Nonlocal, noncommutative diagrammatics and the linked cluster Theorems
Recent developments in quantum chemistry, perturbative quantum field theory, statistical physics or stochastic differential equations require the introduction of new families of Feynman-type diagrams. These new families arise in various ways. In some generalizations of the classical diagrams, the notion of Feynman propagator is extended to generalized propagators connecting more than two vertices of the graphs. In some others (introduced in the present article), the diagrams, associated to noncommuting product of operators inherit from the noncommutativity of the products extra graphical properties.
The purpose of the present article is to introduce a general way of dealing with such diagrams. We prove in particular a “universal” linked cluster theorem and introduce, in the process, a Feynman-type “diagrammatics” that allows to handle simultaneously nonlocal (Coulomb-type) interactions, the generalized diagrams arising from the study of interacting systems (such as the ones where the ground state is not the vacuum but e.g. a vacuum perturbed by a magnetic or electric field, by impurities…) or Wightman fields (that is, expectation values of products of interacting fields). Our diagrammatics seems to be the first attempt to encode in a unified algebraic framework such a wide variety of situations.
In the process, we promote two ideas. First, Feynman-type diagrammatics belong mathematically to the theory of linear forms on combinatorial Hopf algebras. Second, linked cluster-type theorems rely ultimately on Möbius inversion on the partition lattice. The two theories should therefore be introduced and presented accordingly.
Among others, our theorems encompass the usual versions of the theorem (although very different in nature, from Goldstone diagrams in solid state physics to Feynman diagrams in QFT or probabilistic Wick theorems).
The attempts to clarify the mathematical framework underlying quantum chemistry, solid state physics, quantum field theories (QFT) and connected topics such as -for example, the grounds for the functional approaches, the principles underlying renormalization- as well as attempts to deepen our current understanding of widely used techniques (effective Hamiltonians, adiabatic limits…) are often hindered by very basic questions regarding the underlying mathematical methods. This is particularly obvious when it comes to developing new tools, see e.g. our studies of enhanced algorithms and formulas for the computation of eigenstates of Hamiltonians when the (unperturbed) ground state is degenerate [1, 2, 3], or of the combinatorics of one-particle-irreducibility for interacting systems .
The present article focusses on diagrammatics. We argue that Feynman-type diagrammatics belong mathematically to the theory of linear forms on combinatorial Hopf algebras, which allows to generalize the theory to a much wider setting than the classical one. We cover the usual examples corresponding to vacuum expectations over commutative products of fields where the propagators are represented by edges -this includes for example the various Goldstone-type diagrams and the perturbative expansions parametrized by Feynman diagrams in quantum field theories. However we go beyond and cover the case of expectations over general states, which requires the introduction of generalized diagrams [5, 4]. The same combinatorics happens to provide a pictorial description of cumulants in probability. The Hopf algebraic approach also allows to study expectations of products of free fields (Wightman fields) and derive new Feynman diagram expansions in this setting.
In a second part, we prove general linked cluster theorems, covering all these cases and making as explicit as possible the link between the elementary algebra underlying the theorems and the graphical content (that relies on connectedness in the graph-theoretical sense). Notice that we avoid deliberately functional methods (see e.g.): although very efficient to derive the classical QFT linked cluster theorem (using a generating function in terms of an external source), they lack the generality and simplicity of the combinatorial proof. Moreover, they cannot deal with noncommutative Feynman diagrams because functional derivatives commute. In the process, we promote another idea. Namely, linked cluster-type theorems rely ultimately on Möbius inversion on the partition lattice.
We are particularly grateful to R. Stora. Many letters and documents (among which ) he sent us were the initial incentive for the present article -that was conceived to bridge the computations in  with other approaches to quantum field computations and find some suitable mathematical framework for the (much more advanced) problems these documents suggest. In particular, we aimed at developping a mathematical framework to deal with products of Wightman fields and their average values over general states, one of the problems this article addresses.
We use the following convention: since various products will be defined along the article (such as or ), when we want to emphasize what product is used in an exponential, a logarithm or any other operation, we put the product symbol in exponent, so that emans that we compute the logarithm using the product, means that we compute the -th power of using the -product, and so on.
2 Free combinatorial Hopf algebras
The objects we will be interested in are combinatorial Hopf algebras in the sense of Joni and Rota , that is, bialgebras which coproduct is of combinatorial nature (obtained by “splitting” generating symbols according to combinatorial rules encoded by remarkable “section coefficients” –in high-energy physics, these coefficients correspond roughly to the symmetry factors of Feynman graphs). More specifically, we will be interested in families of Hopf algebras corresponding physically to bosonic or fermionic systems, to the usual algebraic structure of quantum fields (equipped with a commutative product such as the normal or time-ordered product) and to Wightman fields.
Since our results are more general than what would be required by applications to quantum systems, we state them in full generality and will show later how they specialize to particular physical systems or mathematical problems. Let be a fixed ordered set, . In most applications, will be infinite and countable, so that the reader may think to as the set of natural numbers.
The notion of combinatorial Hopf algebra, goes back to . The general notion is ill-defined in the litterature (there are many natural candidates, but at the moment no convincing general definition). We choose here a simple and relatively straightforward definition suited for our purposes that reflects some of the natural properties one expects from the notion when the underlying algebra is free commutative or free associative.
We call free commutative (resp. free) combinatorial Hopf algebra any connected graded commutative (resp. associative) and cocommutative Hopf algebra such that:
(Freeness) As an algebra, is the algebra of polynomials (resp. of tensors) over a doubly indexed (finite or countable) set of formal, commuting (resp. noncommuting), variables where runs over finite subsets of and , where and where the sequence of the , is a fixed sequence of integers.
(Equivariance) The structure maps are equivariant with respect to maps induced by substitutions in . In other terms, any substitution (that is, any set automorphism) induces a Hopf algebra automorphism of which action on the generators is defined by .
The are called the generators of , the (commutative or associative) monomials in the form a linear basis of and are therefore called the basis elements.
We do not look for the outmost generality, and many of our constructions and definitions can be extended in a fairly straightforward way to more general systems. For example, one might consider free partially commutative Hopf algebras which generators would satisfy partial commutation rules (e.g. all the would commute for a fixed , but without and commuting for , and so on). The so-constructed Hopf algebras can make sense in various applications and inherit all the properties of free commutative or free combinatorial Hopf algebras that are required for the forthcoming reasonings, we refer e.g. to  for details on the subject.
Some further remarks are in order. Recall first that, by the Leray theorem (see e.g. Prop 4 in ), any connected graded commutative Hopf algebra is free commutative, so that the assumption that is freely generated as a commutative algebra comes for free when one assumes that is connected graded commutative.
The key point that makes the Hopf algebra combinatorial (and, as we shall see, suited to Feynman-type graphical reasonings) is that we assume that a set of polynomial generators is fixed and behaves nicely with respect to substitutions in .
The particular case where for corresponds to the classical situation where the only propagators showing up in diagrams are the ones that describe the free propagation of a particle.
Let us mention at last that a more pedantical definition of combinatorial Hopf algebras could be given in terms of vector species, following the approach to combinatorial Hopf algebraic structures in [11, 12, 13].
A very important consequence of the equivariance condition is the following.
For any subset of , let us write for the subalgebra of generated by the . Then, is a Hopf subalgebra of . Moreover, any automorphism of induces an isomorphism of Hopf algebras from to , where .
The second property is a direct consequence of the first one since (by definition of the induced map) induces an isomorphism of algebras from to . The proof of the first assertion follows immediately from the equivariance condition. Indeed, notice that an element of or of is invariant by any substitution that acts as the identity on a finite subset of if and only if it is a polynomial in the (or, in , a sum of tensor products of such polynomials). Now, since, for , is invariant by any substitution that acts as the identity on , the same property holds also true of , from which the Lemma follows.
3 Some remarkable Hopf algebras
Our favorite examples in view of applications to quantum chemistry, QFT and solid state physics are simple ones. However the generality chosen (allowing for example ) is natural to handle nonlocal interaction terms in the Lagrangians (think for example of the quantum chemistry approach with Coulomb interaction). Our general approach also paves the way to a unification of QFT techniques (Feynman-type diagrammatics), umbral calculus (duality and linear forms on polynomial algebras) and combinatorics of (possibly ordered) set partitions.
For notational simplicity, we treat only commutative algebras in the usual sense, that is we do not treat explicitly the fermionic case (Grassmann or exterior algebras/ Fermi statistics). However, as it is well known, there are no difficulties in switching from a bosonic to a fermionic framework -it just requires adding the right signs in the formulas, see [14, 15], but handling simultaneously the commutative and anticommutative case would have required introducing consistently signs in all our formulas. For notational simplicity, we decided to stick to the bosonic, commutative, case, and to let the interested reader adapt our results to the anticommutative setting. We use the langage of particle physics and quantum chemistry (so that the bosonic algebra coincides with the algebra of polynomials).
Definition 3 (Bosonic algebra)
We write for the algebra of polynomials over the set of (formal, commuting) variables
where . These algebras are naturally equipped with a coproduct that makes them Hopf algebras (that is, is a map of algebras). The coproduct is defined on the generators by requiring them to be primitive, that is:
and extended multiplicatively to ().
The lower index should be thought of as the number of quantum fields showing up (in a very broad sense), whereas the should be thought of as points, momenta or more generally dummy integration variables. For later use, we allow (to encode the countable set of eigenstates of a given many-body Hamiltonian) and will write simply for when no confusion can arise.
Definition 4 (Coulomb algebra)
We write for the algebra of polynomials over the set of (formal, commuting) variables , where . The coproduct is defined by requiring the generators to be primitive.
This algebra is used in non-relativistic many-body physics and quantum chemistry. The Coulomb interaction desribes the force between a charge at point and a charge at point . These two points are linked by the interaction, and a specific variable is used to describe this connection.
Definition 5 (Tensor algebra)
We write for the algebra of noncommutative polynomials over the set of variables
. The coproduct is defined by requiring the generators to be primitive.
Various other free combinatorial Hopf algebras possibly noncommutative but fitting in the general framework of the present article are described (or follow from the results) in [12, 16]. Let us quote the Malvenuto-Reuntenauer Hopf algebra and various Hopf algebras of tree-like structures. Although the decorated version of the Connes-Kreimer Hopf algebra of trees in  may have a use for QFT, a more promissing path in that direction is certainly provided by , the belief of which we do share : “ultimately we think that combinatorics is the right approach to QFT and that a QFT should be thought of as the generating functional of a certain weighted species in the sense of ”. We restrict however the examples in the present article to well-established domains of QFT and leave further extensions to future work.
Let us start by recalling the construction underlying diagrammatic expansions and show how it can be extended naturally in a noncommutative/nonlocal setting. We call this process “graphication”, by analogy with the “arborification” process underlying tree expansions in analysis and dynamical systems [19, 20]. In all the article, as a tribute to the physical motivations the ground field is the field of complex numbers (although the results hold over an arbitrary field of characteristic zero).
The reduced coproduct is defined by . The coproduct and reduced coproduct are coassociative in the sense that (and similarly for ). The iterated coproduct and reduced coproduct maps from to are therefore well-defined and written , resp. .
We assumed in the definition of combinatorial Hopf algebras that the coproduct is cocommutative: with Sweedler’s notation , this means that . Many of our forthcoming results could be adapted to the noncocommutative setting. However, this hypothesis is particularly usefulf when it comes to graphical encodings.
The graphication map is the map from to defined by:
Here, the superscript in means that the elements in the image of are sums of symmetric tensor powers of elements of . We will use the canonical isomorphism from covariants (where stands for the symmetric group of order ) to invariants
to represent invariants using the bar notation (that is, in is written ). Notice that, by definition (since we deal with covariants), for any permutation , .
For example, in the bosonic algebra, abbreviating to (a notation we will use without further comments from now on):
where runs over all partitions , of , with Or (isolating some components in the expansion):
The same formulas hold in the tensor algebra and in the Coulomb algebras. Notice however that, in the tensor algebra, products are noncommutative, so that the order of the products in the monomials does matter. For example, with self-explanatory notation:
We call brackettings the terms such as . The length of a bracketting is the number of vertical bars plus 1. The support of a bracketting is the set of the showing up in . For example,
For each , we write for the total degree of the s in (so that for as above, ). For later use, we also introduce the product of two brackettings, which is simply the concatenation product:
These ideas and notation extend in a self-explanatory way to arbitrary combinatorial Hopf algebras. The only change regards the definition of the support and degree: is included in the support of a bracketting whenever there exists a pair such that and shows up in so that, for example,
Similarly, the degree of in this bracketting accounts for the term and is therefore .
The coefficients in the right hand side of the equation defining the graphication will be referred to as symmetry factors. They are closely related to the structure coefficients for the coproduct (in the basis provided by monomials in the chosen family of generators, e.g. the for the bosonic algebra -these coefficients were called section coefficients by Joni and Rota ) but encode also the symmetries showing up in the coproducts. Whereas, for the bosonic and Coulomb algebras, symmetry factors are defined without ambiguity, for other algebras (the tensor algebra or general free combinatorial Hopf algebras) a given bracketting may appear in the expansion of for various monomials in the generators (for example, in the tensor algebra, appears in the expansion of and of ). In general, we will therefore write for the symmetry factor of a bracketting in the expansion of and will write simply in the particular case of the bosonic and Coulomb algebras (where a unique exists giving rise to such a factor).
5 Graphical representation
Perturbative expansions in particle and solid-state physics are conveniently represented by various families of diagrams. Feynman diagrams are the most popular ones, but there are plenty of other families with construction rules often slightly different from the one underlying Feynman diagrams. Just to mention one interesting feature, Feynman diagrams are usually independent of the time-coordinate of the vertices (this is because of the definition of the so-called Feynman propagators), whereas other families showing up in solid-state physics take into account causality systematically to construct their diagrams. These ideas are particularly well-explained in , to which we refer, also for a comprehensive treatment of the zoology of diagrammatic expansions.
Defining a graphical representation associated to a free combinatorial Hopf algebra depends highly on the structure and particular features of the algebra. We define various such representations, by increasing order of complexity, focussing only on the algebras (tensor, bosonic, Coulomb) we have chosen to investigate in depth. The reader who needs to construct other taylor-made graphical representations will be able to do so easily using our recipes (for notational simplicity and to avoid pointless pedantry, we omit to introduce the most general possible definitions since the process of designing them is straightforward once the leading principles are understood on some examples).
Let us mention that, when two brackettings and have disjoint supports, their product corresponds to the disjoint union of the corresponding graphs (this is the usual product on Hopf algebras of Feynman graphs in QFT, see e.g. ).
5.1 Commutative local case: bipartite graphs
We focus in this section on the bosonic algebra. Each Feynman bracket (say , see fig. 1) can be represented uniquely by a bipartite (non planar) graph with unoriented colored edges (i.e. by a graph with 2-coloured vertices and colored edges) according to the following rule:
For each , recall that we write for the total degree of the s in . Draw a -labelled black vertex with outgoing colored edges (the colors being attributed according to the indices of the , e.g. a 4-edge black vertex for with one 1-colored edge (solid line), two 2-colored edges (dotted line) and a 3-colored edge (dashed line).
Running recursively from the left to the right of , for each term inside brackets and bars (e.g. , then , then ), select randomly according to the colors and powers showing up in the monomials outgoing edges of the corresponding vertices (e.g. select one 1-colored edge from the vertex and two 1-colored edges from the vertex). Connect these edges to a new white vertex (e.g. a new white vertex with 3 outgoing colored edges).
Some simplifications are possible in many cases of interest. For example, when the s are dummy integration variables and can be exchanged freely in any computation (e.g. of scattering amplitudes), the labelling of the black vertices can be omitted.
Another classical situation (encountered with scalar quantum field theories such as or and in statistical physics) is the one where . In that case, there is just one possible color for the edges, so that the coloring of the edges can safely be omitted in the definition of the graphs. In QFT, when several fields coexist (e.g. in QED, where ), instead of using colors, practitioners use often different representations for the edges that depend on the fields involved (typically, in QED, edges associated to electrons are plain lines, whereas edges associated to photon propagation are represented by a succession of small waves). Of course, the use of colors or the use of different shapes for the edges are strictly equivalent and a matter of taste and habits.
Another classical simplification occurs when the only brackettings of interest for practical applications are the ones where the only monomials showing up in the brackettings are products of degree 2 of the form . In the corresponding graphs, the white vertices always have two outgoing edges with the same color, so that these vertices can be erased -what remains is a graph with only black vertices and colored edges (that correspond, physically, to particle types). These are the celebrated Feynman graphs that one encounters in QFT textbooks.
From now on we will identify systematically brackettings and the corresponding interaction graphs.
5.2 Commutative non local case: tripartite graphs
By nonlocal case, we mean that some may be different from zero. The canonical example we have in mind is the one of QED in the solid state picture, that is with instantaneous Coulombian interactions (in the particle physics picture the interactions are local and encoded by products of fields in the Lagrangian; this corresponds to the commutative local case).
For simplicity (and in view of the most natural applications), we assume that the only brackettings of interest are those in which the monomials showing up are either monomials in the , either a (in other terms, no nontrivial products involving a should appear in the bracketting).
Each Feynman bracket (say for example
can be represented uniquely by a tripartite (non planar) graph with two kinds of unoriented edges according to the following rule (see Figure 2):
We distinguish between the two kind of edges in the following way: the first type of edge is drawn as colored lines (the color is an index with ); the second type has no color and is drawn as a sequence of waves (as the photon propagators in QED). We call these edges respectively plain and wavy edges.
For each , draw a -labelled black vertex with outgoing edges. Colors are attributed to the plain edges according to the indices of the , the other edges correspond to s and are wavy edges. For example, we draw a 4-edges black vertex for with one 1-colored edge (represented by a solid line in Fig. 2), two 2-colored edges (dotted lines) and a wavy edge.
Running recursively from the left to the right of , for each term inside brackets and bars (e.g. , then , then ),
If encountering a monomial involving s, proceed as in the commutative local case: select randomly according to the colors and powers showing up in the monomials outgoing edges of the corresponding vertices (e.g. select one 1-colored edge from the vertex and two 1-colored edges from the vertex). Connect these edges to a new white vertex (e.g. a new white vertex with 3 outgoing colored edges).
If encountering a , select randomly a wavy edge outgoing from the black vertex, connect all these edges to new -labelled grey vertex.
These tripartite graphs will be called from now nonlocal interaction graphs. The particular case of solid state physics QED enters in this framework. Moreover, the general case we consider seems new and of interest, since it should allow to treat QED computations in a complex background (e.g. with non trivial vacua). These concrete applications (that originated this work, together with questions and remarks by R. Stora) are left for future work.
Great simplifications occur in many simpler cases of interest. The first situation encountered in pratice is the one where . This corresponds roughly to the case where one class of particles is present (say electrons, up to a change from the bosonic to the fermionic statistics) and where interactions are encoded by nonlocal terms (Coulomb interaction lines). In that case, there is just one possible color for the edges, so that the coloring of the plain edges can be omitted in the definition of the graphs. Besides, since , the grey vertices can also be erased, so that one ends up with bipartite graphs with two types of edges (plain and wavy). If one assumes further that the only graphs of interest are those with two outgoing edges, these edges can be also erased. One ends up with the familiar Feynman diagrams with black vertices only and two types of edges corresponding to electron propagators and Coulombian interactions.
5.3 Noncommutative local case
We focus in this section on the tensor algebra. Each Feynman bracket (say ) can be represented uniquely by a bipartite (non planar) graph with unoriented colored and locally ordered edges (i.e. by a graph with 2-coloured vertices and colored, locally ordered edges) according to the following rule (the rule will make clear the meaning of “local order” that corresponds to an order on the edges reaching a white vertex) (see fig. 3):
Proceed first as in the commutative case: for each , recall that we write for the total degree of the s in . Draw a -labelled black vertex with outgoing colored edges (the colors being attributed according to the indices of the , e.g. a 4-edges black vertex for with one 1-colored edge (solid line), two 2-colored edges (dotted lines) and a 3-colored edge (dashed line).
Running recursively from the left to the right of , for each term inside brackets and bars (e.g. , then …), select randomly according to the colors and powers showing up in the monomials outgoing edges of the corresponding vertices and order them according to the order of the appearance of the colors in the (noncommutative !) monomial (e.g. select a 2-colored edge from the vertex, label it ; a 1-colored edge from the vertex, label it ; a 2-colored edge from the vertex, label it ). Connect these edges to a new white vertex (e.g. a new white vertex with 3 outgoing colored and ordered edges, where the order is defined by the labels).
These edge-colored and locally ordered bipartite graphs will be called from now free interaction graphs.
The usual simplifications are possible in many cases of interest, we do not detail them and simply mention that, when the only monomials showing up in brackettings are of the form , the graphical rule amounts to consider “classical” Feynman graphs with labelled vertices and colored and directed propagators.
The noncommutative nonlocal case could be treated similarly by mixing the conventions for the noncommutative local and commutative local cases. The exercice is left to the reader.
6 Symmetry factors and connectedness
This section is devoted to a technical but fundamental Lemma that connects the symmetry factors of graphs with the topological notion of connectedness. The lemma is the ground for the proof of linked cluster theorems and is particularly meaningful in the noncommutative case (Wightman fields). We assume in this section that the Hopf algebra is an arbitrary free or free commutative combinatorial Hopf algebra which generators are primitive elements (this condition is satisfied by all the combinatorial Hopf algebras we have considered so far).
Let us introduce first a further notation. Let be a basis element (a commutative or noncommutative monomial in the generators) of a combinatorial Hopf algebra which support decomposes into a disjoint union (the support is defined as for brackettings). We write and and call respectively and -components of the two basis elements (possibly equal to zero) defined by , where stands for the projection on orthogonally to all the basis elements that do not belong to .
The reader can check that this definition amounts to the following: to get , replace, in the expansion of as a monomial, all the by a 1 and all the by a zero (and similarly for ).
Let be a basis element and a bracketting or, equivalently, an interaction graph (of any type) such that and (recall that stands for the product of brackettings). Topologically, this amounts to assume that decomposes as a disjoint union of graphs. Then:
The basis elements and are non zero.
In the commutative case, this last identity can be abbreviated to .
Indeed, since the are primitive, the coproduct of a product, say , of s is the sum of the tensor products , where is a (ordered) subset of and its (ordered) supplement. The hypothesis ensures that , and that in , there is no factor with and . The first assertion follows.
To prove the second identity, let us first notice that . Indeed, let us use the same notation as previously and write . The coproduct and its iterations are constructed by extracting disjoint subsequences out of the ordered sequence of the s. On the other hand, the basis elements showing up in and belong to disjoint sets -the relative ordering of the basis elements with support in and in in the expansion of do therefore not matter, which proves the identity.
We can now assume without restriction (because of the cocommutativity) that with . Besides, since is an algebra map, we have:
The multiplicity of in is therefore obtained by summing the coefficients of the tensor products in , where runs over and (here, stands for the stabilizer of in ). However, since the coproduct is cocommutative, these coefficients are all equal and one can restrict the computation to the tensor products , where and run over permutations in and , and multiply the result by the number of -element subsets in . We get finally:
that is, or, in words (and slightly abusively), “the symmetry factor of an interaction graph is the product of the symmetry factors of its connected components”. Notice that the property holds true also in nonlocal and/or noncommutative cases.
7 Amplitudes and Feynman rules
A linear form on a combinatorial Hopf algebra is unital if and infinitesimal if .
Let us recall, for example, how linear forms on , , are usually constructed. Let be an arbitrary finite sequence of integers. For any polynomial in variables, say , we write for the polynomial . One can think of as the interacting part of a Lagrangian. Natural forms should then be thought of as related physical amplitudes. For example, for , a typical is , corresponding to the theory (see  for details). The Green functions of this theory are computed via the formula
where denotes the vacuum expectation value of the time-ordered product of fields (see section 10.1).
Let us treat now a radically different example to show the ubiquity of the approach. Let here the role of be taken by , the interacting Hamiltonian of time-dependent perturbation theory (see e.g. our [1, 2]): . Let be the eigenvectors of with eigenvalues . We assume for simplicity that the ground state is non degenerate (), although the following reasoning holds in full generality due to [1, 2]. The computation of the ground state of the perturbed Hamiltonian relies on the computation of the quantities such as: , where . We set and , . Then, , where . The unital form corresponding to the computation of is given simply by the form on :
where if is even and odd and zero else. The value of the form on odd products is . Of course, this example is purely didactical and for such a computation the use of the formalism developped in the present article is largely pointless. It becomes useful when the situation gets more involved. Actually, the simple requirement of taking efficiently into account the divergences arising from the adiabatic expansion may involve advanced combinatorial techniques, see, besides the articles already quoted, our .
It is well-known that, in many situations, Green functions such as the ones of the theory split into components parametrized by Feynman diagrams. This property also holds for more complex theories and is best explained through Hopf algebraic computations. Recall first that, since is a Hopf algebra, the set of linear forms on is equipped with an (associative; commutative if is cocommutative) “convolution” product:
where we used the Sweedler notation . Notice that, if and are infinitesimal forms, , where we use the notation . By standard graduation arguments, the convolution logarithm of a unital form is a well-defined infinitesimal form on . We extend such a to a linear form (still written ) on (resp. ) by: or .
Summing up, we get, for an arbitrary monomial (basis element) in , and since is an infinitesimal form:
Proposition 8 (Feynman diagrams/rules expansion)
For an arbitrary unital form on , we have:
where runs over all the brackettings (or interaction graphs) in the image of by the graphication map.
Assume that , then:
8 The combinatorial linked cluster theorem
The combinatorial linked cluster theorem expands a linear form on a combinatorial Hopf algebra into “connected” parts closely related to the topological notion of connectedness. In this section, we show that this expansion is a very general phenomenon related to Möbius inversion in the partition lattice. Notations and conventions are in Sect. 6.
We first recall some general facts on the partition lattice and Möbius inversion that are familiar in combinatorics but probably not well-known by practitioners of Feynman-type diagrammatics and linked cluster theorems (the particular case of Möbius inversion for the partition lattice we are interested in here seems due to Schützenberger, we refer to  for further details and references on the subject).
For an arbitrary set , partitions of (that is: , where stands for the disjoint union) are organized into a poset (partially ordered set, this poset is actually a lattice -two elements have a and a , this follows easily from the definition of the order). We write for the length of the partition (so that ) and abbreviate the partitions of minimal and maximal length, respectively and to and . The subsets are called the blocks, and the order is defined by refinement: for any partitions and , if and only if each block of is contained in a block of .
The functions on the partition lattice such that only if form the incidence algebra of the lattice. The (associative) product is defined by convolution: . The identity of the algebra is Kronecker delta function: if and else. The zeta function of the lattice is defined to be equal to 1 if and 0 otherwise. The Möbius function is defined to be the inverse of the zeta function for the convolution product. It can be computed explicitly: for , where , we have:
where is the number of blocks of contained in . Using the identities , we recover in particular the useful combinatorial formulas:
where runs over the partitions of length of and stands for the number of elements in the -th block of and, with the same conventions on ,
The key application of these notions is to inclusion/exclusion computations in the partition lattice. Namely, for an arbitrary function on the lattice, let us set: . This formula defines uniquely and, in the convolution algebra:
so that .
Let now be a basis element of with support written , where is as in Sect. 6. Let be a unital form on . Recall the notation denoting the “T-component” of for an arbitrary . For an arbitrary set partition of , we extend to a function on partitions of and set:
Recall also the decomposition . The s are represented by diagrams, among which some are connected. We set , where the run over connected diagrams. We can then apply the machinery of inclusion/exclusion to and define . The combinatorial linked cluster theorem relates and :
Theorem 10 (Combinatorial linked cluster theorem)
We have, for an arbitrary basis element in a free or free commutative combinatorial Hopf algebra which generators are primitive elements:
We have indeed:
where runs over the partitions of and (with the usual notations) . On the other hand, and . Lemma 9 ensures that , and we get finally:
where is a graph showing up in the expansion of .
Let now be an arbitrary graph with . The graph decomposes uniquely as a union of topologically disjoint graphs , where is the number of connected components of . We write for and . We have to show that the coefficient of in the right hand side of the previous equation is equal to if and to 0 else. The first property is immediate, since if is connected it appears only in the term associated to the trivial partition of and therefore with the coefficient .
The second property is slightly less immediate, but follows from the principles of Möbius inversion together with Lemma 9. Notice first that appears in the right hand side of the equation in association to all partitions of which is a refinement. The coefficient of is therefore: