Exact Semiclassical Evolutions in Relativistic and Nonrelativistic Scalar Quantum Mechanics and Quantum Cosmology

TL;DR

This paper derives conditions for exact semiclassical solutions in relativistic and nonrelativistic scalar quantum mechanics and quantum cosmology, constructing explicit solutions for specific models and analyzing their existence.

Contribution

It provides necessary and sufficient conditions for exact semiclassical solutions and constructs explicit solutions for Wheeler-DeWitt equations in cosmological models.

Findings

Exact semiclassical solutions exist for certain scalar potentials.

No right-going semiclassical solutions for polynomial potentials with p=1, 2, 3.

Develops a new semiclassical perturbation expansion different from traditional methods.

Abstract

The necessary and sufficient conditions for the exactness of the semiclassical approximation for the solution of the Schr\"odinger and Klein-Gordon equations are obtained. It is shown that the existence of an exact semiclassical solution of the Schr\"odinger equation determines both the semiclassical wave function and the interaction potential uniquely up to the choice of the boundary conditions. This result also holds for the Klein-Gordon equation. Its implications for the solution of the Wheeler-DeWitt equation for the FRW scalar field minisuperspace models are discussed. In particular, exact semiclassical solutions of the Wheeler-DeWitt equation for the case of massless scalar field and exponential matter potentials are constructed. The existence of exact semiclassical solutions for polynomial matter potentials of the form $\lambda\phi^{2p}$ is also analyzed. It is shown that for p=1, 2 and 3, right-going semiclassical solutions do not exist. A generalized semiclassical perturbation expansion is also developed which is quite different from the traditional $\hbar$ and $M_p^{-1}$-expansions.