Topological description of pure invariant states of the Weyl $C^*$-algebra

TL;DR

We address the topology of pure invariant states of the Weyl $C^*$-algebra with finite degrees of freedom, focusing on translation-invariant and semi-regular states. By identifying TI and lattice-invariant state spaces with explicit geometric models—including the Bohr compactification $\mathfrak{b}(\mathbb{R}^d)$, the Brillouin zone $\mathbb{B}_\Gamma$, and Grassmann bundles—we obtain complete topological classifications for plane-wave, Bloch-wave, and Zak states. The results reveal path-connectedness and automorphic-equivalence properties (e.g., plane waves are all automorphic to the zero-momentum state $\omega_0$, while Zak states form a $2d$-torus parameter space). This work provides a rigorous, geometrically informed framework for topological phases in finite CCR systems and connects invariant pure states to explicit topological spaces. $

Abstract

In this work we study the topology of certain families of states of the Weyl $C^*$-algebra with finite degrees of freedom. We focus on families of pure states characterized by symmetries and a (semi-)regularity condition, and obtain precise topological descriptions through homeomorphisms with other explicit spaces. Of special importance are the families of pure, semi-regular states invariant under either continuous (plane-wave states) or discrete (Bloch-wave states) spatial translations, and the family of states invariant under discrete, mutually commuting spatial and momentum translations (Zak-wave states), all of which we completely characterize.

Theorems & Definitions (54)

• Remark 2.1
• Definition 2.2: Path-equivalence of states
• Definition 2.3: Automorphic equivalence of states
• Remark 3.1: Presentation by the symplectic structure
• Remark 3.2: Schrödinger representation
• Theorem 3.3
• Proposition 3.4
• Proposition 3.5
• Definition 3.6: Regular and semi-regular states
• Remark 3.7: Group algebra structure