General $O(D)$-equivariant fuzzy hyperspheres via confining potentials and energy cutoffs

Abstract

We summarize our recent construction of new fuzzy hyperspheres $S_{\Lambda }^d$ of arbitrary dimension $d$ covariant under the {\it full} orthogonal group $O(D)$, $D=d+1$. We impose a suitable energy cutoff on a quantum particle in ${\mathbb{R}}^D$ subject to a confining potential well $V(r)$ with a very sharp minimum on the sphere of radius $r=1$; the cutoff and the depth of the well diverge with $\Lambda \in \mathbb{N}$. Consequently, the commutators of the Cartesian coordinates ${\overline{x}}^i$ are proportional to the angular momentum components $L_{ij}$, as in Snyder's noncommutative spaces. The ${\overline{x}}^i$ generate the whole algebra of observables ${\mathcal{A}}_{\Lambda }$ and thus the whole Hilbert space ${\mathcal{H}}_{\Lambda }$ when applied to any state. ${\mathcal{H}}_{\Lambda }$ carries a reducible representation of $O(D)$ isomorphic to the space of harmonic homogeneous polynomials of degree $\Lambda$ in the Cartesian coordinates of (commutative) ${\mathbb{R}}^{D+1}$; the latter carries an irreducible representation ${\pi }_{\Lambda }$ of $O(D+1)\supset O(D)$. Moreover, ${\mathcal{A}}_{\Lambda }$ is isomorphic to ${\pi }_{\Lambda }\left(Uso(D+1)\right)$. We identify the subspace ${\mathcal{C}}_{\Lambda }\subset {\mathcal{A}}_{\Lambda }$ spanned by fuzzy spherical harmonics. We interpret $\{{\mathcal{H}}_{\Lambda }{\}}_{\Lambda \in \mathbb{N}}$, $\{{\mathcal{C}}_{\Lambda }{\}}_{\Lambda \in \mathbb{N}}$ as fuzzy deformations of the space of square integrable functions and the space of continuous functions on $S^d$ respectively, $\{{\mathcal{A}}_{\Lambda }{\}}_{\Lambda \in \mathbb{N}}$ as fuzzy deformation of the associated algebra of observables. $\{{\mathcal{A}}_{\Lambda }{\}}_{\Lambda \in \mathbb{N}}$ yields a fuzzy quantization of a coadjoint orbit of $O(D+1)$ that goes to the classical phase space $T^{\ast }{S}^d$. These models might be useful in quantum field theory, quantum gravity or condensed matter physics.

Latex file, 15 pages. Proceedings of Science Volume 436 - Corfu Summer Institute 2022 "School and Workshops on Elementary Particle Physics and Gravity", more precisely the "Workshop on Noncommutative and generalized geometry in string theory, gauge theory and related physical models", 18-25 September, 2022, Corfu, Greece. arXiv admin note: substantial text overlap with arXiv:2211.13284

References in corpus (1)

• Fuzzy hyperspheres via confining potentials and energy cutoffs