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The Poincaré group is the group of rigid spacetime symmetries of Minkowski spacetime. It is a topological group and as such has unitary representations on infinite-dimensional Hilbert spaces. For any quantum field theory in Minkowski space its space of states therefore decomposes into irreducible representations of the Poincaré group. As was first observed by Hermann Weyl, these irreducible representations encode the particle spectrum of the QFT.
We are interested in the unitary representations of a topological group $G$, i.e., the continuous group homomorphisms
into the unitary group of a Hilbert space $H$, especially those that are irreducible in the usual sense of representation theory. The topology on $U(H)$ here is understood to be the strong operator topology.
In this and related articles, we study such representations in the case where
is the universal cover of the connected component of the identity of the Poincaré group, which is important in the study of quantum field theory. A more physical name for such a representation is “elementary particle”, and we will often use that term in this article. (NB: “elementary particle” will always refer to the formal mathematical
A full rounded account could become large; see the blog discussion, which despite its size was left in a still-nascent state. However, in a nutshell, the basic theorem is that (elementary) particles are classified up to isomorphism according to their mass and helicity; mass is a continuous parameter and helicity is a discrete parameter.
This theorem in commonly ascribed to Eugene Wigner and often refereed to as the Wigner classification.
Wigner classified all irreducible unitary representations of the restricted Poincare group, including the unphysical ones. The latter cannot be used to define a free quantum field theory satisfying the Wightman axioms. Those that can are the physical ones and are characterized by a nonnegative real mass and a nonnegative half-integral spin; the zero component of the momentum has a nonnegative spectrum. Many of these are realized by particles occurring in Nature, though not as ‘elementary particles’‘ but as bound states (in a suitable approximation, e.g., QCD). From the point of view of representation theory, the center of mass of a bound state behavs just like an elementary particle. Thus elementary is meant in this generalized sense.
Relevant topics in a full account will include
Rigged Hilbert spaces and direct integrals?
Stone's theorem?
Mackey theory?
In the first place, physicists tend to be a little carefree with the mathematics, so this account is written from the point of view of a ‘stupid’ mathematician (for the moment Todd Trimble) who wants to get details straight and precise.
For example, physicists tend to talk about “eigenstates” as if they were elements of the Hilbert space, and other states as linear combinations of eigenstates, whereas really we are dealing with some more complicated technology like rigged Hilbert spaces or direct integrals instead of direct sums. Failure to mention such details places hurdles of communication between physicists and mathematicians. In addition, there are stylistic differences in presentation, where a physicist will happily deal with formulas replete with lots of subscripts and superscripts, whereas many mathematicians prefer dealing with more conceptual, less notation-heavy explanations.
There seem to be at least three ways of dealing with spectral theory of (unbounded) self-adjoint operators on a Hilbert space:
The usual Stone theory
Direct integrals of Hilbert spaces
Rigged Hilbert spaces
A rigged Hilbert space
Simultaneous diagonalization.
Fact that Poincaré group is a semidirect product group.
Let $\vec p \in \mathbb{R}^{d-1,1}$ be a given vector in Minkowski spacetime. Write
for its stabilizer subgroup (often called the little group in this context, going back to Wigner). Every unitary irrep of $Iso(\mathbb{R}^{d-1,1})$ of mass $p$ is an induced representation of a finite dimensional representation of the “little group” $Stab_{Iso(\mathbb{R}^{d-1,1}))}(\vec p)$. (recalled concisely e.g. in Dragon 16, p. 2).
The observation that the irreps of the Poincaré group correspond to fundamental particles (Wigner classification) is due to
Review includes
Rudolf Haag, section I.3.1 of Local Quantum Physics -- Fields, Particles, Algebras
Bert Schroer, Wigner Representation Theory of the Poincaré Group, Localization, Statistics and the S-Matrix, 1996 (pdf)
Eberhard Freitag, Unitary representations of the Poincaré group (pdf)
Xavier Bekaert, Nicolas Boulanger, The unitary representations of the Poincare group in any spacetime dimension (arXiv:hep-th/0611263)
Norbert Dragon, Currents for Arbitrary Helicity (arXiv:1601.07825)