Tales from the prehistory of Quantum Gravity. L\'eon Rosenfeld's earliest contribution

TL;DR

This paper examines Le9on Rosenfeld's early 1927 work on quantum gravity, highlighting his approach to unify quantum mechanics and general relativity through a five-dimensional framework and semi-classical approximations.

Contribution

It analyzes Rosenfeld's pioneering 1927 attempt to integrate quantum mechanics with general relativity using a semi-classical approach and a five-dimensional unification framework.

Findings

Rosenfeld introduced a wave equation on a curved background.

He integrated Einstein equations with wave functions as sources.

Obtained the Reissner-Nordstrf6m metric in a semi-classical approximation.

Abstract

The main purpose of this paper is to analyse the earliest work of L\'eon Rosenfeld, one of the pioneers in the search of Quantum Gravity, the supposed theory unifying quantum theory and general relativity. We describe how and why Rosenfeld tried to face this problem in 1927, analysing the role of his mentors: Oskar Klein, Louis de Broglie and Th\'eophile De Donder. Rosenfeld asked himself how quantum mechanics should \textit{concretely} modify general relativity. In the context of a five-dimensional theory, Rosenfeld tried to construct a unifying framework for the gravitational and electromagnetic interaction and wave mechanics. Using a sort of "general relativistic quantum mechanics" Rosenfeld introduced a wave equation on a curved background. He investigated the metric created by what he called `quantum phenomena', represented by wave functions. Rosenfeld integrated Einstein equations in the weak field limit, with wave functions as source of the gravitational field. The author performed a sort of semi-classical approximation obtaining at the first order the Reissner-Nordstr\"om metric. We analyse how Rosenfeld's work is part of the history of Quantum Mechanics, because in his investigation Rosenfeld was guided by Bohr's correspondence principle. Finally we briefly discuss how his contribution is connected with the task of finding out which metric can be generated by a quantum field, a problem that quantum field theory on curved backgrounds will start to address 35 years later.