September 22, 2008 16:30 WSPC/Trim Size: 9in x 6in for Proceedings 1param4

ONE-PARAMETER GROUPS AND COMBINATORIAL PHYSICS

GÉRARD H. E. DUCHAMP

Université de Rouen

Laboratoire d’Informatique Fondamentale et Appliquée de Rouen 76821 Mont-Saint Aignan, France

KAROL A. PENSON Université Pierre et Marie Curie,

Laboratoire de Physique Théorique des Liquides, CNRS UMR 7600 Tour 16, 5^{ième} étage, 4, place Jussieu, F 75252 Paris Cedex 05, France

ALLAN I. SOLOMON The Open University

Physics and Astronomy Department Milton Keynes MK7 6AA, United Kingdom

ANDRZEJ HORZELA

H. Niewodniczański Institute of Nuclear Physics, Polish Academy of Sciences

Department of Theoretical Physics ul. Radzikowskiego 152, PL 31-342 Kraków, Poland

PAWEL BLASIAK

H. Niewodniczański Institute of Nuclear Physics, Polish Academy of Sciences

Department of Theoretical Physics ul. Radzikowskiego 152, PL 31-342 Kraków, Poland

&

Université Pierre et Marie Curie,

Laboratoire de Physique Théorique des Liquides, CNRS UMR 7600 Tour 16, 5^{ième} étage, 4, place Jussieu, F 75252 Paris Cedex 05, France

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In this communication, we consider the normal ordering of operators of the type

Ω = ∑_{α+β=r} c_{α,β}(a^{+})^{α}a(a^{+})^{β}, α, β, r integers

where a (resp. a^{+}) is a boson annihilation (resp. creation) operator; these satisfy [a, a^{+}] ≡ aa^{+} − a^{+}a = 1, and for the purposes of this note may be thought of as a ≡ d/dx and a^{+} ≡ x. We discuss the integration of the one-parameter groups e^{λΩ} and their combinatorial by-products. In particular we show how these groups can be realized as groups of substitutions with prefactor functions.

1. Introduction

This text is the continuation of a series of works on the combinatorial and analytic aspects of normal forms of boson strings[^{1,2,3,4,5,11,13,14,15,20}]. Let w ∈ {a, a^{+}}^{∗} be a word in the letters {a, a^{+}} (i.e. a boson string), and define (as in Blasiak, Penson and Solomon^{1,2,3,4}) by r, s, respectively |w|_{a^{+}} (the number of creation operators), |w|_{a} (the number of annihilation operators) and m = min(r, s) then the normal form of w^{n} is

N (w^{n}) = ∑_{k=0}^{nm} S_{w}(n, nm − k)(a^{+})^{nr−k}a^{ns−k}  (1)

which, according to whether the excess e = r − s is positive or negative, reads

(a^{+})^{ne} (∑_{k=0}^{∞} S_{w}(n, k)(a^{+})^{k}a^{k}) or (∑_{k=0}^{∞} S_{w}(n, k)(a^{+})^{k}a^{k}) (a)^{n|e|} (2)

It has been observed that the numbers S_{w} are rook numbers [^{12,24}] . Consider, as examples, the upper-left corner of the following (doubly

infinite) matrices.

For w = a^{+}a, one gets the usual matrix of Stirling numbers of the second kind.

⌈ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 1 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 1 & 3 & 1 & 0 & 0 & 0 & ⋯ \\ 0 & 1 & 7 & 6 & 1 & 0 & 0 & ⋯ \\ 0 & 1 & 15 & 25 & 10 & 1 & 0 & ⋯ \\ 0 & 1 & 31 & 90 & 65 & 15 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix}  (3)

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For w = a^{+}aa^{+}, we have

⌈ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 2 & 4 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ 6 & 18 & 9 & 1 & 0 & 0 & 0 & ⋯ \\ 24 & 96 & 72 & 16 & 1 & 0 & 0 & ⋯ \\ 120 & 600 & 600 & 200 & 25 & 1 & 0 & ⋯ \\ 720 & 4320 & 5400 & 2400 & 450 & 36 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix}  (4)

For w = a^{+}aaa^{+}a^{+}, one gets

⌈ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 2 & 4 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 12 & 60 & 54 & 14 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ 144 & 1296 & 2232 & 1296 & 306 & 30 & 1 & 0 & 0 & ⋯ \\ 2880 & 40320 & 109440 & 105120 & 45000 & 9504 & 1016 & 52 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix}  (5)

Remark 1.1. In each case, the matrix S_{w} is of staircase form and the “step” depends on the number of a’s in the word w. More precisely, due to equation (1) one can prove that each row ends with a ‘one’ in the cell (n, nr), where r = |w|_{a} and we number the entries from (0, 0). Thus all the matrices are row-finite and unitriangular iff r = 1, which case will be of special interest in the following. Moreover, the first column is (1, 0, ⋯, 0, ⋯, 0, ⋯) iff w ends with an a (this means that N (w^{n}) has no constant term for all n > 0).

In this communication, we concentrate on boson strings and more generally (homogeneous) boson operators involving only one “a”. We will see that this case is closely related to one-parameter substitution groups and their conjugates. The structure of the paper is the following. In section 2 we define the framework for our transformation matrices (spaces, topology and decomposition), then we concentrate on the Riordan subgroup (i.e. transformations which are substitutions with prefactor functions) and adapt the classical theory (Sheffer condition) to the present context. In section 3 we analyse unipotent transformations (Lie group structure and combinatorial examples). The divisibility property of the group of unipotent transformations tells us that every transformation is embedded

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in a one-paramater group. This will be analysed in section 4 from the formal and analytic points of view. Section 5 is devoted to some concluding remarks and further interesting possibilities.

2. The algebra L(C^{N}) of sequence transformations

Let C^{N} be the vector space of all complex sequences, endowed with the Frechet product topology [^{23}]. It is easy to check that the algebra L(C^{N}) of all continuous operators C^{N} → C^{N} is the space of row-finite matrices with complex coefficients. Such a matrix M is indexed by N × N and has the property that, for every fixed row index n, the sequence (M(n, k))_{k≥0} has finite support. For a sequence A = (a_{n})_{n≥0}, the transformed sequence B = MA is given by B = (b_{n})_{n≥0} with

b_{n} = ∑_{k≥0} M(n, k)a_{k}  (6)

Remark that the combinatorial coefficients S_{w} defined above are indeed row-finite matrices.

We may associate a univariate series with a given sequence (a_{n})_{n∈N}, using a sequence of prescribed (non-zero) denominators (d_{n})_{n∈N} , as follows:

∑_{n≥0} a_{n} \frac{z^{n}}{d_{n}} .  (7)

For example, with d_{n} = 1, we get the ordinary generating functions (OGF), with d_{n} = n!, we get the exponential generating functions (EGF) and with d_{n} = (n!)^{2}, the doubly exponential generating functions (DEGF) and so on. Thus, once the denominators have been chosen, to every (linear continuous) transformation of generating functions, one can associate a corresponding matrix.

The algebra L(C^{N}) possesses many interesting subalgebras and groups, such as the algebra of lower triangular transformations T (N, C), the group T_{inv}(N, C) of invertible elements of the latter (which is the set of infinite lower triangular matrices with non-zero elements on the diagonal), the subgroup of unipotent transformations UT (N, C) (i.e. the set of infinite lower triangular matrices with elements on the diagonal all equal to 1) and its Lie algebra N T (N, C), the algebra of locally nilpotent transformations (with zeroes on the diagonal). One has the inclusions (with D_{inv}(N, C), the set of invertible diagonal matrices)

UT (N, C) ⊂ T_{inv}(N, C) ⊂ T (N, C) ⊂ L(C^{N})

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D_{inv}(N, C) ⊂ T_{inv}(N, C) and N T (N, C) ⊂ L(C^{N}).  (8)

We remark that T_{inv}(N, C) = D_{inv}(N, C) ▷◁ UT (N, C) because UT is normalized by D_{inv} and T_{inv} = D_{inv}.UT (every invertible transformation is the product of its diagonal by a unipotent transformation).

We now examine an important class of transformations of T as well as their diagonals: the substitutions with prefactor functions which, from now on, will be called substitutions with prefunctions.

2.1. Substitutions with prefunctions

Let (d_{n})_{n≥0} bet a fixed set of denominators. We consider, for a generating function f, the transformation

Φ_{g,ϕ}[f](x) = g(x)f(ϕ(x)).  (9)

The matrix of this transformation M_{g,ϕ} is given by the transforms of the monomials \frac{x^{k}}{d_{k}} hence

∑_{n≥0} M_{g,ϕ}(n, k)\frac{x^{n}}{d_{n}} = Φ_{g,ϕ} [\frac{x^{k}}{d_{k}} ] = g(x)\frac{ϕ(x)^{k}}{d_{k}} .  (10)

If g, ϕ ≠ 0 (otherwise the transformation is trivial), we can write

g(x) = a_{l} \frac{x^{l}}{d_{l}} + ∑_{r>l} a_{r} \frac{x^{r}}{d_{r}} , ϕ(x) = α_{m} \frac{x^{m}}{d_{m}} + ∑_{s>m} α_{s} \frac{x^{s}}{d_{s}}  (11)

with a_{l}, α_{m} ≠ 0 and then, by (9,10,11),

Φ_{g,ϕ} [\frac{x^{k}}{d_{k}} ] = a_{l}(α_{m})^{k} \frac{x^{l+mk}}{d_{l}d_{m}^{k}d_{k}} + ∑_{t>l+mk} b_{t} \frac{x^{t}}{d_{t}} .  (12)

One then has

M_{g,ϕ} is row − finite ⟺ ϕ has no constant term  (13)

and in this case it is always lower triangular.

The converse is true in the following sense. Let T ∈ L(C^{N}) be a matrix with non-zero two first columns and suppose that the first index n such that T (n, k) ≠ 0 is less for k = 0 than k = 1 (which is, from (10) the case when T = M_{g,ϕ}). Set

g(x) := d_{0} ∑_{n≥0} T (n, 0)\frac{x^{n}}{d_{n}}; ϕ(x) := \frac{d_{1}}{g(x)} ∑_{n≥0} T (n, 1)\frac{x^{n}}{d_{n}}  (14)

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then T = M_{g,ϕ} iff, for all k,

∑_{n≥0} T (n, k)\frac{x^{n}}{d_{n}} = g(x)\frac{ϕ(x)^{k}}{d_{k}} .  (15)

Remark 2.1. Eq. (10) is called the Sheffer condition (see [^{16,20,21,25}]) and, for EGF (d_{n} = n!) it amounts to stating that

∑_{n,k≥0} T (n, k)\frac{x^{n}}{n!} y^{k} = g(x)e^{yϕ(x)}.  (16)

From now on, we will suppose that ϕ has no constant term (α_{0} = 0). Moreover M_{g,ϕ} ∈ T_{inv} if and only if a_{0}, α_{1} ≠ 0 and then the diagonal term with address (n, n) is \frac{a_{0}}{d_{0}} (\frac{α_{1}}{d_{1}} )^{n}. We get

M_{g,ϕ} ∈ UT ⟺ \frac{a_{0}}{d_{0}} = \frac{α_{1}}{d_{1}} = 1.  (17)

In particular for the EGF and the OGF, we have the condition that

g(x) = 1 + higher terms and ϕ(x) = x + higher terms.  (18)

Note 2.1. In classical combinatorics (for OGF and EGF), the matrices M_{g,ϕ}(n, k) are known as Riordan matrices (see [^{16,17}] for example).

3. Unipotent transformations

3.1. Lie group structure

We first remark that n × n truncations (i.e. taking the [0..n] × [0..n] submatrix of a matrix) are algebra morphisms

τ_{n} : T (N, C) → M([0..n] × [0..n], C).  (19)

We can endow T (N, C) with the Frechet topology given by these morphisms. We will not develop this point in detail here, but this topology is metrisable, complete and given by the following convergence criterion:

a sequence (M_{k}) of matrices in T (N, C) converges iff for all fixed n ∈ N the sequence of truncated matrices (τ_{n}(M_{k})) converges.

This topology is compatible with the C-algebra structure of T (N, C). The two maps exp : N T (N, C) → UT (N, C) and log : UT (N, C) →

N T (N, C) are continous and mutually inverse.

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3.2. Examples

3.2.1. Provided by the exponential formula

The “classical exponential formula” [^{7,9,21}] tells us the following: consider a class^{a} of finite labelled graphs C and denote by C^{c} the subclass of connected graphs in C. Then the exponential generating series of C and C^{c} are related as follows:

EGF (C) = e^{EGF (C^{c})}.  (20)

The following examples give us some insight into why combinatorial matrices of the type:

T (n, k) = Number of graphs of C on n vertices having k connected components

give rise to substitution transformations.

Example 3.1. Stirling numbers.

From now on, let v(Γ), c(Γ) denote parameters n, k above, namely the number of vertices of Γ and the number of connected components of Γ. Consider first the class of graphs Γ of equivalence relations. Then using the statistics x^{v(Γ)}y^{c(Γ)} we get

∑_{n,k≥0} S(n, k)\frac{x^{n}}{n!} y^{k} = ∑_{all equivalence graphs Γ} \frac{x^{v(Γ)}}{v(Γ)!}y^{c(Γ)} =

exp ( ∑_{Γ connected} \frac{x^{v(Γ)}}{v(Γ)!}y^{c(Γ)}) = exp (∑_{n≥1} y \frac{x^{n}}{n!} ) = e^{y(e^{x}−1)}  (21)

and we will see that the transformation associated with the matrix S(n, k) is f(x) → f(e^{x} − 1).

Example 3.2. Idempotent numbers. We consider here the class of graphs of endofunctions (i.e. functions from a finite set into itself). Then, using the statistics x^{v(Γ)}y^{c(Γ)} and denoting by I(n, k) the number of graphs of idempotent endofunctions (i.e. being equal to their compositional square) of a given set with n elements and having k connected components, we get

aClosed under relabelling (of the vertices), disjoint union, and taking connected components.

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∑_{n,k≥0} I(n, k)\frac{x^{n}}{n!} y^{k} = ∑_{all graphs Γ of endofunctions in the class (1, 2)} \frac{x^{v(Γ)}}{v(Γ)!}y^{c(Γ)} =

exp ( ∑_{Γ connected} \frac{x^{v(Γ)}}{v(Γ)!}y^{c(Γ)}) = exp (∑_{n≥1} y \frac{nx^{n}}{n!} ) = e^{yxe^{x}}  . (22)

Corresponding to these numbers we get the (doubly) infinite matrix

⌈ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 2 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 3 & 6 & 1 & 0 & 0 & 0 & ⋯ \\ 0 & 4 & 24 & 12 & 1 & 0 & 0 & ⋯ \\ 0 & 5 & 80 & 90 & 20 & 1 & 0 & ⋯ \\ 0 & 6 & 240 & 540 & 240 & 30 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix}  (23)

and we will see that the transformation associated with this matrix is f(x) → f(xe^{x})

Remark 3.1. i) More generally, call the Burnside class with parameters a < b (a, b integers), the class of graphs of (finite) endofunctions such that f^{a} = f^{b} (compositional power) and Burn_{a,b}(n, k), the corresponding numbers as in section (3.2.1). Then, exponential formula shows us that the matrices Burn_{a,b} are all matrices of substitutions. ii) For the special cases a = 1, b = l + 1, the mixed series (EGF in x and OGF in y) is explicit and reads

∑_{n,k≥0} Burn_{1,l+1}(n, k)\frac{x^{n}}{n!} y^{k} = e^{y(∑_{h|l} \frac{x^{h}}{h} e^{hx})}  (24)

3.2.2. Normal ordering powers of boson strings

To get unipotent matrices, one has to consider boson strings with only one annihilation operator. In the introduction, we have given examples with a^{+}a, a^{+}aa^{+} (the matrix of the third string, a^{+}aaa^{+}a^{+}, with two annihilators, is not unipotent). Such a string is then w = (a^{+})^{r−p}a(a^{+})^{p} and we will see shortly that

• if p = 0, S_{w}(n, k) is the matrix of a unipotent substitution

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• if p > 0, S_{w}(n, k) is the matrix of a unipotent substitution with prefunction

To cope with the matrices coming from the normal ordering of powers of boson strings we have to make a small detour to analysis and formal groups.

4. One-parameter subgroups of UT (N, C)

4.1. Exponential of elements of NT (N, C)

Let M = I + N ∈ UT (N, C) (I = I_{N} is the indentity matrix). One has

M^{t} = ∑_{k≥0} ( \begin{matrix}t \\ k \end{matrix} ) N^{k}  (25)

where ( \begin{matrix}t \\ k \end{matrix} ) is the generalized binomial coefficient defined by

( \begin{matrix}t \\ k \end{matrix} ) = \frac{t(t − 1) ⋯ (t − k + 1)}{k!}.  (26)

One can see that, for k ≤ n, due to the local nilpotency of N, the matrix coefficient M^{t}(n, k) is well defined and, in fact, a polynomial of degree n − k in t (for k > n, this coefficient is 0). We have the additive property M^{t_{1}+t_{2}} = M^{t_{1}}M^{t_{2}} and the correspondence t → M^{t} is continuous. Conversely, let t → M_{t} be a continous local one-parameter group in UT (N, C); that is, for some real ϵ > 0

|t_{1}| and |t_{2}| < ϵ ⟹ M_{t_{1}}M_{t_{2}} = M_{t_{1}+t_{2}}  (27)

then there exists a unique matrix H ∈ NT (N, C) such that M_{t} = exp(tH). (This may be proved using the projections τ_{n} and the classical theorem about continuous one-parameter subgroups of Lie groups, see [^{10}], for example). When M_{t} = M^{t} is defined by formula (25) we have H = log(I + N) = ∑_{k≥1} \frac{(−1)^{k−1}}{k} N^{k}.

The mapping t → M^{t} will be called a one parameter group of UT (N, C).

Proposition 4.1. Let M be the matrix of a substitution with prefunction; then so is M^{t} for all t ∈ C.

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The proof will be given in a forthcoming paper and uses the fact that “to be the matrix of a substitution with prefunction” is a property of polynomial type. But, using composition, it is straightforward to see that M^{t} is the matrix of a substitution with prefunction for all t ∈ N. Thus, using a “Zariski-type” argument, we get the result that the property is true for all t ∈ C.

4.2. Link with local Lie groups : Straightening vector fields on the line

We first treat the case p = 0 of subsection (3.2.2). The string (a^{+})^{r}a corrresponds, in the Bargmann-Fock representation, to the vector field x^{r} \frac{d}{dx} defined on the whole line. Now, we can try (at least locally) to straighten this vector field by a diffeomorphism u to get the constant vector field (this procedure has been introduced by G. Goldin in the context of current algebras[^{8}]). As the oneparameter group generated by a constant field is a shift, the one-parameter (local) group of transformations will be, on a suitable domain

U_{λ}[f](x) = f (u^{−1} (u(x) + λ)) .  (28)

Now, we know from section (4.1) that, if two one-parameter groups have the same tangent vector at the origin, then they coincide (tangent paradigm). Direct computation gives this tangent vector :

\frac{d}{dλ}|_{λ=0} f (u^{−1} (u(x) + λ)) = \frac{1}{u′(x)}f′(x)  (29)

and so the local one-parameter group U_{λ} has \frac{1}{u′(x)} \frac{d}{dx} as tangent vector field. Here, we have to solve \frac{1}{u′(x)} = x^{r} in order to get the diffeomorphism u. In the case r ≠ 1, we have (with D =]0, +∞[ as domain)

u(x) = \frac{x^{1−r}}{1 − r} = y; u^{−1}(y) = ((1 − r)y)^{\frac{1}{1−r}}  (30)

and

e^{λx^{r} \frac{d}{dx}} [f](x) = f (\frac{x}{(1 − λ(r − 1)x^{r−1})^{\frac{1}{r−1}}} )  (31)

The substitution factor s_{λ}(x) = _{\frac{x}{(1−λ(r−1)x^{r−1})^{\frac{1}{r−1}}}} has been already

obtained by other means in [^{1}]. The computation is similar for the case

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when r = 1 and, for this case, we get

e^{λx \frac{d}{dx}} [f](x) = f (e^{λ}x)  (32)

with s_{λ}(x) = e^{λ}x as substitution factor. These first examples are summarized in the following table

r s_{λ}(x)  Name
0 x + λ  Shift
1  e^{λ}x  Dilation
2  \frac{x}{1−λx}  Homography
3 \frac{x}{\sqrt{1−2λx^{2}}}  -

Comment 4.1. If one uses classical analysis (i.e. convergent Taylor series), one must be careful about the domain where the substitutions are defined and the one-parameter groups are defined only locally. For each of these examples, one can check by hand that for suitable (and small) values of λ, μ, one has s_{λ}(s_{μ}(x)) = s_{λ+μ}(x) (one-parameter group property). It is possible to avoid discussion of the domains by considering λ, μ as new variables and applying the “substitution principle”; namely by claiming that it is possible to substitute a series without constant term in a series (in the algebra C[[x, λ, μ]]).

Using the same method, one can start with more complicated operators. Examples and substitution factors are given below

Operator  Substitution Factor  Description
(1 + (a^{+})^{2}) a  s_{λ}(x) = \frac{xcos(λ) + sin(λ)}{cos(λ) − xsin(λ)}  One-parameter group

of homographies

\frac{\sqrt{1 + (a^{+})^{2}}}{a^{+}} s_{λ}(x) = \sqrt{x^{2} + 2λ\sqrt{1 + x^{2}} + λ^{2}} Composition of quadratic

direct and inverse functions

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4.3. Case p > 0: another conjugacy trick and a surprising formula

Regarding vector fields as infinitesimal generators of one-parameter groups leads to conjugacy since, if U_{λ} is a one-parameter group of transformation, so too is V U_{λ}V^{−1} (V being a continuous invertible operator). We could formally consider (a^{+})^{r−p}a(a^{+})^{p}; p > 0 as conjugate to ((a^{+})^{r}a) in this context. More generally, supposing all the terms well-defined, if

Ω = u_{1}(x)\frac{d}{dx}u_{2}(x) = \frac{1}{u_{2}(x)} (u_{1}(x)u_{2}(x)\frac{d}{dx}) u_{2}(x)

then

e^{λΩ} = \frac{1}{u_{2}(x)} (e^{λu_{1}(x)u_{2}(x) \frac{d}{dx}} ) u_{2}(x)  (33)

This rather surprising formula (33) may be understood as an operator equality. Now, the tangent paradigm (see section 4.2) tells us that, if we adjust this tangent vector to coincide with x^{r−p} \frac{d}{dx}x^{p} (recall that the original problem was the integration of the operator Ω = (a^{+})^{r−p}a(a^{+})^{p}; p > 0), then we get the right one-parameter group. Using this “conjugacy trick” we get

e^{λΩ}[f](x) = (\frac{s_{λ}(x)}{x}) f(s_{λ}(x)) with s_{λ}(x) = \frac{x}{(1 − λ(r − 1)x^{r−1})^{\frac{1}{r−1}}} (34)

Remark 4.1. (i) It can be checked that, if s_{λ}(x) is a substitution factor (i.e. at least locally s_{λ}(s_{μ}(x)) = s_{λ+μ}(x)) such that s_{λ}(0) = 0 for every λ (which is the case in most of our examples) then the transformations defined by U_{λ}[f](x) = (\frac{s_{λ}(x)}{x}) f(s_{λ}(x)) form a one-parameter (possibly local) group. (ii) It is also possible to use the “ad” operator (Lie adjoint) instead of “Ad” (conjugacy) to obtain integration formulas (see Dattoli ^{6}).

4.4. Characteristic series ↔ one parameter group correspondence

The preceding allows us to extend integration processes to linear combinations of boson strings in the following sense. The algebra W_{1,∞} generated by a^{+}, (a^{+})^{−1}, a is graded by

weight(a^{+}) = 1, weight ((a^{+})^{−1}) = weight(a) = −1  (35)

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and every homogeneous operator of this algebra which is of the form

Ω = ∑_{|w|_{a}=1, weight(w)=e} α_{w}w  (36)

(there is only one annihilation operator in each monomial) can be integrated as above. So one would like to reconstruct the characteristic series

∑_{n,k} S_{Ω}(n, k)\frac{x^{n}}{n!} y^{k}  (37)

from knowledge of the one-parameter subgroup e^{λΩ}. This is the aim of the following paragraph.

For every homogeneous operator as above with e ≥ 0, one defines the coefficients S_{Ω}(n, k) as in the Introduction by

N (Ω^{n}) = (a^{+})^{ne} ∑_{k=0}^{∞} S_{Ω}(n, k)(a^{+})^{k}a^{k}  (38)

One has the following proposition

Proposition 4.2. With the definitions introduced, the following conditions are equivalent:

∑_{n,k≥0} S_{Ω}(n, k)\frac{x^{n}}{n!} y^{k} = g(x)e^{yϕ(x)}  (39)
U_{λ}[f](x) = g(λx^{e})f (x (1 + ϕ(λx^{e})))  (40)

Which solves the problem.

5. Conclusion and remaining problems

We have considered a class of elements of W_{1,∞} (see section 4.4 for a definition) which describe some rational vector fields on the line. For these operators, we have established a correspondence

One-parameter group (=integration of the field) ↔ Characteristic series (=coefficients of the normal ordering)

We have then seen that families of combinatorial matrices give rise through the exponential formula to substitutions. Further work which remains is to study the vector fields associated with these combinatorial matrices. Also it would be desirable to adapt this machinery to other algebras (quons, several boson modes).

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Acknowledgments

We thank Daniel Barsky, Christophe Tollu, Fréderic Toumazet and Jean-Yves Thibon for interesting remarks and observations during earlier stages of this work.

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4. P. Blasiak, K.A. Penson and A.I. Solomon, Combinatorial coherent states via normal ordering of bosons Lett. Math. Phys., (2003) in press, Preprint arXiv:

quant-ph/0311033

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