Non-perturbative spectrum of non-local gravity

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Non-perturbative

spectrum

of

non-local

gravity

Gianluca Calcagni

a

,

∗

,

Leonardo Modesto

b

,

Giuseppe Nardelli

c

,

d a_{InstitutodeEstructuradelaMateria,CSIC,Serrano121,28006Madrid,Spain}

b_{DepartmentofPhysics,SouthernUniversityofScienceandTechnology,Shenzhen518055,China} c_{DipartimentodiMatematicaeFisica,UniversitàCattolicadelSacroCuore,viaMusei41,25121Brescia,Italy} d_{TIFPA–INFNc/oDipartimentodiFisica,UniversitàdiTrento,38123Povo(Trento),Italy}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received11April2019

Receivedinrevisedform9May2019 Accepted18June2019

Availableonline20June2019 Editor:M.Cvetiˇc

We investigatethe non-perturbative degrees of freedom of a classof weakly non-local gravitational theoriesthathavebeenproposed asanultravioletcompletionofgeneralrelativity.Attheperturbative level,itisknownthatthedegreesoffreedomofnon-localgravityarethesameoftheEinstein–Hilbert theoryaround anymaximally symmetricspacetime.We provethat,atthe non-perturbativelevel,the degreesoffreedomareactuallyeightinfourdimensions,contrarytowhatonemightguessonthebasis ofthe “infinite numberofderivatives” present inthe action.It isshown thatsixof thesedegreesof freedomdonotpropagateonMinkowskispacetime,buttheymightplayaroleatlargescalesoncurved backgrounds.Wealsoproposeacriteriontoselecttheformfactoralmostuniquely.

©2019TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

Aquantumtheory ofgravity[1–3] should beable tosolve,or saysomethingconstructiveabout,someproblemsleftopenin gen-eralrelativity,such asthesingularityproblem(there exist space-timepoints where thelawsof physicsbreak down,asinthe big bangatthe beginningof theUniverse orinside blackholes), the cosmologicalconstant problem(two thirds of the contentof our patchof the cosmos is madeof a “dark energy” component not adequatelydescribedbygeneralrelativityorparticlephysics),and themysterysurroundingthe birthandfirststage ofdevelopment oftheUniverse (the actual originoftheinflaton isunknown).In recent years, a new perturbative quantum field theory of grav-ityhasrapidlyemerged asa promisingandaccessibleframework wherethegravitationalforce consistentlyobeysthelawsof quan-tum mechanics and all infinities seem to be tamed [4–15]. This proposal adapts ordinary techniques of perturbative field theory toanactionwithnon-localoperators.Fulfillinginitialexpectations basedonnaïvepower-countingarguments,thetheoryturnsoutto beunitaryandsuper-renormalizableorfinite[10] atthequantum levelthanks to the non-local natureof its dynamics.Causality is notviolated intheusual eikonallimit [16] and oneexpects non-offensive microcausality violations at the non-locality scale. The

*

Correspondingauthor.

E-mailaddresses:g.calcagni@csic.es(G. Calcagni),lmodesto@sustc.edu.cn (L. Modesto),giuseppe.nardelli@unicatt.it(G. Nardelli).

theorymayresolvethebigbang[17–25] andblack-hole singulari-ties[26–31],anditscosmologicalsolutionsmayunravelinteresting bottom-up scenarios in the early universe (inflation) andat late times(darkenergy).

Surprisingly,theseencouraging featuresare accompanied bya numberofappallinggapsofknowledgeonbasicquestionsonthe classicaltheory,suchashowtofindsolutionsofthedynamicsand whether they match the singularity-free geometries found when linearizingtheequationsofmotion.Thedynamicsisusuallysolved withapproximationsorassumptions whichdo notgive accessto all admissible solutions [32]. Also, when considering non-linear interactions(gravityisasnon-linearasitcanbe!)itbecomes un-clearhowmanyinitialconditionsonemustspecifyandhowmany degreesoffreedom(d.o.f.)populatethespectrumofphysical parti-clesofthetheory.ThepurposeofthisLetteristopavethewayto fillthesegapsandinferthetotalnumberofnon-perturbatived.o.f. inminimal non-localgravity[7,9] (super-renormalizable in D

=

4 dimensions;finiteinD

=

5).

We can summarize our findings in three statements. (A) For thetypesofnon-localitygivingrisetoarenormalizabletheory,the numberof fieldd.o.f. is finiteandequal to D

(

D

−

2)

=

8 in four dimensions. Twoof these d.o.f. correspond to the graviton, they propagate onflatspacetime atshortdistances,andarea familiar acquaintanceinthenon-locallinearizeddynamics.Theotherd.o.f. areanoveltybecausetheyemergeonlywhenthefullynon-linear non-local dynamicsis considered. Since they are not visible ina perturbative treatment on Minkowski spacetime, these d.o.f. are https://doi.org/10.1016/j.physletb.2019.06.043

0370-2693/©2019TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

non-perturbative, typical of curved backgrounds, and, hence, may be important in the description of large-scale,long-rangephysics,

suchasatastrophysical orcosmologicalscales.(B)Alsothe num-ber of initial conditions to specify in order to find classical so-lutions is finite. This contributes to, or even settles, a seventy-year-olddebateaboutwhethernon-localtheoriesarepredictiveat all, duetothe peculiaritiesof their problemofinitial conditions. Theanswer isYes, forthespecific non-localitiesconsidered here. Knowing how to construct dynamical solutions makes a funda-mentalstepintheunderstandingofthecapabilitiesofthespecific theory underexamination,bothatthe classicalandthequantum level.(C)Thesystemcanberecastasasetoffinite-order differen-tialequations,andhencethenumberofinitialconditionsisfinite, onlyfortwo specificnon-localformfactorsamongthose circulat-ing inthe literature.This constrains theambiguity onthe choice offormfactors,i.e.,ofnon-localtheory.

Themostimportantconsequence ofidentifyingnewd.o.f. and ofknowinghowtoformulatetheinitial-conditionproblemisthat nowone can,onone hand,constructnontrivialcosmologicaland black-hole classical solutions previously inaccessiblewith current methodsand,ontheotherhand,buildarobust,rigorous,and sys-tematicsetofphenomenologicalmodelsthatcanbetestedagainst experimentsandobservations,avitaltaskforanycandidatetheory ofnaturalphenomena. Therefore,theresults ofthepresentwork maybe ofinterest for theapplied mathematician, the quantum-gravitytheoreticianandphenomenologist,theparticle-physics the-oretician,thecosmologist,andtheastrophysicist.

Aftera brief overview ofthe most prominentnon-local grav-ities, we show that the number ofinitial conditions andd.o.f. is finiteandwe countthemexplicitly.Thelogicto followissimple: (i) we write the non-localdynamics withinfinitelymany deriva-tivesasasystemof“masterequations”whicharesecondorderin spacetimederivatives,hencethe Cauchyproblemis well defined; (ii)fromthisreformulation,weextractthenumberofinitial condi-tionsandthed.o.f. Theproofisself-containedandmaybeskipped bythe readerinterested onlyinthephysicalconsequencesofthe theory,whicharediscussedaboveandinthefinalsection.

2. Non-localdynamics

2.1. Briefoverviewofnon-localquantumgravity

Considertheclassicalaction

S

=

1

2

κ

2



dDx

√

−

g



R

+

G_μν

γ

(

2)

Rμν



,

(1) where Gμν is the Einstein tensor and

γ

(

2)

is a non-local form factor,anentirefunctionof

2

havingspecialasymptoticproperties [5–7,10,12,13,33].Itcanbeparametrizedas

γ

(

2) =

e

H(2)

₋

₁

2

,

(2)

where H(

2)

dependson the dimensionlesscombinationl2

₂

_and

l is afixed length scale.Thefourprincipal theoriesare shownin Table 1.In the firstcase (“pol”: asymptotically polynomial), P

(

z

)

is a real positive polynomial of degree n (2n derivatives) with

P

(0)

=

0,



is Euler function, and

γ

E is the Euler–Mascheroni

constant. In the second case (“exp”) the form factor is asymp-toticallyexponential.QuantumgravitywithKuz’min orTomboulis form factor is renormalizable [5–7]; with the string-related pro-fileHexp

(

2)

= −

l2

2

,itisrenormalizableifperturbativeexpansions withthe resummed propagatorare allowed [11]; withKrasnikov profileHexp

₍

₂₎

₌

_l4

₂

2_{,itisbelievedtoberenormalizable,butthe}

Table 1

Formfactorsinnon-localgravity.

H(2) P(2) f(ω) Form factor name Hpol (2) :=α{ln P(2) +[0,P(2)] +γE} −l2₂ O(2n₎ e−ω Kuz’min[5] Tomboulis[6,7] Hexp (2) :=αP(2) −l2₂ l4₂2 1−ω stringy[9,17] Krasnikov[4]

proof isnot complete [4]. The role of quadraticcurvature opera-tors isto make the theory renormalizable, while the role of the non-localformfactorsistopreserveunitarity.

TheprofileH(

2)

canbedefinedthroughtheintegral H

(

2) =

lim σ→1Hσ

(

2) ,

(3) Hσ

(

2) :=

α

σ

_

P(2) 0 d

ω

1

−

f

(

ω

)

ω

,

(4)

where

α

>

0 is real, P

(

2)

isa generic functionofl2

₂

_{, and f}

_(ω)

isarbitrary.Theparameter

σ

isfictitiousandhasbeenintroduced forlaterconvenience.

2.2. Non-localdynamicsintermsofkernels

The physics stemming from the action (1) is a hard nut to crack. Evenbefore quantizingthe theory,two fundamental ques-tionsarise.Howmanydegreesoffreedomarethere?Canonesolve thedynamicsonceafinitenumberofinitialconditionsaregiven? While it is easy to see that linear systems havea finite number ofd.o.f.andofinitialconditions,thecasewithnon-linear interac-tionsishighlynontrivial.Directlymanipulatingtheinfinitelymany derivatives of(2) makes very difficult toanswer the above ques-tions.

Toputthe mainresultofthisLetterinthemostdirectterms, wetrade(2) forkernelfunctionsthatobeyfinite-orderdifferential equations (masterequations). In general, any operator

γ

(

2)

with finitelyorinfinitelymanyderivativescanbewrittenasanon-local kernelfunction.Considerfirsttheflat-spacetimecase[34].In mo-mentumspace,calling F theFourieranti-transformof

γ

(

−

k2

₎

_(not

tobeconfusedwiththeFouriertransform

γ

˜

(

−

k2

₎

_of

_γ

₍

₂₎

_),fora

generictensor

ϕ

(

x

)

onehas

γ

(

2)

ϕ

(

x

)

=



dDk

γ

(

−

k2

) δ

D

(

kμ

−

i

∇

μ

)

ϕ

(

x

)

=



dDk



dDz F

(

z

)

e−iz·k



δ

D

(

kμ

−

i

∇

μ

)

ϕ

(

x

)

=



dDz F

(

z

)

ez·∇

ϕ

(

x

)

=



dDz F

(

z

)

ϕ

(

x

+

z

)

y:=z+x

=



dDy F

(

y

−

x

)

ϕ

(

y

) .

(5) Thus,asanoperatoridentitywehave

F

(

y

−

x

)

=



dDk

(

2

π

)

De

ik·(y−x)

_γ

₍

₋

_k2

_{) .}

₍₆₎

On a curved spacetime,a similar expression holds after general-izingtheFouriertransformto aninvertiblemomentumtransform where thephasesexp(

±

ik

·

z

)

arereplaced bythe eigenfunctions oftheLaplace–Beltramioperatoronthatspacetime(e.g.,[32]). Dif-ferentoperators

γ

(

2)

willleadtodifferentkernelsF .

Wecanusethisgeneralpropertyoflocalandnon-local opera-torstowrite(1) intermsofkernelfunctions:

S

=

1 2

κ

2



dDx

√

−

g R

+

1 2

κ

2



dDx



−

g

(

x

)

×



dDy



−

g

(

y

)



dDz



−

g

(

z

)

Gμν

(

x

)

×



G(

x

,

y

;

1

)

− [−

g

(

y

)

]

−1/2

δ

D

(

x

−

y

)

˜

G

(

y

,

z

)

Rμν

(

z

) .

(7)

The quantity

G

is the kernel expressing the derivative operator exp H.Oncea givenbackground metric gμν is specified,one can calculate the eigenfunctions of the operator

2

and write down the generalization of (6) [32] for

G

. However, this procedure is inconvenientbecauseitisbackgrounddependent and, ingeneral, the eigenvalue problem of the curved

2

can be challenging. In-stead,firstwe definethekernel

G

formally,tounderstandwhere itcomesfrom,andthenwe findasetofmasterequationswhich can be solved explicitly to find

G.

The formal definition is as a Greenfunctionsolving

e−H(2x)

G(

_x

_,

_y

;

₁

₎

=

δ

D

₍

_y

₋

_x

₎

√

−

g

(

y

)

,

(8)

wherethe number“1” inthe arguments of

G

will be explained shortly.Similarly,thequantityG is

˜

theGreenfunctionsolving[39]

2 ˜

G

(

y

,

z

)

=

δ

D

₍

_y

₋

_z

₎

√

−

g

(

z

)

.

(9)

Inthisway,

2

−1 is expressedin termsof G.

˜

Bydefinition, these kernelsmake(7) fullyequivalentto(1),if(8) and(9) arewelldefined.

Supposetheyare.Thenonecanformallyinvert(8) and(9) toget



dDy



−

g

(

y

)



dDz



−

g

(

z

)

Gμν

(

x

)

×



G(

x

,

y

;

1

)

−

δ

D

₍

_x

₋

_y

₎

√

−

g

(

y

)



˜

G

(

y

,

z

)

Rμν

(

z

)

(9)

=



dDy



−

g

(

y

)



dDz Gμν

(

x

)

×



G(

x

,

y

;

1

)

−

δ

D

₍

_x

₋

_y

₎

√

−

g

(

y

)



δ

D

(

y

−

z

)

1

2

Rμν

(

z

)

=



dDy



−

g

(

y

)

Gμν

(

x

)

×



G(

x

,

y

;

1

)

−

δ

D

₍

_x

₋

_y

₎

√

−

g

(

y

)



1

2

Rμν

(

y

)

(8)

=



dDy G_μν

(

x

)



eH(2x)

−

₁

_δ

D

₍

_x

−

_y

₎

1

2

Rμν

(

y

)

=

Gμν

(

x

)

γ

(

2

x

)

Rμν

(

x

) .

However,theproblemisthat(8) is notwell definedatall!To seethis,letuskeeptheparameter

σ

in(4) genericandwrite(10) asthes

→

1 limitoftheformalexpression

G(

x

,

y

;

σ

)

=

eHσ(2x)

_G(

_x

_,

_y

;

₀

_),

₍₁₀₎

where

G(

x

,

y

;

0)

:= [−

g

(

y

)

]

−1/2

_δ

D

₍

_x

₋

_y

_).

_{Thisexpressionis} espe-ciallydifficulttodealwithbecauseinnon-localquantumgravities thepropagator1_{issuppressedintheultraviolet,sothatitsinverse}

1 _{Thepropagatorofthisclassoftheorieswasamplydiscussedintheliterature;} see,e.g.,[7,8,10–12,38] andreferencestherein.Inparticular,wecanusethe

Feyn-(theformfactorexp H)explodesinEuclideanmomentumspacein allrealisticcases.2Toavoidthis,weconsidertheinverse

F := G

−1, definedimplicitlyas



dDz



−

g

(

z

)

G(

x

−

z

;

σ

)

F(

z

−

y

;

σ

)

=

δ

D

₍

_x

₋

_y

₎

√

−

g

(

y

)

,

(11)

forwhichthenon-localoperatorin

F(

x

,

y

;

σ

)

=

e−Hσ(2x)

_F(

_x

_,

_y

_;

₀

₎

₍₁₂₎

isdamped athighenergies. Once

F

isfound, onecan determine

G

withdeconvolutionmethods[36].

3. Masterequations

Tosummarize,werewrotethenon-localaction(1) as(7).Since (10) isill-definedathighmomenta,wecouldnotfindtheexplicit expression of

G

directly andwehadto introducedits inverse

F

, definedby (11) and obeying (12). Thus, thenon-local system(1) has been fully recast in terms of well-defined kernel functions, equations(7) and(11).Now(12) iswell-definedathighmomenta, butitstill isanon-localequationwithinfinitelymanyderivatives and,ingeneral,we donot knoweitherhowtosolveitorhowto makesenseoftheinitial-valueproblem,orboth.

Toaddressthisissue,wemakeacrucialobservation:anyform factorcanbewrittenintermsofakernel

F

governedbyasimple systemof renormalization-group-likeequations, which determine how

F

variesinthespaceofallpossiblefunctionalsP

(

2)

.Thisspace isparametrizedby

σ

,where

σ

=

1 corresponds totheendofthe flow. The exact form of these master equations depends on the choiceofH(

2)

,whichislimitedbyrenormalizabilityandunitarity. Forthe generalclass f

(ω)

=

exp(

−

ω)

(asymptotically polyno-mial Hpol_{σ ;} includesKuz’min andTomboulis formfactors),we can finallywritethefinite-orderdifferentialequations:

σ

∂

σ

F

(

x

,

y

;

σ

)

=

α

[(

K

F)(

x

,

y

;

σ

)

−

F(

x

,

y

;

σ

)

] ,

(13)

(K

F)(

x

,

y

;

σ

)

:=



dDx

√

−

g

K(

x

,

x

;

σ

)

F(

x

,

y

;

σ

) ,

(14) [

∂

σ

+

P

(

2

x

)

]

K(

x

,

y

;

σ

)

=

0

,

(15)

F(

x

,

y

;

0

)

=

K(

x

,

y

;

0

)

= [−

g

(

y

)

]

−1/2

δ

D

(

x

−

y

) ,

(16) where

K

isthekernelassociatedwiththeoperator f

[

σ

P

(

2)]

.This isequivalenttothesystem(7).Infact,(15) corresponds,in quan-tumgravities,todiffusioninthecorrectdirectionanditssolution

K(

x

,

y

;

σ

)

=

e−σP(2x)

K(

_x

_,

_y

;

₀

₎

₍₁₇₎

iswelldefined.Then,notingthat

∂

σHσ

(

2) =

α

P

(

2)

1

−

f

[

σ

P

(

2)]

σ

P

(

2)

=

α

1

−

e−σP(2)

σ

,

(18)

from(12) onefindstheleft-handsideof(13),

σ

∂

σ

F

= −

σ

∂

σHσ

(

2)

F

=

α



e−σP(2)

−

1

F

.

Thiscoincideswiththeright-handsideof(13),since

manprescription[7,12],whicheventuallygivesrisetoamicro-causalityviolation [38].Thenumberofd.o.f.isnotaffectedbythischoiceanditwillnotbe men-tionedfurtherinthispaper.

2 _{For instance, for Kuz’min Euclidean form factor (see below) H}

σ(−k2)=

α[ln(l2_k2

)+ (0,l2_k2

)+γE]→ +∞ as |k|→ ∞; for the string form factor, Hσ(−k2

K

F

(14

=

) (17)



dDx



−

g

(

x

)

e−σP(2x)

K(

_x

_,

_x

;

₀

₎

F(

_x

_,

_y

;

_σ

₎

(16)

=

e−σP(2x)



dDx

δ

D

(

x

−

x

)

F(

x

,

y

;

σ

)

=

e−σP(2)

F

.

Therefore,(12) isthesolutionof(13).

In the much simpler case of the general class f

(ω)

=

1

−

ω

(exactlymonomialHexp_σ ;includesthestring-relatedandKrasnikov exponentialformfactors),(13)–(16) arereplacedbyjustone mas-terequation:

[∂

σ

+

α

P

(

2

x

)

]

F

(

x

,

y

;

σ

)

=

0

,

(19)

whose solution

F(

x

,

y

;

σ

)

=

e−ασP(2x)

_F(

_x

_,

_y

_;

_{0) is well defined} and coincides with (12). The integro-differential equations (13)–(16) or(19) arethegeneralizationsofthediffusion-equation method[35,37] andofthecasewithan exactlypolynomial H(

2)

treatedinRef. [38].Thus,theoriginal,formidable non-local prob-lemis reducedto onewitha finite numberofderivativeswhere we cancount theinitial conditionsandthefield degreesof free-dom.Bothturnouttobefinite.

4. Initialconditionsforspecialformfactors

Let usconsider the system (13)–(16) with P

(

2)

= −

l2

₂

_{. The}

systemgivenby (16), (13), and

(∂

σ

−

l2

2

x

)

K(

x

,

x

;

σ

)

=

0 is sec-ondorderinspacetimecoordinatesandfirstorderinthediffusion parameter

σ

.In synchronous gauge, themetric in D dimensions

simplifiestods2

= −

dt2

+

hi j

(

t

,

x

)

dxidxj,wherei

,

j

=

1,

. . . ,

D

−

1,

hi j isthemetricofthe spatial section,andthecovariant d’Alem-bertianoperatoronascalartakestheform

2 = −∂

2 t

−

1 2h i j_h

˙

i j

∂

t

+

1

√

h

∂

i

√

hhi j

∂

j



.

(20)

Therefore, to solve the second-order master equations we only need tospecify the spatial metricand its first time derivative. If we insist in havingonly second-order (in spacetime coordinates) differentialequationswithadiffusion-likestructure,thenthereare onlytwochoicesfortheformfactor (2): Kuz’min’s profile Hpol

₍

₂₎

with P

(

2)

= −

l2

₂

_{or the string-related H}exp

₍

_{2) = −}

_l2

₂

_{. By}

re-moving one ambiguityof theproblem[choiceof P

(

2)

] oncethe other [choiceof f

(ω)]

isfixed by requiring a diffusionequation, thelong-standingquestionabouttheuniquenessofnon-local grav-ityissolved.Sincethegeneralizeddiffusionmethod(13)–(16) can beappliedtoany f

(ω),

itpermitstoclassifyallallowedform fac-tors.

Therefore,whenP

(

2)

= −

l2

2

thewholenon-localityinthe ac-tionandintheequationsofmotion(EOM)iscompletelyspecified by second-order differential equations, together with the metric and the first time derivative of the spatial metric. If we impose the retardedboundary condition G

˜

(

x

,

y

)

=

0 for y0

>

x0, (9) de-finestheretardedGreenfunctionG at

˜

timex0_{onlyfromthevalue}

ofhi j anditsfirsttimederivative,fortimes

≤

x0.

5. Equationsofmotion

Thevariationofthenon-localactionwithrespecttothemetric foragenericformfactorisdoablebutcomplicatedforthe nonex-pert.Whilevariationofcurvaturetermsgivesatmosttwo deriva-tives,thesourceofinfinitederivativesintheEOMisthevariation

δγ

(

2)/δ

gμν when

γ

isexpressed (as done in mostapproaches) asa series of

2

powers. There lies the difficulty in understand-ing the Cauchy problemin non-local quantum gravity. However,

thanks to thekernel(instead ofseries) representationand to the master equations(13)–(16) or (19), weare inapositiontocount theinitialconditions.Thestrategyisfirsttocalculatethevariation oftheoriginalsystem(1),whereallmetricdependenceisexplicit, and then to use the kernel representation of the resulting non-localderivativeoperators.Ononehand,variationofthecurvature termsRμν andGμν intheaction(1) givestermswithatmostfour derivativesactingonthesamefield:

(δ

Gμν

)

γ

Rμν

+ (δ

Rμν

)

γ

Gμν

δ

gμν

⇒

4 derivatives

,

(21)

wheretheformfactor

γ

isexpressedintermsoftheaboveintegral kernelsfor

2

−1_{andexp H.Noticethat,contrarytoDeser–Woodard}

theory[39],wedonotcountout twoofthefourderivativesfrom theinverse

2

representedbyG.

˜

3 Thevariationof

γ

is

δ

γ

= −

1

2

(δ

2)

γ

+

α

l2n

2

n



k=1 1



0 dsesHσ σ



0 d

σ

 1



0 ds

×

e−sσP

2

k−1

(δ

2)2

n−ke(s−1)σPe(1−s)Hσ

.

(22) Itinvolvesformfactorsofthetype

2

kwithk

>

0,

2

−1,exp

2

n(for a monomial P

(

2)

= (

l2

₂₎

n_{), andexp H.Theoperators}

₂

k _areleft astheyare,sincetheirkernelrepresentationisthe2k-thderivative oftheDiracdeltaanditdoesnotreducethenumberofderivative operatorsintheequations.Alltheotheroperatorsadmittheabove kernelrepresentations,satisfying(9) inthecaseof

2

−1_,4_{(16) and} (19) inthecaseofexp

2

n,andthemaster equations(13)–(16) in thecaseofexp H.Therefore,usingthekernelrepresentations(22) contributes with 2(k

−

1)

+

2

+

2(n

−

k

)

=

2n derivatives, which actonthetwoderivativesintheRicciorEinsteintensorproducing termswithatmost2n

+

2 derivativesactingonthesamefield:

Gμν

(δ

γ

)

Rμν

δ

gμν

⇒

2n

+

2 derivatives

.

(23)

Thus,theEOMwithrespecttothemetricareoforder2n

+

2.This is one ofthemain findings ofthe presentwork:that there exist suitable kernelrepresentationsofthe formfactorsobeying finite-order equations.The derivative order ofthe systemgiven by the EOMandthemasterequationsisthus2n

+

2.

The conclusion is that theEOM can be expressed in termsof the kernels

K

and

G

livingin the spaceofform factors andthe Green function G

˜

(

x

,

y

).

These kernelsare non-local becausethey depend on two spacetime points, butthey are determined by a set ofequationsindependentofthe actualEOM. Thissetmaybe more or less difficult to solve, but it is well defined. Therefore, the EOMcoming from(7) arenon-localintegral equationsbut fi-nite differentialequations:forP

(

2)

=

O

(

2

n

),

theyareofderivative order 2n

+

2 and we must specify2n

+

2 initial conditions.This resultagrees withthe countingin anon-local scalarfield theory [37,40], and withthe diffusion method in Lagrangian formalism [38]. Deser–Woodardnon-localgravity[39] admitsa welldefined counting, too,although thediffusion methodisnot neededthere duetothedifferentnatureoftheformfactortherein.

3 _{Infact,the roleofthe2}−1 _{operatorisdifferentinthetwotheories.Inthe} presentcase,itcancelsthe O(2)leadingterminthenumeratoroftheform fac-tor(2),makingitentire:inaderivativeexpansion,γ(2)=const+O(2).See[38, section 3.3] formoredetails.

4 _{Thisoperatorshouldbetreatedinthekernelrepresentationratherthan} try-ing to absorbit with positive powers of the 2 to combine it to anoperator 2k−2

δ2. . ..Thisisclearfromthek=1 termin(22),whichis oftheirreducible form2−1

Up to this point, we concentrated on a class of super-renormalizable theories,but themain result isinsensitive to the introductionofother specialoperatorsthatmakethetheoryfinite inevendimension.Here byspecialwemeannon-local“killer op-erators”[10] atleastcubicinthecurvature,namely,

R

γ

k,1

(

2)

R

2,

R

2

_γ

k,2

(

2)

R

2,

. . .,

R

D/2

γ

k,···

(

2)

R

2 and

R

2

R

D2−2

×

γ

_k,···

₍

2)

R

2_,

where

γ

k,··· mayall differ fromoneanother and,again,

R

isthe

generalized curvature. These operators increase the order of the EOM.InD

=

4,killeroperatorsarecubicorquarticintheRiemann tensorandtheorder oftheEOM is thesame,namely, 2n

+

2.In particular,forn

=

1 four-dimensionalfinite non-localgravityonly needsfourinitialconditionatthenon-perturbativelevel.

6. Degreesoffreedom

Havingcountedthenumberofinitialconditions,wealso com-mentonthenumberoffielddegreesoffreedom,i.e.,thenumber of independent components of the tensor fields populating the theory.Weintroducetwoauxiliaryfields,arank-2symmetric ten-sor

φ

μν andascalarfield

ψ,

toinfertheexactnumberofd.o.f. Let usconsidertheLagrangian

2

κ

2

_L

₌

_R

₊

_2G

μν

γ

(

2) φ

μν

− φ

μν

γ

(

2) φ

μν

+

R

γ

(

2) ψ + ψ

γ

(

2) ψ/(

D

−

2

) .

(24)

TheEOMforthetensor

φ

μν andthescalar

ψ

are

φ

μν

=

Gμν

,

ψ

= φ =

G

.

(25)

ψ

is just the trace of

φ

μν and is not an independent degree

offreedom [38]. Eliminatingthe auxiliary fields fromthe action, we end up withthe original action (1). Notice that (25) implies thetransversecondition

∇

μ

φ

μν

=

0 on-shell, asaconsequenceof

Bianchiidentity.TheEOMforthemetricaremoreinvolved,butnot overlyso,andagreewiththecasewithoutauxiliaryfields[38].It turnsout that we only deal withsecond-order differential equa-tions[38], upto

γ

factorsthat, aswe haveshownabove,can be dealtwith the diffusion-equation method without increasing the derivativeorder.Ontheotherhand,byasimplecountofthe inde-pendentcomponentsofthefields,wefindthatthed.o.f.are:

(I) Graviton gμν : symmetric D

×

D matrixwith D

(

D

+

1)/2 in-dependententries,towhichonesubtracts D Bianchiidentities

∇

μ Gμν

=

0 andD diffeomorphisms(thetheoryisfully diffeo-morphisminvariant).Total: D

(

D

−

3)/2.In D

=

4,thereare2 degreesoffreedom,theusualpolarizationmodes.

(II) Tensor

φ

μν :symmetricD

×

D matrixtowhichonesubtractsD

transverseconditions

∇

μ

φ

μν

=

0.Total:D

(

D

−

1)/2.InD

=

4,

thereare6degreesoffreedom.

Thegrandtotalis D

(

D

−

2).Intheminimal casen

=

1 in D

=

4, theEOM are fourthorder andthe degrees offreedom are eight, justlikeinlocalStellequadraticgravity[41].InarbitraryD

dimen-sions,theEOMarestillfourthorder,butthenumberofdegreesof freedomis D

(

D

−

2).Therefore,ingeneralthenumberofd.o.f.is not half the numberof initial conditions, asin local gravity, ex-ceptinfourdimensions.This D

=

4 caseisonlyacoincidence.On onehand,thecountingoffieldd.o.f.(numberofindependentfield components)isthe sameasinStellegravity becauseitisnot af-fectedby the presenceof theform factor.On theother hand, as wesaw,thenumberofinitialconditionsdependsonthechoiceof formfactorbutisindependentofthenumberofdimensions.

We here expand on the nature of these particle d.o.f. in Minkowskispacetime,followingclosely[42].Firstwefocusonthe theoryin D

=

4 dimensions.

In orderto distinguishthe d.o.f. ofthe theory (1), it isuseful toseparatethemassivefromthemasslessfields.Weconsiderthe action(1) atthequadraticorderintheperturbationhμν ,thelatter beingdefinedby gμν

=

η

μν

+

2κhμν :

L

(2)

_{= −}

1

2h

μν

₂

_eH_P(2)

μνρσhρσ

−

hμν

2

eHP(μνρσ0) hρσ

,

(26)

wheretheprojectorsP(2)_{and P}(0)_{aredefinedas}

P_μνρσ(2)

=

1 2

θ

μρ

θ

νσ

+ θ

μσ

θ

νρ



−

P(μνρσ0)

,

Pμνρσ(0)

=

1 3

θ

μν

θ

ρσ

,

θ

μν

=

η

μν

−

∂

μ

∂

ν

2

.

We now proceed asin local quadratic gravity [42] and, in (26), wefirstreplacethegraviton hμν withhμν

+ 

μν and, aftersome

intermediatefieldredefinitions,thetensor



μν with



μν

= 

μν

+

η

μν

χ

−

2

eH

−

1

2

∂

μ

∂

ν

χ

,

(27)

where



μν is a symmetric rank-2 tensorand

χ

is a scalar. The

outcomeatthesecond orderintheperturbationsisthefollowing Lagrangian:

L

(2)

₌

_L

E

(

hμν

)

−

3

∂

μ

χ

∂

μ

χ

+

3

χ

2

eH

₋

₁

χ

−

L

E

(

μν

)

−

1 2



μν

2

eH

₋

₁



μν

+

1 2



ρ ρ_eH

2

₋

₁



σσ

,

(28)

where

2

= ∂

μ

∂

μ and we introduced Einstein’s linearized

La-grangian(foragenericfield Zμν )

L

E

(

Zμν

)

=

1 2Zμν

2

Z μν

₋

1 2Z ρ ρ

2

Zσσ

+

Zμν

∂

μ

∂

νZλλ

−

Zμν

∂

ρ

∂

νZμρ

.

(29)

Selectingthegauge-independenttermsoftheaction,weget

L

(2)

₌

_L

E

(

hμν

)

+

3

χ

2

1

−

e−H

χ

−

1 2



μν

η

μρ

η

νσ

−

η

μν

η

ρσ



2

1

−

e−H



ρσ

.

(30)

Noticethatthelocallimitoftheformfactorsin(30) is

γ

−1eH

=

2

1

−

e−H

2

H

,

(31)

whichreducesto Stellegravity(mass termsfor

χ

and

ψ

μν )only

inthetheorywithstring-related formfactor,whereH

∝ 2

.Inany othercase,thelocallimitofthetheoryisnotStellegravity.

Thespinstructureofthefieldsisnotaffectedbytheform fac-tors,whichareentirefunctions,andisthesameasinStellegravity. Therefore, we do not need to repeat the discussion in [41–43]. However, non-localityradically changes the propagation of these fieldsand,ultimately,thephysicalcontentofthetheory.Thus,on onehandthetheory(1) describesamasslessgravitonfield,a spin-two field5 _{with kinetic term withthe wrong sign (analogous to} Stelle’sPauli–Fierzmassiveghost field),anda scalarfield.Onthe otherhand,thegauge-invarianttermsofthepropagatorsfor

χ

and



μν arebothproportionaltotheinverseof(31),

γ

e−H,whichhas

nopolesbydefinitionofH.Theconclusionisthatthespin-2ghost

5 _{Takingthedivergenceand thetraceoftheequationsofmotion(asdonein} [43] forStellegravity) oneobtains twoconditions ∇μμν=0 and μμ=0 on

Minkowski spacetime.Thefieldμν issymmetric, transverse,andtracelessand

andthescalarpresentinStellelocaltheory[41] donotpropagate attheperturbativelevelinnon-localgravity.

Furthermore,thesixnon-propagating d.o.f.



μν and

χ

are

ex-actly the same of

φ

μν up to a change of basis. We can safely

concludethat theaction (24) describesexactly thesame d.o.f. of Stelle’s theory,that are now harmless thanks to the non-locality oftheaction. Finally,itwas recentlyproved [44] that Minkowski spacetimeisstablenotonlyathigherperturbativeorder,butalso toallordersinthegraviton perturbation.Thisresultwasthen ex-tended to any Ricci-flat spacetime (stable in non-local gravity if stablein general relativity) [45]. Therefore, the field

φ

μν , orthe

fields

χ

and



μν ,neverpropagate atanyarbitrarilyhigh

pertur-bativeorder.

In D dimensions, the number ofindependent components or the fieldschanges, butthe spin decompositioninto a graviton, a spin-2 ghost and a scalar is the same.This was showed in [46] forhigher-derivativelocaltheories,butitistruealsofornon-local theories,sincetheformfactorsonlyaffectthepropagationofthese modes,not theirspin.The non-propagationofthespin-2 andthe scalard.o.f. can becheckedby couplingthe graviton tothe most general energy tensor



μν and computing the transition ampli-tudeinmomentumspace,namely,

A

:= 

μν

₍

_k

_)O

−1

μν,ρσ



ρσ

(

k

) .

(32)

where

O

_μν−1_{,ρσ is}thepropagatorforthetheory(1).Theresultis

A

=



μν



μν

₋

(μμ)2 D−2 k2

−





μν



μν

−

(

μμ

)

2 D

−

1



1

−

e−H k2

+



(

μμ

)

2

(

D

−

1

)(

D

−

2

)



1

−

e−H k2

.

(33)

Again,thegravitonistheonlyone propagatinginanydimension, whileforthemassivespin-2 field andthescalarwe donothave anypole,since

[

1

−

exp(

−

H)

]/

k2 _isentire.

We haveproved that the number ofphysical d.o.f. isat most 8 in D

=

4 dimensions and in the minimal theory, but at the momentwe cannot excludethe possibilitythat other constraints, identities, or symmetries will further reduce this number. How-ever,ourresultalreadyplacesstrongstakesonthetheory.Athigh energy andshort distances, the equivalence principle states that spacetimeisMinkowski.Theextrad.o.f.mightpropagateonother, non-Ricci-flat spacetimes (which, if the extra perturbations have a ghost-likeortachyonicnature, can beunstable andthen decay instantaneously, or in a finite time since the theory is non-local [47–49],intoanotherspacetime)butnotintheultravioletregime. Theliteratureisrepletewithotherexampleswheresomedegrees of freedom are non-physical in certain backgrounds. In Hoˇrava– Lifshitzgravity,some scalarmodesdonot propagateatthelinear perturbation level on a cosmological background due to the ab-senceoftimederivativesintheirequationsofmotion[50–52]. Cer-tain quantizationprescriptions of Lee–Wickgravity theories have modes, calledLee–Wickparticles[53] orfakeons[54], that prop-agate inside Feynman diagrams but decouple from the physical spectrum on Minkowskispacetime. Ourresults are more general becausethey hold atarbitraryperturbative order andfora wide classofbackgrounds,buttheydonot differfromothersituations like these where not all the fields in the theory appear in the physicalspectrum on special backgrounds (in our case, all Ricci-flatspacetimes are ghost-free).Theextra modespropagateinside Feynmanloops(off-shell)butnotinexternallegs(on-shell).

7. Applicationsandconclusions

Some years ago, the diffusion-equation method revealed that a string-motivated non-local scalar field theory with exponen-tial operators has a finite number of initial conditions and non-perturbative degreesoffreedom [37].InthisLetter, weconverted theinfinitenumberofderivativesofrenormalizablenon-local grav-itieswithhighlynontrivialnon-localoperatorsintointegralkernels livinginthespaceofallpossibleformfactorsandthatdonotcarry pathologicalnon-locality.Themainaction(7) mustbeplacedside by side with the finite-order differential equations given by the system (13)–(16) or (19). The systemof EOM forthe metricand the kernels is of finite order 2n

+

2 for the metric, for a form factor (2) with H givenby Hpol _{with P}

₍

₂₎

_{∝ 2}

n_{. Forthe special} formfactorwithn

=

1,wherekernels satisfysecond-order differ-ential equation, theEOM forthe metricare fourthorderandthe number of non-perturbative degreesof freedom is eight in four dimensions. Twoofthese d.o.f. are perturbative andwell known intheliteratureand,thanks toasymptoticfreedom,they describe thetheorycompletelyatsmallscalesonaMinkowskibackground [9], i.e., at the scales of the local inertial frame of the observer where the backgroundis approximately flat.The new degreesof freedom foundinthispaperarenon-perturbativeandmightplay acapitalroleatlargescalesbutnotathighenergy.Bylargescales, onemeansscaleswheretidalforcesbecomeimportant.The diffu-sionmethodproposedherereachesthosesolutionsthatcannotbe found withtheavailablemethods appliedtolinearizedEOM.This is themostdirectandimportantconsequenceofourfindings for thephysicsofnon-localgravity.

These results repair the old and bad reputation of non-local theories for having an ill-defined Cauchy problem. Armed with thegeneralizeddiffusionmethod,onecanconstructunambiguous analytic solutions of cosmologicaland astrophysical backgrounds, previously inaccessible viathe typical Ansatz

2

R

= λ

R in the lit-erature.Fromthesenew,fullynon-perturbativesolutions,onewill be able to extractconclusions on the stability orabsence of big-bang and black-holesingularities, find predictions on the cosmic evolution induced by non-local gravity, and check the theory at thenon-perturbativelevelagainstpresentandnear-future observa-tions,suchasthoseof Planck [55,56] andLIGO[57,58],according toavariedbatteryofphenomenologicalteststhathasalreadybeen effectiveforstringcosmologyandotherquantum gravities[3,59]. Supported by these, other recent, and upcoming data, the phe-nomenologyandtestingofnon-localquantumgravity isaboutto bloom.

Acknowledgements

G.C. and L.M. are supported by the I+D grant FIS2017-86497-C2-2-PoftheSpanishMinistryofScience,Innovationand Univer-sities.

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