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 PFor the first case, we note that the eigenspace (see generalized eigenspaces of unbounded continuous operators) associated with P_{0}>0 and m=0 andP=0.

For the second case, we look at the stabilizer of P_{0}=k, P_{3}=-k, P_{i}=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.