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On a wave-particle in closed and open isotropic universes

### Abstract

The Klein–Gordon equation satisfied by the wave function in general relativity is solved for the metric of the closed and open universe corresponding to Einstein–De Sitter–Friedmann isotropic cosmological model. The angular dependences are specified by spherical harmonics for the longitude and latitude, and for the hyperlatitude by modified spherical harmonics having as variable circular functions for the closed universe and hyperbolic functions for the open universes. The time dependence of the probabilistic wave function is similar for the closed and open universes and is obtained in the following three cases: (I) constant Hubble parameter, (II) constant decceleration parameter, and (III) uniform matter and energy distribution, which corresponds to the Hubble parameter a linear function of time. Thus six solutions are obtained, namely, the three cases I–III each for closed and open isotropic universes. For each of these six solutions is considered: (i) the existence of singularities in space–time including asymptotic time in the future or past, (ii) the square integrability of the wave function over the full extent of the four-dimensional space–time, and (iii) the existence or otherwise of a positive probability density associated with the wave function.

© 2011 American Institute of Physics

Received 07 September 2009
Accepted 13 August 2010
Published online 10 January 2011

Acknowledgments:
The author would like to acknowledge the importance of some issues raised by the referee in connection with the first version of the present work.

Article outline:

I. INTRODUCTION
II. PROBABILISTIC WAVE FUNCTION IN CLOSED AND OPEN UNIVERSES
A. Free quantum particle in general relativity
B. Quantum cosmology for a closed isotropic universe
C. Comparison of closed and open quantum cosmologies
III. DEPENDENCE ON LONGITUDE, CO-LATITUDE AND HYPERLATITUDE
A. Spherical harmonics for longitude and co-latitude
B. Modified spherical harmonics for hyperlatitude in closed universe
C. Modified spherical harmonics of hyperbolic functions in open universe
IV. TIME DEPENDENCE OF THE WAVE FUNCTION
A. Temporal evolution of closed and open isotropic universe
B. Case of constant Hubble parameter for expansion or contraction
C. Constant deceleration parameter versus constant Hubble parameter
V. TIME DEPENDENCES FOR THREE ISOTROPIC COSMOLOGICAL MODELS
A. Mass density, pressure, and energy per unit volume
B. Hubble parameter a linear function of time
C. Wave function without singularities in finite space–time
VI. SQUARE-INTEGRABILITY OF THE OF WAVE FUNCTION OVER SPACE–TIME
A. Separation of the four space–time variables
B. Boundedness of integration over time of square amplitude
C. Hyperlatitude integral in the open universe
VII. POSITIVE PROBABILITIES AND FORWARD WAVES
A. Quadratic identities associated with the Klein–Gordon equation
B. Positive probability density for forward waves
C. Extension from asymptotic to finite time
VIII. DISCUSSION

/content/aip/journal/jmp/52/1/10.1063/1.3512995

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2011-01-10

2016-10-22

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