Elementary States of Quantum Mechanical Systems

Definition 4.9 we know that flw commutes with every p(g). Hence by hypothesis Pw must be a scalar multiple of the identity. If the scalar is nonzero, then W = V. If the scalar is zero, then W = 0.  

We have shown that V (all) and 0 (nothing) me the only invariant subspaces of V. So (G, V, p) is irreducible.  

The following technical proposition will be uselul in Proposition 7.6. 

By Proposition 3.5, since V2 is finite dimensional we know that there is an orthogonal projection 112 with range V2. Because p is unitary, the linear transformation 112 is a homomorphism of representations by Proposition 5.4. Thus by Exercise 5.15 the restriction of 112 to Vi is a homomorphism of representations. By hypothesis, this homomorphism cannot be injective. Hence Schur s lemma (Proposition 6.2) implies that since Vi is irreducible, fl2[Vi] is the trivial subspace. In other words, Vi is perpendicidar to V2.  

We saw in Section 4.5 that a quantum mechanical system with symmetry determines a unitary representation of the symmetry group. It is natural then to ask about the physical meaning of representation-theoretic concepts. In this section, we consider the meaning of invariant subspaces and irreducible representations. 

---