Effective Field Theory for Quantum Gravity from ... - Relativity Group

OutlineIntroduction: motivation, goal of talkSymmetry Trading and Symmetry Doubling: symmetry trading,Shape Dynamics and General Relativity, symmetry doubling andDoubly General RelativityDoubly General Relativity: effective field theory, revised gravity theoryspaceConsequences: new construction principle ⇒ possible observationalconsequences, bulk/bulk duality, renormalizationSummaryT. Koslowski (PI) Quantum Shape Dynamics April 10th, 2012 2 / 23

MotivationGeneral Relativity is not renormalizable:perturbation expansion of Einstein-Hilbert action, unitarity problem withhigher derivative gravity⇒ problem is finding a different universality classImportance of Symmetries:RG flow stays on (evolving) symm. surfaceencoded as BRST-invariance of path integral⇒ Slavnov-Taylor identities for eff. actionuniversality classes are often explained by symmetriesPossibility: Hidden SymmetryFP may not be detected without adapting search to symmetry⇒ important heuristic for finding new universality classT. Koslowski (PI) Quantum Shape Dynamics April 10th, 2012 3 / 23

Main Message“Doubly General Relativity” in one line:There is a hidden BRST-invariance in gravity due to Shape Dynamics.T. Koslowski (PI) Quantum Shape Dynamics April 10th, 2012 4 / 23

BACKGROUND:symmetry trading, Shape Dynamics and BRST-formalismT. Koslowski (PI) Quantum Shape Dynamics April 10th, 2012 5 / 23

Symmetry Trading MechanismT. Koslowski (PI) Quantum Shape Dynamics April 10th, 2012 6 / 23

Construction of Shape DynamicsConstruction on ADM-phase space:T. Koslowski (PI) Quantum Shape Dynamics April 10th, 2012 7 / 23

Relation of Dirac ObservablesIntersecting constraint surfaces in local coordinatesχ α = M β αq β ≈ 0 and σ β = N β α(p α − p α o ) with M β α and N β α intertibleequiv. doubly Abelian set: ˜χ α = q α ≈ 0 and ˜σ β = p β − p β o⇒ vector fields {˜χ α , .} and {˜σ β , .} are Frobenius integrable⇒ for every function f red on intersection Γ red there exists a local functionf on U ⊂ Γ s.t. {˜χ α , f } = 0 = {˜σ β , f } (doubly strong observables)⇒ for these preferred representatives the identification on full Γ is trivialf A ≡ f B .Abelianization generally spoils locality:Nonabelian case: there is still a dictionary for observables through the twophase space reductions of the linking theory.T. Koslowski (PI) Quantum Shape Dynamics April 10th, 2012 8 / 23

BRST-FormalismAbelian constraints χ α• BRST-generator Ω = η α χ α satisfies {Ω, Ω} = 0 (nontrivial: gh(Ω) = 1)⇒ defines graded differential s : f → {Ω, f }, i.e. s 2 = 0• Observables as cohomology of s at gh(.) = 0:gauge invariance: {Ω, f (p, q)} = 0 ⇒ f (strong observable)equivalence: ˜f = f + {Ω, Ψ} = f + σ α χ α + O(η) (weak observable)for gauge fixing Ψ = σ α P α + O(η) with gh(Ψ) = −1⇒ always strong equations on extended phase space• gauge fixed Hamiltonian H BRS = H o + {Ω, Ψ} when {H o , Ω} = 0.Nonabelian constraints ˜χ α = M β αχ βapply canonical transform exp({η α c α β Pβ , .}) to Abelian case˜Ω = η α M β αχ β + O(η 2 ) defines ˜s, cohomology same as of s at gh(.) = 0.T. Koslowski (PI) Quantum Shape Dynamics April 10th, 2012 9 / 23

DOUBLY GENERAL RELATIVITYsymmetry doubling, Extended Shape Dynamics, Doubly General RelativityT. Koslowski (PI) Quantum Shape Dynamics April 10th, 2012 10 / 23

From Symmetry Trading to Symmetry DoublingSymmetry Trading requiresTwo first class surfaces (original and equivalent gauge symmetry)that gauge fix one anotherBRST-gauge-fixingΩ is nilpotent because orig. system is first classΨ can be chosen nilpotent because equiv. system is first classif H o (on shell) Poisson commutes with Ω and Ψ then gauge fixedH BRS = H o + {Ω, Ψ}is annihilated by both s Ω and s ΨSymmetry Doubling (NOT anti-BRST!)Canonical action S = ∫ dt(p i ˙q i + P α ˙η α − H BRST ) is invariant under twoBRST-transformations (up to a boundary term).T. Koslowski (PI) Quantum Shape Dynamics April 10th, 2012 11 / 23

Construction of Doubly General Relativity (I)Extending Shape Dynamicsfixed CMC condition Q(x) = π(x) + λ √ |g|conformal spatial harmonic gaugeF k (x) = (g ab δ k c + 1 3 g ak δ b c )e c α(∇ a − ˆ∇ a )e α bFirst class system: {Q(x), Q(y)} = 0 = {F i (x), F j (y)}as well as {Q(x), F i (y)} = F i (y)δ(x, y)Interpretation as “local conformal system”Q generates spatial dilatations and Poisson brackets resemble C(3) ateach pointGauge fixing ADMgauge fixing operator is elliptic and invertible in a region Rout side R: meager set with finite dimensional kernelT. Koslowski (PI) Quantum Shape Dynamics April 10th, 2012 12 / 23

Construction of Doubly General Relativity (II)BRST-chargesΩ ADM = ∫ ()d 3 x ηS + η a g ac π;d cd + ηb η,b a P a + 1 2 ηa η ,a P + ηη ,c P b g bcΩ ESD = ∫ ()d 3 x P √ π g+ P a F a + 1 P√2 gP a η aGauge-fixed gravity actionS gf = ∫ dt (symp.pot. + {Ω ADM , Ω ESD }) is invariant under usualADM-BRST transformations anda hidden BRST-invariance of S gfunders ESD g ab = P √ gg ab s ESD π ab = “long ′′s ESD η = − √ 1g(π + 1 2 P cη c ) s ESD P = 0s ESD η a = −F a + P2 √ g ηa s ESD P a = P2 √ g P adue to extended Shape Dynamics.T. Koslowski (PI) Quantum Shape Dynamics April 10th, 2012 13 / 23

Construction of Doubly General Relativity (III)InterpretationThe Hamiltonian of Doubly General Relativity isπH DGR = S( √ + λ) + H(F a ) + O(η)|g|The ghost-free part is neither the “frozen Hamiltonain” H = 0nor the CMC-Hamiltonian H = S(N CMC [g, π]),but a generator of dynamics for λ + √ π > 0. |g|T. Koslowski (PI) Quantum Shape Dynamics April 10th, 2012 14 / 23

EFFECTIVE FIELD THEORYstandard reasoning, symmetry doubling, definition of a gravity theoryT. Koslowski (PI) Quantum Shape Dynamics April 10th, 2012 15 / 23

Renormalization Group and Effective Field TheoryRenormalization group: Γ k interpolates between local bare action S Λ (at UVcut-off Λ) and effective action Γ (in IR)Γ k : functional on theory space (field content, symmetries, approx. locality)Asymptotic safety progr.: Find UV-fixed point with few relevant directions⇒ predictive theoryApproximations: Fixed points ↔ broken BRST-symmetries,RG-relevance ↔ dimensional analysis (weak IR coupling)Universality: IR attractive critical manifold (i.e. S Λ unimportant for IR)Two Uses for RG:• Fundamental theory: hard to verify w/o heuristic• Theory Space constr. ppl. for local effective actions(field content, BRST-symmetries, dimensionality, locality)T. Koslowski (PI) Quantum Shape Dynamics April 10th, 2012 16 / 23

Theory Space of GravitySlavnov-Taylor Identities:• assuming an invariant path integral measure ⇒ BRST-variations yield:〈s ADM φ A 〉 δ LΓδφ A= 0 and 〈s ESD φ A 〉 δ LΓδφ A= 0BRST-variations are nonlinear ⇒ difficult Legendre transform• Nonlinearity obstructs use of two Zinn-Justin equations(Γ, Γ) 1 | ˆφ2 =0 = 0 = (Γ, Γ) 2| ˆφ1 =0• in semiclassical approximation (〈F [φ]〉 sc = F [φ sc ] + O()):s ADM Γ = O() and s ESD Γ = O()Refined definition of a gravity theory (from semiclassical reasoning):Gravity = local action for g ab , π ab , η, P, η a , P a at gh. number 0that is invariant under ADM- and ESD- BRST-transformations s ADM , s ESDT. Koslowski (PI) Quantum Shape Dynamics April 10th, 2012 17 / 23

FURTHER DIRECTIONSdualities, experimental implications, renormalization(first baby steps)T. Koslowski (PI) Quantum Shape Dynamics April 10th, 2012 18 / 23

Classical Theory and ObservationsConstruction of classical gravity:effective field theory:revised theory spacedimensional analysis⇒ construction ppl. for classical Doubly General RelativityPossible Observable ConsequencesEffective field theory for GR: all higher derivative curvature invariantsare allowed (just suppressed at low energies)these are generally not compatible with Extended Shape Dynamics⇒ DGR can be experimentally distinguished from usual GR(but only beyond Einstein-Hilbert)This theory space has not been explored!T. Koslowski (PI) Quantum Shape Dynamics April 10th, 2012 19 / 23

Bulk/bulk dualityUsual Shape Dynamics: bulk/boundary dualityHamiltonian at large volume H SD = 〈π〉 2 − 12Λgauge group: diffeomorphisms, vol. pres. conf. trfs. π − 〈π〉 √ |g| ≈ 0⇒ dynamical large CMC-volume CFT-correspondence of gravityDuobly General Relativity:True evolution generated by H BRST ⇒ “bulk”compatible with two equivalent symmetry principles describe gravityRemark:The symmetry doubling mechanism is very generic:possible explanation for dualities like (A)dS/CFTT. Koslowski (PI) Quantum Shape Dynamics April 10th, 2012 20 / 23

RenormalizationImmediate Question:Are there implications of symmetry doubling for counter terms?Possibly yes, but seems unfeasible in metric formulation.Current Directions:Find a formulation of DGR where enough transofrmations are linearlyrealized:This makes prediction about counter terms very feasibleFind a gauge fixing with improved power counting:Idea: part. fkt. Z is independent of gaug.fix. (quant. mast equ.)view action as ESD action and gauge fix with Ω ′ − Ω ADM (in BV)⇒ gives arbitrary gauge fixing of ESDT. Koslowski (PI) Quantum Shape Dynamics April 10th, 2012 21 / 23

Conclusions1 Symmetry trading is generic and gives equivalent gauge theories2 Symmetry trading implies symmetry doubling in BRST formalism3 Equivalence of Shape Dynamics and GR ⇒ Doubly General Relativity4 DGR implies a new theory space for gravity. To explore:◮ are there semiclassical predictions (beyond E-H-action)?◮ universality classes on this revised theory space (FRGE methods)?◮ new view on dualities?5 Just started homogeneous quantum cosmology“Doubly General Relativity” in one line:There is a hidden BRST-invariance in gravity due to Shape Dynamics.T. Koslowski (PI) Quantum Shape Dynamics April 10th, 2012 22 / 23

THANK YOUandmany thanks to my collaborators:H. Gomes, S. Gryb, F. Mercati and J. Barbour.T. Koslowski (PI) Quantum Shape Dynamics April 10th, 2012 23 / 23