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As I understand it, there are representations (reps) of the Poincaré group that are reducible but still indecomposable (i.e., cannot be expressed as a direct sum of two subreps). This would be impossible if the Poincaré group were semisimple, but it isn't. [Edit: Actually, it is possible even for semisimple groups, including the Lorentz group, if one does not restrict to finite-dimensional reps.]

Suppose a physical system or quantum field were described by such a rep. What would be the physical interpretation or description of such a system? Maybe something like an "embellished one-particle system," or a "single particle with additional properties that don't make sense in isolation"? Do any such reps actually appear in physics, and if not, are there good physical reasons for disregarding such reps?

Context: I'm trying to figure out why irreducible representations (irreps) are so important and fundamental, and even assumed to correspond one-to-one with single-particle systems. From what I can tell, there are three arguments, but none is very satisfying.

  • Reducible reps are built up from irreps, so reducible reps can all be understood as multi-particle systems. I think this argument is just wrong, since the Poincaré group is not semisimple. [Edit: Moreover, even for semisimple groups, the argument fails on infinite-dimensional reps, so one should justify why these are not considered.]
  • Irreducibility is sufficient to guarantee that mass and spin are invariant under Poincaré actions. I don't understand the technical details of this argument, claimed here. But at any rate, the claim is only that irreducibility is sufficient, not necessary, which leads to my question above.
  • It is just natural or intuitive that single-particle states should correspond to irreps.* I can't really make sense of this argument without knowing the answer to the question I've asked here.

*This seems to be Weinberg's attitude: "It is natural to identify the states of a specific particle type with the components of a representation of the inhomogeneous Lorentz group which is irreducible, in the sense that it cannot be further decomposed in this way." (Weinberg, p. 63)

WillG
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2 Answers2

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Completely reducible representations are the important ones because unitary representations are always completely reducible.

To see this, note that if $V$ is a unitary representation with an invariant subspace $W \subseteq V$, and orthogonal complement $W^\perp$, then for $|a\rangle \in W$ and $|b\rangle \in W^\perp$, we have that $$0 = \langle a | b \rangle = \langle g \cdot a | g \cdot b \rangle = \langle a' | g \cdot b \rangle$$

where $ | a' \rangle \in W$ by invariance. But that implies $| g \cdot b \rangle \in W^\perp$ because $|a\rangle,|b\rangle$ were arbitrary. Thus, $W^\perp$ is also an invariant subspace, so $V = W \oplus W^\perp$. This argument can be iterated to show that $V$ is completely reducible. By complex conjugating the last two steps above, the same argument holds for anti-unitary representations as well.

So your question becomes equivalent to "What is the physical interpretation of non-(anti)unitary reps of the Poincaré group"? By Wigner's theorem, not much.

11zaq
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''I'm trying to figure out why irreducible representations (irreps) are so important and fundamental, and even assumed to correspond one-to-one with single-particle systems.''

Free systems must be in a unitary representation of the Poincaré group in order that the generators are represented by self-adjoint operators and hence correspond to the traditional observables, 4-momentum and angular momentum.

Reducibility implies that the particle can be decomposed, hence is not elementary. Thus an elementary particle is represented by an irreducible unitary representation of the Poincaré group.

Causality in quantum field theory imposes additional constraints, so that only representations are admissible. where the spectrum of the momentum operator is in the future cone or light cone

Then one generally rules out the continuous spin representations, the most common argument being that these are not found in Nature. But some speculate that they might be candidates for dark matter.

Arnold Neumaier
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