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Constructive field theory can be considered as a reorganization of perturbation theory in a convergent way. In this pedagogical note we propose to wander through five different methods to compute the number of connected graphs of the zero-dimensional

New constructive Bosonic field theory methods have been recently proposed which are based on applying a canonical forest formula to repackage perturbation theory in a better way. This allows to compute the connected quantities of the theory by the same formula but summed over trees instead of forests. (Constructive Fermionic field theory is easier and was repackaged in terms of trees much earlier [

Combining such a forest formula with the intermediate field method leads to a convenient resummation of

Another even more recent constructive point of view [

In this point of view, constructive bounds reduce essentially to the positivity of the universal Hamiltonian operator. The vacuum is the trivial tree and the correlation functions are given by “vacuum expectation values” of the resolvent of that combinatoric Hamiltonian operator. Model-dependent details such as space-time dimension, interactions and propagators enter the definition of the matrix elements of this scalar product. These matrix elements are just finite sums of finite dimensional Feynman integrals.

We were urged to explain these new ideas in a pedagogical way. This is what we do in this short note on the simplest possible example, namely, the connected graphs of the zero dimensional

A forest on

A tree (also called spanning tree) on

Consider

the sum over

the symmetric

One can easily check that for

The complete graph over 3 vertices and its 7 forests.

Loop vertices and a tree on them, or cactus.

A particular variant of this formula (

the sum over

To distinguish these two formulas we call

Scale analysis is the key to renormalization, and scales can be conveniently defined in quantum field theory through the parametric representation of the propagator

Finally notice that various extensions of these formulas should be useful to study further the theory. A quite general theory of such formulas is given in [

Borel summability of a power series

analyticity in a disk tangent to the imaginary axis at the origin lying on the right-hand side of that disk, hence defined by a condition

plus remainder estimates uniform in that disk:

Given any power series

Therefore Borel summability is a perfect substitute for ordinary analyticity when a function is expanded at a point on the frontier of its analyticity domain. Borel summability is just a rigidity which plays the same role than analyticity: it selects a

Very early, both functional integrals and the Feynman perturbative series were introduced to study quantum field theory, but it was realized that the corresponding power series were generically divergent. When a link between both approaches can be established, it is usually through Borel summability or a variant thereof. This is why Borel summability is important for quantum field theory.

In this section we propose to test the evolution of ideas in constructive theory on the simple example of a single-variable ordinary integral which represents the

The normalization or partition function of that theory is the ordinary integral:

We shall review how different methods of increasing sophistication answer this question:

composition of series (XIXth century),

a la Feynman (1950),

“Classical Constructive,” à la Glimm-Jaffe-Spencer (1970s–2000s)

with loop vertices (2007),

with tree vector space (2008).

The first remark is that we know the explicit power series for

To summarize, this method leads to an explicit formula for

Feynman understood that

The theory of combinatoric species is a rich mathematical orchestration of this intuition; see [

In our case the drawings are the Wick contractions of

Usually in the quantum field theory literature there are painful discussions on what is a Feynman graph and what is its combinatoric weight or “symmetry factor.” This is important to make the shortest possible list of independent Feynman amplitudes that one has to compute in practice. But conceptually it is much better to consider a graph as a set of “Wick contractions,” that is a set of pairing of fields, so that no “symmetry factors” are ever discussed. (Fields correspond to half-lines, also called flags in the mathematics literature and the physicists point of view that flags, not lines, are the fundamental elements in graph theory is slowly making its way in the mathematics literature [

Borel summability remains unclear. But as a first fruit of the idea of Feynman graphs clearly, we now see explicitly that

The standard method in Bosonic constructive field theory is to first break up the functional integrals over a discretization of space-time and then test the couplings between the corresponding functional integrals (cluster expansion), which results in the theory being written as a polymer gas with hardcore constraints. For that gas to be dilute at small coupling, the normalization of the free functional integrals must be factored out. Finally the connected functions are computed by expanding away the hardcore constraint through a so-called Mayer expansion [

In the zero-dimensional case there is no need to discretize the single point of space-time; hence it seems that the first step, namely, the cluster expansion is trivial. This is correct except for the fact that what remains from this step is to factorize the “free functional” integral; so a single first-order Taylor expansion with remainder around

The first step (cluster expansion) is therefore

Convergence is now easy because each

Borel summability for

A shortcoming is that “space-time” and functional integrals remains present. Also the cluster step, suitably generalized in non zero dimension by Glimm, Jaffe, Spencer, and followers, heavily relies on locality, hence does not seem to have the potential to work on nonstandard space-times or for nonlocal or matrix-like theories like the Grosse-Wulkenhaar model of noncommutative theory [

The intermediate field representation is a well-known trick to represent a quartic interaction, in terms of a cubic one:

We can introduce again replicas but in a slightly different way. We duplicate the intermediate field into copies,

This does not change the expectation value of any polynomial; hence by the Weierstrass theorem it does not change the theory. But one can now apply the forest formula to the off-diagonal couplings

The main advantage is that the role of propagators and vertices has been exchanged! The result is a sum over trees on loops, or

Since

Once

Convergence is easy because

Borel summability is easy.

This method extends to noncommutative field theory and gives correct estimates for matrix-like models.

The drawback is that the method is more difficult when the interaction is of higher degree, for example,

This last method no longer requires functional integral at all! It is in a way the closest to Feynman graphs, hence looks at first sight like a step backwards in constructive theory. It starts exactly like constructive Fermionic theory.

Within a given quantum field model, the forest formula indeed associates a natural amplitude

In our zero-dimensional case it means that we start with the usual

Then we introduce replicas again but on the ordinary vertices:

Applying the tree formula to that covariance gives in the same vein than before

The zero-dimensional

It seems that little has been achieved at the constructive level by rewriting Feynman graphs simply in terms of an underlying tree, like in Fermionic theories. But there is a hidden

It has indeed be shown in [

This operation induces a natural (semidefinite) scalar product and a natural

For the

The constructive expression for the connected two-point function

This approach does not require any functional integral, since

Also notice that the expressions of the functions

The authors thank A. Abdesselam and P. Leroux for the organization of a very stimulating workshop on combinatorics and physics, during which B. Faris asked for such a pedagogical note. They also thank the IHES where these ideas were presented as part of a course at the invitation of A. Connes. They also thank Jacques Magnen for a lifelong collaboration on the topic of constructive theory, only very slightly touched upon in this pedagogical note. Finally they thank Matteo Smerlak for his critical reading of this manuscript.