Singularity Cancellation in Fermion Loops through Ward Identities

Recently Neumayr and Metzner have shown that the connected N-point density- correlation functions of the two-dimensional and the one-dimensional Fermi gas at one-loop order generically vanish/are regular in the small momentum/small energy-momentum limits. Their result is based on an explicit analysis in the sequel of results of Feldman et al.[2]. In this note we use Ward identities to give a proof of the same fact - in a considerably shortened and simplified way - for any dimension of space.Comment: 11 pages, 2nd corrected and improved version, to appear in Ann. Henri Poincar

Constructive ϕ4\phi^4 field theory without tears

We propose to treat the $\phi^4$ Euclidean theory constructively in a simpler way. Our method, based on a new kind of "loop vertex expansion", no longer requires the painful intermediate tool of cluster and Mayer expansions.Comment: 22 pages, 10 figure

Bosonic Monocluster Expansion

We compute connected Green's functions of a Bosonic field theory with cutoffs by means of a ``minimal'' expansion which in a single move, interpolating a generalized propagator, performs the usual tasks of the cluster and Mayer expansion. In this way it allows a direct construction of the infinite volume or thermodynamic limit and it brings constructive Bosonic expansions closer to constructive Fermionic expansions and to perturbation theory.Comment: 30 pages, 1 figur

Constructive Matrix Theory

We extend the technique of constructive expansions to compute the connected functions of matrix models in a uniform way as the size of the matrix increases. This provides the main missing ingredient for a non-perturbative construction of the $\phi^{\star 4}_4$ field theory on the Moyal four dimensional space.Comment: 12 pages, 3 figure

EPRL/FK Group Field Theory

The purpose of this short note is to clarify the Group Field Theory vertex and propagators corresponding to the EPRL/FK spin foam models and to detail the subtraction of leading divergences of the model.Comment: 20 pages, 2 figure

Bosonic Colored Group Field Theory

Bosonic colored group field theory is considered. Focusing first on dimension four, namely the colored Ooguri group field model, the main properties of Feynman graphs are studied. This leads to a theorem on optimal perturbative bounds of Feynman amplitudes in the "ultraspin" (large spin) limit. The results are generalized in any dimension. Finally integrating out two colors we write a new representation which could be useful for the constructive analysis of this type of models

Constructive Field Theory and Applications: Perspectives and Open Problems

In this paper we review many interesting open problems in mathematical physics which may be attacked with the help of tools from constructive field theory. They could give work for future mathematical physicists trained with the constructive methods well within the 21st century

A Rigorous Proof of Fermi Liquid Behavior for Jellium Two-Dimensional Interacting Fermions

Using the method of continuous constructive renormalization group around the Fermi surface, it is proved that a jellium two-dimensional interacting system of Fermions at low temperature $T$ remains analytic in the coupling constant $\lambda$ for $|\lambda| |\log T| \le K$ where $K$ is some numerical constant and $T$ is the temperature. Furthermore in that range of parameters, the first and second derivatives of the self-energy remain bounded, a behavior which is that of Fermi liquids and in particular excludes Luttinger liquid behavior. Our results prove also that in dimension two any transition temperature must be non-perturbative in the coupling constant, a result expected on physical grounds. The proof exploits the specific momentum conservation rules in two dimensions.Comment: 4 pages, no figure