Geometric and Topological Obstructions to Hermitianization in Quasi-Hermitian Quantum Systems

TL;DR

This paper investigates the geometric and topological barriers to transforming quasi-Hermitian quantum systems into fully Hermitian systems, highlighting conditions for global Hermitianization.

Contribution

It identifies geometric and topological obstructions to global Hermitianization and provides explicit criteria and examples for these barriers.

Findings

Identifies curvature-related geometric obstructions to Hermitianization.

Discovers topological obstructions from non-trivial holonomies in parameter space.

Provides explicit criteria and concrete examples illustrating these obstructions.

Abstract

Quasi-Hermitian quantum systems, including $\mathcal{PT}$-symmetric ones, can be mapped to equivalent Hermitian systems via a similarity transformation that redefines the inner product with a positive-definite metric operator. Although an instantaneous algebraic Hermitianization can be obtained locally from a positive metric operator, a stronger requirement is needed for dynamical equivalence: the similarity transformation must be proper, globally single-valued, and compatible with the modified quasi-Hermitian Schrodinger equation. We identify two distinct obstructions: geometric obstructions arising from the curvature of a metric-induced connection, and topological obstructions originating from non-trivial holonomies around non-contractible loops in parameter space. We derive explicit criteria for these obstructions and illustrate them with concrete examples. Our results establish a geometric and topological foundation for the Hermitianization of quasi-Hermitian systems, clarifying when they can be globally reduced to Hermitian ones and when intrinsic non-Hermitian features persist.