Symétrie conforme

Résumé

En physique théorique, la symétrie conforme désigne la symétrie sous changement d'échelle, on dit aussi sous dilatation, ainsi que sous les transformations conformes spéciales. Sa combinaison avec le groupe de Poincaré donne le groupe de symétrie conforme ou plus simplement, groupe conforme. Voici un exemple de représentation du groupe conforme dans l'espace-temps, ou plus précisément de son algèbre de Lie \begin{align} M_{\mu\nu}&\equiv -i(x_\mu\partial_\nu-x_\nu\partial_\mu)\ P_\mu&\equiv -i\partial_\mu \ D&\equiv -x_\mu\partial^\mu \ K_\mu&\equiv -{i\over2}(x^2\partial_\mu-2x_\mu x_\nu\partial^\nu)\ \end{align} où les M_{\mu\nu}, sont les générateurs associés au groupe de Lorentz, les P_\mu, génèrent les translations de l'espace-temps (les valeurs propres de ces derniers correspondant au quadrivecteur impulsion-énergie), D, engendre la transformation par dilatation et enfin les K_\mu, engendrent les trans

Source officielle

https://fr.wikipedia.org/wiki/Sym%C3%A9trie_conforme

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This thesis is devoted to the study of the local fields in the Ising model. The scaling limit of the critical Ising model is conjecturally described by Conformal Field Theory. The explicit predictions for the building blocks of the continuum theory (spin and energy density) have been rigorously established [HoSm13, CHI15]. We study how the field-theoretic description of these random fields extends beyond the critical regime of the model. Concretely, the thesis consists of two parts: The first part studies the behaviour of lattice local fields in the critical Ising model. A lattice local field is a function of a finite number of spins at microscopic distances from a given point. We study one-point functions of these fields (in particular, their asymptotics under scaling limit and conformal invariance). Our analysis, based on discrete complex analysis methods, results in explicit computations which are of interest in applications (e.g. [HKV17]). The second part considers the behaviour of the massive spin field. In the subcritical massive scaling limit regime first considered by Wu, McCoy, Tracy, and Barouch [WMTB76], we show that the correlations of the massive spin field in a bounded domain have a scaling limit. Furthermore, to this end we generalise the notions and methods of discrete complex analysis in the critical case to the massive regime, and give a new derivation of the formula for the two-point correlation in the full plane in terms of a Painlevé III transcendent.

EPFL2019

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Spontaneously broken conformal symmetry: dealing with the trace anomaly

Roberta Armillis, Alexander Monin

The majority of renormalizable field theories possessing the scale invariance at the classical level exhibits the trace anomaly once quantum corrections are taken into account. This leads to the breaking of scale and conformal invariance. At the same time any realistic theory must contain gravity and is thus non-renormalizable. We show that discarding the renormalizability it is possible to construct viable models allowing to preserve the scale invariance at the quantum level. We present explicit one-loop computations for two toy models to demonstrate the main idea of the approach. Constructing the renormalized energy momentum tensor we show that it is traceless, meaning that the conformal invariance is also preserved.

Springer2013

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Spontaneous conformal symmetry breaking in fishnet CFT

Georgios Karananas

Quantum field theories with exact but spontaneously broken conformal invariance have an intriguing feature: their vacuum energy (cosmological constant) is equal to zero. Up to now, the only known ultraviolet complete theories where conformal symmetry can be spontaneously broken were associated with supersymmetry (SUSY), with the most prominent example being the N=4 SUSY Yang-Mills. In this Letter we show that the recently proposed conformal "fishnet" theory supports at the classical level a rich set of flat directions (moduli) along which conformal symmetry is spontaneously broken. We demonstrate that, at least perturbatively, some of these vacua survive in the full quantum theory (in the planar limit, at the leading order of 1/N-c expansion) without any fine tuning. The vacuum energy is equal to zero along these flat directions, providing the first non-SUSY example of a four-dimensional quantum field theory with "natural" breaking of conformal symmetry. (C) 2020 The Authors. Published by Elsevier B.V.

ELSEVIER2020

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