Why are non-momentum DoFs of single-particle states discretely labeled?

Question

Following the treatment of Weinberg, chapter 2, we consider $\psi_{p,\sigma}$ as single-particle eigenstates of the 4-momentum. Weinberg says that $\sigma$ labels all other degrees of freedom and we take this label to be discrete for one-particle states. So what exactly is the physical implication of discrete and continuous labeling of other degrees of freedom? And why is discrete labeling physically pertinent to single-particle states?

Answer 1

Apart from the spatial translations corresponding to the momentum operator, the other symmetries (that I can think of) that are relevant in particle physics i.e. things like

Spatial Rotations

Phase Transformations

Flavour Transformations

Colour Transformations

are represented by the action of compact Lie groups. The irreducible unitary Hilbert space representations of compact Lie groups are finite dimensional and this is reflected in the discrete labelling.