Invariant mass and spin in deSitter cosmology

[tom.stoer]

A friend of mine was reading Penrose's new book on CCC; I do not want to discuss this story here but a rather interesting detail which could be relevant w/o the whole CCC stuff.

SR and GR rely on (global and local) Lorentz invariance. From these symmetries one can derive invariant mass M² and spin (W² with the Pauli-Lubanski vector W^μ) as 1st and 2nd Casimir operator. Now there are scenarious where it seems to be appropriate to use deSitter invariance instead (inflation, positive cc, q-deformed spin networks, ...). Then one has to use the deSitter group instead - and I expect that the Casimir operators are indeed different and that M² and W² need no longer play a fundamenbtal role.

Does anybody know about ideas how to modify QFT on deSitter spacetime? How to replace M² and W²? What happens with Einstein-Cartan theory when enlarging the underlying symmetry to deSitter?

 

[mitchell porter]

There are lots of papers on QFT in dS space but I find the literature confusing.

I think this thread would be helped by a systematic comparison of how Casimir operators for QFT work in Minkowski, dS, and AdS spaces. To assist such comparison, here are some links.

Tour of dS and AdS (no quantum here, just geometry).

Intro to QFT which discusses Lorentz transformations for fields, Poincare transformations for particles, and the relevance of the Casimir operators. This is all in Minkowski space.

I was unable to find a satisfactory review of QFT in AdS, but this (1977) was a foundational paper.

Gazeau on QFT in dS: review, talk. Some of the complexities of dS Casimir operators start to come out here.

 

[strangerep]

How to replace M² and W²?

The deSitter group contracts to Poincare. Some explicit information on deSitter Casimirs can be found here:

Ion I. Cotaescu, "The physical meaning of the de Sitter invariants",
Available as: http://arxiv.org/abs/1006.1472
Abstract:

We study the Lie algebras of the covariant representations transforming the matter fields under the de Sitter isometries. We point out that the Casimir operators of these representations can be written in closed forms and we deduce how their eigenvalues depend on the field's rest energy and spin. For the scalar, vector and Dirac fields, which have well-defined field equations, we express these eigenvalues in terms of mass and spin obtaining thus the principal invariants of the theory of free fields on the de Sitter spacetime. We show that in the flat limit we recover the corresponding invariants of the Wigner irreducible representations of the Poincare group.

If you search arXiv for other papers by the same author (Cotaescu) you'll find several other papers, some of which deal with field theory in a deSitter context.

It gets tricky because the momentum operators no longer commute, so you can't use ordinary momentum space to help you construct a Hilbert space based on a maximal set of commuting operators in the same way as Poincare-based theory. Cotaescu published another paper a few days ago about representations he'd found (which he says are related to old material by Nachtman), but I haven't studied that stuff yet.

 

[tom.stoer]

thanks for the references; seems to be exactly what I was looking for ;-)

 

[haushofer]

I guess you already looked for it, but googling on papers on the Breitenlohner-Freedman bound (stating that tachyons can be stable in AdS) could help you. Although it's not dS, I can remember a paper about this which treated the representation theory of AdS in a pretty detailed fashion, which could be helpful for you. If I find it, I will post it here.

@strangerep: Isn't the problem really that the AdS algebra is semisimple whereas the Poincare algebra is not, such that the split of the generators of this AdS algebra in (noncommuting) translations and Lorentz transformations is rather artificial? I thought about this because the procedure of applying a gauging procedure to the (A)dS algebra to obtain GR with a cosmological constant is not as straightforward as it is for the Poincaré algebra.

 

[strangerep]

Isn't the problem really that the AdS algebra is semisimple whereas the Poincare algebra is not, such that the split of the generators of this AdS algebra in (noncommuting) translations and Lorentz transformations is rather artificial?

I'm not an expert on this, but although this sounds reasonable when working with dS in a 5D abstract space, I'm really more interested what happens back in the 4D real world. ;-)
Projected back to 4D, translations seem quite distinguishable physically from Lorentz transformations. I'm also not happy with how the extra dS transformations in 4D seem to be implemented with the help of special conformal transformations -- which don't preserve inertial motion in general.

I thought about this because the procedure of applying a gauging procedure to the (A)dS algebra to obtain GR with a cosmological constant is not as straightforward as it is for the Poincaré algebra.

I'm coming at it from the opposite direction: studying possible generalizations of SR kinematics. People have tried this, of course, but they seem to use SCTs to construct a dS space, with CC postulated ad hoc. The use of SCTs disqualifies it as "kinematics" (GR in the absence of matter) since inertial motion is not preserved.

All of the above: IMHO.

 

[Jim Kata]

I worked this out a while back, but forget what I got. There is a paper by Henneaux and Teitelboim called Asymptotically Anti de Sitter Spaces. They do the Anti de-sitter case in one of their appendixes. I'm sure there are papers on the de Sitter case, but here is the gist of what you would do. Solve the Killing equation

$$\xi_{\alpha;\beta} + \xi_{\beta;\alpha}=0$$ but obviously you would use the de - Sitter metric for your covariant derivative not the Lorentz metric. I think maybe how I did it was use the fact that de Sitter space can be embedded in $$\mathbb{R}^{1,4}$$ and worked out the killing vectors from there and then substitute in the embedding.

http://srv2.fis.puc.cl/~mbanados/Cursos/TopicosRelatividadAvanzada/HenneauxTeitelboim.pdf
Anyways here's the Henneaux and Teitelboim paper they do the anti de-sitter case in their in Appendix A you can just tweak what they did for the de - sitter case.

 

[haushofer]

Tom's question concerns representation theory. Killing vectors span the algebra, but we already know what the algebra is, so I don't see how that gives any insight in the questions in the OP.