Quantum Probability, Renormalization and InfiniteDimensional Lie Algebras
Quantum Probability, Renormalization
and InfiniteDimensional Lie Algebras^{⋆This paper is a
contribution to the Special Issue on Kac–Moody Algebras and Applications. The
full collection is available at
http://www.emis.de/journals/SIGMA/KacMoody_algebras.html}
Luigi ACCARDI and Andreas BOUKAS
L. Accardi and A. Boukas
Centro Vito Volterra, Università di Roma “Tor Vergata”, Roma I00133, Italy \EmailD \URLaddressDhttp://volterra.uniroma2.it
Department of Mathematics, American College of Greece,
\EmailD Aghia Paraskevi, Athens 15342, Greece
Received November 20, 2008, in final form May 16, 2009; Published online May 27, 2009
The present paper reviews some intriguing connections which link together a new renormalization technique, the theory of representations of infinite dimensional Lie algebras, quantum probability, white noise and stochastic calculus and the theory of classical and quantum infinitely divisible processes.
quantum probability; quantum white noise; infinitely divisible process; quantum decomposition; Meixner classes; renormalization; infinite dimensional Lie algebra; central extension of a Lie algebra
60H40; 60G51; 81S05; 81S20; 81S25; 81T30; 81T40
1 Introduction
The investigations on the stochastic limit of quantum theory in [25] led to the development of quantum white noise calculus as a natural generalization of classical and quantum stochastic calculus. This was initially developed at a pragmatic level, just to the extent needed to solve the concrete physical problems which stimulated the birth of the theory [23, 24, 25, 30]. The first systematic exposition of the theory is contained in the paper [26] and its full development in [32].
This new approach naturally suggested the idea to generalize stochastic calculus by extending it to higher powers of (classical and quantum) white noise. In this sense we speak of nonlinear white noise calculus.
This attempt led to unexpected connections between mathematical objects and results emerged in different fields of mathematics and at different times, such as white noise, the representation theory of certain famous Lie algebras, the renormalization problem in physics, the theory of independent increment stationary (Lévy) processes and in particular the Meixner classes, . The present paper gives an overview of the path which led to these connections.
Our emphasis will be on latest developments and open problems which are related to the renormalized powers of white noise of degree and the associated Lie algebras. We also briefly review the main results in the case of powers and, since the main results are scattered in several papers, spanning a rather long time period, we include some bibliographical references which allow the interested reader to reconstruct this development.
The content of the paper is the following. In Section 2 we state the Lie algebra renormalization problem taking as a model the Lie algebra of differential operators with polynomial coefficients.
In Section 3 we recall the basic notions on representations of Lie algebras and their connections with quantum probability. Section 4 describes the standard Fock representation (i.e. for first order fields). Sections 5 and 6 recall some notions on current algebras and their connections with (boson) independent increment processes. The transition from first to second quantization (in the usual framework of Heisenberg algebras) can be considered, from the algebraic point of view, as a transition from a Lie algebra to its current algebra over (or ) and, from the probabilistic point of view, as a transition from a class of infinitely divisible random variables to the associated independent increment process. These sections generalize this point of view to Lie algebras more general than the Heisenberg one. Section 7 illustrates the role of renormalization in the quadratic case and shows how, after renormalization, the above mentioned connection between Lie algebras and independent increment processes, can be preserved, leading to interesting new connections. We also quickly describe results obtained in this direction (giving references to existing surveys for more detailed information). Starting from Section 8 we begin to discuss the case of higher (degree ) powers of white noise and we illustrate the ideas that eventually led to the identification of the RHPWN and the Lie algebras (more precisely their closures) (Section 10). We illustrate the content of the nogo theorems and of the various attempts made to overcome the obstructions posed by them. The final sections outline some connections between renormalization and central extensions and some recent results obtained in this direction.
2 Renormalization and differential operators
with polynomial coefficients
Since the standard white noises, i.e. the distribution derivatives of Brownian motions, are the prototypes of free quantum fields, the program to find a meaningful way to define higher powers of white noise is related to an old open problem in physics: the renormalization problem. This problem consists in the fact that these higher powers are strongly singular objects and there is no unique way to attribute a meaning to them. That is why one speaks of renormalized higher powers of white noise (RHPWN).
The renormalization problem has an old history and we refer to [37] for a review and bibliographical indications. In the past 50 years the meaning of the term renormalization has evolved so as to include a multitude of different procedures. A common feature of all these generalizations is that, in the transition from a discrete system to a continuous one, certain expressions become meaningless and one tries to give a mathematical formulation of the continuous theory which keeps as many properties as possible from the discrete approximation. One of the main difficulties of the problem consists in its precise formulation, in fact one does not know a priori which properties of the discrete approximation will be preserved in the continuous theory.
The discovery of such properties is one of the main problems of the theory.
The present paper discusses recent progresses in the specification of this problem to a basic mathematical object: the algebra of differential operators with polynomial coefficients (also called the full oscillator algebra).
More precisely, in the present paper, when speaking of the renormalization problem, we mean the following:

to construct a continuous analogue of the algebra of differential operators with polynomial coefficients acting on the space of complex valued smooth functions in real variables (here continuous means that the space is replaced by the space });

to construct a representation of this algebra as operators on a Hilbert space (all spaces considered in this paper will be complex separable and all associative algebras will have an identity, unless otherwise stated);

the ideal goal would be to have a unitary representation, i.e. one in which the skew symmetric elements of this algebra can be exponentiated, leading to strongly continuous 1parameter unitary groups.
In the physical interpretation, would be the state space of a physical system with infinitely many degrees of freedom (typically a field, an infinite volume gas, ) and the 1parameter unitary groups correspond to time evolutions.
The algebra of differential operators in variables with polynomial coefficients can be thought of as a realization of the universal enveloping algebra of the dimensional Heisenberg algebra and its Lie algebra structure is uniquely determined by the Heisenberg algebra. In the continuous case the renormalization problem arises from this interplay between the structure of Lie algebra and that of associative algebra. The developments we are going to describe were originated by the idea, introduced in the final part of the paper [26], of first renormalizing the Lie algebra structure (i.e. the commutation relations), thus obtaining a new Lie algebra, then proving existence of nontrivial Hilbert space representations.
In the remaining of this section we give a precise formulation of problem above and explain where the difficulty is. In Section 6 we explain the connection with probability, in particular white noise and other independent increment processes.
2.1 1dimensional case
The position operator acts on as multiplication by the independent variable
The usual derivation also acts on and the two operators satisfy the commutation relation
where denotes the identity operator on . Defining the momentum operator by
one obtains the Heisenberg commutation relations
which give a structure of Lie algebra to the vector space generated by the operators , , . This is the Heisenberg algebra . The associative algebra , algebraically generated by the operators , , , is the algebra of differential operators with polynomial coefficients in one real variable and coincides with the vector space
where the are polynomials of arbitrary degree in the indeterminate and almost all the are zero.
There is a unique complex involution on this algebra such that
(2.1) 
The Lie algebra structure on , induced by the commutator, is uniquely determined by the same structure on the Heisenberg algebra and gives the commutation relations
(2.2) 
where is the Pochammer symbol:
(2.3) 
2.2 Discrete case
Let us fix and replace in Section 2.1 by the function space
Thus is replaced by . The notation is more appropriate than because it emphasizes its algebra structure (for the pointwise operations), which is required in its interpretation as a test function space (see the end of this section). For the values of a function we will use indifferently the notations or .
The position and momentum operators are then
which give the Heisenberg commutation relations
(2.4) 
where is the Kronecker delta. The involution is defined as in (2.1), i.e.
(2.5) 
The vector space generated by the operators , , () has therefore a structure of Lie algebra, it is called the dimensional Heisenberg algebra and is denoted
The associative algebra , algebraically generated by the operators , , (), is the algebra of differential operators with polynomial coefficients in real variables and coincides with the vector space
(2.6) 
where the are polynomials of arbitrary degree in the commuting indeterminates , almost all the are zero and, for , by definition
The operation of writing the product of two such operators in the form (2.6) can be called the normally ordered form of such an operator with respect to the generators , , ().
Also in this case the Lie algebra structure on , induced by the commutator, is uniquely determined by the same structure on the Heisenberg algebra and gives the commutation relations
(2.7) 
Notice that, with respect to formula (2.2), the new ingredient is the factor , i.e. the th power of the Kronecker delta. Since
(2.8) 
the power is useless, but we kept it to keep track of the number of commutators performed and to make easier the comparison with the continuous case (see Section 2.3 below). Considering as a test function space and defining the smeared operators
and the scalar product
the commutation relations (2.4) and (2.7) become respectively
Notice that the algebra structure on the test function space is required only when . The above construction can be extended to the case of an arbitrary discrete set : all the above formulae continue to hold with the set replaced by a generic finite subset of which might depend on the test function.
2.3 Continuous case
In this section the discrete space is replaced by and the discrete test function algebra by an algebra of smooth functions from into itself (for the pointwise operations).
As an analogue of the space one can take the space of all smooth cylindrical functions on (i.e. functions for which there exist , , and a smooth function , such that , , ). This space is sufficient for algebraic manipulations but it is too narrow to include the simplest functionals of interest for the applications in physics or in probability theory: this is where white noise and stochastic calculus play a role. The position operators can be defined as before, i.e.
The continuous analogue of the partial derivatives , hence of the momentum operators , can be defined by fixing a subspace of and considering functions whose Gateaux derivative in the direction
exists for any test function and is a continuous linear functional on (in some topology whose specification is not relevant for our goals). Denoting the duality specified by this topology, the elements of can be interpreted as distributions on and symbolically written in the form
The distribution
(2.9) 
is called the Hida derivative of at with respect to . Intuitively one can think of it as the Gateaux derivative along the function at , :
There is a large literature on the theory of Hida distributions and we refer to [46] for more information. The momentum operators are then defined by
and one can prove that the following generalization of the Heisenberg commutation relations holds:
(2.10) 
where now is Dirac’s delta and all the identities are meant in the usual sense of operator valued distributions, i.e. one fixes a space of test functions, defines the smeared operators
and interprets any distribution identity as a shorthand notation for the identity obtained by multiplying both sides by one test function for each free variable and integrating over all variables.
The involution is defined as in (2.5) with the only difference that now and the vector space generated by the operator valued distributions , , () has therefore a structure of Lie algebra. This algebra plays a crucial role in quantum field theory and is called the current algebra of over (see Section 5) or simply the Boson algebra over . In the following, when no confusion can arise, we will use the term Boson algebra also for the discrete Heisenberg algebra (2.4).
One can combine the discrete and continuous case by considering current algebras of over
so that the commutation relations (2.10) become ()
This corresponds to considering dimensional vector fields on rather than scalar fields on . The value of the dimension plays a crucial role in many problems, but not in those discussed in the present paper. Therefore we restrict our discussion to the case . It is however important to keep in mind that all the constructions and statements in the present paper remain true, with minor modifications, when is replaced by .
Up to now the discussion of the continuous case has been exactly parallel to the discrete case. Moreover some unitary representations of the continuous analogue of the Heisenberg algebra are known (in fact very few: essentially only Gaussian – quasifree in the terminology used in physics, see Section 7 below).
However the attempt to build the continuous analogue of the algebra of higher order differential operators with polynomial coefficients leads to some principle difficulties. For example the naive way to define the second Hida derivative of at with respect to (i.e. ) would be to differentiate the “function” for fixed , but even in the simplest examples, one can see that the identity (2.9) defines a distribution so that this map is meaningless.
One might try to forget the concrete realization in terms of multiplication operators and derivatives and to generalize to the continuous case the Lie algebra structure of This can be done for some subalgebras. For example, if the subspace is an algebra for the pointwise operations, then one can extend the Lie algebra structure of the Boson algebra over to first order differential operators (vector fields) by introducing functions of the position operator, which are well defined for any test function by
and using the commutation relation
(2.11) 
which leads to
In terms of test functions and with the notations:
the above commutator becomes, with :
(2.12) 
Another interesting class of subalgebras is obtained by considering the vector space generated by arbitrary (smooth) functions of and first order polynomials in . The test function form of (2.11) is then
(2.13) 
which shows that, for any , the vector space generated by the family where is a complex polynomial of degree and , are arbitrary test functions, is a nilpotent Lie algebra. We will see in Section 14 that the simplest nonlinear case (i.e. ) corresponds to the current algebra on the unique nontrivial central extension of the one dimensional Heisenberg algebra.
The right hand sides of (2.12) and (2.13) are well defined so at least we can speak of the Lie algebra of vector fields in continuously many variables, even if we do not know if some  or unitary representations of this algebra can be built. In the case of the algebra corresponding to (2.12), one can build unitary representations but the interpretation of these representations is still under investigation.
The situation is different with the continuous analogue of the higher order commutation relations (2.7) (i.e. when enters with a power ). Here some difficulties arise even at the Lie algebra level.
In fact the continuous analogue of these relations leads to
(2.14) 
which is meaningless because it involves powers of the delta function.
Any rule to give a meaning to these powers in such a way that the brackets, defined by the right hand side of (2.14) induce a Lie algebra structure, will be called a renormalization rule.
There are many inequivalent ways to achieve this goal. Any Lie algebra obtained with this procedure will be called a renormalized higher power of white noise (RHPWN) Lie algebra.
One might argue that, since in the discrete case the identity (2.8) holds, a natural continuous analogue of the commutation relations (2.7) should be
We will see in Section 7 that this naive approach corresponds, up to a multiplicative constant, to Ivanov’s renormalization or to consider the current algebra, over , of the universal enveloping algebra of .
One of the new features of the renormalization problem, brought to light by the present investigation, is that some subtle algebraic obstructions (nogo theorems) hamper this idea at least as far as the Fock representation is concerned (a discussion of this delicate point is in Section 8.2).
A first nontrivial positive result in this programme is that, by separating first and second powers, one can overcome these obstructions in the case and the results are quite encouraging (see Section 7).
However such a separation becomes impossible for . In fact, in Section 12 we will provide strong evidence in support of the thesis that any attempt to force this separation at the level of a Fock type representation, brings back either to the first or to the second order case.
3 representations of Lie algebras:
connections with quantum probability
Suppose that one fixes a renormalization and defines a RHPWN Lie algebra in the sense specified above. Then, according to the programme formulated in Section 2, the next step is to build representations of it. Since different Lie algebra structures will arise from different renormalization procedures, we recall in this section some notions concerning representations of general Lie algebras and their connections with quantum probability.
A representation of a Lie algebra is a triple
where is a Hilbert space, is a dense subHilbert space of , is a representation of into the linear operators from into itself (this implies in particular that the brackets are well defined on ), and the elements of are adjointable linear operators from into itself satisfying
If moreover the (onemode) field operators
are essentially selfadjoint, the representation is called unitary.
A vector is called cyclic for the representation if:

the vector
(3.1) is well defined (this is always the case if );

denoting the algebraic linear span of the vectors (3.1) the triple is a representation of .
If is a representation of with cyclic vector , one can always assume that . In this case we omit from the notations and speak only of the cyclic representation .
Any cyclic representation of induces a state (positive, normalized linear functional) on the universal enveloping algebra of , namely
where is the representation of induced by . Conversely, given a state on , the GNS construction gives a cyclic representation of hence of . Thus the problem to construct (non trivial) cyclic representations of (we will only be interested in this type of representations) is equivalent to that of constructing (nontrivial) states on hence of . This creates a deep connection with quantum probability. To clarify these connections let us recall (without comments, see [10] for more informations) the following three basic notions of quantum probability:

An algebraic probability space is a pair where is an (associative) algebra and a state on .

An operator process in the algebraic probability space is a selfadjoint family of algebraic generators of (typically a set of generators of ).

For any and any map the complex number
is called a mixed moment of the process of order .
In the above terminology one can say that constructing a representation of a Lie algebra is equivalent to constructing an algebraic probability space based on the universal enveloping algebra of or equivalently, by the Poincaré–Birkhoff–Witt theorem, an operator process in given by any selfadjoint family of algebraic generators of .
In the following section we show that when is the Boson algebra and the Fock state, the resulting algebraic probability space is that of the standard quantum white noise and its restriction to appropriate maximal Abelian (Cartan) subalgebras, gives the standard classical white noise.
4 The Fock representation of the Boson algebra and white noise
In the present section we discuss representations of the Boson algebra introduced in Section 2.3. All the explicitly known representations of this algebra can be constructed from a single one: the Fock representation.
To define this representation it is convenient to replace the generators , , of the Boson algebra by a new set of generators (creator), (annihilator), (central element, often omitted from notations) defined by
(4.1) 
The involution (2.5) and the commutation relations (2.10) then imply the relations
(4.2)  
where is Kronecker’s delta in the discrete case and Dirac’s delta in the continuous case.
The operator valued distribution form of the universal enveloping algebra , of the Boson algebra, is the algebraic linear span of the expressions of the form
where , , and , .
This has a natural structure of algebra induced by (4.2). On this algebra there is a particularly simple state characterized by the following theorem. We outline a proof of this theorem because it illustrates in a simple case the path we have followed to construct analogues of that state in much more complex situations, namely:

to formulate an analogue of the Fock condition (4.3);

to use the commutation relations to associate a distribution kernel to any linear functional , satisfying the analogue of the Fock condition, in such a way that is positive if and only if this kernel is positive definite;

to prove that this kernel is effectively positive definite.
On the algebra , defined above, there exists a unique state satisfying
(4.3) 
Proof.
From the commutation relations we know that is the algebraic linear span of expressions of the form
(4.4) 
(normally ordered products), interpreted as the central element if both . Therefore, if a state satisfying (4.3) exists, then it is uniquely defined by the properties that and for any of the form (4.4) with either or . It remains to prove that the linear functional defined by these properties is positive.
To this goal notice that the commutation relations imply that is also the algebraic linear span of expressions of the form
(antinormally ordered products), interpreted as before. A linear functional on is positive if and only if the distribution kernel
is positive definite. If satisfies condition (4.3), then the above kernel is equal to (in obvious notations)
From this, one deduces that the kernel can be non zero if and only if . Finally, since is a positive definite distribution kernel, the positivity of follows, by induction, from Schur’s lemma. ∎
The unique state on , defined by Theorem 4 above, is called the Fock (or lowest weight) state.
The GNS representation of the pair is characterized by:
(4.5) 
in the operator valued distribution sense.
In the notations of Definition 4 the operator (more precisely, the operator valued distribution) process in , defined by
is called the Boson Fock (or standard quantum) white noise on .
Definition 4 is motivated by the following theorem.
In the notation (4.1), the two operator subprocesses in :
(4.6) 
are classical processes stochastically isomorphic to the standard classical white noise on .
Proof.
The idea of the proof is that the Fock state has clearly mean zero. Using a modification of the argument used in the proof of Theorem 4 one shows that it is Gaussian and deltacorrelated, i.e. it is by definition a standard classical white noise. ∎
Theorem 4 and the relations (4.1) show that the Boson Fock white noise is equivalent to the pair (4.6), of standard classical white noises on . However the commutation relations (2.10) show that these two classical white noises do not commute so that classical probability does not determine their mixed moments: this additional information is provided by quantum probability.
In the following, when no confusion can arise, we omit from the notations the symbol of the representation.
5 Current algebras over
Current algebras are associated to pairs: (Lie algebra, set of generators) as follows. Let be a Lie algebra with a set of generators
where , are sets satisfying
and are the structure constants corresponding to the given set of generators, so that:
(5.1) 
Here and in the following, summation over repeated indices is understood. The sets , can, and in the examples below will, be infinite. However, here and in the following, we require that, in the summation on the right hand side of (5.1), only a finite number of terms are nonzero or equivalently that the structure constants are almost all zero (also this condition is automatically satisfied in the examples below). We assume that
The transition to the current algebra of over is obtained by replacing the generators by valued distributions on
and the corresponding relations by
This means that the structure constants are replaced by
In terms of test functions this can be equivalently formulated as follows.
Let be a Lie algebra with generators
and let be a vector space of functions from to called the test function space. A current algebra of over with test function space is a Lie algebra with generators
such that the maps
are complex linear, the involution satisfies
and the commutation relations are given by:
with the convention:
By restriction of the test function space to real valued functions, or more generally by restricting one’s attention to real Lie algebras, one could avoid the introduction of generators and use an intrinsic definition. We have chosen the nonintrinsic formulation because we want to emphasize the intuitive analogy between the generators and the powers of the creation/annihilation operators and between the generators and the powers of the number operator. Thus, for example, in the RHPWN Lie algebra with generators , the indices are the pairs with and the indices are the diagonal pairs .
6 Connection with independent increment processes
Let be Lie algebra whose elements depend on test functions belonging to a certain space of functions . Then has a natural localization, obtained by fixing a family of subsets of , e.g. intervals, and defining the subalgebra
Suppose that enjoys the following property:
(notice that, if is a current algebra over of a Lie algebra , then this property is automatically satisfied). If this is the case, denoting the algebraic linear span of (the image of) in any representation, the (associative) algebra generated by and is the linear span of the products of the form where (resp. ) is in (resp. ).
A similar conclusion holds if, instead of two disjoint sets, one considers an arbitrary finite number of disjoint sets. Denote the algebraic linear span of in a representation with cyclic vector and , the restriction of the state to . The given cyclic representation and the state are called factorizable if for any finite family , of mutually disjoint intervals of one has
The Fock representation, and all its generalizations we have considered so far, have this property.
By restriction to Abelian subalgebras, factorizable representations give rise to classical (polynomially) independent increment processes. The above definition of factorizability applies to general linear functionals (i.e. not necessarily positive or normalized). In the following we will make use of this remark.
Given a cyclic representation of (we omit from notations), for any interval one defines the subspace of as the closed subspace containing and invariant under .
7 Quadratic powers: brief historical survey
The commutation relations imply that
and the appearance of the term shows that and are not well defined even as operator valued distributions. The following formula, due to Ivanov, for the square of the delta function (cf. [47] for a discussion of its precise meaning)
(7.1) 
was used by Accardi, Lu and Volovich to realize the program discussed in Section 8 for the second powers of WN.
Using this we find the renormalized commutation relation:
(7.2) 
Moreover (without any renormalization!)
(7.3) 
Introducing a test function space (e.g. the complex valued step functions on with finitely many values), one verifies that the smeared operators (see the comments at the beginning of Section 8 about their meaning)
(7.4) 
satisfy the commutation relations
The relations (7.2), (7.3), or their equivalent formulation in terms of the generators (7.4), are then taken as the definition of the renormalized square of white noise (RSWN) Lie algebra. Recalling that is the Lie algebra with 3 generators and relations
one concludes that the RSWN Lie algebra is isomorphic to a current algebra, over , of a central extension of . Notice that this central extension is trivial (like all those of ), but its role is essential because without it, i.e. putting in the commutation relations (7.2), the Fock representation discussed below reduces to the zero representation.
Keeping in mind the intuitive expressions (7.4) of the generators, a natural analogue of the characterizing property (4.3), of the Fock state for this algebra, would be
(7.5) 
(let us emphasize that (7.5) is only in an informal sense a particular case of (4.3) where diagonal terms were not included) or, using test functions and the equivalent characterization (4.5) of the first order Fock state:
Using this property as the definition of the quadratic Fock state, Accardi, Lu and Volovich proved in [26] the existence of the quadratic Fock representation and formulated the programme to achieve a similar result for higher powers, using a natural generalization of the renormalization used for the square (see Section 8.1).
The paper [26] opened a research programme leading to several investigations in different directions. Among them we mention below only those directly related to the representation theory of Lie algebras and we refer, for more analytical and probabilistic directions, to [8, 11, 14, 16, 17, 20, 22, 28]. The latter paper also includes a discussion of previous attempts to give a meaning to the squares of free fields.
Accardi and Skeide introduced in [29] the quadratic exponential (coherent) vectors for the RSWN and noticed that the kernel defined by the scalar product of two such vectors coincided with the kernel used by Boukas and Feinsilver in [38, 40] and [41] to construct unitary representations of the socalled Finite Difference Lie Algebra. Moreover, they proved that the Fock representation of the RSWN Lie algebra, constructed in [26], gave rise to a typeI product system of Hilbert spaces in the sense of Arveson (cf. [31]).
Accardi, Franz and Skeide realized in [21] that the RSWN Lie algebra is a current algebra of over and that the factorization property mentioned in item above naturally suggested a connection with the theory of infinitely divisible stochastic processes along the lines described in the monographs [45] and [56]. In particular they were able to identify the infinitely divisible classical stochastic processes, arising as vacuum distributions of the generalized field operators of the RSWN, with the three nonstandard classes of Meixner laws:

Gamma,

Negative binomial (or Pascal),

Meixner.
Since it was well known that the remaining two classes of Meixner laws, i.e. the Gaussian and Poisson classes, arise as vacuum distributions of the generalized field operators of the usual first order white noise (free boson field), this result showed that the quantum probabilistic approach provided a nice unified view to the 5 Meixner classes which were discovered in 1934 (cf. [54]) in connection with a completely different problem (a survey of this development is contained in [18]).
For a concrete example on how some Meixner laws can appear as vacuum distributions of quantum observables, see Section 13.1 below.
P. Sniady constructed in [59] the free analogue the Fock representation of the RSWN Lie algebra obtained in [26] and proved the first nogo theorem concerning the impossibility of combining together in a nontrivial way the Fock representations of the first and second order white noise Lie algebras. This opened the way to a series of nogo theorems which paralleled, in a quite different context and using different techniques, a series of such theorems obtained in the physical literature.
A stronger form of Sniady’s result, still dealing with the first and second order case, was later obtained in [21]; in [19] this result was extended to the higher powers, defined with the renormalization used in [26], and further extended to the higher powers of the deformed white noise [7].
The attempt to go beyond the Fock representation by constructing more general representations, such as the finite temperature one, related to KMS states, was initiated in [2] where the analogue of the Bogolyubov transformations for the RSWN was introduced (i.e. those transformations on the test function space which induce endomorphisms of the quadratic Lie algebra) and a (very particular) class of KMS states on the RSWN algebra was constructed.
The problem of constructing the most general KMS states (for the free quadratic evolution) on the RSWN algebra was attacked with algebraic techniques in the paper [27] but its solution was obtained later, with a purely analytical approach by Prohorenko [58].
The quadratic Fermi case was investigated by Accardi, Arefeva and Volovich in [3] and led to the rather surprising conclusion that, while the quadratic Bose case leads to the representation theory of the compact form of the real Lie algebra , the corresponding Fermi case leads to the non compact form of the same real Lie algebra.
8 Higher powers of white noise
In order to realize, for the higher powers of white noise, what has been achieved for the square, we define the smeared operators (we will often use this terminology which can be justified only a posteriori by the realization of these objects as linear operators on Hilbert spaces):
(8.1) 
Notice that the above integral is normally ordered in , therefore it always has a meaning as a sesquilinear form on the (first order) exponential vectors with test function in independently of any renormalization rule. This allows to consider the symbols as generators of a complex vector space (in fact a vector space) and also to introduce a topology.
It is only when we want to introduce an additional Lie algebra structure, which keeps some track of the discrete version of the symbolic commutation relations written below (see formula (8.2)), that a renormalization rule is needed.