Quantum mechanics using two auxiliary inner products

TL;DR

This paper explores the mathematical structure of non-Hermitian ${ m PT}$-symmetric Hamiltonians in quantum physics, introducing a second auxiliary inner product to deepen understanding of unitary evolution in such systems.

Contribution

It provides a new mathematical interpretation of ${ m PCT}$-symmetry constraints using two auxiliary inner products, enhancing the theoretical framework of non-Hermitian quantum mechanics.

Findings

Introduces a second auxiliary inner product for ${ m PT}$-symmetric Hamiltonians.

Provides a deeper mathematical understanding of ${ m PCT}$-symmetry.

Clarifies the role of auxiliary inner products in unitary evolution.

Abstract

The current applications of non-Hermitian but ${\cal PT}-$symmetric Hamiltonians $H$ cover several, mutually not too closely connected subdomains of quantum physics. Mathematically, the split between the open and closed systems can be characterized by the respective triviality and non-triviality of an auxiliary inner-product metric $\Theta=\Theta(H)$. With our attention restricted to the latter, mathematically more interesting unitary-evolution case we show that the intuitive but technically decisive simplification of the theory achieved via an "additional" ${\cal PCT}-$symmetry constraint upon $H$ can be given a deeper mathematical meaning via introduction of a certain second auxiliary inner product.