I'm not sure such a thing exists. Usually reps only helps you classify the kind of particles you have (i.e the quantum numbers that identify them) and how they transform under the corresponding group. I believe how to represent this particles mathematically and what is their dynamics is a different matter.

The only thing similar I know about is that some of the Poincare group reps, or actually the vector spaces that carry them, have a correspondence with the Hilbert space of solutions of some wave equation

spin 0 : Klein-Gordon equation

spin 1/2 : Dirac equation

spin 3/2 : Rarita-Schwinger

etc

and you may be able to construct a Lagrangian/Action which gives these as the dynamics. But this is just the usual problem of finding a Hilbert space isomorphic to the Hilbert space of quantum states.

If the solutions can be properly quantised and interpreted as quantum fields is another issue and usually problems appear. For instance, if you try to couple Rarita-Schwinger fields to electromagnetism you encounter superluminal propagation.

Anyone else have any ideas?

This post imported from StackExchange Physics at 2014-04-25 09:28 (UCT), posted by SE-user Julio Parra