Two distinctly different routes to a quantum theory of gravity

 Emergence in condensed matter physics may provide some valuable insights into the elusive search for a quantum theory of gravity. There was a helpful discussion by Bei Lok Hu  in Emergent/quantum gravity: macro/micro structures of spacetime

Hu makes a distinction between two approaches that he characterises as "bottom-up" and "top-down". Both have the common goal of understanding how space-time and Einstein's classical theory of gravity can emerge from some more "fundamental" theory that describes physics at higher energies and shorter distances, such as the Planck scale.

1. Going from the micro- to the macro-

Examples of this approach are string theory (a la Schwarz, Green, and Witten) and loop quantum gravity. The respective microscopic entities are strings and spin foam. This approach is motivated by the success of the standard model of elementary particles and gauge fields. One starts with a well-defined "classical" action inspired by symmetry (and broken symmetry) and uses quantum field theory to calculate observable properties. Generally, one is quantising the classical theory of gravity. Perhaps, we should not be surprised that this approach has not borne fruit as we know from condensed matter that deducing emergent (macro-)properties from microscopic theory is extremely hard.

This picture is taken from a recent Scientific American article

Hu also has the following valuable insight about whether quantising classical theory is the right approach.

[quantising the classical theory of spacetime] will not lead to a microscopic theory of spacetime. In the analogy of a crystal made of atoms quantizing the vibrational models yields phonons, not atoms. Finding the atomic structure of matter does not come from simply quantizing its collective degrees of freedom, but takes a very different path.

àMä?ÍaËdÌMä£ã􏰹î2Ê􏰅à­Ò?ÍEÊVáHu calls this approach "top-down" as it involves going from high energies down to low energies. I found this confusing as I tend to think of this approach means going up in distance, i.e. from the bottom structures (small distances) up to the top structures (long distances). 

2. Going from the macro- to the micro-

This approach is also ambitious. By considering the macroscopic theory (classical space-time and General Relativity) and the associated observed structures the goal is to deduce something about the microscopic theory, even without probes to investigate reality on very short distance scales. 

History suggests this is not completely fanciful. Consider for example the path to the belief that liquids and crystals were actually made of atoms. Einstein's theory of Brownian motion and Perrin's experiments were not at the atomic scale. People had deduced that crystals were made of arrays of atoms before the discovery of x-ray diffraction from crystals.

Space-time and the metric are viewed as collective variables, like order parameters in condensed matter.

Hu calls this approach "bottom-up", advocates it, and explores some possible ways to pursue it.

I thank Gerard Milburn for rekindling my interest in these issues.