Paris QG - th.u-psud.fr · 4xdqwxp *5 dv d 4)7 ()7:h gr kdyh d 4)7 iru *5 ±lw lv dq (iihfwlyh...
Transcript of Paris QG - th.u-psud.fr · 4xdqwxp *5 dv d 4)7 ()7:h gr kdyh d 4)7 iru *5 ±lw lv dq (iihfwlyh...
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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?
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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:
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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
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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
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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
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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
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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
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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 ….
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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
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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
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First: Vacuum polarization of normal light fields- the EFT regime
- with usual prescription
Leads to spin-2 propagator
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Resonance in spin-two propagator
Figure units:
Narrow width pole position:
Written as “ghost-like” plus “opposite sign width”
vs
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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
***
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Resonance in propagator:
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Euclidean/spacelike propagator:
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Green functions:
Partial fractions:
Without any approximation:
It is fair to drop the log. Becomes BW, with funny signs
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Feynman propagator:
Poles:
Both functions have decaying exponential
positive frequency backwards in time
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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
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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
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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
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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
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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
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Verified by complete calculation
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Complete form of propagator
Unitarity still works
Scattering amplitude:
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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
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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
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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
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Beyond the narrow width approximation
Spectral function contribution Diff. (spect. – cc pole)
Note change in scale
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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
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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
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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
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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.
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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
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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
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Preliminary
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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
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Cutting rules (in pictures)
One possible path: Schwartz QFT
After some work, leads to
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When the resonance is the LW ghost:
Leads to
If you use the LW contour:
Cannot be satisfiedCut vanishes
This gives the usual result.
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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
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Microcausality violation Grinstein, O’ Connell, Wise 2009
What is going on:Propagation of positive frequency components
Normal resonance LW resonance
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Aside: Time ordered PT
Decompose Feynman diagram into time orderings:
Could we actually see final particles produced before initial collision?
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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
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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?
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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
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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
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Yang-Mills will induce the Einstein action
Adler-Zee
In QCD we find:
Scale up by 1019 to get correct Planck scale
JFD, Menezes
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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
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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:
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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)
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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
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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
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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
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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