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A non-perturbative approach to non-commutative scalar field theory


LMU-TPS 05/01

A non-perturbative approach to non-commutative scalar ?eld theory

Harold Steinacker?

arXiv:hep-th/0501174v3 1 Apr 2005

Department f¨ ur Physik Ludwig–Maximilians–Universit¨ at M¨ unchen Theresienstr. 37, D-80333 M¨ unchen, Germany

Abstract
Non-commutative Euclidean scalar ?eld theory is shown to have an eigenvalue sector which is dominated by a well-de?ned eigenvalue density, and can be described by a matrix model. This n is established using regularizations of R2 θ via fuzzy spaces for the free and weakly coupled case, and extends naturally to the non-perturbative domain. It allows to study the renormalization of the e?ective potential using matrix model techniques, and is closely related to UV/IR mixing. In particular we ?nd a phase transition for the φ4 model at strong coupling, to a phase which is identi?ed with the striped or matrix phase. The method is expected to be applicable in 4 dimensions, where a critical line is found which terminates at a non-trivial point, with nonzero critical coupling. This provides evidence for a non-trivial ?xed-point for the 4-dimensional NC φ4 model.

?

harold.steinacker@physik.uni-muenchen.de

Contents
1 Introduction 2 NC scalar ?elds and UV/IR mixing 2.1 Matrix regularization of Rn . . . . . . . . . . . . . . . . . . . . . . . . . . θ 3 The 3.1 3.2 3.3 3.4 eigenvalue distribution of the scalar ?eld The free case . . . . . . . . . . . . . . . . . . . . Interactions . . . . . . . . . . . . . . . . . . . . . Angle-eigenvalue coordinates in ?eld space . . . . 3.3.1 Interpretation of the orbits O(φi). . . . . . Relating the matrix model to physical observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 4 5 6 10 12 14 15 16 18 18 19 19 20 21 21 22 25 26 26 28 28 30 32 34

4 Example: the φ4 model 4.1 Phase 1 (“single-cut”): m′2 > 0, or m′2 < 0 with 4.2 Phase 2 (“2 cuts”): m < 0 with
′2 m′4 4g ′

m′4 4g ′

<1

. . . . . . . . .

>1 . . . . . . . . . . . . . . . . . .

5 Weak coupling and renormalization 5.1 2 dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 4 dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 The phase transition. 6.1 4 dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 2 dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 The fuzzy sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Discussion and outlook
n Appendix A: Regularizations of R2 θ 2 2 2 The fuzzy torus TN and TN × TN . . . . . . . . . . . . . . . . . . . . . . . . . . The fuzzy sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fuzzy CP n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Appendix B: Justi?cation of (16)

1

Introduction

The idea of considering quantum ?eld theory on “quantized” or non-commutative spaces (NCFT) was put forward a long time ago [1], and has been pursued vigorously in the past years; see e.g. [2, 3] for a review. A very intriguing phenomenon which was found in this context is the so-called UV/IR mixing [5], linking the usual UV divergences to new singularities in the IR. On a technical level, it arises because of a very di?erent behavior of planar and non-planar diagrams, which must be distinguished on NC spaces. The planar diagrams are essentially the same as in the commutative case. The non-planar diagrams

1

however lead to oscillating integrals, which are typically ?nite as long as the external momentum p is non-zero, but become divergent in the limit p → 0. This leads to serious obstacles to perturbative renormalization [5]. Furthermore, it appears to signal an additional phase denoted as “striped phase” [6–8], which arises as the minimum of the e?ective action is no longer at zero momentum. Because UV/IR mixing is so generic in the NC case, it is necessary to come to terms with it and to ?nd suitably adapted quantization methods. The ?rst step is clearly a suitable regularization of the models. This can be achieved by parametrizing the ?elds in terms of ?nite matrices, which is very natural on NC spaces. Several such methods are available by now, using e.g. using fuzzy spaces, non-commutative tori, etc. The action for scalar ?elds is then a functional of a hermitean matrix φ, where the potential T rV (φ) looks like a “pure” matrix model with U (N ) invariance, which is however broken by the kinetic term. The UV/IR mixing is expected to be recovered in the continuum limit. Such a regularization has been used recently to con?rm numerically the non-trivial phase structure mentioned above in the non-commutative φ4 model [9, 10]. There has also been remarkable progress on the analytical side using matrix techniques: A modi?ed φ4 model with an explicit IR regulator term in the action was shown to be perturbatively renormalizable [11], and certain self-dual models of NC ?eld theory were solved exactly using a matrix model formulation [13]. For gauge theories, the applicability of well-known techniques from random matrix theory has also been shown in simple cases [14, 15]. For the φ4 model, a similar approach using random matrix theory does not seem possible at ?rst sight, lacking U (N ) invariance. Nevertheless, it was conjectured in [9] that the striped phase should be identi?ed with a “matrix phase” for the fuzzy sphere, where the action appears to be dominated by a pure potential model in that phase. Hence a simple analytical approach which allows to study also scalar NCFT’s with non-trivial phase structure and UV/IR mixing is highly desirable. In particular, it seems that the obvious parallels between NCFT and pure matrix models due to UV/IR mixing have not yet been fully exploited, apart from integrable cases [13]. The aim of this paper is to show that there is indeed a simple matrix model description which captures a certain sector of scalar NCFT, due to UV/IR mixing. This suggests a new approach to scalar NCFT which not only provides new insights, but also new tools to study the renormalization of the e?ective potential. The starting point is an appropriate parametrization for the ?elds: since φ is a hermitian matrix, it can be diagonalized as φ = U ?1 diag (φi)U where φi are the real eigenvalues. Hence the ?eld theory can be reformulated in terms of the eigenvalues φi and the unitary matrix U . The main observation of this paper is now the following: the probability measure induced on the (suitably rescaled) eigenvalues φi from the path integral is sharply localized, and described by an ordinary, simple matrix model. This means that only ?elds φ with a particular eigenvalue distribution characterized by a certain function ρ(s) : [?1, 1] → R+ contribute signi?cantly to the (euclidean) path integral. While this is plausible using the above parametrization, it is a nontrivial statement which is only true in the con-commutative regime. It is established ?rst for the free case (which therefore does know about non-commutativity, contrary to a common belief) and extends immediately to the interacting case at least on a perturbative level. This is directly related to UV/IR mixing, since non-planar contributions to the eigenvalue observables are suppressed by the oscillatory factors, while the planar contributions can be described by a simple matrix model without kinetic term. It is quite obvious that this will extend also to 2

the non-perturbative level, in a suitable domain. This suggests that scalar NCFT can be characterized by a single function ρ(s). We then work out some simple applications of this approach, which do not require long computations. In the weak coupling regime, this leads to a very simple method of computing the mass renormalization using matrix model techniques. In particular the standard one-loop result for the mass renormalization in the φ4 model is recovered in a non-standard way, and ?nds a natural interpretation in the matrix model. Unfortunately the running of the coupling does not seem to admit such a simple computation. In any case, we will argue that there exists a scaling limit with a non-trivial correlation length in the continuum limit, suggesting the existence of a renormalized φ4 model in 2 and 4 dimensions. Extending these results to the non-perturbative regime, we ?nd a phase transition for the φ4 model in 2 and 4 dimensions, to a phase which is tentatively identi?ed with the striped or matrix phase of [6, 9]. This can be compared with numerical results available for the fuzzy sphere in 2 dimension, with reasonable agreement which con?rms the overall picture. The method is expected to work better in 4 dimensions, due the stronger divergences which are crucial for our derivation. In the 4-dimensional case, the critical line is found to terminate at a non-trivial point, with nonzero critical coupling gc = 0. This is expected to be a sound prediction, suggesting the existence of a non-trivial ?xed-point for the 4-dimensional NC φ4 model, and hence renormalizability in an appropriate sense. To summarize, while the dominance of planar diagrams in NCFT is very well known, the description of the eigenvalue sector in terms of a simple matrix model (39) appears to be new and is very practical. This provides an alternative approach to some of the results of [6–8] on the phase structure in NCFT, con?rming the rough picture of a phase transition towards a phase which breaks translational invariance. However, the phase transition is predicted to be higher-order, as opposed to [6–8]. We ?nd in addition a critical coupling gc = 0 in 4 dimensions, and it would be very interesting to verify this numerically. This paper is organized as follows. Section 2 provides some background recalling the UV/IR mixing, and introduces the matrix regularizations used later. Section 3 is the core of this paper: after identifying the suitable observables, we show that the eigenvalue distribution of free ?elds is given by Wigner’s semi-circle law, which allows to replace the kinetic term by the matrix model (34). Interactions are included in Section 3.2. The corresponding reformulation of scalar NCFT using eigenvalue and angle coordinates is discussed in general in Section 3.3 and 3.4, including an intuitive semi-classical picture. This is then applied to the φ4 model in section 4, and related to some standard results for hermitean matrix models. These allow in Section 5 to obtain the mass renormalization in a very simple way. The phase transition is studied in detail for 2 and 4 dimensions in section 6, and compared with numerical results for the fuzzy sphere. We conclude in Section 7 with further remarks and an outlook.

2

NC scalar ?elds and UV/IR mixing
Moyal plane Rd θ in even dimensions, g φ?φ?φ?φ . 4 (1)

Consider scalar ?eld theory on the non-commutative with action 1 1 S = dd x ?i φ?i φ + m2 φ2 + 2 2 3

Here φ is a function on Rd , and the ?? product is the standard Moyal product of functions on Rd , which can be written as (a ? b)(x) = d4 k d4 y a(x+ 1 θ·k ) b(x+y ) eik·y , 2 (2π )4 (θ·k )i = θij kj , k ·y = ki yi , θij = ?θji . (2)

This can be understood as a pull-back of the operator product of 2 operators a and b from a representation of the underlying (Heisenberg) algebra [xi , xj ] = iθij , (3)

using a suitable quantization map. We assume that θij is non-degenerate in this paper. The model (1) written down above is not well-de?ned as it stands, and needs regularization. The simplest way to proceed is to use a sharp UV cut-o? Λ, which leads to standard computations and will be justi?ed below. The perturbative quantization of (1) di?ers from the commutative case by the fact that planar and non-planar diagrams must be distinguished. The reason is that commuting 2 plane waves with wavenumbers k and ′ k ′ produces a factor e?ikθk , which makes non-planar loops convergent for generic external momenta. More explicitly, the basic one-loop planar and non-planar self-energy diagrams (without counting symmetry factors) are [5] ΓP := ΓN P (p) :=
(2) (2) (2)

dd k 1 , d 2 (2π ) k + m2 dd k eikθp . (2π )d k 2 + m2

(4)

ΓN P (p) is ?nite as long as p = 0 due to the oscillating term, but has an IR singularity as p → 0 because the k -integral is then divergent as usual. This is known as UV/IR mixing [5], and appears to be a central feature of NC ?eld theories. It is a serious obstacle to perturbative renormalization, which was only overcome recently in a modi?ed φ4 model [11]. In this paper, we shall try to turn this UV/IR mixing into a virtue, and point out that it is closely related to an interesting property of the scalar ?eld φ in the operator formulation, which seems very useful and does not hold for ordinary ?eld theories: The dynamical ?eld φ has a well-de?ned eigenvalue distribution upon quantization, which is governed by a simple matrix model and can be studied using a saddle-point analysis.

2.1

Matrix regularization of Rn θ

Recall that if the non-commutative algebra (3) is represented on a (in?nite-dimensional) Hilbert space H, the integral is given by the suitably normalized trace: √ dd xf (x) = (2π )d/2 det θ T rf = (2πθ)d/2 T rf (5)

where f in the rhs is the operator version of f (x) (as obtained e.g. using the Weyl quantization map). This is a manifestation of the Bohr-Sommerfeld quantization condition, 4

relating the volume of the phase space to the dimension of the Hilbert space. The last line holds for Sp(d)- invariant θij , which we assume in this paper for simplicity. We want to approximate this using some ?nite-dimensional matrix algebra. This can be achieved e.g. using a suitable scaling limit of the fuzzy sphere for d = 2, or more n generally fuzzy CP n for d = 2n, see Appendix A. Indeed fuzzy CP n → R2 θ in a suitable limit, where θij turns out to be invariant under Sp(2n), and the propagator is the usual one with a sharp momentum cuto?. Another possible regularization is using so-called NC lattices which are products of certain (“fuzzy”) NC tori [18], see also Appendix A. These are technically somewhat easier to handle, but lead to a modi?ed behavior of the propagators for large momenta; then the most general θij can be obtained. In all these regularizations, φ is a hermitian N × N matrix in some ?nite matrix algebra Mat(N , C), and the model (1) is replaced by a matrix model (where the kinetic term breaks the U (N ) symmetry). In particular, the trace is now over the N -dimensional Hilbert-space H = CN , where N is related to the cuto? Λ and θ in a speci?c way (154), (118) depending on the regularization. Then V := dd x1 = (2πθ)d/2 N , and we can write 1 V dd xf (x) = 1 T rf. N (6)

In particular, integrals of the type φ2n depend only on the eigenvalues of φ, and these are the observables we want to study. To make the paper most readable, the regularization using fuzzy CP n with sharp momentum cuto? Λ will be understood, while using the conventional language of Rd θ as much as possible. The results for general (non-degenerate) θij and somewhat modi?ed propagators would be qualitatively the same.

3

The eigenvalue distribution of the scalar ?eld

The basic idea is the following: Having regularized the model (1) in terms of a ?nitedimensional hermitean matrix φ, we can diagonalize it as φ = U ? 1 (φ i )U (7)

where U is a unitary N × N matrix, and (φi ) ≡ diag (φ1, ..., φN ) is diagonal with real eigenvalues. The integration measure in the path integral can now be written as D φe?S = dφi ?2 (φi ) dUe?S , where ?2 (φi ) = i<j (φi ? φj )2 is the Vandermonde-determinant and dU the Haar measure for SU (N ). We are interested in the probability measure or e?ective action for these eigenvalues, induced by this path integral. For this purpose, consider e.g. the expectation values dd xφ2n (x) = 1 Z D φ exp(?S )( dd xφ2n (x)). (8)

They are strongly divergent normally but make sense in the regularized (matrix) case; in particular, we do not want to replace the φ2n (x) by some renormalized objects but simply keep track of their dependence on the cuto?. Since they depend only on the eigenvalues of the ?eld φ in the matrix representation, we can determine the e?ective eigenvalue distribution by studying such observables. This turns out to be non-trivial already in the free case: 5

3.1

The free case

We compute the observables (8) for g = 0 with a sharp UV - cuto? Λ using Wicks theorem. This involves in general planar and non-planar diagrams. The simplest case is1 dd xφ2 (x) V 1 dd p = d?1 d/2 d 2 2 2 π (d/2 ? 1)! 0 (2π ) p + m d?2 =: c(m, Λ)V Λ . = V
Λ Λ

dp
0

pd?1 p2 + m2 (9)

=2n This formal result will be fully justi?ed in Appendix A using regularizations of Rd in θ n terms of fuzzy CP (153) or fuzzy tori. Here V denotes the regularized volume of Rd θ, and c = c(m, Λ) is of order 1 for d ≥ 3, and c = O (ln Λ) for d = 2. More precisely, in 4 dimensions we have

d4 φ2 (x) and in 2 dimensions

=

V m2 8π 2

Λ/m

du
0

V Λ2 u3 = u2 + 1 16π 2

1?

m2 Λ ln(1 + ( )2 ) 2 Λ m

(10)

V d φ (x) = 2π
2 2

Λ/m 0

u V Λ2 du 2 = ln(1 + 2 ) u +1 4π m

(11)

(the subleading behavior is modi?ed for the regularization using NC tori, see Appendix A). Therefore 2 Λ2 1 ln(1 + m , d=4 1? m 2) 16π 2 Λ2 c(m, Λ) = (12) 1 Λ2 ln(1 + )) , d = 2 . 4π m2 Next, consider dd xφ(x)4 = = 2V
0

dd xφ(x)4
Λ

P lanar

+

dd xφ(x)4

N on?P lanar Λ 0

1 dd p (2) ΓP ( p ) + V d 2 2 (2π ) p + m

1 dd p (2) ΓN P ( p ) , d 2 2 (2π ) p + m

which is obtained by summing over all complete contractions of a vertex with 4 legs. There are 2 planar and one non-planar such contractions, the latter being given by joining the external legs of the non-planar self-energy diagram (4). The planar contribution is simply dd xφ(x)4
(2) P lanar

= 2V c2 Λ2(d?2) ,

(13)

since ΓP (p) is independent of p. On the other hand, the non-planar contribution is sublead(2) ing (this will be discussed in detail below), since ΓN P (p) is ?nite except for the singularity at p → 0. This is clearly related to UV/IR mixing, even though we are considering only the free case up to now. We therefore expect
1 V 1 V
1

dd xφ(x)4 1 = 2 + O ( d?2 ). d 2 2 cΛ d xφ(x)

(14)

we basically assume that θij is non-degenerate in this paper, which implies that d is even. However, some of the results extend to the degenerate case, which will be pursued elsewhere

6

More generally, consider
1 V 1 V

dd xφ(x)2n = dd xφ(x)2 n

1 V

dd xφ(x)2n P lanar + 1 dd xφ(x)2 n V

1 V

dd xφ(x)2n N on?P lanar . 1 dd xφ(x)2 n V

The ?rst term is of order one, and simply counts the number NP lanar (2n) of planar contractions of a vertex with 2n legs. The non-planar contributions always involve oscillatory integrals, and do not contribute to the above ratio in the large Λ limit. Let us discuss them in more detail: Assume that the cut-o? is much larger than the NC scale, Λ 2 θ ? 1, (15)

which will be understood from now on. By rescaling the momenta k ′ = k/Λ, the above ratios for d ≥ 3 have the form RN P :=
1 V

(16) which vanish for large Λ due to the rapidly oscillating exponential. The integrals are over the unit ball resp. hypercube. This is established more carefully in Appendix B, where we show that RN P = O (1/Λ) (at least) for d ≥ 3. In 2 dimensions, these considerations are more delicate as the above ratio RN P vanishes only logarithmically, and divergences may m is ?xed while Λ → ∞, also arise in the IR. One must then assume furthermore that e.g. m θ where 1 (17) m2 θ := θ is the non-commutative mass scale. Due to this weaker logarithmic behavior, the ?uctuations are expected to be larger for d = 2 than for d = 4. We refrain here from more precise estimates which should be made in the context of the regularized models, see Appendix B. With all these assumptions, we conclude that
1 V 1 V

dd xφ(x)2n N on?P lanar ≈ 1 dd xφ(x)2 n V

1 0

′ ′ dd k1 dd kn 2 ... eiΛ ′ 2 ′ 2 (k 1 ) (k n )

′ Θk ′ ki j

→ 0 as Λ → ∞,

dd xφ(x)2n = dd xφ(x)2 n

1 V

dd xφ(x)2n P lanar = NP lanar (2n) 1 dd xφ(x)2 n V

(18)

in the large Λ limit. Notice that this result is very di?erent from the conventional ?eld theory: the above calculation with a naive cuto? would have the same contributions from planar and non-planar diagrams. Then the total number of contractions of e.g. φ2n is of order 2n n! ? NP lanar (2n), which would invalidate the conclusions below. Next, consider expectation values of products ( dd xφ(x)2n1 )...( dd xφ(x)2nk ) . We claim that this factorizes in the large Λ limit,
1 V 1 V

dd xφ(x)2n1 ... dd xφ(x)2 n1 ...

1 V 1 V

dd xφ(x)2nk dd xφ(x)2 nk

1 dd xφ(x)2n1 dd xφ(x)2nk V ... 1 dd xφ(x)2 n1 dd xφ(x)2 nk V = NP lanar (2n1 ) ... NP lanar (2nk ) (19)

=

1 V 1 V

This can be seen again in terms of contractions, because propagators joining di?erent vertices must satisfy additional momentum constraints as opposed to the disjoint contractions, therefore only the disjoint contributions survive in the large Λ limit. This amounts essentially to the cluster property. 7

Now recall that in the non-commutative case, the integral is given by the suitably normalized trace (6). Hence the integrals above depend only on the eigenvalues of φ, and we obtain statements about the induced eigenvalue distribution. The above observables (19) can be written in the form ( 1 1 T rφn1 )...( T rφnk ), N N (20)

and completely determine the e?ective probability measure for the eigenvalues of φ. We can certainly write any such expectation value as ( 1 1 T rφn1 )...( T rφnk ) = N N dφ1 ...dφn ?(φ1 , ..., φn ) ( 1 N
1 φn i )...(

i

1 N

k φn j )

(21)

j

for some measure ?(φ1 , ..., φn ), which we would like to determine for large Λ. In order to absorb the in?nities we introduce a scaling constant
2 α0 (m)

= 4 cΛ

d?2

=

2 1 Λ2 1 ? m 4π 2 Λ2 Λ2 1 ln(1 + m 2 ), π

ln(1 +

Λ2 ) m2

, d=4 d=2

(22)

and write2 φ = α0 ?. Then (9) gives (23)

1 1 T r?2 = . (24) N 4 1 Now all expectation values N T r?2n are ?nite and have a well-de?ned limit N → ∞. We can then describe the eigenvalues of ? (in increasing order) by an eigenvalue distribution, ? (s ) = ? j , Then e.g. 1 1 Trf (?) = N N
i

s=

j , N

s ∈ [0, 1].
1

(25)

f (? i ) →

ds f (?(s))
0

(26)

in the large N limit. The measure ? now becomes a measure ?[?(s)] on the space of (increasing) functions ?(s) : [0, 1] → R. To ?nd this measure ?[?(s)], we ?rst note that the factorization property (19) implies that the measure ? is localized, i.e. f (?i ) = f (?(s)) (27)

for any function f , where ?(s) is the (sharp and dominant) saddle-point or maximum of ?[?]. This saddle-point ?(s) corresponds to a density of eigenvalues ρ(?) =
2

ds , d?



ρ(p)dp = 1.
?∞

(28)

α0 is not the wavefunction renormalization

8

The expectation value of the above observables is then given by 1 Trf (?) = N
1

dpρ(p) f (p),

(29)

1 1 Tr?n = 0 ds ?n ; for example, . We want to determine ds?(s)2 = ds?(s)2 = 4 or N this saddle-point ?(s). This can be extracted from the above results: (18) implies that 1

ds?(s)2n =
0

1 ρ(p)p2n dp = ( )n NP lanar (2n). 4

(30)

There is a unique eigenvalue distribution with these properties, given by the famous Wigner semi-circle law 2 1 ? p2 p2 < 1 π (31) ρ(p) = 0, otherwise. This means that the eigenvalues of φ are distributed correspondingly in the interval φi ∈ [?α0 , α0 ]. It is fun to verify (30) explicitly for small n; indeed Γ( 1 + n) 1 ρ(x)x dx = 2 = n NP lanar (2n). Γ(2 + n) 4 ?1
1 2n

(32)

(33)

In general, this follows also from the basic properties of matrix models: Consider the Gaussian matrix model with action ?0 = f0 (m) + S
2N α2 0

T rφ2

(34)

where φ = α0 ? is a N × N matrix. Here f0 (m) is some numerical function of m which will ?0 will ?nd another interpretation be determined below, and α0 depends on m via c = c(m). S ?0 is known to reproduce precisely the eigenvalue distribution (31) in the in Section 3.3. S large N limit [19], and (30) follows because again only planar diagrams contribute. Indeed, one ?nds again e.g. 1 1 2 2 1 T rφ2 = α0 T r?2 = α0 (35) N N 4 etc.; we refer to the vast literature available on this subject, e.g. [19–21]. We conclude that if one is only interested in the eigenvalues, the free action S0 = d 1 ?0 in (34). Moreover, one d x 2 (?i φ?i φ + m2 φ2 ) can be replaced by the e?ective action S can determine f0 (m) such that the partition function is also recovered as Z= D φe?S0 = D φe?S0 =
?

D φ exp(?f0 (m) ?

2N T rφ2 ); 2 α0

(36)

this will be understood better in Section 3.3. To summarize, we noted that the observables (19) depend only on the eigenvalues; the factorization property implies that the dominating contribution comes from a well-de?ned eigenvalue distribution, which is via (30) identi?ed as the Wigner-distribution corresponding to a Gaussian matrix model. We can then write 9

down an e?ective Gaussian matrix model which reproduces all the expectation values for these observables. Note that the details of the propagator enter only through α0 , and the basic result depends only on the degree of divergence. It is interesting to compare this with the commutative case, where the eigenvalues are replaced by the values of the ?eld φ(x) at each point, i.e. N d variables for a lattice regularization. Using again a cuto? Λ, we would have
1 V 1 V

dd xφ(x)2n = NP lanar (2n) + NN onplanar (2n) = dd xφ(x)2 n



e?s
?∞

2 /2

s2n ≈ 2n n! ? NP lanar (2n).

?C = 1 2 dd xφ(x)2 . The crucial This could be reproduced by a Gaussian distribution S 0 2α0 d di?erence is that this would describe N independent variables (which is false of course, but ok for these observables) with a very ?at distribution, while in the NC case we consider only N = N d/2 eigenvalues which are collective and intrinsically non-local variables, governed by an e?ective action with a sharp potential3 ; note also the explicit factor N in (34). This will become more explicit in the following sections, which do not apply to the commutative case. Let us try to interpret this result. It may be surprising that already the free scalar NCFT is apparently very di?erent from the commutative case, even though they are supposed to be the same from the star-product point of view. The reason is that we are looking at statistical properties of the operator representation of the wavefunctions, which is related to their point-wise values only for low momenta where the functions are “almost-commutative”. Therefore the non-classical properties of the high-energy modes are responsible for this property. This already indicates UV/IR mixing: the modes with k ≈ 0 are suppressed upon quantization because they tend to have the “wrong” eigenvalue distribution (in particular φk=0 ∝ 1 l has only one eigenvalue). The existence of a well-de?ned eigenvalue distribution and its matrix model description will generalize easily to the interacting case at least on a perturbative level, and is very plausible also in the non-perturbative domain. Furthermore it suggests new and practical insights to the coupling constants and their renormalization. This will be discussed next.

3.2

Interactions

If we include interactions of the form gn gn Sint (φ) = dd xφn (x) = (2πθ)d/2 T rφn , n n the results (34), (36) for the free case generalize at least on a perturbative level: consider Zint = D φe?(S0 +Sint ) = D φe?S0 1 ? gn (2πθ)d/2 T rφn + ... n (37)

Hence the expansion of e?Sint in powers of gn leads to additional terms of the form (19) or (20), which are again evaluated by the rule (29) for the measure ρ of the free case4 . We
recall that the saddle-point approximation for the matrix models is good for the eigenvalues, but not for the matrix elements 4 This is valid as long as the coupling constants gn are small enough so that the eigenvalue distribution remains unchanged. This should be enough to determine the perturbation series, where gn can be arbitrarily small. In particular, it su?ces to determine mass renormalizations etc. as seen below.
3

10

1 T rφ2n g in the presence of can write a similar formula to evaluate observables such as N ?0 in (37). gn . Therefore the results of Section 3.1 apply, and one can simply replace S0 by S We conclude that at least perturbatively, the eigenvalue sector of the NCFT with action

S=

1 dd x (?i φ?i φ + m2 φ2 ) + Sint 2

(38)

for any polynomial interaction is described by the e?ective matrix model ?(φ) = f0 (m) + S
2N T rφ2 α2 0 (m)

+ Sint (φ)

(39)

where α0 = α0 (m) is given by (22), in the large N limit. This is expected to be correct as long as the eigenvalue distribution is close enough to the free one (31). The partition function is similarly given by Zint = D φ exp(?f0 (m) ? 2N T rφ2 + Sint (φ)). 2 α0 (m) (40)

This de?nes by the usual matrix model technology an analytic function in the couplings gn . In the later sections, this will allow us in particular to determine some renormalization properties of the potential is a very simple way. We will also explore non-perturbative implications such as phase-transitions, hoping that this will give at least qualitatively correct results. Scaling and relevant couplings. Let us try to estimate the impact of the interaction terms to the eigenvalue distribution. √ In the matrix-regularizations of NCFT used below, d/2 we will have N ? N and Λ = O ( N ). Therefore the following scaling behavior holds φ ? α0 ? Λ 2 ?1 ,
d

N = Λd

(41)

using (23). We assume this scaling also in the interacting case in the weakly-coupled domain, and check for which gn this is self-consistent. Then (39) essentially becomes ? ? T r (N ?2 + gn αn ?n ) =: T r (N V (?)), omitting constants of order 1 (including θ, which S 0 is not assumed to scale). Here ? is of order one. The resulting eigenvalue distribution is governed by V (?) [19], and it will remain near the Gaussian ?xed point resp. Wigner’s law n provided the bare couplings satisfy gn ≤ N /α0 ? Λδ , where δ =d+n? nd 2 (42)

is just engineering dimension of gn . As usual, this means that relevant or marginal couplings with δ ≥ 0 are “safe” and expected to be renormalizable, while irrelevant couplings with δ < 0 must be ?ne-tuned and are expected to be non-renormalizable. In general, it will be safe to use (39) as long as the shape of the resulting eigenvalue distribution ?(s) is close to the Wigner law. For a qualitatively di?erent shape, one should expect corrections to (39). In particular, it is quite clear that turning on some small coupling g4 must be compensated by a suitable “mass renormalization” in order to preserve the shape of ?(s). This allows to determine the mass renormalization, which will be explored in Section 5. But before discussing these issues, let us look at the above results from a di?erent perspective: 11

3.3

Angle-eigenvalue coordinates in ?eld space

One of the merits of the NC ?eld theory is that it naturally suggests new coordinates in ?eld space, which are very di?erent from the usual “local” ?elds φ(x) in the commutative case. This is particularly obvious using a regularization in terms of ?nite-dimensional matrices φ ∈ Mat(N , C). Then there is a natural action of the SU (N ) group φ → U ?1 φU , which can be seen as NC version of the symplectomorphisms. Even though this is not a symmetry due to the kinetic term, it suggests to parametrize φ in terms of eigenvalues and “angles”, which are very non-local coordinates. This change of variables leads very naturally to the picture we found above. In particular, the non-trivial measure factor in the path integral due to the Jacobian makes the existence of a non-trivial eigenvalue distribution very plausible. We start from the simple fact that any hermitian matrix φ can be diagonalized, φ = U ? 1 (φ i )U (43)

where U ∈ SU (N ) and (φi) is a diagonal matrix, with eigenvalues φi . We can moreover assume that the eigenvalues of (φi) are ordered; then the matrix U is unique up to phase factors K ? = U (1)N ?1 , provided the eigenvalues φi are non-degenerate. This leads to the following de?nition of the orbits O(φ) := {U ?1 (φi )U ; U ∈ SU (N )} ? = SU (N )/K (44)

where K is the stabilizer group of (φi ). These are compact homogeneous spaces. Then the partition function can be written as Z = = = D φ exp(?S (φ))) = dφi dU exp(
i=j

dφi?2 (φi )

dU exp(?S (U ?1 (φi )U ))

log |φi ? φj | ? S (U ?1 (φi )U )) V (φ i ) +
i i=j

? (φi ) exp(?(2πθ)d/2 dφi F

log |φi ? φj |),

(45)

where dU is the integral over N × N unitary matrices; similar manipulations were also done in [9]. We introduced here the function ? (φ) := F dU exp(?Skin (U ?1 (φ)U )) =: e?F (φ) ,
?

(46)

which by de?nition depends only on the eigenvalues of φ. The last form is justi?ed because ? (φ) is positive. It satis?es F ? (φ + c) = F ? (φ ), F ? (?φ) = F ? (φ ). F (47)

? (φi ) is analytic in the φi because the space is compact, and invariant under Moreover, F exchange of the eigenvalues. We can therefore expect that it approaches some nice classical ? [φ(s)] in the large N limit, where φ(s) is the function in one variable which is functional F related to φi as in (25). 12

We can now read o? the induced action for the eigenvalues, ? (φ i ) = F ? (φ i ) ? S log |φi ? φj | + (2πθ)d/2 V (φ i )
i

(48)

i=j

The log - term in (48) could also be absorbed by de?ning ? (φ i ) ? F (φ i ) = F log |φi ? φj |. (49)

i=j

In particular, note that the log-term in (48) strongly suppresses degenerate eigenvalues, and ? (φi ). Therefore the saddle-points of S ? (φ i ) this cannot be compensated by any analytic F corresponds to some non-degenerate eigenvalue distribution. Furthermore the kinetic term ? (φi )) strongly suppresses jumps in this eigenvalue distribution5 , therefore we (encoded in F expect it to approach some smooth function φ(s) in the limit N → ∞. Furthermore, we expect this eigenvalue distribution to have compact support after a suitable rescaling: φ(s) = α?(s) (50)

where α denotes the maximal eigenvalue. This typically happens for matrix models, and ? (φ) only depends appears to be true also in this context as shown below. Indeed since F on the eigenvalues, one can trivially interpret it as a function of any hermitean matrix φ = U ?1 (φi )U , and rewrite the partition function as an ordinary matrix model with formal U (N ) symmetry, Z= ?(φi )) = dφi exp(?S ? (φ) ? (2πθ)d/2 T rV (φ)). D φ exp(?F (51)

The last step is of course completely formal, and we are only allowed to determine observables depending on the eigenvalues with this action. These are determined by the ?(φi), since the degrees of freedom related to U are integrated out. This “e?ective action” S strongly suggests that much of the information about the quantum ?eld theory, in par? (φ i ) ticular the phase transitions and the thermodynamic properties, are determined by S and the resulting eigenvalue distribution in the large N limit. Note that this is essentially a one-dimensional problem, governed by the (unknown) functional F [φ(s)] and V . The advantage of this formulation is that it is very well suited to include interactions, and naturally extends to the non-perturbative domain. ?(φi ), determined Now assume we know F (φi); one can then look for the saddle-points of S by ? δF 1 + (2πθ)d/2 V ′ (φi ) = (52) δφi φj ? φi
j =i

and ask if they are localized enough to dominate the observables. If the existence of a sharp eigenvalue distribution is established, the full path integral in (45) would be dominated by the integral over the corresponding SU (N ) orbit O(φi ). This in turn should allow to recover
At strong coupling however, we will ?nd a phase transition to a distributions with one gap, corresponding to some “striped” phase
5

13

also other properties such as correlation functions, by integrating over the corresponding O(φi ) which is compact. This will be discussed further in Section 3.4. We can now relate this to the results of Section 3.1: All observables of the eigenvalues ?(φi ) above. The result of as considered there are determined by the “e?ective action” S Sections 3.1 and 3.2, in particular the factorization property (19), says that there is indeed a well-de?ned eigenvalue distribution in the non-commutative domain6 , i.e. as long as Λ2 θ ? 1. Comparing with the e?ective action for the free case (34), we ?nd ? (φ) = f0 (m) + 2N T rφ2 F 2 α0 (53)

m2 φ2 . which certainly reproduces all admissible observables for the potential V (φ) = 1 2 ? (φ) should of course be independent of V . This formula may appear strange, since F The reason is that this relation (53) has been established only “on-shell”, for eigenvalue distributions close to the Wigner law for the free case. It is not clear how well this works for large deviations from that case. Later we will use this form also for eigenvalue distributions which are quite di?erent from the free one, where one should expect corrections to (53). The appropriate way to use (53) for some given eigenvalue distribution is therefore to determine m such that the corresponding free distribution matches best the one under consideration. ? (φ). It would be extremely interesting to know more about the functional F The dominance of a given orbit O(φi ) in (45) is clearly related to UV/IR mixing: naively, the action has a minimum at φ = const 1 l; however if the volume-factors from the path integral are taken into account (which happens at one loop), this zero-momentum state is actually highly suppressed, and the dominating ?eld con?gurations have nontrivial position-dependence (i.e. momentum), due to the nontrivial eigenvalue distribution. Note that this argument is completely non-perturbative. Furthermore, depending on the form of the potential V (φ) the dominating eigenvalue distribution may be connected or consist of disjoint pieces. These would clearly correspond to di?erent phases. This picture will be made more quantitative below, and we will be able to identify the “striped” phase of [6]. 3.3.1 Interpretation of the orbits O(φi ).

Consider the reduced model (46) for a given orbit O(φi ) with ?xed eigenvalue distribution, ? (φ). Intuitively, we can interpret this model as follows: whose (free) energy is given by F consider a classical ?uid φ(x) on a compact space (due to the regularization e.g. on CP n or some torus) with prescribed “density” ρ(p) = 1 V dd x δ (φ(x) ? p), (54)

corresponding to the eigenvalue distribution. Then ρ(p) is essentially the density of eigenvalues (28), at least in the semi-classical limit. Note in particular that the action of the classical volume-preserving di?eomorphisms on φ(x) can be approximated by SU (N ), since any con?guration with given ρ can be obtained using SU (N ). Entropy will favor mixing, but the “kinetic energy” suppresses mixing. In a small region of space, the global constraint (54) is quite irrelevant, and the ?uid will behave like a ?uid with the same action
This is not true for conventional ?eld theory with θ = 0, even though the formulation of this section is still possible for e.g. the commutative limit of fuzzy spaces.
6

14

but without the constraint. However, if we ?x the ?eld on a large part of the volume, this must be compensated in the remaining space. This is clearly an IR e?ect, suppressing very large wavelengths. Therefore we expect that the theory behaves like an “ordinary” ?eld theory in small enough regions of space and may hence describe ordinary local physics, however it is certainly di?erent globally. This is quite interesting and encouraging for possible applications in elementary particle physics. In order to go beyond the computations in the following sections, one should therefore study the reduced models (46) in more detail, and see to what extent they approximate a scalar ?eld theory. Note that quite generally if the scale α is increased, the con?gurations with short wavelength will be more strongly suppressed, leading a long correlation length; on the other hand for small α, the correlation length will be short. Therefore there should indeed exist some suitable scaling α(N ) which gives a macroscopic correlation length in the limit N → ∞. The relation with the intuitive picture presented in Section 3.3 and in particular (54) can be seen best using coherent states or projectors (see e.g. [25, 26, 28]), in n particular for the regularizations with CPN . It would be very interesting to combine these methods with the approach in the present paper. The “ground state” of (46) on a given orbit O(φi ) is some very smooth function with the given density, which solves the e.o.m. [?φ, φ] = 0. (55)

This has many interesting solutions: One class is given by solutions of the free wave equation ?φ = cφ. In particular, the (non-commutative) spherical harmonics with suitable eigenvalue density are solutions also on the above orbits O(φi ). However there are other solutions, for example any solution of ?φ = f (φ) for arbitrary f solves (55); in particular any diagonal matrix does (in the usual basis). A careful study of these issues should lead to a better understanding of (46), and hence to improvements of the simple results presented below.

3.4

Relating the matrix model to physical observables

Before analyzing further the matrix models (39), we should try to relate them to the physically interesting quantities such as mass and coupling constants. Recall that the de?nition of mass on a non-commutative space is not obvious, since the Lorentz-invariance n is generally broken. However on Euclidean R2 θ with Sp(2n)- invariant θij one can de?ne a correlation length in terms wave√ of correlation functions for 2 suitably localized Gaussian 7 n packets φ(x) of size 1/ θ, say, or e.g. coherent states for the fuzzy spaces CPN : φ(x1 ) φ(x2 ) =
?1

dφi ?2 (φi )
O (φi )

dUe?S (U
?1

?1 (φ

i )U )

φ(x1 ), U ?1 (φi )U φ(x2 ), U ?1 (φi )U .

(56) Here φ(x), U (φi )U ∝ T r (φ(x)U (φi )U ) denotes the inner product on the NC space. Note again that only the kinetic part of the action is non-trivial here, and we will assume
These “test-functions” can be moved transitively on the NC spaces using the translational symmetry n n which is unbroken. On R2 θ with U (n)- invariant θij resp. their fuzzy versions CPN , the residual unbroken U (n) rotational symmetry is maximal and large enough to ensure that the correlation length is independent of the direction.
7

15

in the following that the full path integral is dominated by some orbit O(φi ). The mass can then be identi?ed as the inverse correlation length, provided it is ? mθ . Similarly, one could de?ne the coupling constants in terms of correlation functions of e.g. 4 such wave-packets. In general, it is plausible that the interesting low-energy observables depend only on the eigenvalue distribution, and can be determined in principle by an integral over the orbit O(φi ). The question of renormalizability is then roughly whether one can scale the (?nitely many) couplings gn with the cuto? Λ → ∞ in such a way that the correlation length and all the other observables at low energy (or at the scale of non-commutativity set by θ) approach a well-de?ned limit. To answer this question of course requires control over all these correlation functions. In this paper, we try to proceed as much as possible without resorting to perturbation theory. In view of the above results on the eigenvalue distribution, it seems very plausible that the scaling of the bare couplings m and gn must be determined such that the shape of the dominating eigenvalue distribution, i.e. the normalized function ?(s) (50) is ?xed as Λ → ∞. These scalings should be easily accessible with matrix model ? (φ). Moreover, it seems plausible techniques if we know the matrix model, in particular F that this should even be su?cient to guarantee “renormalizability” in the above sense, provided the ?eld theory can indeed be reduced to the orbit O(φi) as discussed in Section 3.3. To make explicit computations, we have to use (39) resp. (53) for the time being, i.e. we have to require that ?(s) respectively its related eigenvalue density (28) is close enough to the Wigner law (31). Then the “physical” mass mR resp. correlation length can be identi?ed without computing such correlation functions: it should be given by the bare mass of the corresponding free theory with the same maximal eigenvalue α0 (mR ), for the same cuto? Λ. That is, mR is determined by α = α0 (mR ) (57)

where α (50) will depend on the couplings of the interacting matrix model. This will be elaborated in Section 4. We will apply this prescription (57) even if the function ?(s) is not close to the free one in this paper, hoping that it is mainly the “size” of the eigenvalue distribution which determines the correlation length. This is plausible in view of the classical picture discussed in Section 3.3.1. We want to point out again the following consequence of this picture: (56) is strongly suppressed for zero momentum φ0 ∝ 1 l, since φ0 , U ?1 (φi )U = 0 for the dominating eigenvalue-distribution. On the other hand for non-zero momentum, the eigenvalues are non-degenerate, and the above inner product is non-zero. This suppression of zero momentum is clearly related to UV/IR mixing, and suggests that indeed only localized wavepackets should be used.

4

Example: the φ4 model
dd x 1 1 g ?i φ?i φ + m2 φ2 + φ4 2 2 4 16

Consider now the model S =

=

dd x

1 1 2 ?i φ?i φ + m2 Rφ + 2 2

1 g ?m2 φ2 + φ4 2 4

(58)

2 2 where we have introduced a “renormalized” mass m2 R = m ? ?m . Its eigenvalue sector according to the above results is described by

? = f0 (mR ) + S

2N T rφ2 + (2πθ)d/2 T r 2 α0 (mR )

g 1 ?m2 φ2 + φ4 . 2 4

(59)

While mR is arbitrary in principle, we will adjust it in order to minimize the expected errors due to our only partial knowledge of F (φi). Note that in the regularization using fuzzy CP n , we will ?nd (154) 2N (60) Λ= θ where N = N where N is related to N via (145), and a similar result Λ =
d/2 πN θ

using NC tori (118)

. Therefore

N = O (Nθ(d?2)/2 ) = O (Λ2 θd/2 ) 2 α0 up to log- corrections. To solve this model, we ?rst rescale φ as φ = αg ?

(61)

(62)

such that the saddle-point solution for ? will have an eigenvalue distribution with range8 [?1, 1]. Rewriting the matrix model (59) as ? = f0 (mR ) + N T r S we have
2 αg

m′2 2 g ′ ? + T r?4 . 2 4

(63)

2N d/2 1 + (2 πθ ) ?m2 α0 (mR )2 2 m
2

=

N ′2 m , 2 m2 R + ?m .
2

(64) (65) (66)

4 (2πθ)d/2 gαg = N g ′,

=

The model (63) is well-known and can be solved with standard methods from random matrix theory, see e.g. [19,21]. It is again governed by a single saddle-point in the eigenvalue sector, with eigenvalue distribution ?g (s) resp. ρg (p). Note that now m′2 < 0 is allowed. Assuming that g ′ > 0, there are 2 cases corresponding to distinct phases of the model which we are discussed below. It is interesting to note that even small g ′ < 0 is admissible for this matrix model as long as m′2 > 0. This indicates analyticity in g ′, so that perturbation theory does make sense in the weakly coupled phase.
8

in the disordered phase discussed at present, see Section 4.1

17

4.1

Phase 1 (“single-cut”): m′2 > 0, or m′2 < 0 with
1 ′ 2 g′ (g ? + + m′2 ) 12 ? ?2 . 2π 2

m′4 4g ′

<1

In this case the eigenvalue density is given by [19] ρg (?) = (67)

We have imposed that the range of ? be [?1, 1] as explained above, which using standard formulas [19] implies 2 1 = ′ (?m′2 + m′4 + 12g ′) (68) 3g i.e. 4 (69) g ′ = (4 ? m′2 ) 3 Note that the matrix model (63) has 2 independent parameters, which can be chosen either as m′2 and g ′ , or e.g. αg and g ′. We have chosen to work with αg , therefore m′2 and g ′ are ′4 not independent. In particular, note that m < 1 for g ′ < 16 due to (69), therefore this 4g ′ phase will be the “weakly-coupled” phase in Section 5. g ′ in (64) and using (65) gives Inserting m′2 = 4 ? 3 4
2 2αg α4 3 d/2 2 1 2 d/2 g + (2πθ) αg ?m = 2 ? (2πθ) g. 2 α0 2N 8 N

(70)

So far, mR was arbitrary, and it shouldn’t matter for ?xed “bare parameters” m and g . However, the relation (39) is expected to be good only if the eigenvalue distribution is close to the free one corresponding to mR . We should therefore choose mR accordingly, and a good choice is α0 (mR ) = αg (71) as in (57). This guarantees that the eigenvalue distribution of the interacting model (63) has the same range as the free one obtained from mR , hence it is “close”. Then (70) simpli?es further, and using ?m2 = m2 ? m2 R we have
4 (2πθ)d/2 g α0 (mR ) = N g ′ 3 2 m2 + g α0 (mR ) = m2 R. 4

(72) (73)

These are 2 equations in 4 unknowns (the function α0 (mR ) being known (22)), and we can basically choose any 2 of them to parametrize the model.

4.2

Phase 2 (“2 cuts”): m′2 < 0 with

m′4 4g ′

>1

At 4g ′ = m′4 with m′2 < 0 the eigenvalue density breaks up into 2 disjoint peaks, which are concentrated around the 2 minima. For 4g ′ < m′4 these peaks have ?nite distance, and moreover spontaneous symmetry breaking of T rφ occurs with non-zero “occupation” in both peaks [22]. This clearly describes a di?erent phase of the model. The semi-classical interpretation in the picture of Section 3.3.1 would be a 2-?uid model, with a large surface energy at the contact surface due to the kinetic term. It is very plausible that this will 18

develop some kind of (generalized) striped pattern upon mixing, and we conjecture this to be the “striped” phase of [6], or equivalently the “matrix phase” of [9]. Note that the assumption in [9] of 2 delta-like peaks is qualitatively very close, but strictly ruled out by the log-terms in (48). It is even possible to consider negative g ′ in this model [19], re?ecting the analyticity in ′ g . If g ′ becomes too negative there is another phase transition, which will not be considered any further here.

5

Weak coupling and renormalization

We ?rst consider the weak coupling regime, and try to get some insights into the renormalization of the φ4 model. In particular, one would like to understand better the relation of the above approach with the usual concepts of running coupling constants etc. The natural scaling parameter here is the size of the matrices N ? N d/2 , which is related to the UV-cuto? Λ through (60) for the regularization using fuzzy CP n . We can therefore ask how the bare parameters m2 , g etc. must be scaled with Λ such that the “low-energy physics” remains invariant, and we could de?ne the corresponding beta-functions. In the weak coupling regime, the shape of the eigenvalue distribution ? (50) resp. ρg (67) should certainly be kept close to the free one, given by the Wigner law. As discussed in Section 3.4, the correlation length resp. “renormalized mass” mR can then be obtained simply by matching the size of the eigenvalue spectrum αg with the one for the free case with the same cuto?, αg = α0 (mR ) (74) see (57). Then mR measures the correlation length of the free model which best ?ts the eigenvalue distribution. This should be very reasonable, since we are looking at a weak coupling expansion where g ′ can be taken arbitrarily small. The corresponding running of the original parameters m and g depends on the dimension, and will be worked out below. For the mass, this is very easy in our approach. Unfortunately, it is not obvious how to relate the coupling g resp. g ′ to e.g. 4-point functions or scattering processes; this could be supplemented perturbatively by a conventional RG computation, cp. [11, 12]. Also, note that there is no wavefunction-renormalization in this approach, as there are only 2 coupling constants in the matrix model (63). On the other hand the computations below are much simpler than any standard ?eld-theoretic computations, and provide non-trivial information on the model without having to resort to perturbation theory.

5.1

2 dimensions

To ?nd the scaling of m2 and g we can use (72) and (73), where αg = α0 (mR ) is already built in. The running of the mass can be read o? from (73), 3 2 3 Λ2 2 m2 = m2 ? gα ( m ) = m ? g ln(1 + ) R R R 4 0 4π m2 R (75)

Recall that mR is the “physical”, renormalized mass which is supposed to be ?nite while Λ → ∞. We see explicitly the expected logarithmic divergence of the bare mass. Note that 19

this is an all-order result in g , despite appearance. It is interesting to compare this with a conventional one-loop calculation. Noting that g 4 our interaction term is 4 φ (which is more natural in the matrix model context) rather g 4 than 4! φ , one would ?nd δm2 P +N P = 12 g 4 d2 k 3 Λ2 1 = g ln(1 + ) (2π )2 k 2 + m2 4π m2 R R (76)

from the 8 planar plus 4 non-planar contractions. It is interesting to see that this one-loop computation agrees precisely with our procedure of matching the eigenvalue distribution with the free one, even though we used the non-perturbative matrix model results (67) ?. One might have expected that only the planar diagrams contribute to the mass renormalization; however since the “mass” is obtained as the p → 0 limit of the 1PI 2-point functions (for ?xed Λ), the planar and non-planar diagrams coincide and both contribute. Note that if we use another regularization such as NC tori, we should use the modi?ed propagator (128) rather than the sharp cuto? in (11). This would lead to a somewhat modi?ed formula for α0 (mR ), however the corresponding mass renormalization (75) would still coincide with the one-loop result since the same α0 (mR ) enters in both calculations. Next consider the bare coupling g . Since the matrix model provides no simple relation between g and the 4-point function, the latter would have to be obtained e.g. by a perturbative calculation as usual. However since g is not expected to run in 2 dimensions, we simply interpret g as “physical” coupling constant, ?xing the scale by mθ . The relation with the matrix model coupling g ′ is given by (72), which for large Λ and using (60) is g = Λ2 π g′. Λ 2 )) 16(ln( m R (77)

In particular if we keep g ?nite, we see that g ′ → 0 very rapidly as Λ → ∞. This means that the eigenvalue density ρg (67) is very close to the free one given by Wigner’s law, and we expect that the above relations are reliable in this weak-coupling phase of φ4 in 2 dimensions9 .

5.2

4 dimensions

Using the same procedure, (73) gives now the expected quadratic divergence of the bare mass 3 m2 Λ2 R 2 m2 = m2 1 ? Λ ln( ) g. (78) R ? 16π 2 Λ2 m2 R This is again an all-order result, and agrees again perfectly with a conventional one-loop computation (see e.g. (3.11) of [5] after replacing g → g/6). However the bare coupling g is now also expected to run, being logarithmically divergent at one loop. Recall also that we cannot see any wavefunction-renormalization as there are only 2 parameters in the matrix model. The relation between g and the matrix-parameter g ′ is now given by g′ =
9

2 π2

1?2

m2 Λ R ln( ) Λ2 mR

2

g≈

2 g π2

(79)

Recall however that the non-planar diagrams are suppressed only logarithmically (14) in 2D, which is much weaker than in 4D

20

for Λ ? mR , where (60) and N = N 2 /2 (145) has been used. This means that (keeping in mind some possibly logarithmic renormalization of g ) our procedure to derive (78) is justi?ed in a perturbative sense, since we can make g ′ arbitrarily small so that again ρg is very close to Wigner’s law. It is quite remarkable that according to the above results, the mass is renormalized only at ?rst order in the bare coupling g , even though the derivation is non-perturbative. While this is expected for d = 2, this seems very surprising for d = 4; it would indicate that NC φ4 in 4 dimensions is much better behaved than in the commutative case. This can be traced back to our procedure of determining the mass mR through α = α0 (mR ). While an exact calculation of the correlation length on a given orbit O(φ) would certainly modify this result, the basic picture seems clear and simple. Unfortunately this analysis does not give us the relation between g and the “physical” coupling as obtained by the 4-point function, which probably requires further renormalization. It is tempting here to conjecture that ?xing g ′ and m′ or equivalently the eigenvalue distribution ? in a suitable way su?ces to de?ne a non-trivial NC ?eld theory in 4 dimensions. We will ?nd further evidence for this in Section 6.1. We conclude that while this approach certainly needs further work and thus far provides only a partial window into NCFT, it suggests a very simple and compelling approach to renormalization in the NC case. This may help to overcome the di?culties at higher order found in [5].

5.3

Higher dimensions

It is also illuminating to try to generalize the above considerations to dimensions higher than 4, where the commutative φ4 model is no longer renormalizable. The relations (66) and (72) are still valid, and give
d?2 m2 = m2 ) R ? g O (Λ

(80) (81)

and g ′ = O (Λd?4 ) g. Clearly our matrix model (63) makes sense only for g ′ = O (1), which would require g = O (Λ4?d ). This is just the canonical dimension of g . If we again assume that g ′ and hence the shape of the eigenvalue distribution should be independent of Λ, we see that g must be ?ne-tuned to at least O (Λ4?d ). This is in accord with the expected non-renormalizability.

6

The phase transition.

Now consider the phase transition of the matrix model (63), which is known [21,23] to have a phase transition between the 1-cut and the 2-cut phase at m′4 = 4g ′. (82)

This is expected to be the phase transition between the disordered and striped phase of Gubser and Sondhi. Combining with (69) this gives 1 4 g ′ = (4 ? m′2 ) = m′4 3 4 21 (83)

which has 2 solutions for m′2 ; only the negative one m′2 = ?8 is relevant here and marks the phase transition. The corresponding coupling is10 g ′ = 16. (85) (84)

Note that there is still a free parameter in the model, which we can take to be mR resp. α. Plugging this in (72) gives
4 (2πθ)d/2 gα0 (mR ) = 16N , 3 2 (mR ) = m2 m2 + g α0 R, 4 and solving the ?rst equation for α0 we obtain

(86) (87)

m2 + 3

Ng = m2 R (g ), (2πθ)d/2

(88)

where m2 R is a function of g via (86). This de?nes the critical line. The corresponding phase transition is third order [23] in the variables g ′ and m′ . This is in contrast with [6–8], who argued for a ?rst-order phase transition. Note that m′2 < 0 means that we are quite far from the perturbative domain, and the eigenvalue distribution is signi?cantly changed from Wigner’s law (31). In this regime the replacement (34) for the kinetic term has not been tested, and we cannot expect the results to be exact. Furthermore, we expect the “bare” parameters m, g to have some non-trivial scaling in N ? N d/2 on the critical line corresponding to the RG ?ow, which is not obvious. In any case, it is quite reasonable that this description is at least qualitatively correct. To proceed, we have to discuss the dimensions separately, starting with the most interesting case of 4 dimensions.

6.1

4 dimensions

We consider ?rst the regularization using fuzzy CP 2 discussed in Appendix A, which cor2 responds to a sharp cuto?. In this case we have N = N 2 /2, and α0 = O (Λ2 ) in d = 4 (22). By looking at the equations (86), (87) and Λ2 = 2N θ (89)

which holds for this regularization of R4 θ , it is quite obvious that there should be solutions which scale as m2 ∝ Λ2 ∝ N, g ?xed. (90)

Hence we expect a phase transition at ?nite coupling g and the standard quadratic running of the mass. To ?nd a closed equation for the critical line, we have to use (22) for α0 = α0 (mR ), 2 m2 Λ2 4π 2 α0 (mR ) R = 1 ? ln(1 + ) (91) Λ2 Λ2 m2 R
10

Recall that g ′ < 16 is the weakly coupled single-cut phase as pointed out in section 4.1.

22

This cannot be solved explicitly for mR , but we can use (88) for m2 R which together with 2 N = N /2 and (89) gives m2 3 g m2 R + = . (92) Λ2 4π 2 Λ2 Since m2 R ≥ 0, this makes sense only for 3 m2 >? 2 Λ 4π Plugging this in (91) and using (86)
2 α0 (mR ) =

g . 2

(93)

N πθ g 2

Λ2 2 = √ g π 2g m2 3 + 2 Λ 4π g 2
?1

(94)

we get 2π 2 =1? g 3 m2 + 2 Λ 4π ln 1 +

.

(95)

This is indeed consistent with the scaling (90). Since the rhs is ∈ (0, 1], this has a solution ≤ 1, i.e. only11 for 2π 2 g g ≥ g c = 8π 2 .

(96)

In terms of the dimensionless parameter m ? 2 = m2 /Λ2 , the corresponding critical mass is 3 2 2 m ? c = ? 2 . Expanding g = gc + δg, m ? =m ?2 ? 2 the critical line (95) is given by c + δm δg = ? 32π 2 δm ? 2 + ... 3 (97)

for small variations. This is plotted in Figure 1. Let us try to assess the validity of this result. The basic arguments underlying this result 2 are expected to be good in 4 dimensions as long as m2 R ? Λ (unlike in 2 dimensions, see below and Appendix B). This is satis?ed here since as g → gc from above, (95) implies that m2 3 m2 R = + 2 2 Λ Λ 4π g 2 → 0 (98)

using (92), and in particular mR = 0 at g = gc . Therefore the replacement of the free action with the matrix model (34) is essentially justi?ed, apart from the modi?ed eigenvalue distribution ?(s). It would be very interesting to estimate the e?ects of this modi?ed ?(s) more rigorously, e.g. by estimating the integrals over the compact orbits O(φi ). The existence of gc = 0 can also be understood simply by noting that the critical line is characterized by a speci?c eigenvalue distribution which is di?erent from Wigner’s law, which holds for g = 0. Therefore the critical line12 cannot end at g = 0. While the existence of a critical point should be a sound prediction, the relation between m2 and g
the rhs of (95) is simply c(Λ, mR ) as de?ned in (9), which will very generally satisfy such bounds. note that m2 ∝ Λ2 here, as in the weakly-coupled regime. This is di?erent in 2 dimensions, where the critical line is in a di?erent scaling regime from the weakly-coupled case, and no such conclusion can be drawn.
12 11

23

Figure 1: The critical line in 4D with critical point on the critical line cannot be expected to be precise. If we would use the propagator for the lattice regularization (128), the details of (95) would change but the results would be qualitatively the same. In particular the critical coupling could be evaluated using (132). The existence of a critical point terminating the critical line at gc = 0 is certainly intriguing. Since the critical line will be stable under the RG ?ow, its endpoint should correspond to a non-trivial RG ?xed point for the non-commutative φ4 theory, and the correlation length is expected to diverge13 . That limiting model should be non-trivial, since the eigenvalue distribution ?(s) is di?erent from the free case. It is furthermore plausible following the discussion in [6] that it admits a “continuum” interpretation as a renormalizable NC ?eld theory. The physical content of such a ?xed point would require to study e.g. 4-point functions and the running of the coupling, which is beyond the scope of this work. Note however that if we assume that g becomes larger with increasing cuto? as in the commutative case, this would mean that the low-energy coupling corresponding to gc is small, and hence in a physically interesting regime. These results and in particular the existence of a phase transition are roughly consistent with the results of [6–8], but not precisely; for example [6–8] argue for a ?rst-order phase transition using self-consistent Hartree-Fock approximation resp. a one-loop RG analysis, while we ?nd a higher-order transition using a non-perturbative matrix model result. The approaches are thus quite orthogonal: in the present approach the interaction is treated exactly while the kinetic term is approximated, whereas [6–8] do the opposite. The fact that one always ?nds a phase-transition is quite encouraging. It is also remarkable that θ does not enter our result (95); recall however that we always assume Λ2 θ ? 1 and (154). In particular, one would expect a standard Ising transition to a uniform symmetry breaking state at small θΛ2 ; however our approach is not valid in that regime, and we should therefore not expect to see this transition. 2 In dimensions higher than 4, (86) together with α0 = O (Λd?2) = O (N d/2?1 ) implies that
13

we ?nd indeed mR = 0 at gc , which should however not necessarily be taken at face value

24

g = O (N ? 2 +2 ) must be ?ne-tuned in order to stay in the weakly-coupled phase. This is consistent with the expected non-renormalizability, and we will not pursue this any further.

d

6.2

2 dimensions

In 2 dimensions our approach is more delicate as the planar diagrams are only logarithmically divergent. To be safe we should allow only ?nite mR while Λ → ∞, see Appendix B. Using (22) and (86) for α0 = α0 (mR ) and N = N , we have m2 R Λ2 Λ2 = πα2 . = 2Λ2 e 0 ?1 e2 g ? 1 (99)

for the regularization using the fuzzy Plugging this in (88) and using the relation Λ2 = 2 N θ sphere gives 3 g Λ2 = m2 (100) m2 + Λ= R. 2 2Λ 2 π 2 g ?1 e This would be consistent with a scaling m2 ? Λ2 , g ? Λ2 (101)

on this critical line14 . However, the basic assumptions of Sections 3.1 and Appendix B are 2 no longer valid in this case, since then m2 R ∝ Λ . Our approach should be most reliable Λ 2 if mR remains ?nite, which implies through (99) that g ? 2( ln ) and therefore (100) Λ 3 Λ2 2 m ? ? π ln Λ . Since we cannot solve (100) in closed form, let us assume that g is smaller Λ 2 but not much smaller that g ≈ 2( ln ) , so that we can neglect the term on the rhs of (100). Λ Then the critical line should be given approximately by m2 = ? 3 2 g Λ, π (102)

or a slightly modi?ed formula using the fuzzy torus regularization (due to the di?erent propagator). Unfortunately we cannot compare this with the numerical results of [10], who consider a di?erent scaling. However we can compare it to some extent with the numerical results of [9] on the fuzzy sphere, and ?nd reasonable agreement. This will be done in the following section. If g scales like (101), one can use a simpler argument due to [9] neglecting the kinetic term altogether, and replace the action by the pure potential model T rV (φ); hence this phase was denoted as “matrix phase” in [9]. Note that indeed the scaling (101) is appropriate for the matrix model (58) without the kinetic term. This amounts to identifying m2 g , g ′ = 2πθ , N N m2 θ 2 , i.e. which would predict a phase transition at (2π N ) = 8π gθ N m′2 = 2πθ m2 = ?
14

(103)

1 π

2g Λ .

(104)

note that both m2 and g have dimension of mass2 ; nevertheless, this is a rather strange scaling in 2 dimensions. This phase transition is very far from the weak coupling phase

25

6.2.1

The fuzzy sphere

For the case of the fuzzy sphere, we consider the action S= 4πR2 T r (φ?φ + rφ2 + λφ4 ) N (105)
2

L using the (redundant) parameters r, λ, R following [9]. The eigenvalues of ? = R 2 are 2 2 l(l +1)/R = p , with cuto? Λ = pmax = N/R. Comparing with (58), the above parameters are related to the coupling constants m and g in (58) via

θ=

2R 2 , N

2r = m2 ,

4λ = g.

(106)

2 It turns out that α0 for the fuzzy sphere (141) agrees precisely with the result for R2 θ, 2 α0 (mR ) =

1 Λ2 ln(1 + 2 ). π mR

(107)

Λ 2 ) ; in particular we can ?x λ = 1 as We can therefore apply (102), assuming that λ ? ( ln Λ in [9]. Then the critical line is

3 1 1 r =? √ ≈ ?0.846 . N R 2 π R

(108)

√ Note that R is a free parameter corresponding to θ, and we should take at least R = O ( N ). Let us compare his with Martin’s disordered-matrix transition [9]: he ?nds numerically a phase transition for (see eq. (46) in [9]) r 1 ≈ ?0.56 N R (109)

for large R. This is reasonably close to our analytical result. Recall that we cannot expect our prediction (108) to be exact since the eigenvalue distribution is quite far from Wigner’s law, and moreover the arguments in Section 2 are weaker in 2 dimensions compared to 4 dimensions since the crucial divergences are only logarithmic. Furthermore g strongly Λ 2 ) and therefore mR → 0, so that the replacement of the kinetic term violates g ? 2( ln Λ with (34) is not justi?ed here. The pure matrix model discussed above (103) would give √ 2 1 1 r =? ≈ ?0.45 . (110) N π R R Hence the numerical results of [9] (taken for N ≈ 30) appear to be in between the pure potential model and our approach, and our treatment apparently overestimates the kinetic term. This is not too surprising in 2 dimensions in view of the above remarks.

7

Discussion and outlook

We presented a simple non-perturbative approach to scalar ?eld theory on Euclidean nonn commutative spaces, based on certain matrix regularizations of R2 θ . Starting from a repre?1 sentation of the ?eld φ = U (φi )U in terms of eigenvalues φi and “angles” U , we observe 26

that the di?erent behavior of planar and non-planar diagrams due to UV/IR mixing implies a particular eigenvalue density distribution, which can be reproduced by a simple matrix model. This is shown starting with the case of free ?elds, which can be described by a Gaussian matrix model. Interactions of the form T rV (φ) can then be included very easily, and modify the eigenvalue distribution. This leads to a picture where the basic properties of the QFT such as correlation length and renormalization are related to its eigenvalue density ρ(s), through a “constrained ?eld theory” with compact con?guration space O(φ). It also makes the existence of well-behaved scaling limits of NCFT i.e. renormalizability very plausible. We conclude that the eigenvalue sector of non-commutative scalar ?eld theories is goverened by an ordinary matrix model, which provides a simple and intuitive window into the non-perturbative domain. For weak coupling, this approach provides a new way of computing the renormalization of the potential; in particular, using a very simple approximation we found an expression for the mass renormalization, which coincides with the conventional one-loop calculation. Furthermore, we found a phase transition at strong coupling in the φ4 model in both 2 and 4 dimensions, which is identi?ed with the striped or matrix phase of [6, 9]. This is particularly interesting in the 4-dimensional case since the critical line then terminate at a non-trivial point with gc = 0, which is interpreted as an interacting RG ?xed-point. The existence of this ?xed point can be understood simply by noting that the critical line is characterized by a speci?c eigenvalue distribution which is di?erent from Wigner’s law, which holds for g = 0. Therefore the critical line cannot end at g = 0. All this suggests that such NC ?eld theories may in fact be more acccesible to analytical tools than their commutative counterparts. Perhaps the main shortcoming of this approach is the lack of a precise relation between the eigenvalue distribution and the relevant physical parameters, such as correlation length resp. mass and coupling strength. It is quite clear that the leading parameter is the size α of the maximal eigenvalue, which has been used in this paper to extract physical information. To go beyond this approximation may be di?cult, and may require e.g. perturbative methods. There are many other questions and gaps which should be addressed in future work. For example, the assumption that θij is non-degenerate (or special) is quite clearly not essential, and a similar approach should also work in odd dimension. Furthermore, a more elaborate analysis of the renormalization in the weakly-coupled regime should be attempted. Another interesting question concerns the relation of this approach with the results of [11] on a modi?ed φ4 model; to address this issue, the above analysis should be repeated with a suitably modi?ed propagator. This will be done elsewhere. Perhaps the most interesting perspective is the possibility that careful estimates of the contribution of the kinetic term on the orbits O(φ) (for eigenvalue distributions di?erent from Wigner’s law) should allow to rigorously justify the above picture also in the nonperturbative domain, and in particular the existence of the critical point. One may in particular try to establish renormalizability in this way. This should be facilitated by the fact that O(φ) are compact spaces. Finally, it would of course be extremely interesting to compare all this with numerical results in 4 dimensions, which are not available at this time.

27

Acknowledgements
I would like to thank H. Grosse, I. Sachs, and R. Wulkenhaar for very useful discussions, and C-S. Chu for reading the manuscript. n Appendix A: Regularizations of R2 θ
2 2 2 The fuzzy torus TN and TN × TN

A particularly simple regularization of Rd θ which works in any (even) dimensions was proposed in [18], which we review for convenience. Consider a (toroidal) lattice with lattice constant a and N sites in each dimension. We denote its size with L = Na. (111)

Since we are on a torus, one should not use the unbounded operators xj . Instead consider the unitary generators 2π ZiN = 1. (112) Zj := e L ixj , The commutation relations [xi , xj ] = iΘij then become Zi Zj = exp ? 4π 2 iΘij Zj Zi . L2 (113)

Rather than going through the most general case, we simply consider Θij = θ Qij and work out the 4-dimensional case, where ? 0 ??1 Qij = ? ?0 0 (114)

1 0 0 0 0 0 0 ?1

The generalization to any even dimension is obvious. Periodicity implies a quantization of a resp. θ as Na2 = θ, (116) π assuming that N is odd. The physical momentum is ki = with UV-cuto? at π πN = . a θ This is in qualitative agreement with the scaling (154) obtained using fuzzy CP n . Λ= (118) 2πni π π ∈ (? , ) = (?Λ, Λ), aN a a (117)

? 0 0? ?. 1? 0

(115)

28

In order to write down the action for a scalar ?eld, we also need partial derivatives or shift operators, ? Dj := ea?j , Dj Z i D j = e2πiδij /N Zi . (119) In our case, they can be realized as
? (N +1)/2 ) , D1 = (Z2

D2 = (Z1 )(N +1)/2

(120)

which extends to arbitrary even dimensions by taking tensor products. Hence the ?eld φ is a hermitean N d/2 × N d/2 matrix. The integral is again de?ned as f = (2πθ)d/2 T r (f ). One can then de?ne the “plane waves” 1 φn = d/4 N which satisfy
d

etc. A solution of (113) and ZiN = 1 in 2 dimensions is given by the unitary “clock and shift” operators (recall that N is odd) ? ? ? ? 1 0 1 ? e4πi/N ? ? ? 0 1 ? ? ? ? 2(4πi/N ) ? ? ? ? . . e ? ?, ? ? (121) , Z = Z1 = ? 2 ? ? ? . . . ? ? ? ? ? ? 0 1 ? . ? 1 0

(122)

(Z i )n i (
i=1 j<i

e2πiQij ni nj /N )

(123)

? φn′ ) = δnn′ , T r (φ n

? = φ ?n φn

(124)

for ni ∈ [?(N ? 1)/2, (N + 1)/2]. They form a basis of the “space of functions” Mat(N d/2 , C). Using ? Di φ n Di = exp(2πini /N ) φn (125) one can write down the discretized lattice-version of (1), S [φ] = (2πθ)d/2 Tr For hermitean φ = 1 Tr a2 1 a2
d ? (φ2 ? Dj φDj φ) +

j =1

m2 2 g 4 φ + φ . 2 4

(126)

pn φn with pn = p? ?n , the kinetic term becomes
? Di φDi ?φ 2

= =

i

2 a2

k

| pk | 2
j

j

(1 ? cos(kj a)) (127)

k

| pk | 2 (

2 kj + O (a2k 4 )).

29

The propagator is therefore = δkk′ pk p? k′
2 a2

1 , 2 j (1 ? cos(kj a)) + m
′ ki Qij kj ) φk ′ φk . i<j

(128)

and the phase factor for the nonplanar diagrams is obtained from φk φk′ = exp(?iθ
2 α0 on the fuzzy torus 2 Due to the di?erent behavior of the propagator for large momenta, α0 on the fuzzy tori will be somewhat di?erent from regularizations using a sharp cuto?, such as on fuzzy CP n . In 2 dimensions, we should compute more carefully π/a

(129)

d xφ (x)

2

2

= V = V
0

?π/a π 2

d2 p (2π )2 2

dr (2π )d

1 2 2 i (1 ? cos(ri )) + m a /2

a2 2 2 i (1 ? cos(pi a)) + m a

(130)

and in 4 dimensions
π/a

d4 xφ2 (x)

= V
?π/a

d4 p (2π )4 2
π 0

V 2 Λ = π2

d4 r (2π )4

a2 2 2 i (1 ? cos(pi a)) + m a

1 2 2 i (1 ? cos(ri )) + m a /2

(131)

for the above regularization. In particular, for m = 0 one has
2 α0 (m)



2 α0 (m

4 = 0) = 2 Λ2 π

π 0

d4 r (2π )4

0.31 2 1 Λ = 4π 2 i (1 ? cos(ri ))

(132)

numerically. This is needed e.g. to determine the critical coupling gc for the φ4 model in 4 dimensions.

The fuzzy sphere
2 The algebra SN of functions on the fuzzy sphere [16] is the ?nite algebra generated by Hermitian operators xi = (x1 , x2 , x3 ) satisfying the de?ning relations

[xi , xj ] = iΛN ?ijk xk , 2 2 2 x2 1 + x2 + x3 = R

(133) (134)

where R is an arbitrary radius. The noncommutativity parameter ΛN is of dimension length, and is quantized by R = ΛN N2 ? 1 , 4 30 N = 1, 2 , · · · (135)

This can be easily understood: (133) is simply the Lie algebra su(2), whose irreducible representation have dimension N . The Casimir of the N -dimensional representation is quantized, and related to R2 by (134) and (135). Thus the fuzzy sphere is characterized by its radius R and the “noncommutativity parameters” N or ΛN . The algebra of “functions” 2 is simply the algebra Mat(N ) of N × N matrices. It is covariant under the adjoint action SN of SU (2), under which it decomposes into the irreducible representations with dimensions 2 (1) ⊕ (3) ⊕ (5) ⊕ ... ⊕ (2N ? 1). The integral of a function f ∈ SN over the fuzzy sphere is φ(x) = 4πR2 T r [φ(x)], N (136)

which agrees with the integral on S 2 in the large N limit and is invariant under the SU (2) rotations. The dimensionless coordinates λi = xi /ΛN generate the rotation operators Ji : Ji f = [λi , f ]. One can now easily write down actions for scalar ?elds, such as S= 4πR2 Tr N 1 1 1 φ?φ + m2 φ2 + gφ4 2 2 4 = 2πθ T r 1 1 1 φ?φ + m2 φ2 + gφ4 2 2 4 (138) (137)

were φ is a Hermitian matrix, and ?φ = obtained by de?ning θ through

1 JJφ R2 i i

is the Laplace operator. The last form is

Nθ . (139) 2 There are 2 obvious N → ∞ limits: 1) the conventional, commutative S 2 limit keeping R ?xed, and 2) the limit of the NC plane R2 θ , which can be obtained by keeping θ constant: then the tangential coordinates x1,2 satisfy at the north pole the commutation relations R2 = [x1 , x2 ] = i 2R 2R x3 = i N N
2 R2 ? x2 1 ? x2 ≈ iθ.

(140)

Note that θ?1 now determines the basic (NC) scale of the NC ?eld theory, replacing the radius R. If one considers non-planar loops, the oscillatory behavior due to the 6j symbols sets in for angular momenta l2 ≈ N and was studied in [24]. Note that this corresponds to p2 = O (θ?1 ) in the NC case, but to p → ∞ in the commutative case. Therefore the nonoscillatory domain is divergent in the commutative limit and prevents a matrix behavior. This “low-energy” sector is suppressed in the NC case if θ is kept ?nite.
2 α0 for the fuzzy sphere

The arguments of Section 3.1 for the eigenvalue distribution go through here as well provided the oscillating behavior of the non-planar diagrams is strong enough. Even though there are nontrivial e?ects even for ?nite R resp. θ ∝ 1/N [24], they are su?ciently strong

31

for our purpose only if θ is kept ?nite. To ?nd the appropriate α0 (m), consider for g = 0 φ2
S2

=
l,m N S2

Y lm · Y l?m = l(l + 1)/R2 + m2

N

l=1

2l + 1 l(l + 1)/R2 + m2
N/R

= R
l=1

(2l + 1)/R = 2R 2 l(l + 1)/R2 + m2
2

dx
0

x2

x + m2 (141)

= R2 ln(1 +

V Λ2 N ) = ln(1 + ) m2 R2 4π m2
N . R

/2 in the large N limit, using V = 4πR2 and x = l+1 and Λ = R (11), so that the same α0 (m) as in (22) can be used.

This agrees precisely with

Fuzzy CP n
The construction of fuzzy CP n [27–30] is analogous to the case of fuzzy S 2 ? = CP 1 . Since CP n is an adjoint orbit of SU (n +1), it is a compact symplectic space and can be quantized in terms of ?nite matrix algebras Hom(VN ) where VN are suitable representations of su(n + 1). To identify the correct representations VN of su(n + 1), we must match the space of harmonics on classical CP n C ∞ (CP n ) = ⊕ V(p,0,...,0,p).
p=0 ∞

(142)

with the decomposition of Hom(VN ). It is easy to show that indeed
? ? Hom(VN ) = VN ? VN = ⊕ V(p,0,....,0,p) p=0 N

(143)

for VN := V(N,0,...,0). Here V(l1 ,...,ln) denotes the highest weight irrep of su(n+1) with highest weight l1 Λ1 +...+ln Λn where Λk are the fundamental weights. One can therefore de?ne the algebra of functions on the fuzzy projective space by
n CP N := HomC (VN ) = Mat(N , C)

(144)

(N + n)! Nn ≈ . (145) N !n! n! n The functions on fuzzy CP n have a UV cuto? given by N . Scalar ?elds on CPN are elements in HomC (VN ), and the integral is given by the suitably normalized trace over VN . The coordinate functions xa for a = 1, ..., n2 + 2n on fuzzy CP n are given by suitably rescaled generators of su(n + 1) acting on VN . One ?nds [30] N = [xa , xb ] = iΛN fabc xc , g ab xa xb = R2 , 1 N + )ΛN xc . dab c xa xb = (n ? 1)( n+1 2 32 (146) (147)

with

for ΛN =

R
n N2 2(n+1)

+n N 2

.
2

(148)

For large N , this reduces to the de?ning relations of CP n ? Rn +2n . On the other hand, scaling the radius as n (149) R2 = Nθ n+1 2 n near a given point (the “north pole”) of CPN gives R2 θ with U (n) invariant θij , similarly as for the fuzzy sphere. We refer to [29] for further details. The Laplacian on fuzzy CP n is proportional to the quadratic Casimir of su(n +1) acting on the functions, c ?(φ) = 2 Ja Ja φ, (150) R n . It has eigenvalues where Ja generates the SU (n + 1) rotations and c = n2+1 ?fk (x) = c k (k + n) fk (x) R2 (151)

for fk (x) ∈ V(k,0,...,0,k) according to the decomposition (143). The multiplicity for given k is 2 (k + n ? 1)!2 2k + n ≈ k 2n?1 dim(V(k,0,...,0,k)) = 2 2 2 k ! (n ? 1)! n (n ? 1)! n
N

(152)

for k ? n. To ?nd the appropriate α0 (m), consider φ2
CP n

Y km · Y k?m k 2n?1 2 = 2 2 (n ? 1)!2 n k=1 ck (k + n)/R2 + m2 n ck (k + n)/R + m k,m CP √ √ √ N Λ 2(R/ c)2n?1 x2n?1 (k c/R)2n?1 2(R/ c)2n = dx = (n ? 1)!2 n ck (k + n)/R2 + m2 (n ? 1)!n! 0 x2 + m2 =
k =1

V = 2n?1 n 2 π (n ? 1)!

Λ

0

x2n?1 dx 2 x + m2

(153)

in the large N limit, where we denote the basis again with Y km and used V = V ol(CP n ) = n n √ k 2(n+1) π R2n and x = c R and n n! Λ= √ N c = R 2n N = n+1 R 2N . Θ (154)

This agrees with precisely with (11) and generalizes the results for the fuzzy sphere. Furthermore, putting (154) and (149) together gives :=
n CP N

V T r (.) → (2πθ)n T r (.) N

(155)

in the above scaling limit. Therefore this rescaled fuzzy CP n is a perfect regularization of n R2 θ . It has a sharp mumentum cuto? at Λ, and the same α0 (m) as in (22) can be used. 33

Appendix B: Justi?cation of (16)
Consider ?rst d ≥ 3. Then RN P :=
1 V

dd xφ(x)2n N on?P lanar 1 ≈ 1 c dd xφ(x)2 n V

1 0

′ ′ dd kn dd k1 2 ... eiΛ ′ 2 ′ 2 (k 1 ) (k n )

′ Θk ′ ki j

,

(156)

2 2 and we assume that Λ2 ? m2 θ , Λ ? m . Here the integration domains have been rescaled to be unit balls in momentum space for all diagrams in the numerator and in the denominator; then the denominator can be estimated by the planar contribution which gives a ?nite contribution c (after the rescaling), which will be omitted. To proceed, let v ∝ (k1 ...., knd ) ∈ Rnd denote a unit vector in Rnd with norm v = 1, and consider generalized spherical coordinates such that dd ki ...dd kn = d?(v ) r nd?1 dr in Rnd . Then 1 0 ′ ′ dd k1 dd kn 2 ... eiΛ ′ 2 ′ 2 (k 1 ) (k n ) O (1)

′ Θk ′ ki j

=

d?(v )
0

dr r

nd?1

eiΛ r vΘv ′ 2 ′ )2 (k 1 ) ...(kn

2 2

(157)

in simpli?ed notation. The important point is that the radial integral with increasing frequency. For d ≥ 3, the radial integral behaves as
O (1) 0

dr is oscillatory,

dr r n(d?2)?1 eiΛ

2 r 2 v Θv

O (1)

=
0

du un(d?2)/2?1 eiΛ

2 uv Θv

(158)

where r 2 = u. Clearly the (alternating) contributions increase with u. The integral is therefore estimated by the “last oscillation” which is of order min{1, O (1/(Λ2vΘv ))}. We can exclude the region {vΘv < 1/Λ}, whose volume goes like O (1/Λ) as Λ → ∞ (ignoring log Λ-corrections). Therefore RN P ≤ O (1/Λ) (159) ignoring possible log-corrections15 , which establishes our claim. For d = 2, we apply the same analysis to the numerator of (156). The radial integral has again the form (158), but it is now dominated by small u, i.e. the “?rst half-oscillation”. We therefore have to reintroduce the masses which provide an IR cuto?. To estimate this, we go back to the original form 1 V dd xφ(x)2n
N on?P lanar

d2 kn d2 k1 ... ei ki Θkj 2 + m2 2 + m2 ( k ) ( k ) 1 n 0 2 d k1 d2 kn ≈ ... 2 2 (kn )2 + m2 ki Θkj <π (k1 ) + m m ≈ f (ln( )). mθ =
m mθ

Λ

(160)

Indeed the result can only depend on the ratio m2 θ =
15

where (161)

1 θ

this is probably not the best possible estimate

34

is the NC mass scale, and in 2 dimensions it will depend only logarithmically on this ratio m 2 1 dd xφ(x)4 N on?P lanar = O (ln( m ) ). On the other hand, (for example, V θ 1 V d2 xφ(x)2n
P lanar

= O (ln(

Λ n ) ), m

(162)

m and Λ → ∞, the eigenvalue distribution is again given by the Gaussian Hence for ?xed m θ matrix model (34), however we expect the ?uctuations to be larger in this case than for d = 4. It is interesting to consider also the commutative scaling limit of the fuzzy sphere, in view of the “non-commutative anomaly” found in [24]. In that case, we can set θ = 1/NR2 and therefore m2 θ = O (N ) = O (Λ). Then both planar and non-planar contributions have the same logarithmic behavior, and there is no well-de?ned eigenvalue distribution in that case. This was to be expected since we considered the commutative limit; however the “non-commutative anomaly” indicates some NC behavior in that case too, which is apparently too weak to induce a distinct eigenvalue distribution. The situation is di?erent in 4 dimensions, and a well-de?ned eigenvalue distribution may well exist in the commutative scaling limit.

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