As a preparation for its quantization in the loop formalism, the two-dimensional gravitation model of Jackiw and Teitelboim is analysed in the classical canonical formalism. The dynamics is of pure constraints as is well known. A partial gauge fixing of the temporal type being performed, the resulting second class constraints are sorted out and the corresponding Dirac bracket algebra is worked out. Dirac observables of this classical theory are then calculated.

We consider a scalar field model with a gφ 4 4 interaction and compute the mass correction at next-to-leading order in a large-N expansion to study the summability of the perturbative series. It is already known that at zero temperature this model has a singularity in the Borel plane (a "renormalon"). We find that a small increase in temperature adds two countable sets both with an infinite number of renormalons. For one of the sets the position of the poles is thermal independent and the residue is thermal dependent. In the other one both the position of poles and the residues are thermal dependent. If we consider the model at extremely high temperatures, however, one observes that all the renormalons disappear and the model becomes Borel summable.R 3 × S 1 [20] and the CP N −1 non-linear sigma model on arXiv:1804.10708v2 [hep-th]

We investigate the process of dimensional reduction of one spatial dimension in a thermal scalar field model defined in D dimensions (inverse temperature and D − 1 spatial dimensions). We obtain that a thermal model in D dimensions with one of the spatial dimensions having a finite size L is related to the finite temperature model with just D − 1 spatial dimensions and no finite size. Our results are obtained for one-loop calculations and for any dimension D. For example, in D = 4 we have a relationship between a thin film with thickness L at finite temperature and a surface at finite temperature. We show that, although a strict dimensional reduction is not allowed, it is possible to define a valid prescription for this procedure. * erich@cbpf.br †

In the present paper, we intent to enlarge the axiomatic framework of non-commutative quantum field theories (QFT). We consider QFT on non-commutative spacetimes in terms of the tempered ultrahyperfunctions of Sebastião e Silva corresponding to a convex cone, within the framework formulated by Wightman. Tempered ultrahyperfunctions are representable by means of holomorphic functions. As is well known there are certain advantages to be gained from the representation of distributions in terms of holomorphic functions. In particular, for non-commutative theories the Wightman functions involving the ⋆-product, W ⋆ m , have the same form as the standard form Wm. We conjecture that the functions W ⋆ m satisfy a set of properties which actually will characterize a non-commutative QFT in terms of tempered ultrahyperfunctions. In order to support this conjecture, we prove for this setting the validity of some important theorems, of which the CPT theorem and the theorem on the Spin-Statistics connection are the best known. We assume the validity of these theorems for non-commutative QFT in the case of spatial non-commutativity only.Dedicated to Prof. Olivier Piguet on the occasion of his 65th birthday. Date: August 30, 2018. 1991 Mathematics Subject Classification. 46F15, 46F20, 81T05.Key words and phrases. Non-commutative theory, axiomatic field theory, tempered ultrahyperfunctions. J.A. Lourenço is supported by the Brazilian agency CNPq.

In the present paper, we intend to enlarge the axiomatic framework of non-commutative quantum field theories (QFT). We consider QFT on non-commutative spacetimes in terms of the tempered ultrahyperfunctions of Sebastião e Silva corresponding to a convex cone, within the framework formulated by Wightman. Tempered ultrahyperfunctions are representable by means of holomorphic functions. As is well known there are certain advantages to be gained from the representation of distributions in terms of holomorphic functions. In particular, for non-commutative theories the Wightman functions involving the ⋆-product, , have the same form as the standard form . We conjecture that the functions satisfy a set of properties which actually will characterize a non-commutative QFT in terms of tempered ultrahyperfunctions. In order to support this conjecture, we prove for this setting the validity of some important theorems, of which the CPT theorem and the theorem on the spin-statistics connection are the best known. We assume the validity of these theorems for non-commutative QFT in the case of spatial non-commutativity only.