Representation theory of the Poincaré group

In a classical field theory, the physical states are sections of a Poincaré-equivariant vector bundle over Minkowski space.

In quantum mechanics, the state of the system is determined by the Schrödinger equation, which is invariant under Galilean transformations.

Quantum field theory is the relativistic extension of quantum mechanics, where relativistic (Lorentz/Poincaré invariant) wave equations are solved, "quantized", and act on a Hilbert space composed of Fock states.

There are no finite unitary representations of the full Lorentz (and thus Poincaré) transformations due to the non-compact nature of Lorentz boosts (rotations in Minkowski space along a space and time axis).

However, there are finite non-unitary indecomposable representations of the Poincaré algebra, which may be used for modelling of unstable particles.