Quantum Field Theory on a Discrete Space and Noncommutative Geometry

Abstract

We analyse in detail the quantization of a simple noncommutative model of spontaneous symmetry breaking in zero dimensions taking into account the noncommutative setting seriously. The connection to the counting argument of Feynman diagrams of the corresponding theory in four dimensions is worked out explicitly. Special emphasis is put on the motivation as well as the presentation of some well-known basic notions of quantum field theory which in the zero-dimensional theory can be studied without being spoiled by technical complications due to the absence of divergencies.

Quantizing Yang–Mills theory on a two-point space

“…Note added: After finishing our paper a related article [18] appeared. One of its main purposes is to analyse in depth the counting argument of Feynamn diagrams in the presence of spontaneous symmetry breaking.…”

Non-Perturbative Asymptotic Improvement of Perturbation Theory and Mellin-Barnes Representation

“…It is worth insisting that our final expressions (hyperasymptotic expansions) are initially obtained following a general and simple non-rigourous approach which is completely justified at a later stage in the paper. Zero-dimensional φ 4 field theory has already been used many times to explain new theoretical approaches (see for instance [5,6] and, more recently, [7]) and we will see that it leads here to non-trivial and interesting issues. We would like to add that although zero-dimensional φ 4 field theory cannot be considered, strictly speaking, as a realistic toy model in the context of pure particle physics since, for instance, it cannot mimic some of the pathologies of the Standard Model perturbation theory, it is however very likely that the formalism we describe here, by its generality, can be of use in (other fields of) high energy physics (see our conclusions) or in the study of the resummation of higher order corrections in quantum mechanical models and/or superconvergent quantum field theories that are considered in condensed matter physics (see e.g.…”

“…It is important to note that (2.15) is an exact equality 6 only for c + m < k (since in our case m and k are integers, and since min(k) = n and c < 0, this is equivalent to m ≤ n) 7 . 6 One may in fact also prove that it is an asymptotic equality [10]. 7 If one does not impose this constraint, i.e.…”

Moduli spaces of discrete gravity

“…In the complementary project explained in [19,20] the mathematical structure of a pertubative treatment of a measure dλ 1 · · · dλ n e −S is analyzed in great depth by exploiting the noncommutativity of the underlying manifold.…”

“…| cos ρ e iϕ + sin ρ e i(ϕ+ψ 1 ) | 2 = 1 + 2 sin ρ cos ρ cos ψ 1 and so (19) gives the condition sin ρ cos ρ cos ψ 1 = 0…”