Quantization of bounded symplectic domains associated with compact Lie groups

TL;DR

The paper develops a quantization scheme for bounded phase spaces $D\times G$ with $G$ a compact Lie group and $D\subset \mathfrak{g}^*$ bounded, yielding finite-dimensional Hilbert spaces due to the finite symplectic volume. It combines a prequantization step with a projection/Toeplitz-type mechanism and a norm-based deformation framework, incorporating a Casimir-length filtration on $\mathcal{F}(G)$ and Schmüdgen-type semialgebraic domain quantization to handle boundaries. The approach proves that, for thick, quantizable domains, products and Poisson brackets converge to their classical counterparts on bulk states, while boundary effects become negligible in the classical limit. Physically, the compact momentum space leads to novel phenomena such as supertunneling and a maximal fermion density, offering a consistent setting for systems with both bounded momentum and position spaces and suggesting potential implications for high-density astrophysical contexts and noncommutative geometry.

Abstract

We present a systematic quantization scheme for bounded symplectic domains of the form $D \times G \subset T^\ast G$, where $D \subset \mathfrak{g}^\ast$ is a bounded region defined by algebraic inequalities and $G$ is a compact Lie group with Lie algebra $\mathfrak{g}$. The finiteness of the symplectic volume implies that quantization yields a finite-dimensional Hilbert space, with observables represented by Hermitian matrices, for which we provide an explicit realization. Boundary effects necessitate modifications of the standard von Neumann and Dirac conditions, which usually underlie the correspondence principle. Physically, the compact group $G$ plays the role of momentum space, while $\mathfrak{g}^\ast$ corresponds to the (noncommutative) position space of a particle. The assumption of compact momentum space has profound physical consequences, including the supertunneling phenomenon and the emergence of a maximal fermion density.

Figures (2)

• Figure 1: Left panel: projection of quantizable semialgebraic domains in $\mathfrak{su}(2)^\ast\simeq\mathbb{R}^3$ onto the plane $x_1x_3$, showing a spherical cap, a cone, and a shell. Right panel: eigenstates of these domains on the weight diagram of the right regular representation of $SU(2)$, where $j\in \frac{1}{2}\mathbb{N}_0$ and each weight occurs with multiplicity $2j+1$.
• Figure 2: Left panel: a spherical cap in $\mathfrak{su}(2)^\ast\simeq\mathbb{R}^3$. Right panel: the weight diagram of the right regular representation of $SU(2)$. The green and blue regions indicate the bulk and the boundary states, respectively.

Theorems & Definitions (22)

• proposition 1
• Remark 2.22
• Remark 2.23
• Remark 2.25
• Remark 2.26
• Remark 2.28
• proposition 2
• Example 2.30
• proposition 3
• Theorem 1: Schmüdgen