Is the local Lorentz invariance of general relativity implemented by gauge bosons that have their own Yang-Mills-like action?

TL;DR

This paper proposes that local Lorentz invariance in general relativity might be implemented by independent gauge bosons with a Yang-Mills-like action, potentially contributing to dark matter and violating equivalence principles.

Contribution

It introduces the idea that Lorentz invariance could be gauged by independent bosons with their own Yang-Mills-like dynamics, distinct from the traditional spin connection.

Findings

Lorentz bosons couple to fermions and generate a spin-dependent potential.

Such potentials violate the weak equivalence principle.

Experimental bounds suggest Lorentz bosons could be stable and form dark matter.

Abstract

General relativity with fermions has two independent symmetries: general coordinate invariance and local Lorentz invariance. General coordinate invariance is implemented by the Levi-Civita connection and by Cartan's tetrads both of which have as their action the Einstein-Hilbert action. It is suggested here that local Lorentz invariance is implemented not by a combination of the Levi-Civita connection and Cartan's tetrads known as the spin connection, but by independent Lorentz bosons L_i that gauge the Lorentz group, that couple to fermions like Yang-Mills fields, and that have their own Yang-Mills-like action. A nonsingular 4 x 4 hermitian scalar field h is needed to make the action of the Lorentz bosons invariant under local Lorentz transformations. Lorentz bosons couple to fermion number and generate a spin-dependent static potential that violates the weak equivalence principle. If a Higgs mechanism makes them massive, then the static potential also violates the inverse-square law. Experiments put upper bounds on the strength of such a potential for masses m_L < 20 eV. These upper limits imply that Lorentz bosons, if they exist, are nearly stable and contribute to dark matter.