Mathematical renormalization in quantum electrodynamics via - PowerPoint PPT Presentation

Mathematical renormalization in quantum electrodynamics via noncommutative generating series (Part I) G. Duchamp, Hoang Ngoc Minh, K. Penson, P. Simonnet. Combinatoire, Informatique et Physique, Villetaneuse, 12 Novembre 2013.

Summary 1. Introduction. 2. Algebraic combinatorics of formal power series on noncommutative variables. 3. Algebraic combinatorics of polylogarithms, multiple harmonic sums and polyzetas. 4. Nonlinear differential equations.

INTRODUCTION (Il ´ etait une fois le rˆ eve d’Icare)

Nonlinear dynamical systems Let ∂ z denotes d / dz .  y ( z ) = f ( q ( z )) ,  ( NDS ) ∂ z q ( z ) = A 0 ( q ) u 0 ( z ) + A 1 ( q ) u 1 ( z ) ,  q ( z 0 ) = q 0 , where : ◮ u 0 ( z ) and u 1 ( z ) are “controles”, or “inputs”, ◮ the state q = ( q 1 , . . . , q n ) belongs the complex analytic manifold Q of dimension n and q 0 is the initial state, ◮ the observation f ∈ O , with O the ring of holomorphic functions over Q , n i ( q ) ∂ � A j ◮ For i = 0 .. 1 , A i ( q ) = is an analytic vector field ∂ q j j =1 over Q , with A j i ( q ) ∈ O , for j = 1 , . . . , n , ◮ y ( z ) is the “output” of ( NDS ).

Examples of Nonlinear Dynamical System Example (harmonic oscillator) Let k 1 , k 2 be parameters and ∂ z y ( z ) + k 1 y ( z ) + k 2 y 2 ( z ) = u 1 ( z ) which can be represented by the following state equations ∂ z q ( z ) = A 0 ( q ) u 0 ( z ) + A 1 ( q ) u 1 ( z ) , − ( k 1 q + k 2 q 2 ) ∂ ∂ A 0 = and A 1 = ∂ q , ∂ q y ( z ) = q ( z ) . Example (Duffing’s equation) Let a , b , c be parameters and ∂ 2 z y ( z ) + a ∂ z y ( z ) + by ( z ) + cy 3 ( z ) = u 1 ( z ) which can be represented by the following state equations ∂ z q ( z ) = A 0 ( q ) u 0 ( z ) + A 1 ( q ) u 1 ( z ) , 1 ) ∂ ∂ ∂ − ( aq 2 + b 2 q 1 + cq 3 = + q 2 and A 1 = , A 0 ∂ q 2 ∂ q 1 ∂ q 2 y ( z ) = q 1 ( z ) .

Nonlinear differential equations with three sigularities  y ( z ) = f ( q ( z )) ,  ( NDE ) ∂ z q ( z ) = A 0 ( q ) u 0 ( z ) + A 1 ( q ) u 1 ( z ) ,  q ( z 0 ) = q 0 , where : ◮ u 0 ( z ) = 1 1 z , u 1 ( z ) = 1 − z , ◮ the state q = ( q 1 , . . . , q n ) belongs the complex analytic manifold Q of dimension n and q 0 is the initial state, ◮ the observation f ∈ O , with O the ring of holomorphic functions over Q , n i ( q ) ∂ � A j ◮ For i = 0 .. 1 , A i ( q ) = is an analytic vector field ∂ q j j =1 over Q , with A j i ( q ) ∈ O , for j = 1 , . . . , n .

Particular cases : Fuchsian differential equations (FDE) ∂ z q ( z ) = [ M 0 u 0 ( z ) + M 1 u 1 ( z )] q ( z ) , y ( z ) = λ q ( z ) , q ( z 0 ) = η, where M 0 , M 1 ∈ M n , n ( C ) , λ ∈ M 1 , n ( C ) , η ∈ M n , 1 ( C ), and u 0 ( z ) = z − 1 , u 1 ( z ) = (1 − z ) − 1 . Example (hypergeometric equation) Let t 0 , t 1 , t 2 be parameters and z (1 − z ) ∂ 2 z y ( z ) + [ t 2 − ( t 0 + t 1 + 1) z ] ∂ z y ( z ) − t 0 t 1 y ( z ) = 0 . Let q 1 ( z ) = y ( z ) and q 2 ( z ) = z (1 − z ) ∂ z y ( z ). One has � � �� � � � � � � ∂ z q 1 0 0 0 1 q 1 = u 0 ( z ) − u 1 ( z ) . ∂ z q 2 − t 0 t 1 − t 2 0 t 2 − t 0 − t 1 q 2 � 0 � � 0 � � q 1 ( z 0 ) � 0 1 � 1 0 � λ = M 0 = − M 1 = η = . t 0 t 1 t 2 0 t 2 − t 0 − t 1 q 2 ( z 0 ) A 0 ( q ) = − ( t 0 t 1 q 1 + t 2 q 2 ) ∂ ∂ ∂ and A 1 ( q ) = − q 1 − ( t 2 − t 0 − t 1 ) q 2 . ∂ q 2 ∂ q 1 ∂ q 2

Present work By successive Picard iterations, one get � � y ( z ) = A i 1 ◦ . . . ◦ A i k ( f ( q 0 )) k ≥ 0 i 1 ,..., i k =0 , 1 � z � z 1 � z k − 1 u i 1 ( z 1 ) dz 1 u i 2 ( z 2 ) dz 2 . . . u i k ( z k ) dz k z 0 z 0 z 0 Let X = { x 0 , x 1 } and for any w = x i 1 · · · x i k ∈ X ∗ , A ( w ) = A i 1 ◦ . . . ◦ A i k , � z � z 1 � z k − 1 α z z 0 ( w ) = u i 1 ( z 1 ) dz 1 u i 2 ( z 2 ) dz 2 . . . u i k ( z k ) dz k . z 0 z 0 z 0 Therefore, � � � A ( w ) α z y ( z ) = z 0 ( w ) ( f ( q 0 )) w ∈ X ∗ [( A ⊗ α z = z 0 ) D ]( f ( q 0 )) , � where, D X = w ⊗ w . w ∈ X ∗

Noncommutative generating series ◮ Fliess generating series and Chen series � � α z σ f | q 0 = A ( w )( f ( q 0 )) w and S z 0 � z = z 0 ( w ) w . w ∈ X ∗ w ∈ X ∗ ◮ The duality between σ f | q 0 and S z 0 � z consists on the convergence of � � σ f | q 0 � S z 0 � z � = �A ( w )( f ( q 0 )) | w �� S z 0 � z | w � . w ∈ X ∗ ◮ Divergence : � d � n � � n z d y ( z ) � z → 1 ? and y ( z ) � z → 1 ? dz dz ◮ If theTaylor expansions of ( d / dz ) n y ( z ) and ( zd / dz ) n y ( z ) exist : � d � n � � n z d � � d k z n t k z k y ( z ) = and y ( z ) = dz dz k ≥ 0 k ≥ 0 then k →∞ ? and k →∞ ? d k � t k �

Chen’s iterated integral along a path and polylogarithms Let ω 0 ( z ) = u 0 ( z ) dz and ω 1 ( z ) = u 1 ( z ) dz . The iterated integral over ω 0 ( z ) and ω 1 ( z ) associated to w = x i 1 · · · x i k is defined by � z � z k − 1 α z α z z 0 (1 X ∗ ) = 1 and z 0 ( x i 1 . . . x i k ) = ω i 1 ( z 1 ) . . . ω i k ( z k ) . z 0 z 0 For any w = x s 1 − 1 x 1 . . . x s r − 1 x 1 ∈ X ∗ x 1 , 0 0 z n 1 � α z 0 ( w ) = = Li s 1 ,..., s r ( z ) . n s 1 1 . . . n s r r n 1 >...> n r > 0 � z � s ds dt α z 0 ( x 0 x 1 ) = Li 2 ( z ) = Example 1 − t s 0 0 � z � s ds � t k = dt s 0 0 k ≥ 0 � z ds s k − 1 � = k 0 k ≥ 1 z k � = k 2 . k ≥ 1

Polylogarithms, multiple harmonic sums and polyzetas z n 1 1 � � H s ( N ) = and Li s ( z ) = . n s 1 1 . . . n s r n s 1 1 . . . n s r r r N ≥ n 1 >...> n r > 0 n 1 >...> n r > 0 If s 1 > 1 then by an Abel’s theorem, one has 1 � N →∞ H s ( N ) = lim lim z → 1 Li s ( z ) = ζ ( s ) = n s 1 1 . . . n s r r n 1 >...> n r > 0 else to determine the asymptotic expansion of 1 � H { 1 } r ( N ) = , n 1 . . . n r N ≥ n 1 >...> n r > 0 1 � ,..., s r ( N ) = . H { 1 } k , s k +1 n 1 . . . n k n s k +1 k +1 . . . n s r r ���� N ≥ n 1 >...> n r > 0 > 1 1 � H s ( n ) z n = Fact : one has 1 − z Li s ( z ) = P s ( z ). n ≥ 0 Let Z be the Q -algebra generated by convergent polyzetas.

Encoding the multi-indices by words Y = { y k | k ∈ N + } and X = { x 0 , x 1 } . Y ∗ (resp. X ∗ ) : set of words over Y (resp. X ). s = ( s 1 , . . . , s r ) ↔ w = y s 1 . . . y s r ↔ w = x s 1 − 1 x 1 . . . x s r − 1 x 1 . 0 0 w is said convergent if s 1 > 1. A divergent word is of the form 1 x s k +1 − 1 (1 k , s k +1 , . . . , s r ) ↔ y k 1 y s k +1 . . . y s r ↔ x k x 1 . . . x s r − 1 x 1 . 0 0 z n 1 � ∀ w ∈ Y ∗ , Li w : w �→ Li w ( z ) = , n s 1 1 . . . n s r r n 1 >...> n r > 0 1 � ζ w : w �→ ζ ( w ) = , n s 1 1 . . . n s r r n 1 >...> n r > 0 1 � H w : w �→ H w ( N ) = , n s 1 1 . . . n s r r N ≥ n 1 >...> n r > 0 1 � H w ( N ) z N = P w : w �→ P w ( z ) = 1 − z Li w ( z ) . N ≥ 0

ALGEBRAIC COMBINATORICS OF FORMAL POWER SERIES ON NONCOMMUTATIVE VARIABLES (La conquˆ ete de Mars ...)

Shuffle bialgebra and Sch¨ utzenberger’s factorization Let L ynX be the set of Lyndon words over X . P l = l for l ∈ Y , P l = [ P s , P r ] for l ∈ L ynY \ Y , standard factorization of l = ( s , r ) , P i 1 l 1 . . . P i k for w = l i 1 1 . . . l i k P w = k , l k l 1 > . . . > l k , l 1 . . . , l k ∈ L ynY . S l = 1 for l = 1 X ∗ , S l = xS u , for l = xu ∈ L ynY , ⊔ i 1 ⊔ i k ⊔ ⊔ ⊔ . . . S ⊔ S ⊔ ⊔ l 1 l k 1 . . . l i k for w = l i 1 S w = k , i 1 ! . . . i k ! l 1 > . . . > l k , , l 1 . . . , l k ∈ L ynY . Theorem (Sch¨ utzenberger, 1958) → � � � w ⊗ w = S w ⊗ P w = exp( S l ⊗ P l ) . w ∈ Y ∗ w ∈ Y ∗ l ∈L ynY

Example l P l S l x 0 x 0 x 0 x 1 x 1 x 1 x 0 x 1 [ x 0 , x 1 ] x 0 x 1 x 2 x 2 0 x 1 [ x 0 , [ x 0 , x 1 ]] 0 x 1 x 0 x 2 x 0 x 2 [[ x 0 , x 1 ] , x 1 ] 1 1 x 3 x 3 0 x 1 [ x 0 , [ x 0 , [ x 0 , x 1 ]]] 0 x 1 x 2 0 x 2 x 2 0 x 2 [ x 0 , [[ x 0 , x 1 ] , x 1 ]] 1 1 x 0 x 3 x 0 x 3 [[[ x 0 , x 1 ] , x 1 ] , x 1 ] 1 1 x 4 x 4 [ x 0 , [ x 0 , [ x 0 , [ x 0 , x 1 ]]]] 0 x 1 0 x 1 x 3 0 x 2 x 3 0 x 2 [ x 0 , [ x 0 , [[ x 0 , x 1 ] , x 1 ]]] 1 1 x 2 2 x 3 0 x 2 1 + x 2 0 x 1 x 0 x 1 [[ x 0 , [ x 0 , x 1 ]] , [ x 0 , x 1 ]] 0 x 1 x 0 x 1 x 2 0 x 3 x 2 0 x 3 [ x 0 , [[[ x 0 , x 1 ] , x 1 ] , x 1 ]] 1 1 x 0 x 1 x 0 x 2 3 x 2 0 x 3 1 + x 0 x 1 x 0 x 2 [[ x 0 , x 1 ] , [[ x 0 , x 1 ] , x 1 ]] 1 1 x 0 x 4 x 0 x 4 [[[[ x 0 , x 1 ] , x 1 ] , x 1 ] , x 1 ] 1 1 x 5 x 5 0 x 1 [ x 0 , [ x 0 , [ x 0 , [ x 0 , [ x 0 , x 1 ]]]]] 0 x 1 x 4 0 x 2 x 4 0 x 2 [ x 0 , [ x 0 , [ x 0 , [[ x 0 , x 1 ] , x 1 ]]]] 1 1 x 3 2 x 4 0 x 2 1 + x 3 [ x 0 , [[ x 0 , [ x 0 , x 1 ]] , [ x 0 , x 1 ]]] 0 x 1 x 0 x 1 0 x 1 x 0 x 1 x 3 0 x 3 x 3 0 x 3 [ x 0 , [ x 0 , [[[ x 0 , x 1 ] , x 1 ] , x 1 ]]] 1 1 x 2 0 x 1 x 0 x 2 3 x 3 0 x 3 1 + x 2 0 x 1 x 0 x 2 [ x 0 , [[ x 0 , x 1 ] , [[ x 0 , x 1 ] , x 1 ]]] 1 1 x 2 0 x 2 6 x 3 0 x 3 1 + 3 x 2 0 x 1 x 0 x 2 1 + x 2 0 x 2 1 x 0 x 1 [[ x 0 , [[ x 0 , x 1 ] , x 1 ]] , [ x 0 , x 1 ]] 1 x 0 x 1 x 2 0 x 4 x 2 0 x 4 [ x 0 , [[[[ x 0 , x 1 ] , x 1 ] , x 1 ] , x 1 ]] 1 1 x 0 x 1 x 0 x 3 4 x 2 0 x 4 1 + x 0 x 1 x 0 x 3 [[ x 0 , x 1 ] , [[[ x 0 , x 1 ] , x 1 ] , x 1 ]] 1 1 x 0 x 5 x 0 x 5 [[[[[ x 0 , x 1 ] , x 1 ] , x 1 ] , x 1 ] , x 1 ] 1 1