Searching for Gravity Without a Metric

TL;DR

This paper constructs an affine-invariant Dirac action using infinite-dimensional spinorial representations, proposing a new approach to spontaneous symmetry breaking from affine to Lorentz symmetry as a foundation for gravity.

Contribution

It introduces a systematic method to build $GL(d,\mathbb{R})$ invariant models with potential symmetry breaking, advancing the understanding of gravity without a metric.

Findings

Constructed an affine generalization of the Dirac action.

Developed a procedure for $GL(d,\mathbb{R})$ invariant interactions.

Discussed mechanisms for symmetry breaking to Lorentz invariance.

Abstract

Recently it has been explicitly shown how a theory with global $GL(d,\mathbb{R})$ coordinate (affine) invariance which is spontaneously broken down to its Lorentz subgroup will have as its Goldstone fields enough degrees of freedom to create a metric and a covariant derivative arXiv:1105.5848. Such a theory would constitute an effective theory of gravity. So far however, no explicit theory has been found which exhibits this symmetry breaking pattern, mainly due to the difficulty of even writing down a $GL(d,\mathbb{R})$ invariant actions in the absence of a metric. In this paper we explicitly construct an affine generalization of the Dirac action employing infinite dimensional spinorial representations of the group. This implies that it is built from an infinite number of spinor Lorentz multiplets. We introduce a systematic procedure for obtaining $GL(d,\mathbb{R})$ invariant interaction terms to obtain quite general interacting models. Such models have order operators whose expectation value can break affine symmetry to Poincar\'{e} symmetry. We discuss possible interactions and mechanisms for this symmetry breaking to occur, which would provide a dynamical explanation of the Lorentzian signature of spacetime.