数理解析研究所講究録
第 1904 巻 2014 年 67-104

67

Construction of thequantum field theory on
noncommutative Moyal space$\Phi_{4}^{4}$

Harald GROSSE1 and Raimar WULKENHAAR2
1 Fakult\"at

f\"ur Physik, Universit\"at Wien

Boltzmanngasse 5, A-1090 Wien, Austria
2

Mathematisches Institut der Westf\"alischen Wilhelms-Universit\"atEinsteinstrafle 62, D-48149 M\"unster, Germany

Abstract
We review our recent construction of the -model on four-dimensional
Moyal space. A milestone is the exact solution of the quartic matrix
model $\mathcal{Z}[E, J]=\int d\Phi\exp(trace(J\Phi-E\Phi^{2}-\frac{\lambda}{4}\Phi^{4}))$ in terms of the solution of a non-linear equation for the 2-point function and the eigenvalues
of $E$ . The -function vanishes identically. For the Moyal model, the theory of Carleman type singular integral equations reduces the construction
to a fixed point problem. Its numerical solution reveals a second-orderphase transition at
and a phase transition of infinite order at. The resulting Schwinger functions in position space are symmetric and invariant under the full Euclidean group. They are only sensitive
to diagonal matrix correlation functions, and clustering is violated. The
Schwinger 2-point function is reflection positive iﬀ the diagonal matrix
2-point function is a Stieltjes function. Numerically this seems to be thecase for coupling constants$\phi^{4}$$\beta$$\lambda_{c}\approx-0.396$$\lambda=0$$\lambda\in[\lambda_{c}, 0].$1

Introduction

Perturbatively renormalised quantum field theory is an enormous phenomenological success, a success which lacks a mathematical understanding. The perturbation series is at best an asymptotic expansion which cannot converge at
physical coupling constants. Some physical eﬀects such as confinement are outof reach for perturbation theory. In two and partly three dimensions, methods of constructive physics [GJ87, Riv91], often combined with the Euclidean
approach [Sch59, OS73, OS75], were used to rigorously establish quantum field
theory models.
In four dimensions there was little success so far. It is generally believed that
due to asymptotic freedom, non-Abelian gauge theory (i.e. Yang-Mills theory) has
lharald.grosse@univie.ac.at

2raimar@math.uni-muenster.de

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the chance of a rigorous construction. But this is a hard problem [JW00]. Whatmakes it so diﬃcult is the fact that any simpler model such as quantum electrodynamics or the-model cannot be constructed in four dimensions (Landau
ghost problem $[LAK54a, LAK54b, LAK54c]$ or triviality [Aiz81, Fr\"o82
One of the main diﬃculties is the non-linearity of the models under consideration. Fixed point methods provide a standard approach to non-linear problems,
but they are rarely used in quantum field theory. In this contribution we review
a sequence of papers $[GW12b, GW13b, GW14]$ in which we successfully used
symmetry and fixed point methods to exactly solve a toy model for a quantum
field theory in four dimensions.
1. Following $[GW12b]$ , we show in sec. 2 that the quartic matrix modelis exactly solvable in terms of the solutionof a non-linear equation. This can be traced back to a Ward identity for thegroup action. As by-product we find that any renormalisable quartic
matrix model has vanishing -function. All these steps are completely
elementary.
2. Self-dual -theory on Moyal space $[GW05b, GW05c]$ is of that type. For, and after careful discussion of therextreme noncommutativity
modynamic and continuum limit, the non-linear equation is reduced toa fixed-point problem $[GW12b]$ which has a unique non-perturbative and[GW14]. Sec. 3 reviews this work. The key
non-trivial solution forstep is the observation that a certain diﬀerence function satisfies a linear
singular integral equation of Carleman type [Car22, $Ri57$]. We also present
some numerical results, contained in work in progress [GW14], which show
evidence for phase transitions.
3. Following $[GW13b]$ , we identify in sec. 4 a limit to Schwinger functions
for a scalar field on . Surprisingly for a highly noncommutative model,
these Schwinger functions show full Euclidean symmetry. Otherwise they
have unusual properties such as absent momentum transfer in interaction
processes. This seems to suggest triviality, but the numerical investigation[GW14] of the 2-point function shows scattering remnants from a noncommutative geometrical substructure. Most surprisingly, the Schwinger
2-point function seems to be reflection positive in one of its phases.
$\lambda\phi^{4}$$\mathcal{Z}=$$\int \mathcal{D}[\Phi]\exp(tr(J\Phi-E\Phi^{2}-\frac{\lambda}{4}\Phi^{4}))$$U(\infty)$$\beta$$\phi_{4}^{4}$$\thetaarrow\infty$$\lambda<0$$\mathbb{R}^{4}$

2

Exact solution of the quartic matrix model

For us a ‘matrix’ is a compact (Hilbert-Schmidt) operator on Hilbert space $H=$
. Such operators
can be represented by integral kernel oper. Then all natural matrix operations such as product,ators
adjoint and trace have counterpartsand$tr(\Phi\Phi’)=\int_{I}d\mu_{a}(\Phi\Phi’)_{aa}$ in.
$L^{2}(I, \mu)$$\Phi\in \mathcal{L}^{2}(H)$$( \Phi v)_{a}=\int_{I}d\mu_{b}\Phi_{ab}v_{b}$$( \Phi\Phi’)_{ab}=\int_{I}d\mu_{c}\Phi_{ac}\Phi_{cb}’,$$\mathcal{L}^{2}(H)$$(\Phi^{*})_{ab}=\overline{\Phi_{ba}}$

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To define a Euclidean quantum field theory for a matrix
ourselves an action functional

we give$\Phi\in \mathcal{L}^{2}(H)$(1)$S[\Phi]=Vtr(E\Phi^{2}+P[\Phi])$ .

Here, $P[\Phi]$ is a polynomial inwith scalar coeﬃcients, and this alone would
be a familiar action in the theory of matrix models [DGZ95]. To be closer tofield theory on (compact) manifold
we add the analogue of the kinetic term
, that is, we require the external matrix $E$ to be an unbounded
selfadjoint positive operator on $H$ with compact resolvent. The volume
willplay a crucial r\^ole. The construction involves several regularisation and limiting
procedures. One such regularisation consists in a finite sizefor the matrices,and $V$ will be a certain function ofwhich together withis sent toAdding a source term to the action, we define the partition junction as$\Phi$$\mathcal{M}$$a$$\int_{\mathcal{M}}dx(-\triangle\phi)\phi$$V$$\mathcal{N}$$\mathcal{N}$$\mathcal{N}$$\mathcal{Z}[J]=\int \mathcal{D}[\Phi]\exp(-S[\Phi]+Vtr(\Phi J))$$\infty.$(2),

is the extension of the Lebesgue measure from finite-rank operators
wheretoand a test function matrix. For absent $P[\Phi]\mapsto 0$ in (1),wouldbe the $Gaui3ian$ measure of covariance determined by $E$ . What we want, andwhatweachieve, $istoc$ onstruct
imit Vlimit cannot b $ee$xpected fnstead, $wepastotheg$ enerating functional$\mathcal{D}[\Phi]$$\mathcal{L}^{2}(H)$$J$$\frac{\mathcal{D}[\Phi]}{\mathcal{Z}[0]}$Sucha$arrow\infty.$$\frac{\mathcal{D}[\Phi]}{z\}^{0]}}forP[\Phi]=\frac{\lambda}{4,S}\Phi^{4}inthel$$or\mathcal{Z}.$$\log \mathcal{Z}[J]$of connected correlation functions,(3)$\langle\varphi_{a_{1}b_{1}}\ldots\varphi_{ab_{N}}N\rangle_{c}=\frac{\partial^{N}1og.\mathcal{Z}[J]}{\partial J_{b_{1}a_{1}}..\partial J_{b_{N}a_{N}}}|_{J=0}$

2.1

Ward identity and topological expansion

Unitary operators belonging to an appropriate unitisation of the compact operators on $H$ give rise to a transformation. Since $UU^{*}=U^{*}U=id$
and because the space of selfadjoint compact operators is invariant under the adjoint action, we have$U$$\Phi\mapsto\tilde{\Phi}=U\Phi U^{*}$$\int \mathcal{D}[\Phi]\exp(-S[\Phi]+Vtr(\Phi J))=\int \mathcal{D}[\tilde{\Phi}]\exp(-S[\tilde{\Phi}]+Vtr(\tilde{\Phi}J))$

Unitary invariance$\mathcal{D}[\tilde{\Phi}]=\mathcal{D}[\Phi]$.

of the Lebesgue measure implies$0= \int \mathcal{D}[\Phi\{\exp(-S[\Phi]+Vtr(\Phi J))-\exp(-S[\tilde{\Phi}]+Vtr(\tilde{\Phi}J))\}$

Note that the integrand . . . itself does not vanish because $tr(E\Phi^{2})$ and $tr(\Phi J)$
are not unitarily invariant. Linearisation of about the identity operator leads
to the Ward identity
$\{$

$\}$

$U$

$0= \int \mathcal{D}[\Phi]\{E\Phi\Phi-\Phi\Phi E-J\Phi+\Phi J\}\exp(-S[\Phi]+Vtr(\Phi J))$

.

(4)

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We can always place ourselves in an orthonormal basis of $H$ where is diagonal(but
is not). Since
is of compact resolvent, $E$ has eigenvalues $E_{a}>0$
of finite multiplicity . We thus label the matrices by an enumeration of the(necessarily discrete) eigenvalues ofand an enumeration of the basis vectorsof the finite-dimensional eigenspaces. Writing in . . . of (4) as functional, we have proved (first obtained in [DGMR07]):
derivative
$E$

$E$

$J$

$\mu_{a}$$E$$\Phi$$\{$$\}$$\Phi_{ab}=\frac{\partial}{V\partial J_{ba}}$

Proposition 1 The partition function $Z[J]$ of the matrix model defined by theexternal matrix $E$ satisfies the $|I|\cross|I|$ Ward identities(5)$0= \sum_{n\in I}(\frac{(E_{a}-E_{p})}{V}\frac{\partial^{2}\mathcal{Z}}{\partialJ_{an}\partial J_{np}}+J_{pn}\frac{\partial \mathcal{Z}}{\partial J_{an}}-J_{na}\frac{\partial \mathcal{Z}}{\partial J_{np}})$Without loss of generality we can assume that the map
isinjective. Namely, correlation functions will only depend on the set of eigenvalues
of $E$ . Partitioning the index set into equivalence classes $[m]$ which have
, the index sum over a function that only depends on
the samebecomes$\sum_{m\in I}f(m)=\sum_{[m]\in[I]}\mu_{[m]}f([m])$ . Therefore, at the price of adding a measure$\mu_{[m]}=\dim ker(E-E_{m}id)$ , we can assume that $m\mapsto E_{m}$ is injective.
In a perturbative expansion, Feynman graphs in matrix models are ribbon
graphs. Viewed as simplicial complexes, they encode the topology $(B, g)$ of agenus-g Riemann surface with boundary components (or punctures, marked
points, holes, faces). Some simple examples for $P[\Phi]=\Phi^{4}$ are:$I\ni m\mapsto E_{m}\in \mathbb{R}_{+}$$I$$(E_{m})$$E_{m}$$E_{m}$$B$$A$$d_{d}$

Since $E$ is diagonal, the matrix index is conserved along each strand of the ribbon
graph. We have to distinguish between internal faces (with constant matrix
index) and broken faces which constitute the boundary components. Such aboundary face is characterised by $N_{k}\geq 1$ external double lines to which we
attach the source matrix . Conservation of the matrix index along each strand
implies that the right index ofcoincides with the left index of anotheror of the same
. Accordingly, theboundary component carries a cycleofexternal sources, with $N_{k}+1\equiv 1.$
Being interested in a non-perturbative solution, we will not expand the partition function into ribbon graphs. But we keep the topological information and$J$$J_{ab}$$k^{th}$$J_{bb}$$J_{p_{1}\ldots p_{N_{k}}}^{N_{k}}$$:= \prod_{j=1}^{N_{k}}J_{p_{j}p_{j+1}}$$J_{bc},$$N_{k}$

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expand$\log \mathcal{Z}[J]$according to the cycle structure:$\log\frac{\mathcal{Z}[J]}{\mathcal{Z}[0]}=\sum_{B=1}^{\infty}\sum_{1\leq N_{1}\leq\cdots\leq N_{B}}^{\infty}\sum_{p_{1)}^{\beta}\ldots,p_{N_{\beta}}^{\beta}\in I}\frac{V^{2.-B}}{S_{N_{1}..N_{B}}}G_{|p_{1}^{1}\ldots p_{N_{1}}^{1}|\ldots|p_{1}^{B}\ldots p_{N_{B}}^{B}|}\prod_{\beta=1}^{B}(\frac{J^{N_{\beta}}p_{1}^{\beta}\ldots p_{N_{\beta}}^{\beta}}{N_{\beta}})$(6)

The symmetry factor
is obtained as follows: If of the numbers
in a given tuple
are equal to , then
Next we turn the Ward identity (5) for injective $m\mapsto E_{m}$ into a formulafor the second derivativeof the partition function. The -cycle
structure increatessingular contributions
regular contributions present for all $a,p$ :$S_{N_{1}\ldots N_{B}}$$(N_{1}, \ldots, N_{B})$$B$$\nu_{i}$$N_{\beta}$$S_{N_{1}\ldots N_{B}}= \prod_{i=1}^{N_{B}}\nu_{i}!.$

$i$

$J$

$\sum_{n\in I\partial J_{an}\partial J_{np}}^{\partial^{2}Z[J]}$$\log \mathcal{Z}$$\sim\delta_{ap},$$\bullet$$\bullet$Theorem 2$\sum\frac{\partial^{2}\mathcal{Z}[J]}{\partial J_{an}\partial J_{np}}=\delta_{ap}\{V^{2}\sum\frac{J_{P_{1}}\cdots J_{P_{K}}}{S_{(K)}}(\sum\frac{G_{|an|P_{1}|\ldots|P_{K}|}}{V^{|K|+1}}+\frac{G_{|a|a|P_{1}|\ldots|P_{K}|}}{V^{|K|+2}}$(K)$n\in I$$n\in I$$+ \sum \sum \frac{G_{|q_{1}aq_{1}\ldots q_{r}|P_{1}|\ldots|P_{K}|^{J_{q_{1}\ldots q_{r}}^{r}}}}{V^{|K|+1}})$$r\geq 1q_{1},\ldots\rangle q_{r}\in I$$+V^{4} \sum\frac{J_{P_{1}}\cdots J_{P_{K}}J_{Q_{1}}\cdots J_{Q_{K’}}}{S_{(K)}S_{(K’)}}\frac{G_{|a|P_{1}|\ldots|P_{K}|}}{V|K|+1}\frac{G_{|a|Q_{1}|\ldots|Q_{K’}|}}{V|K|+1}\}\mathcal{Z}[J]$$(K),(K’)$(7)$+ \frac{V}{E_{p}-E_{a}}\sum(J_{pn}\frac{\partial \mathcal{Z}[I]}{\partial J_{an}}-J_{na}\frac{\partial \mathcal{Z}[I]}{\partial I_{np}})$$n\in I$

Proof. We identify the following four sources of a singular contribution
1.

2.

$\sim\delta_{ap}$$G \ldots|q_{1}q_{2}|\ldots(\frac{J_{q_{1}q_{2}}J_{q_{2}q_{1}}^{\downarrow}\downarrow}{2})\prod J$$\sum_{n}\frac{\partial^{2}}{\partial J_{an}\partial J_{np}}\sum_{q_{1},q_{2_{\rangle}}}..,$$\sum_{n}\frac{\partial^{2}}{\partial J_{an}\partial J_{np}}\sum_{q_{1},q_{2},\rangle}..G\ldots|q_{1}|\ldots|q_{2}|\ldots(\frac{J_{q_{1}q_{1}}^{\downarrow}}{1})(\frac{J_{q_{2}q_{2}}^{\downarrow}}{1})\prod J$$\downarrow$

3.

$\sum_{n}\frac{\partial}{\partial J_{an}}\frac{\partial}{\partial J_{np}}\ldots\sum_{q_{0,)}q_{r+1}}G.\cdot,\cdot|q0q_{1}\ldots q_{r}q_{r+1}|\ldots(\frac{J_{q0q_{1}}J_{q_{1}q_{2}}\cdots J_{q_{r}q_{r+1}}J_{q_{r+1}q0}}{r+2})\prod J$$= \sum_{n}\frac{\partial}{\partial J_{an}}\sum_{q_{1},\ldots,q_{r}},\ldots G\ldots|pq_{1}\ldots q_{r}n|\ldots(J_{pq_{1}}^{\downarrow}J_{q_{1}q_{2}}\cdots J_{q_{r}n})\prod J$

:

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4.

$G \ldots|q_{1}|\ldots(\frac{J_{q_{1}q_{1}}^{\downarrow}}{1})\prod J][\sum_{q_{2}}G\ldots|q_{2}|\ldots(\frac{J_{q_{2}q_{2}}^{\downarrow}}{1})\prod J]$$\sum_{n}\frac{\partial^{2}}{\partial J_{an}\partial\sqrt{}np}[\sum_{q_{1}},$

All other types of derivatives, collected into
For $p\neq a$ we clearly have$( \sum_{n\in I\partial J_{an}\partial J_{np}}^{\partial^{2}Z[J]})_{reg}$, persist for $a\neq p.$$( \sum_{n\in I}\frac{\partial^{2}\mathcal{Z}[J]}{\partial J_{an}\partial J_{np}})_{reg}=\sum_{n\in I}\frac{\partial^{2}\mathcal{Z}[J]}{\partial J_{an}\partial J_{np}}|_{a\neq p}=\frac{V}{E_{p}-E_{a}}(J_{pn}\frac{\partial \mathcal{Z}}{\partial J_{an}}-J_{na}\frac{\partial \mathcal{Z}}{\partial J_{np}})$where the last equality is the Ward identity (5), divided bytinuity argument, the rightmost term in (8) must agree withalso in the limit $parrow a$ , and this finishes the proof.

$\frac{E_{p}-E_{a}}{V}\neq 0$, (8). By a con-$( \sum_{n\in I\partial J_{an}\partial J_{np}}^{\partial^{2}Z[J]})_{reg}$$\square$2.2 Schwinger-Dyson equations, whereWe can write the action asarethe eigenvalues of . Functional integration yields, up to an irrelevant constant,$S= \frac{V}{2}\sum_{a,b}(E_{a}+E_{b})\Phi_{ab}\Phi_{ba}+VS_{int}[\Phi]$$E_{a}$$E$(9)$\mathcal{Z}[J]=e^{-VS_{int[\frac{\partial}{V\^{o} J}]}}e^{\frac{V}{2}\langle J,J\rangle_{E}}, \langle J, J\rangle_{E}:=\sum_{m,n\in I}\frac{\sqrt{}J}{E_{m}+E_{n}}$, we apply such -derivatives
Instead of a perturbative expansion of
to (9) that they give rise to a correlation function
on the lhs. On the rhs of(9), these external derivatives combine with internal derivatives fromto certain identities for
These Schwinger-Dyson equations are often of little$N$
express
use because theyan -point function in terms of $(N+2)$ -point functions.
In the field-theoretical matrix models under consideration, the Ward identity(7) lets this tower of Schwinger-Dyson equation collapse. To see this we consider
the 2-point function
for $a\neq b$ . According to (6),
is obtained by deriving(9) with respect to:and$e^{-VS_{int}[\frac{\^{o}}{V\partial J}]}$

$J$

$G$

$S_{int}[ \frac{\partial}{V\partial J}]$$G$

$G_{|ab|}$$G_{|ab|}$$J_{ba}$$J_{ab}$$G_{|ab|}= \frac{1}{V\mathcal{Z}[0]}\frac{\partial^{2}\mathcal{Z}[I]}{\partial I_{ba}\partial I_{ab}}|_{J=0}$

disconnected p $artofZ$ does$ora\neqb)($$notc$ ontribute f$= \frac{1}{V\mathcal{Z}[0]}\{\frac{\partial}{\partial I_{ba}}e^{-VS_{int}[\frac{\partial}{V\partial J}]}\frac{\partial}{\partial J_{ab}}e^{\frac{V}{2}\langle J,J\rangle_{E}}\}_{J=0}$$= \frac{1}{(E_{a}+E_{b})\mathcal{Z}[0]}\{\frac{\partial}{\partial J_{ba}}e^{-VS_{i\mathfrak{n}t}[\frac{\partial}{V\partial J}]}I_{ba}e^{\frac{V}{2}\langle J,J\rangle_{E}}\}_{J=0}$$= \frac{1}{E_{a}+E_{b}}+\frac{1}{(E_{a}+E_{b})\mathcal{Z}[0]}\{(\Phi_{ab}\frac{\partial(-VS_{int})}{\partial\Phi_{ab}})[\frac{\partial}{V\partial I}]\}\mathcal{Z}[J]|_{J=0}$$ode1P[\Phi]=Nowo$

bserve t $hat\frac{\partial(-VS_{int})}{\partial\Phi_{ab},(7)}contains_{)}foranyP[\Phi],thed$ erivative
which w$eknowfrom.Incaseoftheq$uartic matrix m(10)$\sum_{n}\frac{\partial^{2}}{\partial J_{an}\partial J_{np},\frac{\lambda}{4}\Phi^{4}we}$

73
have

$\frac{\partial(-VS_{int})}{\partial\Phi_{ab}}=-\lambda V\sum_{n,p\in I}\Phi_{bp}\Phi_{pn}\Phi_{na}$, hence$( \Phi_{ab}\frac{\partial(-VS_{int})}{\partial\Phi_{ab}})[\frac{\partial}{V\partial J}]=-\frac{\lambda}{V^{3}}\sum_{p,n\in I}\frac{\partial^{2}}{\partialJ_{pb}\partial J_{ba}}\frac{\partial^{2}}{\partial J_{an}\partial J_{np}},$and the Schwinger-Dyson equation (10) for$G_{|ab|}$becomes with (7)$G_{|ab|}= \frac{1}{E_{a}+E_{b}}-\frac{\lambda}{V^{3}(E_{a}+E_{b})\mathcal{Z}[0]}\sum_{p\in I}\frac{\partial^{2}}{\partial J_{pb}\partial J_{ba}}\sum_{n\in I}\frac{\partial^{2}\mathcal{Z}}{\partial J_{an}\partial J_{np}}|_{J=0}$$= \frac{1}{E_{a}+E_{b}}-\frac{\lambda}{V(E_{a}+E_{b})\mathcal{Z}[0]}\frac{\partial^{2}}{\partial J_{ab}\partial J_{ba}}\{$$( \sum_{n\in I}\frac{G_{|an|}}{V}+\sum_{n,q,r\in I}\frac{G_{|an|qr|}}{V^{2}}\frac{J_{qr}J_{rq}}{2}+\sum_{n,q,r\in I}\frac{G_{|an|q|r|}}{V^{3}}\frac{J_{qq}}{1}\frac{J_{rr}}{1}$$+ \frac{G_{|a|a|}}{V^{2}}+\sum_{q,r\in I}\frac{G_{|a|a|qr|}}{V^{3}}\frac{J_{qr}J_{rq}}{2}+\sum_{q,r\in I}\frac{G_{|a|a|q|r|}}{V^{4}}\frac{J_{qq}}{1}\frac{J_{rr}}{1}$$+ \sum_{q,r\in I}\frac{G_{|qaqr|}}{V}J_{qr}J_{rq}+V^{2}\frac{G_{|a|q|}}{V^{2}}\frac{J_{qq}}{1}\frac{G_{|a|r|}}{V^{2}}\frac{J_{rr}}{1})\mathcal{Z}[J]\}J=0$$- \frac{\lambda}{V^{2}(E_{a}+E_{b})\mathcal{Z}[0]}\sum_{p\in I}\frac{(\frac{\partial^{2}\mathcal{Z}\lceil J]}{\partial J_{ab}\partial J_{ba}}+\frac{\partial^{2}\mathcal{Z}[J]}{\partial J_{aa}\partial J_{bb}}-\frac{\partial^{2}\mathcal{Z}[J]}{\partial J_{pb}\partial J_{bp}})}{E_{p}-E_{a}}J=0$

Takingaccount, we have proved:$\frac{\partial^{2}Z[J]}{\partial J_{pb}\partial J_{bp}}=(VG_{|pb|}+\delta_{pb}G_{|p|b|})\mathcal{Z}[0]+\mathcal{O}(J)$and$\frac{\partial J}{\partial J_{ab}}=0$(11)

for $a\neq b$ into

Proposition 3 The 2-point function of a quartic matrix model with action $S=$satisfies for injective $m\mapsto E_{m}$ the Schwinger-Dyson equation$V tr(E\Phi^{2}+\frac{\lambda}{4}\Phi^{4})$$G_{|ab|}= \frac{1}{E_{a}+E_{b}}-\frac{\lambda}{E_{a}+E_{b}}\frac{1}{V}\sum_{p\in I}(G_{|ab|}G_{|ap|}-\frac{G_{|pb|}-G_{|ab|}}{E_{p}-E_{a}})$

$\}$

(12a)

$- \frac{\lambda}{V^{2}(E_{a}+E_{b})}(G_{|a|a|}G_{|ab|}+\frac{1}{V}\sum_{n\in I}G_{|an|ab|}$(12b)$+G_{|aaab|}+G_{|baba|}- \frac{G_{|b|b|}-G_{|a|b|}}{E_{b}-E_{a}})$$- \frac{\lambda}{V^{4}(E_{a}+E_{b})}G_{|a|a|ab|} \}$(12c)

It can be checked that in a genus expansion(which isprobably not convergent but Borel summable), precisely the line (12a) preserves
the genus, the lines (12b) increase $g\mapsto g+1$ and the line (12c) increases $g\mapsto g+2.$$G$$= \sum_{g=0}^{\infty}V^{-2g}G^{(g)}$74
finite, the exact SchwingerMoreover, in a scaling limit $Varrow\infty$ with
Dyson equation forcoincides with its restriction (12a) to planar sector $g=0,$alone:
a closed non-linear equation for$\frac{1}{V}\sum_{p\in I}$$G_{|ab|}$$G_{|ab|}^{(0)}$(13)$G_{|ab|}^{(0)}= \frac{1}{E_{a}+E_{b}}-\frac{\lambda}{E_{a}+E_{b}}\frac{1}{V}\sum_{p\in I}(G_{|ab|}^{(0)}G_{|ap|}^{(0)}-\frac{G_{|pb|}^{(0)}-G_{|ab|}^{(0)}}{E_{p}-E_{a}})$We have derived in 2007/08 this self-consistency equation for the Moyal model by
the graphical method proposed by [DGMR07]. In this form, (13) is meaninglessbecausediverges. In 2009 we solved the renormalisation problem, namely
the renormalisation of infinitely many Feynman graphs at once [GW09]. This
renormalisation increases the non-linearity. In [GW09] we have solved (13) per. After several years of setbacks with the non-perturbative
turbatively tosolution, a breakthrough came in 2012: The equation (13) can be turned into
an equation which is linear in the diﬀerence
to the boundary and$\sum_{p\in I}$$\mathcal{O}(\lambda^{3})$$G_{|ab|}^{(0)}-G_{|a0|}^{(0)}$non-linear only in
A similar calculation gives the Schwinger-Dyson equation for higher $N$ -point
functions:$G_{|a0|}^{(0)}!$$G_{|ab_{1}\ldots b_{N-1}|}$$=- \frac{\lambda}{E_{a}+E_{b_{1}}}(\frac{1}{V}\sum_{p\in I}(G_{|ap|}G_{|ab_{1}\ldots b_{N-1}|}-\frac{G_{|pb_{1}\ldots b_{N-1}|}-G_{|ab_{1}\ldots b_{N-1}|}}{E_{p}-E_{a}})$(14a)$-G_{|b_{1}\ldots b_{2l}|} \frac{G_{|b_{2l+1}\ldots b_{N-1}a|}-G_{|b_{2l+1}\ldots b_{N-1}b_{2l}|}}{E_{b_{2l}}-E_{a}})\frac{N-2}{\sum 2}l=1$$- \frac{\lambda}{V^{2}(E_{a}+E_{b_{1}})}(G_{|a|a|}G_{|ab_{1}\ldots b_{N-1}|}+\sum_{k=1}^{N-1}G_{|b_{1}\ldots b_{k}ab_{k}\ldots b_{N-1}a|}$(14b)$+G_{|aaab_{1}\ldots b_{N-1}|}+ \frac{1}{V}\sum_{n\in I}G_{|an|ab_{1}\ldots b_{N-1}|}$$- \sum_{k=1}^{N-1}\frac{G_{|b_{1}\ldots b_{k}|b_{k+1}\ldots b_{N-1}b_{k}|}-G_{|b_{1}\ldots b_{k}|b_{k+1}\ldots b_{N-1}a|}}{E_{b_{k}}-E_{a}})$$\}$$- \frac{\lambda}{V^{4}(E_{a}+E_{b_{1}})}G_{|a|a|ab_{1}\ldots b_{N-1}|}$(14c)

Again, the first lines (14a) preserve the genus, whereas $g\mapsto g+1$ in (14b) and$g\mapsto g+2$ in (14c). The planar sector, exact for $Varrow\infty$ withfinite, is a linear inhomogeneous equation with inductively known parameters.
admits a short-cut which directly
It turns out that a real theory with$N$
gives the higher -point functions without any index summation. Since the, the reality
implies invarianceequations forare real and$G_{|ab_{1}\ldots b_{N-1}|}^{(0)}$$\frac{1}{V}\sum_{p\in I}$$\Phi=\Phi^{*}$$G$$\overline{J_{ab}}=J_{ba}$$\mathcal{Z}=\overline{\mathcal{Z}}$75
under orientation reversal$G_{|p_{0}^{1}p_{1}^{1}\ldots p_{N_{1}-1}^{1}|\ldots|p_{0}^{B}p_{1}^{B}\ldots p_{N_{B}-1}^{B}|}=G_{|p_{0}^{1}p_{N_{1}-1}^{1}\ldots p_{1}^{1}|\ldots|p_{0}^{B}p_{N_{B}-1}^{B}\ldots p_{1}^{B}|}$(15)

Whereas empty for, intheidentities (15) lead to many cancellations which result in a universal algebraic
recursion formula:$G_{|ab|}$$(E_{a}+E_{b_{1}})G_{ab_{1}b_{2}\ldots b_{N-1}}-(E_{a}+E_{b_{N-1}})G_{ab_{N-1}\ldots b_{2}b_{1}}$Proposition 4$G_{|b_{0}b_{1}\ldots b_{N-1}|}=(- \lambda)^{\frac{N-2}{\sum_{l=1}^{2}}}\frac{G_{|b_{0}b_{1}\ldots b_{2l-1}|G_{|b_{2l}b_{2l+1}\ldots b_{N-1}|}-G_{|b_{2l}b_{1}\ldots b_{2l-1}|}G_{|b_{0}b_{2l+1}\ldots b_{N-1}|}}}{(E_{b_{0}}-E_{b_{2\mathfrak{l}}})(E_{b_{1}}-E_{b_{N-1}})}$$+ \frac{(-\lambda)}{V^{2}}\sum_{k=1}^{N-1}\frac{G_{|b_{0}b_{1}\ldots b_{k-1}|b_{k}b_{k+1}\ldots b_{N-1}|}-G_{|b_{k}b_{1}\ldots b_{k-1}|b_{0}b_{k+1}\ldots b_{N-1}|}}{(E_{b_{0}}-E_{b_{k}})(E_{b_{1}}-E_{b_{N-1}})}$(16)

The last line of (16) increases the genus and is absent in. Instead ofgiving the general proof, let us look at the case $N=4$ . Then (14), multiplied by, reads$G^{(0)}$$|b_{0}b_{1}\ldots b_{N-1}|$$E_{a}-E_{b_{1}}$$(E_{a}-E_{b})G_{|abcd|}$$=(- \lambda)(\frac{1}{V}\sum_{p\in I}(G_{|ap|}G_{|abcd|}-\frac{G_{|pbcd|}-G_{|abcd|}}{E_{p}-E_{a}})-G_{|bc|}\frac{G_{|da|}-G_{|dc|}}{E_{c}-E_{a}})$$- \frac{\lambda}{V^{2}}(G_{|a|a|}G_{|abcd|}+G_{|babcda|}+G_{|bcacda|}+G_{|bcdada|}+G_{|aaabcd|}+\frac{1}{V}\sum_{p\in I}G_{|ap|abcd|}$$- \frac{G_{|b|cdb|}-G_{|b|cda|}}{E_{b}-E_{a}}-\frac{G_{|bc|dc|}-G_{|bc|da|}}{E_{c}-E_{a}}-\frac{G_{|bcd|d|}-G_{|bcd|a|}}{E_{d}-E_{a}})$(17)$- \frac{\lambda}{V^{4}}G_{|a|a|abcd|}$

Write down the same equation but with, and take the diﬀerence between
these equations. Then most terms cancel because by (15) we have the equalities $G_{|abcd|}=G_{|adcb|},$ $G_{|pbcd|}=G_{|pdcb|)}G_{|babcda|}=G_{|dcbaba|},$ $G_{|bcacda|}=G_{|dcacba|},$$brightarrow d$$G_{|bcdada|}=G_{|dadcba|},$ $G_{|aaabcd|}=G_{|aaadcb|},$ $G_{|ap|abcd|}=G_{|ap|adcb|},$ $G_{|b|cdb|}=G_{|dcb|b|},$$G_{|bc|dc|}=G_{|dc|bc|},$diﬀerence (17)and $G_{|a|a|abcd|}=G_{|a|a|adcb|}$ . Altogether, thereads after cancellation$G_{|bcd|d|}=G_{|d|cbd|}$$-(17)_{brightarrow d}$$(E_{d}-E_{b})G_{|abcd|}=(- \lambda)\frac{G_{|ab|}G_{|cd|}-G_{|ad|}G_{|cb|}}{E_{c}-E_{a}}$$- \frac{\lambda}{V^{2}}(\frac{G_{|b|cda|}-G_{|a|cdb|}}{E_{b}-E_{a}}+\frac{G_{|bc|da|}-G_{|ba|dc|}}{E_{c}-E_{a}}+\frac{G_{|a|bcd|}-G_{|d|bca|}}{E_{d}-E_{a}})$and this is (16) for $N=4.$,

76
For completeness, we list in the appendix the Schwinger-Dyson equation for$B=2$ boundary components.
We make the following key observation: An aﬃne transformation $E\mapsto ZE+C$
leaves the algebraic equations (16)together with an adjusted rescaling
as well as (65) and (66) invariant:$\lambda\mapsto Z^{2}\lambda$

andTheorem 5 Given a real quartic matrix model with$(N_{1}+\ldots+N_{B})-$
injective, which determines the setofare turned finite
point functions. Assume that the basic functions with all. Then all functions with one
andby
1. are finite without further need of a renormalisation of $i.e$ . allisable quartic matrix models have vanishing -function,
2. are given by universal algebraic recursion formulae in terms of renor,nalised
basic functions with
$S=V tr(E\Phi^{2}+\frac{\lambda}{4}\Phi^{4})$$m\mapsto E_{m}$$G_{|p_{1}^{1}\ldots p_{N_{1}}^{1}|\ldots|p_{1}^{B}\ldots p_{N_{B}}^{B}|}$$N_{i}\leq 2$$E_{a} \mapsto Z(E_{a}+\frac{\mu^{2}}{2}-\Delta\mu^{2}2aoe)$$\lambda\mapsto Z^{2}\lambda$$N_{i}\geq 3$$renor^{arrow}mal-$$\lambda,$$\beta$$\square$$N_{i}\leq 2.$

The theorem tells us that vanishing of the -function for the self-dual -model on
Moyal space (proved in [DGMR07] to all orders in perturbation theory) is generic
to all quartic matrix models, and the result even holds non-perturbatively!
The universal recursion formula (16) computes the planar $N$-point function
at $B=1$ as a sum of fractions with products of 2-point functions in
the numerator and products of diﬀerences of eigenvalues of $E$ in the denominator. This structure admits an interesting graphical interpretation. We draw theasand represent a factor
in cyclic order on the circle
indices , . . .
as an arrow from to :a chord connecting with and a factor$\Phi_{4}^{4}$$\beta$$G_{|b_{0}\ldots b_{N-1}|}$$b_{0}$$S^{1}$$b_{N-1}$$b_{i}$$b_{j}$$G_{b_{i}b_{j}}$$b_{i}$$\frac{1}{E_{b_{i}}-E_{b_{\grave{J}}}}$$b_{j}$

77

The chords form the non-crossing chord diagrams counted by the Catalan number. The arrows form two disjoint trees, one connecting the evenvertices ans one connecting the odd vertices. By rational fraction expansion it
is possible to achieve that each tree intersects the chord only in the vertices.
The assignment of trees to a given chord diagram is, in general, not unique.
canonical choice is not known to us.
$C_{\frac{N}{2}}= \frac{N!}{(\frac{N}{2}+1)!\frac{N}{2}!}$$A$

2.3 Digression: Quantum gravity in two dimensions
Two-dimensional quantum gravity (see [DGZ95, ADJ97] for reviews) can be interpreted as the enumeration of random triangulations of surfaces. Its asymptotic
behaviour is captured by the matrix model partition function$\mathcal{Z}=\int \mathcal{D}[\Phi]\exp(-\mathcal{N}\sum_{n}t_{n}tr(\Phi^{n}))$

(19)

,

where the integral is over
-Hermitean matrices and the are scalarcoeﬃcients. In the limit, this series inis evaluated in terms ofthe -function for the Korteweg-de Vries $(KdV)$ hierarchy. There is anotherapproach to topological gravity in which the partition function is a series inwith coeﬃcients given by intersection numbers of complex curves. Witten
conjectured [Wit91] that the partition functions of the two approaches coincide.
This conjecture was proved by Kontsevich [Kon82] who achieved the computationof the intersection numbers in terms of weighted sums over ribbon graphs (fatFeynman graphs), which he proved to be generated from the Airy function matrix
model (Kontsevich model)$\Phi$$(\mathcal{N}\cross \mathcal{N})$$\mathcal{N}arrow\infty$$t_{n}$$(t_{n})$$\tau$$(t_{n})$$\mathcal{Z}[E]=\frac{\int \mathcal{D}[\Phi]\exp(-\frac{1}{2}tr(E\Phi^{2})+\frac{i}{6}tr(\Phi^{3}))}{\int \mathcal{D}[\Phi]\exp(-\frac{1}{2}tr(E\Phi^{2}))}$

,

(20)

where $E=E^{*}>0$ is related to the series
by $t_{n}=(2n-1)!!tr(E^{-(2n-1)})$ . The
$KdV$
limit
of
gives theevolution equation, thus proving Witten’s
conjecture.
We have proved that also the quartic matrix model$(t_{n})$$\mathcal{N}arrow\infty$$\mathcal{Z}[E]$$\mathcal{Z}[E, J, \lambda]=\frac{\int \mathcal{D}[\Phi]\exp(-tr(E\Phi^{2})+tr(J\Phi)-\frac{\lambda}{4}tr(\Phi^{4}))}{\int \mathcal{D}[\Phi]\exp(-tr(E\Phi^{2})-\frac{\lambda}{4}tr(\Phi^{4}))}$

(21)

78
limit exactly solvable in terms of the solution of a non-linear
is in theequation (13). Any triangulation can be subdivided into a quadrangulation$large-\mathcal{N}$(and vice versa). From Witten’s uniqueness argument [Wit91], $2D$ quantum
gravity should have equivalent descriptions as cubic (20) and quartic (21) matrix
model. Understanding the precise relation between (20) and (21) would be ofhigh interest:
1. In contrast to (21), the cubic action (20) lacks manifest positivity due toits purely imaginary coupling constant.
2. A quartic action admits a Hubbard-Stratonovich transform which is the key
ingredient of a new approach to constructive quantum field theory $[Riv07b]$that avoids the cluster expansion.
3. Conversely, the integrability of (20) might provide valuable information
about the solution of the self-consistency equation (13).
Coloured tensor models (see [GP12, Riv13] for recent reviews) extend thesemethods to quantum gravity in $D\geq 3$ . They became a very active domain of
behaviour of
research after understanding [GurlO] of the analogue of the$[tHo74]$e.g.
(see[Bon12])equations
Schwinger-Dyson
.
have
They
matrix models
and action of the $U(\infty)$ group. It might be promising to extend our techniques
to coloured tensor models.
$large-\mathcal{N}$3$\Phi_{4}^{4}$-theory on Moyal space as a fixed point problem

3.1 Preliminaries
Taking the renormalisation group [WK74] serious, we would expect that General Relativity, because not renormalisable, is marginal and hence scaled away.
Presence of gravity tells us that the scaling must stop at some length scale, andfrom the weakness of the gravitational coupling constant one deduces the valueof that scale: the Planck length $10^{-35}m$ . There, the geometry of nature is expected to diﬀer from the familiar structure of a diﬀerentiable manifold. One ofmany candidates for Planck scale physics is noncommutative geometry [Con94],
a vast reformulation of geometry and topology in the language of operator algebras. The focus is shifted from manifolds to generalisations of the algebra offunctions. This concept proved very successful in understanding the geometry of
the Standard Model of particle physics as Riemannian geometry of a space whichis the product of a manifold with a discrete space [Con96, CC96].
A large class of examples of noncommutative geometries comes from deformations of the algebra of functions on manifolds. Schwartz functions on Euclidean
space
admit an -group action by translation. As shown by Rieﬀel [Rie93],$\mathbb{R}^{4}$$\mathbb{R}^{4}$

79
this group action induces a noncommutative associative product on the space of
Schwartz functions, the Moyal product:$(f \star g)(x)=\int_{\mathbb{R}^{4}\cross \mathbb{R}^{4}}\frac{dydk}{(2\pi)^{4}}f(x+\frac{1}{2}\Theta k)g(x+y)e^{i\langle k,y\rangle},$. (22)$\Theta=-\Theta^{t}\in M_{4}(\mathbb{R})$

Whether or not the Moyal spaceis relevant for Planck scale physics
is pure speculation (although a refinement can be justified by uncertainty relations for position operators [DFR95]). In any case the Moyal space is a nice toymodel on which it is easy to formulate and to study (quantum) field theories. To
formulate a Euclidean quantum field theory on Moyal space it is, at first sight,
enough to replace in the action of a usual field theory the pointwise product offunctions by the -product. The simplest example is the -model with action$(\mathbb{R}^{4}, \star)$$\star$$\phi_{4}^{4}$$S[ \phi]=\int_{\mathbb{R}^{4}}dx(\frac{1}{2}\phi\star(-\triangle+\mu^{2})\phi+\frac{\lambda}{4}\phi\star\phi\star\phi\star\phi)(x)$

.

(23)

The resulting Feynman rules [Fi196] lead to situations where a multiple insertionof non-planar subgraphs gives rise to divergences of arbitrarily high degree (ultraviolet/infrared mixing [MVS00]). See [CR00] for a thorough investigation ofthis problem. Relativistic quantum field theories on noncommutative Minkowski
space are much more diﬃcult [BDFP02]. Here the $UV/IR$-mixing problem occurs
in diﬀerent types of graphs [BahlO].
The Moyal algebra
has matrix basis [GV88, VG88, GGISV03]$(\mathcal{S}(\mathbb{R}^{4}), \star)$$\phi(x)=\sum_{\underline{m},\underline{n}\in \mathbb{N}^{2}}\Phi_{\underline{m}\underline{n}}f_{\underline{m}\underline{n}}(x) , f_{\underline{m}\underline{n}}(x)=f_{m_{1}n_{1}}(x^{0}, x^{1})f_{m2n_{2}}(x^{3},x^{4})$(24)$f_{mn}(y^{0}, y^{1})=2(-1)^{m} \sqrt{\frac{m!}{n!}}(\sqrt{\frac{2}{\theta}}y)^{n-m}L_{m}^{n-m}(\frac{2|y|^{2}}{\theta})e^{-\frac{|y|^{2}}{\theta}}$

where
are Laguerre polynomials andwe assume the only non-vanishing components of. The functions
satisfy$y\equiv y^{0}+iy^{1}$$L_{m}^{n}$$\Theta_{34}=-\Theta_{43}$,. Without loss of generality
to be $:=\Theta_{12}=-\Theta_{21}=$$\Theta$$\theta$$f_{\underline{m}\underline{n}}$$(f_{\underline{k}\underline{l}} \star f_{\underline{mn}})(x)=\delta_{\underline{ml}}f_{\underline{k}\underline{n}}(x) , \int_{\mathbb{R}^{4}}dxf_{\underline{mn}}(x)=(2\pi\theta)^{2}\delta_{\underline{mn}}.$Therefore, thea function and-interaction in (23) becomes a matrix product (we write
for a matrix):$\phi_{4}^{\star 4}$$\Phi$$\phi$(25)$S[ \phi]=(2\pi\theta)^{2}\sum(\frac{1}{2}\Phi_{\underline{k}\underline{l}}(\triangle_{\underline{kl};\underline{mn}}+\mu^{2}\delta_{\underline{k}\underline{n}}\delta_{\underline{l}\underline{m}})\Phi_{\underline{m}\underline{n}}+\frac{\lambda}{4}\Phi_{\underline{kl}}\Phi_{\underline{l}\underline{m}}\Phi_{\underline{m}\underline{n}}\Phi_{\underline{n}\underline{k}})\underline{k},\underline{l},\underline{m},\underline{n}\in \mathbb{N}^{2}$

The matrix kernelof the Laplacian, viewed as map fromconsists of a local interaction plus nearest neighbour interaction.
$\triangle_{\underline{k}\underline{l};\underline{m}\underline{n}}$$(-\triangle)$$\mathbb{N}^{4}$

for

to

$\mathbb{N}^{4},$80-model in
In $[GW05b]$ we studied the renormalisation group flow of the$[GW05a]$formatrix representation (making use of a power-counting theorem). We noticed that the marginal parts of thematrix models with kernel
have diﬀerent flows. To
local term and of the nearest neighbour term in
relevant/marginal operator in the action funcabsorb these diﬀerent flows ational is necessary. This operator corresponds to a harmonic oscillator potential:$\phi_{4}^{\star 4}$$\triangle_{\underline{k}\underline{l};\underline{m}\underline{n}}$$\Delta_{\underline{kl};\underline{mn}}$$4^{th}$$S[ \phi]=64\pi^{2}\int d^{4}x(\frac{Z}{2}\phi\star(-\triangle+\Omega^{2}(2\Theta^{-1}x)^{2}+\mu_{bare}^{2})\phi+\frac{\lambda Z^{2}}{4}\phi\star\phi\star\phi\star\phi)(x)$. (26)

We proved in $[GW05b]$ that the corresponding Euclidean quantum field theory isrenormalisable to all orders in perturbation theory. This result was reestablishedby various methods, see $[Riv07a]$ for a review.
breaks translation invariance.
Presence of the harmonic oscillator term
Conversely, this term achieves covariance under Langmann-Szabo duality transfollowed
andformation [LS02] which consists in exchanging
transRemarkably,
thisvariables.
original
by Fourier transform back to thewithinvariant, and it exchanges
formation leaves$\Omega\neq 0$$\phi(x)rightarrow\hat{\phi}(p)$$xrightarrow p$$\int dx\phi(-\Delta)\phi$$\int dx\phi\star\phi\star\phi\star\phi$. Presence of the oscillator term gives rise to an interesting spectral noncommutative geometry $[GW13a]$ (see also $[GW12a]$ ) which is conceptuallysimpler than the isospectral deformation [GGISV03] of . Most importantly, theoscillator term cures the Landau ghost problem $[LAK54a, LAK54b, LAK54c]$ ofusual -theory: We have discovered in [GW04] that the one-loop renormalisationgroup flows of and influence each other in such a way that the running coupremains finite at any scale A. Even more, at the self-duality
ling constant$\Omega=1$-coupling vanishes to all orders in perturbpoint
the -function of the$\int dx\phi|2\Theta^{-1}x|^{2}\phi$$\mathbb{R}^{4}$$\phi_{4}^{4}$$\Omega$$\lambda$$\lambda(\Lambda)$$\beta$$\lambda\Phi_{4}^{4}$

ation theory [DGMR07]. This result was obtained by an ingenious combinationof Ward identities and Schwinger-Dyson equations (see [DR07] for an explicit
three-loop calculation). In $[GW12b]$ we have generalised the method of DisertoriGurau-Magnen-Rivasseau [DGMR07] to the whole class of quartic matrix models(reviewed in sec. 2). Vanishing of the -function is often connected with integrability, and together with the absent Landau ghost problem a non-perturbatively
constructed -model on Moyal space came into reach. The first milestone was$\beta$$\phi_{4}^{4}$the derivation of the self-consistency equation (13) and the understanding of its
renormalisation in [GW09]. It took us several years to fully understand thisequation, and it is only recently that we finished the solution/construction of the
Moyal space -model $[GW12b]$ . In the sequel we review this construction.
$\phi_{4}^{4}$

3.2 Renormalisation and integral representationof the Schr\"odinger
At the self-duality point $\Omega=1$ , the matrix kernel
becomes purely local and turns the action (26) inoperator$\triangle_{\underline{kl};\underline{mn}}^{\Omega=1}$$H=-\triangle+\Vert 2\Theta^{-1}x\Vert^{2}$

81
matrix basis (24) into$a$(field-theoretical matrix) quartic model with action$S[ \Phi]=V(\sum_{\underline{m},\underline{n}\in \mathbb{N}_{\mathcal{N}}^{2}}E_{\underline{m}}\Phi_{\underline{m}\underline{n}}\Phi_{\underline{n}\underline{m}}+\frac{Z^{2}\lambda}{4_{\underline{m}}},\sum_{\underline{n},\underline{k},\underline{l}\in \mathbb{N}_{N}^{2}}\Phi_{\underline{mn}}\Phi_{\underline{n}\underline{k}}\Phi_{\underline{kl}}\Phi_{\underline{lm}})$

(27)

,

$E_{\underline{m}}=Z( \frac{|\underline{m}|}{\sqrt{V}}+\frac{\mu_{bare}^{2}}{2}), |\underline{m}|:=m_{1}+m_{2}\leq \mathcal{N}, V=(\frac{\theta}{4})^{2}$

Our general results on quartic matrix models imply that the planar 2-point function
satisfies the self-consistency equation (13),$G_{|\underline{a}\underline{b}|}^{(0)}$(28)$G_{|\underline{a}\underline{b}|}^{(0)}= \frac{1}{E_{\underline{a}}+E_{\underline{b}}}-\frac{Z^{2}\lambda}{E_{\underline{a}}+E_{\underline{b}}}\frac{1}{V}\sum_{\underline{p}\in N_{N}^{2}}(G_{|\underline{ab}|}^{(0)}G_{|\underline{a}\underline{p}|}^{(0)}-\frac{G_{|\underline{p}\underline{b}|}^{(0)}-G_{|\underline{ab}|}^{(0)}}{E_{\underline{p}}-E_{\underline{a}}})$We have introduced a cut-oﬀ
in the matrix size; the index sum diverges for. As usual, the renormalisation strategy consists in adjusting $Z,$insuch a way that the limit
exists. This will be achieved by normalisation
conditions for the 1PI function
defined by, where. We express (28) in terms of$\mathbb{N}_{\mathcal{N}}^{2}$$\mathbb{N}_{\mathcal{N}}^{2}\mapsto \mathbb{N}^{2}$$\mu_{bare}$$\mathbb{N}_{\mathcal{N}}^{2}\mapsto \mathbb{N}^{2}$$G_{|\underline{a}\underline{b}|}^{(0)}=:(H_{\underline{ab}}-\Gamma_{\underline{a}\underline{b}})^{-1}$$\Gamma_{\underline{a}\underline{b}}$$H_{\underline{a}\underline{b}}:=E_{\underline{a}}+E_{\underline{b}}$$\Gamma_{\underline{a}\underline{b}},$$\Gamma_{\underline{a}\underline{b}}=-\frac{\lambda Z^{2}}{V}\sum_{\underline{p}\in \mathbb{N}_{N}^{2}}(\frac{1}{H_{a-\underline{p}}-\Gamma_{\underline{a}\underline{p}}}+\frac{1}{H_{\underline{p}\underline{b}}-\Gamma_{\underline{p}\underline{b}}}-\frac{1}{(H_{\underline{p}\underline{b}}-\Gamma_{\underline{p}\underline{b}})}\frac{z\Gamma}{\sqrt{V}}(|\underline{p}|-|\underline{a}|)\underline{p}\underline{b}^{-\Gamma_{\frac{a}{}\frac{b}{}}})$

and write

as first-order Taylor formula with remainder$\Gamma_{\underline{a}\underline{b}}$$\Gamma_{\underline{a}\underline{b}}=Z\mu_{bare}^{2}-\mu^{2}+\frac{(Z-1)}{\sqrt{V}}(|\underline{a}|+|\underline{b}|)+\Gamma_{\underline{a}\underline{b}}^{ren}$$\Gamma_{\underline{0}\underline{0}}^{ren}=0,$

,

(29)

$\Gamma_{\underline{ab}}^{ren},$$(\partial\Gamma^{ren})_{\underline{00}}=$O.

Equation (29) for
together withandconsti$Z$ .
tute three equations to etermine the three functions
Eliminating$Z$
thus gives rise to a closed equation for renormalised function
alone.
$Z$
For this elimination it is important to note that the equations for
depend on
only via the norms
which parametrise the spectrum of $E.$
Therefore,
is actually a function only of, and consequently the index
sum reduces to.
We study a particular scaling limit in which matrix sizeand volume
are simultaneously sent to such that the ratio
is kept fixed.
$\Gamma_{\underline{a}\underline{b}}[\Gamma_{ab}^{ren}, \mu_{bare}^{2}, Z]$$\Gamma_{\underline{0}\underline{0}}^{ren}=0$$(\partial\Gamma^{ren})_{\underline{0}\underline{0}}$$\overline{\overline{d}}$$\Gamma_{\underline{ab}}^{ren},$$\mu_{bare}^{2},$$\mu_{bare}^{2},$$\Gamma_{\underline{ab}}^{ren}$$\Gamma_{\underline{ab}}^{ren},$$\underline{b}$$|\underline{a}|,$$\underline{a},$$|\underline{b}|$$|\underline{a}|,$$\Gamma_{\underline{a}\underline{b}}$$\mu_{bare}^{2},$$|\underline{b}|$$\sum_{\underline{p}\in \mathbb{N}_{N}^{2}}f(|\underline{p}|)=\sum_{|p|=0}^{\mathcal{N}}(|\underline{p}|+1)f(|\underline{p}|)$$\mathcal{N}$$V$$\infty$$\frac{\mathcal{N}}{\sqrt{V\mu^{4}}}=\Lambda^{2}(1+\mathcal{Y})$Note that $V=$$( \frac{\theta}{4})^{2}arrow\infty$

is a limit of extreme noncommutativity! The newparameter
corresponds to a finite wavefunction renormalisation, identified
later to decouple our equations. The parameter
represents an ultraviolet cutoﬀ which is sent toin the very end (continuum limit). In the scaling
limit, functions ofconverge to functions of “continuous matrix$(1+\mathcal{Y})$$\Lambda^{2}$$\Lambdaarrow\infty$$\frac{|\underline{p}|}{\sqrt{V}}=:\mu^{2}(1+\mathcal{Y})p$82
indices”’

converges to a function
In the same way.
, and the discrete sum converges to a Riemann integral$p\in[0, \Lambda^{2}]$$a,$ $b\in[0, \Lambda^{2}]$.

$\mu^{2}\Gamma_{ab}$$\Gamma_{\underline{ab}}^{ren}$with$\frac{1}{V}\sum_{|\underline{p}|=0}^{\mathcal{N}}(|\underline{p}|+1)f(\frac{|\underline{p}|}{\sqrt{V}})arrow\mu^{4}(1+\mathcal{Y})^{2}\int_{0}^{\Lambda^{2}}pdpf(\mu^{2}(1+\mathcal{Y})p)$

This limit makes the restriction to the planar sector (13) of (12) exact.
, but before elimination of , our equation for
After elimination of
becomes$Z$$\mu_{bare}^{2}$$\Gamma_{ab}$$(Z-1)(1+\mathcal{Y})(a+b)+\Gamma_{ab}$$=- \lambda(1+\mathcal{Y})^{2}\int_{0}^{\Lambda^{2}}pdp(\frac{Z^{2}}{(a+p)(1+\mathcal{Y})+1-\Gamma_{ap}}-\frac{Z^{2}}{p(1+\mathcal{Y})+1-\Gamma_{0p}})$$- \lambda(1+\mathcal{Y})^{2}\int_{0}^{\Lambda^{2}}pdp(\frac{Z}{(b+p)(1+\mathcal{Y})+1-\Gamma_{pb}}-\frac{Z}{p(1+\mathcal{Y})+1-\Gamma_{p0}}$$Z \Gamma_{pb}-\Gamma_{ab}$$-\overline{(b+p)(1+\mathcal{Y})+1-\Gamma_{pb}}(1+\mathcal{Y})(p-a)$(30)$+ \frac{Z}{p(1+\mathcal{Y})+1-\Gamma_{p0}}\frac{\Gamma_{p0}}{p(1+\mathcal{Y})})$(and its derivative). Inserted back one
Applying
we get in terms ofgets a highly non-linear integro-diﬀerential equation. Fortunately we can reduce
the non-linearity by subtracting from (30) the same equation taken at $b=$ O.
. In terms ofThis subtraction eliminates the second line of (30) containing$G_{ab}:=((a+b)(1+\mathcal{Y})+1-\Gamma_{ab})^{-1}$ , this diﬀerence equation reads$Z$$\frac{d}{db}|_{a=b=0}$$\Gamma_{ab}$$Z^{2}$$\frac{Z^{-1}}{(1+\mathcal{Y})}(\frac{1}{G_{ab}}-\frac{1}{G_{a0}})=b-\lambda\int_{0}^{\Lambda^{2}}pdp\frac{\overline{c}_{ab^{--}}^{L^{b}}c_{a}cc_{z_{\frac{0}{0}}}}{p-a}$of (31) yields in terms ofDiﬀerentiation$G’$
can be avoided by adjusting
resulting derivative$Z$$\frac{d}{db}|_{a=b=0}$$G_{ab}$(31)

and its derivative. The$\mathcal{Y}:=-\lambda\lim_{barrow 0}\int_{0}^{\Lambda^{2}}dp\frac{G_{pb}-G_{p0}}{b}$This choice leads to$\frac{Z^{-1}}{(1+\mathcal{Y})}=1-\lambda\int_{0}^{\Lambda^{2}}dpG_{p0}$, which is a perturbatively di-. Inserting
and back into (31) we end up invergent integral for$D_{ab}:= \frac{a}{b}(G_{ab}-G_{a0})$ to thefunction
equationfordiﬀerencethea linear integral
boundary:$\Lambdaarrow\infty$$Z^{-1}$$\mathcal{Y}$$( \frac{b}{a}+\frac{1}{aG_{a0}})D_{ab}+G_{a0}=\lambda\int_{0}^{\Lambda^{2}}dp(\frac{D_{pb}-D_{ab_{G_{a}}^{-}}^{G_{\Delta_{\frac{0}{0}}}}}{p-a})$

(32)

83
The non-linearity restricts to the boundary function
where the second index
is put to zero. Assuming
H\"older-continuous, we can pass to Cauchy
principal values. In terms of the finite Hilbert transform$G_{a0}$$a\mapsto G_{ab}$

(33)

,

$\mathcal{H}_{a}^{\Lambda}[f(\bullet)]:=\frac{1}{\pi}\lim_{\epsilonarrow 0}(\int_{0^{+}}^{a-\epsilon}\int_{a+\epsilon}^{\Lambda^{2}})\frac{f(q)dq}{q-a}$

the integral equation (32) becomes$( \frac{b}{a}+\frac{1+\lambda\pi a\mathcal{H}_{a}^{\Lambda}[G_{0}]}{aG_{a0}})D_{ab}-\lambda\pi \mathcal{H}_{a}^{\Lambda}[D_{b}]=-G_{a0}$

(34)

.

3.3 The Carleman solution
Equation (34) is a well-known singular integral equation of Carleman type [Car22,$Tki57]$ :Theorem 6 $([Ri57],$ transformed from $[-1, 1] to [0, \Lambda^{2}])$ The singular linear integral equation$h(a)y(a)-\lambda\pi \mathcal{H}_{a}^{\Lambda}[y]=f(a)$is for $h(a)$ continuous on ]some $p>1$ (determined by$0,$$\Lambda^{2}$$\theta(O)$,$a\in]O,$[, H\"older-continuous near) solved byand$\Lambda^{2}$

$0,$

[,
$\Lambda^{2}$, and $f\in If$ for$\theta(\Lambda^{2})$$y(a)= \frac{\sin(\theta(a))e^{-\mathcal{H}_{a}^{\Lambda}[\pi-\theta]}}{\lambda\pi a}(af(a)e^{\mathcal{H}_{a}^{\Lambda}[\pi-\theta]}\cos(\theta(a))$(35a)$+\mathcal{H}_{a}^{\Lambda}[e^{\mathcal{H}^{\Lambda}[\pi-\theta]} \bullet f \sin(\theta(\bullet))]+C)$$=* \frac{\sin(\theta(a))e^{\mathcal{H}_{a}^{\Lambda}[\theta]}}{\lambda\pi}(f(a)e^{-\mathcal{H}_{a}^{\Lambda}[\theta]}\cos(\theta(a))$$+ \mathcal{H}_{a}^{\Lambda}[e^{-\mathcal{H}^{\Lambda}[\theta]}f(\bullet)\sin(\theta(\bullet))]+\frac{C’}{\Lambda^{2}-a})$where$\theta(a)=arc,\tan[0\pi](\frac{\lambda\pi}{h(a)})$,$\sin(\theta(a))=\frac{|\lambda\pi|}{\sqrt{(h(a))^{2}+(\lambda\pi)^{2}}}\geq 0$

,

(35b)

and $C,$ $C’$ are arbit-

rary constants.
The possibility of $C,$ $C’\neq 0$ is due to the fact that the finite Hilbert transformhas a kernel, in contrast to the infinite Hilbert transform with integration over
. The two formulae (35a) and (35b) are formally equivalent, but the solutions
belong to diﬀerent function classes and normalisation conditions may (and will)
make a choice.
In principle, (35) provides the solutionof (34), where the angle function$\mathbb{R}$$G_{ab}$$\theta_{b}(a):=arc,\tan[0\pi](\frac{\lambda\pi a}{b+\frac{1+\lambda\pi a\mathcal{H}_{a}^{\Lambda}[G_{0}]}{G_{a0}}})$

(36)

84
plays a key r\^ole. This solution involves multiple Hilbert transforms which are
diﬃcult to control. A better strategy starts from the observation that the angle(36) satisfies, for $b=0$ , again a Carleman type singular integral equation$\lambda\pi\cot\theta_{0}(a)G_{a0}-\lambda\pi \mathcal{H}^{\Lambda}[G_{0}]=\frac{1}{a}$with solution$G_{a0}= \frac{e^{-\mathcal{H}_{a}^{\Lambda}[\pi-\theta_{0}]}\sin(\theta_{0}(a))}{\lambda\pi a}(e^{\mathcal{H}_{a}^{\Lambda}[\pi-\theta_{0}]}\cos(\theta_{0}(a))$(37a)$+\mathcal{H}_{a}^{\Lambda}[e^{\mathcal{H}^{\Lambda}[\pi-\theta_{0}]}\sin(\theta_{0} +C)$$=* \frac{e^{\mathcal{H}_{a}^{\Lambda}[\theta_{0}]}\sin(\theta_{0}(a))}{\lambda\pi}(\frac{e^{-\mathcal{H}_{a}^{\Lambda}[\theta_{0}]}\cos(\theta_{0}(a))}{a}$(37b)$+ \mathcal{H}_{a}^{\Lambda}[^{\underline{e^{-\mathcal{H}^{\Lambda}[\theta_{0}]}\sin(\theta_{0}(\bullet))}}\bullet]+\frac{C’}{\Lambda^{2}-a})$

Tricomi’s identities [Tri57, \S 4.4(28 18)], which can be arranged as$+$$e^{\pm \mathcal{H}_{a}^{\Lambda}[\theta_{b}]}\cos(\theta_{b}(a))\mp \mathcal{H}_{a}^{\Lambda}[e^{\pm \mathcal{H}^{\Lambda}[\theta_{b}]}\sin(\theta_{b}(\bullet))]=1,$

and rational fraction expansion(37) to$\mathcal{H}_{a}^{\Lambda}$$[ \underline{j(}.\cdot)]=\frac{1}{a}(\mathcal{H}_{a}^{\Lambda}[f$$-\mathcal{H}_{0}^{\Lambda}[f(\bullet)]$) simplify(38a)$G_{a0}= \frac{e^{-\mathcal{H}_{a}^{\Lambda}[\pi-\theta_{0}]}\sin(\theta_{0}(a))}{\lambda\pi a}(C-1)$(38b)$=* \frac{e^{\mathcal{H}_{a}^{\Lambda}[\theta_{0}]}\sin(\theta_{0}(a))}{\lambda\pi a}(e^{-\mathcal{H}_{0}^{\Lambda}[\theta_{0}]}\cos(\theta_{0}(0))+\frac{C’a}{\Lambda^{2}-a})$Both lines are formally equivalent, but we have to guarantee the normalisation$\lim_{aarrow 0}G_{a0}=1$

Consequently,. From (36) one concludes$\lim_{parrow 0}\theta_{0}(p)=\{\begin{array}{llll}0 for \lambda \geq 0\pi for \lambda <0\end{array}\}.$$e^{-\mathcal{H}_{0}^{\Lambda}[\theta_{0}]}= \exp(-\int_{0^{-}p}^{\Lambda_{d_{R}}^{2}}\theta_{0}(p))arrow 0\lambda<0$, which means that (38b)reduces forto (38a), with $C’\mapsto C-1$ . Similarly,
O. The normalisation $\lim_{aarrow 0}G_{a0}=1$
so that (38a) is only consistent withhereas (38b) stays
leads w
results c $anbes$ ummarised a$sfo1lows$ :$\lim_{aarrow 0}e^{-\mathcal{H}_{a}^{\Lambda}[\pi-\theta_{0}]}\lambda>0=0,$$\lambda<0$$\lambda<$$ith\lim_{aarrow 0}\frac{\sin\theta_{0}(a)}{T^{1_{hese}^{\lambda|\pi a}}}=ltol-C=e^{-\mathcal{H}_{0}^{\Lambda}[\pi-\theta_{0}]}in(38a),wasitisfor\lambda>0.$

Lemma 7 The angle function$\tau_{b}(a)$$:= arc,\tan[0\pi](\frac{|\lambda|\pi a}{b+\frac{1+\lambda\pi a\mathcal{H}_{a}^{\Lambda}[G_{0}]}{G_{a0}}})$

is for $b=0$

reverted to$G_{a0}= \frac{\sin(\tau_{0}(a))}{|\lambda|\pi a}e^{sign(\lambda)(\mathcal{H}_{0}^{\Lambda}[\tau_{0}(\cdot)]-\mathcal{H}_{a}^{\Lambda}[\tau_{0(\bullet)])}}\{\begin{array}{ll}1 for\lambda<0,(1+\frac{Ca}{\Lambda^{2}-a}) for \lambda>0,\end{array}$

where

$C$

is an arbitrary constant.

(39)

85
Recall that
forms the inhomogeneity in the Carleman equation (34). We
insert (39) into the Carleman solution (35) for (34) and obtain with the addition
theorem $|\lambda|\pi a\sin(\tau_{d}(a)-\tau_{b}(a))=(b-d)\sin\tau_{b}(a)\sin\tau_{d}(a)$ after essentially thesame steps as in the proof of (39):$G_{a0}$

Theorem 8 ([GW14]) The full matrix 2-point functionof self-dual
theory on Moyal space is in the limit
given in terms of the boundary
2-point function
by the equation$G_{ab}$$\phi_{4^{-}}^{4}$$\thetaarrow\infty$$G_{a0}$$G_{ab}= \frac{s\dot{m}a))}{a}e^{sign(\lambda)(\mathcal{H}_{0}^{\Lambda}[\tau_{0}(\cdot)]-\mathcal{H}_{a}^{\Lambda}[\tau_{b}(\cdot)])}\{\begin{array}{ll}1 for\lambda<0,(1+\frac{Ca+bF(b)}{\Lambda^{2}-a}) for \lambda>0,\end{array}$

(40)

where
is a undetermined constant and $bF(b)$ an undetermined function ofvanishing at $b=0.$
$C$

$b$

Some remarks:
We have proved this theorem in 2012 forunder the assumption $C’=0$(35b),
inbut knew that non-trivial solutions of the homogeneous Carlemanequation parametrised by $C’\neq 0$ are possible. That no such term arises for(if angles are redefined) is a recent result [GW14].
An important observation is, at least for. This is a truly nonperturbative results because individual Feynman graphs show no positivity
at all!
As in [GW09], the equation for
can be solved perturbatively. Matchingrequires $C,$ $F$ to be flat functions of . Because ofat, the naive $\arctan$ series is dangerous for
O. Unless there arecancellations, we expect zero radius of convergence!
From (40) we deduce the finite wavefunction renormalisation$\lambda>0$$\bullet$$\lambda<0$$\theta\mapsto\tau$$\lambda<0$$G_{ab}\geq 0$$\bullet$$G_{ab}$$\bullet$$\lambda=0$$\mathcal{H}_{a}^{\Lambda}[G_{0}]\vec{arrow}$$\lambda$$\lambda>$$-\infty$$\bullet$$\mathcal{Y}:=-1-\frac{dG_{ab}}{db}|_{a=b=0}=\int_{0}^{\Lambda^{2}}\frac{dp}{(\lambda\pi p)^{2}+(\frac{1+\lambda\pi p\mathcal{H}_{p}^{\Lambda}[G.0]}{G_{p0}})^{2}}-\{\begin{array}{l}0 for \lambda<0,F(O) for \lambda>0.\end{array}$(41)

The partition function is undefined for. But the Schwinger-Dyson
equations for, extend
and for higher functions, and with themtoO. These extensions are unique but probably not analytic in aneighbourhood of
It remains to identify the boundary function. The Carleman equation$-(30)_{b=0}$(34) for. Consequently, (30)
was obtained from the diﬀerence (30)
gives the second relation between
andfrom which both are determined.
, which in terms
Combining them we obtain a single consistency equation forofreads$\lambda<0$$\mathcal{Z}$$\bullet$$G_{ab}$$\log \mathcal{Z}$$\lambda<$$\lambda=0.$$G_{a0}$$G_{ab}$$G_{ab}$$G_{a0}$$G_{a0}$$\mathcal{T}_{a}:=|\lambda|\pi a\cot\tau_{0}(a)$$\mathcal{T}_{a}=1+a+\lambda\pi a\mathcal{H}_{a}^{\Lambda}[1]$

86

(42)

$+ \int_{0}^{\Lambda^{2}}dp(\frac{p\exp(\mathcal{H}_{a}^{\Lambda}[ar[c\tan\frac{|\lambda|\pi}{p+\mathcal{T}}])}{\sqrt{(\lambda\pi a)^{2}+(p+\mathcal{T}_{a})^{2}}}-\frac{p\exp(\mathcal{H}_{0}^{\Lambda}[ar[c,\tan\frac{|\lambda|\pi}{p+\mathcal{T}}])}{1+p})$This equation is, unfortunately, of little use. The integrals are individually divergent forso that we have to rely on cancellations on which we have nocontrol.
We compensate this lack by a symmetry argument. Given the boundary, the Carleman theory computes the full 2-point function
function
via(40). In particular, we get. But the 2-point function is
as function ofsymmetric, $G_{ab}=G_{ba}$ , and the special case $b=0$ leads to the following selfconsistency equation:$\Lambdaarrow\infty$$G_{a0}$$G_{ab}$$G_{a0}$$G_{0b}$

Proposition 9 The limitof -theory on Moyal space is determined by
the solution of the fixed point equation $G=TG,$$\thetaarrow\infty$$\phi_{4}^{4}$$G_{b0}= \frac{\{\begin{array}{ll}f1or \lambda<01+bF(b)for \lambda>0\end{array}\}}{1+b}\exp(-\lambda\int o^{b}dt\int_{0}^{\Lambda}\frac{2dp}{(\lambda\pi p)^{2}+(t+\frac{1+\lambda\pi p\mathcal{H}_{p}^{\Lambda}[G.0]}{G_{p0}})^{2}})$(43)

At this point we can eventually send
. Any solution of (43) is automatically smooth and for
but $F=0)$ monotonously decreasing. Any solutionof the true equation (30) (without the diﬀerence to $b=0$ ) also solves the masterequation (43), but not necessarily conversely. In case of a unique solution of (43),
it is enough to check one candidate.
Existence of a solution of (43) is established for
but $F(b)=0)$ by the
Schauder fixed point theorem. We consider the following subset of continuously
diﬀerentiable functions onvanishing at :$\Lambdaarrow\infty$$\lambda>0$

$($

$($

$\infty$$\mathbb{R}_{+}$$\mathcal{K}_{\lambda}:=\{$$\lambda>0$$f\in C_{0}^{1}(\mathbb{R}_{+}):f(0)=1,$$0<f(b) \leq\frac{1}{1+b},$$0 \leq-f’(b)\leq(\frac{1}{1+b}+C_{\lambda})f(b)\},$

where
1.
2.

$C_{\lambda}$$\mathcal{K}_{\lambda}$

is defined via
convex,$2\lambda P_{\lambda}^{2}(1+C_{\lambda})e^{C_{\lambda}P_{\lambda}}=1$

at$P_{\lambda}= \frac{\exp(-=_{\lambda\pi}^{1})}{\sqrt{1+4\lambda}}$. Then $[GW12b]$ :$\overline{T\mathcal{K}_{\lambda}}\subset \mathcal{K}_{\lambda},$3. $(Tf)”(b)\leq$for any
4.
is continuous.
The properties 1.-3. imply thatis relatively compact inby a variant ofArzel\’a-Ascoli
thetheorem. Together with 4. the Schauder fixed point theorem
then guarantees that (43) has a solution
This solution provides
via (40) and all higher correlation functions via
the universal algebraic recursion formulae (16), (65), (66), etc, or via the linear$( \frac{23}{4}+\frac{2}{\pi}+\frac{7+8\pi}{2}\frac{1}{(\lambda\pi^{2}P_{\lambda})^{2}})(Tf)(b)$$f\in \mathcal{K}_{\lambda},$$T:\mathcal{K}_{\lambda}arrow \mathcal{K}_{\lambda}$$T\mathcal{K}_{\lambda}$$\mathcal{K}_{\lambda}$$G_{a0}\in \mathcal{K}_{\lambda}.$$G_{ab}$87
equations for the basic $(N_{1}+\ldots+N_{B})$ -point functions such as (63) and (64). The
recursion formula (16) becomes after transition to continuous matrix indices$G_{b_{0}\ldots b_{N-1}}= \frac{(-\lambda)}{(1+\mathcal{Y})^{2}}\frac{G_{b_{0}b_{1}\ldots b_{2/-1}}G_{b_{2l}b_{2l+1}\ldots b_{N-1}}-G_{b_{2l}b_{1}\ldots b_{2l-1}}G_{b_{0}b_{2/+1}\ldots b_{N-1}}}{(b_{0}-b_{2l})(b_{1}-b_{N-1})}\frac{N-2}{\sum 2}l=1$(44)

It involves the finite wavefunction renormalisation
given by
(41). Of particular interest is the eﬀective coupling constant $\lambda_{eff}=-G_{0000}$ . This
limit of coinciding indices is not so easy; therefore we directly solve the integral
equation for
before using the reality condition. We find $[GW12b]$$1+ \mathcal{Y}=-\frac{dG}{d}b\ovalbox{\tt\small REJECT}|_{a=b=0}$$G_{a000}$$\lambda_{eff}=\lambda\{1+\frac{\lambda}{(1+\mathcal{Y})}\int_{0}^{\infty}dp\frac{(\frac{1-G_{p0}}{(1+\mathcal{Y})p}-G_{p0})G_{p0}}{(\lambda\pi pG_{p0})^{2}+(1+\lambda\pi p\mathcal{H}_{p}^{\infty}[G_{0}])^{2}}\}$(45)

arising from (64) is solved in two
The equation for the basic function
steps. A first summation over $b\in I$ in (64) yields after passage to the integral
representation a Carleman equation$G_{ab|cd}$$X_{a|cd} \{1+\lambda\int_{0}dq\infty(G_{aq}-G_{0q})-\lambda\int_{0}^{\infty}dq\frac{G_{aq}\sin\tau_{q}(a)\cos(\tau_{q}(a)-\tau_{0}(a))}{s\dot{m}\tau_{0}(a)}\}$$+7r_{a}[ \frac{X_{|cd}}{\pi}\bullet\int_{0^{qdq}}^{\infty}\sin^{2}\tau_{q}(\bullet)G_{aq}]$$= \lambda\int_{0}^{\infty}qdq(F_{aq|cdcq}+F_{aq|dcdq})+\frac{\lambda}{(1+\mathcal{Y})^{2}}(G_{acdc}+G_{adcd})$

,

where
. Inserted back into (64) gives(after passage to the integral representation) a familiar Carleman equation forwith solution$F_{ab_{1}|c_{1}c_{2}c_{3}c_{4}}:= \frac{G_{abcccc}G_{bc}-G_{bccc}G_{abcc}}{G_{b_{1}c_{1}}G_{b_{1}c_{3}}}$$G_{ab|cd}$$G_{ab|cd}=F_{ab|cdcb}+F_{ab|dcdb}$$- \frac{\sin\tau_{b}(a)}{\lambda\pi a}\cos\tau_{b}(a)G_{ab}X_{a|cd}-G_{ab}\wp_{a}[\frac{\sin^{2}\tau_{b}(\bullet)}{\lambda\pi\bullet}X_{|cd}]$(46)

The $(2+2)$ -point function
turns out to be the most interesting part of the
4-point function in position space (see sec. 4).
$G_{ab|cd}$

3.4

Perturbation theory

The master equation (43) can, for $F(b)\equiv 0$ , be iteratively solved. To lowest, from which the next order becomesorder one has$G_{a0}= \frac{1}{1+a}+\mathcal{O}(\lambda)$$G_{a0}= \frac{1}{1+a}-\lambda\frac{\log(1+a)}{(1+a)}+\mathcal{O}(\lambda^{2})$

.

(47)

88
If we put in$G_{a0}= \frac{1}{(1+a)^{1+\lambda}}+\mathcal{O}(\lambda^{2})$the index$a\mapsto\not\simeq_{2}\mu$, see (58), we get$\int_{\mathbb{R}}\frac{dp}{(2\pi\mu)^{4}}e^{ip(x-y)}G_{*^{2}0}=\frac{2^{-\lambda}}{4\pi^{2}\Gamma(1+\lambda)}\frac{K_{1-\lambda}(\mu\Vert x-y\Vert)}{(\mu\Vert x-y\Vert)^{1-\lambda}}$$x \vec{arrow}\frac{2^{-2\lambda}\Gamma(1-\lambda)}{4\pi^{2}\Gamma(1+\lambda)}\frac{1}{(\mu\Vert x-y\Vert)^{2-2\lambda}}-y0$

We thus conclude that the anomalous dimension is $\eta=-2\lambda$ , i.e. negative for the
stable signof the coupling constant. We shall see in the next section that
this result excludes a Wightman theory for
O. It is worthwhile to mention
that this wrong sign is a consequence of renormalisation. The divergent bare2-point function would lead to the opposite sign. Removing the divergence at$a=0$ overcompensates for $a>0$ and gives $\eta=-2\lambda$ . In two dimensions, would
be non-negative for
From (47) we get:,
Hilbert transform:$\lambda>0$$\lambda>$$\eta$$\lambda>0.$$\lambda\pi W_{a}[G_{0}]=-\lambda\frac{\log(a)}{l+a}+\mathcal{O}(\lambda^{2})$$\bullet$angle function:$\bullet$$\tau_{b}(a)=\frac{|\lambda|\pi a}{1+a+b}(1-\lambda\frac{(1+a)\log(1+a)-a\log a}{(1+a+b)})+\mathcal{O}(\lambda^{3})$wavefunction renormalisation:
Inserted into (40) one finds$\bullet$,$1+\mathcal{Y}=-\underline{d}G_{A}da|_{a=0}=1+\lambda+\mathcal{O}(\lambda^{2})$$G_{ab}= \frac{1}{1+a+b}-\lambda\frac{(1+a)\log(1+a)+(1+b)\log(1+b)}{(1+a+b)^{2}}+\mathcal{O}(\lambda^{2})$

.

.

(48)

This result coincides with renormalised 1-loop ribbon graph computation.
From the action functional (27) one obtains in the infinite volume limit to continuous matrix indices the following Feynman rules:$= \frac{1}{1+(a+b)(1+\mathcal{Y})}$$\bullet$$\frac{a}{\overline{b}:},$.
$=-Z^{2}\lambda$$\bullet$(index conserved at every corner)$J_{A) ’ 0_{p}’}$$\bullet$$=(1+ \mathcal{Y})^{2}\int_{0}^{\Lambda^{2}}pdp$

for every closed face

To lowest order we have $G_{ab}= \frac{1}{1+(a+b)(1+\mathcal{Y})-\Gamma_{ab}^{ren}}$ , where
Taylor remainder of$\Gamma_{ab}^{rn}$is the89(49a)$(Z-1)a$$\Gamma_{ab}^{ren}=(-\lambda)\int_{0}^{\Lambda^{2}}pdp(\frac{1}{1+a+p}-\frac{1}{1+p}+\frac{a}{(1+p)^{2}})+(a\mapsto b)+\mathcal{O}(\lambda^{2})$

,

(49b)

in agreement with (48). This shows that the fixed point solution forand theCarleman solution forprovide the resummation of infinitely many renormalised Feynman graphs!
From (44) and
we obtain for the 4-point function$G_{a0}$$G_{ab}$$\mathcal{Y}=\lambda+\mathcal{O}(\lambda^{2})$$G_{abcd}= \frac{(-\lambda)}{(1+\mathcal{Y})^{2}}\frac{G_{ab}G_{cd}-G_{ad}G_{cd}}{(a-c)(b-d)}=:G_{ab}G_{bc}G_{cd}G_{da}(-\Gamma_{abcd})$,$\Gamma_{abcd}=\lambda(1-\lambda\frac{a-(1+a)\log(1+a)-c+(1+c)\log(1+c)}{a-c}$$- \lambda\frac{b-(1+b)\log(1+b)-d+(1+d)\log(1+d)}{b-d})+\mathcal{O}(\lambda^{3})$

,

(50)

which agrees with$-d(0_{b}^{b^{t}}c\backslash d^{\mathcal{C}}p)’+\mathcal{O}(\lambda^{3})$$j a_{\star}.$$-(- \lambda)^{2}\int_{0}^{\Lambda^{2}}\frac{pdp}{(1+p+a)(1+p+c)}-(-\lambda)^{2}\int_{0}^{\Lambda^{2}}\frac{pdp}{(1+p+b)(1+p+d)}$

The singularities of$Z^{2}$(51)

and of the 4-point graphs cancel exactly!

3.5 Computer simulations [GW14]

A numerical investigation of 43), for $F(b)\equiv 0$ , can reveal interesting propertiesof the -theory on Moyal space. Our strategy is to approximate
as piecewise
linear function on
sampled according to a geometric progression. We view
$G_{a0}^{n+1}=(TG^{n})_{a0}$(43) as iteration
for some initial function . In this way we findnumerically that satisfies, for any, the assumptions of the Banach fixed
point theorem for Lipschitz functions on, i.e.
is contractive andconverges to a fixed point which we view as approximation for. Whereas$G_{a0}$$\phi_{4}^{4}$$[0, \Lambda^{2}]$$G^{0}$$T$$\lambda\in \mathbb{R}$$[0, \Lambda^{2}]$$T$$(G^{n})$$G_{a0}$90converges for any sign of), the necessary(without discontinuity at$G_{ab}=G_{ba}$consistency condition
for (40) turns out to be maximally violated for$\lambda>0$ (assuming $C=0=F(b)$ ) and satisfied (within numerical error bounds). The observed relative asymmetryforof nearly 100% for$(G^{n})$$\lambda=0$$\lambda$$\lambda\leq 0$$\sup_{a,b}|\begin{array}{l}RG-GG_{ab}+Gba\end{array}|$

signals that the parameters $C,$ $F(b)$ in (40) which reflect the non-trivial
solution of the homogeneous Carleman equation are definitely non-zero. Taking$C,$ $F(b)\neq 0$ for $\lambda>0$ into account is not feasible at the moment so that our. Fornumerical results are reliable only forand only 2000 sample
points in, the relative asymmetry foris of the order of 5%.
computer
The most striking outcome of oursimulations concerns the finite
given by (41). Figure 1 shows both andwavefunction renormalisation
the eﬀective coupling constantgiven by (45) as functions of . We find clear$\lambda>0$$\lambda=10^{7}$$\lambda\leq 0$$\lambda\leq 0$$[0, \Lambda^{2}]$$(1+\mathcal{Y})$$\mathcal{Y}$$\lambda$$\lambda_{eff}$

Figure 1:$\mathcal{Y},$$\lambda_{eff}$based on$G_{a0}$for$\Lambda^{2}=10^{7}$

with 2000 sample points.

evidence for a second-order phase transition:is discontinuous at $\lambda_{c}=-0.396,$
and we have in reasonable approximation a critical behaviour$\mathcal{Y}’$$1+\mathcal{Y}=\{\begin{array}{ll}A(\lambda-\lambda_{c})^{\alpha} for \lambda\geq\lambda_{c},0 for \lambda<\lambda_{c},\end{array}$

(52)

91
for some $A,$ $\alpha>0$ . To be precise, we find $1+\mathcal{Y}=0$ only at $\lambda_{0}=-0.455$ , but thisseems to be due to the discretisation. Of course, there cannot be a discontinuity
infor finite , but Figure 1 is strong support for a critical behaviour (52) inthe limit. It is worthwhile to mention that nothing particular happensat the expected pole $\lambda_{b}=-\frac{1}{72}=0.014$ of Borel resummation! Since $1+\mathcal{Y}=0$(within numerical error bounds) in the phase, we see from (44) that higher$N$ -point functions
will not exist for. Most surprisingly, as we discuss at
the end of section 4.2, a key property of the Schwinger 2-point function $S_{c}(x, y)$in position space is precisely realised in, not outside. To be more precise,$0\leqa,$$G_{ab}=0$
Figure 2 suggests, wherefor
increases with$\mathcal{Y}’$$\Lambda$$\Lambda^{2}arrow\infty$$\lambda<\lambda_{c}$$\lambda<\lambda_{c}$$[\lambda_{c}, 0]$$b\leq\Lambda_{0}^{2}$$\lambda_{c}-\lambda>0.$$\Lambda_{0}^{2}$

Figure 2: Plots of $\log G_{a0}$ and $\log G_{aa}$ over $\log(1+a)$ for$\lambda<\lambda_{c}.$This could leave the possibility of meaningful higher functions (44) for matrix
indices, but not for larger indices. Such a picture could have theinterpretation of a maximal momentum cut-oﬀ of the Euclidean particles.
$0\leq a_{i}\leq\Lambda_{0}^{2}$

4

Schwinger functions and reflection positivity

In the previous section we have constructed the connected matrix correlation
functionsof the-limit of -theory on Moyal space.
These functions arise from the topological expansion (6) of the free energy$G_{|\underline{q}_{1}^{1}\ldots\underline{q}_{N_{1}}^{1}|\ldots|\underline{q}_{1}^{B}\ldots\underline{q}_{N_{B}}^{B}|}$$(\thetaarrow\infty)$$\phi_{4}^{4}$$\log\frac{\mathcal{Z}[J]}{\mathcal{Z}[0]}=\sum_{B=11\leq N_{1}}^{\infty}\sum_{\leq\cdots\leq N}^{\infty}\frac{(V\mu^{4})^{2-B}}{BS_{N_{1}\ldots N_{B}}}\sum_{\underline{q}_{i}^{\beta}\in \mathbb{N}^{2}}G_{|\underline{q}_{1}^{1}\ldots\underline{q}_{N_{1}}^{1}|\ldots|\underline{q}_{1}^{B}\ldots\underline{q}_{N_{B}}^{B}|}\prod_{\beta=1}^{B}\frac{1}{N_{\beta}}(\frac{J_{\underline{q}_{1}^{\beta}\underline{q}_{2}^{\beta}}}{\mu^{3}}\cdots\frac{J_{\underline{q}_{N_{\beta}}^{\beta}\underline{q}_{1}^{\beta}}}{\mu^{3}})$

.

(53)

Since

is finite, the limitof thenaturally expected free energy density removes (in addition to the removal ofhigher-genus contributions) all contributions from $B\geq 2$ . As shown in previous
sections, this planar limit is an exactly solvable and (without any doubt) nontrivial matrix model.
$\lim_{V\mu^{4}arrow\infty}G_{|\underline{q}_{1}^{1}\ldots\underline{q}_{N_{1}}^{1}|\ldots|\underline{q}_{1}^{B}\ldots\underline{q}_{N_{B}}^{B}|}$$\lim_{Varrow\infty}\frac{1}{V\mu^{4}}\log_{\mathcal{Z}[0]}^{ZJ}\perp$

92
4. 1

Schwinger functions

We are interested here in another limit to Schwinger functions [Sch59] in position
space. For this end we revert the matrix representation (24) and take the infinite, where we carefully have to pass to densities. Absolutevolume limit
can be used. This means that we
position
have no meaning, only
consider$V\mu^{4}arrow\infty$$x\in \mathbb{R}^{4}$$\mu x$$\langle\phi(\mu x_{1})\ldots\phi(\mu x_{N})\rangle\equiv$$\underline{m}_{1},\underline{n}_{1},\ldots,\underline{m}_{N},\underline{n}_{N}\in N^{2}\sum f_{\underline{m}_{1}\underline{m}_{2}}(\mu x_{1})\cdots f_{\underline{m}_{N}\underline{m}_{N}}(\mu x_{N})\langle\Phi_{\underline{m}_{1}\underline{n}_{1}}\ldots\Phi_{\underline{m}_{N}\underline{n}_{N}}\rangle,$are obtained by derivwhere the matrix correlation functions,...,
We shall see in this section that
atives of (53) with respect togives a meaningful hmit only
the additional index summation over
if we redefine the volume factor in the free energy density toas the volume has its origin in the spectral geometry of
The occurrence ofthe Moyal plane with harmonic propagation $[GW13a, GW12a]$ which has a finite
volume$\langle\Phi_{\underline{m}_{1}\underline{n}_{1}}\ldots\Phi_{\underline{m}_{N}\underline{n}_{N}}\rangle$$\sqrt{}\underline{m}_{1}\underline{n}_{1}$$J_{\underline{m}\underline{n}}NN^{\cdot}$$\underline{n}_{i}\in N^{2}$$\underline{m}_{i},$$\mathcal{F}=\frac{1}{(\mu V)^{2}}\log_{Z[0]}^{ZJ}\perp.$$V^{2}$$( \frac{V}{\Omega})^{2}.$Definition 10 The connected Schwinger functions associated with the action(26) are$\mu^{N}S_{c}(\mu x_{1}, \ldots, \mu x_{N})$$:=V hm\sum_{\underline{m}_{1},\underline{n}_{1},\ldots,\underline{m}_{N},\underline{n}}f_{\underline{m}_{1}\underline{n}_{1}}N\in N^{2}(\mu x_{1})\cdots f_{\underline{m}_{N}\underline{n}_{N}}(\mu x_{N})\frac{\mu^{4N}\partial^{N}\mathcal{F}[J]}{\partial J_{\underline{m}_{1}\underline{n}_{1}}\ldots\partial J_{\underline{m}\underline{n}_{N}}N}J=0$(54)$\mathcal{F}[J]:=\frac{1}{64\pi^{2}V^{2}\mu^{8}}\log(\frac{\int \mathcal{D}[\Phi]e^{-S[\Phi]+V\Sigma_{\underline{a},\underline{b}\in N^{2}}\Phi_{\underline{ab}}J_{\underline{b}\underline{a}}}}{\int \mathcal{D}[\Phi]e^{-S[\Phi]}})_{Z\mu_{bare}^{2}\mapsto\mu^{2}}$where$S[\phi]$

is given by (27) and$f_{\underline{m}\underline{n}}$by (24). By$()_{Z\mu_{bare}^{2}\mapsto\mu^{2}}$we symbolise therenormalisation of sec. 3.2.

Note that by construction the -derivatives, and hence the Schwinger functions,. Applying the -derivatives the the topoare fully symmetric in-cycle for each of thelogical expansion (53) into -cycles produces anboundary components:$J$$\mu x_{1}\ldots,$ $\mu x_{N}$

$J$

$J$

$f_{\underline{m}\underline{n}}$$S_{c}( \mu x_{1,}\mu x_{N})=\lim_{V\mu^{4}arrow\infty}\frac{1}{64\pi^{2}}\sum_{N_{1}+\cdots+N_{B}=N}\sum_{\underline{q}_{i}^{\beta}\in \mathbb{N}^{2}}G_{|\underline{q}_{1}^{1}\ldots\underline{q}_{N_{1}}^{1}|\ldots|\underline{q}_{1}^{B}\ldots\underline{q}_{N_{B}}^{B}|}$$\cross\sum_{\sigma\in S_{N}}\prod_{\beta=1}^{B}\frac{f_{\underline{q}_{1}\underline{q}_{2}}(\mu x_{\sigma(N_{1}+\ldots+N_{\beta-1}+1)})\cdots f_{\underline{q}_{N_{\beta}}\underline{q}_{1}}(\mu x_{\sigma(N_{1}+\ldots+N_{\beta})})}{V\mu^{4}N_{\beta}}$

$B$

93
We compute the sum over the indicesby Laplace-Fourier transform of. For that we temporarily assume that has, for every boundary component,
representation as Laplace transform in the total sum of index norms and Fouriertransform in diﬀerences of index norms. This transform will be reverted in the end
so that the analyticity assumption is not necessary (future analytic continuation
to Minkowski space would imply representation as Laplace transform):$q_{i}^{\beta}\in \mathbb{N}^{2}$$G$$\overline{G}$$a$$G_{|\underline{q}_{1}^{1}\ldots\underline{q}_{N_{1}}^{1}|\ldots|\underline{q}_{1}^{B}\ldots\underline{q}_{N_{B}}^{B}|}$$= \int_{\mathbb{R}_{+}^{B}}d(t^{1}, \ldots, t^{B})\int_{\mathbb{R}^{N-B}}d(\omega_{1}^{1}, \ldots, \omega_{N_{1}-1}^{1}, \ldots, \omega_{1}^{B}, \ldots, \omega_{N_{B}-1}^{B})$$\cross \mathcal{G}(t^{1}, \omega_{1}^{1}, \ldots, \omega_{N_{1}}^{1}|\ldots|t^{B}, \omega_{1}^{B}, \ldots, \omega_{N_{B}}^{B})$$\cross\prod_{\beta=1}^{B}\exp(-\frac{t^{\beta}}{\sqrt{V\mu^{4}}}\sum_{i=1}^{N_{\beta}}|\underline{q}_{i}^{\beta}|+\frac{i}{\sqrt{V\mu^{4}}}\sum_{i=1}^{N_{\beta}-1}\omega_{i}^{\beta}(|\underline{q}_{i}^{\beta}|-|\underline{q}_{i+1}^{\beta}|))$

Note that the 1-norms$|\underline{q}_{i}^{\beta}|=q_{i,1}^{\beta}+q_{i,2}^{\beta}$(55)

imply a factorisation of the exponential,$\exp(\ldots)=\prod_{i}(z_{i}^{\beta}(t^{\beta},\vec{\omega}^{\beta}))^{q_{i,1}^{\beta}}(z_{i}^{\beta}(t^{\beta},\vec{\omega}^{\beta}))^{q_{i,2}^{\beta}}$

For every boundary component $\beta=1$ , . . . ,$B$, we thus need to compute$\sum_{q_{1},\ldots,q_{N’}=0}^{\infty}\frac{f_{q_{1}q_{2}}(\mu\vec{y}_{1})\cdots f_{q_{N’}q_{1}}(\mu\vec{y}_{N’})}{\sqrt{V\mu^{4}N’}}z_{1}^{q_{1}}\cdots z_{N}^{q_{N’}},$$=2^{N’}.

where

,

\sum^{\infty}e^{-\frac{1}{2}(r_{1}+\cdots+r_{N’})}\frac{L_{q^{2}}^{q_{1}-q_{1}}(r_{1})\cdots L_{q_{N’}}^{q_{1}-q_{N’}}(r_{N’})}{\sqrt{V\mu^{4}N’}}(-\tilde{z}_{1})^{q_{1}}q_{1},..,q_{N’}=0\ldots(-\tilde{z})_{N}^{q_{N’}}$$r_{i}= \frac{\mu^{2}|\vec{y}_{i}|^{2}}{\sqrt{4V\mu^{4}}}$$\omega_{0}=\omega_{N’}\equiv 0$

and$\tilde{z}_{j}=\frac{\vec{y}_{j-1}}{\vec{y}_{j}}\exp(-\frac{t-i(\omega_{j}-\omega_{j-1})}{\sqrt{V\mu^{4}}})$, with$\vec{y_{i}}\in \mathbb{C},$$\vec{y}_{0}\equiv\vec{y}_{N’}$

(56)

and

. One hasLemma 11 $([GW13b])$ For$(i.e. N’+j\equiv j)$ is summed to$|\tilde{z}_{j}|<1$, a cyclic product of Laguerre polynomials$\sum_{q_{1},\ldots,q_{N’}=0}^{\infty}\prod_{j=1}^{N’}(-\tilde{z}_{j})^{q_{j}}L_{q_{j}^{j+1}}^{q-q_{j}}(r_{j})=\frac{\exp(-\frac{\sum_{j,k=1}^{N’}r_{j}-\tilde{z}_{k+.j}.).\cdots(-\tilde{z}_{N’+j})}{1-(\tilde{z}_{1}).(-\tilde{z}_{N’})})}{1-(-\tilde{z}\cdot\cdot(-\tilde{z}_{N},)}$(57)

The denominators in (57) become$\frac{N’t}{\sqrt{V\mu^{4}}}$

for $N’$ even,$1-(- \tilde{z}_{1})\cdots(-\tilde{z}_{N’})=1-(-1)^{N’}\exp(-\frac{N’t}{\sqrt{V\mu^{4}}})V\mu^{4}\infty\vec{arrow}\{$

2
Together with the prefactor$\frac{1}{\sqrt{V\mu^{4}N}}$

for $N’$ odd.

, the sum (56) converges for$V\mu^{4}arrow\infty$

tozero if $N’$ is odd, whereas if $N’$ is even the limit is non-zero and finite, depending

94
only onfactors$t$

but no longer on$\omega_{j}$. Recombining the two$\mathbb{N}^{2}$-components we produce$\frac{\exp(-\frac{\Vert\mu X||^{2}}{2Nt})}{(N’ t)^{2}}=\int_{R^{4}}\frac{dp}{4\pi^{2}\mu^{4}}e^{-i\langle_{\mu}^{2},\mu X\rangle}\exp(-\frac{N’t\Vert p\Vert^{2}}{2\mu^{2}})$for every even $N’.$

Altogether we arrive at$\lim_{V\mu^{4}arrow\infty}\sum_{\underline{q}_{1},\ldots,\underline{q}_{N’}\in\mathbb{N}^{2}}\frac{f_{\underline{q}_{1}\underline{q}_{2}}(\mu x_{1})\cdots f_{\underline{q}_{N’}\underline{q}_{1}}(\mu x_{N’})}{V\mu^{4}N’}z_{1}^{q_{1,1}+q_{1,2}}\cdots z_{N}^{q_{N’,1}+q_{N’,2}}$$=\{\begin{array}{ll}\frac{4^{N’}}{N’}\int_{\mathbb{R}^{4}}\frac{dp}{4\pi^{2}\mu^{4}}e^{-i\langle_{\mu^{)}}^{R}\mu(x_{1}-x_{2}+\ldots+x_{N’-1}-x_{N’})\rangle}\exp(-\frac{N’t\Vert p\Vert^{2}}{2\mu^{2}}) for N’ even,0 for N’ odd.\end{array}$Integration of(55) returns to the original function$\mathcal{G}(t^{1}, \omega_{1}^{1}, \ldots, \omega_{N_{1}}^{1}|\ldots|t^{B}, \omega_{1}^{B}, \ldots, \omega_{N_{B}}^{B})$1. for each , all$\beta$2.

$|\underline{q}_{i}^{\beta}|$$\frac{\Sigma_{i=1}^{N_{\beta}}|\underline{q}_{i}^{\beta}|}{\sqrt{V\mu^{4}}}=\frac{N_{\beta}}{2\mu^{2}}\Vert p\Vert^{2}$

against, but with$\exp(-\frac{N_{\beta}t^{\beta}\Vert p^{\beta}\Vert^{2}}{2\mu^{2}})$in$G_{|\underline{q}_{1}^{1}\ldots\underline{q}_{N_{1}}^{1}|\ldots|\underline{q}_{1}^{B}\ldots\underline{q}_{N_{B}}^{B}|}$

coincide (no -dependence),$\omega$, hence$\frac{|\underline{q}_{i}^{\beta}|}{\sqrt{V\mu^{4}}}V\mu^{4}\infty\vec{arrow}(1+\mathcal{Y})q=\frac{|p|^{2}}{2\mu^{2}}$in limit to integral

representation.
We have thus proved:

Theorem 12 The connected $N$ -point Schwinger functions of theare given by
extreme Moyal space$\phi_{4}^{4}$-model on$\thetaarrow\infty$$S_{c}(\mu x_{1}, \ldots, \mu x_{N})$$= \frac{1}{64\pi_{N}^{2}}\sum_{N_{\beta}even}\sum_{\sigma 1^{+..+N_{B}=N}\in \mathcal{S}_{N}}(\prod_{\beta=1}^{B}\frac{4^{N_{\beta}}}{N_{\beta}}\int_{\pi}4\frac{d^{4}p_{\beta}}{4\pi^{2}\mu^{4}}e^{i\langle\frac{p_{\beta}}{\mu},\Sigma_{i=1}^{N_{\beta}}(-1)^{i-1}\mu x_{\sigma(N_{1}+\ldots+N_{\beta-1}+i)}\rangle})$(58)

Some comments:
Only a restricted sector of the underlying matrix model contributes to position space: All strands of the same boundary component carry the same$\bullet$$\bullet$matrix index.
Schwinger functions are symmetric and invariant under the full Euclidean
breaks both translationgroup. This comes truly surprising since
was expected
invariance and manifest rotation invariance. The limit
to make this symmetry violation even worse!
The most interesting sector is the case where every boundary component
has $N_{\beta}=2$ indices. It is described by the $(2+\ldots+2)$ -point functions$\theta\neq 0$$\thetaarrow\infty$$\bullet$$G \frac{||p\iota||^{2}}{2\mu^{2}(1+\mathcal{Y})}\frac{||p_{1}||^{2}}{2\mu^{2}(1+\mathcal{Y})}$$arrow_{2\mu(1}^{||p}\frac{||^{2}}{+\mathcal{Y})}\frac{||p_{B}||^{2}}{2\mu^{2}(1+\mathcal{Y})}$95
This sector describes propagation and interaction of $B$ particles without
any momentum exchange. This is acceptable for a $2D$ -model. In four
dimensions, absence of momentum transfer is a sign of triviality.
However, typical triviality proofs rely on clustering, analyticity in Mandelstam representation or absence of bound states. All this needs verification.
It is already clear that clustering is maximally violated. Looking for instanceat the $(2+2)$ -sector, we have$\bullet$$\bullet$$\lim_{\mu aarrow\infty}S_{c}^{2+2}(\mu x_{1}, \mu x_{2}, \mu(x_{3}+a), \mu(x_{4}+a))$(59)$= \int\frac{dpdq}{4\pi^{6}\mu^{4}}G_{||p||^{2}}\frac{||p||^{2}}{2\mu^{2}(1+\mathcal{Y})}2\mu^{2}(1+\mathcal{Y})|\frac{||q||^{2}}{2\mu^{2}(1+\mathcal{Y})}||q||^{2}$independent of the distance between
on one hand andon theother hand. Absence of clustering means that the vacuum state (of a hypotheticalcontinuation to a Wightman theory) is not a pure state. Non-pure states can be
decomposed into pure states which describe diﬀerent topological sectors.
Let us give an intuitive explanation why the limitof extreme noncommutativity is so close to an ordinary field theory expected for. Theinteraction term in momentum space$\{x_{1}, x_{2}\}$$\{x_{3}, x_{4}\}$$\thetaarrow\infty$$\thetaarrow 0$$\frac{\lambda}{4}\int_{(\pi)^{4}}4(\prod_{i=1}^{4}\frac{dp_{i}}{(2\pi)^{4}})\delta(p_{1}+\cdots+p_{4})\exp(i\sum_{i<j}\langle p_{i}, \Theta p_{j}\rangle)\prod_{\iota’=1}^{4}\hat{\phi}(p_{i})$leads to the Feynman rule
, plus momentum conservation.
For, this converges to zero almost everywhere by the Riemann-Lebesgue
lemma, unlessare linearly dependent. This case of linearly dependent momenta might be protected for topological reasons, and these are precisely theboundary components $B>1$ which guarantee full Lebesgue measure!
$\lambda\exp(i\sum_{i<j}\langle p_{i}, \Theta p_{j}\rangle)$$\thetaarrow\infty$$p_{i},p_{j}$4.2 Reflection positivity
Under conditions found by Osterwalder-Schrader [OS73, OS75], Schwinger functions [Sch59] possess an analytical continuation to Wightman functions [Wig56,
SW64] of a true relativistic quantum field theory. Roughly speaking, the OsterwalderSchrader reconstruction theorem says:

Theorem 13 ([OS73, OS75]) If the Schwinger functions $S(x_{1}, \ldots, x_{N})$ satisfy
0. growth conditions,
1. Euclidean covariance,
2. reflection positivity: for each assignment $N\mapsto f_{N}\in S^{N}$ offunctions,$te\mathcal{S}l$$\sum_{M,N}\int dxdyS(x_{1}, \ldots, x_{N}, y_{1}, \ldots y_{M})\overline{f_{N}(x_{1}^{r},\ldots,x_{N}^{r})}f_{M}(y_{1}, \ldots, y_{M})\geq 0,$

where$(x^{0}, x^{1}, \ldots x^{d-1})^{r}$$:=(-x^{0}, x^{1}, \ldots x^{d-1})$,

96
3. permutation symmetry,, with $\xi_{i}=x_{i}-x_{i+1}$ , are Laplace Fourier transforms
then theof Wightman functions in a relativistic quantum field theory. If in addition thesatisfy
4. clustering
then the Wightman functions satisfy clustering, too.
$S(\xi_{1}, \ldots\xi_{N-1})|_{\xi_{i}^{0}>0}$$S(x_{1}, \ldots, x_{N})$requires analyticity in
Representation as Laplace transform inO.
is a corollary of
For the Schwinger 2-point function (58), such analyticity in]]. We will show that analyticity
inanalyticity of the function
and reflection positivity boil down to Stieltjes functions, i.e. functions :which have a representation as a Stieltjes transform (see [Wid38])${\rm Re}(\xi^{0})>$$\xi^{0}$$\xi^{0}$$a\mapsto G_{aa}$$\mathbb{C}\backslash$$-\infty,$

$0$

$f$

$f(x)=c+ \int_{0}^{\infty}\frac{d(\rho(t))}{x+t}, c=f(\infty)\geq 0$

where$\rho$,$\mathbb{R}_{+}arrow \mathbb{R}$(60)

is non-negative and non-decreasing. We prove:

Proposition 14 The Schwinger function$S_{c}( \mu\xi)=\int_{\mathbb{R}}\frac{dp}{(2\pi\mu)^{4}}e^{ip\xi}G_{||p||^{2}}$identified in (58) is the analytic continuation of a Wightman 2-point function if
and only if
is Stieltjes.
$a\mapsto G_{aa}$Proof. This is verified by explicit calculation. If
terms of$\omega_{\vec{p}}(t)$$a\mapsto G_{aa}$is Stieltjes, we have in$:=\sqrt{p^{-2}+2\mu^{2}(1+\mathcal{Y})t}$$S_{c}( \mu\xi)|_{\xi^{0}>0}=\int_{\mathbb{R}^{3}}\frac{d\vec{p}}{(2\pi\mu)^{3}}\int_{-\infty}^{\infty}\frac{dp^{0}}{2\pi\mu}e^{ip^{0}\xi^{0_{+i\vec{p}\cdot\xi}^{arrow}}}\int_{0}^{\infty}\frac{d\rho(t)}{t+\frac{(p^{0})^{2}+p^{\triangleleft}}{2\mu^{2}(1+\mathcal{Y})}}$$=2 \mu(1+\mathcal{Y})\int_{\mathbb{R}^{3}}\frac{d\vec{p}e^{i\vec{p}\cdot\xi}arrow}{(2\pi\mu)^{3}}\int_{0}^{\infty}\frac{d\rho(t)}{2\omega_{\vec{p}}(t)}\int_{-\infty}^{\infty}\frac{dp^{0}}{2\pi i}(\frac{e^{ip^{0}\xi^{0}}}{p^{0}-i\omega_{\vec{p}}(t)}-pip^{0}\overline{\omega_{\vec{p}}(t)})\xi^{0}$$=2 \mu(1+\mathcal{Y})\int_{\mathbb{R}^{3}}\frac{d\vec{p}e^{-\xi^{0}\omega_{\tilde{p}}(t)+i\vec{p}\cdot\xi}arrow}{(2\pi\mu)^{3}}\int_{0}^{\infty}\frac{d\rho(t)}{2\omega_{\vec{p}}(t)}$$= \int_{0}^{\infty}\frac{2(1+\mathcal{Y})d\rho(t)}{\mu^{4}}\int_{0}^{\infty}dq^{0}\int_{\mathbb{R}^{3}}dq^{arrow}\hat{W}_{t}(q)e^{-q^{0}\xi^{0}+i\vec{q}\cdot\vec{x}}$$\hat{W}_{t}(q):=\frac{\theta(q^{0})}{(2\pi)^{3}}\delta(^{\underline{(q^{0})^{2}-\overline{q}2\mu^{2}(1+\mathcal{Y})t}})$

(61a)

(61b)

The step from the second to third line is the residue theorem. We observe thatis precisely the K\"all\’en-Lehmann spectral representation [K\"a152, Leh54] ofa Wightman 2-point function.
$\hat{W}_{t}(q)$$\square$Remarkably, the Stieltjes property can be tested by purely real conditions:

97
Theorem 15 (Widder [Wid38]) A function:smooth, non-negative and satisfies, where$f$$\mathbb{R}_{+}arrow \mathbb{R}$is Stieltjes iﬀ it is$L_{k,t}[f(\bullet)]\geq 0$$L_{k,t}[f( \bullet)]:=\frac{(-t)^{k-1}}{c_{k}}\frac{d^{2k-1}}{dt^{2k-1}}(t^{k}f(t)) , c_{1}=1, c_{k>1}=k!(k-2)!.$In that case, the measure is recovered by $\rho’(t)=\lim_{karrow\infty}L_{k,t}[f(\bullet)]$ (weakly andalmost everywhere).

The perturbatively established anomalous dimension $\eta=-2\lambda$ implies thatcannot be Stieltjes for. The restriction to negative coupling constantis reminiscent of the planar wrong-sign-model $[tHo82$ , Riv83 . Recall that
our matrix model also reduces to the planar sector, but as result of the infinite
volume limit and not by hand. We nonetheless keep a non-trivial topology inform of $B\geq 1$ boundary components. Moreover, we have an exact solution for$S(x_{1}, \ldots, x_{N})$ , not only an
existence proof.
Whether or not
is a Stieltjes function for
is a highly interestingquestion. A first idea can be obtained by computer simulations, see sec. 3.5. Weshow in Figure 3 interpolation results for near the critical coupling constant.
We find clear evidence thatis not a Stieltjes function for, where$a\mapsto$$\lambda>0$$G_{aa}$$\lambda\phi_{4}^{4}$$]$$\lambda<0$$a\mapsto G_{aa}$$\lambda$$a\mapsto G_{aa}$$\lambda<\lambda_{c}$$\bullet$$\bullet$based on interpolation of discrete
data, noisy for $k\geq 4$
Stieltjes property clearly violatedfor$\lambda<\lambda_{c}$Figure 3: Widder’s criteria$L_{k,a}$[G..]$:= \frac{(-a)^{k-1}}{k!(k-2)!}\frac{d^{2k-1}}{da^{2k-1}}(a^{k}G_{aa})\geq 0$

for$\lambda\approx\lambda_{c}.$-0.3961ocates the discontinuity of. For
the results are notconclusive (as is too small). Since
andshow a very similar behaviour$\lambda_{c}\approx$$\lambda\in[\lambda_{c}, 0]$$\mathcal{Y}’(\lambda)$$k$$G_{aa}$$G_{a0}$98(which are easy to compute) give some(see e.g. Fig. 2), the functions[G..] (which we are interested in). From (43) one can proveindications about
the following identity [GW14]:$L_{k,t}[G_{0}]$$L_{k,t}$$\frac{(\log G_{a0})^{(\ell)}}{(\ell-1)!}=\frac{(-1)^{\ell}}{(1+a)^{p}}+(-1)^{\ell}sign(\lambda)\mathcal{H}_{0}^{\Lambda}[\sin(\ell\tau_{a}$The resulting integrated “mass densities”’
in Figure 4. We find clear evidence for a mass gap,$\tilde{\rho}_{k}(m^{2})=\int_{0}^{m^{2}}dtL_{k,t}[G_{0}]$Figure 4:$\tilde{\rho}_{k}(m^{2})=\int_{0}^{m^{2}}dtL_{k,t}[G_{0}]$(62)$( \frac{\sin\tau_{a}(\bullet)}{|\lambda|\pi\bullet})^{\ell}]$are shownfor$\lim_{karrow\infty}\tilde{\rho}_{k}(\mu^{2})=0$

as approximation for the mass density of. In each row, the left picture is zoomed into small , showing evidence
for a mass gap. The right pictures show the global behaviour, close to a step, close to criticality for
function for$a\mapsto G_{aa}$$\mu$$\lambda\nearrow 0$$\lambda\searrow\lambda_{c}.$. For
the integrated mass density approaches (as expected)
we notice a power-law behaviour typical fora step function, whereas forthere is no further gap in thecritical phenomena. In particular, for(not only from the
support of , which signals scattering right away fromtwo-particle threshold on). We interpret this as scattering of a massive particle
with an infrared cloud. This scattering would be a remnant of the underlying
non-trivial matrix model before the projection to diagonal matrices.
$0\leq\mu^{2}\leq m^{2}$$\lambda\nearrow 0$$\lambda\searrow\lambda_{c}$$\lambda_{c}<\lambda<0$$\tilde{\rho}’$$m^{2}$

99
4.3 Summary
We have shown that the -model on noncommutative Moyal space, consideredin the limitof extreme noncommutativity, is an exactly solvable and nontrivial matrix model. Euclidean symmetry is violated in the beginning, but we
identified a limit which projects to diagonal matrices where Euclidean symmetry
is restored. One would not expect such that such a brutal projection can respect
any quantum field theory axioms. Surprisingly, the first consistency checks, positivity of the lowest Widder criteria[G..], are passed for the only interesting
intervalof the coupling constant!
If these miracles continue and all Osterwalder-Schrader axioms (except forclustering) hold, we would get a relativistic quantum field theory in four dimensions. This theory is somewhat strange as “particles” keep their momenta ininteraction processes. Nevertheless, the theory is not completely trivial. We
find scattering remnants from the noncommutative geometrical (i.e. matricial)substructure. Only the external matrix indices are put ‘on-shell’, internally all
degrees of freedom contribute.
We have seen that clustering is maximally violated. The interaction is insensitive to positions in diﬀerent boundary components. In particular, (particle arenever asymptotically free.
$\phi_{4}^{4}$$\thetaarrow\infty$$L_{k,t}$$[\lambda_{c}, 0]$$s^{)}$A

Schwinger-Dyson equations for $B=2$

We find for the $(1+1)$ -and $(2+2)$ -point functions$\}(63a)$$G_{|a|c|}=- \frac{\lambda}{E_{a}+E_{a}}(\frac{1}{V}\sum_{p\in I}(G_{|ap|}G_{|a|c|}-\frac{G_{|p|c|}-G_{|a|c|}}{E_{p}-E_{a}})-\frac{G_{|cc|}-G_{|ac|}}{E_{c}-E_{a}})$$\}(63b)$$- \frac{\lambda}{V^{2}(E_{a}+E_{a})}(3G_{|a|a|}G_{|a|c|}+G_{|a|cac|}+G_{|c|aaa|}+\frac{1}{V}\sum_{n\in I}G_{|a|c|an|})$$- \frac{\lambda}{V^{4}(E_{a}+E_{a})}G_{|a|a|a|c|}, \}(63c)$$G_{|ab|cd|}$$=- \frac{\lambda}{E_{a}+E_{b}}(\frac{1}{V}\sum_{p\in I}((G_{|ap|}G_{|ab|cd|}+G_{|ab|}G_{|ap|cd|})-\frac{G_{|pb|cd|}-G_{|ab|cd|}}{E_{p}-E_{a}})$(64a)$+G_{|ab|}(G_{|cacd|}+G_{|dadc|})- \frac{G_{|cbcd|}-G_{|cbad|}}{E_{c}-E_{a}}-\frac{G_{|dbdc|}-G_{|dbac|}}{E_{d}-E_{a}})$$- \frac{\lambda}{V^{2}(E_{a}+E_{b})}(G_{|a|a|}G_{|ab|cd|}+G_{ab}G_{|a|a|cd}+\frac{1}{V}\sum_{n\in I}G_{|an|ab|cd|}$(64b)$+G_{|cd|aaab|}+G_{|cd|baba|}+G_{|ab|cacd|}+G_{|ab|cddad|}- \frac{G_{|b|a|cd|}-G_{|b|b|cd|}}{E_{b}-E_{a}})$$- \frac{\lambda}{V^{4}(E_{a}+E_{b})}G_{|a|a|ab|cd|}$

$\}$

(64c)

100
These are basic functions which are not simplified by reality. As before, (63a)and (64a) preserve the genus, whereas $g\mapsto g+1$ in $(63b)+(64b)$ and $g\mapsto g+2$ in$(63c)+(64c)$ . The higher $(N_{1}+N_{2})$ -point functions with one
simplify byreality to universal recursion formulae. For
odd we have$N_{i}\geq 3$$N_{i}$$G_{|b_{0}\ldots b_{2l}|c_{1}\ldots c_{N-2l-1}|}$$=- \lambda\sum_{k=1}^{N-2l-1}\frac{G_{|_{C_{1}}\ldots c_{k}c_{k+1}\ldots c_{N-2l-1}}c_{k-1}b_{0}b_{1}\ldots b_{2l}|-G_{|c_{1}\ldots c_{k-1}c_{k}b_{1}\ldots b_{2l}b_{0}c_{k+1}\ldots c_{N-2/-1}|}}{(E_{b_{1}}-E_{b_{2l}})(E_{b_{0}}-E_{c_{k}})}$$- \lambda\sum_{j=1}^{l}\frac{G_{|b_{0}b_{1}\ldots b_{2j-2}|c_{1}\ldots c_{N-2l-1}|}G_{|b_{2j-1}b_{2j}\ldots b_{2l}|}-G_{|b_{2j-1}b_{1}\ldots b_{2j-2}|c_{1}\ldots c_{N-2l-1}|}G_{|b_{0}b_{2j}\ldots b_{2l}|}}{(E_{b_{1}}-E_{b_{2l}})(E_{b_{0}}-E_{b_{2j-1}})}$$- \lambda\sum_{j=1}^{l}\frac{GG-G}{(E_{b_{1}}-E_{b_{2l}})(E_{b_{0}}-E_{b_{2j}})}$(65)$- \frac{\lambda}{V^{2}}\sum_{k=1}^{2l}\frac{G_{|b_{0}b_{1}\ldots b_{k-1}|b_{k}b_{k+1}\ldots b_{2l}|c_{1}\ldots c_{N-2l-1}|}-G_{|b_{k}b_{1}\ldots b_{k-1}|b_{0}b_{k+1}\ldots b_{2l}|c_{1}\ldots c_{N-2l-1}|}}{(E_{b_{1}}-E_{b_{2l}})(E_{b_{0}}-E_{b_{k}})}$The last line increases the genus and is absent in
one finds$G_{|b_{0}b_{1}\ldots b_{2l}|c_{1}\ldots c_{N-2l-1}|}^{(0)}$. For$N_{i}$even$G_{|ab_{1}\ldots b_{2l-1}|c_{1}\ldots c_{N-2l}|}$$=- \lambda\sum_{j=1}^{l-1}\frac{G_{|b_{1}\ldots b_{2j-1}a|c_{1}\ldots c_{N-2l}|}G_{|b_{2j}b_{2j+1}\ldots b_{2l-1}|}-G_{|b_{1}\ldots b_{2j-1}b_{2j}|c_{1}\ldotsc_{N-2l}|}G_{|ab_{2j+1}\ldots b_{2l-1}|}}{(E_{b_{1}}-E_{b_{2l-1}})(E_{a}-E_{b_{2j}})}$$- \lambda\sum_{j=1}^{l-1}\frac{G_{|b_{1}}\ldots G-G}{(E_{b_{1}}-E_{b_{21-1}})(E_{a}-E_{b_{2j}})}$$- \lambda\sum_{k=1}^{N-2l}\frac{G_{|c_{1}\ldots c_{k-1}ab_{1}\ldots b_{2l-1}c_{k}c_{k+1}\ldots c_{N-2l}|}-G_{|c_{1}\ldots c_{k-1}c_{k}b_{1}\ldots b_{2l-1}ac_{k+1}\ldots c_{N-2l}|}}{(E_{b_{1}}-E_{b_{2l-1}})(E_{a}-E_{c_{k}})}$$- \frac{\lambda}{V^{2}}\sum_{k=1}^{2l-1}\frac{G_{|b_{1}\ldots b_{k-1}a|b_{k}b_{k+1}\ldots b_{2l-1}|_{\mathcal{C}1}\ldots c_{N-2l}|}-G_{|b_{1}\ldots b_{k-1}b_{k}|ab_{k+1}\ldots b_{2l-1}|c_{1}\ldots c_{N-2l}|}}{(E_{b_{1}}-E_{b_{2l-1}})(E_{a}-E_{b_{k}})}$

Again, the last line increases the genus and is absent in(66)$G_{|b_{0}b_{1}\ldots b_{2t-1}|c_{1}\ldots C_{N-2l}|}^{(0)}.$Acknowledgements
RW would like to cordially thank Keiichi R. Ito for the invitation to the RIMS
symposium “Applications of RG Methods in Mathematical Sciences”’ and for theimmense help and hospitality during this visit to Kyoto in September 2013.
In addition to the presentation at the RIMS symposium, this contribution isbased on a lecture series which both of us gave in November 2013 in G\"ottingen.

101
We would like to thank Dorothea Bahns for the arrangement of and hospitality
during this series.

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