History and analogy

There's a lot to talk about. But in this post I just want to mention two perspectives. 

The first is Geometric Unity's place in the spectrum of approaches to quantum gravity. I propose a fourfold way of thinking about it: that Geometric Unity is to Loop Quantum Gravity, as string theory is to general relativity. I like this as a way of thinking about the options in gravitational theory: metric-based versus connection-based approaches to relativistic gravity and the theory of everything. 

The second is to think of Weinstein versus Witten as similar to Schopenhauer versus Hegel. Hegel was the best-known philosopher in Europe, Schopenhauer was a young unknown, yet he dared to schedule his lectures for the same timeslot as Hegel. No-one came and he left academia, and it took thirty years for him to be noticed again; but in the end his work made its mark, not least by helping to create Nietzsche. 

A paradigm for GU's quantum theory

I have struggled to figure out how quantum GU works. By this, I don't mean the various technical challenges that hang over GU, but the more basic question of which parts of GU are "classical" and which parts are "quantum". 

Over the weekend, I was a guest on a GU-themed livestream by Youtuber "Uncompetative", and I tried to explain this problem. Now I think I have an approach. I'm not saying this is the only way to make GU quantum-mechanical, but it is something concrete enough that one can try to make it work. 

The field equations for GU are 14-dimensional, they describe fields living in a 14-dimensional manifold "Y". There is also a 4-dimensional manifold "X", on which we can choose a metric. The first step to connecting X and Y, is that Y is also the metric bundle of X, meaning that to choose a metric on X is to choose a 4-dimensional section of Y. 

So if we have a 14-dimensional solution to the GU field equations, this gives us a way to obtain something 4-dimensional: choose a metric, i.e. a 4-dimensional section of Y, and then consider the restriction of the 14D solution to that 4D submanifold. In effect, this tells us what all the non-metric fields on X (gauge fields, matter fields) are doing. 

The problem with this prescription is that the choice of metric is disconnected from the evolution of the non-metric fields. Everything non-metrical is figured out by dynamics on Y, and then you make an arbitrary choice of metric, to get physics on X. This is unlike general relativity, in which metric and non-metric are in constant mutual interaction. 

But I have omitted part of GU: the "Zorro construction"! 

When I first studied GU, I thought (in effect) that perhaps there was no metric on Y, and that the 14D theory was a topological theory. This is wrong. Also, there is actually a canonical metric for the whole of Y, arising from its interpretation as metric bundle of X. Theoretically, one could study the GU field equations on that background, which is the same no matter what the metric on X is. 

However, in GU, you're also not supposed to use the canonical metric on Y. Instead, you choose a metric for X, corresponding already to a particular section of Y. Then you will extrapolate the Levi-Civita connection on X to a Levi-Civita connection on Y, and from that construct the metric on Y. 

This establishes mutual interaction between X and Y. X determines the metric on Y, which is the background on which the GU field equations must be solved, and then Y determines via pullback, what the non-metric fields are doing on X. 

Now if we're trying to recover general relativity, one has to hope that the energy-momentum distribution of these 4D non-metrical fields, is consistent (via Einstein field equations) with the 4D metric we chose at the start. At present I don't see why that would be so. Maybe some of the GU cohomological complexes are meant to enforce such a relationship. But at least I now see how to "close the loop" and make the choice of metric relevant for the behavior of the non-metrical fields. 

As for the quantum theory, maybe it's still not fully fleshed out. There are still choices to be made. For example, you could treat X as completely classical, and have all the quantum uncertainty on Y. Or, you could have a wavefunction on X as well as on Y. What remains to be checked, is whether any version of this, gives us the desired consistency between metric and non-metric ingredients.