6 1 0 2 g u A 6 Exploring the origin of Minkowski spacetime ] h p - JamesM. Chappell,JohnG. Hartnett,NicolangeloIanella, Azhar n e Iqbaland Derek Abbott g . s c Abstract. FollowingMinkowski’sformulationofspecialrelativity,itisgener- i s allyacceptedthatweliveinafour-dimensionalworldconsistingofthreespace y andonetimedimension.Duetoitsfundamentalimportance,avarietyofargu- h ments have been proposed over the years attempting to derive this spacetime p structure from underlying physical principles. In our approach, we show how [ Minkowski spacetime arises from the geometrical properties of three dimen- 4 sional space. Wedemonstrate thisthrough modeling physical space withClif- v ford’sgeometricalgebraofthreedimensions. Weindeedfindusingthisrepre- 7 sentation that atime-likedimension arises naturallywithinthisspace but also 5 extendsspacetimetoeightdimensionsthroughincorporatingfourspindegrees 8 offreedom.Thisexpandedarenaofspacetimeproducesageneralizedgroupof 4 Lorentztransformationsandprovidesanaturaldescriptionoffundamentalparti- 0 cles.Nearlyallstandardresultsarereturnedinthisexpandedstructureandsoits . 1 mainachievementisthatitprovidesanefficientalternativeformalismtothecon- 0 ventionalfour-vectorformalismbutwewerealsoabletoidentifyseveralareas 5 wherenewphysicsmaybeindicated. 1 : MathematicsSubjectClassification(2010).MSC51B20,83A05. v Keywords.Minkowski,Spacetime,Cliffordgeometricalgebra,Specialrelativ- i X ity,Multivector. r a 1. Introduction Einstein’sseminalpaperof1905[14]introducedarevolutionarynewdescriptionof spaceandtime,thatincludedthepropertiesoftimedilationandlengthcontraction. Minkowskishoweda fewyearslater,thatthesephenomenacouldbesimplyinter- preted as the geometrical properties of a four-dimensional space-time continuum subjecttothemetric ds2 =dt2 dx2, (1.1) − where dx2 = dx dx = dx2 +dx2 +dx2 is the contributionfrom three spatial · 1 2 3 dimensionsandtisthetimeinthisreferenceframe,wherewehaveassumedunits such that the speed of light c = 1. Note, that in this paper we reserve bold fonts, 2Chappell,JohnG.Hartnett,NicolangeloIanella,AzharIqbalandDerekAbbott such as x, to refer to three-vectors.In the Minkowskiformulationwe can write a space-timeeventasthefourvectorX =[t,x]T,wherethecorrespondingeventfor otherinertialframesisfoundthroughtheLorentztransformationX′ = ΛX where Λisa4 4realmatrix,whichpreservesthemetricdistanceinEq.(1.1).Wereiterate × theseelementaryresultstoillustratefirstlytheunifiednatureofspaceandtime,and secondly,thatthemetrichasamixedsignature,asshowninEq.(1.1). Now,duetothefundamentalimportanceofMinkowskispacetime,avarietyof argumentshavebeenproposedovertheyearsseekingtoderivethisspacetimestruc- turefrommoreelementaryconsiderations,suchastheconsistencyofNewton’sFirst Law,spacetimeisotropywithnon-instantaneousinteractionpropagation[31,20],a probabilitydistributionresultingfromquantumfluctuationsofthespacetimemetric, therequirementsfortheexistenceofobservedelementaryparticlesanddynamicsin thecontextofbasicquantummechanicalconsiderations[5,33,34],implicationsof electric/magneticreciprocity[25],andtheeffectiveill-posednatureoftheEinstein fieldequationunlesstheMinkowskimetricholds[35]. WewilladopttheformalismofCliffordgeometricalgebraasithasbeenused previouslybyseveralauthorstoduplicatetheMinkowskispacetimestructure.Two well-knownimplementationsare(i)spacetimealgebra(STA)[22]and(ii)thealge- braofphysicalspace(APS)[2].TheformalismofAPSwillbeshowntobeaspecial case of our approachdetailed shortly, whereas in the case of STA the Minkowski metricisassumedaxiomaticallyratherthandeducedfromthealgebra. WenowbeginwiththeCliffordalgebraofthreedimensionsandproducethe Minkowskimetricasanemergentpropertyofthisalgebraicstructure.Whilerecov- ering standard results we nevertheless generalize the representation of spacetime eventsaswellasthegroupofLorentztransformations.Themetricisfoundtopro- duce a suitable Lagrangian for relativistic processes that indicates possible exten- sionstotheelectromagneticinteraction. 2. Derivation Itisgenerallyacceptedthatweliveinaworldwiththreespacedimensions,ascon- firmedbythepresenceofexactlyfiveregularsolids[11,30]andtheinversesquare lawsofgravityandelectromagnetismthathavebeenexperimentallyverifiedtovery high precision [24], indicating the absence of additional macroscopic dimensions beyondthreespacedimensions.Additionally,overcosmicdistances,thespatialpart oftheuniversehasbeenshowntoapproximateverycloselytobeingEuclidean[12]. Withthesepreliminaryobservations,webeginourinvestigationswiththege- ometricalpropertiesofthreedimensionalEuclideanspace.Inordertodescribethis spaceweadoptClifford’sgeometricalgebra(GA)overthreedimensionsasanap- propriate algebraic representation. This can be expressed using an object called a multivector M =a+x+jn+jb, (2.1) where x = x e + x e + x e a vector, jn = n e e + n e e + n e e a 1 1 2 2 3 3 1 2 3 2 3 1 3 1 2 bivector,andj =e e e trivectorwitha,b,x ,x ,x ,n ,n ,n realscalars[8,6]. 1 2 3 1 2 3 1 2 3 MinkowskiSpacetime 3 A valuable feature of the Clifford algebra description is that it algebraically en- codesthegeometricalentities—points,lines,arealandvolumeelements—foundin three-dimensionalspaceaswellasthephysicalquantitiesidentifiedasscalars,po- lar vectors, axial vectors(pseudovectors)and pseudoscalars(helicity),respectively. WehaveusedasabasisforthreedimensionalClifford’salgebraCℓ 3 ,thethree ℜ unitelementse1,e2,e3subjecttotherelationepeq =δpq+ǫpqrjer,(cid:0)wh(cid:1)ereδisthe Diracdeltafunction,ǫistheantisymmetrictensorandp,q,r 1,2,3 .1Referto ∈ { } AppendixA fora calculationof the multivectorproductusingthese relations.We caneasilyshowthatthetrivectorj commuteswithallelementsofthealgebraand squarestominusone,thusprovidinganaturalsubstitutefortheunitimaginary√ 1 − thusallowingustoremaininacompletelyrealspace.Wecanimmediatelyidentify theCliffordvectorx ofthe multivectorwith conventionalthree-vectorsalso com- monlyreferredtoaspolarvectors.Thebivectorjn,ontheotherhand,hasthetrans- formationalpropertiesappropriatefordescribingaxialvectorquantities.We could also nowattemptto identifya time coordinatewithin the multivectorin Eq. (2.1), however,wewishtoavoidimposinganypre-conceivedideasabouttime,andsowe willrefrainfromdoingthisuntilweclarifyfurtherthepropertiesofthemultivector. Wealsonotethattheeight-dimensionalmultivectorinEq.(2.1)isisomorphictothe complexifiedquaternions,also known as the biquaternionsthat have been studied sincethetimeofHamilton[21,32],however,themultivectorprovidesamorenat- uralgeometricalsetting than the biquaternions,as we can restrict ourselvesto the fieldofrealnumbers.Notethattheoctonionsarealso eight-dimensionalandhave alsobeenutilizedtodescribespacetime[15,17,18,16]. Now,Einstein’sfirstpostulate,thelawsofphysicsarethesameinallinertial framesofreference,leadsustolookfortheinvariantsin three-dimensionalspace, afterapplyinganappropriatetransformationrulesformovingbetweendifferentin- ertialframes. 2.1. Findingtheinvariantsandmetric Definition2.1(Cliffordconjugation). WedefineCliffordconjugationofamulti- vectorM as M¯ =a x jn+jb. (2.2) − − Clifford conjugation is an involution that is an anti-automorphism, so that for a productMN oftwomultivectorsM,N Cℓ 3 ,MN =N¯M¯. ∈ ℜ (cid:0) (cid:1) Cliffordconjugationisanalogoustocomplexconjugationortotheoperation ofraisingandloweringindicesfoundinfour-vectornotation. Definition 2.2 (Multivector amplitude). We define the amplitude squared of a multivectorM throughCliffordconjugationas M 2 =MM¯ =a2 x2+n2 b2+2j(ab x n) (2.3) | | − − − · forming a complex-likenumber. We refer to this as a “complex-like” number be- cause, as already noted, the trivector j is analogous to the unit imaginary and all 1WecandefineaCliffordalgebraCℓ(V,Q)ofaquadraticrealspace(V,Q)whereQisaquadratic formonV.Foranyvectorv∈V,itssquarev2intheCliffordalgebraCℓ(V,Q)equalsQ(v)1,thatis, v2=Q(v)1where1denotestheidentityelementinthealgebra. 4Chappell,JohnG.Hartnett,NicolangeloIanella,AzharIqbalandDerekAbbott other quantities are real scalars. As this is a complex-likenumber the square root is therefore well defined from complex number theory and so we can define the multivectoramplitudeas M = M 2. | | | | p Note that we haveused the conventionaldotproductx n = 1(xn+nx) · 2 whichcannowbedefinedintermsofthealgebraicproductofCliffordvectors,as showninAppendixA. Theorem2.3(Fundamentalmultivectorinvolution). Givenageneralinvolution I inCℓ 3 ,actingonamultivectorI(M) = s a+s x+s jn+s jb,where 0 1 2 3 ℜ s0,s1,s2(cid:0),s3(cid:1)= 1producesapermutationofthesigns,thentheinvolutionofClif- ± fordconjugationthatproducesthemultivectoramplitudeisuniqueinproducinga commutingvalueMI(M). Proof. Wehavetheresultant MI(M) = s a2+s x2 s n2 s b2+j(s +s )ab 0 1 2 3 0 3 − − + j(s xn+s nx)+(s +s )ax (2.4) 2 1 0 1 + (s +s )ajn+(s +s )bjx (s +s )bn. 0 2 1 3 2 3 − Inspectingthisresultantinreverseorder,wecanseethatinordertobecommuting we requires = s , s = s , s = s , s = s and in orderfors xn+ 2 3 1 3 0 2 0 1 2 − − − − s nxtoproducethedotproductandacommutingscalarwerequires = s .This 1 1 2 gives the solution s = s = 1 = s = s that correspondswith Clifford 0 3 1 2 conjugation,showninEq.(2.2).± − − (cid:3) HencewecanconcludethatMI(M) Candsointhecenterofthealgebra. ∈ We will find in the next section that we are constrained to adopt the multivector amplitudeastheonlyviabledefinitionofthemetricoverthespaceofmultivectors duetoitskeypropertyofbeingcommuting. 2.2. Thegrouptransformations Definition2.4(Bilinearmultivectortransformation). Wedefineageneralbilin- eartransformationonamultivectorM as M′ =YMZ, (2.5) whereM,Y,Z Cℓ( 3). ∈ ℜ Wethenfindthetransformedmultivectoramplitude M′ 2 =YMZYMZ =YMZZ¯M¯Y¯ = Y 2 Z 2 M 2, (2.6) | | | | | | | | where we have used the anti-involution property of Clifford conjugation and the commutingproperty of the amplitude. Hence, providedwe specify a unitary con- dition Y 2 Z 2 = 1 for these transformations,thenthe amplitude M will be in- | | | | | | variant. Without loss of generality, it is then convenient to impose the condition Y 2 = Z 2 = 1.ThetransformationinEq.(2.5)isthenthemostgeneralbilinear | | | | ± transformationthatpreservesthemultivectoramplitudeandsoproducesaninvari- antdistanceoverthe space.We will focusonthe specialcase Y 2 = Z 2 = +1 | | | | thatdescribetransformationsthatarecontinuouswiththeidentity. MinkowskiSpacetime 5 WecannowseetheimportanceofCliffordconjugationproducingavaluethat isacommutingscalar,showninEq.(2.3),asunderthegeneraltransformationop- erationinEq.(2.5)thesequantitieswillappearisotropicinspaceandsosatisfythe relativityprincipleofanon-preferredreferenceframeandallowthesimplifications inEq.(2.6).We arethusrequiredtoadoptEq.(2.3)asourdefinitionofthemetric forthespaceofmultivectorsinCℓ( 3). ℜ Now, using the power series expansion of the exponentialfunction,a multi- vectorY canbewritteninanexponentialform[22,23] Y =ec+p+jq+jd, (2.7) provided Y =0,wherec,d andp,q 3.Therefore,wefind | |6 ∈ℜ ∈ℜ YY¯ =ec+p+jq+jdec−p−jq+jd =e2c+2jd. (2.8) Theunitarycondition Y 2 = 1thenrequiresc=0,d=nπ/2,naninteger.Now, | | ± asd = 0simplyaddsacommutingphasetermandbecausethiswillnotaffectthe 6 amplitudeofthemultivectorwe nowsetd = 0 inorderto investigatetheLorentz group.So,alsowritingZ = er+js,withr,s 3,wefinallyproducethegeneral ∈ ℜ transformationoperation M′ =ep+jqMer+js, (2.9) whichwillleavethemultivectoramplitudeinvariant.Thefourthree-vectorsp,q,r,s illustratethatthetransformationisspecifiedbytwelverealparameters,thusgener- alizing the conventional six dimensional Lorentz group, consisting of boosts and rotations. 2.3. ConnectionwiththeconventionalLorentzgroup Now,ifwerepresentspacebythemultivectorinEq.(2.1),thensimplerotationsof thisspacearedescribedbythespecialcaseofEq.(2.9) M′ =e−jw/2Mejw/2, (2.10) whichwillproducearotationofθ = w radiansabouttheaxisw.Itisconvenient || || to have separate notation for the Pythagorean length of a vector w, given by the unboldedsymbolw = w =√w2 = w2+w2+w2. || || 1 2 3 AfurtherspecialcaseofEq.(2.9)pisfoundbyselectingthevectorexponent, whichwillcorrespondtoconventionalLorentzboosts.Thatis M′ =e−φvˆ/2Me−φvˆ/2, (2.11) whereφisdefinedthroughtanhφ = vwherev = v andwherev istherelative || || velocity vector between the two observers. We can rearrange tanhφ = v to give coshφ=γandsinhφ=γv.Hencee−φvˆ =coshφ vˆsinhφ=γ(1 v),where − − γ =1/√1 v2.Inthiscasethevectorvisidentifiedwiththerelativevelocityvec- − torbetweenframeswhereasforrotationsitisidentifiedwiththerotationaxis.These resultsconsistentwiththeknownresultthattheLorentzgroupisasub-manifoldof thePaulialgebra[36]. IfwenowconsidertheeffectofaconventionalLorentzboostonthegeneral- izedeight-dimensionalspacetimecoordinateM =a+x +x +jn +jn +jb, k ⊥ k ⊥ 6Chappell,JohnG.Hartnett,NicolangeloIanella,AzharIqbalandDerekAbbott wherewesplitthespatialcoordinateintocomponentsperpendicularandparallelto theboostdirectionvˆ.WethenfindfromEq.(2.11)that M′ = ae−vˆφ+x e−vˆφ+x (2.12) k ⊥ +jn e−vˆφ+jn +bje−vˆφ k ⊥ = γ a vx +γ x va +x k k ⊥ − − +(cid:0)jγ n (cid:1)vb +(cid:0)jn +j(cid:1)γ b vn , k ⊥ k − − whichnowshowsthetransform(cid:0)ationoft(cid:1)hefullmultive(cid:0)ctorsubj(cid:1)ecttotheconven- tional Lorentz boost operation. We can see that the plane jn orthogonal to the k boostdirectionv isexpandedbytheγ factortojγn .Thisimpliesthatthebivec- k torsdonotrefertoquantitiessuchasareasorangularmomentumofextendedbod- ies2—as the parallelcomponentsare in fact unchangedbysuch boosts—butmust refertootheraxialvector-typequantitiessuchasspinorthemagneticfield.Infact, the bivector and trivector components jn+ jb = j(b + n) are transformed the same as a four-vector therefore have the same transformational properties as the four-spinfour-vector.WecanalsoseethattheconventionalLorentzboosttransfor- mationsplitsthemultivectorspaceintotwofourdimensionalsubspaces 3and 2 3 3 3representedbya+xandjn+jbrespectively,where rℜef⊕erℜstothe ℜ ⊕ ℜ eVxterioraVlgebra.Thefirstfour-vectora+xcanbeidentifiedasconveVntionalspace- timeifweidentifythescalarawiththetimetandthesecondfour-vectorjn+jb asfour-spin.ThusEq.(2.1)appearstodescribeaunifiedformulationofspacetime that includesspin. For example,with respect to the metric in Eq. (2.3), by identi- fyingthescalarawithtimet, andn = 0,b = 0,we obtaintherequiredinvariant spacetimeintervalt2 x2.Thefactthattheconventionalboostoperation,shownin − Eq.(2.12),effectivelysplitsthemultivectorintotwoindependentfour-vectorspaces perhapsillustrateswhythefour-vectoriseffective,eventhoughaunifiedtreatment usingthemultivectorispreferable. Hence,wecanwriteageneralspacetimeeventX,indifferentialform,as dX =dt+dx+jdn+jdb, (2.13) providingageneralizationtoeight-dimensionalevents,wherethespecialcasedX = dt+dxisisomorphictotheconventionalfourvectordX =[dt,dx]T.Referringto Eq.(2.3)thisthereforehasanamplitude dX 2 =dt2 dx2+dn2 db2+2j(dtdb dx dn). (2.14) | | − − − · Ingeneral,theintervalisthereforeacomplex-likenumber. Now,forthespecialcaseforparticlesthathavearestframe,definedasdx= 0,withacorrespondingpropertimedτ,wehave dτ2 = dX 2 =dt2+dn2 db2+2jdt db , (2.15) | 0| 0 0− 0 0 0 where the zero in the subscripts denotes the rest frame. Now as dτ2 is real we thereforerequiredt db = 0, and as we assume dt > 0 then we have db = 0. 0 0 0 0 Hencewehaveintherestframe dτ2 =dt2+dn2 =dt2 dx2+dn2 db2. (2.16) 0 0 − − 2ThedefinitionofangularmomentumforextendedbodiesisconsideredinAppendixB. MinkowskiSpacetime 7 Note that as the imaginary term is now zero in the rest frame and the interval is invariant,thentheimaginarycomponentwillalsoremainzerointheboostedframe. Wecanthenwriteforthepropertime ejudτ =dX =dt +jdn , (2.17) 0 0 0 whereuisathree-vector.Thisthenreturns dX 2 = ejudτ e−judτ = dτ2,as | | required.Notethateju = cosu+juˆsinu,whereu(cid:0)= √u2(cid:1)(cid:0)anduˆ =(cid:1)u/u.Thus thepropertimebecomesidentifiedwiththeevensubalgebraofCℓ( 3),whichcan ℜ be shown to be isomorphic to the quaternions. A connectionis now also made to quantum mechanics, as the Pauli spinors are also isomorphic to the quaternions. Additionally,undera relativisticboostwe willpopulatethefulleight-dimensional multivectorthathasbeenshowntobeisomorphicwiththeeight-dimensionalDirac spinordescribingspin-1 particles[4]. 2 Comparingaprimedrestframewiththeboostedframewethereforefind dt′2+dn′2 =dt2 dx2+dn2 db2. (2.18) − − Now, as we have seen, the two-subspaces of dt+dx and jdn+jdb are disjoint under boosts and so we have dt′2 = dt2 dx2 or dt′ = dt 1 dx2/dt2 = − − dt√1 v2 = dt/γ and so gives the usual time dilation factorpγ = 1/√1 v2, − − wherev = dx.Also,dt′2 > 0thatimpliesdx2 < dt2 andsowehavev < 1,thus dt setting an upper limit speed. Hence, despite the fact that time in the rest frame is quaternionic,neverthelesstheamplitudeofthistimedτ isrealandsowewillfind thatstandardresultsarereturned.However,thisrecognitionoftimeasaquaternion isplannedtobefurtherexploredinasubsequentpaper. 2.4. Momentummultivector DefiningthevelocitymultivectorV =dX/dτ wefindfromEq.(2.13) dX dt dx dt dn dt db dt V = = + +j +j (2.19) dτ dτ dt dτ dt dτ dtdτ = γ(1+v+jw+jh). Now,fromEq.(2.16)assumingtheexistenceofapropertime,thenwehave V 2 = 1. This implies that d V 2 = dVV¯ +V dV¯ = 0, using the product rule|of|dif- dτ| | dτ dτ ferentiation.Also, defining A = dV for an accelerationmultivector,we thuspro- dτ duce an orthogonality condition for the velocity and acceleration multivectors as AV¯+VA¯=0,analogoustoconventionalresultsbutexpandedtoeight-dimensional spacetime. Now,definingthemomentummultivectorP = mV, wherewe assumemis theinvariantrestmass,givingE =γm,p=γmv,s=γmwandH =γmhthen P =mV =E+p+js+jH. (2.20) Theinvariantmassmthusappearshereasascalingfactorofthevelocitymultivec- tor. Note, though, that this assumes that the momentum is parallel to the velocity and the angular momentumvectoris parallelto the angular velocityvector. How- ever,for a classical extendedrigid bodythe angularmomentumis notnecessarily 8Chappell,JohnG.Hartnett,NicolangeloIanella,AzharIqbalandDerekAbbott parallel to its angular velocity vector3. Additionally, the angular momentum of a rigid body does not transform as a four-vectorand so this multivector description cannotbeused to unifythe linearmomentumand angularmomentumofclassical extendedbodies.Now,asalreadynoted,thepropertiesofrelativisticspinforpoint- likeparticlesdoesindeedfollowthistransformationlawoffour-vectorsrequiredby the multivector. Hence it is apparentconfirmation of this approach that the eight- dimensionalmultivectordescriptionrequirestheassumptionofpoint-likeparticles asrequiredbythestandardmodel. Now,wefindtheamplitudeofthemomentummultivector P 2 =E2 p2+s2 H2 =m2, (2.21) | | − − which follows from the fact that V 2 = 1. Referring to Eq. (2.16), we therefore | | haveintherestframe P =E +js . (2.22) 0 0 0 Typically the spin four-vector S = [H,s] is defined to have a zero time compo- nent H in the rest frame, so that S = [0,s ] and so we are consistent with the 0 0 conventionaldefinitionofspin.IfthisreststateP isnowboostedwefind 0 P 2 =E2 p2+s2 H2 =m2 =E2+s2 = P 2. (2.23) | | − − 0 0 | 0| We thus have the expected spin conservation s2 H2 = s2 although the energy − 0 conservation E2 p2 = E2 = m2 s2 implies a rest energy E rather than − 0 − 0 0 the conventionalvalue of m. This can be seen to be due to the fact that the spin energys isnowincludedaspartoftherestenergy.Theinvariantamplitudeofthe 0 momentummultivectorofmthusgivestheexpectedrestenergy.Thusinourcase thetimecomponentofthemomentummultivectordoesnotgivetherestenergy. Now,fromm2 =E2+s2wefind 0 0 γ2 γ2v2 =1 s2/m2, (2.24) − − 0 andtherefore dt 1 s2/m2 =γ = − 0 . (2.25) dτ p√1 v2 − Wenotethatass2 <m2thentherelativetimerateswillremainrealalthoughinthe 0 restframethisexpressionimpliesatimespeedupof dt = 1 s2/m2,relativeto dτ − 0 somehypotheticalspinlessstate.However,itwaspreviousplynotedfromEq.(2.12) that under conventional boost operations the spin components are conserved and so they presumably representintrinsic quantities, so this spinless rest frame is ef- fectivelyunattainable.Thusconventionaltimedilationresultsareretainedbetween framesasshownfromEq.(2.18). Besidesthecaseofmassiveparticleswitharestframe,thesecondveryimpor- tantspecialcase,isthatofmasslessparticles,assumedtobetravelingatthespeed 3Giventheangularvelocityω =w1e1+w2e2+w3e3andtheangularmomentumL=I1w1e1+ I2w2e2+I3w3e3,thenclearlythevectorsωandLareonlyparallelifthemomentsofinertiaI1,I2,I3 areequal. MinkowskiSpacetime 9 of light, in which no rest frameis available to the observer.Hence we expectthat wewillrequirethefullmomentummultivector P =ω+k+js+jH, (2.26) where js+jH describes the angular momentum, assuming units where ~ = 1. Aphotonisassumedtosimultaneouslypossessenergy,linearmomentum,angular momentumandspin,andsoconsistentwiththisdescription. Clearly,manyotherdistinctcasescouldnowbeconsideredsuchasparticles withoutspinorconverselyparticleswithnoenergybutspin.Theapplicationofthe formalismtothesecasescouldthusbeinvestigatedfurther. 2.5. Theaction We require the action integral to involve a Lorentz invariant quantity, and so it is naturaltodefinetheactionbetweentwospacetimeeventsas S = dX , (2.27) Z | | where the distance dX is given by the amplitude of the spacetime multivector, | | givenbyEq.(2.3),whichaswehaveshownisinvariantbetweenobservers.Now,as previously,withtheassumptionofapropertimeinarestframewehave dX =dτ | | andsowehavethespacetimedistance dX 2 = t˙2 x˙2+n˙2 b˙2 dτ2, (2.28) | | (cid:16) − − (cid:17) where we define t˙ = dt, x˙ = dx, n˙ = dn and b˙ = db. If we write the action dτ dτ dτ dτ S = |dX|dτ thenthisimpliesaLagrangian dτ R dX = | | = t˙2 x˙2+n˙2 b˙2 =1, (2.29) L dτ q − − wherewenowextremizetheactionS = dτ. L AswehavenoexplicitcoordinatedRependence, ∂L, ∂L, ∂L and ∂L arecon- ∂t˙ ∂x˙ ∂n˙ ∂b˙ stantsofthemotion.UsingtheEuler-Lagrangeequation[19]fort d ∂ ∂ L = L =0 (2.30) dτ ∂t˙ ∂t thusgivingtheconservedquantity ∂ E L = −1t˙= . (2.31) ∂t˙ L m We have written the conserved quantity as the dimensionless scalar E/m as we expectittorelatetoenergybyNoether’stheorem.Indeed,becauset˙= dt/dτ = γ and −1 = 1, we find equating real components that γ = E/m or E = γm, L whichistheconventionalrelativisticenergyrelation.Thesecondconservedquantity will be the momentump = γmv in agreementwith our previousdefinition. The bivectorcomponentwillproducetheconservationofangularmomentums=γmw asexpectedandthefourthconservedquantitywillbe ∂ H L = −1b˙ = , (2.32) ∂b˙ L m 10Chappell,JohnG.Hartnett,NicolangeloIanella,AzharIqbalandDerekAbbott thatreturnsthehelicityH =mb˙ =γmh=γmdb. dt These relations are consistent with Noether’s theorem [29] that energy con- servation arises from invariance under time translations t, momentum conserva- tionfrominvarianceunderspace translationsx andangularmomentumconserva- tion from rotational invariance jn and a fourth conserved quantity based on the translation over the trivector j representing helicity. The momentum multivector in Eq. (2.20) thus unifies the four conserved quantities into a single principle of theconservationofthemomentummultivector.Thatis,forasetofcollidingparti- cleswith initialmomentummultivectorsP andfinalmomentummultivectorsP , i j we have the conservation condition P = P . Thus the multivector space- i j timestructurenaturallyencodesthefoPurfundamPentalconservationlawsforinertial point-likeparticles. Ifwewishtoincludeinteractionswecanincludetheconventionalinteraction termJ A¯=qV A¯toproduce · · =m VV¯ +qV A¯, (2.33) L · p whereV = dX isthevelocitymultivectorandtheAistheelectromagneticpoten- dτ tial.NaturallyasweexpandV andAtofullmultivectorsthenadditionalinteraction productswillbegenerated,whichcanbefurtherinvestigated. 2.6. Maxwell’sequations UsingCliffordgeometricalgebraitisknownthatMaxwell’sfourequationscanbe writtenwiththesingleequation[6,3] (∂ + )F =ρ J, (2.34) t ∇ − wheretheelectromagneticfield isrepresentedasF = E +jB and = e ∂ + 1 x e ∂ +e ∂ .Now∂ =∂ + andthefour-currentJ =ρ Jareknown∇four-vectors 2 y 3 z t ∇ − andsowecanactwithourgeneraltransformationinEq.(2.9)producing R(∂ + )SS¯FS =R(ρ J)S, (2.35) t ∇ − wherewewererequiredtoinsertS¯andSaroundF forselfconsistency.Thatis,we knowRR¯ =SS¯=1andsomultiplyingEq.(2.35)fromtheleftbyR¯andfromthe rightbyS¯wereproduceMaxwell’sequationsshowninEq.(2.34).Hencewehave therequiredfieldtransformationF′ =S¯FS or (E+jB)′ =e−r−js(E+jB)er+js. (2.36) Thisistheacceptedtransformationfortheelectromagneticfield[6]andsoourgen- eralizedtransformationforspacetimeshowninEq.(2.9)leavesMaxwell’sequations invariantaswellasretainingtheconventionalfieldtransformation. Ifwenowconsidertheeffectofthegeneralizedtransformationsonthefour- currentR(ρ J)S then whereasa conventionalboostwill remainin the formof − a four-current this more general transformation will create bivector and trivector terms R(ρ J)S =ρ′ J′+jK′+jκ′, (2.37) − − wherejκ′ isamagneticmonopoleandK′ isamonopolecurrent.Now,Maxwell’s equation B = 0effectivelystatesthatmonopolesdonotexistandindeed,de- ∇· spitemanyexperiments,haveneverbeenconclusivelydetected.Hence,ifweaccept