A global picture of quantum de Sitter space

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A global picture of quantum de Sitter space

• Donald Marolf
• May 24, 2007

Based on work w/Steve Giddings.
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Perturbative gravity dS
Residual gauge symmetry when both
i. spacetime has symmetries and ii. Cauchy
surfaces are compact.
E.g., de Sitter!
? An opportunity to probe locality in
perturbative quantum gravity!!
Watch out for i) strong gravity
ii) subtle effects on long timescale (e.g., from
Hawking radiation) but keep guesses at non-pert
physics on back burner.
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Framework

• Matter QFT on dS w/ perturbative gravity

Compare with perturbative QED on dS
0th order Consider any Fock state 1st
order Gauss Law includes source iEi r.
Q1 Ei dSi
-Q2
Total charge vanishes!
Restriction on matter states
Qymattergt 0
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Framework
Matter QFT on dS w/ perturbative gravity
Similar linearization stability constraints in
perturbative gravity!

• (Moncrief, Fischer, Marsden, Higuchi, Losic
  Unruh)

Expand in powers of lp w/ canoncial normalization
of graviton.
Matter QFT free gravitons grav.
interactions
Hamiltonian constraints of GR for any vector
field x,
0
(qdS1/2) lp-1(LxqdS)abpab
- (LxpdS)abhab
0
0 Hx
lp0(Tmatter free gravitons)abnaxb
S
A constraint for KVFs x !
Residual gauge symmetry not broken by background.
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Quantum Theory
Requires Qfreex ymatter free gravitonsgt 0
Each ygt is dS-invariant!
If consistent, resolves Goheer-Kleban-Susskindten
sion between dS-invariance and finite number of
states.
Technical Problem In usual Hilbert space, ygt
must be the vacuum!
(But familiar issue from quantum cosmology.)

• Solution introduced by Higuchi Renormalize the
  inner product!

(also Landsmann, D.M.)
dS-invariant!
Consider Ygt dg U(g) ygt
(Not normalizeable, but like ltp )

Fock state (seed)
For such states, define new group averaged
product
(Naïve norm divided by VdS )
lt Y1Y2gtphys dg lty1U(g) y2gt
For compact groups, projects onto trivial rep.
seeds
Vaccum is special case norm finte for n gt 2 free
gravitons in 31
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Results

• dS A laboratory to study locality ( more?) in
  pert. grav.
• Constraints ? each state dS invariant
• Finite of pert states for eternal dS (pert.
  theory valid everywhere) Limit energy of
  seed states to avoid strong gravity. (Any
  Frame)Compact finite F ? finite N. S ln
  N (l/lp) (d-2)(d-1)/d lt SdS
• Simple relational observables (operators) O
  A(x)O,Qx0 Finite matrix elements, but
  (fluctuations)2 VdS. (Boltzmann Brains)
• Solution cut off intermediate states!O P O P
  for P a finite-dim projection e.g. F lt
  F1.Restricts O to region near neck. Heavy
  observer/observable OK for Dt SdS.
• Proto-local physics over volumes exp(SdS)
  Other global projections assoc. w/ non-repeating
  events should work too.
• Picture looks rather different from hot box

Consider F q Tab nanb
neck


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Finite of states?
(Eternal dS)

• L ? acceleration.
• too much r ? collapse!
• As. dS in past and future if small Energy.

At 0th order in lp, consider F q Tab nanb
neck
lt SdS
S ln N (l/lp) (d-2)(d-1)/d
Safe for F lt F0 l d-3/lpd-4 MBH Other
frames? ygt and U(g) ygt group average to same
Ygt no new physical states!
Finite N, dS-invariant
Conjecture for non-eternal dS eSdS states enough
for locally dS observer.
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Observables?
Also dS-invariant to preserve Hphys.
Finite (H0) matrix elements lty1Oy2gt for
appropriate A(x), yigt.
Try O -g A(x)
But fluctuations diverge lty1O1O2y2gt VdS
(vacuum noise, BBs)
Note lty1O1O2y2gt Si lty1O1igtltiO2y2gt .
Control Intermediate States? O P O P for P
a finite-dim projection e.g. F lt F1. dS UV/IR
Use Energy cut-off to control spacetime
volume O is insensitive to details of long time
dynamics, as desired. Tune F1 to control noise
safe for F1 F0.


O is proto-local for appropriate A(x).
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Example Schwarzschild dS
Schwarzschild dS has two black holes/stars/particl
es.
Qx M M 0
x
Solution must be balanced!
No one dS Black Hole vacuum solution.
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II. Why a new picture?The static Hamiltonian is
unphysical.
Qx (qdS1/2) (Tmatter free
gravitons)abnaxb
S
HsR - HsL
But Qx ygt 0
S
ygt dE f(E) ELEgtEREgt
Static Region
Perfect correlations
rR TrL r is diagonal in ER.
HsR generates trivial time evolution
HsR, rR 0
A boost sym of dS
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II. Why a new picture?The static Hamiltonian is
unphysical.
Eigenstates of HsR also unphysical
ER 0gt 0gtRindler
UV divergent no role in low energy effective
theory
S
Static Region
HsR generates trivial time evolution
HsR, rR 0
A boost sym of dS
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Observables?
Also dS-invariant to preserve Hphys.
Finite (H0) matrix elements lty1Oy2gt for
appropriate A(x), yigt.
Try O -g A(x)
Proto-local for appropriate A(x)
Free fields
Expand in modes. Each mode falls off like
e-(d-1)t/2l. Each mode gives finite integral for
A f3, f4, etc. For yigt of finite F, finite
of terms contribute.
Conformal case
maps to finite Dt in ESU F maps to energy Large
conformal weight finite F ? finite integrals!
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But fluctuations diverge!

• Recall 0gt is an attractor.

lty1O1O2y2gt dx1 dx2 lty1A1(x1)A2(x1)y2gt
dx1 dx2
lt0A1(x1)A2(x1)0gt
const(VdS)
Note lty1O1O2y2gt Si lty1O1igtltiO2y2gt .
Control Intermediate States? O P O P for P
a finite-dim projection e.g. F lt F1. dS UV/IR
Use Energy cut-off to control spacetime
volume O is insensitive to details of long time
dynamics, as desired. Tune F1 to control noise
safe for F1 F0.

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Boltzmann Brains?
(Albrecht, Page, etc.)
What do typical observers in dS see?
I am a brain!
dS thermal, vacuum quantum. In large volume, even
rare fluctuations occur.
Detectors or observers (or their brains)arise as
vacuum/thermal fluctuations.
Note Infinity of Boltzmann Brains outnumber
normal observers!!!
V
Our story

• Subtract to control matrix elements ltOgt
• Still dominate fluctuations ltOOgtfor local
  questions integrated over all dS.
• ? Ask different questions (non-local, finite V)
  O P O P

Fits with Hartle Srednicki

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Poincare Recurrences, t eSdS?
(L. Dyson, Lindesay, Kleban, Susskind)

• Finite N, Hs Hot Static Box
• Global dynamics of scale factor
• Unique neck defines zero of time, never
  returns. States relax to vacuum

Relational Dynamics
neck
E 0
time-dependent background.
No recurrences relative to neck.
Local relational recurrences?
No issue local observers destroyed or decay
after t eSdS
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Summary

• dS symmetries are gauge ? constraints!
• Hs, No Hot Static Box picture.
• Future and Past As. dS ?Finite N (F lt F0), each
  ygt dS-invariant
• Relational dynamics
• neck gives useful t0states relax to vacuum,
  no recurrences.
• O samples finite region R (relational, e.g.,
  set by F1).
• For moderate R, Boltzmann brains give small
  noise term.Recover approx. local physics in R.

Vol(R) lt l (d-1) exp(SdS), details to come!!
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What limits locality in dS?
Need reference marker to select event.
Possible limits from

• Vacuum noise (Boltzmann Brains) V exp(SdS)
• Quantum Diffusion t l SdS1/2
• Marker Decay/Destruction t exp(SdS)
• Regulate avoid eternal inflation, or Short
  Time Nonlocality t l SdS (Arkani-Hamed)
• Grav. Back-reaction t l SdS (Giddings)
• l ln l ?

Confusion
Durability
Other