Weyl geometry

Weyl geometry. Weyl geometry is a generalization of Riemannian geometry where the length of a vector is not preserved under parallel transport, introducing a concept of scale or gauge. Proposed by the mathematician Hermann Weyl in 1918, it was an ambitious attempt to unify gravitation with electromagnetism by geometrizing the electromagnetic field. Although initially rejected as a physical theory, its core ideas profoundly influenced the development of gauge theory and modern theoretical physics, finding renewed interest in contexts like scale-invariant gravity and quantum gravity.

Introduction

The foundational insight emerged from Hermann Weyl's critique of the inherently local nature of standards in Albert Einstein's general relativity. In Riemannian geometry, the angle between vectors remains fixed during parallel transport along a curve, but their absolute lengths are compared only at the same point. Weyl proposed a more flexible geometry where the length of a vector could change during transport, governed by an additional differential form field. This introduced a local scale or gauge symmetry, allowing the comparison of magnitudes at different points only after a choice of gauge. His seminal paper, presented to the Prussian Academy of Sciences, argued this new geometric object could be identified with the electromagnetic potential, offering a unified field theory. The initial reception, particularly from Einstein and Paul Ehrenfest, was critical, as the theory seemed to predict observable effects like spectral line broadening that contradicted experimental evidence from atomic clocks.

Mathematical formulation

A Weyl manifold is defined by the triple \((M, g, \varphi)\), where \(M\) is a differentiable manifold, \(g\) is a metric tensor, and \(\varphi\) is a differential 1-form called the Weyl connection 1-form. The fundamental law governing the change of length \(l\) of a vector under parallel transport along a curve with tangent vector \(v\) is given by \(\nabla_v l = \varphi(v) l\). This implies the covariant derivative of the metric is non-zero: \(\nabla g = 2 \varphi \otimes g\). The curvature of the Weyl connection decomposes into two parts: the Riemann curvature tensor of the metric and a contribution from the field strength \(F = d\varphi\) of the Weyl form. When \(F\) vanishes identically, the geometry is called integrable, and the connection is locally a Levi-Civita connection of a conformally related metric. Key mathematical investigations were later advanced by figures like Élie Cartan and Shiing-Shen Chern, placing it within the broader framework of Cartan geometry and fiber bundle theory.

Physical interpretations

Weyl's original physical interpretation identified the 1-form \(\varphi\) with the electromagnetic four-potential \(A_\mu\), with its exterior derivative \(F\) corresponding to the electromagnetic field tensor. This promised a purely geometric origin for Maxwell's equations. The theory's apparent failure, highlighted by Einstein's objection concerning non-integrability of length, led to its abandonment as a classical unified theory. However, the principle of local gauge invariance was resurrected and transformed in quantum mechanics by Chen Ning Yang and Robert Mills, leading to Yang–Mills theory. In modern physics, Weyl geometry resurfaces in scale-invariant theories of gravity, such as those explored by Pascual Jordan and later in Brans–Dicke theory. It also provides a geometric framework for discussing conformal transformations in quantum field theory and investigations into the early universe within cosmology.

Relation to other geometries

Weyl geometry is a direct extension of Riemannian geometry, reducing to it when the Weyl 1-form is a pure gauge or zero. It is intimately connected to conformal geometry, as the theory is invariant under local scale transformations of the metric \(g \rightarrow e^{2\lambda} g\) accompanied by a shift in the Weyl form \(\varphi \rightarrow \varphi + d\lambda\). This places it within the study of conformal manifolds. Its mathematical structure is a special case of a Cartan connection, specifically a \((CO(n), \mathbb{R}^n)\)-geometry. Furthermore, it is a precursor to the modern theory of gauge connections on principal bundles, where the Weyl form becomes the connection on a \(\mathbb{R}^+\)-bundle. In the context of supergravity and string theory, certain dilatonic couplings can be interpreted through a Weyl-geometric lens.

Historical development and significance

Introduced in 1918 in Weyl's book *Raum, Zeit, Materie*, the theory was a bold step beyond Einstein's work. The famous critique from Einstein, published in the Sitzungsberichte der Preussischen Akademie der Wissenschaften, centered on the "clock paradox," which seemed to violate the consistency of atomic spectra, a point Weyl conceded. Despite this, the conceptual framework of local gauge symmetry was its lasting legacy. The mathematician Hermann Weyl himself revisited the idea in the 1920s, connecting it to the emerging quantum theory and the work of Erwin Schrödinger. The geometry was later rigorously formalized within differential geometry. Its significance was fully recognized decades later when Chen Ning Yang noted that the key idea of gauge invariance originated with Weyl's work. Today, it remains an active area of research in alternative theories of gravity, studies of the Planck scale, and the geometry of spacetime. Category:Differential geometry Category:Theoretical physics Category:Gauge theories