[Paper Review] Spontaneously Broken Erlangen Program Offers a Bridge Between the Einstein and the Yang-Mills Theories

Yi Yang, Wai Bong Yeung|arXiv (Cornell University)|May 11, 2012

Particle physics theoretical and experimental studies1 references3 citations

TL;DR

This paper proposes a unified framework for gravity and gauge theories by gauging the local affine group GL(4,R) as a Yang-Mills symmetry, with spontaneous symmetry breaking to local Lorentz invariance. The resulting theory describes classical gravity via 16 spin-1 vector bosons, deriving the Schwarzschild metric as a solution, and offers a geometric bridge between Einstein’s general relativity and Yang-Mills gauge theory without introducing spin-2 gravitons.

ABSTRACT

We shall give dynamics to our spacetime manifold by first identifying the local affine symmetry as the characterizing symmetry for our geometry á la Felix Klein, and then by prescribing 16 gauge vector bosons to this symmetry á la Yang and Mills. The locally affine symmetric Yang-Mills Lagrangian in the presence of a background world metric, and the corresponding equations of motion, are respectively constructed and derived. Spontaneous breaking of the local affine symmetry to the local Lorentz symmetry is achieved by classical solutions to the equations of motion. In these classical solutions, the 16 gauge vector bosons are shown to select the Schwarzschild metric as one among the admissible background world metrics. Classical gravity is expressed by a spontaneously broken Erlangen program.

Motivation & Objective

• To unify Einstein’s general relativity and Yang-Mills gauge theory through a geometric framework based on the Erlangen Program.
• To address the fundamental incompatibility between gravitational and gauge field theories by assigning dynamics to spacetime geometry via local affine symmetry.
• To show that classical gravity emerges as a spontaneously broken gauge theory of the affine group GL(4,R), with the Schwarzschild metric as a solution.
• To demonstrate that the observed spacetime metric is not fundamental but arises as a classical solution to the Yang-Mills equations of the affine gauge group.
• To propose a theory of vector gravity where 16 spin-1 gauge bosons mediate gravity, avoiding the need for spin-2 gravitons.

Proposed method

• Identify the local affine group GL(4,R) as the symmetry underlying spacetime geometry, motivated by the laws of inertia and causality.
• Construct a Yang-Mills Lagrangian for 16 gauge vector bosons associated with the affine group, using a non-dynamical background world metric for measuring distances.
• Introduce vierbein fields to relate the world metric to a local Minkowskian frame, ensuring compatibility with local Lorentz invariance.
• Impose the compatibility ansatz to ensure that the Levi-Civita connection and metric remain invariant under local geometric redefinitions.
• Derive the equations of motion from the Yang-Mills Lagrangian and show that classical solutions spontaneously break GL(4,R) to the local Lorentz group.
• Demonstrate that the Schwarzschild metric is selected as a solution due to the dynamics of the 16 gauge bosons, with no need for a fundamental metric.

Experimental results

Research questions

• RQ1 Can a spontaneously broken gauge theory of the affine group GL(4,R) reproduce classical gravity without introducing spin-2 gravitons?
• RQ2 How does the choice of local affine symmetry as the underlying geometric structure unify Einstein’s spacetime geometry with Yang-Mills gauge theory?
• RQ3 Why does the Schwarzschild metric emerge as a solution in this framework, and what role do the 16 gauge vector bosons play in selecting it?
• RQ4 What is the role of the non-dynamical background world metric in this theory, and how does it differ from the dynamical metric in general relativity?
• RQ5 Can a theory based on GL(4,R) gauge symmetry avoid the pathologies of higher-derivative gravity while still describing known gravitational phenomena?

Key findings

• The theory constructs a Yang-Mills Lagrangian for 16 gauge vector bosons associated with the local affine group GL(4,R), with no dynamical terms for the world metric.
• Spontaneous symmetry breaking of GL(4,R) to the local Lorentz group occurs via classical solutions, leaving the remaining symmetry unbroken.
• The Schwarzschild metric is derived as a classical solution to the equations of motion, selected by the dynamics of the 16 gauge bosons.
• The resulting theory describes gravity via 16 spin-1 vector bosons, eliminating the need for spin-2 gravitons.
• The metric and Levi-Civita connection remain invariant under local geometric redefinitions due to the compatibility ansatz, preserving physical observables.
• Restricting the gauge group to the Lorentz subgroup yields a standard Yang-Mills theory with 6 gauge bosons, but at the cost of reintroducing second-derivative terms in the Riemann tensor and potential higher-derivative pathologies.