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

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

2002-09-06

arxiv.org/abs/math-ph/0209014v3

Class.Quant.Grav. 20 (2003) 155-172

Physics

Mathematical Physics

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

Scientific paper

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.

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