# Torsion in GR and QG from fermions?

I know that GR is essentially a Riemann theory w/o Torsion; the Levi-Cevita-Connection is symmetric and therefore the torsion vanishes.

What happens when fermions are included?
Does the spin-connection still guarantue that the covariant derivative comes with a torsion-free connection?

What happens when this theory is quantized?
I've never seen a QG approach talking about torsion (except for LQG where a topological term involving torsion appears; but at the same time they claim that this term does not modify the classical equations of motions)

=> Is there a torsion operator in LQG?

Gold Member
Dearly Missed
Here's a relevant Wiki quote:
"LQG is a quantization of a classical Lagrangian field theory which is equivalent to the usual Einstein-Cartan theory in that it leads to the same equations of motion describing general relativity with torsion. As such, it can be argued that LQG respects local Lorentz invariance. Global Lorentz invariance is broken in LQG just as in general relativity. A positive cosmological constant can be realized in LQG by replacing the Lorentz group with the corresponding quantum group."

http://en.wikipedia.org/wiki/Loop_quantum_gravity

By itself, Wikipedia can't be taken as authoritative but in this case my impression is that this agrees with other sources, and it's handy. They also have an short article on Einstein-Cartan.

I don't trust Wikipedia when it comes to those subtleties.

What is "general relativity with torsion"? In GR you do not have torsion at all. The metric connection (Levi-Cevita-Symbols) leads to a torsion-free theory.

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Dearly Missed
Tom, I have never heard of a torsion operator in LQG. That's hardly conclusive though.

You asked about fermions. A natural question is how are they represented in LQG? According to Rovelli's book (section 7.2.2) basic LQG extends in a simple way to include fermions.
Here's a link to the online pdf draft copy:
http://www.cpt.univ-mrs.fr/~rovelli/book.pdf
Section 7.2.2 is on page 207.

Section 7.2, about including matter of several kinds, starts on page 206, and is quite brief. Maybe you can get something out of it.

I'll take a look around for something on torsion in LQG. I vaguely recall a paper by Laurent Freidel from a couple of years back, on that topic.
It just took a moment and I came up with a couple of things. I'm not suggesting you should read these or that the answers are here just keeping you abreast of what I found.
http://arXiv.org/abs/gr-qc/0505081
Physical effects of the Immirzi parameter
Alejandro Perez, Carlo Rovelli
"The Immirzi parameter is a constant appearing in the general relativity action used as a starting point for the loop quantization of gravity. The parameter is commonly believed not to show up in the equations of motion, because it appears in front of a term in the action that vanishes on shell. We show that in the presence of fermions, instead, the Immirzi term in the action does not vanish on shell, and the Immirzi parameter does appear in the equations of motion. It determines the coupling constant of a four-fermion interaction. Therefore the Immirzi parameter leads to effects that are observable in principle, even independently from nonperturbative quantum gravity."

http://arxiv.org/abs/hep-th/0507253
Quantum Gravity, Torsion, Parity Violation and all that
Laurent Freidel, Djordje Minic, Tatsu Takeuchi
"We discuss the issue of parity violation in quantum gravity. In particular, we study the coupling of fermionic degrees of freedom in the presence of torsion and the physical meaning of the Immirzi parameter from the viewpoint of effective field theory. We derive the low-energy effective lagrangian which turns out to involve two parameters, one measuring the non-minimal coupling of fermions in the presence of torsion, the other being the Immirzi parameter..."

Again I stress this is just preliminary search. I'd like to find something more recent since LQG has taken on a new look since 2006. Now seems to be just an adjunct to the type of spinfoam QG that came along in 2007. I'd like to find something about fermions in the spinfoam context.

BTW something in the Rovelli-Perez paper's abstract reminded me of what you said in your original post. Your intuition may have been in the right direction. Sorry I can't be more definite.

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"Does the spin-connection still guarantue that the covariant derivative comes with a torsion-free connection?"

In order to form a spin connection, you have to work in the Vielbien formulation of GR. There is a certain freedom in doing this. The answer to your question is both, depending on what you choose to include or not in your theory.

The minimal spin connection is free of torsion, but typically you might be interested in additional fields (coming from the matter sector for instance) in which case the spin connection is enlarged. The degree by which it is enlarged is parametrized by the torsion tensor, which of course will only depend on the spin connection and not the christoffel symbols. For instance in supergravity you might be required to have certain tensor fields other than the metric tensor, and these will act like a torsion tensor from the GR point of view.

In general absent some motivation or physical criteria (like a symmetry principle) for including a torsion tensor for some physical or phenomenological purpose there is nothing fundamental that requires its inclusion.

Torsion can be absent even if I work in the vielbein formalism. So if torsion is to be included due to physical reasons the usual spin-connection has to be enlarged. I will study the papers just to make sure that I got the point right: "presence of fermions may force the presence of torsion."

Let me add on paper: " A possible topological interpretation of the Barbero-Immirzi parameter" from Simone Mercuri. http://arxiv.org/abs/0903.2270v2

There's no focus on torsion in the connection; he uses torsion in the topological part of the action. This action is a generalization of the Holst action Rovelli uses in the paper mentioned above. That's were I first wondered how and why torsion comes in. This new action with the Nieh-Yan density replacing the Holst part shows that LQG and QCD have one common aspect, namely a topological superselection sector resultiung in the presence of the theta-angle and the Immirizi-parameter, respectively.

Thanks for the support; things are much clearer now!

Now I remember that I studied a paper regarding Einstein-Cartan theory several years ago. Thanks again!

Finbar
Hi
I have a related question. If i want to do QG with an action that includes fermions can i still use the metric formulism? I'm assuming not because i would need to define a spin connection that can only be defined in Vielbien formulation of GR. In that case how can one define the metric as the gravitational field? Surely the vielbien has to be the field if fermions are included?

Does this make sense? Or can one still define the gravitational field as the metric and include fermions in the action?

The metric is just the "square" of the vierbein with tangent indices contracted. The spin connection can be solved for as a function of the vierbein.

You should check
http://arxiv.org/abs/hep-th/0507253
Quantum Gravity, Torsion, Parity Violation and all that
Laurent Freidel, Djordje Minic, Tatsu Takeuchi
for details.

In the meantime the loopy guys seem to agree that the most general action is the Einstein-Hilbert action + fermion coupling (the fermions couple to the vierbeins via the gamma matrices defined in tangent space) + the Nieh-Yan piece. The latter one provides an interpretation of the Barbero-Immirzi parameter similar to the theta angle in QCD.

It's not clear to me in all details but at a first glance it seems to be quite natural.

Finbar
Just read in "Field theory: a modern primer, Pierre Ramond" that gravity "..coupled to a spinor yields a non-zero value for the torsion." Essentially when you derieve GR as a gauge theory you have 16 vierbiens and 24 spin connections and you need to vary the action with respect to both to to obtain your eq of motion. In the absence of matter or with minimally coupled scalars your equation of motion for the connections implies that the torsion is zero and that the connection can be completely determined by the vierbien (in other words the connections are just auxiliary fields).