A global picture of quantum A global picture of quantum de Sitter spacede Sitter space

Donald MarolfDonald Marolf

May 24, 2007May 24, 2007

Based on work w/Steve Based on work w/Steve Giddings.Giddings.

Perturbative gravity & dSPerturbative gravity & dSResidual gauge symmetry when Residual gauge symmetry when bothboth

i. spacetime has symmetries i. spacetime has symmetries andand

ii. Cauchy surfaces are compact.ii. Cauchy surfaces are compact.

An An opportunityopportunity to probe locality in perturbative quantum gravity!! to probe locality in perturbative quantum gravity!!

E.g., de Sitter!E.g., de Sitter!

Watch out for Watch out for i) strong gravityi) strong gravity ii) subtle effects on long timescale (e.g., from Hawking radiation) ii) subtle effects on long timescale (e.g., from Hawking radiation)

but keep guesses at non-pert physics on back burner.but keep guesses at non-pert physics on back burner.

FrameworkFrameworkMatter QFT on dS w/ perturbative gravityMatter QFT on dS w/ perturbative gravity

Compare with perturbative QED on dS: Compare with perturbative QED on dS:

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QQ11= E= Eii dS dSi i = -Q= -Q22Total charge vanishes!Total charge vanishes!

Q|Q|mattermatter

00thth order: Consider any order: Consider any Fock state Fock state

11stst order: Gauss Law order: Gauss Law includes source includes source iiEEii = =..

Restriction on matter states:Restriction on matter states:

Matter QFT on dS w/ perturbative gravityMatter QFT on dS w/ perturbative gravity

(Moncrief, Fischer, Marsden, …Higuchi, Losic & Unruh) (Moncrief, Fischer, Marsden, …Higuchi, Losic & Unruh) Similar “linearization stability constraints” in perturbative gravity!Similar “linearization stability constraints” in perturbative gravity!

Hamiltonian constraints of GRHamiltonian constraints of GR: for any vector field : for any vector field , ,

Expand in powers of Expand in powers of llpp w/ canoncial normalization of graviton. w/ canoncial normalization of graviton.

Matter QFT & free gravitons + grav. interactionsMatter QFT & free gravitons + grav. interactions

0 = H[0 = H[]] = (q= (qdSdS1/21/2) ) {{llpp

-1-1[[((LLqqdSdS))abababab - ( - (LLdSdS))ababhhabab]] + + llpp

00(T(Tmatter + free gravitonsmatter + free gravitons))ababnnaabb +… +…}}

A constraint for KVFs A constraint for KVFs ! !

0 0

Residual gauge symmetry not broken by background.Residual gauge symmetry not broken by background.

FrameworkFramework

Quantum TheoryQuantum Theory

If consistent, resolves Goheer-Kleban-SusskindIf consistent, resolves Goheer-Kleban-Susskindtension between dS-invariance and finite number of states.tension between dS-invariance and finite number of states.

Technical Problem: In usual Hilbert space, |Technical Problem: In usual Hilbert space, |> must be the vacuum!> must be the vacuum!(But familiar issue from quantum cosmology….)(But familiar issue from quantum cosmology….)

Requires: QRequires: Qfreefree[[] |] |matter + free gravitonsmatter + free gravitons> = 0> = 0

Each |Each |> is dS-invariant!> is dS-invariant!

Solution introduced by Higuchi: Solution introduced by Higuchi: Renormalize the inner product!Renormalize the inner product!

Consider |Consider |> = dg U(g) |> = dg U(g) |>>g dSg dS

For such states, define new “group averaged” product:For such states, define new “group averaged” product:

< < 11||>>physphys := dg < := dg <|U(g) ||U(g) |>>

g dSg dS

(Naïve norm (Naïve norm “divided by V“divided by VdSdS” )” )

For compact groups, projects onto trivial rep.

(also Landsmann, D.M.)

{{Fock state (seed)Fock state (seed)

seedsseeds

(Not normalizeable, (Not normalizeable, but like <p| )but like <p| )

dS-invariant!dS-invariant!

Vaccum is special case; norm finte for n Vaccum is special case; norm finte for n >> 2 free gravitons in 3+1 2 free gravitons in 3+1

ResultsResults• dS:dS: A laboratory to study locality (& more?) in pert. grav. A laboratory to study locality (& more?) in pert. grav.

• Constraints Constraints eacheach state dS invariant state dS invariant

• Finite # of pert states for Finite # of pert states for eternaleternal dS (pert. theory valid everywhere) dS (pert. theory valid everywhere) Limit ``energy’’ of seed states to avoid strong gravity. Limit ``energy’’ of seed states to avoid strong gravity. (Any Frame)(Any Frame)

Compact & finite F Compact & finite F finite N. S = ln N ~ ( finite N. S = ln N ~ (l/ll/lpp)) (d-2)(d-1)/d (d-2)(d-1)/d < S< SdSdS

• Simple relational observables (operators): Simple relational observables (operators): OO = A(x) = A(x)[[OO,Q,Q]=0; Finite matrix elements, but (fluctuations)]=0; Finite matrix elements, but (fluctuations)22 ~ V ~ VdSdS. . (Boltzmann Brains)(Boltzmann Brains)

• Solution: cut off intermediate states!Solution: cut off intermediate states!

OO = = P O P P O P for for PP a finite-dim projection; e.g. F < F a finite-dim projection; e.g. F < F11..

Restricts Restricts OO to region near neck. Heavy observer/observable OK for to region near neck. Heavy observer/observable OK for t ~ St ~ SdSdS..

• Proto-local physicsProto-local physics over volumes ~ exp(S over volumes ~ exp(SdSdS) )

Other Other globalglobal projections assoc. w/ non-repeating events should work too. projections assoc. w/ non-repeating events should work too.

• Picture looks rather different from “hot box…”Picture looks rather different from “hot box…”

neckneckConsider F = q TConsider F = q Tabab n naannbb

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Finite # of states?Finite # of states?(Eternal dS)(Eternal dS)

Conjecture for non-eternal dS: eConjecture for non-eternal dS: eSSdSdS states enough for “locally dS” observer. states enough for “locally dS” observer.

• acceleration.acceleration.

• too much too much collapse! collapse!

• As. dS in past and future if small “Energy.”As. dS in past and future if small “Energy.”

At 0At 0thth order in order in llpp,,

consider F = q T consider F = q Tabab n naannbb

neckneck

Safe for F < FSafe for F < F00 ~ ~ l l d-3d-3//llppd-4 d-4 ~ M~ MBHBH ; ;

Other frames?Other frames?

||> and U(g) |> and U(g) |> group average to same |> group average to same |>; >; no new physical states!no new physical states!

Finite N, dS-Finite N, dS-invariantinvariant

S = ln N ~ (S = ln N ~ (l/ll/lpp)) (d-2)(d-1)/d(d-2)(d-1)/d < S< SdSdS

Observables?Observables?

Try Try O O = -g A(x) = -g A(x)x dSx dS

Finite (Finite (HH00) matrix elements <) matrix elements <11||OO||22> >

for appropriate A(x), |for appropriate A(x), |ii>.>.

OO is is protoproto-local for appropriate A(x).-local for appropriate A(x).

Also dS-invariant to preserve Also dS-invariant to preserve HHphysphys..

But fluctuations diverge: <But fluctuations diverge: <11||OO11OO22||22> ~ V> ~ VdS dS (vacuum noise, BBs)(vacuum noise, BBs)

Note: <Note: <11||OO11OO22||22> = > = ii < <11||OO11|i><i||i><i|OO22||22> .> .

Control Intermediate States?Control Intermediate States?

OO = = P O P P O P for for PP a finite-dim projection; e.g. F < F a finite-dim projection; e.g. F < F11..

dS UV/IR: Use “Energy” cut-off to control spacetime volumedS UV/IR: Use “Energy” cut-off to control spacetime volume

OO is insensitive to details of long time dynamics, as desired. is insensitive to details of long time dynamics, as desired.

Tune FTune F11 to control “noise;” safe for F to control “noise;” safe for F11 ~ F ~ F00..

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Example: Schwarzschild dSExample: Schwarzschild dS

Schwarzschild dS has two Schwarzschild dS has two black holes/stars/particles.black holes/stars/particles.

Q[Q[] = M – M = 0] = M – M = 0

Solution must be `balanced’! Solution must be `balanced’!

No “one dS Black Hole” vacuum solution.No “one dS Black Hole” vacuum solution.

II. Why a II. Why a newnew picture? picture?The static Hamiltonian is unphysical.The static Hamiltonian is unphysical.

A “boost” sym of dSA “boost” sym of dS

Q[Q[] = (q] = (qdSdS1/21/2) (T) (Tmatter + free gravitonsmatter + free gravitons))ababnnaabb

= H= Hss

RR - H - HssLL

But Q[But Q[] |] |> = 0> = 0

||> = dE f(E) |E> = dE f(E) |ELL=E>|E=E>|ERR=E>=E>

Perfect correlations…Perfect correlations…

RR = Tr = TrLL is diagonal in E is diagonal in ERR..

[[ H HssRR, , R R ]] = 0 = 0

HHssRR generates trivial time evolution: generates trivial time evolution:

Static Static RegionRegion

A “boost” sym of dSA “boost” sym of dS[[ H Hss

RR, , R R ]] = 0 = 0HHss

RR generates trivial time evolution: generates trivial time evolution:

Eigenstates of HEigenstates of HssRR also unphysical also unphysical

|E|ERR= 0> ~ |0>= 0> ~ |0>RindlerRindler

UV divergent: UV divergent: no role in low energy no role in low energy

effective theoryeffective theory

II. Why a II. Why a newnew picture? picture?The static Hamiltonian is unphysical.The static Hamiltonian is unphysical.

Static Static RegionRegion

Observables?Observables?

Try Try O O = -g A(x) = -g A(x)x dSx dS

Finite (Finite (HH00) matrix elements <) matrix elements <11||OO||22> >

for appropriate A(x), |for appropriate A(x), |ii>.>.

ProtoProto-local for appropriate A(x)-local for appropriate A(x)

Expand in modes. Expand in modes.

Each mode falls off like Each mode falls off like ee-(d-1)t/2-(d-1)t/2ll. .

Each mode gives finite integral for A ~ Each mode gives finite integral for A ~ 33, , 44, etc., etc.

For |For |ii> of finite F, finite # of terms contribute.> of finite F, finite # of terms contribute.

Free fields:Free fields:

Conformal case:Conformal case: maps to finite maps to finite t in ESUt in ESU

F maps to energyF maps to energy

Large conformal weightLarge conformal weight

& finite F & finite F finite integrals! finite integrals!

Also dS-invariant to preserve Also dS-invariant to preserve HHphysphys..

But fluctuations diverge!But fluctuations diverge!Recall: |0> is an attractor….Recall: |0> is an attractor….

<<11||OO11OO22||22> = dx> = dx11 dx dx22 < <11|A|A11(x(x11)A)A22(x(x11)|)|22> >

~ dx~ dx11 dx dx22 < <|A|A11(x(x11)A)A22(x(x11)|)|>> ~ const(V~ const(VdSdS))

Note: <Note: <11||OO11OO22||22> = > = ii < <11||OO11|i><i||i><i|OO22||22> .> .

Control Intermediate States?Control Intermediate States?

OO = = P O P P O P for for PP a finite-dim projection; e.g. F < F a finite-dim projection; e.g. F < F11..

dS UV/IR: Use “Energy” cut-off to control spacetime volumedS UV/IR: Use “Energy” cut-off to control spacetime volume

OO is insensitive to details of long time dynamics, as desired. is insensitive to details of long time dynamics, as desired.

Tune FTune F11 to control “noise;” safe for F to control “noise;” safe for F11 ~ F ~ F00..

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Boltzmann Brains? Boltzmann Brains?

dS thermal, vacuum quantum.dS thermal, vacuum quantum.In large volume, even rare In large volume, even rare

fluctuations occur….fluctuations occur….

What do typical observers in dS see?What do typical observers in dS see?

Detectors or observers (or their brains)Detectors or observers (or their brains)arise as vacuum/thermal fluctuations. arise as vacuum/thermal fluctuations.

Note:Note: Infinity of ``Boltzmann Brains’’ Infinity of ``Boltzmann Brains’’ outnumber `normal’ observers!!! outnumber `normal’ observers!!!

(Albrecht, Page, etc.)(Albrecht, Page, etc.)

I am a brain!

Our story: Our story: • Subtract to control matrix elements <Subtract to control matrix elements <OO>>

• Still dominate fluctuations <Still dominate fluctuations <OOOO>>for local questions integrated over all dS.for local questions integrated over all dS.

• Ask different questions (non-local, finite V): Ask different questions (non-local, finite V): OO = = P O PP O P~~

Fits with Hartle & SrednickiFits with Hartle & Srednicki

V

Poincare Recurrences, t ~ ePoincare Recurrences, t ~ eSSdSdS??

• Finite N, HFinite N, Hss: Hot Static Box: Hot Static Box

• Global dynamics of scale factorGlobal dynamics of scale factor

• Unique neck defines zero of time, never returns.Unique neck defines zero of time, never returns. States relax to vacuum; States relax to vacuum;

Relational DynamicsRelational Dynamics

neckneck

Local relational recurrences?Local relational recurrences?

No recurrences relative to neck.No recurrences relative to neck.

(L. Dyson, Lindesay, Kleban, Susskind) (L. Dyson, Lindesay, Kleban, Susskind)

E = 0E = 0

““time-dependent background.”time-dependent background.”

No issue: local observers destroyed or decay No issue: local observers destroyed or decay after t ~ eafter t ~ eSSdSdS

SummarySummary• dS symmetries are gauge dS symmetries are gauge constraints! constraints!

• HHss, No “Hot Static Box” picture., No “Hot Static Box” picture.

• Future and Past As. dS Future and Past As. dS Finite NFinite N (F < F (F < F00), ), eacheach | |> dS-invariant> dS-invariant

• Relational dynamicsRelational dynamics

• ““neck” gives useful t=0neck” gives useful t=0states relax to vacuum, no recurrencesstates relax to vacuum, no recurrences..

• O O samples finite region samples finite region RR ( (relationalrelational, e.g., set by F, e.g., set by F11).).

For moderate For moderate RR, Boltzmann brains give , Boltzmann brains give smallsmall noise term. noise term.Recover approx. local physics in Recover approx. local physics in RR..

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Vol(Vol(RR) < ) < ll (d-1)(d-1) exp(S exp(SdSdS), details to come!!), details to come!!

What limits locality in dS?What limits locality in dS?

Possible limits fromPossible limits from

• Vacuum noise (Boltzmann Brains) V ~ exp(SVacuum noise (Boltzmann Brains) V ~ exp(SdSdS))

• Quantum Diffusion t ~ [Quantum Diffusion t ~ [ll S SdSdS]]1/21/2

• Marker Decay/Destruction t ~ exp(SMarker Decay/Destruction t ~ exp(SdSdS))

• Regulate & avoid eternal inflation, orRegulate & avoid eternal inflation, or Short Time Nonlocality t ~ Short Time Nonlocality t ~ ll S SdS dS (Arkani-Hamed)(Arkani-Hamed)

• Grav. Back-reaction t ~ Grav. Back-reaction t ~ ll S SdS dS (Giddings)(Giddings)

• ll ln ln l l ??

Confusion:Confusion:

Durability:Durability:

Need “reference marker” to select event.Need “reference marker” to select event.

Other:Other: