Twisted Grosse-Wulkenhaar ϕ⋆4\phi^{\star 4} model: dynamical noncommutativity and Noether currents

This paper addresses the computation of Noether currrents for the renormalizable Grosse-Wulkenhaar (GW) $\phi^{\star 4}$ model subjected to a dynamical noncomutativity realized through a twisted Moyal product. The noncommutative (NC) energy-momentum tensor (EMT), angular momentum tensor (AMT) and the dilatation current (DC) are explicitly derived. The breaking of translation and rotation invariances has been avoided via a constraint equation

Non-perturbative renormalization for the neural network-QFT correspondence

Abstract In a recent work (Halverson et al 2021 Mach. Learn.: Sci. Technol. 2 035002), Halverson, Maiti and Stoner proposed a description of neural networks (NNs) in terms of a Wilsonian effective field theory. The infinite-width limit is mapped to a free field theory while finite N corrections are taken into account by interactions (non-Gaussian terms in the action). In this paper, we study two related aspects of this correspondence. First, we comment on the concepts of locality and power-counting in this context. Indeed, these usual space-time notions may not hold for NNs (since inputs can be arbitrary), however, the renormalization group (RG) provides natural notions of locality and scaling. Moreover, we comment on several subtleties, for example, that data components may not have a permutation symmetry: in that case, we argue that random tensor field theories could provide a natural generalization. Second, we improve the perturbative Wilsonian renormalization from Halverson et al (2021 Mach. Learn.: Sci. Technol. 2 035002) by providing an analysis in terms of the non-perturbative RG using the Wetterich-Morris equation. An important difference with usual non-perturbative RG analysis is that only the effective infrared 2-point function is known, which requires setting the problem with care. Our aim is to provide a useful formalism to investigate NNs behavior beyond the large-width limit (i.e. far from Gaussian limit) in a non-perturbative fashion. A major result of our analysis is that changing the standard deviation of the NN weight distribution can be interpreted as a renormalization flow in the space of networks. We focus on translations invariant kernels and provide preliminary numerical results.</jats:p

Renormalization group flow of coupled tensorial group field theories: Towards the Ising model on random lattices

International audienceWe introduce a new family of tensorial field theories by coupling different fields in a nontrivial way, with a view towards the investigation of the coupling between matter and gravity in the quantum regime. As a first step, we consider the simple case with two tensors of the same rank coupled together, with Dirac like a kinetic kernel. We focus especially on rank-3 tensors, which lead to a power counting just-renormalizable model, and interpret Feynman graphs as Ising configurations on random lattices. We investigate the renormalization group flow for this model, using two different and complementary tools for approximations, namely, the effective vertex expansion method and finite-dimensional vertex expansion for the flowing action. Due to the complicated structure of the resulting flow equations, we divided the work into two parts. In this first part, we only investigate the fundamental aspects on the construction of the model and the different ways to get tractable renormalization group equations, while their numerical analysis will be addressed in a companion paper

Noncommutative Dirac and Klein-Gordon oscillators in the background of cosmic string: spectrum and dynamics

From a study of an oscillator in a $4D$ NC spacetime, we establish the Hamilton equations of motion. The formers are solved to give the oscillator position and momentum coordinates. These coordinates are used to build a metric similar to that describing a cosmic string. On this basis, Dirac and Klein-Gordon oscillators are investigated. Their spectrum and dynamics are analysed giving rise to novel interesting properties