We intuitively have a sense of what a particle means in the conventional sense. But is it possible to have a group theoretical definition of a particle, I mean in terms of irreducible representations etc.?
Answer
An elementary particle is defined as an irreducible representation of the Poincar\'e group. These were classified by Wigner in 1939. This was done via the little group construction. The important representations are (metric signature $(-,+,+,+)$
$p^2 = 0$, $p^0 < 0$ - The little group is ISO(2). All finite dimensional representations of this group are one-dimensional and labelled by a single number $h$ (called helicity). Topological considerations require that $h$ be a half-integer. Under parity, the representation $h$ is rotated to $-h$. Thus, a massless particle that has parity and proper Lorentz invariance, has two degrees of freedom and is labelled by its helicity $|h|$.
$p^2 = - m^2 < 0$, $p^0 < 0$ - The little group is $SO(3)$. All representations of this are finite dimensional and are labelled by a single number $j$ with dimension $2j+1$ ($j$ is called spin). Thus, a massive particle of spin $j$ has $2j+1$ degrees of freedom.
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