Boundaries in the Moyal plane

Abstract

We study the oscillations of a scalar field on a noncommutative disc implementing the boundary as the limit case of an interaction with an appropriately chosen confining background. The space of quantum fluctuations of the field is finite dimensional and displays the rotational and parity symmetry of the disc. We perform a numerical evaluation of the (finite) Casimir energy and obtain similar results as for the fuzzy sphere and torus.

Thermodynamics in the NC disc

“…In this section we rederive the spectrum of the NC disc following the lines of [9], where the formal aspects are considered in more detail. Independently of the statistics, we will study massive spinless particles, which we describe by a real scalar field φ(t, x) on Moyal plane, whose coordinates we denote by x = (x 1 , x 2 ); we consider Minkowski time t as an ordinary commuting parameter.…”

“…In this article we define the NC disc as an appropriate limit of a confining background in Moyal plane and, in this way, avoid the ambiguities inherent to the implementation of boundary conditions. Previous work on the NC disc has been performed in [9] and other NC manifolds with boundaries have been studied in [10,11,12,13,14,15,16].…”

“…In [9] quantum fields have been confined in a disc of radius R as the result of a limiting process in which the fields interact with a rotationally invariant background that, in the limit, becomes infinite at distances larger than R. The spectrum of oneloop oscillations has been explicitly determined and some peculiar properties have been exposed.…”

Formalismo de línea de mundo en teorías no conmutativas

“…El resultado principal, contenido en el trabajo publicado en conjunto con Bonezzi et al (2012), es una fórmula maestra para el desarrollo, a tiempo propio pequeño, del NdC de operadores con potenciales no locales. Además, tomamos diversos modelos no conmutativos para ejemplificar las peculiaridades que en general ofrecen; entre ellos, el modelo del disco no conmutativo, estudiado junto a Falomir, Franchino Viñas et al (2013).…”

“…No reviste importancia el conocer la forma precisa de la relación de recurrencia; intuitivamente se puede comprender de (V.B.2) que para Λ → ∞ las soluciones son posibles sólo si los argumentos de las funciones Heaviside son negativos o lo que es lo mismo, recordando cómo actúan N ± sobre las funciones φ ±l n (x), si l < N y las componentes c l n (λ) se anulan para n ≥ N −l. El resultado exacto es (Falomir, Franchino Viñas et al 2013)…”