Exact solution of a 4 D field theory

Transcription

1 Exact solution of a 4 D field theory Harald Grosse Physics Department, University of Vienna (based on joint work with Raimar Wulkenhaar, arxiv: , , & ) Harald Grosse Exact solution of a 4 D field theory 0

2 Introduction Motivation: Improve local QFT, add quantum gravity effects. There are no nontrivial local QFTs in 4 D constructed (obeying the Axioms). Candidates are: perturbative renormalizable fields, Φ 4 (UV? Landau ghost). YM (asymptotic free) IR not understood (confinement, renormalons). We show: 4D-φ 4 -model can be constructed on Euclidean ST. Involves NCG at an intermediate step, but finally lives on standard R 4. β function vanishes, model asymptotic safe. Infinitely many renormalisable Feynman graphs summed up. Model solved (in terms of solution of nonlinear integral equation). Analytic continuation to Minkowski ST under study. Harald Grosse Exact solution of a 4 D field theory 1

3 Regularisation & renormalisation 1 We start from a Euclidean partition function. 2 To get it well-defined we need two regulators: finite volume and finite energy density. 3 Pass to densities which have limits. Symmetry The regulated theory usually has less symmetry. We: Use regulator which has more symmetry, so constraining, allows to solve the model. With some luck, a limit procedure gives a constructive QFT on standard R 4. With even more luck, it satisfies OS? Harald Grosse Exact solution of a 4 D field theory 2

4 A toy model for 4D non-linear QFT Regulariseφ 4 4 on nc Moyal space M θ with critical oscillator potential. Is a matrix model, with infinite number of WI s from action of U( ) group. These Ward id s, and the theory of singular integral equations, turn the Schwinger-Dyson eq s into a fixed point problem. 1 For θ we prove existence of a solution. 2 Surprisingly,θ describes Schwinger functions on R 4! 3 They satisfy (OS0) boundedness, (OS1) invariance and (OS3) symmetry. 4 Numerical evidence for phase transitions. 5 Numerical evidence for (OS2) reflection positivity of the 2-point function in one of the phases. Harald Grosse Exact solution of a 4 D field theory 3

5 φ 4 4 on Moyal space with harmonic propagation ( ZΛ S[φ]= dx R 4 2 φ ( +Ω 2 (2Θ 1 x) 2 +µ 2 bare with Moyal product(f g)(x) = R 4 R 4 takes at Ω = 1 in matrix basis: φ = mn fmnφmn f mn (y 0, y 1 ) = c mn L n m m ) φ+ λz 2 Λ 4 φ φ φ φ ) (x) 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 m,n N 2 N E mφ mnφ nm + Z Λλ 2 4 ) Φ mnφ nk Φ kl Φ lm m,n,k,l N 2 N ( m ) E 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 Exact solution of a 4 D field theory 4

6 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 functionz[j] = D[Φ] exp( S[Φ] + V tr(φj)) Observe: Z is covariant, but not invariant under Φ UΦU : [ ] 0 = DΦ EΦΦ ΦΦE JΦ+ΦJ exp( S[Φ]+V tr(φj))... choose E (but not J) diagonal, use Φ ab = V J ba : Ward identity [Disertori-Gurau-Magnen-Rivasseau, 2007] 0 = n I ( (Ea E p) V 2 Z J an J np + J pn Z J an J na Z J np ) For E of compact resolvent we can assume that m E m > 0 is injective! Harald Grosse Exact solution of a 4 D field theory 5

7 Topological expansion Feynman graphs in matrix models are ribbon graphs. a b a p b p q q a d d c c a b b Viewed as simplicial complexes, they encode the topology(b, g) of a genus-g Riemann surface with B boundary components (or punctures, marked points, holes, faces). The k th boundary component carries a cycle J N k p 1...p Nk N k external sources, N k Expand logz[j] = 1 V 2 B G S p p N 1... p B pB N B according to the cycle structure. B β=1 JN β := N k j=1 Jp j p j+1 of p β 1...pβ N β The G p p N 1... p B pB become (smeared) Schwinger functions. N B QFT of matrix models determines the weights of Riemann surfaces with decorated boundary components compatible with (1) gluing and (2) symmetry. Harald Grosse Exact solution of a 4 D field theory 6

8 For E of compact resolvent, the kernel of E p E a can be determined from the J-cycle structure in log Z: Theorem (2012): Ward identity for E of compact resolvent n I 2 Z[J] { = δ ap V 2 J P1 J PK ( J an J np S K (K) n I +V 4 (K),(K ) + r 1 q 1...q r I J P1 J PK J Q1 J QK S K S K G an P1... P K + G a a P 1... P K V K +1 V K +2 G q1 aq 1...q r P 1... P K Jq r ) 1...q r V K +1 G a P1... P K V K +1 + V ( Z[J] Z[J] ) J pn J na E p E a J an J np n I G } a Q1... Q K Z[J] V K +1 J-derivatives of Z[J] = e VS int [ ] V J e V 2 J,J E, where J, J E := J mnj nm m,n I E m+e n, lead to Schwinger-Dyson equations. The Theorem lets the usually infinite tower collapse Harald Grosse Exact solution of a 4 D field theory 7

9 WI SD Interpretation Insertion of special vertex V ins ab := n (E an E nb )φ bn φ na into external face equals the difference between the exchanges of external sources J nb J na and J an J bn gives for 2 point function: Z Λ ( a b )G ins [ab]... = G b... G a... Z 2 Λλ q G aq Z Λ λ p (G ab ) 1 G bp G ba p a = H ab G 1 ab. Harald Grosse Exact solution of a 4 D field theory 8

10 Schwinger-Dyson equations (for S int [φ] = λ 4 tr(φ4 )) In a scaling limit V and 1 V 1. A closed non-linear equation for G (0) ab G (0) ab = 1 E a + E b p I finite, we have λ 1 ( (E a + E b ) V p I G (0) ab G(0) 2. For N 4 a universal algebraic recursion formula G (0) b 0 b 1...b N 1 = ( λ) N 2 2 l=1 ap G(0) pb G(0) ab E p E a ) G (0) b 0 b 1...b 2l 1 G(0) b 2l b 2l+1...b N 1 G(0) b 2l b 1...b 2l 1 G(0) b 0 b 2l+1...b N 1 (E b0 E b2l )(E b1 E bn 1 ) 2. uses realityz = Z scaling limit corresponds to restriction to genus g = 0 similar formulae for B 2 Harald Grosse Exact solution of a 4 D field theory 9

11 Graphical realisation G (0) b 0 b 1 b 2 b 3 G (0) b 0...b 5 = λ2 G(0) b = ( λ) 0 b 1 G(0) b 2 b 3 G(0) b 0 b 3 G(0) b 2 b 1 (b 0 b 2 )(b 1 b 3 ) { + + = λ { + } + ( ) ( + + )} b i b j = G (0) b i b j leads to non-crossing chord diagrams; these are counted by the Catalan number C N= 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 Exact solution of a 4 D field theory 10

12 Renormalisation theorem Theorem (2013) Given a real quartic matrix model S = V tr(eφ 2 + λ 4 Φ4 ) with E of compact resolvent. Assume that the selfconsistency equation for G (0) ab has a finite solution after affine renormalisation E Z(E + C1I) and λ Z 2 λ. Then All higher functions G (0) b 0 with N 4 are automatically finite...b N 1 without further need of a renormalisation of λ. All quartic matrix models (with renormalisable G (0) ) have vanishing ab β-function (i.e. are almost scale-invariant). The perturbative observation β = 0 for Moyal [Disertori-Gurau-Magnen-Rivasseau, 2007] is generic! (Similar statements hold for B 2) Harald Grosse Exact solution of a 4 D field theory 11

13 φ 4 4 on Moyal space Moyal product(f g)(x) = R d R d dx dk (2π) d f(x+ 1 2Θk) g(x+y) ei k,y Z S[φ]= d x( 4 2 φ ( ) +Ω 2 (2Θ 1 x) 2 +µ 2 λz 2 ) bare φ+ 4 φ φ φ φ (x) renormalisable as formal power series in λ [HG+R.Wulkenhaar, 2004] (renormalisation of µ 2 bare,λ, Z R + and Ω [0, 1]) means: well-defined perturbative quantum field theory Langmann-Szabo duality (2002): theories at Ω and Ω = 1 are the Ω same; self-dual case Ω = 1 is matrix model β-function vanishes to all orders in λ for Ω = 1 [Disertori-Gurau-Magnen-Rivasseau, 2006] means: almost scale-invariant Is the self-dual (critical) model integrable? Harald Grosse Exact solution of a 4 D field theory 12

14 Integral equs Matrix indices become continuous p V µ 2 p with p [0,Λ 2 ] Normalised planar 2-point function G ab = µ 2 G (0) ab, a, b [0,Λ2 ] Difference of eqns for G ab and G a0 cancels worst divergence Renormalisationµ bare µ and Z 1 (1+Y) by normalisation conditions G 00 = 1 and dg ab db a=b=0 = (1+Y) Integral equation for Hölder-continuous G ab and Λ ( [ ] b a + 1+λπaHa G 0 ) [ ] D ab λπh a D b = Ga0 ag a0 where D ab := a b (G ab G a0 ), Hilbert transform Y = λ 0 H a[f( )] := 1 π lim ǫ 0 dp p D p0 ( a ǫ + 0 a+ǫ ) f(q) dq q a Harald Grosse Exact solution of a 4 D field theory 13

15 The Carleman equation Theorem [Carleman 1922, Tricomi 1957] The singular linear integral equation h(x)y(x) λπh x[y] = f(x), x [ 1, 1] is for h(x) continuous + Hölder near ±1 and f L p solved by y(x)= sin(ϑ(x)) ( f(x) cos(ϑ(x)) λπ ]+ CeHx[ϑ] +e Hx[ϑ] H x [e H [ϑ] f( ) sin(ϑ( )) 1 x ( ϑ(x) = arctan λπ ) λπ, sin(ϑ(x)) = [0,π] h(x) (h(x))2 +(λπ) 2 where C is an arbitrary constant. Assumption: C = 0 ) Harald Grosse Exact solution of a 4 D field theory 14

16 Solution ( angle ϑ b (a) := arctan λπa [0,π] b + 1+λπaHa[G 0] G a0 G a0 is solved for ϑ 0 (a): ) G a0 = sin(ϑ 0(a)) e Ha[ϑ 0( )] H 0 [ϑ 0 ( )] λ πa Addition theorems and Tricomi s identity ] e Ha[ϑ b ] cos(ϑ b (a))+h a [e H [ϑ b ] sin(ϑ b ( ) = 1 give: Theorem G ab = sin(ϑ b(a)) e Ha[ϑ b ] H 0[ϑ 0 ] = λ πa e Ha[ϑ b ( )] H 0[ϑ 0 ( )] (λπa) 2 + ( b+ 1+λπaHa[G 0] G a0 ) 2 Consequence: G ab 0! Y = λ 0 dp (λπp) 2 + ( 1+λπpH p[g 0 ] G p0 ) 2 Harald Grosse Exact solution of a 4 D field theory 15

17 The self-consistency equation Given boundary value G a0, Carleman computes G ab, in particular G 0b symmetry forces G b0 = G 0b Master equation Λ 2 0 b G ab a The theory is completely determined by the solution of the fixed point equation G = TG ( G b0 = 1 b ) 1+b exp dp λ dt 0 0 (λπp) 2 + ( t + 1+λπpHp[G ) 0] 2 G ρ0 Λ 2 Harald Grosse Exact solution of a 4 D field theory 16

18 Computer simulation: evidence for phase transitions 1.0 Ga 1 a 2...a N singular λ 0 λ c λ G ab G ba (winding number neglected) Y= 1 dg 0b db b=0 λ eff = G 0000 G ab for Λ 2 =10 7 with 2000 sample points G 0b(λ) discontinuous at λ c = 0.39 λ eff singular at λ 0 = where G 0b(λ) = 0 Nothing particular at pole λ b = 1 = of Borel 72 resummation A key property for Schwinger functions is realised in subinterval of [λ c, 0], not outside! Harald Grosse Exact solution of a 4 D field theory 17

19 Osterwalder-Schrader reconstruction theorem (1974) Assume for Schwinger functions S(x 1,..., x N ): S0 growth rate: dx f(x 1,..., x N )S(x 1,..., x N ) c 1 (N!) c 2 f Nc3 S1 Euclidean invariance: S(x 1,..., x N ) = S(Rx 1 +a,..., Rx N +a) S2 reflection positivity: for each (f 0,..., f K ) with f N S(R Nd ), K dxdy S(x N,..., x 1, y 1,...y M )f N ( r x 1,..., r x N )f M (y 1,..., y M ) 0 M,N=0 where r (x 0, x 1,...x d 1 ) := ( x 0, x 1,...x d 1 ) S3 permutation symmetry: S(x 1,..., x N ) = S(x σ(1),..., x σ(n) ) Then the S(ξ 1,...,ξ N 1 ) ξ 0 i >0, with ξ i = x i x i+1, are inverse Laplace-Fourier transforms of FT Ŵ(q 1,..., q N 1 ) of Wightman distributions in a relativistic QFT. If in addition the S(x 1,..., x N ) satisfy S4 clustering then the Wightman QFT has a unique vacuum state Harald Grosse Exact solution of a 4 D field theory 18

20 From matrix model to Schwinger functions on R 4 folding G... with eigenbasis of 4D harmonic oscillator: Theorem (2013): S c (µx 1,...,µx N ) = 1 ( B 64π 2 N N B =N N β even connected Schwinger functions σ S N β=1 4 N β N β R 4 d 4 p pβ ) β 4π 2 µ 4 ei µ, N β i=1 ( 1)i 1 µx σ(n N β 1 +i) G p1 2 2µ 2 (1+Y),, p p B 2 2µ 2 (1+Y) 2µ }{{} 2 (1+Y),, p B 2 2µ 2 (1+Y) }{{} N 1 N B Schwinger functions are symmetric S3 and invariant under full Euclidean group S1 (completely unexpected for NCQFT) growth conditions S0 established clustering S4 is violated: The (N N B )-point functions are insensitive to the distance of different boundaries. remains: reflection positivity S2 Harald Grosse Exact solution of a 4 D field theory 19

21 Osterwalder-Schrader reflection positivity Reflection positivity S2 gives spectrum condition which guarantees representation as Laplace transform in ξ 0, hence analyticity in Re(ξ 0 ) > 0. Proposition (2013) S(x 1, x 2 ) is reflection positive iff a G aa is a Stieltjes function, d(ρ(t)) G aa = 0 a+t with ρ positive and non-decreasing. Proof: Källén-Lehmann Excluded for any λ > 0 (due to renormalisation)! The Stieltjes property is a particularly strong positivity in mathematics. Is positivity in quantum field theory (Hilbert space scalar product and spectrum condition) Harald Grosse Exact solution of a 4 D field theory 20

22 Classes of strongly positive smooth functions 1 C = completely monotonic functions: ( 1) n f (n) 0 implies rep n as Laplace transform f(z) = 0 dµ(t) e tz related to Bernstein and Pick/Nevanlinna functions and Hausdorff moment problem 2 L C = logarithmically completely monotonic functions: ( 1) n (log f) (n) 0 3 S L C Stieltjes functions: L k,t [f( )] 0 where [Widder,1938] L k,t [f( )] := ( t)k 1 c k d 2k 1 dt 2k 1 ( t k f(t) ), c 1 = 1, c k>1 = k!(k 2)! imply analyticity in cut plane C\], 0] with Im(f(z)) < 0 for Im(z) > 0 (anti-herglotz function) measure recoved from ρ (t) = lim k L k,t [f( )] Harald Grosse Exact solution of a 4 D field theory 21

23 Positivity of G ab : Widder s L k,t [G ] 0.30 key step: integral formula for n+l G ab n a l b L L L L λ = λ = L 11 L L L λ = λ = Harald Grosse Exact solution of a 4 D field theory 22

24 An order parameter log(1+a) log(1+a) log G a0 4 log G a0 3 6 λ = λ c log G aa 4 λ = log G aa log(1+a) log G a0 log G aa λ = Solution G 0b = 1 becomes stable for λ < λ c A := sup{b : G 0b = 1} is order parameter; A = 0 for λ > λ c and A > 0 for λ < λ c. Higher correlation function ill-defined for matrix indices A Harald Grosse Exact solution of a 4 D field theory 23

25 An analogy 2D Ising model 4D nc φ 4 -theory temperature T, K = J k B T frequencyω 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 1, n d 2+η η = 1 4 G φφ n0 1, n 1+λ λ ]λ c, 0] Virasoro algebra, CFT, subfactors,...??? Harald Grosse Exact solution of a 4 D field theory 24

26 Summary 1 The quartic matrix model Z = dm exp(tr(jm EM 2 λ 4 M4 )) is exactly solvable in terms of solution of a non-linear equation. 2 Similar to Kontsevich model in 2D quantum gravity. 3 λφ 4 4 on nc Moyal space is, at infinite noncommutativity, exactly solvable in terms of a fixed point solution stable non-perturbative solution for λ < 0 planar wrong-sign λφ 4 4-model [t Hooft; Rivasseau, 1983] phase transitions and critical phenomena, hence interesting statistical physics model non-trivial as a matrix model 4 Projection to Schwinger functions for scalar field on R 4 : S3 automatic, full Euclidean symmetry S1, control about S0 no clustering S4 Scattering remnants from NCG substructure 5 Reflection positivity S2 does not fail immediately. Why? Harald Grosse Exact solution of a 4 D field theory 25