The changing faces of the Problem of Space in the work of Hermann Weyl

TL;DR

This paper explores Hermann Weyl's evolving perspectives on the nature of space, focusing on affine connections and the symmetry groups that define physical space, highlighting his contributions across different periods.

Contribution

It analyzes Weyl's changing views on affine connections and the role of symmetry groups in characterizing space, emphasizing his defense of torsion-free connections and their significance.

Findings

Weyl emphasized the unique determination of affine connections in his early work.

He defended the importance of torsion-free affine connections despite Cartan's general framework.

Posthumously, Cartan's approach to torsion gained renewed interest in gravity theories.

Abstract

During his life Weyl approached the problem of space (PoS) from various sides. Two aspects stand out as permanent features of his different approaches: the {\em unique determination of an affine connection} (i.e., without torsion in the terminology of Cartan) and the question {\em which type of group} characterizes physical space. The first feature came up in 1919 (commentaries to Riemann's inaugural lecture) and played a crucial role in Weyl's work on the PoS in the early 1920s. He defended the central role of affine connections even in the light of Cartan's more general framework of connections with torsion. In later years, after the rise of the Dirac field, it could have become problematic, but Weyl saw the challenge posed to Einstein gravity by spin coupling primarily in the possibility to allow for non-metric affine connections. Only after Weyl's death Cartan's approach to infinitesimal homogeneity and torsion became revitalized in gravity theories.