field renormalization in qg - heidelberg universityeichhorn/ambjorn.pdffield renormalization in qg...

Field Renormalization in QG

Jan Ambjorn

Niels Bohr Institute, Copenhagen, Denmarkand

Radboud University, Nijmegen, The Netherlands

Talk at Quantum Spacetime and the Renormalization Group,Bad Honnef , June 18th, 2018

Work done in collaboration with Y. Makeenko

J. Ambjørn quantum spacetime

Lattice field theory has provided us with a non-perturbative toolfor defining and calculating certain observables. The mostimpressive results are obtained in QCD.

If we want to extend this success to QG we are facing twomajor obstacles:

(1) No clear Euclidean-Lorentzian signature dictionary

(2) Not obvious that there exists a QG theory “an sich”.

I will ignore (1) and (2) and assume we have a lattice Euclideanfield theory of some type.


The exists one example of a theory of fluctuating Euclideangeometries coupled to matter fields: non-critical string theory,which can be solved analytically in the continuum, can beformulated as a lattice theory, solved analytically on thelattice(!) and which can studied by MC simulations. So,seemingly this is the proof of concept that some QG theoriessatisfy (1) and (2). But even here the devil is in the detail, andthis will be the topic of my talk which could also be called “isthere a bosonic lattice string theory ?”

The whole enterprise of DT, non-critical strings and matrixmodels started with the desire to have a non-perturbativeformulation of the bosonic string theory.


One has two formulations of the bosonic string theory in RD,which are believed to be equivalent.

Z =∫DS e−kA[S], Nambu-Goto

Z =∫D[g]

∫DgX e−S[X ,g], Polyakov

S[X ,g] = k∫

d2ξ√

g(

gab∂aX∂bX + Λ).

The Polyakov formulation is just 2d Euclidean QG coupled to Dmassless Gaussian fields Xµ which live in taget space RD.


In the case of the Nambu-Goto action the natural lattice theoryconsists of a hypercubic lattice with link lenth a in RD and thestring worldsheet is a plaquette surface, the area being thenumber of plaquettes times a2. For the Polyakov string anatural lattice theory is one where∫

D[g]→∑

T

Discretisation

where the summation is over equilateral triangulations with linklength a and where the Gaussian action of Xµ has a naturalimplementation on the triangulation T as NT Gaussian fields.


The hypercubic lattice theory is beautiful in its extremesimplicity. When formulated on a lattice in dimensionless units(lattice spacing of length 1) we have

Z (µ) =∑

S

e−µA(S), , A(S) = # plaquettes in S

S denote connected plaquette surfaces with a fixed topologyand fixed boundaries. The partition function has a critical pointµc . For µ < µc the partition function is not defined and thecontinuum limit is obtained for µ→ µc where large surfaces willdominate.


Here are the results , formulated in dimensionless lattice units.We consider the correlation function G(n) of two plaquetteboundaries separated a lattice distance n, and the partitionfunction G(m,n) for a rectangular “Wilson loop” spanned by aminimal surface of m × n plaquettes.

One can prove that for µ > µc one has for large n,m anexponential fall off:

Gµ(n) ∼ e−m(µ)n, Gµ(m,n) ∼ e−σ(µ) m n.

G(n) and G(m,n) only exist in the continuum limit µ→ µc if

m(µ)→ 0 and σ(µ)→ 0, µ→ µc


However, the scaling of m(µ) and σ(µ) are

m(µ) ∼ (µ−µc)1/4 σ(µ) = σ(µc)+ c̃(µ−µc)1/2, σ(µc) > 0.

The last result implies that the physical string tension goes toinfinity:

m(µ) = mph a(µ), σ(µ) = kph a2(µ)

Thuskph ∼

σ(µc)√µ− µc

m2ph →∞ for µ→ µc .

e−m(µ)n =e−mphL, L = n a,

e−σ(µ)nm =e−kphA, A = mn a2


This seems in contradiction to the continuum calculation fromstring theory. Bosonic strings have tachyons, but we can avoidthe tachyons if we consider, say, closed strings where onespace dimension is compactified to a circle of radius β and thestring wraps once around this dimension. We now consider theclosed string correlator where two closed strings are separateda distance L. The minimal worldsheet surface has area L× β.This situation can clearly be implemented on the hypercubiclattice too and one would have a minimal surface with n ×mlinks,

L = n · a(µ), β = m · a(µ)

The non-scaling of σ(µ) is still valid, being a bulk property, i.e.Kph is infinite. On the lattice one would have a minimal surfacewith a few excitations carrying no area (so-called branchedpolymer excitations).


In contrast continuum string calculations state that the groundstate energy for L� β, and where we use notation d := D − 2,is given by

Emin = kph

√β2 − πd

3kph, Fmin = LEmin(β)

and the average area of the world sheet can be calculated

〈A〉 = Lβ2 − πd6kph√β2 − πd3kph

In these formulas kph is just the continuum finite string tensionwhich appears is the Nambu-Goto action. The formula for 〈A〉 isweird...


The difference between the lattice and continuum calculationsis somewhat of a mystery, in the sense that it should not makea difference whether one uses a lattice as a regularization orone uses another regularization with a cut-off which plays therole of the lattice link length a. We decided to address thequestion, using continuum string calculations, but carefullykeeping the dimensionful cut-off at all steps in the calculations.The starting point is the Nambu-Goto action

S(X ) = k0∫

d2ω√

det ∂aX · ∂bX

= k0∫

d2ω√ρ+

k02

∫d2ω λab (∂aX · ∂bX − ρab) .

where ρab(ω) and λab(ω) are Lagrange multipliers.


Choosing the worldsheet parameter space as [ωL, ωβ], theminimal action worldsheet configuration is

X 1cl =LωLω1, X 2cl =

β

ωβω2, X⊥cl = 0,

[ρab]cl = diag

(L2

ω2L,β2

ω2β

),

λabcl = diag(βωLLωβ

,LωββωL

)= ρabcl

√ρcl,

S[Xcl] = k0L β


We now write X = Xcl + Xq and integrate over Xq in

Z (k0,L, β) = e−F (k0,L,β) =∫DXµDρabDλab e−S[X ,ρ,λ]

and determine the mean field values of the Langrangemultipliers ρ and λ from the corresponding effective action. Weexpect this mean field approximation to be exact in the d →∞limit.

This approximation can be compared to the use of Lagrangemultipliers in the O(N) non-linear sigma model. Also there oneobtains very good results for using mean field for the Lagrangemultiplier.


A cut-off Λ enters when we perform the Gaussian integral overXq and we need to calculate the determinant of the operatorO = − 1√ρ∂aλ

ab∂b. We use proper-time regularization

tr logO = −∫ ∞

a2

dττ

tr e−τO, a2 ≡ 14πΛ2

.

a is seemingly a worldsheet cut-off rather than a cut-off intarget space, but its eigenvalues only depend on L, β (bydiffeomorphism invariance) and

a2 = (∆s)2 = ρab∆ωa∆ωb = (∆X )2

Thus it is also a cut-off in target space, like the hypercubiclattice.


The minimum of the effective action is reached at

ρ̄11 =L2

ω2L

(β2 − β

20

2c

)(β2 − β

20

c

) c2c − 1

,

ρ̄22 =1ω2β

(β2 −

β202c

)c

2c − 1,

λ̄ab = cρ̄ab√ρ̄, c =

12

+

√14− dΛ

2

2k0,

where

β20 =πd3k0

, c = 1− dΛ2

k0+ O

(Λ4k20

)


For these mean field values of ρ and λ we have (for L� β)

Fmf = k0c L√β2 − β20/c

and the average area of the world sheet will be

〈A〉 =∫

d2ω√ρ̄11ρ̄22 = L

(β2 − β20/2c

)√β2 − β20/c

c(2c − 1)

.

For k0 →∞ we obtain the minimal surface. For c = 1 weobtain the former string result. However, the only physicalreasonable result is that when the cut-off Λ→∞ then c → 1/2:

c =12

+12

k̃phdΛ2

k0 = 2dΛ2 +k̃2phdΛ2


With these definitions we can define a non-tachyonic two-pointfunction where the two infinitesimal loops are separated adistance L by choosing β as small as possible without√β2 − β20/c becoming imaginary for any k0:

β = βmin =2πa√

3.

Note that βmin is of the cut-off scale a and thus the loop aroundthe compactified dimension is of the size of a minimal boundaryplaquette loop on the lattice.

G(L, k0) ∼ e−mphL, m2ph =πd6

k̃ph.

This defines a tunable lowest mass of our string !


Similarly we can now define the string tension kph in the limitwhere L� β � βmin:

Z (k0,L, β) = e−F (k0,L,β) = e−kphAmin+O(L,β), Amin = Lβ

We conclude that the physical string tension kph is

kph = k0c = dΛ2 +12

k̃ph +O(1/Λ2).

Thus the physical string tension as defined above diverges asthe cutoff Λ is taken to infinity. However, the first correction isfinite and behaves as we would have liked kph to behave,namely as k̃ph ∝ m2ph/d .


We can now formulate the results in dimensionless quantitiesand we see that we have reproduced the lattice results:

µ = k0a2, µc = d/2π, n = L/a, σ(µ) = kpha2

m(µ) n = mphL, m(µ) ∼ (µ− µc)1/4,

σ(µ) = σ(µc) + c̃(µ− µc)1/2, σ(µc) =µc2> 0, c̃ =

12√µc.

After 30 years we have finally managed to reproduce the latticeresults by a continuum standard string calculation. However,the problem is that these are of course not the standard stringresults and this is where the link to lattice quantum gravitybecomes interesting. How to connect to “ordinary” stringtheory?


Fmf (L, β, k0) = k0c L

√β2 − dπ

3k0c, c =

12

+

√14− dΛ

2

2k0

Since k0 ≥ 2dΛ2 one would be tempted to think that one canignore the ”tachyonic” term under the square root. It is of theorder of the cut off and should not play any role in the limitwhere Λ→∞. This is precisely true in the limit we have justbeen discussing.

However, if we just consider Xµ as two-dimensional quantumfields (the 2d QG point of view), the decompositionX = Xcl + Xq is a decomposition in a background field andquantum fluctuations, and in field theory, integrating out thequantum field will leave us with an effective action dependingon the background fields which in general need to berenormalized for the effective action to be finite.


ThusXcl = Z 1/2XR =⇒ (L, β) = Z 1/2(LR, βR)

Is there a choice of Z (k0) and a renormalization of k0 such that

F (L, β, k0) = FR(LR, βR, kR), LR, βR, kR fixed for Λ→∞.

YES! Z =2c−1

c, kR = k0c Z = k̃ph.

Z = 1− dΛ2

2k0+O(k−20 ), Z (k0)→ 0 for c →

12.

1k̃ph

=1k0

+dΛ2

k20+O(k−30 )


With this renormalization we find the continuum string results

F (LR, βR, kR) = kRLR

√β2R −

πd3kR

〈A〉 = LR

(β2R −

πd6kR

)√β2R −

πd3kR

,

However, this seems impossible to obtain using the latticereguarization: on the lattice we have L = n · a(µ) and thescaling limit is such that L is fixed for a→ 0 while n→∞. ThusL� a.This is conventional LFT. However the background fieldrenormalization amount to writing

L = a ·√

2kR/µc LR, β = a ·√

2kR/µc βR.

Thus the scaling limit where KR, LR and βR are finite as a→ 0is a limit where L and β are of the order of the cutoff a incontradiction to the LFT limit.


On the lattice it is difficult to understand the existence of atachyon.

nnn 1 2>

G(n) ≥ G(n1)G(n2) =⇒ − limn→∞log G(n)

n≥ 0

Our continuum calculation is in accordance with this before werenormalize the lengths L, β. However when we go from aGulliver world to a Lilliputian world we introduce a tachyon:

k0c L

√β2 − πd

3k0c= kRLR

√β2R −

πd3kR

, m2tac =3kRπd

.


〈AGulliver〉 = Lβ2 − β20/2c√β2 − β20/c

c(2c − 1)

∼ LβkRa2

〈ALilliput〉 = LRβ2R −

πd6kR√

β2R −πd3kR

∼ LRβR

In the Gulliver world the wordsheet is unphysical, like thelength of a path in the path integral of the free particle isunphysical (and diverge like L/(mpha)). However, in theLilliputian world the worldsheet has become a physical 2d worldwhich does not fluctuate too much (unless β → 1/mtac).


This situation seems generic in QG if we want to definecorrelation functions depending on some kind ofdiffeomorphism invariant distance.

GV (R) ≡1V

∫D[g]

∫Dgφ e−S[g,φ] δ

(∫ √g−V

)∫∫

dx dy√

g(x)√

g(y) φ(x)φ(y) δ(R−Dg(x , y)).

Here the field [gµν ] corresponds to Xµ and like L, β are relatedto X then R is related to [g] via the geodesic distance Dg , but ina much more complicated way. In fact the geodesic distanceoperator D̂g(x , y) is so complicated that one could doubt that itmakes any sense.


Again 2d Euclidean quantum gravity comes to our help via DTand via computer simulations and offers some hope that thegeodesic distance operator might make sense.

As the very simpest example, let us consider 2d Euclideanquantum gravity and put φ = 1. This can be done in two ways:either there is no field coupled to gravity at all (pure 2d gravity),or we have a field coupled to gravity but we choose to put φ = 1in the correlator. First we consider pure 2d QG.

GV (R) = 〈S(R)〉V ∼ Rdh−1, R � V 1/dh ,

GV (R) =1V

〈∫dxdy

√g(x)

√g(y) δ(Dg(x , y)−R)

〉V


One can calculate GV (R) in the case of 2d Euclidean QG usingDT as the lattice theory. Fix Λ rather than V :

GΛ(R) =∫ ∞

0dV e−Λ V GV (R)

The lattice result is (close to the critical point)

Gµ(r) ∼ (µ− µc)3/4cosh

[(µ− µc)1/4r

]sinh3

[(µ− µc)1/4r

]while

µ− µc = Λ a2(µ), V = NT a2, L = `T a.


Thus, in order for Gµ(r) to have a continuum limit for a(µ)→ 0R has to have a anomalous scaling dimension:

GΛ(R) ∼ Gµ(r), R = r√

a(µ) rather than R = r a(µ).

What happens when a matter field φ is coupled to 2dEuclidean quantum gravity? It is not known analytically even in2d Euclidean quantum gravity, but for for a number of fieldtheories can be addressed by computer simulations.

One observes that the anomalous scaling of R is related to thecentral charge of the conformal field theories couple to 2dquantum gravity and if such a global anomalous scaling existsthen it is related to a non-trivial Hausdorff dimension viaV ∼ Rdh


The fractal dimension can be ”seen” nicely if the topology isthat of a torus, which has the virtue that the shorestnon-contractable loop is automatically a geodesic curve.Further, for the torus we have for analytic manifolds harmonicforms which have very nice discretized analogies, and we canuse the these to construct a conformal mapping from theabstract triangulation to the complex plane (Timothy Budd andJA)


Since the shortest non-contractable loop is a geodesic:

〈L〉N ∼ N1/dh(c)

left figure c = 0, i.e. dh = 4, right figure c = −2, dh = 3.56


dh(c) = 2√

49− c +√

25− c√25− c +

√1− c

, dh(0) = 4.

(formula of Y. Watabiki)

In the case of a CFT (mass =0) it is indeed a global scale.


However, how should we really think about a massive correlatorof the generic form

〈ΦΦ(R)〉V ∼ R−αe−mphR R �1

V 1/dh

For large distances one would expect R to be a distance like ingravity without matter (but R in pure gravity already scalesanomalous, so mph has the wrong dimension). For shortdistances one would expect the propagator to behave like amassless propagator corresponding to a certain central charge,but then the dimension of R is different from the R at longdistances. Thus we are seemingly led to a scale dependentHausdorff dimension. One might feel uncomfortable with thissince the Hausdorff dimension was used to define the lengthscale in the first place.

Maybe the lattice definition of gravity is more subtle than onenaively would have expected approaching the problem with anWilsonian attitude ?


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