Hilbert Space Structures on the Solution Space of Klein-Gordon Type Evolution Equations

TL;DR

This paper explores how to construct positive-definite inner products on solution spaces of Klein-Gordon type equations using pseudo-Hermitian operators, enabling a consistent quantum dynamics framework for various physical models.

Contribution

It introduces a method to classify and construct invariant inner products on solution spaces of Klein-Gordon equations using pseudo-Hermitian operator theory.

Findings

Established Hilbert space structures for simple harmonic oscillator solutions

Derived invariant inner products for free Klein-Gordon equations

Applied the framework to Wheeler-DeWitt equations in quantum cosmology

Abstract

We use the theory of pseudo-Hermitian operators to address the problem of the construction and classification of positive-definite invariant inner-products on the space of solutions of a Klein-Gordon type evolution equation. This involves dealing with the peculiarities of formulating a unitary quantum dynamics in a Hilbert space with a time-dependent inner product. We apply our general results to obtain possible Hilbert space structures on the solution space of the equation of motion for a classical simple harmonic oscillator, a free Klein-Gordon equation, and the Wheeler-DeWitt equation for the FRW-massive-real-scalar-field models.