Here, Z ¯ ¯ α α is the complex conjugate of Z α α . The appearance of ⟨ ⟨ ψ ψ | ψ ψ ⟩ ⟩ in the denominator is a reminder that | ψ ψ ⟩ ⟩ and likewise | φ φ ⟩ ⟩ were not normalized to unit length; thus the normalization is made explicit here. In Hilbert space, the metric can be interpreted as the angle between two vectors; thus it is occasionally called the quantum angle. The angle is real-valued, and runs from 0 to π π / 2 .