Post on 23-Mar-2020
Matricial quantum field theory:renormalisation, 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: renormalisation, 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〉 areboundary values of holomorphic functions
their restriction to real subspace of Euclidean points(minus diagonals) defines Schwinger functionsSchwinger 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: renormalisation, integrability & positivity 1
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〉 areboundary values of holomorphic functions
their restriction to real subspace of Euclidean points(minus diagonals) defines Schwinger functionsSchwinger 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: renormalisation, integrability & positivity 1
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〉 areboundary values of holomorphic functions
their restriction to real subspace of Euclidean points(minus diagonals) defines Schwinger functionsSchwinger 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: renormalisation, integrability & positivity 1
Introduction Matricial QFT Schwinger functions
Selected techniquesexactly solvable 2D-models (e.g. Thirring, Schwinger)
candidate Schwinger functions as moments of perturbedGaußian measure (e.g. P[φ]2, φ4
3, probably not φ44)
fermionic summation techniques (e.g. Gross-Neveu2)
for all realistic models (e.g. QED4, Standard Model4):renormalised perturbation theory – BPHZ(L)
Z= Wolfhart ZimmermannBPHZ(L) has two aspects:
1 Renormalisation amounts to normalisation conditions forrelevant/marginal correlation functions. These conditionsare of non-perturbative nature.
2 When restricted to graphs, these conditions boil down tomomentum space Taylor subtraction and forest formula.
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, integrability & positivity 2
Introduction Matricial QFT Schwinger functions
Selected techniquesexactly solvable 2D-models (e.g. Thirring, Schwinger)
candidate Schwinger functions as moments of perturbedGaußian measure (e.g. P[φ]2, φ4
3, probably not φ44)
fermionic summation techniques (e.g. Gross-Neveu2)
for all realistic models (e.g. QED4, Standard Model4):renormalised perturbation theory – BPHZ(L)
Z= Wolfhart ZimmermannBPHZ(L) has two aspects:
1 Renormalisation amounts to normalisation conditions forrelevant/marginal correlation functions. These conditionsare of non-perturbative nature.
2 When restricted to graphs, these conditions boil down tomomentum space Taylor subtraction and forest formula.
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, integrability & positivity 2
Introduction Matricial QFT Schwinger functions
Selected techniquesexactly solvable 2D-models (e.g. Thirring, Schwinger)
candidate Schwinger functions as moments of perturbedGaußian measure (e.g. P[φ]2, φ4
3, probably not φ44)
fermionic summation techniques (e.g. Gross-Neveu2)
for all realistic models (e.g. QED4, Standard Model4):renormalised perturbation theory – BPHZ(L)
Z= Wolfhart ZimmermannBPHZ(L) has two aspects:
1 Renormalisation amounts to normalisation conditions forrelevant/marginal correlation functions. These conditionsare of non-perturbative nature.
2 When restricted to graphs, these conditions boil down tomomentum space Taylor subtraction and forest formula.
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, integrability & positivity 2
Introduction Matricial QFT Schwinger functions
Selected techniquesexactly solvable 2D-models (e.g. Thirring, Schwinger)
candidate Schwinger functions as moments of perturbedGaußian measure (e.g. P[φ]2, φ4
3, probably not φ44)
fermionic summation techniques (e.g. Gross-Neveu2)
for all realistic models (e.g. QED4, Standard Model4):renormalised perturbation theory – BPHZ(L)
Z= Wolfhart ZimmermannBPHZ(L) has two aspects:
1 Renormalisation amounts to normalisation conditions forrelevant/marginal correlation functions. These conditionsare of non-perturbative nature.
2 When restricted to graphs, these conditions boil down tomomentum space Taylor subtraction and forest formula.
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, integrability & positivity 2
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 approximations
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, 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 approximations
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, 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 approximations
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, integrability & positivity 3
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: renormalisation, integrability & positivity 4
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 + n
V2D
)δmn), V = ( θ4)D/2
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, 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 + n
V2D
)δmn), V = ( θ4)D/2
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, integrability & positivity 5
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
2 ] ∈ 2,4 2[D2 ] = 6 2[D
2 ] > 6Φ4 D < 2 2[D
2 ] = 2 2[D2 ] = 4 2[D
2 ] > 4
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, 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
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: renormalisation, integrability & positivity 6
Introduction Matricial QFT Schwinger functions
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!
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, integrability & positivity 7
Introduction Matricial QFT Schwinger functions
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!
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, integrability & positivity 7
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 function
Here we are rescued:1 G|a|a| comes with 1
V 2 , goes away in limit V 2/D ∼ θ →∞2 G|am| expressable in terms of G|a|,G|m| thanks to
Ward-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: renormalisation, 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 terms 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: renormalisation, 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 terms 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: renormalisation, integrability & positivity 8
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 equation from Virasoro constraint in Kontsevich model: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 ))
Ansatz for G(x) =: 12λ(W (X )−
√X )
W (X ) =
√X + c√
Z− ν +
12
∫ b
a
dY ρ(Y )
(√
X + c +√
Y + c)√
Y + cnormalisation conditions on G... translate toW (1) = 1︸ ︷︷ ︸
D≥2
, W ′(1) =d
dX
√X∣∣∣X=1
=12︸ ︷︷ ︸
D≥4
, W ′′(1) =d2
dX 2
√X∣∣∣X=1
= −14︸ ︷︷ ︸
D=6
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, 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 equation from Virasoro constraint in Kontsevich model: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 ))
Ansatz for G(x) =: 12λ(W (X )−
√X )
W (X ) =
√X + c√
Z− ν +
12
∫ b
a
dY ρ(Y )
(√
X + c +√
Y + c)√
Y + cnormalisation conditions on G... translate toW (1) = 1︸ ︷︷ ︸
D≥2
, W ′(1) =d
dX
√X∣∣∣X=1
=12︸ ︷︷ ︸
D≥4
, W ′′(1) =d2
dX 2
√X∣∣∣X=1
= −14︸ ︷︷ ︸
D=6Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, integrability & positivity 9
Introduction Matricial QFT Schwinger functions
Solution of renormalised equation for D = 6
1√Z [Λ]
=√
1 + c +12
∫ Λ
1dT
ρ(T )
(√
1 + c +√
T + c)2√
T + c
⇒ Z ∈ [0, 1]for λ ∈ R(see LSZ)
−c =
∫ ∞1
dT ρ(T )
(√
1 + c +√
T + c)3√
T + c
W (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
βλ := Λ2 dλbare(Λ(Λ))
dΛ2 =2λ3Λ6(√
1+c +√
(2e(Λ2)+1)2+c)2√
(2e(Λ2)+1)2+c> 0
Perturbative expansion for e(x) = x , ρ(T ) = λ2(√
T−1)2
4√
T
c = −2 log 2− 14
λ2 +(2 log 2− 1)(4 log 2− 3)
32λ4 +O(λ6)
G(x) =λ
4(2x + 1)
(2(1 + x)2 log(1 + x)− x(2 + 3x)
)+
λ3
16(2x + 1)3
(x3(2 + 3x)(2 log 2− 1)2)+O(λ5)
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, integrability & positivity 10
Introduction Matricial QFT Schwinger functions
Solution of renormalised equation for D = 6
1√Z [Λ]
=√
1 + c +12
∫ Λ
1dT
ρ(T )
(√
1 + c +√
T + c)2√
T + c
⇒ Z ∈ [0, 1]for λ ∈ R(see LSZ)
−c =
∫ ∞1
dT ρ(T )
(√
1 + c +√
T + c)3√
T + c
W (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
βλ := Λ2 dλbare(Λ(Λ))
dΛ2 =2λ3Λ6(√
1+c +√
(2e(Λ2)+1)2+c)2√
(2e(Λ2)+1)2+c> 0
Perturbative expansion for e(x) = x , ρ(T ) = λ2(√
T−1)2
4√
T
c = −2 log 2− 14
λ2 +(2 log 2− 1)(4 log 2− 3)
32λ4 +O(λ6)
G(x) =λ
4(2x + 1)
(2(1 + x)2 log(1 + x)− x(2 + 3x)
)+
λ3
16(2x + 1)3
(x3(2 + 3x)(2 log 2− 1)2)+O(λ5)
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, 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
3
Fa = renormalisation of Ea
= λN1+···+NB−BN1∑
k1=1
· · ·NB∑
kB=1
G|a1k1|...|aB
kB|
B∏β=1
Nβ∏lβ=1
lβ 6=kβ
1F 2
aβkβ− F 2
aβlβ
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 ) = limΛ→∞
( 1√Z (λ)
−∫ Λ
1
dTρ(T )√T + c
1
(√
T + c +√
T + c − 2t)√
T + c − 2t
)Proof: ansatz for recursion and experience with Bell polynomials
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, integrability & positivity 11
Introduction Matricial QFT Schwinger functions
Higher correlation functions
. . . satisfy linear integral equations, easily reduced to (1+ . . .+1):
G|a11...a
1N1|...|aB
1 ...aBNB| = λN1+···+NB−B
N1∑k1=1
· · ·NB∑
kB=1
G|a1k1|...|aB
kB|
B∏β=1
Nβ∏lβ=1
lβ 6=kβ
1F 2
aβkβ− F 2
aβlβ
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 ) = limΛ→∞
( 1√Z (λ)
−∫ Λ
1
dTρ(T )√T + c
1
(√
T + c +√
T + c − 2t)√
T + c − 2t
)
Proof: ansatz for recursion and experience with Bell polynomials
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, integrability & positivity 11
Introduction Matricial QFT Schwinger functions
Higher correlation functions
. . . satisfy linear integral equations, easily reduced to (1+ . . .+1):
G|a11...a
1N1|...|aB
1 ...aBNB| = λN1+···+NB−B
N1∑k1=1
· · ·NB∑
kB=1
G|a1k1|...|aB
kB|
B∏β=1
Nβ∏lβ=1
lβ 6=kβ
1F 2
aβkβ− F 2
aβlβ
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 ) = limΛ→∞
( 1√Z (λ)
−∫ Λ
1
dTρ(T )√T + c
1
(√
T + c +√
T + c − 2t)√
T + c − 2t
)Proof: ansatz for recursion and experience with Bell polynomials
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, integrability & positivity 11
Introduction Matricial QFT Schwinger functions
Simplest 6D-ribbon graph with overlapping divergence
y2y3
x ••
•=
(−λ)3
(2x+1)
∫ ∞0
y23 dy3
2
∫ ∞0
y22 dy2
2
1
(x+y3+1)2(y3+y2+1)(x+y2+1)
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, integrability & positivity 12
Introduction Matricial QFT Schwinger functions
Zimmermann’s forest formula
y2y3
x ••
•=
(−λ)3
(2x+1)
∫ ∞0
y23 dy3
2
∫ ∞0
y22 dy2
2
[1
(x+y3+1)2(y3+y2+1)(x+y2+1)
]∅
+
[(− 1
(y3+1)3
) 1x+y2+1
]3
+
[1
(x+y3+1)2
(− 1
(y2+1)2 +y3 + x
(y2+1)3
)]2
+
[1
(y3+y2+1)
(− 1
(y3+1)2(y2+1)+
2x(y3+1)3(y2+1)
+x
(y3+1)2(y2+1)2
− 3x2
(y3+1)4(y2+1)− x2
(y3+1)2(y2+1)3 −2x2
(y3+1)3(y2+1)2
)]1
+
[(− 1
(y3+1)3
)(− 1
y2+1+
x(y2+1)2 −
x2
(y2+1)3
)]13
+
[((− 1
(y3+1)2 +2x
(y3+1)3 −3x2
(y3+1)4
)(− 1
(y2+1)2 +y3
(y2+1)3
)+(− 1
(y3+1)2 +2x
(y3+1)3
)( x(y2+1)3
))]12
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, integrability & positivity 12
Introduction Matricial QFT Schwinger functions
Zimmermann’s forest formula
y2y3
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
)
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, integrability & positivity 12
Introduction Matricial QFT Schwinger functions
Zimmermann’s forest formula
y2y3
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: renormalisation, integrability & positivity 12
Introduction Matricial QFT Schwinger functions
Schwinger functionsundo the passage to the fmn-matrix basis of Moyal space:
Theorem [HG+RW, 2013]: connected Schwinger functions
ScN(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)
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, integrability & positivity 13
Introduction Matricial QFT Schwinger functions
Schwinger functionsundo the passage to the fmn-matrix basis of Moyal space:
Theorem [HG+RW, 2013]: connected Schwinger functions
ScN(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)
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, integrability & positivity 13
Introduction Matricial QFT Schwinger functions
Schwinger functionsundo the passage to the fmn-matrix basis of Moyal space:
Theorem [HG+RW, 2013]: connected Schwinger functions
ScN(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)
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, integrability & positivity 13
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.
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 measure3 F is completely monotonic, (−1)k1+···+kN∂k1
t1 . . . ∂kNtN F (t) ≥ 0
∗This is 60% of the proof of the Osterwalder-Schrader theorem.
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, 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.
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 measure3 F is completely monotonic, (−1)k1+···+kN∂k1
t1 . . . ∂kNtN F (t) ≥ 0
∗This is 60% of the proof of the Osterwalder-Schrader theorem.
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, 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.
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) ≥ 0
2 F is the joint Laplace transform of a positive measure3 F is completely monotonic, (−1)k1+···+kN∂k1
t1 . . . ∂kNtN F (t) ≥ 0
∗This is 60% of the proof of the Osterwalder-Schrader theorem.
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, 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.
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 60% of the proof of the Osterwalder-Schrader theorem.
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, 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.
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 measure3 F is completely monotonic, (−1)k1+···+kN∂k1
t1 . . . ∂kNtN F (t) ≥ 0
∗This is 60% of the proof of the Osterwalder-Schrader theorem.
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, 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.
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 60% of the proof of the Osterwalder-Schrader theorem.Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, integrability & positivity 14
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: renormalisation, 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: renormalisation, 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: renormalisation, integrability & positivity 15
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: renormalisation, 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: renormalisation, 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: renormalisation, integrability & positivity 16
Introduction Matricial QFT Schwinger functions
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,∞[
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, integrability & positivity 17
Introduction Matricial QFT Schwinger functions
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,∞[
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, integrability & positivity 17
Introduction Matricial QFT Schwinger functions
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: renormalisation, integrability & positivity 18
Introduction Matricial QFT Schwinger functions
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: renormalisation, integrability & positivity 18
Introduction Matricial QFT Schwinger functions
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: renormalisation, integrability & positivity 18
Introduction Matricial QFT Schwinger functions
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: renormalisation, integrability & positivity 18
Backup: 2-point function G(x , y) of Φ44
after renormalisation in large-(V ,N ) limit:
1 λx∫ ∞
0
G(x ,0)G(p, y)−G(p,0)G(x , y)
p − x= (1 + yG(x ,0))G(x , y)− (1 + y)G(x ,0)G(0, y)
2 1 + λ∫∞
0 dp(G(p, y)−G(p,0)) = (1 + y)G(0, y)
3 G(x , y) = G(y , x)
using Riemann-Hilbert techniques we solved (1)+(2) up toone unknown function
one-sided Hilbert transform Ha(f ) =1πP∫ ∞
0
f (p) dpp−a
arises
remains (3): a single integral equation G(x ,0) = G(0, x)
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, integrability & positivity 19
Backup: 2-point function G(x , y) of Φ44
after renormalisation in large-(V ,N ) limit:
1 λx∫ ∞
0
G(x ,0)G(p, y)−G(p,0)G(x , y)
p − x= (1 + yG(x ,0))G(x , y)− (1 + y)G(x ,0)G(0, y)
2 1 + λ∫∞
0 dp(G(p, y)−G(p,0)) = (1 + y)G(0, y)
3 G(x , y) = G(y , x)
using Riemann-Hilbert techniques we solved (1)+(2) up toone unknown function
one-sided Hilbert transform Ha(f ) =1πP∫ ∞
0
f (p) dpp−a
arises
remains (3): a single integral equation G(x ,0) = G(0, x)
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, integrability & positivity 19
Solution of λφ44 on extreme Moyal space
Theorem (2012/13)
Given boundary function G(x ,0),
define τy (x) := arctan[0, π]
(|λ|πx
y + 1+λπxHx [G(•,0)]G(x,0)
). Then
G(x , y)=sin(τy (x))
|λ|πxesign(λ)(H0[τ0(•)]−Hx [τy (•)])
1 λ<0(
1+ Cx+yF (y)
Λ2−x
)λ>0
From symmetry G(x ,0) = G(0, x):
Fixed point equation for boundary function (assuming λ < 0)
G(x ,0)=1
1+xexp
(−λ∫ x
0dt∫ ∞
0
dp
(λπp)2 +(t+ 1+λπpHp[G(•,0)]
G(p,0)
)2
)
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, integrability & positivity 20
Fixed point theorem
Reflection positivity = Stieltjes property is excluded for λ > 0
Theorem [H.Grosse+RW, 2015]
Let −16 ≤ λ ≤ 0. Then the equation has a C1
0 -solution1
(1+x)1−|λ| ≤ G(x ,0) ≤ 1
(1+x)1− |λ|
1−2|λ|
5 10 15 20
0.2
0.4
0.6
0.8
1.0
λ = − 12π
proof via Schauder fixed pointtheoremcompactness via Arzela-AscoliBanach is slightly missed:‖Tf − Tg‖ ≤(1 + 1
e +O(λ))‖f − g‖need exact asymptotics!
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, integrability & positivity 21
Approximation by 4F3 hypergeometric functionansatz G(x ,0) = 4F3(a,b1,b2,b3
c1,c2,c3| − x); matching a,bi , ci at one
point x result in global error supx | . . . | ≈ 10−8 in fixed point eq.
G(x , 0) = 4F3(. . . |−x)G( x
2,x2 )
Stieltjes measure ρfor G(x , 0) =
∫∞0 dt ρ(t)/(t + x)
at λ = −0.1
··········································
x x
λ = −0.1
reflection positivity equivalent to existence of a blue curve on theright whose Stieltjes transform is G(x
2 ,x2 ) on the left
measure for G(x ,0) (and almost surely for G(x2 ,
x2 )) has
mass gap [0,1[, but no further gap (remnant of UV/IR-mixing)absence of the second gap (usually ]1,4[) circumventstriviality theorems
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, integrability & positivity 22
Approximation by 4F3 hypergeometric functionansatz G(x ,0) = 4F3(a,b1,b2,b3
c1,c2,c3| − x); matching a,bi , ci at one
point x result in global error supx | . . . | ≈ 10−8 in fixed point eq.
G(x , 0) = 4F3(. . . |−x)G( x
2,x2 )
Stieltjes measure ρfor G(x , 0) =
∫∞0 dt ρ(t)/(t + x)
at λ = −0.1
··········································
x x
λ = −0.1
reflection positivity equivalent to existence of a blue curve on theright whose Stieltjes transform is G(x
2 ,x2 ) on the left
measure for G(x ,0) (and almost surely for G(x2 ,
x2 )) has
mass gap [0,1[, but no further gap (remnant of UV/IR-mixing)absence of the second gap (usually ]1,4[) circumventstriviality theorems
Raimar Wulkenhaar (Munster) Matricial quantum field theory: renormalisation, integrability & positivity 22