In what sense can fields or states form *representations* of a group?

Question

My understanding of a representation is that it is a map from the group to a set of, say, matrices or operators \(D(g)\), such that the mapping "preserves" the group multiplication law, so that:\[D(g_1*g_2)=D(g_1)D(g_2).\]Based on this notion of representations, I don't see how states \(|p,h>\) or fields \(\phi\) can be thought of as forming representations of, say, the Poincare group. Shouldn't the representation be the thing that acts on these states? Why is it that when an article talks of "finding representations" of a group, it actually discusses what to me seem to be states, not matrices (which I currently understand to be the representations)?

Answer 1

A unitary representation of a group $G$ is a mapping $U$ from $G$ to the algebra of unitary linear operators on a Hilbert space $V$. This space is called the representation space, and one says that $G$ acts unitarily (or is unitarily represented) on $V$. Thus for every $g\in G$, the representation defines a mapping $\psi\to U(g)\psi$ with the natural compatibility properties. In particular, unit vectors are mapped to unit vectors. In quantum mechanics, the elements of $V$ (or only the unit vectors, depending on the author) are referred to as the states, and the thing that acts on the states is the group element. 

Saying that states of a certain form form a representation is loose talk for saying that all states of this certain form form a Hilbert space (with inner product taken from the context) on which $G$ acts unitarily (in the obvious way, or in the way defined by the context).

Finding a representation means finding a Hilbert space $V$ and the action of $G$ on it. Typically, one pieces $V$ together from constituents already known. This defines the states of interest. When the construction is elegant, the notation used for the states is such that the group action is obvious; otherwise the group action has to be defined explicitly and one must prove that products behave correctly.

Matrices are just the special case when $V$ is the Hilbert space of complex column vectors. The representation of states as column vectors is appropriate for an $N$-level system and the unitary group $U(n)$, but for other groups it is usually preferable to use a different notation for the states that is adapted to the group.