Dimensionless physics

TL;DR

This paper explores two models of emergent gravity where the metric has a dimension of inverse length squared, leading to dimensionless physical quantities and revealing topological effects like quantum Hall phenomena and anomalies.

Contribution

It introduces the concept of dimensionful metrics in emergent gravity scenarios, highlighting their implications for diffeomorphism invariance and topological operators.

Findings

Metrics in emergent gravity have dimension of inverse length squared.

Physical quantities become dimensionless due to this metric dimension.

Topological operators emerge, affecting quantum Hall effects and anomalies.

Abstract

We discuss two scenarios of emergent gravity. In one of them the quantum vacuum is considered as superplastic crystal, and the effective gravity describes the dynamical elastic deformations of this crystal. In the other one the gravitational tetrads emerge as the bilinear form of the fermionic fields. In spite of the essentially different mechanisms of emergent gravity, these two scenarios have one important common property: the metric field has dimension of the inverse square of length $[g_{\mu\nu}]=1/[l]^2$, as distinct from the conventional dimensionless metric, $[g_{\mu\nu}]=1$, in general relativity. As a result the physical quantities, which obey diffeomorphism invariance, become dimensionless. This takes place for such quantities as Newton constant, the scalar curvature, the cosmological constant, particle masses, fermionic and scalar bosonic fields, etc. This may suggest that the dimensionless physics can be the natural consequence of the diffeomorphism invariance, and thus can be the general property of any gravity, which emerges in the quantum vacuum. One of the nontrivial consequences of the shift of dimensions is related to topology. Due to the shift of dimensions some operators become topological, and contain the integer or fractional prefactors in the action. This in particular concerns the intrinsic 3+1 quantum Hall effect and the Nieh-Yan quantum anomaly in terms of torsion.