The problem of quantising gravity, and how to …B.Knorr/images/Freiburg2016.pdfThe problem of...

The problem of quantising gravity, and how topossibly overcome it

Benjamin Knorrin collaboration with H. Gies, S. Lippoldt and F. Saueressig

based on PRL 116, 211302 (2016) and work in progress

Theoretisch-Physikalisches InstitutFriedrich-Schiller-Universitat Jena

Seminar on Fundamental Interactions

Research Training GroupQuantum and Gravitational Fields

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Outline Gravity 101 QG in PT AS and FRG QG beyond PT Summary

Outline

Gravity 101Quantum Gravity in Perturbation TheoryAsymptotic Safety and Functional Renormalisation GroupQuantum Gravity beyond Perturbation Theory

B. Knorr TPI Uni JenaThe problem of quantising gravity, and how to possibly overcome it

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Gravity 101


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What is gravity about?

Gravity describes the curvature of a spacetime. Geometry iscaptured by a symmetric tensor, the metric gµν . The centralsymmetry group is the diffeomorphism group, which containsarbitrary coordinate transformations.Curvature can be described in a coordinate-independent way. If vρ

is a vector field, then

[Dµ, Dν ] vρ = Rµνρσvσ .

The rank-4 tensor R is the Riemann tensor, and is a diff-invariantfunction of the metric and its first and second derivative.


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An action functional

The Riemann tensor is the basic diff-invariant building block. Itscontractions are the Ricci tensor, Rµν = Rρ

µρν , and the Ricciscalar, R = Rµ

µ.A possible action for classical gravity, the Einstein-Hilbert action,is

SEH =1

16πGN

∫ √|g | (R − 2Λ) .

Here, GN is the Newton’s constant and Λ the cosmologicalconstant. The (vacuum) field equations read

Rµν + gµν

(Λ − 1

2R)= 0 .


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A tensor basis

It will be convenient to introduce a trace-free tensor basis,consisting of the Weyl tensor Cµνρσ, the trace-free Ricci tensor Sµν

and the Ricci scalar R. Invariants can be sorted according to theirmass dimension:

order 0: “1”order 1: Rorder 2: R2, SµνSµν , CµνρσCµνρσ

order 3: R□R, Sµν□Sµν , R3, R SµνSµν , SµνSν

ρSρµ,

SµνSρσCµρνσ, R CµνρσCµνρσ, CµνρσCρσ

τωCτωµν

. . .


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Field equations in the trace-free tensor basis

As an illustration, the field equations in the trace-free basis are

Sµν + gµν

(Λ − 1

4R)= 0 .

The general solution is

Sµν = 0 , R = 4Λ .

From a QFT point of view, these are the on-shell conditions.Only divergences that don’t vanish on-shell have to berenormalised.


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Quantum Gravity in Perturbation Theory


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Degrees of freedom

To quantise a theory, we first have to specify what are the relevantdegrees of freedom. A natural candidate is the metric, but thereare other options (vielbein, connection, . . . ).Diffeomorphisms are generated by the Lie derivative, whichtransports a tensor along a given vector field. For the metric,

Lv gµν = Dµvν + Dνvµ .

This symmetry has to be taken care of by a suitable gauge fixingprocedure.


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Analysing the action

What can we learn about the quantisation of gravity based on theEinstein-Hilbert action, without getting our hands dirty?

SEEH =

116πGN

∫ √|g | (−R + 2Λ) .

The mass dimension of Newton’s coupling is negative,[GN ] = −[M2

Pl] = −2. From standard power counting arguments,one would expect that every order in perturbation theoryintroduces new terms to the action, resulting in an infinite numberof couplings that have to be fixed experimentally.


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1-loop quantum gravity I

At 1-loop, second order invariants will be generated. The divergentpart of the effective action can thus be written as

∆Γdiv ∼ 1ϵ

∫ √|g |

(αR2 + βSµνSµν + γCµνρσCµνρσ

).

In 4 dimensions, the Euler characteristic,

χ =1

32π2

∫ √|g |

(16R2 − 2SµνSµν + CµνρσCµνρσ

),

is a topological invariant, and thus plays no role forrenormalisation.


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1-loop quantum gravity II

Using the Euler characteristic, we can eliminate the C2 term in thedivergent part,

∆Γdiv ∼ 1ϵ

∫ √|g |

(α′R2 + β′SµνSµν

).

Recall that on-shell, Sµν = 0 and R = 4Λ. Hence, if we assumethat Λ = 0, gravity based on the Einstein-Hilbert action is 1-loopfinite.


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2-loop quantum gravity

At 2-loop, there is exactly one basis monomial which doesn’tvanish on-shell, the “Goroff-Sagnotti term”,

CµνρσCρσ

τωCτωµν

Goroff and Sagnotti (1985) and van de Ven (1991) showed thatthe divergent part of the 2-loop effective action doesn’t vanish,

∆Γdiv = (32πGN)−2092880

1(4π)4

1ϵ

∫ √|g | Cµν

ρσCρστωCτω

µν .


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The failure (?) of perturbative quantum gravity

The appearance of the Goroff-Sagnotti term is taken as a hallmarkof the breakdown of perturbative quantisation of quantum gravity.One expects that the same problem reappears at any higher order.It is easy to write down terms that don’t vanish on-shell, andwhich should be generated at n-loop level.

Any proposal of a quantum theory of gravity has to clarifythe fate of the Goroff-Sagnotti counterterm.


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Asymptotic Safety and FunctionalRenormalisation Group


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Asymptotic Safety

Asymptotic Safety generalises the notion of asymptotic freedom.If a theory is asymptotically safe, some couplings have a finitevalue in the UV. If the number of relevant couplings is finite, thetheory is as predictive as an asymptotically free theory.

This idea was first formulated by Weinberg (1976). Clearly, whencouplings are finite, non-perturbative techniques are necessary.For this reason, first investigations of the conjecture that gravitymight be asymptotically safe only started with the advent of exactrenormalisation group equations, in particular the Wetterichequation.


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Functional Renormalisation Group (FRG)

A modern, non-perturbative formulation of the renormalisationgroup was formulated by Wetterich (1993), and first applied togravity by Reuter (1996). It relies on the notion of an effectiveaverage action Γk , which interpolates between the bare orclassical action S in the UV, and the full quantum effective actionΓ in the IR.

From the UV to the IR, modes are integrated out shell-by-shell. Aregulator function acts like a momentum-dependent mass term,and ensures finiteness of every RG step.


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Wetterich equation

The effective average action fulfills an exact RG equation:

k∂kΓk =12STr

[(Γ(2)k + Rk

)−1k∂kRk

].

b

b

S

Γ

Γk


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Quantum Gravity beyond PerturbationTheory


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Asymptotic Safety - status

Numerous approximations have been used to see whether gravitycan be asymptotically safe:

EH (Reuter ’96, Benedetti+ ’11)higher derivative gravity+f(R) (Benedetti+ ’09, Groh+ ’11,Dietz+ ’12,’13, Falls+ ’13,’14, Demmel+ ’15, Ohta+ ’15)gravity+matter (Harst+ ’11, Folkerts+ ’11, Dona+ ’13,’15,Percacci+ ’15, Labus+ ’15, Meibohm+ ’15,’16)

There are also results onbackground/diff invariance (Manrique+ ’10, Donkin+ ’12,Christiansen+ ’12,’14,’15, Demmel+ ’14, Becker+ ’14)unitarity (Nink+ ’15)


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-∞ -2 -1 - 34 - 1

2 - 14 - 1

8 0 18

14

12

34 1 2 ∞

36π145

72π145

108π145

144π145

gλ

g

H. Gies, BK, S. Lippoldt (2015)


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Asymptotic Safety meets Goroff-Sagnotti

A long-standing question was whether Asymptotic Safety canactually overcome the difficulties perturbation theory faces. Forthis, we studied a minimal approximation:

Γk =

∫ √|g |

[ 116πGk

(−R + 2Λk) + σkCµνρσCρσ

τωCτωµν

].

The central questions are:Does this system have a UV fixed point?If so, is the coupling σk relevant, or predicted by the theory?


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Some analytical findings

One can show several properties of the betafunctions of thissystem without calculation:

The betafunctions of Gk and Λk are independent of σk .The betafunction of σk is cubic in σk , and the prefactor of thecubic term is strictly positive for positive Newton’s constant.

With this at hand, one can already conclude that there will be atleast one fixed point in the combined system, and theGoroff-Sagnotti term is irrelevant there.


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Fixed point structure

We considered two scenarios: to make contact with PT, weartificially set Λk to zero (A), and the compare it to the full flow ofall three couplings (B).

scenario A: Λk = 0g∗ σ∗ θg θσ

FP 1 1.64 -0.226 2.09 -77.4FP 2 1.64 -0.0186 2.09 5.62FP 3 1.64 -0.00227 2.09 -6.06

scenario B: “full” flowg∗ λ∗ σ∗ θ1,2 θσ

0.707 0.193 -0.305 1.48 ± 3.04 i -79.4


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Phase diagram


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Summary

By now, there is accumulating evidence that gravity might benon-perturbatively renormalisable. In all approximations so farstudied (including higher derivative theories or f (R)), a suitableinteracting fixed point as a UV completion was found.

A long-standing critique was that the Asymptotic Safety scenariomight not survive the inclusion of a “proper perturbativecounterterm”. In this talk, I gave evidence that this doesn’t seemto be the case, and the Goroff-Sagnotti term seems even to bestrongly irrelevant.


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