General Relativity as an Effective Field Theory

General Relativity as an Effective Field Theory 1) Gravity is very much like the rest of our fundamental interactions - can calculate quantum effects at present energies 2) Gravity from a particle physics perspective 3) Quantum corrections to the Newtonian Potential

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4) Other examples 5) Limitations 6) Some applications to cosmology John Donoghue

Zuoz 2014

Why this calculation? 1) The bad news: Quantum correction is too small to observe - or is this related to good news? BUT 2) Calculation is fundamental and has interesting features:

large mass or v2/c2 (Precession of Mercury) - why did this only get done in 1994/2002?

Newtonian realm

shorter distances

3) Reliable calculation of quantum general relativity! But quantum mechanics and gravity not compatible?????

What is the problem with quantum gravity? “Quantum mechanics and relativity are contradictory to each other and therefore cannot both be correct.” “The existence of gravity clashes with our description of the rest of physics by quantum fields” “Attempting to combine general relativity and quantum mechanics leads to a meaningless quantum field theory with unmanageable divergences.” “The application of conventional field quantization to GR fails because it yields a nonrenormalizable theory” “Quantum mechanics and general relativity are incompatible”

These statements are old-fashioned and misleading

Standard lore: Quantum gravity doesn’t exist We know interactions at present energies: QCD Electroweak Gravity All have beautiful field theory descriptions – SM + GR All described by quantum theory – except gravity?? But this really is not correct: Need to reshape the way that we think of quantum gravity - effective field theory is the tool

Rather: Quantum general relativity at E<
Gravity without Einstein: Perhaps try Yukawa/Higgs theory

Yields potential

But this fails for many reasons - masses not the whole story - binding energies - E=mc2 - bending of light

Role of the equivalence principle

Want coupling to total energy and momentum

Yang-Mills

Gauge theory logic: global symmetries correspond to conserved charges local symmetries generate forces coupled to the charges Local symmetry - requires gauge field and covariant derivative

- transforms covariantly

- generates covariant field strength tensor

Gauged spacetime symmetry = GR Charges of energy and momentum are associated with invariance under time and space translations If we gauge these, we will get forces for which the sources are the energy and momentum Global space time transformations (Lorentz plus translations)

Local version – different at each point of spacetime:

or

General coordinate invariance

New field – the metric tensor To make this an invariance requires a new field

which transforms as

We can make invariant actions using a covariant derivative - e.g for vector fields the covariant derivative should obey

This can be solved via with

Note: 1 derivative

Curvatures: - like field strengths - covariant strengths of “non-flatness” Basic defintion: Riemann tensor Note: 2 derivatives

and

The scalar curvature R is completely invariant

The Einstein Action Lets construct a gauge invariant theory

The variation of this yields Einstein’s equation

with the energy-momentum tensor as the source

The weak field limit: -shows that this is really a theory of gravity Expand Need to fix a gauge (harmonic gauge here)

yields

For a point mass

Homework: Scalar field with metric:

In weak field limit, find EL equation, do NR reduction, find Schrodinger equation in gravitational potential

Gravity as an effective theory

Weinberg JFD

Both General Relativity and Quantum Mechanics known and tested over common range of scales Is there an incompatibility at those scales ? Or are problems only at uncharted high energies? Need to study GR with a careful consideration of scales

Recall general procedure of EFT 1) Identify Lagrangian -- most general (given symmetries and low energy DOF) -- order by energy expansion 2) Calculate and renormalize -- start with lowest order -- renormalize parameters 3) Phenomenology -- measure parameters -- residual relations are predictions

Aspects of GR that we will use: 1) The gravitational field –deviation from flat space

2) Symmetry – general coordinate invariance - restricts Lagrangian to invariant terms

3) Gravity couples to energy and to itself – Einstein’s equation

The general Lagrangian The Einstein action:

But this is not the most general lagrangian consistent with general covariance. Key: R depends on two derivatives of the metric ï Energy expansion – expansion in number of derivatives

Parameters

1) L = cosmological constant

-this is observable only on cosmological scales -neglect for rest of talk -interesting aspects 2) Newton’s constant

3) Curvature –squared terms c1, c2 - studied by Stelle - modify gravity at very small scales -essentially unconstrained by experiment

Matter couplings Spinless heavy particle:

Parameters di like charge radii – non-universal

Quantizing general relativity Feynman quantized gravity in the 1960’s Quanta = gravitons (massless, spin 2) Rules for Feynman diagrams given Subtle features: hmn has 4x4 components – only 2 are physical DOF! -need to remove effects of unphysical ones Gauge invariance (general coordinate invariance) - calculations done in some gauge -need to maintain symmetry In the end, the techniques used are very similar to other gauge theories

Quantization “Easy” to quantize gravity: -Covariant quantization Feynman deWitt -gauge fixing -ghosts fields -Background field method ‘t Hooft Veltman -retains symmetries of GR -path integral

Background field: Expand around this background:

Linear term vanishes by Einstein Eq.

Recall the need for gauge fixing and ghosts Gauge invariance causes problems - no propagator in pert. theory - infinite gauge copies in path integral We proceed by fixing gauge But some gauges include non-physical DOF Removed by Feynman-DeWitt-Fadeev-Popov trick - exponentiating gauge constraint as ghost particles - remove unphysical decrees of freedom

Gauge fixing: -harmonic gauge

Ghost fields:

vector fields anticommuting, in loops only Interesting note: Feynman introduced ghost fields in GR before F-P in YM

Quantum lagrangian:

with

and

Propagator around flat space:

Feynman rules:

Vacuum state Depends on physical setting -in general non-equilibrium -global vacuum state generally not possible - initial conditions and particle production Applications have been near flat space -use flat space vacuum state DeSitter space (inflation) seems to be quite tricky We will see some cosmology later

Performing quantum calculations Quantization was straightforward, but what do you do next? - calculations are not as simple Next step: Renormalization - divergences arise at high energies - not of the form of the basic Lagraingian Solution: Effective field theory and renormalization - renormalize divergences into parameters of the most general lagrangian (c1,c2…) Power counting theorem: (pure gravity, L=0) -each graviton loopï2 more powers in energy expansion -1 loop ï Order (∑g)4 -2 loop ï Order (∑g)6

Renormalization One loop calculation:

‘t Hooft and Veltman dim. reg. preserves symmetry

Divergences are local:

Renormalize parameters in general action: Pure gravity “one loop finite” since Rmn=0

Note: Two loop calculation known in pure gravity

Order of six derivatves

Goroff and Sagnotti

More formal study – Gomis and Weinberg “Are non-renormalizable gauge theories renormalizeable” Gauge theories could present separate problems -gauge fixing -are there potential coefficients in general Lagrangian to renormalize all divergences? Proven for Yang-Mills and gravitation - structural constraints and cohomology theorems Undecided for theories with U(1) symmetry -no counter examples exist

What are the quantum predictions? Not the divergences -they come from the Planck scale -unreliable part of theory Not the parameters -local terms in L -we would have to measure them Amp ~ q 2 ln( q 2 ) ,

 q2

Low energy propagation -not the same as terms in the Lagrangian - most always non-analytic dependence in momentum space -can’t be Taylor expanded – can’t be part of a local Lagrangian -long distance in coordinate space

Example- Corrections to Newtonian Potential JFD 1994 JFD, Holstein, Bjerrum-Bohr 2002 Khriplovich and Kirilin Other references later

Here discuss scattering potential of two heavy masses.

Potential found using from

Classical potential has been well studied

Iwasaki Gupta-Radford Hiida-Okamura

Lowest order: one graviton exchange

Non-relativistic reduction:

Potential:

What to expect: General expansion:

Classical expansion parameter Relation to momentum space:

Quantum expansion parameter

Short range

Momentum space amplitudes: Classical

Non-analytic

quantum

analytic

short range

Parameter free and divergence free Recall: divergences like local Lagrangian ~R2 Also unknown parameters in local Lagrangian ~c1,c2 But this generates only “short distance term” Note: R2 has 4 derivatives

Then: Treating R2 as perturbation

R2

Local lagrangian gives only short range terms

The calculation: Lowest order:

Vertex corrections:

Vacuum polarization: (Duff 1974)

Box and crossed box

Others:

Results: Pull out non-analytic terms: -for example the vertex corrections:

Sum diagrams:

Gives precession of Mercury, etc (Iwasaki ; Gupta + Radford)

Quantum correction

Comments 1) Both classical and quantum emerge from a one loop calculation! - classical first done by Gupta and Radford (1980) 1) Unmeasurably small correction: - best perturbation theory known(!) 3) Quantum loop well behaved - no conflict of GR and QM 4) Other calculations (Radikowski, Duff, JFD; Muzinich and Vokos; Hamber and Liu; Akhundov, Bellucci, and Sheikh ; Khriplovich and Kirilin ) -other potentials or mistakes 5) Why not done 30 years ago? - power of effective field theory reasoning

Aside: Classical Physics from Quantum Loops: JFD, Holstein 2004 PRL

Field theory folk lore: Loop expansion is an expansion in Ñ “Proofs” in field theory books This is not really true. - numerous counter examples – such as the gravitational potential - can remove a power of Ñ via kinematic dependence

- classical behavior seen when massless particles are involved

Aside: Coordinate redefinitions: Possibility of further coordinate changes

which changes the classical potential

However, in Hamiltonian treatment this is compensated by change of other terms

Einstein Infeld Hoffmann coordinates

Quantum terms in potential: Similar coordinate ambiguity

Independent from classical terms at this order:

Terms which mix under this transformation:

But quantum corrections do not generate second term - single power of G -if involving inter-particle separation, r, then quantum effects are of order G2 -quantum potential then unique – -coordinates where coefficient of Hqp is zero

Quantum corrections to Reissner-Nordstrom and KerrNewman metrics JFD Holstein Garbrecht Konstantin

Metric around charged bodies, without (RN) or with (KN) angular momentum Quantum Electrodynamics calculation -gravity is classical here -but uses EFT logic Metric determined by energy momentum tensor:

Logic: looking for non-analytic terms again: -long range propagation of photons

harmonic gauge

Calculation: Boson:

Fermion:

Results: -reproduce classical terms (harmonic gauge) -quantum terms common to fermions, bosons

Physical intepretation: - classical terms are just the classical field around charged particle

-reproduced in the loops expansion - quantum terms are fluctuations in the electromagnetic fields

Graviton –graviton scattering Fundamental quantum gravity process Lowest order amplitude: Cooke; Behrends Gastmans Grisaru et al

One loop: Incredibly difficult using field theory Dunbar and Norridge –string based methods! (just tool, not full string theory)

JFD + Torma

Infrared safe: The 1/e is from infrared -soft graviton radiation -made finite in usual way 1/e -> ln(1/resolution) (gives scale to loops) -cross section finite

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Beautiful result: -low energy theorem of quantum gravity

finite

Hawking Radiation Exploratory calculation -remove high energy contributios -Pauli Villars regulators -flux from local limit of Green’s function

-dependence on regulator vanishes exponentially -radiation appears to be property of the low energy theory

Hambli, Burgess

Limitations of the effective field theory Corrections grow like

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Amp ~ A0 1  Gq 2  Gq 2 ln q 2

Overwhelm lowest order at q2~ MP2 Also sicknesses of R+R2 theories beyond MP (J. Simon) Effective theory predicts its own breakdown at MP -could in principle be earlier Needs to be replaced by more complete theory at that scale (String theory??) Treating quantum GR beyond the Planck scale is likely not sensible

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The extreme IR limit Singularity theorems: -most space times have singularities -EFT breaks down near coord singularity Can we take extreme IR limit? -wavelength greater than distance to nearest singularity? -rö¶ past black holes? Possible treat singular region as source -boundary conditions needed deSitter horizon in IR

Consider horizons: - locally safe – we could be passing a BH horizon right now - local neighborhood makes a fine EFT - can be small curvature - no curvature singularity - locally flat coordinates in free fall through horizon - but cannot pass information to spatial infinity - EFT cannot be continued to very long distances (!) - also, when far away, horizon appears source of thermal radiation - incoherent, non-unitary EFT has some failure at long distance - but long distance is where the EFT is supposed to work - what is the parameter governing the problem?

Singularities could be even more problematic - can you consider wavelengths past the nearest singularity?

Diagnosis: Consider Reimann normal coordinates Taylor expansion in a local neighborhood:

Even for small curvature, there is a limit to a perturbative treatment of long distance: << 1

Hawking-Penrose tell us that this is not just a bad choice of coordinates But this is not the usual EFT expansion: R y2 ~ R / q2 -

gets worse at long distance

IR Quantum gravity issues -gravitational effects build up – horizons and singularities -Apparantly new IR expansion parameters enter in GREFT - integrated curvature

Basic issues: - in GR, can choose coordinates to be locally flat - apply EFT in any local neighborhood with small curvature But, 1) Curvature can build up over long distances – horizons 2) Horizons associated with non-unitary behavior – i.e. thermal radiation 3) Hawking Penrose singularity theorems – “all” spacetimes contain singularities This leads to novel issues in the extreme IR – not well understood at present

Integrated curvature qualitatively explains IR issues: - curvature builds up between horizon and spatial infinity - singularities due to evolution of any curvature to long enough distance

But how do we treat this in EFT?

Summation of pert theory? (Duff 73) – summation of trees = Schwarzschild Maybe singularities can be treated as gravitational sources - excise a region around the singularity - include a coupling to the boundary - analogy Skyrmions in ChPTh But distant horizons? - perhaps non-perturbatively small??

Reformulate problem of quantum gravity Old view: GR and Quantum Mechanics incompatible Unacceptable New view: We need to find the right “high energy” theory which includes gravity Less shocking: -not a conflict of GR and QM -just incomplete knowledge THIS IS PROGRESS!

A Modern Viewpoint: A lot of portentous drivel has been written about the quantum theory of gravity, so I'd like to begin by making a fundamental observation about it that tends to be obfuscated. There is a perfectly well-defined quantum theory of gravity that agrees accurately with all available experimental data. Frank Wilczek Physics Today August 2002

Another thoughtful quote: “I also question the assertion that we presently have no quantum field theory of gravitation. It is true that there is no closed, internally consistent theory of quantum gravity valid at all distance scales. But such theories are hard to come by, and in any case, are not very relevant in practice. But as an open theory, quantum gravity is arguably our best quantum field theory, not the worst. …. {Here he describes the effective field theory treatment} From this viewpoint, quantum gravity, when treated –as described above- as an effective field theory, has the largest bandwidth; it is credible over 60 orders of magnitude, from the cosmological to the Planck scale of distances.” J.D. Bjorken

Summary of EFT treatment We have a quantum theory of general relativity -quantization and renormalization -perturbative expansion It is an effective field theory -valid well below the Planck scale -corrections are very well behaved Effective field theory techniques allow predictions -finite -parameter free -due to low energy (massless) propagation EFT may be full quantum content of pure GR -points to breakdown by E = MP Need full theory at or before Planck scale -many interesting questions need full theory -not conflict between QM and GR, but lack of knowledge about fundamental high energy theory

Possible applications Singularity theorems Cosmology – early universe Possible tests??? -long distance propagation -quantum effects on photons -frequency dispersion Comparison to numerical methods -lattice gravity EFT in presence of L -quantization and divergences known (Christensen and Duff) -power counting modified de Sitter “instability”: -understanding Tsamis Woodard effect -”screening” of L

Classical EFT treatment of GR - use EFT techniques to do classical gravity - binary inspiral and gravity waves - Lagrangian treatment for NR particles - ahead of traditional methods on spin effects now - working on templates now

Goldberger and Rothstein 2003

Rothstein Porto, Ross

JFD, El-Menoufi

Motivations for studying quantum effects in cosmology New types of effects (non-locality) not present in classical GR Emergence of classical behavior Singularity avoidance?? -Hawking Penrose singularity theorems - quantum effects avoid ingredients of singularity theorems - loop quantum cosmology sees bounce -at scales below MP - if really below Planck scale, EFT should see it also Can quantum effects be large? - large N

Interesting limit - Large number of matter fields Han, Willenbrock Anber, Aydemir,JFD

Scattering violates tree unitarity below the Planck scale

But theory heals its unitarity violation

Unitarity is restored:

But note form of correctons

Can easily do pure gravity correction also

What are effects of logs on cosmology? I told you action was R + R2 +.. But really also has logs Recall E&M where really In gravity there will be or really

The general form

Here the coefficients depend on the number and type of fields

For the Standard Model:

Causal propagation: - Variation of Leff would yield a source to Eq. of motion - But – non-causal – half retarded , half advanced - Scattering (in-out) solution Need (causal, in-in, closed time path, Schwinger-Keldysh) formalism

-expectation values -can reformulate using closed time path

FLRW cosmology: scale of universe a(t)

Einstein equation becomes

If time is the only variable, the non-local function

Non-local FLRW equations:

with

and the time-dependent weight:

For scalars: Memory of the past scale factor

Quantum memory

Emergence of classical behavior:

Collapsing universes – singularity avoidance

Independent of all local parameters

But there are some cases where singularity is not overcome:

Single non-conformal scalar field

Local terms overwhelm non-local effect:

Summary of cosmology section Quantum effect give non-local (in time) behavior Remembrances of scales past Does not disrupt classical behavior in the appropriate regime Does allow avoidance of singularities Other types of non-locality need to be explored

Overall summary: We have a quantum theory of General Relativity It has the structure of a non-linear EFT Is capable of parameter free predictions But it does not answer questions above MP The low energy limit is only partially explored Infrared Quantum Gravity needs to be understood as one of our fundamental theories