Exploring Quantum Gravity as a Renormalizeable Quantum Field Theory

John DonoghueParisApril 16, 2019

Work with Gabriel MenezesarXiv:1712.04468 , arXiv:1804.04980, arXiv:1812.03603…

All other fundamental interactions are r-QFTs

But each with variations

QG as QFT will need another variation

Is there a fundamental obstacle?

Quantum GR as a QFT/EFT

We do have a QFT for GR – it is an Effective Field Theory-High Energy parts unknown but local-Shift focus to the IR – nonlocal

Long distance / low energy propagation – non-analytic behavior:

Leads to rigorous quantum predictions:

Sample Predictions:

Potential:

Light bending

Unitarity techniques well suited for EFT

JFD, Bjerrum Bohr, HolsteinKriplovich, Kirilin

UniversalJFD, Bjerrum Bohr, Vanhove

Not universal – non-geodesicJFD, Bjerrum Bohr, Holstein, Plante, Vanhove

Light cone ill-defined

Power Counting

EFT quantizes lowest order action ( R )

Expansion in the energy/curvature

Tree ~ O(E2) , R1 loop ~ O(E4) , R2

2 loops ~ O(E6) , R3

Both finite partsand divergences

Basic Diagnosis for UV complete QFT:

Matter loops renormalize R2 terms- at one loop – and at all loop order- dimensionless coupling constant

R2 terms must be in a fundamental QFT action

R2 terms lead to quartic propagators

With quartic propagators, graviton loops also stop at R2

- all orders- dimensionless coupling constant – scale invariant- R2 theories renormalizeable (Stelle)

QFT “variation” needs quartic propagators

Overview:

Comes with expectations of ghosts:

For quadratic gravity, dangerous ghost in spin-two propagator

But, coupling to light particles makes ghost unstable

Not part of asymptotic spectrum

We will see how theory maintains unitarity

But, causality uncertainty

Outline:The spin-2 resonance

Propagators

Stability

Unitarity 1.0

Modified Lehmann representation

Lee Wick Theories

Unitarity with unstable ghosts

Reinterpreting the Lee Wick contour

Causality uncertainty

Gauge Assisted Quadratic Gravity

Connections:

Early pioneers: Stelle, Fradkin-Tsetlyn, Adler, Zee, Smilga, Tomboulis, Hasslacher-Mottola, Lee-Wick, Coleman, Boulware-Gross….

Present activity: Einhorn-Jones, Salvio-Strumia, Holdom-Ren,Donoghue-Menezes, Mannheim, AnselmiOdintsov-Shapiro, Narain-Anishetty…

In the neighborhood: Lu-Perkins-Pope-Stelle, ‘t Hooft,Grinstein-O’Connell-Wise ….

Quadratic gravity

Free-field mode decomposition depends on gauge fixing.-all contain a scalar mode and tensor mode

Scalar has massive non-ghost and massless ghost – due to R2

Spin 2 mode has massive ghost

Quick notes:

1) Easiest to consider weak coupling

- nonperturbative effects small- scattering remains always weak

2) Different normalizations- most often

- but sometimes I use conventional normalization

First: Vacuum polarization of normal light fields- the EFT regime

- with usual prescription

Leads to spin-2 propagator

Resonance in spin-two propagator

Figure units:

Narrow width pole position:

Written as “ghost-like” plus “opposite sign width”

vs

Unusual features:

Norm and sign of imaginary part are opposite normal

vs

The two signs are important:

Equivalently stated, overall imaginary propagator carries same sign

***

Resonance in propagator:

Euclidean/spacelike propagator:

Green functions:

Partial fractions:

Without any approximation:

It is fair to drop the log. Becomes BW, with funny signs

Feynman propagator:

Poles:

Both functions have decaying exponential

positive frequency backwards in time

Retarded Green Function

Modified BC and pole location:

Retarded GF now non-zero for t<0- exponentially close to t=0

Advanced GF has similar behavior – switched pole positions

StabilityNo growing exponentials in any of the Greens functions

When using retarded GF:

Planck scale acausality

Stability studies in curved backgrounds:Study via equations of motion:Salvio – fluctuations in deSitter

Chapiro, Castardelli, Shapiro – Bianchi I

Both find stability for low energy/curvature,but do not address imaginary part, particle production or high curvature

Unitarity in the spin two channel

Do these features cause trouble in scattering? - consider scattering in spin 2 channel

First consider single scalar at low energy:

Results in

Satisfies elastic unitarity:

This implies the structure

for any real f(s)

Signs and magnitudes work out for

Multi-particle problem:- just diagonalize the J=2 channel- same result but with general N

Detour – gravitons in the vacuum polarization- Graviton coupling in EFT region (from Einstein action)

acts just like matter loops. Adds 7/2 to Neff

- But quadratic coupling is different

No imaginary part generated from quadratic gravity

-on shell triple graviton coupling vanishes in Minkowski

-on shell condition is R=0

Or regularize with

Logs then come with

Verified by complete calculation

Complete form of propagator

Unitarity still works

Scattering amplitude:

Modified Lehman representation

plus

Physical propagator evaluated with

Cut along real axisAnother pole found on other side of cut using

Then

leads to a representation

Coleman

Modified Lehman representation:

Spectral function has resonant behavior:

“Explains” unitarity calculation- imaginary part from coupling to stable on-shell states

Imaginary parts live hereSum is real

This is really three resonance structures

Fix spectral function by matching

Previously we found a single resonance in propagator

In Lehman representation we have three

The spectral portion and the cc pole essentially cancel- exact in narrow width approximation G.O’C.W

Beyond the narrow width approximation

Spectral function contribution Diff. (spect. – cc pole)

Note change in scale

Demonstration of cancelation within three pole representation

The sum of the c.c.-pole and the spectral function has no pole left over

Note change in scale

Lee-Wick theories:

Introduce new particle with negative metric- like dynamical Pauli-Villars

Hamiltonian quantization of free field theory-“indefinite metric”- stable ground state

Turn on interactions- heavy particle becomes unstable- not part of asymptotic spectrum

~1969

See also:Bender-MannheimSalvio-Strumia

Lessons from Lee-Wick

1) “micro-causality” violation - non-causal propagation of order the ghost width

2) Unitarity seems satisfied- but with “LW contour”

Modern overview of unitarity and causality -Grinstein, O’Connell, Wise (2009)

Strong similarity to quadratic gravity

The Lee-Wick contour:

Deformed contour – “LW contour”

Introduced to pick up the poles found in Hamiltonian quantization

This is most puzzling feature of LW theories- new physics principle?- definition from the Euclidean PI?

We will return to this.

Unitarity 2.0Veltman 1963 Unitarity with unstable particles

Issue: We start using free field theory – identify spectrum-then “turn on interactions”- some particles unstable – not in asymptotic spectrum- who counts for unitarity, cutting rules?- should only involve asymptotic states

Answer: only stable particles count in unitarity sum- no cuts across unstable particle lines

But, corollary: If width is small, can get the same result bycutting resonance and ignoring stable particles

- Narrow Width Approximation

Key technical point:

Using Källén–Lehmann representation for both stable and unstableBreaks into time-ordered components

with

such that

Diff. of stable and resonance, - only for stable particle – long time propagation

Then largest time method gives cutting rulesgives

Preliminary

LW contour as “fix” for NWA

But in this case, narrow width approximation does not givethe same result

Cut resonance propagator instead of stable particles- pole sits on wrong side of the cut

Can recover right answer if using LW contour!

Example: three body intermediate state- Described by onshell massless graviton and two particles in loop

- Using NWA gives zero with usual contour

Cutting rules (in pictures)

One possible path: Schwartz QFT

After some work, leads to

When the resonance is the LW ghost:

Leads to

If you use the LW contour:

Cannot be satisfiedCut vanishes

This gives the usual result.

New interpretation:

Find stable states from pole in propagator- only m=0 graviton- no need to quantize free-field ghost

Ghost like resonance as a feature in propagator

Unitarity satisfied with only normal cuts and stable states- no ghosts in unitarity sum- normal integration path

If we pretend that ghost resonance is physical via NWA- then need LW contour to get the same answer

Microcausality violation Grinstein, O’ Connell, Wise 2009

What is going on:Propagation of positive frequency components

Normal resonance LW resonance

Aside: Time ordered PT

Decompose Feynman diagram into time orderings:

Could we actually see final particles produced before initial collision?

Aside: Time ordered PT

Decompose Feynman diagram into time orderings:

Could we actually see final particles produced before initial collision?

Uncertainty Principle to the rescue:- time difference in propagators is ~ 1/ E- need to form wavepacket for initial state- minimum time uncertainty is also ~ 1/ E

Collision/observation times are all uncertain > 1/E

Causality uncertainty principle for quadratic gravity?

“Form a wavepacket” is an idealization

Initial state is also prepared via scattering processes- these will also carry time delays and advances

Gravity near the resonance will be causally uncertain

Is this a common feature of UV QG?

My favorite variation:

“Gauge assisted quadratic gravity” also with Gabriel Menezes

1) Start with pure quadratic gravity- classically scale invariant- no Einstein term

2) Weakly coupled at all scales

3) Induce Einstein action through “helper” gauge interaction- Planck scale not from gravity itself but from gauge sector

Overview:

Consider Yang-Mills plus quadratic gravity:

Classically scale invariant

“Helper” YM becoming strong will define Planck scale

Take quadratic gravity to be weakly coupled at Planck scale- in particular

Yang-Mills will induce the Einstein action

Adler-Zee

In QCD we find:

Scale up by 1019 to get correct Planck scale

JFD, Menezes

Quick derivation:

Weak field limit:

Gravitaional coupling:

To second order in the gravitational field:

For convenience, consider special case (see also Brown, Zee)

Slowly varying fields (on QCD scale):

Zee

This results in the effective Lagrangian:

The first term is part of the cosmological constant, the second is the Einstein action. Identify via:

This gives the Adler-Zee formula:

Induced G – in SU(N) theories

Construct the sum rule for QCD – lattice data, OPE and pert. theory

Separate long and short distance techniques at x=x0

Perturbative: (Adler)

OPE: (NSVZ,Bagan, Steele)

Lattice: (Chen, et al.2006)

Note also:

Cosmological constant:

With standard values, Λ = -0.0044 GeV4 (return to this later)

Induced R2 term

We find (low energy only):

Brown-Zee

Three regions:

1) Beyond Planck mass: YM plus quadratic gravity

2) Intermediate energy- interplay of quadratic and quartic terms- dominated by resonance

3) Low energy EFT regionquartic terms subdominant

The cosmological constant problem Embarrassment for general picture- vacuum energy generated from scale invariant start

Some options:1) Unimodular gravity – vacuum energy is not relevant

2) Caswell-Banks-Zaks – beta function vanishes in IR

3) Strong coupled gravity – perhaps gravity cancels gauge

4) Spin connection as helper gauge theory – no vev

5) Supersymmetric helper gauge theory – no c.c. at this scale

6) Abandon our principles – just add a constant to cancel it

JFD’16

Summary:

Quadratic gravity is a renormalizeable quantum field theory

Positive features:- massless graviton identified through pole in propagator- ghost resonance is unstable – does not appear in spectrum- formal quantization schemes (free field) exist but not needed- seems stable under perturbations- unitarity with only stable asymptotic states- LW contour as shortcut via narrow width approximation

Surprising (?) feature:- causality violation/uncertainty near Planck scale

More work needed, but appears as a viable option for quantum gravity