Representation theory of the Poincaré group
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In mathematics, the representation theory of the double cover of the Poincaré group is an example of the theory for a Lie group, in a case that is neither a compact group nor a semisimple group. It is important in relation with theoretical physics.
For unitary representations, see Wigner's classification.
For more general (linear) representations of the double cover, see below.
A rep of the Poincaré is called a covariant field rep if there is an injective intertwiner from it to a rep consisting of the space of sections of a fiber bundle over Minkowski spacetime where if a is mapped to b under the action of an element of the double cover, then the base point of a maps to the base point of b as the (affine) Minkowski rep of the same group.
In general, single particle reps are unitary irreps while operators form covariant field reps (see operator valued distribution). This is because single particle reps form a Hilbert space while if we have any operator A let's say, looking at the operator valued function over the (double cover of the) Poincaré group defined by g[A] for each element g of the double cover of the Poincaré group, we can decompose them under harmonic analysis and each component of the decomposition would correspond to a linear covariant field rep. More precisely, Minkowski space is a homogeneous space of the Poincaré group, with the Lorentz group as the stabilizer, which means that under harmonic analysis, the operator valued function over the Poincaré group can be rewritten as operator valued rep of the Lorentz group function over Minkowski space.
In general, the injective intertwiner needn't be surjective. In other words, the space of all sections (forget about topology for the moment) could be a reducible representation. A covariant intertwiner is an intertwiner mapping one covariant field rep to another. The derivative (since Minkowski space is flat, we don't need a covariant derivative) for example is a nonzero intertwiner from a covariant field rep to another. Also, if the fiber is a reducible rep, we could have a (pointwise) intertwiner projecting out one of its components for example.
Such an analysis doesn't explain the spacelike commutativity/anticommutativity properties of the fields though. This form of locality is an additional assumption.
Things like resonances are actually more complicated.
This injective intertwiner needn't necessarily be unique. Ignoring topological issues like continuity, smoothness, boundary conditions, topology, etc, for example, the rep of all scalar field configurations satisfying the Klein-Gordon equation for mass m is equivalent to a subrep of all smooth closed 1-forms satisfying the Klein-Gordon equation for mass m.
Also, the reps, and hence the fibers of the fiber bundle needn't necessarily be linear. It could be affine or even nonlinear, like for example nonlinear sigma models.
A rep is called a quotient covariant field rep if it is equivalent to the quotient representation of a covariant field rep by a subrep of it which is also a covariant field rep (with respect to the same fiber bundle). This is the case, for example in free Abelian Yang-Mills theories.
