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
Kevin Falls (Heidelberg) T alk at ERG 2016, ICTP , T rieste.
In quantum gravity we would like to compute observables:
meaningful result. Then the parameters of the theory should depend on a cutoff scale such that observables are renormalisation group (RG) invariants:
hOi = X
geometries
O eiS Λ d dΛhOi = 0
T ypically beta functions are derived from the RG invariance of correlation functions:
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
Λ d dΛhφa1φa2...φani = 0 Λ d dΛhOi = 0
One loop beta function for Newton’s constant (Weinberg ’79):
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).
dimensions i.e. simplest approximation that the continuum limit of Gravity can be studied.
βG = (D − 2)G − b G2 ,
S = 1 16πG ✓Z ddx√−gR + 2 Z
Σ
dD−1y√γK ◆
Here I consider Einstein theory within a semi-classical regime
The functional measure should be the one obtained by canonical quantisation giving the functional integral:
SΛ ≈ SEH = − 1 16πG Z √g(R − 2¯ λ) + ...
Z = Z dM(φ) e−S[φ]
φA = gµν , φA = gµν , φA = √ggµν etc.
gµν = ¯ gµν + φµν , gµν = ¯ gµρ(eφ)ρ
ν
The measure must be re-parameterisation invariant in order to manifestly preserve the invariance of the functional integral.
invariant volume element. Fields are just coordinates in the space of geometries. Invariant line element:
dM(φ) = V −1
diff
Y
a
dφa (2π)1/2 p | det Cab(φ)| δl2 = Cabδφaδφb
DeWitt notation:
e.g. φa = gµν(x)
Correct form of the measure can be determined by BRST invariance (Fujikawa ’83) or canonical quantisation (Fradkin and Vilkovisky ’73, T
Use Fujikawa’s measure which agrees with T
the DeWitt type:
Cabδφaδφb = µ2 32πG 1 2 Z dDx√g(gµρgνσ + gµσgνρ − gµνgρσ)δgµνδgρσ
Where does the gauge and parameterisation dependence come from? Standard approach: Faddeev-Popov functional integral with sources
Effective action: Gauge and parameterisation independence only realised by going on shell or computing an
Illustrative example: quantum corrections to the trajectory of a test particle (Dalvit and Mazzitelli ’97; KF 2015).
Γ(1)
n [ ¯
ϕ] = Jn
eW [J] = Z Y
n
dϕn (2π)1/2 p |sdet Cnm(ϕ)| e−S[ϕ]+Jnϕn
Γ[ ¯ ϕ] = −W[J] + ¯ ϕnJn
e.g. ϕa = {gµν(x), ¯ ηµ(x), ηµ(x)}
Where does the gauge and parameterisation dependence come from? Standard approach: Faddeev-Popov functional integral with sources
Effective action: Gauge and parameterisation independence only realised by going on shell or computing an
Illustrative example: quantum corrections to the trajectory of a test particle (Dalvit and Mazzitelli ’97; KF 2015).
Γ(1)
n [ ¯
ϕ] = Jn
eW [J] = Z Y
n
dϕn (2π)1/2 p |sdet Cnm(ϕ)| e−S[ϕ]+Jnϕn
Γ[ ¯ ϕ] = −W[J] + ¯ ϕnJn
e.g. ϕa = {gµν(x), ¯ ηµ(x), ηµ(x)}
One-loop beta functions from the Legendre effective action effective action
Hessian has the form: Considering a ultra-local re-parameterisation: The coefficient of the Laplacian transforms as a metric of the space of geometries:
Γ[gµν] = S[gµν] + 1 2STr log ⇣ C−1 · S(2)⌘
S(2)
nm = cno(r2δo m Eo m) ⌘ cno∆o m
˜ cnm = δϕr δ ˜ ϕn crs δϕs δ ˜ ϕm
˜ S(2)
nm = δϕo
δ ˜ ϕn S(2)
δϕp δ ˜ ϕm + δϕo δ ˜ ϕnδ ˜ ϕm S(1)
T ypically only the super-trace
One either uses the correct measure or one has additional UV divergencies which are ignored in the effective average action approach.
1 2STr log (∆)
∼ STr log(C−1 · c) = δ(0) Z dDx str log(C−1 · c)
Cnm = cnm
Standard effective average action scheme (Reuter ’96)
parameterisations.
Γ[gµν] = S[gµν] + 1 2STr log
2STr [k∂kR · (∆ + Rk)−1]
One-loop beta functions from the Legendre effective action effective action
Hessian has the form: Considering a different parameterisation: The second term is proportional to the equation of motion and is the origin
Gauge dependence has the same origin since only the on shell hessian is guaranteed to be gauge invariant (Benedetti 2011; KF 2015) .
Γ[gµν] = S[gµν] + 1 2STr log ⇣ C−1 · S(2)⌘
S(2)
nm = cno(r2δo m Eo m) ⌘ cno∆o m
˜ S(2)
nm = δϕo
δ ˜ ϕn S(2)
δϕp δ ˜ ϕm + δϕo δ ˜ ϕnδ ˜ ϕm S(1)
Generating function:
eW (λJ,κJ) = V −1
diff,Λ
Z Y
a
dφa (2π)1/2 q | det CΛ
ab(φ)| exp
⇢ −(λJ + δΛλ) Z dDx√g +(κJ + δΛκ) Z dDx√gR + δΛS[φ]
dDx√g
⌧Z dDx√gR
κ0 = 1 16πG0 λ0 = ¯ λ0 8πG0
∂ ∂ΛW(λJ, κJ) = 0
Perturbation theory around a saddle point: Saddle point geometry dependent on the couplings:
are left out of the vector trace (see e.g. Volkov and Wipf ’00).
φa = ¯ φa(κJ, λJ) + δφa
Rµν(¯ φ) = gµν(¯ φ) 1 D − 2 λJ κJ
−W(λJ, κJ) = SΛ[¯ φ] + 1 2Tr2 log(∆2/µ2) − Tr0
1 log ∆1/µ2 + log Ω(µ)
∆1✏µ = ✓ r2 R D ◆ ✏µ , ∆2hµν = r2hµν 2Rµ
ρ ν σhρσ .
Proper-time regulator implemented as a modification of the measure One-loop flow equation: Heat kernel expansion:
Λ∂ΛSΛ = Tr2[e−∆2/Λ2] − 2Tr1[e−∆1/Λ2]
Λ∂ΛSΛ = ΛD Ng (4π)
D 2
Z dDx√g + 1 6 (Ng − 18) (4π)
D 2
ΛD−2 Z dDx√gR + ... Ng ≡ D(D − 3) 2
Number polarisations of the graviton βG = (D − 2)G − b G2 ,
b = 1 6 16π (4π)
D 2 (18 − Ng)
On spacetime manifolds with boundaries we can consider amplitudes:
Generically there is a lack of boundary conditions in quantum gravity which are diffeomorphism invariant and lead to a well defined heat kernel. On boundaries with extrinsic curvature: Moss and Silva ’97 have found suitable boundary conditions.
hφ1|φ2i = Z[φ1, φ2]
Kij = 1 D − 1K γij , ∂iK = 0 ,
hin = 0 = ✏n ˙ ✏i − Kj
i ✏j = 0
˙ hnn + Khnn − 2Kijhij = 0 ˙ hij − Kijhnn = 0
δφa = hµν(x)
Results can be generalised to manifolds with two disjoint boundaries with the addition of the Gibbons-Hawking-York term in the action.
and boundary terms. Saddle point boundary geometry:
eW (λJ,κJ,λΣ1
J ,λΣ2 J ) = V −1
diff,Λ
Z Y
a
dφa (2π)1/2 q det CΛ
ab(φ) exp
⇢ −(λJ + δΛλ) Z dDx√g +(κJ + δΛκ) ✓Z dDx√gR + 2 Z
Σ1
dD−1y√γK + 2 Z
Σ2
dD−1y√γK ◆ −(λΣ1
J + δΛλΣ1)
Z
Σ1
dDy√γ − (λΣ2
J + δΛλΣ2)
Z
Σ2
dDy√γ + δΛS[φ]
D − 2 λΣ1,2
J
2κJ
Results can be generalised to manifolds with two disjoint boundaries with the addition of the Gibbons-Hawking-York term in the action.
Gibbons 1977) and the functional integral doesn’t have the composition properties of an amplitude (Hawking 1980). All previous calculations have found this is not possible after renormalisation. However diffeomorphism invariance has been broken either by the action or the boundary conditions (or both). Jacobson and Satz (2014) showed the balance can be achieved on shell in four dimensions. Here we preserve diffeomorphism invariance…
eW (λJ,κJ,λΣ1
J ,λΣ2 J ) = V −1
diff,Λ
Z Y
a
dφa (2π)1/2 q det CΛ
ab(φ) exp
⇢ −(λJ + δΛλ) Z dDx√g +(κJ + δΛκ) ✓Z dDx√gR + 2 Z
Σ1
dD−1y√γK + 2 Z
Σ2
dD−1y√γK ◆ −(λΣ1
J + δΛλΣ1)
Z
Σ1
dDy√γ − (λΣ2
J + δΛλΣ2)
Z
Σ2
dDy√γ + δΛS[φ]
Flow equation derived from:
are generated:
Universal result near two dimensions:
Λ∂ΛSΣ
Λ =
1 (4π)
D 2
Z
Σ
dD−1y√γ ✓√π 2 1 2(D − 4)(D − 3)ΛD−1 + 1 6(Ng − 18)ΛD−2 · 2 K ◆ + ...
βG = εG − 38 3 G2 ∂ ∂ΛW(λJ, κJ, λΣ1
J , λΣ2 J ) = 0
D = 2 + ε
Gauge and parameterisation dependent beta functions come from looking at correlation functions (even if we take care of the measure). This prevents a direct physical interpretation of fixed points. We can avoid these problems by looking at observables. At one-loop three important problems are solved: