What does braket subscript vector mean?

Reading some quantum theory lecture notes, I came across a notation that really confuses me: $\langle a \rangle_{\vec{u}}$, for $a \in \{-1, 1\}$ and a unit 3D vector $\vec{u}$.

See equation (15) in these notes for context.

• Could you please give some explanation as well? I understand that this is supposed to denote an expectation value, but I don't understand how $a$ can be considered to be an observable, plus how can a vector in those notes represent the state of the quantum system? – gen Sep 26 '17 at 11:54
• $a$ is not an observable (operator), it's an eigenvalue; and $\vec a$ is just a vector, not a state. – md2perpe Sep 26 '17 at 18:37
• This is how the Wikipedia article you linked defines the notation: Consider an operator $A$. The expectation value is then $\langle A\rangle =\langle \psi |A|\psi \rangle$ in Dirac notation with $\psi \rangle$ a normalised state vector. Which contradicts what you are saying just above... I'm sure you understand this, but could you help me understand it as well by making your answer more verbose? – gen Sep 26 '17 at 19:38