The correspondence principle in quantum field theory and quantum gravity

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The correspondence principle in quantum field theory and quantum gravity

Damiano Anselmi

To cite this version:

Damiano Anselmi. The correspondence principle in quantum field theory and quantum gravity. 2018.

�hal-01900207v2�

In Quantum Field Theory

And Quantum Gravity

Damiano Anselmi

Dipartimento di Fisia Enrio Fermi, Università di Pisa

and INFN, Sezione di Pisa,

Largo B. Ponteorvo 3, 56127 Pisa, Italy

damiano.anselmiunipi.it

Abstrat

Wedisussthefateoftheorrespondene priniplebeyondquantummehanis,speif-

ially in quantum eld theory and quantum gravity, in onnetion with the intrinsi lim-

itations of the human ability to observe the external world. We onlude that the best

orrespondene priniple is made of unitarity, loality, proper renormalizability (a rene-

ment of strit renormalizability), ombined with fundamental loal symmetries and the

requirement of having a nite number of elds. Quantum gravity is identied in an es-

sentially unique way. The gauge interations are uniquely identied in form. Instead,

the mattersetor remains basially unrestrited. The majorpredition isthe violation of

ausalityat smalldistanes.

Bohr's orrespondene priniple is a guideline for the seletion of theories in quantum

mehanis. It is useful to guess the right Hamiltonians of several mirosopi systems.

Basially, it states that the laws of lassial physis must be obtained from the laws of

quantumphysisinasuitablelimit,whihistypiallythe limitoflargequantumnumbers.

More generally, the limitis alledlassiallimit.

Itmaysoundaredundantrequirement. Anynewlydisoveredlawsofnaturehavetobe

onsistentwith the knowledgealreadyavailable,sine natureisone andannotontradit

itself. At the same time, if we view the priniple as a guideline for the preseletion of

theories, it ould be argued that antiipating the laws of physis might be dangerous. It

ouldmisleadintodisarding possibilitiesthatmightturnout toberight. Thismay delay

or prevent the disoveries of new laws. In view of these arguments, it might be better

to put the idea of a orrespondene prinipleaside and proeed as in every other domain

of siene, whihmeans making as many experiments as possible and expressing the laws

of physis in mathematial language. Quantum physis will exhibit an approximatively

lassial behavior whenever naturesays so.

Thewholepoint,however, isthatthis proedureisnot suientatthe quantum level,

beause our possibilities of observing the mirosopi world are severely handiapped by

the laws of physis themselves. Certainly, they are muh smaller than our possibilitiesto

observe the marosopi world.

This fat is hidden in the meaning of the word quantization, whih understands that

a quantum theory is not built from srath, but instead guessed from another theory,

typially a lassial one, whih is later quantized. So, there must be some sort of orre-

spondene between the two. Per se, this way of proeeding is kind of awkward, sine the

quantum theory issupposed tobethe rightone and the lassialtheory issupposed tobe

anapproximationof it. Howan weget thenal theoryorret, ifwe guessitfromalimit

of it? There must be many theorieswith the same limit. Howan we deidewhih one is

the right one?

Yet, for a variety of reasons that we are going to emphasize here, this guesswork is

the only thing we an do. And it gets worse when we plan to explore smaller sales

of magnitude, where quantum eld theory plays a key role. There, the orrespondene

between quantum physis and lassial physis beomes weaker. For example, quantum

hromodynamis has a lassial limit that has little to do with lassial physis. The

problem beomes even worse when we move tosmaller distanes, where quantum gravity

the lassial world willeventuallyfade away and disappear, leaving uspowerless.

We think that the lessons learned from the standard model and ertain reent devel-

opments in quantum gravity make us ready to take stok of the situation and properly

addressthe problemof the orrespondene priniplebeyond quantum mehanis. Collet-

ingthe pieesof thepuzzle,we see thatthere isindeedaneedtosupplementthe sienti

method with some sort of guidelines, although we know in advane that their power is

doomed tobeomeweakerand weaker when the distanes get smaller and smaller.

In this paper we onsider assumptions, priniples and requirements of various types

that have been proved to be important inquantum eld theory and high-energy physis.

We ommenton their ranges of validity and use them tooer anupgraded version of the

orrespondene priniplethat better ts the knowledge gathered sofar.

To begin with, we stress that we take quantum eld theory for granted. We might as

well add

0. Quantum eld theory

as the zeroth assumption of the upgraded orrespondene priniple. We think that the

suess of quantum eld theory in partile physis is a suient reason to justify the as-

sumptionwithoutfurtheromment. Moreover, forthe reasonsbetterexplainedinthenext

setion, we have tobe asonservative aspossible. Our ability to observe the mirosopi

world is intrinsially limited, so it is not onvenient to depart too muh from the kind of

orrespondenethathasworkedsofar,whihistheonlysoureoflightwehaveinarelative

darkness. In this respet, we think that approahes alternative to quantum eld theory,

or even olorful twists of quantum eld theory, like holography, have very few hanes of

suess.

2 Do we need a orrespondene priniple?

In this setion wedisuss the need of a orrespondene priniplein quantum eld theory,

whatitsmeaningissupposedtobeandwhatitshouldbeusefulfor. Westartbyonsidering

the intrinsibiologiallimitationsofthe humanbeing,whihneessarilyaet the way we

pereive and desribe the world. Withobviousmodiations, the argumentsapply toany

living being.

First, we must take into aount that we do have a size. Our body has a size, like

our brains and ells, and the atoms of whih we are made. Seond, most of our diret

for larger and smaller distanes, are indiret,mediated by the instrumentswe build.

Therangeofourdiretpereptionsdenes thesizeoftheenvironmentweareplaedin.

Forexample,wehaveeyesthatpereiveuptoaertainresolution,whihis

3 · 10 ⁻⁴

^radians

[1℄. Getting loser to an objet, we an improve the auray of our visual pereption.

Combining our mobility with our eye resolution, we an stillpereive only a very limited

portionofthe universe,whihwealldiretpereptionrange (DPR).Forthe rest,wemust

relyonindiretmeasurementsandobservations. Inthe longrun,indiretpereptions may

introdue and propagate errors, partiularly in onnetion with the notions of time and

ause.

Foronreteness andforvariousargumentsthatweplantodevelop inthenext setion,

we fous on ausality. Everyday experiene tells us that many events are onneted by

relations of ause and eet, that time has an ordering (past

→

^present

→

^{future), that}

the past inuenes the present and the future, but the future annot aet the past and

the present. However, time isanextremely deliatepereption. Stritlyspeaking, wehave

never seentime. Wean see the threespaedimensions, but timeremainsonned toour

imagination. Insome sense, time is the way we organizememories. This makesthe entire

onept kindof fuzzy.

Speially,thehumanbrainanproessanimagepereivedforaround

10 ⁻³

^s[2℄(being

optimisti). Weallitthe diret pereption timeresolution (DPTR). Basially, wesee the

world ataround 1000fpsand establishdiret relationsof ause and eetbetween pairs of

events that are separated by atleast the DPTR.Yet, very unlikelythe ausalitypriniple

breaks down rightbelowthe DPTR: thatwould meanthat the universe onspiredto trik

us. We haveto assume that ausality doeshold for shorter time intervals.

Ifwehelpourselveswithinstruments,weanresolvetimeintervalsthatarewayshorter

than thelimitsofour diretpereption. Thereexist

5 · 10 ¹²

^{fps ameras,whihanapture}

light in motion [3℄. The shortest time interval ever measured is about

10 ⁻¹⁸

^{s [4℄. There}

are elementarypartiles with meanlifetimes of about

10 ⁻²⁵

^s.

If we think a moment, we build our instruments on the work hypothesis that the

validity of the laws of nature, in partiular ausality, an be extended below the DPTR.

When we use the instruments and ross hek that everything works as expeted, we get

an a-posteriori validation of the assumption. This allows us to onlude that, indeed,

ausality holds well below the DPTR. Yet, having heked that it extends to, say, one

billionthof abillionthof theDPTRisstillnotenoughtoprovethatitholdsforarbitrarily

short time intervals orlarge energies. Eventually, itmay break down.

laws relatively easily, beause we an turn on the light, i.e. throw a huge number of

photons onthe objet of our observation without disturbing it. Our eyes then olletthe

photons reeted/emitted by the objet, whih are also a huge number. In pratie, we

make a very large numberof experimental observations at one and pay no prie. This is

a very luky situation. Instead, our exploration of the innitesimally small distanes is

neessarilyhandiapped. Weannotturn onthe lightthere, beauseeven a singlephoton

disturbs the objet we want to detet. It is like trying and detet a rok by throwing a

building atitand hekingthe resultsof the sattering.

Lukily,whenthe distanes weexploreare not toosmall, thelaws ofnaturekeep some

similarity with the lassial laws we are austomedto. The orrespondene priniple, as

wenormallyunderstandit,dealswiththiskindofsimilaritydown tothe atomidistanes,

whihare therealmof quantummehanis. Asvagueasthe priniplemaysound,areipe

rather than a true priniple, it is useful. It desribes the relationthat linksthe world we

liveintothe onewewishto explore,the world thatshapesour thinkingtothe world that

makes us what wephysially are.

However, we expet that exploring smaller and smaller distanes, the orrespondene

will beome weaker and weaker and the devies we build will not help us indenitely.

Atually, it may happen, as quantum mehanis told us, that the possibilities of our

experimentsarelimitedbytheverysamelawsofphysis. Forexample,wehavetolivewith

the fatthat we annotmeasure the positionand the veloityof apartilesimultaneously

with arbitrary preision.

Therstdesenttosmallerdistanesafterquantummehanisisquantumeldtheory.

What is alled lassial limit there is not neessarily related to lassial phenomena, as

quantumhromodynamisshows. Nevertheless,anupgradedversionoftheorrespondene

priniplehasworkedsuessfully,sofar,atleastintheabseneofgravity,whereasimilarity

with a (sort of) lassial world has more orless survived. The upgraded priniplean be

odied by means of the requirements of unitarity, loality and renormalizability.

Theseonddesent isquantum gravity,whihmightrequiretoreonsider orrenethe

orrespondene prinipleina nontrivialway.

We may view the matter like this. Our thought is shaped by our interations with

the environment that surrounds us. In some sense, it is a lassial thought. The very

same keywords of our logi (existene, origin, priniple, onsequene, ause, eet, time,

et.) are inherited from that environment, whih means that they might just be useful

apply themtothe restof the universe,weassume thatour knowledgeisuniversal,whih

is farfromjustied. Atsome point,we might have toradiallymodify the laws of physis

and even the basi priniples of our thinking. Like it or not, the indeterminay priniple

is stillthere, despite somany people whined, tirelessly objeted and desperately searhed

for hidden variables. Atually, we should onsider ourselves luky that we did manage to

get somewhere by renouning determinism. But that means preisely that the priniples

suggested by our lassial experienes were not priniples. And that there is nopriniple

that an betrusted tothe very end.

In suh a situation we might not have muh more at our disposal than some sort of

orrespondene, even if weknowinadvanethat theorrespondene willget weakerand

weaker atsmaller and smallerdistanes. What happens ata billionthof a billionthofthe

DPTR? How would the world appear to us if we ould see it at a speed of

10 ²⁶

^{fps? Is}

it aurate to talk about auses and eets down there? And what about past, present

and future? The same an be said about every other onepts and notions we normally

use, sine they are inherited from experienes made in a radially dierent environment

and derived from a very approximate and rough pereption, if ompared with the one

that would be required. These handiaps limit our possibilities dramatially. A vague

orrespondene with what we understand better may be all we an hope for. And we

annot aord tobepiky.

Summarizing, the reason why a orrespondene priniple may be useful in quantum

eld theory and quantum gravity is rooted in howthe quantization works, sine the right

quantum theory must be identied by starting from a non quantum theory. The envi-

ronment we wish to explore is so dierent from the environment we are plaed in, that

a orrespondene between the two may be all we an get. It may help us organize the

guesswork to make progress in the relativedarkness we have tofae.

3 The orrespondene priniple

In this setion,we examinepriniples, assumptions,properties and requirements that are

relevantto quantum eldtheory. We ompare the versionof the orrespondene priniple

that ts the standard model in at spae to the versions that t quantum gravity and

the ombination of the two. The most important properties are unitarity, loality and

renormalizability, whih we separate from the rest, sine they enode a great part of the

The standard model of partile physis is a quantum eld theory of gauge elds and

matterinatspae. Sofar,ithasbeenverysuessful. Alargenumberofpreditionshave

been onrmed, in some ases with high preision. No serious ontradition has emerged

and,ifright-handedneutrinosareinluded,even theneutrinomassesanbeaountedfor,

without having to modify the fundamental priniples. Quantum gravity has been elusive

for deades, but reently a onsistent theory was formulated[5℄ (see also [6,7℄)by means

of a new quantization presription that turns ertain poles of the free propagators into

fakepartiles, orfakeons [8℄,whihan beonsistentlyprojetedaway fromthephysial

spetrum. The theory an be oupledto the standard model withno eort.

Thefakeonsbehavelikephysialpartilesinseveralsituations, e.g. whenthey mediate

interations or deay into physial partiles. However, they annot be deteted diretly,

sinethe rosssetionofeveryproessthatinvolvesthemasinitialornalstatesvanishes.

Thanks tothis property, thefakeons an beonsistently projeted away fromthe physial

spetrum. The projeted theory isunitary.

The fakeons annot be eliminated ompletely from the orrelation funtions, though,

sine they are mediators of interations. This fat leads to an important physial pre-

dition: the violation of ausality at energies larger than the fakeon masses. The masses

of the fakeons annotvanish, otherwise ausality would be broken atarbitrary distanes,

ontraryto evidene. Atually,they must besuiently large toavoidontradition with

the experimentaldata.

The violationof miroausality survives the lassiallimit[9℄, whihopens the way to

study its eets in nonperturbative ongurations, rather than insisting with elementary

proesses. If we add that the masses of the fakeons are not known and ould be muh

smaller that the Plank mass, we infer that there is hope to detet the eets of the

violation inthe foreseeable future. They ould bethe rst signs of quantum gravity.

Westress againthatinquantum mehanis andquantum eldtheory thequantization

is in some sense a dynamial logial proess that builds the right theory starting from a

limit of it. When gravity is swithed o, the starting lassial ation is also the ation

that desribes the lassial limit

~ → 0

^{(whih might have nothing to do with lassial}

physis, as in the ase of quantum hromodynamis). In quantum gravity, instead, the

starting ation, whih is (3.4), is just an interim ation, beause it misses the projetion

that throws away the fakedegrees of freedom fromthe physialspetrum. The projetion

is determined by the quantization proess itself. The orret lassial limit

~ → 0

^an

be obtained by lassiizing the quantum theory and is enoded in the nalized lassial

ationandthenalizedlassialationoinideinatspae,wheretheprojetionistrivial.

Dierent quantizationpresriptions an lead toinequivalentphysial preditions from

the same interim lassial ation. Some quantization presriptions may even lead to un-

aeptable onsequenes. Forexample, if the interim lassialation (3.4) is quantized in

the standard way, ithas ghosts, insteadof fakeons. In that ase, unitarity isviolated [10℄.

Thus, a requirement that we may inlude in a temporary version of the orrespondene

prinipleisthat anaeptablequantumeld theory shouldonlyontainphysialpartiles

and possibly fakeons, but not ghosts.

Ifthe fakeonsareexludedaswell,itispossibletohaveausalityatallenergies. Then,

however, itisnot possible toexplain quantum gravity bymeans ofaquantum eldtheory

that isloal,unitary andrenormalizable. Forthe reasonsoutlinedinthe previoussetion,

enforing miroausality is a streth, given the limitationsof our apabilities to pereive

the external world. So, the rst priniple that has to be saried when we desend to

the realm of quantum gravity is miroausality (together with the presumption that we

an impose restritions on nature based on our personal tastes). Moreover, the violation

of miroausality might atually be a bonus, instead of a prie to pay, sine, as stressed

above, there are hopesto detet it.

Priniples

Now we examine the various andidate ingredients of the orrespondene priniple, orga-

nized in dierent tiers. The rst tier ollets the most important priniples, whih are

almost suient toenode the upgraded orrespondene priniple.

1 Unitarity

It is the statement that the sattering matrix

S

^{is unitary, i.e.}

SS ^† = 1

^{. In partiular,}

there exists a physial Fok spae

V

^{suh that, if}

| n i

^{denotes an} orthonormal basis of

V

^{, the identity}

X

|ni∈V

h α | S | n ih n | S ^† | β i = h α | β i

^(3.1)

holds for every states

| α i , | β i ∈ V

^{. In some ases, the physial subspae is identied}

by means of asuitableprojetion

W → V

^{starting fromalarger, oftenunphysial Fok}

spae

W

^{. Example of unphysial elds that an be} onsistently projeted away are the Faddeev-Popov ghosts and the temporal and longitudinalomponents of the gauge

ghosts of higher-derivative propagators (whih typially appear when the poles due to

thehigherderivativesarequantizedusingtheFeynmanpresription). Examplesofelds

whih are neither physial nor unphysial and an be onsistently projeted away are

the fakeons.

The requirement enoded by unitarity is the onservation of probabilities. It ensures

that the theory is in some sense omplete, i.e. a state annot appear from nowhere

or disappear into nowhere. Basially, there annot be a preexisting, external soure or

reservoir. The osmologial onstant

Λ C

^{is an example of} preexisting onditions, so we annotdemandunitarityinastritsense inthe preseneof anonvanishingosmologial

onstant, where a meaningful

S

^{matrix might not even exist [11℄. In the far infrared}

limit (one the massive elds have been integrated out)

Λ C

^{ows to a onstant value}

Λ ^∗ _C

^{. If}

Λ ^∗ _C = 0

^{unitarity is exat, otherwise it is anomalous. Note that}

Λ ^∗ _C

^{is not the}

measured value of the osmologial onstant (indeed, we annot reah the far infrared

limit), so

Λ ^∗ _C = 0

^{is not in} ontradition with the observations. The measured value (

Λ C = 4.33 · 10 ⁻⁶⁶

^eV

²

^{)ould beduetothe rstradiativeorretions(indeed,}

m ⁴ _ν /M

Pl

² ∼ 7 · 10 ⁻⁶⁵

^eV

²

^{for neutrino masses}

m _ν

^{of the order of}

10 ⁻²

^eV).

1a Perturbative unitarity

Perturbativeunitarityisthe unitarityequation

SS ^† = 1

^{expressed orderby orderin}

the perturbative expansion. It isenoded inaset ofdiagrammatiidentities, alled

utting equations [12℄. It is often useful to refer to perturbative unitarity rather

than unitarity, sine the resummations of the perturbative series typially lead to

nontrivial widths, whih make the partiles deay (like the muon in the standard

model). Perturbative unitarity allows us to onsider proesses where the deaying

(physial) partilesare deteted diretly whilethey are stillalive.

2 Loality

Loalityisprobablytheassumptionthatismoreintrinsiallyrelatedtothequantization

proess and the orrespondene priniple. The nal quantum theory is enoded in its

S

^{matrix, orthegenerating funtional}

Γ

^{of the}one-partile irreduibleorrelationfun- tions. Both are nonloal. The nalizedlassial ation is also nonloal, unless fakeons

are absent. Thus, the requirement of loality an only apply to the interim lassial

ation. Ifwe relax this assumption, the quantizationproess loses most of its meaning.

We would have toguess the generating funtional

Γ

^{diretly and run intothe problems}

world, we have nopossibility of making innitelymany observations ina nite amount

of time and/or without disturbing the system. This means that we have no way to

determine atheory with aninnitedegree of arbitrariness.

The limitationspointed out inthe previous setion suggest that nature isnot arranged

tobefully understood orexplained by us humans. Therefore, asfar as we an tell,the

ultimatetheory of the universe mightwellbe innitely arbitrary. Some signs that this

istheaseare alreadyavailable(hek thedisussionbelowabout uniquenessversusthe

arbitrariness of the matter setor). Nevertheless, the suess of quantum eld theory

and the reent results about quantum gravity give us reasons to believe that we might

stillhave something interesting tosay, aslong aswe donotrenoune loality. Forthese

reasons, we regard the loality of the interim lassial ation as a ornerstone of the

orrespondene prinipleand the quantizationproess.

2a Perturbative loality

Perturbative loalityis the version of loality that appliestothe nonrenormalizable

theories, where the lassialation ontains innitely many terms and an arbitrary

numberofhigherderivatives. Resumming thosetermsleadsingeneraltoanonloal

lassialation. Theusualperturbativeexpansionisdenedinombinationwiththe

expansionof the lassial ationin powers of the elds and their derivatives. Every

trunation of the latter is obviously loal. Perturbative loality is the assumption

thatitmakessensetoworkwithsuhtrunationsasapproximationsoftheomplete

ation.

A nonrenormalizable theory (like Einstein gravity equipped with the ounterterms

turnedon by renormalization[13℄)is preditive atlowenergies. Forthe reasons al-

readystressed, the ultimatetheoryof theuniverse mightwellbenonrenormalizable.

However,sinewehaveabetteroptionforthemoment,whihistheinterimlassial

ation (3.8) quantized as explained in ref. [5℄, i.e. a loal, unitary and renormaliz-

able theory that explains both quantum gravity and the standard model, we think

that we an postpone this possibility and fous onloalityand renormalizability.

3 Renormalizability

Renormalizabilityhas todowith the fatthat aninterim lassialationisnot guaran-

teedtobestable withrespettotheradiativeorretionsgeneratedbytheperturbative

expansion, in partiular the removal of their divergent parts. If this kind of stablility

lassial ation in order to stabilize it. The theory is nonrenormalizable if the only

hane tostabilizeitis toinlude innitelymany terms(whihtypially means: allthe

loaltermsoneanbuild, uptoeldredenitionsandnonanomaloussymmetry require-

ments),multipliedby independentparameters. It isrenormalizableif anitenumberof

termsorindependentparametersissuient. Nonrenormalizabletheoriesare preditive

at low energies, where perturbative loality ensures that only a nite number of terms

are important (those partiipating in the trunation). Renormalizable theories an in

priniplebe preditive atall energies.

3a Strit renormalizability

Strit renormalizability means that the physial parameters have nonnegative di-

mensions in units of mass, with respet to the power ounting that governs the

high-energy behavior of the theory. The ounterterms are of nitely many types.

3b Super-renormalizability

Super renormalizability means that all the physial parameters have stritly posi-

tive dimensions in units of mass. Then the divergenes are nitely many. Super

renormalizabilitydoes not seem to befavored todesribehigh-energy physis.

3 Proper renormalizability

We introdue this renement of the notion of strit renormalizability, beause it

is partiularly useful, in ombination with other key requirements, to single out a

uniquetheory of quantum gravity(see below). Properrenormalizabilitymeans that

the gauge ouplings (inluding the Newton onstant) must be dimensionless (with

respet tothe power ounting governing the ultravioletbehaviors ofthe orrelation

funtions) and the other physial parameters must have nonnegative dimensions in

units of mass.

4 Fundamental symmetry requirements

The symmetries an be global or loal; exat, expliitly broken, spontaneously broken

oranomalous. Theloalsymmetriesare alsoalled(generalized)gauge symmetriesand

mediateinterations. They must be exat orspontaneously broken, otherwise unitarity

is violated. They inludeboth the gauge (i.e. Yang-Mills) symmetries as suh and the

loal symmetries of gravity (i.e. invariane under general hanges of oordinates and

loalLorentz invariane).

Lorentz invariane is a global symmetry in at spae. There, it an be expliitly

broken without violating unitarity, in whih ase many verties that are normally

nonrenormalizable beome renormalizable [14℄, even if the fakeons are forbidden.

However, in quantum gravity Lorentz invariane is aloalsymmetry, whih annot

bedynamiallyorexpliitlybroken withoutviolatingunitarity. Thus, wheregravity

exists, Lorentz invariane must be exat or spontaneously broken. Sine gravity

exists at low energies, Lorentz invariane an be violated only at high energies. If

this ourred, gravity would be a low-energy eetive phenomenon, emerging from

a radially dierent high-energy piture, of whih, however, there is at present no

idea. For these reasons, in most arguments of this paper we assume that Lorentz

symmetry isexat orspontaneously broken.

4b General ovariane

4 Loal Lorentz invariane

4d Gauge invariane

The seond tier of properties olletsonsequenes of the rst tier and supplementary

requirements.

5 Uniqueness

By uniqueness, or essential uniqueness, we mean that the theory is determined up to a

nite number of independent physial parameters (whih need to be measured experi-

mentally) and a nite number of options for the quantization presription. We ould

speify that the total number of possibilitiesmust be small (with respet to our bio-

logialand physiallimits),but thereisnoobjetivedenition ofsmallnessthat wean

use here, so we prefer to leave this point unresolved. Yet, it is important to emphasize

that the requirement of uniqueness exludes the nonrenormalizable theories and most

theorieswithinnitelymanyelds,unlesstheirphysialparametersaresomehowrelated

toone another and ultimatelyjust depend ona nite number of independent ones.

Investigations about onsistent redutions on the number of independent physial pa-

rameters in renormalizable theories dates bak to Zimmermann and Oheme [15℄. The

redutions in nonrenormalizable theories in at spae have been studied in refs. [16℄.

The extension of the redution to the nonrenormalizable theory of quantum gravity

(whih is the Hilbert-Einstein theory equipped with all the orretions turned on by

redenitions [13℄) poses serious problems, sine there is no way to have a perturbative

ontrolon it. However, the redution has a hane in higher-derivative gravity, due to

the presene of the higher-derivative quadratiterms, whih play a ruialrole [17℄.

A orrespondene priniple that points to a unique theory would overome our handi-

apped pereption ofthe mirosopiworld. However, varioussignals,likethearbitrari-

nessofthemattersetorofthestandard model,tellusthatwemustprobablyopewith

the fat that this goalis utopian. Nevertheless, we an have uniqueness in gravityand,

to some extent, within the gauge interations, as expressed by the following points 5a

and 5b.

5a Uniqueness in form of the gauge interations

Theombinationofunitarity,1,loality,2,properrenormalizability,3 ,andLorentz

invariane, 4a, is a very powerful orrespondene priniple in at spae. Indeed, if

wealsoforbidthe presene offakeons,10d, the set oftheserequirementsimplies4d,

8 and 10, i.e. it determines the gauge transformations [18℄, the form of the ation

and even that the spaetime dimension

D

^{must be equal tofour. The ationis the}

Yang-Mills one,

S

^YM

= − 1 4

Z

d ⁴ x √

− gF _µν ^a F ^aµν ,

^(3.2)

where

F _µν ^a

^{denotes the eld strength.}

Ifwereplae3with3aor3,theallowed dimensionsare4,3,2,and1. However, the

universe predited by quantum eld theory is too simple below four dimensions, so

weregard four asthe minimumvalue. Inthis sense, 3, 3aand 3 an be onsidered

equivalent at this level.

Ifwe relax10d by allowingmassive fakeons, whih is requirement 10, then we just

have maroausality,10a, insteadof ausality, and there are solutionsinevery even

spaetime dimensions

D > 6

^{. Their interim lassialations are}

S

YM

^D = − 1

4 Z

d ^D x √

− g

F _µν ^a P (D−4)/2 ( D ² )F ^aµν + O (F ³ )

, (3.3)

where

P n (x)

^{is a real polynomialof degree}

n

ⁱⁿ

x

^and

D

^{is the ovariant}derivative, while

O (F ³ )

^{aretheLagrangiantermsthathavedimensionssmallerthanorequalto}

D

^{andare builtwithatleastthreeeldstrengths and/ortheir ovariant}derivatives.

Observe that by 3 the gauge oupling is dimensionless. We have used Bianhi

of the polynomial

P (D−6)/2

^{must satisfy a few} restritions, so that, after projeting awaythegaugemodes(byworking,forexample,intheCoulombgauge),thepolesof

the propagators have squared masses with nonnegative real parts and the massless

poles have positive residues.

The poles with negativeoromplex residues, as wellasthose with positiveresidues

butomplexmasses, mustbequantized asfakeons. The poleswith positiveresidues

andnonvanishingrealmassesanbequantizedeitherasfakeonsorphysialpartiles.

Those with vanishingmasses must bequantized as physialpartiles.

In all the situations just desribed, the gauge group remains essentially free, as

long as it is unitary and (toghether with the matter ontent) satises the anomaly

anellationonditions (whih are other onsequenes of unitarity). The knowledge

we have today does not explain why the gauge group of the standard model is

preiselytheprodutofthethreesimplestgroups,

U (1)

^,

SU(2)

^and

SU (3)

^,andwhy,

say,

SU (37)

^,

SU(41)

^{, et., are absent. We annot antiipate if other gauge groups}

will be disovered (possibly ompletely broken or onned). There ould even be

a sort of periodi table of the gauge groups and we might just have grabbed the

threesimplest representatives. This issue istied insome way tothe non uniqueness

of the matter setor(see 5below).

5b Uniqueness of the gravitational interations

Gravity does not have this problem, beause its loal symmetry (invariane under

dieomorphisms times loal Lorentz invariane) is unique. The requirements 1, 2,

3, 4b, 4, 8and 10 lead tothe unique interimlassial ation

S

^QG

= − 1 2κ ²

Z √

− g

2Λ C + ζR + α

R µν R ^µν − 1 3 R ²

− ξ 6 R ²

(3.4)

infourdimensions,andalsoseletthequantizationpresriptionsthatare physially

aeptable. In formula(3.4)

α

^,

ξ

^,

ζ

^and

κ

^{are real positive onstants, and}

Λ C

^an

be positive or negative. The ation must be quantized as explained in ref. [5℄.

It propagates the graviton, a salar

φ

^{of squared mass}

m ² _φ = ζ/ξ

^{(whih an be}

quantized as a physial partile or a fakeon) and a spin-2 fakeon

χ µν

^{of squared}

mass

m ² _χ = ζ/α

^1.

We reallthat if we use the Feynmanquantization presriptionfor allthe elds, we

obtainthe Stelletheory[10℄,where

χ _µν

^{isaghost. Inthatase,unitarityisviolated.}

1

Inthisformula,wearenegletingasmallorretionduetotheosmologialonstant[7℄.

itly. Ifwe eliminate it,the requirements 1, 2, 3, 4b, 4 and 10 admit solutionsin

every even dimensions greater than orequal tofour. Theirinterimlassial ations

read

S

QG

^D = − 1 2κ ²

Z √

− g

2Λ C + ζR + R µν P ^(D−4)/2 ( D ² )R ^µν +R P (D−4)/2 ^′ ( D ² )R + O (R ³ )

,

^(3.5)

where

P ⁿ

^and

P n ^′

^{denote other real} polynomials of degree

n

^and

O (R ³ )

^{are the}

Lagrangian terms that have dimensions smaller than or equal to

D

^{and are built}

with at least three urvature tensors and/or their ovariant derivatives. Again, the

squared masses must have nonnegative real parts and the poles of the propagators

must bequantized as explainedbefore.

5 Non uniqueness of the matter setor

As far as we know today, quantum eld theory annot predit the matter ontent

of the theory that desribes nature. For example, we an enlarge the standard

modeloupledtoquantumgravitybyinludingnewmassivepartilesand/ormassive

fakeons, as long as they satisfy the anomaly anellation onditions and are heavy

enough, so that their presene does not aet the experimental results available

today. The ultimatetheoryould even ontaininnitelymanymatterelds. This is

a point where the orrespondene priniple has been almost ompletely powerless.

Probably, it is a sign of the fading orrespondene and it might be impossible to

remedyinthe future. Letusremarkthateveryattempttorelatethe matterontent

to the interations beyond the anomaly anellation onditions (grand uniation,

supersymmetry, stringtheory and soon) has failed.

6 Analytiity

Analytiity ensures that it is suient to alulate an amplitude, or a loop diagram,

in any open subset of the spae

P

^{of the omplexied external momenta to derive it}

everywhere in

P

^{by means of the analyti} ontinuation. It holds if the theory ontains onlyphysial partiles.

6a Regionwise analytiity

Regionwiseanalytiityisthegeneralizationofanalytiitythatholdswhenthetheory

ontainsfakeonsinadditiontophysialpartiles. Thespae

P

^{isdividedintodisjoint}

but the relation between the results found in dierent regions is not analyti. The

main region is the Eulidean one, whih ontains neighborhoods of the imaginary

energies. It is suient to alulate an amplitude, or a loop diagram, in any open

set of the Eulidean region toderive it everywhere in

P

^{by means of a nonanalyti}

operation, alled average ontinuation. The average ontinuation is the arithmeti

average of the two analyti ontinuationsthat irumvent a branh point[19, 8℄.

7 Existene of interations

7a Existene of the gauge interations

7b Existene of gravity

8 Four spaetime dimensions

Quantum eld theory predits that if the spaetime dimensions of the universe were

smaller than four, then the universe would be too simple. So, they must be at least

four. It is an experimental fat that they are four at large distanes, but in priniple

they ould be more at small distanes. If that were that ase, they would have to be

ompatiedat lowenergies. This makes the higher-dimensional theoriesequivalent to

four-dimensionaltheorieswith innitesetsof matterelds. Moreover, theompatia-

tioninvolvesmanifoldsof even dimensions

> 2

^{, whihausethe appearane ofinnitely}

many independent parameters (the moduli of the ompatiation). Sine we do not

have robust arguments to restrit the matter ontent, as already pointed out, these

possibilitiesremain open.

9 Finite numbers of elds and independent physial parameters

A restritionon thematter ontent isto demand that theset ofelds be atleast nite.

Only in

D = 4

^{the theory an have nitely many elds and nitely many} parameters, beause the need of a ompatiation to

D = 4

^{introdues an innite} arbitrariness in the matter setorof the theorieswith

D > 4

^.

In the lasttier, we inludeother more or less important onsequenes and properties.

9 Consisteny requirements

9a Well-dened Hilbert (Fok) spae (positive denitenorms)

10 Causality

Everyday experiene tells us that the future is determined by the past and eets are

determined by auses. On a loser look, however, these statements are rather vague,

to say the least. In a deterministi framework, for example, it is true that the initial

onditions uniquely determine the future, but we ould turn the argument aroundand

laimthatthefutureuniquelydeterminesthepast,forthesamereason. Isitappropriate

tospeak about auses, when the future is already determined?

To avoid paradoxes like this, we should dene the notions of ause and eet with

preision. However, it isnot that easy. At most,wean replaethe intuitiveideas with

more formal or mathematialdenitions on whih we an agree. The drawbak of this

approah is that we might have todelassify them to seondary properties rather than

fundamentalpriniples. This is a riskthat all priniples fae, atually, beause atthe

end theyare formalrequirements(see setion4formore ommentsonthis),sowhihis

prinipleand whihis side property is amatter of howeetive eah of themis.

At the lassiallevel,for example, ausality an be formulated asthe requirement that

theeldequationsnot involvethesouresofinterationsloatedinthe futurelightone

and atspaelike separations.

If we adopt this denition, quantum eld theory predits that the laws of nature are

ausalin the lassiallimit,inat spae. On the other hand, quantum gravity predits

orretions to the (lassial) eld equations of general relativity that do require the

knowledge of interations at future times, albeitrestrited to nite rangesof spaetime

separations[9℄. Wean saythattheorreted Einsteinequationsviolatemiroausality,

but satisfy maroausality.

We an dene ausality in quantum eld theory by adding the requirement that the

ommutators of spaelike separated observables vanish.

10a Maroausality

At the lassial level, we dene maroausality as the requirement that there exist

nite positive thresholds

τ

^and

σ

^{suh that the eld equations in}

x

^{not involve the}

souresofinterationsloatedin

y ∈ U ^x,τ S

V ^x,σ

^,where

U ^x,τ = { y : (x − y) ² < − 1/τ }

and

V x,σ { y : x ⁰ − y ⁰ < 0, (x − y) ² > 1/σ }

^. Alternatively, they involve those soures by negligibleamounts that tend tozero when

| (x − y) ² | → ∞

^.

ommutators of any two loalobservables

O (x)

^and

O (y)

^{are negligiblefor every}

x

and every

y ∈ U ^x,τ

^{and tend tozero for}

| (x − y) ² | → ∞

^.

Maroausality is supported by experimental evidene. In quantum eld theory,

it follows as a bonus if we do not ompromise the other major requirements (in

partiular, we do not renoune the loality of the interim lassial ation), as long

asa muh weaker formof maroausality holds, whih we denebelow.

10b Miroausality

It is the mirror of maroausality, where

U ^x,τ

^{is replaed by}

U ¯ ^x,τ = { y : − 1/τ 6 (x − y) ² < 0 }

^and

V ^x,σ

^{isreplaed by}

V ¯ ^x,σ = { y : x ⁰ − y ⁰ < 0, 0 6 (x − y) ² 6 1/σ }

^.

Wearenotawareofwaysofviolatingitwithoutalsoviolatingmaroausality,unless

weompromisethe basiarhiteture of quantum eld theory, inpartiularloality.

10 Weak maroausality

Itis useful todenea weaknotion ofmaroausality,whihisthe requirement that

the theory not ontain massless fakeons. Indeed, in most ases maroausality as

dened above follows for from free, when weak maroausality is assumed. Note

that weak maroausality allows the theory toontain massive fakeons.

10d Weak miroausality

It an be dened as the requirement that the theory haveno fakeons.

11 Ultraviolet behavior

Sometheories have partiularlynie ultravioletbehaviors. Forexample,quantum hro-

modynamisis asymptotiallyfree. Long ago, Weinberg [20℄ suggested ageneralization

of asymptoti freedom, whih is asymptoti safety, where the ultraviolet limitis an in-

terating onformal eld theory with a nite dimensional ritial surfae. Evidene of

asymptotisafety inquantum gravity has been found inrefs. [21℄. Thesepropertiesare

interesting from the theoretial point of view. However, the standard model does not

seem to have a nie ultraviolet behavior, so requirements like asymptoti freedom and

asymptotisafety sound very restritive.

11a Asymptoti freedom

11b Asymptoti safety

12 Positive deniteness of the (bosoni setor of the) Eulidean theory

nalized lassial ations oinide. When fakeons are present, it is not meaningful to

demand thatthe Eulidean version ofthe interim lassialation, whihis unprojeted,

be positive denite, even when itis purely bosoni.

Combinations of priniples

Nowwe omment onthe propertiesimplied by variousombinationsof requirements. We

start from the standard model in at spae, whih suggests a orrespondene priniple

made of (I) 1, 2, 3, 4a, 4d, 7a and 10d. These requirements imply 5a, 6, 8, 10 and 12,

but they are not ompatible with the existene of gravity, 7b. We obtainanother orret

orrespondene if (II) we replae the assumption 10d with 6. If (III) we drop 6 and

replae 10d with 10, we may inlude massive fakeons and have Yang-Mills theories (3.3)

in arbitraryeven dimensions.

Comingtoquantum gravity, the theory (3.4) and itshigher dimensionalversions (3.5)

suggest a orrespondene priniple made of (IV) 1, 2, 3, 4a, 4b, 4, 9 and 10, whih

implies 5b, 6a, 8 and 10a, but not 6, 10b and 10d. If we drop the assumption 9 we have

to renoune the impliation8.

Quantum gravity oupled to the (ovariantized) standard model suggests the orre-

spondene priniple made of (V) 1, 2, 3, 4, 9 and 10, whih implies 5a, 5b, 6a, 8 and

10a, but not 6, 10b and 10d. Again, (VI) if we drop the assumption 9 we renoune the

impliation8.

The resultsfound sofar are summarized inthe table

Assumptions Impliations Missing impliations

(I) 1, 2, 3,4a, 4d, 7a,10d 5a, 6, 8,10, 12 7b

(II) 1, 2, 3,4a, 4d, 7a,6 5a, 8, 10, 12 7b

(III) 1, 2, 3,4a, 4d, 7a,10 5a, 6a, 10a 6, 7b, 8, 10b, 10d

(IV) 1, 2, 3, 4a, 4b, 4, 7b, 9, 10 5b, 6a, 8,10a 6, 10b, 10d

(V) 1, 2, 3, 4, 7a,7b, 9, 10 5a, 5b, 6a,8, 10a 6, 10b, 10d

(VI) 1, 2, 3, 4, 7a,7b, 10 5a, 5b, 6a,10a 6, 8, 10b, 10d

(3.6)

Ifwe relax the requirement3 of properrenormalizabilityby onverting itinto simple

renormalizability, 3, then we lose the uniqueness properties 5a and 5b, beause in every

spaetime dimensions

D

^innitelymanysuper-renormalizabletheoriesofquantum gravity and gauge elds withfakeons are admitted. Their interim ationsare obtained from (3.3)

and (3.5) by raisingthe degrees

n

^ofthe polynomials

P ⁿ

^and

P n ^′

^.

unitarity

loality

properrenormalizability

fundamentalsymmetries

(3.7)

and therequirementsofhaving(

i

^{)nitelymanyeldsandparametersand(}

ii

^)nomassless

fakeons. Thisombinationimpliesquantum gravity oupledtothe standard modelinfour

dimensions, with interim lassial ation

S

^QG

+ S m ,

^(3.8)

where

S m

^{is the} ovariantized ation of the standard model (or one of its extensions), equipped with the nonminimalterms ompatible with renormalizability.

If we drop the assumption 9, solutions with analogous properties exist in every even

spaetime dimension

D

^{, made of (3.3), (3.5) and the matter setor. In that ase,}

D

^plays

the role of anadditionalphysialparameter that must be measured experimentally.

With or without 9, miroausality is violated, analytiity is replaed by regionwise

analytiity,thegravitationalinterationsareessentiallyunique,theYang-Millsinterations

are unique inform and the mattersetor remains basiallyunrestrited.

With respet to the version of the orrespondene priniple that is suessful in at

spae, the only upgrade required by quantum gravity amounts to renouning analytiity,

6, in favor of regionwise analytiity, 6a, and settle for maroausality, 10, instead of full

ausality,10. Aswewanted, thenalsolutionisasonservativeaspossible. Moreover, the

violation of miroausalityis turnedintoa physial predition, whih might be onrmed

experimentallyif a suitableampliation mehanism is found.

4 Conlusions

The fatesof determinismand possiblyausalityaretheretoremindusthat theorrespon-

denebetweenthe environmentwelivein,whihshapesour thinking,andthe mirosopi

worldisdoomedtobeomeweakerandweakerasweexploresmallerandsmallerdistanes.

At some point, we fae the intrinsi limitationsof our ability to understand the universe.

Maybe the impossibility to eetively restrit the matter setor and the gauge group of

the standard model is afurther sign of the fadingorrespondene.

but mostly formal ones. Unitarity, as said, is a requirement of ompleteness. Loality is

intrinsially tied to the dynamis of the logial proess that builds the quantum theory

fromaninterimlassialtheory,thenal theorybeingnonloalanyway. Asfaras(proper)

renormalizabilityis onerned,it ithard to viewit asmore than formal.

This is an interesting turn of events. Perhaps ironi, but onsistent with what we

have been remarking allover this paper. Insisting onphysial requirements would belike

requesting thatnatureadapttous, ratherthanopingwiththefatthat wehavetoadapt

to nature. What are the odds that a physialintuition shaped by a lassial environment

gets itright,when itomestothe phenomenaof the innitesimallysmall? Notmany. We

have been knowing that for a fatsine the birth of quantum mehanis. And whatif we

annotunderstandwhy theorrespondene prinipleismadeof(3.7)insteadof something

else? Even better, we might argue. Indeed, if we understood that, the priniple would

probablybeinadequate,beauseitwould bettodesribeourworld,but unttodesribe

the innitesimal world.

Nevertheless, it isimportanttostress thatthe ornerstones of(3.7) are not hosen be-

ause they are appealing, but simplybeause they work. The oppositeattitude hoose

something appealing and pretend it works isvery fashionablenowadays, but spetau-

larlyunsuessful [22℄. The failuresaumulatedduringthe past deades inthepursuit of

beauty, mathematial elegane, symmetry, supersymmetry, uniation, granduniation,

theories of everything and theories with no parameters, and then holography and who

knows what will be next, remind us of the innite monkey theorem. Are humans that

stupid? We annot exlude it. The possibility that humans are ill equipped to embrae

the hallenges demanded by the exploration of the smaller and smaller distanes is high.

We are already witnessing a slow, relentless involution into arealm where the main judg-

mentriteriaare the authority prinipleoritsmoderndistorsions, suhasthe ounting of

likes, alled itations. Sadly, this turn of events is not even new: it is alledMiddle Ages.

But the puzzling question is: how long willitlast this time?

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