Noncommutative QFT and Renormalization · point of the parameter space the model becomes self-dual,...

Quantum GravityB. Fauser, J. Tolksdorf and E. Zeidler, Eds., 315–326c© 2006 Birkhauser Verlag Basel/Switzerland

Noncommutative QFT and Renormalization

Harald Grosse and Raimar Wulkenhaar

Abstract. Since the two pillars of modern physics: Quantum Field Theory andgeneral relativity are incompatible, one tries to take fluctuating geometriesinto account through deforming space-time. The resulting noncommutativeQuantum Field Theory shows the IR/UV mixing. We modify the action fora scalar model in 4 dimensions and show, that a renormalizable field theoryresults. For the proof we fist transform to a matrix model and use the Wilson-Polchinski approach to renormalization. An efficient power-counting theoremallows to eliminate all higher genus contributions. By taking finite differenceswe reduce the infinite number of possible two point and four point functions toonly two relevant/marginal operators, thus completing the proof. At a specialpoint of the parameter space the model becomes self-dual, the beta functionvanishes and the model connects to integrable systems.

Mathematics Subject Classification (2000). Primary 81T15; Secondary 81T75.

Keywords. Noncommutative quantum field theory, renormalization, infrared/ultraviolet mixing, θ-deformation, matrix models.

1. Introduction

Four-dimensional quantum field theory suffers from infrared and ultraviolet diver-gences as well as from the divergence of the renormalized perturbation expansion.Despite the impressive agreement between theory and experiments and despitemany attempts, these problems are not settled and remain a big challenge for the-oretical physics. Furthermore, attempts to formulate a quantum theory of gravityhave not yet been fully successful. It is astonishing that the two pillars of mod-ern physics, quantum field theory and general relativity, seem incompatible. Thisconvinced physicists to look for more general descriptions: After the formulationof supersymmetry (initiated by Bruno Zumino and Julius Wess) and supergravity,string theory was developed, and anomaly cancellation forced the introduction ofsix additional dimensions. On the other hand, loop gravity was formulated, and ledto spin networks and space-time foams. Both approaches are not fully satisfactory.A third impulse came from noncommutative geometry developed by Alain Connes,

316 Harald Grosse and Raimar Wulkenhaar

providing a natural interpretation of the Higgs effect at the classical level. Thisfinally led to noncommutative quantum field theory, which is the subject of thiscontribution. It allows to incorporate fluctuations of space into quantum field the-ory. There are of course relations among these three developments. In particular,the field theory limit of string theory leads to certain noncommutative field theorymodels, and some models defined over fuzzy spaces are related to spin networks.

The argument that space-time should be modified at very short distancesgoes back to Schrodinger and Heisenberg. Noncommutative coordinates appearedalready in the work of Peierls for the magnetic field problem, and are obtainedafter projecting onto a particular Landau level. Pauli communicated this to Op-penheimer, whose student Snyder [27] wrote down the first deformed space-timealgebra preserving Lorentz symmetry. After the development of noncommutativegeometry by Connes [8], it was first applied in physics to the integer quantumHall effect. Gauge models on the two-dimensional noncommutative tori were for-mulated, and the relevant projective modules over this space were classified.

Through interactions with John Madore the first author realized that suchFuzzy geometries allow to obtain natural cutoffs for quantum field theory [13].This line of work was further developed together with Peter Presnajder and Cti-rad Klimcık [12]. At almost the same time, Filk [11] developed his Feynman rulesfor the canonically deformed four-dimensional field theory, and Doplicher, Fre-denhagen and Roberts [9] published their work on deformed spaces. The subjectexperienced a major boost after one realized that string theory leads to noncom-mutative field theory under certain conditions [25, 26], and the subject developedvery rapidly; see e.g. [10, 19, 30].

2. Noncommutative Quantum Field Theory

The formulation of Noncommutative Quantum Field Theory (NCFT) follows adictionary worked out by mathematicians. Starting from some manifold M oneobtains the commutative algebra of smooth functions overM, which is then quan-tized along with additional structure. Space itself then looks locally like a phasespace in quantum mechanics. Fields are elements of the algebra resp. a finitelygenerated projective module, and integration is replaced by a suitable trace oper-ation.

Following these lines, one obtains field theory on quantized (or deformed)spaces, and Feynman rules for a perturbative expansion can be worked out. How-ever some unexpected features such as IR/UV mixing arise upon quantization,which are described below. In 2000 Minwalla, van Raamsdonk and Seiberg real-ized [21] that perturbation theory for field theories defined on the Moyal planefaces a serious problem. The planar contributions show the standard singularitieswhich can be handled by a renormalization procedure. The nonplanar one loopcontributions are finite for generic momenta, however they become singular at

Noncommutative QFT and Renormalization 317

exceptional momenta. The usual UV divergences are then reflected in new sin-gularities in the infrared, which is called IR/UV mixing. This spoils the usualrenormalization procedure: Inserting many such loops to a higher order diagramgenerates singularities of any inverse power. Without imposing a special structuresuch as supersymmetry, the renormalizability seems lost; see also [6, 7].

However, progress was made recently, when we were able to give a solutionof this problem for the special case of a scalar four-dimensional theory defined onthe Moyal-deformed space R4

θ [16]. The IR/UV mixing contributions were takeninto account through a modification of the free Lagrangian by adding an oscillatorterm with parameter Ω, which modifies the spectrum of the free Hamiltonian. Theharmonic oscillator term was obtained as a result of the renormalization proof.The model fulfills then the Langmann-Szabo duality [18] relating short distanceand long distance behavior. Our proof followed ideas of Polchinski. There areindications that a constructive procedure might be possible and give a nontrivialφ4 model, which is currently under investigation [23]. At Ω = 1 the model becomesself-dual, and we are presently studying this model in greater details.

Nonperturbative aspects of NCFT have also been studied in recent years.The most significant and surprising result is that the IR/UV mixing can lead toa new phase denoted as “striped phase” [17], where translational symmetry isspontaneously broken. The existence of such a phase has indeed been confirmedin numerical studies [4,20]. To understand better the properties of this phase andthe phase transitions, further work and better analytical techniques are required,combining results from perturbative renormalization with nonperturbative tech-niques. Here a particular feature of scalar NCFT is very suggestive: the field canbe described as a hermitian matrix, and the quantization is defined nonperturba-tively by integrating over all such matrices. This provides a natural starting pointfor nonperturbative studies. In particular, it suggests and allows to apply ideasand techniques from random matrix theory.

Remarkably, gauge theories on quantized spaces can also be formulated in asimilar way [1, 2, 5, 28]. The action can be written as multi-matrix models, wherethe gauge fields are encoded in terms of matrices which can be interpreted as“covariant coordinates”. The field strength can be written as commutator, whichinduces the usual kinetic terms in the commutative limit. Again, this allows anatural nonperturbative quantization in terms of matrix integrals.

Numerical studies for gauge theories have also been published including the4-dimensional case [3], which again show a very intriguing picture of nontrivialphases and spontaneous symmetry breaking. These studies also strongly suggestthe nonperturbative stability and renormalizability of NC gauge theory, adding tothe need of further theoretical work.


3. Renormalization of φ4-theory on the 4D Moyal plane

We briefly sketch the methods used by ourselves [16] in the proof of renormaliz-ability for scalar field theory defined on the 4-dimensional quantum plane R4

θ, withcommutation relations [xµ, xν ] = iθµν . The IR/UV mixing was taken into accountthrough a modification of the free Lagrangian, by adding an oscillator term whichmodifies the spectrum of the free Hamiltonian:

S =∫

d4x(1

2∂µφ ∂µφ +

Ω2

2(xµφ) (xµφ) +

µ2

2φ φ +

λ

4!φ φ φ φ

)(x) . (1)

Here, xµ = 2(θ−1)µνxν and is the Moyal star product

(a b)(x) :=∫

d4yd4k

(2π)4a(x+ 1

2θ·k)b(x+y) eiky , θµν = −θνµ ∈ R . (2)

The harmonic oscillator term in (1) was found as a result of the renormalizationproof. The model is covariant under the Langmann-Szabo duality relating shortdistance and long distance behavior. At Ω = 1 the model becomes self-dual, andconnected to integrable models. This leads to the hope that a constructive proce-dure around this particular case allows the construction of a nontrivial interactingφ4 model, which would be an extremely interesting and remarkable achievement.

The renormalization proof proceeds by using a matrix base, which leads to adynamical matrix model of the type:

S[φ] = (2πθ)2∑

m,n,k,l∈N2

(12φmn∆mn;klφkl +

λ

4!φmnφnkφklφlm

), (3)

where

∆m1m2

n1n2 ; k1

k2l1l2

=(µ2+ 2+2Ω2

θ (m1+n1+m2+n2+2))δn1k1δm1l1δn2k2δm2l2

− 2−2Ω2

θ

(√k1l1 δn1+1,k1δm1+1,l1 +

√m1n1 δn1−1,k1δm1−1,l1

)δn2k2δm2l2

− 2−2Ω2

θ

(√k2l2 δn2+1,k2δm2+1,l2 +

√m2n2 δn2−1,k2δm2−1,l2

)δn1k1δm1l1 .

(4)

The interaction part becomes a trace of product of matrices, and no oscillations oc-cur in this basis. The propagator obtained from the free part is quite complicated,


in 4 dimensions it is:

Gm1m2

n1n2 ; k1

k2l1l2

=θ

2(1+Ω)2

m1+l12∑

v1= |m1−l1|2

m2+l22∑

v2= |m2−l2|2

B(1+µ2θ

8Ω +12 (m1+k1+m2+k2)−v1−v2, 1+2v1+2v2

)

× 2F1

(1+2v1+2v2 , µ2θ

8Ω −12 (m1+k1+m2+k2)+v1+v2

2+µ2θ8Ω +1

2 (m1+k1+m2+k2)+v1+v2

∣∣∣∣ (1−Ω)2

(1+Ω)2

)(1−Ω1+Ω

)2v1+2v2

×2∏

i=1

δmi+ki,ni+li

√(ni

vi+ni−ki

2

)(ki

vi+ki−ni

2

)(mi

vi+mi−li

2

)(li

vi+ li−mi

2

). (5)

These propagators (in 2 and 4 dimensions) show asymmetric decay properties:

-10-5

0

5

10

0

5

10

15

20

0

0.05

0.1

0.15

0

5

θ−1∆ 100

10+α0 ; l+α

0l0

α

l

Ω = 0.1 µ = 0

(6)

They decay exponentially on particular directions (in l-direction in the picture),but have power law decay in others (in α-direction in the picture). These decayproperties are crucial for the perturbative renormalizability of the models.

Our proof in [15,16] then followed the ideas of Polchinski [22]. The quantumfield theory corresponding to the action (3) is defined — as usual — by the partitionfunction

Z[J ] =∫ (∏

m,n

dφmn

)exp

(−S[φ]−

∑m,n

φmnJnm

). (7)

The strategy due to Wilson consists in integrating in the first step only those fieldmodes φmn which have a matrix index bigger than some scale θΛ2. The result isan effective action for the remaining field modes which depends on Λ. One cannow adopt a smooth transition between integrated and not integrated field modesso that the Λ-dependence of the effective action is given by a certain differentialequation, the Polchinski equation.

Now, renormalization amounts to prove that the Polchinski equation admitsa regular solution for the effective action which depends on only a finite numberof initial data. This requirement is hard to satisfy because the space of effective


actions is infinite dimensional and as such develops an infinite dimensional spaceof singularities when starting from generic initial data.

The Polchinski equation can be iteratively solved in perturbation theorywhere it can be graphically written as

Λ∂

∂Λ

............. . . . . .

n1

m1

n2m2

mN

nN

=12

∑m,n,k,l

N−1∑N1=1

m1

n1

nN1

mN1mN1+1

nN1+1

nNmN

k

l

n

m

− 14πθ

∑m,n,k,l

m1

n1

ni−1

mi−1

mi

ni

nNmN

n m

k l

(8)

The graphs are graded by the number of vertices and the number of external legs.Then, to the Λ-variation of a graph on the lhs there only contribute graphs witha smaller number of vertices and a bigger number of legs. A general graph is thusobtained by iteratively adding a propagator to smaller building blocks, startingwith the initial φ4-vertex, and integrating over Λ. Here, these propagators are dif-ferentiated cut-off propagators Qmn;kl(Λ) which vanish (for an appropriate choiceof the cut-off function) unless the maximal index is in the interval [θΛ2, 2θΛ2]. Asthe field carry two matrix indices and the propagator four of them, the graphs areribbon graphs familiar from matrix models.

We have then proven that the cut-off propagator Q(Λ) is bounded by CθΛ2 .

This was achieved numerically in [16] and later confirmed analytically in [23].A nonvanishing frequency parameter Ω is required for such a decay behavior.As the volume of each two-component index m ∈ N2 is bounded by C′θ2Λ4 ingraphs of the above type, the power counting degree of divergence is (at firstsight) ω = 4S − 2I, where I is the number of propagators and S the number ofsummation indices.

It is now important to take into account that if three indices of a propagatorQmn;kl(Λ) are given, the fourth one is determined by m+k = n+ l, see (5). Then,for simple planar graphs one finds that ω = 4 − N where N is the number ofexternal legs. But this conclusion is too early, there is a difficulty in presence ofcompletely inner vertices, which require additional index summations. The graph


m

n

l

k

q

p1+m

p1+q

p2+l

p2+q

p3+q

p3+lp3+m

(9)

entails four independent summation indices p1, p2, p3 and q, whereas for thepowercounting degree 2 = 4−N = 4S−5 ·2 we should only have S = 3 of them. Itturns out that due to the quasi-locality of the propagator (the exponential decayin l-direction in (6)), the sum over q for fixed m can be estimated without theneed of the volume factor.

Remarkably, the quasi-locality of the propagator not only ensures the cor-rect powercounting degree for planar graphs, it also renders all nonplanar graphssuperficially convergent. For instance, in the nonplanar graphs

m4

n4

m1n1

n2

m2

m3n3

q

q′

∣∣∣∣∣∣∣∣∣q′=n1+n3−q

m2

n2 r′ r

m1n1

q

q′

∣∣∣∣∣∣∣∣∣∣∣ q′ = m2 + r − qr′ = n2 + r − m1

(10)

the summation over q and q, r, respectively, is of the same type as over q in (9) sothat the graphs in (10) can be estimated without any volume factor.

After all, we have obtained the powercounting degree of divergence

ω = 4−N − 4(2g + B − 1) (11)

for a general ribbon graph, where g is the genus of the Riemann surface on whichthe graph is drawn and B the number of holes in the Riemann surface. Both aredirectly determined by the graph. It should be stressed, however, that although thenumber ω (11) follows from counting the required volume factors, its proof in ourscheme is not so obvious: The procedure consists in adding a new cut-off propagatorto a given graph, and in doing so the topology (B, g) has many possibilities to arisefrom the topologies of the smaller parts for which we have estimates by induction.The proof that in every situation of adding a new propagator one obtains (11) goesalone over 20 pages in [15]. Moreover, the boundary conditions for the integrationhave to be correctly chosen to confirm (11), see below.

The powercounting behavior (11) is good news because it implies that (incontrast to the situation without the oscillator potential) all nonplanar graphs


are superficially convergent. However, this does not mean that all problems aresolved: The remaining planar two- and four-leg graphs which are divergent carrymatrix indices, and (11) suggests that these are divergent independent of thematrix indices. An infinite number of adjusted initial data would be necessaryin order to remove these divergences.

Fortunately, a more careful analysis shows that the powercounting behavioris improved by the index jump along the trajectories of the graph. For example, theindex jump for the graph (9) is defined as J = ‖k−n‖1+‖q−l‖1+‖m−q‖1. Then,

the amplitude is suppressed by a factor of order(

max(m, n . . . )θΛ2

) J2

compared with

the naive estimation. Thus, only planar four-leg graphs with J = 0 and planartwo-leg graphs with J = 0 or J = 2 are divergent (the total jumps are even). Forthese cases, we have invented a discrete Taylor expansion about the graphs withvanishing indices. Only the leading terms of the expansion, i.e. the reference graphswith vanishing indices, are divergent whereas the difference between original graphand reference graph is convergent. Accordingly, in our scheme only the referencegraphs must be integrated in a way that involves initial conditions. For example,if the contribution to the rhs of the Polchinski equation (8) is given by the graph

Λ∂

∂ΛA

(2)planar,1PImn;nk;kl;lm [Λ] =

∑p∈N2

⎛⎜⎝

m

mk

k

n n

l l

p p

⎞⎟⎠ (Λ) , (12)

the Λ-integration is performed as follows:

A(2)planar,1PImn;nk;kl;lm [Λ]

= −∫ ∞

Λ

dΛ′

Λ′∑p∈N2

⎡⎢⎣

m

mk

k

n n

l l

p p −

m

mk

k

n n

l l0 0

0 0

p p

⎤⎥⎦[Λ′]

+

m

mk

k

n n

l l ⎡⎢⎣∫ Λ

ΛR

dΛ′

Λ′∑p∈N2

⎛⎜⎝

0

0

0

0

0 0

0 0

p p

⎞⎟⎠[Λ′] + A(2,1,0)1PI00;00;00;00[ΛR]

⎤⎥⎦.(13)

Only one initial condition, A(2,1,0)1PI00;00;00;00[ΛR], is required for an infinite number of

planar four-leg graphs (distinguished by the matrix indices). We need one furtherinitial condition for the two-leg graphs with J = 2 and two more initial conditionfor the two-leg graphs with J = 0 (for the leading quadratic and the subleadinglogarithmic divergence). This is one condition more than in a commutative φ4-theory, and this additional condition justifies a posteriori our starting point ofadding one new term to the action (1), the oscillator term Ω.


This being established, it was straightforward to derive beta functions for thecoupling constant flow. To one-loop order we have found [14]

βλ =λ2

phys

48π2

(1−Ω2phys)

(1+Ω2phys)3

, βΩ =λphysΩphys

96π2

(1−Ω2phys)

(1+Ω2phys)3

, (14)

βµ = −λphys

(4N ln(2) +

(8+θµ2phys)Ω

2phys

(1+Ω2phys)

2

)48π2θµ2

phys(1+Ω2phys)

, γ =λphys

96π2

Ω2phys

(1+Ω2phys)3

. (15)

Together with the differential equation for the β-functions,

limN→∞

(N ∂

∂N + Nγ + µ2βµ∂

∂µ20

+ βλ∂

∂λ+ βΩ

∂

∂Ω

)Γm1n1;...;mN nN [µ, λ, Ω,N ] = 0 ,

(16)

(14) shows that the ratio of the coupling constants λΩ2 remains bounded along

the renormalization group flow up to first order. Starting from given small valuesfor ΩR, λR at NR, the frequency grows in a small region around ln N

NR= 48π2

λRto

Ω ≈ 1. The coupling constant approaches λ∞ = λR

Ω2R

, which can be made small forsufficiently small λR. This leaves the chance of a nonperturbative construction [24]of the model.

In particular, the β-function vanishes at the self-dual point Ω = 1, indicatingspecial properties of the model.

4. Matrix-model techniques

Our recent interests turned towards dynamical matrix models, which are closelyconnected to integrable models. We briefly explain this method. Consider e.g.the scalar field theory defined by (3). Since φ is a hermitian matrix, it can bediagonalized as φ = U−1diag(φi)U where φi are the real eigenvalues. Hence thefield theory can be reformulated in terms of the eigenvalues φi and the unitarymatrix U . The main idea is now the following: consider the probability measure forthe (suitably rescaled) eigenvalues φi induced by the path integral by integratingout U :

Z =∫Dφ exp(−S(φ))) =

∫dφi∆2(φi)

∫dU exp(−S(U−1(φi)U))

=∫

dφi exp(−F(φ)− (2πθ)d/2∑

i

V (φi) +∑i=j

log |φi − φj |), (17)

where the analytic function

e−F(φ) :=∫

dU exp(−Skin(U−1(φ)U)) (18)

is introduced, which depends only on the eigenvalues of φ. The crucial point isthat the logarithmic terms in the effective action above implies a repulsion of theeigenvalues φi, which therefore arrange themselves according to some distribution


similar as in the standard matrix models of the form S =∫

dφ exp(TrV (φ)). Thisis related to the fact that nonplanar diagrams are suppressed. The presence of theunknown function F(φ) in (17) cannot alter this conclusion qualitatively, since it isanalytic. The function F(φ) can be determined approximately by considering theweak coupling regime. For example, the effective action of the eigenvalue sectorfor the φ4 model in the noncommutative regime 1

θ * Λ2 becomes essentially

S(φ) = f0(m) +2N

α20(m)

Trφ2 + gφ4, (19)

where α20(m) depends on the degree of divergence of a basic diagram [29].

This effective action (19) can now be studied using standard results fromrandom matrix theory. For example, this allows to study the renormalization ofthe effective potential using matrix model techniques. The basic mechanism is thefollowing: In the free case, the eigenvalue sector follows Wigner’s semicircle law,where the size of the eigenvalue distribution depends on m via α0(m). Turning onthe coupling g alters that eigenvalue distribution. The effective or renormalizedmass can be found by matching that distribution with the “closest” free distribu-tion. To have a finite renormalized mass then requires a negative mass countertermas usual.

This approach is particularly suitable to study the thermodynamical proper-ties of the field theory. For the φ4 model, the above effective action (19) impliesa phase transition at strong coupling, to a phase which was identified with thestriped or matrix phase in [29]. Based on the known universality properties ofmatrix models, these results on phase transitions are expected to be realistic, andshould not depend on the details of the unknown function F(φ). The method isapplicable to 4 dimensions, where a critical line is found which terminates at anontrivial point, with finite critical coupling. This can be seen as evidence for anew nontrivial fixed-point in the 4-dimensional NC φ4 model. This is in accor-dance with results from the RG analysis of [14], which also point to the existenceof nontrivial φ4 model in 4 dimensions.

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Harald GrosseUniversitat Wien, Institut fur Theoretische PhysikBoltzmanngasse 5A-1090 Wien, Austriae-mail: [email protected]

Raimar WulkenhaarUniversitat Munster, Mathematisches InstitutEinsteinstraße 62D-48149 MunsterGermanye-mail: [email protected]

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