The Quantum Field Theory of Quadratic Gravity

John DonoghueHeidelbergSept 10, 2019

Work with Gabriel MenezesarXiv:1712.04468 , arXiv:1804.04980, arXiv:1812.03603arXiv:1908.02416, arXiv:1908.04170, …

1) Causality and the arrow of causality

2) Dueling arrows of causality

3) Quartic propagators in interacting theories

4) Stability

5) Unitarity

6) Quadratic gravity

Brief motivation/orientation:

Quadratic gravity:

Renormalizeable QFT for quantum gravity- the most conservative version of quantum gravity

BUT:

Higher derivative theories have “issues” and mythology

Explore the QFT for such theories.

Bottom line: Find unitary theory, appears stable near Minkowski-but with Planck scale causality violation/uncertainty

Coupling to matter makes high mass ghost decay

This makes all the difference:- not part of the asymptotic spectrum- don’t need to discuss free field quantization- not part of summation in unitarity relations- signs on decay width important for unitarity- stability in propagators- limits problems with causality

Insight dates to Lee and Wick ~1969

Causality is not really “cause before effect”

Decompose into time orderings:

Positive energies propagate forward in time

Operators commute for spacelike separation

Note: metric is(+,-,-,-)

But also – Arrow of causality

LHCb

What if we used e-iS instead of eiS?Consider generating functions:

Need to make this better defined – add

Solved by completing the square:

Yield propagator with specific analyticity structure

Result is time-reversed propagator

Positive energy propagates backwards in time

Use of this generating functional yields time reversed scattering processes

Opposite arrow of causality

Time reversal is anti-unitary

Lagrangian can be invariant, but PI is not

Note also, with t = - τ ,

and canonical quantization changes

to

Time as a parameter is a convention

Time measured by decreasing amounts

Time measured by increasing amounts

Countdown:

But, basic conclusion: Quantum physics is causal and has an arrow of causality

“Arrow of time”:

Typical motivation:"The laws of physics at the fundamental level don’t distinguish between the past and the future."

But this is not correct

The laws of quantum physics have an arrow of causality

Buried in the factors of i in the quantization procedures

Our time convention uses Z+- if reverse time convention used, use Z-

Note: Arrow of thermodynamics follows arrow of causality

Dueling arrows of causality

Quartic propagators have opposing arrows

Who wins?-massive state decays-stable states win

No tachyons allowed

The all in one propagator (including self-energy)

Above some threshold

If threshold is above m2 , stable particle at q2=m2

If threshold is below m2, this is a normal resonance

The high mass pole carries two minus sign differences:

This is a finite width version of D-F

In all cases, same imaginary part:

Propagation in both directions:

Quartic propagators carry two arrows of causality- stable states win over large times- massive states decay- backwards propagation over scale of width

Merlin modes:-Merlin (the wizard in the tales of King Arthur) ages backwards

At the Lagrangian level:

Sample interacting theory (notation for later convenience)

Use auxiliary field

Then shift

Two modes propagating in opposite directions (coupled by matter)

Or keep single field descriptionOverall propagator (notation to match quadratic gravity)

Neff = 20 N (or 20(N+1) if we include self interaction)

PhenomenologyLee, WickColemanGrinstein, O’Connell, WiseAlvarez, Da Roid, Schat, Szynkman

Vertex displacements: (ADSS)- look for final state emergence- before beam collision

Form wavepackets – early arrival (LW, GOW)- wavepacket description of scattering process- some components arrive at detector early

Resonance Wigner time delay reversal- normal resonaces counterclockwise on Argand diagram

- Merlin modes are clockwise resonance

For gravity, all are Planck scale- no conflict with experiment

Causality UncertaintyWavepackets are an idealization:-really formed by previous interactions

Likewise beam construction from previous scattering- and measurement due to final scattering

The timing of scattering will become uncertain

Stability:

Consider propagator with retarded BC:

Again propagation in both directions:

Backwards perturbations have finite lifetime:

See also Salvio;Reis, Chapiro, Shapiro

Unitarity:

Who counts in unitarity relation?- Veltman 1963 - only stable particles count- they form asymptotic Hilbert space- do not make any cuts on unstable resonances

This looks funny from free-field quantization- interaction removes states from the Hilbert space

Also, we know some states are almost stable - can treat them as essentially stable- Narrow Width Approximation (NWA)

Nevertheless, Veltman is correct

Cutkosky cutting rules

Obtain discontinuity by replacing propagator with:

Also on far side of cut, use:

Example – self energy

Also can repackage this inserting:

To have the equivalent expression:

Doing the integrals now yields

Or

The discontinuity is equivalent to the decay width at q2

Cuts in a resonance propagator:

Bubble sum on each side of propagator:- will c.c. propagators on the far side

This is true no matter if normal resonance or Merlin modes- recall “all in one propagator”

Three particle cut = resonance + stable cut

Identify matrix element

and play similar games, to get expected unitarity relation

Again result is independent of type of resonance

Bottom line: discontinuities come from cuts on stable particles

Narrow width approximation

Lessons ala Veltman

Heuristic proof of unitarity

Unitarity works with stable particle as external states

Cuts through stable particle loops same for normal and Merlin resonances

Both normal and Merlin resonances described by same propagator- all-in-one propagator

Veltman proved normal resonances satisfy unitarity to all orders

The Merlins will then also satisfy unitarity

Formal proof of unitarity with unstable ghosts

Follows Veltman:- circling rules- largest time equation- turns into derivation of cutting rules

Only difference is energy flow

Important point - all steps in Minkoswki space- no analytic continuation employed

arXiv:1908.02416

Narrow Width Approximation with Merlin modesThis path follows M. Schwartz: QFT +SM

Convert DF to advanced propagator

Product of advanced propagators vanishes

Play some games and pick out Im part

Normal Ghost

Take same path

But delta functions cannot besatisfied

Lee-Wick contour

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

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 properties

Can be kept weakly coupled at Plankian energies.

Renormalizeable

Shares properties described above- unitary- stable, at least at moderate curvatures- contains Merlin mode ghosts - causality uncertainty

Unitarity in the spin two channelDo 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

Scattering amplitude at weak coupling:

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

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

Most unusual feature:- causality violation/uncertainty near Planck scale

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