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Diana Vaman Physics Dept, U Emergent gravity Diana Vaman Physics Dept, U. Virginia September 24, U Virginia, Charlottesvile, VA What is gravity? Newton, 1686: Universal gravitational attraction law M M F = G 1 2 2 R12 Einstein, General Relativity, 1915: Gravity is a manifestation of the geometry/curvature of spacetime. Rμν − gμνR = 8π G Tμν − Λgμν Is gravity one of the fundamental forces, besides electromagnetism, weak and strong nuclear forces? Fundamental forces are described by fields. For gravitation, the field is the metric of spacetime g μ ν . Distances between spacetime points x μ , x μ + d x μ are measured by 2 μ ν ds = dx dx gμν(x) which is invariant under a change of coordinates α β μ μ ∂ξ ∂ξ x ⟶ ξ (x) g (x) = g˜ (ξ) . μν αβ ∂xμ ∂xν gμν(x) ⟶ g˜αβ(ξ) diffeomorphisms/general coordinate/Einstein transformations This is the symmetry of General Relativity. Weak field limit of Einstein’s equation: gμν(x) = ημν + hμν(x) metric fluctuation Linearized Einstein equation, without sources: ρ ρ □ hμν + ∂μ∂νh − ∂μ∂ hρν − ∂ν∂ hμρ = 0 is invariant under Einstein transformations δhμν = ∂μϵν + ∂νϵμ 1 Fix a gauge: h¯ = h − η h, ∂ρh¯ = 0 μν μν 2 μν ρν Gravitational waves □ h¯ μν = 0 Weak field limit, with sources Massive objects curve spacetime. Rμν − gμνR = 8π G Tμν − Λgμν In a curved spacetime fee particles/planets move on geodesics. Gravity! Weak field limit, with sources Gravity couples to the energy-momentum tensor 1 S = S [ϕ] + S [h ] − T (x) hμν(x) + . matter gravity μν ∫ 2 μν 00 00 ⃗ With T = M1δ(r⃗) , T = M2δ(r⃗ − R ) M M V(R) = − G 1 2 R Graviton (massless spin 2 particle) exchange/propagator Electromagnetism analogy: Currents couple to the electromagnetic potential/field μ μ j Aμ , j = (charge density, current density) ∫x The field sourced by a particular current Massless pole in q-space ν Aμ(x) = Propagatorμν(x, y)j (y) ∫y An electric charge would interact with the electromagnetic field of the other charge μ ν J (x)Propagatorμν(x, y)j (y) Coulomb potential ∫x,y Faraday, 1850: “On the possible relation of gravity to electricity” “Gravity. Surely this force must be capable of an experimental relation to electricity, magnetism, and other forces, so as to bind it up with them in reciprocal action and equivalent effect.” “I have been arranging certain experiments in reference to the notion that gravity itself may be practicaly or directly related by experiment to the other powers of matter, and this morning proceeded to make them.” “Here end my trials for the present. The results are negative. They do not shake my strong feeling of the existence of a relation between gravity and electricity, though they give no proof that such a relation exists.” Weinberg-Witten: A NO-GO Theorem Can massless spin two states (gravitons) arise as bound states in a quantum field theory of particles with spin less or equal to one? If yes, then gravity is emergent. Weinberg-Witten Theorem: There can be no massless states with helicity greater than one in theories with a conserved, Lorentz covariant energy-momentum tensor , with nonzero values of Tμν the energy and/or momentum. Corolary: the Standard Model (or any relativistic theory) cannot accommodate a composite massless graviton. Emergent gravity fom the vanishing of Tμν Consider the folowing possibility to evade the WW theorem: a field theory which is diffeomorphism invariant, in particular a theory for which Tμν = 0 Carone, Erlich, DV: consider N+D scalars coupled to an auxiliary metric gμν 1 N D−1 S = dDx |g| gμν ∂ ϕa ∂ ϕa + ∂ XI∂ XJη − V(ϕa) ∫ ∑ μ ν ∑ μ ν IJ 2 ( a=1 I=0 ) The metric is non-dynamical. The energy momentum tensor is zero. δS = 0 Tμν = 0 δgμν Solve for the auxiliary metric fom the vanishing of Tμν . Substitute into the action. Obtain: D/2−1 D − 2 N D−1 S = dDx det ∂ ϕa∂ ϕa + ∂ XI∂ XJη ∫ ( ) (∑ μ ν ∑ μ ν IJ) 2V(ϕ) a=1 I,J=0 Reminiscent of the action for a fee particle · · S = − m dτ |Xμ(τ)Xν(τ)η | ∫ μν or, the Nambu-Goto string action 1 2 μ ν S = − d ξ det (∂αX (ξ)∂βX (ξ) ημν 2πα′! ∫ ( ) Emergent gravity model: non-metric action, D/2−1 D − 2 N D−1 S = dDx det ∂ ϕa∂ ϕa + ∂ XI∂ XJη ∫ ( ) (∑ μ ν ∑ μ ν IJ) 2V(ϕ) a=1 I,J=0 which is invariant under reparametrizations, just like the fee particle/string actions. Matter fields Clock -and-ruler fields A large energy scale I μ I Gauge fix: X ∝ x δμ 1 μν|ρσ S = Sfree[ϕ] + tμνΠ tρσ + . ∫x V0 Energy-momentum tensor of fee scalar fields P a P c P1 a P c P a P c 1 3 3 1 3 Emergent gravitons = + P2 a P4 c P a P c a c 2 4 P P4 1 2 1 N 1 V0 V 0 V0 P 1 a P c P a c 3 1 P3 + + + ... P a 2 1 1 1 P4 c NN P a 1 1 1 1 P c 2 NNN 4 V0 V0 V0 V0 V0 V0 V0 (a) P 1 a P c P a P c P a P c 3 1 3 1 3 = + P a P c P a a 2 4 2 P4 c P P c 2 4 (b) Graviton exchange in a 2-2 scalar scattering process P1 a P c P1 a P3 c 3 h Pole = . P a c a c 2 P4 P2 P4 = t (q) t (q) μν q2 ρσ (c) Planck mass/Newton’s constant is determined in terms of the UV regulator. What about emergent graviton self-interactions? Cubic graviton interaction obtained fom Einstein’s action: 1 d4x gR 16πG ∫ Carone, Claringbold, DV: This is accounted for by expanding the scalar action to 6th order in fields and computing a 6-scalar amplitude. q = -k-p l+p p l-k k l q = -k-p l+p q = -k-p p p k k l (a) (b) Emergent graviton cubic interactions match those of GR. Beyond flat space-time linearized gravity gμν = ημν + hμν ⟨hμν⟩ = 0 Energy momentum Want to have a geometry which is non-flat? tensor in flat spacetime Add a conserved source to the non-metric action D αβ μν δS = d xJ (x)Pαβ|μνt [ϕ] ∫ AMinkowski metric bi-tensor ⟨hμν⟩J = ... (a) Cubic graviton vertex oo Pole Σ ... n=0 n (b) Past attempts at composite gravitons Start with some generic quartic interaction T.. T.. and constrain such that a massless spin 2 pole is found. Moving beyond linearized gravity was a problem. : Here we start with a theory that has the same symmetries as General Relativity. It was predetermined that the composite graviton, if found at the linearized level, wil have the same interactions as in GR. Sakharov: Induced gravity 1967: Sakharov considered the possibility that gravity is like elasticity of spacetime. He considered matter fields in a curved spacetime. The quantum dynamics of matter fields generates an effective action for the background metric, which includes the GR Einstein action. h h + h h = Seffective[h] = SEinstein−Hilbert + cosmo const 2nd order h 2nd order h Bjorken: Emergent electromagnetism 1963: “A dynamical origin for the electromagnetic field” Electromagnetism is not a fundamental force: start with a Nambu Jona-Lasinio model with a 4-fermi interaction. Introduce an auxiliary 4-vector field A and do a Hubbard-Stratonovich transformation. Integrate out the fermions. Get an effective action for the auxiliary field S [A] = S + #A ⋅ A + #(A ⋅ A)2 effective Maxwell ∫ Then break Lorentz symmetry by giving a vev to the field A. The massless Nambu-Goldstone mode is be interpreted as the photon. Gravity fom entanglement in AdS/CFT AdS/CFT correspondence/holography 1998: Maldacena formulated a duality between a strongly coupled field theory (of spin less or equal to one) and a weakly coupled gravity/superstring theory in a curved space time with one extra infinite-extent direction, caled the radial or holographic direction. The flat-space field theory lives at the boundary of the gravity bulk space. Dictionary: AdS = Anti de Sitter space 2 2 2 2 2 2 2 −X−1 − X0 + X1 + X2 + X3 + X4 = − R CFT: conformal field theory SO(2,4) Escher Supergravity CFT 5-dim AdS 5th dimension 4-dim flat spacetime What is it used for? Compute correlation functions of operators in the boundary theory using the supergravity partition function. Entanglement Have the field theory be described by some density matrix/operator state ρ . Define a subregion A of the field theory space-time. Perform measurements of observables with support only in A. ⟨OA⟩ = Tr(ρOA) = TrA(ρAOA) ρA : reduced density matrix ρA = TrAc(ρ) Entanglement entropy: measure of the information loss regarding the degrees of feedom in A c SA = − Tr(ρA ln ρA) 1 Example: ρ = (| ↑ ↓ ⟩ − | ↓ ↑ ⟩)(⟨ ↑ ↓ | − ⟨ ↓ ↑ |) 2 1 ρ = (| ↑ ⟩⟨ ↑ | + | ↓ ⟩⟨ ↓ |) 1 2 Focus on the reduced density matrix and define the entanglement Hamiltonian: ρA = exp(−HA) Casini, Huerta, Myers Then, for a spherical region A in a CFT, Energy density HA = β(x)T00(x) ∫A Inverse entanglement temperature cπ Entanglement entropy SA = ∫A 3β(x) Proof of the CHM formula Klich,Wong,Pando-Zayas, DV Generator of s-evolution − + −1 ⟨ϕ(x) |ρA |ϕ(x) ⟩ = ZA exp − K(s) ( ∫s ) 2dim QFT, A=half-line : 1 0 1 1 K = ang mom = (x T00 − x T01) = x T00 β = 2πx ∫Σ ∫A cπ c L Just for curiosity’s sake, S = dx = ln A ∫ 6x 6 ϵ Entanglement entropy of excited states and first law for EE Consider a smal change ρ ⟶ ρ + δρ The entropy change is δSA = − Tr(δρA ln(ρA)) = β(x)δ⟨T00(x)⟩ ∫A For a smal region A δS = β0 δEA TdS = dE Klich, Wong, Pando-Zayas, DV Ryu-Takayanagi formula for holographic EE 2006: The entanglement entropy for a CFT state, in a subregion A, is computed in the holographic dual by the minimal area of a surface which is anchored at the boundary onto ∂A .
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