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Macroscopic Effects of the Quantum Trace Anomaly BlackBlack Holes,Holes, DarkDark EnergyEnergy && CondensateCondensate StarsStars MacroscopicMacroscopic EffectsEffects ofof thethe QuantumQuantum TraceTrace AnomalyAnomaly E. Mottola Los Alamos & CERN w. P. O. Mazur Proc. Natl. Acad. Sci., 101, 9545 (2004) w. R. Vaulin, Phys. Rev. D 74, 064004 (2006) w. P. Anderson & R. Vaulin, Phys. Rev. D 76, 024018 (2007) Review Article: w. I. Antoniadis & P. O. Mazur, N. Jour. Phys. 9, 11 (2007) w. M. Giannotti, Phys. Rev. D 79, 045014 (2009) w. P. Anderson & C. Molina-Paris, Phys. Rev. D 80, 084005 (2009) OutlineOutline • Classical Black Holes in General Relativity • Black Hole Paradoxes & Enigmas • Entropy & the Second Law of Thermodynamics • Negative Heat Capacity & the ‘Information Paradox’ • Vacuum Polarization & Backreaction Effects near the Horizon • Effective Theory of Low Energy Gravity: Role of the Trace Anomaly • Massless Scalar Poles in Flat Space Amplitudes, Chiral Anomaly • General Form of Effective Action of the Anomaly • New Massless Scalar Degrees of Freedom in Low Energy Gravity • The Quantum Endpoint of Collapse: Gravitational Condensate Stars • Effect of the Anomaly Scalars near the Classical Event Horizon • Non-Singular Λ Dark Energy Interior • Astrophysical Properties • Cosmological Dark Energy as Finite Size Effect SchwarzschildSchwarzschild MaximalMaximal AnalyticAnalytic ExtensionExtension Carter-Penrose Conformal Diagram Interior T=X World Line of typical infalling particle Exterior T= -X BlackBlack HolesHoles andand EntropyEntropy • A fixed classical solution usually has no entropy : (What is the “entropy” of the Coulomb potential Φ = Q/r ?) … But if matter/radiation disappears into the black hole, what happens to its entropy? (Only M, J, Q remain) • Is the horizon area A--which always increases when matter is thrown in--(Christodoulou, 1972) “entropy”? To get units of entropy need to divide Area, A by (length)2 … But there is no fixed length scale in classical Gen. Rel. 2 3 • ! = "G /c • Planck length Pl involves ! 2 • Bekenstein suggested SBH = γ kB A/LPl with γ ~ O(1) • Hawking (1974) argued black holes emit thermal radiation at ! 3 Apparently then the first law, !c T = dE = T dS fixes γ = 1/4 H 8 Gk M ! H BH " B Great ! But … ! AA FewFew ParadoxesParadoxes && EnigmasEnigmas …… • TH requires trans-Planckian frequencies at the horizon • ħ cancels out of dE = TH dSBH : Quantum or Classical? • In the classical limit TH → 0 (cold) but SBH → ∞ (? !) • SBH ∝ A is non-extensive and HUGE ! • E ∝ T-1 implies negative heat capacity: 〈(H-E)2〉 < 0 (? !) dE << 0 ⇒ highly unstable dT Equilibrium Thermodynamics cannot be applied (? !) • Information Paradox: Where does the information go? ! (Pure states → Mixed States? Unitarity ? !) • What is the statistical interpretation of SBH ? Boltzman asks: What about S = kB ln W ? ! AA SecondSecond ProblemProblem ofof SomeSome GravityGravity Requires both ħ and G different from zero QuantumQuantum EffectsEffects inin Gravity:Gravity: MacroscopicMacroscopic oror MicroscopicMicroscopic ?? • In both black holes and cosmological vacuum energy the UV quantum nature of matter apparently has macroscopic effects • We deal with UV divergences by renormalization, and now understand most (all?) QFT’s as Effective Theories • Λ is a free parameter of the Low Energy Effective Theory Just because something is infinite does not necessarily mean that it is zero – W. Pauli • The Standard Model has Spontaneous Symmetry Breaking When the ground state changes, so does its energy – so we should expect generically Λ ≠ 0 now. • More Symmetries at Very High Energy (UV) Cannot Help • This is a problem of fixing the infrared Quantum Vacuum State of Macroscopic Gravity ConstructingConstructing thethe QuantumQuantum EFTEFT ofof GravityGravity • Assume Equivalence Principle (Symmetry) • Metric Order Parameter Field gab • Only two strictly relevant operators (R, Λ) • Einstein’s General Relativity is an EFT • But EFT = General Relativity + Quantum Corrections • Semi-classical Einstein Eqs. (k << Mpl): Gab+ Λ gab = 8π G 〈 Tab〉 • But there is also a quantum (trace) anomaly: 2 a 2 〈 Ta 〉 = b C + b' (E - 3 R ) + b" R • New (marginally) relevant operator(s) needed EffectiveEffective FieldField TheoryTheory && QuantumQuantum AnomaliesAnomalies • Expansion of Effective Action in Local Invariants assumes Decoupling of Short Distance from Long Distance Modes • But Massless Modes do not decouple • Chiral, Conformal Symmetries are Anomalous • Special Non-local Additions to Local EFT • IR Sensitivity to UV degrees of freedom • Macroscopic Effects of Short Distance (high energy) physics • Conformal Symmetry & its Breaking controlled by the Conformal Trace Anomaly ChiralChiral AnomalyAnomaly inin QCDQCD • QCD with Nf massless quarks has an apparent U(Nf) ⊗ Uch(Nf) Symmetry µ • But Uch(1) Symmetryµ is Anomalous 2 • Effective Lagrangian in Chiral Limit has Nf - 1 2 (not Nf ) massless pions at low energies • Low Energy π0 → 2 γ dominated by the anomaly µν q µ5 2 ~ µν 2 π0 γ5 q ∂µ j = e Nc Fµν F /16π q • No Local QCD Action in chiral limit -- but Non-local IR Relevant Operator with massless pole coupling at low E • Measured decay rate verifies Nc = 3 in QCD Anomaly Matching IR ↔ UV MasslessMassless Poles:Poles: AA GenericGeneric FeatureFeature ofof AnomaliesAnomalies Eg. QED in an External EM Field Aµ e2 T µ = F µ" F µ 24! 2 µ" Triangle One-Loop Amplitude as in Chiral Case abcd 2 ab a b cd c d 2 Γ (p,q) = (k g - k k ) (g p•q - q p ) F1(k ) + (traceless terms) 2 In the limit of massless fermions, F1(k ) must have a massless pole: Jc p ab k = p + q T q Jd Corresponding Imag. Part Spectral Fn. has a δ fn This is a new massless scalar degree of freedom in the two-particle correlated spin-0 state <TJJ><TJJ> TriangleTriangle AmplitudeAmplitude inin QEDQED Determining the Amplitude by Symmetries and Its Finite Parts M. Giannotti & E. M. Phys. Rev. D 79, 045014 (2009) Γabcd: Mass Dimension 2 Use low energy symmetries: Jc 1. By Lorentz invariance, can be p expanded in a complete set of ab abcd k = p + q T 13 tensors ti (p,q), i =1, …13: q Jd abcd abcd ΓΓ (p,q)(p,q) == ΣΣi FFi tti (p,q)(p,q) abcd abcd 2. By current conservation: pcti (p,q) = 0 = qdti (p,q) All (but one) of these 13 tensors are dimension ≥ 4, so dim(Fi) ≤ -2 so 2 2 2 these scalar Fi(k ; p ,q ) are completely UV Convergent <TJJ><TJJ> TriangleTriangle AmplitudeAmplitude inin QEDQED Ward Identities ab ab 3. By stress tensor conservation Ward Identity: ∂b〈T 〉A = eF 〈Jb〉 ⇒ abcd ac c a bd ad d a bc kb" (p,q) = (g pb #$b p )% (q) + (g qb #$b q )% (p) 4. Bose exchange symmetry: Γabcd (p,q) = Γabdc (q,p) 2 2 2 Finally all 13 scalar functions Fi(k ; p , q ) can be found in terms of ! finite (IR) Feynman parameter integrals and the polarization, Πab(p) = (p2gab - papb) Π(p2) abcd 2 ab a b cd c d 2 2 2 Γ (p,q) = (k g - k k ) (g p•q - q p ) F1(k ; p , q ) + … (12 other terms, 11 traceless, and 1 with zero trace when m=0) e2 % 1 1#x (1# 4xy)( F (k 2; p2,q2) 1 3m2 dx dy Result: 1 = 2 2 & # $ $ ) 18" k ' 0 0 D * with D = (p2 x + q2 y)(1-x-y) + xy k2 + m2 UV Regularization Independent ! <TJJ><TJJ> TriangleTriangle AmplitudeAmplitude inin QEDQED Spectral Representation and Sum Rule Numerator & Denominator cancel here 2 Im F1(k = -s): Non-anomalous,vanishes when m=0 obeys a finite sum rule independent of p2, q2, m2 and as p2, q2 , m2 → 0+ Massless scalar intermediate two-particle state analogous to the pion in chiral limit of QCD MasslessMassless AnomalyAnomaly PolePole 2 2 2 For p = q = 0 (both photons on shell) and me = 0 the pole at k = 0 describes a massless e+ e - pair moving at v=c collinearly, with opposite helicities in a total spin-0 state ⇒ a massless scalar 0+ state which couples to gravity Effective vertex µν µ ν αβ hµν (g - ∂ ∂ )ψ´ ϕ F Fαβ Effective Action special case of the general form ScalarScalar PolePole inin GravitationalGravitational ScatteringScattering • In Einstein’s Theory only transverse, tracefree polarized waves (spin-2) are emitted/absorbed and propagate between sources T´µν and Tµν • The scalar parts give only non-progagating constrained interaction (like Coulomb field in E&M) • But for me = 0 there is a scalar pole in the 〈TJJ〉 triangle amplitude coupling to photons • This scalar wave propagates in gravitational scattering between sources T´µν and Tµν µ • Couples to trace T´ µ • 〈TTT〉 triangle of massless photons has similar pole • Two new scalar degrees of freedom in EFT TraceTrace AnomalyAnomaly inin CurvedCurved SpaceSpace a 2 2 2 〈 Ta 〉 = b C + b’ (E - 3 R ) + b’’R + cF (for me = 0 ) 〈Tab〉 is the Stress Tensor of Conformal Matter a 2 • 〈Ta 〉 is a sum of Geometric Invariants E=E4, C =F4 • One-loop amplitudes similar to previous examples • State-independent, independent of GN • No local effective action in terms of curvature tensor But there exists a non-local effective action which can be rendered local in terms of two new scalar degrees of freedom Macroscopic Quantum Modification of Classical Gravity EffectiveEffective ActionAction forfor thethe TraceTrace Anomaly:Anomaly: LocalLocal AuxiliaryAuxiliary FieldField FormForm • Two New Scalar Auxiliary Degrees of Freedom • Variation of the action with respect to ϕ, ψ -- the anomaly auxiliary fields -- leads to the equations of motion, with StressStress TensorTensor ofof thethe AnomalyAnomaly Variation of the Effective Action with respect to the metric gives stress-energy tensor • QuantumQuantum VacuumVacuum PolarizationPolarization inin TermsTerms ofof (Semi-)(Semi-) ClassicalClassical AuxiliaryAuxiliary potentialspotentials •• ϕϕ,, ψψ areare newnew scalarscalar degreesdegrees ofof freedomfreedom inin lowlow energyenergy gravitygravity whichwhich ddepenepen uponupon thethe globalglobal topologytopology ofof spacetimesspacetimes andand itsits boundaries,boundaries, horizonshorizons SchwarzschildSchwarzschild SpacetimeSpacetime ReconsideredReconsidered dr 2 ds 2 = !(1! 2 M )dt 2 + + r 2d"2 r 2 M (1! r ) ϕ = → ∞ solves homogeneous auxiliary field eq.
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