Construction of a quantum field theory in four dimensions Raimar - PowerPoint PPT Presentation

Construction of a quantum field theory in four dimensions Raimar Wulkenhaar Mathematisches Institut, Westf¨ alische Wilhelms-Universit¨ at M¨ unster a (based on joint work with Harald Grosse, arXiv: 1205.0465, 1306.2816, 1402.1041 & 1406.7755) Raimar Wulkenhaar Construction of a quantum field theory in four dimensions 0

Φ 4 Introduction 4 on Moyal space Integral representation Schwinger functions Summary Introduction axiomatic settings for rigorous quantum field theories by Wightman [1956] 1 Haag-Kastler [1964] 2 Osterwalder-Schrader [1974] 3 today: numerous examples in dimension 1,2,3; not a single non-trivial example in 4 dimensions We have got a candidate: Construction of 4D Euclidean QFT is achieved (2012/13). Find phase transitions and critical phenomena. Osterwalder-Schrader axioms are under investigation. So far everything is OK. Non-triviality is open, but not impossible. Ideally, we can get the 4D-analogue of factorising S -matrices [Iagolnitzer, 1978]; [Zamolodchikov-Zamolodchikov, 1979] Raimar Wulkenhaar Construction of a quantum field theory in four dimensions 1

Φ 4 Introduction 4 on Moyal space Integral representation Schwinger functions Summary Historical notes on 4D QFT Perturbative argument that QED cannot exist as 4D QFT 1 [Landau-Abrikosov-Khalatnikov, 1954] (this almost killed renormalisation theory) Same argument (sign of β -function) for λφ 4 4 . 2 λφ 4 4 + ǫ is trivial: [Aizenman, 1981]; [Fr¨ ohlich, 1982] Asymptotic freedom in QCD 3 [Gross-Wilczek, 1973]; [Politzer, 1973] Construction of Yang-Mills theory is Millennium Prize 4 problem. Having one example of a rigorously constructed 4D QFT, even with S = e i α 1 I , would be something. . . Raimar Wulkenhaar Construction of a quantum field theory in four dimensions 2

Φ 4 Introduction 4 on Moyal space Integral representation Schwinger functions Summary Regularisation & renormalisation We follow the Euclidean track, starting from a partition function. 1 To make this rigorous we need two regulators: 2 finite volume and finite energy density. Pass to quantities (densities and with certain normalised 3 functions) which have infinite volume & energy limits. Symmetry The regulated theory usually has less symmetry. Proving that symmetry is restored in the end is part of the game. We propose another strategy: Search for a regulator which has more (or very different) symmetry, so constraining that it completely solves the model. With some luck, a limit procedure gives a constructive QFT on standard R 4 . With even more luck, it satisfies OS. Raimar Wulkenhaar Construction of a quantum field theory in four dimensions 3

Φ 4 Introduction 4 on Moyal space Integral representation Schwinger functions Summary A regularisation of φ 4 4 � φ + λ Z 2 � Z � � − ∆+Ω 2 ( 2 Θ − 1 x ) 2 + µ 2 � S [ φ ]= 64 π 2 dx 2 φ⋆ 4 φ⋆φ⋆φ⋆φ ( x ) bare R 4 � dy dk 2 Θ k ) g ( x + y ) e i � k , y � ( 2 π ) 4 f ( x + 1 with Moyal product ( f ⋆ g )( x ) = R 4 × R 4 takes at Ω = 1 in matrix basis f mn ( x ) = f m 1 n 1 ( x 0 , x 1 ) f m 2 n 2 ( x 3 , x 4 ) � �� � n − m � � e − | y | 2 2 | y | 2 f mn ( y 0 , y 1 )= 2 ( − 1 ) m m ! 2 L n − m θ y θ m n ! θ � dx f mn ( x ) = 64 π 2 V δ mn the form due to f mn ⋆ f kl = δ nk f ml and � � E m Φ mn Φ nm + Z 2 λ � � S [Φ] = V Φ mn Φ nk Φ kl Φ lm 4 m , n ∈ N 2 m , n , k , l ∈ N 2 N N � | m | + µ 2 � bare E m = Z √ | m | := m 1 + m 2 ≤ N , 2 V � θ � 2 is for Ω = 1 the volume of the nc manifold. V = 4 Raimar Wulkenhaar Construction of a quantum field theory in four dimensions 4

Φ 4 Introduction 4 on Moyal space Integral representation Schwinger functions Summary More generally: field-theoretical matrix models Euclidean quantum field theory action S [Φ] = V tr ( E Φ 2 + P [Φ]) for unbounded positive selfadjoint operator E with compact resolvent, and P [Φ] a polynomial � D [Φ] exp ( − S [Φ] + V tr (Φ J )) partition function Z [ J ] = Observe: Z is covariant, but not invariant under Φ �→ U Φ U ∗ : � � � exp ( − S [Φ] + V tr (Φ J )) 0 = D Φ E ΦΦ − ΦΦ E − J Φ + Φ J ∂ . . . choose E (but not J ) diagonal, use Φ ab = V ∂ J ba : Ward identity [Disertori-Gurau-Magnen-Rivasseau, 2007] ∂ 2 Z � ( E a − E p ) ∂ Z ∂ Z � � − J na 0 = + J pn V ∂ J an ∂ J np ∂ J an ∂ J np n ∈ I Raimar Wulkenhaar Construction of a quantum field theory in four dimensions 5 For E of compact resolvent we can always assume that

Φ 4 Introduction 4 on Moyal space Integral representation Schwinger functions Summary Topological expansion c d d c Feynman graphs in matrix models b a are ribbon graphs. J ab J bc J cd J da Encode genus- g Riemann surface a b with B boundary components G | abcd | ef | gh | The k th boundary component carries p 1 ... p Nk := � N k a cycle J N k j = 1 J p j p j + 1 of N k J ef J fe e f f e external sources, N k + 1 ≡ 1. J gh J hg g g h h Expand log Z [ J ] = � 1 � B β = 1 J N β S V 2 − B G | p 1 1 ... p 1 N 1 | ... | p B 1 ... p B NB | p β 1 ... p β N β according to the cycle structure. The G | p 1 NB | become (smeared) Schwinger functions. 1 ... p 1 N 1 | ... | p B 1 ... p B QFT of matrix models determines the weights of Riemann surfaces with decorated boundary components compatible with (1) gluing and (2) symmetry. Raimar Wulkenhaar Construction of a quantum field theory in four dimensions 6

Φ 4 Introduction 4 on Moyal space Integral representation Schwinger functions Summary 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 ∂ 2 Z [ J ] J P 1 · · · J P K G | an | P 1 | ... | P K | + G | a | a | P 1 | ... | P K | � � � � V 2 � = δ ap ∂ J an ∂ J np S K V | K | + 1 V | K | + 2 n ∈ I ( K ) n ∈ I G | q 1 aq 1 ... q r | P 1 | ... | P K | J r � � � q 1 ... q r + V | K | + 1 r ≥ 1 q 1 .... q r ∈ I J P 1 · · · J P K J Q 1 · · · J Q K ′ G | a | P 1 | ... | P K | G | a | Q 1 | ... | Q K ′ | � + V 4 � Z [ J ] V | K ′ | + 1 S K S K ′ V | K | + 1 ( K ) , ( K ′ ) V � ∂ Z [ J ] ∂ Z [ J ] � � + J pn − J na E p − E a ∂ J an ∂ J np n ∈ I ∂ V J -derivatives of Z [ J ] = e − VS int [ V ∂ J ] e 2 � J , J � E , where � J , J � E := � J mn J nm E m + E n , lead to Schwinger-Dyson equations. m , n ∈ I The Theorem lets the usually infinite tower collape: Raimar Wulkenhaar Construction of a quantum field theory in four dimensions 7

Φ 4 Introduction 4 on Moyal space Integral representation Schwinger functions Summary 4 tr ( φ 4 ) ) Schwinger-Dyson equations (for S int [ φ ] = λ � In a scaling limit V → ∞ and 1 p ∈ I finite, we have V 1. A closed non-linear equation for G | ab | G | ab | G | ap | − G | pb | − G | ab | 1 λ 1 � � � G | ab | = − E a + E b ( E a + E b ) V E p − E a p ∈ I For N ≥ 4 a universal algebraic recursion formula 2. G | b 0 b 1 ... b N − 1 | N − 2 2 G | b 0 b 1 ... b 2 l − 1 | G | b 2 l b 2 l + 1 ... b N − 1 | − G | b 2 l b 1 ... b 2 l − 1 | G | b 0 b 2 l + 1 ... b N − 1 | � = ( − λ ) ( E b 0 − E b 2 l )( E b 1 − E b N − 1 ) l = 1 2. uses reality Z = Z scaling limit corresponds to restriction to genus g = 0 similar formulae for B ≥ 2 ⇒ no index summation in G | abcd | β -function zero! Raimar Wulkenhaar Construction of a quantum field theory in four dimensions 8

Φ 4 Introduction 4 on Moyal space Integral representation Schwinger functions Summary 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 a 0 Theorem (2012/13) (for λ < 0, using G b 0 = G 0 b ) � Λ 2 a ( f ) = 1 f ( p ) dp Λ Let H be the finite Hilbert transform . Then π P p − a 0 G ab = sin ( τ b ( a )) e sign ( λ )( H Λ Λ 0 [ τ 0 ( • )] −H a [ τ b ( • )]) | λ | π a � � | λ | π a where τ b ( a ) := arctan and G 0 b solution of Λ b + 1 + λπ a H a [ G 0 • ] [ 0 , π ] G 0 a � � � Λ 2 � b G 0 b = 1 dp 1 + b exp − λ dt � Λ � 2 ( λπ p ) 2 + 1 + λπ p H p [ G 0 • ] t + 0 0 G 0 p Raimar Wulkenhaar Construction of a quantum field theory in four dimensions 9

Φ 4 Introduction 4 on Moyal space Integral representation Schwinger functions Summary 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 � � � b � Λ 2 G 0 b = 1 dp 1 + b exp − λ dt Λ � 1 + λπ p H p [ G 0 • ] � 2 ( λπ p ) 2 + 0 0 t + G 0 p For λ > 0 solution exists by Schauder fixed point theorem 1 (but ambiguity due to winding number) For λ < 0 and Λ 2 → ∞ one exact solution is G 0 b = 1 2 Formula can be put on a computer and solved by iteration. 3 Shows that G 0 b = 1 is unstable, but attractive solution G 0 b 4 exists for all λ ∈ R . Raimar Wulkenhaar Construction of a quantum field theory in four dimensions 10