Resurgence, Phase Transitions and Large N Gerald Dunne University of Connecticut Yukawa Institute/RIKEN iTHEMS Conference 2020 , September 2020 review: arXiv:1601.03414 ; winter school lectures
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) phase transitions in cold atom systems • quantum systems in extreme background fields • transition to hydrodynamics • quantum gravity extreme systems are extremely difficult to analyze quantitatively extreme = strongly-coupled; high density; ultra-fast driving; ultra-cold; strong fields; strong curvature; heavy ion collisions; … • perturbation theory is of limited use • 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) phase transitions in cold atom systems • quantum systems in extreme background fields • transition to hydrodynamics • quantum gravity extreme systems are extremely difficult to analyze quantitatively extreme = strongly-coupled; high density; ultra-fast driving; ultra-cold; strong fields; strong curvature; heavy ion collisions; … • perturbation theory is of limited use • 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) phase transitions in cold atom systems • quantum systems in extreme background fields • transition to hydrodynamics • quantum gravity extreme systems are extremely difficult to analyze quantitatively extreme = strongly-coupled; high density; ultra-fast driving; ultra-cold; strong fields; strong curvature; heavy ion collisions; … • perturbation theory is of limited use • 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
Stokes and the Airy Function: “Stokes Phenomenon” Ai(z) z 3 z 3 / 2 e − 2 8 z → + ∞ , 2 √ π z 1 / 4 Z + ∞ > > Ai( z ) = 1 < 3 x 3 + z x ) dx ∼ e i ( 1 2 π sin ( 2 3 ( − z ) 3 / 2 + π 4 ) −∞ > > , z → −∞ : √ π ( − z ) 1 / 4
Stokes and the Airy Function: “Stokes Phenomenon” Ai(z) 1.0 0.5 x - 4 - 2 2 4 z - 0.5 - 1.0 3 z 3 / 2 e − 2 8 z → + ∞ , 2 √ π z 1 / 4 Z + ∞ > > Ai( z ) = 1 < 3 x 3 + z x ) dx ∼ e i ( 1 2 π sin ( 2 3 ( − z ) 3 / 2 + π 4 ) −∞ > > , z → −∞ : √ π ( − z ) 1 / 4 • integral cannot be evaluated without contour deformation • “Stokes transition” at z=0 • fluctuation expansions about saddles must be divergent, and must be related • underlies optics and WKB analysis
Analytic Continuation of Path Integrals since we need complex analysis and contour deformation to make sense of oscillatory 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 )] idea: seek a computationally viable constructive definition of the path integral as a resurgent trans-series
Resurgent Trans-Series resurgence: “new” idea in mathematics Dingle 1960s, Ecalle, 1980s; 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 mathematics: • trans-series is well-defined under analytic continuation • 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 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: universal large orders of fluctuation coefficients: ( F nm ≡ f m − f n ) " # ( 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: universal large orders of fluctuation coefficients: ( F nm ≡ f m − f n ) " # ( 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 ! r ∼ ( r − 1)! ✓ 4 1 ✓ 4 1 T + 1 − ( r − 1) + ( r − 1)( r − 2) − . . . � 4 � r 3 48 3 4608 (2 π ) 3 generic “large-order/low-order” resurgence relation
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 ! r ∼ ( r − 1)! ✓ 4 1 ✓ 4 1 T + 1 − ( r − 1) + ( r − 1)( r − 2) − . . . � 4 � r 3 48 3 4608 (2 π ) 3 generic “large-order/low-order” resurgence relation amazing fact: this 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 path integrals
Decoding a Path Integral as a Trans-Series Z ~ S [ A ] = i i X ~ S [ A thimble ] × (fluctuations) × (logs) D A e e thimbles logarithms • expansions along different axes must be quantitatively related • expansions about different saddles must be quantitatively relate d
Perturbation Theory perturbation theory works, but it is generically divergent this is actually a very good thing ! and there is a lot of interesting physics behind this
unstable
The Struggle to Make Sense of Divergent Series ∞ ( − 1) n n ! x n = ??? X n =0 L. Euler, De seriebus divergentibus , Opera Omnia, I, 14, 585–617, 1760.
The Struggle to Make Sense of Divergent Series factorial: “Borel summation” Z ∞ ∞ 1 ( − 1) n n ! x n = X dt e − t f ( x ) = 1 + x t 0 n =0 convergent for all x > 0
The Struggle to Make Sense of Divergent Series factorial: “Borel summation” Z ∞ ∞ 1 ( − 1) n n ! x n = X dt e − t f ( x ) = 1 + x t 0 n =0 convergent for all x > 0 Z ∞ ∞ 1 n ! x n = X dt e − t f ( − x ) = 1 − x t 0 n =0 Im[ f ( − x )] ∼ e − 1 /x nonperturbative imaginary part !
QM Perturbation Theory: Zeeman & Stark Effects Zeeman : divergent, alternating, asymptotic series a n ∼ ( − 1) n (2 n )! physics: magnetic field causes energy level shifts (real) Stark : divergent, non-alternating, asymptotic series a n ∼ (2 n )! physics: • electric field causes energy level shifts (real) • and ionization (imaginary, exponentially small)
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