[Paper Review] The unexpected resurgence of Weyl geometry in late 20-th century physics

Erhard Scholz|arXiv (Cornell University)|Mar 9, 2017

Quantum Mechanics and Applications128 references4 citations

TL;DR

This paper surveys the revival of Weyl geometry in late 20th-century physics, highlighting its resurgence in scalar-tensor gravity, quantum foundations, and cosmology. It demonstrates how Weyl's scale geometry—originally abandoned in 1920s—reemerged as a framework for unifying gravity, quantum mechanics, and elementary particle physics, offering potential solutions to the hierarchy problem and dark matter via scale symmetry at quantum and cosmological levels.

ABSTRACT

Weyl's original scale geometry of 1918 ("purely infinitesimal geometry") was withdrawn by its author from physical theorizing in the early 1920s. It had a comeback in the last third of the 20th century in different contexts: scalar tensor theories of gravity, foundations of gravity, foundations of quantum mechanics, elementary particle physics, and cosmology. It seems that Weyl geometry continues to offer an open research potential for the foundations of physics even after the turn to the new millennium.

Motivation & Objective

• To document and analyze the unexpected revival of Weyl geometry in theoretical physics from the 1970s onward, despite its initial rejection in the 1920s.
• To clarify the role of scale covariance and scalar fields in gravity, quantum mechanics, and particle physics through a historical and systematic survey.
• To examine how Weyl geometry contributed to alternative approaches in cosmology, including MOND-like phenomenology and dark matter candidates.
• To assess the potential of Weyl geometry for addressing foundational problems in physics, such as the hierarchy problem and the naturalness of the Higgs mass.
• To identify open research directions where Weyl geometry may offer new insights into unification and quantum gravity.

Proposed method

• Tracing the historical development of Weyl geometry from its 1918 formulation to its re-emergence in the 1970s and 1980s in distinct research communities.
• Analyzing key works by Ehlers-Pirani-Schild, Dirac, Omote/Utiyama, and Cartan-Weyl approaches to understand their use of scale connections and conformal structures.
• Surveying applications in scalar-tensor gravity, particularly in relation to Jordan-Brans-Dicke theory and its geometric interpretation.
• Examining proposals by Santamato and others to geometrize quantum mechanics using Weyl geometry, especially in the context of Bohmian mechanics.
• Investigating the role of Weyl scaling in the Standard Model, including mass generation and the Higgs mechanism, and its implications for quantum field theory.
• Reviewing cosmological models based on Weyl geometry, including the Brazilian approach and Palatini variations, to assess their viability and redshift interpretations.

Experimental results

Research questions

• RQ1 Why did Weyl geometry experience a resurgence in theoretical physics despite its initial rejection in the 1920s?
• RQ2 How do scale-covariant scalar fields in Weyl geometry relate to Jordan-Brans-Dicke gravity and what are their physical implications?
• RQ3 Can Weyl geometry provide a geometric foundation for quantum mechanics, particularly in the context of Bohmian mechanics?
• RQ4 What role does Weyl scaling play in the Standard Model, especially in mass generation and the Higgs mechanism?
• RQ5 Can Weyl geometric models offer alternative explanations for cosmological phenomena such as dark matter, MOND-like behavior, and the cosmological redshift?

Key findings

• Weyl geometry re-emerged independently in the 1970s through three distinct research lines: foundational gravity (EPS), scale-invariant gravity (Dirac/Omote/Utiyama), and Cartan-Weyl gauge gravity, all within a short span (1971–1974).
• The scalar field in Weyl geometry, particularly in Jordan-Brans-Dicke-type theories, plays a central role in extending gravitational dynamics and enabling scale covariance.
• Santamato’s proposal to geometrize quantum mechanics via Weyl geometry offers a potential bridge between quantum configurations and spacetime geometry, though it remains isolated from mainstream developments.
• Weyl scaling in the 1980s–1990s at Munich linked gravity and quantum physics through scale-invariant field theories, suggesting deeper geometric unification.
• Recent work on field quantization in Weyl geometry preserves scale symmetry at the quantum level, offering a potential resolution to the hierarchy problem via cancellation of quadratically divergent Higgs corrections.
• Weyl geometric cosmological models suggest that part of the cosmological redshift may stem from the scale connection rather than spacetime expansion, and some models exhibit MOND-like behavior under non-standard scalar field kinematics.