Time-Dependent and/or Nonlocal Representations of Hilbert Spaces in Quantum Theory

TL;DR

This paper reviews recent innovations in quantum theory involving multiple Hilbert space formulations, especially non-Hermitian models with real spectra, and explores their nonlocal and dynamic representations.

Contribution

It introduces a broadened four-Hilbert-space formulation and discusses non-Hermitian Hamiltonians with real spectra and their nonlocal representations in quantum theory.

Findings

Four-Hilbert-space formulation extends the three-Hilbert-space approach.

Non-Hermitian Hamiltonians can have real spectra and physical relevance.

Nonlocal and moving-frame representations of Hilbert spaces are feasible.

Abstract

A few recent innovations of applicability of standard textbook Quantum Theory are reviewed. The three-Hilbert-space formulation of the theory (known from the interacting boson models in nuclear physics) is discussed in its slightly broadened four-Hilbert-space update. Among applications involving several new scattering and bound-state problems the central role is played by the models using apparently non-Hermitian ("crypto-Hermitian") Hamiltonians with real spectra. The formalism (originally inspired by the topical need of mathematically consistent description of tobogganic quantum models) is shown to admit certain unusual nonlocal and/or "moving-frame" representations of the standard physical Hilbert space of wave functions.