The paper deals with an extended classification of general scalar tensor theories in the background of Homogeneous and isotropic FLRW space time geometry. Noether symmetry analysis has been used to investigate the classical and quantum cosmological description of a non-minimally coupled scalar field. By this symmetry analysis, the symmetry vector and the conserved charge associated with the symmetry have been determined. Further, identifying the cyclic coordinate using Cartan one form is possible to have the classical solution. Subsequently, in quantum cosmology, a canonical quantization scheme has been described by formulating the Wheeler-DeWitt(WD) equation and it is possible to have the probability amplitude from the solution of the WD equation. Finally, the quantum trajectories have been determined by using the Hamilton-Jacobi approach with quantum potential.
Mondal et al. (Wed,) studied this question.
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