Review of Barnett's "Locality and Exceptional Points in Pseudo-Hermitian Physics" (2023)

Jacob L. Barnett's thesis explores the theoretical framework of pseudo-Hermitian physics by examining locality, exceptional points, and the symmetries and perturbative features of non-Hermitian operators. The study extends traditional quantum mechanics by including quasi-Hermitian operators maintaining real-valued measurement outcomes and unitary time evolution.

Barnett's work addresses the modification of the inner product assumption in standard quantum frameworks. In quasi-Hermitian theory, this assumption is altered through a metric operator with nontrivial Schmidt rank, affecting the structure of local observable algebras and expectation values. The thesis delves into exceptional points, unique to non-Hermitian operators, identifying a novel correspondence between higher-order exceptional points and cusp singularities in algebraic curves. Thus contributing to the mathematical understanding of these phenomena.

Additionally, it investigates one-dimensional lattice models with non-Hermitian defect potentials, exploring their spectral properties and phase transitions. For defects at nearest neighbours, the entire spectrum becomes complex beyond a threshold identified as a second-order exceptional point. Furthermore, Barnett utilises chiral symmetry and representation theory to construct pseudo-Hermitian operators with closed-form intertwining operators, which is crucial for understanding these systems' algebraic structure and symmetries.

The thesis has significant implications for theoretical physics, particularly in understanding systems with non-Hermitian dynamics, such as open quantum systems and optical systems exhibiting gain and loss. By providing a detailed mathematical framework and exploring the perturbative features of pseudo-Hermitian matrices, Barnett's research opens new avenues for both theoretical exploration and practical applications in quantum mechanics and related fields. Consequently, his work extends the boundaries of quantum mechanics, offering valuable insights into the nature of locality, the role of exceptional points, and the algebraic structure of non-Hermitian systems.

Reference reading: https://arxiv.org/pdf/2306.04044