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Hilbert space structures on the solution space of Klein–Gordon-type evolution equations

Ali Mostafazadeh

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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.


PACS

03.65.-w Quantum mechanics

98.80.Es Observational cosmology (including Hubble constant, distance scale, cosmological constant, early Universe, etc)

04.60.-m Quantum gravity

MSC

83F05 Cosmology

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other quantum-mechanical equations

83Cxx General relativity

47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)

47A70 (Generalized) eigenfunction expansions; rigged Hilbert spaces

Subjects

Gravitation and cosmology

Quantum information and quantum mechanics

Astrophysics and astroparticles

Dates

Issue 1 (7 January 2003)

Received 27 September 2002

Published 12 December 2002



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