenbythecrossproductoftwovectors:

Whathappensifwe“mirror”thevectorsbyreversingthesignofthethreeaxes?

So,e1->-e1,e2->-e2,e3->-e3

Thenv->-vandu->-u,butw->w!

Thus,wisadifferenttypeofvectorthanvandu.InphysicssuchenGGesarecalledaxialvectorsincontrasttothe(ordinary)polarvectors.Exampleofaxialvectorsareangularvelocity,torque,magne6cfield

SomepreparaGons–3

polar

axial

Polarvsaxialvector(from[1])

Cliffordalgebra

Requirements(WilliamK.Clifford,1878):

1.  v*v:shouldcorrespondtothesquaredlengthofv2.  Basicalgebraiclaws(associaGve,distribuGve)shouldapply3.  MulGplicaGonoperaGonshouldbedefinedforany-dimensionalvector4.  Thereshouldbeawell-definedinverse(whenv≠0)

From1:v*v=v•v.Replacingv=u+wgives:(u+w) * (u+w)=(u+w)•(u+w)From1and2:u2+w2+u*w+w*u=u2+w2+2u•wThusu*w+w*u=2u•wThisisobviouslyfulfilledwhen*correspondstoscalarmulGplicaGon.Butcanitbefulfilledforothercases?Assumeitcan,whataretheconsequences?E.g.ifu,waremutuallyothogonal:u*w=-w*u(leadingtonon-commutaGvity).Wehavealreadyseenthisforthecrossproduct!

CliffordmulGplicaGonMulGplicaGonrules1):(ei)2=1,eiej=-ejei,i≠ j.Thisgives2):vu=a1b1+a2b2+a3b3+(a2b3-a3b2)e2e3-(a1b3-a3b1)e1e3+(a1b2-a2b1)e1e2Thusascalarpartandacrossproductpart,butalongthenewaxes(e1e2),(e2e3),(e3e1).Specialcase:v2=vv=a12+a22+a32(ascalar,aswealsoassumediniGally)Thisallowstodefinetheinverseofavector:v-1=v/v2(wheneverv2≠ 0).

1)Inthefollowingtheunitvectorswillnotbeshownasbold2)JuxtaposiGonofvectorsdenotesCliffordmulGplicaGon

Wenotethat(e1e2)2=e1e2e1e2=-e1e2e2e1=-e1e1=-1

Similarly(e2e3)2=(e3e1)2=-1Itisalsoseenthat(e1e2e3)2=-1Denote:e1e2e3=jfromwhichweseethatje1=e2e3etceterasothatwecanwrite:vu=v•u+j(v×u)thusresemblingan“extendedcomplex”enGty.vΛu=j(v×u)iscalledthe“wedgeproduct”.Itismoregeneralthanthecrossproductasitiswell-definedforboth2D,3Dandhigherdimensions.Note1:Ifv,uarecolinearthenvu=uv,iftheyareperpendicular,vu=-uv

Note2:WecanwriteeiejaseiΛej(Anotherterminologyiseiej=eij)

InCliffordalgebrawethushavethefollowingdimensions/unitvectors:

1:Scalars(0thgrade)e1,e2,e3:vectors(1stgrade)

e1e2,e2e3,e3e1:bivectors(2ndgrade)e1e2e3:trivector(3rdgrade)

Allinall,8dimensions!Thesearedenoted“blades”tobedisGnguishfromthe3Dvectorspacespannedbye1,e2,e3.AgeneralCliffordalgebraelement(“mulGvector”)isthus:M=a+v1e1+v2e2+v3e3+w1e1e2+w2e2e3+w3e3e1+be1e2e3==a+v+jw+jbThedifferenttermscanbeseparatedaccordingtotheirgradeasfollows:<M>0=a,<M>1=v,<M>2=jw,<M>3=jb

M=a+v+jw+jbSpecialcase1:(firstandlastterms)Z=a+jbisisomorphwiththecomplexnumbersSpecialcase2:(firstandthirdterms)Senngi=-e2e3,k=-e3e1,l=-e1e2Wenotethatik=l,kl=i,li=k,ikl=-1ThusH=a+jwisisomorphwithquaternionsCliffordalgebracombinescomplexnumbers,Cartesianvectorsandquaternionsinonecompoundelement!

ProperGesofbivectorsandthetrivectorConsidertwoCartesianvectorsu,vin2D:u=u1e1+u2e2,v=v1e1+v2e2TheirwedgeproductbecomesuΛv=(u1v2–u2v1)e1e2A=(u1v2–u2v1)isa(signed)measureoftheareathatthetwovectorsspan:

Thus,(e1e2)canbeconsideredasaunitforthe“areadimension”

u

v

A

Similarly,in3D,thetrivectorrepresentsaunitforthe“volumedimension”ThisisthereasonwhyCliffordAlgebraisnamed“GeometricAlgebra”.Allthe8dimensionsrelatetoproperGesofthebasic3DCartesianspace.

TheCliffordAlgebraunitvectors(exceptthescalarunit1)

(Figurefrom[1])

ThereversionofamulGvectorDefinethereversionMrofMas:Mr=a+v-jw–jbItisthenseenthatMMr=(a+v+jw+jb)(a+v-jw-jb)=

a2+av-ajw-ajb+ va+v2–vjw-vjb+ jwa+jwv–(jw)2-jwjb+ jba+jbv-jbjw-j2b2=

=a2+2av+v2–jvw+jwv-(jw)2+2bw+b2=a2+b2+v2-(jw)2+2av+2bw–2vΛwForbivectorsjw=w1e2e3+w2e3e1+w3e1e2wehavethefollowinggeneralproperty:(jw)2=(w1e2e3+w2e3e1+w3e1e2)(w1e2e3+w2e3e1+w3e1e2)=-w1

2-w22-w3

2thusanon-posiGvescalar.(Thiscanalsobededuceddirectlysinceforvectorwweknowthat:w2=w1

2+w22+w3

2)Wedefine||M||2=<MMr>0=a2+b2+v2+w2(thesquarednormofM)

TheconjugaGonofamulGvectorDefinetheconjuga6onMcofMas:Mc=a-v-jw+jbItisthenseenthatMMc=(a+v+jw+jb)(a-v-jw+jb)=

a2–av-ajw+ajb+ va–v2–vjw+vjb+ jwa–jwv–j2w2+jwjb+ jba-jbv-jbjw+j2b2=

=a2+2jab-v2–jvw–jwv+w2-b2=a2-b2–v2+w2+2j(ab-v•w)ThusMMc=scalar+j(scalar)Wedefine|M|2=MMc:thesquaredamplitudeofM.(Observethatthisisnotascalar,rathera“complex-type”enGty)TheinverseofamulGvectorDefineM-1=Mc/MMcastheinverseofM.(NotallmulGvectorshaveaninverse.)

ApplicaGons1)  Algebra:Complexnumbers(a+jb,usede.g.toanalyzelinearACcircuits)

2)  Geometry:Quaternions(a+jw,usede.g.torotate3Dobjectsincomputergraphics)

3)  Geometry:ReflecGons,rotaGonsofvectorsandplanes

4)  Physics:RadialandaxialenGGes

5)  Physics:MaxwellsequaGons,SpecialrelaGvity

GeometricapplicaGon:reflecGonandrotaGon

Assumenormalizedvectorn.Forthereflectedvectoruwehave:v´=nvn

Proof:writev=vc+vp(composantswhicharecolinearandperpendicularton)

Thenwegetv´=n(vc+vp)n=n2vc-n2vp=vc-vpThus,v´isvectorvreflectedonn.RotaGoncanbeobtainedbysecondreflecGononm:v”=mnvnm

n

v=vc+vp

v´=nvn

-vp vp

vc vc

ReflecGonofbivectorA<2>onvectorn

GeometricapplicaGon:reflecGonofplane

Alsobivectors(planes)canbereflected(androtated)inthesameway

Proof:Assumetwoarbitrarynon-colinearvectors,v,uintheplane.ThenA<2>=vΛu.AnypointintheplanecanbedescribedasalinearcombinaGonofthesevectors:p=av+bu.ThereflecGonbecomesp´=n(av+bu)n=anvn+bnun=av´+bu´,alinearcombinaGonofthereflecGonofthearbitraryvectors.Thus,wecanwritethecorrespondingreflectedplaneA´<2>=v´Λu´=(nvn)Λ(nun)=n(vΛu)n=nA<2>n(seeextraslide).

Proofthat(nvn)Λ(nun)=n(vΛu)n

Asearlierwecanwritev=vc+vpandu=uc+up.Then:(nvn)Λ(nun)=(n(vc+vp)n)Λ(n(uc+up)n)==n(-nvp+nvc)Λ(-upn+ucn))=(vc–vp)Λ(uc-up)Similarly:n(vΛu)n=n((vp+vc)Λ(up+uc))n==n(vpΛup)n+n(vpΛuc)n+n(vcΛup)n+n(vcΛuc)n==n(vpup–vp•up)n+n(vpuc–vp•uc)n+n(vcup–vc•up)n+n(vcuc–vc•uc)n==(vpup-vp•up)-(vpuc+vp•uc)-(vcup+vc•up)+(vcuc–vc•uc)==vpΛup–vpΛuc+2vp•uc-vcΛup+2vc•up+vcΛuc=(vc–vp)Λ(uc-up)+2(vp•uc+vc•up)==(vc–vp)Λ(uc-up).

Physics

BivectorexamplesTorque:xΛ FAngularvelocity:xΛ vMagneGcfield:B1e2e3+B2e3e1+B3e1e2=e1e2e3(B1e1+B2e2+B3e3)=jB

(Tablefrom[1])

Physics:Maxwell’sequaGons

(Gauss’law)(Ampère’slaw)(Faraday’slaw)(Gauss’lawofmagneGsm)

∇ = e1∂∂x+ e2

∂∂y+ e3

∂∂z

where UsingtheGAproductrule:vu=v•u+j(v×u)

∇E =∇⋅E + j∇×Egives

∇B=∇⋅B+ j∇×Bsothat

1c∂∂t+∇

⎛

⎝⎜

⎞

⎠⎟ E + jcB( ) = ρ

ε0− cµ0J

Definefieldvariable:F=E+jcB,sourceJ=(ρ/ε)–cµJand∂ =(1/c)(∂/∂t)+∇

=>∂F=J

∇⋅E = ρε0

∇×B− 1c2∂E∂t

= µ0J

∇×E + ∂B∂t

= 0

∇⋅B= 0

Usefulformulas

a•b=½(ab+ba)aΛb=½(ab-ba)aΛbΛc=(abc+bca+cab-cba-bac-acb)/6(ifa,b,clinearlyindependentthengrade=3,otherwisegrade=0)Ageneralizeddotproduct(Contrac6oninnerproduct,Lounestoinnerproduct):a•(bΛc)=½[a(bΛc)-(bΛc)a]a•(bΛc)=(a•b)c−(a•c)b(aΛb)•A<k>=a•(b•A<k>)k>1

ReadingmaterialReference1:J.M.Chappelletal,GeometricAlgebraforElectricalandElectronicEngineers,ProceedingsoftheIEEE,Vol.102,No.9,September2014.Webmaterial:JaapSuter,GeometricAlgebraPrimer,h|p://www.jaapsuter.com/geometric-algebra/JohnDenker,Introduc6ontoCliffordAlgebra,h|ps://www.av8n.com/physics/clifford-intro.htmE.Hitzeretal.Applica6onsofClifford’sGeometricAlgebra,h|p://arxiv.org/abs/1305.5663J.WGibbs,E.B.Wilson,VectorAnalysish|p://www.archive.org/details/vectoranaysiste00gibbiala(ScannedversionofthefirsttextbookonVectorAnalysis,MIT1901)h|ps://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introducGon/h|ps://en.wikipedia.org/wiki/MulGvector