Matricial aquantum field theory: renormalisation, integrability & positivity

Transcription

1 Matricial aquantum field theory: renormalisation, integrability & positivity Raimar Wulkenhaar Mathematisches Institut, Westfälische Wilhelms-Universität Münster based on arxiv: & with Harald Grosse and Akifumi Sako and arxiv: , , , & with Harald Grosse Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 0

2 Goal: Quantum Field Theory satisfying axioms 932: axioms for quantum mechanics [von Neumann] 950 s: unique extension to quantum fields [Wightman] = unbounded op.-valued distributions f Φ(f ) : D D H Theorem: vacuum expectation values Ω, Φ(x ) Φ(x N )Ω are boundary values of holomorphic functions Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity

3 Goal: Quantum Field Theory satisfying axioms 932: axioms for quantum mechanics [von Neumann] 950 s: unique extension to quantum fields [Wightman] = unbounded op.-valued distributions f Φ(f ) : D D H Theorem: vacuum expectation values Ω, Φ(x ) Φ(x N )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, Euclidean invariance, complete symmetry and reflection positivity Theorem [Osterwalder-Schrader 974] These properties are sufficient to reconstruct Wightman theory! Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity

4 Goal: Quantum Field Theory satisfying axioms 932: axioms for quantum mechanics [von Neumann] 950 s: unique extension to quantum fields [Wightman] = unbounded op.-valued distributions f Φ(f ) : D D H Theorem: vacuum expectation values Ω, Φ(x ) Φ(x N )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, Euclidean invariance, complete symmetry and reflection positivity Theorem [Osterwalder-Schrader 974] These properties are sufficient to reconstruct Wightman theory! So far no non-trivial QFT model in 4 dimensions... Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity

5 Selected techniques exactly solvable 2D-models (e.g. Thirring, Schwinger) candidate Schwinger functions as moments of perturbed Gaußian measure (e.g. P[φ] 2, φ 4 3, probably not φ4 4 ) fermionic summation techniques (e.g. Gross-Neveu 2 ) Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 2

6 Selected techniques exactly solvable 2D-models (e.g. Thirring, Schwinger) candidate Schwinger functions as moments of perturbed Gaußian measure (e.g. P[φ] 2, φ 4 3, probably not φ4 4 ) fermionic summation techniques (e.g. Gross-Neveu 2 ) for all realistic models (e.g. QED 4, Standard Model 4 ): renormalised perturbation theory BPHZ(L) Z= Wolfhart Zimmermann BPHZ(L) has two aspects: Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 2

7 Selected techniques exactly solvable 2D-models (e.g. Thirring, Schwinger) candidate Schwinger functions as moments of perturbed Gaußian measure (e.g. P[φ] 2, φ 4 3, probably not φ4 4 ) fermionic summation techniques (e.g. Gross-Neveu 2 ) for all realistic models (e.g. QED 4, Standard Model 4 ): renormalised perturbation theory BPHZ(L) Z= Wolfhart Zimmermann BPHZ(L) has two aspects: Renormalisation amounts to normalisation conditions for relevant/marginal correlation functions. These conditions are of non-perturbative nature. Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 2

8 Selected techniques exactly solvable 2D-models (e.g. Thirring, Schwinger) candidate Schwinger functions as moments of perturbed Gaußian measure (e.g. P[φ] 2, φ 4 3, probably not φ4 4 ) fermionic summation techniques (e.g. Gross-Neveu 2 ) for all realistic models (e.g. QED 4, Standard Model 4 ): renormalised perturbation theory BPHZ(L) Z= Wolfhart Zimmermann BPHZ(L) has two aspects: Renormalisation amounts to normalisation conditions for relevant/marginal correlation functions. These conditions are of non-perturbative nature. 2 When restricted to graphs, these conditions boil down to momentum space Taylor subtraction and forest formula. Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 2

9 Matricial quantum field theory... is the marriage of matrix models for 2D quantum gravity 2 QFT on noncommutative spaces Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 3

10 Matricial quantum field theory... is the marriage of matrix models for 2D quantum gravity 2 QFT on noncommutative spaces Kontsevich model (992) designed to prove Witten s conjecture that hermitean one-matrix model computes intersection numbers of stable cohomology classes on the moduli space of complex curves Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 3

11 Matricial quantum field theory... is the marriage of matrix models for 2D quantum gravity 2 QFT on noncommutative spaces Kontsevich model (992) designed to prove Witten s conjecture that hermitean one-matrix model computes intersection numbers of stable cohomology classes on the moduli space of complex curves 2 Space-time should become a noncommutative manifold at short distances. Euclidean scalar field φ A (noncommutative algebra) A often has finite-dimensional approximations Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 3

12 The Kontsevich model defined by partition function ( dφ exp Tr ( EΦ 2 + i 6 Φ3)) Z(E) := ( dφ exp Tr ( EΦ 2)) Asymptotic expansion in coupling constant i 6 gives rational function of eigenvalues e i of E. This rational function generates the intersection numbers. Related to Hermitean one-matrix model Z(E)[[t n ]] = DM exp( N t n tr(m n )) n where t n := (2n )!!tr(e (2n ) ) Large-N limit gives KdV evolution equation. Exact solution related to Virasoro algebra. Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 4

13 QFT on noncommutative geometries Example: Moyal algebra = Rieffel deformation of C (R 2 ) ( ) dη dk 0 θ (f g)(ξ) = R 2 R (2π) 2 f (x+ 2 Θk) g(ξ+η) ei k,η Θ = θ 0 2 matrix basis φ(ξ) = m,n=0 Φ mnf mn (ξ) ( ) n ml ( f mn (ξ) = 2( ) m m! 2 n! θ ξ n m 2 ξ 2 +iξ 2 m θ )e ξ 2 θ satisies f mn f kl = δ nk f ml and dξ 8π f mn(ξ) = θ 4 δ mn Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 5

14 QFT on noncommutative geometries Example: Moyal algebra = Rieffel deformation of C (R 2 ) ( ) dη dk 0 θ (f g)(ξ) = R 2 R (2π) 2 f (x+ 2 Θk) g(ξ+η) ei k,η Θ = θ 0 2 matrix basis φ(ξ) = m,n=0 Φ mnf mn (ξ) ( ) n ml ( f mn (ξ) = 2( ) m m! 2 n! θ ξ n m 2 ξ 2 +iξ 2 m θ )e ξ 2 θ satisies f mn f kl = δ nk f ml and dξ 8π f mn(ξ) = θ 4 δ mn Consider scalar field theories on Moyal space ( S(φ) := (8π) D/2 dξ R D 2 φ ( +4Ω2 Θ ξ 2 ) φ + tr(pol(φ))) f mn -expansion at Ω = yields Kontsevich-type matrix model S(Φ) = V tr(eφ 2 + pol(φ)), E = ( ( µ2 2 + n V 2 D )δ mn ), V = ( θ 4 )D/2 Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 5

15 Two independent dimensions Topological dimension 2 from expansion of matrix models into ribbon graphs, i.e. simplicial 2-complexes. dual to triangulations (Φ 3 ) or quadrangulations (Φ 4 ) of 2D-surfaces partition function counts them = 2D quantum gravity non-planar ribbon graphs suppressed in large-n limit Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 6

16 Two independent dimensions Topological dimension 2 from expansion of matrix models into ribbon graphs, i.e. simplicial 2-complexes. dual to triangulations (Φ 3 ) or quadrangulations (Φ 4 ) of 2D-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 the unbounded positive operator E, D = inf{p R + : tr(( + E) p 2 ) < } ignored in 2D quantum gravity highly relevant for renormalisation of matricial QFT polynomial finite super-ren just ren. not ren. Φ 3 D < 2 2[ D 2 ] {2, 4} 2[ D 2 ] = 6 2[ D 2 ] > 6 Φ 4 D < 2 2[ D 2 ] = 2 2[ D 2 ] = 4 2[ D 2 ] > 4 Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 6

17 Φ 3 6 matricial QFT log Z(J) Z(0) = action S(Φ) = V tr(zeφ 2 + (κ+νe+ζe 2 )Φ + λ barez Φ 3 ) ( µ 2 for E = bare 2 + µ 2 e ( n ) ) δmn, m, n N D/2 µ 2 V 2/D µ bare, λ bare, Z, κ, ν, ζ to be fixed by normalisation conditions partition function Z(J) = dφ exp( S(Φ) + V tr(φj)) B= N B N V 2 B S N...N B G p...p N... p B...pB N B B β= ( N β j β = J p β j β p β j β + ) cycl Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 7

18 Φ 3 6 matricial QFT log Z(J) Z(0) = action S(Φ) = V tr(zeφ 2 + (κ+νe+ζe 2 )Φ + λ barez Φ 3 ) ( µ 2 for E = bare 2 + µ 2 e ( n ) ) δmn, m, n N D/2 µ 2 V 2/D µ bare, λ bare, Z, κ, ν, ζ to be fixed by normalisation conditions partition function Z(J) = dφ exp( S(Φ) + V tr(φj)) B= N B N V 2 B S N...N B G p...p N... p B...pB N B Strategy Z(J) is meaningless for λ R! B β= ( N β j β = J p β j β p β j β + Z(J) is only used as tool to derive identities (Schwinger-Dyson equations) between G p...p N... p B...pB N B Forget Z, declare SD-equations as exact and search for rigorous solutions G... of them! Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 7 ) cycl

19 Schwinger-Dyson equations Inserting Z(J) = exp ( Z 3/2 λ bare ) J kl J lm J Z 2 mk (J) into G a log Z[J] J=0 V J aa gives equation quadratic in G a, linear in m G am and G a a typical feature: SD-equation for n-point function depends on (m > n)-point function 3V 2 3 Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 8

20 Schwinger-Dyson equations Inserting Z(J) = exp ( Z 3/2 λ bare ) J kl J lm J Z 2 mk (J) into G a log Z[J] J=0 V J aa gives equation quadratic in G a, linear in m G am and G a a typical feature: SD-equation for n-point function depends on (m > n)-point function 3V 2 3 Here we are rescued: G a a comes with V 2, goes away in limit V 2/D θ Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 8

21 Schwinger-Dyson equations Inserting Z(J) = exp ( Z 3/2 λ bare ) J kl J lm J Z 2 mk (J) into G a log Z[J] J=0 V J aa gives equation quadratic in G a, linear in m G am and G a a typical feature: SD-equation for n-point function depends on (m > n)-point function 3V 2 3 Here we are rescued: G a a comes with 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) 2 Z[J] = ( ) V J an J nb Z[J] J n bn J na Z (E n a E b ) J bn J na V Z (ν + ζ(e a + E b )) Z[J] (for a b) J ba Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 8

22 Scaling limit N, V with N V 2/D = µ 2 Λ 2 fixed Non-linear integral equation for G(x) = µ D/2 G a a =V 2/D µ 2 x similar to equation from Virasoro constraint in Kontsevich model: Theorem [Makeenko-Semenoff 99] W 2 (X) + b a W (X) W (Y ) dy ρ(y ) = X + const X Y b a is solved by W (X) = X + c + 2 together with a consistency condition on c. dy ρ(y ) ( X+c+ Y +c) Y +c Identification X = (2e(x) + ) 2, ρ(y ) = 2λ2 (e ( Y )) 2 D/2 Γ(D/2) Y e (e ( Y )) 2 Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 9

23 Scaling limit N, V with N V 2/D = µ 2 Λ 2 fixed Non-linear integral equation for G(x) = µ D/2 G a a =V 2/D µ 2 x similar to equation from Virasoro constraint in Kontsevich model: Theorem [Makeenko-Semenoff 99] W 2 (X) + b a W (X) W (Y ) dy ρ(y ) = X + const X Y b a is solved by W (X) = X + c + 2 together with a consistency condition on c. dy ρ(y ) ( X+c+ Y +c) Y +c Identification X = (2e(x) + ) 2, ρ(y ) = 2λ2 (e ( Y )) 2 D/2 Γ(D/2) Y e (e ( Y )) 2 Ansatz for G(x) =: 2λ (W (X) X) X + c W (X) = ν + b dy ρ(y ) Z 2 a ( X + c + Y + c) Y + c normalisation conditions on G... translate to W () =, W () = d X= X =, W () = d 2 X= X = }{{}} dx {{ 2 }} dx 2 {{ 4 } D 2 D 4 D=6 Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 9

24 Solution of renormalised equation for D = 6 = + c + 2 Z [ Λ] c = Λ ρ(t ) dt ( + c + T + c) 2 T + c dt ρ(t ) ( + c + T + c) 3 T + c W (X)= (X+c)(+c) c + 2 Z [0, ] for λ R (see LSZ) dt ρ(t ) ( X+c +c) 2 ( X+c+ T +c)( +c+ T +c) 2 T +c β λ := Λ 2 dλ bare ( Λ(Λ)) 2λ 3 Λ 6 = dλ 2 ( +c + (2e(Λ2 )+) 2 +c ) 2 (2e(Λ 2 )+) 2 +c > 0 Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 0

25 Solution of renormalised equation for D = 6 = + c + 2 Z [ Λ] c = Λ ρ(t ) dt ( + c + T + c) 2 T + c dt ρ(t ) ( + c + T + c) 3 T + c W (X)= (X+c)(+c) c + 2 Z [0, ] for λ R (see LSZ) dt ρ(t ) ( X+c +c) 2 ( X+c+ T +c)( +c+ T +c) 2 T +c β λ := Λ 2 dλ bare ( Λ(Λ)) 2λ 3 Λ 6 = dλ 2 ( +c + (2e(Λ2 )+) 2 +c ) 2 (2e(Λ 2 )+) 2 +c > 0 Perturbative expansion for e(x) = x, ρ(t ) = λ2 ( T ) 2 4 T c = 2 log 2 λ 2 (2 log 2 )(4 log 2 3) + λ 4 + O(λ 6 ) 4 32 λ ( G(x) = 2( + x) 2 log( + x) x(2 + 3x) ) 4(2x + ) λ 3 ( + x 3 (2 + 3x)(2 log 2 ) 2) + O(λ 5 ) 6(2x + ) 3 Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 0

26 Higher correlation functions... satisfy linear integral equations, easily reduced to (+... +): G a...a N... a B...aB N B = λ N + +N B B N k = N B G a k... a B k B k B = β= W ak if B= B N β F 2 F 2 l β = a β a β k l β k β β l β F a = renormalisation of E a Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity

27 Higher correlation functions... satisfy linear integral equations, easily reduced to (+... +): G a...a N... a B...aB N B = λn + +N B B N k = Proposition 4λ 2 G(X Y ) = X + c Y + c ( X + c + Y + c) 2 B N B N β G a k... ak B B F 2 k B = β= l β = a β k l β k β β F 2 a β l β G(X... X B ) = d B 3 dt B 3 ( ( 2λ) 3B 4 (R(t)) B 2 X +c 2t 3 X B +c 2t 3 ) t=0 R(T ) = lim Λ Λ ) dt ρ(t ) Z ( λ) T + c ( T + c + T + c 2t) T + c 2t ( Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity

28 Higher correlation functions... satisfy linear integral equations, easily reduced to (+... +): G a...a N... a B...aB N B = λn + +N B B N k = Proposition 4λ 2 G(X Y ) = X + c Y + c ( X + c + Y + c) 2 B N B N β G a k... ak B B F 2 k B = β= l β = a β k l β k β β F 2 a β l β G(X... X B ) = d B 3 dt B 3 ( ( 2λ) 3B 4 (R(t)) B 2 X +c 2t 3 X B +c 2t 3 ) t=0 R(T ) = lim Λ Λ ) dt ρ(t ) Z ( λ) T + c ( T + c + T + c 2t) T + c 2t ( Proof: ansatz for recursion and experience with Bell polynomials Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity

29 Simplest 6D-ribbon graph with overlapping divergence x y 3 y 2 = ( λ)3 (2x+) 0 y 2 3 dy y 2 2 dy 2 2 { } (x+y 3 +) 2 (y 3 +y 2 +)(x+y 2 +) Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 2

30 Zimmermann s forest formula x y 3 y 2 = ( λ)3 y3 2dy { [ 3 y2 2dy ] 2 (2x+) (x+y 3 +) 2 (y 3 +y 2 +)(x+y 2 +) [ ( ) ] [ ( + (y 3 +) 3 + x+y (x+y 3 +) 2 (y 2 +) 2 + y 3 + x (y 2 +) )]2 3 [ ( + (y 3 +y 2 +) (y 3 +) 2 (y 2 +) + 2x (y 3 +) 3 (y 2 +) + x (y 3 +) 2 (y 2 +) 2 3x 2 (y 3 +) 4 (y 2 +) x 2 (y 3 +) 2 (y 2 +) 3 2x 2 (y 3 +) 3 (y 2 +) )] 2 [ ( )( + (y 3 +) 3 y x (y 2 +) 2 x 2 (y 2 +) )]3 3 +[ (( (y 3 +) 2 + 2x (y 3 +) 3 3x 2 ( + )( (y 3 +) 4 (y 3 +) 2 + 2x (y 2 +) 2 + y 3 (y 2 +) 3 ) (y 3 +) 3 )( x (y 2 +) 3 ))]2 Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 2 }

31 Zimmermann s forest formula x y 3 y 2 λ 3 { = 4(2x+) 3 (x+)(2x+)(3x+2) log(+x) + (x+) 3 (3x+)(log(+x)) 2 ( ( )) + x(+x)(+3x+3x 2 ) (log(+x)) 2 2 log(+x) log x + 2Li 2 +x } 3x 3 (2+3x)ζ(2) + λ3 x ( ζ(2) + x ) 2(2x+) 2 Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 2

32 Zimmermann s forest formula x y 3 y 2 λ 3 { = 4(2x+) 3 (x+)(2x+)(3x+2) log(+x) + (x+) 3 (3x+)(log(+x)) 2 ( ( )) + x(+x)(+3x+3x 2 ) (log(+x)) 2 2 log(+x) log x + 2Li 2 +x } 3x 3 (2+3x)ζ(2) + λ3 x ( ζ(2) + x ) 2(2x+) 2 x adding: y y x y 2 y2 y 2 x y gives the λ 3 -order of the exact formula for G(x)! Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 2

33 Schwinger functions undo the passage to the f mn -matrix basis of Moyal space: Theorem [HG+RW, 203]: connected Schwinger functions SN c (µξ,..., µξ N ) := lim f m n (ξ ) f mn n V µ 2 N (ξ N ) (V µ2 ) 2 µ 3N N log Z(J) J m n... J mn n N m i,n i =0 J=0 Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 3

34 Schwinger functions undo the passage to the f mn -matrix basis of Moyal space: Theorem [HG+RW, 203]: connected Schwinger functions SN c (µξ,..., µξ N ) := lim f m n (ξ ) f mn n V µ 2 N (ξ N ) (V µ2 ) 2 µ 3N N log Z(J) J m n m i,n i =0... J mn n N J=0 = ( B 2 DN β ) 2 dp β e i p β, N β i= ( )i ξ σ(n +...+N β +i) N N +...+N B =N β σ S N β= R D (2πµ 2 ) D 2 N β even 2 G( p,, p ) 2 (8π) D 2µ 2 S 2 2µ... p B 2,, p B 2 2 2µ N...N B }{{} 2 2µ }{{ 2 } N N B Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 3

35 Schwinger functions undo the passage to the f mn -matrix basis of Moyal space: Theorem [HG+RW, 203]: connected Schwinger functions SN c (µξ,..., µξ N ) := lim f m n (ξ ) f mn n V µ 2 N (ξ N ) (V µ2 ) 2 µ 3N N log Z(J) J m n m i,n i =0... J mn n N J=0 = ( B 2 DN β ) 2 dp β e i p β, N β i= ( )i ξ σ(n +...+N β +i) N N +...+N B =N β σ S N β= R D (2πµ 2 ) D 2 N β even 2 G( p,, p ) 2 (8π) D 2µ 2 S 2 2µ... p B 2,, p B 2 2 2µ N...N B }{{} 2 2µ }{{ 2 } N N B Confinement of noncommutativity: have internal interaction of matrices; commutative subsector propagates to outside world Schwinger functions are symmetric and invariant under full Euclidean group (completely unexpected for NCQFT!) remains: reflection positivity (... and non-triviality) Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 3

36 Reflection positivity S( f r f ) 0 f stands for sequences of test functions of complicated support f r (τ, ξ) = f ( τ, ξ) is time reflection Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 4

37 Reflection positivity S( f r f ) 0 f stands for sequences of test functions of complicated support f r (τ, ξ) = f ( 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. Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 4

38 Reflection positivity S( f r f ) 0 f stands for sequences of test functions of complicated support f r (τ, ξ) = f ( 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, 92-92/28-94) For a [smooth] function F on (R + ) N t = (t,..., t N ) are equivalent: F is positive definite, i.e. K i,j= c ic j F(t i + t j ) 0 Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 4

39 Reflection positivity S( f r f ) 0 f stands for sequences of test functions of complicated support f r (τ, ξ) = f ( 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, 92-92/28-94) For a [smooth] function F on (R + ) N t = (t,..., t N ) are equivalent: F is positive definite, i.e. K i,j= c ic j F(t i + t j ) 0 2 F is the joint Laplace transform of a positive measure Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 4

40 Reflection positivity S( f r f ) 0 f stands for sequences of test functions of complicated support f r (τ, ξ) = f ( 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, 92-92/28-94) For a [smooth] function F on (R + ) N t = (t,..., t N ) are equivalent: F is positive definite, i.e. K i,j= c ic j F(t i + t j ) 0 2 F is the joint Laplace transform of a positive measure 3 F is completely monotonic, ( ) k + +k N k t... k N t N F(t) 0 Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 4

41 Reflection positivity S( f r f ) 0 f stands for sequences of test functions of complicated support f r (τ, ξ) = f ( 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, 92-92/28-94) For a [smooth] function F on (R + ) N t = (t,..., t N ) are equivalent: F is positive definite, i.e. K i,j= c ic j F(t i + t j ) 0 2 F is the joint Laplace transform of a positive measure 3 F is completely monotonic, ( ) k + +k N k t... k N t N F(t) 0 This is 60% of the proof of the Osterwalder-Schrader theorem. Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 4

42 Stieltjes functions Prototype for N = e ip0 t = ( ) 2πt (p 0 )+ p 2 +m 2 p 2 +m 2 2 K 2 (t p 2 + m 2 ) = πe t p 2 +m 2 p 2 +m 2 Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 5

43 Stieltjes functions Prototype for N = e ip0 t = ( ) 2πt (p 0 )+ p 2 +m 2 p 2 +m 2 Theorem 2 K 2 (t p 2 + m 2 ) = πe t p 2 +m 2 p 2 +m 2 Up to integration in m 2 with positive measure, (p 0 )+ p 2 +m 2 is the only function with positive definite Fourier transform for N =. ϱ(m 2 )dm 2 p 2 +m 2 p 2 0 forms the class of Stieltjes functions in QFT, ϱ(m 2 ) is the Källén-Lehmann spectral measure Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 5

44 Stieltjes functions Prototype for N = e ip0 t = ( ) 2πt (p 0 )+ p 2 +m 2 p 2 +m 2 Theorem 2 K 2 (t p 2 + m 2 ) = πe t p 2 +m 2 p 2 +m 2 Up to integration in m 2 with positive measure, (p 0 )+ p 2 +m 2 is the only function with positive definite Fourier transform for N =. Is ϱ(m 2 )dm 2 p 2 +m 2 p 2 0 forms the class of Stieltjes functions in QFT, ϱ(m 2 ) is the Källén-Lehmann spectral measure G( p 2 2µ 2, p 2 2µ 2 ) Stieltjes? We work on this for Φ 4 4 since 203. Have some analytic evidence, confirmed by computer, but no complete proof. For Φ 3 D we have the answer: Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 5

45 Reflection positivity of the 2-point function Theorem (Grosse-Sako-W 206) The Φ 3 D-matricial QFT is not reflection positive for λ ir. Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 6

46 Reflection positivity of the 2-point function Theorem (Grosse-Sako-W 206) The Φ 3 D-matricial QFT is not reflection positive for λ ir. 2 The Φ 3 D two-point function is reflection positive for D {4, 6} and some range of λ R, but not in D = 2. Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 6

47 Reflection positivity of the 2-point function Theorem (Grosse-Sako-W 206) The Φ 3 D-matricial QFT is not reflection positive for λ ir. 2 The Φ 3 D 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: { 2 log(+σ) ( p 2 ) G 2µ 2, p 2 6D= λ 2 σ + σ(σ ) tan 2 φ π tan φ ( +σ 2 tan 2 φ )( arctan [0,π] (σ tan φ) φ )} 2µ 2 4π(σ 2 dφ ) 0 σ 2 σ cos φ + p 2 µ 2 + λ2 t(t 2)/(t )3 dt, 4 2 t + p 2 µ 2 where σ := +c [, 2W ( 2 ) ] is the e inverse solution of λ 2 4(σ = 2 ) σ 2 2σ+2 log(+σ) [, 8W ( 2 e ) +2W ( 2 )] e Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 6

48 Källén-Lehmann measure: plots 3.0 D= legend σ λ ϱ() supp(ϱ) [0.9955,.0045] [2, [ [0.9859,.04] [2, [ [0.9553,.0447] [2, [ [0.8596,.404] [2, [ [0.7604,.2396] [2, [ [0.5834,.466] [2, [ [0.360,.6390] [2, [ [0.685,.835] [2, [ [0.0826,.974] [2, [ Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 7

49 Källén-Lehmann measure: plots 3.0 D= legend σ λ ϱ() supp(ϱ) [0.9955,.0045] [2, [ [0.9859,.04] [2, [ [0.9553,.0447] [2, [ [0.8596,.404] [2, [ [0.7604,.2396] [2, [ [0.5834,.466] [2, [ [0.2546,.7454] [2, [ [0.0835,.965] [2, [ [0.033,.9669] [2, [ Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 7

50 Reflection positivity of higher Schwinger functions? Connected Schwinger functions SN 4 c are not positive! Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 8

51 Reflection positivity of higher Schwinger functions? Connected Schwinger functions SN 4 c are not positive! Anyway too much, one needs positivity of FT of full functions p e.g. G( 2, p 2 q 2 ) G(, q 2 p 2 ) + G(, p 2 q 2, q 2 ) 2µ 2 2µ 2 2µ 2 2µ 2 2µ 2 2µ 2 2µ 2 2µ 2 Difficult for N = 4, but G(2 2 2) + G(2)G(2)G(2) is not positive. Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 8

52 Reflection positivity of higher Schwinger functions? Connected Schwinger functions SN 4 c are not positive! Anyway too much, one needs positivity of FT of full functions p e.g. G( 2, p 2 q 2 ) G(, q 2 p 2 ) + G(, p 2 q 2, q 2 ) 2µ 2 2µ 2 2µ 2 2µ 2 2µ 2 2µ 2 2µ 2 2µ 2 Difficult for N = 4, but G(2 2 2) + G(2)G(2)G(2) is not positive. Very probable conclusion The Φ 3 D matricial QFT does not satisfy Osterwalder-Schrader. Reason: Higher functions too much localised in p-space! p 2 already G(, p 2 ) C log( p 2 +µ 2 )+C 2 almost fails 2µ 2 2µ 2 p 2 +µ 2 Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 8

53 Reflection positivity of higher Schwinger functions? Connected Schwinger functions SN 4 c are not positive! Anyway too much, one needs positivity of FT of full functions p e.g. G( 2, p 2 q 2 ) G(, q 2 p 2 ) + G(, p 2 q 2, q 2 ) 2µ 2 2µ 2 2µ 2 2µ 2 2µ 2 2µ 2 2µ 2 2µ 2 Difficult for N = 4, but G(2 2 2) + G(2)G(2)G(2) is not positive. Very probable conclusion The Φ 3 D matricial QFT does not satisfy Osterwalder-Schrader. Reason: Higher functions too much localised in p-space! p 2 already G(, p 2 ) C log( p 2 +µ 2 )+C 2 almost fails 2µ 2 2µ 2 p 2 +µ 2 For Φ 4 p 2 4 we expect G(, p 2 C ) 2µ 2 2µ 2 ( p 2 +µ 2 ) π arcsin( λ π) Keeps us busy for the next time! (hope!) Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 8

54 Backup: 2-point function G(x, y) of Φ 4 4 after renormalisation in large-(v, N ) limit: G(x, 0)G(p, y) G(p, 0)G(x, y) λx 0 p x = ( + yg(x, 0))G(x, y) ( + y)g(x, 0)G(0, y) 2 + λ 0 dp(g(p, y) G(p, 0)) = ( + y)g(0, y) 3 G(x, y) = G(y, x) Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 9

55 Backup: 2-point function G(x, y) of Φ 4 4 after renormalisation in large-(v, N ) limit: G(x, 0)G(p, y) G(p, 0)G(x, y) λx 0 p x = ( + yg(x, 0))G(x, y) ( + y)g(x, 0)G(0, y) 2 + λ 0 dp(g(p, y) G(p, 0)) = ( + y)g(0, y) 3 G(x, y) = G(y, x) using Riemann-Hilbert techniques we solved ()+(2) up to one unknown function one-sided Hilbert transform H a (f ) = π P f (p) dp 0 p a arises remains (3): a single integral equation G(x, 0) = G(0, x) Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 9

56 Solution of λφ 4 4 on extreme Moyal space Theorem (202/3) Given boundary function ( G(x, 0), λ πx define τ y (x) := arctan [0, π] y + +λπxhx [G(,0)] G(x,0) G(x, y)= sin(τ y(x)) e sign(λ)(h 0[τ 0 ( )] H x [τ y ( )]) λ πx From symmetry G(x, 0) = G(0, x): ). Then { λ<0 ) λ>0 ( + Cx+yF(y) Λ 2 x Fixed point equation for boundary function (assuming λ < 0) G(x, 0)= ( +x exp λ x 0 dt 0 dp (λπp) 2 + ( t+ +λπphp[g(,0)] G(p,0) ) 2 ) Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 20

57 Fixed point theorem Reflection positivity = Stieltjes property is excluded for λ > 0 Theorem [H.Grosse+RW, 205] Let 6 λ 0. Then the equation has a C 0 -solution G(x, 0) (+x) λ λ (+x) 2 λ λ = 2π proof via Schauder fixed point theorem compactness via Arzelà-Ascoli Banach is slightly missed: Tf Tg ( + e + O(λ)) f g need exact asymptotics! Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 2

58 Approximation by 4F3 hypergeometric function ansatz G(x, 0) = 4 F 3 ( a,b,b 2,b 3 c,c 2,c 3 x); matching a, b i, c i at one point x result in global error sup x in fixed point eq. λ = 0. Stieltjes measure ρ for G(x, 0) = 0 dt ρ(t)/(t + x) at λ = 0. G(x, 0) = 4 F 3 (... x) G( x 2,x 2 ) x reflection positivity equivalent to existence of a blue curve on the right whose Stieltjes transform is G( x 2, x 2 ) on the left x Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 22

59 Approximation by 4F3 hypergeometric function ansatz G(x, 0) = 4 F 3 ( a,b,b 2,b 3 c,c 2,c 3 x); matching a, b i, c i at one point x result in global error sup x in fixed point eq. λ = 0. Stieltjes measure ρ for G(x, 0) = 0 dt ρ(t)/(t + x) at λ = 0. G(x, 0) = 4 F 3 (... x) G( x 2,x 2 ) x reflection positivity equivalent to existence of a blue curve on the right whose Stieltjes transform is G( x 2, x 2 ) on the left measure for G(x, 0) (and almost surely for G( x 2, x 2 )) has mass gap [0, [, but no further gap (remnant of UV/IR-mixing) absence of the second gap (usually ], 4[) circumvents triviality theorems Raimar Wulkenhaar (Münster) Matricial quantum field theory: renormalisation, integrability & positivity 22 x