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A classification of the nonnegative energy irreducible representations of the Poincaré group which have sharp mass eigenvalues.
The double cover of the Poincaré group admits no central
extensions.
Note: This leaves out tachyonic solutions, solutions with no fixed mass, infraparticles with no fixed mass,
etc..
is a Casimir invariant of the Poincaré group. So, we can classify the irreps into whether m>0, m=0 but P0>0 and m=0 and P=0.
For the first case, we note that the eigenspace (see generalized eigenspaces of unbounded operators) associated with
P0=m and Pi=0 is a representation of SO(3). In the ray interpretation, we can go over to Spin(3) instead. So, massive states are
classified by a Spin(3) unitary irrep and a positive mass, m.
For the second case, we look at the stabilizer of P0=k,
P3=-k, Pi=0, i=1,2. This is the double cover of
SE(2) (see again unit ray representation). We have two case, one where irreps are described by an integral multiple
of 1/2, called the helicity and the other
called the "continuous spin" representation.
The last case describes the vacuum. The only finite dimensional unitary solution is
the trivial representation called the vacuum.
See also the method of induced representations.
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