3jg865camacukyjonathangwolframcom1fortheoverallmultiwaysystemFinallywediscussvariousconsequencesofthismultiwayrelativityincludingthederivationofthepathintegralthederivationofparticlelikeexcitationsan ID: 861504
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1 SomeQuantumMechanicalPropertiesoftheWolf
SomeQuantumMechanicalPropertiesoftheWolframModelJonathanGorard1,21UniversityofCambridge2WolframResearch
2 ,Inc.yApril21,2020AbstractThisarticlebui
,Inc.yApril21,2020AbstractThisarticlebuildsuponthetechniquesdevelopedwithinourpreviousinvestigationoftherel
3 ativisticandgravitationalpropertiesofthe
ativisticandgravitationalpropertiesoftheWolframModel-anewdiscretespacetimeformalismbasedonhy-pergraphtransf
4 ormationdynamics-inordertostudyclassesof
ormationdynamics-inordertostudyclassesofsuchmodelsinwhichcausalinvarianceisexplicitlyviolated,asaconsequenc
5 eofnon-con uenceoftheunderlyingrewriting
eofnon-con uenceoftheunderlyingrewritingsystem.Weshowthattheevolutionoftheresultingmultiwaysystem,whiche
6 1;ectivelycontainsallpossiblebranchesofe
1;ectivelycontainsallpossiblebranchesofevolutionhistory(correspondingtoallpossiblehypergraphupdatingorders)
7 ,isanalogoustotheevo-lutionofalinearsupe
,isanalogoustotheevo-lutionofalinearsuperpositionofpurequantumeigenstates;observersmaythenimpose\e ecti
8 ve"causalinvariancebyperformingaKnuth-Be
ve"causalinvariancebyperformingaKnuth-Bendixcompletionoperationonthisevolution,thuscollapsingdistinctmultiw
9 aybranchesdowntoasingle,unambiguousthrea
aybranchesdowntoasingle,unambiguousthreadoftime,inamanneranalogoustotheprocessesofdecoherenceandwavefunctio
10 ncollapseinconventionalquantummechanics(
ncollapseinconventionalquantummechanics(andwhichweproveiscompatiblewithamultiwayanalogoftheuncertaintyprinc
11 iple).Bydeningtheobservermathematic
iple).Bydeningtheobservermathematicallyasadiscretehypersurfacefoliationofthemultiwayevolutiongraph,wed
12 emonstratehowthisnovelinterpretationofqu
emonstratehowthisnovelinterpretationofquantummechanicsfollowsfromageneralizedanalogofgeneralrelativityinthe
13 multiwaycausalgraph,withtheFubini-Studym
multiwaycausalgraph,withtheFubini-Studymetrictensorplayingtheroleofthespacetimemetric,thequantumZenoe e
14 ctplayingtheroleofgravitationaltimedilat
ctplayingtheroleofgravitationaltimedilation,etc.Werigorouslyjustifythiscorrespondencebyproving(usingvarious
15 combinatorialandorder-theoretictechnique
combinatorialandorder-theoretictechniques)thatthegeometryofthemultiwayevolutiongraphconvergestothatofcomple
16 xprojectiveHilbertspaceinthecontinuumlim
xprojectiveHilbertspaceinthecontinuumlimit,andproceedtousethisinformationtoderivetheanalogoftheEinstein
17 ;eldequations jg865@cam.ac.ukyjonath
;eldequations jg865@cam.ac.ukyjonathang@wolfram.com1 fortheoverallmultiwaysystem.Finally,wediscussvario
18 usconsequencesofthis\multiwayrelativity"
usconsequencesofthis\multiwayrelativity",includingthederivationofthepathintegral,thederivationofparticle-li
19 keexcitationsandtheirdynam-ics,theproofo
keexcitationsandtheirdynam-ics,theproofofcompatibilitywithBell'stheoremandviolationoftheCHSHinequality,thed
20 erivationofthediscreteSchrodingerequati
erivationofthediscreteSchrodingerequation,andthederivationofthenon-relativisticpropagator.Connectionstoman
21 yeldsofmathematicsandphysics-includ
yeldsofmathematicsandphysics-includingmathematicallogic,abstractrewritingtheory,au-tomatedtheorem-prov
22 ing,universalalgebra,computationalgroupt
ing,universalalgebra,computationalgrouptheory,quantuminformationtheory,projectivegeometry,ordertheory,latti
23 cetheory,algorithmiccomplexitytheory,adv
cetheory,algorithmiccomplexitytheory,advancedcombinatorics,superrelativity,twistortheoryandAdS/CFT-correspo
24 ndence-arealsodiscussed.1IntroductionIno
ndence-arealsodiscussed.1IntroductionInourpreviouspaper[1],weformallyintroducedtheWolframModel[2]-anewdiscr
25 etespacetimeformalisminwhichspaceisrepre
etespacetimeformalisminwhichspaceisrepresentedbyahypergraph,andinwhichlawsofphysicsaremodeledbytransformati
26 onrulesonsetsystems-andinvestigateditsva
onrulesonsetsystems-andinvestigateditsvariousrelativisticandgravitationalpropertiesinthecontinuumlimit,as&#
27 12;rstdiscussedinStephenWolfram'sANewKin
12;rstdiscussedinStephenWolfram'sANewKindofScience(NKS)[3].Ourcentralresultwastheproofthatlargeclassesofsuc
28 hmodels,withtransformationrulesobeyingpa
hmodels,withtransformationrulesobeyingparticularconstraints,weremathematicallyconsistentwithdiscreteformsof
29 bothspecialandgeneralrelativity.Anexampl
bothspecialandgeneralrelativity.AnexampleofsuchatransformationruleisshowninFigure1,andanexampleofitsevoluti
30 onisshowninFigures2and3. Figure1:Anexamp
onisshowninFigures2and3. Figure1:Anexampleofapossiblereplacementoperationonasetsystem,herevisualizedasatran
31 sformationrulebetweentwohypergraphs(whic
sformationrulebetweentwohypergraphs(which,inthisparticularcase,alsohappentobeequivalenttoordinarygraphs).Ad
32 aptedfromS.Wolfram,AClassofModelswithPot
aptedfromS.Wolfram,AClassofModelswithPotentialtoRepresentFundamentalPhysics[2].Inparticular,weintroducedthe
33 notionofcausalinvariance(i.e.theconditio
notionofcausalinvariance(i.e.theconditionthatallcausalgraphsbeisomorphic,independentofthechoiceofupdatingor
34 derforthehypergraphs),provedittobeequiva
derforthehypergraphs),provedittobeequivalenttoadiscreteversionofgeneralcovariance,withchangesinupdatingorde
35 rcorrespondingtodiscretegaugetransformat
rcorrespondingtodiscretegaugetransformations,andlaterusedthisfacttodeducediscreteanalogsofbothLorentzandloc
36 alLorentzcovariance.Havingderivedthephys
alLorentzcovariance.HavingderivedthephysicalconsequencesofdiscreteLorentztransformationsinthesemodels,wesub
37 sequentlyprovedvariousresultsaboutthegro
sequentlyprovedvariousresultsaboutthegrowthratesofvolumesofspatialballsinhypergraphs,2 Figure2:Anexampleevo
38 lutionoftheabovetransformationrule,start
lutionoftheabovetransformationrule,startingfromaninitial(multi)hypergraphconsistingofasinglevertexwithtwose
39 lfloops.AdaptedfromS.Wolfram,AClassofMod
lfloops.AdaptedfromS.Wolfram,AClassofModelswithPotentialtoRepresentFundamentalPhysics.andofspacetimeconesin
40 causalgraphs,ultimatelyconcludingthatbot
causalgraphs,ultimatelyconcludingthatbothquantitiesarerelatedtodiscreteanalogsoftheRiccicurvaturetensorfor(
41 hyper)graphs.Weusedthisfacttoprovethatth
hyper)graphs.Weusedthisfacttoprovethattheconditionthatthecausalgraphshouldlimittoamanifoldofxeddimensi
42 onalityisequivalenttotheconditionthatthe
onalityisequivalenttotheconditionthatthediscreteEinsteineldequationsaresatisedinthecausalgraph,an
43 dthereforethatgeneralrelativitymusthold.
dthereforethatgeneralrelativitymusthold.Wewentontodiscusssomemorespeculativeproposalsregardingageneralrelat
44 ivisticformalismforhypergraphsofvaryingl
ivisticformalismforhypergraphsofvaryinglocaldimensionality,andafewofthecosmologicalconsequencesthatsuchafor
45 malismwouldentail.Thepresentarticlebegin
malismwouldentail.Thepresentarticlebeginsbybrie yrecappingthetheoryofabstractrewritingsystemsandtheircon-ne
46 ctionstotheWolframModelinSection2.1,befo
ctionstotheWolframModelinSection2.1,beforeproceedingtointroducetheKnuth-Bendixcompletionalgorithmfor\collap
47 sing"distinctmultiwayevolutionbranchesdo
sing"distinctmultiwayevolutionbranchesdowntoasingle,unambiguousthreadoftime,thusobtaininge ectivecausal
48 invariancefromanon-con uentrewritingsyst
invariancefromanon-con uentrewritingsystem,inSection2.2.WegoontoshowinSection2.3thattheevolutionofthemultiw
49 aysystemismathematicallyanalogoustotheev
aysystemismathematicallyanalogoustotheevolutionofalinearsuperpositionofpurequantumeigenstates,andthereforet
50 hatKnuth-Bendixcompletionisanalogoustoth
hatKnuth-Bendixcompletionisanalogoustotheprocessofdecoherenceandwavefunctioncollapsethatoccursduringtheacto
51 fmea-surementwithinstandardquantummechan
fmea-surementwithinstandardquantummechanicalformalism(indeed,weprovethatthisprocessisconsistentwithamultiwa
52 yanalogoftheuncertaintyprinciple).Wealso
yanalogoftheuncertaintyprinciple).Wealsodiscusssomemathematicalconnectionsto3 Figure3:Thenalstateofthe
53 aboveWolframModelevolution.AdaptedfromS.
aboveWolframModelevolution.AdaptedfromS.Wolfram,AClassofModelswithPotentialtoRepresentFundamentalPhysics.un
54 iversalalgebraandcomputationalgrouptheor
iversalalgebraandcomputationalgrouptheory,aswellasvariousimplicationsofthisnewformalismforquantuminformatio
55 ntheory,inSection2.4.InSection3.1weintro
ntheory,inSection2.4.InSection3.1weintroduceanewmathematicaldenitionofaquantumobserverasadiscretehyper
56 -surfacefoliationofthemultiwayevolutiong
-surfacefoliationofthemultiwayevolutiongraph,andproceedtooutlinehowthenovelinterpretationofquantummechanics
57 presentedintheprevioussectionthereforefo
presentedintheprevioussectionthereforefollowsfromageneralizedvariantofgeneralrelativityinthemultiwaycausalg
58 raph,withtheFubini-Studymetrictensorplay
raph,withtheFubini-Studymetrictensorplayingtheroleofthespacetimemetric.Wegoontoprovethiscorrespondencerigor
59 ouslyinSection3.2,byrstprovingthatt
ouslyinSection3.2,byrstprovingthatthegeometryofthemultiwayevolutiongraphconvergestothatofcomplexprojec
60 tiveHilbertspaceinthecontinuumlimit(usin
tiveHilbertspaceinthecontinuumlimit(usingvarioustechniquesfromcombinatorics,ordertheoryandlatticetheory,and
61 byexploitingvonNeumann's\continuousgeome
byexploitingvonNeumann's\continuousgeometry"formalismforcomplexprojectivegeometry),andthenlaterbyexplicitly
62 derivingthemultiwayvariantoftheEinstein&
derivingthemultiwayvariantoftheEinsteineldequationsusingthemethodsofsuperrelativity.Section3.3outlines
63 afewgeometricalandphysicalfeaturesofthem
afewgeometricalandphysicalfeaturesofthemultiwaycausalgraph,andmakesaconjectureregardingitsconnectiontotheco
64 rrespondencespaceoftwistortheory.Finally
rrespondencespaceoftwistortheory.Finally,inSection3.4wediscussvariousconsequencesof\multiwayrelativity",inc
65 ludingformalderivationsofthepathintegral
ludingformalderivationsofthepathintegral,theexistenceofparticle-likeexcitations,thediscreteSchrodingerequa
66 tion,andthenon-relativisticpropagator,as
tion,andthenon-relativisticpropagator,aswellasaproofofcompatibilitywithBell'stheoremandtheviolationoftheCHS