A Beginners Guide to Resurgence and Trans-series in Quantum Theories - PowerPoint PPT Presentation

A Beginners’ Guide to Resurgence and Trans-series in Quantum Theories Gerald Dunne University of Connecticut Recent Developments in Semiclassical Probes of Quantum Field Theories UMass Amherst ACFI, March 17-19, 2016 GD & Mithat Ünsal, reviews: 1511.05977, 1601.03414 GD, lectures at CERN 2014 Winter School GD, lectures at Schladming 2015 Winter School

Lecture 1 ◮ motivation: physical and mathematical ◮ trans-series and resurgence ◮ divergence of perturbation theory in QM ◮ basics of Borel summation ◮ the Bogomolny/Zinn-Justin cancellation mechanism ◮ towards resurgence in QFT ◮ effective field theory: Euler-Heisenberg effective action

Physical Motivation • infrared renormalon puzzle in asymptotically free QFT • non-perturbative physics without instantons: physical meaning of non-BPS saddles • "sign problem" in finite density QFT • exponentially improved asymptotics Bigger Picture • non-perturbative definition of non-trivial QFT, in the continuum • analytic continuation of path integrals • dynamical and non-equilibrium physics from path integrals • uncover hidden ‘magic’ in perturbation theory

Physical Motivation • what does a Minkowski path integral mean? � i � � � � � − 1 D A exp � S [ A ] versus D A exp � S [ A ]  e − 2 3 x 3 / 2  , x → + ∞ 2 √ π x 1 / 4  � ∞   1 e i ( 1 3 t 3 + x t ) dt ∼ 2 π −∞  sin ( 2 3 ( − x ) 3 / 2 + π 4 )   , x → −∞ √ π ( − x ) 1 / 4 

Physical Motivation • what does a Minkowski path integral mean? � i � � � � � − 1 D A exp � S [ A ] versus D A exp � S [ A ] 1.0 0.5 � 10 � 5 5 10 � 0.5 � 1.0  e − 2 3 x 3 / 2  , x → + ∞ 2 √ π x 1 / 4  � ∞   1 e i ( 1 3 t 3 + x t ) dt ∼ 2 π −∞  sin ( 2 3 ( − x ) 3 / 2 + π 4 )   , x → −∞ √ π ( − x ) 1 / 4 

Mathematical Motivation Resurgence: ‘new’ idea in mathematics (Écalle, 1980; Stokes, 1850) resurgence = unification of perturbation theory and non-perturbative physics • perturbation theory generally ⇒ divergent series • series expansion − → trans-series expansion • trans-series ‘well-defined under analytic continuation’ • perturbative and non-perturbative physics entwined • applications: ODEs, PDEs, fluids, QM, Matrix Models, QFT, String Theory, ... • philosophical shift: view semiclassical expansions as potentially exact

Resurgent Trans-Series • trans-series expansion in QM and QFT applications: ∞ ∞ k − 1 � � �� k � � �� l − c ± 1 � � � f ( g 2 ) = c k,l,p g 2 p exp ln g 2 g 2 � �� � p =0 k =0 l =1 perturbative fluctuations � �� � � �� � k − instantons quasi-zero-modes • J. Écalle (1980): closed set of functions: ( Borel transform ) + ( analytic continuation ) + ( Laplace transform ) − 1 g 2 , ln( g 2 ) , are familiar • trans-monomial elements : g 2 , e • “multi-instanton calculus” in QFT • new: analytic continuation encoded in trans-series • new: trans-series coefficients c k,l,p highly correlated • new: exponentially improved asymptotics

Resurgence resurgent functions display at each of their singular points a behaviour closely related to their behaviour at the origin. Loosely speaking, these functions resurrect, or surge up - in a slightly different guise, as it were - at their singularities J. Écalle, 1980 n m

Perturbation theory • perturbation theory generally → divergent series e.g. QM ground state energy: E = � ∞ n =0 c n ( coupling ) n ◮ Zeeman: c n ∼ ( − 1) n (2 n )! ◮ Stark: c n ∼ (2 n )! ◮ cubic oscillator: c n ∼ Γ( n + 1 2 ) ◮ quartic oscillator: c n ∼ ( − 1) n Γ( n + 1 2 ) ◮ periodic Sine-Gordon (Mathieu) potential: c n ∼ n ! ◮ double-well: c n ∼ n ! note generic factorial growth of perturbative coefficients

Asymptotic Series vs Convergent Series N − 1 � c n ( x − x 0 ) n + R N ( x ) f ( x ) = n =0 convergent series: fixed | R N ( x ) | → 0 , N → ∞ , x asymptotic series: | R N ( x ) | ≪ | x − x 0 | N fixed x → x 0 , , N “optimal truncation”: − → truncate just before the least term ( x dependent!)

Asymptotic Series: optimal truncation & exponential precision ∞ � 1 � ( − 1) n n ! x n ∼ 1 � 1 x E 1 x e x n =0 optimal truncation: N opt ≈ 1 x ⇒ exponentially small error √ Ne − N ≈ e − 1 /x | R N ( x ) | N ≈ 1 /x ≈ N ! x N � N ≈ 1 /x ≈ N ! N − N ≈ √ x � � � 0.920 0.90 � � � � 0.918 � 0.85 � 0.916 � � � � � � � � � � � � � � � 0.914 0.80 � � � 0.912 0.75 20 N N � �� � 0 5 10 15 2 4 6 8 ( x = 0 . 1) ( x = 0 . 2)

Borel summation: basic idea � ∞ 0 dt e − t t n write n ! = alternating factorially divergent series: � ∞ ∞ 1 � ( − 1) n n ! g n = dt e − t (?) 1 + g t 0 n =0 integral convergent for all g > 0 : “Borel sum” of the series

Borel Summation: basic idea � ∞ ∞ 1 � ( − 1) n n ! x n = dt e − t 1 + x t 0 n =0 1.2 1.1 1.0 0.9 0.8 0.7 0.4 x 0.0 0.1 0.2 0.3

Borel summation: basic idea � ∞ 0 dt e − t t n write n ! = non-alternating factorially divergent series: � ∞ ∞ 1 � n ! g n = dt e − t (??) 1 − g t 0 n =0 pole on the Borel axis!

Borel summation: basic idea � ∞ 0 dt e − t t n write n ! = non-alternating factorially divergent series: � ∞ ∞ 1 � n ! g n = dt e − t (??) 1 − g t 0 n =0 pole on the Borel axis! ⇒ non-perturbative imaginary part ± i π g e − 1 g but every term in the series is real !?!

Borel Summation: basic idea � ∞ � � ∞ � 1 � 1 − x t = 1 1 � x e − 1 n ! x n dt e − t Borel x Ei ⇒ R e = P x 0 n =0 2.0 1.5 1.0 0.5 3.0 x 0.5 1.0 1.5 2.0 2.5 � 0.5

Borel summation Borel transform of series f ( g ) ∼ � ∞ n =0 c n g n : ∞ c n � n ! t n B [ f ]( t ) = n =0 new series typically has finite radius of convergence. Borel resummation of original asymptotic series: � ∞ S f ( g ) = 1 B [ f ]( t ) e − t/g dt g 0 warning: B [ f ]( t ) may have singularities in (Borel) t plane

Borel singularities avoid singularities on R + : directional Borel sums: � e iθ ∞ S θ f ( g ) = 1 B [ f ]( t ) e − t/g dt g 0 C + C - go above/below the singularity: θ = 0 ± non-perturbative ambiguity: ± Im[ S 0 f ( g )] − → challenge: use physical input to resolve ambiguity

Borel summation: existence theorem (Nevanlinna & Sokal) � � � < R � z − R f ( z ) analytic in circle C R = { z : 2 } 2 N − 1 � a n z n + R N ( z ) | R N ( z ) | ≤ A σ N N ! | z | N f ( z ) = , n =0 Borel transform R/2 ∞ a n � n ! t n B ( t ) = n =0 Im(t) analytic continuation to S σ = { t : | t − R + | < 1 /σ } 1/ σ � ∞ Re(t) f ( z ) = 1 e − t/z B ( t ) dt z 0

Borel summation in practice ∞ � c n ∼ β n Γ( γ n + δ ) c n g n f ( g ) ∼ , n =0 • alternating series: real Borel sum � � t � t � � 1 /γ � � ∞ � � δ/γ f ( g ) ∼ 1 dt 1 − exp γ t 1 + t βg βg 0 • nonalternating series: ambiguous imaginary part � � t � t � � 1 /γ � � ∞ � � δ/γ 1 dt 1 Re f ( − g ) ∼ γ P exp − t 1 − t βg βg 0 � 1 � 1 � � 1 /γ � � δ/γ ± π Im f ( − g ) ∼ − exp γ βg βg

Resurgence and Analytic Continuation another view of resurgence: resurgence can be viewed as a method for making formal asymptotic expansions consistent with global analytic continuation properties “the trans-series really IS the function” ⇒ (question: to what extent is this true/useful in physics?)

Resurgence: Preserving Analytic Continuation • zero-dimensional partition functions � 1 � ∞ � √ 1 dx e − 1 λ x ) = 1 2 λ sinh 2 ( 4 λ K 0 √ Z 1 ( λ ) = e 4 λ λ −∞ � π ∞ ( − 1) n (2 λ ) n Γ( n + 1 2 ) 2 � Borel-summable ∼ � 1 � 2 2 n ! Γ n =0 2 √ � 1 � π/ � λ √ λ x ) = π dx e − 1 2 λ sin 2 ( e − 1 4 λ I 0 √ Z 2 ( λ ) = 4 λ λ 0 � π ∞ (2 λ ) n Γ( n + 1 2 ) 2 � non-Borel-summable ∼ � 1 � 2 2 n ! Γ n =0 2 • naively: Z 1 ( − λ ) = Z 2 ( λ )

Resurgence: Preserving Analytic Continuation • zero-dimensional partition functions � 1 � ∞ � √ 1 dx e − 1 λ x ) = 1 2 λ sinh 2 ( 4 λ K 0 √ Z 1 ( λ ) = e 4 λ λ −∞ � π ∞ ( − 1) n (2 λ ) n Γ( n + 1 2 ) 2 � Borel-summable ∼ � 1 � 2 2 n ! Γ n =0 2 √ � 1 � π/ � λ √ λ x ) = π dx e − 1 2 λ sin 2 ( e − 1 4 λ I 0 √ Z 2 ( λ ) = 4 λ λ 0 � π ∞ (2 λ ) n Γ( n + 1 2 ) 2 � non-Borel-summable ∼ � 1 � 2 2 n ! Γ n =0 2 • naively: Z 1 ( − λ ) = Z 2 ( λ ) • connection formula: K 0 ( e ± iπ | z | ) = K 0 ( | z | ) ∓ i π I 0 ( | z | ) Z 1 ( e ± iπ λ ) = Z 2 ( λ ) ∓ i e − 1 ⇒ 2 λ Z 1 ( λ )