Decoding the Path Integral: Resurgent Asymptotics and Extreme QFT Gerald Dunne University of Connecticut Theoretical Physics Colloquium, Arizona State, October 28, 2020
Physical Motivation 300 250 Temperature (MeV) Quark-Gluon Plasma 200 150 100 Hadron Gas Color 50 Superconductor 0 0 200 400 600 800 1000 1200 1400 1600 Baryon Doping – � B (MeV)
Physical Motivation: Quantum Physics in Extreme Conditions QCD phase diagram • non-equilibrium physics at strong-coupling • quantum systems in extreme background fields • back-reaction physics • transition to hydrodynamics • extreme = strongly-coupled; high density; ultra-fast driving; ultra-cold; strong fields; strong curvature; heavy ion collisions; … perturbation theory • non-perturbative semi-classical methods: “instantons" • non-perturbative numerical methods: Monte Carlo • asymptotics •
Physical Motivation: Quantum Physics in Extreme Conditions QCD phase diagram • non-equilibrium physics at strong-coupling • quantum systems in extreme background fields • back-reaction physics • transition to hydrodynamics • extreme = strongly-coupled; high density; ultra-fast driving; ultra-cold; strong fields; strong curvature; heavy ion collisions; … perturbation theory • non-perturbative semi-classical methods: “instantons" • non-perturbative numerical methods: Monte Carlo • asymptotics • “resurgence”: new form of asymptotics that unifies these approaches
Physical Motivation: Quantum Physics in Extreme Conditions QCD phase diagram • non-equilibrium physics at strong-coupling • quantum systems in extreme background fields • back-reaction physics • transition to hydrodynamics • extreme = strongly-coupled; high density; ultra-fast driving; ultra-cold; strong fields; strong curvature; heavy ion collisions; … perturbation theory • non-perturbative semi-classical methods: “instantons" • non-perturbative numerical methods: Monte Carlo • asymptotics • “resurgence”: new form of asymptotics that unifies these approaches technical problem: what does a quantum path integral really mean?
The Feynman Path Integral i � Z QM: D x ( t ) exp ~ S [ x ( t )] i � Z QFT: D A ( x µ ) exp g 2 S [ A ( x µ )] • stationary phase approximation: classical physics • quantum perturbation theory: fluctuations about trivial saddle point • other saddle points: non-perturbative physics • resurgence: saddle points are related by analytic continuation, so perturbative and non-perturbative physics are unified
<latexit sha1_base64="LniK6RuOldbifUpm4aziNV4E61c=">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</latexit> Stokes and the Airy Function: “Stokes Phenomenon” Ai ( x ) Z + ∞ Ai( x ) = 1 dt e i ( 1 3 t 3 + x t ) 0.4 2 π 0.2 −∞ x - 10 - 8 - 6 - 4 - 2 2 - 0.2 - 0.4
<latexit sha1_base64="I0gCwmyIzur325MjdPsE8tE1LE=">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</latexit> <latexit sha1_base64="LniK6RuOldbifUpm4aziNV4E61c=">ACaHicbVFdb9MwFHXCx0bHR8aHEOLFWoXUMaiSDgl4QBrwuOQ6DapbivHuWmtOU6wb9AiK+I/8sYP4IVfgdtFaGxcyfLROf42sdpaTFOP4ZhNeu37i5sXmrt3X7zt170fb9I1vWRsBYlKo0Jym3oKSGMUpUcFIZ4EWq4Dg9/bjSj7+BsbLUX7CpYFrwhZa5FBw9NY+sxQWUjv4qrkxvHneOmYK+l62g7Pdyw3XLikdSNWyZJjXP30m85Nu3M7XWIZsheUJg5yRTkOPjr2m9xtr935kVkRi6WuNv2GOjswrR51I+H8broVZB0oE+6OpxHP1hWiroAjUJxaydJXOHUcYNSKPDn1xYqLk75AiYeal6Anbp1UC195pmM5qXxSyNdsxcdjhfWNkXqOwuOS3tZW5H/0yY15m+mTuqRtDifFBeK4olXaVOM2lAoGo84MJIf1cqltzHhP5vej6E5PKTr4Kj0TB5NXz7edQ/+NDFsUmekh0yIAl5TQ7IJ3JIxkSQX8FW8DB4FPwOo/Bx+OS8NQw6zwPyT4U7fwC+bLqb</latexit> Stokes and the Airy Function: “Stokes Phenomenon” Ai ( x ) Z + ∞ Ai( x ) = 1 dt e i ( 1 3 t 3 + x t ) 0.4 2 π 0.2 −∞ x - 10 - 8 - 6 - 4 - 2 2 Stokes transitions occur in the complex x plane • - 0.2 - 0.4 non-perturbative connection • formulae connect sectors z √ r Z dz e − r 3 / 2 ( 1 3 z 3 − e i θ z ) Ai( x ) = 2 π i γ
Analytic Continuation of Path Integrals since we need complex analysis and contour deformation to make sense of oscillatory exponential integrals, it is natural to explore similar methods for (infinite dimensional) path integrals i � Z − 1 � Z D x ( t ) exp ~ S [ x ( t )] D x ( t ) exp ~ S [ x ( t )] goal: a satisfactory formulation of the functional integral should be able to describe Stokes transitions idea: seek a computationally viable constructive definition of the path integral using ideas from resurgent trans-series
Resurgent Trans-Series resurgence: “new” idea in mathematics Ecalle 1980s; Dingle 1960s; Stokes 1850 perturbative series “trans-series” ~ ~ p (ln ~ ) l c [ kpl ] e − k X X X X f ( ~ ) = c [ p ] ~ p f ( ~ ) = p k l p physics: • unifies perturbative and non-perturbative physics • trans-series is well-defined under analytic continuation mathematics: • expansions about different saddles are related • exponentially improved asymptotics
Resurgent Functions “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. Ecalle, 1980 implication: fluctuations about different singularities are related conjecture: this structure occurs for all “natural problems’’
Resurgence in Exponential Integrals steepest descent integral through saddle point “n”: 1 Z ~ f ( x ) = ~ f n T ( n ) ( ~ ) i i I ( n ) ( ~ ) = dx e e p 1 / ~ C n all fluctuations beyond the Gaussian approximation: ∞ X T ( n ) ( ~ ) ∼ T ( n ) ~ r r r =0
Resurgence in Exponential Integrals steepest descent integral through saddle point “n”: 1 Z ~ f ( x ) = ~ f n T ( n ) ( ~ ) i i I ( n ) ( ~ ) = dx e e p 1 / ~ C n all fluctuations beyond the Gaussian approximation: ∞ X T ( n ) ( ~ ) ∼ T ( n ) ~ r r r =0 straightforward complex analysis implies: ( F nm ≡ f m − f n ) universal large orders of fluctuation coefficients: " # ( F nm ) 2 ∼ ( r − 1)! ( ± 1) F nm T ( m ) ( r − 1) T ( m ) ( r − 1)( r − 2) T ( m ) X T ( n ) + + + . . 0 1 2 ( F nm ) r r 2 π i m
Resurgence in Exponential Integrals steepest descent integral through saddle point “n”: 1 Z ~ f ( x ) = ~ f n T ( n ) ( ~ ) i i I ( n ) ( ~ ) = dx e e p 1 / ~ C n all fluctuations beyond the Gaussian approximation: ∞ X T ( n ) ( ~ ) ∼ T ( n ) ~ r r r =0 straightforward complex analysis implies: ( F nm ≡ f m − f n ) universal large orders of fluctuation coefficients: " # ( F nm ) 2 ∼ ( r − 1)! ( ± 1) F nm T ( m ) ( r − 1) T ( m ) ( r − 1)( r − 2) T ( m ) X T ( n ) + + + . . 0 1 2 ( F nm ) r r 2 π i m fluctuations about different saddles are quantitatively related
Resurgence in Exponential Integrals canonical example: Airy function: 2 saddle points � r + 1 � � r + 5 � ⇢ � r = ( ± 1) r Γ Γ 1 , ± 5 48 , 385 4608 , ± 85085 T ± 6 6 = 663552 , . . . � 4 � r r ! (2 π ) 3
Resurgence in Exponential Integrals canonical example: Airy function: 2 saddle points � r + 1 � � r + 5 � ⇢ � r = ( ± 1) r Γ Γ 1 , ± 5 48 , 385 4608 , ± 85085 T ± 6 6 = 663552 , . . . � 4 � r r ! (2 π ) 3 large orders of fluctuation coefficients: ◆ 5 ◆ 2 385 ! ✓ 4 ✓ 4 r ∼ ( r − 1)! 1 1 T + 1 − ( r − 1) + ( r − 1)( r − 2) − . . . � 4 � r 3 48 3 4608 (2 π ) 3 generic “large-order/low-order” resurgence
Resurgence in Exponential Integrals canonical example: Airy function: 2 saddle points � r + 1 � � r + 5 � ⇢ � r = ( ± 1) r Γ Γ 1 , ± 5 48 , 385 4608 , ± 85085 T ± 6 6 = 663552 , . . . � 4 � r r ! (2 π ) 3 large orders of fluctuation coefficients: ◆ 5 ◆ 2 385 ! ✓ 4 ✓ 4 r ∼ ( r − 1)! 1 1 T + 1 − ( r − 1) + ( r − 1)( r − 2) − . . . � 4 � r 3 48 3 4608 (2 π ) 3 generic “large-order/low-order” resurgence amazing fact: this resurgent large-order/low-order behavior has been found in matrix models, QM, QFT, string theory, … the only natural way to explain this is via analytic continuation of functional integrals
Perturbation Theory perturbation theory works, but it is generically divergent perturbation theory encodes non-perturbative information
unstable
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