Classical and quantum parts in Madelung variables: Splitting the source term of the Einstein equation into classical and quantum parts

Tamás Bı́ró ,P Ván

doi:10.1051/epjconf/20147802003

Abstract

Postulating a particular quantum correction to the source term in the classi- cal Einstein equation we identify the conformal content of the above action and obtain classical gravitation for massive particles, but with a cosmological term representing off- mass-shell contribution to the energy-momentum tensor.

Highlights

In this short paper we report an interesting observation based on a conformal treatment of the Schrödinger equation: as if the quantum mechanical problem of obtaining wave functions could be splitted to a massive and a conformal part in line with a classical – quantum partition
Identifying the quantum part as belonging to a traceless relativistic energy-momentum tensor, we suggest to modify its Bohm-Takabayashi form [1] and connect the remaining classical part to Einstein’s gravity equation in form of a dust matter source of massive point particles moving on Bohm trajectories
In a metric view this dispersion relation is: gμνuμuν = 1 +. This formula, multiplied by R2/2 can be rewritten as a full divergence of the Bohm-Takabayashi energy-momentum tensor

Summary

Introduction

In this short paper we report an interesting observation based on a conformal treatment of the Schrödinger equation: as if the quantum mechanical problem of obtaining wave functions could be splitted to a massive and a conformal part in line with a classical – quantum partition. Identifying the quantum part as belonging to a traceless relativistic energy-momentum tensor, we suggest to modify its Bohm-Takabayashi form [1] and connect the remaining classical part to Einstein’s gravity equation in form of a dust matter source of massive point particles moving on Bohm trajectories. The details of the outlined calculations are given in [2]

Energy-momentum tensors of the Klein-Gordon equation

General relativistic quantum mechanics