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A global picture of quantum de Sitter space

A global picture of quantum de Sitter space. Donald Marolf May 24, 2007. Based on work w/Steve Giddings. Perturbative gravity & dS. Residual gauge symmetry when both. i. spacetime has symmetries and ii. Cauchy surfaces are compact. E.g., de Sitter!.

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A global picture of quantum de Sitter space

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  1. A global picture of quantum de Sitter space Donald Marolf May 24, 2007 Based on work w/Steve Giddings.

  2. 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.

  3. + + + + - - - - 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: Q|ymatter> = 0

  4. Framework Matter QFT on dS w/ perturbative gravity (Moncrief, Fischer, Marsden, …Higuchi, Losic & Unruh) Similar “linearization stability constraints” in perturbative gravity! 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 = H[x] S + lp0(Tmatter + free gravitons)abnaxb +…} A constraint for KVFs x ! Residual gauge symmetry not broken by background.

  5. g dS g dS Quantum Theory Requires: Qfree[x] |ymatter + free gravitons> = 0 Each |y> is dS-invariant! Solution introduced by Higuchi: Renormalize the inner product! If consistent, resolves Goheer-Kleban-Susskindtension between dS-invariance and finite number of states. Technical Problem: In usual Hilbert space, |y> must be the vacuum! (But familiar issue from quantum cosmology….) (also Landsmann, D.M.) dS-invariant! Consider |Y> = dg U(g) |y> (Not normalizeable, but like <p| ) { Fock state (seed) For such states, define new “group averaged” product: (Naïve norm “divided by VdS” ) < Y1|Y2>phys := dg <y1|U(g) |y2> For compact groups, projects onto trivial rep. seeds Vaccum is special case; norm finte for n > 2 free gravitons in 3+1

  6. 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 < SdS • Simple relational observables (operators): O = A(x)[O,Qx]=0; 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 < 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 ~ ~

  7. 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 < SdS S = ln N ~ (l/lp)(d-2)(d-1)/d Safe for F < F0 ~ l d-3/lpd-4 ~ MBH ; Other frames? |y> and U(g) |y> group average to same |Y>; no new physical states! Finite N, dS-invariant Conjecture for non-eternal dS: eSdS states enough for “locally dS” observer.

  8. x dS Observables? Also dS-invariant to preserve Hphys. Finite (H0) matrix elements <y1|O|y2> for appropriate A(x), |yi>. Try O = -g A(x) But fluctuations diverge: <y1|O1O2|y2> ~ VdS (vacuum noise, BBs) Note: <y1|O1O2|y2> = Si <y1|O1|i><i|O2|y2> . Control Intermediate States? O = P O P for P a finite-dim projection; e.g. F < 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).

  9. Example: Schwarzschild dS Schwarzschild dS has two black holes/stars/particles. Q[x] = M – M = 0 x Solution must be `balanced’! No “one dS Black Hole” vacuum solution.

  10. II. Why a new picture?The static Hamiltonian is unphysical. Q[x] = (qdS1/2) (Tmatter + free gravitons)abnaxb S = HsR - HsL But Q[x] |y> = 0 S |y> = dE f(E) |EL=E>|ER=E> Static Region Perfect correlations… rR = TrLr is diagonal in ER. HsR generates trivial time evolution: [ HsR, rR ] = 0 A “boost” sym of dS

  11. II. Why a new picture?The static Hamiltonian is unphysical. Eigenstates of HsR also unphysical |ER= 0> ~ |0>Rindler 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

  12. x dS Observables? Also dS-invariant to preserve Hphys. Finite (H0) matrix elements <y1|O|y2> for appropriate A(x), |yi>. 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 |yi> 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!

  13. But fluctuations diverge! Recall: |0> is an attractor…. <y1|O1O2|y2> = dx1 dx2 <y1|A1(x1)A2(x1)|y2> ~ dx1 dx2 <0|A1(x1)A2(x1)|0> ~ const(VdS) Note: <y1|O1O2|y2> = Si <y1|O1|i><i|O2|y2> . Control Intermediate States? O = P O P for P a finite-dim projection; e.g. F < 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. ~

  14. 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 <O> • Still dominate fluctuations <OO>for local questions integrated over all dS. •  Ask different questions (non-local, finite V): O = P O P Fits with Hartle & Srednicki ~

  15. 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

  16. Summary • dS symmetries are gauge  constraints! • Hs, No “Hot Static Box” picture. • Future and Past As. dS Finite N (F < F0), each |y> dS-invariant • Relational dynamics • “neck” gives useful t=0states 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) < l(d-1) exp(SdS), details to come!!

  17. 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 SdS]1/2 • 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:

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