Time's Arrow in a Quantum Universe I

31 Pages·2017·0.26 MB·English

Time’s Arrow in a Quantum Universe I:
On the Simplicity and Uniqueness of the Initial Quantum State
ddy eming hen
E K C *
December 4, 2017
Abstract
In a quantum universe with a strong arrow of time, we postulate a low-
entropyboundarycondition(thePastHypothesis)toaccountforthetemporal
asymmetry. Inthispaper,IshowthatthePastHypothesisalsocontainsenough
information to significantly simplify the quantum ontology and clearly define
auniqueinitialconditioninsuchaworld.
First,IintroduceDensityMatrixRealism,thethesisthatthequantumuniverse
is described by a fundamental density matrix (a mixed state) that corresponds
tosomephysicaldegreesoffreedomintheworld. Thisstandsincontrastwith
Wave Function Realism, the thesis that the quantum universe is described by a
wavefunction(apurestate)thatrepresentssomethingphysical.
ffi
Second,IsuggestthatthePastHypothesisissu cienttodetermineaunique
and simple density matrix. This is achieved by what I call the Initial Projection
Hypothesis: the initial density matrix of the universe is the projection onto the
speciallow-dimensionalHilbertspace.
Third, because the initial quantum state is unique and simple, we have a
strongcasefortheNomologicalThesis: theinitialquantumstateoftheuniverse
iscompletelyspecifiedbyalawofnature.
Thisnewpackageofideashasseveralinterestingimplications,includingon
thedynamicunityoftheuniverseandthe subsystems, thetheoreticalunityof
statisticalmechanicsandquantummechanics,andtheallegedconflictbetween
Humeansupervenienceandquantumentanglement.
Keywords: time’s arrow, Past Hypothesis, Statistical Postulate, reduction, typicality,
foundationsofprobability,quantumstatisticalmechanics,wavefunctionrealism,quantum
ontology,densitymatrix,WeylCurvatureHypothesis,HumeanSupervenience
*DepartmentofPhilosophy,106SomersetStreet,RutgersUniversity,NewBrunswick,NJ08901,
USA.Email: [email protected]
1
Contents
1 Introduction 2
2 FoundationsofQuantumMechanicsandStatisticalMechanics 4
2.1 QuantumMechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 StatisticalMechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 ElementsofClassicalStatisticalMechanics . . . . . . . . . . . . . 6
2.2.2 ElementsofQuantumStatisticalMechanics . . . . . . . . . . . . 8
2.2.3 CSM,QSM,andMicrostateDispensability . . . . . . . . . . . . . 11
3 DensityMatrixRealism 11
3.1 Example: W-BohmianMechanics. . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 OtherExamples: EverettianandGRWTheories . . . . . . . . . . . . . . 13
3.3 SomeIntepretationsofW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4 TheInitialProjectionHypothesis 15
4.1 StandardVersionsofthePastHypothesis . . . . . . . . . . . . . . . . . . 15
4.2 IntroducingtheInitialProjectionHypothesis . . . . . . . . . . . . . . . . 16
4.3 ConnectionstotheWeylCurvatureHypothesis . . . . . . . . . . . . . . 18
5 Unification 19
5.1 DynamicUnityoftheUniverseandtheSubsystems . . . . . . . . . . . . 19
5.2 UnificationofStatisticalMechanicsandQuantumMechanics . . . . . . 21
6 TheNomologicalThesis 22
6.1 TheHumeanConflictwithQuantumEntanglement . . . . . . . . . . . . 23
6.2 ANewVersionofQuantumHumeanism . . . . . . . . . . . . . . . . . . 24
6.3 OtherVersionsofQuantumHumeanism . . . . . . . . . . . . . . . . . . . 24
7 Conclusion 27
1 Introduction
In recent debates about the metaphysics of the wave function, it is standard to
assume that the universal wave function represents something physical. It may be
interpreted as a field on the configuration space, a multi-field on physical space,
something like a physical law, or an entity of a completely novel kind. Let us call
thisviewWaveFunctionRealism.1
1SeeAlbert(1996),Loewer(1996),WallaceandTimpson(2010),Ney(2012),North(2013),Maudlin
(2013),GoldsteinandZanghì(2013),Miller(2014),Esfeld(2014),BhogalandPerry(2015),Esfeldand
2
However, we may reject the assumption that there is a universal wave function
that represents something physical. Indeed, it has been rejected by many people,
notablybyquantumBayesians,andvariousanti-realistsandinstrumentalists. Asa
scientificrealist,Idonotfindtheirargumentsconvincing. Inpreviouspapers,Ihave
assumed and defended Wave Function Realism. However, in this paper I want to
ff
argueforadi erentperspective,forreasonsrelatedtotheoriginoftime-asymmetry
inaquantumuniverse.
Tobesure,realismabouttheuniversalwavefunctionisquitenaturalgivenstan-
dard quantum mechanics and various realist quantum theories such as Bohmian
mechanics, GRW spontaneous collapse theories, and Everettian quantum mechan-
ics. Inthosetheories,theuniversalwavefunctionisindispensabletothekinematics
andthedynamicsofthequantumsystem. However,asIwouldliketoemphasizein
thispaper,ourworldisnotjustquantum-mechanical. Wealsoliveinaworldwitha
strongarrowoftime(entropygradient). Therearethermodynamicphenomenathat
we hope to explain with quantum mechanics and quantum statistical mechanics.
A central theme of this paper is to suggest that quantum statistical mechanics is
highly relevant for assessing the fundamentality and reality of the universal wave
function.
Wewilltakeacloselookattheconnectionsbetweenthefoundationsofquantum
statistical mechanics and various solutions to the quantum measurement problem.
Whenwedo,werealizethatwedonotneedtopostulateauniversalwavefunction.
Weonlyneedcertain“coarse-grained”informationaboutthequantummacrostate,
which can be represented by either a class of universal wave functions or a density
matrix. A natural question is: can we describe the universe with a fundamental
densitymatrixinsteadofawavefunction?
The first step of of this paper is to argue that we can. I call this view Density
MatrixRealism,thethesisthattheuniversalquantumstateisgivenbyafundamental
densitymatrixthatrepresentssomethingphysical. DensityMatrixRealismmightbe
unfamiliartosomepeople,asweusuallyusemixedstatestorepresentmacrostates
underepistemicuncertainties: wedonotknowtheprecisemicrostate. Butitseems
to be an implicit attitude among some working physicists and has been previously
discussedinfoundationsofquantummechanics.
The second step is to point out that Density Matrix Realism allows us to unify
quantum ontology with time-asymmetry in a new way. In classical and quantum
statisticalmechanics,thermodynamictime-asymmetryarisesfromaspecialbound-
aryconditionthatisnowcalledthePastHypothesis.2 Isuggestthattheinformation
ffi
in the Past Hypothesis is su cient to determine a unique and simple fundamental
density matrix. This can be done by postulating what I call the Initial Projection
Hypothesis: thequantumstateoftheuniverseatt isgivenbytheuniqueprojection
0
Deckert(2017),Chen(2017a,b,ms). NoticethatthisisnothowAlbert,Loewer,andNeyusetheterm.
Forthem,tobeawavefunctionrealististobearealistaboutthewavefunctionandafundamental
high-dimensionalspace—the“configurationspace.” Forthepurposeofthispaper,letususeWave
Function Realism to designate just the commitment that the wave function represents something
physical.
2SeeAlbert(2000).
3
onthespeciallow-dimensionalHilbertspace.
The third step is to show that, because of the simplicity and the uniqueness of
the initial quantum state (now given by a fundamental density matrix), we have
a strong case for the Nomological Thesis: the initial quantum state of the world is
exactlyspecifiedbyalawofnature.
As we shall see, this package of views has interesting implications for the dy-
namicunityoftheuniverseandthesubsystems,reductionofstatisticalmechanical
probabilities to quantum mechanics, and Humean supervenience in a quantum
world.
Here is the roadmap of the paper. First, in §2, I review the foundations of
quantum mechanics and quantum statistical mechanics. In §3, I introduce the
framework of Density Matrix Realism and illustrate it with a concrete example. In
§4,IformulatetheInitialProjectionHypothesisintheframeworkofDensityMatrix
Realism. In§5,Idiscusstheirimplicationsfordynamicalandtheoreticalunification.
In§6,IsuggestthattheyprovideastrongcasefortheNomologicalThesisandanew
solutiontotheconflictbetweenquantumentanglementandHumeansupervenience.
2 Foundations of Quantum Mechanics and Statistical Mechanics
Inthissection,wefirstreviewthefoundationsofquantummechanicsandstatistical
mechanics. As we shall see in the next section, they suggest an alternative to Wave
FunctionRealism.
2.1 Quantum Mechanics
Standard quantum mechanics is often presented with a set of axioms and rules
about measurement. Firstly, there is a quantum state of the system, represented by
a wave function ψ. For a N-particle quantum system in R3, the wave function is
a (square-integrable) function from the configuration space R3N to the spin space
Ck. Secondly, the wave function evolves in time according to the the Schrödinger
equation:
∂ψ
ih̵ =Hψ (1)
∂
t
Thirdly, the Schrödinger evolution of the wave function is supplemented with col-
lapserules. Thewavefunctiontypicallyevolvesintosuperpositionsofmacrostates,
suchasthecatbeingaliveandthecatbeingdead. Thiscanberepresentedbywave
functions on the configuration space with disjoint macroscopic supports X and Y.
During measurements, which are not precisely defined processes in the standard
theory, the wave function undergoes collapses. Moreover, the probability that it
collapsesintoanyparticularmacrostateX isgivenbytheBornrule:
P(X)=∫ ∣ψ(x)∣2dx (2)
X
4
As such, quantum mechanics is not a candidate for a fundamental physical
theory. It has two dynamical laws: the deterministic Schrödinger equation and
the stochastic collapse rule. What are the conditions for applying the former, and
what are the conditions for applying the latter? Measurements and observations
areextremelyvagueconcepts. Takeaconcreteexperimentalapparatusforexample.
When should we treat it as part of the quantum system that evolves linearly and
when should we treat it as an “observer,” i.e. something that stands outside the
quantum system and collapses the wave function? That is, in short, the quantum
measurementproblem.3
Various solutions have been proposed regarding the measurement problem.
Bohmianmechanics(BM)solvesitbyaddingparticlestotheontologyandanaddi-
tional guidance equation for the particles’ motion. Ghirardi-Rimini-Weber (GRW)
theoriespostulateaspontaneouscollapsemechanism. Everettianquantummechan-
ics(EQM)simplyremovesthecollapserulesfromstandardquantummechanicsand
suggestthattherearemanyemergentworlds,correspondingtoemergentbranches
of the wave function, which are all real. My aim here is not to adjudicate among
ffi
these theories. Su ce it to say that they are all quantum theories that remove the
centralityofobservationsandobservers.
Tosimplifythediscussions,IwilluseBMasakeyexample.4 InBM,inaddition
to the wave function that evolves linearly according to the Schrödinger equation,
, ,...,
there are particles with precise locations, Q Q Q , which follow the guidance
1 2 N
equation:
dQ h̵ ψ∗∇ψ
i = Im i (3)
dt m ψ∗ψ
i
Moreover, the initial particle distribution is given by the Quantum Equilibrium
Hypothesis:
ρ (x)=∣ψ(x,t )∣2 (4)
t0 0
Bytheequivariancetheorem,ifthisconditionholdsattheinitialtime,thenitholdsat
all time. Consequently, BM agrees with standard quantum mechanics with respect
to the Born rule predictions (which are all there is to the observable predictions of
quantummechanics).
ψ
In BM, the wave function is central to the quantum system. It not only has
its own dynamics described by (1) but also guides particle motion via (3). Its
connection to the empirical predictions of quantum mechanics is manifested in (2)
and(4). ForauniversewithN elementaryparticles,letuscallthewavefunctionof
the universe the universal wave function and denote it as Ψ(q ,q ,...q ). Therefore,
1 2 N
Ψ
at least prima facie, the universal wave function seems central to the description
ofthekinematicsandthedynamicsoftheuniverseasawhole.
3SeeBell(1990),Myrvold(2017)forintroductionstothequantummeasurementproblem.
4SeeDürretal.(1992)forarigorouspresentationofBManditsstatisticalanalysis.
5
2.2 Statistical Mechanics
Letusnowconsidermacroscopicsystemssuchasgasinabox. Thiscanbedescribed
by a system of N particles, with N > 1020. If the system is governed by classical
ffi
mechanics,althoughitisdi culttosolvethesystemofequationsexactly,wecanstill
useclassicalstatisticalmechanics(CSM)todescribeitsstatisticalbehaviors,suchas
approachtothermalequilibriumsuggestedbytheSecondLawofThermodynamics.
Similarly, if the system is governed by quantum mechanics, we can use quantum
statisticalmechanics(QSM)todescribeitsstatisticalbehaviors. Generallyspeaking,
ff
there are two di erent views on CSM: the individualistic view and the ensemblist
view. We will first illustrate the two views with CSM, which is more familiar and
willbehelpfulforunderstandingthetwoviewsinQSM.
2.2.1 ElementsofClassicalStatisticalMechanics
Let us review the basic elements of CSM on the individualistic view.5 For con-
creteness, let us consider a classical-mechanical system with N particles in a box
Λ=[0,L]3 ⊂R3 andaHamiltonianH.
1. Microstate: atanytimet,themicrostateofthesystemisgivenbyapointona
6N-dimensionalphasespace,
X =(q ,...,q ;p ,...,p )∈Γ ⊆R6N, (5)
1 N 1 n total
Γ
where isthetotalphasespaceofthesystem.
total
2. Dynamics: the time dependence of X = (q ,...,q ;p ,...,p ;t) is given by the
t 1 N 1 n
Hamiltonianequationsofmotion:
∂q ∂H ∂p ∂H
i = , i =− . (6)
∂ ∂ ∂ ∂
t p t q
i i
3. Energyshell: thephysicallyrelevantpartofthetotalphasespaceistheenergy
shellΓ⊆Γ definedas:
total
Γ={X ∈Γ ∶E≤H(x)≤E+δE}. (7)
total
Γ.
Weonlyconsidermicrostatesin
4. Measure: the measure µ is the standard Lebesgue measure of volume ∣⋅∣ on
R6N.
Γ
5. Macrostate: with a choice of macro-variables, the energy shell can be parti-
Γ
tionedintomacrostates ν:
Γ=⋃Γν. (8)
ν
5HereIfollowthediscussioninGoldsteinandTumulka(2011).
6
Γ
6. Uniquecorrespondence: everyphasepointX belongstooneandonlyone ν.
Γ
7. Thermalequilibrium: typically,thereisadominantmacrostate thathasthe
eq
µ
mostvolumewithrespectto :
µ(Γ )
eq ≈1. (9)
µ(Γ)
AsystemisinthermalequilibriumifitsphasepointX ∈Γ .
eq
8. Boltzmann Entropy: the Boltzmann entropy of a classical-mechanical system
inmicrostateX isgivenby:
S (X)=k log(µ(Γ(X))), (10)
B B
where Γ(X) denotes the macrostate containing X. The thermal equilibrium
statethushasthemaximumentropy.
9. Low-Entropy Initial Condition: when we consider the universe as a classical-
mechanical system, we postulate a special low-entropy boundary condition,
whichDavidAlbertcallsthePastHypothesis:
X ∈Γ ,µ(Γ )≪µ(Γ )≈µ(Γ), (11)
t0 PH PH eq
Γ
where is the Past Hypothesis macrostate with volume much smaller than
PH
that of the equilibrium macrostate. Hence, S (X ), the Boltzmann entropy
B t0
of the microstate at the boundary, is very small compared to that of thermal
equilibrium.
10. AcentraltaskofCSMistoestablishmathematicalresultsthatdemonstrate(or
suggest)that forµ−mostmicrostates satisfyingthe PastHypothesis, theywill
approachthermalequilibrium(inreasonabletime).
AboveistheindividualisticviewofCSMinanutshell. Incontrast,theensemblist
ff
view di ers in several ways. First, on the ensemblist view, instead of focusing on
themicrostateofanindividualsystem,thefocusisontheensembleofsystemsthat
have the same statistical state ρ.6 ρ is a distribution on the energy shell, and it also
ff
evolves according to the Hamiltonian dynamics. The crucial di erence lies in the
definition of thermal equilibrium. On the ensemblist view, a system is in thermal
equilibriumif:
ρ=ρ orρ=ρ , (12)
mc can
whereρ isthemicrocanonicalensembleandρ isthecanonicalensemble.7
mc can
6Someensemblistswouldfurtherinsistthatitmakesnosensetotalkaboutthethermodynamic
stateXofanindividualsystem.
7InsteadofusingtheBoltzmannentropy,someensemblistsusetheGibbsentropy:
SG(ρ)=−kB∫ ρlog(ρ)dx.
Γ
7
2.2.2 ElementsofQuantumStatisticalMechanics
ff
The foundations of QSM have important similarities to and di erences with the
foundationsofCSM.Forconcreteness,letusconsideraquantum-mechanicalsystem
withN fermionsinaboxΛ=[0,L]3 ⊂R3 andaHamiltonianHˆ.8
1. Microstate: atanytimet,themicrostateofthesystemisgivenbyanormalized
(andanti-symmetrized)wavefunction:
ψ(q ,...,q )∈H =L2(R3N,Ck), ∥ψ∥ =1, (13)
1 N total L2
whereH =L2(R3N,Ck)isthetotalHilbertspaceofthesystem.
total
2. Dynamics: the time dependence of ψ(q ,...,q ;t) is given by the Schrödinger
1 N
equation:
∂ψ
ih̵ =Hψ. (14)
∂
t
3. Energy shell: the physically relevant part of the total Hilbert space is the
subspace(“theenergyshell”):
H ⊆Htotal ,H =span{φα ∶Eα ∈[E,E+δE]}, (15)
Thisisthesubspace(ofthetotalHilbertspace)spannedbyenergyeigenstates
φα whoseeigenvaluesEα’sbelongtothe[E,E+δE]range. LetD=dimH ,the
numberofenergylevelsbetweenEandE+δE.
ψ H
Weonlyconsiderwavefunctions ’sin .
µ
4. Measure: the measure is given by the standard Lebesgue measure on the
unitsphereintheenergysubspaceS(H ).9
5. Macrostate: withachoiceofmacro-variables(suitably“rounded”àlaVonNeu-
H
mann(1955)),theenergyshell canbeorthogonallydecomposedintomacro-
spaces:
H =⊕νHν , ∑dimHν =D (16)
ν
H
Each νcorrespondsmoreorlesstosmallrangesofvaluesofmacro-variables
thatwehavechoseninadvance.
6. Non-unique correspondence: typically, a wave function is in a superposition
ofmacrostatesandisnotentirelyinanyoneofthemacrospaces. However,we
Since SG(ρt) is stationary under the Hamiltonian dynamics, it is not the right kind of object for
understanding the approach to thermal equilibrium in the sense of the Second Law, as we would
liketohaveanobjectthatcanchange,and,inparticular,increasewithtime.
8HereIfollowthediscussionsinGoldsteinetal.(2010a)andGoldsteinandTumulka(2011).
9In cases where the Hilbert space is infinite-dimensional, we should use a Gaussian measure,
whichisnottranslation-invariant.
8
ψ
canmakesenseofsituationswhere is(intheHilbertspacenorm)veryclose
H
toamacrostate ν:
⟨ψ∣Pν∣ψ⟩≈1, (17)
where Pν is the projection operator into Hν. This means that almost all of ∣ψ⟩
H
liesin ν.
H
7. Thermal equilibrium: typically, there is a dominant macro-space that has
eq
adimensionthatalmostequaltoD:
H
dim
eq ≈1. (18)
H
dim
ψ ψ
Asystemwithwavefunction isinequilibriumifthewavefunction isvery
closetoH inthesenseof(17): ⟨ψ∣P ∣ψ⟩≈1.
eq eq
Simple Example. Consider a gas consisting of n = 1023 atoms in a box Λ ⊆ R3.
Thesystemisgovernedbyquantummechanics. Weorthogonallydecompose
the Hilbert space H into 51 macro-spaces: H ⊕H ⊕H ⊕...⊕H , where
0 2 4 100
H
ν isthesubspacecorrespondingtothemacrostatethatthenumberofatoms
in the left half of the box is between (ν − 1)% and (ν + 1)% of n. In this
H
example, has the overwhelming majority of dimensions and is thus the
50
H
equilibriummacro-space. Asystemwhosewavefunctionisverycloseto
50
isinequilibrium.
8. BoltzmannEntropy: theBoltzmannentropyofaquantum-mechanicalsystem
ψ ν
withwavefunction thatisveryclosetoamacrostate isgivenby:
SB(ψ)=kBlog(dimHν), (19)
H ψ
where ν denotesthesubspacecontainingalmostallof inthesenseof(17).
Thethermalequilibriumstatethushasthemaximumentropy:
S (eq)=k log(dimH )≈k log(D)=S (mc), (20)
B B eq B B
where eq denotes the equilibrium macrostate and mc the micro-canonical en-
semble.
9. Low-EntropyInitialCondition: whenweconsidertheuniverseasaquantum-
mechanical system, we postulate a special low-entropy boundary condition
on the universal wave function—the quantum-mechanical version of the Past
Hypothesis:
Ψ(t )∈H ,dimH ≪dimH ≈dimH (21)
0 PH PH eq
H
where isthePastHypothesismacro-spacewithdimensionmuchsmaller
PH
than that of the equilibrium macro-space.10 Hence, the initial state has very
lowentropyinthesenseof(19).
10ThePastHypothesismacro-spaceisthusfinite-dimensional. SowecanusetheLebesguemeasure
ontheunitsphereasthetypicalitymeasurefor#10.
9
10. AcentraltaskofQSMistoestablishmathematicalresultsthatdemonstrate(or
suggest) that for µ−most (maybe even all) wave functions satisfying the Past
Hypothesis,theywillapproachthermalequilibrium(inreasonabletime).
AboveistheindividualisticviewofQSMinanutshell. Incontrast,theensemblist
ff
view of QSM di ers in several ways. First, on the ensemblist view, instead of
focusingonthewavefunctionofanindividualsystem,thefocusisonanensemble
of systems that have the same statistical state Wˆ , a density matrix.11 As a statistical
density matrix, Wˆ can be defined from a uniform distribution on the unit sphere in
theHilbertspace:
Wˆ =∫ µ(dψ)∣ψ⟩⟨ψ∣. (22)
S(H)
ItevolvesaccordingtothevonNeumannequation:
ih̵dWˆ (t) =[Hˆ,Wˆ ]. (23)
dt
ff
The crucial di erence between the individualistic and the ensemblist views of
QSMlies,again,inthedefinitionofthermalequilibrium. Ontheensemblistview,a
systemisinthermalequilibriumif:
W =ρ orW =ρ , (24)
mc can
ρ ρ
where isthemicrocanonicalensembleand isthecanonicalensemble.
mc can
ψ
FortheQSMindividualist,ifthemicrostate ofasystemisclosetosomemacro-
H H
space ν in the sense of (17), we can say that the macrostate of the system is ν.
However, we can also represent the macrostate by a density matrix Wˆν generated
fromtheunitsphereinHν withauniformdistributionµ(dψ).
Wˆ ν =∫ µ(dψ)∣ψ⟩⟨ψ∣. (25)
S(Hν)
In(25),thereisaclearsensethatWˆ νisdefinedwithachoiceofmeasureandfromthe
H ff
wave functions on the unit sphere of the Hilbert space ν. But di erent measures
cangiverisetothesamedensitymatrix. Whatisessentialandintrinsictoadensity
matrixisitsgeometricalmeaningintheHilbertspace—aprojectionoperator. When
Hν is finite-dimensional, we can simply think of Wˆ ν as the normalized identity
H
operatoron ν anddefineitwithoutusingameasure:
Wˆ ν = IνH , (26)
dim ν
H
whereIν istheidentityoperatoron ν.
11SimilarlytothesituationinCSM,someensemblistswouldfurtherinsistthatitmakesnosense
totalkaboutthethermodynamicstateofanindividualsystem.
10