The Existence and Stability of Noncommutative Scalar Solitons

TL;DR

This paper proves the existence and stability of noncommutative scalar solitons in even-dimensional spaces, showing their dependence on the noncommutativity parameter and characterizing stability conditions.

Contribution

It establishes the existence, smooth dependence on the noncommutativity parameter, and stability criteria of scalar solitons in noncommutative field theories, extending previous results to higher dimensions.

Findings

Existence of rotationally invariant solitons for large noncommutativity parameter

Stability of certain solitons depending on spectral projections

Non-existence of smoothly dependent solitons at small noncommutativity

Abstract

We establish existence and stabilty results for solitons in noncommutative scalar field theories in even space dimension $2d$. In particular, for any finite rank spectral projection $P$ of the number operator ${\mathcal N}$ of the $d$-dimensional harmonic oscillator and sufficiently large noncommutativity parameter $\theta$ we prove the existence of a rotationally invariant soliton which depends smoothly on $\theta$ and converges to a multiple of $P$ as $\theta\to\infty$. In the two-dimensional case we prove that these solitons are stable at large $\theta$, if $P=P_N$, where $P_N$ projects onto the space spanned by the $N+1$ lowest eigenstates of ${\mathcal N}$, and otherwise they are unstable. We also discuss the generalisation of the stability results to higher dimensions. In particular, we prove stability of the soliton corresponding to $P=P_0$ for all $\theta$ in its domain of existence. Finally, for arbitrary $d$ and small values of $\theta$, we prove without assuming rotational invariance that there do not exist any solitons depending smoothly on $\theta$.