Renormalisation & resurgent transseries in quantum field theory - PowerPoint PPT Presentation

Introduction Analysable functions & transseries Renormalisation as a game changer Results from Dyson-Schwinger equations Renormalisation & resurgent transseries in quantum field theory Lutz Klaczynski, Humboldt University Berlin Paths to, from and in renormalisation Potsdam February 12th, 2016 Lutz Klaczynski, Humboldt University Berlin Renormalisation & resurgent transseries in quantum field theory

Introduction Analysable functions & transseries Renormalisation as a game changer Results from Dyson-Schwinger equations Outline 1 Introduction Euclidean scalar field Coupling dependence in Euclidean QFT 2 Analysable functions & transseries ´ Ecalle’s analysable functions Resurgent transseries (in QFT) 3 Renormalisation as a game changer Perturbation theory (Super)renormalisation Transseries inconceivable? 4 Results from Dyson-Schwinger equations Whirlwind introduction to Dyson-Schwinger equations Dyson-Schwinger equations and transseries Lutz Klaczynski, Humboldt University Berlin Renormalisation & resurgent transseries in quantum field theory

Introduction Analysable functions & transseries Euclidean scalar field Renormalisation as a game changer Coupling dependence in Euclidean QFT Results from Dyson-Schwinger equations Basic ingredients 1 finite lattice: Γ = ε Z d / L Z d with L 2 ε ∈ N (discrete torus) 2 real scalar field φ : Γ → R Euclidean action of φ 4 theory � � S Γ ( φ, λ ) = 1 φ [ − ∆ + m 2 ] φ + λ φ 4 (1) 2 Γ Γ � Γ F := ε d � x ∈ Γ F ( x ) for F ∈ R Γ and where d − ∆ φ ( x ) = 1 � [2 φ ( x ) − φ ( x + ε e j ) − φ ( x − ε e j )] ε 2 j =1 lattice Laplacian Lutz Klaczynski, Humboldt University Berlin Renormalisation & resurgent transseries in quantum field theory

Introduction Analysable functions & transseries Euclidean scalar field Renormalisation as a game changer Coupling dependence in Euclidean QFT Results from Dyson-Schwinger equations Partition function Partition function as path integral: � D Γ φ e − S Γ ( φ,λ )+ � Γ J · φ Z Γ ( J , λ ) = (2) where J ∈ R Γ external field and � Lebesgue measure in R | Γ | : D Γ φ = d φ ( x ) x ∈ Γ All quantities obtained from Z Γ ( J , λ ), eg Correlators � 1 δ δ � � φ ( x 1 ) . . . φ ( x n ) � λ = δ J ( x 1 ) . . . δ J ( x n ) Z Γ ( J , λ ) � Z Γ (0 , λ ) � J =0 Lutz Klaczynski, Humboldt University Berlin Renormalisation & resurgent transseries in quantum field theory

Introduction Analysable functions & transseries Euclidean scalar field Renormalisation as a game changer Coupling dependence in Euclidean QFT Results from Dyson-Schwinger equations Coupling dependence Continuum limit: ε → 0 and/or L → ∞ Question Given continuum limit exists, does � λ �→ Z Γ ( J , λ ) 1 Γ φ 4 + � � D Γ φ e − S Γ ( φ, 0) − λ Γ J · φ Z Γ (0 , λ ) = (3) Z Γ (0 , λ ) belong to ´ Ecalle’s class of analysable functions in this limit? 1 Is there a valid transseries representation? 2 If so, what is it and is it accelero-summable? Lutz Klaczynski, Humboldt University Berlin Renormalisation & resurgent transseries in quantum field theory

Introduction ´ Analysable functions & transseries Ecalle’s analysable functions Renormalisation as a game changer Resurgent transseries (in QFT) Results from Dyson-Schwinger equations Analysable functions: ’field with no escape’ Class of analysable functions is stable under algebraic operations of a field (like C ) composition and inversion (if injective) integration and differentiation Generators Take C -linear span of 1 , z , e z , log z and perform all of the above operations. � log z = z � � z − 1 e z = e z � 1 k ! k ≥ 0 k ! z − k − 1 , example: (log z ) k +1 , k ≥ 0 e e z = . . . � Lutz Klaczynski, Humboldt University Berlin Renormalisation & resurgent transseries in quantum field theory

Introduction ´ Analysable functions & transseries Ecalle’s analysable functions Renormalisation as a game changer Resurgent transseries (in QFT) Results from Dyson-Schwinger equations Transseries Grid-based transseries formal series of the form � � c ( l 1 ,..., l k ) m l 1 1 . . . m l k . . . ( α j ∈ Z ) k l 1 ≥ α 1 l k ≥ α k with transmonomials m 1 , . . . , m k , no convergence required Group of transmonomials: examples z − 1 , e z , z 4 e − z , e e z � j ≥ 0 z − j , e − z + z 2 , log z , log ◦ log z , . . . Accelero-summation of height-one transseries B L formal transseries − → convergent transseries − → analysable fct Lutz Klaczynski, Humboldt University Berlin Renormalisation & resurgent transseries in quantum field theory

Introduction ´ Analysable functions & transseries Ecalle’s analysable functions Renormalisation as a game changer Resurgent transseries (in QFT) Results from Dyson-Schwinger equations Semi-classical expansion I | Γ | 1 I := λ − 1 2 φ 2 J 2 D Γ φ Rescaling: ϕ := λ D Γ ϕ = λ partition function, rescaled 1 � 2 I , λ ) Z Γ ( λ 1 D Γ ϕ e − 1 � λ S Γ ( ϕ, 1)+ Γ I · ϕ = (4) Z Γ (0 , λ ) Z Γ (0 , λ ) Semi-classical expansion around critical points 1 2 I , λ ) Z Γ ( λ � � e − 1 e − 1 ∼ λ S Γ ( ϕ c , 1) F ϕ c ( I , λ ) λ S Γ ( ϕ c , 1) C [[ λ ]] = ∈ Z Γ (0 , λ ) ϕ c ϕ c � �� � transseries (fixed I ) connection to transseries: z = λ − 1 Lutz Klaczynski, Humboldt University Berlin Renormalisation & resurgent transseries in quantum field theory

Introduction ´ Analysable functions & transseries Ecalle’s analysable functions Renormalisation as a game changer Resurgent transseries (in QFT) Results from Dyson-Schwinger equations Topical transseries ans¨ atze Currently used ans¨ atze of the form Height-1, depth-1 transseries � � z c · σ e − ( b · σ ) z P σ (log z ) c ( σ, s ) z − s f = σ ∈ N r s ≥ 0 0 c , b ∈ C r , P σ (log z ) ∈ C [log z ] polynomial, z = (coupling) − 1 in quantum mechanics, toy model and SUSY QFTs, toy model and SUSY string theories Sectors of the transseries subseries for fixed σ ∈ N r 0 : � z c · σ e − ( b · σ ) z P σ (log z ) c ( σ, s ) z − s . s ≥ 0 Lutz Klaczynski, Humboldt University Berlin Renormalisation & resurgent transseries in quantum field theory

Introduction Perturbation theory Analysable functions & transseries (Super)renormalisation Renormalisation as a game changer Transseries inconceivable? Results from Dyson-Schwinger equations Perturbation theory (path integral approach) partition function revisited � Z Γ ( J , λ ) 1 � Γ φ 4 + � D Γ φ e − S Γ ( φ, 0) − λ Γ J · φ Z Γ (0 , λ ) = Z Γ (0 , λ ) Idea of perturbation theory expand interaction exponential in λ � � s � � Γ J · φ = Γ φ 4 e e − λ � � φ 4 � Γ J · φ λ s − e Γ s ≥ 0 generate polynomials in φ and compute Gaussian expectations Lutz Klaczynski, Humboldt University Berlin Renormalisation & resurgent transseries in quantum field theory

Introduction Perturbation theory Analysable functions & transseries (Super)renormalisation Renormalisation as a game changer Transseries inconceivable? Results from Dyson-Schwinger equations The need for renormalisation Problem: divergent expectations already at order O ( λ ), � d µ Γ ( φ ) φ ( x ) 4 | → ∞ | as ε → 0 or L → ∞ , where d µ Γ ( φ ) = Z Γ (0 , 0) − 1 D Γ φ e − S Γ ( φ, 0) Solution in d = 2: Wick ordering : φ ( x ) 4 : = φ ( x ) 4 − 6 C Γ ( x , x ) φ ( x ) 2 + 3 C Γ ( x , x ) 2 � with C Γ ( x , y ) = � φ ( x ) φ ( y ) � 0 = d µ Γ ( φ ) φ ( x ) φ ( y ), then � d µ Γ ( φ ) : φ ( x ) 4 : = 0 ∀ ε > 0 Lutz Klaczynski, Humboldt University Berlin Renormalisation & resurgent transseries in quantum field theory

Introduction Perturbation theory Analysable functions & transseries (Super)renormalisation Renormalisation as a game changer Transseries inconceivable? Results from Dyson-Schwinger equations Superrenormalsation I: Z factor in d = 2 (super)renormalised Euclidean action ( d = 2) � � R 2 [ S Γ ]( φ, λ ) = 1 : φ 4 : φ [ − ∆ + m 2 ] φ + λ 2 Γ Γ In physics: replace m 2 by m 2 Z m ( λ ) := m 2 (1 + c 1 λ ) to obtain mass-corrected Euclidean action ( d = 2) ( ∼ Wick ordered) � � R 2 [ S Γ ]( φ, λ ) = 1 φ [ − ∆ + m 2 Z m ( λ )] φ + λ φ 4 2 Γ Γ renormalisation Z factor: Z m ( λ ) = 1 + c 1 λ = 1 − 12 m − 2 C Γ (0 , 0) λ where C Γ (0 , 0) = C Γ ( x , x ) constant ( → ∞ in continuum limit) Lutz Klaczynski, Humboldt University Berlin Renormalisation & resurgent transseries in quantum field theory

Introduction Perturbation theory Analysable functions & transseries (Super)renormalisation Renormalisation as a game changer Transseries inconceivable? Results from Dyson-Schwinger equations Superrenormalsation II: Z factor in d = 3 Wick ordering not sufficient in d = 3! One additional counterterm necessary. Mass-renormalisation Z factor in spacetime dimension d = 3: Z m ( λ ) = 1 + c 1 λ + c 2 λ 2 mass-corrected Euclidean action ( d = 3) � � R 3 [ S Γ ]( φ, λ ) = 1 φ [ − ∆ + m 2 Z m ( λ )] φ + λ φ 4 2 Γ Γ Lutz Klaczynski, Humboldt University Berlin Renormalisation & resurgent transseries in quantum field theory