Non-commutative Wick polynomials Nikolas Tapia (WIAS/TU Berlin) FG6 - PowerPoint PPT Presentation

Non-commutative Wick polynomials Nikolas Tapia (WIAS/TU Berlin) FG6 based on ongoing work with K. Ebrahimi-Fard (NTNU), F . Patras (U. Nice) and L. Zambotti (Sorbonne U.) Weierstraß-Institut für angewandte Analysis und Stochastik October 3, 2019 @ Rencontre GDR, Calais.

Goals 1. Motivation 2. Moments and cumulants 3. Free Wick polynomials 4. If time permits: 4.1 Modification of products 4.2 Relation to power series 2/27 Free Wick polynomials

Classical Wick polynomials: A probabilist’s approach Definition Let X be a r.v. with � X n < ∞ for all n > 0 . Recursive definition: W ′ n ( x ) = nW n − 1 ( x ) , � W n ( X ) = 0 . W 2 ( x ) = x 2 − 2 x � X + 2 ( � X ) 2 − � X 2 , . . . For example: W 1 ( x ) = x − � X , Definition (Multivariate Wick polynomials) ∂ W n ( x 1 , . . . , x n ) = W n − 1 ( x 1 , . . . , x i − 1 , x i +1 , . . . , x n ) , � W n ( X 1 , . . . , X n ) = 0 . ∂x i 3/27 Free Wick polynomials

Classical Wick polynomials: A naïve physics approach Let F ◦ ≔ �Ω ⊕ H ⊕ H ◦ 2 ⊕ · · · be the symmetric Fock space over H . For each f ∈ H we have (bosonic) annihilation and creation operators a ( f ) , a † ( f ) on F ◦ such that a † ( f ) Ω = f a ( f ) Ω = 0 , and n � a ( f )( f 1 ◦ · · · ◦ f n ) = � f , f j � f 1 ◦ · · · ◦ f j − 1 ◦ f j +1 ◦ · · · ◦ f n , j =1 a † ( f )( f 1 ◦ · · · ◦ f n ) = f ◦ f 1 ◦ · · · ◦ f n . They satisfy the (canonical) commutation relation a ( f ) a † ( g ) − a † ( g ) a ( f ) = � f , g � 1 . 4/27 Free Wick polynomials

Classical Wick polynomials: A naïve physics approach The normal order operator N puts creation operators to the left of annihilation operators. For example N ( a † ( f ) a ( g )) = a † ( f ) a ( g ) and N ( a ( f ) a † ( g )) = a † ( g ) a ( f ) , etc. The (unnormalized) position operators p ( f ) ≔ a ( f ) + a † ( f ) satisfy N ( p ( f )) = p ( f ) and p ( f ) p ( g ) = a ( f ) a ( g ) + a ( f ) a † ( g ) + a † ( f ) a ( g ) + a † ( f ) a † ( g ) = a ( f ) a ( g ) + a † ( g ) a ( f ) + a † ( f ) a ( g ) + a † ( f ) a † ( g ) + � f , g � 1 = N ( p ( f ) p ( g )) + � f , g � 1 , i.e. N ( p ( f ) p ( g )) = p ( f ) p ( g ) − � f , g � 1 . Denoting X = p ( f ) , Y = p ( g ) and � ( b ) ≔ � b Ω , Ω � we see that N ( XY ) = XY − � ( XY ) = W 2 ( X , Y ) . By definition � [ N ( p ( f 1 ) · · · p ( f n ))] = 0 . 5/27 Free Wick polynomials

Classical Wick polynomials: Link to Dyson–Schwinger Suppose we have a measure of the form µ ( d x ) = e − S ( x ) d x and set I [ f ] = ∫ f d µ . Integrating by parts we get, for any nice function f , the Dyson-Schwinger equation I [ ∂ i f − ( ∂ i S ) f ] = 0 . The measure µ is characterized by the values I [ f ] , for nice f , usually polynomials are enough. For a 1-D Gaussian weight the equation is simply I ( f ′ − xf ) = 0 which entails I ( x 2 n ) = ( n − 1 ) !! and zero else. 2 J 2 , or I [ e Jx − 1 1 2 J 2 ] = 1 . Observe also that ( J − d This ultimately means that Z [ J ] ≔ I x ( e Jx ) = e d J ) Z [ J ] = 0 . n − xH n ) = 0 , so that re-expanding x n in Actually, the Dyson-Schwinger equaition implies that I ( H n +1 ) = − I ( H ′ the H n basis: n � I ( x n ) = α k I ( H k ) = α 0 k =0 6/27 Free Wick polynomials

Classical Wick polynomials: Link to Dyson–Schwinger 2 g ij x i x j a similar argument gives that More generally, for a multidimensional Gaussian weight S ( x ) = 1 n � I ( x j x i 1 · · · x i n ) = g ij k I ( x i 1 · · · ˆ x i k · · · x i n ) . k =1 But, since we have that I ( H n ) = 0 we can again re-expand any monomial in the H n basis and recover the above formula from our knowledge of H n . In our annihilation–creator operator example: � p ( f 1 ) p ( f 2 ) Ω , Ω � = � f 1 , f 2 � � p ( f 1 ) p ( f 2 ) p ( f 3 ) p ( f 4 ) Ω , Ω � = � f 1 , f 2 �� f 3 , f 4 � + � f 1 , f 3 �� f 2 , f 4 � + � f 1 , f 4 �� f 2 , f 3 � 7/27 Free Wick polynomials

Classical Wick polynomials: Hopf-algebraic approach Definition A noncommutative probability space is a tuple ( A , ϕ ) where A is an associative algebra and ϕ : A → k is unital, i.e. ϕ ( 1 A ) = 1 . n > 0 A ⊗ n define ∆ : T ( A ) → T ( A ) ⊗ T ( A ) by On T ( A ) ≔ � � ∆ ✁ ( a 1 · · · a n ) ≔ a S ⊗ a [ n ]\ S . S ⊆[ n ] This induces a product on T ( A ) ∗ : µ ✁ ν ≔ ( µ ⊗ ν ) ∆ ✁ . 8/27 Free Wick polynomials

Classical Wick polynomials: Hopf-algebraic approach Define φ : T ( A ) → k by φ ( a 1 · · · a n ) ≔ ϕ ( a 1 · A · · · · A a n ) and extend to T ( A ) ≔ k 1 ⊕ T ( A ) by φ ( 1 ) = 1 . There is c : T ( A ) → k with c ( 1 ) = 0 such that φ = exp ✁ ( c ) . In particular � � φ ( a 1 · · · a n ) = c ( a B ) . B ∈ π π ∈ P ( n ) Definition Since φ is invertible, we set W ≔ ( id ⊗ φ − 1 ) ∆ ✁ . Theorem The map W : T ( A ) → T ( A ) is the unique linear map such that ∂ a ◦ W = W ◦ ∂ a and φ ◦ W = ε . Its inverse is given by W − 1 = ( id ⊗ φ ) ∆ ✁ . Observe also that trivially ε ◦ W − 1 = φ = exp ✁ ( c ) . 9/27 Free Wick polynomials

Moments and cumulants We have other notions of independence: freeness, boolean idependence, monotone independence, etc. . . Each is characterised by a set of cumulants: κ , β , ρ resp. On the double tensor algebra T ( T ( A )) consider � ∆ ( a 1 · · · a n ) ≔ a S ⊗ a J S 1 | · · · | a J S k . S ⊆[ n ] This splits as � ∆ ≺ ( a 1 · · · a n ) ≔ a S ⊗ a J S 1 | · · · | a J S k , 1 ∈ S ⊆[ n ] � ∆ ≻ ( a 1 · · · a n ) ≔ a S ⊗ a J S 1 | · · · | a J S k . 1 � S ⊆[ n ] 10/27 Free Wick polynomials

Moments and cumulants Therefore, the convolution product µ ∗ ν ≔ ( µ ⊗ ν ) ∆ also splits: µ ≺ ν ≔ ( µ ⊗ ν ) ∆ ≺ , µ ≻ ν ≔ ( µ ⊗ ν ) ∆ ≻ . Consider Φ : T ( T ( A )) → k the unique character extension of φ . Theorem (Ebrahimi-Fard,Patras; 2014, 2017) The cumulants κ , β , ρ are the unique infinitesimal characters of T ( T ( A )) such that Φ = ε + κ ≺ Φ = ε + Φ ≻ β and Φ = exp ∗ ( ρ ) . 11/27 Free Wick polynomials

Moments and cumulants: Some known results Theorem (Speicher; 1997) � � ϕ ( a 1 · A · · · · A a n ) = κ ( a B ) . B ∈ π π ∈ N C ( n ) Theorem (Speicher, Woroudi; 1997) � � ϕ ( a 1 · A · · · · A a n ) = β ( a B ) . B ∈ π π ∈ I nt ( n ) Theorem (Hasebe, Saigo; 2011) 1 � � ϕ ( a 1 · A · · · · A a n ) = ρ ( a B ) | π | ! B ∈ π ( π , λ )∈ M ( n ) 12/27 Free Wick polynomials

Moments and cumulants We write Φ = E ≺ ( κ ) = E ≻ ( β ) = exp ∗ ( ρ ) . Every character has an inverse for ∗ . For Φ we have Φ − 1 = E ≻ (− κ ) = E ≺ (− β ) = exp ∗ (− ρ ) . In fact, characters on T ( T ( A )) form a group denoted by G . Observe that ∆ : T ( A ) → T ( A ) ⊗ T ( T ( A )) , i.e. we have a coaction. Thus, the character group G acts on End ( T ( A )) . 13/27 Free Wick polynomials

Wick polynomials Definition By analogy, define W : T ( A ) → T ( A ) by W ≔ ( id ⊗ Φ − 1 ) ∆ . Examples: W ( a ) = a − φ ( a ) 1 W ( ab ) = ab − aφ ( b ) − bφ ( a ) + ( 2 φ ( a ) φ ( b ) − φ ( a · b )) 1 W ( abc ) = abc − ϕ ( c ) ab − ϕ ( b ) ac − ϕ ( a ) bc − [ φ ( b · c ) − 2 φ ( b ) φ ( c )] a + φ ( a ) φ ( c ) b − [ φ ( a · b ) − 2 φ ( a ) φ ( b )] c − [ φ ( a · b · c ) − 2 φ ( a ) φ ( b · c ) − 2 φ ( c ) φ ( a · b ) − φ ( b ) φ ( a · c ) + 5 φ ( a ) φ ( b ) φ ( c )] 1 14/27 Free Wick polynomials

Wick polynomials By definition Φ ◦ W = ( Φ ⊗ Φ − 1 ) ∆ = ε that is, Φ ( W ( a 1 . . . a n )) = 0 for any a 1 , . . . , a n ∈ A . It’s easy to check that W is invertible with W − 1 = ( id ⊗ Φ ) ∆ and so Φ = ε ◦ W − 1 . In particular � a 1 · · · a n = W ( a s ) Φ ( a J S 1 ) · · · Φ ( a J S k ) . S ⊆[ n ] Theorem (Anshelevich, 2004) � � (− 1 ) | π | � W ( a 1 · · · a n ) = κ ( a B ) . a S B ∈ π S ⊆[ n ] π ∈ Int ([ n ]\ S ) π ∪ S ∈ N C ( n ) 15/27 Free Wick polynomials

Wick polynomials Theorem The Wick polynomials satisfy the recursion n − 1 � W ( a 1 · · · a n ) = a 1 W ( a 2 · · · a n ) − W ( a j +1 · · · a n ) κ ( a 1 · · · a j ) . j =0 Proof. W = ( id ⊗ Φ − 1 ) ∆ = id ≺ Φ − 1 + id ≻ Φ − 1 = id ≺ Φ − 1 − id ≻ ( Φ − 1 ≻ κ ) = id ≺ Φ − 1 − W ≻ κ . � 16/27 Free Wick polynomials

Wick polynomials Now consider the full Fock space F = �Ω ⊕ H ⊕ H ⊗ 2 ⊕ · · · . We have annihilation and creator operators a ∗ ( f )( f 1 ⊗ · · · ⊗ f n ) = f ⊗ f 1 ⊗ · · · ⊗ f n . a ( f )( f 1 ⊗ · · · ⊗ f n ) = � f , f 1 � f 2 ⊗ · · · ⊗ f n , This time they satisfy a ( f ) a ∗ ( g ) = � f , g � 1 . Set as before p ( f ) = ( a ( f ) + a ∗ ( f )) . We have � p ( f 1 ) p ( f 2 ) Ω , Ω � = � f 1 , f 2 � � p ( f 1 ) p ( f 2 ) p ( f 3 ) p ( f 4 ) Ω , Ω � = � f 1 , f 2 �� f 3 , f 4 � + � f 1 , f 4 �� f 2 , f 3 � . We get a free version of Wick’s theorem (Effros, Poppa; 2003). 17/27 Free Wick polynomials