TheodoreSt.John TheHolomorphicQuanta June10,2018 1

TheHolomorphicQuanta

Part2:ProjectionTheodoreSt.John

stjohntheodore@gmail.com

“Weliveinahouseofmirrorsandthinkwearelookingoutthewindows” –FrederickSalomonPerls

AbstractThisisthesecondpartofafour-partpresentation.Inpart1Iintroducedarelationalmodelthatallowedmetodemonstratetheequivalenceofspaceandtimeas𝑆 = 𝑇𝑐!andshowedthatSrepresentsenergyastheproductofscalarspacewithspatialfrequencyandTrepresentsenergyastheproductoftimeunitswithtemporalfrequency.DoingsorevealedtheequationsforquantumenergyofaparticletobetheinversedomainsscaledbyPlanck’sconstant.Inthiscontext,theyservedastwocomponents(basevectors)ofaquantumwavefunction(acompositespace-timevector).Inthispart,Iwillcontinuetodevelopthemodelanddiscusshowtheprojectionofaunifiedconceptontoaplanethatrepresentstheirseparationcreatesascalingproblemthatcanbedealtwithadifferentways.Abstract..........................................................................................................................................................1Introduction.................................................................................................................................................1ADROPLETofEnergy..............................................................................................................................3TheDROPLETApp................................................................................................................................5ProjectionCreatesaScalingProblem..........................................................................................6

TheBackground......................................................................................................................................10Conclusion..................................................................................................................................................10Bibliography..............................................................................................................................................11

IntroductionInpart1ofthispaperIintroducedthespace-time-motiondiagram,avisual

toolforrepresentingtherelationshipsbetweenspaceandtimeastwoequivalentaspectsofmotionthatareconceptuallyseparated,butinessencetwodifferentaspectsofthesamething.Toanalyzethis,Isuperimposedtherelativisticspace-versus-timeplotinthelineardomainwiththeplotofinversetime-versus-inversespaceasthefrequencydomain,linkedbytheplotofmotionitself.Thisproducedthespace-time-motionorSTMdiagram,shownhereasFigure1.

TheodoreSt.John TheHolomorphicQuanta June10,2018 2

Figure1Thespace-time-motionorSTMdiagraminterposestheinverse-spaceandinverse-timedomains,scaledtounitsofhinthelinearaxis,revealsthedeBroglierelationsforenergy.Sincetheyarebothquantumunits,theyarerepresentedasbaseenergyvectorsinHilbertspace.

Thisshowedenergyexpressedinthetemporalfrequencydomainasaquantumunitinspaceandenergyexpressedinthespatialfrequencydomainasaquantumunitintime.Theeffectofthisflippingofdomainsistherealizationthattheyareasymmetricreflectionsofeachother;thelineardomainbeingrepresentedastherelativisticpartoftheplotandtheinversedomainrepresentedasbasevectors,orunitvectors𝑠inthespacedomainand𝑡inthetimedomain.Inotherwords,withouteverhavingtomentionaparticle,waveorparticle-waveduality,aquantizedunitofemptyspaceandaquantizedunitoftimeareshowntobeequivalentunitsofenergythemselves.

TheSTMdiagramvisuallyillustratedhowseparation,thefirststepinthetransformationprocess,splitstheunifiedconceptofenergyintotwopairsofconceptuallyindependentdomainsthatarehyperlinkedataboundarycalledtheeventreference.Comparingthistoquantuminformationtheory,theseunitsareequivalenttoquantumbitsor“qbits”sinceenergy,aunifiedconcept,wasseparatedintotwoorthogonalyetequivalentmeasures:space–quantifiedasscalarunitsofdisplacement(digitalincrementsof“1”),andtime–quantifiedasscalarunitsof“1clockincrement”,whichtransitionsto“0”ateachevent.InthissectionIwillusetheSTMdiagramtoillustratethesecondstepinthetransformationprocess,projection,andshowthatmotionisasuperpositionofthebasevectors.Motionprojectsthisenergyfromthequantumdomainintotherelativisticdomainproducingadistortionthatcanbeviewedasacurvatureofeitherspaceortime.

Itgetsa“bitconfusing”becausethelabelscrossoverdomains,soyouhavetokeepthemseparateinyourmind.Normallyweintroducenewvariablestohelpkeepthemseparate,butthisintroducesadditionaldistortionfactorssothatwillbe

TheodoreSt.John TheHolomorphicQuanta June10,2018 3

avoided.Instead,Ioccasionallyreflectandremindthereaderoftheintendedmeaning.

ADROPLETofEnergyTheunitvector𝑠isthetemporalfrequencydomainassociatedwith𝐸 = ℎ𝑓

interposedonthespatialaxisand𝑡isthespatialfrequencydomaininterposedonthetemporaldomain.Theyareintegralpartsoftheenergydomainthatisthevectorsumofthetwobasevectors,𝑆 = 𝑠 + 𝑡.

Temporalfrequencyiswhatwenormallythinkofasfrequency.Thetermspatialfrequencyisnotusedinquantumphysicsandrarelyinclassicalphysicsexceptforoptics,especiallyholography.Instead,itisreferredtoaswavenumber(𝑘 = !

!)becauseitisjustanumberthatindicatesacycle.InMedicalPhysics,itis

usedasameasureofimagequalityinlinepairspercm.Butinelectrostatics(FieldTheory),!

!= ∅iscalledthescalarpotentialofaunitcharge.Sothespatialfrequency

couldbeusedtorepresentthepotentialfieldofachargeorthepotentialfieldofgravitysurroundingaunitmass.Infact,apotentialfieldisascalesoitdoesn’tevenneedamassorchargetoexist.AccordingtoQuantumFieldTheorythereisnoneedforapointchargetoprovidethefield,onlyapoint.Instead,the“fieldquanta”islikeanorboradropletofenergy,apartofanunderlyingfieldthatissomehowexcited.

Imagineafieldofabsolutenothingnessandimaginepointsinthisfieldlaidoutinagridwithscales.Scalesthemselves(s=0,1,2,…or𝑓 = 1, !

!, !!…)represent

agradient,∇∅ = !∅!"

𝑜𝑟 !∅!"i.e.somethingischanging,evenifitjustthescaleitself.If

thesizeoftheincrementsischanging,asintheinversescale,thechangeinthegradient–calledthegradientofpotential–representsaforce,𝐹 = −∇∅.

Lookingoutward(positiveonthescale)thechangeislinear,andthereisnochangeintheunit∆betweenthelinearscalemarkssothatgradient∇ ∆s ,isconstant.Butthereisstillagradientanditstillhasasenseofdirection,so∇ ∆s =𝑠.However,thechangeinthegradientiszerounlesssomeoutsideactionattemptstochangeit,soitpresentsasinertiaormomentuminclassicalandquantumphysics.Butinrelativisticphysics,itisrecognizedtobethecurvatureofspacethatprovidesadownwardslopeforgravitation.WiththeSTMmodel,nowaveorgravitonisneeded.

Spaceistheimaginaryscale,∅,whichservesasthebaseoftransformation(offormlessenergyintoform).Combiningthelinearscalewiththeinversescaleresultsintwodifferentgradientsinspace.Lookingoutward,inthelineardomainwherewelive,thecurvatureorgradientofstaticspaceisconstantsoitappearsflat,∇∅ = !

!"𝑠 = 1yetdivergesinalldirections.Soeventhoughpotentialisjustthe

inversescale,i.e.ascalar,ithasaninherentdirectioninwhichitchangesandisthusapotentialforce,𝑭 = −∇∅,(thetemporalcomponentofpotentialenergy)thatwillbecomearealforceifactedon.Butmotioncreatesaseparationofspaceandtimeresultinginagradientmadeupoftwopairsofdifferentgradients.Asavector,callit𝐶,thegradienthastwoparts,asshowninFigure2.

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∇∅⟹ 𝐶 = 𝐶!𝑠 + 𝐶!𝑡 (1)

where𝐶! = 𝐶𝑐𝑜𝑠 𝜗 and𝐶! = 𝐶𝑠𝑖𝑛 𝜗 arethemagnitudeofeachunitvector,definedasoneunit.InDiracnotation,thiswouldbe

|𝜓 = 𝜓!|𝑠 + 𝜓!|𝑡 . (2)Intermsofderivatives,thisis

∇∅ = !∅

!"!"!∅+ !∅

!"!"!∅. (3)

Figure2Scalesthemselvesrepresentgradients.Combiningthelinearscalewiththeinversescaleresultsintwodifferentgradientsoneachaxis.

Theterms!"

!∅and!"

!∅representthemagnitudeofthescaleinthegivendomain,which

isalwaysoneunitbydefinitionofaunitregardlessofwhichdomainyouarein.SotheyarenotneededandEquation(3)becomes

∇∅ = !∅!"+ !∅

!". (4)

alsoshowninFigure2.

Thefirsttermisachangewithrespecttospace,soitreferstosandisthusscaledbythelineardomain,soitisascalarpotentialdivergentinalldirections,i.e.∇ ∙ ∅.Butthesecondterm,thechangewithrespecttotime,istheinversespatialterminterposedinthetemporalaxis.Soitisthederivativewithrespecttotheinverseofspace.FromEquation(1),thisis𝐶𝑠𝑖𝑛(𝜗),whichisthecross-productand

TheodoreSt.John TheHolomorphicQuanta June10,2018 5

comparestoavectorpotentialinFieldTheoryasthecurlofthevector,∇×∅,since𝐶isthevectornotationforthegradient.

∇∅ = !∅

!"+ !∅

! ! != ∇ ∙ ∅+ ∇×∅. (5)

ThisisaformofHelmholtzTheorem(Wangsness1986,pg.37)rearrangedas

𝑭 = −∇∅+ ∇×𝚨 (6)

whereFisthedivergentfield–arealforcesimplyduetothefieldofmotion,pointinginwardtransformingthesquaregridintoacirculardroplet.AccordingtoDavidHestenes,Equation(5)isthefundamentaldecompositionofthegeometricproduct(Hestenes2003,pg.15).Firsthetreatsthecurltermasanimaginarycomponentandthenremovestheimaginarylabelandcallsitthe“outerproduct”,atermfromGeometricAlgebra,andproceedstoderiveallofMaxwell’sequationsasasingleunifiedequation.

RememberthatFigure1representedtheflashoflightexpandingin3Dspaceandtime,andtheflashbulbonlyflashedoncesothelightspherewouldactuallybeashell.Theboundaryconditionsoneachaxisarerepresentedbythetipsofthevectors,wheretheboundaryconditionsaresatisfied.Theyarethehyperlinkstothescalardomain,theoneunambiguouspointineitherdomain.Theyallrepresentthesamefield∅,andtheboundaryiswhere∅ = !

∅= ∅!.Onthespatialandtemporal

axestheyarecalledthe“Surfacereference”and“Clockreference”andonthemotionvectoritiscalledthe“Eventreference”.Itistheonlyplacethatactuallyrepresentsthesurfaceofthelightsphere–thepresent(hereandnow).The“Surfacereference”and“Clockreference”areback-projectionsofthepresent.

TheDROPLETAppIfweweretowriteanAppforthat,youcouldclickonahyperlinktoselect

theperspectiveyouwanttoobserve.Clickatthesurfacereference,unitnumber1onthespaceaxis,andanotherwindowwouldopenshowingtheaxisunfoldedwiththeunitenfoldedintoa3-Davatarofthedroplet.Apopupmessagewouldcallthis“Here”.Wecouldclickanddragoutalongthespatialaxis“scrollbar”tozoomoutandmaketheavatarcollapsetoatinyicon(aunitdropofenergyout“There”)andtheS-Tcoordinatesystemexpandinthebackground.Wecouldscrollallthewayintotheorigin,ordoubleclickontheicontoputtheviewpointontheinside(theinverse-timedomainsectionofthespatialaxis)andthescreenwouldgodark.Adda“Direction”buttontoswitchviewinganglefrominwardtooutward,butthescreenwouldstillbedark.Apopupmessagewouldread“Stillness”.

Wecoulddothesamethinginthetimedomain,clickingontheclockreference–unit1onthetemporalaxis.Thisnewwindowwouldopenwiththeinwarddirectionselectedtounfoldthespatialfrequencydomainsectionofthetimeaxisandwewouldbelookingfromtheinsideoutagain.Anotherbuttonlabeled“Viewgridlines”wouldturnonspatialfrequencygridlinesatthesurface.Rather

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thanzoomingout,thescrollbarwouldjustmovethetimescaletotheleftontheSTMplot.

Sowecouldclicktheplaybuttonandletitrun.Nowifyoulookatthespatialdomainwindow,zoombackoutandchange

viewingdirectiontoinward,witheachincrementoftimeyouwouldseegridlinescollapsinginfromouterspacelabeled“Future”untiltheyreachedthesurface,wheretheywouldflashtheword“Present”.Halfoftheenergy(theywouldchangetoalowerfrequencycolor,sayred)wouldreflectoffthesurfaceandtheotherhalfwouldpaintthesurface.Thelabelwouldchangefrom“Future”asthegridmovedin,flashtheword“Present”foraninstant,thentochangeto“Past”asitmovedoutward.

Sowhatwouldbedisplayedifweclickoff-axis,ontheS-Tplane?Becausethediagonal(composite)vectorstretchesoutbeyondthedashedarcinFigure1andFigure2,thedisplaywouldbeaconformalprojectionofamotionvectorprojectedoutoftheenergydomainandontothebackgroundscalarplane.Soifweclickonthediagonal,anotherslightlylargeravatarwouldappeartoenvelopethefirsticon,onaflat,veryfinebackgroundgrid,linearandevenlyspaced.Ifweclicktheplaybutton,thegridwouldappeartogetcloserandcloser.Thegridspacingwouldgrow(decreasingtemporalfrequency)butremainlinearuntilthedropfitinsideoneunitofthegrid.Theentiregridwouldsuddenlycollapseintothedrop,butanothergridwouldfollowandthenitwouldcollapse.Doubleclickonthesurfacereferencetolookinwardandnow,ratherthanbeingdark,wewouldseethegrid,non-linearlyshrinking,likethefocalpointofalens,towardthesmallericoninside.Again,partofitwouldreflectoffthesurfaceandonespatialunitofredwouldfitbetweenthetwospheres.Thisfrequencysplitwillbepresentedgraphicallyandmathematicallyinpart3.

TomaketheAppdisplayrelativemotionIwouldjusthidethecollapsinggridlinesandshowthelineargridlinesfromanotherparticle,movinglinearlyacrossthebackgroundinwhateverdirectiontheparticlewasmoving.Iftheotherparticlewerenotmovingrelativetothescreen,wewouldseegridlinescollapsingintoit.Andsincewesharethesamespace,itwouldgravitatetowardthescreen.

SoIwouldcalltheApp,TheDigitalRoundOpticalParticlewithLinearEnergyTransportorDROPLET.

Asaprojectionoutoftheenergydomain,thecompositevectorpresentedasaswollenparticle–stretchedoutincomparisonwiththebasevectors.Relativetoit,thebasevectorsappeartobecontracted.ThisisduetoaparallaxthatisthesamerelationastheLorentzfactor(explainedinthenextsection).

ProjectionCreatesaScalingProblem

AsImentionedbefore,physicistsandmathematicianscanhandlevectorswiththeireyesclosed,soaconformalprojectionisnotaproblem.Butforastudent,itisimportanttorememberthatoneunitofmeasureintheenergydomainisdifferentfromoneunitinthescalardomain.Toillustratethis,IrefertoFigure3.Theareaofthelargesquare(𝑐!)representsoneunitofenergy,𝐸 = 𝑐!(sooneunit

TheodoreSt.John TheHolomorphicQuanta June10,2018 7

ofmass).Itisequalto2unitsofmeasure(oneeachfor𝑠!and𝑡!)inthescalarS-Tdomain.SowhenprojectedontothelinearscalarS-Tdomain,thehypotenuseofthetriangle(c)is𝑐 = 𝑠! + 𝑡!.

Figure3Thesquareofthehypotenuseofatrianglerepresentstheareaofthelargesquare,

whichisfour-timestheareaofthetriangle.

Nowrecallthatthehyperlinkbetweenthetwodomainsistheslope(c)ofthegraphintheS-Tplane,whichisequaltotheratioofthespaceunitwithrespecttothetimeunit.Takentothelimit,speedisthederivative𝑣 = !"

!".Ifthisexample

representsanobjectofunitmass,m,andvelocity𝑣,wewouldcalculatethekineticenergybyusingtheWork-KineticenergyTheoremichangingvariablesandintegratingalongthehypotenuse(unitsofvelocity),sothat𝐾𝐸 = 𝑚 𝑣𝑑𝑣 = !

!𝑚𝑣!!

! .Butthatisonlyhalftheareaofthelargesquareii(𝑚𝑣!).Therefore,togettotalenergyweeitherhavetocorrecttheresultbyascalingfactororbyaddingaconstantofintegrationiii.Thescalingfactorisjusttheratioofoneofthesmallareas(𝑠!sincevelocityisalwaysdenominatedto1unitoftime)tothelargearea.Inthiscase,sincetheareasareequal,theyareeachhalfofthetotal.Sothescalingfactoris2.Butthatwon’tworkforthetotalenergyofanobjectbecauseitstotalenergyincludesrestenergy,whichisnotafunctionofkineticenergy.

Vectorsprovideasolutionbecausetheyallowustorepresenttheentirearea(theintegrationofmorethanoneconcept,inthiscasetwo)inonesymbol.Tofindthetotalenergywecanrepresenttheareaofthelargesquareasavectorofmagnitude𝑐! = 1,asshownontheverticalaxisinFigure4.Itisontheverticalaxistorepresentnorelativemotion,i.e.time-independent;theenergyvectorhasnotbeenseparatedintospaceandtimeiv.Thenintroducetimebyrotatingthevectortoalignwiththehypotenuse(toseparateenergyintotwomeasuresofspaceandtimewitha1:1relation)whilekeepingitsmagnitudeconstant.Comparingitsback-projectiononthespatialaxistotheback-projectionofthe“stretchedout”composite-vector,itappearstobe“contracted”byavalue,callit𝑣!.Theratioofthe

TheodoreSt.John TheHolomorphicQuanta June10,2018 8

actualvector,𝑐!tothecontractedvector,𝑐! − 𝑣!isamagnificationfactor,whichisthesquareofLorentzfactor,𝛾.

𝛾! = !!

!!!!!= !

!!!!

!!

. (7)

Ifthebaseofthesmalltriangleisscaledbymass,thenit’sareaisthesameequationaskineticenergy.

Figure4TheLorentzfactorisamagnificationfactorthatresultsfromusingscalarquantitiestosetthescaleforavectorquantity.Thecompositevectorismagnifiedto𝒄𝟐𝜸𝟐.Inthiscase𝒄𝟐 = 𝟏.

Ifweapplythistotheexpandinglightsphereexample,butforgetthatthese

vectorsaremapsymbolsthatneedtobescaledtofitthehyperlinkedcoordinatesystem,thisfactormightbeinterpretedasmeaningthattheradiusoftheexpandinglightsphere(themeasureoftheprojectionofthevectorontothespatialaxisinFigure4)issmallerthanitreallyisandthatthereissomeextraenergythatisunaccountedfor.Andwecouldapplythiscorrectionfactorsothatourmeasurementsfittherelativisticmodel,toincludeakineticenergyterm.Thiscanbedonebyscalingthemagnitudeoftotalenergyvectorby𝛾𝑐,whichcanbeseparatedintoasumas

𝛾𝑐 = 𝑐 − 𝑐 + 𝛾𝑐 = 𝑐 + 𝑐(𝛾 − 1). (8)

Wethenusethistoscalemomentum,(i.e.multiplyitbymc)togettheHamiltonian(equationfortotalenergyofaparticle)

TheodoreSt.John TheHolomorphicQuanta June10,2018 9

𝐸!"# = 𝑚𝑐 𝑐 + 𝑐 𝛾 − 1 = 𝑚𝑐! +𝑚𝑐! 𝛾 − 1

(9)

asshowninFigure5(a).The“extraenergy”istherelativistic(kinetic)energythatthelightspherewouldhaveifitwereaparticleitselfmovingrelativetosomethingelseinthescalarplane.

(a) (b)

Figure5(a)Vectorrepresentationofthelightspherescaledbyunitsofspaceandtime.ThesametriangleandrelationsareintheEnergydiagramofFigure1becausetherestenergyEoisequaltothedeBroglieenergyEd.

(b)Arelationaltriangleprovidedbyatextbook(Halliday,ResnickandWalker1993)asamnemonicdevicetohelpthestudentremembertherelativisticrelationsbetweenthetotalenergy(Hamiltonian)andtherestenergy,kineticenergyandmomentum.Thearcinthefigureismeanttoillustratethatthemagnitudeof𝒎𝒄𝟐onthehypotenuseisthesameasthatonthehorizontalleg,regardlessoftheangle𝜽.

Sothisscalingproblemisdealtwithbyapplyingacorrectionfactor.Figure5(a)and(b),showhowthiswasillustratedinaFundamentalsofPhysicstext(Halliday,ResnickandWalker1993).Itisusedasamnemonicdevicetohelpremembertherelativisticrelationsamongthetotalenergy“Hamiltonian”(𝐸!),restenergy(𝐸!),kineticenergy(𝐾𝐸)andmomentum(𝑝).Theangles𝜃and𝜑arerelatedto𝛽 = 𝑣𝑐and𝛾as𝑠𝑖𝑛𝜃 = 𝛽and𝑠𝑖𝑛𝜑 = !

!.Thequantumenergyineither

form(𝐸 = ℎ𝑓or𝐸 = 𝑝𝑐)isequaltotherestenergy,𝐸! = 𝑚𝑐!ofaquantumparticlesoitismorethanamnemonicdevice.ItisanSTMdiagram.FromthelegsofthetriangleinFigure5(a)wegettherelativisticenergydispersionrelation

𝐸!"#! = 𝑝𝑐 ! + 𝑚𝑐! ! (10)

ThetwodiagramsinFigure5(a)and(b)representtheexactsamegeometricrelationships.Thedifferenceisonlyinscalev,since𝑚𝑐! = ℎ𝑓 = !

!,thetimeaxisis

scaledby

TheodoreSt.John TheHolomorphicQuanta June10,2018 10

𝑡 = !!!!

, (11)

whichisaPlanck-secondtimes2𝜋,i.e.onecycle(periodorwavelength).Inpart3ofthispaper,thiswillbeaccountedforbyusingyetanotherdomainandcoordinatesystem:polarcoordinates.

TheBackgroundThereisnosuchthingasaparticleatrest,isolatedfromtherelativistic

background.Anyandeveryparticlecanpotentiallybeprojectedontothebackgroundandcomparedtoanyothermovingparticleintheuniverse,seeminglygivingitkineticenergyinstantly.Whatwenormallycallpotentialenergyisactually“potential-energy-of-motion”.Nothinghappenstotheparticlewhenyoudecidetoincludethemovingbackground;itisjustadifferenceinperspectivethatchangesourmethodofquantification.

Inclassicalphysics,wetendtothinkofthebackgroundasnothingmorethanascale,separatefromtheparticle.Butinmechanics,thebackgroundaroundmassisthegravitationalpotentialfield.Inelectromagnetics,thebackgroundisthepotentialfieldofapointcharge,anditiswhatgivestheparticleitsforminspace(asmomentum 𝑝 = !

!),whichisresistanttode-form 𝐹 = !"

!"= ℎ !!!

!",whereFisforce.

AccordingtotheSTMdiagram,energyisprojectedontothebackgroundbymotion,andthatenergyisthenreflectedbacktothequantumdomainattheeventreferenceasaunitofspatialfrequency.Thisgivesenergytotheboundaryandformtothedroplet.Soinessence,thebackgroundistrulypartoftheparticle.

ConclusionOurperceptionoftheworldisbothquantumandrelativistic,sotheSTM

modelrepresentsbothperspectives.Equalrepresentationoftimeandspaceaswellastheirmirrorimages,spatialandtemporalfrequency,allowsthemodeltomorphbetweenthetwoperspectives.

Understandingthisopensthedoortoabetterunderstanding(part3ofthispresentation)ofhowthescalarquantitiespersistintheformofthewaveequationandhowstatisticalequationsusedinquantummechanicsgivethesameresultsasvectoroperations.Italsoprovidesinsightintowhythespeedoflightisnotrelativetothespeedofitssource,howanelectronseemstotakeformasawavepatterninthedouble-slitexperiment,andhowquanta(pairsofquantumparticles)canbedescribedasasphericalstandingwave(TheHolomorphicQuanta)discussedinpart4.

TheodoreSt.John TheHolomorphicQuanta June10,2018 11

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