Wigner classfied elementary particle as unitary irreducible representations (UIR) of the Poincaré group.

Suppose the spacetime is curved with symmetries $G$. Should the elementary particles in this spacetime always be UIR of $G$?

Specifically: Our universe has a positive cosmological constant, so shouldn't we look for unitary irreducible representations of the de Sitter group (as the next best approximation)? Or given a Schwarzschild spacetime: Should the elementary particles in this spacetime be UIR of its symmetries?

(It is clear to me that in general spacetimes have no symmetries at all. I'm wondering why it is the Poincaré group that has physical significance for our elementary particles and not, say, the de Sitter group. Or why do we even believe that elementary particles are UIR of Poincaré even though realistic spacetime of our universe do not have Poincaré symmetries.)

  • $\begingroup$ Ordinary quantum field theory - which is the context of Wigner's classification as applied to particles - always happens on Minkowski space. What framework for QFT in curved spaces are you using here to talk about "particles in curved space"? $\endgroup$
    – ACuriousMind
    May 2, 2020 at 16:02
  • $\begingroup$ @ACuriousMind: I guess there are various frameworks for QFT in curved spacetimes in which there is a notion of a particle (no?). I did not really want to tie my question a specific framework. Via the orbit method, I would have guessed that in any of them the UIR of the relevant symmetries of the underlying spacetime might be the relevant ones, but maybe that is a misconception. Is it too vague in the current form? $\endgroup$
    – ungerade
    May 2, 2020 at 16:23
  • $\begingroup$ Or said differently: Suppose I want to built QFT on a curved spacetime (without knowing how to). Would one expect that Wigner's logic still applies? I.e., should I start by looking at the UIR of the isometries of the spacetime or would that be a bad idea (and why)? $\endgroup$
    – ungerade
    May 2, 2020 at 16:40
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    $\begingroup$ The (imprecise) definition of particles in QFT does not rely on any symmetries, but if spacetime symmetries are present, then we can expect them to preserve the set of single-particle states of any given species (which is really part of how we determine "species"). Using symmetries to define particles is a short-cut of very limited utility, but when it does make sense, it's much easier than trying to use the more general (and imprecise) appraoch. The general approach requires finding observables that work like (quasi)local detectors, which is usually difficult (except in free models). $\endgroup$ May 25, 2020 at 19:45
  • 2
    $\begingroup$ 1) I don't have enough experience with QFT in AdS or dS to give a definitive answer, but any time a spacetime symmetry exists, I expect that single-particle states should form irreps, as in Minkowski. 2) Chapter VI in Haag's book Local Quantum Physics describes the idea. Basically, any operator that annihilates the vacuum state is a candidate for a detector-observable, and Haag's chapter explains how to narrow those candidates down to single-particle detectors. It can't be precise, because particles are often transient phenomena. I guess that's why there isn't much literature about it. $\endgroup$ May 25, 2020 at 22:46

1 Answer 1


I'm at around the same point as you, so I'm not sure how well I'll be able to answer, but I'll try. I don't know if you've come across the Einstein-Cartan formulism for GR yet, but it's basically treating gravity as a gauge theory, in an analogous way to Yang-Mills. Gravity is the gauge theory of the Poincare group so spacetime does have a Poincare symmetry, but it's a local one. I suppose, you could maybe also think about it like any particle is so small, that it is in an inertial reference frame and so just has the minkowski metric.

The symmetries of the spacetime I think you're thinking about are connected more to conserved quantities in the spacetime. As you're probably aware, in Schwarzschild, we have spherical symmetry so the angular momentum is conserved. I'm unsure as to why it is we wouldn't get extra particles, not just conserved quantities, from those UIR from the extra spacetime symmetries. Although, the stuff I covered in Wigner's classification, not all irreps actually show up as physical particles (e.g the continuous spin ones for massless particles) and I was given no real reason for this. Hopefully someone can else can answer that. Also, remember the FRW-metric is only a 0th order approximation too, so not accurate once you come zoom in to some normal distance.


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