Why is it interesting?

Layman?

Why not start with quarks? Teachers investigate a learning unit on the subatomic structure of matter with 12-year-olds by Gerfried J. Wiener, Sascha M. Schmeling and Martin Hopf

Student

Researcher

First of all, for a Lorentz-invariant theory the space of single-particle states for a given particle species must be the basis for an irreducible representation of the Lorentz group. Indeed, if the representation were reducible, there would be different states that cannot be brought into one another by any Lorentz transformation. In this case it is natural to talk of different particle species rather than of different states of the same particle. […] Massless particles have no rest-frame—thus there is no reference frame where we can apply our non-relativistic QM knowledge about rotations and spin.

http://phys.columbia.edu/~nicolis/GR_from_LI.pdf

Particles can be elementary or composite. An elementary particle cannot be decomposed into parts. We do not have an exact criterion for elementary particles. However, Wigner has proposed a necessary but not sufficient condition. A free electron should be a free electron in all relativistic frames. Any two states of a.free elementary particle should be connectible by a transformation in the Poincare group. Thus all its states are representable by superposition of states obtained by relativistic transformations of a single state. In other words,there must be no relativistically invariant subspaces for the state space of a free elementary particle, otherwise we would call the invariant subspaces elementary. The state space of a free elementary particle is the Hilbert space for an irreducible representation of the Poincare group.

From "How is Quantum Field Theory possible" by Auyang

Examples

Example1

Example2:

History