We investigate the relationship between universal topological polynomials for graphs in mathematics and parametric representation of Feynman amplitudes quantum field theory. In this first article we consider translation invariant theories with usual heat-kernel-based propagator. show how Symanzik theory are particular multivariate versions Tutte polynomial, new noncommutative special Bollobás–Riordan polynomials.

In this paper we extend the method of loop vertex expansion to interactions with degree higher than 4. As an example provide through explicit proof that free energy Phi^2k scalar theory in zero dimension is Borel-Le Roy summable order k-1. We detail computations case a Phi^6 interaction.

The purpose of this short letter is to clarify which set pieces Feynman graphs are resummed in a Loop Vertex Expansion, and formulate conjecture on the $ϕ^4$ theory non-integer dimension.

This paper is an extended erratum to Rivasseau and Wang [J. Math. Phys. 53, 042302 (2012); e-print arXiv:1104.3443 [math-ph]], in which the classic construction Borel summability of ϕ24 Euclidean quantum field theory was revisited combining a multi-scale analysis with constructive method called Loop Vertex Expansion (LVE). Unfortunately we discovered important error [math-ph]]. We explain mistake, provide new, correct according LVE.

In this paper, we construct the two-dimensional Euclidean ϕ4 quantum field theory using method of loop vertex expansion. We reproduce results standard constructive theory, for example, Borel summability Schwinger functions in coupling constant. Our should be also suitable future construction Grosse-Wulkenhaar models on non-commutative space-time.

We present an oscillator realization of discrete series representations group SU(2,2). give formulas for the coherent state star-product quantization a Bergman domain $D$. A formulation (regularized) non-commutative scalar field on quantized $D$ is given.

In the Luttinger model with non-local interaction we investigate, by exact analytical methods, time evolution of an inhomogeneous state a localized fermion added to non interacting ground state. absence averaged density has two peaks moving in opposite directions constant velocities. If is evolved Hamiltonian main effects appear. The first that have velocities which are not but vary between minimal and maximal value. second dynamical `Landau quasi-particle weight' appears oscillating part...

This paper is an extended erratum to J. Math. Phys.53, 042302 (2012) and arXiv:1104.3443, in which the classic construction Borel summability of phi^4_2 Euclidean quantum field theory was revisited combining a multi-scale analysis with constructive method called Loop Vertex Expansion (LVE). Unfortunately we discovered important error (2012). We explain mistake, provide new, correct according LVE.

In this paper we construct the noncommutative Grosse-Wulkenhaar model on 2-dimensional Moyal plane with method of loop vertex expansion. We treat renormalization new tool, adapt Nelson's argument and prove Borel summability perturbation series. This is first non-commutative quantum field theory to be built in a non-perturbative sense.

We describe the construction of noncommutative complex ball whose commutative analog is Hermitian symmetric space D = SU(m, 1)/U(m), with method coherent state quantization. In limit, we obtain standard manifold. also consider a quantum field theory model on

In this talk we briefly report the recent work on construction of 2-dimensional Grosse-Wulkenhaar model with method loop vertex expansion. We treat renormalization new tool, adapt Nelson's argument and prove Borel summability perturbation series. This is first non-commutative quantum field theory to be built in a non-perturbative sense.

We investigate, in the Luttinger model with fixed box potential, time evolution of an inhomogeneous state prepared as a localized fermion added to noninteracting ground state. proved that, if is evolved interacting Hamiltonian, averaged density has two peaks moving opposite directions, constant but renormalized velocity. also that dynamical `Landau quasi-particle weight' appears oscillating part density, asymptotically vanishing large time. The results are Mattis-Lieb diagonalization method....