Minisuperspace model of quantum geometrodynamics in the Madelung-Bohm formalism

TL;DR

This paper draws an analogy between quantum mechanics and quantum cosmology using the Madelung-Bohm formalism, deriving equations for cosmic evolution influenced by quantum effects in simplified models.

Contribution

It introduces a hydrodynamic approach to quantum geometrodynamics, calculating the quantum potential and explicit scale factor dependences in minisuperspace models.

Findings

Quantum potential affects cosmic fluid properties.

Explicit scale factor solutions with quantum corrections.

Quantum effects modify cosmological evolution equations.

Abstract

An analogy between non-relativistic quantum mechanics in the Madelung formulation and quantum geometrodynamics in the case of the maximally symmetric space is drawn. The equations equivalent to the continuity equation and the hydrodynamic Euler equation describing the evolution of the velocity introduced for the case of hypothetical fluid flow characterizing the cosmological system are obtained. It is shown that the perfect nature of the fluid is broken by the quantum Bohm potential. The quantum potential is calculated in semi-classical approximation for different forces acting in the system both in standard quantum mechanics and in minisuperspace model of quantum geometrodynamics. The explicit dependences of the cosmic scale factor on the conformal time, which take into account the quantum additive, are found for empty space with spatial curvature and for a spatially flat universe with dust and radiation.