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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?

Qmechanic
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Panopticon
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  • I doubt that Einstein-Cartan theory is any easier to quantize, as the problems of defining the metric at plank length scales likely resurfaces there also. I have heard that Einstein-Cartan theory is largely a mathematical curiosity due to the "torsional" characteristics of the covariant derivative on such manifolds. I can't wait to see what the experts here have to say. – Albertus Magnus Dec 20 '23 at 23:40
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    "Since the presence of singularities is one of the most vexing parts of the theory of quantum gravity" is it? Standard GR is simply non-renormalizable in its naive quantization, which has nothing to do with singularities. Since Einstein-Cartan has a similar action (which therefore by power-counting is similarly non-renormalizable), why would the "status" of its quantization be any different from that of standard GR? – ACuriousMind Dec 20 '23 at 23:41
  • @ACuriousMind The sort of thing that makes me wonder about the details of this sort of thing beyond power counting is that the behavior of black holes is often used as a "deeper" justification for why building a theory of QG is hard. – Panopticon Dec 21 '23 at 01:54
  • @ACuriousMind For instance the "pedagogical" argument for the nonrenormalizability of quantum gravity in https://arxiv.org/abs/0709.3555 argues that the UV fixed point of QG cannot be a CFT (and thus QG cannot be a renormalizable field theory in the traditional sense) because a theory with a high-energy spectrum dominated by black holes with the standard Bekenstein-Hawking entropy produces entropies inconsistent with the entropy of a CFT. – Panopticon Dec 21 '23 at 01:55
  • @ACuriousMind But it seems like the entropy computation of a black hole in EC gravity might not be as straightforward as in standard semiclassical gravity, since in EC the theory suggests that on the other side of the event horizon a new expanding universe may be formed. Such a thing is by no means a proper argument that EC has any advantage over standard GR when it comes to quantization but is enough to make me curious about the overall status of the quantum theory. – Panopticon Dec 21 '23 at 02:00
  • The paper you link has never been published in a peer-reviewed journal and its argument makes no sense to me (being the opposite of "pedagogical"!) - the "entropy" argument for some reason connects a hand-waving dimensional analysis for the entropy of a finite temperature QFT to the classical (in the sense of $g$ being definite and not a quantum field, this is at least semi-classical) Bekenstein-Hawking entropy of a black hole, but it is not obvious to me at all that these two kinds of entropies are commensurable. – ACuriousMind Dec 21 '23 at 02:23
  • Related: https://physics.stackexchange.com/q/387/2451 and links therein. – Qmechanic Dec 21 '23 at 03:21
  • If a black hole has entropy, that is a signal that there are hidden degrees of freedom inside, so GR is usually thought as an effective theory – Pato Galmarini Dec 21 '23 at 03:26
  • Aren't EC and Hehl's PGT superseded by SuGra? – DanielC Dec 21 '23 at 08:56

1 Answers1

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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 it's 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.

NinjaDarth
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