Hehl on gauge aspects of spacetime - beyond Riemann

[tom.stoer]

http://arxiv.org/abs/1204.3672

Gauge Theory of Gravity and Spacetime

Friedrich W. Hehl (U Cologne and U of Missouri, Columbia)
(Submitted on 17 Apr 2012)
The advent of general relativity settled it once and for all that a theory of spacetime is inextricably linked to the theory of gravity. From the point of view of the gauge principle of Weyl and Yang-Mills-Utiyama, it became manifest around the 1960s (Sciama--Kibble) that gravity is closely related to the Poincare group acting in Minkowski space. The gauging of this external group induces a Riemann-Cartan geometry on spacetime. If one generalizes the gauge group of gravity, one finds still more involved spacetime geometries. If one specializes it to the translation group, one finds a specific Riemann-Cartan geometry with teleparallelism (Weitzenbock geometry).

 

[Paulibus]

It seems to me remotely possible that the geometry of generalised spaces, specifically spaces that do not preserve the property of distant parallelism, may be related to what Hehl talks of as a "small but decisive step beyond the established Riemannian spacetime structure of GR".

In a doggy and uncomprehending way I suspect that the association with gravity of Riemannian geometry (geometry sans torsion) may have been only a first step in describing other interactions of nature in geometrical terms. Crazy thought!

But it might prove worthwhile for some clever, mathematically literate person to glance at Chapter VIII of Frank Nabarro's Theory of Crystal Dislocations (Oxford University Press 1967) to see if the geometry of dislocated solids could help in taking a second step 'beyond Riemann', as it were.

 

[tom.stoer]

It's rather confusing; studying fermions makes indicates that Riemann geometry is not sufficient and that Riemann-Cartan geometry (which cannot be ruled out experimentally) is much more natural (geometrically); but b/c torsion is non-propagating the equations are purely algebraically and therefore torsion can be integrated out resulting in a pure fermionic contact interaction.

Of course this is fine geometrically we may assume (based on our experience with other interactions) that these contact terms are just low-energy effective interactions which should be relaplaced by a gauge interaction at higher energies (this reasoning has nothing to do with spacetime geometry but relies on aspects of gauge theories).

So there may be another step, now beyond Einstein-Cartan, and this is the point where I do not see a straighforward principle how to proceed.

 

[Haelfix]

Isn't this just describing the usual Sciama-Kibble-Cartan gauge theory? It gets confusing as people call it many different things in the literature, but as far as I know this is more or less the standard way quantum gravity people describe gravitational systems with fermions. It's a ubiquitous formulation in string theory, supergravity and LQG (at least the old theory), right?

Hehls papers have often confused me, as he is very impressed by the role of torsion, and sometimes he modifies the dynamics so that it is a propagating degree of freedom.

 

[tom.stoer]

It became standard in SUGRA and LQG, but afaik the origin is geometry, not physics.

I don't think that he is 'impressed'; the statement is that from a geometrical perspective it seems natural to study EC instead of EH - w/o ever referring to 'modern' developments like SUGRA, LQG, ...

 

[Haelfix]

I had some time to read the paper. It's really designed for a conference on classical GR and its the many variants, where they are looking for some sort of grand selection principle.. Perhaps a unifying theory that has all the well studied variants as special cases.

Hehl mentions the gauge principle, whereby if you take a classical gravitational system with rigid symmetries and gauge it, depending on how you do it, you get a number of differe GR like theories. He is most impressed with the usual Kibble-Sciama-Cartan theory where you gauge the Poincare group, although he mentions several other methods (for instance by going after the translation group instead which leads to Weitzenbrock spaces and teleparrelism etc)
I don't think this preference has so much to do with geometry perse, but more the fact that it is the simplest variant that successfully allows a cherished principle (the gauge principle) to go through in a straightforward manner. That it also allows fermions to be incorporated into GR is of course another reason.

Anyway, all this is old hat. What's relatively new (but still old in spirit, since he's been playing around with this for years) is the generalization of the EC theory to theories with propagating torsion. He parametrizes the ensuing quadratic theory (eq 37) with 15 undetermined constants, where certain choices reduce to EC as well as suitable strong gravity modifications.

All of this is fine, but it seems to me to be somewhat academic given the fact that all of those classical "strong gravity" components are empirically tiny, and relegated to energy regimes around the Planck scale. Even if whatever classical theory is in fact the final 'classical' theory, the quantum version will instantly generate all those action terms anyway, provided they respect the symmetries of the system.

Which goes right back to Weinbergs old comment about propagating torsion. It's perfectly reasonable to include the terms, but then they are completely indistinguishable from any other set of fields that respect the same symmetries.

 

[tom.stoer]

I agree on the "old hat". Hehl just explains his usual idea: EC s preferred due to fermions and naturaleness; but solving the constraint leaves us with a 4-fermion interaction which is unnatural, therefore we should look for theories with propagating torsion.

I think the lesson to be learned is that we should study asymptotic with fermions in the EC context and let the renormalization approach decide an the higher order terms.