Renormalized powers of white noise, infinitely divisible processes, - PDF document

Renormalized powers of white noise, infinitely divisible processes, Virasoro–Zamolodchikov hierarchy Luigi Accardi Email: accardi@volterra.mat.uniroma2.it WEB page: http:volterra.mat.uniroma2.it Talk given at the: Symposium on Probability and Analysis Institute of Mathematics, Academia Sinica Taipei, August 10-12, 2010

Renormalization problem Emergence in physics: remove singularities of interactions (QED). Perturbative renormalization. Constructive renormalization: Give a meaning to nonlinear functionals (powers) of free quantum fields. Bridge with probability theory: free (Boson, Fock) quantum fields ⇔ (Boson, Fock) white noise Common difficulty: transition from discrete to continuum. Usual procedure: introduce cut–off (discretization) and try to remove it (passage to continuum) using various limit procedures.

Obstructions to the transition from discrete to continuum are typical in probability theory: infinite divisibility . One can take arbitrary natural integer convolution powers of any probability measure P on the real line: P ∗ n ; n ∈ N some probability measures have an n –th convolution root (for some n ∈ N ) P ∗ n 1 /n = P Therefore to any probability measure P on the real line: one can associate two characteristic lengths: ρ discr := inf { n ∈ N : P has an n –th conv. root } P := inf { r ∈ R : ∀ t ≥ r, P ∗ t is a prob. meas. } ρ cont P ( P ∗ t is defined by Fourier transform).

A measure is called infinitely divisible if ρ discr := 0 P In this case automatically ρ cont := 0 P Meaning of inf. div.: X r.v., law P . I ⊆ R interval, | I | Lebesgue measure. If | I | ≥ ρ cont there ∃ a r.v. X I such that P E ( e iX I ) = E ( e iX ) | I | Therefore, if ( I n ) is a partition of R s.t. ∀ n , | I n | = | I | (lattice), then one can define a sequence of r.v. ( X I n ) s.t. E ( e iX In ) = E ( e iX ) | I n | ; n ∈ N ρ cont is an obstruction to the transition P from discrete lattice to continuum: | I | → 0 In QFT we will meet similar obstructions.

1999 new approach to the renormalization problem (motivated by the stochastic limit of quantum theory) based on the following steps: 1 –st renormalize commutation relations, then build a representation More precisely: (i) introduce renormalization at heuristic level (ii) use heuristic renormalization to define a ∗ –Lie algebra (iii) construct a unitary representation of this ∗ –Lie algebra (iv) check that this unitary representation “gives the correct statistics” In step (iii) one meets some algebraic obstructions to the transition discrete → continuum (no–go theorems)

probabilistic renormalization problem ⇔ 3 sub–problems : Problem I : to construct a continuous analogue of the differential operators in d variables with polynomial coefficients DOPC ( R d ) := P n ( x ) ∂ n ; x ∈ R d � x n ∈ N acting on the space C ∞ ( R d ; C ) of complex valued smooth functions in d ∈ N real variables continuous means that the space R d ≡ { functions { 1 , . . . , d } → R } is replaced by some function space { functions R → R }

DOPC ( R d ) has two basic algebraic structures: 1) associative ∗ –algebra 2) ∗ –Lie–algebra In the continuous case the renormalization problem arises from this interplay between the structure of Lie algebra and that of associative algebra.

Problem II : construct ∗ –representations of this ∗ –Lie–algebra as operators on a Hilbert space H Problem III : prove the unitarity of these representations, i.e. that the skew symmetric elements of this ∗ –Lie–algebra can be exponentiated, leading to strongly continuous 1–parameter unitary groups.

Plan of the present lecture : to give a precise formulation of the above problems and explain where the difficulty is. Basic idea: – transition from 1 point (1 degree of freedom) to discrete lattice ( n degree of freedom) always possible n ∈ N ∪ {∞} – transition from discrete lattice to continuum obstructions arise (renormalization) – deep connection with the theory of infinitely di- visible probability measures

Quantum interpretation of DOPC ( R d ) 1–dimensional case ( d = 1) – q position operator ≡ multiplication by x ( qf )( x ) : = xf ( x ) ; x ∈ R , f ∈ C ∞ ( R ; C ) – ∂ x derivation – commutation relations [ q, ∂ x ] = − 1 ; the other ones - zero – momentum operator p : = 1 i ∂ x ; ( pf )( x ) : = 1 � d f � ( x ) i dx

Heisenberg ( ∗ –Lie) algebra Heis ( R ) generators { q , p , 1 } relations [ q, p ] = i ; [ q, q ] = [ p, p ] = 0 (1) q ∗ = q ; p ∗ = p ; 1 ∗ = 1 (2)

The following identifications take place: DOPC ( R ): the ∗ –algebra of differential operators with polynomial coefficients in one real variable ≡ the vector space of differential operators of the form P n ( q ) p n ∼ P n ( x ) ∂ n � � (3) = x n ∈ N n ∈ N where p 0 := q 0 := 1 and: – the P n ( X ) are polynomials of arbitrary degree in the indeterminate X – almost all the P n ( X ) are zero.

The operation of writing the product of two such operators P n ( q ) p n )( Q m ( q ) p m ) � � ( n ∈ N m ∈ N in the form R n ( q ) p n � n ∈ N can be called the ( q, p ) –normally ordered form of such a product normal order with respect to the generators p , q , 1

∗ –Lie algebra structure on DOPC ( R d ) n � � n k ( h ) q k − h p n − h [ p n , q k ] = ( − i ) h � (4) h h =1 k ( h ) ≡ Pochammer symbol (decreasing partial factorial): x (0) = 1 x ( y ) = x ( x − 1) · · · ( x − y + 1) ; x ( y ) = 0 ; if y > x More generally using δ h s,t = δ s,t ǫ n,k := 1 − δ n,k one finds for all n, k, N, K ∈ N [ b † n b k , b † N b K ] = � � k N ( l ) b † n b † N − l b k − l b K − � = ǫ k, 0 ǫ N, 0 l l ≥ 1

� � K N b † n − L b K − L b k n ( L ) b † � − ǫ K, 0 ǫ n, 0 s t L L ≥ 1 Important point: the commutator on U ( Heis ( R )) is uniquely determined by the commutator on DOPC ( R d ) ≡ Heis ( R ). (PBW theorem) Renormalization breaks this connection.

Arbitrary finite dimensional case – fix N ∈ N (degrees of freedom, modes) – replace R by the function space R N : = { x ≡ ( x 1 , . . . , x N ) : x j ∈ R , ∀ j } ≡ ≡ { Functions { 1 , . . . , N } → R } =: F { 1 ,...,N } ( R ) – Notation: x ( s ) =: x s – replace C ∞ ( R ; C ) by C ∞ ( R N ; C ). – position and momentum operators: For s, t ∈ { 1 , . . . , N } , f ∈ C ∞ ( R N ; C ) ( q s f )( x ) : = x s f ( x ) � ∂f p s : = 1 ∂ ( p s f )( x ) = 1 � ; ( x ) i ∂x s i ∂x s

from 1–point lattice to discrete lattice generators q s , p t , 1 ( s, t ∈ { 1 , . . . , N } ) with relations: – involution 1 ∗ = 1 q ∗ p ∗ s = q s ; s = p s ; (5) – Heisenberg commutation relations [ q s , p t ] = iδ s,t · 1 ; [ q s , q t ] = [ p s , p t ] = 0 (6) – δ s,t is the Kronecker delta current algebra on a discrete lattice

commutation relations n � � n k ( h ) δ h s,t q k − h [ p n s , q k ( − i ) h p n − h � t ] = (7) s t h h =1 New ingredient: the h –th power of the Kronecker delta δ h s,t = δ s,t (8) keeps track of the number of ( s, t ) commutators performed This power is the source of all troubles in the passage to the continuous case

Continuous case Replace: I ≡ { 1 , . . . , N ( ≤ ∞ ) } → R R N ≡ F { 1 ,...,N } ( R ) → F R ( R ) C ∞ ( R N ; C ) → C ∞ ( F R ( R ); C ) Define the position operators as in the discrete case: ( q s f )( x ) = x s f ( x ) ; x ∈ F R ( R ) ; f ∈ C ∞ ( F R ( R ); C ) To define the momentum operators p s , we need the continuous analogue of the partial derivatives ∂ ∂x s

This is the Hida derivative of f at x with respect to x s : f ′ ( x )( s ) =: ∂f ( x ) (9) ∂x s Intuitively: Hida derivative ≡ Gateaux derivative along the δ –function at s : δ s ( t ) := δ ( s − t ) ∂f = D δ s f ∂x s usual Gateaux derivative in the direction of S (a test function) 1 ε ( f ( x + εS ) − f ( x )) = � f ′ ( x ) , S � D S f ( x ) = lim ε → 0 Hida derivative of f ≡ distribution kernel of Gateaux derivative: ∂f � � � f ′ ( x ) , S � = f ′ ( x )( s ) S ( s ) ds = ( x ) S ( s ) ds ∂x s

momentum operators, defined by p s = 1 ∂ = 1 i D δ s i ∂x s

Connection with quantum field theory Restrict to the sub–algebra of 1 –st order polynomials generators: { q s , p t } continuous analogue of the Heisenberg algebra: the current algebra of Heis ( R ) over R or simply the Boson algebra over R . Commutation relations of a boson FT (in momentum representation): ; s, t ∈ R [ q s , p t ] = iδ ( s − t ) · 1 ; [ q s , q t ] = [ p s , p t ] = 0 (10) now: – δ ( s − t ) is Dirac’s delta – all the identities are meant in the sense of operator valued distributions Remark No problem in the transition

from scalar fields on R to N –dimensional vector fields on R d . This solves Problem (I) in the 1–st order case.

Change generators: q s , p t ≡ b t , b + t q t = b + p t = b t − b + t + b t t √ ; √ 2 i 2 Oscillator algebra: { q s , p t , b t b + t } ≡ { b s , b + t , b t b + t }