Representation theory of the Poincaré group
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In mathematics, the representation theory of the Poincaré group is an example of the representation theory of a Lie group that is neither a compact group nor a semisimple group. It is important in theoretical physics.
In a physical theory having Minkowski space as the underlying spacetime, the space of physical states is typically a representation of the Poincaré group. (More generally, it may be a projective representation, which amounts to a representation of the double cover of the group.)
In a classical field theory, the physical states are sections of a Poincaré-equivariant vector bundle over Minkowski space. The equivariance condition means that the group acts on the total space of the vector bundle, and the projection to Minkowski space is an equivariant map. Therefore the Poincaré group also acts on the space of sections. Representations arising in this way (and their subquotients) are called covariant field representations, and are not usually unitary.
In quantum physics the space of states is a Hilbert space and the representation of the Poincaré group is unitary. For a discussion of such unitary representations, see Wigner's classification.
