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

• n the real line:
• ne can associate two characteristic lengths:

ρdiscr

P

:= inf{n ∈ N : P has an n–th conv. root} ρcont

P

:= inf{r ∈ R : ∀t ≥ r, P ∗t is a prob. meas.} (P ∗t is defined by Fourier transform).

A measure is called infinitely divisible if ρdiscr

P

:= 0 In this case automatically ρcont

P

:= 0 Meaning of inf. div.: X r.v., law P. I ⊆ R interval, |I| Lebesgue measure. If |I| ≥ ρcont

P

there ∃ a r.v. XI such that E(eiXI) = E(eiX)|I| Therefore, if (In) is a partition of R s.t. ∀n, |In| = |I| (lattice), then one can define a sequence of r.v. (XIn) s.t. E(eiXIn) = E(eiX)|In| ; n ∈ N ρcont

P

is an obstruction to the transition 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

• f this ∗–Lie algebra

(iv) check that this unitary representation “gives the correct statistics” In step (iii) one meets some algebraic

• bstructions 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(Rd) :=

• n∈N

Pn(x)∂n

x

; x ∈ Rd acting on the space C∞(Rd; C)

• f complex valued smooth functions in d ∈ N real

variables continuous means that the space

Rd ≡ {functions {1, . . . , d} → R}

is replaced by some function space {functions R → R}

DOPC(Rd) 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

• bstructions arise (renormalization)

– deep connection with the theory of infinitely di- visible probability measures

Quantum interpretation of DOPC(Rd) 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 i

d

f dx

• (x)

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

• n∈N

Pn(q)pn ∼ =

• n∈N

Pn(x)∂n

x

(3) where p0 := q0 := 1 and: – the Pn(X) are polynomials of arbitrary degree in the indeterminate X – almost all the Pn(X) are zero.

The operation of writing the product

• f two such operators

(

• n∈N

Pn(q)pn)(

• m∈N

Qm(q)pm) in the form

• n∈N

Rn(q)pn can be called the (q, p)–normally

• rdered form of such a product

normal order with respect to the generators p, q, 1

∗–Lie algebra structure on DOPC(Rd) [pn, qk] =

n

• h=1

(−i)h

• n

h

• k(h)qk−hpn−h

(4) 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 δh

s,t = δs,t

ǫn,k := 1 − δn,k

• ne finds for all n, k, N, K ∈ N

[b†nbk, b†NbK] = = ǫk,0ǫN,0

• l≥1
• k

l

• N(l) b†n b†N−l bk−l bK−

−ǫK,0ǫn,0

• L≥1
• K

L

• n(L) b†

s N b†n−L bK−L bk t

Important point: the commutator on U(Heis(R)) is uniquely determined by the commutator on DOPC(Rd) ≡ 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

RN : = {x ≡ (x1, . . . , xN) : xj ∈ R , ∀j} ≡

≡ {Functions {1, . . . , N} → R} =: F{1,...,N}(R) – Notation: x(s) =: xs – replace C∞(R; C) by C∞(RN; C). – position and momentum operators: For s, t ∈ {1, . . . , N} , f ∈ C∞(RN; C) (qsf)(x) : = xsf(x) ps : = 1 i ∂ ∂xs ; (psf)(x) = 1 i

∂f

∂xs

• (x)

from 1–point lattice to discrete lattice generators qs, pt, 1 (s, t ∈ {1, . . . , N}) with relations: – involution q∗

s = qs ;

p∗

s = ps ;

1∗ = 1 (5) – Heisenberg commutation relations [qs, pt] = iδs,t · 1 ; [qs, qt] = [ps, pt] = 0 (6) – δs,t is the Kronecker delta current algebra on a discrete lattice

commutation relations [pn

s , qk t ] = n

• h=1

(−i)h

• n

h

• k(h)δh

s,tqk−h t

pn−h

s

(7) New ingredient: the h–th power of the Kronecker delta δh

s,t = δs,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

RN ≡ F{1,...,N}(R) → FR(R)

C∞(RN; C) → C∞(FR(R); C) Define the position operators as in the discrete case: (qsf)(x) = xsf(x) ; x ∈ FR(R) ; f ∈ C∞(FR(R); C) To define the momentum operators ps, we need the continuous analogue

• f the partial derivatives

∂ ∂xs

This is the Hida derivative of f at x with respect to xs: f′(x)(s) =: ∂f ∂xs (x) (9) Intuitively: Hida derivative ≡ Gateaux derivative along the δ–function at s: δs(t) := δ(s − t) ∂f ∂xs = Dδsf usual Gateaux derivative in the direction of S (a test function) DSf(x) = lim

ε→0

1 ε (f(x + εS) − f(x)) = f′(x), S Hida derivative of f ≡ distribution kernel of Gateaux derivative: f′(x), S =

• f′(x)(s)S(s)ds =
• ∂f

∂xs (x)S(s)ds

momentum operators, defined by ps = 1 i ∂ ∂xs = 1 i Dδs

Connection with quantum field theory Restrict to the sub–algebra

• f 1–st order polynomials

generators: {qs, pt} continuous analogue

• f the Heisenberg algebra:

the current algebra of Heis(R) over R

• r simply the Boson algebra over R.

Commutation relations of a boson FT (in momentum representation): [qs, pt] = iδ(s−t)·1 ; [qs, qt] = [ps, pt] = 0 ; s, t ∈ R (10) now: – δ(s − t) is Dirac’s delta – all the identities are meant in the sense

• f operator valued distributions

Remark No problem in the transition

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

Change generators: qs, pt ≡ bt, b+

t

qt = b+

t + bt

√ 2 ; pt = bt − b+

t

i √ 2 Oscillator algebra: {qs, pt, btb+

t } ≡ {bs, b+ t , btb+ t }

Problems (II) and (III) in the 1–st order case What can we say of unitary representations

• f the continuous analogue
• f the Heisenberg algebra?

Unfortunately, even in the 1–st order case, not so much! Some unitary representations are explicitly known: in fact essentially only Gaussian ones (quasi–free in the terminology used in physics)

Higher order commutation relations both qt and pt with powers ≥ 2, difficulties arise even at the Lie algebra level! the continuous analogue of the CR leads to [pn

s , qk t ] =

=

n

• h=1

(−i)h

• n

h

• k(h)δ(s − t)hqk−h

t

pn−h

s

powers of the delta function!

Any rule to give a meaning to powers of the delta function will be called a renormalization rule. There are many inequivalent renormalization rules. Any Lie algebra obtained with this procedure will be called a renormalized higher power of white noise (RHPWN) ∗–Lie algebra. Algebraic constraints After the renormalization the brackets, defined by the right hand side of the following identity should induce a ∗–Lie algebra structure, ǫn,k := 1 − δn,k x(y) = x(x − 1) · · · (x − y + 1) ; x(0) = 1 for all t, s ∈ T and n, k, N, K ≥ 0 one has [a†

t nak t , a† s NaK s ] =

= ǫk,0ǫN,0

• l≥1
• k

l

• a†

t n a† s N−l ak−l t

aK

s δl(t − s)

−ǫK,0ǫn,0

• L≥1
• K

L

• n(L) a†

s N a† t n−L aK−L s

ak

t δL(t − s)

(Naive) Definition A representation of any renormalized higher power of white noise (RHPWN) ∗–Lie algebra over a Hilbert space H with cyclic unit vector Φ (vacuum) and generators (b+)k

t bn t

is called a Fock representation if ∀n > 0 (b+)k

t bn t Φ = 0

Ivanov renormalization gives the obvious generalization, up to a multiplicative constant,

• f the prescription used in the discrete case:

δ(t)l = c l−1 δ(t) , c > 0 , l = 2, 3, .... (11) With this choice the continuous analogue

• f the discrete commutation relations

[pn

s , qk t ] = δs,t n

• h=1

(−i)h

• n

h

• k(h)qk−h

t

pn−h

s

is [pn

s , qk t ] = δ(s − t) n

• h=1

(−i)h

• n

h

• k(h)qk−h

t

pn−h

s

The resulting algebra is a current algebra, over R,

• f the universal enveloping algebra of Heis(R).

Theorem This is a ∗–Lie algebra.

Definition Let H be an algebra of test functions (by the pointwise operations). The ∗–Lie algebra with generators {b+n

t

bk

t : k, n ∈ N, b+0 t

b0

t := 1 := central element, t ∈ Rd}

and relations: (b+k

s

bn

t )+ = (b+n t

)bk

s

[bn

t , b+ s k] = ǫn,0ǫk,0

• l≥1
• n

l

• k(l) c l−1 b+

s k−l bn−l t

δ(t−s) ǫn,k := 1 − δn,k meant in the sense of operator–valued distributions

• n H, is called the IRHPdWN ∗–Lie algebra

(Ivanov renormalized higher powers of d–dimensional (boson ) white noise). No–go Theorem The IRHPdWN ∗–Lie algebra does not admit a Fock representation over a measure space (Rd, µ) unless the measure µ is purely atomic and the mea- sure of the atoms is larger than the inverse of the renormalization constant.

Definition of the RSWN ∗–Lie algebra The smallest ∗–Lie algebra containing both b+

t

and bt with powers ≥ 2 is the square–oscillator ∗–Lie algebra generators {b+2, b2, b+a, 1} is canonically isomorphic to (a central extension of) the ∗–Lie algebra sl(2, R), i.e. the ∗–Lie algebra with 3 generators B− , B+ , M and relations (B−)∗ = B+ ; N∗ = N (12) [B−, B+] = M (13) [M, B±] = ±2B± (14)

Smeared quadratic fields b+

ϕ “=”

• dtϕ(t)b2

t

; bϕ = (b+

ϕ )+

nϕ“=”

• dtϕ(t)b+

t bt

The Ivanov renormalization rule δ2 = γδ leads to the commutation relations: [bϕ, b+

ψ ] = γϕ, ψ + nϕψ

[nϕ, bψ] = −2bϕψ [nϕ, b+

ψ ] = 2b+ ϕψ

(b+

ϕ )+ = bϕ

; n+

ϕ = nϕ

Construction of the Fock representation for the quadratic (I)RSWN ∗–Lie algebra Theorem (Lu, Volovich, A. 1999) The Fock rep- resentation of the second order white noise exists and can be explicitly constructed. Franz U., Skeide M. (Comm. Math. Phys. 228 (2002) 123–150, Preprint Volterra, N. 423 (2000)) – study classical sub–processes – establish connections with the Meixner classes – realization of the RSWN in terms of countably many standard white noises

quadratic second quantization ⇒ strong stimulus to apply the new renormalization technique to attack the higher powers of WN. Problem: can we extend to the renormalized higher powers

• f quantum white noise what has been achieved for

the second powers?

No go theorems necessity to go beyond the Ivanov renormalization Theorem Let L be a Lie ∗–algebra with the following prop- erties: (i) L contains Bn

0, and B2n

(ii) the BN

K satisfy the higher power commutation

relations Then L does not have a Fock representation if the interval I is such that µ(I) ≤ 1 c (15)

A new renormalization motivated by the no go theorems a different renormalization rule was introduced (Boukas, A. 2006) δl(t − s) = δ(s) δ(t − s), l = 2, 3, 4, . . . (16) where the right hand side is defined as a convolu- tion of distributions. In appropriate test function space f(0) = g(0) = 0 (17) the singular terms vanish and the commutation re- lations become: [Bn

k (¯

g), BN

K(f)]R := (k N − K n) Bn+N−1 k+K−1(¯

gf) (18) which no longer include ill defined objects. The involution is: Bn

k (f)∗ := Bk n(f)

Theorem The commutation relations (18) and the involution define, on the family of symbols Bn

k (f), a structure

• f ∗–Lie algebra.

One mode case The commutation relations (18) imply that, fixing a sub–set I ⊆ Rd, not containing 0, and the test function χI(s) = { 1 , s ∈ I0 , s / ∈ I (19) the (self–adjoint) family {Bn

k := Bn k (χI) : n, k ∈ N , n, k ≥ 1 , n + k ≥ 3}

(20) satisfies the commutation relations [Bn

k , BN K]R := (k N − K n) Bn+N−1 k+K−1

(21)

Emergence of the Virasoro–Zamolodchikov al- gebra Definition The w∞ − ∗–Lie–algebra is the infinite dimensional Lie algebra spanned by the generators ˆ Bn

k , where

n, N ∈ N , n, N ≥ 2 and k, K ∈ Z with commutation relations: [ ˆ Bn

k , ˆ

BN

K]w∞ = (k (N − 1) − K (n − 1)) ˆ

Bn+N−2

k+K

(22) and involution

• ˆ

Bn

k

∗ = ˆ

Bn

−k

(23) For N = n = 2, one finds [ ˆ B2

k, ˆ

B2

K]V ir := (k − K) ˆ

B2

k+K

(24) defining the centerless Virasoro (or Witt) Lie alge- bra

The elements of the w∞–∗–Lie–algebra are interpreted as area preserving diffeomorphisms of 2–manifolds, Both w∞ and a quantum deformation of it, de- noted W∞, have been studied extensively in con- nection to two-dimensional Conformal Field The-

• ry and Quantum Gravity.

Similarities One mode case relations for the RHPWN ∗–Lie algebra: [Bn

k , BN K]R := (k N − K n) Bn+N−1 k+K−1

(25) relations for the w∞ ∗–Lie algebra: [ ˆ Bn

k , ˆ

BN

K]w∞ = (k (N − 1) − K (n − 1)) ˆ

Bn+N−2

k+K

(26)

dissimilarities – index set for the RHPWN–case n, k ∈ N – index set for the w∞–case n ∈ N , n ≥ 2 ; k ∈ Z – involution for the RHPWN–case Bn

k (f)∗ := Bk n(f)

– involution for the w∞–case

• ˆ

Bn

k

∗ = ˆ

Bn

−k

(27) The basic idea to identify the two algebras arose from an analysis of their classical realizations in terms of Poisson brackets.

Classical representations of the RHPWN and w∞ Lie algebras

• Definition. A Lie algebra G

with generators (lα)α∈F (F a set) and structure constants (cγ

α,β) admits a classical

representation if there exists a space of functions ˆ G from some Rd (with even d) and with values in C such that (i) ˆ G, as a linear space, has a set of algebraic gen- erators (ˆ lα)α∈F (i.e. any element of ˆ G is a linear combination of a finite sub–set of the (ˆ lα)α∈F). (ii) ˆ G is a Lie algebra with brackets given by the Poisson–brackets: [f, g] ˆ

G :=

i {f, g} = i

• ∂f

∂x · ∂g ∂y − ∂f ∂y · ∂g ∂x

• (28)

(iii) the structure constants of ˆ G in the basis (ˆ lα)α∈F are the (cγ

α,β), i.e.

[ˆ lα,ˆ lβ] ˆ

G = cγ α,βˆ

lγ

equivalently: the map lα → ˆ lα extends to a Lie algebra isomorphism between (G, (lα)α∈F) and (ˆ G, (ˆ lα)α∈F)

Renormalized Boson expression

• f the w∞–generators

(A. Boukas and LA.) ˆ Bn

k (f) =

(29) 1 2n−1

n−1

• m=0
• n − 1

m

• ∞
• p=0

∞

• q=0

(−1)p kp+q p! q! Bm+p

n−1−m+q(f)

The series on the right hand side (29) is conver- gent in a natural topology. This gives a natural meaning to its analytic con- tinuation in a neighborhood of k = 0. This is is used in the following inversion formula: Bn

k (f) = k

• ρ=0

n

• σ=0

(30)

• k

ρ n σ

• (−1)ρ

2ρ+σ ∂ρ+σ ∂zρ+σ|z=0 ˆ Bk+n+1−(ρ+σ)

z

(f)

The following classical representation of the w∞ Lie algebra was known in the literature: ˆ w∞ := linear span of (31) {fn,k(x, y) := eikx yn−1 : n, k ∈ Z, n ≥ 2, x, y ∈ R} In fact one verifies that: {fn,k(x, y), fN,K(x, y)} = i (k(N − 1) − K(n − 1)) fn+N−2,k+K(x, y) Notice that, for n = N = 2

• ne recovers a classical representation, of the Witt–

Virasoro algebra, in which the space ˆ G is a space of trigonometric polynomials in two real variables. The usual realization of the Witt–Virasoro algebra is in terms of vector fields on the unit circle.

The analogue classical representation of the RH- PWN Lie algebra was introduced Boukas and A. 06

• RHPWN := linear span of {

(32) gn,k :=

• x + iy

√ 2

n

x − iy √ 2

k

: n, k ∈ N, x, y ∈ R} In fact one verifies that: {gn,k(x, y), gN,K(x, y)} = = i (kN − nK) gn+N−1,k+K−1(x, y)

Comparing the classical realizations of the two al- gebras one realizes that, although they are differ- ent their closures in many natural topologies are the same. In fact the monomials are dense in the trigonomet- ric polynomials. Therefore it is natural to conjecture that a similar relationship holds also in the quantum case.

Conclusion: as suggested by the analogy with the classical case, the two ∗–Lie algebras w∞ and RHPWN are quite different from a purely algebraic point of view, but their closure in a natural topology coincide. Moreover the explicit representations given above provide a concrete realization of the RHPWN as sesquilinear forms on the space of the first order white noise. The problem of realizing them as bona fide closable

• perators on some Hilbert space is largely open.

aaa

Alternative no go theorems In the case of 2 − d order noise and of higher or- ders with a single renormalization constant, the current algebra restricted to a single block Lie– span {Bh

k(χ[0,1])} is isomorphic to the 1–mode Lie

algebra Lie–span–{a+hak} Lemma Let (b±

t ) be the Boson Fock scalar white noise.

Suppose that the k–th power of white noise ex- ists for some natural integer k and admit a Fock

• representation. then the process

{W k

[s,t], Φ, −∞ < s < t < +∞}

defined formally by some renormalization of W k

[s,t] =

t

s (b+ u + bu)kdu

should be a stationary additive independent incre- ment process on R. Lemma The map ak →

1

0 bk t dt =: B(k) [0,1]

with the [AcBuFr] renormalization, is a Lie algebra isomorphism. Proof . Check Corollary. The Fock statistics of (B(k)

[0,1]) is the

same as that of (ak). Proof . This statistics is uniquely determined by the Lie algebra structure. Lemma The vacuum distribution of

1

0 (b+ t + bt)ndt =

1

0 wn t dt

coincides with that of (a+ + a)n Proof . The statistics is uniquely determined by the Lie algebra structure Corollary. If the Fock representation of the n– th power of white noise exists, then the vacuum distribution of (a+ + a)n must be infinitely divisible. Proof . From Lemma () it follows that the distri- bution of (a+ + a)n is the same as the distribution

• f

1

0 dt(b+ t +bt)n and from Lemma () we know that

this is infinitely divisible. Theorem A necessary condition for the existence

• f the n–th power of white noise, renormalized as

in [AcBouFr] is that the n–th power of a classical gaussian random variable is infinitely divisible. Conclusion. The n–th powers of the Gaussian random variable and their distributions have been widely studied. It is known that, ∀k ≥ 1: γ2k (γ ∼ (0, 1) standard) is infinitely divisible. It is not known if, ∀k ≥ 1: γ2k+1(γ ∼ (0, 1) standard) is infinitely divisible. aaa

∗–Lie sub–algebras: when precisely the renor- malization problem steps in Example 1: The algebra (Funct(q), p) the generators are {u(qs) , pt : s, t ∈ R} – arbitrary (smooth) functions of q – linear functions of p (no mixed products, no higher powers of p) functions u(qs) of the position operator are defined by: (u(qs)f)(x) : = u(xs)f(x) x ∈ FR(R) ; f : FR(R) → C the brackets are [u(qs), pt] = iδ(t − s)u′(qs) (33) This gives a ∗–Lie algebra with involution (u(qs))∗ = ¯ u(qs)

p∗

t = pt

Smeared commutation relations Fix an algebra of test functions for the pointwise

• perations.

For any smooth function u and for any pair of test functions a, b the smeared vector fields (generators) are {u(q; a) , p(b) : a, b ∈ F0

R(R)}

where: u(q; a) : =

• R bsu(qs)ds

; p(b) : =

• b(s)psds

and the brackets [u(q; a), p(b)] = iu′(q; ab) (34) This gives a nilpotent ∗–Lie algebra with involution (u(q; a))∗ = ¯ u(q; a) p(b)∗ = p(b) (a and b are real valued).

Example 1b: Simplest nonlinear case: n = 2 the generators are {q2

s , qs , pt : s, t ∈ R}

corresponds to the current algebra over R

• f the unique nontrivial central extension
• f the one dimensional Heisenberg algebra.

Example 2: the Boson algebra of smeared vector fields

• ver R

(first order differential operators with arbitrary smooth coefficients) – Generators: {u(qs)pt : u is a smooth function} – commutation relation [u(qs1)pt1, v(qs2)pt2] = = iu(qs1)v′(qs2)δ(t1−s2)pt2−iv(qs2)u′(qs1)δ(s1−t2)pt1 The vector fields over R form a Lie algebra but not a ∗–Lie algebra. Since the (natural ) involution on them is given by: (u(qs)pt)∗ = ¯ u(qs)pt + iδ(s − t)v′(qs) · 1 in order to obtain a ∗–Lie algebra, we have to en- large the set of generators as follows: {u(qs)pt , δ(s−t)v(qs) : u, v are smooth functions}

and the set of commutation relation as follows: [v(qs), pt] = iδ(s − t)v′(qs) · 1 (35) Introducing a space of test functions a, b ∈ F0

R(R)

and the corresponding smeared operators: u(q; a) :=

• R bsu(qs)ds

; p(b) :=

• b(s)psds

[u(qs)pt1, v(qt2)] = −iδ(t1 − t2)u(qs)v′(qt1) the smeared vector fields are u(q; a)p(b) and the above commutator becomes: [u(q; a)p(b), v(q; c)p(d)] = (36) = iv′(q; bc)u(q; a)p(d) − iu′(q; ad)v(q; c)p(b) [u(q; a)pb, v(q; c)] = −iδ(t1 − t2)u(q; a)v′(q; bc) The smeared involution is given by: (u(q; a)p(b))∗ = ¯ u(q; a)p(b) + ¯ u(q; ab)′ · 1 Also here: at least a ∗–Lie algebra structure exists.

Important remark For these sub–algebras no renormalization is re- quired! The brackets and the involution are well defined, so Problem (I) is automatically solved for them. But we do not know if some unitary

• r even ∗–representations of these algebras

can be built, so Problem (II) is open even for these simplest al- gebras. Fact: In the simplest next step, the algebra of local vector fields, problems begin to arise even at the level of Problem (I).

Example 3: The algebra of local vector fields Loc−Funct(q)p (first order differential operators with arbitrary smooth diagonal coefficients) the generators are {u(qs)ps : s ∈ R} where u arbitrary is an (smooth) function of q The Lie algebra structure is give by the brackets [u(qs)ps, v(qt)pt] = iδ(s−t)

• v(qs)u′(qs) − u(qs)v′(qs)
• ps

=: iδ(s − t)σ(v, u)(qs)ps Remark Notice that ps always remains at the first power. For these sub–algebras no renormalization is re- quired! The brackets are well defined, so Problem (I) is automatically solved for them at the Lie algebra level. The situation changes if we want a solution at a ∗–Lie algebra level.

In fact any reasonable involution should satisfy (v(qs)ps)∗ = ps¯ v(qs) In order to write this as a linear combination of the generators, we have to enlarge the set of generators so to include all the operators of degree 1: {u(qs)ps , v(qs) : s ∈ R} where u, v are arbitrary (smooth) functions of q and the corresponding brackets [u(qs)ps, v(qt)] = −iδ(s − t)(uv′)qs) [u(qs)ps, pt] = −iδ(s − t)u′(qs)ps After that one can define the involution by (v(qs)ps)∗ := −i¯ v′(qs) + ¯ v(qs)ps One can check that this is indeed an involution, i.e. ((v(qs)ps)∗)∗ = v(qs)ps [u(qs)ps, v(qt)pt]∗ = [(v(qt)pt)∗, u(qs)ps)∗]

The algebra of local vector fields in terms of test functions a, b ∈ F0

R(R)

and with the notations : u(q)p(a) :=

• R bsu(qs)psds

the smeared vector fields are u(q)p(a) and the commutation relations becomes: [u(q)p(a), v(q)p(b)] = iσ(v, u)(q)p(ab) (37) = i

• (v′u) − (u′v)
• (q)p(ab)

Also here at least a ∗–Lie algebra structure exists.