Renormalisation and Observables in Quantum Gravity Kevin Falls - PowerPoint PPT Presentation

Renormalisation and Observables in Quantum Gravity Kevin Falls (Heidelberg) T alk at ERG 2016, ICTP , T rieste.

Introduction In quantum gravity we would like to compute observables: � X O e iS h O i = � geometries This formal expression needs to be regulated in order to obtain a meaningful result. Then the parameters of the theory should depend on a cutoff scale such that observables are renormalisation group (RG) invariants: Λ d d Λ h O i = 0

Introduction T ypically beta functions are derived from the RG invariance of correlation functions: Λ d d Λ h φ a 1 φ a 2 ... φ a n i = 0 � These break diffeomorphism and re-parameterisation invariance and as consequence beta functions depend on the gauge fixing and the parameterisation of the fields. Instead I consider the RG invariance of diffeomorphism invariant observables directly: Λ d d Λ h O i = 0

Beta function for Newton’s constant One loop beta function for Newton’s constant (Weinberg ’79): β G = ( D − 2) G − b G 2 , � The beta function depends on the gauge and parameterisation (talk by A. Pereira). Furthermore different beta functions are found if the Einstein-Hilbert or Gibbons- Hawking-York boundary term are considered. This breaks the required balance between the two terms (Gastmans, R. Kallosh, and C. T ruffin 1978; Becker and Reuter 2012; Jacobson and Satz 2014). ✓Z ◆ Z 1 d d x √− gR + 2 d D − 1 y √ γ K S = � 16 π G Σ These problems are acute when we consider asymptotic safety close to two spacetime dimensions i.e. simplest approximation that the continuum limit of Gravity can be studied.

Functional measure Here I consider Einstein theory within a semi-classical regime Z √ g ( R − 2¯ 1 � S Λ ≈ S EH = − λ ) + ... 16 π G with the ellipsis denoting required boundary terms. The functional measure should be the one obtained by canonical quantisation giving the functional integral: Z � d M ( φ ) e − S [ φ ] Z = What is the field? φ A = g µ ν , φ A = g µ ν , φ A = √ gg µ ν etc . � g µ ρ ( e φ ) ρ g µ ν = ¯ g µ ν + φ µ ν , g µ ν = ¯ ν Choice should not affect the physics.

Functional measure The measure must be re-parameterisation invariant in order to manifestly preserve the invariance of the functional integral. � d φ a Y p d M ( φ ) = V − 1 | det C ab ( φ ) | di ff (2 π ) 1 / 2 � a Involves a metric on the ‘space of geometries’ which provides the invariant volume element. DeWitt notation: Fields are just coordinates in the space of geometries. e . g . φ a = g µ ν ( x ) δ l 2 = C ab δφ a δφ b Invariant line element:

Functional measure Correct form of the measure can be determined by BRST invariance (Fujikawa ’83) or canonical quantisation (Fradkin and Vilkovisky ’73, T oms ’87). Use Fujikawa’s measure which agrees with T oms. The metric is of the DeWitt type: µ 2 1 Z C ab δφ a δφ b = d D x √ g ( g µ ρ g νσ + g µ σ g νρ − g µ ν g ρσ ) δ g µ ν δ g ρσ 32 π G 2

Origin of gauge and parameterisation dependence Where does the gauge and parameterisation dependence come from? Standard approach: Faddeev-Popov functional integral with sources � d ϕ n Z Y e W [ J ] = p | sdet C nm ( ϕ ) | e − S [ ϕ ]+ J n ϕ n � (2 π ) 1 / 2 n Fields now include ghosts and the diffeomorphisms are factored out e . g . ϕ a = { g µ ν ( x ) , ¯ � η µ ( x ) , η µ ( x ) } Source term breaks re-parameterisation and diffeomorphism invariance. Effective action: Γ (1) Γ [ ¯ ϕ ] = − W [ J ] + ¯ ϕ n J n n [ ¯ ϕ ] = J n Gauge and parameterisation independence only realised by going on shell or computing an observable. Illustrative example: quantum corrections to the trajectory of a test particle (Dalvit and Mazzitelli ’97; KF 2015).

Origin of gauge and parameterisation dependence Where does the gauge and parameterisation dependence come from? Standard approach: Faddeev-Popov functional integral with sources � d ϕ n Z Y e W [ J ] = p | sdet C nm ( ϕ ) | e − S [ ϕ ]+ J n ϕ n � (2 π ) 1 / 2 n Fields now include ghosts while the diffeomorphisms are factored out e . g . ϕ a = { g µ ν ( x ) , ¯ � η µ ( x ) , η µ ( x ) } Source term breaks re-parameterisation and diffeomorphism invariance. Effective action: Γ (1) Γ [ ¯ ϕ ] = − W [ J ] + ¯ ϕ n J n n [ ¯ ϕ ] = J n Gauge and parameterisation independence only realised by going on shell or computing an observable. Illustrative example: quantum corrections to the trajectory of a test particle (Dalvit and Mazzitelli ’97; KF 2015).

Origin of gauge and parameterisation dependence One-loop beta functions from the Legendre effective action effective action Γ [ g µ ν ] = S [ g µ ν ] + 1 ⇣ C − 1 · S (2) ⌘ 2STr log � Contribution from the action and the measure S (2) nm = c no ( �r 2 δ o Hessian has the form: m ) ⌘ c no ∆ o m � E o m nm = δϕ o δϕ p δϕ o Considering a ultra-local re-parameterisation: S (2) ϕ n S (2) ϕ m S (1) ˜ ϕ m + op o δ ˜ δ ˜ δ ˜ ϕ n δ ˜ The coefficient of the Laplacian transforms as a metric of the space of geometries: c nm = δϕ r δϕ s ˜ ϕ n c rs δ ˜ δ ˜ ϕ m

Origin of gauge and parameterisation dependence T ypically only the super-trace 1 2STr log ( ∆ ) � is regulated. Which leaves behind a divergent part: Z ∼ STr log( C − 1 · c ) = δ (0) d D x str log( C − 1 · c ) � However for the correct BRST measure one has C nm = c nm One either uses the correct measure or one has additional UV divergencies which are ignored in the effective average action approach.

Origin of gauge and parameterisation dependence Standard effective average action scheme (Reuter ’96) � Γ [ g µ ν ] = S [ g µ ν ] + 1 C − 1 · c · ( ∆ + R k ( �r 2 )) � � 2STr log � Regardless of the measure we get the same flow equation: � k ∂ k Γ = 1 2STr [ k ∂ k R · ( ∆ + R k ) − 1 ] � The measure is not the origin of differences in beta functions for different parameterisations.

Origin of gauge and parameterisation dependence One-loop beta functions from the Legendre effective action effective action Γ [ g µ ν ] = S [ g µ ν ] + 1 ⇣ C − 1 · S (2) ⌘ � 2STr log Contribution from the action and the measure S (2) nm = c no ( �r 2 δ o m � E o m ) ⌘ c no ∆ o Hessian has the form: m nm = δϕ o δϕ p δϕ o Considering a different parameterisation: S (2) ˜ ϕ n S (2) ϕ m S (1) ϕ m + δ ˜ op δ ˜ δ ˜ ϕ n δ ˜ o The second term is proportional to the equation of motion and is the origin of parameterisation dependence. Gauge dependence has the same origin since only the on shell hessian is guaranteed to be gauge invariant (Benedetti 2011; KF 2015) .

Generating function for observables Generating function: d φ a ⇢ Z Y Z q e W ( λ J , κ J ) = � V − 1 d D x √ g | det C Λ ab ( φ ) | exp − ( λ J + δ Λ λ ) di ff , Λ (2 π ) 1 / 2 a � Z � d D x √ gR + δ Λ S [ φ ] +( κ J + δ Λ κ ) Observables obtained by taking derivatives with respect to couplings: � ⌧Z � ⌧Z � d D x √ g = − W (1 , 0) ( λ 0 , κ 0 ) , d D x √ gR = W (0 , 1) ( λ 0 , κ 0 ) etc. � 1 ∂ κ 0 = Derive RG flow from: 16 π G 0 ∂ Λ W ( λ J , κ J ) = 0 ¯ λ 0 λ 0 = 8 π G 0

One-loop flow equation φ a = ¯ φ a ( κ J , λ J ) + δφ a Perturbation theory around a saddle point: Saddle point geometry dependent on the couplings: 1 λ J R µ ν (¯ φ ) = g µ ν (¯ φ ) � D − 2 κ J Gauge and parameterisation independent φ ] + 1 1 log ∆ 1 /µ 2 + log Ω ( µ ) − W ( λ J , κ J ) = S Λ [¯ 2Tr 2 log( ∆ 2 /µ 2 ) − Tr 0 � Last term is the contribution of Killing vector diffeomorphisms which are left out of the vector trace (see e.g. Volkov and Wipf ’00). ✓ ◆ �r 2 � R ∆ 2 h µ ν = �r 2 h µ ν � 2 R µ ∆ 1 ✏ µ = ρ σ h ρσ . ✏ µ , ν D

One-loop flow equation Proper-time regulator implemented as a modification of the measure One-loop flow equation: Λ ∂ Λ S Λ = Tr 2 [ e − ∆ 2 / Λ 2 ] − 2Tr 1 [ e − ∆ 1 / Λ 2 ] Heat kernel expansion: d D x √ g + 1 ( N g − 18) N g Z Z d D x √ gR + ... Λ ∂ Λ S Λ = Λ D Λ D − 2 � D D 6 (4 π ) (4 π ) 2 2 N g ≡ D ( D − 3) Number polarisations of the graviton � 2 Beta function for the gravitational coupling: b = 1 16 π β G = ( D − 2) G − b G 2 , 2 (18 − N g ) � D 6 (4 π ) Agrees with the previous gauge independent result (KF 2015)

Amplitudes On spacetime manifolds with boundaries we can consider amplitudes: h φ 1 | φ 2 i = Z [ φ 1 , φ 2 ] � We need to provide diffeomorphism invariant boundary conditions. Generically there is a lack of boundary conditions in quantum gravity which are diffeomorphism invariant and lead to a well defined heat kernel. 1 On boundaries with extrinsic curvature: K ij = D − 1 K γ ij , ∂ i K = 0 , Moss and Silva ’97 have found suitable boundary conditions. h in = 0 = ✏ n ✏ i − K j δφ a = h µ ν ( x ) ˙ i ✏ j = 0 ˙ h nn + Kh nn − 2 K ij h ij = 0 ˙ h ij − K ij h nn = 0