Abstract
We overview a new mechanism whereby classical Riemannian geometry emerges out of the differential structure on quantum spacetime, as extension data for the classical algebra of differential forms. Outcomes for physics include a new formula for the standard Levi-Civita connection, a new point of view of the cosmological constant as a very small mass for the graviton of around $10^{-33}$ev, and a weakening of metric-compatibility in the presence of torsion. The same mechanism also provides a new construction for quantum bimodule connections on quantum spacetimes and a new approach to the quantum Ricci tensor.
Highlights
It has been argued and is widely accepted that somehow classical gravity should emerge from an as yet unknown theory of quantum gravity
More recently we found the emergence in some models of curvature as being forced by the algebraic constraints of non-commutative geometry not-directly visible in the classical limit[5]
What we show is that the second line on the right – the concept of differential structure – implies the third line on the left – classical Riemannian geometry
Summary
It has been argued and is widely accepted that somehow classical gravity should emerge from an as yet unknown theory of quantum gravity. This new way of thinking about Riemannian geometry as emerging out of differential algebra suggests an interpretation of the cosmological constant or ‘dark energy density’ as mass of the gravitational field This is an alternative point of view in massive gravity but which tends to lead there to physically unrealistic theories[7]. We find that a BV algebra equipped with a differential d and where the BRST operator is symmetric in a certain sense (which we will explain), more or less is a Riemannian manifold This seems to me a remarkable confluence of a structure in quantum field theory (for the quantisation of ghosts) with gravity. This new approach provides for the first time a somewhat general definition of the quantum Ricci tensor, see Section 4
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