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 I motivation: physical and mathematical I trans-series and resurgence I divergence of perturbation theory in QM I basics of Borel summation I the Bogomolny/Zinn-Justin cancellation mechanism I towards resurgence in QFT I 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 A exp S[A] versus A exp S[A] D D − Z ~ Z ~ − 2 x3=2 ep3 ; x + 1 2 π x1=4 ! 1 1 i 1 t3+x t e ( 3 ) dt 8 2π −∞ ∼ > 2 3=2 π Z > sin( 3 (−x) + 4 ) < p ; x π (−x)1=4 ! −∞ > :> Physical Motivation what does a Minkowski path integral mean? • i 1 A exp S[A] versus A exp S[A] D D − Z ~ Z ~ 1.0 0.5 -10 -5 5 10 -0.5 -1.0 − 2 x3=2 ep3 ; x + 1 2 π x1=4 ! 1 1 i 1 t3+x t e ( 3 ) dt 8 2π −∞ ∼ > 2 3=2 π Z > sin( 3 (−x) + 4 ) < p ; 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: • 1 1 k−1 k l 2 2p c 1 f(g ) = ck;l;p g exp ln −g2 ±g2 p=0 k=0 l=1 X X X perturbative fluctuations k−instantons quasi-zero-modes | {z } J. Écalle (1980): closed set of functions:| {z } | {z } • (Borel transform) + (analytic continuation) + (Laplace transform) − 1 trans-monomial elements: g2, e g2 , ln(g2), are familiar • “multi-instanton calculus” in QFT • new: analytic continuation encoded in trans-series • new: trans-series coefficients ck;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 • ! 1 n e.g. QM ground state energy: E = n=0 cn (coupling) P n I Zeeman: cn ( 1) (2n)! ∼ − I Stark: cn (2n)! ∼ 1 I cubic oscillator: cn Γ(n + ) ∼ 2 n 1 I quartic oscillator: cn ( 1) Γ(n + ) ∼ − 2 I periodic Sine-Gordon (Mathieu) potential: cn n! ∼ I double-well: cn n! ∼ note generic factorial growth of perturbative coefficients Asymptotic Series vs Convergent Series N−1 n f(x) = cn (x x ) + RN (x) − 0 n=0 X convergent series: RN (x) 0 ;N ; x fixed j j ! ! 1 asymptotic series: N RN (x) x x ; x x ;N fixed j j j − 0j ! 0 “optimal truncation”: −! truncate just before the least term (x dependent!) Asymptotic Series: optimal truncation & exponential precision 1 n n 1 1 1 ( 1) n! x e x E − ∼ x 1 x n=0 X optimal truncation: N 1 exponentially small error opt ≈ x ) −1=x N −N −N e RN (x) N! x N!N pNe j jN≈1=x ≈ N≈1=x ≈ ≈ ≈ px æ æ æ 0.920 æ 0.90 æ æ 0.918 æ æ 0.85 æ æ 0.916 æ æ æ æ æ æ æ æ æ æ æ æ æ 0.914 æ 0.80 æ æ 0.912 0.75 æ ææ N æ N 0 5 10 15 20 2 4 6 8 (x = 0:1) (x = 0:2) Borel summation: basic idea 1 −t n write n! = 0 dt e t alternatingR factorially divergent series: 1 1 1 ( 1)n n! gn = dt e−t (?) − 1 + g t n=0 0 X Z integral convergent for all g > 0: “Borel sum” of the series Borel Summation: basic idea 1 1 1 ( 1)n n! xn = dt e−t − 1 + x t n=0 0 X Z 1.2 1.1 1.0 0.9 0.8 0.7 x 0.0 0.1 0.2 0.3 0.4 non-perturbative imaginary part ) i π − 1 e g ± g but every term in the series is real !?! Borel summation: basic idea 1 −t n write n! = 0 dt e t non-alternatingR factorially divergent series: 1 1 1 n! gn = dt e−t (??) 1 g t n=0 0 X Z − pole on the Borel axis! Borel summation: basic idea 1 −t n write n! = 0 dt e t non-alternatingR factorially divergent series: 1 1 1 n! gn = dt e−t (??) 1 g t n=0 0 X Z − pole on the Borel axis! non-perturbative imaginary part ) i π − 1 e g ± g but every term in the series is real !?! Borel Summation: basic idea 1 1 n −t 1 1 − 1 1 Borel e n! x = dt e = e x Ei )R P 1 x t x x "n=0 # 0 X Z − 2.0 1.5 1.0 0.5 x 0.5 1.0 1.5 2.0 2.5 3.0 -0.5 Borel summation 1 n Borel transform of series f(g) cn g : ∼ n=0 1P cn [f](t) = tn B n! n=0 X new series typically has finite radius of convergence. Borel resummation of original asymptotic series: 1 1 f(g) = [f](t)e−t=gdt S g B Z0 warning: [f](t) may have singularities in (Borel) t plane B Borel singularities avoid singularities on R+: directional Borel sums: eiθ1 1 −t=g θf(g) = [f](t)e dt S g B Z0 C+ C- go above/below the singularity: θ = 0± non-perturbative ambiguity: Im[ f(g)] −! ± S0 challenge: use physical input to resolve ambiguity Borel summation: existence theorem (Nevanlinna & Sokal) R R f(z) analytic in circle CR = z : z < f − 2 2 g N−1 n N N f(z) = an z + RN (z) ; RN (z) A σ N! z j j ≤ j j n=0 X Borel transform 1 R/2 an B(t) = tn n! n=0 X analytic continuation to Im(t) + Sσ = t : t R < 1/σ f j − j g 1/σ 1 1 Re(t) f(z) = e−t=z B(t) dt z Z0 Borel summation in practice 1 n n f(g) cn g ; cn β Γ(γ n + δ) ∼ ∼ n=0 X alternating series: real Borel sum • 1 1 dt 1 t δ=γ t 1/γ f(g) exp ∼ γ t 1 + t βg − βg Z0 " # nonalternating series: ambiguous imaginary part • 1 1 dt 1 t δ=γ t 1/γ Re f( g) exp − ∼ γ P t 1 t βg − βg Z0 − " # π 1 δ/γ 1 1/γ 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?) connection formula: K (e±iπ z ) = K ( z ) i π I ( z ) • 0 j j 0 j j ∓ 0 j j ±iπ − 1 Z (e λ) = Z (λ) i e 2λ Z (λ) ) 1 2 ∓ 1 Resurgence: Preserving Analytic Continuation zero-dimensional partition functions • 1 p − 1 sinh2( λ x) 1 1 1 Z1(λ) = dx e 2λ = e 4λ K0 −∞ pλ 4λ Z 1 π Γ(n + 1 )2 ( 1)n(2λ)n 2 Borel-summable ∼ 2 − 1 2 r n=0 n!Γ 2 X p π= λ p − 1 sin2( λ x) π − 1 1 Z2(λ) = dx e 2λ = e 4λ I0 0 pλ 4λ Z 1 π Γ(n + 1 )2 (2λ)n 2 non-Borel-summable ∼ 2 1 2 r n=0 n!Γ 2 X naively: Z ( λ) = Z (λ) • 1 − 2 Resurgence: Preserving Analytic Continuation zero-dimensional partition functions • 1 p − 1 sinh2( λ x) 1 1 1 Z1(λ) = dx e 2λ = e 4λ K0 −∞ pλ 4λ Z 1 π Γ(n + 1 )2 ( 1)n(2λ)n 2 Borel-summable ∼ 2 − 1 2 r n=0 n!Γ 2 X p π= λ p − 1 sin2( λ x) π − 1 1 Z2(λ) = dx e 2λ = e 4λ I0 0 pλ 4λ Z 1 π Γ(n + 1 )2 (2λ)n 2 non-Borel-summable ∼ 2 1 2 r n=0 n!Γ 2 X naively: Z ( λ) = Z (λ) • 1 − 2 connection formula: K (e±iπ z ) = K ( z ) i π I ( z ) • 0 j j 0 j j ∓ 0 j j ±iπ − 1 Z (e λ) = Z (λ) i e 2λ Z (λ) ) 1 2 ∓ 1 Resurgence: Preserving Analytic Continuation Borel summation • 1 π 1 − t 1 1 Z (λ) = dt e 2λ F ; ; 1; t 1 2 2λ 2 1 2 2 − r Z0 directional Borel summation: • iπ −iπ Z1(e λ) Z1(e λ) − 1 π 1 − t 1 1 1 1 = dt e 2λ F ; ; 1; t i" F ; ; 1; t + i" 2 2λ 2 1 2 2 − − 2 1 2 2 r 1 Z 1 π 1 − 1 − t 1 1 = (2i) e 2λ dt e 2λ F ; ; 1; t − 2 2λ 2 1 2 2 − r Z0 − 1 = 2 i e 2λ Z (λ) − 1 (Im F 1 ; 1 ; 1; t i" = F 1 ; 1 ; 1; 1 t ) 2 1 2 2 − 2 1 2 2 − ±iπ − 1 connection formula: Z (e λ) = Z (λ) i e 2λ Z (λ) • 1 2 ∓ 1 reflection formula: (1 + z) (1 z) = 1 π cot(π z) • − − z − 1 1 π Im (1 + iy) + + π e−2π k y ) ∼ −2y 2 k X=1 “raw” asymptotics inconsistent with analytic continuation resurgence fixes this Resurgence: Preserving Analytic Continuation d Stirling expansion for (x) = dx ln Γ(x) is divergent 1 1 1 1 174611 (1 + z) ln z + + + + ::: ∼ 2z − 12z2 120z4 − 252z6 ··· 6600z20 − functional relation: (1 + z) = (z) + 1 • z formal series Im (1 + iy) 1 + π ) ∼ − 2y 2 Resurgence: Preserving Analytic Continuation d Stirling expansion for (x) = dx ln Γ(x) is divergent 1 1 1 1 174611 (1 + z) ln z + + + + ::: ∼ 2z − 12z2 120z4 − 252z6 ··· 6600z20 − functional relation: (1 + z) = (z) + 1 • z formal series Im (1 + iy) 1 + π ) ∼ − 2y 2 reflection formula: (1 + z) (1 z) = 1 π cot(π z) • − − z − 1 1 π Im (1 + iy) + + π e−2π k y ) ∼ −2y 2 k X=1 “raw” asymptotics inconsistent with analytic continuation resurgence fixes this Transseries Example: Painlevé II (matrix models, fluids ..