What is the current status (December 2023) of the quantization of Einstein-Cartan Theory?

Question

Einstein-Cartan theory at a classical level is able to "smooth out" some of the singularities in general relativity. Since the presence of singularities is one of the most vexing parts of the theory of quantum gravity, I am curious if the Einstein-Cartan theory is more easily amenable to quantization, although I assume some obstruction must arise otherwise such a formalism would be much more popular today. Thus: what is the current status (December 2023) of efforts to quantize Einstein-Cartan theory?

Answer 1

One approach, that is not available to any theory that constrains the connection to be the Levi-Civita connection, is to quantize the connection, but keep the metric classical. Effectively, that makes a quantized gauge theory on a classical background, where the gauge group is either Lorentz, or an extension of it. A brief search reveals Drechsler's Quantized De Sitter (i.e. SO(4,1)) Gauge Theory With Classical Metric And Axial Torsion as one example that falls within this broad class.

This is a source that discusses Coupling Of Quantum Fields To Classical Gravity, in general, though I don't think it's been adapted to handle the case of mixed classico-quantum gravity field of the kind that I just described. In principle, alone, it is possible to consistently pair classical gravity with quantum systems in a way that dodges the known No Go theorems for classico-quantum hybridization ... that's been established by Oppenheim's framework A Post-Quantum Theory Of Classical Gravity?, but it looks like it has only Einstein-Hilbert in its scope, or Palatini, if you're in a Riemann-Cartan geometry. It can probably be adapted to Einstein-Cartan on a Riemann-Cartan geometry, but its treating both connection and metric as classical. I don't know if it can be generalized and adapted to work with a quantized connection and classical metric.

If you have a non-zero cosmological coefficient, then you can remake the Einstein-Cartan action as a Yang-Mills action for either $SO(3,2)$ or $SO(4,1)$, depending on the sign of the cosmological coefficient. I think positive coefficient goes with $SO(4,1)$ - which might fall in line with the very first link cited above. Here's the trick that does it: Yang-Mills'ing Up Gravity.

For zero cosmological coefficient, there is Carmeli's SL(2,C) Gauge Gravity. I forgot what the details of it were (particularly, if the metric - or whatever passes for a metric in it - is classical or not), but its Lagrangian is quadratic like Yang-Mills too. The reason it didn't take off as "The Actual Solution" is because he couldn't pair it with matter. It can only be used to quantize pure gravity.