A “Minus 1” Mystery

There remains another quantum state detail that we should discuss. In manipulating quantum state vectors, we will often find an expression like $-\ket\downarrow$. What does the negative of a quantum state vector mean? For ordinary vectors $−\vec{F}$ is a vector that has the same length (magnitude) as the vector $\vec{F}$ but points in the opposite direction. The “obvious” generalization to quantum states is that in state space $−\ket\downarrow$ points in the direction opposite to that of $\ket\downarrow$. Note that $−\ket\downarrow\neq \ket\uparrow$. The direction is reversed in state space, not in physical space. In physical space, spin-up is in the direction opposite to spin-down, but not in state space.