Asymptotic expansions and Dyson-Schwinger equations Michael Borinsky - PowerPoint PPT Presentation

Asymptotic expansions and Dyson-Schwinger equations Michael Borinsky 1 Humboldt-University Berlin Departments of Physics and Mathematics Paths to, from and in renormalisation, Potsdam 2016 1 borinsky@physik.hu-berlin.de M. Borinsky (HU Berlin) Asymptotic expansions and Dyson-Schwinger equations 1

We will analyze a class of power series F α β ⊂ R [[ x ]] with α, β > 0 , ∞ � f n x n ∈ F α f ( x ) = β n =0 with coefficients which satisfy, f n lim α n Γ( n + β ) = C n →∞ and ˜ f n = f n − C α n Γ( n + β ) ∞ � f n +1 x n ∈ F α ˜ β . n =0 M. Borinsky (HU Berlin) Asymptotic expansions and Dyson-Schwinger equations 2

These are the power series which admit an asymptotic expansion of the form, � � c 0 + c 1 c 2 f n = α n + β Γ( n + β ) n + n ( n − 1) + . . . f n including power series with lim n →∞ α n Γ( n + β ) = 0 ⇒ c k = 0 for all k ≥ 0. These power series appear in Graph and permutation counting problems in combinatorics. Perturbation expansions in physics. Subclass of gevrey-1 -power series. M. Borinsky (HU Berlin) Asymptotic expansions and Dyson-Schwinger equations 3

Consider a power series f ( x ) ∈ F α β : � � c 0 + c 1 c 2 f n = α n + β Γ( n + β ) n + n ( n − 1) + . . . Idea: Interpret the coefficients c k of the asymptotic expansion as a new power series. Definition A maps a power series to its asymptotic expansion: F α A : → R [[ x ]] β ∞ � c k x k f ( x ) �→ γ ( x ) = k =0 M. Borinsky (HU Berlin) Asymptotic expansions and Dyson-Schwinger equations 4

Theorem 1 A is a derivation on F α β : ( A f ( x ) g ( x ))( x ) = f ( x )( A g )( x ) + ( A f )( x ) g ( x ) Follows from the log-convexity of Γ. ⇒ F α β is a subring of R [[ x ]]. Proof sketch With h ( x ) = f ( x ) g ( x ), R − 1 R − 1 n − R � � � h n = f n − k g k + f k g n − k + f k g n − k k =0 k =0 k = R � �� � � �� � High order times low order O ( α n Γ( n + β − R )) . M. Borinsky (HU Berlin) Asymptotic expansions and Dyson-Schwinger equations 5

Derivative ∂ Analyze ∂ , the ordinary derivative on power series, F α F α ∂ : → β +2 , β ∞ � f ′ ( x ) = nf n x n − 1 f ( x ) �→ n =1 where the β + 2 comes from ( n + 1) f n +1 ∼ Γ( n + β + 2). M. Borinsky (HU Berlin) Asymptotic expansions and Dyson-Schwinger equations 6

We have the commutative diagram, F α F α ∂ β +2 β A A R [[ x ]] R [[ x ]] ∂ A with ∂ A = α − 1 − x β + x 2 ∂ where ∂ A is a bijection, because ker ∂ ⊂ ker A ! M. Borinsky (HU Berlin) Asymptotic expansions and Dyson-Schwinger equations 7

What happens for composition of power series ∈ F α β ? Theorem 2 Bender [1975] If f ( x ) is a power series of a function analytic at the origin, i.e. | f n | ≤ C n , then, for g ∈ F α β with g (0) = 0: f ◦ g ∈ F α β ( A f ◦ g )( x ) = f ′ ( g ( x ))( A g )( x ) Bender considered much more general power series, but this is a direct corollary of his theorem in 1975. M. Borinsky (HU Berlin) Asymptotic expansions and Dyson-Schwinger equations 8

What happens if f / ∈ ker A ? A fulfills a general ‘chain rule’: Theorem 3 MB [2016] β with g (0) = 0 and g ′ (0) = 1: If f , g ∈ F α f ◦ g ∈ F α β � � β x g ( x ) − x α xg ( x ) ( A f )( g ( x )) ( A f ◦ g )( x ) = f ′ ( g ( x ))( A g )( x ) + e g ( x ) ⇒ We can solve for asymptotics of implicitly defined power series! M. Borinsky (HU Berlin) Asymptotic expansions and Dyson-Schwinger equations 9

Theorem 3 MB [2016] If f , g ∈ F α β with g (0) = 0 and g ′ (0) = 1: f ◦ g ∈ F α β � � β x g ( x ) − x ( A f ◦ g )( x ) = f ′ ( g ( x ))( A g )( x ) + α xg ( x ) ( A f )( g ( x )) e g ( x ) g ′ (0) = 1 not a real restriction. Scaling maps spaces β → F α ′ F α trivially. β g ( x ) − x α xg ( x ) generates ‘funny exponentials’: Typical prefactors of e the form g 2 e α in asymptotic expansions. M. Borinsky (HU Berlin) Asymptotic expansions and Dyson-Schwinger equations 10

Differential equations ∂ f ( x ) = F ( f ( x ) , x ) with F ( x , y ) analytic at (0 , 0). Apply A : A ∂ f ( x ) = ∂ F ∂ f ( f ( x ) , x )( A f )( x ) Use ∂ A with ∂ A A = A ∂ : ⇒ ∂ A ( A f )( x ) = ∂ F ∂ f ( f ( x ) , x )( A f )( x ) Linear differential equation for ( A f )( x ). M. Borinsky (HU Berlin) Asymptotic expansions and Dyson-Schwinger equations 11

Applications Action on Dyson-Schwinger-Equations Let p , g , f ∈ F α β and p ∈ ker A , then the functional equation, p ( g ( x )) = x + f ( g ( x )) � � β x g ( x ) − x α xg ( x ) ( A f )( g ( x )) ( A g )( x ) = g ′ ( x ) implies e g ( x ) � � β g − 1( x ) − x x ( A f )( x ) = g − 1 ′ ( x ) α xg − 1( x ) ( A g )( g − 1 ( x )) . and e g − 1 ( x ) where g ( g − 1 ( x )) = x . ⇒ Solving the DSE ‘perturbativly’ to n terms gives an asymptotic expansion up to order n − 2! A maps low order expansions to high order expansions. Asymptotic expansion independent of p . M. Borinsky (HU Berlin) Asymptotic expansions and Dyson-Schwinger equations 12

Example: Simple permutations Let π ∈ S simple ⊂ S n such that π ([ i , j ]) � = [ k , l ] for all n i , j , k , l ∈ [0 , n ] with 2 ≤ | [ i , j ] | ≤ n − 1, then π is a simple permutation, which does not map an interval to another interval. With S ( x ) = � ∞ | x n and F ( x ) = � ∞ n =0 | S simple n =1 n ! x n : n Albert et al. [2003] F ( x ) − F ( x ) 2 = x + S ( F ( x )) 1 + F ( x ) F ( x ) ∈ F 1 1 and ( A F ) = 1 ⇒ even though S ( x ) is only given implicitly, we have an asymptotic expansion! M. Borinsky (HU Berlin) Asymptotic expansions and Dyson-Schwinger equations 13

Generating function for asymptotic coefficients of S ( x ): � � β F − 1( x ) − x x ( A S )( x ) = F − 1 ′ ( x ) α xF − 1( x ) e F − 1 ( x ) � � 1 − 4 2 40 s n = e − 2 n ! n + n ( n − 1) − 3 n ( n − 1)( n − 2) + . . . Generating function for asymptotic coefficients ⇒ can analyse asymptotics of asymptotics. M. Borinsky (HU Berlin) Asymptotic expansions and Dyson-Schwinger equations 14

Conclusions F α β forms a subring of R [[ x ]] closed under composition, differentiation* and integration. A is a derivation on F α β which can be used to obtain asymptotic expansions of implicitly defined power series. Nice closure properties under asymptotic derivative A . Generalizations possible to multiple α 1 , . . . , α l ∈ C with | α i | = α . Suitable for resummation of perturbation series ⇒ applications in QFT and QM! There are probably many connections to resurgence! M. Borinsky (HU Berlin) Asymptotic expansions and Dyson-Schwinger equations 15

‘Asymptotic calculus’ Action under transformation with A -operator f ( x ) g ( x ) → ( A f )( x ) g ( x ) + f ( x )( A g )( x ) ( α − 1 − x β + x 2 ∂ )( A f )( x ) ∂ f ( x ) → � � β g ( x ) − x α xg ( x ) ( A f )( g ( x )) f ′ ( g ( x ))( A g )( x ) + x f ( g ( x )) → e g ( x ) M. Borinsky (HU Berlin) Asymptotic expansions and Dyson-Schwinger equations 16

MH Albert, M Klazar, and MD Atkinson. The enumeration of simple permutations. 2003. Edward A Bender. An asymptotic expansion for the coefficients of some formal power series. Journal of the London Mathematical Society , 2(3):451–458, 1975. MB. Power series asymptotics power series (in preparation). 2016. M. Borinsky (HU Berlin) Asymptotic expansions and Dyson-Schwinger equations 16