How does LQG handle discreteness?

[Vorde]

Hi,

The explanation I have been given for why GR breaks down at the quantum level is that GR requires spacetime to be mathematically smooth, and quantum foam is 'un-smooth'. If this is correct, then is a reworking of GR in a mathematics that does not require 'smoothness' one possible solution to this problem? Or is this impossible/I have been mislead?

Thank you.

 

[tom.stoer]

There are attempts to formulate QG using discrete structures; the most prominent one ist LQG.

But the reasoning is slightly different: one knows that standard quantization techniques break down for GR b/c the theory is not perturbatively renormalizable. Whether this requires a new fundamental structure of spacetime (like in LQG), a unification of 'matter and geometry' (like in string theory) or nothing else but a 'slightly modified quantization technique' (like in asymptotic safety') is still a matter of debate.

It is correct that many people think that spacetime at the fundamental level shows some discreteness; but it is by no means clear how this discreteness shall be introduced or generated. In LQG it is used at a fundamental level (model building) whereas in other approaches it seems to be a kind of emergent phenomenon (in string theory nearly all approaches are based on a continuous structure - manifolds - and discreteness may emerge for physical entities like observables, spectra etc.).

Compare this to the standard approach for angular momentum in QM: the measurable quantities (spectra of observables) are discrete, nevertheless the basic variables (angles) are continuous. Therefore it is by no means clear if a discrete structure must already be present at the level of model building or if it is 'only' a result for certain quantities.