>>11379964so each particle has some helicity and or a mass, by the poincare group,
then the step is to determine ''the relativistic wave equation'' for each field and also the state space of each ''particle ie some pair [mass, helicity]."
It turns out that for the photon, ie m=0 h=1,, the equation is the usual maxwell equation. For spin 0+mass is the klein gordon equation.
So far there is no QM. So far there is only Lorentz transformations and things invariant under Lorentz transformations.
physical states = particle states = definite mass and helicity=unitary irreducible representation of poincare group.
wave functions and fields are solutions to the wave equations, wave functions and fields may or may not be physical. This is the most important thing to have in mind. wave functions and fields are not tied to unitary reprensation. wave functions and fields are indexed by finite representations of the spacetime symmetry group + whatever internal symmetry the system has.
wave functions and fields transform as finite dimensional representations of the Lorentz group
physical particle states transform as unitary representations of the
Poincare group
in QFT, for instance the ghost fields are not physical.
so now that there is a physical state of m=0 and helicity 1, there is a state space , and there is a wave equation to this physical state [purely from group theoretic consideration] and this equation has wave solutions which are grouped into a hilbert space. Then those solutions transform again under the Lorentz group and this is done by some finitary matrices on the waves. It is those waves which appear in the lagrangian and the action.
