Projective unitary structure

In this section we have studied the shadow downstairs (in projective space) of the complex scalar product upstairs (in the linear space). We have found that although the scalar product itself does not descend, we can use it to define angles and orthogonality. Up to a phase factor, we can expand kets in orthogonal bases. We will use this projective unitary structure to define projective unitary representations and physical symmetries. 

Not every linear-subspace-preserving function on projective space descends from a complex linear operator. However, when we consider the unitary structure in Section 10.3 we find an imperfect but still useful converse — see Proposition 10.9. 

If the kets label individual states, i.e.. points in projective space, and if addition makes no sense in projective space, what could this addition mean The answer lies with the unitary structure (i.e., the complex scalar product) on V and how it descends to P(y). If V models a quantum mechanical system, then there is a complex scalar product ( , ) on V. Naively speaking, the complex scalar product does not descend to an operation on P(V). For example, if v, w e V 0 and v, w Q v/e have u 2v but (v, w) 2 v, w) = 2v, w). So the bracket is not well defined on equivalence classes. Still, one important consequence of the bracket survives the equivalence orthogonality. 

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