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

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.

29 pages, slightly revised version, accepted for publication in Class. Quantum Grav

References in corpus (2)

• Pseudo-Hermiticity and Generalized PT- and CPT-Symmetries
• Pseudo-Hermiticity for a Class of Nondiagonalizable Hamiltonians

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