Doing physics with quaternions

110 Pages·1998·0.597 MB·English

by  Sweetser D.

Contents
1 UnifyingTwoViewsofEvents 2
2 ABriefHistoryofQuaternions 3
I Mathematics 4
3 MultiplyingQuaternionstheEasyWay 5
4 Scalars,Vectors,TensorsandAllThat 6
5 InnerandOuterProductsofQuaternions 10
6 QuaternionAnalysis 12
7 TopologicalPropertiesofQuaternions 19
II ClassicalMechanics 23
8 Newton’sSecondLaw 24
9 OscillatorsandWaves 26
10 FourTestsforaConservativeForce 28
III SpecialRelativity 30
11 RotationsandDilationsCreatetheLorentzGroup 31
12 AnAlternativeAlgebraforLorentzBoosts 33
IV Electromagnetism 36
13 ClassicalElectrodynamics 37
14 Electromagneticfieldgauges 40
15 TheMaxwellEquationsintheLightGauge: QED? 42
i
16 TheLorentzForce 45
17 TheStressTensoroftheElectromagneticField 46
V QuantumMechanics 48
18 ACompleteInnerProductSpacewithDirac’sBracketNotation 49
19 MultiplyingQuaternionsinPolarCoordinateForm 53
20 CommutatorsandtheUncertaintyPrinciple 55
21 UnifyingtheRepresentationofSpinandAngularMomentum 58
22 DerivingAQuaternionAnalogtotheSchro¨dingerEquation 62
23 IntroductiontoRelativisticQuantumMechanics 65
24 TimeReversalTransformationsforIntervals 67
VI Gravity 68
25 Einstein’svisionI:ClassicalunifiedfieldequationsforgravityandelectromagnetismusingRiemannian
quaternions 69
26 Einstein’svisionII:Aunifiedforceequationwithconstantvelocityprofilesolutions 78
27 StringsandQuantumGravity 82
28 AnsweringPrimaFacieQuestionsinQuantumGravityUsingQuaternions 85
29 LengthinCurvedSpacetime 91
30 ANewIdeaforMetrics 93
31 TheGravitationalRedshift 95
VII Conclusions 97
32 Summary 98
ii
Doing Physics with Quaternions
DouglasB.Sweetser
http://quaternions.com
1 UNIFYINGTWOVIEWSOFEVENTS 2
1 Unifying Two Views of Events
An experimentalist collects events about a physical system. A theorists builds a model to describe what patterns of
eventswithinasystemmightgeneratetheexperimentalist’sdataset. Withhardworkandluck,thetwowillagree!
Events are handled mathematically as 4-vectors. They can be added or subtracted from another, or multiplied by a
scalar. Nothing else can be done. A theorist can import very powerful tools to generate patterns, like metrics and
grouptheory.Theoristsinphysicshavebeenabletoconstructthemostaccuratemodelsofnatureinallofscience.
I hope to bring the full power of mathematics down to the level of the events themselves. This may be done by
representingeventsasthemathematicalfieldofquaternions.Allthestandardtoolsforcreatingmathematicalpatterns
- multiplication, trigonometric functions, transcendental functions, infinite series, the special functions of physics -
shouldbeavailableforquaternions.Nowatheoristcancreatepatternsofeventswithevents.Thismayleadtoabetter
unificationbetweentheworkofatheoristandtheworkofanexperimentalist.
AnOverviewofDoingPhysicswithQuaternions
It has been said that one reason physics succeeds is because all the terms in an equation are tensors of the same
rank. This work challenges that assumption, proposing instead an integrated set of equations which are all based
on the same 4-dimensional mathematical field of quaternions. Mostly this document shows in cookbook style how
quaternionequationsareequivalenttoapproachesalreadyinuse. AsFeynmanpointedout,”whateverweareallowed
toimagineinsciencemustbeconsistentwitheverythingelseweknow.”Freshperspectivesarisebecause,inessence,
tensorsofdifferentrankcanmixwithinthesameequation.ThefourMaxwellequationsbecomeonenonhomogeneous
quaternionwave equation, and the Klein-Gordon equation is partof a quaternionsimple harmonic oscillator. There
is hope of integrating general relativity with the rest of physics because the affine parameter naturally arises when
thinking about lengths of intervals where the origin moves. Since all of the tools used are woven from the same
mathematicalfabric,theinterrelationshipsbecomemorecleartomyeye. Hopeyouenjoy.
2 ABRIEFHISTORYOFQUATERNIONS 3
2 A Brief History of Quaternions
Complex numbers were a hot subject for research in the early eighteen hundreds. An obvious question was that if
aruleformultiplyingtwonumbers togetherwasknown,whataboutmultiplyingthreenumbers? Foroveradecade,
thissimplequestionhadbotheredHamilton,thebigmathematicianofhisday.Thepressuretofindasolutionwasnot
merelyfromwithin. Hamiltonwrotetohisson:
”Everymorningintheearlypartoftheabove-citedmonth[Oct. 1843]onmycomingdowntobreakfast,yourbrother
WilliamEdwinandyourselfusedtoaskme,’Well,Papa,canyoumultiplytriplets?’ WheretoIwasalwaysobligedto
reply,withasadshakeofthehead,’No,Icanonlyaddandsubtractthem.’”
WecanguesshowHollywoodwouldhandletheBroughamBridgesceneinDublin. StrollingalongtheRoyalCanal
withMrs. H-,herealizesthesolutiontotheproblem,jotsitdowninanotebook. Soexcited,hetookoutaknifeand
carvedtheanswerinthestoneofthebridge.
Hamilton had found a long sought-after solution, but it was weird, very weird, it was 4D. One of the first things
Hamiltondidwasgetridofthefourthdimension,settingitequaltozero,andcallingtheresulta”properquaternion.”
Hespenttherestofhislifetryingtofindauseforquaternions.Bytheendofthenineteenthcentury,quaternionswere
viewedasanoversoldnovelty.
Intheearlyyearsofthiscentury,Prof. GibbsofYalefoundauseforproperquaternionsbyreducingtheextrafluid
surroundingHamilton’sworkandaddingkeyingredientsfromRodriguesconcerningtheapplicationtotherotationof
spheres. Heendedupwiththevectordotproductandcrossproductweknowtoday.Thiswasausefulandpotentbrew.
Ourinvestmentinvectorsisenormous,eclipsingtheirplaceofbirth(Harvardhad>1000referencesunder”vector”,
about20under”quaternions”,mostofthosewrittenbeforetheturnofthecentury).
In the early years of this century, Albert Einstein found a use for four dimensions. In order to make the speed of
lightconstantforallinertialobservers,spaceandtimehadtobeunited. Herewasatopictailor-madefora4Dtool,
but Albert was not a math buff, and built a machine that workedfrom locally available parts. We can say now that
Einstein discovered Minkowski spacetime and the Lorentz transformation, the tools required to solve problems in
specialrelativity.
Today, quaternions are of interest to historians of mathematics. Vector analysis performs the daily mathematical
routinethatcouldalsobedonewithquaternions. Ipersonallythinkthattheremaybe4Droadsinphysicsthatcanbe
efficientlytraveledonlybyquaternions,andthatisthepathwhichislaidoutinthesewebpages.
4
Part I
Mathematics
3 MULTIPLYINGQUATERNIONSTHEEASYWAY 5
3 Multiplying Quaternions the Easy Way
(cid:0) (cid:0)
Multiplyingtwocomplexnumbersa bIandc dIisstraightforward.
(cid:1) (cid:1) (cid:1)
a,b(cid:2) c,d(cid:2)(cid:4)(cid:3) ac (cid:5) bd, ad (cid:6) bc(cid:2)
(cid:0) (cid:0)
For two quaternions, b I and d I become the 3-vectors B and D, where B (cid:7) x I y J z K and similarly for D.
Multiplicationofquaternionsislikecomplexnumbers,butwiththeadditionofthecrossproduct.
(cid:8) (cid:8) (cid:8) (cid:8) (cid:8) (cid:8) (cid:8) (cid:8)
a,B c,D (cid:3) ac (cid:5) B.D, aD (cid:6) Bc (cid:6) BxD
Notethatthelastterm,thecrossproduct,wouldchangeitssigniftheorderofmultiplicationwerereversed(unlikeall
theotherterms). Thatiswhyquaternionsingeneraldonotcommute.
(cid:0) (cid:0)
If a is the operator d/dt, and B is the del operator, or d/dx I d/dy J d/dz K (all partial derivatives), then these
operatorsactonthescalarfunctioncandthe3-vectorfunctionDinthefollowingmanner:
d (cid:9)(cid:8) (cid:8) (cid:10)(cid:11)(cid:11)(cid:11)(cid:11) dc (cid:9)(cid:8) (cid:8) d(cid:8)D (cid:9)(cid:8) (cid:9)(cid:8) (cid:8) (cid:13)(cid:14)(cid:14)(cid:14)(cid:14)
dt, c,D (cid:3) (cid:12) dt (cid:5) .D, dt (cid:6) c(cid:6) xD(cid:15)
Thisonequaternioncontainsthetimederivativesofthescalarand3-vectorfunctions,alongwiththedivergence,the
gradientandthecurl. Densenotation:-)
4 SCALARS,VECTORS,TENSORSANDALLTHAT 6
4 Scalars, Vectors, Tensors and All That
Accordingtomymathdictionary,atensoris...
”Anabstractobjecthavingadefinitelyspecifiedsystemofcomponentsineverycoordinatesystemunderconsideration
and such that, under transformation of coordinates, the components of the object undergoes a transformation of a
certainnature.”
To make this introduction less abstract, I will confine the discussion to the simplest tensors under rotational trans-
formations. A rank-0 tensor isknownasa scalar. It doesnotchangeat allundera rotation. It contains exactlyone
number,nevermoreorless. Thereisazeroindexforascalar. Arank-1tensorisavector.Avectordoeschangeunder
rotation. Vectorshaveoneindexwhichcanrunfrom1tothenumberofdimensionsofthefield,sothereisnowayto
knowapriorihowmanynumbers(oroperators,or...) areinavector. n-ranktensorshavenindices. Thenumberof
numbersneededisthenumberofdimensionsinthevectorspaceraisedbytherank. Symmetrycanoftensimplifythe
numberofnumbersactuallyneededtodescribeatensor.
There are a variety of important spin-offsof a standard vector. Dual vectors, when multiplied by its corresponding
vector, generate a real number, by systematically multiplying each component from the dual vector and the vector
togetherandsummingthetotal. Ifthespaceavectorlivesinisshrunk,acontravariantvectorshrinks,butacovariant
vectorgetslarger. Atangentvectoris,well,tangenttoavectorfunction.
Physics equations involvetensors of the same rank. There are scalar equations, polar vector equations, axial vector
equations,andequationsforhigherranktensors.Sincethesameranktensorsareonbothsides,theidentityispreserved
underarotationaltransformation. Onecoulddecidetoarbitrarilycombinetensorequationsofdifferentrank,andthey
wouldstillbevalidunderthetransformation.
There are ways to switch ranks. If there are two vectors and one wants a result that is a scalar, that requires the
interventionofametrictobrokerthetransaction. Thisprocessinknownasaninnertensorproductoracontraction.
The vectorsin question must have the same number of dimensions. The metric defines how to form a scalar as the
indicesareexaminedone-by-one. Metricsinmathcanbeanything,butnatureimposesconstraintsonwhichonesare
importantinphysics. Anaside: mathematiciansrequirethedistanceisnon-negative,butphysicistsdonot. Iwillbe
usingthephysicsnotionofametric. Inlookingateventsinspacetime(a4-dimensionalvector),theaxiomsofspecial
relativityrequiretheMinkowskimetric,whichisa4x4realmatrixwhichhasdownthediagonal1,-1,-1,-1andzeros
elsewhere. Somepeoplepreferthesignstobeflipped,buttobeconsistentwitheverythingelseonthissite,Ichoose
thisconvention. AnotherpopularchoiceistheEuclideanmetric, whichisthesameasanidentitymatrix. Theresult
ofgeneralrelativityforasphericallysymmetric,non-rotatingmassistheSchwarzschildmetric,whichhas”non-one”
termsdownthediagonal,zeroselsewhere,andbecomestheMinkowskimetricinthelimitofthemassgoingtozero
ortheradiusgoingtoinfinity.
Anoutertensor productisawaytoincrease therankoftensors. Thetensorproductoftwovectorswillbea2-rank
tensor. Avectorcanbeviewedasthetensorproductofasetofbasisvectors.
WhatAreQuaternions?
Quaternions could be viewed as the outer tensor product of a scalar and a 3-vector. Under rotation for an event in
spacetimerepresentedbyaquaternion,timeisunchanged,butthe3-vectorforspacewouldberotated. Thetreatment
ofscalarsisthesameasabove,butthenotionofvectorsisfarmorerestrictive,asrestrictiveasthenotionofscalars.
Quaternions can only handle 3-vectors. To those familiar to playing with higher dimensions, this may appear too
restrictivetobeofinterest. Yetphysicsonboththequantumandcosmologicalscalesisconfinedto3-spatialdimen-
sions. NotethattheinfiniteHilbertspacesinquantummechanicsafunctionoftheprinciplequantumnumbern,not
thespatialdimensions. Aninfinitecollectionofquaternionsoftheform(En,Pn)couldrepresentaquantumstate. The
HilbertspaceisformedusingtheEuclideanproduct(q*q’).
4 SCALARS,VECTORS,TENSORSANDALLTHAT 7
(cid:0)
Adualquaternionisformedbytakingtheconjugate,becauseq*q (cid:7) (tˆ2 X.X,0). Atangentquaternioniscreated
byhavinganoperatoractonaquaternion-valuedfunction
(cid:16) (cid:9)(cid:8) (cid:1) (cid:1) (cid:8) (cid:1) (cid:10)(cid:11)(cid:11)(cid:11)(cid:11) (cid:16) f (cid:9)(cid:8) (cid:8) (cid:16) (cid:8)F (cid:9)(cid:8) (cid:9)(cid:8) (cid:8) (cid:13)(cid:14)(cid:14)(cid:14)(cid:14)
(cid:16) t, f q(cid:2) ,F q(cid:2)(cid:17)(cid:2)(cid:18)(cid:3) (cid:12) (cid:16) t (cid:5) .F, (cid:16) t (cid:6) f(cid:6) XF(cid:15)
Whatwouldhappentothesefivetermsifspacewereshrunk?The3-vectorFwouldgetshrunk,aswouldthedivisorsin
theDeloperator,makingfunctionsactedonbyDelgetlarger.Thescalartermsarecompletelyunaffectedbyshrinking
space, because df/dthas nothingto shrink, and theDel and Fcanceleachother. Thetime derivativeofthe 3-vector
isacontravariantvector,becauseFwouldgetsmaller. Thegradientofthescalarfieldisa covariantvector,because
ofthe work oftheDel operator in thedivisormakesitlarger. The curlat firstglance might appearasa draw,but it
isacovariantvectorcapacitybecauseoftheright-anglenatureofthecrossproduct. Notethatiftimewheretoshrink
exactlyasmuchasspace,nothinginthetangentquaternionwouldchange.
Aquaternionequationmustgeneratethesamecollectionoftensorsonbothsides. Considertheproductoftwoevents,
qandq’:
(cid:8) (cid:8) (cid:8) (cid:8) (cid:8) (cid:8) (cid:8) (cid:8)
t,X t(cid:19) ,X(cid:19) (cid:3) tt(cid:19)(cid:20)(cid:5) X.X(cid:19) , tX(cid:19) (cid:6) Xt(cid:19)(cid:21)(cid:6) XxX(cid:19)
(cid:8) (cid:8)
scalars (cid:22) t, t(cid:19) , tt(cid:19) (cid:5) X.X(cid:19)
(cid:8) (cid:8) (cid:8) (cid:8)
polarvectors (cid:22) X, X(cid:19) , tX(cid:19) (cid:6) Xt(cid:19)
(cid:8) (cid:8)
axialvectors (cid:22) XxX(cid:19)
Whereistheaxialvectorforthelefthandside? Itisimbeddedinthemultiplicationoperation,honest:-)
(cid:8) (cid:8) (cid:8) (cid:8) (cid:8) (cid:8) (cid:8) (cid:8)
t(cid:19) ,X(cid:19) t,X (cid:3) t(cid:19) t (cid:5) X(cid:19) .X, t(cid:19) X (cid:6) X(cid:19) t(cid:6) X(cid:19) xX
(cid:8) (cid:8) (cid:8) (cid:8) (cid:8) (cid:8)
(cid:3) tt(cid:19)(cid:23)(cid:5) X.X(cid:19) , tX(cid:19) (cid:6) Xt(cid:19)(cid:21)(cid:5) XxX(cid:19)
Theaxialvectoristheonethatflipssignsiftheorderisreversed.
Termscancontinuetogetmorecomplicated.Inaquaterniontripleproduct,therewillbetermsoftheform(XxX’).X”.
Thisiscalledapseudo-scalar,becauseitdoesnotchangeunderarotation,butitwillchangesignsunderareflection,
due to the cross product. Youcan convinceyourself of this by noting that the cross product involvesthe sine of an
angleandthedotproductinvolvesthecosineofanangle. Neitherofthesewillchangeunderarotation,andaneven
functiontimesanoddfunctionisodd.Iftheorderofquaterniontripleproductischanged,thisscalarwillchangesigns
forateachstepinthepermutation.
Ithasbeenmyexperiencethatanytensorinphysicscanbeexpressedusingquaternions. Sometimesittakesabitof
effort,butitcanbedone.
Individualpartscanbeisolatedifonechooses. Combinationsofconjugationoperatorswhichflipthesignofavector,
andsymmetricandantisymmetricproductscanisolateanyparticularterm. Hereareallthetermsoftheexamplefrom
above
(cid:8) (cid:8) (cid:8) (cid:8) (cid:8) (cid:8) (cid:8) (cid:8)
t,X t(cid:19) ,X(cid:19) (cid:3) tt(cid:19) (cid:5) X.X(cid:19) , tX(cid:19) (cid:6) Xt(cid:19) (cid:6) XxX(cid:19)
(cid:8) (cid:8) (cid:1)
q(cid:6) q(cid:24) q(cid:19) (cid:6) q(cid:19) (cid:24) qq(cid:19) (cid:6) qq(cid:19) (cid:2)(cid:25)(cid:24)
scalars (cid:22) t(cid:3) , t(cid:19) (cid:3) ,tt(cid:19) (cid:5) X.X(cid:19) (cid:3)
2 2 2
(cid:8) (cid:8)
q(cid:5) q(cid:24) q(cid:19) (cid:5) q(cid:19) (cid:24)
polarvectors (cid:22) X (cid:3) , X(cid:19) (cid:3) ,
2 2
(cid:8) (cid:8) (cid:1) (cid:1) (cid:1) (cid:1)
qq(cid:19) (cid:6) q(cid:19) q(cid:2)(cid:17)(cid:2)(cid:26)(cid:5) qq(cid:19) (cid:6) q(cid:19) q(cid:2)(cid:17)(cid:2) (cid:24)
tX(cid:19) (cid:6) Xt(cid:19) (cid:3)
4
(cid:8) (cid:8) (cid:1)
qq(cid:19) (cid:5) q(cid:19) q(cid:2)
axialvectors (cid:22) XxX(cid:19) (cid:3)
2
ThemetricforquaternionsisimbeddedinHamilton’sruleforthefield.
4 SCALARS,VECTORS,TENSORSANDALLTHAT 8
(cid:8) (cid:8) (cid:8) (cid:8) (cid:8) (cid:8)
2 2 2
i (cid:3) j (cid:3) k (cid:3) ijk (cid:3)(cid:4)(cid:5) 1
Thislookslikeawaytogeneratescalarsfromvectors,butitismorethanthat. Italsosaysimplicitlythatij(cid:7) k,jk(cid:7)
i,andi,j,kmusthaveinverses.Thisisanimportantobservation,becauseitmeansthatinnerandoutertensorproducts
canoccur inthesame operation. Whentwo quaternionsaremultiplied together,a newscalar (innertensor product)
andvector(outertensorproduct)areformed.
Howcanthemetricbegeneralizedforarbitrarytransformations? Thetraditionalapproachwouldinvolveplayingwith
Hamilton’srulesforthefield. Ithinkthatwouldbeamistake, sincethatruleinvolvesthefundamentaldefinitionof
a quaternion. Change the rule of what a quaternion is in one context and it will not be possible to compare it to a
quaternioninanothercontext.Instead,consideranarbitrarytransformationTwhichtakesqintoq’
q (cid:27) q(cid:19) (cid:3) Tq
Tisalsoaquaternion,infactitisequaltoq’qˆ-1.Thisisguaranteedtoworklocally,withinneighborhoodsofqandq’.
Thereisnopromisethatitwillworkglobally,thatoneTwillworkforanyq.Undercertaincircumstances,Twillwork
foranyq.TheimportantthingtoknowisthatatransformationTnecessarilyexistsbecausequaternionsareafield.The
twomostimportanttheoriesinphysics, generalrelativityandthestandardmodel,involvelocaltransformations(but
thetechnicaldefinitionoflocaltransformationisdifferentthantheideapresentedherebecauseitinvolvesgroups).
ThisquaterniondefinitionofatransformationcreatesaninterestingrelationshipbetweentheMinkowskiandEuclidean
metrics.
LetT (cid:3) I, theidentitymatrix
(cid:1) (cid:8) (cid:8)
IqIq (cid:6) IqIq(cid:2)(cid:25)(cid:24)
(cid:3) t2(cid:5) X.X, 0
2
(cid:8) (cid:8)
(cid:1)
Iq(cid:2) (cid:24) Iq (cid:3) t2(cid:6) X.X, 0
Inordertochangefromwristwatchtime(theintervalinspacetime)tothenormofaHilbertspacedoesnotrequireany
changeinthetransformationquaternion, onlyachangeinthemultiplicationstep. Thereforeatransformationwhich
generatestheSchwarzschildintervalofgeneralrelativityshouldbeeasilyportabletoaHilbertspace,andthatmight
bethestartofaquantumtheoryofgravity.
SoWhatIsthe Difference?
Ithinkitissubtlebutsignificant. ItgoesbacktosomethingIlearnedinagraduatelevelclassonthefoundationsof
calculus. Tomakecalculusrigorousrequiresthatitisdefinedoveramathematicalfield. Physicistsdothisbesaying
thatthescalars,vectorsandtensorstheyworkwitharedefinedoverthefieldofrealorcomplexnumbers.
What arethenumbers usedbynature? There areevents,which consistofthe scalartime andthe 3-vectorofspace.
There is mass, which is defined by the scalar energy and the 3-vector of momentum. There is the electromagnetic
potential,whichhasascalarfieldphianda3-vectorpotentialA.
Todocalculuswithonly informationcontained ineventsrequiresthata scalarand a3-vectorforma field. Accord-
ing to a theorem by Frobenius on finite dimensional fields, the only fields that fit are isomorphic to the quaternions
(isomorphicisasophisticatednotionofequality,whosesubtletiesareappreciatedonlybypeoplewithadeepunder-
standingofmathematics).Todocalculuswithamassoranelectromagneticpotentialhasanidenticalrequirementand
anidenticalsolution. Thisisthelogicalfoundationfordoingphysicswithquaternions.
Canphysicsbedonewithoutquaternions? Ofcourseitcan! Eventscanbedefinedoverthefieldofrealnumbers,and
thentheMinkowskimetricandtheLorentzgroupcanbedeployedtogeteveryresulteverconfirmedbyexperiment.
QuantummechanicscanbedefinedusingaHilbertspacedefinedoverthefieldofcomplexnumbersandreturnwith
everyresultmeasuredtodate.