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
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