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Matricial quantum field theory:Integrability. Positivity?
Clay Math Institute Millenium Prize Problem:Prove existence of Yang-Mills4 with mass gap
Raimar Wulkenhaar (Munster) Matricial quantum field theory: Integrability. Positivity? 2
Introduction Matricial QFT Schwinger functions
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
Introduction Matricial QFT Schwinger functions
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
Raimar Wulkenhaar (Munster) Matricial quantum field theory: Integrability. Positivity? 4
Introduction Matricial QFT Schwinger functions
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.
Raimar Wulkenhaar (Munster) Matricial quantum field theory: Integrability. Positivity? 5
Introduction Matricial QFT Schwinger functions
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
Raimar Wulkenhaar (Munster) Matricial quantum field theory: Integrability. Positivity? 6
Introduction Matricial QFT Schwinger functions
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
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!
Raimar Wulkenhaar (Munster) Matricial quantum field theory: Integrability. Positivity? 8
Introduction Matricial QFT Schwinger functions
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)
Raimar Wulkenhaar (Munster) Matricial quantum field theory: Integrability. Positivity? 9
Introduction Matricial QFT Schwinger functions
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 )
Raimar Wulkenhaar (Munster) Matricial quantum field theory: Integrability. Positivity? 10
Introduction Matricial QFT Schwinger functions
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
Introduction Matricial QFT Schwinger functions
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?
Raimar Wulkenhaar (Munster) Matricial quantum field theory: Integrability. Positivity? 12
Introduction Matricial QFT Schwinger functions
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)
Raimar Wulkenhaar (Munster) Matricial quantum field theory: Integrability. Positivity? 13
Introduction Matricial QFT Schwinger functions
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)
+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
))
Raimar Wulkenhaar (Munster) Matricial quantum field theory: Integrability. Positivity? 13
Introduction Matricial QFT Schwinger functions
Simplest 6D-ribbon graph with overlapping divergence
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)
Raimar Wulkenhaar (Munster) Matricial quantum field theory: Integrability. Positivity? 14
Introduction Matricial QFT Schwinger functions
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.
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
Introduction Matricial QFT Schwinger functions
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
Introduction Matricial QFT Schwinger functions
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)]
Raimar Wulkenhaar (Munster) Matricial quantum field theory: Integrability. Positivity? 17