Choose the representation category
A finite-dimensional field multiplet and a one-particle Hilbert space use different representation categories. Keep the object, spacetime signature, dimension, and global cover explicit.
Lorentzian fields in four dimensions
The identity component of the Lorentz group has spin cover SL(2,ℂ). Finite-dimensional irreducible complex representations of this real Lie group carry labels (jL,jR). Weyl spinors are (½,0) and (0,½). These representations need not be unitary on a positive-definite finite-dimensional Hilbert space.
Particle states
Unitary representations of the Poincaré group use momentum orbits and little groups. The usual massive sectors are labeled by mass and spin; standard finite-helicity massless sectors use helicity. These familiar sectors should not be mistaken for the complete unitary dual.
Euclidean geometry
The Euclidean rotation group SO(4) has spin cover Spin(4), which is a product of two compact SU(2) factors. This is a different real group from the Lorentzian spin group SL(2,ℂ), even though pairs of spin labels appear in both settings.
The AdS and conformal pathway
The conformal and AdS guides distinguish real forms and covering groups, finite-dimensional field indices, and infinite-dimensional state multiplets. Worked examples derive a scalar unitarity bound and the AdS scalar mass–dimension relation. These are scoped entry points, not a complete noncompact representation database.
Continue through spacetime symmetry
Follow the spacetime learning path, work through spinors and reality conditions, then explore conformal primaries and unitarity and AdS isometries and holographic symmetry.