A solvable quantum field theory in 4 dimensions

A solvable quantum field theory in 4 dimensions Harald Grosse Faculty of Physics, University of Vienna (based on joint work with Raimar Wulkenhaar, arxiv: 1205.0465, 1306.2816, 1402.1041, 1406.7755 & 1505.05161) Harald Grosse (Vienna) A solvable quantum field theory in 4 dimensions 1

Introduction Prove that a non-trivial toy model for a quantum field theory on R 4 exists and satisfies [O-S, Wightman]. Φ 4 4 is renormalizable, Φ4 4+ǫ trivial Φ 4 4 on Moyal space is nonrenormalizable due to IR/UV mixing (Φ 4 4 ) modified on Moyal space is renormalizable (HG+R Wulkenhaar, 2004) β function is perturbative zero (Rivasseau et al, 2006) Ward identities allow to decouple SD equs. Can we construct it? Harald Grosse (Vienna) A solvable quantum field theory in 4 dimensions 2

Regularisation of λφ 4 4 on noncommutative space ( 1 S[φ]= dx R 4 2 φ ( +µ 2 ) λ ) φ+ 4 φ φ φ φ (x) Harald Grosse (Vienna) A solvable quantum field theory in 4 dimensions 3

Regularisation of λφ 4 4 on noncommutative space ( 1 S[φ]= dx R 4 2 φ ( +Ω 2 ( x) 2 +µ 2 ) λ ) φ+ 4 φ φ φ φ (x) Harald Grosse (Vienna) A solvable quantum field theory in 4 dimensions 3

Regularisation of λφ 4 4 on noncommutative space ( 1 S[φ]= dx R 4 2 φ ( +Ω 2 (2Θ 1 x) 2 +µ 2 ) λ ) φ+ 4 φ φ φ φ (x) with Moyal product (f g)(x) = R 4 R 4 dy dk (2π) 4 f(x+ 1 2Θk) g(x+y) ei k,y Harald Grosse (Vienna) A solvable quantum field theory in 4 dimensions 3

Regularisation of λφ 4 4 on noncommutative space ( ZΛ S[φ]= dx R 4 2 φ ( +Ω 2 (2Θ 1 x) 2 +µ 2 ) λz 2 ) bare φ+ Λ 4 φ φ φ φ (x) with Moyal product (f g)(x) = R 4 R 4 dy dk (2π) 4 f(x+ 1 2Θk) g(x+y) ei k,y Harald Grosse (Vienna) A solvable quantum field theory in 4 dimensions 3

Regularisation of λφ 4 4 on noncommutative space ( ZΛ S[φ]= dx R 4 2 φ ( +Ω 2 (2Θ 1 x) 2 +µ 2 ) λz 2 ) bare φ+ Λ 4 φ φ φ φ (x) with Moyal product (f g)(x) = R 4 R 4 matrix basis f mn(x) = f m1 n 1 (x 0, x 1 )f m2 n 2 (x 3, x 4 ) f mn(y 0, y 1 )= 2( 1) m ( ) m! 2 n ml n! θ y n m m dy dk (2π) 4 f(x+ 1 2Θk) g(x+y) ei k,y ( 2 y 2 θ )e y 2 θ due to f mn f kl = δ nk f ml and dx f mn (x) = Vδ mn Harald Grosse (Vienna) A solvable quantum field theory in 4 dimensions 3

Regularisation of λφ 4 4 on noncommutative space ( ZΛ S[φ]= dx R 4 2 φ ( +Ω 2 (2Θ 1 x) 2 +µ 2 ) λz 2 ) bare φ+ Λ 4 φ φ φ φ (x) with Moyal product (f g)(x) = R 4 R 4 takes at Ω = 1 in matrix basis f mn(x) = f m1 n 1 (x 0, x 1 )f m2 n 2 (x 3, x 4 ) f mn(y 0, y 1 )= 2( 1) m ( ) m! 2 n ml n! θ y n m m dy dk (2π) 4 f(x+ 1 2Θk) g(x+y) ei k,y ( 2 y 2 θ )e y 2 θ due to f mn f kl = δ nk f ml and dx f mn (x) = Vδ mn the form ( S[Φ]= V H m Φ mn Φ nm + Z Λ 2λ ) Φ mn Φ nk Φ kl Φ lm 4 m,n N 2 N m,n,k,l N 2 N ( m ) H m = Z Λ + µ2 bare, m := m 1 + m 2 N V 2 V = ( θ 4) 2 is for Ω = 1 the volume of the nc manifold. Harald Grosse (Vienna) A solvable quantum field theory in 4 dimensions 3

More generally: field-theoretical matrix models Euclidean quantum field theory action S[Φ] = V tr(eφ 2 + P[Φ]) for unbounded positive selfadjoint operator E with compact resolvent, and P[Φ] a polynomial partition function Z[J] = D[Φ] exp( S[Φ] + V tr(φj)) For P[Φ] = i 6 Φ3 this is the Kontsevich model which computes the intersection theory on the moduli space of complex curves. We choose P[Φ] = λ 4 Φ4. Perturbative expansion e V tr(p[φ]) = ( 1) n n=0 n! (V tr(p[φ])) n leads to ribbon graphs. They encode genus-g Riemann surface with B boundary components. We avoid the expansion, but keep the topological structure: Harald Grosse (Vienna) A solvable quantum field theory in 4 dimensions 4

Topological expansion Choosing H = diag(h a ), matrix index conserved along every strand. d a d c c b The k th boundary component carries a cycle J N k p 1...p Nk := N k j=1 J p j p j+1 of N k external sources, N k + 1 1. a b e f Expand logz[j] = 1 S V 2 B G p 1 1...pN 1... p B 1 1...pB N B according to the cycle structure. f e g h B β=1 JN β QFT of matrix models determines the weights of Riemann surfaces with decorated boundary components compatible with 1 gluing (of fringes) p β 1...pβ N β 2 covariance (under Φ U ΦU, which is not a symmetry!) g h Harald Grosse (Vienna) A solvable quantum field theory in 4 dimensions 5

Topological expansion Choosing H = diag(h a ), matrix index conserved along every strand. The k th boundary component carries a cycle J N k p 1...p Nk := N k j=1 J p j p j+1 of N k external sources, N k + 1 1. d a a d c J ab J bc J cd J da b G abcd ef gh c b J ef J fe e f Expand logz[j] = 1 S V 2 B G p 1 1...pN 1... p B 1 1...pB N B according to the cycle structure. f e g J ghj hg h B β=1 JN β QFT of matrix models determines the weights of Riemann surfaces with decorated boundary components compatible with 1 gluing (of fringes) p β 1...pβ N β 2 covariance (under Φ U ΦU, which is not a symmetry!) g h Harald Grosse (Vienna) A solvable quantum field theory in 4 dimensions 5

Ward identity Matrix Base S[Φ] = n,m 1 2 Φ nmh nm Φ mn + λ 4 H nm = Z Λ ( (n+m) V nmpq Z 2 Λ Φ nmφ mp Φ pq Φ qn inner automorphismφ U ΦU of M Λ, infinitesimally, not a symmetry of the action, but invariance of measure Interpretation + µ2 bare 2 ) Insertion of special vertex Vab ins := (H an H nb )φ bn φ na n into external face equals the difference between the exchanges of external sources J nb J na and J an J bn The dots stand for the remaining face indices. Harald Grosse (Vienna) A solvable quantum field theory in 4 dimensions 6

SD equation 2 vertex is Z 2 Λ λ, connected two-point function is G ab: first graph equals Z 2 Λ λ q G aq open p-face in Σ R and compare with insertion into connected two-point function gives for 2 point function: Z 2 Λ λ q G aq Z Λ λ p (G ab ) 1 G bp G ba p a = H ab G 1 ab. Harald Grosse (Vienna) A solvable quantum field theory in 4 dimensions 7

Schwinger-Dyson equations (for S int [Φ] = λ 4 tr(φ4 )) In a scaling limit V and 1 V p I finite, we have: 1. A closed non-linear equation for G ab 1 λ 1 ( G ab = G E a + E b (E a + E b ) V ab G ap G pb G ) ab E p E a p I 2. For N 4 a universal algebraic recursion formula G b0 b 1...b N 1 = ( λ) N 2 2 l=1 G b0 b 1...b 2l 1 G b 2l b 2l+1...b N 1 G b 2l b 1...b 2l 1 G b 0 b 2l+1...b N 1 (E b0 E b2l )(E b1 E bn 1 ) scaling limit corresponds to restriction to genus g = 0 similar formulae for B 2 no index summation in G abcd β-function zero! Harald Grosse (Vienna) A solvable quantum field theory in 4 dimensions 8

Graphical realisation G b0 b 1 b 2 b 3 = ( λ) G b 0 b 1 G b2 b 3 G b0 b 3 G b2 b 1 = λ (b 0 b 2 )(b 1 b 3 ) { G b0...b 5 = λ 2 + + + ( ) ( + + + + { } + + )} b i b j = G bi b j leads to non-crossing chord diagrams; these are counted by the Catalan number C N= 2 ( N 2 +1)! N 2! b i b j = 1 b i b j leads to rooted trees connecting the even or odd vertices, intersecting the chords only at vertices Harald Grosse (Vienna) A solvable quantum field theory in 4 dimensions 9

Back to λφ 4 4 on Moyal space Infinite volume limit (i.e. θ ) turns discrete matrix indices into continuous variables a, b, R + and sums into integrals Need energy cutoff a, b, [0,Λ 2 ] and normalisation of lowest Taylor terms of two-point function G nm G ab Carleman-type singular integral equation for G ab G a0 Theorem (2012/13) (for λ < 0, using G b0 = G 0b ) Let H Λ a (f) = 1 Λ 2 π P f(p) dp 0 p a G ab = sin(τ b(a)) where τ b (a) := arctan [0,π] λ πa ( G b0 = G 0b = 1 1+b exp be the finite Hilbert transform. e sign(λ)(hλ 0 [τ 0( )] H Λ a[τ b ( )]) ) λ πa b + 1+λπaHΛ a[g 0 ] G ( a0 λ b 0 dt Λ 2 0 and G a0 solution of dp (λπp) 2 + ( t+ 1+λπpHΛ p[g 0 ] G p0 ) 2 Harald Grosse (Vienna) A solvable quantum field theory in 4 dimensions 10 )

Discussion Together with explicit (but complicated for G ab cd, G ab cd ef,... ) formulae for higher correlation functions, we have exact solution of λφ 4 4 on extreme Moyal space in terms of ( G b0 = G 0b = 1 b ) Λ 2 1+b exp dp λ dt 0 0 (λπp) 2 + ( t+ 1+λπpHΛ p[g 0 ]) 2 G p0 Possible treatments 1 perturbative solution: reproduces all Feynman graphs, generates polylogarithms and ζ-functions 2 iterative solution on computer: nicely convergent, find interesting phase structure 3 rigorous existence proof of a solution 4 work in progress: (guess); should give uniqueness Harald Grosse (Vienna) A solvable quantum field theory in 4 dimensions 11

Computer simulation: evidence for phase transitions piecewise linear approximation of G 0b, G ab for Λ 2 =10 7 and 2000 sample points. Consider 1+Y:= dg 0b db b=0 Ga1a2...a N singular 1.5 1+Y 1.0 0.5 G ab G ba (winding number neglected) 1.0 0.5 0.5 1.0 λ c 0 1+Y= dg 0b db b=0 λ (1+Y) (λ) discontinuous at λ c = 0.39 order parameter b λ = sup{b : G 0b =1} non-zero for λ < λ c A key property for Schwinger functions is realised in ]λ c, 0], outside? Harald Grosse (Vienna) A solvable quantum field theory in 4 dimensions 12

Fixed point theorem Theorem (2015) Let 1 6 λ 0. Then the equation has a C1 0 -solution 1 1 G (1+b) 1 λ 0b λ 1 (1+b) 1 2 λ 1.0 0.8 0.6 0.4 λ = 1 2π Proof via Schauder fixed point theorem. This involves continuity and compactness of a certain operator (in norm topology) 0.2 5 10 15 20 Harald Grosse (Vienna) A solvable quantum field theory in 4 dimensions 13

An analogy 2D Ising model 4D nc φ 4 -theory temperature T, K = J k B T frequency Ω Kramers-Wannier duality Langmann-Szabo duality sinh(2k) sinh(2k ) = 1 ΩΩ = 1 solvable at K = K solvable at Ω = Ω scale-invariant almost scale-invariant CFT minimal model (m = 3) matrix model operator product expansion Schwinger-Dyson equation Virasoro constraints Ward identities critical exponents critical exponents Gn0 σσ, n d 2+η η = 1 4 G φφ n0 1, n 2+η λ ]λ c, 0] Virasoro algebra, CFT, subfactors,...??? Harald Grosse (Vienna) A solvable quantum field theory in 4 dimensions 14

Relativistic and Euclidean quantum field theory We define a QFT by Wightman distributions W N (x 1,...,x N ) = W N (x 1 x 2,...,x N 1 x N ) Theorem: The W N are boundary values of holomorphic functions (on permuted extended forward tube C 4(N 1) ) Euclidean points (minus diagonals) defines Schwinger functions Schwinger functions inherit properties such as real analyticity, Euclidean invariance and complete symmetry Hence, moments of probability distributions provide candidate Schwinger functions (link to statistical physics) Theorem (Osterwalder-Schrader, 1974) One additional requirement, reflection positivity, leads back to Wightman theory Harald Grosse (Vienna) A solvable quantum field theory in 4 dimensions 15

From matrix model to Schwinger functions on R 4 reverting harmonic oscillator basis Theorem (2013): S c (µx 1,...,µx N ) = 1 ( B 64π 2 N 1 +...+N B =N N β even σ S N, 1+Y:= dg 0b db connected Schwinger functions β=1 4 N β N β R 4 b=0... d 4 p pβ ) β 4π 2 µ 4 ei µ, N β i=1 ( 1)i 1 µx σ(n1 +...+N β 1 +i) G p1 2 2µ 2 (1+Y),, p 1 2... p B 2 2µ 2 (1+Y) 2µ }{{} 2 (1+Y),, p B 2 2µ 2 (1+Y) }{{} N 1 N B Hidden 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 finally: Is it non-trivial? Harald Grosse (Vienna) A solvable quantum field theory in 4 dimensions 16

Connected (4+2+2)-point function σ S 8 q x σ(1) p b= q 2 2µ 2 (1+Y) x σ(6) p x σ(5) q x σ(4) p a= p 2 2µ 2 (1+Y) x σ(2) G aaaa bb cc r x σ(8) x σ(3) p c= r 2 2µ 2 (1+Y) x σ(7) r 1 individual Euclidean symmetry in every boundary component (no clustering) 2 particle scattering without momentum exchange in 4D a sign of triviality (mind assumptions!) familiar in 2D models with factorising S-matrix a consequence of integrability Is there a precise link between exact solution of our 4D model and traditional integrability known from 2D? Harald Grosse (Vienna) A solvable quantum field theory in 4 dimensions 17

Osterwalder-Schrader reflection positivity Proposition (2013) S(x 1, x 2 ) is reflection positive iff a G aa is a Stieltjes function, d(ρ(t)) G aa =, ρ positive measure. a+t 0 Excluded for any λ > 0 (unless rescued by winding number) naïve anomalous dimension η positive for λ > 0, renormalisation oversubtracts: η ren,λ of opposite sign p-space 2-point function 1 (p 2 +m 2 ) 1 η/2 Positivity and convergence contradict each other! Need (analytical?) continuation between one regime where existence can be proved and another regime where positivity holds. Harald Grosse (Vienna) A solvable quantum field theory in 4 dimensions 18

Reflection positivity simplifies the problem If G x0 is Stieltjes, then Hilbert transform can be avoided: ( G xy = exp 1 ( )) dt G x0 π 1 t+x arctan y Im(G (t+iǫ),0 ) [0,π] 1 λt ds G s0 0 t+s + y Re(G (t+iǫ),0) Which class of functions has desired analyticity+holomorphicity and manageable integral transforms? ( ) hypergeometric functions G x0 = n F a,b1,...b n 1 x n 1 c 1,...,c n 1 if a [0, 1] and c i > b i > a holomorphicity at y > 0: determine a, b i, c i by G (k) 0y find: a = 1+ 1 π arcsin(λπ), n i=1 c i 1 b i 1 = arcsin(λπ) λπ critical coupling constant is λ c = 1 π = 0.3183... = G(k) y0 Harald Grosse (Vienna) A solvable quantum field theory in 4 dimensions 19

Källén-Lehmann spectrum Numerics makes it completely clear (but doesn t prove) that G x0 is Stieltjes reflection positivity equivalent to G xx a Stieltjes function the shape makes this plausible: λ = 0.1 Stieltjes measure ρ for G x0 = 0 dt ρ(t)/(t + x) at λ = 0.1 G x 2, x 2 G x0 = 4 F 3 (... x) x measure for G x0 has mass gap [0, 1[, but no further gap (remnant of UV/IR-mixing) absence of the second gap (usually ]1, 4[) circumvents triviality theorems Harald Grosse (Vienna) A solvable quantum field theory in 4 dimensions 20 x

Summary 1 λφ 4 4 on nc Moyal space is, at infinite noncommutativity, exactly solvable in terms of a fixed point problem theory defined by quantum equations of motion (= Schwinger-Dyson equations) existence proved for 1 6 <λ 0 phase transitions and critical phenomena 2 Projection to Schwinger functions for scalar field on R 4 = hidden noncommutativity full Euclidean symmetry (completely unexpected) no momentum exchange (close to triviality), possibly a consequence of integrability numerical approach with tiny error: leaves no doubt that Schwinger 2-point function is reflection positive for 1 π <λ 0 Harald Grosse (Vienna) A solvable quantum field theory in 4 dimensions 21

Summary 1 λφ 4 4 on nc Moyal space is, at infinite noncommutativity, exactly solvable in terms of a fixed point problem theory defined by quantum equations of motion (= Schwinger-Dyson equations) existence proved for 1 6 <λ 0 phase transitions and critical phenomena 2 Projection to Schwinger functions for scalar field on R 4 = hidden noncommutativity full Euclidean symmetry (completely unexpected) no momentum exchange (close to triviality), possibly a consequence of integrability numerical approach with tiny error: leaves no doubt that Schwinger 2-point function is reflection positive for 1 π <λ 0 3 ready to embark on higher Schwinger functions Harald Grosse (Vienna) A solvable quantum field theory in 4 dimensions 21