The Problem of Time: Quantum Mechanics Versus General Relativity

917 Pages·2017·14.86 MB·English

by  Edward Anderson

Fundamental Theories of Physics 190
Edward Anderson
The Problem
of Time
Quantum Mechanics Versus
General Relativity
Fundamental Theories of Physics
Volume 190
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Edward Anderson
The Problem of Time
Quantum Mechanics Versus General
Relativity
EdwardAnderson
DAMTP
CentreforMathematicalSciences
Cambridge,UK
ISSN0168-1222 ISSN2365-6425(electronic)
FundamentalTheoriesofPhysics
ISBN978-3-319-58846-9 ISBN978-3-319-58848-3(eBook)
DOI10.1007/978-3-319-58848-3
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Preface
ThisbookconcernsthefoundationsofQuantumGravity,inparticularfromacon-
ceptualpointofview.Itprovidesaself-containedintroductiontothistopic,resting
on particular features of the accepted Paradigms of Physics: Newtonian Physics,
SpecialRelativity(SR),QuantumMechanics(QM),QuantumFieldTheory(QFT)
andGeneralRelativity(GR).InapproachingQuantumGravity,manyconceptualis-
suesturnouttoberelatedtonotionsoftime.Thisoccursbecausenotionsoftimeare
substantiallydifferentacrosstheseParadigms.Afirstexampleinwhichthisoccurs
isQMversusGR.IshamandKuchaˇrformalizedthestudyofsuchdiscrepanciesbe-
tweennotionsoftime.Theydidsobygivingaconceptualclassificationofthemany
time-relatedreasonswhyawiderangeofattemptedapproachestoQuantumGravity
fail to be satisfactory, in two seminal Reviews in the early 1990’s [483, 586]. The
currentbook’stitular‘ProblemofTime’referstothisconceptualclassification.This
isamulti-facetedcollectionofveryinterestingproblemswhichturnouttobeheav-
ilyinterlinked.Quiteafewoftheseproblemswerefirstglimpsedinthepioneering
worksofWheelerandDeWittinthe1960’s[237,897,899]onthegeometrodynam-
icalformulationofGR.
The Problem of Time is, in greater generality, a consequence of the mismatch
betweenBackgroundDependentandBackgroundIndependent[12,363]Paradigms
ofPhysics.NewtonianPhysics,SR,QM,andQFTareallBackgroundDependent,
whereas GR is Background Independent and many approaches to Quantum Grav-
ity expect this to be Background Independent as well. So, whereas there has been
quite widespread belief among theoretical physicists that the Problem of Time is
a quantum matter, this is a misconception since clearly also Classical Physics can
exhibitmismatchesbetweenBackgroundDependentandBackgroundIndependent
Paradigms. Once this is taken into account, models exhibiting classical versions
of the Problem of Time turn out to provide substantial conceptual insight into the
harderquantumversionsoftheProblemofTime.
Itisthusclearthatfurtherexplanationofwhatthisbook(and[483,586])takes
theProblemofTimetoconsistofisbestdoneafterthefollowing.
v
vi Preface
A) Presenting the standard Paradigms of Physics and explaining how notions of
timedifferacrossthese.
B) OutliningwhateachofQuantumGravityandBackgroundIndependenceare.
N.B. that A)—in Chaps. 1 to 8’s account of notions of time and of space and of
the diversity of physical laws across accepted Paradigms in Physics—serves as a
preamble.Itisnottobemistakenforintroductionofthematerialwhichtherestof
thebookgreatlyexpandsupon,whichis,rather,Chaps.9,10and12ontheProblem
ofTimeandBackgroundIndependenceissueswhichunderliethis.Chapters1to8
enter,rather,intoassemblingcheckliststotestfoundationalandQuantumGravita-
tionalcandidatetimesagainst,toseeifthesemerittobecalledtimefunctions,and
towardbuildinguptowardplausibleQuantumGravitationallawsinChap.11.While
these may be somewhat unexpected and indirect uses of Chaps. 1 to 8’s material,
thisistheintendeduseoftheseChaptersinwritingthisbook.Bywayofexplana-
tion, this book’s main topic happens to benefit from a preliminary presentation of
thetypesoflawandnotionsoftimeandspacethateachoftheestablishedtheories
has. This is prudent given that this book’s main topic is a systematic analysis of a
widerangeofmorespeculativefoundationalandQuantumGravitationalprograms
in which only subsets of standard theories’ laws and temporal and spatial notions
arekept.
FromtheAcceptedParadigmsofPhysicstoQuantum Gravity
Thisbookthusbeginsbyconsideringtimeandclockconcepts,alongsidesupporting
notionsofspace,length-measuringdevices,spacetimeandframes.Chapter1givesa
largelytheory-freeconceptualoutlineofthese,intendedforaverywideanddiverse
multidisciplinaryaudience.
Each of the Newtonian Paradigm, SR, QM, QFT and GR are then covered in
turn,inChaps.2to7.ThistreatmentincludesinoutlinehowtheseParadigmShifts
affecttime,clock,space,length-measuring,spacetimeandframeconcepts.
This theory by theory development has the further complication of not being a
linearventure:theseParadigmsofPhysicsfanoutfromNewtonianMechanicsasin-
dicatedinFig.1.a).Threedistincttheoreticaldevelopmentseachbringinoneofthe
threeknownfundamentalconstantsofNature:1 Newton’sgravitationalconstantG,
thereciprocalofthespeedoflightc,andPlanck’sconstant(cid:2),asfollows.
1Valuesof—anduncertaintiesin—thesefundamentalconstantsareasfollows[661].Thespeedof
lightinvacuocisdefinedtobeexactly299,792,458ms−1duetothemetreitselfbeingdefinedin
termsofc(seeChap.1.13).Planck’sconstant(cid:2)=1.054571800(13)×10−34kgm2s−1.Newton’s
gravitationalconstantG=6.67408(31)×10−11 m3kg−1s−1.TheerroranalysisforthePlanck
unitsisverystraightforward:sinceGisbyfartheleastaccuratelyknownofthefundamentalunits,
theerrorinthisswampstheothers.
Preface vii
Fig.1 a)Planckiancubeoffundamentalphysicaltheories.Here,NMstandsforNewtonianMe-
chanicsand(Q)NGstandsfor(Quantum)NewtonianGravity.b)indicatesthe‘Newton–Einstein’
(alias‘classical’,inthesenseof‘non-quantum’)plane,andthe‘ParticlePhysics’(i.e.‘non-grav-
itational’)plane.c)Gordiancube:hereitisperceivedthatdifferentroutesalongtheedgestothe
‘finalQuantumGravityvertex’maynotcommute(inthealgebraicsense,uponviewingitsedges
asmaps).d)CuttingtheGordiancube?[Or‘thinkingoutsideofthebox’thatisa)?]Here,(Q)RPM
standsfor(Quantum)RelationalParticleMechanics;seethenextSubsectionforanoutlineofwhat
(Quantum)Gestaltmeans.e)IndicatesthecorrespondingcoveragebyChaptersinPartI
1) G is significant when gravitational force is non-negligible in comparison to
whichever forces dominate the physics. G was originally formulated in New-
tonianGravity,whichlieswithinNewton’sParadigmofPhysics,whereas2)and
3)eachadditionallyrepresentintroducinganewParadigm.
2) cisnon-negligiblyfiniteinSR[736];thisisrelevanttoobjectswhosevelocities
varenon-negligiblecomparedtoc.
3) (cid:2) is significant in Quantum Mechanics (QM) [599], due to certain quantities
cominginminimum-sizedpackets.Forinstance,angularmomentumcomesin(cid:2)
(or(cid:2)/2)sizedpackets.Thisisrelevantinsituationsinvolvingquantitiescompa-
rableinsizetothecorrespondingminimumpackets.
Pairwiseincorporationsoftheseconstantsarefurthermoreasfollows(Fig.1.a).
4) RelativisticQFT[712]involvescand(cid:2)together,correspondingtotheCompton
wavelength
l =(cid:2)/mc (1)
C
viii Preface
length-scalefora‘particle’ofmassm.
5) GR [874]—in the sense of a Relativistic Theory of Gravitation—considers c
and G together, corresponding to strongly gravitating fast-moving objects e.g.
confinedtoaroundthescalegivenbytheSchwarzschildradius,
r =GM/2c2. (2)
Schw
Each of the ‘Particle Physics’ and ‘Newton–Einstein’ planes indicated in
Fig.1.b)areself-consistenttwo-stepParadigmShifts.
6) Considering(cid:2)andGtogethergives‘QuantumNewtonianGravity’;thisishow-
evermuchlessrelevant(ExVI.0).Acharacteristiclengthscaleherewouldbe
l :=(cid:2)2/2GMm2. (3)
g
7) Finally, ‘Quantum Gravity’ [75, 154, 194, 237–239, 385, 471, 474, 475, 483,
485, 552, 586, 746, 845] is often held to be ‘the’ triple combination at the last
vertexofFig.1.a)’s‘cube’oftheories.Thethreefundamentalconstantscombine
heretoformthePlanckunits:
(cid:2)
l = (cid:2)G/c3(cid:2)1.616228(38)×10−35m, (4)
Pl
(cid:2)
t = (cid:2)G/c5(cid:2)5.39116(13)×10−44s, (5)
Pl
(cid:3)
m = (cid:2)c/G(cid:2)2.176470(51)×10−8kg. (6)
Pl
Thefirsttwooftheseareverysmallcomparedto‘ordinaryphysicalquantities’.
[Compare l with the atomic (cid:2)10−10 m and nuclear (cid:2)10−15 m lengthscales,
Pl
and with the maximum precision of displacement detection in existing gravi-
tational wave detectors corresponding to displacements of (cid:2)10−18 m. Com-
pare also the ratio of t to the timescales of observational Physics with the
Pl
maximally accurate clock precision of currently around 1 part in 1016, as per
Chap. 1.] On the other hand, m is very large upon considering its interpre-
Pl
tation as a single ‘fundamental particle’ mass: compare e.g. the proton mass
1.672621898(21)×10−27 kg.BySR’sE=mc2,m correspondstoanenergy
Pl
scaleE =1.220910(29)×1019 GeV.N.B.thisismuchlargerthanthe102 to
Pl
104 GeVrangeofthemostpowerfulparticleacceleratortodate:CERN’sLarge
Hadron Collider (LHC). Moreover, as detailed in Chap. 11, the Planck regime
is expected to feature in some parts of Black Hole Physics and Early-Universe
Cosmology. In particular, this book covers the Quantum Cosmology arena and
simplermodelarenasthatexhibitfeaturesofthis.
DifferingRolesofTimeand SpaceThroughout theParadigms
ofPhysics
Space and especially time are moreover not consistently conceived of throughout
‘the Planckian cube of theories’. Due to this, consideration of which units can be
Preface ix
builtoutoffundamentalconstantsmaynotsufficeasaconceptualframeworkwithin
whichtoreconciletheParadigmShiftsofPhysics.Indeed,thisbookexpoundsthat
GRinvolvesqualitativelydistinctconceptsoftime,space,spacetimeandframefrom
thoseusedinParticlePhysics.
On these grounds, this book contends that conceiving of ‘Quantum Gravity’
solely in terms of ‘the Planckian cube’ is a misleading simplification. This con-
ceptualdisparitypointsinsteadtodifferentpathsaroundthis‘cube’notcommuting
(Fig.1.c).Bythisdisparityandthesubsequentnotoriousdifficultywithitsresolu-
tion, it might be more apt to name the cube not ‘Planckian’ but ‘Gordian’: after a
notoriousknotoftheancientworldthatwaspresentedtoAlexandertheGreat.Heis
supposedtohavedealtwiththisknotby‘thinkingoutofthebox’.Accountsdiffer,
however,astowhetherthisinvolvedcuttingitorremovingitfromthewoodenpole
itwasmountedupon.Indeed,suggestionsforapproaching‘QuantumGravity’differ
amongstthemselvesaswell.
A further interplay is that the Theoretical Physics literature often pays little at-
tentiontothepropertiesentailedincallinganentityatimeoraclock.Thisisunfor-
tunate,becauseanumberofsuchpurportedtimequantitiesdonotstandupagainst
a suitable list of temporal properties. We emphasize this point in this book firstly
byattributingmathematicalpropertiesto‘timefunctions’,andoperationalcharacter
to ‘clocks’, in contradistinction to physical, philosophical and conceptual discus-
sionofaspectsoftime.2 Secondly,werefertocandidatetimes,timefunctionsand
clocksuntilenoughpropertiesofthesehavebeenestablished.Thereismoreovernot
auniquelistofpropertiestocheckagainst,sincedifferentphysicaltheoriesinvolve
differentlistsofproperties,asthe‘Gordiancube’inFig.1.cforeshadows.
Background Independence inMechanics,GRand Quantum
Gravity
Following Einstein, a second perspective on the nature of—and motivation for—
GR is as a freeing from absolute or background structures. From this perspective,
GR is more than just a Relativistic Theory of Gravitation. Such perspectives have
subsequentlybeendubbedBackgroundIndependence[12,363,483,485,552,752];
contrastwithhowBackgroundDependentabsolutestructurespervadeallsixofthe
othernon-finalverticesofthecube.ThisbookconsidersGRasembodyingbothof
theseperspectivesatonce,phrasingthisintheshorthandthatGRisa‘gestalt’ofa
RelativisticTheoryofGravitationandofBackgroundIndependence.
2Thisbookprioritizesconceptualmattersconcerningtimeovergivingdetailedspecificsofaccu-
ratetimekeeping[82,783].Itdoescontainsomecommentsonsiderealandephemerisastronom-
icaltimes,atomicclocks,andSRandGRimplicationsfortimekeeping.Someofthetheoretical
conceptsoutlinedinthisbook,moreover,mayeventuallybecomerelevanttotimekeeping:space
clocks, extending Earth or Solar System based reference systems to galactic and cosmological
scales,andclocksinphysicallyextremeregimes.