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Matricial quantum field theory:Integrability. Positivity?

Raimar Wulkenhaar

Mathematisches Institut, Westfalische Wilhelms-Universitat Munster

abased on arXiv:1610.00526 & 1612.07584 with Harald Grosse and Akifumi Sako

and arXiv: 1205.0465, 1306.2816, 1402.1041, 1406.7755 & 1505.05161with Harald Grosse

Raimar Wulkenhaar (Munster) Matricial quantum field theory: Integrability. Positivity? 0

Introduction Matricial QFT Schwinger functions

Goal: Quantum Field Theory satisfying axioms

1932: axioms for quantum mechanics [von Neumann]

1950’s: unique extension to quantum fields [Wightman]= unbounded op.-valued distributions f 7→ Φ(f ) : D → D ⊂ H

Theorem: vacuum expectation values 〈Ω,Φ(x1) · · ·Φ(xN)Ω〉are boundary values of holomorphic functions

their restriction to real subspace of Euclidean points(minus diagonals) defines Schwinger functions

Schwinger functions inherit real analyticity, Euclideaninvariance, complete symmetry and reflection positivity

Theorem [Osterwalder-Schrader 1974]These properties are sufficient to reconstruct Wightman theory!

So far no non-trivial QFT model in 4 dimensions . . .Raimar Wulkenhaar (Munster) Matricial quantum field theory: Integrability. Positivity? 1


State of the artConstructed models (selection)

1 exactly solvable 2D-models (e.g. Thirring, Schwinger)2 φ4

2 by hard work [Glimm-Jaffe 1968–72]3 P[φ]2: candidate Schwinger functions as moments of

Feynman-Kac perturbation of Gaußian measure[Glimm, Jaffe, Simon, . . . 1974]

4 φ43 similar techniques [Feldman-Osterwalder 1976]

5 Gross-Neveu2: Fermionic summation techniques[Gawedzki-Kupiainen 1985,Feldman-Magnen-Rivasseau-Seneor 1986]

probable non-example: φ44 [Aizenman 1981, Frohlich 1982]

Clay Math Institute Millenium Prize Problem:Prove existence of Yang-Mills4 with mass gap



Standard model of particle physicsconfimed by all experiments of last 40 years

It is a NONCOMMUTATIVE GEOMETRY[Alain Connes 1986–now, with collaborators:Lott, Chamseddine, Marcolli, van Suijlekom, Mukhanov, . . . ]

No rigorous mathematical treatment available; instead:1 Feynman graphs and pertubative renormalisation theory

[Bogoliubov-Parasiuk-Hepp-Zimmermann-Lowenstein]2 Monte Carlo simulation on supercomputers

Nonetheless mathematically interesting:1 zero-locus of Symanzik polynomials is algebraic variety2 amplitudes evaluate to special numbers

(polylogarithms, multiple zeta values)3 renormalisation is Birkhoff-decomposition of a loop in the

group of characters of the Connes-Kreimer Hopf algebraRaimar Wulkenhaar (Munster) Matricial quantum field theory: Integrability. Positivity? 3


Matricial quantum field theory

. . . is the marriage of1 matrix models for 2D quantum gravity2 QFT on noncommutative spaces

1 Kontsevich model (1992)designed to prove Witten’s conjecture that hermiteanone-matrix model computes intersection numbers of stablecohomology classes on the moduli space of complex curves

2 Space-time should become a noncommutative manifold atshort distances.

Euclidean scalar field φ ∈ A (noncommutative algebra)A often has finite-dimensional approximationsFor converse convergence of matrices to A,see Marc Rieffel’s talk



The Kontsevich modeldefined by partition function

Z(E) :=

∫dΦ exp

(− Tr

(EΦ2 + i

6Φ3))∫dΦ exp

(− Tr

(EΦ2))

Asymptotic expansion in ‘coupling constant’ i6

gives rational function of eigenvalues ei of E .This rational function generates the intersection numbers.

Related to Hermitean one-matrix modelZ(E)[[tn]] =

∫DM exp(−N

∑n

tn tr(Mn))

where tn := (2n − 1)!!tr(E−(2n−1))

Large-N limit gives KdV evolution equation.Exact solution related to Virasoro algebra.



QFT on noncommutative geometries

Example: Moyal algebra = Rieffel deformation of C∞(R2)

(f ? g)(ξ) =

∫R2×R2

dη dk(2π)2 f (x+ 1

2 Θk) g(ξ+η) ei〈k,η〉 Θ =

(0 θ

−θ 0

)matrix basis φ(ξ) =

∑∞m,n=0 Φmnfmn(ξ)

fmn(ξ) = 2(−1)m√

m!n!

(√2θ ξ1+iξ2

)n−mLn−m

m

(2‖ξ‖2

θ

)e−‖ξ‖2

θ

satisies fmn ? fkl = δnk fml and∫ dξ

8π fmn(ξ) = θ4δmn

Consider scalar field theories on Moyal space

S(φ) :=1

(8π)D/2

∫RD

dξ(1

2φ?(−∆+4Ω2‖Θ−1ξ‖2)?φ+ tr(pol(φ))

)fmn-expansion at Ω = 1 yields Kontsevich-type matrix model

S(Φ) = V tr(EΦ2 + pol(Φ)), E =((µ

2

2 + nV )δmn

), V = ( θ4)D/2



Two independent dimensions1 Topological dimension 2 from expansion of matrix models

into ribbon graphs, i.e. simplicial 2-complexes.dual to triangulations (Φ3) or quadrangulations (Φ4) of2D-surfaces

partition function counts them = 2D quantum gravity

non-planar ribbon graphs suppressed in large-N limit

2 Dynamical dimension D encoded in spectrum of theunbounded positive operator E ,

D = infp ∈ R+ : tr((1 + E)−p2 ) <∞

ignored in 2D quantum gravity

highly relevant for renormalisation of matricial QFT

polynomial finite super-ren just ren. not ren.Φ3 D < 2 2[D

2 ] ∈ 2,4 2[D2 ] = 6 2[D

2 ] > 6Φ4 D < 2 2[D

2 ] = 2 2[D2 ] = 4 2[D

2 ] > 4Raimar Wulkenhaar (Munster) Matricial quantum field theory: Integrability. Positivity? 7


Φ36 matricial QFT

action S(Φ) = V tr(ZEΦ2 + (κ+νE+ζE2)Φ + λbareZ32

3 Φ3)

for E =(µ2

bare2 + µ2e

( |n|µ2V 2/D

)δmn

), m,n ∈ ND/2

µbare, λbare,Z , κ, ν, ζ to be fixed by normalisation conditions

partition function Z(J) =∫

dΦ exp(−S(Φ) + V tr(ΦJ))

logZ(J)

Z(0)=∞∑

B=1

∑NB≥···≥N1≥1

V 2−B

SN1...NB

G|p11 ...p

1N1|...|pB

1 ...pBNB|

B∏β=1

( Nβ∏jβ=1

Jpβjβpβjβ+1

)cycl

StrategyZ(J) is meaningless for λ ∈ R!Z(J) is only used as tool to derive identities(Schwinger-Dyson equations) between G|p1

1 ...p1N1|...|pB

1 ...pBNB|

Forget Z, declare SD-equations as exact and search forrigorous solutions G... of them!



Schwinger-Dyson equationsInserting Z(J) = exp

(− Z 3/2λbare

3V 2

∑ ∂3

∂Jkl∂Jlm∂Jmk

)Z≤2(J) into

G|a| ≡ 1V∂ logZ[J]∂Jaa

∣∣∣J=0

gives equation quadratic in G|a|, linear in∑m G|am| and G|a|a|

typical feature: SD-equation for n-point function dependson (m > n)-point functionHere we are rescued:

1 G|a|a| comes with 1V 2 , goes away in limit V 2/D ∼ θ →∞

2 G|am| expressable in term of G|a|,G|m| thanks toWard-Takahashi identity for U(∞)-group action:

Theorem (Disertori-Gurau-Magnen-Rivasseau 2006)∑n

∂2Z[J]

∂Jbn∂Jna=∑

n

VZ (Ea − Eb)

(Jan

∂

∂Jbn− Jnb

∂

∂Jna

)Z[J]

− VZ

(ν + ζ(Ea + Eb))∂Z[J]

∂Jba(for a 6= b)



Scaling limit N ,V →∞ with NV 2/D = µ2Λ2 fixed

Non-linear integral equation for G(x) = µ1−D/2G|a|∣∣|a|=V 2/Dµ2x

similar to the string equation:Theorem [Makeenko-Semenoff 1991]

W 2(X ) +∫ b

a dYρ(Y )W (X)−W (Y )X−Y = X + const

is solved by W (X ) =√

X + c + 12

∫ ba

dY ρ(Y )

(√

X+c+√

Y +c)√

Y +ctogether with a consistency condition on c.

Identification X = (2e(x) + 1)2, ρ(Y ) =2λ2(e−1(

√Y−12 ))D/2−1

Γ(D/2)√

Ye′(e−1(√

Y−12 ))

Solution of renormalised equation for D = 6W (X ) =

√X + c

√1 + c − c

+ 12

∫∞1

dT ρ(T ) (√

X+c−√

1+c)2

(√

X+c+√

T +c)(√

1+c+√

T +c)2√

T +c,

−c =∫∞

1dT ρ(T )

(√

1+c+√

T +c)3√

T +c

gives G(x) = 12λ(W (X )−

√X )



Higher correlation functions. . . satisfy linear integral equations, easily reduced to (1+ . . .+1):

G|a11...a

1N1|...|aB

1 ...aBNB|

W|ak |if B=1

= λN1+···+NB−BN1∑

k1=1

· · ·NB∑

kB=1

G|a1k1|...|aB

kB|

B∏β=1

Nβ∏lβ=1

lβ 6=kβ

1E2

a1k1

− E2a1

l1

Proposition

G(X |Y ) =4λ2

√X + c ·

√Y + c · (

√X + c +

√Y + c)2

G(X 1| . . . |X B) =dB−3

dtB−3

( (−2λ)3B−4

(R(t))B−21

√X 1+c−2t

3 · · ·1

√X B+c−2t

3

)∣∣∣∣∣t=0

R(T )6D=√

1+c −∫ ∞

1

dT ρ(T )√

1+c(2√

T +c +√

1+c)(√

T +c−2t +√

T +c)+t(√

T +c−2t + 2√

T +c)

√T +c(

√1+c +

√T +c)2(

√T +c +

√T +c−2t)2√T +c−2t

D<6= 1−

∫ ∞1

dTρ(T )√T + c

1

(√

T + c +√

T + c − 2t)√

T + c − 2t

Proof: ansatz for recursion and experience with Bell polynomialsRaimar Wulkenhaar (Munster) Matricial quantum field theory: Integrability. Positivity? 11


Consistency: (new?) identity for Bell polynomialsFor any l ,n0, . . . ,np ∈ N, the Bell polynomials satisfy

(2l+5)!!

(l + 2)!

∑K≥0

(N−2+K )!BN−M−l−4,K (xr)

(N−M−l−4)!

−∑K≥0

(N−3+K )!BN−M−l−4,K (xr)

(N−M−l−4)!

p∑i=0

ni(2l + 2i + 3)!!(2i + 1)i!

(2i + 1)!!(l + i + 1)!

=∑j≥1

∑K≥0

(N−2+K )!(2j+2l+5)!!(j+1)!

(2j+1)!!(j+l+2)!·

xj

j!·

BN−M−l−j−4,K (xr)(N−M−l−j−4)!

+12

l∑l′=0

n0∑n′0=0

· · ·np∑

n′p=0

∑K ′,K ′′≥0

(2l ′+1)!!(2l ′′+1)!!

l ′! l ′′!

(n0

n′0

)· · ·(

np

n′p

)

×(N ′−2+K ′)!BN′−M′−l′−2,K ′(xr)

(N ′−M ′−l ′−2)!(N ′′−2+K ′′)!

BN′′−M′′−l′′−2,K ′′(xr)(N ′′−M ′′−l ′′−2)!

where l ′′ := l − l ′, N ′′ := N − N ′ and M ′′ := M −M ′ and for? = ∅,′ ,′′ : n?

0 + · · ·+ n?p = N? and 0n?

0 + 1n?1 + · · ·+ pn?

p = M?



Simplest 6D-ribbon graph with overlapping divergence

y2y1

x ••

•=

(−λ)3

(2x+1)

∫ ∞0

y21 dy1

2

∫ ∞0

y22 dy2

2

1

(x+y1+1)2(y1+y2+1)(x+y2+1)




y2y1

x ••

•=

(−λ)3

(2x+1)

∫ ∞0

y21 dy1

2

∫ ∞0

y22 dy2

2

1

(x+y1+1)2(y1+y2+1)(x+y2+1)

+1

x+y2+1

(− 1

(y1+1)3

)+

1(x+y1+1)2

(− 1

(y2+1)2 +y1 + x

(y2+1)3

)+

1(y1+y2+1)

(− 1

(y1+1)2(y2+1)+

2x(y1+1)3(y2+1)

+x

(y1+1)2(y2+1)2

− 3x2

(y1+1)4(y2+1)− x2

(y1+1)2(y2+1)3 −2x2

(y1+1)3(y2+1)2

)+(− 1

(y1+1)3

)(− 1

y2+1+

x(y2+1)2 −

x2

(y2+1)3

)+((− 1

(y1+1)2 +2x

(y1+1)3 −3x2

(y1+1)4

)(− 1

(y2+1)2 +y1

(y2+1)3

)+(− 1

(y1+1)2 +2x

(y1+1)3

)( x(y2+1)3

))




y2y1

x ••

•

=−λ3

4(2x+1)3

(x+1)(2x+1)(3x+2) log(1+x) + (x+1)3(3x+1)(log(1+x))2

+ x(1+x)(1+3x+3x2)(

(log(1+x))2 − 2 log(1+x) log x + 2Li2( 1

1+x

))− 3x3(2+3x)ζ(2)

+

λ3x2(2x+1)

(ζ(2) + 1− x

2

)




y2y1

x ••

•

=−λ3

4(2x+1)3

(x+1)(2x+1)(3x+2) log(1+x) + (x+1)3(3x+1)(log(1+x))2

+ x(1+x)(1+3x+3x2)(

(log(1+x))2 − 2 log(1+x) log x + 2Li2( 1

1+x

))− 3x3(2+3x)ζ(2)

+

λ3x2(2x+1)

(ζ(2) + 1− x

2

)

adding: y1 y2

x • • •

y2

y1x • •• HH y1

y2x • ••

gives the λ3-order of the exact formula for G(x)!Raimar Wulkenhaar (Munster) Matricial quantum field theory: Integrability. Positivity? 13


Schwinger functionsundo the passage to the fmn-matrix basis of Moyal space:

Theorem [HG+RW, 2013]: connected Schwinger functionsSc

N(µξ1, . . . , µξN)

:= limVµ2→∞

∞∑mi ,ni =0

fm1n1 (ξ1) · · · fmN nN (ξN)(Vµ2)−2µ3N∂N logZ(J)

∂Jm1n1 . . . ∂JmN nN

∣∣∣∣J=0

=∑

N1+...+NB =NNβ even

∑σ∈SN

( B∏β=1

2DNβ

2

Nβ

∫RD

dpβ(2πµ2)

D2

ei⟨

pβ ,∑Nβ

i=1(−1)i−1ξσ(N1+...+Nβ−1+i)

⟩)

× 1(8π)

D2 SN1...NB

G(‖p1‖2

2µ2 , · · · , ‖p1‖2

2µ2︸︷︷︸N1

∣∣ . . . ∣∣‖pB‖2

2µ2 , · · · , ‖pB‖2

2µ2︸︷︷︸NB

)

Confinement of noncommutativity: have internal interaction ofmatrices; commutative subsector propagates to outside world

Schwinger functions are symmetric and invariant under fullEuclidean group (completely unexpected for NCQFT!)remains: reflection positivity (. . . and non-triviality)



Reflection positivity S(f r ⊗ f ) ≥ 0

f stands for sequences of test functions of complicatedsupportf r1(τ, ~ξ) = f1(−τ, ~ξ) is time reflection

Implies for very special f :The temporal Fourier transform of S (in all independent energies)is, for any spatial momenta, a positive definite function.

Theorem (Hausdorff-Bernstein-Widder, 1921-1912/28-1941)

For a [smooth] function F on (R+)N 3 t = (t1, . . . , tN) areequivalent:

1 F is positive definite, i.e.∑K

i,j=1 cicjF (ti + tj) ≥ 02 F is the joint Laplace transform of a positive measure∗

3 F is completely monotonic, (−1)k1+···+kN∂k1t1 . . . ∂

kNtN F (t) ≥ 0

∗This is 80% of the proof of the Osterwalder-Schrader theorem.Raimar Wulkenhaar (Munster) Matricial quantum field theory: Integrability. Positivity? 15


Stieltjes functionsPrototype for N = 1∫ ∞−∞

eip0t

(p0)+~p2+m2 =( 2πt√

~p2+m2

) 12 K 1

2(t√~p2 + m2) = πe−t

√~p2+m2

√~p2+m2

Theorem

Up to integration in m2 with positive measure, 1(p0)+~p2+m2 is the

only function with positive definite Fourier transform for N = 1.

p2 7→∫∞

0%(m2)dm2

p2+m2 forms the class of Stieltjes functionsin QFT, %(m2) is the Kallen-Lehmann spectral measure

Is G(‖p‖2

2µ2 ,‖p‖2

2µ2 ) Stieltjes?

We work on this for Φ44 since 2013. Have some analytic

evidence, confirmed by computer, but no complete proof.For Φ3

D we have the answer:Raimar Wulkenhaar (Munster) Matricial quantum field theory: Integrability. Positivity? 16


Reflection positivity of the 2-point functionTheorem (Grosse-Sako-W 2016)

1 The Φ3D-matricial QFT is not reflection positive for λ ∈ iR.

2 The Φ3D two-point function is reflection positive for

D ∈ 4,6 and some range of λ ∈ R, but not in D = 2.

measure supported on fuzzy mass shell plus scattering part:

G(‖p‖2

2µ2 ,‖p‖2

2µ2

)6D=

λ2

4π(σ2−1)

∫ π

0dφ

2 log(1+σ)

σ −1 + σ(σ−1) tan2 φ

− tanφ(1+σ2 tan2 φ

)(arctan[0,π](σ tanφ)−φ

)1−√σ2−1σ cosφ+ ‖p‖2

µ2

+λ2

4

∫ ∞2

dtt(t − 2)/(t − 1)3

t + ‖p‖2

µ2

,

where σ := 1√1+c∈ [1,−2W−1(− 1

2√

e )− 1] is the

inverse solution of λ2 = 4(σ2−1)σ2−2σ+2 log(1+σ)

∈ [1,8W−1(− 1

2√

e)

1+2W−1(− 12√

e)]



Kallen-Lehmann measure: plots

0 2 4 6 8 10 12 14

0.0

0.5

1.0

1.5

2.0

2.5

3.0 D=6legend σ λ %(1) supp(%)

1.00001 0.01439 71.174 [0.9955, 1.0045] ∪ [2,∞[

1.0001 0.04550 22.502 [0.9859, 1.0141] ∪ [2,∞[

1.001 0.14376 7.0975 [0.9553, 1.0447] ∪ [2,∞[

• • • 1.01 0.45038 2.1885 [0.8596, 1.1404] ∪ [2,∞[

1.03 0.76434 1.1971 [0.7604, 1.2396] ∪ [2,∞[

1.10 1.30416 0.5544 [0.5834, 1.4166] ∪ [2,∞[

N N N 1.30 1.91093 0.2357 [0.3610, 1.6390] ∪ [2,∞[

H H H 1.80 2.29629 0.1339 [0.1685, 1.8315] ∪ [2,∞[

FFF 2.51286 2.36470 0.1251 [0.0826, 1.9174] ∪ [2,∞[



Kallen-Lehmann measure: plots

0 1 2 3 4 5 6

0.0

0.5

1.0

1.5

2.0

2.5

3.0

D=4legend σ λ %(1) supp(%)

1.00001 0.00571 71.176 [0.9955, 1.0045] ∪ [2,∞[

1.0001 0.01805 22.506 [0.9859, 1.0141] ∪ [2,∞[

1.001 0.05704 7.1114 [0.9553, 1.0447] ∪ [2,∞[

• • • 1.01 0.17907 2.2315 [0.8596, 1.1404] ∪ [2,∞[

1.03 0.30525 1.2676 [0.7604, 1.2396] ∪ [2,∞[

1.10 0.52847 0.6621 [0.5834, 1.4166] ∪ [2,∞[

N N N 1.50 0.92552 0.2726 [0.2546, 1.7454] ∪ [2,∞[

H H H 2.50 1.09666 0.1922 [0.0835, 1.9165] ∪ [2,∞[

FFF 3.92155 1.12027 0.1843 [0.0331, 1.9669] ∪ [2,∞[



Reflection positivity of higher Schwinger functions?

Connected Schwinger functions ScN≥4 are not positive!

Anyway too much, one needs positivity of FT of full functions

e.g. G(‖p‖2

2µ2 ,‖p‖2

2µ2 )G(‖q‖2

2µ2 ,‖q‖2

2µ2 ) + G(‖p‖2

2µ2 ,‖p‖2

2µ2 |‖q‖2

2µ2 ,‖q‖2

2µ2 )

Difficult for N = 4,but G(2|2|2) + G(2)G(2)G(2) is not positive.

Very probable conclusion

The Φ3D matricial QFT does not satisfy Osterwalder-Schrader.

Reason: Higher functions too much localised in p-space!already G(‖p‖

2

2µ2 ,‖p‖2

2µ2 ) ∝ C1 log(‖p‖2+µ2)+C2‖p‖2+µ2 almost fails

For Φ44 we expect G(‖p‖

2

2µ2 ,‖p‖2

2µ2 ) ∝ C(‖p‖2+µ2)1− 1

π arcsin(|λ|π)(hope!)

Keeps us busy for the next time!Raimar Wulkenhaar (Munster) Matricial quantum field theory: Integrability. Positivity? 19

Bon anniversaireAlain!
