Feynman graphs and related Hopf algebras

G H E Duchamp^{a}, P Blasiak^{b}, A Horzela^{b}, K A Penson^{c} and A I Solomon^{c,d}

a Institut Galilée, LIPN, CNRS UMR 7030 99 Av. J.-B. Clement, F-93430 Villetaneuse, France

b H. Niewodniczański Institute of Nuclear Physics, Polish Academy of Sciences ul. Eliasza-Radzikowskiego 152, PL 31342 Kraków, Poland

c Laboratoire de Physique Théorique de la Matière Condensée Université Pierre et Marie Curie, CNRS UMR 7600 Tour 24 - 2ième ét., 4 pl. Jussieu, F 75252 Paris Cedex 05, France

e The Open University, Physics and Astronomy Department Milton Keynes MK7 6AA, United Kingdom

E-mail:

ghed@lipn-univ.paris13.fr, pawel.blasiak@ifj.edu.pl,
andrzej.horzela@ifj.edu.pl, penson@lptl.jussieu.fr,
a.i.solomon@open.ac.uk

Abstract. In a recent series of communications we have shown that the reordering problem of bosons leads to certain combinatorial structures. These structures may be associated with a certain graphical description. In this paper, we show that there is a Hopf Algebra structure associated with this problem which is, in a certain sense, unique.

Contents

1. Introduction

In a relatively recent paper Bender, Brody and Meister [?] introduced a special Field Theory described by a product formula (a kind of Hadamard product for two exponential generating functions - EGF) in the purpose of proving that any sequence of numbers could be described by a suitable set of rules applied to some type of Feynman graphs. Inspired by this idea, we have worked out combinatorial consequences of the product and exponential formulas in a recent series of papers [?, ?, ?, ?, ?, ?, ?].

Here, we consider two aspects of the product formula for formal power series applied to combinatorial field theories. Firstly, we remark that the case when the functions involved in the product formula have a constant term (equal to one) is of special interest as often these functions give rise to substitutional groups. The groups arising from the normal ordering problem of boson strings are naturally associated with explicit vector fields, or their conjugates, in the case when there is only one annihilation operator [?, ?]. We also consider one-parameter groups of operators when several annihilators are present. Secondly, we discuss the Feynman-type graph representation resulting from the product formula. We show that there is a correspondence between the packed integer matrices of the theory of noncommutative symmetric functions and the labelled version of these Feynman-type graphs.

We thus obtain a new Hopf algebra structure over the space of matrix quasisymmetric functions that is a natural cocommutative Hopf algebra structure on the space of diagrams themselves which originates from the formal doubling of variables in the product formula.

Aknowledgements We would like here to express our gratitude to Jean-Louis Loday, Jean-Bernard Zuber, Jean-Yves Thibon and Florent Hivert for stimulating interactions on this subject.

2. Single and double exponentials

2.1. One parameter groups and the connected graph theorem

2.1.1. Substitutions The Weyl algebra W is the -associative algebra (with unit) defined by two generators a and a^{+} and the unique relation [a, a^{+}] = 1. This algebra is of Gelfand-Kirillov dimension 2 and has a basis consisting of the following family {(a^{+})^{k}a^{l}}_{k,l≥0}. It is known that it is impossible to represent faithfully a, a^{+} by bounded operators in a Banach space, but one often uses the representation a ↦ \frac{d}{dx}; a^{+} ↦ x as operators acting “on the line” or, better said, on the space of polynomials [x]. Through this representation (faithful and coined under the name “Bargmann-Fock”), one sees that we can define a grading on W by the weight function w(a) = −1; w(a^{+}) = 1. A homogeneous operator (under this grading) Ω ∈ W is then of the form

Ω = ∑_{k,l; k−l=e} c(k, l)(a^{+})^{k}a^{l} (1)

According to whether the excess e is positive or negative, the normal ordering of Ω^{n} reads

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

We get combinatorial quantities with two indices i.e. an infinite × matrix {S_{Ω}(n, k)}_{n,k≥0} which we will call the generalized Stirling matrix of Ω. In fact, it is easily checked that, if the coefficients c(k, l) of Ω are non-negative integers, so are the entries (S_{Ω}(n, k)) of this matrix.

Let us give some examples of these generalized Stirling matrices. For Ω = a^{+}a, one gets the usual matrix of the 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)

For Ω = a^{+}aa^{+} + a^{+}, we obtain

⌈ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 2 & 1 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 6 & 6 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ 24 & 36 & 12 & 1 & 0 & 0 & 0 & ⋯ \\ 120 & 240 & 120 & 20 & 1 & 0 & 0 & ⋯ \\ 720 & 1800 & 1200 & 300 & 30 & 1 & 0 & ⋯ \\ 5040 & 15120 & 12600 & 4200 & 630 & 42 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} (4)

and for w = a^{+}aaa^{+}a^{+}

⌈ \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)

Let Ω = ∑_{k,l≥0} c(k, l)(a^{+})^{k}a^{l} (finite supported sum) be a general term of W in normal form and let us call a dominant term, the sum of monomials with maximum length k + l. It is not difficult to prove that, if Ω is homogeneous, the dominant term consists of a single monomial c(k_{0}, l_{0})(a^{+})^{k_{0}}a^{l_{0}}. Thus, the dominant term of Ω^{n} must be c(k_{0}, l_{0})^{n}(a^{+})^{nk_{0}}a^{nl_{0}}. Then, for example in the case when e = k_{0} − l_{0} ≥ 0, in the generalized Stirling matrix of Ω, the rightmost non-zero coefficient of the line n has address (n, n.l_{0}) and bears the coefficient c(k_{0}, l_{0})^{n}. All these matrices are row-finite and triangular iff l_{0} = 1 (which means that no monomial possesses more than one a).

Remark 2.1 i) There is a beautiful combinatorial expression of the normal form of w^{n} in case w is a string in a and a^{+}. The normal form of w is

N (w) = ∑_{k≥0} r(B, k)(a^{+})^{r−k}a^{s−k} (6)

where r(B, k) is the kth rook number of a certain board B constructed after w (see [?, ?], and r = |w|_{a^{+}}; s = |w|_{a} are the number of occurences of a^{+} and a in w. ii) To each matrix M ∈^{×} of this kind and more generally “row finite” matrices (which means that, for each n, the family (M(n, k))_{k∈} is finite supported), one can associate a transformation of EGFs (see [?, ?]) f ↦ f̂ such that, if f = ∑_{n≥0} a_{n} \frac{z^{n}}{n!} then f̂ = ∑_{n≥0} b_{n} \frac{z^{n}}{n!} (with b_{n} = ∑_{k≥0} M(n, k)a_{k}). iii) It can be shown that, if no monomial of Ω possesses more than one a, the action of the transformation induced by Ω (through the Bargmann-Fock representation) can be expressed in terms of vector fields or their conjugates, thus the one-parameter group e^{λΩ} acts by substitutions and products [?, ?].

2.1.2. Combinatorial matrices and one-parameter groups One can also draw generalized Stirling matrices from another source, namely from the combinatorial graph theory. Let C be a class of graphs such that

Γ ∈ C iff every connected component of Γ is in C (7)

For these classes of graphs, one has the exponential formula [?, ?, ?] saying roughly that

EGF(all graphs) = e^{EGF(Connected Graphs)} (8)

This implies, in particular, that the matrix

M(n, k) = number of graphs with n vertices and having k connected components(9)

is the matrix of a substitution (see [?, ?]). One can prove, using a Zariski-like argument (a polynomial vanishing for every integer vanishes everywhere), that, if M is such a matrix (with identity diagonal) then, all its powers (positive, negative and fractional) are substitution matrices and form a one-parameter group of substitutions, thus coming from a vector field on the line which can be computed. But no nice combinatorial principle seems to emerge. For example, beginning with the Stirling substitution z ↦ e^{z} − 1, we know that there is a unique one-parameter group of substitutions s_{λ}(z) such that, for λ integer, one has the value (s_{2}(z) ↔ partition of partitions)

s_{2}(z) = e^{e^{z}−1} − 1 ; s_{3}(z) = e^{e^{e^{z}−1}}